^'■^f^-f--'- . id', (X^. \^[ vni((5^ With the Compliments of YALE UNIVERSITY LIBRARY NEW HAVEN, CONN., U. S. A. ®0tMl Utttetsiitg Jibmg THE GIFT OF h^....%'].k'\.3L3. !..s:|57ni\.\^.v3.. 6561 DATE DUE i^N^ ^^ JtWt^ '^ ^ »»»»■• ^=mi tiiMl ^A982_^ W 2 |/M- / 1/ [y APn3 ^0 * I GAYLORD PRINTEDINU.S.A. Cornell University Library QB 901.C19 Stellar motions, with special reference t 3 1924 005 024 652 Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924005024652 YALE UNIVERSITY MRS. HEPSA ELY SILLIMAN MEMORIAL LECTURES STELLAR MOTIONS SILLIMAN MEMOEIAL LECTUEES PUBLISHED BY YALE TJNIVEESITY PEESS ELECTEICITY AND MATTEE. By Joseph John Thomson, b.sc, ll.d., PH.D., F.E.S., Fellow of Trinity College, Cambridge, Cavendish Professor of Experimental Physics, Cambridge. Price $1.S5 net; postage 10 cents extra. THE INTEGEATIVE ACTION OF THE NEEVOUS SYSTEM. By Charles S. Shekeington, d.sc, m.d., hon. ll.d., tor., f.r.s., Bolt Professor of Physiology in the University of Liverpool. Price $3.50 net; postage SS cents extra. EADIOACTIVE TEANSPORMATIONS. By Ernest Eutherford, d.sc, LL.D., F.R.S., Macdonald Professor of Physics, McGill University. Price $3.60 net; postage SS cents extra. EXPERIMENTAL AND THEOEETICAL APPLICATIONS OF THER- MODYNAMICS TO CHEMISTRY. By Dr. Walther Neenst, Professor and Director of the Institute of Physical Chemistry in the University of Berlin. Price $1.26 net; postage 10 cents extra. THE PROBLEMS OF GENETICS. By William Batbson, m.a., f.r.s., Director of the John Innes Horticultural Institution, Merton Park, Surrey, England. Price $4.00 net; postage B5 cents extra. STELLAE MOTIONS. With Special Reference to Motions Deter- mined BY Means of the Spectrograph. By William Wallace Campbell, sc.D., LL.D., Director of the Lick Observatory, University of California. Price $4.00 net; postage 30 cents extra. THEORIES OF SOLUTIONS. By Svante August Arrhenius, ph.d., sc.D., M.D., Director of the Physico-Chemical Department of the Nobel Insti- tute, Stockholm, Sweden. Price $S.S6 net; postage 16 cents extra. IRRITABILITY. A Physiological Analysis of the General Effect of Stimuli in Living Substances. By Max Verworn, Professor at Bonn Physiological Institute. Price $3.60 net; postage SO cents extra. Sir Isaac Newton, 1642-1727 Stellar Motions WITH SPECIAL REFERENCE TO MOTIONS DETERMINED BY MEANS OF THE SPECTROGRAPH BY William Wallace Campbell, Sc. D., LL. D. Director of the Lick Observatory, University of California New Haven : Yale University Press London : Henry Fbowde Oxford University Press MCMXIII ^:'i-^<^'^'i'^ COPYRIGHT, 1913 By YALE UNIVERSITY First Pkinted June, 1913, 1000 Copies i^E THE SILLIMAN FOUNDATION In the year 1883 a legacy of eighty thousand dollars was left to the President and FeUows of Yale College in the city of New Haven, to be held in trust, as a gift from her children, in memory of their beloved and honored mother, Mrs. Hepsa Ely Silliman. On this foundation Yale College was requested and directed to establish an annual course of lectures designed to illustrate the presence and providence, the wisdom and goodness of God, as manifested in the natural and moral world. These were to be designated as the Mrs. Hepsa Ely Silliman Memorial Lectures. It was the belief of the testator that any orderly presentation of the facts of nature or history contributed to the end of this foundation more effectively than any attempt to emphasize the elements of doctrine or of creed ; and he therefore provided that lectures on dogmatic or polemical theology should be excluded from the scope of this foundation, and that the subjects should be selected rather from the domains of natural science and history, giving special prominence to astronomy, chemistry, geology, and anatomy. It was further directed that each annual course should be made the basis of a volume to form part of a series constituting a memorial to Mrs. Silliman. The memorial fund came into the possession of the Corporation of Yale University in the year 1901 ; and the present volume constitutes the seventh of the series of memorial lectures. PREFACE The contents of this book formed the Silliman Lectures in Yale University for the academic year 1909-1910. They were deliv- ered in the period January 24 to February 4, 1910. Numerous modifications of an entirely minor character have been made in the manuscript, in order to briag out points, by means of text and printed illustrations, which were presented in the lectures with the help of lantern slides. All significant changes or addi- tions made subsequent to the delivery of the lectures are duly indicated in the text. The following paragraph, which concluded the series of eight lectures, has been brought forward to the preface. "The lecturer is well aware that the great subject of stellar motions, limited as far as practicable to stellar motions spec- trographically determined, has been presented incompletely and imperfectly, but he hopes that he has been able to furnish a glimpse into a surprisingly rich field of astronomical investiga- tion. When we recall that stellar radial velocities afford perhaps our best method of determining the scale on which the solar system is constructed (the solar parallax), and when combined with proper-motion data the best method of determining the scale on which the stellar system exists, certainly the most fruitful method of studying the evolution of double star systems, and a most promising method of studying the evolution of stars in general (illustrated by the relation existing between spectral classes and average radial velocities), we are quite prepared to acknowledge their almost unlimited power. It is a safe prophecy that the possession and study of the radial velocities of the brighter stars will but strengthen future demands for a knowledge of the motions of fainter and fainter stars. The spectrographic methods of observation referred to have been developed almost to the point of standardization, though the future is expected to introduce many and important improve- ments; and there is abundant justification for entering at once X PREFACE upon the observation of extensive programs embracing all stars down to visual magnitudes approximating 6^. The task is, however, too great for any one institution, and too great in each hemisphere for any one institution. It is hoped that a consid- erable number of observatories equipped with powerful tele- scopes may soon agree upon cooperative plans for securing the desired observations, following somewhat the ideas of the great organization {Die astronomische Oesellschaft) which is rapidly extending over the whole sky the accurate meridian determi- nations of stellar positions down to the ninth visual magnitude. ' ' For a large share of the observational materials which have been utilized in these lectures I am under obligations to the late Mr. D. 0. Mills, to the Carnegie Institution of Washington, and to essentially all of my colleagues on Mount Hamilton and on Cerro San Cristobal. W. W. Cambpell. June 1, 1912. CONTENTS PAGE Chaptee I. Historical and introductory. Theory of spectroscopic measurements of the radial velocities of celestial bodies. Types of spectra to be dealt with 1 Chapter II. Development of the photographic method. Conditions required to obtain accurate results. Proofs that observed motions are correct. Conditions other than motion affect the positions of spectral lines 40 Chapter III. Eotational velocities of members of the solar system. Eadial velocities obtained for individual stars. Accuracy attain- able for stars of various magnitudes and spectral types. Veloci- ties of groups of stars in different areas of the sky. Introduction to the solar-motion problem .90 Chapter IV. The solar motion as determined from stellar proper motions : consideration of the principal methods and results. Systematic motions of the stars. Distribution of the brighter stars with reference to the Milky Way .... ... 127 Chapter V. The spectrographic determination of the solar motion: advantages of the method; development of the method. Selection of materials for solution of the problem. Motions of related groups of stars. Eecent results for direction and speed of solar motion 163 Chapter VI. Studies of the stellar system. Eelation between the numbers and numerical magnitudes of radial velocities, by spectral types. Apparent systematic errors of radial velocities with respect to spectral types. Eolations between average observed radial velocities ami average space and tangential velocities. Stellar radial velocities with reference to angular distances from the Kapteyn preferential vertices. The average distances of classes of stars. Eadial velocities of eighty-eight large proper-motion stars .197 Chapter VII. Visual and spectroscopic binary stars. Eelation of periods of revolution to spectral classes. Eelation of eccentricities of orbits to periods of spectral classes. Stars with composite spectra. Eelative masses of primary and secondary components. The solar system may be an extreme type and not the prevailing type 234 Chapter VIII. Variable stars. Bearing of radial velocity observa- tions upon the interpretation of variable star systems. Applica- tion of stellar radial velocities to determinations of the solar parallax ... 282 STELLAR MOTIONS STELLAR MOTIONS CHAPTER I HISTORICAL AND INTEODUCTORY When we look at the sky on a clear night we see two distinct classes of objects. The early astronomers called one class the planets, or wanderers ; and the other class, the fixed stars. The term, "fixed star," is a misnomer and is becoming obsolete. We are learning to speak only of "the planets" and "the stars." There are no stars whose positions are fixed: all are in motion, with reference to any point, line or plane we may define in gen- eral terms. The planets of our solar system, in their orbits around the Sun, are travelling from 5 km. per second for Neptune up to 55 km. per second in the case of Mercury; yet the refined observations and calculations of the past two decades have established that the average star is moving even more rapidly than the average planet. Our Sun, as one of the ordinary stars, is no exception to the rule, in that it has been found to be travel- ling rapidly through space, carrying its family of planets along with it.i 1 The astronomers of today are so familiar with the ideas of stellar and solar motions that it is difficult to realize how recently these ideas were developed. Giordano Bruno, martyred liy the Inquisition in the year 1600 on account of his original views concerning scientific subjects, was perhaps the first to question the immovability of the stars. He said that we had no right to assume the fixity of the stars with reference to each other because, on account of their enormous distances from us, we could not hope to detect changes of position until after the lapse of long ages. Robert Hooke (1635-1703) was of the opinion that the stars do not occupy fixed positions, but that all the stars, including our Sun, may well be in relative motion.. — Posthumous WorTcs, p. 506. An important paper by James Bradley, presented to the Eoyal Society in 1748, remarked: "When the causes which affect the places of all the stars in general are known; such as the precession, aberration, and nutation; it 2 STELLAR MOTIONS The problems of motions within the solar system, and the problems of motions in the stellar system, possess widely variant orders of difficulty. This condition arises from our relative near- ness to the Sun and to the other planets, and from the inconceiv- ably greater distances of the stars. The crude instruments and methods of three hundred years ago, in the hands of Tycho Brahe and Kepler, were immeasurably better able to solve the problems of our own system than are the delicate instruments and the refined methods of today to solve the problems of the distant stars. In fact, our ability to make serious headway in the study of the stellar system is solely because the number of stars is so great that we can apply statistical methods and the doctrine of averages to them. The energies of astronomers prior to 1750 were all but exclusively devoted to the investigation of the solar system, in large part because they felt powerless to attack the problems of the distant stars. From that date on to the present, as telescopes became more perfect and more power- may be of singular use to examine nicely the relative situations of particular stars; and especially of those of the greatest lustre, which it may be pre- sumed lie nearest to us, and may therefore be subject to more sensible changes; either from their own motion, or from that of our system." — Phil. Trans. (Abridged Ed.), 9, 437-438, 1748. It was a favorite subject with Thomas Wright of Durham that our Sun and its planets, in common with all the stars, are in motion. — .471 Original Theory or New Hypothesis of the Universe, London, 1750, p. 52. We are perhaps justified in saying that the first comprehensive discus- sion of the subject in the literature of astronomy is due to Lalande, in the year 1776. From mathematical considerations he concluded, among other things, that our Sun, having a motion of rotation about its axis, could scarcely avoid having a motion of translation. His thoughtful statement follows : "II me reste h dire un mot sur un effet de la rotation solaire, dont les Physiciens n'ont point encore parll, mais qui sera peut-etre un jour un ph^nom&ne bien remarquable dans la Cosmologie; c'est le mouvement de translation du Soleil & de tout notre systfeme planetaire. "Le mouvement de rotation, consid^re comme I'efEet physique d'une cause quelconque, est produit par une impulsion communiqufie hors du centre. Jean Bernoulli calcule pour chaque PlanSte le point oil cette force doit avoir 6te appliquee, & proportion de la vitesse de sa rotation {Opera, 4, p. 283); mais une force quelconque imprimfie k un corps, & capable de le f aire tourner HISTORICAL AND INTBODVGTOBY 3 ful, and as the nature of the stellar problems was better compre- hended, the energies of a constantly increasing proportion of investigators have been turned to the stars. Our knowledge of the internal motions of the solar system is marvellously accurate. It has become a matter of almost routine computation to predict the positions of the planets, the times and places of solar eclipses, and other phenomena, a century in advance. Halley's comet, whose last appearance occurred in 1835, was rediscovered a few weeks ago within seven minutes of arc of its predicted place ; and it will reach its closest approach to the Sun on April 19, within three days of the predicted time. It is true that there are minute discrepancies in the motions of Mercury, Venus, the Earth, and Mars, but it is possible that Seeliger's researches^ on the attracting power of the finely autour de son centre, ne pent manquer aussi de d^placer le centre, & I'on ne sauroit concevoir I'un sans 1 'autre. II paroit done tris-vraisemblable que le Soleil a un mouvement T§el dans I'espace absolu; mais comme il entraine nficessairement la Terre, de meme que toutes les Planets & les Comdtes qui tournent autour de lui, nous ne pouvons nous apercevoir de ce mouvement, h, moins que par la suite des si6eles le Soleil ne soit arrive sensiblement plus pr6s des ifitoiles qui sont vers une region du Ciel, que de celles qui sont opposees; alors les distances apparentes des :fitoiles entr'elles auront augmente d'un c6te & diminue de 1 'autre; ce qui nous apprendra de quel c6t6 se fait le mouvement de translation du systSme solaire; mais il n'y a pour ainsi-dire que quelques instans d'6coul6s depuis que I'on observe; & la distance des iltoiles est immense; il est done assez naturel qu'on n'ait fait jusqu'ici aucune remarque k ee sujet "Si les positions des :fitoiles, observes par Hipparque il y a pr6s de deux mUle ans, avoient plus de precision, on pourroit commencer £l voir si les differences de longitudes sont plus grandes d'un c6te & plus petites de 1 'autre que celles qui avoient lieu de son temps; mais un jour viendra oil cette comparaison pourra nous apprendre quelque chose sur la question dont il s'agit. " — M^moires de I'AcadSmie Soyale des Sciences, Paris, 1776, p. 513. Herschel strongly suspected, as early as 1783, to quote his own words, "that there is not, in strictness of speaking, one fixed star in the heavens; .... there can hardly remain a doubt of the general motion of all the starry systems, and consequently of the solar one amongst the rest." — Phil. Trans. (Abridged Ed.), 15, 397, 1783. ^Sitmngsber. der Tc. Bayer, Alcad. Wiss., 36 (III), 595, 1906. A clear estimate of the discrepancies, and a most interesting discussion of 4 STELLAR MOTIONS divided zodiacal-light material will explain and eventually remove these discrepancies. The only other noteworthy dis- crepancy in the whole solar system relates to our Moon, for which there exists a considerable difference between prediction and observation. It is under the auspices of Yale University that a distinguished astronomer is placing our knowledge of the Moon's motions on an improved basis. The story of Newton's law of gravitation, as applied to the Sun, to the planets and their moons, and to the comets and meteors, if written by a worthy pen, would constitute the world's real epic: an epic of intel- lectual struggles, rather than of physical prowess. We scarcely exaggerate in saying that successive generations of astronomers, spanning two centuries, fully expected that Newton's law would ultimately be found to account for the general motions of all celestial bodies. However, the recent discoveries of forces pos- sessing quite different natures have weakened this point of view. For example, radiation pressure, discovered theoretically by Maxwell,' and established experimentally by Lebedew,* and by Nichols and Hull,' may be far-reaching in its consequences upon cosmical motions. It is too early to forecast the propelling effects of magnetic forces, whose existence in the Earth and Sun points strongly to their existence also in other celestial bodies. And we must not assume that other propelling forces, of natures entirely unknown, will not manifest themselves to future inves- tigators. However, we cannot doubt that all the stars, all the nebulse, all the dark and invisible bodies which must exist in profusion throughout space — in brief, all tangible bodies making possible explanations, are contained in Newcomb's Astronomical Constants, 1895, pp. 109, 123. [Note added March, 1912. — Brown makes interesting comments on Seeliger's results, in Mon. Not. B. A. S., 70, 342-344, 1910; and de Sitter likewise, in Mon. Not. S. A. S., 71, 408-409, 1911. Wacker {Inaug. Dissertation, Tubingen, 1909), Lorentz (Phys. Zeitschrift, 11, 1239, 1910) and de Sitter {Mon. Not. S. A. S., 71, 405, 1911) show that the Principle of Eelativity may be an important factor in explaining the pro- gression of planetary perihelia.] 3 A Treatise on Electricity and Magnetism, Oxford, 2, 391, 1873. i Ann. der FhysiJe, 6, 433, 1901. 5 Ap. J., 17, 315, 1903. HISTORICAL AND INTBODUGTOBT 5 up our sidereal universe — are moving in accordance with definite laws. "Will astronomers ever be able to tell their fellow men how each bright star in turn is moving, and how groups of stars, great groups as well as small ones, are related to each other? Will the starry heavens be reduced to a system, as the Sun, planets, satellites and comets have been fitted into the solar system? The methods of today, truly remarkable in their accu- racy, are contributing to this purpose and ambition; but time alone can tell the outcome. Figure 1 The motion of a star resolves itself naturally into two compo- nents: one measured along the line (called the "line of sight") drawn from the star to the observer, and the other at right angles to this line. The former component is known as the star's radial motion, or motion in the line of sight; and the latter as its proper motion. Thus, if S, S, S (Figure 1), represent three stars on the celestial sphere, each moving in a direction and with a speed indicated by its vector SA : then, the observer being in the direc- tion SO, the component BA represents the linear value of the proper motion in each case and SB the radial motion or radial velocity. We cannot describe proper motion in linear values except in the cases of the few stars whose distances are known, 6 STELLAR MOTIONS and it is customary to define proper motions in terms of the angles SOA through which the stars appear to move in the unit of time — a year or a century. It is customary and we have the power to define radial motions in absolute units, as, for example, kilometers per second. In the figure one of the stars has proper motion to the right, and the others to the left. Two of the stars have radial motions of approach toward the observer, and one has radial motion of recession from him. As aU stellar bodies are in motion, they are changing both their apparent positions on the celestial sphere and their dis- tances from us. That is, they have both proper motion and radial motion. Edmund Halley was the first to recognize that certain stars had actually moved. He announced in 1718 that Sirius, Aldeb- aran, Betelgeux, and Arcturus were certainly occupying posi- tions appreciably different from those assigned in Ptolemy's Almagest, as based upon the observations of Timocharis and Aristyllus about 300 years before Christ, and by Hipparchus about 130 years before Christ. Commenting upon these changes, he said : ' ' These stars being the most conspicuous in Heaven are in aU probability the nearest to the earth; and if they have any particular motion of their own, it is most likely to be perceived in them.'" When their instruments had reached a considerable degree of perfection, in the latter half of the eighteenth century, astron- omers began to determine the proper motions of the brighter stars, in this manner: Their positions on the celestial sphere — their right ascensions and declinations — were measured with great precision; at a later date, — twenty, forty, sixty years later, — the positions of the same stars were remeasured in the same manner; a comparison of the positions at the two epochs showed that some of the stars had moved appreciably in the interval, and at what angular rates; these rates being their so- called proper motions. As early as 1783, Maskelyne had determined the proper motions of seven of the first-magnitude stars in the northern a Phil. Trans. (Abridged Ed.), 6, 329-330, 1718. Sir Friedrich Wilhelm Herschel, 1738-1822 8 STELLAR MOTIONS sky, and Mayer of twelve of the brighter stars. Six of these were common to both lists, and the motions of thirteen stars were thus known to a fair degree of accuracy. To illustrate at once the great value of proper-motion knowledge, let us recall that in 1783 Sir William Herschel used these thirteen proper motions to support the theory that our Sun and its system of worlds must be travelling rapidly through space in the general direction of the constellation Hercules.'' As the accuracy of instruments and methods of observation improved, and espe- cially as the interval between the old observations and the new grew longer, the power to measure these minute stellar motions increased. The famous observations of star positions made by Bradley at Greenwich between 1750 and 1762 are noteworthy for their value as starting points in determining stellar motions. Today we know the proper motions of several thousand of the brighter stars to a fair degree of accuracy. These have fur- nished the basis for investigations of great importance as to the structure of the stellar universe. Nevertheless, proper- motion data, standing alone, have serious limitations: they are expressed in angular measurement; and to know that a star is changing its apparent position on the celestial sphere by one- tenth of a second of arc per year is to know next to nothing about the actual motion of that individual star. We need to know three other factors : First, the component of the star's motion toward or away from us: this component may be negligible, in some cases, or it may be a dozen-fold greater than the proper-motion component, in other cases. Second, the star's distance from us: a given angular motion, that is, proper motion, may mean the rapid motion of a very distant star, or the very slow motion of a close star. Third, the extent to which the star's apparent motion is affected by the observer's motion. To illustrate the last point, let us suppose that the motion of the Earth is carrying the observer due east with a speed of 19 km. per second. If a certain star is in reality also moving due east 19 km. per sec- ond, it will seem to us not to move at all; and another star in 7 See Chapter IV. HISTORICAL AND INTBODUGTOBY 9 reality moving due west 19 km. per second will seem to be moving with twice its true speed. In the same manner the observed motion of every star in the sky is not its true motion, for it is affected by the observer's motion. A moment's con- sideration will make plain that a full knowledge of the direc- tion and speed of our own star's motion is a sine qua non to a satisfactory study of the motions of the other stars. Twenty-two years ago we did not know the radial motion of any star in the heavens. Observed radial velocities for several scores of stars were published, it is true, but they could not be depended upon to be even approximations to the truth. We felt that we knew within 20° the direction of the solar motion, but we knew nothing as to the velocity with which it carried the observer on through space. This was variously estimated at from 10 to 70 km. per second. We knew the distances of not more than two score stars : measures of stellar distances presented difficulties so great that even today we possess reliable knowledge on the approximate distances of not over a hundred stars. At no point in astronomical science is fuller knowledge more desirable, more pressingly urgent, than in the subject of stellar distances; or, speaking technically, of stellar parallaxes. Sixty-two years ago the human mind had no conception that we should ever be able to measure the radial motions of the stars. Yet in the past twenty-two years the problem has been given practical solution; the radial velocities of more than a thousand stars have been determined. These velocities will enable us, both alone and in combination with proper motions and parallaxes, to solve many of the fundamental problems of stellar astronomy; not suddenly, but by rapid approximations to the truth. The unexpected by-products of the observations are scarcely less important than the foreseen results. In brief, the field of discovery here opened up has proved to be of super- lative richness. It is chiefly to this field that these lectures refer. In his celebrated work, Cours de philosophie positive (Paris, 6 vols., 1830-1842), Auguste Comte declared that "We shall 10 STELLAR MOTIONS never be able to study the chemical composition of the celestial bodies ; . . . . Our positive knowledge with regard to them will necessarily be limited to their geometrical and mechanical phenomena. It will be impossible, by any means, to include investigations of their physical, chemical (and other) prop- erties." Fraunhofer had discovered,* a quarter of a century earlier, that the visual solar spectrum is not merely a band of variously colored light, passing by insensible gradations from violet at one end to deep red at the other, but that this band is crossed by a multitude of dark lines (at least 600) ; and that the spectra of the brightest stars are crossed by many similar lines. Laboratory investigations on a few of the chemi- cal elements had shown that their spectra consist character- 8 Wollaston had indeed observed seven lines in the solar spectrum, in 1802, but they were given a most peculiar interpretation, and the subject was not further pursued. Wollaston described the observations as follows: ' ' I cannot conclude these observations on dispersion, without remarking that the colours into which a beam of white light is separable by refraction, appear to me to be neither seven, as they usually are seen in the rainbow, nor reducible by any means (that I can find) to three, as some persons have conceived; but that, by employing a very narrow pencil of light, four primary divisions of the prismatic spectrum may be seen, with a degree of distinctness that, I believe, has not been described nor observed before. "If a beam of daylight be admitted into a dark room by a crevice %0 of an inch broad, and received by the eye at the distance of 10 or 12 feet, through a prism of flint glass, free from veins, held near the eye, the beam is seen to be separated into the four following colours only, red, yellowish green, blue, and violet; .... "The line A that bounds the red side of the spectrum is somewhat confused, which seems in part owing to want of power in the eye to converge red light. The line B, between red and green, in a certain position of the prism, is perfectly distinct; so also are D and E, the two limits of violet. But C, the limit of green and blue, is not so clearly marked as the rest; and there are also, on each side of this limit, other distinct dark lines, / and g, either of which, in an imperfect experiment, might be mistaken for the boundary of these colours." — Phil. Trans., 1802, p. 378. Fraunhofer 's remarkable observations (in 1814 and later) were made possible by the use of a narrow slit to admit the light into his spectroscope, as well as of a telescope to collect the light of a star and form its image on the slit. — Venlcschriften der Tc. ATcad. Munchen, 5, 1817. HISTORICAL AND INTRODUCTORY 11 istically of bright lines, certain of which occupy the same posi- tions in these spectra as are occupied by prominent dark lines in the solar and stellar spectra. The significance of these spectral features, since shown to be fimdamental and almost unique in their far-reaching power for analysis, were then GuSTAV Robert Kirchhoff, 1824-1887 unknown, as may be inferred from Comte's famous dictum. Twenty-four years after the publication of Comte's conclusion, the fundamental principles of spectroscopy were discovered by Kirchhoff,* whereupon it became a comparatively simple matter B Monatsierichte fc. Akad. Berlin, 1859, p. 664. Lack of space prevents 12 STELLAR MOTIONS to determine the chemical compositions of the stars, within limits set by their brightness and their physical conditions. But we pass by this great field of astronomy and physics, as not directly concerning our subject. It was soon recognized that the spectroscope supplies, in theory at least, the long- hoped-for method of measuring the components of stellar motions in the line of sight — their radial velocities. The effect of the approach or recession of a light source, such as a star, upon the spectrum was first considered by Christian Doppler, of the University of Prague, whose conclusions were announced^" in 1842. He explained the now familiar fact that a full and painstaking description of the fundamental principles of spectro- scopy, but we may state them briefly, as follows: 1. When a solid body, a liquid, or a highly condensed gas is heated to incandescence, its light when passed through a spectroscope forms a con- tinuous spectrum: that is, a band of light, red at one end and violet at the other, uninterrupted by either dark or bright lines. 2. The light from the incandescent gas or vapor of a chemical element passed through a spectroscope forms a iright-line spectrum: that is, one consisting entirely of isolated bright lines, distributed differently through- out the spectrum for the different elements, or of bright lines superimposed upon a relatively faint continuous spectrum. 3. If radiations from a continuous-spectrum source pass through cooler gases or vapors before entering the spectroscope, a dark-line spectrum results: that is, the positions which the bright lines in the spectra of the vapors and gases would have are occupied by dark or absorption lines. These are frequently spoken of as Fraunhofer lines. To illustrate: The gases and vapors forming the outer strata of the Sun's atmosphere would in themselves produce hright-line spectra of the elements involved. If these gases and vapors could in effect be removed, without changing underlying conditions, the remaining condensed body of the Sun should have a continuous spectrum. The overlying gases and vapors absorb those radiations which the gases and vapors would themselves emit, and thus form the dark-line spectrum of the Sun. The stretches of spectrum between the dark lines are of course continuous-spectrum radiations. There is an endless variety of radiation and absorption spectra of the elements (see Kayser's Eandbuch der Spectroscopic, Baude 2 u. 3). There is also a great variety of stellar spectra; for example, there are many stars whose strong continuous spectra are crossed by both bright and dark lines and bands of various widths. (See Ap. J., 2, 177, 1895.) 10 Abhandlungen d. Tc. Bohmischen Gesell. d. Wiss., 2, 467, 1841-1842. THEORY OF SPECTROSCOPIC MEASUREMENTS 13 when a source of sound waves, such as the whistle of a loco- motive, is moving rapidly away from or toward the listener, the pitch of the sound perceived is not the normal pitch : it is lowered in the case of recession and raised iu the case of approach. The sound waves are, in effect, lengthened and the pitch lowered for a receding locomotive ; the waves are, ia effect, shortened and the pitch raised for an approaching locomotive. Reasoning analogously upon light as a phenomenon of waves in the ether, he correctly concluded that these waves, as perceived by an observer, would in effect be shortened if the light source is approaching him and lengthened if receding from him. It is practically immaterial with velocities thus far observed whether the light source is approaching the observer, or the observer approaching the light source — the observed wave lengths would be shortened in either case; and similarly for the recession of the light source from the observer or of the observer from the light source. As the red rays of the spectrum are the visible result of long waves, speaking popularly, and the violet rays at the other end of the spectrum the result of short waves, Doppler concluded, erroneously, that a star mov- ing very rapidly toward us would be changed in color to a more violet tinge, and one moving rapidly away to a redder tinge. He overlooked the fact that there are stretches of invisible spectrum to the red and to the violet of the visible spectrum, which would be drawn upon to compensate for any loss from the cause described, and thus leave the color sensibly unchanged. Practically, the stellar motions of approach and recession are so small in comparison with the velocity of light that no change of color would be perceptible to the eye, even aside from the compensating principle. It appears to have been Fizeau, in 1848, who first enunciated the principle, correctly, that motions of approach and reces- sion must cause corresponding shiftings of the entire spectrum, including the dark lines of Fraunhofer, toward the violet and toward the red, respectively, but without change of color. He outlined methods for applying the principle to measuring the motions of celestial bodies toward and away from the observer. 14 STELLAR MOTIONS While these methods were sound theoretically, they were unpractical. All matters spectroscopic were still mysterious, and Fizeau's statements attracted no serious attention. In fact, his lecture on the subject, in 1848, before a minor society, in Paris, was not published" until 1870. Following the impetus Christian Doppler, 1803-1853 given to spectroscopic investigations by KirchhofE in 1859, Dr. (now Sir William) Huggins and Professor Miller, jointly engaged in observing the spectra of stars in 1862-1863, realized^^ 11 Ann. de Chimie et de Physique, 19, 217-220, 1870. 12 PM. Trans., 158, 529, 1868. THEORY OF SPECTROSCOPIC MEASUREMENTS 15 that stellar motions to and from the observer should displace the lines in the spectra, and unsuccessful efforts to measure the displacements were made in 1866 by Dr. Huggins; but Clerk Maxwell was the first to present the subject in definite form.^^ None of these eminent investigators realized the tremendous importance which the Doppler-Fizeau principle was later to attain in practical astronomy, if we may judge from the tardiness of publication characteristic of all. It may not be definitely known what caruses the phenomenon called light; but, speaking popularly, and according to the mechanical theory, we may say that waves of energy, of an infinite variety of lengths, travel outward in every direction from the light source. In the ordinary image of a star, whether formed by the eye alone, or by an achromatic telescope and the eye combined, the light waves of all lengths fall in a confused heap upon the same minute point, and the observer is unable to say that rays corresponding to any given wave lengths are present or absent. "When the star's light has been passed through the prisms or diffracted from the grating of a spectroscope, these rays are separated, one from another, and arranged, side by side, in perfect order, ready for the observer to survey them, and to determine which ones are present in superabundance, and which ones are lacking wholly or in part. The following comparison is a fair one : The ordinary point image of a star is as if all the books in the University library were thrown together in a disorderly but compact pile in the centre of the reading room: we could say little concerning the contents and characteristics of that library. The spectrum of a star is as the same library when the books are arranged on the shelves in complete perfection and simplicity, so that he who looks may appraise its contents at any or all points. The retina of the human eye is affected only by those waves whose lengths lie between approximately 0.00078 and 0.000,38 mm. Waves of the former length, at the extreme red end of the visible spectrum, are 1300 to the mm. ; and of the latter length, at the extreme violet end of the visible spectrum, are 2700 to the mm. The wave 13 PM. Trans., 158, 532, 1868. 16 STELLAR MOTIONS length varies continuously from one end of the spectrum to the other, and every shade of pure color — more conveniently, every point in the spectrum — has its own definite wave length. Thus, a point can be selected in the yellow whose wave length is %ooo of a mm. ; a point in the orange whose wave length is %7oo of a mm. ; and so on for every wave length between the limits noted for the retina. Extending apparently indefinitely into the red of the spectrum of the Sun are rays which, as carriers of heat, have been investigated by Langley and Abbot as far as to wave length 0.00534 mm. ; and extending into the violet, appar- ently indefinitely, is the ultra-violet region, invisible, but investigated photographically by Lyman for the element hydro- gen as far as to wave length 0.000103 mm. It has been found convenient to define a given point in the spectrum in terms of the number of ten-millionths of a milli- meter which measure the wave length of the radiations reach- ing that point. The imit, formerly called the tenth-meter, is now known as the Angstrom. Thus the hydrogen line in the green- blue, whose position corresponds to wave length .00048615 mm., is said to have a wave length of 4861.5 Angstroms (formerly 4861.5 tenth-meters), or 4861.5 A, or, simply, the line is said to be at 4861.5 A ; and similarly for all other lines throughout the spectrum. For a monochromatic ray emitted by a light source whose distance from the observer is not changing, let A denote the number of Angstroms in the wave length, and n the number of waves received by the observer in a mean solar second. Let X' and n' be the changed values of A. and /; resulting from a velocity of the light source with reference to the observer, or of the observer with reference to the light source, amounting to ± V km. per second ; -\- V for motions which further separate the light source and observer, and — V for motions which bring them nearer together. Theoretically, it makes an extremely slight difference whether the light source or the observer is moving, but practically the difference is easily negligible for all known cosmical motions, as onlj^ the higher orders of minute- ness are involved. Eecalling that the velocity of light through THEORY OF SPECTROSCOPIC MEASUREMENTS 17 interstellar space is very nearly 299,860 km. per second, we shall have: ^, ^ 299,860 ,^, n'=^n 1 ; (1) 299,860 ±F the sign of the Y term being + for recessions and — for approaches. Since the effective wave length is inversely propor- tional to the number of waves received per second, or in any given unit of time, we may write: ,,^,299,860 ±F^^ F \ 299,860 299,860/ or A.'-X = AA.= ± _AZ_. (3) 299,860 In other words, the change, AX (in Angstrom units), corre- sponding to a relative velocity of recession or approach of ± F km. per second, is equal to the normal value of the wave length, multiplied by the ratio of the relative velocity of the light source and observer to the velocity of light. To find the velocity F, required to alter the apparent wave length by 1 A, it is but necessary to let A \ = 1, and we have : y^^^ 299,860 _ ^^^ A Thus at 3000 A, a velocity of ± 100 km. per second will change the apparent wave length to 3001 A, and 2999 A, respectively; at 7500 A, a velocity of only ±40 km. is required to change the value of X to 7501 A and 7499 A, respectively. In practice Fi is tabulated for uniformly distributed points throughout the spectrum. Transforming equation (3) into F^± ^99,860 ^,^ (5) X expressing A X in terms of 1 A, and replacing the fraction by its value from (4), we obtain: F=±F,.AX; (6) 18 STELLAR MOTIONS that is, a velocity, V. of approach or recession, will change any k by amount AX, expressed as Angstroms, to the extent F/Fj. Conversely, if it be found from observation that a V7ave length has been changed by the amount AX, we may compute from (6) the radial velocity which produced this change. The problem of determining the radial velocities of the stars con- sists, in outline, in measuring the displacement AX by means of a suitable spectroscope; or, if photographic methods are employed, by a spectrograph. In practice the displacement is measured with a micrometer, and it is expressed directly in terms of one revolution, r, of the micrometer screw. Equation (6) is arranged thus: r=±rr..Ai. (7) /■ rF, is tabulated for definite points in the spectrum, and is the linear displacement as read directly with the microm- eter." 1* The derivation of radial velocities from stellar spectrograms is most conveniently made through the use of standard reduction tables constructed for each spectrograph concerned. If a solar spectrogram be secured, under good conditions, on a fine-grained plate, and a considerable number of the best lines in the field of good definition be measured with a micrometer microscope, an equation expressing the relationship between the assumed wave lengths of the lines and the micrometer readings on the lines can be determined empirically by means of the simple method first suggested by Cornu and later discovered independently and developed by Hartmann. {Fuhl. Astroph. Ohs. Potsdam, 12 {Anhang), 3-25, 1898; Ap. J., 8, 218, 1898.) If the region of spectrum does not cover too large a range of wave- length values, the deduced equation will probably reproduce the wave lengths of micrometer readings within the limits of unavoidable error. By means of the equation we may compute the standard micrometer readings for all the lines of definitely assigned wave lengths which we desire to use. Further, it is a simple matter to compute the radial velocity values, or factors, rV„ corresponding to displacements of the separate lines through one revolution, r, of the micrometer screw. Here are extracts from the standard table used in reducing the spectrograms obtained with the original Mills spectrograph. This table, constructed before Hartmann 's method was available, is based upon an equation determined empirically from a Mills spectrogram of the Sun, as described in Ap. J., 8, 142-144, 1898. THEORY OF SPECTROSCOPIC MEASUREMENTS 19 The spectroscope as applied to stars is used in connection with a telescope. The latter serves to collect a great quantity of the star's light and to deliver this light properly to the spectroscope. The human eye sees a star by virtue of the rays which enter the pupil, estimated to be 5 mm. in diameter at night. The telescope collects more light than the eye does, in proportion as the area of the object-glass is greater than that of the pupil, and delivers this light, neglecting great losses by reflection at the surfaces of the lenses and by absorption within the lenses, to the spectroscope. Even for first-magnitude stars, as observed with the largest telescopes^^ existing, this is none TABLE I Miorometer Beading X O rV, 4238. 188A 0.033 188.6 km. 38.970 0.326 188.8 4337.216 34.363 215.9 37.414 34.427 216.0 37.725 34.528 216.0 38.084 34.642 216.1 4441.881 65.399 245.5 42.510 65.572 245.6 15 The prospects are not favorable for the erection of refracting tele- scopes larger than the 36-inch Lick and 40-inch Yerkes telescopes, mounted according to the same system as they are, chiefly because there is reason to believe that a lens greater than 40 inches in diameter, supported at the edge, would bend sufficiently under its own weight to make imperfect images. Again, it must not be overlooked that lenses larger in diameter mean thicker lenses, and greater thicknesses of glass mean increased loss of light by absorption. On the contrary, there appears to be no important reason, save that of cost, why reflecting telescopes should not be increased in diameter several fold. K < § H O W B< CO ►J Peraei and p Morwcerotis are fairly well defined. 38 STELLAR MOTIONS rate determinations of their radial velocities is merely a question of their brilliancy; neglecting the fact, for the moment, that other factors than velocity may affect apparent wave lengths. The period of existence succeeding that of Secchi's Type IV stars has illustrations near at hand, apparently, in Jupiter and the other planets of the solar system, invisible save by borrowed light; and the radial velocities of all such (in our system) are apparently obtained more readily and accurately by means of the spectrograph, whose results must here be based upon reflected light, than through the indirect methods based upon solar parallax (see Chapter VIII, p. 313). When the internal heat of a body shall have become impotent, which is apparently the case with the planets, the future promises nothing save the slow levelling influences of its own gravitational and meteorological elements. It is true that a collision may occur to transform a dark body's energy of motion into heat sufiieient in quantity to convert the body into a glowing nebula, and start it once more over the path of evolution. This is a beautiful theory, but the facts of observation do not give it extensive support. There is little doubt that the principal new stars of recent years have been the results of collisions or of close approaches, either of two dark bodies, or of a dark massive body and invisible resist- ing materials. The suddenness with which intense brilliancy is generated would seem to call for the former, but the latter is vastly more probable, in view of many facts. The typical nova spectrum, of very broad bright and dark bands, gives way to a nebular spectrum of broad bright bands in a few months ; but in every ease thus far observed the bright nebular bands grow faint very rapidly, and in the course of a few years leave a continuous spectrum, apparently that of an ordinary faint star. Either the masses involved in the phenomena are extremely small, for stars, or the disturbances are but skin deep. In any ease, the novae have afforded little evidence as to the complete renebularization of dark bodies. The spectra of novse, in their brighter stages, have contained exceedingly broad and diffuse bright and dark bands, with only one partial exception — that of Nova Persei, in which, for a time, there were well-defined narrow lines of calcium TYPES OF SPECTRA 39 and sodium. This was the only nova out of six or eight, in the last two decades, whose radial velocity^" could be measured at all. The comets, which remain the most mysterious of all sidereal objects, have composite spectra. The denser materials forming the nucleus send us a reflected solar spectrum, in which the Fraunhofer lines have been photographically recorded with great success. The most prominent features of comet spectra, however, are the bright bands due to carbon and cyanogen, which have their source chiefly in the more or less condensed head, the carbon monoxide bands which have been observed to proceed chiefly from the tails of the brighter recent comets, and the orange sodium lines which have been seen in the spectra of a few comets whose orbits carried them relatively near to the Sun. There is little doubt that the radial velocities of the brightest comets could be observed, with limited accuracy, by means of the spectrograph, but velocities obtained by computation from orbital data would undoubtedly possess higher orders of accuracy. We have described the principal classification systems of stellar spectra, and have sketched with exceeding brevity the more generally accepted order of stellar evolution, referring at the same time to the adaptability of the various spectral types for radial velocity determinations. In the following chapter we shall consider the methods which have enabled us to measure the radial velocities of more than one thousand objects with a satisfactory degree of accuracy. i^LicTc Ohs. Bull, 1, 50-51, 1901. CHAPTER II DEVELOPMENT OF THE PHOTOGRAPHIC METHOD The problem of determining stellar radial velocities in the manner described, so exceedingly simple in theory, has been in practice one of the most difficult in the history of astronomy. Using visual methods only, the best efforts of the most expe- rienced observers met with signal failure^ for more than twenty years, and doubts even as to ultimate success were generally felt. The lines, so distinct and capable of accurate measurement in the brilliant solar spectrum, appear indistinct and are difficult to measure in the spectra of the brightest and most favorable stars. Again, the displacements of the lines, even for high velocities, are really very minute. With the average dispersion employed, a speed of 10 km. per second (6 miles) caused a displacement of the order of 0.01 mm. (0.0004 inch), and speeds much smaller than this were to be observed. The imperfections of instruments and methods, incident to a subject so delicate, introduced errors many fold larger than the quantities to be measured. The efforts of several observers of great skill, in England and in Germany, between 1863 and 1887, some of these efforts continuing through many years, were unproductive of a single trustworthy result for the velocity of any star. The probable error of a single observed velocity was much greater than the average velocities of the stars. As Vogel, the ablest of observers, has remarked, it was often necessary to observe for hours, even under the best conditions, before an estimate of a displacement, and still less a 1 It is furthest from my purpose to convey the meaning that these efforts were useless; in fact, I should like to be among the first to express respect for and appreciation of the early struggles to measure radial -velocity by visual methods. Every effort of the pioneers, whether a success or a failure, is an index pointing the way of success to the observers who follow them. DEVELOPMENT OF PHOTOGRAPHIC METHOD 41 measurement, could be attempted. Visual methods failed com- pletely until the year 1890, when the combination of the 36-ineh Lick refractor, an efficient spectroscope, good atmospheric con- ditions, and the comprehending observer Keeler, gave us our first reliable visually determined stellar velocity. In that year Keeler found for Arcturus an approach of 6 km. per second ; for Aldeb- aran a recession of 55 km. per second; and for Betelgeux a recession of 16 km. per second r results shown by many later photographic observations to be substantially correct. Illustrating the untrustworthiness of earlier visual observa- tions, we may say that two observers published the speed of Arcturus, probably the star most favorable for this purpose in the entire northern sky, as 73 and 89 km. per second, recession, respectively. The speeds of only three stars altogether were measured by Keeler. So difficult were the observations, even with the unequalled combination of favorable circumstances, that the number of stars capable of passably accurate visual measure- ment was seen to be very limited. At the same time Keeler made a magnificent contribution to the general subject by measuring the velocities of approach and recession of thirteen planetary nebulae and the Great Nebula in Orion. Although these objects are very faint as compared with stars, — only one of the fourteen being visible without a tele- scope, — they are capable of observation in this manner because nearly all their light is condensed into a few monochromatic bright lines, which are not widened under increased dispersion ; whereas a star's light is distributed thinly over a large area of spectrum. Drawings of three of the planetary nebulee and of their visual spectra (with the slit of the spectroscope passing through the nebular nuclei) are reproduced in Figure 2. Keeler obtained a recession of 17.7 km. per second for the densest part of the Orion nebula, and the results for the thirteen planetary nebulee varied from 50 km. per second recession to 65 km. approach. Keeler 's work in 1890-1891 represents the high- water mark in stellar nebular spectroscopy by visual methods. It seemed at the time that perhaps fewer than fifty stars in the 2Pm6L LicTc Ois., 3, 395-196, 1894. 42 STELLAR MOTIONS entire sky could be observed satisfactorily by visual spectroscopy, even with the assistance of the most powerful existing telescopes, and it was considered improbable that the number of nebulae in the whole sky, north and south, whose velocities could be meas- ured in this manner, exceeded forty. Plans were formed to apply visual measurement to as many stars and nebulae as pos- sible; but before further progress could be made an incom- parably better method was discovered and announced. This method is based upon the application of the photographic dry- plate to the problem. XV iio seo m 600 ojo niP /ho +-*- i»^i5 mJ^ _^- jIl^S^ ^^ •"^'^'^ ■ :.^ipi",,,^.,, ^, , ^^ ^^^^ B ■ " ^ ''"'-- '.''f^wBB^St '." ■.-/■'!-.■>■■■ --^oliKiaii^' The New Mills Spectrograph REQUIREMENTS FOR ACCURATE RESULTS 49 an opening of rectangular cross section in the casting at the base of the prism box is pivoted at the centre of the casting in such a way that it is free to rotate about the pivot through an angle of several degrees. The ends of this bar are attached to the sup- porting trusses. A cylindrical ring bearing near the upper end of the spectrograph receives a spherical flange of the spectro- graph casting in such a way that the spherical flange has uni- versal freedom of motion within the cylindrical ring. This form of support ensures that any strains generated in the truss system cannot induce corresponding strains in the spectrograph. Neither of the two spectrographs shows any trace of differential flexure.' In the original Mills spectrograph, as in similar instruments elsewhere, the collimator section alone was moved with reference to the remainder of the spectrograph, in order to place the slit in the focus of the telescope objective. Attention was called in Chapter I to the large variations of focal positions at different or with changing temperatures. In the MiUs spectrograph at Santiago and in the new Mills spectrograph at Mount Hamilton, the ring bearings near the slits and the sliding supports for the pivoted bar bearings near the lower ends enable the spectro- graphs as a whole to be moved parallel to the axes of their colli- mators easily and quickly, in order to place the slits in the focal plane. [Note added in 1911. — The simple devices described above, for supporting spectrographs at points near their two ends, and for moving the instruments as a whole into the proper focal posi- tions, as well as the construction of the immediate spectrograph mountings in box form from thin plates, have been quite widely adopted in other instruments, apparently with entire success; for example, in the 1-prism spectrographs of the Allegheny," Ottawa and Detroit observatories.] The method of impressing the comparison spectrum on either side of the stellar spectrum is an important matter. If tempera- ture changes or flexure effects occur during the time of an » A more detailed description of the supporting system, with illustrative photographs, may be found in Puil. LicTc Ohs., 9, 50-53, ] 907. dPuU. Allegheny Ois., 2, 3, 7, 1911. 50 STELLAR MOTIONS exposure to the stellar spectrum, the spectral lines will be more or less blurred, and be slightly displaced, as many observers have pointed out. It is essential that the comparison spectrum be per- mitted to fall upon the plate several times at stated intert'-als throughout the exposure, if the exposure is of more than fifteen The New Mills Spectrograph REQUIREMENTS FOR ACCURATE RESULTS 51 or twenty minutes ' duration. Wright 's device" for sending the light from two comparison-spectrum sources through two minute totally reflecting prisms, whose adjoining edges define the length of the slit, permits the throwing in of the comparison spectrum as often as is desired without interrupting the exposure on the star spectrum, and does not require dangerous handling of the delicate mechanism in the vicinity of the slit-plates. The device has been used on all the spectrographs of the Lick Observatory since 1900. It was adopted in designing the 1-prism spectro- graph of the Allegheny Observatory, and is probably used at a few other places; but that it has not been generally adopted by stellar spectroscopists is to me a surprising fact. The errors due to variations of temperature during exposures of considerable length, which were reduced in amount by wrap- ping 'the more vital parts of the spectrograph with heavy insu- lating materials, were later almost entirely eliminated by the adoption of devices for maintaining the spectrograph at a sen- sibly constant temperature. It was well known that the increase and decrease in the indices of refraction of the prism glass with rising and falling temperatures were the principal sources of error. The expansion and contraction of the metallic mounting with varying temperature, though less effective in displac- ing the spectrum, were nevertheless far from negligible. Des- landres^^ was the first observer who attempted to eliminate this source of disturbance. He mounted long strips of metal around and near the prisms and lenses of the Paris Observatory spectro- graph, connecting them with a metallic thermometer in such a way that a slight fall in the temperature would produce an electric contact, and pass a current over the metallic strips. This would heat the air in the spectrograph, which in turn would raise the temperature of the thermometer, and break the contact at the proper point. In this manner the temperature of the air around the prisms and lenses would oscillate between two close limits. Deslandres made more extensive use of another very ingenious device. The outer walls of the spectrograph were 10 4p. J., 12, 274, 1900. 11 Bull. Astr., 15, 57-61, 1898. 52 STELLAR MOTIONS hollow, and so designed that water from the city supply could be circulated continuously through the channels in the walls. In this manner temperature variations within the spectrograph were decreased to about one-third their natural amount. Des- landres further suggested that it would theoretically be possible to render temperature variations essentially ineffective, by con- structing the mounting of Guillaume's nickel-iron composition known as invar, whose temperature coefficient of expansion is about one-fiftieth that of ordinary steel, and by selecting glass for the prisms whose indices of refraction would be independent of temperature. About the same time Lord^^ attempted to maintain the prisms of the Columbus spectrograph at a fairly constant temperature by placing coils of resistance wire outside of the prism box, the whole being wrapped in a layer of felt. A thermometer, whose bulb was within the prism box, was read from time to time, and, following its indications, the heating current was turned on for short intervals by hand. Wright^^ made the useful suggestion that the entire spectro- graph should be enclosed in a tight-fitting box on whose inner surfaces resistance wires would be disposed. This idea was adopted for the Mills spectrograph and applied in 1900, as shown in the illustration. One-half of the constant-temperature C£ise is shown in position on the instrument, and the other half is on the floor. It is lined with thick hair-felt. The two halves fit together by a tight tongued-and-grooved joint, and the metal clamps which bind them together serve also to conduct the elec- tric current between the wire systems in the two halves. An open-ended thermometer, forming a delicate thermostat, mounted on the inner surface of the case, operates a relay which is con- nected with the heating circuit. The sensitiveness of the system is such that when the temperature of the air in the case falls 0°.05 C. the heating current is turned on. When the heat thus generated has brought the temperature of the air in the case up to the level selected for the night, the current is .automatically 12 Ap. J., 8, 66, 1898. is^p. J., 11, 259, 1900. REQUIREMENTS FOR ACCURATE RESULTS 53 turned off. An electric fan within the case prevents stratifica- tion of the air. The heating current is turned on and off, on and off, every few minutes automatically throughout the night, keep- ing the temperature in the case nearly constant without attention on the part of the observer. The superiority of this system of control lies in the fact that the air outside of the prism box is kept nearly constant, giving the assurance that the temperatures within the prism box, and especially those prevailing in the slowly conducting glass prisms, are still more nearly constant. (To be accurate, it should be said that the heating current was turned on and off by hand during about one year, according to the readings of a sensitive thermometer within the wooden case, before the adoption of the automatic thermostat control.) A comprehensive treatment of this subject, including a description of the elaborate constant-temperature control of the Potsdam 3-prism spectrograph, has been given by Hartmann.^* With the flexure and temperature factors essentially elimi- nated, as described in the preceding paragraphs, there is no apparent reason why the radial velocities of fifth-magnitude stars cannot be observed as accurately with exposures of eighty minutes as first-magnitude stars with exposures of two minutes, using the same spectrograph and sensitive plates; especially as any slight residual disturbances from these sources are in prac- tice reduced almost to the vanishing poiat by inserting the com- parison spectrum, not all at once, but at previously determined intervals throughout the exposures. To guard against misunderstanding, it is well to repeat that we cannot measure the speeds of all stars, even of all bright stars, equally well. As explained in Chapter I, there are some stars with which we can do nothing because their spectra do not contain measurable lines. Other spectra have only broad and ill-defined lines which cannot be measured accurately ; and so on, through a great variety of spectra measurable with increasing accuracy, up to the stars whose spectra contain a few sharply liAy. J., 15, 172, 1902. o H O o H a » ^^<^ ffU^ory. -.., .^jy.,^.^^,^ -. Sir William Huggins, 1824-1910 TABLE n Velocities Corresponding to Displacements of One Angstrom Unit Line V. Log r. Wave Length V. Log V, km. km. Nel). 3727. 80.46 1.9056 3000A 99.95 1.9998 Hf 3889. 77.10 1.8871 3200 93.71 1.9718 K 3933.82 76.23 1.8821 3400 88.19 1.9454 H 3968.62 75.56 1.8783 3600 83.29 1.9206 He 3970.2 75.53 1.8781 3800 78.91 1.8971 He 4026.4 74.47 1.8720 3900 76.88 1.8858 H8 4101.9 73.10 1.8639 4000 74.96 1.8748 Fe 4308.08 69.60 1.8426 4100 73.13 1.8641 H7 4340.63 69.08 1.8394 4200 71.38 1.8536 He 4471.7 67.06 1.8264 4300 69.73 1.8434 Mr 4481.4 66.91 1.8255 4400 68.14 1.8334 Neb. 4686. 63.99 1.8061 4500 66.64 1.8237 HP 4861.5 61.68 1.7901 4600 65.18 1.8141 Neh. 4959.0 60.47 1.7815 4800 62.46 1.7956 Ne"b. 5007.0 59.89 1.7773 5000 59.97 1.7779 Ill 5183.79 57.85 1.7623 5200 57.66 1.7609 E, 5269.72 56.90 1.7551 5400 55.53 1.7445 Cor. 5303. 56.55 1.7.524 5600 53.54 1.7287 T), 5875.9 51.03 1.7078 5800 51.70 1.7135 D. 5890.19 50.91 1.7068 6000 49.98 1.6988 Di 5896.35 50.86 1.7063 6500 46.13 1.6640 Ha 6563.04 45.69 1.6598 7000 42.84 1.6318 B 6867.6 43.66 1.6401 7500 39.98 1.6019 A 7594.06 39.49 1.5964 8000 37.48 1.5738 TABLE ni The Earth's Orbital Velocity F„ and the Deviation i when the Sun's Longitude is © Va Log Va i Va Log F„ i 0° 29.87km 1.4752 +56' .3 180° 29.68km 1.4724 —56'. 7 10 29.78 1.4740 +57 .5 190 29.76 1.4737 —57 .6 20 29.70 1.4727 +57 .0 200 29.85 1.4749 —56 .8 30 29.61 1.4715 +54 .8 210 29.93 1.4762 —54 .2 40 29.53 1.4703 +50 .8 220 30.01 1.4773 —50 .1 50 29.46 1.4692 +45 .4 230 30.08 1.4783 —44 .4 60 29.40 1.4683 +38 .4 240 30.14 1.4792 —37 .5 70 29.34 1.4675 +30 .3 250 30.20 1.4800 —29 .4 SO 29.30 1.4669 +21 .2 260 30.24 1.4805 —20 .5 90 29.28 1.4666 + 11 .4 270 30.26 1.4808 —11 .0 100 29.27 1.4664 + 1 .2 280 30.27 1.4810 — 1 .2 no 29.28 1.4665 — 8 .9 290 30.26 1.4809 + 8 .6 120 29.30 1.4668 —18 .8 300 30.24 1.4806 +18 .2 130 29.33 1.4673 —28 .1 310 30.21 1.4801 +27 .3 140 29.38 1.4681 —36 .5 320 30.16 1.4794 +35 .6 150 29.44 1.4690 —43 .8 330 30.10 1.4786 +42 .8 160 29.51 1.4700 —49 .7 340 30.03 1.4776 +48 .8 170 29.59 1.4712 —54 .0 350 29.95 1.4764 +53 .4 180 29.68 1.4724 —56 .7 360 29.87 1.4752 +56 .3 1 ■4-1 oinoiraoinomoinoioo '^ M^c:lCQ(^!<^Jr^I-lOOOiOioooo (McMC0Ji-(CqC0' OOrHrHiHCqCqCMCOiMCOCOCO o o O oiooTt oT 3 ST § g lo o ro 5 m ^ CO 00 (1) ° f -w -. J rQ „ CS ja =" ^ ^ CS I^- ^ -H S ^ =" ,a o o ■S a a O 1— I -2 a I J I ^ O h & ^ -a «> 03 ID rj o cS » •9 5l a a 2 S 2 § 5 --S a " •j3 a 2 S 0} CO i w > o -g 1^ -a « CO cS o S § ^ la a> -3 a ® 3 70 STELLAR MOTIONS corrections are taken from a simple table as functions of the distances that the selected points of observation on the compari- son lines are from the central line of the star spectrum.^^ By using a suitably curved slit, instead of a straight slit, the spectral lines are straight, and the so-called curvature correction is completely eliminated. The Mills spectrographs at Mount Hamilton and in Chile have been provided with such curved slits since about 1900, but nearly all spectroscopic observers con- tiaue to use straight slits, requiring the application of curvature corrections to the results. To be accurate, it should be said that the curvature of the slit is computed to give straight lines in the middle of the photo- graphed spectrum. To the red of the middle the stellar and comparison lines will be very slightly concave toward the red, and to the violet of the middle the lines will be very slightly concave toward the violet. The departures from straight lines in the regions of good definition will be so slight as not to be troublesome in making the measures, and the slight errors due to curvatures in the two directions will usually balance each other. To fix in mind the foregoing principles, and as an introduction to another phase of the subject, we shall examine some of the actual results of observation. Here are three check observations of Mars and three of Venus. The spectrographic "observed velocities," after correcting for curvature of the lines, are given in the fifth column. The next three columns contain, respectively, the computed velocities of these planets with reference to the Earth's centre, of the planets with reference to the Sun as the source of light, and of the observer's motion to or from the planets due to the diurnal rota- tion of the Earth. The sum of these three terms is given in the column ' ' computed velocity ' ' ; and with this the ' ' observed velocity" should agree. The last column quotes the discrepan- cies between Observation and Computation, and these we call the (at present) unavoidable errors of observation. I was espe- cially gratified that the mean of these errors is so small, only a 25 See Ap. J., 8, 145, 1898. ni g 7a,, J4 1^ 3 1 o o w 9 99 S 1 1 1 'S >> § So IC CI CO 0.0 »o (M ira a;3 ^> 1 1 + t^ B "ca 1 M ^^ i-i 1-i (X) iH O O s| O O O 1 + 1 > >, ■|r^ 1 ^i -* (M O OS iH Oi 1^ 1-1 cq .-1 + + + >^^ a ^ P CO IC w >i CO CO I> O O rooo ^ 1 1 + § ^ "f^ OiOiOi r- S CO CO CO SO o o o o 1 1 1 ^^" 1 3> O CD Oi «D OO t- ca o ® o ^1 I— ( 1 1 + (D l-g h3 S3 =S i CO CO 1-1 iS » &^ B 1-1 ^ o s a 1-1 1-1 o o g 1-H 1-1 ca w at^ Pi -4^ IBOt? :o t- C3 ro S 00 CC 1— 1 T^ to 1 1^ M- CO iH lO ^i o Oi o .-1 o -rti so o o o o o o M + + + + 1 1 -2.-B 1 , ■3 S CO CD 1-1 iH -t^ -tH C o O O lO IC .-H iH §^ CO CO rH iH Gi Ol Ob* 1-t rH iH 1-1 O + + + +I 1 t^ B •3 1 M S r^ CO »H MH 3 -*^ CO CO CO d o d 1 1 + !> oOD 1 1-^ t- -^ 00 i-( (M O o d o 1 1 + -*3 -g 1 «D ?D CO lO O lO CO (M en rt rH kI + + 1 1^ 1 fcl s O 05 (M CO CD iH S o 1-1 OS im CD CD CO 00 rr! S^ CO CO 1—1 I— 1 OS OS r-~\ 1-^ r^ y-i ++++I 1 -^ 1 CI iH (M O C3 O ^ 3 OO QO CO l>- CO OS SO O O O O O O o 1 1 1 1 1 1 1 CI O ^ CD '«t< iH M .t5 OS 00 iH CO O L^ C8 O a> o eo ^' CI CI OS oo' s| ,-1 i-i iH 1-1 , , ++++ 1 1 □Q s-g ^^ t- d ^ t- .H d »i CO CO CO c^ CO cq og iz; E-i B OS OS CQ CI 00 CO O O in lO lO lO S ^ CO CD CD CD b- t^ d iH iH 1-H -H O O o -»^ •a (M CI CO CO 00 00 1 CI CI cq CI t-^ t-» <1 m <1 g O (g o o ^ '-"■.■ :-J^'- - "* 'i^^^lB^^^^^ vfB^- !^^''. ... 1^. 3 , ■-• ' pSKi } ^^^s^ <^^mH|^Wcj m ^J^^.;'- 'y^>^ ' H«B»gW[|^^^r^. ».'\\ ■ .»'«^ ^" S?«T7^j J^ ws^ ft p: ■ •v^.v. ■> 1 ffe^' - WJ^ '- |-'t ^f M: ^1 '5? / ;^i Lewis Morris Rutherfurd, 1816-1892 CONDITIONS AFFECTING WAVE LENGTHS 77 perfect plates. The plate to be measured must be illuminated uniformly; no more strongly from one direction than from another, or the effect will be to displace the star lines and com- parison lines differently. It must be measured with the violet end to the right and with the red end to the right, for the two sets of measurements will, in general, not agree.^' Constant judg- ment is called for in selecting lines suitable for measurement. One side of a given line, the other side of another line, or one end of a third line, may be made irregular by the presence of silver grains in the film that are especially sensitive or insensi- tive, and such lines must be avoided. A good line in one spec- trum may not be usable in another, or it may have an appreciably different position in a third, due to the coming in of a close companion line in the latter. My assistant and colleague, Albrecht, has shown'^ that certain lines, as observed with 3-prism dispersion, systematically increase in wave length as we pass from the white stars through the yellow to the red stars ; that other lines decrease their wave lengths in a similar manner; whereas still other lines change their positions in one direction in passing through certain types of spectra and in the other direction in passing through the remaining types. It is thought that this shifting of a line with reference to neighboring lines is due to the fact that the line in question, single with 3-prism dispersion, would in reality be seen double or multiple under much higher dispersive power ; and that the relative intensities of the two or more components vary in different spectral types. The use of a fixed wave length for such a composite line will frequently give an erroneous veloc- ity for that line, amounting to 2, 3 or even 5 kms. per second, though the velocity for all the measured lines of a spectrogram will be made erroneous by but a small fraction of this amount ; and it will usually happen that errors due to this source are pres- ent with opposite signs, thus tending to correct one another. Such lines need not necessarily be avoided, but should be allowed 27 See articles by Lord, A'p. J., 6, 425-426, 1897; Eeese, L. 0. Bulletin, 1, 126, 1901; Hasselberg, Ap. J., 15, 208, 1902. 28 X. 0. Bulletin, 4, 90, 1906. 78 STELLAR MOTIONS for, on the basis of an investigation into their behavior in the various spectral types. For the wave lengths of stellar lines, it has been customary to employ Rowland's values of the corresponding lines in the Sun's spectrum,^^ as observed with the high dispersion of a powerful grating spectrograph, in the case of all stellar lines which appear in the solar spectrum. The wave lengths of several lines, in Classes B and A, which do not appear in the solar spec- trum, have been adopted as determined in the laboratory from arc and spark sources; as examples, the principal lines in the spectra of helium, magnesium, carbon and oxygen. Rowland's determinations of wave lengths, incomparably better than pre- vious results, were originally thought to form standards so accurate that they would remain sufficient and satisfactory for the spectroscopic work of several generations. However, it has been found, by Fabry and Perot,^" and by Kayser,^^ that Row- land's wave lengths contain errors, in long stretches of the spec- trum, so large as to limit their usefulness. Similar conditions prevail as to Rowland's and others' wave lengths of the labora- tory lines of iron, titanium, etc., used as comparison spectra in the line-of -sight problem. Some observers use Kayser's more accurate wave lengths for the iron comparison lines, and Row- land's wave lengths for the stellar lines; but the apparent slightly greater accuracy of resulting velocities is in my opinion more than counterbalanced by the confusion which results from the mixing of two systems. This part of the subject is at present in an unsatisfactory state, though the matter is not now vital, as it is not the absolute error which enters, but only a differ- ential; and it is a question, in nearly all cases, of fractions of 1 km. in resulting velocities. Michelson's invention of the interferometer and the applications of it by himself,"^ Fabry and 29 A Preliminary Table of Solar Spectrum Wave-Lengths, in Ap. J., Vols. I to VI, 1895-1897. soAp. J., 15, 272, 1902. 31 Ap. J., 19, 157, 1904, and later articles. 32 Tron. et Mem. du Bureau intern, des Poids et Measures, 11, 1895; Zeitschrift filr Instrumentenkunde, 22, 293, 1902. CONDITIONS AFFECTING WAVE LENGTHS 79 Buisson,^^ Pfund^* and Eversheim,"^ to the very accurate deter- minations of wave lengths, has led to comprehensive plans, under the auspices of The International Union for Co-operation in Solar Research,^" for reconstructing wave-length tables to a marvellous degree of accuracy. It will be many years before these tables become available in our problem,' through the solar spectrum as a basis. Further, the completion of the tables will not provide the line-of -sight observer with all the wave lengths necessary and sufficient for his purpose. From the standard tables, as a basis, he will probably find it desirable in many, or most, cases to construct tables of wave lengths in stellar spectra to conform to his own special requirements. To illustrate, the high dispersion of Rowland's solar spectrum separates two, three, or more lines in a group, whereas, the relatively low dispersion of the Mills 3-prism spectrograph and similar instruments causes these lines in many eases to blend into an apparently single line. A slight uncertainty would exist as to the wave length to be assigned to the composite line, no matter how perfectly the wave lengths of the individual lines might be known. Again, in pass- ing from the solar stars in one direction toward the white stars, or in the other direction toward the red stars, the components of the blend referred to will in many cases change their rela- tive intensities, and thus change the effective wave length of the blended result. The errors arising from these and similar sources will be dependent most largely upon the dispersive power of the spectrograph employed. The ideal method of procedure in the future will, I believe, be this : Let observers in this field select for use as comparison lines a list of carefully determined single lines in the spectra of iron, titanium, etc., as may be required, in the region of spectrum to be observed. With these as a basis of wave lengths, let all observers using instruments of substan- tially equal dispersive power, by cooperation, determine the effective solar and stellar wave lengths corresponding to that 33 Jp. J., 28, 169, 1908. 34 Ap. J., 28, 197, 1908. 35 Ann. der PhysiTc, 30, 815, 1909. 36 Transactions of the Intern. Union, 1, 153, 1906. 80 STELLAR MOTIONS power ; one set of wave lengths for 3-prism instruments, possibly another set for 2-prism instruments, and a third set for 1-prism instruments. The two or three systems will differ appreciably. Following this suggestion, the methods of constructing such tables, beginning with the spectrum of the Sun, planet, or other body whose velocity is known, and with the isolated lines of hydrogen, helium, magnesium, etc., in the simpler stellar spectra, will readily appear to those concerned. Spectroscopists have held in mind the question of eliminating the influence of errors in assumed wave lengths; and this, in effect, has been accomplished for spectra of purely solar type. Vogel placed a spectrogram of the Sun and one of a solar-type star obtained with the same instrument, side by side, and film to film, in the measuring microscope, making the lines iu the two spectra, in effect, to coincide. The reference lines on the star spectrogram were then compared micrometrically with the corre- sponding absorption lines ia the solar spectrum. This method^^ eliminated wave-length values, theoretically, but there was noth- ing gained, practically; for flexure effects and temperature effects existing in the star spectrogram were larger than errors in assigned wave lengths; and, in fact, the solar lines finally measured — those corresponding to the comparison liaes — were in nearly all eases the lines least sharply defined in the spec- trum. One of my students, Curtiss, made an advance^* on Vogel's method by constructing reduction tables, from micrometer meas- ures of Sun or sky spectrograms, arbitrarily giving such values to the wave lengths of the absorption lines that each line would reproduce the correct velocity of the Sun with reference to the observer, as computed from the Earth's orbitaP^ and diurnal motions. These values of the wave lengths were assumed, then, to be the same in all strictly solar types. Similar reduction S7 Publ. Astroph. Ois. Potsdam, 7 (I), 36, 1892. 38 i. 0. Bulletin, 3, 22, 1904. 39 The radial velocity of the Earth 's centre with reference to the Sun varies between -)- 0.51 km. per second, about April 8, and — 0.50 km. about October 13. CONDITIONS AFFECTING WAVE LENGTHS 81 tables could be constructed for any type of spectrum, but such tables would be liable to yield results systematically in error because the radial velocities of the celestial objects upon which they were based would be unknown. The degree of systematic error to which such results would be liable would depend upon the number of well-defined lines in the basal spectra, such as those of helium, hydrogen, magnesium, etc., whose normal wave lengths could be assumed as known. Tables such as these would, of course, be useful in measuring differences in the velocities of the same object, such as a star revolving in an elliptic orbit. The greatest contribution to this subject is Hartmann's spec- tro-comparator,*" an instrument which compares, in one eye- piece, a standard solar spectrogram with the stellar spectrogram of solar type ; each spectrogram having the usual reference spec- trum on either side of it. The micrometer screw moves one of the plates until the two reference spectra of bright lines are in coincidence ; and again until the solar and stellar lines coincide. The micrometer difference of the two positions of coincidence gives at once, by simple computation, the difference of the radial velocities of Sun and star. That of the Sun being known, the velocity of the star becomes known. The result is free from wave-length error provided the star's spectrum is a duplicate of the Sun's. The method is not applicable to other spectral types ; but within this type the instrument is of great utility. The Hartmann comparator offers a splendid method of deter- mining the differences in the velocities of the same object, by selecting one spectrogram of the series to serve as standard of reference in the measurement of all the other spectrograms of that object. Repeated measurement of the spectrogram selected as standard, by several observers, using the original form of measuring microscope, will give the basis for converting the measures of all the plates from relative to absolute velocities. The same considerations enable a selected spectrogram of any spectral class, such as Class A, B, or M, to serve as a basis of comparison for all other stellar spectra of the same class. ioPull. Astroph. Ois. Potsdam, 18, No. 53, 1906. 82 STELLAR MOTIONS Of a different nature are questions relating to changes of wave lengths of individual lines relatively to neighboring lines, due to changed conditions in the source of the light radiations. It was noticed by Jewell, of Johns Hopkins University, that certain solar lines were shifted toward the red, in comparison with laboratory standards, not by the same amount for lines of dif- ferent elements nor for the different lines of the same element, and therefore, clearly not as Doppler-Fizeau effects. A little later Humphreys and Mohler, of Rowland's laboratory, discov- ered that the wave lengths of lines in the arc spectra of the ele- ments are functions, to a small but easily measurable degree, of the pressures of the atmosphere in which the arc is burning. This effect has been investigated under varying pressures from nearly zero up to 101 atmospheres, by Humphreys, Duffield, and many others; and several simple laws governing the displace- ment have been formulated.*^ In brief, an increase of pressure in the light source has the result that all the isolated lines — that is, lines not appearing in banded spectra — are displaced toward the red. For simple lines of the same element, these displace- ments are proportional to the increase of pressure and to the wave lengths of the lines. Further, they are functions of the temperature coefficients of expansion and of the atomic weights of the elements to which the lines belong; and, in many cases, of the melting points of the metals. Such line displacements in the Sun are small, due to pressures, according to Jewell, up to only two or three atmospheres, as a maximum, except in the case of the very broad lines. The situation is of concern in our line-of-sight problems. If lines in the solar spectrum are dis- placed in this manner, we cannot doubt that lines in other stars are similarly affected. Our Sun is believed to be an average- sized star. It may readily occur, in the stars vastly larger than our Sun, that the pressures under which the lines are formed are greater than in the case of our Sun; that the wave lengths are greater, in consequence, than we assume them to be; and that resulting radial velocities based upon the assumed wave lengths will be estimated slightly greater than they really are. For stars *i Humphreys, Jahrbuch der BadioaMivitdt und ElecMronik, 5, 324, 1908. CONDITIONS AFFECTING WAVE LENGTHS 83 less massive than our Sun, on the contrary, the observed veloci- ties may be smaller than they really are. However, it seems equally probable that the lines of the same element, in stars of widely varying masses but of the same spectral type, may be formed under essentially equal pressures in all, at the various depths in their atmospheres which equalize the pressures. It has not been found possible to test stellar spectra for this pres- sure effect by direct methods; our telescopes have had too little power to let us use the high dispersion required. It is hoped that the large reflecting telescopes recently completed may be successfully applied to the problem. Unfortunately, while it is the pressure displacements which will affect our results, it is only the differences of these displacements for different closely related lines throughout the spectrum which will be observable ; and from these very minute differences we should have to work back to the greater quantities wanted. Knowledge is lacking, but the displacements for all stars approximating the solar type are believed to be small. Systematic errors from this source cannot be considered as wholly absent from line-of-sight results. As to pressure effects in spectra very different from the Sun's spectrum, such as Classes B and A in the one direction, and Classes M and N in the other, nothing is known; but it would be surprising if accurate knowledge when finally obtained would not decidedly require us to take these effects into account. Eadial velocities assigned to the stars in general may be appre- ciably in error from the pressure effect alone. We shall refer agaia to this important subject in the discussion of recent obser- vations, as I think it quite probable we have strong evidence that the observed radial velocities for certain classes of stars are systematically too large on this account. The velocity measurements on the sources of canal rays in vacuum tubes made by Stark and others afford an interesting example of the pressure effect, and at the same time illustrate the wide field of application of the Doppler-Fizeau principle. With one end of the tube pointed toward the slit of the spectrograph, the lines were observed strongly displaced toward the violet ; and with the tube reversed in direction, the lines were correspond- The 36-inch Refractor with New Mills Spectrograph Attached CONDITIONS AFFECTING WAVE LENGTHS 85 ingly displaced toward the red. With the axis of the tiibe placed at right angles to the axis of the eoUimator, the lines were observed in their normal positions. Taking space for only one result, the velocities of the canal-ray light sources in a helium tube were observed to be 399 km. per second for the 4471.7 A line and 343 km. per second for the Ds line.*^ These velocities corre- spond only to the conditions under which the experiment was made, as the observed displacements appear to be sensitive func- tions of the gas pressure within the tube. It was thought for a time that the positions of the laboratory lines of the elements vary slightly with the voltage, amperage, and other constants of the electric current used in forming the spark or arc source. Eesearehes in the Johns Hopkins physical laboratory*^ have recently shown that this is not the case, at least within the limits of accuracy of existing instruments to detect. The studies of spark spectra by Schuster and Hemsalech, by Schenck, and by others indicate that there are Doppler effects within the spark structure,** due probably to the diffusion of generated vapors in the general direction from the cathode toward the anode; but as the line joining cathode and anode is always parallel to the slit-plate of the spectrograph, in radial velocity observations, any Doppler effects from this source may be considered negligible. Whether such changes from normal positions of the lines as accompany the Zeeman effect will have to be taken into account when dealing with stellar spectra is a question for the future. Gmelin*^ has shown that the central component of a bright line trebled in a magnetic field is shifted toward the red, minutely, in proportion to the square of the magnetic force. Hale has 42 A condensed resume of the results obtained by many investigators on eleven of the principal elements is given in Ann. der Physik, 26, 829, 1908. 43Kilby, Ap. J., 30, 263-266, 1909. Eeferences are given in Kilby's paper to the more important literature of the subject. nFhil. Trans., 193, 189, 1900; Ap. J., 14, 116, 1901; Conduction of Elec- tricity through Gases, 2d Ed., 520, 1906. 45 Phys. Zeitschrift, 9, 212-214, 1908. 86 STELLAR MOTIONS made the brilliant discovery of Zeeman effects in the spectra of sun spots;** but in the case of a star for which an entire hemi- sphere is integrated into a point image, we now have no reason to believe that this effect would be appreciable. If our Sun is situated in a strong magnetic field — an attractive subject which has been carefully considered by many physicists — it is quite possible that the Fraunhofer lines should show appre- ciable Zeeman effects; but as no such effects have yet been observed it seems hopeless to expect that they could be observed in the distant stars with the low dispersive power which must be used upon the point images. The bright hydrogen lines in the spectrum of the well-known variable o Ceti are som.etimes trebled, and it was suggested by Miss Gierke more than a decade ago that this might possibly be a Zeeman effect. At the recent maxi- mum of this star, my colleague, Wright, photographed the triple Hy bright line through a variety of analyzing optical pieces, but no traces of polarization effects could be detected in any one of the three components. The most recent question calling for consideration in connec- tion with radial velocity determinations is that of a possible dispersion of light in its passage through interstellar space, announced independently and almost simultaneously by Nord- mann and by Tikhoff. Observing certain variable stars, they were convinced that the recorded minima of brightness were more and more retarded in point of time as they observed these stars in light of shorter and shorter wave lengths. Belopolsky and Tikhoff" report that the minima of p Aurigce occur 0.015 day earlier in blue light than in violet light. Albrecht has observed for T VuJpeculm that "Wilkens's" light curve, determined by a photographic method, gives the epoch of maximum 0.4 day earlier than the curves which were determined by visual methods." The minimum in Albrecht 's radial velocity curve for this star falls between the times of visual and photo- graphic maximum light, but nearer to the photographic maxi- ioAp. J., 28, 315, 1908. 47 Publ. School of Mines Jekatermoslaw, 32-33, 1905. 48 i. 0. Bulletin, 4, 137, 1907. CONDITIONS AFFECTING WAVE LENGTHS 87 mum. Nordmann" estimates that in j8 Persei the X6800 mini- mum, precedes the \4300 minimum by 16 minutes, and that the A.5400 minimum precedes the X4300 minimum by 9 minutes. Schlesinger and Curtiss^" observed that the photometric mini- mum of j3 Persei in visual rays lags from one and a half to two hours behind the time required by the velocity determinations based upon blue rays ; and Schlesinger finds that Belopolsky 's" observations of the same star give a result in accord with his own. Schlesinger^- has just reported a similar effect amounting to at least an hour in the system of 8 Librm. The evidence quoted seems to be somewhat contradictory. Nordmann and Tikhoff reached the same conclusion, that the velocity of stellar light is a function of its wave length, owing to the retardation of an interplanetary medium. Such an effect, if existent, would enter seriously into the observed velocities of many and perhaps all distant celestial objects. The question has been discussed at length, and current opinion inclines strongly to the view that the thesis has not been main- tained. It is not certain even that the observed differences of phase have not their explanation in purely photographic or allied causes, or in the conditions existing in the variable stars themselves. Going directly to Nordmann 's results for one variable star, and to Tikhoff 's results for another variable star, we find, as Lebedew^^ pointed out, that one set of observations yields 30 times as much retardation as the other. Under ordinary cir- cumstances this 30-fold discrepancy would justify dropping the subject; but here the uncertainties in the distances of the two stars enter directly; and if there is any real lag of the minima of variable stars, with decreasing wave lengths of light used, the general subject remains important and should be investi- gated. Lebedew recalls that the electro-magnetic theory of 49 Comptes Bendus, 146, 266 and 383, 1908. ^0 Puhl. Allegheny Ohs., 1, 32, 1908. ^ciMitt. Fullc. Stern., 1, 103, 1906. ^^TuU. Allegheny Ols., 1, 127-129, 1909. 53 .4p. J., 29, 105, 1909. 88 STELLAR MOTIONS dispersion, or, in fact, any known theory of dispersion, demands also absorption; and he deduces the result, unquestionably cor- rect, that the minimum dispersion in space reported to exist by Nordmann and by Tikhoff would also require so heavy an absorp- tion of light in space that not only would the stars be invisible, but the Sun itself would be snuffed out. Frost has given negative evidence on the same question. Meas- uring the radial velocities of the spectroscopic double star, (i Orionis, which revolves at high speed in a short-period orbit, — only 0.77 day, — he was unable to detect any differences between the velocities afforded by the separate lines distributed through- out a long range of spectrum, as should have been the case if interstellar dispersive effects were large; though, if Lebedew is correct, a dispersive effect of the magnitude which could have been detected by this method would have left no unabsorbed light to form the spectrum. The subject is in need of accurate and experienced observations on variable star minima, as given by widely different parts of the spectrum. W. Michelson'* has shown that the spectral lines of a light source, even absolutely at rest in reference to the observer, must be changed from their normal positions if an absorbing medium lying between the light source and the observer is changing its thickness or its indices of refraction during the progress of observations. This fact has been favorably considered by some students of the Sun in explanation of apparently high or rapidly changing velocities'^ observed for certain solar details of struc- ture. It would seem that the integrated spectra of point-image stars would not be subject to frequent disturbances from this source. Julius has developed very extensively the possible bearing of anomalous dispersion upon the interpretation of astronomical phenomena. He is of the opinion,'^ for example, that anomalous dispersion may cause the lines in the spectrum of the Sun's edge to be displaced slightly toward the red with reference to their ^iAp. J., 13, 192, 1901. ssTenyi, Ap. J., 19, 70, 1904. 56 Le Badmm, 7, 281, 1910. CONDITIONS AFFECTING WAVE LENGTHS 89 positions in the spectrum of the Sun's centre. Whether this effect, if real, would be appreciable in the integrated spectrum of a distant star is doubtful. "While the pressure and other effects which we have described in the last few pages may be troublesome in the problems imme- diately before us, we should not regard their existence as causing tmfortunate complications, for in them we have promise of future means of analysis of great power and value in studies on conditions existing in the stars. Going back over the history of science, we recall that apparent complications, finally reduced to law, were the source of much of our present power to interpret conditions existing in the stars. CHAPTER III ROTATIONAL VELOCITIES IN THE SOLAR SYSTEM RADIAL VELOCITIES OBTAINED FOR INDIVIDUAL STARS Before passing on to considerations of the motions of the dis- tant stars, let us note several interesting applications of the Doppler-Pizeau principle to studies on members of the solar system. In Chapter II (pp. 56-57) reference was made to radial veloc- ity measures of the Sun 's rotation. Of unusual merit is Duner 's'^ solution of the problem, based upon observations made between 1887-1901, inclusive. From the relative displacements of a few lines in the spectra of the two limbs of the Sun, with reference to neighboring telluric liaes of oxygen, and with the slit of his instrument directed to latitudes differing fifteen degrees suc- cessively, he observed average velocities of approach and reces- sion of the east and west limbs, respectively, as quoted in the second column of the accompanying table. 11 0-C P 0°.4 2.09 km. 0.00 km. 24.2 days 15 .0 1.97 —0.01 24.7 30 .0 1.70 +0.02 26.1 44 .9 1.27 +0.01 28.3 60 .0 0.80 ^0.01 31.1 75 .0 0.39 0.00 33.4 (90 .0) (34.3) Faye had earlier succeeded in representing the law of solar rotation, as based upon sun-spot observations, by an equation of the form : I cos <^ = « cos <^ + & cos^ <^, '^Nova Acta Soc. Sci. Upsaliensis, 1 (IV), 1907. ROTATIONAL VELOCITIES IN 80LAB SYSTEM 91 in whicli ^ is the angular rotational velocity per mean solar day at any latitude (^, and a and & are constants. Determining the values of a and 6 by the method of least squares, from the six mean spectrographic velocities, Duner deduced the follow- ing equation: f cos f = h (23) which is the equation to the curve of which the lines in the spectrum of the ring are a part. The curve is represented by the dotted line in the figure; it is symmetrical with respect to the axis of x, but only the upper branch has a physical meaning, and the curve corresponding to the other half of the image is obtained by taking both x and y with negative values. ' ' In the equation V = — ^; , ^/ B ■ k = 3.7992 for the Satumian system, B being expressed in kilometers and V in kilometers per second. The com- puted motions of different parts of the system are given in the following table. The gauze ring is not considered, as its spectrum does not appear on the photographs; the rings known as A and B are not separately dis- tinguishable. Period of a Velocity in Velocity in Object B Satellite at Equatorial Line of Sight Distance B Plane Apr. 10, 1895 km. h km. km. Outer edge of ring 135,100 13.77 17.14 16.35 Middle of ring 112,500 10.46 18.78 17.91 Tntier fidgfi of ring 89,870 7.47 21.01 20.04 Limb of planet 60,340 4.11 Rotation 25.64 24.46 Limb of planet 60,340 10k. 23 (A. Hall) 10.29 9.82 ' ' With the values given in the above table, and others which do not correspond to actual points in the system, the dotted curves were platted. For the ordinates, however, twice the values in the last column were taken, since the displacement of a line, due to motion in the line of sight, is doubled in a case of a body which shines by reflected and not by inherent light, provided (as in this case) the Sun and the Earth are in sensibly the same direction from the body. . . "The most accurate method (due to Deslandres) of measuring the rela- tive displacement of the opposite ends of a line in the spectrum of the planet is to measure the angle (j>. "The value of depends upon the dispersion and other constants of the spectroscope employed, as well as upon quantities which are independent of the instrument. If we let L = the velocity of light in kilometers per ROTATIONAL VELOCITIES IN SOLAR SYSTEM 99 second = 299,860; X = the wave length of the measured line in tenth-meters ; -D = the linear dispersion of the photographed spectrvim at the position of the same line, expressed in tenth-meters per millimeter ; p = half the width of the spectrum in mUlimeters ; — we have by Doppler's principle, allowing for the double effect already mentioned, y = a;tan0= , DL or, I) = a: tan D , 2X from which we obtain the velocity in the line of sight at the limb ( V„ cos ;3) by placing x = p. That is, Y^ _ pDLta,n _ 2^^ 2 \ cos ;8 "The value of p is computed from the angular semi-diameter of the planet, and the ratio of the focal lengths of the camera and collimator of the spectrograph "The relative displacement of a line in the spectra of the ansae is, measured directly, the micrometer wire having first been placed parallel to the lines of the comparison spectrum. If 5 is this measured interval, the mean velocity of the ring is F/ = ^^. (25) 4 X cos ;3 "The results of all the measurements on the spectrogram of April 9, 1895, are given in the following tables ; X D (t> Velocity of Limb C — S Mean Velocity of Ring C — A. A. km. km. mm. km. km. 5324.3 27.55 3° 36' 10.92 —0.63 0.0456 18.54 -1-0.24 5328.4 27.65 4 24 13.39 —3.10 0.0464 18.92 -0.14 5371.6 28.77 3 11 9.99 +0.30 0.0404 17.01 + 1.77 5383 . 5 29.09 3 20 10.56 —0.27 0.0362 15.37 H-3.41 5429.9 30.37 3 8 10.27 -1-0.02 0.0402 17.67 -t-l.ll 11.03 —0.74 17.50 -1-1.28 [The spectrogram of April 10, 1895, yielded 9.58 km. for the velocity of the limb and 18.52 km. for the mean velocity of the ring.] ' ' The results from both photographs are Observed velocity of the limb 10.3 ± 0.4 km. per second Mean velocity of ring 18.0 ± 0.3 km. per second; the computed values being 10.29 and 18.78 km. per second respectively." 100 STELLAR MOTIONS Immediately following the annoimcement of Keeler's results, they were confirmed by Belopolsky,^^ hy Deslandres," and by Campbell." The scale of Keeler's photographs was so small on account of the small telescope employed that he did not attempt to measure the inclinations of the lines in the spectra of the ansae. He merely determined that the inner edge of the ring system is travelling more rapidly than the outer edge. The large scale of the photographs secured with the 36-inch refractor and Mills spectrograph enabled me to measure the inclination of the ansae lines in the direction opposite to that of the planet's lines. The excess of the velocity of the inner edge over that of the outer edge was found by measuring the ineUnation <^ of the lines in the ring spectrum to the lines in the lunar spectrum, and reducing to kilometers per second by the formula pBL tan 4> (26) 2A.eoSi8 ' in which p is the width of the ansa in millimeters. The inclina- tions of ten lines in both ansae and the corresponding excess velocity for the inner edge, as determined from the spectrogram of May 10, 1895, are quoted in the following table : X D Excess for Inner Edge C— A. A. km. km. 4340.6 12.40 1° 8' 3.03 -1-0.84 4359.8 12.86 1 30 4.14 —0.27 4367.8 13.02 1 29 4.14 —0.27 4369.9 13.05 1 7 3.12 -1-0.75 4371.3 13.08 59 2.75 -1-1.12 4395.3 13.56 50 3.03 . -t-0.84 4404.9 13.76 1 9 3.34 -1-0.53 4415.3 13.94 49 2.41 -fl.46 4425.6 14.12 1 12 2.84 -Hi. 03 4427.5 14.16 .56 2.79 -1-1.08 Means 3.16 -0.71 15 Astr. NacK, 139, 1, 1895. 16 Comptes Bendus, 120, 1155, 1895. 17 Ap. J., 2, 127, 1895. §■ ^ s 102 STELLAR MOTIONS The computed excess of a supposed satellite at the inner edge of the ring system over that of a supposed satellite at the outer edge was 3.87 km., the value of p for the date being 0.3410 mm. The excess of the computed velocity over the observed velocity, as determined from the ten lines, is given in the last column of the table. The excess velocities, computed from three spectro- grams of dates May 10, May 14, and May 16, 1895, respectively, were 3.16, 3.06 and 3.17 km. per second. The mean of the three values, 3.13 km., differs 0.75 km. from the computed value. In the year 1900, Belopolsky^* applied the speetrographic method to the study of the rotation period of Venus. The aver- age equatorial velocity observed by him, 0.5 km. per second, corresponds to a rotation period of approximately twenty-two hours, from west to east. A quite different result was obtained by Slipher^' in 1903. His extensive observations led with remark- able accordance to a rotational velocity vanishingly small. The corresponding rotation period would exceed several days; it could, in fact, equal the period of revolution around the Sun, 225 days, which Schiaparelli, from his long-continued observa- tions of surface markings on the planet, announced to be the planet's rotation period. It is desirable that the speetro- graphic rotational velocity of Venus be re-observed with higher dispersion, if possible. As a check upon the accuracy of his results for the rotation of Venus, Slipher^" used the same instrument and methods in measuring the known rotational velocity of Mars. The period from the speetrographic observations came out 25'' 35", or just one hour longer than the true period. Attempts have been made to measure spectrographically the rotation periods of Uranus and Neptune, at present unknown; but practical difficulties, in part due to the intrinsic faintness of these planets, have prevented definite results. Successful applications of the spectrograph to determinations of the rota- laAstr. Nach., 152, 263, 1900. 18 Lowell Obs. Bull, 1, 9, 1903. 20 Lowell Ols. Bull., 1, 19, 1903. RADIAL VELOCITIES OF STABS 103 tional periods of Mercury, Uranus, and Nepture remain as future problems. It is extremely desirable, in current studies of the stellar sys- tem, that we know, as soon as practicable, the radial velocities of all stars bright enough for observation with 3-prism spectro- graphs, such as that described in the first chapter. The larger telescopes now engaged in this work can observe satisfactorily down as far as the fifth visual magnitude ; that is, in the entire sky, about 1600 stars, of which 1100 are withia reach of northern observers. In measuring the radial velocity of a star, as in other departments of physical science, it is advisable and necessary to repeat an observation four or five times, taking the mean of the values obtained on different nights, in order to reduce the effects of unavoidable errors, and to give confidence in the observed result. Now astronomical observation is dependent upon the weather, over which we have no control. Not all clear nights, even, are workable. Freedom of the air from confused currents of unequal temperatures is more important than great trans- parency of the air. Some nights on which the stars shine with great brilliancy are absolutely useless to observers with tele- scopes, on account of the non-homogeneous qualities of the air strata. Because of atmospheric imperfections and other vicissi- tudes not necessary to describe, the securing of four satisfactory spectrograms of a star will, on the average, necessitate five or six exposures. The exposure times vary from a few minutes for the first-magnitude stars up to 2^/^ hours for the fifth visual magnitudes, of the solar type, with average exposure of fully 1% hours — recalling that the faint stars are much the more numerous. Six plates constitute an average night's work, from sunset to sunrise. If the telescope is available three nights per week to the spectroscopist, he can count on ninety usable nights per year. Four good spectrograms of each of the 1100 stars referred to thus require from ten to twelve years for observation alone. The dark-room work on the plates, the measurement and mathematical reduction of the plates, and the keeping of the apparatus in adjustment, not to mention extensive and frequent experiments for developing new ideas and improving the meth- 104. STELLAR MOTIONS ods, consume fully threefold as much time as the night work of observation. Further, the unexpected discovery, during the progress of the investigation, that at least one star on the average out of every four is a double star with components so close together as to be invisible, each such double requiring twenty- five or more spectrograms for its proper preliminary study — this unexpected development has more than doubled the labor, both night and day. Fortunately, the pressing need for a knowl- edge of radial velocities of the stars, and the untold fruitfulness of this field of research, have brought several of the largest tele- scopes into the problem. In this country, the Allegheny, Lick, Lowell, and Yerkes Observatories ; in Canada, the Ottawa Obser- vatory; in Germany, the Bonn and Potsdam Observatories; and in Russia, the Pulkowa Observatory ; these eight institutions have been contributing a part of their resources for several years to this labor. The observatories at Columbus, Ohio, at Cambridge, England, and at Paris, France, were valued contributors for a short time, but their energies have been turned in other directions. In 1892 I called attention to the important fact that we should not be able to make a satisfactory start in the study of the stellar system upon the basis of stellar radial velocities until we should have determined "the velocities of several hundred stars dis- tributed fairly uniformly over the celestial sphere. "^^ As no other plans seemed to be forming for the purpose of securing these observations of the southern stars, I looked forward, in 1894, to the organizing and conducting of an expedition to the Southern Hemisphere for this purpose. When the subject was presented to the late Mr. D. 0. Mills in 1900, he was pleased to provide funds for the equipment and maintenance of a suitable observing station in the Southern Hemisphere. The D. 0. Mills Observatory, located on the summit of Cerro San Cristobal, in the suburbs of Santiago, Chile, is equipped with a 92 cm. (36%- inch) reflecting telescope, and with spectrographs of 3-prism, 2-prism and 1-prism dispersions, for investigations exclusively in the line-of-sight field. Since 1903, successively in charge of 21 Astr. and AstropK, 11, 321, 1892. RADIAL VELOCITIES OF STABS 105 my colleagues, Wright, Curtis, and Moore, this expedition has measured the radial velocities of some 500 stars south of Declina- tion — 30°. In the past two years the Cape of Good Hope Observatory has become a second southern contributor. Each of these ten observatories — eight in the Northern and two in the Southern Hemisphere — is pushing some phase of this work with energy. It is hoped that all stars down to the fifth visual magni- tude, through the efforts of these ten observatories in both hemi- spheres, will have been observed in two years from date, except in the cases of perhaps 300 binary systems whose investigation must extend over many additional years, partly because of the numerous observations required and partly because it will be necessary to await the completion of revolution periods covering several years. Let us examine the radial velocity results secured for repre- sentative stars, in order to gain an idea of the accuracy attainable for stars of different magnitudes and spectral types, and of the magnitudes of the speeds to be expected. The following are col- lected for a Cassiopeim, visual magnitude 2.4, whose spectrum, of Class K, is excellent for measurement. (A negative velocity indicates approach ; a positive velocity, recession, from the solar system.) a CASSIOPEIA Date Eadial Velocity Observatory 1885 +90 km. Greenwich (visual) 1887 +58 Greenwich (visual) 1890 Feb. 20 — 15.9 Potsdam (photographic) Feb. 21 — 14.5 Potsdam (photographic) Mean —15.2 1896 Nov. 12 — 4.1 km. Lick Dec. 8 — 4.1 Lick Dee. 17 — 4.9 Lick Dec. 24 — 4.2 Lick 1898 Nov. 1 — 3.0 Lick 1900 Sept. 18 — 3.6 Lick 1901 Oct. 22 — 3.0 Lick 1907 Aug. Mean 12 — 4.2 to. Lick — 3.9; 15 km. 1-1 « o o~ o < w ■>! n •o H o M O > o H ■< % n O ►J O W RADIAL VELOCITIES OF STABS 107 Date Eadial Velocity Observatory 1897 Sept. 4 —2.8 km. Columbus Sept. 14 +1.6 Columbus 1898 Nov. 2 —4.0 Columbus Nov. 7 —2.3 Columbus Nov. 11 +5.7 Columbus Nov. 20 —2.1 Columbus Dec. 8 +0.6 Columbus 1903 Sept. 24 +0.7 Columbus Oct. 12 —3.5 Columbus Oct. 18 —2.7 Columbus Oct. 24 —2.9 Columbus Oct. Mean 25 —4.4 Columbus —1.3 1904 Oct. 19 —2.4 km. Bonn Oct. 27 —1.9 Bonn 1905 Nov. 30 —3.1 Bonn 1906 Dec. 23 —2.5 Bonn Mean — 2.5 Systematic correction — 1.0 Corrected mean - — 3.5 The Greenwich velocities were measured visually, and they illustrate the futility of visual methods, using small telescopes, in this delicate field of observation. The results obtained photographically at the other four obser- vatories differ considerably, but there is no apparent reason to suspect that the velocity changes. The pioneer observations at Potsdam were accordant as to each other, but a systematic error of 11 or 12 km. seems probable. The Columbus observations are discordant amongst themselves, and the mean result is perhaps in error both fortuitously and systematically. The Lick and Bonn series are very accordant within themselves, but their means differ 1.4 km. If we apply the systematic correction — 1.0 km. to all observations of this and other stars made at Bonn, as recommended^^ by the Bonn observers, the Lick and Bonn means, — 3.9 km. and — 3.5 km., are in remarkably close agreement. 22 Ap. J., 27, 324, 1908. 108 STELLAR MOTIONS The next observations are of the second brightest star in the sky, Canopus (a Garinm), whose spectrum, of Class F, is probably as favorable for accurate measurement as any we could select. CANOPUS (a CABINS) 1903 Sept. 21 +20.7 km. Chile, D. 0. Mills 1904 Jan. 19 +20.9 Sept. 6 +20.1 Sept. 11 +20.0 1905 Jan. € +20.3 Jan. 6 +20.7 Jan. 6 +20.1 Jan. 6 +21.2 Nov. 9 +20.0 Nov. 24 +21.1 Mean +20.6 No other results are available for this star. The extreme range is 1.2 km., and the probable error of the mean result is ± 0.10 km. Here are results for the third-magnitude star e Pegasi, whose spectrum, of Class K, is favorable for accurate measurement. The means are taken for each observatory, the number of individual results being radicated by the subscript. t PEGASI Mean Date Mean Velocity Observatory 1888.9 V^= +8.0 km. Potsdam 1901.5 V^=+7.5 Columbus 1901.7 V^=+5.0 Lick 1902.7 V^=+6.2 Yerkes 1903.4 V=+5.9 Pulkowa 1903.8 V-+3.3 Cambridge 1905.6 V-+6.1 Flagstaff 1905.8 V-+5.0* Bonn Mean ^3s=+5-8 ■ Corrected by application of — 1.0 km. RADIAL VELOCITIES OF STABS 109 Here are the results available for p Aquilm, a fourth-magnitude star of Class K: j3 AQUILiE 1896 Aug. 25 — 39.3 km . Lick Aug. 31 —38.5 Lick 1897 May 12 —40.9 Lick 1898 Sept. 9 —39.2 Lick 1905 June 11 —39.4 Lick 1907 June Mean 27 —40.0 Lick —39.6 ± 0.22 1905 July 26 —37.3 Bonn Aug. 17 —37.4 Bonn 1906 July 17 —40.3 Bonn Sept. Mean 19 —39.0 Bonn —38.5 Syste matie correction •rected mean — 1.0 Coi —39.5 Following are Lick observations of /* Cygni, a flfth-magnitude star of favorable spectral type, Class F5 : /iCYGOT July 81 +18.2 km. July 31 +19.1 1900 Aug. 1 +18.6 1905 Aug. 1907 July Aug. 7 5 1 +19.7 +18.7 +18.7 Mean +18.8 ± 0.13 The results for the fifth-magnitude star appear to be essen- tially as accurate as those for Canopus, as the range is but 1.5 km. Selecting some large velocity results, we have for X Aurigce of the fifth magnitude, Class G : X AURIGA +67.1 km. Lick +67.5 +64.1 +66.5 +67.5 1899 1900 1904 Dec. 18 Dee. 25 Jan. 22 Oct. 9 Oct. 26 Mean +66.5 110 STELLAR MOTIONS The third plate, giving a somewhat discrepant result, is under- exposed, owing to poor atmospheric conditions. For r) Cephei, of the fourth magnitude, Class K : 1897 Sept. 29 1898 July 20 Aug. 21 Aug. 25 1901 Aug. 24 1904 July 6 Mean 1904 Aug. 29 Oct. 14 Oct. 15 1906 Aug. 30 Mean Systematic correction Corrected mean ■n CEPHEI -87.4 km. -86.2 -86.9 -86.2 -86.8 Lick Lick Lick Lick Lick Lick —87.0 —86.3 Bonn —85.3 Bonn —86.0 Bonn —86.4 Bonn —86.0 — 1.0 -87.0 Again, for the fourth-magnitude star, 8 Leporis, of Class K : S LEPORIS Lick 1900 Dec. 24 +98.5 km. Dec. 25 +99.1 Dec. 30 +99.7 1904 Dec. 13 +99.8 1906 Nov. 8 +99.1 Mean +99.2 It is instructive to compare the velocities of the Great Nebula in Orion, Class P, probably the most favorable nebula for such observation in the whole sky, as determined visually by Keeler with the 36-ineh refractor, and photographically at Potsdam, Lick, and Yerkes Observatories. RADIAL VELOCITIES OF STARS 111 ORION NEBULA Keeler * Liekf Potsdam X Yerkes § Visual Photo. Photo. Photo. 1890-91 1901 1901-02 1903-04 -1-17.5 km. -fl7.1 km. -j-18.0 km. -(-19 km. 19.3 16.1 14.1 23 15.6 17.0 17.0 18 6.4 14.8 17.8 21 13.5 18.1 19 23.8 18.9 16 8.5 18.4 19 34.6 19.8 16 20.0 16.2 19 19.0 15.7 19 14.0 14 21.7 16.4 Means -(-17.7 4-16.2 -fl7.4 -(-18.5 Mean^ -1-17.4 Keeler 's individual values differ considerably amongst them- selves, showing a total range of 28 kilometers, but he would be a skillfid observer indeed who could improve upon them, using visual methods. The photographic results exhibited in the last three columns are very accordant, showing total ranges of but 2.3, 5.7 and 9 km., the latter being somewhat large because the low dispersion of a 1-prism spectrograph was employed, whereas the Lick and Potsdam observers used 3-prism instruments. The means of the four series are remarkably accordant. The advantage of the photographic method is clearly apparent. Other factors being equal, the accuracy of radial velocity determinations is roughly proportional to the dispersion em- ployed in the spectrograph. In the case of such a bright star * FuU. LicJc Ohs., 3, 197, 1894. t Wright, LicTc Ols. Bull, 1, 155, 1902. tVogel and Eberhard, Ap. J., 15, 303, 1902; and SiUungsher. Kgl. AJcad. Wiss. Berlin, p. 260, 1902. § Frost & Adams, Ap. J., 19, 354, 1904. 112 STELLAR MOTIONS as Ganopus, for example, an instrument giving five times as much dispersion as the Mills spectrograph could be designed and applied, thereby reducing the accidental and unavoidable errors of observation to perhaps a third their present dimensions; and similarly for several other first-magnitude stars. A very con- siderable gain in accuracy may be accomplished by using fine- grained (slow) photographic plates on all stars bright enough to permit their employment. On the other hand, if it is a question of observing the speeds of stars fainter than the fifth visual magnitude, an instrument with 2-prism dispersion can be used to advantage. Decreasing the dispersion by a third decreases the accuracy nearly a third, but this enables us to measure velocities with quite satisfactory accuracy for stars nearly a magnitude fainter than the limit for three prisms, with exposures of the same lengths. Going yet further, it is possible to measure the speeds of stars considerably fainter, certainly down to the seventh visual magnitude, sufiieiently accurately for many pur- poses, with 1-prism instruments. For example, here are 1-prism observations of Lacaille 661 ^ R. H. P. 637, of 6.3 visual magnitude. Class G: LACAILLE NO. 661 1908 Oct. 7 +51 km. Chile, D. 0. Mills Oct. 11 +48 Oct. 18 +52 Dec. 12 +47 Mean +49 And again. for Weisse I, 4M189=-B. H. P. 1614, ma Class K. 1908 Oct. 18 +33 km. Chile, D. 0. Mills Oct. 21 +31 Oct. 24 +31 Dec. 12 +28 Mean +31 Less difficult — easy, in fact — ^was it to measure the speed of the 9.3 magnitude star B. D. + 30°.3639, of the Wolf-Kayet type, for its light is condensed principally into a few bright lines. RADIAL VELOCITIES OF STARS 113 Measures of tlie positions of the Hy, H8 and He hydrogen bright lines by Duncan yielded speeds as follows"' : B. D. + 30°.3639. 1908 July 7 —25 km. Lick July 17 —32 July 21 —33 Mean — 30 My principal purpose in quoting these results for faint stars is to introduce and support the statement that 2000 stars, more or less, fainter than those already observed with 3-prism dis- persion, are within reach of 2-prism spectrographs; and that 5000 still fainter stars, more or less, are easily within practicable reach of 1-prism spectrographs. Although the velocities deter- mined with such instruments would not be so accurate as those obtained with 3-prism instruments, they would serve admirably in the study of individual stars in very many cases, and be of tremendous importance in statistical studies as to the distribu- tion and motions of the stars throughout the stellar system. The carrying out of this suggestion, using 2-prism dispersion on stars sufficiently bright, and 1-prism dispersion on the fainter stars, would constitute at least a decade's exceedingly fruitful labor for ten or twelve well-manned observatories, in the two hemi- spheres ; and a beginning on such a cooperative program should not be long delayed. The reasons will appear, forcibly, in the discussion (Chapters VI, VII and VIII) of recent observations. I have in nowise indicated the limiting possibilities of the 1-prism instrument. Dr. Curtis, when in charge of the D. 0. Mills Expedition, measured the radial speed of a 9.2 visual magnitude star, whose spectrum is approximately of the solar type, by making exposures on the same plate for four consecutive nights, twenty-nine hours in all. There was special reason for observing this star, Cordoba Zones 5'^.243, as it possesses the largest proper motion of any known star, 8".7 per year. In 225 23 Liclc Ols. Bull., 6, 59, 1910. lU STELLAR MOTIONS years its position on the surface of the sphere changes through an angle equal to the Moon's diameter. Is its motion in the line of sight of unusual magnitude ? Here are the Mills observations : C. Z. 511.243 1908 Dec. 2 (mean of 2 nights) +240 km. Dec. 9 (mean of 4 nights) -\-2i4: Mean -|-242 The foregoing references to the use of 1-prism dispersion have been chiefly in connection with faint stars whose spectra contain sharply defined lines. One-prism spectrographs have in the past eight years done important service at several observatories in measuring the speeds of those stars, both bright and faint, whose spectra contain unusually broad and poorly defined lines. For such stars, in many cases, 3-prism spectrograms are unmeas- urable : the hazy lines are magnified too highly, and the contrasts between the absorption lines and the continuous spectrum back- ground are not sufficient to define the positions to be measured. Reducing the dispersion to one-third reduces the widths of the broad lines to one-third, and the contrasts are increased suffi- ciently to let the boundaries of the lines be estimated. The accuracy is not so great as for stars with good lines, but it is sufficient for the solution of a large class of most interesting problems. Here are remarkably accordant observations^* of three Pleiades stars made at the Yerkes Observatory with a 1-prism instrument. Their spectra are of Classes B8p, B5, and B5, respectively. ATLAS (27 TAURI) ALCYONE (25 TAURI) MEEOPE (23 TAURI) 1903 Oct. 30 -1-14 km. 1903 Oct. 30 +17 km. 1903 Dec. 27 +6 km. Dec. 25 +12 Dec. 4 +14 1904 Mar. 19 +5 1904 Jan. 29 +15 Dec. 25 +13 Apr. 16 +8 Feb. 26 +10 Means +13 +15 +6 2iAp. J., 19, 340-341, 1904. TABLE V STELLAR MOTIONS EXCEEDING ± 50 KM. PER SECOND Object a [1900) S (1900) Spm. Dk.t. Obs'd V Corr'd V km. km. Serpentis 15 14 - - 2 9 G 133 .6 + 53.8 + 65.4 X Serpentis 15 41 ^6 _ - 7 40 G 137 .6 - 65.6 - 51.9 6 Vulpeeulffi 19 24 .5 -\ -24 28 Ma 139 .5 - 85.0 - 68.0 37 LibrsB 15 28 .7 - 9 43 K 139 .9 + 48.9 + 59.5 *N. G. C. 6891 20 10 .4 +12 26 Neb. 141 .9 + 40.7 -- 55.7 A. G. C. 27600 20 4 .6 -36 21 K5 144 .8 -132. -125.8 A. G. C. 24321 17 49 .5 -44 19 K 147 .2 + 46. + 51.5 *N. G. C. 6790 19 17 .9 + 1 19 Neb. 158 .9 -- 48.5 + 63.8 j'2 Sagittarii 18 49 .1 -22 47 K 165 .9 -106. - 94.9 a Seuti 18 29 .8 - 8 19 K 174 .3 + 36,0 + 50.4 [Elements of solar motion used: V„ = — 17.77 km., a„ = 272°, 5„ = -I- 27°.5. See Chapter V, p. 189.] *Nebul8e observed by Keeler. **These three stars were added to the table after the date of this lecture, February 1, 1910. 116 STELLAR MOTIONS These useful results, and a long list of similar ones, could not have been obtained so accurately with 3-prisni dispersion; and there is the added advantage that exposure-times with 1-prism instruments are not more than one-fifth to one-eighth as long as with 3-prism instruments. The output of results can be correspondingly augmented. There are occasional stars, perhaps from 1 per cent to 2 per cent of the entire number, whose spectra contain such extremely broad and imperfectly defined lines or bands as to be at present incapable of measurement to any satisfactory degree of accu- racy. Once in a great while we photograph a spectrum which seems to be entirely devoid of lines; and with such spectra we can, of course, do nothing ; but there is promise that for spectra containing only very wide and very hazy lines we shall be able, by using slow plates for the exposure, and by purely photo- graphic manipulation of the original negatives, to increase the contrasts within the spectrum enough to let fairly satisfactory measures be made. It is not that we dislike to pass by a certain number of stars as unmeasurable, but that we prefer not to pass by a certain class of stars as indicated by their spectra. The stellar velocities quoted on preceding pages include a few which equal 100 km. or more per second. Speeds of this order are few in number, but can scarcely be called exceptional. The accompanying table contains a list of thirty-seven stars on my program and three planetary nebulte whose observed radial velocities are equal to or greater than 50 km. per second, as con- tained in the last column, after correcting the observed velocities contained in the next to the last column for the direction and speed of the solar motion. The fifth column of the table defines the angular distances of the stars from the Kapteyn vertex of preferential stellar motion, to which reference will be made in a subsequent chapter. The methods of determining and elimi- nating the solar motion effects are described in Chapter V, and it is sufficient to say that the results in the last column of Table V are the velocities of the stars themselves toward and away from the position which the solar system occupies at any one instant. They present interesting considerations. In the RADIAL VELOCITIES OF STABS 117 first place, they are not the real velocities of the stars in space but merely the projections of those velocities on the line of sight ; for example, the faint star, Groombridge 1830, which is known to he one of our near neighbors (parallax, n- = 0" .10 ± ) and to possess high proper motion (/* = 7".05 per annum), must have a speed in space of approximately 250 km. per second in order to harmonize these observational data. Curtis^^ finds that the star Cordoia Zones 5^.24:3, whose observed radial velocity is 242 km. per second (page 114) and whose parallax is 0".32, must have a velocity in space of approximately 260 km. per second. If the observed parallax of the latter star is approximately cor- rect, it must be an exceedingly small star, for it is of the ninth visual magnitude. It would be interesting to supplement this list of high radial velocities with other stars whose spectro- graphic velocities are small or medium but whose cross motions, as determiued by their parallax and proper motions, must be very great. A case in point is Arcturus, with parallax supposed to be approximately 0".07, and a well-determined proper motion of 2".26 per year, whose radial velocity is but 6 km. per second approach. To make these elements harmonize, the velocity of Arcturus in space must be more than 150 km. per second. It is conceivable that the parallaxes of the three stars referred to in this paragraph may be in error by 25 to 40 per cent ; and, if so, the velocities attributed to them would be somewhat in error. These high-velocity stars are sometimes described as run- aways, because they seem to be quite beyond the control of the gravitational power of the universe on any reasonable assump- tion. Newcomb^' has calculated that the maximum velocity attainable by a body starting with velocity zero at an infinite distance and passing through a stellar system containing one hundred million stars, each five times as massive as our Sun and distributed throughout a disk-like spheroid whose maximum radii correspond to 15,000 light years, cannot exceed 40 km. per second. Oroombridge 1830 has a speed nearly nine times this value, and the massive star Arcturus a speed probably four times 25 LicTc Ols. Bull, 5, 133, 1909. 28 Popular Astronomy, Eevised 4th Ed., p. 499. 118 STELLAR MOTIONS this value. If existing velocities owe their magnitudes to the gravitation of the system, the quantity of attracting matter in the system would have to be at least eighty times that assumed by Newcomb, as, other factors being equal, the velocities are proportional to the square roots of the attracting masses. The most comprehensive investigation on this subject is that of Kelvin.^^ Assuming the universe to be composed of gravita- tional matter such as we are acquainted with, in quantity equal to one thousand million times the Sun's mass (twice the mass in Newcomb 's assumption), uniformly distributed throughout a sphere whose radius would correspond to 3300 light years (parallax = 0".001), he finds that the velocity acquired by a body starting originally at rest from the surface of this sphere would, in five million years, be about 20 km. per second, and in twenty- five million years would be about 108 km. per second, provided that the acceleration remained sensibly constant throughout these intervals; or, if conditions were such that the concrete bodies were now about equally spaced throughout the assumed sphere, their mean velocity would be about 50 km. per second. I shall show later that the mean velocity of the stars included on my program of radial velocity determinations is 27 km. per second, as against an average of 50 km. required by Kelvin's assumption of mass and other conditions. Kelvin's assumed mass of the attracting matter in our stellar universe may be regarded as a superior limit, if we agree that velocities are purely gravitational effects, in the ordinary sense ; but it is certain that many stellar velocities are greatly in excess of 108 km. per sec- ond. If the Sun is a star of average mass, and one hundred mil- lion suns are observable in our greatest telescopes, it is clear that by far the major quantity of gravitational matter exists in a form rendering it invisible to us. When we consider the quan- tity of interplanetary matter within the limits of our own solar system, manifesting itself in the zodiacal light phenomena, and in the form of meteors ; and the possibilities of interstellar space as a reservoir for similar minutely divided matter; we should have no difficulty in seeing the reasonableness of Kelvin's results. 27 Beport B. A. A. S., p. 563, 1901. RADIAL VELOCITIES OF STARS 119 We pass to another phase of the subject, beginning by way of illustration with the well-known double-star system, a Gentauri. This double star is our nearest neighbor, so far as known, and is an especially interesting system on that as well as on other accounts. The two stars of the pair, u.^ and u^, of magnitudes 1.7 and 0.3, and of spectral Classes K5 and G, respectively, are known to revolve about their mutual centre of mass in a period of 81.2 years, and their masses, m^ and m^, are supposed to be to each other as about 1.00 to 1.04.=* Our present problem is to determine whether the system as a whole, in other words, whether the centre of mass of the system, is approaching or receding from us, and at what rate. It is a well-known fact that ia every such binary system the two bodies are always on precisely opposite sides of the centre of mass and moving in pre- cisely opposite directions, with speeds inversely proportional to their respective masses, m^ and m^. Let these speeds projected upon the line of sight be V^ and F,. Knowing the ratio of the masses in this case, as stated above, if we measure the speeds F„j and V^^ of the two components in the line of sight, we shall be able to determine the speed y„ of the system as a whole in the line of sight. We shall have : 01 ' n^ ' 1 ' 02 '2 If we multiply these equations through by m^ and m^, respec- tively, we shall have : m, F„, = m, y„ + m^ V^ ■»h ^02 = »»», T^o - »«, ^9 Now the mass of the second times the velocity of the first is always equal to the mass of the first times the velocity of the second. Therefore the first members are equal; and from the equated second members we have : y^ ^ w'2 T^o. + m, y,^ (27) "', + ">, 28 There exists some doubt as to their relative masses : see Chaptek VII, Table XXX. 120 STELLAR MOTIONS Observations secured by my colleague, Mr. Wright of the D. 0. Mills Expedition, on the radial velocities of the two stars are: u CENTAUEI Toi ro2 1904 Feb. 21 —24.4 km. Feb. 25 —19.2 km. Mar. 4 —18.8 —24.4 June 23 —20.1 —24.8 1905 Jan. 27 —19.4 —25.4 Mar. 7 —20.3 —24.9 Apr. 17 —19.0 —25.0 Means —19.5 —24.8 Substituting in (27) we find that the system is approaching us with a speed T"„ = — 22.2 km. per second. The two stars of this system are far enough apart to permit their spectra to be observed separately. Suppose they were so close together that they could not be observed individually, but that the light of both stars entered the slit of the spectrograph at the same time; what should we find on the photographic plate? A composite spectrum, with two sets of lines, corre- sponding to the two speeds observed as above. In this particular case, however, with relative speeds differing only 5 km. per second, the two sets of lines, on the basis of 3-prism dispersion, would not be separated; the composite lines would be only slightly broadened. If the two bodies composing a system are relatively close together, so that they revolve around their common centre of mass very rapidly, say in a few days or a few weeks, the two systems of lines corresponding to the two stars will in general shift rapidly with reference to one another. As the two stars continue to revolve, the two sets of spectral lines will swing past one another very much as two pendulums of equal lengths would, if one pendulum were mounted directly in front of the other and they differed a half cycle in phase. Conversely, if a spectrum is seen to be composite, containing two sets of RADIAL VELOCITIES OF STARS 121 lines which shift from violet toward red and red toward violet, respectively, in continued succession, this is conclusive evidence that we are dealing with a double star whose two components are not very unequal in brightness. A series of spectrograms of such a composite star, measured, will tell us the form of the orbits of the two bodies about their common centre of mass, and, if the usual comparison spectrum is photographed on the plates, the relative masses of the two stars. We can likewise deduce the speed of the centre of mass of the system ; that is, the speed of the system as a whole, toward or from the observer. The first system of this kind to be discovered, in 1889 by Pick- ering, was t, Ursce llajoris,'" the star at the bend in the handle of the Big Dipper. Its spectrum is of a simple type, the principal lines present being those of hydrogen, magnesium, and calcium. It was observed that these lines were at times widely doubled, again they were single and at other times narrowly doubled. There should be no doubt that this star consists of two com- ponent stars nearly equal in magnitude, revolving around their centre of mass, as Vogel later determined,^" in 20.54 mean solar days. By means of a long series of observations Ludendorff^^ has found that the masses of the two bodies are as 1.01 to 1.00, and that the system as a whole is approaching us with a speed of 12.6 km. per second. Suppose, now, that one of the components of such a double- star system is considerably fainter than the other component, because it is a smaller body, or because its efficiency as a light radiator is less. The brighter body will revolve in its orbit around the centre of mass of the system, its velocity of approach and recession will vary regularly through a cycle, and the spec- trum lines will accordingly shift from red toward violet and from violet toward red of their average positions, in proportion to the changes of radial speed. Conversely, if the measured positions of the lines in the spectrum of any star are appreciably different at different times, indicating a variable velocity, we conclude 29 Amer. Jour. Sci., 39, 46, 1890. 30 Ap. J., 13, 328, 1901. 31 Astr. Nach., -ISO, 276, 1909. 122 STELLAR MOTIONS that this star is attended by an invisible companion, massive enough to swing the observed star around in an elliptic orbit. I found in 1898 that rj Pegasi is such a system. The following seven observations, selected from twenty-nine secured altogether, exhibit the alterations of radial velocity with reference to the solar system: Date Velooity Greenwich M. T. Km. 1896 August 27.8 + 7.10 1897 July 8.9 — 6.37 1898 September 4.7 +16.46 1899 January 23.6 — 0.84 June 21.0 — 8.02 1900 September 25.7 +21.40 1901 May 9.0 — 0.18 The period of revolution of the bright star around the centre of mass of itself and its invisible companion proved to be 818 days. Making use of this period, the twenty-nine observations were charted as in Figure 4, with times as abscissae and kilometers as ordinates. Crawford's determination of the orbital elements Figure 4 RADIAL VELOCITIES OF STABS 123 fixed the eccentricity at 0.155 and the velocity of the centre of mass of the system at 4.3 km. per second recession.^"^ In this manner, by means of the synchronous shiftings of two sets of lines, or the shifting of one set of lines, through recur- ring cycles of change, about 300 such double-star systems, known as spectroscopic binary stars, have been discovered to date. How- ever, at this point in the development of our subject, we are interested in obtaining the velocities of as many stars, or sys- tems of stars, as possible, with a view to using these velocities in solving certain fundamental problems of the stellar system. I have, therefore, introduced the subject of binary systems in this chapter only to illustrate the methods of getting at the veloci- ties of such systems as a whole, i.e., of their centres of mass, and a study of their interesting features will be taken up in another chapter. Of the 300 binary systems known, about sixty-five have had their orbits computed, and, consequently, the velocities of their centres of mass determined. These sixty-five velocities are available for use in the fundamental problems immediately before us, precisely as the unchanging velocities of solitary stars are; but the variable observed velocities of the remaining 240 binary stars have not been fully investigated, though the quite approximate values of the systemic velocities are known in eighty of these cases. We now have available, from observations made with the Mills spectrographs at Mount Hamilton and at Santiago, the radial velocities of 1340 stars : of these, 160 are binary stars whose veloci- ties, as explained above, are variable, and for which the veloci- ties of the centres of mass remain as yet unknown. About 160 stars have been observed only once, or are on the list of sus- pected binaries. Deducting these stars, and adding 40 stars observed elsewhere and not at Mount Hamilton or at Santiago, we have 1060 stars whose radial velocities we know.''^ "When these velocities are critically examined and compared, certain interesting facts stand out. 32Ltcfc Ols. Bull., 1, 26-30, 1901. 33 It should be said, however, that many of these stars will undoubtedly prove to be spectroscopic, binaries of long period or small range. 124 STELLAR MOTIONS If a group of neighboring stars be considered, their velocities will seem, in general, to be unrelated to one another; as, for example, the following for a group of twenty-five stars near the vernal equinox: STAES NEAR THE VERNAL EQUINOX — 5.5 km. —10.8 km. — 5.4 —16.0 +12.9 + 4.3 +15.9 — 8.5 —44.9 —22.1 + 6.0 —14.9 + 6.9 + 6.0 + 4.8 +18.6 +10.5 +15.0 + 1.3 —19.0 +11. — 4.6 — 2.0 +32.4 +16.1 V =■ 25 —0.1 km. Here are the velocities of twenty-five neighboring stars near the autumnal equinox: STARS NEAR THE AUTXJMNAL EQUINOX + 6.0 km. + 4.0 km. +16.9 + 4.8 — 8.6 —29.4 — 5.1 +43.5 — 9.9 + 4.2 — 8.9 + 3.0 +21.0 —19.2 — 4.4 —19.0 + 2.0 +17.3 — 4.1 —17.6 +51.2 — 1.2 — 6.0 —13.2 + 4.9 V =+1.3 km. 25 I It will be noticed that positive and negative velocities are nearly equal in number in these two lists, and that their algebraic sums differ little from zero. There is a large region of the sky in which apparent motions RADIAL VELOCITIES OF STABS 125 of approach plainly predominate. Here is an illustrative group of twenty-five velocities in or near the constellations Hercules and Lyra: STARS NEAR HERCULES AND LYRA —14.5 km. — 58:0 IfiTi —26.1 —14.0 —16.1 —26.0 —15.6 +22.6 —26.8 —16.4 — 1.5 — 0.8 —21.6 —26.7 —21.8 —25.5 + 0.4 —51.3 —36.0 —20.3 —14.0 — 9.0 —32.7 —30.3 —22.0 \.= —19.9 km. The differences of their velocities are as great as before, but the negative sign (indicating approach) plainly predominates, only two being positive; and the mean for the group is large, 19.9 km. per second. Here is another group of twenty-five velocities in exactly the opposite region of the sky, in or near the constellations Canis Major and Golumha: STARS NEAR CANIS MAJOR AND COLUMBA —13.7 km. + 2.7 km. — 4.6 — 1.1 +25.0 — 7. +35. + 8. — 9.1 +22.5 +99.2 +37. +89.4 —15. 0.0 +40. +63.5 +28. +21. +21.9 +18.7 +49. +22. +28.5 +34. V =+23.6 km. 126 STELLAR MOTIONS Here again, the velocities seem as unrelated individually, as before ; their mutual differences are as large as in other groups, but the positive sign (indicating recession) plainly predominates. The mean velocity is + 23.6 km. per second. Three of the veloci- ties are very large. If we reject them, the average for the remaining twenty-two stars becomes + 15.3 km. If we form a series of such groupings of velocities, passing in any ■"' .ction from Lyra-Hercules, as the point of departure, ov' * spherical surface of the sky, to the opposite point, we s Jl ii have the mutual differences of velocity within the grOitps approximately as large as ever, but the means for the groups will increase somewhat continuously from approximately — 20 km. up through zero to approximately + 20 km. The significance of these facts is plain: first, the stars have their individual motions, in a large measure apparently at random, but, as we shall see later, not entirely so; second, the stellar sj'stem as a whole has an apparent motion or drift away from the Lyra-Hercules region. This is as we had expected, for a long list of investigators, from Herschel to the present generation, have determined that the solar system must be carrying terres- trial observers toward the general region of sky that includes Lyra-Hercules. If we determine the direction and speed of the solar motion, and correct each observed stellar speed for the effect of the solar motion, the corrected velocities of neighboring stars will differ from each other as widely as ever, but the alge- braic sum of the velocities in any small area of the sky, say areas containing twenty-five neighboring stars each, should be approximately zero, provided the motions of the stars are at random as to direction and magnitude of speed. We now take up the consideration of the solar-motion problem, approaching it from the proper-motion side, but pursuing this phase of the subject only far enough to supply a natural introduction for the radial velocity method of solving the problem. A comprehensive presentation of the proper-motion determinations of the Sun's course would demand the capacity of a separate volume. CHAPTER IV THE SOLAR MOTION AS DETERMINED FROM STELLAR PROPER MOTIONS ,, "We shall gain a better comprehension of the solai-'-, ..t;il;:»l problem if we assume, first, that the distant stars are really fixed; that is, that they are at rest relatively to one another; and that our star alone is in motion. What should be the observed effects? Using Herschel's original diagram,^ Figure 5, J*B let the Sun have been at S at some past epoch, and let its motion in the interval have carried it along the line SB to its present position C. Let the stars be distributed through surrounding space, in all directions and at all distances from the solar system, as at s, s, s, When the Sun was at S, the stars were seen iPM. Trans. (Abridged Ed.), 15, 1783, Fig. 8, PI. VI. 128 STELLAR MOTIONS :■ projected upon the celestial sphere at a, a, a From the Sun now at C, the same stars are seen at b, b, b, That is, each star in the sky will appear to have moved away from B, along the great circle drawn from B through the star to A, over the angular distance, ab, ab, ab, The point B toward which the solar system is assumed to move is known as the apex of the Sun's way, and the point A is the antapex. The apparent motion of every star in the sky away from the apex and toward the anta- pex, due to the observer's motion toward the apex, is variously called, for convenience, the parallactic component of the star's motion, or, more briefly, the star 's parallactic motion, parallactic displacement, or secular parallax. It is clear that this parallactic motion is a function : 1. Of the Sun's speed: a doubling of the Sun's velocity would double the corresponding displacement of every star; 2. Of the star 's distance : a doubling of the star 's distance woidd divide the parallactic displacement by two ; and 3. Of the star's angular distance from the apex: a star exactly in the apex or antapex would suffer no apparent displacement, and a star 90° from the apex would suffer the maximum displacement. Other conditions being equal, the displacement would vary as the sine of the star's apical distance. Giving mathematical expression to these relations, in any one of the triangles sSC, let SC, the Sun's motion in the unit of time, be called q; let D be the angular distance sSB of the star from apex B ; let u be the parallactic displacement, CsS of the star s ; and let p = sC be the star's distance from the solar system. Then we have : g : p : : sin v : sin I) (28) As V is always a small angle, we may replace it by v" sin 1". If we express q as the angular speed of the Sun when viewed at right angles from distance unity, we may place q" sin 1" for q and (28) becomes - sin 2* = - (29) P This equation expresses the relation which always holds between 80LAB MOTION FROM PROPER MOTIONS 129 the Sun's motion, the star's direction and distance, and the resulting parallactic motion of the star. Let us consider the reverse problem. If observation should show that all the stars are moving toward a common point in the sky, with speeds as the sine function of their distances from that point, we should come to but one reasonable explanation: there must be a motion of our Sun away from that point. In this case, how simple the problem of determining the elements of the solar motion. If for any two stars whose positions (right ascension and declination) are known, we should determine by observa- tion the directions of their apparent motion, the amount of the parallactic motion, v, of one of the stars, and the distance, p, of that star, we should be able to solve absolutely and completely the solar-motion problem by a simple calculation : the great circles containing the parallactic motions would intersect in the apex and antapex; the value of sin D in (29) would come from simple computation ; and q, the speed of the Sun toward the apex, being the only unknown quantity in (29), would be determined at once. This simple illustrative hypothesis gives way, in the actuality, to a situation so complex as to exceed the bounds of present com- prehension. A century and a quarter of investigation, almost continuous since the days of Herschel, has taught us much con- cerning the Sun's motion; but the chief result has been to make more acute our sense of the difficulties besetting the problem. From the proper-motion side of approach, at least, the last five years have been especially remarkable for the unexpected com- plications shown to exist in this problem. To these we shall refer later. The stars are not at rest, but each has its own motion, not on the surface of the celestial sphere, but in space of three dimen- sions. These motions, in amount and direction, we shall con- sider, for the present, as at random, showing no preference for any speed or direction. Let S, S, S, . . . ., in Figure 6,= repre- sent the positions of stars at some past time, as projected upon the celestial sphere. Let the stars have had actual motions in 2 From Kobold 's Bau des Fixsternsystems, p. 86. 4 130 STELLAR MOTIONS space from that given time to the present, such that the pro- jectioji8 of those motions upon the celestial sphere are repre- sented by the ares SM, SM, SM, These are known as the real or peculiar motions {motus peculiares) of the stars. Repre- sent the position of the antapex by A and the stars' parallactic motions in the given time interval by SN, SN, SN, Then the apparent or observed motions of the stars are the resultants of the peculiar and the parallactic motions, SR, SB, SR ; and these are the star's proper motions in the time interval. i a<. Figure 6 Let the proper-motion arcs be prolonged, two by two, until they intersect in the points a, a, a, None of these points coincides with the antapex A, but the points will cluster around A; how closely around A will depend upon the magnitudes of the parallactic components as compared with the peculiar motion components. It was this graphical method of locating A, as the approximate centre of gravity of the intersections a, a, a, . . . ., that HerscheP used with the proper motions of thirteen stars in sPhil. Trans. (Abridged Ed.), 15, 402, 1783. SOLAR MOTION FROM PROPER MOTIONS 131 1783 — all the proper motions then known — to fix the position of the apex at a„ = 262°, So = + 26°, near the star A. SercuUs.* A few months after the presentation of Herschel's results to the Eoyal Society, Prevost" presented to the Berlin Academy of Sciences (July, 1783) the results of his discussion of the same proper motions. He concluded that the most probable position of the solar apex was a^ = 231°, 8„ = + 25° (reduced to equinox of 1900.0). A critical discussion of Herschel's first results was made by KliigeP in 1789. He concluded that Herschel's position of the apex as depending upon the proper motions of the stars employed could not be improved upon. In the Figure the proper-motion arcs pass at distances Ad, Ad, Ad, . . . ., on one side or the other of A. That position of the antapex is best, in general, which makes these distances Ad, . . . . , as small as possible. Herschel, in 1805, used, first, the proper motions of six of the brightest stars, Sirius, Arcturus, Capella, Vega, Aldebaran, and Procyon, to determine a new posi- tion of the apex, such that the sum of the six distances. Ad, would be a minimum. The position defined by the six was tested, but not modified, by the proper motions of thirty other and fainter stars — ^the thirty-six in Maskelyne's catalogue being all that were then regarded as satisfactorily determined. His result was' a„ = 246°.5, S,, = 49°.4. This position differs 27° from that adopted in 1783, from thirteen proper motions. The discrepancy represents fairly well the uncertainty remaiuing in each solution. Herschel was well aware of the weak foundation on which his results rested; for, in reporting those of the year 1783 to the Eoyal Society, he said, "that we have already some reasons to guess {sic) which way the solar system is probably tending its course." [Phil. Trans. (Abridged Ed.), 15, 402, 1783.] That he came so close to the truth was not the result of a guess, but of the philosophical * The coordinates of the apex are reduced to the equinox of 1900.0. 5 Mem. de I 'Acad, de Prusse, 1781, p. 445. 6 Berliner Jahrlueh, 1789, p. 214. f Phil. Trans., 1805, p. 256. 132 STELLAR MOTIONS ability, amounting to real genius, with whieli lie considered all known facts bearing upon the questions before him. The 1805 apex has not stood the test of time so well as the 1783 apex. Hersehel employed a purely "cut and try" method for locat- ing the antapex, which would be impracticable if the number of proper motions employed were large. A direct analytical solu- tion for the most probable position of the antapex is obtained as follows : Let xj/ be the known position angle of the proper-motion arc SR, x the unknown position angle of the parallactic motion SN, and D the imknown angular distance of the star from the apex. Then in the right triangle dSA, we shall have, for any star, sin D sin (^ — i/f) = sin Ad ; each equation containing as unknowns Xj ^j ^-nd Ad. Now any value of D and of x '^^^ ^ ^ position for the apex, but the resulting values of Ad will be different for every star. If the number of stars is very large and their motions are entirely at random, the distances Ad will follow the laws of accidental errors, and a solution of all the equations by the method of least squares, making the sum of the squares of all the distances. Ad, a minimum, will give the most probable position of the antapex. That is, the complete solution, in principle, is contained in the equation : 2 ] arc sin [ sin D sin (x-'A)] ['=5 Ad' = Minimum. (30) In practice, this equation woidd have to be transformed consider- ably before solution. Of course Hersehel did not make use of the least-squares principle, as he preceded Gauss, the developer of the method, by a quarter of a century. In effect, he omitted the exponent 2 and made the sum of the Ad's a minimum. Thus far Hersehel gave no concern to the different distances of the stars from our system — a factor far from negligible — ^nor to the speed of the solar motion. His generation was without knowledge of the distance of a single star, and the making of assumptions was unavoidable. He assumed that the six stars, SOLAR MOTION FROM PROPER MOTIONS 133 Sirius, Arcturus, Capella, Vega, Aldebaran, and Procyon, are of equal absolute brightness, and, therefore, that their distances are inversely proportional to the square roots of their apparent brightness. In this manner, letting Sirius be at unit distance and of unit brightness, he deduced relative distances for the other five stars. Assuming a position for the antapex. A, in Figure 6, he computed for each star the angle NSR = x — "A? ^-s before, between the observed direction, i/f, of the proper motion, SR — As and the great circle SA. (SN is the parallaetie com- ponent of the star's apparent motion, and SM = A s sin (x — ij/), the star's own or peculiar motion.) Herschel's idea was that the antapex, A, and speed, q, of the solar motion should be so chosen as to make the component remaining as each star's own motion as small as possible. Evidently the quantities A s sin (x — f) for the several stars are not homogeneous, as the bodies are at differ- ent distances; and in order that aU may enter the problem with equal weight, A s must be multiplied by p before using. Thus, the expression to be given as small a value as possible is A s p sin (x — "A)- By trial of various values of the Sun's speed, q, originally with the six first-magnitude stars, and later by permitting the observed proper motions of thirty other fainter stars to check and modify his conclusions, Herschel decided in favor of 3 = 0".89. The speed, q, is expressed in terms of the average distance of the six first-magnitude stars. That is, there being 206,265 seconds in the unit of length, or distance, the solar motion in one year would carry it ' = 1/230,000 the distance of a 206,265 first-magnitude star. Herschel did not know the distances of the first-magnitude stars, and, therefore, he could not translate this result into linear velocity. From the parallaxes now assigned to the six first-magnitude stars used by Herschel, giving average distance thirty-three light years, we find the linear value of 3 = 0".89 to be 43 km. per second.' 8 To be exact, Herschel expressed the Sun 's speed in terms of the distance of Sirius taken as unity: g=:l".117. His assumed relative distances of the six stars were, respectively: 1.00, 1.20, 1.25, 1.30, 1.40 and lAO.—Phil. Trans., 1806, p. 233. 134 STELLAR MOTIONS Although the enormous accumulation of proper-motion obser- vations since Herschel's day has rendered his results obso- lete, one cannot pass on without registering high admiration for the genius which opened this field of research. It is necessary that we define more exactly the term "solar motion," for all motion is relative, and we must describe the datum to which it relates. It would be meaningless, for example, to say simply that the solar system is moving toward a point whose E. A. is 270° and Decl. + 30°, with a speed of 19 km. per second. Knowing, as we now do, the motions of many stars, it would be possible to select a list of special stars relatively to which our solar system would be moving in a direction opposite to that quoted above, with a speed of 10 km. ; or other groups of stars could be formed, by selection, such that the solar motion with reference to them would be at right angles to the direction quoted, with still different speeds. The unusual interest taken by astronomers in the solar-motion problem is almost negligibly in that motion on its own account, though this is of wide human interest. We want to know the direction and speed of the solar motion in order that we may eliminate its effects from the apparent motions of the other stars, and thus have left the real motions of the stars. The purpose of our investigations fixes, therefore, the datum to which we would have the solution refer. We would know the elements of the solar motion — the direction and speed — with reference to the centre of mass of the stellar universe and to some definite system of coordinate directions. The universe, at least that part of it which the telescope shows, and for which we have present means of study, is quite generally believed to be finite in extent, and it is here thus assumed. The problem, at first sight, appears to be definite, but it is so only to a limited degree ; for we must not overlook the dark and invisible bodies which space near and far is thought to contain. The complete solution of the problem involves a knowledge of the mass, and of the direction, distance, direction of motion and speed of motion, at the desired instant of time, of every body in the system, whether visible or invisible. Such a solution is beyond our present powers, and beyond our SOLAR MOTION FROM PROPER MOTIONS 135 present conception of future powers; and we must content our- selves with approximate solutions, improving them with the advance of time and the development of better methods. Fortu- nately, we need not know even the approximate locus of the centre of mass of the system in order to determine and define the present direction and speed of our motion, though this knowl- edge would be essential in predicting our motion in the distant future. The directions of rectangular coordinate axes in and perpendicular to the ecliptic, or in and perpendicular to the equator, are determined by the geometry of the Earth 's motions. A set of axes anywhere in space, parallel to, or definitely related in directions to, these axes will serve as a basis for studies of the stellar system, provided also their intersection, as the origin of coordinates, remains in a fixed, but it may be entirely unknown, relation to the real centre of mass of the system; for the direc- tion and speed of our motion relative to the origin and axes described are identically the same as with reference to axes actually passing through the unlocated centre of mass of the system. Our available methods of determining the solar motion are necessarily deductive, and dependent upon the observed effects of the solar motion on the apparent motions of the stars. The results must refer the solar motion exclusively to the system of stars utilized in the solution. In this and similar problems astronomers should ever hold in mind two salient points: 1. To select stars as a basis for the solution such that their motions, as a small system, will be representative of the entire stellar system, as far as possible. 2. To employ such mathematical processes of solution as shall best eliminate the effects of the individual motions of the stars employed as the basis of the investigation, and leave the parallactic motion as the residuum sought. Herschel's first solution, described above, based upon the observed proper motions of the thirteen available stars, referred the solar motion to the system composed of those thirteen stars, and no other stars ; and it was satisfactory only in so far as those 136 STELLAR MOTIONS stars were representative of the hundreds of millions of bodies in the stellar system ; and similarly for his later solution, based upon the known proper motions of thirty-six stars. After Herschel, the greatest of observers, came Bessel, the greatest of practical astronomers, with quite a different method of attack. He determined the poles of the great circles which pass, respectively, through the stars in the directions of their known proper motions. If the observed motions of the stars were entirely parallactic, that is, directed exactly toward the antapex, the poles of their proper-motion circles would all lie on one great circle (parallactic circle) whose poles .in their turn would be the antapex and apex. In reality, as the observed motions are compounded of the peculiar motions and the paral- lactic motions, the poles of their great circles would be scattered more or less widely on both sides of the real parallactic circle. Bessel had hoped to draw a great circle amongst the plotted polar positions which would make the sum of the distances of the indi- vidual proper-motion poles from this circle a minimum; or, according to more modern methods, the sum of the squares of these distances a minimum. He found on the contrary that the plotted poles did not define the position of such a great circle: the proper-motion poles of the seventy-one stars utilized by him appeared to be distributed at random over the entire sphere. He, therefore, came to the conclusion that there was no justifi- cation for assigning a position to the solar apex, as Herschel and others had done. Bessel 's position in the profession was so high that his views prevailed for two decades.' It appeared nearly a century later that Bessel 's negative result was due to his selection of an unfortunate method. New life was given to the subject in 1837 by Argelander's solution,^" based upon the well-determined proper motions of 390 stars. He assumed a position of the apex and computed for each 9 Many authors {e.g., Chauvenet, Spher. and Prac. Astr., 1, 705, Fifth Ed.) credit Gauss with a solution of the solar motion, leading to the position of the apex o„ = 260°, So = -(- 31° ; but I have not been able to find Gauss's paper, nor is it listed in Houzeau's Vade-Mecum. ■LOMem. de I 'Acad. St. Petersbourg, 3, 590, 1837; Astr. Nach., 16, 45, 1838. SOLAR MOTION FROM PROPER MOTIONS 137 star the value of the expression sin (x — 1 . Porter 282 .0 - -53 .7 576 " -^O Porter 280 .8 - -40 .1 533 " . Porter 285 .3 - -34 .0 142 " . Porter 277 .1 - -34 .9 70 " >1 Newcomb 276 .9 -31 .4 644 " large /i Neweomb 272 .5 - -31 .O 2527 " small m Kapteyn 273 .8 -29 .5 Extensive Boss 283 .5 -44 .1 273 stars, Dee. = Boss 288 .9 - -51 .5 247 " " Kapteyn 275 .0 - -29 .2 1809 " Kapteyn 274 .6 - -27 O All Bradley stars Kapteyn 267 .6 -29 .4 U li it Kapteyn 275 .0 -28 .5 tt 11 cc Kobold 275 .0 - h ,4 213 stars, southei Kobold 269 .0 - 2 .9 1579 stars Kobold 270 .3 .0 1579 " in 122 g Kobold 270 O - 2 .3 2262 " Kobold 271 .0 - .2 2262 " Kobold 270 .4 +16 .5 2262 " in 122 g Kobold 265 .1 - 8 .1 144 " large /i Kobold .23 Stars near Vernal Kobold .79 " " Antape Kobold 79 " " Autunu Kobold .25 " " Apex Kobold .22 " inHemisph. Kobold .82 U ti it Weersma 270 o +30 .8 Bradley stars Weersma 268 .0 +31 .4 14.9 km. All available (190 [OTION DETERMINATIONS, 1783-1908 Fsed References Phil. T,-a„>i., 1783 (Abridged Ed.), p. 402 Mein. de I'Acad. de Prusse, 1781, p. 445 Phil. Traiix., 1805, p. 256 Phil. Trans., 1806, p. 233 Asti: Xach., 16, 48, 1838 Astr. Nach., 21, 73, 1843 Dm-pat Beoh., 14, 227, 1856 Mem. R. A. S., 28, 160, 1860 ilmn. li. A. S., 32, 27 and 31, 1864 Mem. de VAeud. St. Petershaurg (VII), 35, p. 24 to0".:i2 A.ftr. Xuch., 125, 42.i, 1890 !toO .6t 11 ti li a it t tol ,28 i ) il 11 u u Cl ti 11 ti ii a Astr. Jour., 12, 93, 1892 ) to 60 li li 11 11 a )tol .20 ) n a 11 a li ^i a 11 it a " " 20, 4, 1899 li ii (i ii ii Astr. Nach., 156, 17, 1901 °+ to +5° Astr. Jour., 9, 165, 1890 " to " ii ti ii 11 ii Astr. Nach., 161, 347, 1903 " " 156, 15, 1901 ii 11 li ii ii li 11 ii ii ii " " 144, 39, 1897 " 144, 40, 1897 )ups it li ii ii ii " " 150, 269, 1899 " " 150, 279, 1899 )ups " 150, 272, 1899 " 166, 3, 1904 ijuinox Ban des Fixsternsystems, p. 131 (( It It It it it 1 Equinox tl it (( :( (( a tl it a tt tl 11 E Vernal Eq. 11 It it tt tl ii Autum. Eq. tt ti ii tt tt tl PuU. Astr. Lab. Gron., No. 21, p. 21, 1908 li's Publ. Astr. Lab. Gron., No. 21, p. 57, 1908 SOLAR MOTION FROM PROPER MOTIONS 143 would contain fully twice as many names and more than twice as many entries. Numerous results by Kapteyn and Kobold for special lists of proper-motion stars are not quoted. The refer- ences to their more comprehensive results will lead the reader to the omitted ones. We stop for a few comments on these results. The right ascensions vary from Herschel's 247° to Boss's 289°, or 42° ; the declinations range from Porter's -f 54° to Kobold 's — 8°, total 62°. The determinations based upon only a few stars, upon stars distributed over a small part of the sphere, or upon stars whose proper motions are of selected dimensions, should have small weight. Some of the discordances in the table must be due in part to systematic errors in the catalogues of star places as well as to small errors in the precession factors. Kapteyn has shown, for example, that the discordant apical declinations assigned by several determinations of greatest weight are made more accord- ant by allowing for systematic errors in the declination compo- nents of the proper motions, and by the use of a corrected value for the precession constant. The values of q, the annual solar motion, are in terms of the average distance of a 1.0 magnitude star. Results for the speed of solar motion obtained by several computers, but not quoted in the table, are very discordant, some of the results being ten times as large as others. The uncertainties in the assigned values are perhaps indicated in fair measure by Kobold 's results: the stars in one hemisphere of the sky assign a solar velocity three and a half times as great as that assigned by the stars in another hemisphere of the sky. The weighted mean of the determinations would seem to place the apex in Right Ascension 271°, Declination -(-31°. It is a convenience, however, to assume it in position i*o = 270°, So = -j- 30° ; and this is certainly well within the limits of prob- able error. The definitive proper motions of more than six thousand stars contained in Professor Boss's Preliminary Gen- eral Catalogue, now almost through the press, should form the basis for a solution of the solar-motion problem which will undoubtedly render existing solutions in a sense obsolete. 144 STELLAR MOTIONS Kapteyn'^ announced, in 1904-1905, that his researches on proper motions during preceding years not only confirmed Kobold's conclusions as to the non-random character of stellar motions, but justified him in announcing the directions to which stellar motions give preference. He had used again the proper motions of more than 2400 Bradley- Auwers stars, extending from the North Pole to Declination — 31°. Dividing this three- quarters of the sky into twenty-eight compact areas, averaging nearly 100 stars to each area, he determined the prevailing ten- dencies in each area, by a method which we shall not here take time to present. The outcome was that these prevailing direction- tendencies divided themselves quite clearly into two groups ; one group pointing to a certain not very large area of the sky and the other group to another well-defined area. These two areas, widely separated, represent the convergent points of the prefer- ential proper motions. Kapteyn assigned the position of one area at u = 85°, 8 = — 11°, and of the other area u, = 260°, 8 = — 48°. While the apparent motions of the stars favored one or the other of these centres, we must not overlook the effect of the solar-motion component. When this component is elimi- nated the mathematical principles involved leave us no recourse but to assume that the stars, with reference to the stellar system as a whole, have a preference for motion in diametrically oppo- site directions, either toward a point at a = 91°, 8 = -|- 13° in the northern edge of Orion, or toward the antipodal point at a = 271°, 8 = — 13°. These points Kapteyn has denominated the "vertices of preferential motion." It must not be under- stood that the individual stars according to this theory are actually moving parallel to the straight line joining these ver- tices, but simply that their components of motion parallel to this line are considerably greater, on the average, than in any other direction. We may visualize these ideas in the following manner : Assume the existence of a great cluster of stars, spherical or otherwise, distributed through space uniformly or otherwise, IS Congress of Arts and Sciences, St. Louis, 4, 413, 1904; Meport B. A. A. S., 1905, p. 257. PREFERENTIAL MOTION 145 whose individual motions were at random in both magnitude and direction. Suppose, further, that an entirely similar group of stars, whose members were moving at random, occupied another volume of space. Let these two groups of stars approach each other and more or less completely interpenetrate. Necessarily the resultant preferential motion in the combined system is in the direction of the line of approach of the two systems. There are stars moving in all directions, with speeds of all dimensions withiu certain limits, and yet there exists a preference in the combined system for motion along and parallel to the line which originally joined the centres of the two groups. This is the line joining the vertices. Assume now that our Sun is moving through the combined group in a direction making a considerable angle with the line of preferential motion. The motions of the indi- vidual stars as observed from the solar system would have pref- erences for two directions very different from the line joining the vertices. We should then have in all essential matters the conditions which Kapteyn, working backwards so to speak, has described. Neither Kapteyn nor any other investigator, so far as I know, has expressed or holds the belief that the stellar system is the result of the combination of two primordial star groups which, originally separate, are now combined, and the description is intended to be merely a hypothetical help to the imagination. To this tendency of the stars to move in opposite directions, Kapteyn has applied the term "star streaming," and it is cus- tomary to speak of two "streams" of stars which have prefer- ential motions in opposite directions. Eddington has used the term, two "star drifts," in this connection. Eddington, of Greenwich, gave substantial confirmation to Kapteyn 's conclusions, based upon a study of the proper motions of about 4500 stars lying within 52° of the North Pole, which had been observed by Groombridge early in the nineteenth century and re-observed at Greenwich late in that century. He found strong evidence for the existence of two star drifts, substan- tially alike as to details of composition, the directions of the drifts agreeing closely with Kapteyn 's directions. U6 STELLAR MOTIONS Dyson, astronomer royal for Scotland, based a study of the subject upon about 1900 stars whose proper motions are greater than 20" of arc per century. The characteristics of the two streams were brought out with great clearness. In fact, for the stars whose proper motions are greater than 100" per century, he thought he was able to assign a majority definitely to one or to the other of the two streams. The theory was well illustrated by means of his tabulation of the apparent directions of the indi- vidual proper motions with reference to lines drawn from the separate stars to the assumed vertices of the two streams. The proper motions whose directions make small angles with the lines through the vertices are most numerous, and as the devia- tions from these directions increase more and more the number of corresponding stellar motions decreases almost continuously. Calling the speed of the solar system unity, Dyson's investigation led him to the conclusion that the relative speed of the two streams is 2.6; that is, for a solar speed of 19 km. per second, the separation-speed of the streams would be 48 km. per second. Schwarzschild, of Gottingen, introduced a promising hjrpothe- sis in connection with this problem. There is a fair chance that it has greater probability of conforming to the reality. He leaves the stars in one system instead of dividing them into two streams, or drifts, and assumes that the components of the real stellar motions are on the average greater in one direction than in any other; and that the actual stellar motions, as functions of their directions, can be represented in amount and in direction by all the radii of an ellipsoid whose longest axis coincides with the direction of relative motion in Kapteyn's two-stream theory. Assuming this hypothesis to be correct, and that our solar system is travelling through the stellar system not at right angles to the long axis of the ellipsoid, we should have another representation of the preferences of the observed proper motions for the two directions which Kaptejni had determined. The difference between Kapteyn's and Schwarzschild 's hypotheses may be more apparent than real. So far as we can now see, two streams of stars, thoroughly intermingled, with preferential motions in opposite directions, are essentially equiv- K. A. Dec. E.A. Dec. 91° +13- 266° +31° 95 + 3 266 +33 93 + 6 281 +42 88 +24 281 +36 86 +24 109 + 6 268 +26 96 + 7 87 +11 PREFERENTIAL MOTION 147 alent to opposite prevailing motions in one system, at least for periods of time which are in efiEect instantaneous. Below are solutions for the apex of the Sun's motion and for the vertices of the two star streams, by Kapteyn, Eddington, and Dyson; and for the positions of the solar apex and of the ver- tices of Schwarzschild's ellipsoid of preferential motion, by Sehwarzschild, Beljawsky, and Rudolph. Apex Vertex K. Kapteyn — ^Bradley stars* Eddington — Groombridge stars** Sehwarzschild — Groombridge stars t Dyson— Stars of large proper motion tt Beljawsky — Porter's starst Eddington — ^Zodiacal stars** Eudolph — Bradley stars§ Budolph — Bradley starsj Means 272 +34 93 +12 The vertex, as Kapteyn noted, is in the Milky Way ; that is, the mutual axis of the two streams lies in the plane of the Milky "Way. This is not out of harmony with the view that prevail- ing stellar motions, if gravitational effects, should have their governing centres in the plane of the Milky Way. It was thought by some, for a time, that the two apparent star streams might not have an objective existence, but that they could be explained on the basis of systematic errors in star cata- logues of position, yielding erroneous steUar proper motions. However, the question has been investigated from so many points of view, upon the basis of observational data taken from inde- pendent sources, all leading to positions for the vertices of the streams in remarkable accord, that we cannot hesitate to accept * Report B. A. A. S., 1905, p. 257. ** Mon. Not. B. A. S., 68, 602, 1908. t Gottingener Nach., 1907, pp. 628, 631. tt Proc. Soy. Soc. Edinburgh, 28, 231, 1908, and 29, 376, 1909. t Astr. NacK, 179, 298, 1908. § Astr. Nach., 183, 5-6, 1909. 148 STELLAR MOTIONS the theory as representing a cosmical truth. Later, we shall consider the subject in the light of radial velocity methods and results. We should inquire as to whether and to what extent the stars whose proper motions have been utilized in determining the solar motion are representative of the general stellar system. It is well known that our sidereal universe appears to have vastly greater extension in the plane of the Milky "Way than in any other direction. The prevailing idea of the form of the uni- verse is, that it is roughly an ellipsoid whose minor axis at right angles to the plane of the Milky Way is extremely short in com- parison with the major axes in the plane of the Milky Way. Bearing vitally upon this question are the "star gauges" of the Hersehels, father and son ; and we may recall in passing that it was Sir William Hersehel who first brought the question of the form of the universe under quantitative consideration. Using a reflecting telescope of 18 inches aperture, 20 feet focus, and magnifying power of 180 diameters, Hersehel counted the total number of stars visible in the field of view, 15' in diameter, in about 5000 regions distributed over the northern sky. It was his habit to average the results for one to ten or more closely neighboring areas into one result, and in this manner the counts formed 1088 star gauges.^" Sir John Hersehel employed identi- cally the same instrument at the Cape of Good Hope in counting the number of stars in 2299 areas uniformly distributed over the southern sky, suitable precautions being taken to avoid or to eliminate the effects of fields which happened to be extraordi- narily rich or poor in stars. The limit of visibility for the tele- scope used was probably in the vicinity of 13% visual magnitude. These invaluable data have been made the basis for several investigations of stellar distribution, with reference to the Milky Way as a plane of symmetry. Wilhelm Struve^" made a statis- tical study of Sir William Hersehel 's star gauges. He expressed i^Publ. Washburn Ohs., 2, 113-173, 1888; 405 "gauges" were here published for the first time. 20 Etudes D'Astron. Stellaire, 1847, pp. 71-72; using the 683 gauges pub- lished by Hersehel in Phil. Trans., 1785. o bo o O ■< Hi O H h < ■=> 5: 00 § Of 1 OS _3C == o ^ o '^ f^ o d a « 8 S 8 < fe erttTs PREFERENTIAL MOTION 149 the law of their distribution, in terms of galactic latitudes, as follows : North Galactic Latitude 90° 75 60 45 30 15 Average number of stars per field 15' in diameter 4.15 4.68 6.52 10.36 17.68 30.30 122.00 Sir John Herschel published" his results for the Southern Hemisphere not for definite latitude values, but as averages for latitude zones 15° wide. By means of Struve's formula he expressed Sir William Hersehel's results in the same manner. In the following table the first six entries contain Sir "William's data for the zones north of the central plane of the Milky Way, and the last six Sir John's for the six zones south of the Milky Way. North Gralactic Average number Latitude of stars per field Zones 15' in diameter _)-90°-+75° 4.32 _|_75 -+60 5.42 +60 -+45 8.21 +45 -+30 13.61 +30 -+15 24.09 +15 - 53.43 15 59.06 —15 30 26.29 —30 45 13.49 — 45 60 9.08 —60 75 6.62 —75 90 6.05 A glance at the table shows that similar laws of density prevail in the two hemispheres with reference to the plane of the Milky 21 Outlines of Astronomy, 1849, p. 535. 150 STELLAR MOTIONS Way, though the number of stars in the Southern Hemisphere appears to be slightly greater. Suitable allowance having been made for the greater transparency of the air at the southern observing station, the generally accepted explanation of the dis- crepancy is that the solar system is not in the central plane of the Milky "Way, but that it lies a little to the north of it. Struve's statistical studies led him to the conclusion that the effective central line of the Milky "Way is not a great circle of the sphere but is a small circle lying at a distance of 92° from the North Pole of the galaxy and 88° from the South Pole.^^ The extremely rapid increase of star density with decreasing galactic latitudes is undoubtedly due to the greater extent of the stellar universe in the direction of the Milky Way, though Struve believed that the stars are somewhat more closely crowded together in space as we approach the Milky Way. The distribution of naked-eye stars with reference to the Milky Way as the plane of symmetry was well brought out by Houzeau, as in the following table,^' in which the first six magnitudes are quoted for nine zones of galactic latitude each 20° wide. Galactic Numbers of Stars of Latitude Visual Magnitudes Zones 1 2 3 4 5 6 Total Density +90° -+70° 1 4 8 42 86 141 0.113 +70 -+50 3 6 18 42 74 295 438 0.122 +50 -+30 1 4 21 70 148 439 683 0.124 +30 -+10 4 7 34 114 190 625 974 0.145 +10 10 7 11 46 108 243 730 1145 0.160 —10 30 3 11 34 111 257 619 1035 0.154 —30 50 7 28 65 141 465 706 0.129 —50 70 2 2 13 65 81 281 444 0.124 —70 90 2 2 12 37 100 153 0.125 Total 20 51 200 595 1213 3640 5719 0.139 The totals for the nine zones should, of course, differ, even for uniform distribution, because the areas of the zones are unequal, 22 Etudes D 'Astron. Stellaire, 1847, p. 61. 23 Annales de I'Observatoire Bruxelles, 1, 51, 1878. ij Cannce Nova? Nova Nova Nwmw Nova Sagittarii N. G. C. 7662 Gaseous Nebula N. G. C. 7009 Gaseous Nebula jK. Centauri Hydrogen lines bright Secondary Hy- drogen lines Ti Gruis Spectrum Peculiar Peculiar Stellar Spectra.- Observatory -Harvard College 152 STELLAR MOTIONS but the increase of star density with decreasing latitudes is demonstrated in the last column. A further condensation of the table, ^* as below, is illuminating. Galactic Latitude Zones Magnitudes 1+2+3 Magnitudes 4+5 + 6 Number of Stars Density Number of Stars Density +90°- +30° +30 30 —30 90 58 157 56 0.00563 0.00761 0.00543 1204 2997 1247 0.117 0.145 0.121 From the relative densities, we see that the stars in both groups show a moderate preference for low galactic latitudes. The central zone, -)- 30° to — 30°, embracing one-half of the sky, con- tains 3154 stars, whereas the other two zones together, embracing the other half of the sky, contain only 2565 stars. Investigations analogous to those described have been made by several astronomers, for both the brighter and the fainter stars, but we have space only for a few of Seeliger's results, as published in numerous papers presented, for the most part, to the Munich Academy of Sciences. Among other sources of information, he used the Herschel star gauges, and the Bonner Durchmusterung, which Argelander and Schonfeld intended should include all stars, to the ninth magnitude, from the North Pole of the heavens to Declination — 24°. Seeliger divided the sky into nine galactic zones, each 20° wide, as in the first column of the following table. The average numbers of Durchmusterung stars down to magnitude 9.0 per square degree in the several zones are in the second column, headed "DM. Density." The third column contains the Durchmusterung densities after due allowance has been made for the fact that the brightnesses were overestimated, in effect, in the regions containing few stars, and underestimated in the denser regions lying in the Milky Way. The last column contains the Herschelian densities. 2i Annales de I'O'bservatoire Bruxelles, 1, 52, 1878. DISTRIBUTION OF BRIGHTER STARS 153 Galactic Latitude DM. Density Corrected Herschelian Zones per sq. degi-ee DM. Density Gauges -)_90°-+70° 3.06 2.78 107 +70 -+50 3.24 3.03 154 +50 -+30 3.80 3.54 281 +30 -+10 5.34 5.32 560 +10 10 7.36 8.17 2019 —10 30 5.94 6.07 672 —30 50 3.99 3.71 261 —50 70 3.56 3.21 154 —70 90 3.51 3.14 111 Extremely important facts concerning the stellar universe are apparent from Houzeau's and Seeliger's tables. Stars to the sixth magnitude have a small but certain preference for the galaxy. Stars down to the ninth magnitude have threefold greater density of distribution in the zone containing the plane of the Milky Way than in the zones containing the galactic poles. The stars down to the 13 1/^ magnitude are shown by Herschel's gauges to have nearly twenty times greater density of distribu- tion LQ the Milky "Way than at the galactic poles. These facts make the principal basis for existing ideas of the form of the stellar universe. Professor Pickering's studies on the distribution of the spec- tral classes, especially with reference to the Milky Way, have been very extensive and fruitful. Not to enter upon the details, he wrote in 1891 : "It appears that the number of stars of the (Secchi) second and third type is nearly the same in the Milky Way as in other parts of the sky. Considering, therefore, only the stars whose spectra resemble that of our Sun, we should find them nearly equally distributed in the sky. The stars of Class A on the other hand are twice as numerous in Region M (the Milky Way areas) as in Region N (outside the Milky Way), and in the case of Class B this ratio exceeds four. The Milky Way is, there- fore, due to an aggregation of stars of the first type, a class to which our Sun seems to bear no resemblance as regards its spec- trum. Spectra of Class B seem to conform still more closely to the region of the Milky Way, although probably they are not sufficiently numerous to materially affect its light. The Milky Way must, therefore, be described as a distinct cluster of stars to 154 STELLAR MOTIONS which, from its composition or age, the Sun does not seem to belong. ' ' The proportion of stars of Class A in the Milky Way appears to be greater for the faint than for the bright stars. ' '^^ Pickering has tabulated the numbers of stars of the different spectral classes, brighter than visual magnitude 6.25, in terms of their galactic latitudes, as in Table VII. By way of expla- nation, the latitudes quoted in the first column are the average latitudes of each of eight approximately equal zones, beginning with that which surrounds the north galactic pole and ending with that which surrounds the soxith galactic pole. If the stars of the different spectral classes were uniformly distributed over the sky, the numbers for the eight zones, in each column, should be equal. It is seen that they are very unequal, most of all for Class B stars; being more nearly equal as we pass through the advancing spectral classes to Class M. Pickering draws these conclusions from the table: "It seems probable that its (the Milky Way's) effect extends over the entire sky. Even in Class A, to which the Milky Way is mainly due, the falling off in numbers, as the latitude increases numeri- cally, shows itself in all except the first line. This would indi- cate that the Milky Way consisted of a layer of stars rather than a ring. In all the classes, the number of stars is greater when the latitude is small, or all are affected by the Milky Way. ' '"^ TABLE vn GALACTIC LATITUDES OP STAES BEIGHTEE THAN 6.25 Latitude B A P G K M All -l-62°.3 8 189 79 61 176 56 569 +41 .3 28 184 58 69 174 49 562 -f21 .0 69 263 83 70 212 57 754 + 9 .2 206 323 96 99 266 77 1067 — 7 .0 161 382 116 84 239 45 1027 —22 .2 158 276 117 100 247 69 967 —38 .2 57 161 94 59 203 59 633 —62 .3 29 107 77 67 202 45 527 Sums 716 1885 720 609 1719 457 6106 Mean 90 236 90 76 215 57 763 25 Annals E. C. O., 26 (1), 152, 1891. 26 Annals E. C. 0., 64, 143-144, 1909. DISTRIBUTION OF BRIGHTER STABS 155 I have been interested in tabulating the numbers of stars of Classes B, A and K contained in the Revised Harvard Pho- tometry, with the following results : Vis. Mag. B A K > 5.01 350 318 403 5.01-5.50 217 346 318 5.51-6.00 163 790 560 6.01-6.50 164 1423 939 The rapid decrease in the number of Class B stars with de- creasing brightness, and the rapid increase of Class A stars with decreasing brightness are striking. Pickering has said of the Class B stars, that "of the bright stars one out of four belongs to this class [B], while of the stars of the sixth magnitude there is only one out of twenty; and that few [of Class B], if any, will be found fainter than the seventh or eighth magnitude."^' The figures are, of course, in complete harmony with Pickering's statement quoted above, that the Milky Way is composed largely of Class A stars. The solar-motion solutions described above have necessarily been limited to those stars which have moved perceptibly in the interval covered by accurate observations of their positions. Do the available proper motions refer only to those stars which are relatively close to us, or may we consider them as representative also of the distant stars in the Milky "Way? The best available test for nearness, aside from direct measures of parallax for the nearest hundred stars, more or less, lies in the proper motions. Large proper motions are in general indicative of proximity, though there are known exceptions to the rule. Kapteyn is of the opinion that the stars whose proper motions exceed 5" per century are distributed nearly uniformly over the sky; that is, with little tendency for clustering in or near the Milky "Way ; but for proper motions smaller and smaller, there is a continuous tendency for increasing numbers as we approach the Milky "Way. Newcomb, writing ten years ago,^* questions the latter 2T Annals E. C. 0., 56 (II), 37, 1905. 28 See Newcomb 's The Stars, 1901, pp. 252-256. Simon Nkwcomb, 1835-1909 DISTRIBUTION OF BRIGHTER STABS 157 statement, and is convinced that "if we should blot out from the sky all the stars having no proper motion large enough to be detected, we should find remaining stars of all magnitudes ; but they would be scattered almost uniformly over the sky and show little or no tendency to crowd toward the galaxy unless, perhaps, in the region at nineteen hours of right ascension. From this again it follows that the stars belonging to the galaxy lie further away than those whose proper motions can be detected." The term "Milky Way stars," or "stars belonging to the galaxy," is understood to have the following significance: Let the form of the sidereal system be assumed to be an ellipsoid; with one relatively very short axis, 2a, passing through the present position of the solar system at right angles to the plane of the Milky Way, and with relatively very long axes in the plane of the Milky Way. All stars at a distance from the solar system greater than a, i. e., all stars lying outside of the sphere of radius a, whose centre is at the solar system, may be described in position as "Milky Way stars." Eddington's evidence as to the mean distances of the Groom- bridge stars^^ affords interesting testimony on the question. "The mean parallax steadily increases with the distances from the galaxy, a result which is in accordance with the generally accepted ideas of the distribution of stars, viz., that the increased number of stars in the low galactic latitudes is due to additional more distant stars being visible, and not to any crowding among the nearer stars." These differences should be resolved to a considerable extent when the proper motions of Boss's catalogue become available^ It must be regarded as uncertain to what extent the proper- motion stars, extending down to the 9.5 magnitude in some eases, can be considered as representative of the great majority of stars. Certainly, to take extreme but actual cases, a very distant star of unusually great mass, though possessing average real motion, cannot enter the solar-motion solution, for its proper motion is imperceptible, whereas a small star close to us, yielding a large proper motion, may enter the solution powerfully. It is difficult 29 Mon. Not. B. A. S., 68, 104, 1907. 158 STELLAR MOTIONS to realize the inequality of stellar brightness. In the days of Herschel it was supposed that Rigel and Canopus, for example, were our near neighbors, but the fact is they are immeasurably distant. Sir David Gill has been unable to obtain an appreciable or certain parallax for them. The parallaxes can scarcely exceed .01 or .02 of a second as a maximum. We can scarcely doubt that Canopus is radiating certainly 1000 and perhaps 10,000 times as much light as the Sun. If the effective radiating power of its surface equals that of the Sun, the surface must be fully 1000 times as great as the Sun's. Its corresponding volume would be 31,000 solar volumes. Its mass must greatly exceed the Sun's mass, probably between 1000- and 30,000-fold. At the other extreme is the large proper-motion star C. Z. 5''.243, with parallax 0".32. It is one of our nearest neighbors, yet its visual magnitude is but 9.2. Its output of light — ^visual radiations — cannot well exceed %oo of our Sun's. Its spectrum is of the solar type, and its surface area is, therefore, probably related to our Sun's surface area, roughly, as their luminosities. On this siipposition, its radius is but M.o and its volume but %ooo of the Sun 's. Its mass is probably less than M.000 of the solar mass, for its internal forces of gravitation may be very small. There is scarcely any doubt that this star is decidedly less massive than the planet Jupiter. In Canopus we, therefore, have a star whose mass may be 1000 times 1000, or 1,000,000 times as great as that possessed by the little neighbor referred to. It is of course unsafe to make a positive statement on this or other subjects involving a relationship between mass and lumi- nosity. For example, the companion of Sirius is perhaps the star of least relative luminosity thus far recognized. Its output of light is certainly not %oo that of our Sun, yet according to Auwers its mass is 4 per cent greater than our Sun's mass. Unfortu- nately we know nothing as to its spectrum. We cannot overlook the fact that the question of stellar masses must some day be taken into account in statistical researches on the solar motion. Up to the present time, investigators have depended upon the doctrine of averages to smooth away the influences of excessive stellar mass and excessive proper motion. DISTRIBUTION OF BRIGHTER STARS 159 and this condition will no doubt continue into the future. As our knowledge of proper motions increases, and as information concerning stellar distances, densities, and masses becomes avail- able — it is now pouring in from many directions — the solutions can be made more and more representative of the stellar system, and therefore possess increasing weight. Kapteyn has investigated'" the question of stellar luminosities, in terms of the absolute brightness of the Sun as unity. He assumed that the faint stars are more distant than the bright ones, and that the stars with small proper motions are more distant than those with large motions. His classification as to relative distances, according to these two criteria, was calibrated, so to speak, by means of the magnitudes, proper motions, and dis- tances of those stars whose parallaxes are fairly well determined. Assuming further that the mixture of stars of different lumi- nosities is everywhere the same throughout a sphere whose radius is 555 light years (comprising all stars whose parallaxes are equal to a greater than 0".006), he obtained the following table of relative luminosities for the brighter stars in this sphere : No. of Stars Tim«s more luminous than Sun 1 46 1300 22000 140000 430000 650000 0.1 to 0.01 Kapteyn has not intended that these numbers shall be taken literally, but merely as a rough approximation to the truth. The smaller stars are much the more numerous. Data concern- ing the numbers, proper motions, and parallaxes of still fainter stars are wanting, but it is not probable that another line, in continuation of the table, should be added. It should be noted that a parallax of 0".006 confines the classification to the nearer stars — to those well within the minor dimension of the oblate 80 Congress of Arts and Sciences, St. Louis, 4, 407, 1904. 100,000 to 10,000 10,000 to 1,000 1,000 to 100 100 to 10 10 to 1 1 to 0.1 160 STELLAR MOTIONS spheroid of stars, and has little connection with the stars com- posing the extension of the universe in the direction of the Milky Way. Newcomh estimates'^ that a sphere of radius 3300 light years (7r=0".001) must include essentially all stars in the direction of the poles of the Milky "Way. On the surface of such a sphere a star moving with average speed would have a mean proper motion of about 0".6 per century. The proper motions of such distant stars cannot be known satisfactorily for one or two centuries, at least through present methods. We may remark, in passing, that the calibration of magnitudes and proper motions by means of known parallaxes is a weak point in Kapteyn's and Newcomb's structures, for most of our known parallaxes are of stars selected for observation on account of large proper motions and presumed nearness. They are thus not representative of the stellar system as a whole, for their large proper motions are in many cases due to linear speeds greater than the average for the stars in general. It is interesting to compare Newcomb's and Kapteyn's esti- mates of density of distribution of the stars; Newcomb's based chiefly upon determined parallaxes and general considerations; and Kapteyn's upon visual magnitudes and proper motions, as calibrated by known parallaxes. Neweomb estimates'*^ that there is one visible star for each volume of space equal to a sphere of radius corresponding to a parallax 0".5, in our region of the universe. This sphere is of GYz light-year radius. The volume of Kapteyn's sphere is 555 \' = 625,000 times Newcomb's unit sphere; and this would 6.5 be Newcomb's number of stars in Kapteyn's sphere. Kapteyn's luminosity table sets down 1,250,000 ± for this volume, not including possible stars of less than .01 the Sun's luminosity. These numbers are not tmreasonably discordant, considering that neither investigator defines clearly the limiting magnitudes included in the calculation. They can be usefully revised in 31 The Stars, p. 3] 6. 32 The Stars, p. 310. DISTRIBUTION OF BRIGHTER STARS 161 the light of information as to stellar distances supplied by line- of -sight results soon to be available. Another idea, that there may be a slow rotation of our stellar system around an axis passing through the centre of mass of the system at right angles to the plane of the Milky Way, has long been held, and several experienced investigators have given it careful consideration. This idea is a priori not an unreason- able one. The spiral nebulae, such as the Andromeda nebula or those in Ursa Major and Canes Venatici, are with little doubt in slow rotation about axes through their centres at right angles to the planes passing through their major dimensions; they are in general form somewhat the same as the supposed form of our universe ; and, in harmony with spectroscopic obser- vations, they either are now or in the distant future will be great systems of stars. May not our system, as viewed from tremen- dously distant space, preferably at right angles to the plane of the Milky Way, present the appearance of a great spiral nebula ? To Sir John Herschel we owe a general statement of the problem.^^ Schonfeld was, perhaps, the first to give it mathe- matical expression.'* It readily appears that a study of such minute rotational effects as are in question involves the true value of the precession constant as well as the speed and direc- tion of the solar motion. The observed proper motions of the stars have been utilized by half a dozen astronomers in efforts to uncover the first indications of a rotation effect. Kobold'^ collected eleven results for the deduced angular speed of rota- tion; but as six of these indicate rotation toward the east, and five rotation toward the west, all being of. minute value, we can say that the evidence in support of a rotatory motion of the stellar system is negligible. However, the problem is not an unpromising one for the distant future. In this connection we merely mention Madler's study of stellar motions in search of a great central sun, about which the indi- vidual stars may revolve. He named Alcyone in the Pleiades as 33 Outlines of Astronomy, 1849, p. 588. 34 V. J. S. Astron. Gesell, 17, 255, 1882. 35 Ban des Fixsternsystems, p. 118. 162 STELLAR MOTIONS the central body. Madler's idea appealed strongly to man's imagination, and it has become widely circulated in the more popular literature of science. More complete studies of stellar motions showed that the idea is without foundation. In fact, the position of Alcyone, far outside the plane of the Milky Way, as a centre about which the myriads of stars in the Milky Way are in revolution, is sufficient to stamp the idea as absurd. CHAPTEE V THE SPECTROGEAPHIC DETERMINATION OF THE SOLAR MOTION We now take up the solar motion and related problems from the radial velocity side. This method has several weighty advan- tages over the proper-motion methods, with apparently but one serious disadvantage; and this one, we hope, is of a temporary and passing nature. 1. Proper motions are measured and expressed in angle, and for linear motions of given dimensions these angles vary inversely as the stellar distances. Now the distances are unknown except in relatively few eases, and this lack of knowledge is a constant source of embarrassment, and a limitation, in the proper-motion methods of determining the Sun's way and speed. The spectrographic method, on the contrary, is independent of the distances of the stars ; their velocities in the line of sight are determinable as readily and as accurately for an extremely dis- tant star as for a near one, provided they are both bright enough, and provided further their spectra contain measurable lines. Distance is eliminated, save as distance governs the quantity of light delivered to the spectrograph. 2. The determination of proper motions requires that a long interval of time elapse between two accurate position observa- tions. Auwers 's revision of Bradley 's great star catalogue, based upon meridian observations of 3222 stars extending from the North Pole down to Declination — 31°, made at Greenwich about 1755, with accuracy wonderful for that time, serves as the invalu- able starting point for the proper motions of the brighter stars. However, the interval of 155 years to date is all too short as a base line for determining the proper motions of a consider- able proportion of the stars in Bradley's catalogue. Radial velocities, on the contrary, can be- measured accurately at once. 16i STELLAR MOTIONS and expressed in definite and absolute units, say in kilometers per second. 3. Proper-motion determinations in the Southern Hemisphere are more than a half century behind those in the northern sky based upon Bradley's catalogue. This is a weakness we cannot overcome except as we give the stars in the Southern Hemisphere time to move. The contributions to this subject by Professor Boss under the auspices of the Carnegie Institution of "Wash- ington, in both the Southern and the Northern Hemispheres, should prove exceedingly valuable not only in the distant future but within a few years, though nothing except the lapse of time, so far as we now are able to say, can make up for the deficit of knowledge concerning the proper motions of southern stars. 4. Proper motions have been determined for the nearer stars only, excepting a few distant ones whose lateral motions are abnormally large, and they are, therefore, not representative of distant stars. The great mass of the stars are so far away that generations must come and go before they will have changed their directions appreciably. Solutions of the solar motion based upon proper motions refer the Sun's motion to the stars immediately around us. If the cloud-like forms, which we are accustomed to call the Milky "Way, were swept aside, we could still mark out its position by the larger number of naked-eye stars located near one great circle of the sphere. This great circle would make an angle of not over 5° with the real galactic circle. The excess of bright stars as we approach the Milky "Way is made up of distant stars, of great luminosity, and presumably of great mass, whose proper motions are minute or uncertainly determined, and, therefore, of little or no influence in proper-motion solutions. Eadial velocity determinations, on the contrary, include these stars as readily as fainter close stars, and are, therefore, more representative of the general stellar system. 5. If there is an absorption or obstruction of light in its transmission through interstellar space, as a function of the dis- tances of the stars, any results involving assumptions as to the relations between stellar distances, stellar magnitudes, and M W H7 H^ «4^i:^ t^« I Solar Spectrum for comparison Andromeda Nebula Spiral Nebula N. G, C. 1068 1 fmm^ I Spectra of Sun, Nebula and Cluster. — Fath Hydrogen Spectrum for comparison Globular Cluster N. G. C. 7078 166 STELLAR MOTIONS proper motions will be vitiated, unless the effects of such an absorption or obstruction can be allowed for. This subject, though an old one, has come prominently into the foreground within the past three years, for it is clearly recognized that proper-motion researches have reached the point where inter- stellar absorption should be taken into account, or shown to be non-existent. One should expect a priori that the transmission of energy through space would require that toll be paid, to some extent, but the insensitive methods thus far applied have not certainly detected such an effect. Radial velocity methods appear to have the advantage, unless, perhaps, a dispersing effect in interstellar space really exists, in accordance with the announcements of Nordmann and Tikhoff, already considered (page 87). If space absorption is in reality space obstruction by concrete particles of considerable size, it does not seem that radial velocities are affected with error on this account. 6. Only the brighter stars are thus far amenable to radial velocity measurement, it is true, for the light is weakened by spreading it over the large area of the spectrum, whereas the point image of a very faint star is available for proper-motion determination. The number of well-determined radial velocities is now in the neighborhood of 1100, and is rapidly increasing. Inside of ten years the accurately determined radial motions may well exceed the number of known proper motions of corre- sponding accuracy. The processes are straightforward, more powerful telescopes can be constructed, and the number is merely a function of the energy devoted to the subject. However, the greatest value of radial velocities lies not in their advantages over proper motions, but in aiding proper motions to come into their own strategic worth. The radial and cross motions are mutually supplementary, and when they are skill- fully combined, as perhaps 1500 should be within the next five years, after Boss's proper motions in both Northern and South- ern Hemispheres shall have been published, and current pro- grams of radial velocities shall have been carried through and published, the results for the solar motion should be of a weight far surpassing any hitherto deduced. SOLAR MOTION FROM RADIAL VELOCITIES 167 Several determinations of the solar motion have been made in the last quarter century from radial velocity determinations, with results as helow : SPECTKOSCOPIC DETERMINATIONS "^ So Vo Kovesligethy* (261°) (+35°) 64km. Homann** 320 +41 39.8 ± 4.2 310 +70 48. ±23.1 279 +14 24.5+15.8 Kempft 206 +46 18.6+ 3.0 160 +50 13.0+ 3.3 (267) (+31) 12.3 ± 3.0 Eisteentt 218 +45 17.5 Campbell} 277.5 +20 19.9+ 1.5 About 70 stars, visual Greenwich observations, visual Huggins's observations, visual Seabroke 's observations, visual 51 stars, photographic 41 stars, photographic 41 stars, photographic 42 stars, photographic 280 stars, photographic Here V^ is equivalent to q in Table VI (Chap. IV) of proper- motion solutions. Kovesligethy 's solution is founded upon the visual measurements of radial velocities made at Greenwich prior to 1881. The right ascension and declination of the apex (261°), (+ 35°), do not depend upon the radial velocity results, but were assumed from previous proper-motion solutions. The failure of the visual method is apparent from the deduced value of the Sun 's velocity, 64 km. per second, which is certainly three times too great. The details of Homann 's investigation, published in his inau- gural dissertation, Berlin, 1885, are not accessible to me; but the solution depending upon Seabroke 's observations is certainly remarkable for its agreement with recent solutions based upon a grefft mass of spectrographic observations. Whether the agree- ment is accidental, or indicates freedom from systematic errors, is uncertain, especially in view of the large probable error assigned to the velocity, Fo. * Astr. Nach., 114, 327, 1886. ** Astr. Nach., 114, 25-26, 1886. t Astr. Nach., 132, 81-82, 1893. U Astr. Jour., 13, 75, 1893. t Ap. J., 13, 83, 1901. 168 STELLAR MOTIONS Kempf's solutions, based upon the Potsdam spectrographic velocities obtained in 1888-1891, place the apex of the solar motion approximately midway between the positions of the apex and antapex assigned by proper-motion solutions. Under this condition, the fair agreement of the solar velocities 18.6 and 13.0 km. with recent determinations of greater weight has little sig- nificance. Kempf's first solution depends upon all of the fifty- one Potsdam velocities. In the second solution the radial veloci- ties of five Orion stars were combined into one resulting velocity ; and similarly for five Ursa Major stars and for three Leo stars ; inasmuch as the individual stars in each of the three groups seemed to have related and essentially equal velocities. Assuming the proper-motion position of the apex as in the parentheses, Kempf's third determination of the solar speed is 12.3 km. per second. Risteen's solution is based in effect upon forty -two Potsdam spectrographic velocities. The velocities of five Ursa Major stars were combined and used as the velocity of the centre of gravity of the group. The velocities of Sirius and Procyon were rejected on account of known orbital irregularities not yet investigated. Three uncertain velocities were also rejected. Up to the end of the year 1900, the radial velocities of some- what over three hundred stars had been measured at Mount Hamilton, by means of the Mills spectrograph attached to the 36-inch refractor. Rejecting about twenty-five stars whose velocities were variable under the attractions of unseen massive companions, there remained 280 stars whose velocities were apparently constant.^ The 280 stars, in order to form the basis of a solution of the solar motion, were divided into eighty groups, each group representing the mean right ascension, declination, and observed velocity of from two to seven neighboring stars ; on the average, three and a half stars in each group. These means are exhibited in Table VIII. It will be noticed that the observed velocities are prevailingly positive, indicating recession, in the hemisphere from to 12 hours of right ascension ; and prevailingly negative 1 A great many of these have since been found to be variable. SOLAR MOTION FROM RADIAL VELOCITIES 169 in the other hemisphere. In the former only ten groups out of forty have negative velocities, and in the latter only five out of forty are positive. TABLE vrn No. of Stars Mean R. A. Mean Dec. Mean Observed Velocity No. of Stars Mean E. A. Mean Dec. Mean Observed Velocity h m km. h m km. 2 26.5 -14°. +17.0 3 12 20.4 +43°. - 4.7 6 40.0 +58 .8 -10.3 3 12 35.7 -22 .5 - 1.7 "5 45.0 +28 .2 -24.2 3 12 48.1 + 4 .9 -17.0 3 1 23,0 -10 .1 +12.7 2 13 6.2 +23 .2 - 8.2 4 1 35.5 +43 .9 + 0.2 2 14 9.2 - 7 .6 + 3.0 3 1 43.7 +10 .6 + 7.3 5 14 14.9 +18 .1 - 3.9 2 1 47.4 -19 .0 + 2.5 2 14 29.5 -25 .6 +11.8 2 2 2.6 +24 .2 -19.0 7 14 58.0 - -29 .6 -13.1 9 2 38.5 —14 .1 - 3.2 4 15 2.4 - -46 .0 -28.5 3 2 54.4 1-0.1 h21.7 4 15 23.2 _ - 4 .8 - 7.0 5 2 55.4 -50 .2 -13.6 4 15 27.4 - -65 .8 -14.2 4 3 3.8 -41 .2 -4.5 4 15 41.4 -12 .9 - 1.0 3 3 16.9 -13 .6 - - 6.7 4 16 17.1 -20 .8 -11.1 3 3 45.8 - 9 .2 -10.7 4 16 32.8 - 8 .5 - 7.0 3 4 3.2 -17 .3 +34.8 3 16 38.0 - -13 .3 -26.0 5 4 16.6 hl8 .0 +35.5 3 16 49.5 - -35 .9 -29.0 2 4 29.7 -79 .8 - 3.0 3 17 44.7 J -53 .6 -22.3 4 4 46.8 -44 .7 h 8.2 4 17 53.6 H -29 .5 - 9.9 3 5 15.7 -37 .4 -30.7 3 17 53.8 ^ -5.3 - 7.5 2 5 17.0 - 7 .1 -22.0 6 18 18.6 - 7 .2 - 5.4 4 5 23.5 -20 .9 - 1.8 6 18 36.4 -23 .5 -23.5 3 5 32.6 - 7 .5 - 0.7 3 18 38.6 _. 1-19 .0 -28.5 3 5 51.5 +57 .7 \- 5.0 2 18 43.0 - -41 .2 -20.5 4 6 18.1 +25 .0 -24.8 6 18 48.9 _ -69 .6 + 6.4 2 6 41.0 -15 .6 -50.5 4 19 42.1 _ - 6 .2 -19.9 3 6 43.3 +16 .7 - 6.3 5 19 42.6 _ -23 .6 -11.6 3 7 32.4 +27 .0 -11.3 4 20 3.6 - -55 .4 -44.2 2 7 33.8 -25 .1 -42.0 3 20 19.3 - 9 .1 - 8.7 2 7 47.0 + 9.3 -34.0 3 20 48.9 - 1-43 .3 + 1.0 2 7 50.6 - 6 .0 -21.0 5 20 58.5 -12 .8 —28.2 4 8 56.6 - -10 .5 -26.9 5 21 1.3 - -31 .9 - 2.6 2 8 57.8 -32 .0 -1 ^25.2 4 21 24.6 -15 .8 - 2.9 4 8 58.0 -65 .7 - 4.0 3 22 0.2 +12 .7 - 7.3 2 9 10.2 - -47 .2 +19.0 2 22 13.6 +54 .7 -14.2 3 9 27.2 - 3 .7 +12.3 4 22 32.6 - 6 .7 -11.3 2 9 57.2 +22 .3 -15.2 5 22 52.9 --25 .4 - 0.2 5 10 40.2 -15 .2 +19.6 2 23 10.6 --71 .4 -26.5 5 10 44.7 h38 .1 - 2.8 4 23 10.7 -18 .7 + 5.6 2 11 11.6 -66 .1 ± 0.0 2 23 16.0 +43 .8 - 2.0 6 11 32.8 - ^ 6 .2 + 3.7 4 23 31,0 -- 5 .0 - 0.9 170 STELLAR MOTIONS The determination of the elements of the solar motion from radial velocity data, such as those contained in Table VIII, has the great merits of directness and extreme simplicity. "We shall now develop the simple equations required for the solution. Antapex Apex Figure 7 Let V„ be the Sun's unknown speed with reference to the entire system of observed stars; V the observed speed of a star with reference to the solar system ; and D the unknown angular distance of the star from the apex of the solar motion. Now, if the star is assumed to be at rest with reference to the whole system of stars, its apparent velocity with reference to the observer in the solar system will be To cos D. But every star has a motion of its own, v, of unknown amount. Let us suppose in Figure 7 that the small region of sky at S contains n stars, whose individual observed radial velocities SOLAR MOTION FROM RADIAL VELOCITIES 171 are F^, F,, . . . F„, and wliose radial velocities with reference to the stellar system are «,, v^^ . . . v„. Then we may write the n equations Fq cos I)^= V^ — v^ Fo cos D, = F, — « Fo cos -D„ = F„ — -y^ Now the chances are assumed to be equally in favor of motions of approach and recession with reference to the stellar system, and small motions are assumed to be more numerous than large ones. As these are the conditions governing the occurrence of accidental errors, we cannot at present do better than to express the total radial motion of the n stars by the equation F„ S cos D = 2F, obtained by summing the n individual equations. If it is desired to write one equation for each observed velocity, we may assume that the equation F„ cos J* = F (31) is true except for the accidental error which represents the radial velocity of the star with reference to the system; and as there wiU be similar equations for neighboring stars, each affected by a corresponding assumed error of observation, the combination of the individual equations into a normal equation will eliminate, more or less successfully, the errors involved in the equations for the separate stars. Let a„, 8„ be the coordinates of the unknown apex, and a and 8 the known coordinates of a star. Then the value of cos D for the star will be defined by the well-known equation expressing the distance between two points, cos D = cos So cos S cos (a„ — a) -f sin So sin 8. If we place x= Vo cos tto cos So, y =zVo sin Oo cos S„, z—Yo sin 8„, 172 STELLAR MOTIONS equation (31) takes the form: cos a cos 8 a;-}- sin a cos 8 y -\- sin 8s— F= 0, (32) from which the values of x, y and z may be determined. The values of Vo, ao; and 8„ may then be found from the relations r,^ = x^ + f + z% tan u,o = - , (33 sm bo = — . ' o Forming the equation of condition (32) for each observed star or group of stars, and combining these into the three usual normal equations, we may solve the values of x, y and z, and thence by (83) for the direction of the solar motion as defined by a;, and So and for the speed of the solar motion, V^. The right ascensions, declinations, and observed radial veloci- ties of the 280 stars would have enabled us to construct the same number of equations of condition on the basis of equation (32). However, as explained in connection with proper-motion solu- tions of the solar motion, it is better to combine the results for a group of neighboring stars into a mean result, in order to eliminate in good measure the individualities of stellar motions and leave the mean result more truly representative of the system. Eighty equations formed from the mean results in the table were solved for the most probable values of a„, So and V^, with the results recorded in the last line in the table on page 167. The right ascension of the apex, 277°. 5, is in fair accord with the proper-motion results, but the declination of the apex, + 20°, places it further south than the proper-motion solutions, except- ing Kobold's. The right ascension value has fair weight, for the observations were well distributed completely around the northern sky, east and west, so as to give symmetry to the solu- tion in this coordinate. The declination result, while it may be near the truth, cannot be assigned so great a weight, for the southern third of the sky was completely unrepresented in the solution, no radial velocity observations having up to that time SOLAR MOTION FROM RADIAL VELOCITIES 173 been attempted on the far south stars. It was to remedy this foreseen defect in the solution and to secure data for a later solution, as uniformly distributed over the sky as possible, that the D. O. Mills Expedition to the Southern Hemisphere was organized. It was stated in Chapter III that the radial velocities of 1020 stars were known up to January 1, 1910, as measured with the Mills spectrograph at Mount Hamilton in the Northern Hemi- sphere, and by the D. 0. Mills Observatory in the Southern Hemisphere. This list does not include about 200 stars which have been discovered in the progress of the observations to be spectroscopic binaries whose systemic velocities have not yet been determined; nor does it include about 200 other stars whose spectra contain lines of quality so poor that the spectrograms with moderate and high dispersion are not accurately measur- able. Velocities of other stars not observed as yet by us, but measured at other observatories and published to date, number about forty. Adding thirteen of the results^ for nebulae obtained by Keeler, using visual methods, we have a total of 1073 radial velocities available for a first approximation in the solution of certain fundamental problems of the stellar system. Practical questions exist as to the proper weights to assign to results of different degrees of accuracy when it is desired to combine them statistically. The apparent speeds of the brighter solar type stars, or those containing numerous well-defined lines, have been determined well within a kilometer per second; whereas the speeds of stars containing only broad and hazy lines may be in error up to a maximum of 6 or 8 km. per second; it being understood that stars containing only extremely broad and indefinite absorption bands, whose observed velocities might be in error 10 or 15 km., have been rejected from this investigation. Again, the speeds of the fainter stars, determined with 2-prism and 1-prism instruments, considered individually, have smaller weights than 3-prism results. When we combine the observa- tions for all the stars, shall the weights assigned to individual 2 Omitting the single rough measure of G. C. 5851 (E. A. = 17li8m). — Publ. LicTc Ols., 3, 205, 1894. 174 STELLAR MOTIONS results be proportional to the inverse squares of their probable errors, as is usually the ease ? I think not, for we are dealing with special conditions. We desire that the solar motion shall refer to an observed program of stars which shall be as repre- sentative as possible of the entire stellar system. If we were to divide the stellar system into three sub-systems, one containing the bright solar type stars, a second the bright stars with hazy lines in their spectra, and a third containing the great mass of faint stars, and give greater weights to the observed velocities of the stars in one of these groups, the solar motion deduced would have discriminated agaiast the other groups. The solution must refer to stars with hazy lines, or to faint distant stars, as truly as to bright solar type stars. One poorly determined result for stellar velocity used alone should have small weight, but a large number of such determinations should be given consider- able weight, proper care being taken to avoid systematic error. Inasmuch as the results, even for the poorer stars, appear, at this stage of our knowledge, to be sensibly free from systematic error which can ajBfect the solar-motion problem, and the number of such stars is much smaller than the number whose spectra admit of accurate measurement, I have assigned equal weights to all the velocities admitted into the solution. A study of the results of the present investigation will no doubt serve as a basis for a possibly more logical assignment of weights, when the time arrives for making another solution based upon a greater number of observed stars. Before proceeding to apply this method to the data of obser- vation at hand, it is necessary to decide whether .any of the velocities for individual stars should be rejected, for one reason or another. For example, if there are groups or families of stars travelling along parallel lines, toward the same goal, and the computer should let each star, or several stars of such a group, enter his solution for the Sun's motion, his results would be vitiated in a small and corresponding degree : the veloci- ties of such stars would not be independent of each other, as demanded by the basis of the solution. To illustrate by an extreme case : If one-tenth the stars used individually in the solu- MOTIONS OF GROUPS OF STARS 175 tion were members of a closely related system, moving through space along parallel or approximately parallel lines, with a speed equal to the average speed of the stars, whereas the remaining nine-tenths were moving in a truly haphazard manner, the one- tenth might influence the solution more than the other nine- tenths combined. Now several systems of related stars, aside from ordinary and extremely wide double stars, are known to exist, and the investigator of stellar problems should consider them. Proctor^ called attention to such a system, comprising five out of the seven bright stars in the Big Dipper. He found that all but the first and last of the seven stars possess nearly equal and parallel proper motions, and he concluded that they are, with little doubt, travelling along lines very nearly parallel, with linear speeds essentially equal. Klinkerfues, before the days of accurate radial velocity determinations, had investigated this system. Hoffler repeated the investigation when the Potsdam photographic radial veloci- ties of four of the stars, j8, y, e and f Ursce. Majoris, became available." Vogel and Scheiner had assigned to /8, y, c and ^ radial velocities in close accord, averaging — 30 km. per second, but they were unable to determine the velocity of 8 because of the poor quality of the lines in its spectrum. Hoffler found that the observed radial velocities and the observed proper motions conform, within the limits of unavoidable observa- tional errors, to the hypothesis that these stars are travelling with equal linear velocities along parallel lines whose apparent "radiant" — the point at infinite distance from which the parallel lines seem to radiate — is at a = 123°. 7, 8 = + 34°.9, and that they have an average parallax, v = 0".0165. Following the improvements in spectrographs and spectro- graphic methods, Ludendorff has recently secured long series of radial velocity measurements of /3 and e Ursce Majoris, both of which he found to be spectroscopic binary systems, and of ^ Ursce Majoris, which is a well-known spectroscopic binary. As 3Proc. Royal Soc, 18, 169, 1869. 4 Astr. Nach., 144, 369, 1897. O Q Q O « « O »-5 < CO p I d < » ■^S CO r^ »; o fi M . . o ^ 1 3 g > ^ g > s g i^^-gse p.a 4ii . . 5 t^ MH (j3 (^ rH M^ s^ . 0) B . . in a> P,P, P, P II II 1 1 J< QO •• CD 00 1 «D CO CI «— 1 T-< -— ( CC iJ i 1 1 1 ^ CO -* «D (M O ao a (o in '^ CO (m' X -p o 1— 1 .— 1 T— ( 1— 1 iH O i 1 1 1 1 1 7 1 ;0 !>■ t- CO t- CC » o 1 W i-H O i-H i-H o o o 1 1 1 1 1 + 1 i^i^i^ 1 1 t^ OD Ci O O b- CO a o C< (O (>. (D CO 00 o O t^ OO 00 Oi o CO CD I— I CI CI CO IC «0 «D 00 in ci S O S CI QO CO lO 1-1 o o L- CO CO O) O CO CD 1— ( C1C3 QO c^i in ci CI ^ b- .-( lO CO rH i-( 1—1 rH o 1 o o o o o O O o o o o o O O o o o o o o o o + 1 + 1 + 1 + CO CD O 03 CO 00 Ir^ in Gi CD ira CO CI in i CO C5 O iH CI ■^ i-i O O 1— 1 iH 1— 1 rH rH o ^ o o o o o O O --( -H in o CI -t 't^ 1-, -H crj 'f^^ LO rH |>. CO 00 o: o 1-1 CI -^ rH pO O O iH iH I— 1 O o o o o o O O CI m cj o in Cs ^ o ci CI in «n -+1 CI CO o o 1— I CO '^ t>- CD lO CI OS in in in in in CD-tH _| 1 1 1 |_ ++ _( 1 1 1 [_ ^zr in '^ « (o tJ ?:- MOTIONS OF GROUPS OF STABS 177 the radial velocities of the three stars from the later observations are less than one-half those used in Hoffler's study of the Ursa Major system, Ludendorff has made a complete re-investigation of the subject.^ Assuming that the five stars are travelling along parallel lines with equal velocities, he finds that all observed data are satisfied within the limits of unavoidable error by an apparent radiant point at u, = 123°.2 ± 3.0, 8 = + 36°.6 ± 3.5 (mean error) ; an average parallax of the system IT = 0".0352 ± 0" .022 ; and an actual velocity of the system (relative to the Sun), 20.7 km. per second. The position of the radiant is in remarkably good accord with that assigned by Hoffler, but the parallax is more than twice Hoffler's value, chiefiy because the radial velocities used by Ludendorff were less than one-half those used by HoflSer. Table IX exhibits the extent to which the hypothesis satisfied the observational data. The significance of the various columns will be clear from their headings. The differences between observation and computation in both proper motion and radial velocity are a little larger than we should expect in the hypothesis of motions absolutely parallel and equal. A share of the discordances may readily be due to the imavoidable errors of observation. Another share may arise from an error in the assumed position of the radiant, or in the assigned parallaxes; and of course the chief source of discrepancy may lie in the fact that the motions are not strictly eiqual and parallel. Under the heading, Eadial Velocities, are given, respectively, the theo- retical velocities as computed by Ludendorff, and the velocities as observed by Ludendorff, by Vogel and Scheiner (1888-1891), and by the Lick Observatory (up to 1912). There can exist no doubt that the five TJrsa Major stars form a closely related system. The last two lines of the table contain corresponding data for u, and rj Ursce Majoris. Ludendorff' s studies of the motions of sAstr. Nach., 180, 265, 1909. 178 STELLAR MOTIONS these two stars led him to conclude that they are probably travel- ling with equal velocities along another set of parallel lines whose radiant point is situated at a = 270°.2, 8 = + 36°.5 ; in which case the average parallax of the two stars is •^ = 0".0360, agreeing almost perfectly with the parallax of the other system of five stars ; and the actual velocity in space of the two stars (with reference to the Sun) is 20.8 km. However, some doubt must exist as to the close relationship of u and 17, for the radial velocity of 77 is essentially unknown ; and, as Ludendorff pointed out, the radiant practically coincides with the solar apex, the velocity of the two stars is nearly equal to the solar velocity, and the observed motions of u. and 7; Ursce Majoris are but a reflex of the Sun's motion. From a consideration of the visual magnitudes and the par- allaxes of the Ursa Major stars Ludendorff concluded that all are exceedingly brilliant stars. Their luminosities are to that of our Sun as assigned in the following table : a Ursw |3 '• Majitris 126 72 y '■ •• 66 S " il 32 € " l( 105 f •■ ■" 87 71 '• •' 95 Inasmuch as the five stars composing one system are moving as one star, it would not be permissible to let all enter as indi- vidual stars into the solar-motion problem. However, they represent five stars in mass — much larger stars than the average, we have strong reason to believe — and it would be unjust to let but one-fifth of this mass enter. Again, there are undoubtedly many cases of closely related stars, as yet undetected, in other MOTIONS OF GROUPS OF STABS 179 parts of the sky for which the individual velocities will be used in the solution. A fair compromise, I have decided, is to let two of the five stars enter the solution; 8, with undetermined velocity, and two others, y and e, are rejected. Since some uncertainty existed as to whether a and -q have motions in common, both were utilized in the solution. Shortly after the publication of Ludendorff's paper, Hertz- sprung called attention" to the fact that eight other promuient stars, including Sirius, appear to be moving along lines parallel to those followed by the five Ursa Major stars, and with equal linear speeds. By trial he determined that position of a radiant point on the great circle defined by the proper motion of Sirius, near the intersection of that circle and the great circle repre- senting the average proper motions of the five Ursa Major stars, which would make the sum of the angular deviations of the remaining proper-motion circles (omitting that of /3 Eridani) a minimum. This placed the radiant at a = 127°.8, 8 = -f 40°.2 (1900). Table X shows the agreement between the observed and computed values of the proper -motion position angles, f, and between the observed and computed radial velocities, the latter by LudendorfE, miscellaneous observers, and the Lick Observa- tory, respectively. The parallaxes and the computed radial velocities were deduced, by intention, to agree as exactly as possible with Ludendorff's observed velocities. In the cases of the first two stars on the list, j8 Eridani and /8 Aurigce, the discordances in position angle are large enough to raise a question as to whether these stars belong to the Ursa Major system. On the basis of probabilities, we should have to expect that a certain number of stars whose proper motions and radial velocities are known would have proper-motion position angles differing not more than 7° from any assigned position angle; but the probabilities would be small that two stars as 6 Ap. J., 30, 135, 1909. m tsj H O CS O CO I 1 Spectrum v. v. poor Sp. Bi., Harv. Col. Obs. Vis. and Sp. Bi. Sp. Bi., Ludendorff Spectrum v. poor Spectrum v. v. poor Spectrum v. v. poor Sp. Bi., Ludendorff Sp. Bi., Harv. Col. Obs. Sp. Bi., Hartmann i II 4 OOCOOOOOOOOOOO as ooooooooooooo d 4 ^ ::^^coio " ". .. o CO .3 r-l rt WH t-H 1 1 1 1 1 1 1 1 1 1 1 o o m §00* -^ 00 o 7 T 1 + a a to CO cj ■^ 1 II a o O . IC O UC to i-H -+ O 'ct* ^ Tl O CI C-1 §t-cDa;«DcD^iO':^i:ofococic-i 1 7 1 777777777 i 1 O OOiHOt^COCDrHOmCQlOt-^ O <»iraO.-H(?lCOf-(.-ICo«Di>(riQOQOcjOioioco Cl Cl Cl I— 1 1— 1 I— ( 1 T-(rH(DXCOIOlO«DlO'r>in)OOlO O CKOMi— IClC^*lOtO00 W-t^Clt-OCOClOt-i-Hi-iCllC OOi-IOOOOOOOOOO o o CI «3 ci-+^aoocqt-utiOit-o.-iioo cic:ir:itc>oiocMioooin>aj-+ocDccimt- 1 _L 1 _| l_J 1 1 1 1 1 1 |_ 1 I I 1 1 tn in 1 1 1 r o o Cs 8 COlOCiCOIOO-+'C100.— 1.— (OOi— ( ira-xot^cot-t^cii-HCi-t^csci t-QOOmcDCDt-ODCiClClCirQ rH.-Hi-HWi-l.-'.-l.-H.-H.-ICl W to "y T 1' I + a s ■'. CI c: ^ iH ^.^ I ^ I tm ^^ I + qq I + 1— I C"i ro T^ MOTIONS OF GROUPS OF STABS 181 bright as these, whose observed and computed radial velocities are in close accord, should deviate only five and seven degrees respectively from the positions of great circles passing through the Ursa Major radiant. It is indeed remarkable that stars so widely separated in space should be travelling along parallel lines with equal speeds. The two stars ^ and t, Ursce, Majoris are separated by an angle of approximately 20°, which means that the minimum distance between /3 and ^ must exceed one-third the average distance of the system from us; that is, the distance between these two stars is such that light requires thirty years as a minimum to travel over it. The angular separation of Sirius and ^ Ursce Majoris is approximately 90°, which means that the distance between the two stars is greater than the distance of either from the solar system. The large cluster of stars known as the Pleiades concerns our problem in the same manner. The relative positions not only of the naked-eye stars but of many between the sixth and ninth magnitudes, in this group, were accurately determined by BesseF in the years 1829-1841. Subsequent observations by Gould, Jaeoby, and especially by Elkin' in 1884-1885, failed to show any certain changes in their relative positions. Fully three score members of the cluster have a common proper motion; they appear to move on through space with a uniform speed of 5".3 per century. Only the half dozen most brilliant members of the group are brighter than 5.01 visual magnitude, and the spectra of all these contain only a few poorly defined lines. It was expected that their velocities would be essentially equal. This expectation was not fully realized when the speeds of six of the brighter stars in the group were measured recently by Adams' at the Yerkes Observatory. The measures were made with 1-prism dispersion, on account of the hazy character of their lines, and the results are therefore subject to larger errors than usual. 7 Astr. Nach., 18, 353, 1841. 8 Trans. Yale Col. Obs., 1, 1, 1887. sAp. J., 19, 338, 1904. 182 STELLAR MOTIONS BRIGHTER PLEIADES STARS a (1900) 8(1900) Spm. r fia II.S Electra (17 Tauri) 3ii38ni.9 +23°.8 B5 +15 km. +0S.0014 — 0".050 Taygeta (19 Tauri) 3 39 .8 +24 .2 B5 +3 8 48 Maia (20 Tauri) 3 39 .9 +24 .1 B5 +21to- -7 21 45 Merope (23 Tauri) 3 40 .4 +23 .6 B5 +6 17 54 Alcyone (25 Tauri) 3 41 .5 +23 .8 B5 +15 14 48 Atlas (27 Tauri) 3 43 .2 +23 .7 B8p.+13 +0 .0014 — .050 Average +10 +0 .0015 —0 .049 The parallax of the Pleiades group is unknown, beyond the fact that it is small. It can scarcely exceed 0".02. The annual proper motions of the six bright stars, in right ascension and declination, are quoted in the last two columns of the table (changed in March, 1910, by the substitution of the proper motions from Boss's Preliminary General Catalogue). The average value of the six resultant proper motions is 0".053, and the proper-motion vectors point nearly to the antapex of the Sun's way. The mean of Adams's radial velocities for the six stars, -(- 10 km. per second, is almost exactly the Sun's velocity with reference to the point occupied by the Pleiades, for they are approximately 60° from the antapex. Several writers have noted that the Pleiades group must be nearly fixed in position, with reference to the stellar system. The values of the proper motions and radial velocities quoted in the table are in harmony with this view. If we assume the group to be at rest in the sidereal system, equation (29) enables us to compute the distance of the group from the point occupied by the solar system as 215 light years, or parallax 0".015. The Pleiades stars are certainly of great luminosity. Corresponding to the distance which we have computed, Alcyone must be nearly 200 times as luminous as our Sun, and the average luminosity of the six bright stars must be in the neighborhood of 100 times that of the Sun. The velocity of Maia is variable. Differences between the velocities assigned to other stars in the group seem to be real in good part. However, if these differences, of the order of 10 km. per second, are a fair measure of differential motions existing MOTIONS OF GROUPS OF STARS 183 within the Pleiades system, then the Pleiades must have a par- allax much smaller than 0".015, in order to account for the essen- tial absence of relative motions in the sixty years between Bessel's and Elkin 's surveys. If we increase the estimate of distance, we must increase the estimated luminosities even more rapidly. Here again we must not use the radial velocities of all the indi- vidual stars, for several of them are approximately equal, and are no doubt the related velocities of a system ; nor must we con- fine our solution to one star-mass. We decide to let Electra, Alcyone, and Atlas enter as one star ; to let Taygeta and Merope enter as one star having their mean velocity ; and to omit Maia, whose speed is variable and whose systemic velocity is as yet unknown. Thus the massive Pleiades cluster enters as only two stars. Figure 8 More remarkable than either of these groups, in some ways, is an approximately globular cluster, about 15° in diameter, con- taining thirty-nine or more stars whose proper motions are such as to carry them toward a common converging point, as dis- covered^" by Boss of Albany. Figure 8 shows these stars in their present relative positions as round dots, and the arrows loAstr. Jour., 26, 31, 3908. 184 STELLAR MOTIONS represent by their directions and lengths the apparent motions of these stars in 50,000 years. The stars, at their present dis- tances from us, are ranked as of the third to the seventh magni- tudes. In the distant future, after 65,000,000 years elapse. Professor Boss computes, this cluster, now 15° in diameter, will have condensed into a cluster not over one-third of a degree in diameter, with the stars reduced to the ninth to twelfth magnitudes, and occupying the relative positions indicated in Figure 9. It is not expected that they will be nearer each other than they are now; but, moving along parallel to each other, as they are supposed to do, they will converge as parallel lines drawn to infinity. The radial velocities of these stars are of interest. Three were observed at Bonn and seven at Mount Hamilton, in the prosecution of their regular programs ; and at the Yerkes Observatory the task of observing all of the stars . known to be in this group has recently been assumed. The following are available to date (1910) : a (1900) 5 (1900) Bonn Lick Spm. p-a fJ-d y Tauri 4i>14ni.l + 1.5°.4 +38.6kin. +38.0km. G +0s.0081 — 0".027 d Tauri 4 17 .2 +17.3 +39.8 +38.6 K 77 33 68 TflHri 4 19 .7 +17.7 +35. A 75 24 e Tauri 4 22 .8 +19 .0 +38.4 +39.2 K 80 38 01 Tauri 4 22 .9 +15.7 +37.5 K 72 28 $2 Tauri 4 23 .0 + 15 .6 Variable A5 72 25 c Tauri 4 32 .6 -|- 12 .3 Variable A5 + 0.0071 -0.010 The Yerkes Observatory has found that a large proportion of the stars in this group have variable velocities; that is, are spectroscopic binaries: but for most of them — ^perhaps for all — the speeds of the centres of mass appear to be approximately 35 to 40 km. per second, recession. These radial velocities therefore support the proper-motion indications that the stars in question have equal and parallel motions. Professor Boss was able on this hypothesis to determine all the principal elements of the motion of this group, as well as the mean distance of the group. The observed proper motions and radial velocities were har- monized within the limits of unavoidable error by assuming: MOTIONS OF GROUPS OF STABS 185 1. A common convergent point at a = 92°, 8 = -|- 7° ; 2. The angle (at present) between the line of sight and the direction of the group motion, 28°.9 ; 3. The mean parallax of the group, 0".025 ; 4. The linear speed of the group (with reference to the solar system), 45.6 km. per second. Figure 9 This cluster includes and surrounds the Hyades cluster. Kapteyn has recently determined the parallax of the Hyades group by ordinary methods, and finds it to be 0".023." The agreement with Boss's value, 0".025, is remarkably close. Our question is again, To what extent shall we let these stars enter into the determination of the solar motion ? Of the thirty- nine stars at present thought to be members of the cluster, four- teen are as bright as the 5.0 magnitude, which is the limit for our present purpose. Compromising between the equal and parallel motions of these stars on the one hand, and their large 11 Pui. Astr. Lab. Groningen, No. 23, 1909. 186 STELLAR MOTIONS masses on the other, we have decided to enter three individual radial velocities as representative of the fourteen bright members of the system. There are several cases, throughout the sky, of two stars, fairly close together in appearance, whose proper motions are seem- ingly equal and parallel, and whose radial velocities we have found to be equal; such as ^, and (^ Jteticuli," in the Southern Hemisphere. a (1900) 3(1900) ju.a f^S V Spm. ft Betimli SnS'^.G -62° 57' -|-0s.l947 -|-0".677 +13. km. G ^2 Sethuli 3 16 .0 -62 53 +0.1924 +0.692 +12. P8 The two stars, 5' apart, are closely related, and their radial velocities were combined and used as if for one star in. our present problem. It will happen occasionally, without violating the probabilities, that two or more neighboring stars will have essentially equal and parallel proper motions, as viewed from the position of the solar system, whereas if viewed from a quite different direction in space they would clearly be unrelated. The spectrograph has supplied the equivalent of the different viewpoints for several other supposedly related groups, only to prove them unrelated. In the case of the few double stars for which we measured the radial velocities of both components, the clear course to pursue is to use each pair of stars as one star whose velocity is the velocity of the centre of mass of the double system. Such was the method followed for a Centaur i, y Yiryinis, Castor, and one or two others. There is another kind of discrimination which we must exer- cise in selecting and rejecting stars in connection with the solar- motion problem. Certain stars are travelling at very high speeds, both in and across the line of sight. Some of them appear to be sporadic eases, not representative of general prevailing condi- tions. For example, the star Cordoba Zones 5''.243, moving away from the Sun at the rate of 242 km. per second, is one of our 12 The proper -motion equality of the two stars was pointed out by Mr. E. J. Stone.— afo?i. Not. B. A. S., 40, 26, 1879. REGENT RESULTS FOR SOLAR MOTION 187 nearest neighbors; the fifth star from us, to the best of our knowledge. Its visual brightness is of only the 9.2 magnitude, and it must in reality be a very small body. Now if we were to let this negligible mass with abnormal velocity enter the prob- lem it might influence the result more powerfully than twenty stars of normal masses and velocities. It should clearly not be used. Another runaway star, of relatively small mass, is Groom- bridge 1830, moving toward us 97 km. per second, and across the line of sight perhaps 250 km. per second. This, too, should be excluded as abnormal. However, it will not do to reject arbi- trarily, and a definite rule of rejection has been formulated and followed. My preliminary solution of the Sun's motion, in the year 1900, based on the velocities of 280 stars, gave a speed F„ — 19.9 km. per second, in direction ao=277°.5, So = + 20° (see page 167). Combining this determination of direction with the many determinations based on proper motions, we assumed, for the present purpose, that the solar motion is toward a^ = 275°, S„ = + 30°. If D is the angular distance of a star (a, 8) from this apex, VaS the star's radial motion with reference to the stellar system, and V the star's observed radial motion with reference to the solar system, then we have the observed velocity of the star V= VaS - 19.9 cos D ; or the star 's motion with reference to the stellar system is y„s= "r+ 19.9 cos -D. Before beginning the present solution for the solar motion, based upon the materials now available, we applied the term -|- 19.9 cos D to all the large observed velocities and obtained a closely approximate value of these individual stellar velocities with reference to the stellar system. Further, from the early solution, the velocities of the stars to and from the solar system, freed from the solar-motion component, were as follows : 151 positive velocities, average +17.01 km. 129 negative velocities, average — 17.10 km. 280 numerical average 17.06 km. 188 STELLAR MOTIONS It has seemed to me that all stars whose radial speeds with reference to the stellar system exceed four times the average observed speed 17, or 68 km., would unduly influence the com- puted value of the solar motion and that they should be excluded from the solution. This is substantially in accord with Peirce's criterion and other criteria for the rejection of discordant observations. Further, there are indications that the stars having extremely rapid motions are in general of small mass, though there are exceptions, and stars of known small mass should not enter with their full value. Accordingly, I decided to reject all radial velocities (freed from the correction for solar motion) which equal or exceed three and a half times the average velocity ; that is, all greater than 60 km. per second. My original aim was to have all stars brighter than the 5.01 visual magnitude, Revised Harvard Photometry, included in the solution. A small number of such stars, chiefly those winter stars whose spectra contain poorly defined lines, have not yet been sufficiently observed. On the other hand, especially in the Southern Hemisphere, a number of stars fainter than the fifth magnitude were observed because their proper motions are large. Now large proper motion means, in part, proximity to the solar system, and, in part, great linear speed. An examination of these stars shows that if we exclude all whose speeds are more than 60 km., according to the above ruling, the remainder of the proper-motion stars fainter than the fifth magnitude have radial speeds only slightly larger than ordinary. In many of these cases the greater than average proper motions must be due to proximity, in which case their masses are relatively small. Because of their presumably small masses and because they were added to the observing program for special reasons, I have let only about one-half of these stars enter into the solution. Deducting twenty-six velocities, rejected for reasons described in the preceding paragraphs, from the original 1073 observed velocities, we have left as a basis for the solution 1047 individual results. These are distributed not uniformly but nevertheless quite satisfactorily over the entire sphere. They present stars of essentially all spectral classes, including thirteen nebulae REGENT RESULTS FOB SOLAR MOTION 189 observed by Keeler; but stars of spectral Classes B and A have perhaps the greatest lack of homogeneity. The 1047 stars were combined into 172 groups of neighboring stars, each group repre- senting on the average slightly more than six stars. It was then a question of forming the equation of condition (32), from the average data for each of the groups; 172 equations in all, each involving three unknown quantities. Combining the 172 equations into three normal equations and solving, we obtained the following elements of the solar motion : a„ = 272° 0' ± 2°.50 8„ = +27°26' ± 3°.00 y„ = -17.77km. ± .62km. I was surprised at the smallness of the resulting V„. I had expected the value of the velocity to exceed rather than to be smaller than the value — 19.9, deduced in the year 1900. Yet the observational data cannot be made to yield an appreciably different result, as we shall see in the next paragraphs. Another solution was made with an equation of condition for each of the observed velocities. That is, there were 1047 sepa- rate equations. Solving them by the method of least squares we obtained the following elements: a, = 273°..5 S„ = 28°.0 T„ = —17.73 km. per second. The two solutions, one based on individual velocities and the other upon group velocities, are in remarkable accord. StiU another form of solution for the velocity of the solar motion was employed, as follows : Neglecting the thirteen nebulae, there remained the observed velocities of 1034 stars. Three hundred and thirty of these are of spectral Classes 0, B, A and F-F4, inclusive, which we may say constitute Secchi's Type I. The remaining 704 are of spec- tral Classes F5 to M, inclusive. These conform to Secchi's Type II, except" that Class K5 and Class M stars fall in Secchi's Type III. The angular distances B of these stars from the 13 Only about % of the 704 stars are of Type III. & l?4 a ml -- ^ ^ '-' T— I ,— ( rH "—I O O O ^ cn Cl Oi >-i ^ ^ « :! " ^ 1 ti - o - ~* ^ ^ o (U p4 OQ --H C^ CQ -+ RECENT RESULTS FOR SOLAR MOTION 191 deduced position of the apex a — 272°. 0, 8 = + 27° 26', were computed. The observed velocity of each star was freed from the solar-motion component by applying correction — 17.77 cos D km. per second. The stars were then tabulated as in the accompanying table, in terms of their apical distances and cor- rected radial velocities. For instance, in the zone whose limiting apical distances are 60° and 65°, there are eleven stars of Secchi's Type I, whose average apical distance is 62°. 3, and whose average radial velocity with reference to the solar system (freed from the solar-motion component) is -\-l.% km. per second ; and in the same zone there are twenty stars of Secchi's Type II whose average apical distance is 62°. 6, and whose average radial velocity with reference to the solar system is + 6.1 km. per second. If the elements of the solar motion used as a basis for correcting the velocities here tabulated can be regarded as satis- factory, then the residual velocities in the hemisphere whose apical distances are between 0° and 90° should "balance" the residual velocities in the hemisphere whose apical distances are between 90° and 180°. It is noticed immediately that the signs of the average velocities quoted in the table are prevailingly positive; a small positive average for the Type II stars, but a large positive average for the Type I stars. However, the mean velocity of the 506 stars in the hemisphere surrounding the apex of the Sun's way agrees well with the mean velocity of the 528 stars in the hemisphere surrounding the antapex. Each average residual velocity is entitled to have weight in determining a cor- rection to the deduced velocity of the solar system, — 17.77 km. per second, in proportion to the number of velocities which have combined to form the mean, and to the cosine of its distance from the apex. The expression for the correction to the velocity forms the left member of the following equation. The numerical values of the corrections supplied by the data for Type I stars and for Type II stars are as set down respectively in the right members of the equation. Z " ^ COB D f + 0.08 km. for Classes B to r4G stars (Secchi's Type I) COS D = J — = ;;; — - 0.29 km. for Classes F5G to M stars (Secchi's Type II) S H cos U \ TABLE XI COREECTED VELOCITIES IN TERMS OP APICAL DISTANCES (a„ = 272° C, «o = + 27° 26', and V„ = - 17.77 km.) Apical Distances n Type I Stars D V n Type n Stars D V km. km. 0°— 5° 1 4°.4 — 3.9 2 4°.0 +17.2 5 — 10 2 8 .4 +10.2 6 7 .4 —10.8 10 — 15 4 12 .4 — 2.9 4 11 .3 + 4.3 15 — 20 5 17 .3 + 0.1 9 17 .9 — 7.7 20 — 25 3 23 .9 — 3.4 12 23 .2 + 2.7 25 — 30 4 27 .8 + 3.8 19 28 .0 — 3.6 30 — 35 9 33 .0 + 4.6 13 32 .1 — 4.8 35 — 40 8 36 .4 — 0.6 23 37 .2 + 0.8 40 — 45 8 41 .6 + 4.1 17 42 .7 + 4.5 45 — 50 10 48 .0 + 6.9 17 47. .8 + 1.5 50 — 55 13 52 .7 — 1.1 28 52 .6 + 2.4 55 — 60 8 58 .2 + 1.9 27 57 .1 + 2.5 60 — 65 11 62 .3 + 7.9 20 62 .6 + 6.1 65 — 70 12 67 .3 + 4.0 31 67 .6 + 0.8 70 — 75 11 72 .8 + 4.6 30 72 .2 + 0.8 75 — 80 15 77 .5 — 2.8 21 77 .4 — 3.5 80 — 85 17 83 .0 + 2.9 33 82 .3 + 2.6 85 — 90 20 87 .4 + 1.0 33 87 .8 + 1.6 90 — 95 13 92 .6 + 0.5 27 92 .4 + 5.6 95 —100 10 97 .6 — 1.8 30 97 .6 — 6.7 100 —105 8 101 .4 + 4.2 28 102 .3 + 0.8 105 —110 11 107 .7 + 0.7 25 107 .3 + 1.2 110 —115 12 112 .6 + 0.8 24 112 .3 + 3.4 115 —120 23 117 .2 + 2.7 26 117 .4 + 2.3 120 —125 10 122 .9 + 0.3 24 122 .3 — 1.8 125 —130 11 127 .1 + 1.8 34 127 .5 + 0.4 130 —135 12 132 .6 + 2.4 22 132 .4 + 6.5 135 —140 9 137 .0 + 2.5 23 137 .2 — 3.0 140 —145 7 142 .2 — 5.6 26 142 .1 + 9.1 145 —150 7 147 .5 + 2.5 15 146 .6 — 1.7 150 —155 10 153 .3 + 4.2 13 152 .7 + 3.4 155 —160 11 157 .1 + 3.0 21 157 .1 — 1.1 160 —165 6 162 .8 + 8.2 5 162 .8 + 3.2 165 —170 6 167 .0 • +10.0 14 167 .9 + 0.1 170 —175 3 170 .4 + 3.5 2 172 .1 —11.2 175 —180 Totals 330 704 RECENT RESULTS FOR SOLAR MOTION 193 The 330 stars of Secchi's Type I yield a corrected value of the solar velocity To = - 17.77 + 0.08 = - 17.69 km. The 704 stars of Secchi's Type II yield a corrected value of Vo— — 17.77 — 0.29 = — 18.06 km. Weighting these results in proportion to the number of component velocities, we obtain F„ = —17.94 km. per second. The most probable value of the solar velocity obtainable from the observational data is approximately To = — 17.85 km. per second. However, the discussions which foUow in the next chapter have been based upon the velocity — 17.77 km. as determined by the group method of solution. "We may say that the position of the apex as determined from the radial velocity data agrees satisfactorily in right ascension with the average of the best proper-motion apices; but that the radial velocity apex is three or four degrees south of the proper- motion apex of greatest weight. The radial velocity data are, perhaps, a little nearer homogeneity in right ascension than in declination, for the Mount Hamilton observations have been made throughout the twenty-four hours of right ascension with the same spectrographs, and, in general, with the same personal equations in the plate measurements; and the same conditions hold for the observations in the Southern Hemisphere, obtained by the D. 0. Mills Expedition. It is not impossible that small systematic differences in the personal equations of the observers in the two hemispheres may be responsible for a part of the dis- crepancy between the radial velocity and proper-motion declina- tions of the apex, but there is an even stronger probability that systematic differences in the proper motions assigned to stars in the Northern Hemisphere and the Southern Hemisphere may be responsible for an appreciable share of the discrepancy. Again, it seems certain that the radial velocity data extend out further amongst the Milky "Way stars than do the proper-motion data. The proper-motion apex and the radial velocity apex may both be correct, for they refer only to the systems of stars actually used in the solutions; but the radial velocity solution may be the more representative of the stellar universe. "We must wait for the future to decide. 194 STELLAR MOTIONS The deduced speed, — 17.77 km. per second, carries the solar system a distance of 560,000,000 km. per year, or 3.75 times the Earth's mean distance from the Sun. These are frequent and legitimate questions: Is the solar system moving in a simple orbit, such as a conic section ? Will it eventually complete a circuit in this orbit and return to the part of its orbit where it is now? The idea of affirmative answers to these questions appears to be prevalent in the human mind. It is natural to think that we must be moving on a great curve — perhaps closed like an ellipse, or open like a parabola — -the centre of mass of the universe being in the curve 's principal focus. The attraction which any individual star is exerting upon us is cer- tainly slight, owing to its enormous distance, and the resultant attraction of all the stars may not be very much greater; for since we are believed to be somewhere near the centre of our stellar system, the attractions of the stars in the various direc- tions should nearly neutralize one another, in accordance with the principle that a body situated within a concentrically homo- geneous sphere is efiEeetively acted upon only by the gravitational matter nearer the centre of the sphere than itself. Even though we may be following a definite curve at the present time, there is, in my opinion, little doubt that we shall be prevented from con- tinuing upon it indefinitely. In the course of our travels we should be carried, sooner or later, relatively close to some indi- vidual star whose attraction would be vastly more powerful than that of aU the other stars combined. This would draw us more or less from our present curve and cause us to follow a different curve. At a later date our travels might carry us into the sphere of attraction of some other great sun which would send us away in a still different direction. Thus, the chances are, in my opinion, that our path would, in time, be made up of a succession of unrelated curves. The results deduced above define the direction and speed of the solar motion along a straight line ; and, as a single line does not fix the position of a plane, we are without knowledge as to the plane in which the solar system is moving. It is of great interest that the present line of motion lies nearly in the plane RECENT RESULTS FOR SOLAR MOTION 195 of the Milky "Way, making in fact an angle of about 17° with the central line of the Milky Way. "We need not concern our- selves at present with the question of the plane of our orbit, for the curvature of our path is undoubtedly so slight that we may consider it as a straight line for many generations of astronomers to come. When my solution for the solar motion, as based upon 1047 radial velocities, was under way and nearly concluded, there appeared a paper^* on the same subject by Hough and Halm of the Cape of Good Hope Observatory. It is based upon their velocities of 166 of the brightest southern stars, for 50 per cent of which they had secured more than one spectrogram ; plus the radial velocities of forty-five stars, mostly northern, published by various observatories in the past seven years ; plus the radial velocities of 280 stars, four or more spectrograms each, north of Declination — 30°, which I published as 80 mean velocities nine years ago (Table VIII) ; a total of about 460 velocities, deducting duplicates. There has not been an opportunity since receiving and reading the paper to make a critical analysis of the results, for this would require rather extensive computations ; and this is impossible, as the velocities of the individual stars have not been published. Their results are quoted here in order that the subject may be brought up to date (January, 1910) . Hough and Halm's deduced speed of the solar motion, — 20.85, is 3 km. per second greater than mine. The explana- tion of a part of the discrepancy seems to me to be clear. The observational data are very far from homogeneous. The forty- five miscellaneous velocities include twenty stars of the Orion type, as observed at the Yerkes Observatory, and a large pro- portion of the remaining twenty-five are Class B spectroscopic binary systems. We shall show in the next chapter that the radial velocities of Class B stars are, for some unknown reason, observed too great to the extent of about 5 km. per second, positive. Half of the forty-five stars concerned are situated relatively near the antapex, where, in common with stars near the apex, they have the maximum weight in determining the w Mon. Not. E. A. S., 70, 85, 1909. 196 STELLAR MOTIONS solar velocity. As the observed velocities of these stars are on the average abnormally great, a large value of the solar speed naturally foUows. The right ascension and declination of the apex, o^ = 271°.2 ± 3°. 3, S„ = + 25°. 6 ± 3°. 7, are in remarkably close accord with my results depending upon 1047 velocities. CHAPTER VI STUDIES OF THE STELLAR SYSTEM We recall that the equations used in solving for the elements of the solar motion were developed on the assumption that the motions of the stars, with reference to the sidereal system, follow the laws of accidental errors. This procedure is, of course, not permissible unless the radial velocities are distributed in accord- ance with such laws. However, this assumption cannot lead to results seriously in error provided that in each small area of sky we consider the motions of approach equal to motions of recession, on the average, even though the average approach and average recession in different areas of the sky may be quite different, as must certainly be the case if Kapteyn's conception of two star streams is correct. Further, it has been shown by Weersma^ that the three normal equations resulting from the individual equations of condition for observed stars, as devel- oped and used in Chapter V, are in reality independent of the supposition that the radial velocities are distributed according to the laws of accidental errors. It was not my purpose to regard the assumption referred to as final; but it was intended to investigate the law according to which the stellar radial veloci- ties, with reference to the sidereal system, are really distributed. This we shaU now undertake. Bach observed stellar velocity was freed from the solar-motion component by applying the correction — 17.77 cos D km. per second. The results are the velocities of the individual stars with reference to the system of stars employed. These residual velocities have been arranged as in Table XII, with reference to their numerical magnitudes and their spectral types. Stars of spectral Classes to F4 inclusive have been tabulated as 1 Pull. Astr. Lai. Groningen, 21, 59, 1908. 198 STELLAR MOTIONS TABLE xn MSTEIBUTION OF STELLAE VELOCITIES WITH RESPECT TO SPECTRAL TYPES Residual I n 1+] Bad. Vel. n 11 tc Above +80 Vm 5 5 -|-70 to +80 3 3 +60 to +70 1* 3 4 +50 to +60 7 7 +40 to +50 18 18 +35 to +40 1 7 8 +30 to +35 4 14 18 +25 to +30 1 19 20 +20 to +25 17 32 49 +15 to +20 21 53 74 +10 to +15 48 71 119 + 5 to +10 49 61 110 + to + 5 73 78 151 — to — 5 50 84 134 — 5 to —10 19 70 89 —10 to —15 17 53 70 —15 to —20 13 49 62 —20 to —25 9 20 29 —25 to —30 5 20 25 —30 to —35 5 19 24 —35 to — 40 1 10 11 —40 to —50 2 13 15 —50 to —60 1 4 5 —60 to —70 4 4 — 70 to — 80 1 1 Below — 80 5 5 Totals 337 723 1060 Column I includes Classes B to r4 (Seechi's Type I). ColumTi II includes Classes F5 to M (Seechi's Type II). The nebular velocities are not included in this table. * Onlj two spectrograms of this star secured ; a good chance that the velocity is variable. RADIAL VELOCITY AND SPECTRAL TYPE 199 Secchi's Type I, and stars of Classes F5 to M inclusive as Secchi's Type II. Strictly, the Classes K5 and M stars belong to Secchi's Type III, but as their number is relatively small we have in these preliminary studies entered them in Type II. The significance of the table can, perhaps, be most definitely stated by describing the contents of one horizontal line; for example, of stars whose residual radial velocities lie between + 20 and + 25 km. per second there are seventeen of Type I and thirty- two of Type II ; and of stars whose residual velocities lie between — 20 and — 25 km. per second, there are nine of Type I and twenty of Type II. Two facts appear prominently on the face of the table. 1. The number of positive velocities is considerably greater than the number of negative velocities for both types, but espe- cially in the case of Type I stars. Of Type II stars, 371 have positive velocities and 352 have negative, velocities. Of Type I stars, 215 have positive velocities and 122 have negative veloci- ties. "We shall consider this discrepancy a little later, in greater detail. 2. There are no velocities amongst the Type I stars exceed- ing + 70 or — 70 km. per second, whereas there are fourteen Type II stars with residual velocities greater than these limits. Exceeding the limits ± 50 km., there are two stars of Type I and thirty-two stars of Type II. Exceeding the limits ± 25 km., there are twenty-one stars of Type I and one hundred and fifty- two stars of Type II. The proportion of small velocities of Type I stars is much greater than in the case of Type II stars. The data in Table XII have been plotted as in Figure 10, first multiplying the number of Type I stars in the different compartments by the ratio of the number of stars of the two types, 723/337, in order to make the data for the two types comparable. The ordinate of each black circle represents the number of Type I velocities whose arithmetical mean velocity is the abscissa of that circle, as determined from the individual velocities which lie in the corresponding compartment of the table. Bach open circle represents the corresponding number and average arithmetical velocity of Type II stars whose indi- 200 STELLAR MOTIONS vidual velocities fall in the corresponding compartment of the table. Ill ■to -si -It -to -S9 -no _j^ -^jg -loL^ +to^ fJO *Ja *vo *fo *it ^70 tfo 4jo Figure 10 If these stellar velocities are distributed according to the laws of accidental errors we should be able to represent them reason- ably well by means of probability curves with suitable constants. The dotted curve in Figure 10 corresponds to the equation 2/ = 2.5e-''-'°^" (34) It represents extremely well the radial velocities of Type II stars with reference to the stellar system. Apparently the use of the method of least squares, so far as the Type II stars are concerned, was entirely justifiable. The full curve in Figure 10 corresponds to the equation It does not represent the velocities of Type I stars very satis- factorily; yet it would be difficult and perhaps impossible to find a symmetrical and reasonably simple curve which would represent them better. It may be that the number of Type I stars observed, 337, is too small to serve as a statistical basis, or, more probably, that the data included under Type I are not RADIAL VELOCITY AND SPECTRAL TYPE 201 homogeneous and comparable. The sudden decrease in the number of stars of Type I whose velocities are between — 5 and — 10 km., as compared with the number in the to — 5 compartment, is very striking. The value of the unit in the two equations and their curves is 20 km. The numerical term, — 0.17, in the exponent of the Type I equation corresponds to a positive displacement of the curve amounting to 3.33 km. per second. It is apparent from Figure 10, as well as from Table XII, that the numerical average velocity of the Type II stars is much in excess of the average velocity of the Type I stars. The num- ber of stellar velocities upon which the results rest, 1060, is so large that the discrepancy in the average velocity of the two types can scarcely be otherwise than a fact of nature. In order to explain, if possible, the prevailing positive ten- dencies of the mean velocities in Table XII, especially for the Type I stars, the mean residual velocities were arranged in greater detail as in Table XIII. The stars were divided into four spectral classes, B to B9, A, A2 to F8, and G to M. The numbers of stars of the four classes in each zone of apical dis- tance, 5° wide, and the mean residual velocities of these stars, are as quoted in the table. The numbers of stars in the four divisions and the corresponding averages of the residual veloci- ties are given at the foot of the table. We see that the averages are prevailingly positive: very small for Class A velocities; + 0.60 km. for Classes A2 to F8 ; -f 0.91 km. for Classes G to M; and -|- 4.93 for the 138 stars of Classes B to B9 inclusive. These positive tendencies prevail alike in the hemispheres of the apex and of the antapex. The mean residuals for the Class A and for the Classes A2 to F8 are perhaps no greater than could be ascribed to casual velocity distribution and personal equation of measurement ; but the residual, + 0.91 km., for Classes G to M can scarcely be explained in this manner; and the residual, + 4.93 km., for the Classes B to B9 must certainly seek some other explanation. Let us review some possible explanations of the Classes B to B9 discrepancy. TABLE Xin STELLAR VELOCITIES WITH REFERENCE TO SPECTRAL CLASSES AND APICAL DISTANCES a„ = 2; TOO 0', So = +27 ° 26', To = -17.77 km. Classes Class Classes Classes Apical B-B9 A A2-F8 G-M Distance n V n r n V n V km. km. km. km. 0° - 5° 2 + 7.2 2 +17.7 5 - 10 2 +10.2 6 -10.8 10 - 15 1 — 3.9 2 + 0.4 2 +15.8 3 - 7.7 15 - 20 3 + 6.6 -12.1 1 - 7.1 9 - 7.7 20 - 25 1 +12.2 -16.6 2 -11.3 11 + 4.5 25 - 30 2 + 1.0 + 8.7 6 - 1.9 14 — 3.8 30 - 35 1 +21.2 + 6.0 6 +12.3 10 -10.0 35 - 40 3 — 2.5 - 8.5 7 - 2.0 20 + 2.2 40 - 45 1 +24.6 4 + 2.4 5 + 0.5 15 + 4.9 45 - 50 3 — 2.4 4 + 9.4 8 + 4.2 12 + 1.2 50 - 55 5 + 0.4 2 - 5.2 9 - 1.6 25 + 3.1 55 - 60 2 + 1.9 2 + 8.1 7 + 2.3 24 + 1.9 60 - 65 4 + 8.0 3 - 4.0 7 +15.5 17 + 4.8 65 - 70 6 + 9.7 2 + 2.4 10 - 5.0 25 + 2.4 70 - 75 6 + 2.4 3 + 8.2 7 +10.3 25 - 1.4 75 - 80 4 — 2.0 5 -11.3 8 + 5.9 19 - 5.1 80 - 85 7 + 4.4 2 - 1.2 16 + 1.5 25 + 3.4 85 - 90 11 — 2.6 3 +14.4 9 + 1-4 30 + L5 90 - 95 3 + 6.6 1 - 5.5 12 - 2.8 24 + 7.4 95 -100 3 + 1.0 3 - 3.5 5 + 0.7 29 - 7.3 100 -105 1 + 8.3 3 + 5.8 8 - 1.9 24 + 1.8 105 -110 5 + 1.0 3 - 9.6 3 +10.5 25 + 1-2 110 -115 9 + 3.2 4 — 4.0 22 - 0.9 115 -120 6 + 5.5 6 + 1.9 15 + 0.5 22 + 3.2 120 -125 5 + 5.2 3 - 3.2 4 -10.2 22 - 0.7 125 -130 4 + 9.7 1 + 9.3 12 - 8.2 28 + 2.3 130 -135 7 + 7.5 3 -13.7 5 + LO 19 + 8.2 135 -140 5 +11.3 1 -23.7 5 - 0.3 21 - 3.7 140 -145 3 +10.6 2 + 3.7 6 -12.2 22 +10.2 145 -150 5 + 0.7 1 + 9.0 6 + 6.1 10 - 5.7 150 -155 5 + 8.8 1 + 8.0 6 + 4.1 11 + 0.8 155 -160 4 + 6.8 3 + 7.9 7 - 1.6 18 — 1.1 160 -165 5 + 8.2 1 + 8.0 1 +44.1 4 - 7.0 165 -170 3 +21.6 2 - 6.3 4 - 6,0 11 + 3.0 170 -175 3 + 3.5 1 -26.7 1 + 4.4 175 -180 Means 138 + 4.93 74 + 0.18 216 + 0.60 605 + 0.91 The nebular velocities are not included in above table. The twenty-six velocities rejected for reasons explained in Chapter V are not included. One additional velocity was omitted by mistake. RADIAL VELOCITY AND SPECTRAL TYPE 203 1. If we give a literal interpretation to this result, it signifies that the universe of Classes B to B9 stars is expanding, with reference to the instantaneous position of the solar system as a centre, at the rate of 4.93 km. per second. Pickering has shown that the Class B stars are strongly clustered in the Milky Way and vicinity, and quite irregularly in galactic longitudes, though all parts of the galaxy and vicinity are fairly well represented. It is exceedingly improbable that the Class B stars in all the con- siderable areas of the sky where they are found are travelling outwardly from the point in space which we happen to occupy, as shown by the average residual velocities in Table XIII. 2. A personal equation in the measurement of the spectro- grams, systematically positive, amounting to 5 km. per second, cannot be regarded as possible. Observations of a few Class B star spectra at other observatories have been published, and the results are either in good agreement with those obtained at Mount Hamilton and in Chile, or these published velocities are in general larger than ours. 3. A more probable explanation, it seems to me, is that the wave lengths of the lines in the Class B spectra, adopted by the radial velocity observers, err in being too small. An average increase of 0.07 A in the wave lengths of all the lines utilized would fully explain the phenomenon. Unfortunately, there is no apparent means of testing this question directly; but the question of causes is an interesting one. It is recognized that high pressures in radiating or absorbing media not only broaden spectral lines but shift their apparent centres in general in the direction of greater wave lengths. The absorption lines — or absorption bands preferably — in Class B spectra are usually of considerable breadth. It appears that axial rotations of the stars can be but minor factors in the broadening of lines. It seems not impossible that the conditions existing in Class B stars are such that the absorptions are effective at great depths in their atmospheres under high pressures, as well as in the surface strata under low pressures. If we grant the efficiency of this factor, the systematic positive tendency given to observed radial velocities of Class B stars is in a fair way to be explained. 204 STELLAR MOTIONS 4. Another hypothesis, perhaps simpler, should be mentioned, but it is not considered of great weight. Many of the helium lines, which are the most prominent lines in Class B stars, occur in pairs in the laboratory spectrum of helium f as examples, the lines at 4026 A, 4120 A, 4471 A, and 4713 A, in the region utilized by radial velocity observers. In every ease the more refrangible component is stronger than the less refrangible. The laboratory line at 4471.646 A has intensity 6, and its companion at 4471.858 A has intensity less than 1 ; and somewhat similarly in other cases. The interval between the components of the pairs corresponds approximately to a radial velocity difference of 15 km. In most of the radial velocity determination, at the Yerkes Observatory, at the Lick Observatory, at Santiago, Chile, and perhaps elsewhere, the wave length of the helium line in this region of spectrum of the Class B stars has been assumed to be 4471.676 A, obtained by giving weight 6 to the wave length of that component whose intensity is 6, and weight 1 to that of the component whose intensity is about 1. If conditions in Class B stars are such that the relative intensities of the red components of the helium pairs are considerably augmented, so that the effective wave length of a pair is greater than we have assumed it to be, from laboratory measurements, it is possible we should not need to look further for the explanation of the positive dis- crepancy. It is known to experienced radial velocity observers that the systems of wave lengths adopted for the lines of differ- ent elements may be satisfactory for one Class B spectrum, and apparently quite unsatisfactory for another Class B spectrum. Here may exist a fruitful field for investigation. Adopted wave lengths for the helium lines must of course harmonize with wave lengths adopted for the hydrogen, oxygen, silicon, and other lines existing in the same spectrum, so that all the lines in a given spectrum will yield equal radial velocities. 5. There can be little doubt that the Class B stars of the Orion region are or have been intimately associated with the great nebulous structures which we know to exist there. The observed velocities for the densest part of the Orion nebula, as 2 Eunge and Paschen, Ap. J., 3, 11, 1896. RADIAL VELOCITY AND SPECTRAL TYPE 205 obtained by Keeler, Vogel and Bberhard, "Wright, Frost and Adams, are in excellent agreement, with mean value + 17.4 km. per second. The observed radial velocities of the Class B stars in the Orion region, though differing in essentially the usual amounts from one another, average about + 221^ km. Here again we have an indication, more or less weighty, that the observed radial velocities of Class B stars are for some unknown reason about 5 km. too great. Of all the explanations suggested, that of a pressure effect in the extensive atmospheres of Class B stars appears to be by far the most probable one. It is not improbable that the excess of positive velocity, 0.91 km. for the stars of Classes G to M, is due to the same cause. The peculiarities 'of the data for Type I and Type II stars in Table XII, as further illustrated in Figure 10, are clearly in confirmation of previous indications that the stars of early spectral classes are travelling more slowly than those of later classes. To test this question, and at the same time that of stellar velocities as a function of visual magnitudes, the residual radial velocities were tabulated as in Table XIV. The visual magnitudes in the first column are from the Revised Harvard Photometry. The two columns under Secchi's Type I include, for stars of Classes 0, B, A and F to F4 inclusive, both the number of stars in each magnitude division and their average residual radial velocity. The two columns under Secchi's Type II contaiu corresponding data for Classes F5 to F9, G, K and M. The last two columns combine the data for Type I and Type II stars. It appears, in brief, that 330 stars of Type I have an average residual radial velocity of 10.25 km. per second; and that 704 Type II stars have an average residual radial velocity of 15.08 km. per second ; that is, the Type II stars in the present list have radial velocities nearly 50 per cent greater than those of the Type I stars. Eecalling that the 280 radial velocities published by me in the year 1901, which were chiefly of the G, K and M types, averaged 17.06 km. per second; that the radial velocities of twenty Class 206 STELLAR MOTIONS B stars, published by Frost and Adams of the Yerkes Observa- tory in the year 1904, averaged only 7 km. per second ; and con- sidering the additional fact that a hasty tabulation of about sixty Class M velocities (Secchi's Type III) in my present list shows an average velocity of about 17 km. per second : I am led to the remarkable conclusion that the velocities of the stars must be functions of their spectral types; that is, of their effective ages. To recapitulate: we have the average radial velocities of twenty Class B stars published by Frost and Adams, 7 km. per second; of 330 Secchi's Type I stars, 10.25 km. per second; of TABLE XIV AVERAGE VELOCITIES EST TERMS OF VISUAL MAGNITUDES AND SPECTRAL TYPES Vis. Mag. Type I h' r Typen H r Types landn » V Vis. Mag. n Avg. r km. km. km. km. Above 1.50 9 10.5 8 16.8 17 12.2 1.51 to 2.50 26 9.2 22 12.3 48 10.7 ■ Br. than 3.50 179 12.3 2.51 to 3.50 40 9.4 74 15.0 114 13.0 J 3.51 to 4.50 4.51 to 5.50 116 8.9 126 11.3 239 15.1 336 15.1 355 13.0 462 14.1 } 3.50 to 5.50 817 13.e 5.51 to 6.50 Below 6.50 9 19.1 4 11.3 18 18.5 8 12.5 27 18.7 12 12.1 } Ftr.than5.50 39 16.7 Means 330 10.25 704 15.08 1034 13.51 704 Secchi's Type II stars, 15.08 km. per second; of the Class M stars in the present list, numbering about sixty, 17 ± km. per second ; and of the 280 stars considered in 1900, consisting mostly of Classes G, K and M, and from which no rejections were made on account of abnormally high velocities, 17.06 km. per second. The progression of average velocity with advancing spectral type is clear and unmistakable.^ 3 Footnotes added after the date of the lecture : (A) As the question of priority in making this discovery is of interest to some writers, I make the following statement: Aside from the presentation of all the above results, including Tables RADIAL VELOCITY AND SPECTRAL TYPE 207 The table of radial velocities whicli exceed ± 50 km. per second, in Chapter III, page 115, bears strongly upon this subject. With the exception of one star each of Classes B8, F and F8, all the stars in that table are of Classes G, K and M ; and the chances are reasonably strong that the high velocity of the Class B8 star, v Pavonis, R. A. = 18''22".0, will prove to be orbital, rather than systemic, as only two spectrograms of this star have been secured. It would scarcely be possible to secure XII, XIII and XIV, and Figure 10, in the SUliman Lecture of January 31, 1910, they were discussed with those of my colleagues who had assisted in the computations and in forming the tables and the figure, all before January 17, 1910; on January 18 and 19, in San Francisco, I informed certain high officials of the University of California of the discovery that the motions of the stars increase in speed with increasing age; and Figure 10 and Tables XII, XIII and XIV were shown and their significance explained to leading astronomers in the eastern part of the United States between February 6 and February 10, 1910. (B) Eeturning to Mount Hamilton on February 24, 1910, I had the velocities with reference to the stellar system tabulated on the basis of the Harvard classification of spectra, as follows: leotral Classes Number of Stars Average Badial Velocities and B 141 8.99 km. A 133 9.94 F 159 13.90 G and K 529 15.15 M 72 16.55 Nebulffi (Keeler) 13 23.4 The increase of stellar velocity with advancing type was seeu to hold on the Harvard classification, as well as for the Secchi classification upon which the discovery had been made. It was a surprise to find the average velocity of the thirteen nebulse, as observed by Keeler, in excess of the averages for the stellar spectral types. This may or may not indicate that the number of nebulae available is too small to serve as a basis for averages. If from the list of thirteen nebulae we remove the Orion nebula, which has radial velocity nearly zero, the average residual radial velocity for the remaining twelve nebulse is in excess of 25 km. Here we may have evidence of great strength and impor- tance, in support of a hypothesis that the planetary nebulse have been 208 STELLAR MOTIONS a stronger bit of evidence that high stellar velocities appertain to the later spectral classes and abhor the early spectral classes. The velocities of the nebulae, as observed by Keeler, have not been included in the results thus far described in this chapter. Here is a list of these nebula, with the observed velocities in the next to the last column, and the velocities with reference to the stellar system in the last column. Keeler noted that the observed velocity of the Great Nebula in Orion must be due formed from stars through processes arising from colliBions with or close approaches to other massive bodies. The zero velocity of the Great Nebula in Orion is not out of harmony with the hypothesis. (C) In February and March, 1911, the elements of the solar motion were re-determined according to a rather extensive program, as outlined in Table XV. The number of stars for which spectrograms had been secured with the Mills spectrographs on Mount Hamilton and on Cerro San Cristdbal, plus additional stars whose radial velocities have been observed and pub- lished elsewhere, was in excess of 1700. Excluding those stars whose spectral lines are too indefinite for high dispersion measurement, and those spectroscopic binaries whose systemic velocities either had not been deter- mined or could not be estimated from the available data, but including thirteen nebular velocities, there were available for this investigation the radial velocities of 1193 objects. Twelve solutions for the velocity of the solar motion as a function of spectral classes, as described in the first twelve lines of the following table, were made. Two solutions for the direction and speed of the solar motion, based upon the radial velocities of all spectral classes, are described in the last two lines. [The last two solutions, based upon the 1193 velocities, include the stars used in the first twelve solutions, and in addition, 13 nebulae, 3 Wolf-Eayet stars, 1 Class G star, 1 Class K star, and 5 Class O stars.] In the first thirteen solutions a constant term K was introduced to represent any systematic tendency of the velocities, such as that which is apparent for the Class B stars in Figure 10 and Table XIII. The position of the apex quoted in the table as o„ =272°.. 5, S„ = -\-Zi°.^ is assumed from Professor Boss's proper-motion solution {Astr. Jour., 26, 112, 1910). The apical position a„ = 270°, 5„ = -)-30° was assumed as a satisfactory mean of the positions determined from proper-motion and spectrographie data. The average radial velocities of the stars of different spectral classes, with reference to the sidereal system, are as quoted in the last column of the table. The resulting velocity of the solar motion, V„ appears to be a function of the spectral class of the stars upon which it rests, at least as far as the brighter stars are concerned. RADIAL VELOCITY AND SPECTRAL TYPE 20.9 H 02 Eh O CM CC W I EH O O OQ o 1^ ^; o t— I P 1-5 O m. 2i B-B5 (177 stars) 47 B-B5 (180 stars) 68 B8-B9 (45 stars) 66 B8-B9 (45 stars) 48 A-A9 (177 stars) 95 A-A9 (177 stai's) 37 97 9 8 1 14 4 r^ • CD CO (CUD O O ^ M^ Cq CD lO t- ira f=^ rCiL'-O CD 050 o ^ CO CO CO CO CO CO CO CO CO CO CO CO t> l> t- l> t~ t^ t~ t^ t^ l> CD CD CI c = the angle which a plane passing through the observer and a velocity vector makes with a reference plane through the line of sight. The number of velocity vectors which we can draw to an element of the spherical surface is 7'' sin i di d. RELATIONS BETWEEN VELOCITY COMPONENTS 215 The average value of the velocity components parallel to the line of sight is defined by the expression Jo Jo sin i cos i di d4> — ^ y2 r. That is, for an infinite number of stars moving at random in direction and speed the average radial velocity, F^, is equal to one-half the average space velocity, Y^ ; or F, = 2y^. (36) The average component of velocity at right angles to the line of sight, %.€,., tangential to the celestial sphere, is defined by the expression ■K IT Jo Jo r^ sin" i di d = 1 radian = 57 °.3. ( 38 y2 -r r' That is, if an infinite number of stars are moving at random, the average of the angles which their motion vectors make with the line of sight is 57°. 3. 216 STELLAR MOTIONS Applying equation (36) to the average radial velocities quoted above, we obtain the corresponding average stellar velocities in space with reference to the sidereal system : 20 Class B stars (Frost & Adams), 14. km. 330 Type I stars (Campbell), 20.5 km. 704 Type II stars (Campbell), 30.2 km. 70± Type III stars (Campbell), 33. km. Our Sun having a speed of approximately 18 km. per second with reference to the sidereal system is thus one of the slow- moving stars; its speed is only 60 per cent of the average speed of the brighter solar type stars. It is not easy to explain why the velocities of stars should increase with their effective ages, for we are accustomed to think of all matter as equally old gravitationally. Why should not the materials composing a nebula or a Class B star have been acted upon by gravitational forces as long and as effectively as the materials in the Class M stars ? Are stellar materials in the ante-stellar state subject to Newton's law of gravitation? Does gravitation become effective only after the processes of combi- nation are well under way? Is it possible that the gaseous matter composing a nebula is acted upon as effectively by radia- tion pressure as by gravitational attraction? The observed fact of the dependence of stellar velocity upon the spectral class is so new that these comments and questions make no pretensions to the status of a solution ; but I am unable to suggest any other directions in which we should seek for the explanation. It is of interest to examine the residual radial velocities to see how they would bear upon the question of two stellar drifts, which Professor Kapteyn had announced in 1905 to exist. The proper-motion investigations of Kapteyn, Eddington, and others, place the vertices of preferential proper motions at Right Ascen- sion 93°, Declination -|- 12°, and at the antipodal point of the celestial sphere. If the stars seen in projection upon the various large areas of the sphere have in reality equal average velocities in space, but with proper-motion preferences for the two vertices, the residual radial velocities of stars near the vertices should PREFERENTIAL RADIAL VELOCITIES 217 on the average be greater than for the stars in the zone midway between these vertices. The 1034 residual radial velocities were tabulated in terms of the angular distances of the stars from the vertex at Right Ascension 93°, Declination -|- 12°, as in Table XVII. The numbers of stars in the zones 10° wide, and their average velocities, plus and minus, are tabulated in the second, third, fourth, and fifth columns, and the average velocities, irrespective of sign, for all the stars in the several zones are given in the sixth column. The significance of the results is brought out more clearly by combining the eighteen zones into six zones, each 30° wide. The numbers of stars in each of the six zones and their average velocities are assigned in the seventh and eighth columns. Plotting the sis velocities and constructing a curve representing them, the ordinates to the curve corresponding to the two vertices and to the great circle TABLE XVn COBRECTED VELOCITIES IX TERMS OF ANGULAR DISTANCE FROM ASSUMED Vertex at E. A. = 93°, Dee. = + 12° No. of Distance No. km. No. km. km. Stars v„ km. 0°- 9° 2 +19.7 2 —13.2 16.4 (16.7) 10 - 19 14 +13.8 10 —11.3 12.6 - 74 16.1 20 - 29 31 +16.9 15 —19.1 18.0 30 - 39 23 +14.3 19 —18.7 16.5 ■ 40 - 49 34 +15.6 17 — 7.6 11.6 \ 164 14.0 50 - 59 40 +13.2 31 —14.0 13.6 60 - 69 44 +11.5 35 —12.1 11.8 ■ 70 - 79 42 +10.8 33 —14.2 12.5 y 249 12.6 80 - 89 51 +14.6 44 —12.2 13.4 90 - 99 46 +11.7 48 —12.6 12.2 1 (12.4) 100 -109 46 +11.9 34 —11.9 11.9 . 259 12.9 110 -119 49 +13.9 36 —14.8 14.4 J 120 -129 37 +11.3 43 -10.3 10.8 ■ 130 -139 36 +14.7 45 —14.1 14.4 . 219 14.0 140 -149 34 +15.8 24 —16.8 16.3 150 -159 26 +14.1 17 —17.3 15.7 160 -169 17 +14.2 11 —15.3 14.8 . 80 15.7 170 -180 6 +19.6 3 —17.5 18.6 (16.3) 218 STELLAR MOTIONS midway between the two vertices are as eontaiaed within paren- theses in the last column. The average radial velocity, 12.5 km., of the stars midway between the two vertices is about 25 per cent less than the average radial velocity, 16.5 km., at the ver- tices. The indications for preferential velocities toward and away from the vertices are therefore fairly clear; but quantita- tively the preference is very much smaller than we were expect- ing to find, in view of the proper-motion results. Professor Dyson,* for example, assigned a relative velocity of 2.6 times the velocity of the solar motion, or approximately 48 km. per second, to the two drifts into which his important investigation divided the stars having large proper motions. TABLE xvni Residual Velocities in Terms of Angular Distance From Assumed Vertex at B. A. =93°, Decl. = + 12° Distance from Vertex Number of Stars Average Radial Velocity 0° to 30° 94 13.72 km. 30 to 60 188 14.56 60 to 90 269 12.78 90 to 120 296 11.92 120 to 150 251 14.51 150 to 180 99 14.55 This table includes all the velocities in Table XIII and the velocities of the twelve planetary nebulae in addition. It does not include the stars whose radial velocities were rejected because in excess of ± 60 km. per second. If the stars with velocities in excess of 60 km. per second had been included in Table XVII, the average velocities in the different zones would have been appreciably greater, but the ratios of the velocities would not have been changed more than we should fortuitously expect.' 8 Proe. B. S. Edinburgh, 29, 391, 1909. 9 Let us examine corresponding data obtained in March, 1911, from 1192 residual velocities, as in Table XV. Indications for preferential velocities to and from the vertex are not so strong as those presented in the Silliman Lecture (Table XVII). The DISTANCES OF STARS 219 Table V, in Chapter III, page 115, contains a list of all observed radial velocities, with reference to the stellar system, which are in excess of ± 50 km., including the velocities of three planetary nebulee. (Three stars were added to the list after the date of the lecture.) On this list there are 11 stars within 60° of the vertex at a = 93°, 8 = + 12° ; and 15 stars within 60° of the opposite vertex. These two areas make up one-half the area of the sky and contain 26 stars on the ± 50 km. list. The other half of the sky, comprising vertex distances lying between 60° and 120°, contains 14 stars. We have therefore a tolerably clear and strong indication that the high velocity stars have a preference for motion toward the Kapteyn vertices. There is promise that the observed radial velocities of large numbers of stars afford the best means of determining the aver- age distances of groups of stars ; say of the fifth-magnitude stars or of the stars of a given class of spectrum and magnitude ; and thus to determine the scale on which the stellar universe is con- structed. "We shall assume, again, that the real motions of the stars are at random as to speed and direction. After the effects of the solar motion have been eliminated from the measured radial velocities of the stars in question, we should find that the number of approaching stars and the number of receding stars are essentially equal. Likewise the mean corrected velocities of approach and recession should be nearly equal. If perchance they are affected by an apparent systematic error, as in the case of the Class B to B9 stars. Table XIII, such effect can be eliminated by subtracting the algebraic mean of the velocities from the individual velocities, and forming the arithmetical mean of the results. Let this mean velocity, without regard to algebraic sign, be V„. "We have shown (page 215) that the corresponding average velocity of these stars in space must be 2F^, and their average velocity at right angles to the line of differences are probably due largely to the addition, in the intervening four- teen months, of a greater proportion of Classes B, A, and F velocities, which are low, than of G, K, and M velocities, which are high. The large reduction of the average velocity in the zone 0° to 30° is undoubtedly owing to the addition of many Classes B and A velocities in the region of the constellation of Orion, which includes Kapteyn 's northern vertex of preferential motion. 220 STELLAR MOTIONS sight, that is, on the surface of the celestial sphere, - V„,. These velocities, we repeat, are free from the effects of the solar motion. If we could free the observed proper motions of the same stars from the effects of the solar motion, obtaining thus the displacements of the stars on the surface of the sphere due to their own motions, as they would be determined by an observer at rest with reference to the sidereal system, it is evident that the average of these corrected proper motions in angular measure should be very closely equivalent to the linear cross motion, - Y^, as determined by means of the spectrograph, provided the distances of the stars in the group are not too un- equal. If the distances were very unequal an error of appre- ciable size could ensue, for the distances of the stars and their proper motions measured in angle are reciprocal relations; and it is a well-known principle that the reciprocal of the mean of several quantities is not equal to the mean of their several reciprocals. . Strictly, the application of the principle would require us to know the law of stellar distances, which would leave us about where we started. Practically, the proper motions being, in general, simple reciprocal functions of the stellar dis- tances, we could combine in one solution those stars of nearly equal magnitudes and nearly equal proper motions and deter- mine their mean distances; and so on for other stars nearly equal in brightness and in proper motion. Now the effects of a known solar motion upon the observed proper motions can be eliminated in part for individual stars, and for the combined stars in a large group, provided, as we assumed, that the motions are at random. Following Kapteyn's practice and notation, let us draw a great circle through each star and the apex of the Sun's way, and resolve the star's proper motion into its compo- nent, u, along this circle, and its component, t, at right angles to this circle. The former component, v, involves the whole of the parallactic effect and a certain part of the motion of the star itself. The other component, t, is independent of the solar motion and is due entirely to the motion of the star with refer- DISTANCES OF STABS 221 ence to tlie stellar system. The values of t are obtainable from the observed proper motions, by computation, assuming the position of the apex as known. Let these be determined for the group of stars in which we are interested, and let their mean value be t„. From the spectrographic velocities in the line of sight we found Y^ to be the average radial velocity of the group, after correcting for the Sun's motion. Now the average velocity of the stars toward us — i.e., at right angles to the sur- face of the celestial sphere — is equal to their average lineai' velocity at right angles to any other plane, say the plane to which the r component is perpendicular. Therefore V^ in linear measure must be equivalent to the t^ component in angular measure, within limits depending upon the homogeneity of the group of stars. If p^ is the average distance of the group of stars in kilometers, we shall have, remembering that t^ is expressed in terms of one mean solar year as the unit of time, and Vjh in km. per mean second, p^ sin r^ = V^ 86400-365.25; or, in kilometers, — the brackets indicating logarithms, — [7.4991] „ [12.8135] Pm = - ^^^m = - -V^. (39) For convenience, as well as to assist the comprehension, we shall convert the distance in kilometers given by this equation into light years. Dividing both sides of (39) by the number of kilo- meters traversed by light waves in one year, we obtain p^ in light years, _ [9.8373] _a6875 Pm — * rn — ' m • \^^} r It T II If IT is the parallax of a star we have P sin T = 149,500,000 km. ; or, expressing p in light years and replacing sin tt with it" sin 1", we have „ 3.257 ,.,, it" — . (41) 222 STELLAR MOTIONS Therefore, the number of light years, p^, expressing the aver- age distance of a fairly homogeneous group of stars having been obtained from (40), the mean parallax of the close group is given very nearly by . 3.257 (42) Prom (42) and (40) we have and therefore _ 3.257 _ 0.6875 Pm — — ' m ? ,r„" = 4.738^. (43) Recalling again the random distribution of the proper motions, according to assumption, it must occur that the real motions of the stars are carrying as many in one direction along the circles through the apex-star-antapex as in the opposite direc- tion. The effect of the Sun's motion overcomes the real motions of many of those actually moving toward the position of the apex and makes them seem to move away from the Sun's apex. If we form the arithmetical mean value of the " components of the proper motions of a large number o/ neighboring stars, this value must be approximately the average actual angular motion of these stars along the great circles intersecting at the Sun's apex. Therefore v^ in angle must correspond to F„j in km. per second ; whence we shall have, nearly. 0.6875 ^^^ 0.6875 ^^^^^ ^^^ in light years ; and ^J' = 4.738 ^^^ = 4.738 >- . (45) At the apex and antapex the solar components of the proper motion would be nil, and the t and v components would be alike. DISTANCES OF STARS 223 Elsewhere the solar effect on each proper motion would vary as the sine of the angular distance from the apex. In a similar manner it can be shown that the algebraic mean value of the v components of a group of stars reasonably homogeneous would be due almost entirely to the solar-motion displacements of the stars. We deduce, very simply, an equation defining the approxi- mate mean distances of the stars, on this basis. For any star, at angular distance D from the apex, whose parallactic compo- nent of proper motion is v, we have from (28) q •■ P " sin V : sin D ; and for the mean of a large number of stars not too different in magnitude and proper motion, sin u^: sin D^ :: q : p (approximately), in which v^ is the algebraic mean of the separate v values ; or F„ sin D^ F„ . 86400 . 365.25 sin D^ Pm — : — : — ; sm v^ v„" sm 1" _ [12.8135] Yq sin D^ V ' in kilometers. Transferring to light years, as before, _ 0.6875 Fq sin J^ Replacing V^ by its value, 17.77 km., we have, in light years, 12.22 sin D^ Pm = • (46) If it is found that the real motions of the stars are not at random, and the laws of distribution of the motions become known, these equations would require changes, more or less simple, to make them correspond. To illustrate, the Kapteyn 224 STELLAR MOTIONS drift theory would require that these relations be applied with different constants to different parts of the sky.^" Great caution must be used in attempting to transform from average stellar distance to average parallax, or vice versa, for, as stated above, distances and parallaxes are reciprocal func- 10 For convenience we deduce the formulse trhich are employed in com- puting the values of t and v. Let •*oi ^o = the right ascension and declination of the assumed apex ; a, 5 = the right ascension and declination of a star ; X = the position angle of the apex as viewed from the star ; that is, the angle pole-star-apex, measured from the pole in the direction of increasing position angles ; ]) = the angular distance of the star from the apex ; f-a = the right ascension component of the star's proper motion expressed in seconds of time per annum, as quoted in star catalogues ; /J-a, = the right ascension component of the star's proper motion converted to the arc of a great cii-cle ; /xS = the declination component of the star's proper motion expressed in seconds of arc per annum ; ^ = the total resultant proper motion of the star ; tp =: the position angle of the star's proper-motion vector. For the spherical triangle whose vertices are at the pole star and vertex we may write cos D = sin So sin S -\- cos 5o cos 5 cos (a — a^) sin D cos X = sin So cos S — cos So sin S cos (a — Oo) sin D sin x = — cos S„ sin (a — Oo) Let /( sin X r= sin So n cos J\' = cos So cos {a — Oo) Then tan So tan X = cos (a — tto) cos D = n cos (S — X) sin D cos X = ~ " sin (S — X) sin D sin x = — cos 5„ sin (a — Oo) tan (a — ao) cos X tan X — (47) sin (S - -V) ^ ' tan (5-,V) tan Z) = ^^ (48) cosx DISTANCES OF STABS 225 tions, and if the stars are very irregularly distributed as to distance a large error would result. The reciprocal of the mean of several stellar distances may be quite different from the mean of the reciprocals of the same distances. It is safer to express average stellar distances in terms of mean parallax rather than in kilometers or light years. If we are dealing with the t com- ponents of proper motion it does not matter how irregularly the stellar distances may be distributed: equation (43) will give the correct value for the average parallax of the group, whereas equation (44) might give a highly erroneous value of the average distance of the stars in light years. We also have li-a. = 15 cos 5 ina' Mo tan !/■ = — (49) /*« ^^=->^=J^ = V~;J+^i (50) sin ^ cos f T = ;nsin(x-'/') (51) u = ^cos(x-\t') (52) The angle N will always lie between 0° and 180°, inasmuch as the apex is certainly in the Northern Hemisphere; the angle x ™8'y vary from 0° to 360°; and the angular distance D will lie between 0° and 180° If jJ-a is plus, \j/ will lie between 0° and 180° ; but if /".o is minus, ^ will lie between 180° and 360°. ii is always positive. The sign of t is determined by the value of x — ^- The sign of " is deter- mined in the same manner; it is negative when the proper motion increases the star's distance from the apex. If the apex is assumed to lie in the convenient position a„ = 270°, S„ = + 30°, the equations for determimng D and % take the form H sin .V = [9.6990] II coaN = — [9.9375] sin a [9.7614] tan X = tan X = — -7- -— r- (53) tan Z» = - ^ (54) cos X and the accuracy of the computations may be checked in part by the equation [0.0625] sin D sin x sec a = — 1. sin a cot a cos N sin (5- -N) tan (5 - -N) + + pa DISTANCES OF STABS 227 "We shall now illustrate the application of the foregoing prin- ciples in determining the average parallaxes of the Type I and Type II stars whose proper motions and radial velocities are known. We shall utilize only the t components of the observed proper motions. The results are condensed in Table XIX, in terms of visual magnitude. The proper-motion data were taken for the most part from Kapteyn's valuable lists in Publi- cations of the Astronomical Laboratory at Groningen, No. 7, and the rest were collected from various sources. Stars whose r components of proper motion exceed 0".30 per annum were excluded. The numbers of stars of Secehi's Type I, Type II, Types I and II combined, the average t components, and the average residual radial velocities, are given in the table. TABLE xrx Parallaxes by Magnitudes and Types Fkom Badial Velocities and T Components of Proper Motion Type I Type n Types I and II Vis. Mag. n km. n Tm km. km. 2.51-3.50 36 0".0148 9.2 50 0".0508 15.3 86 0".0357 12.7 3.51-4.50 76 .0279 8.8 150 .0545 15.3 226 .0455 13.1 4.51-5.50 58 .0266 8.6 162 .0352 14.0 220 .0329 12.5 7r„, Type I T„, Type n Tr„ , Types I and n Vis. Mag. L.O. Kapteyn L.O. Kapteyn L.O. 2.51-3.50 0".0076 0".0223 0".0157 0".O515 0".0133 3.51-4.50 .0150 .0157 .0169 .0357 .0163 4.51-5.50 .0147 .0111 .0119 .0253 .0125 [Note added in 1911.— The substantial aeeuraey of these results is confirmed by a much more extensive investigation, made in March, 1910, based upon proper motions from Boss 's Preliminary General Catalogue. See LicTc Ols. Bull, 6, 132, 134, 1911.] Assuming that the average radial velocities in the various compartments of the table are equal to the corresponding aver- age linear t components of motion, the average parallaxes were computed by equation (43). For comparison, the mean parallaxes assigned by Professor Kapteyn to stars of Secehi's 228 STELLAR MOTIONS Type I and Type II, with reference to the integral stellar magnitudes, are quoted iu the table, from Publications of the Astronomical Laboratory at Groningen, No. 8, page 24. We note that the third-, fourth- and fifth-magnitude stars on our list (a large proportion of stars between 5.0 and 5.5 not being included in the discussion) are not at distances in the order of their magnitudes, the brightest group being appar- ently as far away as the faintest. This, to some extent, is prob- ably a result arising from the limited number of proper motions utilised : 86 for the brighter stars, and 220 for the fainter ; but the apparent mixture -of the different magnitudes in space is more complete than I had expected to find. In general, the average parallaxes are much smaller than those assigned in Kapteyn's parallax tables. The conclusions appear to be : 1. That the stars of these three magnitudes are more thor- oughly mixed than we had supposed. 2. That the brighter stars of Secchi 's Type I are not so much farther away than the brighter stars of Secchi 's Type II as we had supposed. 3. That the scale on which the universe of brighter stars is constructed is apparently a great deal larger than we had supposed. We shall not consider the v components of proper motion at the present time, and in other respects also we shall regard the investigation merely as preliminary, chiefly for the reason that Professor Boss's more accurate and more extensive proper motions will be available in a few weeks. Kobold has published a list^^ of 307 stars whose proper motions are greater than 0".50 per annum. We have obtained spectro- graphic velocities for eighty-eight of these stars. They are arranged in Table XX, in the order of their angular distances, D, from Kapteyn's preferential vertex at a — 93°, 8 = + 12°. The observed velocities are contained in the next to the last column, and the velocities reduced to the stellar system, on the basis of Vo— — 17.77 km. per second, in the last column. 11 Bau des Fixsternsystems, p. 232. TABLE XX Radial Velocities op 88 Stabs with Propeb Motions Gkeater than 0".50, Tabulated with Repebence to Preperential Veetex at u = 93°, 5 = 12° (Corrected for o„ = 272°, 5o = + 27°. 5, F„ -17.77 km.) Kobold No. Star a (1900) 8 (1900) Sp. D Obs'd r Corrected V Z> = 0°- 30° h. m. km. km. 69 A. G. C. 5700 4 55.9 - 5° 52' K 26°. 1 -1- 31. ± -1- 15.4 75 ■Weisse5i'.592 5 26.3 - 3 39 G 19 .4 -1- 7. - 8.8 84 a Can. Maj. 6 40.7 -16 35 A 29 .6 - 7.4 - 24.5 93 a Can. Min. 7 34.1 + 5 29 Fs 21 .3 - 3.5 - 16.7 94 /3 Gemin. 7 39.2 --28 16 K 26 .0 -1- 3.9 - 4.2 X» = 30° - 60° 47 I Persei 3 1.8 4-49 14 G 53 .9 -- 50.5 - 50.1 57 e Eridani 3 28.2 - 9 48 K 46 .2 -- 16.4 - 2.9 58 10 Tami 3 31.8 + 5 Gs 41 .5 -- 29.0 - 16.7 60 d Eridam 3 38.5 -10 6 K 44 .1 - 5.5 - 19.4 62 T6 Eridani 3 42.5 -23 33 Fs 51 .0 -I- 7.3 - 7.9 66 02 Eridani 4 10.7 - 7 49 Gs 36 .1 - 41.6 - 56.4 71 Cord. Zones 5ii.243 5 7.7 -44 58 G-K 58 .8 -1-242. h225.1 72 X Aurigce 5 12.1 -1-40 1 G 31 .0 -- 66.5 - 60.7 80 S Leporis 5 47.0 -20 53 K 33 .5 -- 99.2 - 81.6 97 A. G. C. 10120 7 41.9 -33 59 Fs 50 .7 --105. - 88.5 105 A. G. C. 11070 8 13.7 -12 18 F 38 .8 -- 24. - 9.5 106 A. G. C. 11499 8 29.0 -31 11 G2 54 .2 -- 34. - - 16.9 111 I Ursw Maj. 8 52.4 - -48 26 As 49 .3 -- 8. - 7.2 112 10 TJrsae Maj. 8 54.2 - -42 11 Fs 46 .3 -- 27.3 - 25.0 118 6 Ursce Maj. 9 26.2 - -52 8 Fs 55 .9 -- 15.7 - 16.8 D = 60° - ■ 120° 2 /3 CassiopeicB 3.8 +58 36 Fs 80 .8 -1- 12.8 - 20.2 4 f TuoaiuB 14.9 -65 28 Fs 100 .6 -- 9.3 - 0,8 7 fiJBydri 20.5 -77 49 G 101 .3 -- 22.8 - 13.8 9 A. G. C. 544 32.2 -25 19 K 90 .6 -- 18.2 - 12.5 16 ■n CaasiapeuB 43.0 +57 17 Fs 75 .7 -- 10.0 - 15.9 19 H Casaiopeice 1 1.6 -1-54 26 Gb 73 .1 - 97.4 - 92.6 22 V Plumicia 1 10.6 -46 4 G 88 .7 -- 11.8 -1- 2.2 29 41 S. Androm. 1 35.7 -1-42 7 F 66 .5 -- 4.9 -I- 6.2 31 T Ceti 1 39.4 -16 28 K 73 .1 - 15.7 - 24.4 34 X Eridani 1 52.0 -52 7 Gs 84 .8 - 5.7 - 17.2 36 A. G. C. 2201 2 6.3 -51 19 G 82 .5 -I- 49.5 -1- 37.5 49 a Fomacis 3 7.8 -29 23 F 60 .7 - 20.6 - 34.8 51 fi Beticuli 3 15.6 -62 58 G 82 .3 -- 13.5 -1- 0.3 52 e Eridani 3 15.9 -43 27 Gs 68 .4 -- 87.3 -1- 72.7 53 ft Betieuli 3 16.0 -62 53 Fg 82 ,2 -- 12.2 - 1.0 56 K Betieuli 3 27.6 -63 18 Fs 81 .6 -- 12.1 - 1.3 79 T Mensce 5 45.1 -80 34 Gs 92 .6 -- 12.2 - 0.9 82 A. G. C. 7066 5 53.4 -63 8 K 75 .2 -- 25.4 - - 10.5 130 a Crateris 10 54.9 -17 46 K 75 .9 -- 47.9 — - 41.1 Kobold No. Star o (1900) 5 (1900) Sp. D Obs'd r Correeted V 134 1 TJrsoe Maj. 138 20 Hydn 141 A. G. C. 16103 142 |8 Leonis 143 p Virginia 144 Gromnhr. 1830 155 /3 Can. Ven. 156 7 Virginis 160 S Virginis 166 /3 CanUB 167 61 Virginia 172 i Centauri 177 8 Centauri 178 a Bootis 182 a Centauri 204 XI Herculis 235 X Draamis 245 a Draconis 203 T) Ceplwi 269 61 Cygnipr. 275 7 Pavonis 278 e Indi 281 A. G. C. 29191 283 V Indi 285 iPegasi 286 a Pegasi 289 A. G. C. 31353 296 A. G. C. 31584 297 7 Piscium 300 I Piscium 305 85 Pegasi 306 A. G. C. 32416 197 5 Serpentis 205 7 Serpentis 232 fi Herculis 246 a Aquilce 248 A. G. C. 27380 250 5 Pavonis 254 A. G. C. 27600 256 A. G. C. 27708 259 02 Pavonis 264 A. G. C. 28703 OTO A. G. C. 29191 I» = 60 -120°(Cont.) h. m. km. 11 12.9 +32° 6' G 71° .2 - 14.3 11 29.6 -32 18 G 87 .7 - 24. 11 41.8 -39 57 G 92 .1 + 17.7 11 44.0 - -15 8 A2 80 .2 -- 1.3 11 45.5 - - 2 20 F8 83 .0 -- 4.9 11 47.2 - -38 26 G-K 77 .7 - 97. 12 29.0 - -41 54 G 85 .1 + 6.5 12 36.6 - 54 P 96 .2 - 20.0 12 50.6 + 3 56 Ma 98 .6 - 17.6 13 7.2 +28 23 G 96 .7 + 5.8 13 13.2 -17 45 K 108 .0 - 6.6 13 40.0 -32 33 Fs 114 .9 - 14.6 14 0.8 -35 53 K 118 .9 + 1.8 14 11.1 +19 42 K 112 .8 - 4.7 14 32.8 -60 25 G 117 .4 - 22.8 15 49.2 +42 44 r 116 2 - 55.6 18 22.9 +72 41 Fs 95 .3 + 32.4 19 32.6 +69 29 Gs 97 .3 + 25.5 20 43.3 +61 27 K 100 .8 - 87.0 21 2.4 +38 15 K5 115 .9 — 62. 21 18.2 -65 49 Fg 117 .7 - 31.1 21 55.7 -57 12 Kj 118 .1 - 38.7 22 11.7 -54 7 G 117 .1 - 13.4 22 16.0 -72 44 F 109 .8 + 25.6 22 41.6 +11 40 Fs 109 .0 - 4.6 22 47.4 + 9 18 P 108 .3 + 13.4 22 59.4 -36 26 K 111 .6 + 12. 23 11.9 -14 22 F 107 .3 - 5.5 23 12.0 - - 2 44 K 104 .1 - 13. 23 34.8 - - 5 5 Ps 98 .0 + 6.0 23 56.8 - -26 34 G 88 .0 - 37. 23 59.5 - -37 51 G-K 99 .8 + 23. D = 120° - 150° 15 14 2 15 51 8 17 42 5 19 45 9 19 55 5 19 58 9 20 4 6 20 9 1 20 31 8 20 50 8 21 11 5 D = 150° 180° H 1-2 9 G 133 .6 + 53.8 -15 59 F« 135 .4 -- 7.3 - -27 47 G 139 .7 - 15.6 - - 8 36 A,, 148 .9 - 33. -67 35 G,, 121 .8 - 13.5 -66 26 6,, 122 .7 - 21.7 -36 21 K,, 144 .9 -132. -27 20 G, 148 .5 - 48.5 -60 53 F8 124 .9 - 32.0 -44 29 f 133 .1 - 16. - -39 15 G 132 .0 + 13. km. - 12.2 - 30.5 - - 11.1 - - 2.6 - - 4.3 - 92.1 + 13.9 — 17.7 - 13.6 + 14.0 - 4.7 - 13.9 + 2.9 -- 6.1 - 25.8 - 40.0 -- 45.5 -- 38.8 - 73.6 - 48.0 - 35.1 - 41.9 - 16.5 - - 19.1 - - 2.3 - 19.6 - - 10.0 - 4.6 - 9.4 + 8.5 - 32.8 + 17.6 1- 65.7 - - 22.3 - - 2.1 - 18.0 - 16.6 - 24.5 -126.6 - 41.0 — 33.9 - 14.1 - \- 15.4 217 ( Scarpa 16 43.7 -34 7 K 150 .2 2 2 H h 4.9 ooo A Ophiuchi 17 9.2 -26 27 K 159 .3 + 0.8 \- 10.3 223 A. G. C. 23370 17 10.1 -26 24 G 159 .6 - 8. J h 1.5 224 A. G. C. 23422 17 12.1 -34 53 Ks 153 .3 - 4.5 I h 2.8 233 70 OpMucki 18 0.4 + 2 31 K 165 .2 - 7.4 _ h 8.4 234 77 Serpentis 18 16.1 - 2 55 K 171 .0 + 9.4 - - 24.3 VELOCITIES OF PROPER MOTION STARS 231 TABLE XXI Average Eadial Velocities op the 88 Obseeved Staes with Refeeencb TO Angular Distances feom Peepbeential Vertex D km. km. km. 5 stars 15 " 51 " 11 " 6 " 0°- 30° 30 - 60 60 -120 120 -150 150 -180 +15.40 50.08 16.10 26.38 8.70 -13.55 27.90 29.73 39.24 13.92 45.65 22.25 34.56 8.70 88 " +24.02 -29.63 26.38 11 " 26 " 51 " (0-30) (150-180) (30-60) (120-150) (60-120) 9.66 44.16 16.10 13.55 35.84 29.73 11.07 40.96 22.25 5 stars 9 " 47 " 9 " _6 " 76 '■ 11 " 18 " 47 " Velocities «= 60.0 km. 5 stars 11 " 47 " 9 " 6 " 30 30 60 60 -120 120 -150 150 -180 +15.40 18.14 14.01 13.23 8.70 -13.55 27.90 21.27 24.68 13.92 20.80 17.10 20.88 8.70 78 " +14.02 -21.56 17.21 11 " 20 " 47 " (0-30) (150-180) (30-60) (120-150) (60-120) Velocit 9.66 16.81 14.01 lES < 50.0 kn 13.55 25.76 21.27 a. 11.07 20.84 17.10 30 30 ■ 60 60 -120 120-150 150-180 (0-30) (150-180) (30-60) (120-150) (60-120) +15.40 13.57 14.01 13.23 8.70 +13.20 9.66 13.48 14.01 —13.55 13.65 21.27 24.68 -20.47 13,55 21.92 21.27 13.92 13.59 17.10 20.88 8.70 16.26 11.07 17.23 17.10 The means of all the positive velocities and of all the negative velocities, the numerical mean of the eighty-eight velocities, the numerical mean velocity of seventy-eight stars after rejecting ten velocities greater than ± 60 km. per second, and the numeri- cal mean velocity of seventy-six stars after rejecting twelve velocities greater than ± 50 km. per second, are entered in Table XXI. 232 STELLAR MOTIONS Let us note next the spectral classes of the stars. Nearly all are of Classes F, G and K; only four are of Class A, one of Class M, and none of Class B or Class 0. The numbers of Class A and Class M stars are too small to have any statistical'weight, but the arithmetical mean velocities of the Classes F, G and K stars on the list are much greater than the normal averages foi* these classes as quoted in Table XIV and the accompanying text; even after we remove the velocities in excess of ± 60 -km. or dz 50 km., approximately 15 per cent greater. Inasmuch as these stars have in effect been selected at random from a complete list of large proper-motion stars, it must be that the average inclination of their motion vectors with the line of sight is considerably greater than the normal average value, 57°. 3. This being true, it follows that the average radial velocities of these stars should be less than the normal averages for their spectral classes, provided the space velocities of the stars are of average dimensions. "We have foimd, on the contrary, that the radial velocities of these stars are considerably above normal; and we conclude that the large proper motions of these stars are due not merely to their prox- imity to the solar system but, in addition, to their possession of space velocities greater than the average. Of course all the stars on the list are comparatively near us. We have also tabulated the resulting velocities of the eighty- eight proper-motion stars with reference to their angular dis- tances from Kapteyn's preferential vertex as averages for zones 30° wide. Fifty-one of the eighty-eight stars are situated at dis- tances between 60° and 120° from the preferential vertex, and only thirty-seven in the equal areas lying within 60° of the two antipodal vertices. It was expected that the average velocities corresponding to the zones nearest the vertices would exceed those in the zones midway between the vertices, but this expectation has not been realized. Observing programs for determining the parallaxes of indi- vidual stars have usually been made up of stars with large proper motions, and it has necessarily followed that an undue proportion of individual parallaxes refer to stars which are not VELOCITIES OF PROPER MOTION STARS 233 strictly representative of the stellar system. The resulting par- allaxes may be quite correct ; but if they and the corresponding proper-motion data are used as samples with which to calibrate the distances of stars in general whose proper motions are of certain magnitudes, the results must necessarily contain an appreciable factor of error. In other words, the resulting scale attributed to the stellar universe must be in error. These facts, and the great increase in our estimate of the scale of the uni- verse resulting from the combination of proper-motion and radial velocity data, in Table XIX, make clear some of the difficulties which beset the investigator of stellar distances in the days when he was obliged to depend entirely upon individual parallax and proper-motion data. CHAPTER VII VISUAL AND SPECTROSCOPIC BINARY STARS A study of the composition of the stellar system, based upon the observed motions of the stars, must not overlook the double and multiple stars. These are not sporadic cases in the domain of stellar development; on the contrary, they are so numerous that they must be regarded as direct results of the simplest processes of sidereal evolution. This department of astronomy has had a glorious history. The first discovery of a double star, Mizar, at the bend in the handle of the Big Dipper, was made by Riecioli of Bologna in 1650. Quite by accident, in the prose- cution of other observations, Hooke in 1665 found y Arietis to consist of two stars. Richaud found a Centauri, in 1689, to be composed of a first-magnitude and a second-magnitude star, and Bradley about 1750 noted Castor, y Virginis, /8 Cygni, and 61 Cygni to be double. Apparently the earliest serious search for doubles was that made by Mayer of Mannheim, in 1776, which led to the discovery of thirty-three pairs. The sys- tematic searches of Sir William Herschel, beginning iu the year 1779, gave life to the subject. Strange to relate, the nature of double stars was completely misunderstood by Herschel during most of the years he was engaged in observing them. He thought of them as accidentally doubled: two stars at great distances from each other, and from us, which happen to lie nearly in line with us, and are therefore seen as a close double by projection. In fact, Herschel's first paper on determining the solar motion spoke enthusiastically of the valuable assistance we could soon expect, in solving this problem, from the relative parallactic dis- placement of the two components of the double stars : the nearer star of two stars almost in line would be displaced more than the distant star, and be displaced away from the solar apex. A study of these displacements, he said, would indicate the goal VISUAL DOUBLE STABS 235 of the solar motion.^ It was not until 1803 that he was able to detect a motion of revolution of one star around another — ^to be exact, around their common centre of mass — ^in. the ease of Castor.^ That Herschel's philgsophical mind should have been so slow to grasp the truth about double stars is all the more surprising when we recall that the contents of Michell's papers were known to him. As early as 1767 Michell applied the doc- trine of probabilities to a study of the distribution of the stars over the celestial sphere, and came to the conclusion that the very close proximity of two stars must in general be due to a physical connection existing between them.^ Again, in a paper read before the Royal Society in 1784,* he upheld the proba- bility that some of the great number of double and triple stars which had been observed by Herschel are ia reality systems of bodies revolving about each other, and he expressed the opinion that in the distant future we should be able to determine their revolution periods. If the six thousand stars visible to good eyes were distributed at random over the celestial sphere, that is, if they were physically unrelated, the mathematical probabilities would be exceedingly strong against more than two or three of the six thousand lyiag so nearly in the same directions as two or three others, at the same time, that the unassisted eye could not separate them ; and the probabilities that six or eight of the six thousand stars would form three or four pairs as close as the closer pairs in Herschel's list would be so small as to be prac- tically impossible. Of the seven or eight thousand known double stars whose components are at this moment less than five seconds of arc apart, and whose fainter components go down even to the thirteenth magnitude, it can scarcely be true that more than five pairs in the entire list are doubled merely as a projection effect. With not more than this number of exceptions, all the pairs are probably in mutual revolution around their respective centres of mass. iPM. Trans. (Abridged Ed.), 15, 403-404, 1783. 2 TUl. Trans., 1803, p. 339. ^PUl. Trans. (Abridged Ed.), 12, 428, 1767. iPUl. Trans. (Abridged Ed.), 15, 465, 1784. 236 STELLAR MOTIONS Sir William Herschel's discoveries of double stars, extending over several decades, included about eight hundred of the brighter stars visible from English latitudes. In 1825-1827, Wilhelm Struve, of Dorpat, made a systematic survey of the northern three-fifths of the sky and found 2200 additional pairs. He devoted the following ten years to careful observation of these and other known pairs by means of the micrometer. The results, published in 1837 as the Mensurce Micrometricce, in- clude the position angles, distances, estimated colors, and esti- mated brightness of 2640 double and multiple stars. This vol- ume is the magnum opus of double-star astronomy in the nine- teenth century. In order that the surveys begun by his father should extend over the entire sky. Sir John Herschel sailed with his telescope to the Cape of Good Hope in 1834, where he devoted several years to searching the southern heavens for double stars and nebulae. Following the two Herschels and Struve, other observers studied double stars with increasing success, but none more successfully than Burnham. It was he who first developed the fuU power of modern telescopes in double-star discovery and measurement, and his fruitful researches still continue. The telescopes of today are much larger and in quality better than those of the Herschels and Struve, and the urgent need for a modern resurvey of the whole sky has been recognized for a full generation. In 1899, my colleagues Aitken and Hussey began this double- star survey, with our excellent Alvan Clark telescopes, in accord- ance with most carefully prepared plans.^ Up to June, 1905, when Hussey assumed other duties, he had discovered and measured about 1300 pairs whose combined light in each case is equal to or brighter than that of a 9.1 magnitude star. Aitken 5 To be exact, it should be said that the survey was originated in all its essential features by Aitken early in 1899, and the systematic observations were begun by him in April, 1899 (Astr. Nach., 152, 161, 1900). Coopera- tive observations on the survey were begun by Aitken and Hussey in July, 1899. VISUAL DOUBLE STARS 237 has discovered and measured nearly 2300 close pairs whose com- biaed light is in each case equal to or brighter than that of a 9.0 magnitude star. All previous observers combined had found that one star in thirty-six in the northern two-thirds of the sky, on the average, is double. The survey by Aitken and Hussey, observing the same regions of the sky, shows that one star in about eighteen is a visual double. In other words, for every double star found by former observers, Aitken is adding another in. the same region. In three years from date, Aitken expects to complete his sur- vey of all the stars, down to the niuth magnitude, that can be observed to advantage from Mount Hamilton. This comprises seven-tenths of the sky. The survey of the remaining three- tenths demands a capable observer in the Southern Hemisphere, equipped with a suitable telescope. This great piece of work should be completed as promptly as possible. If carried out along present lines, it is probable that 500 additional pairs will be found in the northern section, from the North Pole of the sky to — 22° declination, and 2000 or more iu the remaining southern skies. It should be said that the mere discovery of a double star, or a thousand double stars, is a fact of little eon- sequence. The principal value lies iu the opportunity for profit- able study of double stars, both individually and as a class, iu order that our knowledge of them may be made ready for use in solving the stiU greater problems of the stellar system. As telescopes increased in size, and especially as they grew in excellence, the definition of a double star changed. The Hersehels catalogued large numbers of pairs whose components were very wide apart. Comparatively few of them have shown change in relative positions, or possess special interest. Wilhelm Struve catalogued no pairs with separations exceeding 32", and Otto Struve set the Umit at 16". Bumham set a narrower limit ; for the fainter stars, at 5". Aitken and Hussey do not record pairs whose combined magnitude is 7, 8 or 9, if the separation is in excess of 5". The reason for this can be made clear. The angular separation of the two components of a double-star sys- tem, as viewed by terrestrial observers, depends upon three 238 STELLAR MOTIONS factors: (1) the true linear distance between the components; (2) the distance of the system from the Earth, and (3) the angle between the liae joining the components and our line of sight. For our present purpose the third factor may be neglected, which amounts to assuming the true dimensions of the orbits to be identical with their projections on the celestial sphere. Let us assume that the apparent distances are the mean dis- tances, and also that the combined mass of each double- star system is twice the mass of the Sun; then, if we take the year, and the Earth's distance from the Sun, as units of time and distance, we may reduce the formula to P' = -^^ (55) in which P is the period of the binary system in years and D is the separation of the two components in astronomical units. From this formula we may compute the roughly approximate revolution periods to be expected for binary systems of difiEerent angular distances and different parallaxes. Thus we derive the following table: Approximate Peeiods op Binaey Staks, with the Abguments Angtilae Separation and Parallax Angular Separation Re 0".l volution Peri 0".05 od for Parallax 0".01 0".005 0*.25 .50 2V2 JTS. 8 8 yrs. 20 90 yrs. 250 250 yrs. 700 1 .00 25 65 700 2,000 2 .00 65 180 2,000 5,600 5 .00 250 700 7,900 22,000 If the combined mass be assumed to be eight times the Sun's mass, the periods quoted would be divided by two. The figures are, of course, given in round numbers only. From the results a a a w c3 3 240 STELLAR MOTIONS in Table XIX we may assume that the parallax of the average double-star system (magnitude 6.5 to 9.0 for the eombiiied components) will be considerably less than a 0".01. Hence it is not to be expected, in general, that a double star fainter than 6.5 magnitude, with distance in excess of 2", will show decisive orbital motion in less than a century. And it is clearly not worth while, at present at any rate, to catalogue systems with distances in excess of 5" unless one component be very bright or the proper motion be large, and the system therefore prob- ably relatively near us {e.g., Sirius, a Centauri or 61 Gygni). However, the naked-eye stars in the northern sky have been so thoroughly examined that it is improbable that any wide pair whose secondary component is reasonably bright remains undis- covered. Hence the limit of distance for present discovery with existing telescopes may well be taken at five seconds for all stars. No doubt this limit will here and there cut out individual stars of special interest, but there should be satisfactory compensation in making the observational data more homogeneous. The chief interest in double stars thus far has been the deter- mination of their periods. This is simplicity itself in the case of such an easy double as a Centauri, our nearest neighbor in the stellar system, whose components of the first and second magnitudes have been followed through nearly three complete revolutions since the system was discovered in 1689. The period is 81.2 years. However, the situation is far different with the great majority of doubles. For example, the components of y Arietis, each of 4.8 visual magnitude, 8".6 apart, have given no evidence of relative motion since Bradley observed them in 1755; yet we cannot doubt that the two stars are physically related, for they have been moving with equal angular speeds, eleven seconds per century, along lines apparently parallel. The linear distances apart of the components must in reality be enormous ; yet the linear separation of the two bodies is undoubt- edly small in comparison with the distances which separate the members of the Ursa Major group of stars, which are known to have essentially equal and parallel motion (Table IX). Only twenty-six known double-star systems have completed at least VISUAL DOUBLE STARS 241 one revolution each since their discovery. This number includes two systems with periods of more than one hundred years, which were discovered by Herschel in the eighteenth century. The double stars Sirius and Procyon are in many respects the most interesting of all known pairs, but chiefly because of the mode of their discovery. Along with Neptune they were near the starting point, so to speak, of the astronomy of the invisible. Bessel noticed as early as 1834 that the proper motion of Sirius is variable, and in 1840 he observed a similar irregularity in the apparent motion of Procyon. Following their motions care- fully, he was able to announce, in 1844, that these irregularities in the motions of the two stars are due to the attractions of in- visible companions, one in each system, the period of revolution in each case being about half a century. He wrote to Humboldt : ' ' I adhere to the conviction that Procyon and Sirius are genuine binary systems, each consisting of a visible and an invisible star. "We have no reason to suppose that luminosity is a necessary property of eosmical bodies. The visibility of numberless stars is no argument against the invisibility" of numberless others.'" The motions of Sirius were investigated by Peters,' in 1851, who found in favor of an unseen companion. An exhaustive investigation of all the observed positions of Sirius, by Auwers,' placed the question beyond doubt, by determining the orbits and relative masses of the bright star and the invisible companion; but before the results were published, Mr. Alvan G. Clark discov- ered the companion, in 1862, near its predicted place. It has all but completed an entire revolution, meanwhile, and in the predicted period of fifty years. The companion is remarkable for its relatively great mass and relatively feeble radiating power. It is one-half as massive as the bright component, but sends out only %oooo as much light. The bright star is only twice as massive as our Sun, but it radiates twenty times as much 6 Except for the presence of the brilliant primaries, the secondaries in the systems of Sirius and Procyon would be easily visible in telescopes of moderate size. — W. W. C. 7 Wolf's GeschicMe der Astron., p. 743, 1877. 8 Astr. NacK, 32, 1-58, 1851. sAstr. Nach., 58, 35, 1862; 129, 185, 1892. 242 STELLAR MOTIONS light. The faint component is as massive as our Sun and radiates only M.500 as much light — a condition which we are powerless at present to explain. A similar investigation of Procyon by Auwers,^" in 1862, assigned a period of forty years to the system. More than half a century following Bessel's prediction, Sehaeberle^^ found the companion, iii 1896, with the Lick telescope. It is of the thir- teenth magnitude and exceedingly difficult to observe even under the best atmospheric conditions, although the distance separating primary and secondary is 5". The companion is usually lost to sight in the glare of the bright body. Here, as in the system of Sirius, the companion is very massive in proportion to its brightness. The application of the Doppler-Fizeau principle to the study of the stars by photographic means has led to the discovery of a different class of stellar systems, known as spectroscopic binaries. This term is applied, in general, to those stars which are apparently single when viewed through our most powerful telescopes, but which the spectrograph has shown to be accom- panied by invisible companions. Just as the variable proper motions of Sirius and Procyon led Bessel to suspect, and Peters and Auwers finally to prove, the existence of invisible com- panions to these stars, so the variations in observed radial motions are attributed to the attractions of invisible companions. The presence of two members in a system implies and demands, as we well know, a mutual revolution of the separate bodies, in elliptical orbits, around their common centre of mass; and, if the orbital plane is not at right angles to the line of sight, there must be radial motions alternately of approach and recession, with reference to the velocity of the system as a whole. The corresponding spectral lines of each body must swing alter- nately toward the red and toward the violet from their mean positions; and if the spectra of both bodies are visible, the two sets of lines must move in opposite directions from their mean positions. 10 Astr. Nach., 58, 35, 1862. 11 Astr. Jour., 17, 37, 1896. SPECTROSCOPIC BINARY STARS 243 Tlie first spectroscopic binary, ^ Ursce, Majoris, was dis- covered by Pickering in 1889. The photographic spectrum of this star shows only a few lines, chiefly those of hydrogen, helium, calcium, etc. It was noted, first, that the calcium K line was alternately double and single. Long-continued observation by Pickering and, later, by Vogel, established that the period of revolution of the two stars, of essentially equal brightness, is 20.5 days. A few months later the Harvard photographs led Miss Maury to the discovery" that j8 Aurigm is a binary system of almost exactly the same sort. Two stars essentially equal in luminosity are in this case revolving around a common centre of mass, in the short period of 3.96 days. Another discovery made in the same year by Vogel,^^ of Potsdam, relates to Spica. The spectrograms of this star yielded velocities varying through about 200 km. per second. Mathe- matical treatment of these velocities proved that Spica is attended by a massive companion, the two bodies, one bright and the other relatively dark, forming a binary system which completes one revolution in a period of 4.01 days. Vogel sus- pected that the spectrum of the fainter component was faintly visible on some of the photographs. Baker has made a thor- ough investigation of the system, securing velocity observations of the fainter component as well as of the brighter.^* He deter- mined the elements P = 4.01416 days a sin i = 6,930,000 km. a, sin i = 11,400,000 km. e = 0.10 m sin' i = 9.6 O wij sin' i = 5.8 O in which the subscript " 1 " refers to the fainter component. We may gain some idea of the dimensions of this interesting system. If we were exactly in the plane of the orbit, we should 12 Mon. Not. B. A. S., 50, 296, 1890. 13 Sitsungsber. der Tcgl. Akad. Wiss. Berlin, p. 401, 1890. i*PuU. Allegheny Ols., 1, 65, 1909. 244 STELLAR MOTIONS of course observe eclipses partial or complete; once in. each period, if the companion star is a dark object, or twice in each period if the companion is fairly luminous iu comparison with the brighter star. This condition does not hold, for eclipses have not been observed. We have reason to believe that the orbit is only slightly inclined to the line of sight because of the high radial velocities observed. "We are therefore fairly safe in assuming that the maximum distance separating the two compo- nents is not greatly in excess of 18,000,000 km. Granting that the parallax of Spica is as great as 0".15, the corresponding maximum separation of the two components would be 0".02. Now we know that the parallax of Spica can scarcely exceed 0".02, for Gill's heliometer value of the parallax is — 0".02 ± 0".01," with reference to comparison stars of magnitude 8.7. It is therefore impossible to hope that the two members of the system, even if of equal brightness, could be seen as an ordinary double star with a telescope of existing forms unless its objective should have a separating power at least thirty- or forty-fold greater than the 36-uich refractor. In successive years additions to the list of spectroscopic binaries were made by Pickering, Vogel, Belopolsky, Miss Maury, Mrs. Fleming, and Bailey, until, in 1898, thirteen spec- troscopic binary systems had been discovered. Since 1898 the number has increased ■\^th great rapidity. A Second Catalogue of Spectroscopic Binaries, to be issued in a few weeks, will con- tain fully 300 entries. These exhibit a great variation in lengths of revolution periods, relative luminosities of components, forms of the orbits, etc. There are a few in which the two components must be almost precisely of the same luminosity and spectral type; a great majority are of those for which the spectrum of but one component records itself upon the plates; and there exists a continuous gradation of luminosity-ratios between the two extremes just described. The proportion of spectroscopic binaries discovered to stars observed is extraordinarily large. The 300 binaries thus far announced are from observation pro- grams which probably total fewer than 1600 stars. "We may be ^5 Ann. Cape Ols., 8 (Part 2), 135 B, 1900. SPECTROSCOPIC BINARY STARS 245 sure, when we look at the stars, that at least one in five, on the average, is attended by a companion, invisible and of mass suffi- ciently great to swing the bright member of the system rapidly around in a large orbit. Every radial velocity observer has a list of stars whose velocities are suspected to vary, but whose binary characters are awaiting confirmation through continued obser- vation. The number of stars on existing lists of suspects which will eventually reveal their binary nature can hardly be less than 100. These would increase the proportion of variable velocities to constant velocities up to one in four. The observed proportion is plainly greater for certain spectral types than for others. Frost has made a specialty of measuring the radial velocities of those stars whose spectra are of the so-called Orion or helium type, known as Class B stars. He has found that one star in two and a half on his program is a spectroscopic binary. His observations of the stars in Boss's Taurus cluster (page 183) have led to the surprising result, as far as the work has been carried, that one star in two, on the average, is a spectroscopic binary." It is certainly a most interesting fact, for a consider- able group of stars, to have the chances equal that any star in the group will be double or single. To bring out certain characteristics of visual and speetro- graphic methods and limitations, let us consider the well-known double star a Gentauri." 16 Ay. J., 29, 237, 1909. IT A critical review of methods employed in the determination of the orbits of spectroscopic binaries is given by Plummer in Ap. J., 28, 212, 1908, with references to the principal original papers on the subject. A very practical paper by Curtis, in Publ. Astron. Soc. Pac, 20, 133, 1908, discusses the points of advantage presented by several of the leading methods of determining spectroscopic binary orbits. Schlesinger has published con- venient and efficient methods of improving the elements of preliminary orbits by the application of the method of least squares, in Fubl. Allegheny Ois., 1, 33, 1908. A brief discussion of the relations existing between the elements of the orbits of visual and spectroscopic binary systems is given by Campbell in Lielc Ohs. Bull., 3, 80, 1905. The elements of spectroscopic orbits as used in the Second Catalogue of Spectroacopic Binary Stars, now in course of preparation and soon to be published in Lich Observatory Bulle- 246 STELLAR MOTIONS tin, Volume VI, are as defined below, including the significance given to the symbol T by the computers in order to meet various conditions. P^apparent period of revolution in mean solar days, unless specified in mean solar years. The true period of revolution, P^, may be found from the apparent period by means of the equation K P„=P/(l + — ). ' ^ 300,000 (The apparent period, P, is more frequently used than Po in dealing with binary systems; but for triple systems, such as that of Polaris, P is variable, and the variations must be taken into account.) T=:Greenwich mean time of periastron passage, expressed in Julian days; except (1) that for variable star orbits the quantity in the column T is the mean solar interval after maximum or minimum brightness, as specified in each case; and (2) that for circular orbits T has been given significance by the computers, as described in the column ' ' Remarks. ' ' fci = angular distance of periastron from the ascending node, measured in the direction of orbital motion. (At the ascending node the radial velocity of the observed body has its maximum value. Spectrographic observations enable the computer to distinguish between the ascending and descending nodes; but micrometer observations of visual double stars do not distinguish between the two nodes, and therefore leave the value of w uncertain by 180°.) e^eccentricity of the orbit. £^::^semi-amplitude of velocity curve of the primary member of the system, with reference to the centre of mass of the system. Z^— semi-amplitude of velocity curve of the secondary member of the system, with reference to the centre of mass of the system. £^-)-2'— semi-amplitude of the velocity curve of one member of the system, with reference to the other member of the system. a^semi-major axis of the orbit of the primary member of the system, with reference to the centre of mass of the system. li =semi-major axis of the orbit of the secondary member of the system, with reference to the centre of mass of the system. a-)-o— semi-major axis of the orbit of one member of the system, with reference to the other member of the system. ^inclination of the orbit plane, conveniently defined as the angle between the line of sight and the normal to the orbit plane. (Spectrographic observations leave the value of i undetermined; micrometer observa- tions of a visual binary system leave the quadrant of i undetermined; spectrographic and micrometer observations combined determine i completely, but the number of known systems available for both classes of observation is at present very limited.) SPECTROSCOPIC BINARY STARS 247 The orbit of a Gentauri as a double star is probably better deter miaed^* than that of any other. The period of revolution, the form of the orbit, and the position of the orbit in one of two alternative but definitely stated planes, are known from the micrometer measurements of the relative positions of the two m^mass of primary member of the system. »ii=mas8 of secondary member of the system. »i-|-mi = mass of the system. O ^maas of the Sim. V ^radial velocity of the centre of mass of the system. F^radial velocity of the primary member of the system at any instant. V ^radial velocity of the secondary member of the system at the same instant. r;=radius of the primary member, r ^radius of the secondary member, demean density of the primary member. d =:mean density of the secondary member, d — mean density of the system. J)o=mean density of the Sun. Certain relations existing between the semi-major axes, the semi-ampli- tudes, the velocities, the masses, the periods of revolution, and the inclina- tions of the orbit planes, are defined in the eight equations which follow. I. One stellar spectrum observed, with comparison spectra: a sin i = [4.13833] (1 - e^) ' KP, (56) = [3.01642-10] (l-«2)^ iT^PQ. (57) mi" sm" I (m-|-»ii)' II. Two stellar spectra observed, either without comparison spectra, as by means of objective-prism spectrographs, or with comparison spectra: (a + di) sin i = [4.13833] (1 - 62)^ (K+Ki) P, (58) (m + mi) sis? I - [3.01642 - 10] (1 - e2) ' (K-\- KiY P © . (59) III. Two stellar spectra observed, with comparison spectra: m? sin' i a = [3.01642-10] (l-e2) 5 iTiSPO, (60) (m-|-mi)2 m sin3 i = [3.01642 - 10] (1 - e^)^ (JT-f JS:i)2 J"iP ©, (61) mi sin3 i = [3.01642 - 10] (1 - e^) * (K+ K^f KP ©, (62) m Ki (Vi-r„) mi K (V-Vo) isEoberts, Astr. Nach., 133, 106, 1893. (63) 248 STELLAR MOTIONS companions. Measurements of the varying distances of the two companions from neighboring stars have determined the relative sizes of the orbits, and therefore the relative masses of the components^** as 51 to 49. Grill's and Elkin's heliometer deter- minations"" of the parallax of the system give ir — 0".15 ± 0".01. This paraUax is of course relative to the comparison stars, of average magnitude 7.6. Making the usual allowance for the paraUax of a 7.6 magnitude star, the absolute parallax of a Centauri becomes 0".76. From these observational results we know the distance of the system to be 4.4 light years, the linear size of the orbit and therefore the masses of the two bodies. The only remaining orbital uncertainty is as to which of the two orbital planes is correct. Eadial velocity observations of the two components by the D. 0. Mills Expedition have enabled my colleague "Wright to decide between the two planes, keeping one and rejecting the other, and to make a valuable independent determination^^ of the parallax. These measures give a velocity of approach for the system amounting to 22.2 km. per second, and at the epoch 1904.8 a differential velocity of 5.30 km. for the two stars. Now it is apparent that this differential velocity is a direct function of the linear size of the orbit; and as the angular dimensions of the orbit are known from the ordinary double-star observations, it rema,ined only to deduce that value of the distance or parallax which will harmonize the linear and angular dimensions of the system. The speetrographic parallax came out 0".76 ± 0".02, agreeing perfectly with the Cape heliometer parallax. The linear dimensions of the orbit having been determined in this way, and the relative masses being known, it is a simple matter to determine that the combined mass of the two stars is 1.9 O." In such favorable eases as a Centauri the speetrographic method of determining parallax possesses striking advantages over the heliometer method. The degree of accuracy is inde- 19 Roberts, Astr. Nach., 139, 10, 1895; see also Table XXX. 20 Ann. Cape Ols., 8 (Part 2), 135B, 1900. 21 Liclc Obs. Bull, 3, 4, 1904. 22 Lich Ols. Bull., 3, 4, 1904. SPECTROSCOPIC BINARY STABS 249 pendent of the distance of the star, except as this factor influ- ences the apparent brightness of the two components and the micrometer determination of the angular dimensions of the orbit. No assumption as to the relative distance of primary stars and comparison stars is involved. The absolute parallax is deter- mined directly. The differential radial velocity is large in com- parison with the observed radial velocity; in this case about 25 per cent. With the heliometer, large angular distances are usually measured, and they involve corresponding uncertainties as to refraction and other similar corrections. A series of twenty-five spectrograms would apparently reduce the probable error of the resulting parallax to equality with that of the heliometer value. However, it must be made clear that the spectrographic method of determining distances of individual stars is applicable to only a few known systems, in which the velocities of both components are measurable and the visual orbits are well determined ; unless, in systems with well-defined orbits, we wait for the bright members to pass over large parts of their orbits. To illustrate, the orbit of Sirius as a double star is extremely well determined, but the velocity of the companion cannot be measured spectrographically. The spectrograph has decided between the two possible orbit planes, accepting the positivie value of the inclination^^ and rejecting the negative, and has given us a fair value of the radial velocity of the system, — 7.4 km. per second; but we have not sufficient basis for making a spectrographic determination of the parallax of Sirius, except by waiting for a large variation of velocities to occur. The spectrograph has made important contributions to our knowledge of Castor as a stellar system, this being the double star in which Herschel first detected orbital motion. Fourteen years ago Belopolsky, of Pulkowa, fovmd that the fainter com- ponent of Castor is a spectroscopic binary, period 2.93 days, and orbit nearly circular.^* My colleague Curtis found five years ago that the brighter component of Castor is also a spectroscopic 23 LicTc Ohs. Bull., 3, 83, 1905. 24 Mem. Acad. St. Petershourg, 11, No. 4, 83, 1900. 250 STELLAR MOTIONS binary, period 9.22 days, and eccentricity 0.50.^^ Castor is thus a quadruple system. Curtis has determiaed the radial velocities of the centres of mass of the two binary members, and finds a difference in their radial speeds amounting to 7 km. per second. Unfortunately, the visual orbit of Castor is completely unknown. Numerous published orbits assign periods varying from 232 up to 1001 years. All we can now say is that the period must be several hundred years in length. Not knowing the period and the other elements of the orbit, the spectroscope cannot at present determine the parallax of this system. We have been interested in measuring the radial velocities of Procyon from time to time since 1896 ; that is, through one-third of the period of the visual companion. The radial velocity appears not to have varied systematically in this interval, though there is perhaps a secondary variation amounting to only li^ km. per second, in a period of seven years; thus Auwers's con- clusion is probably correct, to the effect that the orbital plane of the visual Procyon system is tangent to the celestial sphere. The small secondary variation is not certainly established, on account of its smallness and the fact that it has but recently been suspected. Spectrographic binary stars present a great variety of orbital elements. There is time for only the briefest reference to the more interesting individual features. The binary of shortest known period is yS Gephei, period only 4 hours 34.2 minutes, discovered by Prost.^° The orbit is nearly circular and there is a total variation of 34 km. per second in radial velocity. Another interesting short-period system is that of j8 Ganis Majoris, for which my colleague Albrecht has determined a period of almost exactly six hours.^' In this system also the orbit is nearly circular. Figure 4, Chapter III, reproduces the velocity curve for r] Pegasi, a solar-type binary, the spectrum of only one component being visible; period 2^ years, eccentricity 0.15. 25 Liclc Ols. Bull, 4, 64, 1906. 26 Ap. J., 24, 259, 1906. 27 Lick Ohs. Bull, 6, 22-23, 1910. SPECTROSCOPIC BINARY STARS 251 We have recently found that u Ursce Majoris, the first star in the Big Dipper, has a variable velocity.^' Observations extending from 1896 to the present time have been needed in order to show that the velocity is certainly variable. So slow is the change and so long the period that it is impossible to predict the length of period at present, but it may iaelude several decades, and perhaps several scores of years. The visual com- panion discovered by Burnham may be involved. The Mills spectrograph determined^" in 1899 that the first- magnitude star Capella is a binary through the fact that it has a composite spectrum, consisting of two sets of spectral lines which shift alternately back and forth over each other. The binary character of Capella was also discovered three months later, quite independently, by Professor Newall,^" at Cambridge. The period is 104.2 days. The system as a whole is receding from us at the rate of 30 km. a second. The orbit is nearly circular, eccentricity equal to 0.02. The more massive star varies in radial velocity from + 4 to +56 km. and the lesser from + 63 to — 3 km. We thus determine that the masses of the two bodies are as 1.26 to 1. The brighter star, of solar type, is a half magnitude more brilliant photographically than the fainter star, of Sirian type. This is equivalent to the solar-type component being a full visual magnitude brighter than the companion. The position of the orbital plane of Capella is totally unknown. If we were exactly in the plane, we should expect to observe an eclipse every fifty-two days. In this case the maximum distance between the two stars would be 83,000,000 km., a little more than one-half the distance separating Earth and Sun. This dis- tance is the value of (a + aj sin i. Blkin's value of the parallax, 0".08, would give in this case an angular separa- tion of the two companions amounting to 0".045. As no eclipses have been observed, we conclude that we are not in the orbital plane; but the high velocities observed lead us to believe that 28 Liclc Ols. Bull, 5, 174, 1910. 29 Ap. J., 10, 177, 1899; LicJc Ohs. Bull, 1, 34, 1901. 30 Mon. Not. B. A. S., 60, 2, 1899. 252 STELLAR MOTIONS the plane does not pass far to one side or the other of our posi- tion in space. If the inclination of the orbital plane to the sur- face of the celestial sphere were 30°, the corresponding value of a+tti would be doubled, and the maximum separation would be 0".09. Of course, if the orbital plane were nearly tangent to the celestial sphere, the dimensions of the orbit would have to be very great in order to give the observed radial velocities. In that case the observation of the spectroscopic binary also as a visual binary should have been accomplished before this, for the star has been most carefully examined. I should say that ten observers at Greenwich, usiag a 28-inch telescope, made a series of observations'^ of Capella as a double star shortly fol- lowing the announcement of its spectroscopic binary character, the visual observations seeming to be fairly easy to secure, inasmuch as they were made when Capella was from 3 hours to 6y2 hours from the meridian, and in part by observers inex- perienced with double stars. However, the series has been dis- continued, and there is little doubt that the observers were misled by elongated images, possibly due either to the lack of adjust- ment of the object glass of the telescope or to the refraction of the star's light in our atmosphere. Perhaps as interesting a spectrographic system as we have thus far found is that of the North Pole star, Polaris. This system appeals to me no doubt more than to others, for Polaris was the first star observed with the Mills spectrograph — in 1896. The velocity in that year was thought to be constant, as the obser- vations, compared with one another, were suprisingly consistent, yielding a probable error of but ^4 ^^- for ^ single result. The observed velocities of that year were published as of constant character. There were six of them, with values about — 19 km. and — 20 km., as plotted near the bottom of Figure 11. Re- observing the star in 1899 to test the adjustments of the appa- ratus, a quite different velocity was obtained. Following up the indications of variable velocity, by means of an extensive series of observations, we found'^ that in Polaris we have a triple 31 Mon. Not. B. A. S., 60, 595, 1900. 32 ^p. J., 10, 180, 1899; LicTc Ohs. Bull, 6, 18-19, 1910. SPECTROSCOPIC BINARY STARS 253 system, composed of one star bright enough to impress its spectrum upon the plate in a few minutes, and of two invisible companions which leave no traces whatever upon the spectro- grams. The salient features of the system are illustrated by means of the velocity curves in Figure 11. It was found that the velocities observed in 1899 repeated themselves in a cycle of 3.9681 days, the variation in velocity amounting to about 6 km. The velocity of the centre of mass of the binary system in 1899, d 2 1 1 1 4 6 8 1 1 1 1 1 lOd 1 1 1 o 8 ' Tmn, r \ r\ /* 0^ -10 - f \ f \ / \ / r^\ \ / *'**» \ / y'N / • / ^ V ' \ \ Y- - -f-/* — \-V— _ >>i9— -L .A- - A -\ —J-L^\- / ''' ^'^ A // ^^A 1 1 z -■i r_ii 1. _\ lis' fil ^ Y -f-f" 2 o / / \ \.° y 1 ^ V° -15 - V o / ' ! r\ \ ' 1 ' \ / / / * \ ' 1 > 4 / / > / \ / V f' / \ *-' ^ / \ /• \ /• X .^____.J,. jrj,___I_____\ _ --/---^-- -20 - / » 1 • /• * • A / / / / ^ J » / ^_^ • ^---^ V^^ 1 1 1 1 1 1 1 1 1 1 1 Figure 11. — Some velocity curves of Polaris represented by the straight line YY, amoimted to — 11.5 km. per second. When the well-determined curve of 1899 was moved down to coincide with the plotted observations of 1896, it was found that the early observations were well represented by a curve of the same form. It had happened that a seventh spectro- gram secured in 1896 had been seriously damaged in the dark- room manipulation, the film having been badly torn, and this plate had not been measured. It was measured in 1899, with 254 STELLAR MOTIONS result y = — 17 km., and found to conform to the 1896 curve. The velocity of the centre of mass of the binary system in 1896, represented by the line XX, was — 17.9 km. per second. The velocity curve of the binary system in 1901 occupied a still dif- ferent position, with systemic velocity represented by ZZ, — 13.4 km. per second. Series of observations in succeeding years have shown that the systemic velocity has been approach- ing that of 1896, and that the period of revolution of the binary system around the centre of mass of itself and a third member of the system appears to be in the vicinity of twelve years. Two neighboring binary systems in the constellation of Orion, discovered at the Yerkes Observatory, have recently been shown by Plaskett and Harper {Ap. J., 30, 373, 1909) to possess orbits remarkable for the similarity of their elements, and especially for their high eccentricities, as may be seen from the following tabulation. ( ORio>as B. D. - 1°.1004 Eight Ascension St 30"" 5''36n' Declination -5° 59' -1° 11' Spectrum Oe5 B3 Period 29.136 days 27.160 days Eccentricity 0.74 0.76 Velocity Variation 227 km. 186 km Long, of Periastron 112°.4 87°.0 Velocity of System +21.5 km. +26.1 km. a sin i 28,907,000 km. 22,380,000 km. How serious a complication is injected into the general prob- lems of stellar radial velocities by the discovery of so many spectroscopic binary systems is well illustrated by the stars in the Big Dipper. We referred above to Alpha as a long period binary. LudendorfP' has announced Beta to be a spectroscopic binary, with period a little over 27 days, though the Mount Hamilton observations do not seem to be confirmatory. Adams'* has suggested that Epsilon may have variable velocity, and 33A8tr. Nach., 177, 235, 1908; 180, 271, 1909. siAp. J., 18, 68, 1903. SPECTROSCOPIC BINARY STARS 255 Ludendorffi suspects^' that it varies in a period of two years, more or less. We have noted that Zeta was the first spectro- scopic binary system to be discovered by Pickering, period 20.5 days. Zeta and its near neighbor, Alcor, form an exceedingly interesting multiple group. It is well known that Zeta is a visual double star, magnitudes two and four, with angular separation about 14", which was discovered by Sir William Herschel. The brighter component is the Pickering spectroscopic binary. The fourth-magnitude component was discovered to be a spectro- scopic binary by Frost and Lee,'" and it was independently sus- pected of variation by Ludendorff.'^ The near neighbor, Alcor, was found by Frost to be a spectroscopic binary.'* Thus the spectrograph has established that the three stars visible in the telescope are in reality six stars. Of the other bright stars in the Big Dipper — Gamma, Delta, and Eta — accurate observations have not been secured, as their spectral lines are very poorly defined ; but it would not be surprising to find that some of them are travelling through space accompanied by massive invisible companions. The number of spectroscopic binary systems discovered to date is more than three hundred, but the orbits have been deter- mined, by various observatories and astronomers, for only sixty- five of the systems, though the approximate periods are known for many additional systems. A study of the relations existing between spectral classes, lengths of periods, eccentricities, and other constants is an engrossing occupation. My colleagues and I have been noticing, for several years, that for the binaries of early spectral classes there is a tendency toward short periods and orbits nearly circular ; and for binaries of the older spectral classes, a tendency toward periods relatively long and orbits of considerable eccentricity.'" Thanks to the shortness of period 35 Astr. Nach., 180, 273, 1909. seAstr. Nach., 177, 173, 1908. 37 Astr. Nach., 177, 9, 1908. 38 Astr. Nach., 177, 172, 1908. 89 Professor See has recently called attention to the fact that short period spectroscopic binaries have small eccentricities, and longer period binaries have larger eccentricities, in Mon. Not. E. A. S., 68, 201, 1908. TABLE XXn Spectroscopic Binaries op Classes O and B H. E. Star Class Period g a sin i mi^ sin.^ i Til sin^ i 7ti\ sin^ i wi : m km. (m + mi)2 ''V tJAJA V 779 1088 1347 2781 4118 4662 S Ceti Ts Eridani vi Jiridani 29 Can. Maj. S Antlice 7 Coiri B2 B8 B9 Oe B9 B8 short* short short short short short 6027 V Scorpii B3 short 7248 Y Aquilce B8 short 8238 p Cephei Bl Od.ig small 45,000 0.0001© 2294 |8 Can. Maj. Bl .25 0.1± 33,510 0.00002 1567 ITS Orionis B3 .87 3129 YPitppis Blp 1 .45 (0.) 6,100,000 17.10 17. lO 1.0 6247 HI Scwpii B3p 1 .45 4,586,000 7.2 7.2 1.0 5944 w Scorpii B2p 1 .57 6431 u Herculk B3 2 .05 0.05 2,800,000 0.205 6.8 2.6 0.39 6431i u Serculis 2 .05 0.05 7,120,000 3.43 ISll ^ Ononis B2 2 .53 0.06 4,995,100 0.780 5.53 4.19 0.76 ISlli ^ Ononis 2 .53 0.06 6,570,000 0.343 8523 2 Lacertce B5 2 .62 0.02 2,890,000 0.141 0.87 0.71 0.82 8523i 2 Lacertce 2 .62 0.02 3,550,000 0.026 8001 57 Cygni B3 2 .85 0.96 936 Algol B8 2 .87 0.04 1,600,000 0.020 0.44 0.22 0.50 1239 X Tauri B3 3 .95 0. 3,300,000 0.089 5056 a Viri/inis B2 4 .01 0.10 6,930,000 0.833 9.6 5.8 0.60 5056i a. Vircjinis 4 .01 0.10 11,400,000 3.68 226 r Androm. B3 4 .28 0. 4,299,000 0.173 1131 Persei Bl 4 .42 0. 7,172,000 0.754 6527 \ Scorpii B2 5 .6 1852 5 Ononis B 5 .73 0.10 7,906,600 0.601 8926 BD +57°. 2748 B3 6 .07 1542 9 Camelopard. B 6 .33 3623 K Cancri B8 6 .39 0.15 5,890,000 0.200 3659 a Carince B3 6 .74 0.18 1,960,000 0.007 5984 P Scoipii Bl 6 .9 1788 1) Ononis Bl 7 .99 0.02 15,901,000 2.51 11.2 10.6 0.95 17S8i 7) Orionis 7 .99 0.02 16,750,000 2.15 5231 f Centauri B2p 8 .02 1552 TTi Orionis B3 9 .52 0.03 3,393,000 0.017 7790 a Pavonis B3 11 .75 0.01 1,170,000 0.0005 7106 p lAjrw B2p 12 .91 0.07 33,000,000 9.7 9.6 20.9 2.2 7106i p Lyrce 12 .91 0.07 15,800,000 1.0 1713 ^ Orionis B8p 21 .90 0.30 1,108,900 0.0001 2653 02 Can. Maj. B5p 24 .3± 1,670,000± 0.0003± 1952 BD-1°.1004 B3 27 .16 0.76 22,380,000 0.607 1899 I Orionis Oe5 29 .14 0.75 28,907,000 1.14 6084 a Scorpii Bl 30 .± 5190 V Centauri B2 31 .± 3734 K Velorum B3 116 .65 0.19 73,200,000 1.15 496 Persei Bp 126 .5 0.43 42,298,000 0.189 2159 V Orionis B2 131 .4 1910 ^ Tauri B3 138 0.18 27,900,000 0.046 154 T Androm. B3 143 .72 0.58 76,790,000 0.876 6812 /J. Sagittarii B8p 180 .2 0.44 143,500,000 3.63 936i Algol B8 ly.90 0. 89,000,000 0.060 * The periods marked "short" are unknown. SPECTROSCOPIC BINARY STARS 257 TABLE XXin Spectroscopic Binaeibs op Class A H. R. Star Class Period e a sin ( km. mi^ sin^ i m sin^ i TOi sin^ i mi : m (m + mi)2 104 BD4-43°.92 A2 short* 897 $1 Eridani A2 short 1380 64 TauH A2 short 5971 I Cor. Bar. A short 2124 H Orionis A2 O'i.77 815 KZ Cas»iop. A 1 .20 1,170,000 0.05±O 0.65O 0.36© 0.55 1497 T Tauri A 1 .50 0.08 916,130 0.0136 5586 S Librw V Cephei A A 2 .33 2 .49 0.05 2,450,000 0.110 0.8 0.6 0.7 2890 ai Gemin. A 2 .93 0.01 1,279,000 0.0097 7326 U Sagittce A 3 .38 1568 7 Camelopard. A2 3 .88 2088 (3 AurigcB Ap 3 .96 0.00 6,000,000 2.16 2.16 1.0 2088i ^ Aurigoe Ap 3 .96 0.00 6,000,000 6324 6 Hei-culis A 4 .02 0.07 2,900,000 0.061 0.37 0.25 0.7 6324i e Eei'cuUs 4 .02 0.07 4,286,000 0.194 2891 a.2 Gemin. A 9 .22 0.50 1,485,000 0.0015 4072 BD +66° 664 A 11 .6± large 7710 6 Aquilw A 17 .11 0.69 7,665,000 0.061 31 0,28 0.89 7710i 6 Aquilw 17 .11 0.69 8,610,000 0.087 5793 a Cor. Bor. A 17 .36 0.33 7,600,000 0.059 5054 ft Vrs. MnJ. Ap 20 .54 0.50 17,500,000 3.96 2.0 2.0 1.0 5054i h Vi-x. Maj. Ap 20 .54 0.50 17,500,000 8178 P Equidei A 22 .7± 7178 7 Lyrw A 25 .6± 4295 /3 Urx. Maj. A 27 .16 0.79 1,774,000 0.0003 622 /3 Ti-iaii. A5 37 .± 5291 a Draconis A 51 .38 0.40 30,000,000 0.42 4689 ri Tii-fiinis A 71 .9 0.33 25,500,000 0.126 15 a Aiidrom. A 96 .7 0.51 33,000,000 0.155 ' 553 P Arictis A5 107 .0 0.88 22,880,000 0.042 0.17 0.17 1 2421 7 Gemin. A 3y.5± large 2491 Siriits A 49y.3 0.59 *The periods marked "short" are unknown. in most spectroscopic binary systems, and to the energy and skill of spectrographic observers, the time has come to study in greater detail and more accurately the apparent relationships existing between spectral classes, periods of revolution, and eccentricities. The available data on spectroscopic binary orbits are contained in Tables XXII, XXIII, XXIV and XXV. (These tables have been extended, after the delivery of this 258 STELLAR MOTIONS lecture, to include all results available up to March 15, 1910, the epoch of the Second Catalogue of Spectroscopic Binary Stars.) The tabulations are self-explanatory, and it remains only to draw from them certain evident and simple conclusions of apparently great importance. TABLE XXIV Specteoscopic Binaries of Class F (Cepheid variables not included) H. B. Star Class Period a sin i mi' sin^ i in sin^ I JHi sin'^ / W!i ; wi ^^ j-MJOk:? km. (m + »«i)2 2788 R Call. Maj. F l'i.14 142 13 Cefi F 2 .08 0.06 981,460 0.0087© 5986 Draconw F8 3 .07 0.01 990,000 0.0041 424 a Urs. Min. Z Herculis F8 F 3 .97 3 .99 0.13 164,500 0.00001 7056 iilAjrw F 4 .30 0.00 3,030,000 0.060 6596 u DrOAxmis F5 5 .28 0.01 2,632,000 0.0261 8315 K PegaM F5 6 .± 8430 i Pegad F5 10 .21 0.01 6,740,000 0.117 3852 Leonis F5p 14 .50 <0.02 10,775,000 0.238 1.30© 1.12© 0.86 3852i LeoiiiH F± 14 .50 <0.02 12,571,000 0.378 2693 S Can. Maj. F8p ^yrs.± 5,650,000 0.0001 6927 X Ih-amnis F8 281d.8 0.42 63,000,000 0.126 8123 5 Equulei F5 5y.7 0.36 424i a Urs. Min. F8 lly.9 0.35 166,800,000 0.0098 3482 £ Hj/drce F8 155-. 7 0.6 550,000,000 0.18 3185 I Argus F5 long* * The period marked "long" is unknown. Table XXII contains the orbital data for those spectroscopic binaries of Classes and B (stars thought to be effectively young) whose periods of revolution have been fairly well deter- mined. These binaries are arranged in the order of the length of period. Except in the case of the last star on the list, which refers, apparently with considerable uncertainty, to the centre of mass of the binary system ot Algol in revolution around the centre of mass of itself and a suspected third body, the periods are all under one-half year ; and two-thirds of them are under ten days. An inspection of column e shows that the eccentricities SPECTROSCOPIC BINARY STARS 259 are, in general, small for the short periods and relatively large for the longer periods ; that is, that the orbits are more nearly circular for the short-period binaries than for those whose periods are relatively long. TABLE XXV Specteoscopio Binaries op Classes G to M (Cepheid variables not included) H. E. Star Class Period e a sin i km. mi' sin' i m sin' i mi sin'i mi : m (m + mi)2 8961 X Androm. K 20^.54 0.11 1,900,000 0.0006© 1708 Capella G 104 .02 0.02 36,847,900 0.184 1.19© 0.94© 0.79 1708i Capella F+ 104 .02 0.02 46,430,000 0.369 429 7 Pluenic. K5 190 .± 6148 j3 HerculU K 410 .58 0.55 60,280,000 0.052 5235 7) BooUs G 489 .14 0.18 56,010,000 0.0293 4375 1 Ura. Maj. G It. 8 8650 ■q Pegasi G 2y.24 0.16 157,800,000 0.234 7776 j3 Capric. Gp 3y.77 0.44 377,000,000 1.13 6134 a Scorpii Map 5y.8 0.20 60,490,000 0.0020 6212 iBermlis G 34y. 0.46 5459 a\ Centauri G 81y.l8 0.53 5460 02 Centauri K5 81y.l8 0.53 6752 70 Ophiuchi K 87y.86 0.50 99 a Phcmic. K long* 539 ^Persei K long 549 1 Pisdum K long 854 T Peraei Gp long 915 7 Peraei Gp long 1066 fTaun K long 2216 1) Gemin. Ma long 2296 S Colum. G5 long 2553 T Puppia K long 2854 7 Can. Min. K long 4301 a Ura. Maj. K long 8079 1 Cygni K5 long 8115 } Cygni K long * The periods marked "long" are unknown. Table XXIII contains corresponding facts for the Class A stars ; that is, those stars which are believed to be further along in their processes of development than the and B types. The last star on the list, Sirius, is the well-known visual binary ; and some of the elements of the visual orbit are included here, for the 260 STELLAR MOTIONS reason that the spectrograph has measured the velocity of the brighter component in the system and found it to vary in accord- ance with the requirements of the visual orbit. In this list, also, the eccentricity is seen to be, in general, a function of the period of revolution. Table XXIV contains corresponding data for Class F stars, supposedly of greater effective age. Again the short periods have orbits of small eccentricity, and the long periods have orbits of great eccentricity. The Cepheid-Geminid variables, many of which have Class F spectra, are not included in this tabulation, for these variables appear to belong in a class by themselves. Table XXV contains orbital data for stellar systems of Classes G, K and M, which are believed to represent advanced stellar age. The eccentricities are again a function, in general, of the length of period. This table also does not include the Cepheid- Geminid variables, of Classes G, K and M, for the reason stated in the preceding paragraph. TABLE XXVI Spectroscopic Binaries.— General Summary Periods ' 'Short" Od-5'J 5'i-lO'i 10 = 1. The average value of sin' i, however, would not be sin' 57°. 3 (^.65), but approximately 0.59, in accordance with the formula TT TV ^ ? /"2 /'2 sin^ idid=—T = 0.59. sm ( It would not be permissible to use this as the average value of sin' i for the observed systems, as there is the practical considera- tion that binary systems whose orbital planes have large inclina- tions are more readily discoverable than those whose inclinations are small. For example, the maximum variation of radial velocity in an orbit whose inclination is 5° will be but one-twelfth the corresponding variation of velocity in the orbit. It is clear, therefore, that many systems whose orbital inclinations are less SPECTROSCOPIC BINARY STARS 263 than 10°, and even less than 20°, will escape detection, in the present state of speetrographic observation. The number of inclinations less than 10° would, however, be relatively small, for the same reason that the area of the surface of a sphere lying within 10° of one of its poles is small in comparison with the area 4250 4383 Class Fo. I Pegasi 1. 1899, October 3. 2. " " 17. 3. " " 25. Measured displacement, —39 km. " " +53 km. " " +1 km- of a hemisphere. Under ordinary circumstances, and when deal- ing with a considerable number of orbits, a compromise value of sin° i = 0.65 might in fairness be applied. We should use a still larger value in dealing with the eighteen eases before us, for six of the stars are Algol or /8 Ltjrm variables whose orbital inclina- tions are in every case probably between 60° and 90°. Assuming 264 STELLAR MOTIONS sin^ i = 0.75 for the present list, we find the average mass of the eighteen principal components to be 5.6 times the Sun's mass, and of the eighteen secondary components 5.7 times the Sun's mass. Excluding j8 Lyrce, on account of very considerable un- certainty in the interpretation of its spectrum, these values reduce to 5.1 and 4.4. The Classes and B systems appear to be the more massive, but the data are too meagre to venture this as a conclusion. "With only one exception the principal (brighter) component is more massive than its secondary. The exception relates to j8 Lyne, in which system the fainter member is accredited with a mass 2.2 times that of the brighter member. The two members are believed to be nearly in contact, and Myers has suggested*" that the greater brightness of the less massive companion may be due to the influence of a heavy absorbing atmosphere which completely encloses both members of the system; the more mas- sive component drawing this atmosphere more densely around itself, leaving the atmosphere overlying the less massive com- ponent thinner and less absorptive, thereby permitting "the smaller (less massive) to appear brighter even though it might be intrinsically darker than the larger (more massive) body." There arises the question whether the masses deduced for the seventeen systems involved are representative of the masses in spectroscopic binary stars in general, discovered and undis- covered. The data are too meagre to answer this question satis- factorily. The spectroscopic binary systems thus far detected and investigated are on the whole those most readily discover- able. They are of stars brighter than the average, and there- fore of masses above the average. They are systems in which the range of radial velocity is large, and systems whose periods of revolution are short. Systems with small velocity amplitudes and systems with long periods remain undiscovered in propor- tions unduly large. It will be seen from equations (61), (62) and (59) that, in the systems to be discovered in the future, the smaller values of K will decrease the corresponding values of m, nil S-iid m+wii, and the larger values of P will increase the 40 Ap. J., 7, 21-22, 1898. SPECTROSCOPIC BINARY STARS 265 corresponding values of m, m^ and m + m^. The prevailing ten- dencies of the K's and P's in future systems will therefore counterbalance one another in these equations, to some extent, though attention should be called to the fact that the K's enter as cubes and the P's enter to the first power. In the great majority of systems thus far investigated we have orbital elements for the brighter components only. There is a relationship expressed in equation (57) which we have computed and entered in the first four tables; namely, m/sin'^/(w^ + wii)^- These quantities convey little definite information. They range from 9.7, in the ease of p Lyrce, down to .00001 for Polaris. If we assume that the average value of sin^ i which has entered into these quantities is 0.65, we shall see that the secondary members of the systems are in general of considerably smaller mass than their primaries ; and this would appear to be a reliable conclusion, though there are a few prob- able exceptions. The larger values of this quantity, as may be seen from the tables, seem not to show preference for any spectral type or types. Dr. Aitken has prepared a list of fifty doubles, arranged in the order of their periods, according to the knowledge of today as in Table XXVII. These periods vary from 5.7 years for S Uquulei, discovered by my colleague Hussey, up to 194 ± years for y Virgin is. Other columns of the table give the computed eccentricities of the orbits, the spectral classes, the mean angular distances of the components, and the visual magnitudes of the components. The number of the pair in Bumham's General Catalogue of Double Stars is given in the first column. Up to date (1910), orbits have been published for 102 binary systems with periods ranging up to 1578 years. At least half of these are too uncertain to be of value, and in only about forty cases have we a fairly exact knowledge of the true orbits. Several of the assigned periods and eccentricities in the tabular list of fifty will undoubtedly be changed as a result of future observations, but it is not expected that a radical change will be required in the elements assigned to the first forty stars on the list. 266 STELLAR MOTIONS TABLE XXVn Visual Double Stars* Bumham's No. Star Period e Spectrum a Vis. Mag's. 10829 S EquuM, 5y .70 0.36 P5 Qii.'i^ 4.5- 5.0 314 13 Ceti 7 .42 0.74 F .21 5.5- 6.2 11222 K Pegam 11 .37 0.40 P5 .29 4.5- 5.3 4771 e Hydrm 15 .70 0.68 P8 .24 4.0- 6.0 2381 ^883 16 .35 0.48 ? .24 7.0- 7.0 8965 f Sagittarii 21 .17 0.18 A2 .56 3.4- 3.6 6578 /3 612 22 .8 0.48 ' A .24 6.4- 6 5 4310 9 Argus 23 .3 0.68 P8 .61 5.7- 6.3 335 /3 395 24 .00 0.15 K .66 6.3- 6.4 6406 42 Co»i«e 25 .56 0.46 P5 .64 6.0- 6.0 12701 85 Pegam 25 .70 0.43 G .78 5.8-11.0 10363 /3 Delphini 27 .66 0.36 P5 .54 4.6- 5.0 5005 S3121 34 .00 0.33 P? .67 7.2- 7.5 7717 ^ Herculin 34 .53 0.46 G 1 .36 3.1- 6.5 1471 20 Persei 36 0.75 P .16 5 6-64 4414 (3 581 41 .2 0.53 ? .61 8.0- 8 7251 ri Cm: Bor. 41 .51 0.28 G .89 5.5- 6 7487 1 Scorpii 44 .5 0.77 P8 .70 5.0- 5.2 8162 lii Herculis 45 .39 0.21 ? 1 .37 10.0-10.1 7929 /3 416 45 .90 0.62 K5 1 .93 6.0- 8.0 8038 Z2173 46 .0 0.20 G 1 .14 6.0- 6.4 3596 Sirius 49 .4 0.50 At 7 .59 -1.4-10.0 7332 OS 298 52 .0 0.58 ? .80 7.0- 7.3 1036 ,8 513 53 .0 0.35 A2 .61 5.0- 7.5 1070 7 Audr. BC 55 .0 0.82 Kl .35 5.0- 6 2 * Note added May 1, 1912. Additional observations secured since this table was prepared have effected improvements in the orbits of some of the stars listed. Thus, for ;3 581 we now have, P = 46>^ years, e = 0.40, a = 0".53 ; and for 02 235, P = 71.9 years, e — 0.40, a — 0".78. Good orbits have been computed for two other pairs with periods under one hundred years, viz. : Secohi 2 (S 2481 BC, Bumham No. 9114), P = 58 years, e — 0.50, a=0".40, Mags. 8.0-8.0; and OS 79 (Bumham No. 2134), P= 88.9 years, 6 = 0.62, a = 0".57, Mags. 7.0-8.8. Minor improvements have been effected in a number of other orbits. f Star A = Class A. Star B — yellow? I Star A = 2.28, Class K. Star BC = blue. VISUAL BINARY STARS 267 TABLE XXVII {Cmitinued) Burnliam's No. Star Period e Spectrum a Vis. Mag's. 10846 T Cygni 53 ■ ± 0.31± F 1 .16 3,9-10.0 4477 f CaiicH 59 .1 0.38 F .86 5.5- 6.2 5734 £ U)-8. Maj. 59 .8 0.41 G 2 .5 4.0- 4.9 8372 99 Hei-culis 64 .5 0.81 F8 1 .28 6.0-13.7 5811 OS 235 66 .0 0.50 F .83 6.0- 7 3 5235 8 Sextantis 68 .8 0.60 A2 .35 5.8- 6.1 7368 7 Cor. Bor. 73 .0 0.48 A .74 4.2- 7.0 5805 OS 234 77 .0 0.30 1 .35 7.0- 7.4 9979 OS 400 81 .0 0.46 ? .47 7.2- 8.2 Centauri 7 Centauri 70 OpUwchi 81 88 88 .2 .0 .4 0.53 0.80 0.50 G, K5 A K 17 1 4 .71 .02 .55 0.1- 1.9 3.2- 3.2 "'8340" 4.1- 6.1 9650 02 387 90 .0 0.60 ? .66 7.2- 8.2 7001 OS 285 97 .9 0.60 ? .34 7.1- 7.6 5223 XJrs. Maj. 99 .7 0.44 A2 .32 5.0- 5.6 12755 S3062 104 .6 0.45 F 1 37 6.9- 8.0 5103 01 Leonis 116 2 0.54 G .88 6.2- 7.0 1144 S228 123 .1 0.31 ? .90 6.7- 7.6 7034 1 Bootis 148 .5 0.54 K5p 4 .99 4.7- 6.6 7 Cor. ^Ms. S2 152 166 .7 .2 0.42 0.40 F8 A 2 .45 .55 5.1- 5.1 21" 6.3- 6^6 2109 02 Eridani BC 180 .0 0.13 G5§ 4 .79 9.2-10.9 6566 25 Ca». Few. 184 .0 0.75 F 1 .13 5,0- 8.5 7783 S2107 186 0.39 N? 1 .0 6.5- 8.0 6243 7 Virginis 194 !o 0.90 F 3 .99 3.6- 3.6 Summaries for Visual Binaries Spectra No. of Stars 0-B A 9 F 18 G-K 14 M-N s? Star A = 4.5, Class G5. Star BC = = Wue. No. of Stars P„ fio 25 32y.8 25 108 .1 0.48 0.51 268 STELLAR MOTIONS The spectral designations are from the Revised Harvard Photometry, excepting that of the forty-ninth star on the list, which is from the Draper Catalogue. The class is unknown for ten of the fainter systems. Summarizing the data by spectral class, two facts are striking : First, there are no visual binaries of known periods, either short or long, belonging to the Classes and B. This is in strong contrast with the very large proportion of spectroscopic binaries of Classes and B. It is apparent that the component stars in or B binary systems are in general too close together to be separated by direct telescopic observation. Second, there are no visual binary systems of determined periods belonging to the Classes M and N, but the apparent explanation is a very different one. The periods in such systems are so long, in comparison with the interval covering accurate micrometer studies of double stars, that we have no reliable information concerning the period of revolution for even one system. Dr. Aitken has tabulated the spectral classes of 164 double stars on his observing list of the more rapidly moving systems. All assignments of class but one, that of y Lupi. were taken from the Draper Catalogue. The results are as follows : stars B 4 stars A-F 131 stars G-K 28 stars il-N 1 star? Total 164 stars Visual double stars clearly abhor the Classes and B, and visual double stars of relatively short periods clearly abhor Classes M and N. In Table XXVIII let us bring together data for the spectro- scopic and visual binaries. As in Table XXVI, the parentheses contain the numbers of individual eccentricities which enter into the mean values of the eccentricities; and the unbracketed numbers denote the numbers of periods represented in the average periods. No. of Stars Po Co 31(19) 2.59 days 0.04 13(7) 6.90 " 0.14 33(24) 73.5 ' ' 0.36 15(13) 20.5 years 0.38 25(25) 32.8 " 0.48 25(25) 108.1 " 0.51 VISUAL AND SPECTBOSCOPIC BINARIES 269 TABLE XXVni Spectroscopic and Visual Binaries Spectroscopic binaries Spectroscopic binaries Spectroscopic binaries Spectroscopic binaries Visual binaries Visual binaries The period of revolution in a binary system is, in general, a function of the spectral class ; and the eccentricity is, in general, a function of the period. What is the significance of these facts ? Darwin and Poincare have studied the origin of binary stars from theoretical considerations. Not to review their work in technical detail, nor to define the underlying assumptions, they came to the conclusion that a condensing nebulous mass, rotating about an axis, constantly faster and faster, to keep pace with loss of heat through radiation, should eventually separate into two nebulous masses revolving around their mutual centre of mass. These two masses would in the beginning be revolving in contact, in orbits essentially circular. With advancing time, tidal disturbances within the more or less viscous bodies would cause them to draw apart, rapidly at first and less rapidly later. Jeans*^ and others have called attention to certain limitations in these investigations, which their authors recognize. Darwin has, in fact, stated*^ that the assumed conditions in the parent mass are necessarily not in strict accord with probable distribu- tion of density and other circumstances. However, confidence prevails that the deductions are substantially correct. See 's valuable study of visual double stars. Die Entwickelung der Doppelstern Systeme, 1892, taking note of the high eccen- tricities of their orbits, established that their periods and their eccentricities should increase as a gravitational effect resulting from tidal friction in the viscous components of the system. i^Fhil. Trans., 199 A, 1, 1902. 42 Darwin and Modern Science, pp. 548-549, Cambridge, 1909. 270 STELLAR MOTIONS In the spectroscopic and visual binary systems here described in the present paper we have a tolerably complete sequence of systems illustrative and confirmative of the Darwin-Poincare- See hypotheses. Short-period orbits should be circular or nearly so, and they should appertain preferentially to stars of early spectral classes; long-period orbits should, in general, attach to the more eccentric orbits and to the older spectral classes. These are the facts unquestionably established by spectrographic and micrometer observations of actual binary systems. There are many lines of supporting evidence which we shall consider. The Harvard College Observatory has announced the dis- covery of stars with composite spectra, as in Table XXIX. (The Harvard list has been revised by Miss Cannon to date, 1911.) The slit spectrograph has confirmed the expectation that the stars with composite spectra would eventually prove to have variable radial velocities, in nearly all cases. Of the thirty-four stars in this list {Annals H. C. 0., 28, 93 and 229 ; 50, 200-25 ; and 56, 113) : Six are moderately close visual double stars whose component spectra overlap on the original photographs, and the resulting spectra may appear to be composite for this reason. Two are fainter than visual magnitude 5.0, and seem not to have been placed on existing programs for radial velocity measurement. One has been observed only once for radial velocity. Of the twenty-five remaining, twenty have been found to be spectroscopic binaries, and the remaining five, observed several times each, have not certainly shown variable radial velocity, but this may be due to unfavorable distribution of observation times, or more probably to long periods of revolution. It is probable that the radial velocities of these five stars will be observed to vary in the future. Omitting the visual doubles in the list, all the repeatedly observed stars of Classes B or A have been shown to be spectro- scopic binaries. Those whose velocities appear to be constant, or to have varied but slightly, are of F5 or more advanced spectral classes. TABLE XXIX Stars with Composite Spectra (Objective Prism) H. E. Stai- R. A. Br. Sp. Ftr. Sp. Vis. Mag. Bemarks 595-6 o Piseium Ih 56m .9 A2p I type 5.23-4.33 Visual double, s = 3".6, p = 336 °§ 603-4 7 Androm. 1 57 .8 K I type 2.28-4.96 Visual triple, BC-A, s = 10", p = 62° Perhaps small variation in V 854 T Pet-aei 2 47 .2 Gp I-n type 4.06 Speetroseopic binary 915 7 Peraei 2 57 .6 Gp I type 3.08 Speotroscopio binary 1129 — Camelop. 3 37 .3 G A 4.96 Velocity apparently oonstant§ 1211-2 w Eridani 3 49 .2 G5 B? 6.33-4.95 Visual double, s = 6". 7, p = 347° Br. component perhaps variable V 1230 — Cephel 3 53 .3 F8p 5.25 Observed V approx. constant 1252 36 TauH 3 58 .4 A G 5.67* No observations 1306 f Fermi 4 8 .1 K A 4.89 Velocity apparently constant 1612 f Aungce 4 55 .5 Kp Orion 3.94 Spectroscopic binary 3307 e Carince 8 20 .5 Kp Orion 1.74 Perhaps small variation in V 3619 f Ura. Maj. 9 1 .9 A3** F5 4.54 No observations || 3624 T Ura. Maj. 9 2 .7 F8p I-n type 4.74 Spectroscopic binary 3852 Leonia 9 35 .8 F5p I type 3.76 Spectroscopic binary 4707 12 Crnn. Bear. 12 17 .5 G A 4.78 Spectroscopic binaryf 5440 ri Centauri 14 29 .2 B3p A2 2.65 Spectroscopic binary 5505-6 eBootia 14 40 .6 K A 5.12-2.70 Visual double, s = 2".6, p = 321° Br. component perhaps variable V 5704 7 Cirdni 15 15 .4 B5p F8 4.54 1 observation only Approx. equal photo, magnitudes 6134 a Scorpii 16 23 .3 Map A3 1.22-7 Spectroscopic binary Visual double, s = 2".6, p = 273° 6497 — OphvucM 17 21 .5 F5p 5.98 No observations 6729-0 95 Herculia 17 57 .2 A3 G5 5.21-5.13 Visual double, s = 6".l, p = 262° 6918 d Serpent. 18 22 .1 A G 5.33 Spectroscopic binary 7342 V Sagittar. 19 16 .0 B8p B± 4.58 Spectroscopic binary 7417 (3 Oygni 19 26 ,7 Kp I-n type 3.24 Br. component of visual double Velocity apparently constant 7536 S Sagittw 19 42 .9 Map I type 3.78 Spectroscopic binary 7735 31 Cygni 20 10 .5 Kp 3.95 Spectroscopic binary 7751 02 Cygni 20 12 .3 G5 I type 4.16 Spectroscopic binary 7776 p Capricor. 20 15 .4 Gp I type 3.25 Spectroscopic binary 7866 47 Cygni 20 30 .0 K5 B? 4.85 Spectroscopic binaryf 8131 a Equttlei 21 10 .8 A8p I type 4.14 Spectroscopic binary 8278 7 Capricar. 21 34 .6 Fp F± 3.80 Perhaps small variation in V§ 8417 f Cephei 22 .9 A3t G 4.57-6.47 Visual double, s = 5".6, p = 289° Perhaps small variation in V 8762 Androm. 22 57 .3 B3 Orion 3.63 Spectroscopic binary 8817 C3 Aquarii 23 4 .5 Gp A2 4.94 Observed V approx. constant * Visual companion, 12^ mag., too faint to affect spectrograms. ** "Spectrum is peculiar and appears to be composite." Ann. H. C. 0., 56 (IV), 106, 1908. f Variable velocity found at Lick Obsei^vatory after January 1, 1910. i Burnham, No. 11483, assigns colors yellow and blue to br. and ftr. components. § In 1910 found to have var. radial velocity. I Four observations in 1910-1911 vary only 1.8 km. I + I fe SPECTROSCOPIC BINARY STARS 273 For sixty-two spectroscopic binary systems catalogued in the tables, the spectra of both components have been either observed on the spectrograms, or strongly suspected. These systems are distributed among the spectral classes as follows : Classes and B 30 systems Class A 23 systems Class F 5 systems Class G 3 systems Class K 1 system Classes M and N system Total 62 systems From the published descriptions of the double spectra, it is fairly well established that when the two spectra are substantially equal in brightness they are identical in class ; and when one spectrum is considerably fainter than the other, the spectrum of the sec- ondary is apparently of a slightly earlier class than the spec- trum of the primary. There appear to be no exceptions to this rule, though the difficulty in the way of giving accurate descrip- tions of the fainter spectrum must be recognized. Another fact, seemingly of great significance, holds for every one of these systems, so far as they have been investigated, excepting pos- sibly in the very uncertain case of p Lyrce: the less massive member of the system is the fainter member, and has the earlier class of spectrum. The wide visual and spectroscopic binary, u Cetitmiri, appears to controvert this rule, slightly, in that the fainter component has the older class of spectrum ; but according to the data in Table XXX, a doubt exists as to which of the components is the more massive. Attempts have been made to determine the relative masses of the two components in several well-known visual double stars. The most of these investigations are based upon the apparent orbit of the primary, as determined from meridian observations, and upon micrometer observations of the secondary with refer- ence to the primary. Lewis's Struve Double Stars, xxi, 1906, contains results for nineteen systems. These were obtained for 274 STELLAR MOTIONS the most part by himself and other Greenwich investigators, but for very little of this Greenwich work have I been able to find published details. Estimates of the reliability of the results are therefore difficult. In general, they attribute much greater mass to the secondary members than to the primaries, in the case of a Geminornm twenty times ; but to this I think we need not attach any weight, for would not another value of the assigned proper motion in declination change the resulting value of m-^/m? In fact, was it not necessary to assume a position for the centre of mass of the system and therefore the ratio of the masses before a value of the proper motion could be deduced ? Neither does it appear that sufficient attention has been paid to the elimination of the systematic errors of observation, which we should expect to be large and variable for the same observer, as well as for different observers. Doubts^' as to the reliability of many of the results quoted in Lewis's table arise on comparing them with those secured by other investigators. For example, in the system of c Hi/drm, Lewis assigns to mjm the value 6; Seeliger's value is 0.9. In the system of o- Goronm Borealis, Lewis's value of m^/m is 4; Hadley's is 1.1. Prey's value of ynjm for 70 Ophiuchi is 4; whereas, from substantially the same data, Comstock obtains 1.0, and Lau 0.5. In the system of 85 Pegasi, three Greenwich observers, using two methods, assign to m^/m the value 3%, whereas Comstock obtains 1.6. Schorr's value for the system | Scorpii, 1.36, appears to be uncertain, for it is based upon P = 105.2 years, e = 0.12 ; and we now know the correct values to be approximately P = 44.5 years, e = 0.77. To the system of X Ophiuchi Lewis assigns mjm = 4.3, but Burn- ham's comment is, "Nothing whatever is known concerning the orbit" {General Catalogue, II, 717, 1906). On Lewis's list of nineteen systems there are assigned «;, > ni in thirteen systems, m^ = m in three systems, m^ Ceutauri, out of 3000 examined, Bailey found' 125 stars, approximately of the thirteenth and fourteenth magnitudes, which vary in brightness a full magnitude or more in short periods ; 98 at least having periods less than twenty-four hours. In cluster Messier 3, Bailey found 132 variables out of 900 stars examined; in cluster Messier 5, 85 variables out of 900 stars examined. Of the latter, Bailey has determined the periods of 63. All are short, 39 of the 63 completing their cycle of change in from 10% hours to 15 hours. In the Magellanic Clouds are fully a thousand of these variables, all faint and probably all of short period. The brightness curves of all the cluster variables thus far investigated repeat themselves with apparent faithfulness and on time, and all exhibit the same general form of curve, descending slowly from maximum to minimum, remain- ing almost if not quite stationary at minimum for a few hours, and ascending from minimum to maximum with extreme , rapidity. There are other types of variable stars, such as the so-called "new stars," which appear with great suddenness at points where previously no star of catalogue brightness (that is, as bright as the ninth magnitude) was known to exist, and occa- sionally, according to photographic observations, where no star as bright as the twelfth magnitude was recorded. They reach maximum brilliancy in a few days or a few weeks, pulsate through a considerable range of brightness for a few additional weeks, and thereafter decline more or less continuously until they become comparatively faint stars. In some cases they assume approximate constancy as faint stars, and in others they seem to go beyond the reach of telescopic power, and later become visible again as faint objects. There is the case of rj Garince, 3 Ann. H. C. 0., 38, 1902. t<--c ^ L^ CD It- Oi O CO lO CD in m t- 1 1 CD t- p: ^ CO in t- i-( ^ lo CO CO CO 05 lOOOO OO OS 00C3 00CO.-I TtHCOOTf =: o O .-H O iH i-H -^ O -* I-H C (M '—J ^ «3 CD Oi CM L"- OCllCO i-HOCOLO m m t^ CI o GO I-H o u: l> L- t^ O Oi CO 31 CD lO CO in lO CO CO 00 CO iH CD CO irt CD CO CD CD CC jg I— 1 1 , o :d «o (M lo t- IMinOO --HOCDO ix ^ -fH CO CTi iH oin^+^co^-tcocom .-< o ^ ^ o m m -Tt< CO c^ [5^ 1 1 g PLh fQ >r ^C!p p>. pq>H Ix m [X fin , W WW o O ^ pq ^ >^^ ^ KHp^ t>H ^ ^ >^ >H !>< >^t>( 1 — 1 CO CO -* 00 -H t- ci in m lO ^ CO CO O Ol 00 OOCiOlCmoOd^Oi-t y-t t^ 00 ■^ o o ^ ^ -^< -rrt^O-^COCDi-Hi-ICDCDOOCSCNQO 00 CO '^ -t^ ■•?»* t- CO CO cr- CI (M T-H ,-H ^ CO 1—1 .— ( CO CO I-H lO cq CD 1 — 1 OS ca ro m OS I-H t- CO t^ GO ^ -H-H ■ • -H • GO CD t- CD LO o o o cq CO I— 1 m O CO .-H T-H o t- t-- (M --H mm OS CD O 00 00 OS OS OS in < CO o o 00 -^ CD i-H CO CO CO CI CO -^ I— I Tf T+l i-H CO CO ^ \c:i i-\ CO CO M Si o ^ CD l>- t^ OO CC .-1 I-H CO T-H CO -f I-H 1—1 I-H m CD CD CD I— 1 »— ( -H I— 1 o _j_ W« pq eg M c8<^ ■« fc 1 1 IS 1 m^ •5 1 la ii p:- o" « s a ^^*> M* ■•Jg' ?- cq a Me JOj b /< Kn "s :3 ocsoons^ooo o h:; pq Q p^ hJ eq ffl pq SPECTROSCOPIC STUDY OF VARIABLES 291 as determined by the yellow and orange rays which make up the visual image of the star, by approximately an hour. This is just the opposite result from those obtained by Nordmann and Tikhoff for two other variable stars, for they found that the visual photometric minima preceded the photographic velocity minima. As explained in Chapter II, no explanation of the dis- crepancy in the times of arrival of the two minima is at present apparent. There are about 72 Algol variables known up to the present time ; and the accomplished understanding of the original Algol is the key which is rapidly unlocking the mysteries of many members of this class. The results which have been fairly well established for /8 LyrcB are even more interesting than those for Algol. This is a star whose brightness varies through a period of 12.9 days, apparently without maintaining a stationary brilliancy even for a moment, except as this passes through a maximum or mini- mum value. Photographic spectra by Pickering^^ led him to the conclusion that here again were two revolving stars. The lines in two spectra shifted their relative positions in accordance with the requirements of orbital motions. Belopolsky'^ verified this conclusion and obtained quantitative values of the velocities in the system. Myers^* correlated the photometric and spectro- graphic information and established some of the salient features of the system. The two stars are enormous in size, but of very low density. They are so close together as to be almost in con- tact. The two bodies are in form approximately prolate ellip- soids, with their longer dimensions in the line joining the two bodies. The immediate cause of the variable brightness is due, in large part, to the eclipsing of one body by the other, but there are probably other factors entering to a minor degree, such as tidal ebb and flow, which must exist, as the orbit seems to be slightly eccentric. Belopolsky assigns a value of 0.07 to the eccentricity, and 12 Astr. Nach., 128, 40, 1891. 13 Ap. J., 6, 328, 1897. 14 Ap. J., 7, 1, 1898. 292 STELLAR MOTIONS Myers, 0.11. Both bodies are luminous, but the larger one sup- plies only 40 per cent as much light as the smaller and more intensely luminous body. The long axis of the larger and fainter star is nearly 50,000,000 km. — thirty-five times our Sun's diame- ter ; and the long diameter of the smaller body is 39,000,000 km. The mean distance of their centres is roughly 50,000,000 km. They appear to be, in fact, like two eggs, one slightly larger than the other, with the small ends almost in contact. The masses of the two bodies, according to Myers, assuming that we are in or nearly in the orbital plane, are respectively ten and twenty-one times the solar mass. Their average densities are comparable with that of our atmosphere at sea level. The spec- trum of one of the bodies at least contains strong bright lines, and many of these bright lines have corresponding dark lines in the spectrum of the other body ; clear evidence that they are not far removed from nebular conditions. When the smaller and brighter body is eclipsed by the larger and fainter body, we have the light minimum ; when both bodies are broadside to us, we have the maximum; when the smaller body is between us and the larger body, we have the secondary minimum; and so on. As the two bodies are nearly in contact, there is almost continuous eclipse. A short stretch of uniform brightness may well be masked by tidal surgings in the atmospheres. Roberts obtained^^ a better representation of the photometric curve by assuming that the inclination of the orbit plane is 83°, so that we have only a partial eclipse. He assigns an eccentricity value of 0.02, a density equal to 0.0003 Z>„, and a combined mass of the system equal to 640. In view of the complexity reported to exist in the spectra, by several of the observers, it must be recognized that considerable uncertainty attaches to the published elements of the orbits. [Note added May, 1912. — Prom a very extensive investigation of the spectrum of |8 Lyrw, CurtissW concludes that the composite spectrum of the system consists of two dark-line spectra and a bright -line spectrum. He 15 Beport B. A. A. S., 254-256, 1905. 16 Puhl. Allegheny Ohs., 2, 73, 1911. SPECTROSCOPIC STUDY OF VARIABLES 293 has attempted to satisfy the observations on the basis of two very different hypothetical binary systems, each of which is consistent with the photo- metric curve. It seems clear from the radical difference of the two systems that the proper interpretation of the composite spectrum remains in serious doubt.] E LyncAa Mdl S Carinm Md4 R Carinm Md6 Ceti Md9 E Eydrm MdlO Class Md Spectra— Harvard College Observatory Much light is thrown upon conditions existing in the Algol and )8 Lyra systems by the data colleeted in Table XXXI, into which the photometric work of Roberts, Duner, Myers, Hartwig, Wendell, and others enters. The results collected in this table •=8 • 00 CO o o ■■BA iH - tH O • o o M El IS B O o Eh O U a o • CO • o • o CO lO i-H O t^ ?0 CO CO • >^ O 00 I-l en O) OS t- Ttl 1-H CO CS OS Id >-l CO t- t- 1— I «0 lO C> rH i-H '<1^ OS O t> t* CS OS o o t> rH O I-H lO CO o to T P •g g SD ^ II 1 r^i Ct, go ft. cd & ^x > ^ I'll § 03. < Eh 02 1-5 o a r QQ 00 stf in sfT rH OO Oi 00 cq -s 00 ^■g ^ WOO "^ b gH O o d CO OS rH O a, S-1 1 . o O -: °" . t- OS Oi o^ . 05 Oj r-( o OS rsi •t cq^ .H iH -"IJ ,-( O Qr lO CO ^ c5 tH -1^ o 4 o'l-i - ^ rH ^ WrH rH 3 1 1? f^H si mm . ■ CO . CO CO 4* >>o ^CO . ^CO CO §,^~l^ i S ^•s ^ » II g II II 11 ,0,0 rO ^c§ « e c i -,H -i-H <|3 w II CI II + +« H QP S Q 5. e e e Izico J •i s .00 ;S ■2 1 -2 ll ll 1 ^S' 1 1 1 i 11 1 1 O^ HcQ W ft^ P o c; m ^ CO O m ^> ^U Ph t2 X > s >W>H N to. 296 STELLAR MOTIONS do not depend in any way upon observed stellar velocities, and on this account no effort has been msde to have the data com- plete. For convenience, all known /S Lyrae stars are listed. There are many interesting facts brought out concerning the eccentricities (e) of orbits, relative brightness (briibr), and relative size {r^:r) of the component stars, average density (do), distances between components («j -(- a), etc. ; though- it should be said, I thiak, that studies of this kind, based exclu- sively upon photometric data without the illuminating assist- ance of radial velocity measurements for orbital proportions and scale values, must be considered as roughly approximate. The relatively high eccentricities, 0.14 to 0.25, assigned to three Algol stars, with periods under four days, are surprising, but prob- ably not seriously in error ; and the distances between the com- ponents, as determined by Duner, in two of these systems, afford abundance of space for revolution without direct interference between the primaries and their secondaries. The diameters of the secondaries are large relatively to their luminosity, and their mean densities are very low. These facts, however, do not appear to be out of harmony with characteristics supplied per- haps more definitely by other means of binary investigation. Table XXXI includes all known /3 Lyrae variables, to date 1910. It is customary to treat the Algol and j8 Lyrae variables sepa- rately, but there is scarcely a doubt that they are closely related. The y3 LyrsB stars vary in luminosity without pause, except as they pass through maxima or minima, and they seem to repeat their photometric curves with absolute fidelity as to form and period, neglecting possible -secular changes. Praiseworthy inves- tigations by Eoberts, Myers, and others appear to have estab- lished that the observed variations of brightness may be accounted for, in all eases studied, within the unavoidable errors of observation, by the mutual eclipses of two attenuated com- ponents nearly or quite in contact. This being true, the y8 Lyrae variables are Algols of an extreme type. It is not cer- tain, however, that minor brightness variations, due to tidal surgings reasonably to be expected, may not be involved. The SPECTROSCOPIC STUDY OF VARIABLES 297 72 Algol stars listed in V. J. S. Astr. Gesell, 44, 391-392, 1909, are assorted according to the lengths of their periods in the first two columns of Table XXXII. Similar data for the fi Lyrse stars from the same publication, pages 322-365, are tabulated in columns three and four. TABLE XXXn Vabiables of Algol and 13 Ltk^ Types Algols 72 p Lyr^ No. of Periods in No. of Periods in Stars Days Stars Days 9 to 1 3 or 4 to 1 n 1 to 2 2 or 1 1 to 2 17 2 to 3 2 to 3 12 3 to 4 3 to 4 4 4 to 5 13 4 5 to 6 37 4 6 to 7 1? 38 1 7 to 8 79 1 8 to 9 ? 2 9 to 10 10 10 to 12 11 2 12 1 13 1 31 1 32 1 35 1 262 Total S IPECTRA 1. of Stars P in Days Class B 8 stars 56 to 4 Class A 15 stars 16 4 to 10 Class F 3 stars 12 10 to 272 Classes G-K-M stars Unknown class 58 stars 84 84 298 STELLAR MOTIONS Densities (Algols) Spectrographic Orbits Roberts Eoberts No. of Stars 4 6 Average Density 0.13 Do 0.06 jS Persei X Tauri Y Puppis e .03 V. small V. small P 24.87 3 .95 1 .45 Russell 17 <0.20 S Librce .05 2 .33 Ristenpart 10 0.07 /i HercuUs |8 Lyrce .05 .05 ± 2 .05 12 .91 The great preponderance of short periods is striking. Of the eighty-four stars in both classes, fifty-six have periods less than four days long ; sixteen have periods between four and ten days ; and only twelve have periods between ten and two hundred sixty-two days. Most of the stars concerned are faint, and their classes of spectra have not been determined at Harvard College Observa- tory or elsewhere. Of the twenty-six spectra described, eight are of Class B, fifteen are of Class A, three are of Class F, and none are of Classes 6-K-M-N. The preference for early spectral classes is marked, which fact has often been commented upon. Roberts^' has computed, upon reasonable assumptions, the densities in eight Algol systems ; Russell,^* in seventeen systems ; and Ristenpart,^* in ten systems. The mean of the densities assigned by the three investigators is 0.13 of the Sun's mean density. Considering the preference of these stars for short periods, early spectral classes, and low densities, it is easy to reach eon- elusions as to why we have Algol and /8 Lyrse variables. The two members in Algol and ^ Lyrse systems are, in gen- eral, so near each other and so great in diameter that the eclipse phenomena for any one system are observable throughout a wide volume of sidereal space, and the eclipses are of long duration in reference to the periods of revolution, so that even the un- systematic observations of the past have readily detected variable brightness. It is not necessary that observers be in or near the orbital plane of the binary eclipsing system, nor that !■! Ap. J., 10, 314, 1899, and Bep. B. A. A. S., p. 256, 1905. 18 Ap. J., 10, 317, 1899. ISA. N., 178, 31, 1908. 8 Ve».orum >D S! s o u a s MM. 7-7 8-0 6 9 9-a s £!Seia£!Soc£J!S o 1 -H -H+f -H -TtHO'— " 1— ICO CDCl-f OOO O iH OOO + + +1 I++ o 1 + + > a "H 00 ~H "iH -H CT C-1 CO t- CI IC O W CI OOO oooooo +++ +11 11 CT CO "H ^ Cl 1 1 M CO "H(» L': o "H CO lo "H t^ Lt 1-- o Ci O OOO .— CI .-- Tl ^ rt CI iH .-( CI CO 00 t- CI o o "H CI o :c ^ -f ir: L- L^ o -tH CO u- +T+I+1 +7+ tr- io 1 C30 O 1— 1 + Lt -f :-: o "H CI CD m — re .— a; -f Lt' -r 00 ci +7 M +7 777 QO + o 1 5* + -H +1 COOOGOt^OCO COOO ^ LC '-' C-; -f i-H -h ic -tH OOOOOO OOO OOOOOO OOO o o' o o o' o o' o o CO CI o .—1 o -H -H OOOOOO OOO OOCCOOO OOO LtO.-iOOO OOO o^o^cT— "irr-f" co'cTcT LC irr o I- Xi r-3 i> co o X :-: cr. :■: -+■ re t* o o o X o o cT r-T s^ -H -H t; « r; p ■- o = ;J L^ Cl o o 1 (2 r-: -p -+ t- t- rt CI 00 o ■» C: L^ X -t- r- t' O ^ ^ CO r-: o .-1 CO ^ O iH ro r-: -JH u.^ Lt t> t- t- (> 00 c: o 1—1 CO t- t- to — .— r Cl CO trt CO lO i-C cs Pi. &, c5 o fe e o &< o &< c^ s ■^ Mil t ■ ■rt CI X cc -- re :o CI o CI o t- re ^ X t^ O --1 O I- h O O rc LQ o irr X CO -f ic 1-- CO — Cl t>- L^ X CO CO t^ t>- O t- l^ o CO Cl Cl 1-1 O -f :0 CD ^ X O. CD CO X L^ CD Cl CO SPECTROSCOPIC STUDY OF VARIABLES 305 atmospheres of the two bodies, though we should not expect so great a range of brightness variation to go with so short a period. However, the irregularities in the periods of all these stars substantially deny the existence of companion stars of appreci- able masses ; and, as explained, the only radial velocity evidence at hand is in harmony with this view. Unfortunately, we cannot assign with much confidence the reasons for variability of the red stars. Fully as interesting as the Algol stars are the Cepheids and Geminids. These we shall treat briefly together, for the only apparent distinction is that in one case the brightness increases to a maximum more rapidly than it falls off to minimum, whereas the reverse is the case for the other sub-class. Fifty-three of these stars are known, and all have perfectly definite periods. Eleven of them have been extensively investigated on the basis of their spectrographic velocities, ten of these studies having been made with the Mills spectrographs of the Lick Observatory. The principal known facts are contained in Table XXXIII. In every case investigated, the radial velocity is variable, indicating clearly, I think, that we are again dealing with binary systems. The period of velocity variation, studied for sixteen stars, is in every case equal in length to the photometric period. We cannot doubt, therefore, that the invisible companion star is in some way, perhaps only indirectly, responsible for the light variations. In the column V^ of the table are given the velocities of the centres of mass of the systems. In the next column are these velocities after correcting for the solar motion. It is interesting to note in passing that while these systems have their individual velocities of approach and recession, their average is almost exactly zero, being in fact + 1 km. The range of orbital veloci- ties in these systems is very much less than in the Algol systems. K is the single amplitude of the velocity curve. The correspond- ing amplitudes for Algols will be several times as great. The orbits have considerable eccentricities, as indicated by the column e. Some of the most interesting and apparently impor- tant conclusions for any variable stars are those indicated by the figures in the last two columns. The first of these gives the 306 STELLAR MOTIONS interval for each system between the instants of minimum velocity and maximum light. The second, the time interval between maximum velocity and minimum light. These intervals are small in comparison with the lengths of the periods. Inter- preted, we have the astonishing result that every star investi- gated has its maximum brilliancy at or very near the time of greatest velocity of approach toward the solar system, and the minimum brilliancy at or very near the time when the bright star in the system has its maximum velocity of recession from the solar system. These are keys which give promise of unlock- ing many secrets of the Cepheids and Geminids. What can be more remarkable than that variable stars of this class should be at their brightest when they are moving rapidly toward the observer and at their faintest when they are moving rapidly away from him ? Let us hold this fact in mind for consideration a little later. Other features should now be made familiar. The quantity of light received from one of these systems varies con- tinuously throughout the period, without stopping, except as the brightness passes through a maximum or minimum. The photometric curves are usually not simple or smooth curves but include irregularities, more or less prominent. The amount of variation from maximum to minimum and vice versa is usually about one magnitude. These stars, so far as investigated, are of spectral classes approximately solar, whereas the Algol and 13 Lyras stars, apparently without exception, are of the newer and simpler classes. No case has been found in which the spectrum of more than one of the two stars has recorded itself upon the spectrograms. The values of asini are all less than two million km., which is evidence that the primaries revolve in orbits whose dimensions may be described as minute. The values of m^' sin^ i/{m -\- m^)^ are also abnormally small, the largest being about 0.005. "We have here a tolerably clear indication that the masses of the companion stars are very small in proportion to the masses of their primaries, for it is not probable that smallness of the angle of inclination, i, is a peculiarity of these systems. SPECTROSCOPIC STUDY OF VARIABLES 307 Here, as always, the purpose of investigations is to determine why the stars vary. We may dismiss most promptly any eclipse theory accounting for the change in brightness, for the relative positions of the two bodies as determined by the spectrograph are such that eclipses can have no connection with the time of minimum brightness. Nevertheless, the light variation raust be connected in some way with orbital revolution in the system. Several hypotheses have been advanced to explain the observed phenomena; but on the whole it must be said that these systems are still involved in mystery. A very interesting hypothesis, one which may have great merit, is that of a resisting medium in which the binary system is supposed to be enveloped, so that the impacts of the advancing side of the bright star with the elements of the resisting medium generate heat, and in this manner build up the maximum of brightness. We can readily see that as a star revolves in its orbit, if it moves at different times with different speeds, or through portions of the medium having different densities, there could result corresponding differences in the rate of light generation and emission ; or, looking at the question from a different point of view, if the bright star presents always the same face to its companion, as it probably does, the preceding or forward face of the bright body will by the impacts of the particles be kept constantly hotter than the following side, so that the observed fact of maximum brilliancy at the time of maximum velocity of approach would be explained. The related fact of minimum brilliancy when the following side of the star is turned toward us and receding from us would likewise be explained. The objections to this hypothesis present some interesting considerations. If the companion's mass is less than the bright star's mass, as it probably or undoubtedly is, its orbital velocity must be greater than that of the bright star. Why is it not visible also, not separately, but in the combined light curve, as a result of correspondingly more severe impacts? We certainly should expect this result. Again, it must occur that the work done by the stars in giving momentum to resisting particles will tend to shorten the period of revolution. On reasonable o H > 10 a O M o SPECTROSCOPIC STUDY OF VARIABLES 309 assumptions as to the elements of such collisions, the correspond- ing diminution in the length of period has been computed by Duncan.^' He finds for one of the stars, of period six days, that the observed excess of radiation at maximum over that at mini- mum would probably demand a diminution of the period by as much as 0.04 second in each revolution. This, of course, may be excessive, as it rests upon some uncertain assumptions, but such a shortening of the period could not long escape detection. A diminution of this kind has been noted by Chandler for one and only one star in this class of variables, and the observed rate is very small indeed. Another apparently important fact noted by several observers of these stars is that the spectrum at maximum is relatively strong in the violet and at minimum relatively weak in the violet. Further, certain of the lines change their apparent wave lengths, with reference to the wave lengths of neighboring lines, in the same way that the wave lengths change as one passes from solar type stars toward the red stars and vice versa. "We can scarcely doubt that variable absorption of the bright star 's light is here involved. Duncan has suggested that the rapid motion of the bright star through a resisting medium may cause a partial brushing aside of some of the absorbing strata of the star's atmosphere on the advancing side, that is, the side presented to us at maximum brightness ; and a building up of the atmosphere on the following side, that is, the side presented to us at the time of maximum recession and minimum light. It is conceivable that the changes plainly visible in the spectrum, corresponding to these two positions of the revolving body, may have a partial explanation in this theory. The further study of the stars in this class is an exceedingly promising one, but unfortunately the forty-odd uninvestigated stars are for the most part faint; and, as their spectra are approximately of the solar type, implying comparative weakness in the blue and violet regions, the dispersion employed must be regrettably low. It seems tolerably certain that the problem of interpreting these variables is not exclusively a geometrical one, 29 lAcTc Obs. Bull, 5, 90, 1908, and 6, 154, 1911. 310 STELLAR MOTIONS and it may be that radial velocity methods are not the most efficient, yet we cannot doubt that the observed phenomena have their origin in the influence of the invisible companion over the brighter member of the system. TABLE XXXIV Cepheid-Geminid Variables No. of Period No. of Stars Days Spectrum Stars P„ 7 0- 1 0-B 1- 2 A 1 0.6 days 1 2- 3 r 14 8.2 days 5 3- 4 G-K5 26 11.4 days 5 4- 5 Unknown 12 7.9 days 5 5- 6 • — 5 6- 7 53 8 7- 8 ] 8- 9 3 9- 10 1 10 2 12 1 14 1 16 2 17 1 20 1 27 1 35 1 38 1 39 1 41 11 orbits computed — P^^=7.3 days 53 e^-0.31 The Cepheid-Geminid variables were not included in the tables of spectroscopic binaries in Chapter VII, for the reason that they appear to stand apart from other binary and variable-star systems. Fifty-three of the Cepheid-Geminid variables were known up to the beginning of the present year. These are classified in Table XXXIV, in the order of their lengths of period. To seven are assigned periods less than one day. It is SPECTROSCOPIC STUDY OF VARIABLES 311 probable that some of these extremely short-period variables may later be assigned to the Cluster variables, or it is not impossible that the Cluster variables are, in fact, entitled to classification with the Cepheid-Geminid variables. None of the many hundred Cluster variables discovered by Harvard College Observatory have yet been submitted to radial velocity measurements, owing to their faintness. There are twenty-eight Cepheid-Geminid variables with periods between three and eight days. The small IMag. V ^s \ r \ 13'5 14-0 \ s f > \ ^v. * Qd . ■ 2 -, 3 • » > . > • 13-0 13-5 140 ^^ «Ni d ^ \ ^^ / Od .1 .2 .3 A .5 .6 .7 .8 .9 Figm-e U. — Typical Light Curves of Cluster Variables number with periods between one and three days may or may not be significant of different conditions existing ia the systems with periods on opposite sides of this minimum. Only eighteen are known to have periods longer than eight days. The numbers and periods corresponding to the different spectral classes are quoted ia the table. There are none of Classes 0, B, M or N, so far as known. There is only one of Class A, and there are forty of Classes F to K5. The length of period seems from this tabu- 312 STELLAR MOTIONS lation to increase slowly with advancing spectral class, but the relationship cannot be regarded as established, because of the small number of stars involved. There is no evidence that the eccentricities are functions of the length of period, as in all other classes of stars discussed in these lectures. We take a moment for brief reference to the remaining impor- tant class of variables, namely, those in the star clusters. Un- fortunately the 2000 stars now known to belong to this class are nearly all faint, usually of the twelfth, thirteenth, fourteenth, and fifteenth magnitudes, and are, therefore, quite beyond our present spectrographio power of attack, except in possibly two or three cases. The photometric curves for two stars in this class are given in Figure 14.^° The curves of all investigated members of this class have the same general features. They resemble the curves for the Cepheid variables in that the decrease from maximum to minimum brightness is relatively slow and the increase from minimum to maximum is relatively rapid. The photometric curves for many of the Cluster variables indicate that the light is constant for a time at minimum, but on account of the faintness of these stars and the consequent diffi- culties in the way of accurate measurements, it should probably be said that the question of constancy at minimum remains in doubt. Inasmuch as the Cluster variables repeat their photo- metric cycles exactly on time, I think we cannot seriously doubt that in all these stars we are observing binary systems whose members interact upon each other in such a direct or indirect way as to vary the output of radiations. Roberts has said that in the study of Algol and /8 Lyrte systems we are searching for facts concerning stars in their early youth. The investigations of spectroscopic binary systems of all spectral classes confirm and emphasize this view. Our knowl- edge of these wonderful systems is covered by a span of but twenty years. That it exceeds in fact our knowledge of visual binary systems is due not so much to the energy and admirable instrumental equipment and methods of spectrographic ob- servers, but chiefly to the condition that the periods of revolu- 30 Taken from The System of the Stars, Second Ed., p. 119. SOLAR PARALLAX 313 tion are short. There is danger that the fruitfulness of the newer field of lahor will withhold attention from the equally- important older field of visual binaries. It is of the utmost desirability that available instruments be applied to the accurate observation of the positions of the primary members in double- star systems, with reference to the surrounding stars, by pho- tography, micrometer, heliometer, or other applicable means. It is equally desirable that spectroscopic observers measure the radial velocities in as many visual systems as practicable; the velocities of both members where possible, and of the primaries in other cases. Appreciable variations in the positions and speeds, in many cases, may not occur for centuries; but the securing of these observations, for the benefit of posterity, is a pressing duty. The wide range of problems to which stellar radial velocity results may be applied is admirably illustrated by the fact that such data may furnish our most valuable method of determining the solar parallax. In the year 1892, I published''^ the following statement concerning this promising method: "By assuming the Earth's mean distance from the Sun to be 92,500,000 miles, which corresponds to a solar parallax of 8".838, it is probable that the resulting orbital velocities [of the Earth] will not be in error by more than 0.1 mile per second. There is reason to hope that the probable errors of spectroscopic observations will soon reach this low limit, in which case the problem will be reversed and the spectroscope will be used to measure the Earth's orbital motion and thus to determine the solar parallax. ' ' The principles involved are of extreme simplicity. If we assume, on the basis of observations continued through several years, that the radial velocity of a star situated in or near the ecliptic is constant with reference to the solar system, and we observe the radial velocity of this star in the evening when its longitude is about 90° greater than the Sun's, and again in the morning when its longitude is about 270° greater than the Sun's, the observed velocities of the evening and morning series should siAstr. and Astroph., 11, 320, 1892. 314 STELLAR MOTIONS differ by approximately twice the orbital velocity of the Earth; that is, by about 60 km. per second. Assuming that the velocity of light, the diameter of the Earth, and the times of observation are knovm, and that the orbital velocity of the Earth is unknown : since the form and position of the Earth's orbit are knovrai to a satisfactory degree of accuracy, the solution of a simple equation will determine that value of the Earth's orbital velocity which will harmonize the evening and morning observed stellar veloci- ties. Inasmuch as a variation in the Earth's assigned velocity implies a corresponding variation in the linear dimensions of the Earth's orbit, the resulting alteration of the solar parallax value follows at once. Plummer has called attention^^ to the fact that the constant of aberration is involved with the solar parallax in this form of solution. It is true that all radial velocity determinations by spectrographic methods depend upon an assumed value of the velocity of light ; but as this is thought to be known within narrow limits of error, the uncertainty in a value of the solar parallax determined spectrographically should be correspondingly slight. The first systematic effort to determine the value of the solar parallax by this method was made by Kiistner at Bonn in 1904- 1905. Prom eighteen spectrograms of Arcturus he deduced the value^^ •n- = 8". 844 ± 0".017. It is a fair conclusion, judging from the minuteness of the probable error here assigned from only eighteen spectrograms, that a really extensive radial velocity program for determining the solar parallax would yield a result comparable with and perhaps superior ia value to the best determinations hitherto based upon any of the conventional methods, not excepting the value of the parallax derived from the extremely numerous observations of Eros. We might even question whether astron- omers will be justified in carrying through another Eros paral- lax program, some fifteen years from now, when that asteroid 32 The Observatory, 31, 239, 1908. ^^Astr. Nach., 169, 262, 1905. SOLAR PARALLAX 315 shall again be in a fayorable position; except that it is always desirable to arrive at conclusions by two or more independent methods, if they are fortunately available. [Note added April 1, 1910. — ^After the delivery of this lecture, there appeared (March, 1910) A Spectrographic Determination of the Constant of Aberration and of the Solar Parallax,^* made by the Eoyal Observatory, Cape of Good Hope, as based upon 302 measured radial velocities of seven prominent stars situated not far from the ecliptic. The value assigned to the solar parallax by this investigation is T - 8".800 ± 0".006. This is in remarkable accord with the results of greatest weight deduced from observations of 'Eros, viz.: 7r = 8".807 ± 0".003; and the small amount of labor involved in reaching the spectrographic result is in most striking contrast with the Herculean labor devoted to the deter- mination based upon observations of 'Eros.'\ 34 Ann. Cape Ohs., 10 (III), 56C, 1909. INDEX Absorption lines, discovery of, 10. Acceleration of velocity, cause of, 216. Accuracy attainable in observationB for radial velocities, 53. Adams, measures of solar rotation, 91. radial velocities of Pleiades stars, 181. and Frost, average radial veloc- ity of 20 Class B stars, 206. K line in the spectrum of 9 Camelopardalis, 277. observations of Orion Nebula, 111. proportion of binary systems in Class B stars, 278. Airy, solution of solar-motion prob- lem, 137. Aitken, list of short-period double stars, 266, 267. modem double-star survey, 236. progress made in double-star survey, 237. Albrecht, minima of T Vulpeculce, 86. orbit of j3 Canis Majoris, 250. variations of apparent wave lengths with spectral type, 77. Aldeharan, radial velocities for, 72, 73. Algol, orbit of, 289. secondary minimum in light curve of, 289. studies of, 287-289. suggestion to measure radial velocity of, 288. system of, 289. variation in period of, 287, 288. variables, densities of, 298. variables, light curves of, 300. variables, spectra of, 297. Angstrom unit, definition of, 16. Antapex of Sun's way, definition of, 127. Apex of Sun's way, definition of, 127. Argelander, solution of solar -motion problem, 136. Auwers, mass of companion of Sirius, 158. motion and binary character of Procyon, 242. motion and binary character of Sirius, 241. Bailey, cluster variables, 285. Baker, orbit of Spica, 243. Beljawsky, preferential motions of stars, 147. Belopolsky, Doppler-Pizeau principle, 55. duplicity of one component of Castor, 249. radial velocities of stars, 44. rotation of Jupiter, 94. rotation of Saturn and its rings, 100. rotation of Venus, 102. system of Algol, 289. and Tikhoff, minima of p Aurigce, 86. Bessel, discovery of binary character of Sirius and Procyon, 241. method of solving solar-motion problem, 136. positions of Pleiades stars, 181. Binary systems, evolution of, 269, 270. Boss, proper motions of Pleiades stars, 182. star group in Taurus, 183. Bradley, discoveries of double stars, 234. observations of star positions, 8. recommended search for stellar proper motions, 1. Bravais, method of solving solar- motion problem, 140. Brightness, relative, of components of spectroscopic binaries, 278. Bruno, motion of the stars, 1. Burnham, advances in double star astronomy, 236. 318 INDEX Burnham, relative masses in system of X Ophiuchi, 274. Buisson and Fabry, interferometer determination of wave lengths, 79. Calcium, binary system giving fixed K line of, 277. envelope aroimd system of p LyrcB, 278. K line in spectra of binary systems, 277. K line in spectrum of 9 Cam- elopardalis, 277. spectra containing sharp and narrow lines of, 277. Campbell, rotation of Saturn and its rings, 100. rotation of solar corona, 92. spectrographic determination of solar motion, 167. Canopus, parallax of, 158. and Cordoba Zones 511.243, ratio of masses, 158. Capella, discovery of binary charac- ter of, 251. Greenwich visual observations of, 252. Castor, duplicity of one component of, 249. Cepheid variable, variation in period of a, 309. Cepheid-Geminid variables, orbits of, 304. variables, relative masses of components of, 306. variables, spectral classes of, 311. variables, theory of, 307-312. o Ceti, see Mira. Chandler, catalogue of variable stars, 282. relation of redness to variation period in variable stars, 284. studies of Algol, 289. variation in period of Algol, 287. variation in period of a Cepheid variable, 309. Clark, discovery of companion to Sirius, 241. Cluster, variables, definition of, 285. typical light curves of, 311, 312. Comet, Halley's, return of, 3. Comets, spectra of, 38. Comparison spectrum,' device for in- serting, 51. Comte, limitations to knowledge of stars, 9. Comstock, relative masses in system of 70 Ophiuchi, 274. relative masses in system of 85 Pegasi, 274. Cordoba Zones 5i».243, observations of, 114, 117. Zones 5'».243 and Canopus, ratio of masses, 158. Corona, rotation of, 92. Correcting lens, use of a, 22. Crawford, orbit of i) Pegasi, 122. Curtis, dupKcity of one component of Castor, 249. in charge of D. O. MUls Expedi- tion to ChOe, 105. observations of Cordoba Zones 511.243, 114, 117. Curtiss, method of standard reduc- tion tables, 80. system of Algol, 289. system of /3 Lyroe, 292, 293. and Schlesinger, minima of /3 Persei, 87. Curvature of spectral lines, 66. D Dark bodies, existence of, 26. Darwin, evolution of binary systems, 269, 270. Deslandres, rotation of Jupiter, 94. rotation of Saturn and its rings, 100. rotation of solar corona, 92. temperature control of spectro- graphs, 51. Direction of motion, average, 215. of motion, Kapteyn's preferen- tial, evidence of radial velocities upon, 216, 219. Displacements of certain lines in solar spectrum, 82. Dispersion, anomalous, 88. effect of variation in thickness or index of refraction of medium traversed, 88. of light in interstellar space, evidence of, 86-88. INDEX 319 Distribution of brighter stars of dif- ferent spectral classes, 153. of naked-eye stars, 150, 152. of stars, 152. Doppler, effect of a moving light source, 13. Doppler-Pizeau, principle, 13, 15, 55. effects in canal rays, 83. Double star astronomy, advances in, 286. astronomy, beginnings of, 234, 235. survey, 236, 237. systems, need for radial velocity of, 313. Double stars, definitions of, 234, 235. discovery of, 234. discovery of binary character of Sirius and Procyon, 241. discovery of companion to Pro- cyon, 242. discovery of companion to Sirius, 241. discovery of y Arietis, 234. discovery of a Centauri, 234. discovery of many, 234. discovery of Mizar, 234. evolution of, 269. ideas on, Miehell's, 235. list of short-period, 266, 267. motion and binary character of Procyon, 242. motion and binary character of Sirius, 241. period of 5 Equulei, 265. relations between periods, paral- laxes and angular separa- tions, 238, 240. relative masses of components of 273-277. relative masses in system of a Coronw Borealis, 274. relative masses in system of e Hydrm, 274. relative masses in system of X Ophiuchi, 274. relative masses in system of 70 Ophiuchi, 274. relative masses in system of 85 Pegasi, 27 i. relative masses in system of | Scorpii, 274. relative masses in systems of, 273, 274, 276. Double stars, searches for, 234, 236. visual, relation to spectral classes, 268. and spectroscopic binaries, rela- tion to periods and eccen- tricities, 268. Draper, photographic record of spec- tral lines, 43. Duffield, wave lengths as affected by pressure, 82. Duncan, observations of B. D. + 30°.3639, 113. theory of Cepheid-Geminid vari- ables, 309. Duner, elements of variable star orbits, 295. solar rotation, 56, 90. Dyson, preferential motions of stars, 146, 218. E Eberhard and Vogel, observations of Orion Nebula, 111. Eddington, mean distances of Groom- bridge stars, 157. preferential motions of stars, 145. EUdn, parallax of o Centauri, 248. positions of Pleiades stars, 181. Bros, parallax program, 314. Evershed, motions in and near sun- spots, 92. Eversheim, interferometer determi- nation of wave lengths, 79. Evolution of binary systems, 269, 270. of double stars, 269. Fabricius, discovery of first variable star, 282. Fabry and Buisson, interferometer determination of wave lengths, 78. and Perot, errors in wave lengths, 78. Faye, law of solar rotation, 90. Field for 1- and 2-prism work, 113. Fizeau, see Doppler. Fraunhofer, discovery of absorption lines, 10. description of spectral types, 26. 320 INDEX Frost, discovery of spectroscopic binary of shortest known period, 250. evidence on question of inter- stellar dispersion, 88. number of spectroscopic binaries in Boss's Taurus cluster, 245. number of spectroscopic binaries in Class B stars, 245. spectra with sharp and narrow calcium lines, 277. and Adams, average radial velocity of 20 Class B stars, 206. K line in the spectrum of 9 Camelopardalis, 277. observations of Orion Nebula, m. proportion of binary systems in Class B stars, 278. G Galitzen and Wilip, Doppler-Fizeau principle, 55. Gauss, solution of solar-motion prob- lem, 136. Gill, parallax of Canopus, 158. parallax of o Centauri, 248. parallax of Bigel, 158. parallax of Spica, 244. Gould, positions of Pleiades stars, 181. Gravitational power of Universe, 117, 118. Greenwich, visual observations of Capella, 252. Groombridge stars, mean distances of, 157. H Hadley, relative masses in system of a CoroncB Borealis, 274. Hale, Zeeman effects in sun-spot spectra, 86. Ilalley, observed proper motions of 4 stars, 6. Halley's Comet, return of, 3. Habn, solar rotation, 91. and Hough, solution of solar- motion problem, 195. Harper and Plaskett, two similar binaries, 254. Hartmann, binary system giving fixed K line, 277. spectro-comparator, 81. temperature control of spectro- graphs, 53. Hartwig, elements of variable star orbits, 295. Harvard College Observatory, classi- fication of stellar spectra, 31. principal discoverer of variable stars, 282. stars with composite spectra, 270, 271, 273. see Observatory, photometry, relations of its stars, as to magnitudes and spectra, 155. Hastings, solar rotation, 56. Herschel, Sir John, possible rotation of stellar system, 161. southern star gauges, 148. survey of southern sky, 236. Herschel, Sir Wm., determination of Sun's motion, 8, 131. discussion of Herschel 's solution of solar motion by Kliigel, 131. methods of solving solar-motion problem, 131. motion of all stars, 3. new solution of solar motion, 132. northern star gauges, 148. searches for double stars, 234, 236. Hertzsprung, elements of enlarged Ursa Major group, 179, 180. High velocity stars of Classes G, K and M, 207. Hoffler, related motions of stars in Big Dipper, 175. Holden, Director of Lick Observa- tory, 44. Holwarda, early observations of Mira, 282. Homann, spectroscopic determination of solar motion, 167. Hooke, discovery of duplicity of 7 Arietis, 234. Sun and stars moving, 1. Hough and Halm, solution of solar- motion problem, 195. Houzeau, distribution of naked-eye stars, 150, 152. INDEX 321 Huggins, photography with wet plates, 42. pioneer attempts to measure radial velocities, 15. and Miller, stellar spectra, 27. Hull, measured radiation pressure, 4. Humphreys and Mohler, wave lengths as affected by pres- sure, 82. Hussey, observations in double-star survey, 236, 237. period of 5 Equulei, 265. Hyades cluster, parallax of, 185. International Union for Co-operation in Solar Eesearch, 79. Jacoby, positions of Pleiades stars, 181. Jewell, displacements of certain lines in solar spectrum, 82. Julius, effect of anomalous disper- sion, 88. Jupiter, rotation of, 94. Kapteyn, density of stellar distribu- tion, 160. numbers of stars of given lumin- osities, 159. preferential motion of stars, 140, 144. relations of proper motions to Milky Way, 155. results for solar motion, 139. solution of solar -motion problem, 137. Kapteyn 's preferential direction of motion, evidences of radial velocities upon, 216-219. Kayser, errors in wave lengths, 78. Keeler, observations of Orion Neb- ula, 111. photography of nebulae, 36. radial velocities of 3 stars, 41. radial velocities of 14 nebulae, 41. rotation of Saturn and its rings, 95. use of a correcting lens, 22. Kelvin, gravitational power of the Universe, 118. Kempf, spectroscopic determination of solar motion, 167, 168. Kirchhoff, discovery of principles of spectrum analysis, 11. Klinkerfues, motions of 5 stars in Big Dipper, 175. Kliigel, Herschel's solution for solar motion, 131. Kobold, results for solar motion, 139, 140. rotation of stellar system, 161. Kovesligethy, spectroscopic determi- nation of solar motion, 167. Kiistner, determination of solar par- allax, 314. Lalande, Sun's motions, 2. Langley, solar rotation, 56. Lebedew, interstellar dispersion of light, 87, 88. measured radiation pressure, 4. Lewis, relative masses in double-star systems, 273, 274. Lines, absorption, discovery of, 10. observed, 10. Lockyer, classification of stellar spectra, 31. Lord, temperature control of spectro- graphs, 52. Ludendorff, elements of Ursa Major group, 176. luminosities of Big Dipper stars, 178. motions of a and 17 Ursw Majoris, 177. Luminosities of Big Dipper stars, 178. of Pleiades, 182. (3 Lyrce, system of, 291-293. variables, spectra of, 297. M Madler, revolution of stars around Alcyone, 161. Mars, radial velocities of, 71. rotation of, 102. Maskelyne, proper motions of 7 stars, 6. Masses, relative, see Double stars. Maunder, rotation of Jupiter, 94. 322 INDEX Maury, discovery of binary character of p Aurigce, 243. Maxwell, composition of Saturn's rings, 95. Doppler-Fizeau, principle, 15. radiation pressure, 4. Mayer, discoveries of double stars, 234. proper motions of 12 stars, 8. Method of computing relative plane- tary velocities, 59. of measuring radial velocity dis- placements, 80. of standard reduction tables, 80. Michell, ideas on double stars, 235. Michelson, A. A., interferometer determination of wave lengths, 78. Michelson, W., effect of variation in thickness or index of refrac- tion of medium traversed, 88. Milky Way, relations of proper motions to, 155, 157. stars, definition of, 157. studies of, 149, 150. Miller and Huggins, stellar spectra, 27. Mills, D. 0., Expedition to Chile, 104. gift of spectrograph, 44. spectrograph, description of original, 21. spectrograph in Chile, descrip- tion of, 47. Mira, discovery of variability of, 282. early observations of, 282. radial velocities of, 303. search for Zeeman effects in, 86. Mohler and Humphreys, wave lengths as affected by pressure, 82. Moore, in charge of D. O. Mills Ex- pedition to Chile, 105. Motion, proper, definition of, 5. radial, definition of, 5. Motions of stars, 1. Myers, elements of variable star orbits, 295, 296. system of '^ Lyres, 291, 292. N Nebulee, photography of, 36. radial velocities of, 208, 210. Newall, discovery of binary charac- ter of Capella, 251. Newall, rotation of solar corona, 92. use of a correcting lens, 22. Newcomb, density of stellar distri- bution, 160. gravitational power of the Uni- verse, 117. minor dimensions of stellar sys- tem, 160. relations of proper motions to Milky Way, 155. New stars, definition of, 285. spectra of, 38. Nichols, measured radiation pressure, 4. Nordmann, dispersion of light in in- terstellar space, 86, 87, 166. minima of /3 Persei, 87. O Observatory, Allegheny, 51, 104, 289. Bonn, 104. Cambridge, 104. Cape of Good Hope, 105, 195, 314. Columbus, 104. D. 0. Mills, 23, 46, 104, 173. Greenwich, 104. Harvard College, facing 30. 31, 43, 151, 270, 293, 311. Lick, 44, 104, 305. Lowell, 104. Mt. Wilson Solar, 23. Ottawa, 104. Paris, 51, 104. Potsdam, 43, 104. Pulkowa, 44, 104. Terkes, 104, 114, 181, 184, 206, 278. Observatories engaged in measuring radial velocities, 104. Orbit of spectroscopic binary, Algol, 289. of spectroscopic binary, /3 Canis Majoris, 250. of spectroscopic binary, Spica, 243. Orion Nebula, observations of. 111. Parallactic components of motion, 128. Parallax of Canopus, 158. of a Centauri, 248. of Hyades cluster, 185. INDEX 323 Parallax of Pleiades stars, 182. of Bigel, 158. of Spica, 244. of Taurus group, 185. program of Eros, 314. spectrographic determination of solar, 313-315. Parallaxes, proper motions, and radial velocities, relations among, 219-228. of stars, programs for observing the, 232, 233. Perihelia, motions of, perhaps ex- plained by Principle of Ee- lativity, 4. Perot and Fabry, errors in wave lengths, 78. Perrine, photography of nebulse, 36. Peters, motion and binary character of Sirius, 241. Pfund, interferometer determination of wave lengths, 79. Pickering, addition to Secchi's star classification, 31. advances spectrum photography, 43. discovery of first spectroscopic binary, 121, 243. distribution of brighter stars of different spectral classes, 153. suggestion to measure radial velocity of Algol, 288. system of '/S Lyrce, 291. Plaskett and Harper, two similar binaries, 254. Pleiades stars, luminosities of, 182. observations of, Yerkes Observa- tory, 114. parallax of, 182. positions of, 181. proper motions of, 182. radial velocities of, 181. Plummer, solar parallax problem, 314. Poincare, evolution of binary sys- tems, 269, 270. Polaris, a triple star, 252, 254. Prey, relative masses in system of 70 Ophiuchi, 274. Proctor, related motions of 5 stars in Big Dipper, 175. Procyon, discovery of companion to, 242. Procyon, motion and binary char- acter of, 241. radial velocities of, 250. and Sirius, binary systems,, 241. Proper motion, definition of, 5. motions, how determined, 6. motions of 4 stars, 6. motions of 7 stars, 6. motions of 12 stars, 8. motions, parallaxes and radial velocities, relations among, 219-228. motions relative to Milky Way, 155, 157. Preferential motion of stars, 140, 144-147. Prevost, determination of solar motion, 131. E Radial motion, definition of, 5. Eadial velocities, accuracy attainable in observations for, 53. velocities, algebraic means of, with reference to spectral classes, 201. velocities as functions of spec- tral types, 206. velocities, average, by spectral classes, 207, 209. velocities, average for 20 Class B stars, 206. velocities, average for 280 stars, 205. velocities, averages for Type I and Type II stars, 205. velocities, averages with refer- ences to magnitudes and spectral types, 198, 206, 212, 214. , velocities, effect of solar motion on, 126. velocities, exceeding ± 50 km., 115. velocities, Huggins 's pioneer attempts to measure, 15. velocities, observatories engaged in measuring, 104. velocities of Class B stars, 203. velocities of nebulse, 208, 210. velocities of 14 nebulae, 41. velocities of planets for tests of accuracy, 59. velocities of stars, 41, 43, 44. 324 INDEX Eadial velocities of stars in Big Dipper, 175. velocities of stars near Autum- nal Equinox, 124. velocities of stars near Canis Major and Columba, 12.5. velocities of stars near Herculis and Lyra, 125. velocities of stars near Vernal Equinox, 124. velocities of stars with large proper motions, 228-232. velocities, planetary, method of computing, 59. velocities, relative, in solar sys- tem, 59. velocities, relative to space velocities, 215. velocities, relative to tangential velocities, 215. velocities, reduction of, to Sun, 64. Eadial velocity of Alcyone (25 Tauri), 114. Aldebaran, 72, 73. ^ AquilcB, 109. Arciurus, 117. Atlas (27 Tauri), 114. \ Aurigce, 109. B. D. -j-30°.3639, 113. tt Cassiopeice, 105. u CarincE, 108. a Centauri, 120. 1) Cephei, 110. C. Z. 5h.243, 114, 117. fi. Cygni, 109. Groombridge 1830, 117. LacaOle 661, 112. S Leporis, 110. Mars, 71. Merope (23 Tauri), 114. Orion Nebula, 111. e Pegasi, 108. Frocyon, 250. E. H. P. 1614, 112. Venus, 71. velocity, relation to displace- ments in spectrum, 16. velocity with reference to Sun, definition of, 65. Eadiation pressure, 4. Eeduction table, 19. Eelations of Harvard Photometry stars as to magnitude and spectra, 155. Eelativity, Principle of, may explain motions of perihelia, 4. Eevolution of stars around Alcyone, 161. Eiccioli, discovery of duplicity of Mizar, 234. Eichaud, discovery of duplicity of o Centauri, 234. Bigel, parallax of, 158. Eisteen, spectrographic determina- tion of solar motion, 167, 168. Eoberts, cross section of X Carincs system, 301. elements of variable star orbits, 295, 296. light curves of Algol variables, 300. system of /3 Lyrce, 292. system of V Vulpecuke, 302. Eotation of stellar system, 161. Eowland, errors in wave lengths, 78. Eudolph, preferential motions of stars, 147. Eutherfurd, stellar spectra, 27. S Saturn and its rings, rotation of, 95, 100. Saturn's rings, composition of, 95. Scale of the universe, 228. Schseberle, discovery of companion to Frocyon, 242. Scheiner and Vogel, radial velocities of stars, 43. radial velocities of stars in Big Dipper, 175. Schiaparelli, rotation of Venus, 102. Schlesinger, minima of S Litres, 87. orbit of Algol, 289. system of S Lihr(E, 289, 290. and Curtiss, minima of /3 Fersei, 87. Schoenfeld, catalogue of variable stars, 282. possible rotation of stellar sys- tem, 161. Schorr, relative masses in system of f Scorpii, 274. Schwarzschild, preferential motions of stars, 146. INDEX 325 Secchi, classification of stellar spec- tra, 27. classification of stellar spectra, Pickering's addition to, 31. See, evolution of binary systems, 269, 270. Seeliger, distribution of stars, 152. relative masses in system of e Hydrce, 274. zodiacal light material, 3. Sirius, discovery of companion to, 241. mass of companion to, 158. motion and binary character of, 241. and Procyon, binary systems, 241. Slipher, K calcium line in spectra of binary systems, 277. rotation of Mars, 102. rotation of Venus, 102. Solar motion, advantages of spectro- graphic determination of, 163. belief in, 1, 2. derivation of equations for solv- ing problem of, 170. determined, 8, 131. determined from 1193 radial velocities, 209. difference between proper motion and radial velocity results for, 193. early estimates of, 9. effect of, on radial velocities, 126. first determination of, by Her- schel, 131. Kliigel, on Herschel's first solu- tion, 131. Lalande, 2. new solution of, by Herschel, 132. problem, methods of solving, 132, 136. problem, solution of, 136-142, 195. spectrographic determination of, 167, 168. solution from 1047 radial veloci- ties, 189. solution, rejection of high veloci- ties from, 187, 188. Solar motion solutions, table of, fac- ing 142. velocity from observations of Type I and Type II stars, 191. parallax, spectrographic deter- mination of, 313-315. rotation, 56, 90. rotation, law of, 90, 91. rotation, measures of, 91. spectrum, displacements of cer- tain lines in, 82. system, an extreme type, 281. Spectra, Class Md, 293. comparison, how formed, 24. composite, stars with. Harvard College Observatory, 270, 271, 273. containing sharp and narrow calcium lines, 277. devoid of measurable lines, 116. double, 273. limits of observed, 16. of comets, 39. of novse, 38. of spectroscopic binary systems, 120, 121. stellar, 27. stellar, classification of, 27, 31. Spectral lines, photographs of, by Draper, 43. types, Fraunhofer's description of, 26. types, variation of apparent wave lengths with, 77. Spectro-comparatoT, Hartmann, 81. Spectrograph, improved support for, 47. original Mills, description of, 21. Mills, gift of, 44. Mills in Chile, description of, 47. temperature control of, 51-53. Spectroscopic and visual binaries, periods and eccentricities, 269. binaries, Cepheid-Geminid stars as, 304. binaries, discovery of ';8 Aurigce, 243. binaries, discovery of first sys- tem, 121, 243. binaries, discovery of Spica, 243. binaries in Big Dipper, 254, 255. binaries, masses in, 18, 264. binaries, number of, 244, 245. 326 INDEX Spectroscopic binaries, number of in Class B stars, 245. binaries, number of in Taurus group, 245. binaries of Class A, 257. binaries of Class F, 258. binaries of Classes G to M, 259. binaries of Classes and B, 256. binaries of shortest known period, 250. binaries, orbit of ri Pegasi, 122. binaries, period of f VrscB Majoris, 121, 243. binaries, proportion of, in Class B stars, 278. binaries, proportion of, observed Tvith Mills spectrograph, 280. binaries, relations between spec- tral classes, periods and ec- centricities in, 255-261. binaries, relative brightness of components of, 278. binaries, relative masses of pri- mary and secondary mem- bers of, 265. binaries, spectra of, 120, 121. binaries, systemic velocities of, 119. binaries, two similar, 254. Spectrum, advantage of, over point image of a star, 15. limits of visual, 15. relation of radial velocity to dis- placements in, 16. analysis, discovery of principles of, 11. analysis, principles of, 12. photography, advances in, 43. Spica, orbit of spectroscopic binary, 243. parallax of, 244. Standard reduction tables, method of, 80. Star gauges, northern, 148. gauges, southern, 148. Star positions, observations of, 8. Stark, Doppler-Fizeau, effects in canal rays, 83. Stebbins, observation of secondary . minimum in light curve of Algol, 289. Stellar distances, proper motions and radial velocities, relations between, 219-228. Stellar distribution, density of, 160. luminosities relative to distances of stars, 159. motions, average directions of, 215. motions, belief in, 1, 2. motions in Big Dipper, 175. motions in Taurus, 183, 184. motions in Ursa Major, 176, 179, 180. motions of all stars, 3. motions of a and t) UrscB Majoris, 177. motions recommended search for, 1. motions, revolution of stars about Alcyone, 161. motions, rotation of stellar sys- tem, 161. system, minor dimensions of, 160. system, possible rotation of, 161. Struve, W., double-star survey, 236. studies of Milky Way, 149, 150. Sun, probable form of orbit of, 194. Sun spots, motions in and near, 92. Sun 's way, antapex of, definition of, 127. apex of, definition of, 127. Taurus, group of stars, 1S3. motions of, 184. parallax of, 185. Telescope, functions of a, 19. Telescopes, limits on size of, 19. reflecting, effects of temperature on foci of, 23. refracting, chromatic aberra- tions of, 22. refracting effects of temperature on foci of, 23. Temperature control of spectro- graphs, 51. Tikhoff, dispersion of light in inter- stellar space, 86, 87, 166. and Belopolsky, minima of /3 Aurigce, 86. Tisserand, variation of period of Algol, 288. Types, spectral, Fraunhof er 's de- scription of, 26. INDEX 327 V Universe, scale of, 228. Ursa Major group, elements of, 176, 178, 180. Variables, Algol, definition of, 284. Cepheid, definition of, 284. Cepheid, variation in period of a, 309. Cepheid-Geminid, orbits of, 304. cluster, 284. cluster, typical light curves of, 311, 312. Gemiuid, definition of, 284. of Algol type, 297-302. of Algol and /3 Lyrse types, spectra of, 297. of Algol type, densities of, 298. of Algol type, light curves of, 300. of ^ LyTEB type, 297-302. Variable star orbits, 295, 296. stars, catalogue of, 282. stars, Cepheid-Geminid, relative masses of components of, 306. stars, Cepheid-Geminid, spectral classes of, 311. stars, Cepheid-Geminid, theory of, 307-312. stars, o Ceti, radial velocity of, 303. stars, cross section of X Carinm system, 301. stars, periods of, 283. stars, principal discoverer of, 283. stars, red, theory of, 303, 305. stars, minima of /3 Aurigce, 86. stars, minima of 5 Libra;, 87. stars, minima of ^ Fersei, 87. stars, minima of T Vulpeculce, 86. stars, system of S Libroi, 289, 290. stars, system of /3 Lyras, see p LyrcB. stars, system of V Vulpeculce, 302. stars, studies of Algol, see Algol. Velocities, average linear, in space, 216. relation between radial and space, 215. relation between radial and tan- gential, 215. in space, of Arcturus, 117. in space, of C. Z. 5i.243, 117. in space, of Groombridge 1830, 117. Venus, radial velocities of, 71. rotation of, 102. Vogel, classification of stellar spec- tra, 31. discovery of binary character of Spica, 243. method of measuring radial velocity displacements, 80. period of f TJrsce Majoris, 121, 243. solar rotation, 56. study of Algol system, 288. and Eberhard, observations of Orion Nebula, 111. and Scheiner, radial velocities of stars, 43. velocities of stars in Big Dipper, 175. von Hepperger, elements of variable star orbits, 295. W Wave lengths, apparent, variation with spectral type, 77. as affected by changes in elec- trical constants employed, 85. as affected by pressure, 82. errors in, 78. interferometer determination of, 78, 79. standard, in comparison spectra, 79. Weersma, solution for solar motion, 142. Wendell, elements of variable star orbits, 295. Wollaston, absorption lines observed by, 10. Wright, device for inserting com- parison spectrum, 51. in charge of D. O. Mills Expedi- tion to Chile, 105. observations of a Centauri, 120. 328 INDEX Wright, observations of Orion Ne- bula, 1]1. parallax of a Centauri, 248. search for Zeeman eflfeets in Ceti, 86. suggests improved support for spectrographs, 47. temperature control of spectro- graphs, 52. Wright of Durham, Sun and stars moving, 2. Y Yerkes Observatory, see Observatory. Z Zeeman effects in o Ceti, 86. effects in sun-spot spectra, 86. Zodiacal light, material of, 3.