t

MACMII.LAX AND CO., LIMITED

LONDON . ROMHAY . C.MXIII A MELBOURNE

THK MA< "Ml LI. AN COMPANY

Ni:\V YORK . BOSTON . CHICAGO

DALLAS . <AN KKANCISCO

THE MACMILLAN CO. OF CANADA, LTD.

TOR'

Plate I,

GR,KAT STAR-CLOUD IN SAGITTARIUS.

STELLAR MOVEMENTS

AND THE STRUCTURE

OF THE UNIVERSE

BY

A. S. EDDINGTON

'\

M.A. (CANTAB.), M.Sc. (MANCHESTER), B.Sc. (Lo.\n.), F.R.S.

Plnmian Professor of Astronomy, University of Cambridge

MACMILLAN AND CO., LIMITED ST. MARTIN'S STREET, LONDON

1914

COPYR1GH7*

PREFACE

THE purpose of this monograph is to give an account of the present state of our knowledge of the structure of the sidereal universe. This branch of astronomy has become especially prominent during the last ten years ; and many new facts have recently been brought to light. There is every reason to hope^that the next few years will be equally fruitful ; and it may seem hazardous at the present stage to attempt a general discussion of our knowledge. Yet perhaps at a time like the present, when investigations are being actively prosecuted, a survey of the advance made may be especially helpful.

The knowledge that progress will inevitably lead to a readjustment of ideas must instil a writer with caution; but I believe that excessive caution is not to be desired. There can be no harm in building hypotheses, and weaving explanations which seem best fitted to our present partial knowledge. These are not idle speculations if they help us, even temporarily, to grasp the relations of scattered fjicts, and to organise our knowledge.

No attempt has been made to treat the subject histori- cally. I have preferred to describe the results of inves-

300340

vi PREFACE

tigations founded on the most recent data rather than early pioneer researches. One inevitable result I particularly regret ; many of the workers who have prepared the way for recent progress receive but scanty mention. Sir W. Herschel, Kobold, Seeliger, Newcomb and others would have figured far more prominently in a historical account. But it was outside my purpose to describe the steps by which knowledge has advanced ; it is the present situation that is here surveyed.

So far as practicable I have endeavoured to write for the general scientific reader. It was impossible, without too great a sacrifice, to avoid mathematical arguments altogether ; but the greater part of the mathematical analysis has been segregated into two chapters (VII and X). Its occasional intrusion into the remaining chapters will, it is hoped, not interfere with the readable character of the book.

I am indebted to Prof. G. E. Hale for permission to re- produce the two photographs of nebulae (Plate 4), taken by Mr. G. W. Ritchey at the Mount Wilson Observatory, and to the Astronomer Royal, Dr. F. W. Dyson, for the three re- maining plates taken from the Franklin-Adams Chart of the Sky. This represents but a small part of my obligation to I )r. Dyson ; at one time and another nearly all the subjects treated in this book have been discussed between us, and I make no attempt to discriminate the idcns wliir-h I owe to him. There are many other astronomers from whose conversation, consciously or unconsciously, I have drawn material for this work.

PREFACE vii

I have to thank Mr. , P. J. Melotte of the Royal Observatory, Greenwich, who kindly prepared the three Franklin- Adams photographs for reproduction.

Dr. S. Chapman, Chief Assistant at the Royal Observa- tory, Greenwich, has kindly read the proof-sheets, and I am grateful to him for his careful scrutiny and advice.

I also desire to record my great indebtedness to the Editor of this series of monographs, Prof. R. A. Gregory, for many valuable suggestions and for his assistance in passing this work through the press.

A. S. EDDINGTON.

THE OBSERVATORY, CAMBRIDGE. April, 1914.

CONTENTS

CHAPTER I

PAOF.

THE DATA OF OBSERVATION 1

CHAPTER II

GENERAL OUTLINE . 30

CHAPTER III

THE NEAREST STARS 40

CHAPTER IV

-MOVING CLUSTERS 54

CHAPTER V

THE SOLAR MOTION 71

CHAPTER VI

THE TWO STAR STREAM- 86

CHAPTER VII

THE TWO STAR STREAMS — MATHEMATICAL THEORY 127

CHAPTER VIII

PHENOMENA ASSOCIATED WITH SPECTRAL TYPE 154

CHAPTER IX

COUNTS OF STARS 184

x CONTENTS

CHAPTER X

GENERAL STATISTICAL INVESTIGATIONS 201

CHAPTER XI

THE MILKY WAY, STAR-CLUSTERS AND NEBULAE 232

CHAPTER XII

DYNAMICS OF THE STELLAR SYSTEM 246

INDEX 263

LIST OF ILLUSTRATIONS

PLATES

/

PLATE I. The Great Star-cloud in Sagittarius Frontispiece

,, II. Nebulous dark space in Corona Austrina facing paye 236 ,, III. The Greater Magellanic Cloud ,, 240

(Nebula, Canes Venatici. N.G.C. 5194-5 \ 249

' iNebula, Coma Berenices. H.V. 24 /

FIGURES

PAGI

1. Hypothetical Section of the Stellar System 31

2. Moving Cluster in Taurus. (Boss.) 57

3. Geometrical Diagram 59

4. Moving Cluster of " Orion "Stars in Perseus 64

5. Simple Drift Curves 88

6. Observed Distribution of Proper Motions. (Groombridge Cata-

logue—R. A. 14*> to 18h, Dec. +38° to +70:.) .... 89

7. Calculated Distributions of Proper Motions 92

8. Observed and Calculated Distributions of Proper Motions.

(Boss, Region VIII.) 93

9. Diagrams for the Proper Motions of Boss's " Preliminary

General Catalogue " 95, 96, 97, 98

10. Convergence of the Directions of Drift I. from the 17 Regions . 99

11. Convergence of the Directions of Drift II. from the 17 Regions . 99

12. Geometrical Diagram 101

13. Distribution of Large Proper Motions (Dyson) .... 105

14. Mean Proper Motion Curves 112

xi

xii LIST OF ILLUSTRATIONS

Hi;. PAGE

15. Mean Proper Motion Curve (Region of Boss's Catalogue) . . 116

16. Comparison of Two-drift and Three-drift Theories . . .122

17. Distribution of Drifts along the Equator (Halm) .... 123

18. Comparison of Two-drift and Ellipsoidal Hypotheses . . . 135

19. Absolute Magnitudes of Stars (Russell) 171

20. Number of Stars brighter than each Magnitude for Eight Zones

(Chapman and Melotte) 188

21. Curves of Equal Frequency of Velocity 220

22. Geometrical Diagram 249

STELLAR MOVEMENTS AND THE STRUCTURE OF THE UNIVERSE

CHAPTER I

THE DATA OF OBSERVATION

IT is estimated that the number of stars which could be revealed by the most powerful telescopes now in use amounts to some hundreds of millions. One of the princi- pal aims of stellar astronomy is to ascertain the relations and associations which exist among this multitude of individuals, and to study the nature and organisation of the great system which they constitute. This study is as yet in its infancy ; and, when we consider the magnitude of the problem, we shall scarcely expect that progress towards a full understanding of the nature of the sidereal universe will be rapid. But active research in this branch of astronomy, especially during the last ten years, has led to many results, which appear to be well established. It has become possible to form an idea of the general dis- tribution of the stars through space, and the general character of their motions. Though gaps remain in our knowledge, and some of the most vital questions are as yet without an answer, investigation along many different lines has elicited striking facts that may well be set down. It is our task in the following pages to coordinate these results, and review the advance that has been made.

B

MOVEMENTS CHAP.

Until recent years the study of the bodies of the solar system formed by far the largest division of astronomy ; but with that branch of the subject we have nothing to do here. From our point of view the whole solar system is only a unit among myriads of similar units. The system of the stars is on a scale a million-fold greater than that of the planets ; and stellar distances exceed a million times the distances of the comparatively well-known bodies which circulate round the Sun. Further, although we have taken the stars for our subject, not all branches of stellar astronomy fall within the scope of this book. It is to the relations between the stars — to the stars as a society — that attention is here directed. We are not con- cerned with the individual peculiarities of stars, except in so far as they assist in a broad classification according to brightness, stage of development, and other properties. Accordingly, it is not proposed to enter here into the more detailed study of the physical characters of stars ; and the many interesting phenomena of Variable Stars and Novae, of binary systems, of stellar chemistry and temperatures are foreign to our present aim.

The principal astronomical observations, on which the whole superstructure of fact or hypothesis must be based, may be briefly enumerated. The data about a star which are useful for our investigations are : —

1. Apparent position in the sky.

2. Magnitude.

3. Type of spectrum, or Colour.

4. Parallax.

5. Proper motion.

6. Radial velocity.

In addition, it is possible in some rather rare cases to find the mass or density of a star. This is a matter of import- ance, for presumably one star can only influence another by means of its gravitational attraction, which depends on the masses.

1 THE DATA OF OBSERVATION 3

This nearly exhausts the list of characteristics which are useful in investigating the general problems of stellar distribution.* It is only in the rarest cases that the com- plete knowledge of a star, indicated under the foregoing six heads, is obtainable ; and the indirect nature of most of the processes of investigation that are adopted is due to the necessity of making as much use as possible of the very partial knowledge that we do possess.

The observations enumerated already will now be con- sidered in order. The apparent position in the sky needs no comment ; it can always be stated with all the accuracy requisite.

Magnitude. — The magnitude of a star is a measure of its apparent brightness ; unless the distance is known, we are not in a position to calculate the intrinsic or absolute brightness. Magnitudes of stars are measured on a logarithmic scale. Starting from a sixth magnitude star, which represents an arbitrary standard of brightness of traditional origin, but now fixed with sufficient pre- cision, a star of magnitude 5 is one from which we receive

2 '5 12 times more light. Similarly, each step of one magni- tude downwards or upwards represents an increase or decrease of light in the ratio 1 : 2*5 12. f The number is so chosen that a difference of five magnitudes corresponds to a light ratio of 100 = (2*5 12)5. The general formula is

= - --

where

.Lj, L2 are the intensities of the light from two stars mj, m2 are their magnitudes.

Magnitude classifications are of two kinds : photometric (or visual), and photographic ; for it is often found that of

* We should perhaps add that the separations and periods of binary stars are also likely to prove useful data.

f It is necessary to warn the reader that there are magnitude systems still in common use — that of the Bonn Durchmusterung, that used by many double-star observers, and even the magnitudes of Boss's Preliminary General Catalogue (1910) — which do not conform to this scale.

B 2

4 STELLAR MOVEMENTS CHAP.

two stars the one which appears the brighter to the eye leaves a fainter image on a photographic plate. Neither of the two systems has been defined very rigorously; for, when stars are of different colours, a certain amount of personality exists in judging equality of light by the eye, and, if a photographic plate is used, differences may arise depending on the colour-sensitiveness of the particular kind of plate, or on the chromatic correction of the telescope object- glass. As the accuracy of magnitude-determinations im- proves, it will probably become necessary to adopt more precise definitions of the visual and photographic scales ; but at present there appears to be no serious want of uniformity from this cause. But the distinction between the photometric and photographic magnitudes is very important, and the differences are large. The bluer the colour of a star, the greater is its relative effect on the photographic plate. A blue star and a red star of the same visual brightness may differ photographically by as much as two magnitudes.

The use of a logarithmic scale for measuring brightness possesses many advantages ; but it is liable to give a misleading impression of the real significance of the numbers thus employed. It is not always realised how very rough are the usual measures of stellar brightness. If the magnitude of an individual star is not more than 0IUT in error, we are generally well satisfied ; yet this means an error of nearly 10 per cent, in the light-intensity. Interpreted in that way it seems a rather poor result. An important part of stellar investigation is concerned with counts of the number of stars within certain limits of magnitude. As the number of stars increases about three-fold for each successive step of one magnitude, it is clear that all such work will depend very vitally on the absence of systematic error in the adopted scale of magnitudes; an error of two- or three- tenths of a magni- tude would affect the figures profoundly. The establish-

i THE DATA OF OBSERVATION 5

ment of an accurate magnitude-system, with sequences of standard stars, has been a matter of great difficulty, and it is not certain that even now a sufficiently definitive system has been reached. The stars which coine under notice range over more than twenty magnitudes, corre- sponding to a light ratio of 100,000,000 to 1. To sub-divide such a range without serious cumulative error would be a task of great difficulty in any kind of physical measurement.

The extensive researches of the Harvard Observatory, covering both hemispheres of the sky, are the main basis of modern standard magnitudes. The Harvard sequence of standard stars in the^neighbourhood of the North Pole, extending by convenient steps as far as magnitude 21, at the limit reached with the 60-inch reflector of the Mount Wilson Observatory, now provides a suitable scale from which differential measures can be made. The absolute scale of the Harvard sequence of photographic magnitudes has been independently tested by F. H. Scares, at Mount Wilson, and S. Chapman and P. J. Melotte, at Greenwich. Both agree that from the tenth to the fifteenth magnitudes the scale is sensibly correct. But according to Scares a correction is needed between the second and ninth magnitudes, lm'00 on the Harvard scale being equivalent to lm*07 absolute. If this result is correct, the error introduced into statistical discussions must be quite appreciable.

For statistical purposes there are now available deter- minations of the magnitudes of the stars in bulk made at Harvard, Potsdam, Gottingen, Greenwich and Yerkes Observatories. The revised Harvard photometry gives the visual magnitudes of all stars down to about 6m'5 ; the Potsdam magnitudes, also visual, carry us in a more limited part of the sky as far as magnitude 7m'5. The Gottingen determinations, which are absolute determinations, inde- pendent of but agreeing very fairly with the Harvard

6 STELLAR MOVEMENTS CHAP.

sequences, provide photographic magnitudes over a large zone of the sky for stars brighter than 7m<5. The Yerkes investigation gives visual and photographic magnitudes of the stars within 17° of the North Pole down to 7m>5. A series of investigations at Greenwich, based on the Harvard sequences, provides statistics for the fainter stars extending as far as magnitude 17, and is a specially valuable source for the study of the remoter parts of the stellar system.

So far we have been considering the apparent brightness of stars and not their intrinsic brightness. The latter quantity can be calculated when the distance of the star is known. We shall measure the absolute luminosity in terms of the Sun as unit. The brightness of the Sun has been measured in stellar magnitudes and may be taken to be — 26m'l, that is to say it is 26*1 magnitudes brighter than a star of zero magnitude.* From this the luminosity L of a star, the magnitude of which is m, and parallax is &", is given by

Iog10 L = 0'2 - 0'4 x w -2 loglo Z.

The absolute magnitudes of stars differ nearly as widely as their apparent magnitudes. The feeblest star known is the companion to Groombridge 34, which is eight magni- tudes fainter than the Sun. Estimates of the luminosities of the brightest stars are usually very uncertain ; but, to take only results which have been definitely ascertained, the Cepheid Variables are on an average seven magnitudes brighter than the Sun. Probably this luminosity is exceeded by many of the Orion type stars. There is thus a range of at least fifteen magnitudes in the intrinsic brightness, or a light ratio of 1,000,000 to 1.

* The most recent researches give a value -26*5 (Ceraski, Annul* of the Obwvatory of Moscow, 1911). It is best, however, to regard the unit of luminosity as a conventional unit, roughly representing the Sun. ami defined by the formula, rather than to keep changing tin- measures of stellar luminosity every time a better determination of the Sun's stellar magnitude is mad*-.

i THE DATA OF OBSERVATION 7

Type of Spectrum. — For the type of spectrum various systems of classification have been used by astro- physicists, but that of the Draper Catalogue of Harvard Observatory is the most extensively employed in work on stellar distribution. This is largely due to the very complete classification of the brighter stars that has been made on this system. The classes in the supposed order of evolution are denoted by the letters : —

O, B, A, F, G, K, M, N.

A continuous gradation is recognised from 0 to M, and a more minute sub-division is obtained by supposing the transition from one class to the next to be divided into tenths. Thus B5A, usually abbreviated to B5, indicates a type midway between B and A ; G2 denotes a type between Gr and K, but more closely allied to the former. It is usual to class as Type A all stars from AO to A9 ; but presumably it would be preferable to group together the stars from B6 to A5, and this principle has occasionally been adopted.

For our purposes it is not generally necessary to consider what physical peculiarities in the stars are represented by these letters, as the knowledge is not necessary for discussing the relations of motion and distribution. All that we require is a means of dividing the stars according to the stage of evolution they have attained, and of grouping the stars with certain common characteristics. It may, however, be of interest to describe briefly the principles which govern the classification, and to indicate the leading types of spectrum.

Tracing an imaginary star, as it passes through the successive stages of evolution from the earliest to the latest, the changes in its spectrum are supposed to pursue the following course. At first the spectrum consists wholly of diffuse bright bands on a faint continuous back- ground. The bands become fewer and narrower, and faint

8 STELLAR MOVEMENTS CHAP.

absorption lines make their appearance ; the first lines seen are those of the various helium series, the well- known Balmer hydrogen series, and the " additional " or "sharp" hydrogen series.* The last is a spectrum which had been recognised in the stars by E. C. Pickering in 1896 but was first produced artificially by A. Fowler in 1913. The bright bands now disappear, and in the remaining stages the spectrum consists wholly of absorp- tion lines and bands, except in abnormal individual stars, which exceptionally show bright lines. The next phase is an enormous increase in the intensity of the true hydrogen spectrum, the lines becoming very wide and diffused ; the other lines disappear. The lines H and K of calcium and other solar lines next become evident and gain in intensity. After this the hydrogen lines decline, though long remaining the leading feature of the spectrum ; first a stage is reached in which the calcium spectrum becomes very intense, and afterwards all the multitudinous liues of the solar spectrum are seen. After passing the stage reached by our Sun, the chief feature is a shortening of the spectrum from the ultraviolet end, a further fading of the hydrogen lines, an increase in the number of fine absorption lines, and finally the appearance of bands due to metallic compounds, particularly the flutings of titanium oxide. The whole spectrum ultimately approxi- mates towards that of sunspots.

Guided by these principles, we distinguish eight leading types, between which, however, there is a continuous series of gradations.

TYPE 0 (WOLF-RAYET TYPE) — The spectrum consists of bright bands on a faint continuous background ; of these the most conspicuous have their centres at \\ 4633, 4651, 4686, 5693, and 5813. The type is divided into

* The practical worU of A. Fowler and the theoretical researches of N. I5« tin- leave little doubt that this spectrum is due to helium, notwithstanding its simple numerical relation to the hydrogen spectrum.

i THE DATA OF OBSERVATION 9

five divisions, Oa, Ob, Oe, Od, and Oe, marked by varying intensities and widths of the bands. Further, in Od and Oe dark lines, chiefly belonging to the helium and helium- hydrogen series, make their appearance.

TYPE B (ORION TYPE) — This is often called the helium type owing to the prominence of the lines of that element. In addition there are some characteristic lines the origin of which is unknown, as well as both the " sharp " and Balmer series. The bright bands seen in Type 0 are no longer present; in fact, they disappear as early as the sub-division OeoB, which is therefore usually reckoned the starting point of the Orion type.

TYPE A (SiRiAN TYPE). — The Balmer series of hydro- gen is present in great intensity, and is far the most conspicuous feature of the spectrum. Other lines are present, but they are relatively faint.

TYPE F (CALCIUM TYPE). — The hydrogen series is still very conspicuous, but not so strong as in the preceding type. The narrow H and K lines of calcium have become very prominent, and characterise this spectrum.

TYPE G (SOLAR TYPE). — The Sun may be regarded as a typical star of this class, the numerous metallic lines having made their appearance.

TYPE K. — The spectrum is somewhat similar to the last. It is mainly distinguished by the fact that the hydrogen lines, which are still fairly strong in the G stars, are now fainter than some of the metallic lines.

TYPE M. — The spectrum is now marked by the appear- ance of flutings, due to titanium oxide. It is remarkable that the spectrum should be dominated so completely by this one substance. Two successive stages are recognised, indicated by the sub-divisions Ma and Mb. The long- period variable stars, which show bright hydrogen lines in addition to the ordinary characteristics of Type M, form the class Md.

TYPE N. — The regular progression of the types

io STELLAR MOVEMENTS CHAP.

terminates with Mb. There is no transition to Type N, and the relation of this to the foregoing types is uncertain. It is marked by characteristic flu tings attributed to com- pounds of carbon. The stars of both the M and N types are of a strongly reddish tinge.

It is sometimes convenient to use the rather less detailed classification of A. Secchi. Strictly speaking his system relates to the visual spectrum, and the Draper notation to the photographic ; but the two can well be harmonised.

Secchi's Type I. includes Draper B and A

„ II. „ „ F, G, andK

„ III. „ „ M

TV N

>j »»-"-'• » » -^

As comparatively few of the stars in any catalogue belong to the last two types, this classification is practically a separation into two groups, which are of about equal size. This is a very useful division when the material is too scanty to admit of a more extended discussion.

From time to time there are indications that the Draper classification has not succeeded in separating the stars into really homogeneous groups. According to Sir Norman Lockyer, there are stars of ascending temperature and of descending temperature in practically every group ; so that, for example, the stars enumerated under K are a mixture of two classes, one in a very early, the other in a late, stage of evolution. In the case of Type B, H. Ludendorff1 has found considerable systematic differences in the measured radial motions of the stars classed by Lockyer as ascending and descending respectively (pointing, however, to real differences not of motion but of physical state, which have introduced an error into the spectroscopic measurements). E. Hertzsprung 2 has pointed out that the presence or absence of the c character on Miss Maury's classification (i.e., sharply defined absorption-lines) corre- sponds to an important difference in the intrinsic luminosities of the stars. Hitherto, however, it has

THE DATA OF OBSERVATION

ii

been usual to accept the Draper classification as at least the most complete available for our investigations.

Colour-Index. — Stars may be classified according to colour as an alternative to spectral type. Both methods involve dividing the stars according to the nature of the light emitted by them, and thus have something in common. Perhaps we might not expect a very close correspondence between the two classifications ; for, whilst colour depends mainly on the continuous background of the spectrum, the spectral type is determined by the fine lines and bands, which can have little direct effect on the colour. Nevertheless a close correlation is found between the two characters, owing no doubt to the fact that both are intimately connected with the effective temperature of the star.

The most convenient measure of colour is afforded by the difference, photographic minus visual magnitude ; this is called the colour-index. The relation between the spectral type and the colour-index is shown below.

	Colour index according to
Spectral Type.
	King.	Schwarzschild.
	m	m
Bo	-0-31	-0-64
Ao	o-oo	-0-32
Fo	+ 0-32	o-oo
Go	+ 0-71	+ 0-32
Ko	+ 1-17	+ 0-95
M	+ 1-68	+ 1-89

I

King's results 3 refer to the Harvard visual and photographic scales ; Schwarzschild's 4 to the Gottingen photographic and Potsdam visual determinations. Allowing for the constant difference, depending on the particular type of spectrum for which the photographic and visual magnitudes are

12 STELLAR MOVEMENTS CHAP.

made to agree, the two investigations confirm one another closely.

The foregoing results are derived from the means of a considerable number of stars, but the Table may be applied to individual stars with considerable accuracy. Thus the spectral type can be found when the colour- index is known, and conversely. At least in the case of the early type stars, the spectral type fixes the colour- index with an average uncertainty of not more than Om'l ; for Types G and K larger deviations are found, but the correlation is still a very close one.

Yet another method of classifying stars according to the character of the light emitted is afforded by measures of the "effective wave-length.'* If a coarse grating, consist- ing of parallel strips or wires equally spaced, is placed in front of the object-glass of a telescope, diffraction images appear on either side of the principal image. These diffraction images are strictly spectra, and the spot which a measurer would select as the centre of the image will depend on the distribution of intensity in the spectrum. Each star will thus have a certain effective wave-length which will be an index of its colour, or rather of the appreciation of its colour by the photographic plate. For the same telescope and grating the interval between the two first diffracted images is a constant multiple of the effective wave-length. The method was first used by K. Schwarz- schild in 1895 ; and an important application of it was made by Prosper Henry to determine the effect of atmospheric dispersion on the places of the planet Eros. It has been applied to stellar classification by E. Hertzsprung.5

Parallax. — The annual motion of the earth around the Sun causes a minute change in the direction in which a star is seen, so that the star appears to describe a small ellipse in the sky. This periodic displacement is superposed on the uniform proper- motion of the star,

i THE DATA OF OBSERVATION 13

which is generally much greater in amount ; there is, however, no difficulty in disentangling the two kinds of displacement. Since we are only concerned with the direction of the line joining the two bodies, the effect of the earth's motion is the same as if the earth remained at rest, and the star described an orbit in space equal to that of the earth, but with the displacement reversed, so that the star in its imaginary orbit is six months ahead of the earth. This orbit is nearly circular ; but, as it is generally viewed at an angle, it appears as an ellipse in the sky. In any case the major axis of the ellipse is equal to the diameter of the earth's orbit ; and, since the latter length is known, a determination of its apparent or angular magnitude affords a means of calculating the star's distance. The parallax is defined as the angle subtended by one astronomical unit (the radius of the earth's orbit) at the distance of the star, and is equivalent to the major semiaxis of the ellipse which the star appears to describe.

The measurement of this small ellipse is always made relatively to some surrounding stars, for it is hopeless to make absolute determinations of direction with the necessary accuracy. The relative parallax which is thus obtained needs to be corrected by the amount of the average parallax of the comparison stars in order to obtain the absolute parallax. This correction can only be guessed from our general knowledge of the distances of stars similar in magnitude and proper motion to the comparison stars ; but as it could seldom be more than 0"*01, not much uncertainty is introduced into the final result from this cause.

The parallax is the most difficult to determine of all the quantities which we wish to know, and for only a very few of the stars has it been measured with any approach to certainty. Until some great advance is made in means of measurement, all but a few hundreds of the nearest stars must be out of range of the method. But so

i4 STELLAR MOVEMENTS CHAP.

Laborious are the observations required, that even these will occupy investigators for a long while. In general, the published lists of parallaxes contain many that are ex- tremely uncertain, and some that are altogether spurious. Statistical investigations based on these are liable to be very misleading. Nevertheless, it is believed that by rejecting unsparingly all determinations but those of the highest refinement, some important information can be obtained, and in Chapter III. these results are discussed. In addition, determinations which are not individually of high accuracy may be used for finding the mean parallaxes of stars of different orders of magnitude and proper motion, provided they are sensibly free from systematic error ; these at least serve to check the results found by less direct methods.

The measurements are generally made either by photo- graphy or visually with a heliometer. The former method now appears to give the best results owing to the greater focal-length of the instruments available. It has also the advantage of using a greater number of comparison stars, so that there is less chance of the correction to reduce to absolute parallax being inaccurate. Some early determin- ations with the heliometer are, however, still unsurpassed. The meridian-circle is also used for this work, and consider- able improvement is shown in the more recent results of this method ; but we still think that meridian parallaxes are to be regarded with suspicion, and have deemed it best not to use them at all in Chapter III.

A convenient unit for measuring stellar distances is the parsec* or distance which corresponds to a parallax of one

* The parsec, a p<>i •tin.-uitrau-namr sujj^i-strd by H. H. Turner, will be used throughout this book. Several different units of stellar distance have, however, been employed by investigators. Kobold's Sternweite is identical with the parsec. Seeliger's Siriusu'eite corresponds to a parallax 0"'2, and ( 'h;irli»-r's Xiriometer to a million astronomical units or purallax 0"'206. The light-year which, notwithstanding its inconvenience and irrelevance, has sometimes crept from popular use into technical investigations, is e<]ual to 0'31 parsecs.

i THE DATA OF OBSERVATION 15

second of arc. This is equal to 206,000 astronomical units or about 19,000,000,000,000 miles. The nearest fixed star, a Centauri, is at a distance of 1*3 parsecs. There are perhaps thirty or forty stars within a distance of five parsecs, and, of course, the number at greater distances will increase as the cube of the limiting distance, so long as the distribution is uniform. But these nearest stars are not by any means the brightest visible to us ; the range in intrinsic luminosity is so great that the apparent magni- tude is very little clue to the distance. A sphere of radius thirty parsecs would probably contain 6000 stars ; but the 6000 stars visible to the average eye are spread through a far larger volume of space. It appears indeed that some of the naked -eye stars are situated in the remotest parts of the stellar system.

A parallax-determination may be considered first-class if its probable error is as low as 0"'01. If, for instance, the measured parallax is 0"'05±0"*01, it is an even chance that the true value lies between 0"'06 and 0"'04, and we probably should not place much confidence in any nearer limits than 0"'07 and 0"'03. This is equivalent to saying that the star is distant something between fourteen and thirty-three parsecs from us. It will be seen that, when the parallax is as low as 0"'05, even the best measures give only the very roughest idea of the distance of the star, and for smaller values the information becomes still more vague. Clearly, to be of value a parallax must be at least 0"'05. It may be estimated that there are not more than 2000 stars so near as this, and a very large proportion of these will be fainter than the tenth magnitude. The chances are that, of five plates of the international Carte du Ciel taken at random, only one will be fortunate enough to pick up a serviceable parallax, and even that is likely to be a very inconspicuous star, which would evade any but the most thorough search. The prospect of so overwhelming a proportion of negative results suggests that, for the present

1 6 STELLAR MOVEMENTS CHAP.

at any rate, work can be most usefully done on special objects for which an exceptionally large proper motion affords an a priori expectation of a sensible parallax. A star of parallax 0"'05 may be expected to have a proper motion of 20" per century or more, and that seems to be a reasonable limit to work down to.

It will generally be admitted that a most valuable extension of our knowledge will result from precise measures of the distances of as many as possible of the individual stars that come within the range above mentioned. But many investigators have also sought to determine the mean parallaxes of stars of different magnitudes or motions. When the individual distances are too uncertain, the means of a large number may still have some significance. Whilst some useful results can be and have been obtained by this kind of research, its possibilities seem to be very limited. Generally speaking, this class of determination requires even greater refine- ment than the measurement of individual parallaxes ; refinement which is scarcely yet within reach. For example, the mean parallax of stars of the sixth magnitude is 0"*014 (perhaps a rather high estimate) ; that of the comparison stars would probably be about half this, so that the relative parallax actually measured would be 0"'007. The possible systematic errors depending on magnitude and colour (the mean colour of the sixth magnitude is perhaps different from the ninth) make the problem of determining this difference one of far higher difficulty than that of measuring the parallax of an individual star. It means gaining almost another decimal place beyond the point yet reached. We need not dwell on the enormous labour of observing the necessary fifty or one hundred sixth magnitude stars to obtain this mean with reason- able accuracy ; it might well be thought worth the trouble ; but there is no evidence that systematic errors have as yet been brought as low as 0"-001

i THE DATA OF OBSERVATION 17

even in the best work, and indeed it seems almost inconceivable.

From these considerations it appears that parallax- determinations should be directed towards : t (l) Individual stars with proper motions exceeding 20" a century. This will yield many negative results, but a fair proportion of successes.

(2) Classes of stars with proper motions less than 20", but still much above the average. These parallaxes will have to be found individually, but for the most part only the mean result for a class will be of use.

(3) A possible extension to classes of stars not dis- tinguished by large proper motion, provided it is realised that a far higher standard is required for this work, and that a freedom from systematic error as great as 0"*001 can be ensured.

Proper Motion. --For stellar investigation the proper motions, i.e.. the apparent angular motions of the stars, form most valuable material. For extension in our knowledge of magnitudes, parallaxes, radial velocities, and spectral classification, we shall ultimately come to depend on improved equipment and methods of observation ; but the mere lapse of time enables the proper motions to become known with greater and greater accuracy, and the only limit to our knowledge is the labour that can be devoted, and the number of centuries we are content to wait.

Proper motions of stars differ greatly in amount, but in general the motion of any reasonably bright star (e.g., brighter than 7m'0) is large enough to be detected in the time over which observations have already accumulated. Whilst it is quite the exception for a star to have a measurable parallax, it is exceptional for the proper motion to be insensible. It may be useful to give some idea of the certainty and trustworthiness of the proper motions in ordinary use, though the figures are^ necessarily only

C

1 8 STELLAR MOVEMENTS CHAP.

approximate. When the probable error is about 1" per century in both coordinates, the motion may be considered to be determined fairly satisfactorily ; the Groombridge and Carrington Catalogues, largely used in statistical investiga- tions, are of about this order of accuracy. A higher standard — probable error about 0" 6 per century — is reached in Boss's "Preliminary General Catalogue of 6188 Stars," which is much the best source of proper motions at present available. For some of the fundamental stars regularly observed at a large number of places throughout the last century the accidental error is as low as 0*'2 per century ; but the inevitable systematic errors may well make the true error somewhat larger. Various sources of systematic error, particularly uncertainties in the constant of precession and the motion of the equinox, may render the motions in any region of the sky as much as 0"*5 per century in error ; it is unlikely that the systematic error of the best proper motions can be greater than this, except in one or two special regions of southern declination, where exceptional uncertainty exists.

"We may thus regard the proper motions used in statistical researches as known with a probable error of not more than 1" per century in right ascension and declination. Koughly speaking, an average motion is from 3" to 7" per century. A centennial motion of more than 20" is considered large, although there are some stars which greatly exceed this speed. The fastest of all is C.Z. 5h 243 — a star of the ninth magnitude found by J. C. Kapteyn and R. T. A. Innes on the plates of the Cape Photographic Durchmusterung — which moves at the rate of 870" per century. This speed would carry it over an arc equal to the length of Orion's belt in just above a thousand years. Table 1 shows the stars at present known of which the centennial speed exceeds 300". The number of faint stars on this list is very striking ; and, as our information practically stops at the ninth magnitude, it

THE DATA OF OBSERVATION

may be conjectured that there are a number of still fainter stars yet to be detected.

TABLE 1. Stars with large Proper Motion.

Name. »j&	Dec. £nnual 1900. P™Per Motion.	Magnitude.
h. m. C.Z. 5h 243 5 8 Groombridge 1830 ... 11 47 Lacaille 9352 . . . 22 59	0 -45-0 870 + 38-4 7-07 -36'4 7-02	8-3 6-5 7-4
Cordoba 32416 0 0 611 Cygni 21 2	-37-8 6-07 + 38-3 5-25	8-5 5*6
Lalonde 21185 . 10 58	+ 36 '6 477	7 '6
f Indi . . . . 21 56	-57 -2 4-67	47
Lalonde 21258 . 11 0	+ 44-0 4-46	8-9
o2 Eridani 4 11	- 7'8 4-08	4-5
fO.A. (s.) 14318 15 5 \O.A. (s.) 14320. 15 5	-16-0 3-76 -15-9 376	9-6 9'2
p. Cassiopeiae . . 12	+ 54-4 375	5'3
a1 Centauri 14 33	-60-4 3*66	0*3
Lacaille 8760 ... . 21 11 e Eridani .... 3 16	-39-2 3-53 -43'4 3-15	7-3 4 '3
O.A. (x.) 11677 .... 11 15	+ 66-4 3-03.	9-2

Comparatively little is known of the motions of stars fainter than the ninth magnitude. The Carrington proper motions, discussed by F. W. Dyson, carry us down to 10m'3 for the region within 9° of the North Pole. A number of the larger proper motions of faint stars in the Oxford Zone of the Carte du Ciel have been published by H. H. Turner and F. A. Bellamy.6 Further, by the reduc- tion of micrometric measurements,^ G. C. Comstock 7 has obtained a number of proper motions extending even to the thirteenth magnitude. There is no difficulty to be expected in securing data for faint stars ; but the work has been taken up comparatively recently, and the one essential is — lapse of a sufficient time.

Radial Velocity. — The velocity in the line of sight is measured by means of a spectroscope. In accordance with Doppler's Principle, the lines in the spectrum of a star are displaced towards the red or the violet

c 2

20 STELLAR MOVEMENTS CHAP.

(relatively to a terrestrial comparison spectrum) according as the star is receding from or approaching the earth. Unlike the proper motion, the radial motion is found directly in kilometres per second, so that the actual linear speed, unmixed with the doubtful element of distance, is known. Hitherto it has scarcely been possible to measure the velocities of stars fainter than the fifth magnitude, but that limitation is now being removed. The main difficulty in regard to the use of the results is the large proportion of spectroscopic binary stars, about one in three or four of the total number observed. As the orbital motion is often very much larger than the true radial velocity, it is essential to allow sufficient time to elapse to detect any variation in the motion, before assuming that the measures give the real secular motion, of which we are in search. Another uncertainty arises from possible systematic errors affecting all the stars belonging to a particular type of spectrum. There is reason to believe that the measured velocity of recession of the Type B stars is systematically 5 km. per sec. too great.8 This may be due to errors in the standard wave-lengths employed, or to a pressure-shift of the lines under the physical conditions prevailing in this kind of star. Smaller errors affect the stars of other types.

Apart from possible systematic error, a remarkable accuracy has been attained in these observations. For a star with sharp spectral lines a probable error of under 0'25 km. per sec. is well within reach. Stars of Types B and A have more diffuse lines and the results are not quite so good ; but the accuracy even in these cases is far beyond the requirements of the statistician. The observed velocities range up to above one hundred km. per sec., but speeds greater than sixty km. per sec. are not very common. The greatest speed yet measured is that of Lalande 19 GO, viz. , 325 km. per sec. The next highest is C.Z. 5h 243, already mentioned as having the greatest apparent motion acr<>—

i THE DATA OF OBSERVATION 21

the sky ; it is observed to be receding at the rate of 242 km. per sec., or 225 km. per sec. if we make allowance for the Sun's own motion. As these figures refer to only one component of motion, the total speeds of stars are sometimes considerably greater.

Radial motions of about 1400 stars have now been published, the great bulk of the observations having been made at the Lick Observatory. Most of this material only became accessible to investigators in 1913, and there has scarcely yet been time to make full use of the new data.

There are a few systems which can be observed both as visual and as spectroscopic binaries. In such cases it is possible to deduce the distance of the star by a method quite independent of the usual parallax determinations. From the visual observations, the period and the other elements of the orbit can be found. The dimensions, however, are all expressed in arc, i.e., in linear measure divided by the unknown distance of the star. From these elements we can calculate for any date the relative velocity in the line of sight of the two components ; but this also will be expressed as a linear velocity divided by the unknown distance. By comparing this result with the same relative velocity measured spectrographically, and therefore directly in linear measure, the distance of the system can be derived. This method is of very limited application ; but in the case of a Centauri it has given a very valuable confirmation of the parallax determined in the ordinary way. It increases our confidence that the usual method of measuring stellar distances is a sound one.

Mass and Density. — Knowledge of the masses and densities of stars is derived entirely from binary systems. The sources of information are of three kinds :—

(1) From visual binaries.

(2) From ordinary spectroscopic binaries.

(3) From eclipsing variables.

22

STELLAR MOVEMENTS

CHAP.

The combined mass of the two components of a binary system can be found from the length of the major semi- axis of the orbit a and the period P by the formula

Here the masses are expressed in terms of the Sun's mass as unit, and the astronomical unit and the year are taken as the units of length and time.

In a well-observed visual orbit, all the elements are known (except for an ambiguity of sign of the inclination), but the major axis is expressed in arc. This can be converted into linear measure, if the parallax has been determined ; and hence m^ + m2 can be found. When further, besides the relative orbit, a rough absolute orbit of one of the components has been found, by meridian observations or otherwise, the ratio fml\im.2 is determinate, and m1 and m2 are deduced separately. Owing to the difficulty of determining parallaxes, cases of a complete solution of this kind are rare. They are, however, sufficient to indicate the fact that the range in the masses of the stars is not at all proportionate to the huge range in their luminosities.

TABLE 2.

Well-detei-mined Masses of Stars.

	Combined System	Brighter Com- ponent
Star
Mass (Sun=l)	Period Years	a	Parallax	Luminos- ity (Sun-1)	Spectral Type
( Herculis . .	1-8	::i D 1-35	0-14	5-0	G
Procyon . . .	1-3	39-0	4-05	0-32	97	F5
Sirius ....	3"4	48-8	7-59	0-38	48-0	A
a Centauri . .	1-9	81-2 1771	0-76	2-0	G
7'i uphiuchi .	2-5	88-4	4-55	0-17	1-1	K
o2 Eridani .	n-7	lHO-n 479	0-17	0-84	G
rj Cassiopeiae * .	J-0	328-0 9-48	0-20	1-4	F5
					8

* Another published orbit P = 508y «-12'2" gives the mass =0'9. The great uncertainty of the orbit appears to have little effect on the result.

i THE DATA OF OBSERVATION 23

Table 2 contains all the systems, of which the masses can be ascertained with reasonable accuracy, i.e., systems for which good orbits 9 and good parallax-determinations 10 have been published. Possibly some of the more doubtful orbits would have been good enough for the purpose, but I doubt if the list could be much extended without lowering the standard.

Another fact which appears is that the ratio of the masses of the two components of a binary is generally not far from equality, notwithstanding considerable differences in the luminosity. Thus Lewis Boss11 in ten systems found that the ratio of the mass of the faint star to the brighter star ranged from 0'33 to IT,* the mean being 071. The result is confirmed by observations of such spectroscopic binaries as show the lines of both components, though in this case the disparity of luminosity cannot be so great.

Even wrhen the parallax is not known, important information as to the density can be obtained. Consider for simplicity a system in which one component is of negligible mass ; the application to the more general case requires only slight modifications, provided mjm.2 is known or can be assumed to have its average value. Let

d be the distance of the star

6 its radius

S its surface brightness

L, I its intrinsic and apparent luminosities

M its mass

p its density

y the constant of gravitation

Then

L = nV

and

* Excluding one very doubtful result.

24 STELLAR MOVEMENTS CHAP.

From these

-7 is the semi-axis of the orbit in arc, and I and P are a

observed quantities ; consequently the coefficient of S% is known. We have thus an expression for the density in terms of the surface brightness, and can at least compare the densities of those stars which, on spectroscopic evi- dence, may be presumed to have similar surface conditions.

The deusity is found to have a large range, many of the stars being apparently in a very diffused state with densities perhaps not greater than that of atmospheric air.

The spectroscopic binaries also give some information as to the masses of stars. The formula (mx + m2) = a3/P2 is applicable, and as a is now found in linear measure it is not necessary to know the parallax. The quantity deduced from the observations is, however, in this case a sin i, where i is the inclination of the plane of the orbit. The inclination remains unknown, except when the star is an eclipsing variable * or in the rare case when the system is at the same time a visual and a spectroscopic binary. For statistical purposes, such as comparing the masses of different types of stars, we may assume that in the mean of a sufficient number of cases the planes of the orbits will be distributed at random, and can adopt a mean value for sin i. Thus from spectroscopic binaries the average masses of classes, but not of individual stars, can be found.

In the case of eclipsing variable stars, the densities of the two components can be deduced entirely from the light- curve of the star. Although these are necessarily spectro- scopic binary systems, observations of the radial velocity are not needed, and are not used in the results. The actual procedure, which is due to H. N. Russell and H. Shapley,12 is too complicated to be detailed here, as

* In this case it is evident that * must be nearly 90°, and accordingly sin i may be taken as unity.

i THE DATA OF OBSERVATION 25

it bears only incidentally on our subject ; but the general principle may be briefly indicated, it will be easily realised that the proportionate duration of the eclipse and other features of the light curve do not depend on the absolute dimensions of the system, but on the ratio of the three linear quantities involved, viz., the diameters of the two stars and the distance between their centres. By strictly geometrical considerations therefore, we find the radii rlt r2 of the stars expressed in terms of the unknown semi-axis of the orbit a as unit. Now the relation between the mass and density of a star involves the cube of the radius, and the dynamical relation between the mass and the period involves the cube of a. Thus, on division, the absolute masses and the unknown unit a disappear simul- taneously, and we are left with the density expressed in

7* /T»

terms of the period and the known ratios — , — . Tiie

a a

key to the solution is that in astronomical units the Dimensions of density are (time)"2; the density thus depends on the period, and on the ratios, but not the absolute values, of the other constants of the system.

The densities found in this way are not quite rigorously determined. It is necessary to assume a value of the ratio of the masses of the two stars ; as already explained, this ratio does not differ widely from unity, but in extreme cases the results may be as much as fifty per cent, in error from this cause. Further, the darkening at the limb of the star has some effect on the determination, and the assumed law of darkening is hypothetical. By taking different assumptions, between which the truth is bound to lie, it can be shown that these uncertainties do not amount to anything important, when regard is had to the great range in stellar densities which is actually found.

We may conclude this account of the nature of the observations, on which our knowledge of the stellar

26 STELLAR MOVEMENTS CHAP.

universe is based, by a reference to J. C. Kapteyn's " Plan of Selected Areas." When the study of stars was confined mainly to those brighter than the seventh magnitude, and again when it was extended as far as the ninth, or tenth, a complete survey of all the stars was not an impossible aim, and indeed all the data obtainable could well be utilised. But investigations are now being pushed towards the fifteenth and even lower magnitudes. These fainter stars are so numerous that it is impossible and unneces- sary to do more than make a selection. As the different kinds of observation for parallax, proper motion, magnitude, spectral type, and radial velocity are highly specialised and usually carried out at different observatories, some co-opera- tion is necessary in order that so far as possible the observations may be concentrated on the same groups of stars. Kapteyn's13 plan of devoting attention to 206 selected areas, distributed all over the sky, so as to cover all varieties of stellar distribution, has met with very general support. The areas have their centres on or near the circles of declination, 0°, ±15°, ± 30°, ± 45°, =t 60°, ±75°, ± 90°. The exact centres have been chosen with regard to various practical considerations ; but the distribution is very nearly uniform. In addition to the main " Plan," 46 areas in the Milky Way have been chosen, typical of its main varieties of structure. The area proper consists of a square 75' x 75', or alternatively a circle of 42' radius ; but the dimensions may be extended or diminished for investigations of particular data.

The whole scheme of work includes nine main sub- divisions : (1) A Durchmusterung of the areas. (2) Standard photographic magnitudes. (3) Visual and photovisual magnitudes. (4) Parallaxes. (5) Differential proper motions. (6) Standard proper motions. (7) Spectra. (8) Radial velocities. (9) Intensity of the background of the sky. The Durchmusterung is well advanced ; it will include all stars to 17m, the positions being given with a

i THE DATA OF OBSERVATION 27

probable error of about 1" in each co-ordinate, and the magnitudes (differential so far as this part of the work is concerned) with a probable error of Om*l. Considerable progress has been made with the determination of sequences of standard photographic magnitudes for each area. The work of determining the visual magnitudes has been partly accomplished for the northern zones. For parallaxes, most of the areas have been portioned out between different Observatories ; the greatest progress has been made at the Cape Observatory for the southern sky, but the plates have not yet been measured. From what has already been stated with regard to the practical possibilities of parallax-determinations, it will be seen that there is some doubt as to the utility of this part of the Plan. For the proper motions of faint stars, the work has necessarily been confined mainly to obtaining plates for the initial epoch. At the Radcliffe Observatory, 150 plates have been stored away undeveloped, ready for a second exposure after a suitable interval, but in most cases it is intended to rely on the parallax plates for giving the initial positions. For standard proper motions in the northern sky, observations are shortly to be started at Bonn ; these will serve for com- parison with older catalogues, but they may also be regarded as initial observations for more accurate determinations in the future. Determinations of spectral type as far as the ninth magnitude, made at Harvard, will shortly be available for these areas and, indeed, for the whole sky. The extension to the eleventh magnitude is very desirable, and is one of the most urgent problems of the whole Plan. Radial velocity determinations are being pressed as far as 8m*0 at Mount Wilson, but rapid progress is not to be expected until the completion of the 100- inch reflector. A valuable, though unofficial, addition to the programme is E. A. Fath's Durchmusterung u of all the nebulae in the areas from the North pole to Dec. -15°.

28 STELLAR MOVEMENTS CHAP.

REFERENCES. — CHAPTER I.

1. Ludendorff, Astr. Nach., No. 4547.

2. Hertzsprung, Astr. Nach., No. 4296.

3. King, Harvard Annalx, Vol. 59, p. 152.

4. Schwarzschild, " Aktinometrie, " Teil B, p. 19.

5. Hertzsprung, Potsdam Publications, No. 63 ; Astr. Nach., No. 4362 (contains a bibliography).

6. Monthly Notices, Vol. 74, p. 26.

7. Comstock, Pub. Washburn Observatory, Vol. 12, Pt. 1.

8. Campbell, Lick Bulletin, No. 195, p. 104.

9. Aitken, Lick Bulletin, No. 84.

10. Kapteyn and Weersma, Groningen Publications, No. 24.

11. Boss, Preliminary General Catalogue of 6188 Stars, Introduction,

p. 23.

12. Russell and Shapley, Astrophysical Journal, Vol. 35, p. 315, et seq.

13. Kapteyn, "Plan of Selected Areas"; ditto, ''First and Second

Reports " ; Monthly Notices, Vol. 74, p. 348.

14. Fath, Astronomical Journal, Nos. 658-9.

BIBLIOGRAPHY.

Magnitudes. — The Harvard Standard Polar Sequence is given in Harvard Circular, No. 170. For an examination by Scares, see Astrophysical Journal, Vol. 38, p. 241.

The chief catalogues of visual stellar magnitudes are : — ** Revised Photometry," Harv. Ann., Vols. 50 and 54. Miiller and Kempf, Potsdam Publications, Vol. 17.

For photographic magnitudes : —

Schwarzschild, "Aktinometrie," Teil B (Gottingen Abhandlungen,

Vol. 8, No. 4).

Greenurich Astrographic Catalogue, Vol. 3 (advance section). Parkhurst, Astrophysical Journal, Vol. 36, p. 169.

A useful discussion of the methods of determining photographic magni- tude will be found in an R.A.S. " Council Note," Monthly Notices, Vol. 73, p. 291 (1913).

Spectral Type*. — The most extensive determinations are to be found in Jltirv. Ann., Vol. 50, which gives the type for stars brighter than 6m>5. Scattered determinations of many fainter stars also exist. It is understood that a very comprehensive catalogue containing the types of 200,000 stars will shortly be issued from Harvard.

A description of the principles of the Draper classification is given in Harv. Ann., Vol. 28, pp. 140, 146.

Parallaxes. — The principal sources are : —

Peter, Abhandlunyen kb'niglichsiichsische Gesell. der Wissenschaften,

Vol. 22, p. 239, and Vol. 24, p. 179. Gill, Cape Annals, Vol. 8, Pt. 2 (1900).

i THE DATA OF OBSERVATION 29

Schlesinger, Aatrophyaical Journal, Vol. 34, p. 28. Russell and Hinks, Astronomical Journal, No. 618-9. Elkin, Chase and Smith, Yale Transactions, Vol. 2, p. 389. Slocum and Mitchell, J-sfro^/j^/m^ Jmrrwd, Vol.38, p. 1.

Very useful compilations of the parallaxes taken from all available sources are given by Kapteyn and Weersma, Qroningen Publications, No. 24, and by Bigourdan, Bulletin Axtronmiiiiiue, Vol. 26. Fuller references are given in these publications.

Proper Motions. — Lewis Boss's Preliminary General Catalogue of 6188 Stars, which includes proper motions of all the brighter stars, supersedes many earlier collections.

Dyson and Thackeray's New Reduction of Groombridge's Catalogue contains 4243 stars, including many fainter than the eighth magnitude, within 50' of the North Pole.

Still fainter stars are contained in the Greenwich-Carrington proper motions discussed by Dyson. Some of these are published in the Second Nine-Year Catalogue (1900), but certain additional corrections are required to the motions there given (see Monthly Notices, Vol. 73, p. 336). The complete list (unpublished) contains 3735 stars.

Porter's Catalogue, Cincinnati Publications. No. 12, gives 1340 stars of especially large proper motion.

Radial Velocities. — Catalogues containing a practically complete summary of the radial velocities at present available for discussion are given by Campbell in Lick Bulletin, Nos. 195, 211, and 229. These contain about 1350 stars.

CHAPTER II

GENERAL OUTLINE

THIS chapter will be devoted to a general description of the sidereal universe as it is revealed by modern researches. The evidence for the statements now made will appear gradually in the subsequent part of the book, and minor details will be filled in. But it seems necessary to presume a general acquaintance with the whole field of knowledge before starting on any one line of detailed investigation. At first sight it might seem possible to divide the subject into compartments — the distribution of the stars through space, their luminosities, their motions, and the characters of the different spectral types — but it is not possible to pursue these different branches of inquiry independently. Any one mode of investigation leads, as a rule, to results in which all these matters are involved together, and no one inquiry can be worked out to a conclusion without frequent reference to parallel investiga- tions. We have therefore adopted the unusual course of placing what may be regarded as a summary thus early in the book.

In presenting a summary, we may claim the privilege of neglecting many awkward difficulties and uncertainties that arise, promising to deal fairly with them later. We can pass over alternative explanations, which for the moment are out of favour ; though they need to be kept

30

CH. ii GENERAL OUTLINE 31

alive, for at any moment new facts may be found, which will cause us to turn to them again. The bare outline, devoid of the details, must not be taken as an adequate presentation of our knowledge, and in particular it will fail to convey the real complexity of the phenomena discussed. Above all, let it be remembered that our object in building up a connected idea of the universe from the facts of observation is not to assert as unalterable truth the views we arrive at, but, by means of working hypotheses, to assist the mind to grasp the interrelations of the facts, and to prepare the way for a further advance. When we look back on the many transformations that theories in all departments of science have undergone in the past, we

m =»*:•

-Galactic Plane

FIG. 1.— Hypothetical Sect on of the Stellar System.

shall not be so rash as to suppose that the mystery of the sidereal universe has yielded almost at the first attack. But as each revolution of thought has contained some kernel of surviving truth, so we may hope that our present representation of the universe contains something that will last, notwithstanding its faulty expression.

It is believed that the great mass of the stars with which we are concerned in these researches are arranged in the form of a lens- or bun-shaped system. That is to say, the system is considerably flattened towards one plane. A general idea of the arrangement is given in Fig. 1 , where the middle patch represents the system to which we are now referring. In this aggregation the Sun occupies a fairly central position, indicated by -K The median plane of the lens is the same as the plane marked out in the sky by the

32 STELLAR MOVEMENTS CHAP.

Milky Way, so that, when we look in any direction along the galactic plane (as the plane of the Milky Way is called), we are looking towards the perimeter of the lens where the boundary is most remote. At right angles to this, that is, towards the north and south galactic poles, the boundary is nearest to us ; so near, indeed, that our telescopes can penetrate to its limits. The actual position of the Sun is a little north of the median plane ; there is little evidence as to its position with respect to the perimeter of the lens ; all that we can say is that it is not markedly eccentric.

The thickness of the system, though enormous com- pared with ordinary units, is not immeasurably great. No definite distance can be specified, because it is unlikely that there is a sharp boundary ; there is only a gradual thinning out of the stars. The facts would perhaps be best expressed by saying that the surfaces of equal density resemble oblate spheroids. To give a general idea of the scale of the system, it may be stated that in directions towards the galactic poles the density continues practically uniform up to a distance of about 100 parsecs ; after that the falling off becomes noticeable, so that at 300 parsecs it is only a fraction (perhaps a fifth) of the density near the Sun. The extension in the galactic plane is at least three times greater. These figures are subject to large uncertainties.

It seems that near the Sun the stars are scattered in a fairly uniform manner ; any irregularities are on a small scale, and may be overlooked in considering the general architecture of the stellar system. But in the remoter parts of the lens, or more probably right beyond it, there lies the great cluster or series of star-clouds which make up the Milky Way. In Fig. 1 this is indicated (in section) by the star-groups to the extreme left and right. These star-clouds form a belt stretching completely round the main flattened system, a series of irregular agglomera-

ii GENERAL OUTLINE 33

tions of stars of wonderful richness, diverse in form and grouping, but keeping close to the fundamental plane. It is important to distinguish clearly the two properties of the galactic plane, for they have sometimes been con- fused. First, it is the median plane of the bun-shaped arrangement of the nearer stars, and, secondly, it is the plane in which the star-clouds of the Milky Way are coiled.

Not all the stars are equally condensed to the galactic plane. Generally speaking, the stars of early type con- gregate there strongly, whereas those of late types are distributed in a much less flattened, or even in a practically globular, form. The mean result is a decidedly oblate system ; but if, for example, we consider separately the stars of Type M, many of which are at a great distance from us, they appear to form a nearly spherical system.

In the Milky Way are found some vast tracts of absorbing matter, which cut off the light of the stars behind. These are of the same nature as the extended irregular nebulae, which are also generally associated with the Milky Way. The dark absorbing patches and the faintly shining nebulae fade into one another insensibly, so that we may have a dark region with a faintly luminous edge. Whether the material is faintly luminous or not, it exercises the same effect in dimming or hiding the bodies behind it. There is probably some of this absorb- ing stuff even within the limits of the central aggregation. In addition to these specially opaque regions, it is probable that fine particles may be diffused generally through interstellar space, which would have the effect of dimming the light of the more distant stars ; but, so far as can be ascertained, this " fog " is not sufficient to produce any important effect, and we shall usually neglect it in the investigations which follow.

In studying the movements of the stars we necessarily leave the remoter parts of space, confining attention mainly to the lens-shaped system, and perhaps only to the inner

D

34. STELLAR MOVEMENTS CHAP.

parts of it, where the apparent angular movements are appreciable. Researches on radial motions need, not be quite so limited, because in them the quantity to be measured is independent of the distance of the stars ; but here too the nearer parts of the system obtain a preference, for observations are confined to the bright stars. Although thus restricted, our sphere of knowledge is yet wide enough to embrace some hundreds of thousands of stars (considered through representative samples) ; the results that are deduced will have a more than local importance.

The remarkable result appears that within the inner system the stars move with a strong preference in two opposite directions in the galactic plane. There are two favoured directions of motion ; and the appearance is as though two large aggregates of stars of more or less independent origin were passing through one another, and so for the time being were intermingled. It is true that such a straightforward interpretation seems to be at variance with the plan of a single oblate system, which has just been sketched. Various alternatives will be considered later ; meanwhile it is sufficient to note that the difficulty exists. But, whatever may be the physical cause, there is no doubt that one line in the galactic plane is singled out, and the stars tend to move to and fro along it in preference to any transverse directions. We shall find it convenient to distinguish the two streams of stars, which move in opposite directions along the line, reserving judgment as to whether they are really two independent systems or whether there is some other origin for this curious phenomenon. The names assigned are

Stream I. moving towards R. A. 94°, Dec. +12'. II. ,, „ R.A. 274% Dec. -12°.

The relative motion of one stream with respect to the other is about 40 km. per sec.

ii GENERAL OUTLINE 35

The Sun itself has an individual motion with respect to the mean of all the stars. Its velocity is 20 kilo- metres per second directed towards the point R.A. 270° Dec. + 35°. The stellar movements that are directly observed are referred to the Sun as standard, and are consequently affected by its motion. This makes a considerable alteration in the apparent directions of the two streams ; thus we find

Stream I. moving towards R.A. 91C, Dec. - 15° ^ relatively to „ II. „ „ R.A. 288°, Dec. -64° / the Sun ;

and moreover the velocity of the first stream is about 1'8 times that of the second (probably 34 and 19 km. per sec., respectively). Stream I is sometimes therefore referred to as the quick-moving stream, and Stream II as the slow- moving one ; but it must be remembered that this description refers only to the motion relative to the Sun. The stars which constitute the streams have, besides the stream-motion, individual motions of their own ; but the stream-motion sufficiently dominates over these random motions to cause a marked general agreement of direction.

Stream I contains more stars than Stream II in the ratio 3 : 2. Though this ratio varies irregularly in different parts of the sky, the mixture is everywhere fairly complete. Moreover, there is no appreciable difference in the average distances of the stars of the two streams. It is not a case of a group of nearer stars moving in one direction across a background of stars moving the opposite way ; there is evidence that the two streams thoroughly permeate each other at all distances and in all parts of the heavens.

A more minute investigation of this phenomenon shows that it is complicated by differences in the behaviour of stars according to their spectral type. An analysis which treats the heterogeneous mass of the stars as a whole

D 2

36 STELLAR MOVEMENTS CHAP.

without any separation of the different types will fail to give a complete insight into the phenomenon. But, until a great deal more material is accumulated, this interrelation of stream motions and spectral type cannot be worked out very satisfactorily. The outstanding feature, however, is that the stars of the Orion Type (Type B) seem riot to share to any appreciable extent in the star-streaming tendency. Their individual motions, which are always very small, are nearly haphazard, though the apparent motions are, of course, affected by the solar motion. They thus form a third system, having the motion of neither of the two great streams, but nearly at rest relatively to the mean of the stars. This third system is not entirely confined to the B stars. In the ordinary analysis into two streams we always find some stars left over — com- paratively few in number yet constituting a distinct irregularity — which evidently belong to the same system. These stars may be of any of the spectral types. There is something arbitrary in this dissection into streams (which may be compared to a Fourier or spherical harmonic analysis of observations), and we can, if we like, adopt a dissection which gives much fuller recognition to this third system.

At one time it seemed that the third stream, Stream 0 as it is called, might be constituted of the very distai.t stars, lying beyond those whose motions are the main theme of discussion. If that were so, it would not be surprising to find that they followed a different law, and were not comprised in the two main streams. But this explanation is now found to be at variance with the facts. \\ «• have to recognise that Stream 0 is to be found even among the nearer stars.

The smallness of the individual movements of the B stars is found to be part of a much more general law. Astrophysicists have by a study of the spectra arranged the stars in what they believe to be the successive stages

ii GENERAL OUTLINE 37

of evolution. Now it is found that there is a regular progression in the size of the linear motions from the youngest to the oldest stars. It is as though a star was born without motion, and gradually acquires or grows one. The average individual motion (resolved in one direction) increases steadily from about 6*5 km. per sec. for Type B to 1 7 km. per sec. for Type M.

We may well regard this relation of age and velocity as one of the most startling results of modern astronomy. For the last forty years astrophysicists have been studying the spectra and arranging the stars in order of evolution. However plausible may be their arguments one would have said that their hypotheses must be for ever outside the possibility of confirmation. Yet, if this result is right, we have a totally distinct criterion by which the stars are arranged in the same order. If it is really true that the mean motion of a class of stars measures its progress along the path of evolution, we have a new and powerful aid to the understanding of the steps of stellar development.

It is not at all easy to explain why the stellar velocities increase with advancing development. I am inclined to think that the following hypothesis offers the best explanation of the facts. In a primitive state the star- forming material was scattered much as the stars are now, that is, densely along the galactic plane up to moderate distances, and more thinly away from the plane and at great distances. Where the material was rich, large stars, which evolved slowly, were formed ; where it was rare, small stars, which developed rapidly. The former are our early type stars ; not having fallen in from any great distance they move slowly and in the main parallel to the galactic plane. The latter — our late type stars — have been formed at a great distance, and have acquired large velocities in falling in ; moreover since they were not necessarily formed near the galactic plane, their motions are not so predominantly parallel to it.

38 STELLAR MOVEMENTS CHAP.

Whilst the individual motion of a star gradually increases from type to type, the stream- motion appears with remark- able suddenness. Throughout Type B, even up to B 8 and B 9, the two star-streams are unperceived ; but in the next type, A, the phenomenon is seen in its clearest and most pronounced form. Through the remaining types it is still very prominent, but there is an appreciable falling off. There is no reason to believe that this decline is due to any actual decrease in the stream-velocities ; it is only that the gradual increase of the haphazard motions renders the systematic motions less dominant.

Among the most beautiful objects that the telescope reveals are the star-clusters, particularly the globular clusters, in which hundreds or even thousands of stars are crowded into a compact mass easily comprised within the field of a telescope. Recent research has revealed several systems, presumably of a similar nature to these, which are actually in our neighbourhood, and in one case even surrounding us. Being seen from a short distance the concentration is lost, and the cluster scarcely attracts notice. The detection of these systems relatively close to us is an important branch of study ; they are distinguished by the members having all precisely equal and parallel motions. The stars seem to be at quite ordinary stellar distances apart, and their mutual at traction is too weak to cause any appreciable orbital motion. They are not held together by any force ; and we can only infer that they continue to move together because no force has ever intervened to separate them.

These " moving clusters" are contained within the central aggregation of stars. Many of the globular clusters, though much more distant, are probably also contained in it ; others, however, may be situated in the star-clouds of the Milky Way. Their distribution in the sky is curiously uneven ; they are nearly all contained in one hemisphere. They are most abundant in Sagittarius and Ophiuchus,

ii GENERAL OUTLINE 39

near a brilliant patch of the Milky Way, which is undoubtedly the most extraordinary region in the sky. This might be described as the home of the globular clusters.

We shall also have to consider the nebulae, and their relation to the system of the stars. At this stage it may be sufficient to state that under the name "nebulae" are grouped together a number of objects of widely differing constitution ; we must not be deceived into supposing that the different species have anything in common. There is some reason for thinking that the spiral or "white" nebulas are objects actually outside the whole stellar system, that they are indeed stellar systems coequal with our own, and isolated from us by a vast intervening void. But the gaseous irregular nebulae, and probably also the planetary nebulae, are more closely associated with the stars and must be placed among them.

It is now time to turn from this outline of the leading- phenomena to a more detailed consideration of the problems. The procedure will be to treat first the nearest stars, of which our knowledge is unusually full and direct. From these we pass to other groups that happen to be specially instructive. From this very limited number of stars, a certain amount of generalisation is permissible, but our next duty is to consider the motions of the stars in general ; this will occupy Chapters V. — VII. After considering the dependence of the various phenomena on spectral type, we pass on to the problems of stellar distribution. This comes after our treatment of stellar motions, because the proper motions are, when carefully treated, among the most important sources of information as to the distances of stars. In Chapter XL we pass to subjects — the Milky Way and Nebulae — of which our knowledge is even more indefinite. The concluding Chapter attempts to introduce the problem of the dynamical forces under which motions of the stellar system are maintained.

CHAPTER III

THE NEAREST STARS

MOST of our knowledge of the distribution of the stars is derived by indirect methods. Statistics of stellar magni- tudes and motions are analysed and inferences are drawn from them. In this Chapter, however, we shall consider what may be learnt from those stars the distances of which have been measured directly ; and, although but a small sample of the stellar system comes under review in this way, it forms an excellent starting point, from which we may proceed to investigations that are usually more hypothetical in their basis.

For a parallax-determination of the highest order of accuracy, the probable error is usually about 0"'01. Thus the position of a star in space is subject to a compara- tively large uncertainty, unless its parallax amounts to at least a tenth of a second of arc. How small a ratio such stars bear to the whole number may be judged from the fact that the median parallax of the stars visible to the naked eye is only 0"'008 ; as many naked-eye stars have parallaxes below this figure as above it. We are therefore in this Chapter confined to the merest fringe of the surrounding universe, making for the present no attempt to penetrate into the general mass of the stars.

Table 3 shows all the stars that have been found to have

40

CH. Ill

THE NEAREST STARS

parallaxes of 0"*20 or greater.* Only the most trustworthy determinations have been accepted, and in most cases at least two independent investigators have confirmed one another. The list is based mainly on Kapteyn and Weersma's com- pilation.1

TABLE 3.

The Nineteen Nearest Stars. (Stars distant less than five parsecs from the Sun.)

Star.	Magni- tude.	Spectrum.	Parallax.	Luminos- ity (Sun = l).	Remarks.
Groombridge 34 . TJ Cassiopeiae . . . T Ceti	8-2 3-6 3-6	Ma F8 K	0-28 0-20 0-33-	o-oio 1-4 0-50	Binary Binary
f Eridani . ... CZ5h243 . . . Sirius	3-3 8-3 - 1-6	K G-K A	0-31 0-32 0'38	0-79 0-007 48-0	Binary
Procyon . ... Lai. 21185 . . . Lai. 21258 . . . OA (N.) 11677 . . a Centauri .... 0 A (N.) 17415 . . Pos. Med. 2164 . . v Draconis .... a Aquilne .... 61 Cygni f Indi	0-5 7-6 8-9 9-2 0-3 9-3 8-8 4-8 0-9 5-6 4'7	F5 Ma Ma G, K5 F K -rr A5 K5 K5	0-32- 0-40- 0-20 0-20 0-76' 0-27 0-29 0-20 0-24 0-31 0-28	97 0-009 0-011 0-008 J2-0 | 10-6 J 0-004 0-006 0-5 12-3 o-io 0-25	Binary Binary Binary Binary
Kriiger 60 .... Lacaille 9352 . . .	9-2 7'4	Ma	0-26 0-29	0-005 0-019	Binary

It is of much interest to inquire how far this list of nineteen stars is exhaustive. Does it include all the stars in a sphere about the Sun as centre with radius five parsecs ? In one respect the Table is admittedly incom- plete ; for stars fainter than magnitude 9'5 (on the B.D. scale), determinations are entirely lacking. A 9"U5 star with a parallax 0"'2 would have a luminosity O'OOG, the

* In using a table of this kind I am following the Astronomer Royal, F. \\ . Dyson, who first showed me its importance.

42 STELLAR MOVEMENTS CHAP.

Sun being the unit ; so that, in general, stars giving less than l/200th of the light of the Sun could not be included in the list. The distribution of the luminosities in column five of the Table leads us to expect that these very feeble stars may be rather numerous.

Admitting, then, that Table 3 breaks off at about luminosity O'OOG, and that in all probability numerous fainter stars exist within the sphere, how far is it complete above this limit ? Generally speaking, stars are selected for parallax-determinations on account of their large proper motions. Most of the very bright stars have also been measured, but in no case has a parallax greater than 0"'2 been found which was not already rendered probable by the existence of a large proper motion. To form some idea of the completeness with which the stars have been surveyed for parallax, consider those stars the motions of which exceed 1" per annum. F. W. Dyson has given a list of ninety-five of these stars,'2 and it is probable that his list is nearly complete, at least as far as the ninth magnitude ; the Durchmusterungs and Meridian Catalogues of most parts of the sky have involved so thorough a scrutiny, that it would be difficult for motions as large as this to remain unnoticed. Of these ninety-five stars, sixty-five may be considered to have well-determined parallaxes, or at least the determinations have sufficed to show that they lie beyond the limits of our sphere ; among the former are seventeen of the nineteen stars of Table 3. For the remaining thirty, either measurements have been unat- tempted, or the determinations do not negative the possi- bility of their falling within the sphere. This remainder is not likely to be sa rich in large parallaxes, because it includes a rather large proportion of stars which only just exceed the annual motion of 1" ; but there are some notable exceptions. The star Cordoba 32416, mag. 8*5, having the enormous annual motion of G"'07 seems to have been left alone entirely. It may be expected that

in THE NEAREST STARS 43

further examination of these thirty stars will yield four or five additional members for our Table.

Of stars with annual proper motions less than 1" the Table contains only two. It is not difficult to show that this is an inadequate proportion. The median parallax of stars distributed uniformly through the sphere (of radius 5 parsecs) is 0"'25 ; now for a star of that parallax an annual motion of 1" would be equivalent to a linear transverse motion of 20 km. per sec. Approximately then for our sphere

No. of PM's > 1" No. of transverse motions > 20 km. /sec.

No. of PM's < 1" No. of transverse motions < 20 km. /sec.

Now our general knowledge of stellar velocities, derived from other sources, is probably sufficiently good to give a rough idea of the latter ratio ; for it may be expected that the distribution of linear velocities within the sphere will not differ much from the general distribution outside. Taking the average radial velocity of a star as 17 km. per sec. (the figure given by Campbell for types K and M, which constitute the great majority of the stars), a Maxwellian distribution would give 64 per cent, of the transverse motions greater than 20 km. per sec. and 36 per cent, less — a ratio of 1*8 : 1. The addition of the solar motion will increase the proportion of high velocities ; and in those parts of the sky where it has its full effect the ratio is nearly 3:1. Probably we shall not be far wrong in assuming that there will be two-fifths as many stars with motions below 1" per annum as above it.

Having regard to these considerations, the calculation stands thus —

No. of stars in the Table with proper motion greater than 1" . 17

Proportionate allowance for stars not yet examined 5

Due proportion with proper motion less than 1", say .... 9

The Sun 1

Total 32

44 STELLAR MOVEMENTS CHAP.

To this must be added an unknown but probably consider- able number of stars the luminosity of which is less than 1 /200th of the Sun.

In round numbers we shall take thirty as the density of the stars within the sphere (tacitly ignoring the intrinsically faint stars). As twenty of these are actually identified, the number may be considered to rest on observation with very little assistance from hypothetical considerations.

This short list of the nearest stars well repays a careful study. Many of the leading facts of stellar distribution are contained in it; and, although it would be unsafe to generalise from so small a sample, results are suggested that may.be verified by more extensive studies.

Perhaps the most striking feature is the number of double stars. It will be seen that eight out of the nine- teen are marked " binary." Why some stars have split into two components, whilst others have held together, is an interesting question ; but it appears that the fission of a star is by no means an abnormal fate. The stars which separate into two appear to be not much less numerous than those which remain intact. The large number of discoveries of variable radial velocity made with the spectroscope • confirms this inference, though the spectro- scopists generally do not give quite so high a proportion. \V. W. Campbell3 from an examination of 1600 stars concludes that one-quarter are spectroscopic binaries. But this proportion must be increased if visual binaries are included (for these are not usually revealed by the spectro- scope) ; and, in addition, there must be pairs too far separated to be detected as spectroscopic binaries, but too distant from us to be recognised visually. E. B. Frost, examining the stars of Type B, found that two-fifths of those on his programme were binary ; he also found that in Boss's Taurus cluster the proportion was one-half.4

in THE NEAREST STARS 45

In the Ursa Major cluster nine stars out of fifteen are known to be binary.' Apparently the division into two bodies takes place at a very early stage in a star's history or in the pre-stellar state, as is evidenced by the high proportion found in the earliest spectral type. As time passes the components separate further from each other and the orbital velocity becomes small, so that in the later types an increasing proportion escapes detection. There thus seems no reason to doubt that the proportion eight out of nineteen very fairly represents the general average, but at the very lowest it cannot be less than one in three.

The luminosities of the stars in the Table range from 48 to 0*004, that of the Sun being taken as unit. We have already seen that the lower limit is due to the fact that our information breaks off near this point, and it is natural to expect that there must be a continuous series of fainter bodies terminating with totally extinct stars. At the other end of the scale a larger sample would un- doubtedly contain stars of much greater luminosity ; but these are comparatively rare in space. Arcturus, for example, is from 150 to 350 times as bright as the Sun, An tares at least 180 times, whilst Eigel and Canopus can scarcely be less than 2000 times as bright, allowing in each case a wide margin for the possible uncertainty of the measured parallaxes. There is little doubt that these estimates err on the side of excessive caution.

For a star of the same intrinsic luminosity as the Sun to appear as bright as the sixth magnitude, its parallax must be not less than 0"'08. Since there is no doubt that the majority of stars visible to the naked eye are much further away than this, it follows that the great majority of these stars must be much brighter, in fact more than a hundred times as bright as the Sun. We might hastily suppose that the Sun is therefore far below the average brilliancy. But Table 3 reveals a very different state of

4 6 STELLAR MOVEMENTS CHAP.

things. Of the nineteen stars, only five exceed it, whilst fourteen are fainter. The apparent paradox directs attention to a fact which we shall have occasion to notice frequently. The stars visible to the naked eye, and the stars enumerated in the catalogues, are quite unrepresentative of the stars as a whole. The more intensely luminous stars are seen and recorded in numbers out of all propor- tion to their actual abundance in space. It is of great importance to bear in mind this limitation of statistical work on star- catalogues ; we ought to consider whether the results derived from the very special kind of stars that appear in them may legitimately be extended to the stars as a whole.

In the same way, the Table gives a very different idea of the proportions in which the different types of spectra occur from the impression we should gather by examining the catalogues. Four stars of Type M are included, although in the catalogues this class forms only about a fifteenth of the whole number. On the other hand, Type B stars (the Orion type), which are rather more numerous than Type M in the catalogues, have not a single repre- sentative here. The explanation is that these M stars are usually very feebly luminous objects (as may be seen in the Table), and can rarely be seen except in our immediate neighbourhood. Type B stars on the other hand are intensely bright, and although they occur but sparsely in space, we can record even those near the limits of the stellar system, making a disproportionately large number. Again in the catalogues the Sirian Type stars (A) and the Solar Type (F, G, K) are about equally numerous, but in this definite volume of space the latter outnumber the former by ten to two.

In order to obtain more extensive data as to the true proportions of the spectral types, and the relation of spectral type to luminosity, Table 4 has been drawn up, containing the stars with fairly well-determined parallaxes

Ill

THE NEAREST STARS

47

between 0"'19 and 0"T1. These are not generally so trust- worthy as the parallaxes of Table 3, because chief atten- tion has naturally been lavished on those stars known to be nearest to us ; but the standard is fairly high, a great many measured parallaxes being rejected as too uncertain. Those marked with an asterisk are the best determined and may be considered equal in accuracy to the parallaxes of Table 3 ; but as the parallaxes are smaller, the propor- tionate uncertainty of the calculated luminosity is greater.

TABLE 4. Stars distant between 5 and 10 parsecs from the Sun.

Star. ^ludT	Spectrum.	Annual Proper Motion.	Parallax.	Luminos- ity (Sun = l).
f Toucanse . . . 4 "3	F8	2-07	0-15	1-3
*,•* Hydri . ... 2'9	G	2-24	0-14	5-4
54 Piscium ... O'l	K	0-59	0-15	0-26
Mayer 20 ... 5 '8	K	1-34	0-16	0-28
*p. Cassiopeire ... 5 '3	G5	375	0-11	1-0
8 Trianguli ... 5'1	G	1-16	0-12	1-0
Pi. 2* 123 ... 5-9	G5	2-31	0-14	0-33
e Eridani ... 4 '3	G5	3-15	0-16	1-15
*5 Eridani ... 3'3	K	075	0-19	2-1
o2 Eridani .... 4 '5	G5	4-08	0-17	0-84
X Aurig&e .... 4'8	G	0-85	0-11	1-5
*Weisse 5h 592 . . 8 '9	Ma	2-23	0-18	0-013
Pi. 5h 146 .... 6-4	G2	0-55	0-11	0-32
*Fed. 1457-8 ... 7 "9	Ma	1*69	0-16	0-042
*Groombridge 1618. 6 '8	K	1-45	0-18	0-09
43 Coma* .... 4'3	G	1-18	0-12	2-2
*Lalande 25372 . . 87	K	2-33	0-18	0-017
Lalande 26196 . . 7'6	G5	0-68	0-14	0-074
*Pi. 14h 212 ... 5-8	K	2-07	0-17	0-26
*Gronin^en VII.,
No. 20 .... 107	—	1-22	0-13	0-005
fHerculis .... 3'0	G	0-61	0-14	5-0
*Weisse 17h 322 . . 7 '8	Ma	1-36	0-12	0-08
70 Ophiuchi ... 4'3	K	1-15	0-17	1-1
*17 Lyrae C. ... 11 '3	—	175	0-13	0-003
Fomalhaut .... 1*3	A3	0-37	0-14	25-0
*Bradley 3077 - . . 5-6	K	2-11	0-14	0-45
*Lalande 46650 . . 8'9	Ma	1-40	0-18	0-013

Table 4 is far from being a complete list of the stars within the limits. There should be about 200 stars in

48 STELLAR MOVEMENTS CHAP.

this volume of space, but only 27 are given here. However, the incompleteness will not much affect the present inquiry, except that a number of the faintest stars, which are especially of Types K and Ma, will naturally be lost. This is noticeable when the list is compared with Table 3.

Collecting the stars of different spectral types, we have the following distribution of luminosities from Tables 3 and 4 :

Luminosities.

Type of Parallaxes Parallaxes

Spectrum. greater than 0'20". from 0'19" to O'll".

A ... 48-0

A3 . . 25-0

Ao . . 12-3

F . . . 0-004

F5 . . 9-7

F8 . . 1-4 1-3

G . . . 2-0 *5-4, 5-0, 2-2, 1-5, I'O

G2 . . 0-32

G5 . . 1-15, *1-0, 0-84, 0-33, 0'074

K /0-79 0-5 0-5 0-OOfi I *2'1» r1' *°'45' °'28' *°'26

. . O < J, 0 5, 0 o, 0 . *. *.

K5 . . 0-6, 0-25

Ma . . 0-019, 0-011 0-010, 0*009 *0'08, *0'042, *0'013, *0'013

This summary shows a remarkable tendency towards equality of brightness among stars of the same type, and there is a striking progressive diminution of brightness with advance in the stage of evolution. The single star of Type F (strictly FO) makes a curious exception. This star, OA (N) 17415, was measured with the heliometer by Kriiger as early as 1863 ; apparently the determination was an excellent one. It would perhaps be desirable to check his result by observations according to more modern methods ; but we are inclined to believe that the exception is real.

It would be tempting to conclude that the great range in absolute luminosity of the stars is mainly due to differences of type, and thnt within the same spectral

parallaxes.

in THE NEAREST STARS 49

the range is very limited. We might apply this in searching for large parallaxes ; for it would seem that, since stars of Type Ma have so far been found to be of very feeble luminosity, any star of that class which appears bright must be very close to us. This hope is not fulfilled. The bright Ma stars which have been measured — Betelgeuse, Antares, 77 Geminorum and 8 Vir- ginis — have all very small parallaxes, and are certainly much more luminous even than Sirius, the brightest star in the Tables. It is perhaps unfortunate that these brilliant exceptions force themselves on our notice, whilst the far greater number of normal members of the class are too faint to attract attention ; we must not be led into an exaggerated idea of the number of these luminous third type stars. But it is clear that, notwithstanding the tendency to equality, if a large enough sample is taken, the range of luminosity is very great indeed. We shall have to return to this subject in Chapter VIII.

Table 5 contains some details as to the motions of the nineteen nearest stars. The transverse velocities are formed by using the measured parallaxes to convert proper motions into linear measure.* The radial motions from spectroscopic observations are added when available. These motions are relative to the Sun ; if we wished to refer them to the centroid of the stars, we should have to apply the solar motion of 20 km. per sec., which might either increase or decrease the velocities according to circumstances, but would usually somewhat decrease them. After allowing for this, the commonness of large velocities is still a strik- ing and most surprising feature. The ordinary studies of stellar motions do not lead us to expect anything of the kind ; and in fact it is not easy to reconcile the general

* The formula, which is often useful, is

Linear speed = annual pjgpermotion_ x 4.;4 km per sec parallax

E

STELLAR MOVEMENTS

CHAP.

investigations with the results of this special study of a small collection of stars. Taking the fastest moving stars, Type M, Campbell found an average radial motion of 17 km. per sec. Assuming a Maxwellian distribution of velocities, this would give for a transverse motion (i.e., motion in two dimensions)

Speed greater than

60 km. per sec. 80 ,,

100 ,,

1 star in 53 1 star in 1,100 1 star in 60,000

The presence of 3 stars in the list with transverse velocities of more than 1 00 km. per sec. (and therefore certainly more than 80 km. per sec., when the solar motion is removed) is wholly at variance with the above statistical scheme.

TABLE 5. Motions of the nineteen nearest stars.

Star	Proper 5	lotion.	Radial
	Arc.	Linear.	Velocity.
Groombridge 34 .... r\ Cassiopeiae	2-85 1-25	km. per sec. 48 30	km. per sec. + 10	I. I.
T Ceti .	1-93	28	-16	II.
f Eridani	1-00	15	+ 16	II.
CZ 5h 243 ..	870	129	+ 242	II.
Sirius	1-32	16	— 7	II
Prooyon Lalande 21185 Lalande 21258 OA(N) 11677 a Centauri	1-25 477 4-46 3-03 3-66	19 57 106 72 23	-3 -22	1.1 II. I. I. I.
OA (N) 17415 .... Pos. Med. 2164 . . . <r Draconis	1-31 2-28 1-84	23 37 43	+ 25	II. I. II.
a Aquilae	0-65	T3	-33	I.
61 Cygni ( Indi . . . . .	5-25 4'67	80 79	-62 -39	I. I
Kriiger 60 Lacaille 9352	0-92 7-02	17 115	+ 12	II. I.

We cannot attribute the result to errors in the paral- laxes, for if it should happen that the parallax has

in THE NEAREST STARS 51

been overestimated, the speed will have been under- estimated ; and it is scarcely likely that any of these stars have parallaxes appreciably greater than those assigned in the Table. A possible criticism is that these stars have been specially selected for parallax-measurement, because they were known to have large proper motions ; but the objection has not much weight unless it is seriously suggested that there are, in this small volume, the hundreds or even thousands of stars of small linear motion that the statistical scheme seems to require. Moreover we have already shown that on the ordinary view as to stellar motions, seven additional stars would supply the loss due to the neglect of stars with motions less than 1" per annum.

Nor can we help matters by throwing over the Max- wellian law. Originally used as a pure assumption, this law has been confirmed in the main by recent study of the radial motions. We should, however, be prepared to admit that it may not give quite sufficient very large motions. It has long been known that certain stars, as Arcturus and Groombridge 1830, had excessive speeds that seemed to stand outside the ordinary laws. But we find that the average transverse motion of the nineteen stars is fifty km. per sec. ; this considerably exceeds the average speed we

should deduce from the stars that come into the ordinary

«/

investigations. In round numbers, a mean speed of thirty km. per sec. relative to the Sun would have been expected.*

Thus once more it is found that the survey of stars in the limited volume of space very near to the Sun leads to results differing from those derived from the stars of the catalogues. We may fall back on the same explana-

* An average radial speed of 17 km. per sec. gives an average transverse speed of 26 '5 km. per sec., the factor being ^ whatever the law (Maxwel-

lian or otherwise) of stellar motions. This does not include the solar motion, which, however, would not increase the result very greatly.

E 2

52 STELLAR MOVEMENTS CHAP.

tion as before, that the catalogues give a very untypical selection of the stars. But this time the result is more surprising ; it would scarcely have been expected that the catalogue-selection, which is purely by brightness, would have so large an effect on the motions. Yet this seems to be the case. We notice that the three stars with transverse speeds of more than 100 km. per sec. have luminosities 0'007, O'Oll and 0'019. These would be far too faint to come into the ordinary statistical investigations. The five stars brighter than the Sun have all very moderate speeds. Setting the nine most lum- inous stars against the ten feeblest, we have —

Luminosity. Mean transverse speed.

9 Brightest stars . . 48 '0 to 0'25 29 km. per sec.

10 Faintest „ . . O'lO „ 0'004 68

It is the stars with luminosity less than l/10th that of the Sun, with which we are scarcely ever concerned in the ordinary researches on stellar motions, that are wholly responsible for the anomaly. The nine bright stars simply confirm our general estimate of thirty km. per sec. for the average speed.

The stars of Table 4 also add a little evidence pointing in the same direction. The parallaxes are scarcely accurate enough (proportionately to their size) to be used for this purpose ; but we give the result for what it is worth. It may be noted that the parallaxes are likely to be a little overestimated and therefore both luminosities and speeds will be underestimated. On the other hand, the influence of selection (on account of large proper motion) will be greater than in Table 3, tending to increase the mean speed unduly. There are nine stars given in Table 4 as having luminosities less than O'l ; their mean speed is forty-eight km. per sec. Thus these stars have speeds con- siderably in excess of the thirty km. per sec. originally expected. They do not, however, include any excessive speeds.

in THE NEAREST STARS 53

It would be very desirable to have more evidence on this point before drawing a general conclusion ; but our task is to sum up the present state of our knowledge, however fragmentary. The stars, of which the proper motions and radial velocities are ordinarily discussed, are almost exclusively those at least as bright as the Sun. It is generally tacitly assumed that the motions of the far more numerous stars of less brilliance will be similar to them. But the present discussion affords a strong sus- picion that there exists a class of stars, comprising the majority of those having a luminosity below OT, the speeds of which are on the average twice as great as the fastest class ordinarily considered. Apparently the progressive increase of velocity with spectral type does not end with the seventeen km. per sec. of the brilliant members of Type M, but continues for fainter stars up to at least twice that speed.

In the final column of Table 5. each star has been assigned to its respective star-stream according to the direction in which it is moving. We see that eleven stars probably belong to Stream I, and eight probably to Stream II. This is in excellent accordance with the ratio 3 : 2 derived from the discussion of the 6000 stars of Boss's Catalogue. We are not able to detect any significant difference between the luminosities, spectra, or speeds of the stars constituting the two streams. The thorough inter-penetration of the t\vo star-streams is well illustrated, since we find even in this small volume of space that members of both streams are mingled together in just about the average proportion.

REFERENCES. — CHAPTER III.

1. Kapteyn and Weersma, Groningen Publications, No. 24.

2. Dyson, Proc. Roy. Sac. Edtnluruh, Vol. 29, p. 378. :5. Campbell, Stellar Motions, p. 245.

4. Frost, Axh-nit/iit.-ii'rtd J<»i,-,i<i.l, Vol. 29, p. 237.

5. Hertzsprung, Attrophyrical Journal, Vol. 30, p. 139.

CHAPTER IV

MOVING CLUSTERS

THE investigation of stellar motions has revealed a number of groups of stars in which the individual members have equal and parallel velocities. The stars which form these associations are not exceptionally near to one another, and indeed it often happens that other stars, not belonging to the group, are actually interspersed between them. We may perhaps arrive at a better understanding of these systems by recalling a few elementary considerations regarding double stars.

In only a small proportion of the double stars classed as " physically connected " pairs, has the orbital motion of one component round the other been detected. In most cases the connection is inferred from the fact that the two stars are moving across the sky with the same proper motion in the same direction. The argument is that, apart from exceptional coincidences, the equality of angular motion signifies both an equality of distance and an equality of linear velocity. Accordingly the two stars must be close together in space, and their motions are such that they must have remained close together for a long period. Having established the fact that they are permanent neighbours, we may rightly deduce that their mutual gravitation will involve some orbital motion, though it may be too slow to detect ; but that is a subsidiary

CH. iv MOVING CLUSTERS 55

matter, and, in speaking of physical connection, we are not thinking of two stars tied together by an attractive force. The connection, if we try to interpret it, appears to be one of origin. The components have originated in the same part of space, probably from a single star or nebula ; they started with the same motion, and have shared all the accidents of the journey together. If the path of one is being slowly deflected by the resultant pull of the stellar system, the path of the other is being deflected at the same rate, so that equality of motion is preserved. It is true that the mutual attraction in these widely separated binaries may help to prevent the stars separating ; but it is a very feeble tie, and, in the main, community of motion persists because there are no forces tending to destroy it.

From this point of view we may have physically con- nected pairs separated by much greater and even by ordinary stellar distances, remembering, however, that the greater the distance the more likely are they to lose their common velocity by being exposed to different forces. It is known that exceedingly wide pairs do exist The case of A Ophiuchi and Bradley 2179 may be instanced ; these stars are separated by about 14', but have the same unusually large motion of lr/<24 per annum in the same direction. In general it would be difficult to detect pairs of this kind ; for unless the motion is in some way remarkable, an accidental equality of motion must often be expected, and it would be impossible to distinguish the true pairs from the spurious. It is only when there is something unusual in the amount or in the direction of the motion that there are grounds for believing the equality is not accidental.

In the Moving Clusters we find a closely similar kind of physical connection. They are considerable groups of stars, widely separated in the sky, but betraying their association by the equality of their motions. The most

56 STELLAR MOVEMENTS CHAP.

thoroughly investigated example of a moving cluster is the Taurus-stream, which comprises part of the stars of the Hyades and other neighbouring stars. The existence of a great number of stars with associated motions in this region was pointed out by R. A. Proctor ; but the researches of L. Boss have shown the nature of the connection in a new light. Thirty-nine stars are recognised as belong- ing to the group, distributed over an area of the sky about 15° square; there can be no doubt that many additional fainter stars l in the region also belong to the cluster but, until better determinations of their motions have been made, these cannot be picked out with certainty.

The first criterion in such a case as this is that the motions should all appear to converge towards a single point in the sky ; in Fig. 2 the arrows indicate the observed motions of these stars, and the convergence is well shown. From this it may be deduced that the motions are parallel ; for lines parallel in space appear, when projected on a sphere, to converge to a point. It is true that the same appear- ance would be produced if the motions were all converging to or diverging from a point, but either supposition is obviously improbable. Theoretically there may be a slight degree of divergence, the cluster having been originally more compact ; but calculation shows that on any reasonable assumption as to the age of the cluster the divergence must be quite negligible. As the stars are not all at the same distance from the Sun, the fact that the speeds are all equal cannot be demonstrated in the same exact way ; but, allowing for the foreshortening of the apparent motion in the front of the cluster as compared with the rear, the proper motions all agree with one another very nearly. The divergences are just what we should expect if the cluster extends towards the Sun and away from it to the same distance that it extends laterally.

It is clear that in a cluster of this kind the equality ;md

IV

MOVING CLUSTERS

57

c

J C

si °C sJ C

} °c a *

i

3 °T

-c

J -c

3 °0

0 "u

3

\

"T CJ

\

ID ^

\

I

\

\ \\

I

f

ID

V

s

\ \

5

-\M

\ . \

T

*! *

\ V

CM CM

i

. 1 \

|

I

CO 0

\\

\

T oc

'

\

\ \

\

TT U3

\

\

\

I

i 1

in >

\

\

1

CV

\

t

!

0

\

\

1 \

C'J oc

\

\

i

CJ ur

\

1 1 44 \ \ \

1

CO -r

Vx \

rr CM

\ \

;

in

1

>

58 STELLAR MOVEMENTS CHAP.

parallelism of the motions must be extremely accurate, otherwise the cluster could not have held together. Suppose that the motion of one member deviated from the mean by one km. per sec. ; it would draw away from the rest of the cluster at the rate of one astronomical unit in 4f years. In ten million years it would have receded ten parsecs (the distance corresponding to a parallax of O^'IO). We shall see later that the actual dimensions of the cluster are not so great as this ; the remotest member is about seven parsecs from the centre. According to present ideas, ten million years is a short period in the life even of a planet like the earth ; the age of the Taurus-cluster, which contains stars of a fairly advanced type of evolution, must be vastly greater than this. From the fact that it still remains a compact group we deduce that the individual velocities must all agree to within a small fraction of a kilometre per second.

The very close convergence of the directions of motion of these stars supports this view ; the deviations may all be attributed to the accidental errors of observation. In fact the mean deviation (calculated - observed) in position angle is ±1°'8, whereas the expected deviation, due to the probable errors of the observed proper motions, is greater than this, — a paradox which is explained by the fact that stars for which the accidental error is especially great would not be picked out as belonging to the group.

To complete our knowledge of this cluster, one other fact of observation is required ; namely, the motion in the line of sight of any one of the individual stars. Actually six have been measured, and the results are in satisfac- tory accordance. The data are now sufficient to locate completely not only the cluster but its individual members and also to determine the linear motion, which, as has been shown, must be the same for all the stars to a very close approximation. This can be done as follows : —

IV

MOVING CLUSTERS

59

The position of the -convergent point, shown at the extreme left of Fig. 2, is found to be

R.A. 6h7m'2 Dec. +6' 56' (1875'0)

with a probable error of ±1°*5 chiefly in right ascension. If 0 is the observer (Fig. 3), let OA be the direction of this con- vergent point.

Consider one of the stars S of the cluster. Its motion * in space ST must be parallel to OA. Re- solve ST into transverse and radial components SX and SY. If SY has been measured by the spec- troscope, we can at once find ST for

ST = SY sec TSY O

= SY sec ACS FIG. 3.

and, since A and S are known points on the celestial sphere, the angle AOS is known.

Since the velocity ST is the same for every star of the cluster, it is sufficient to determine it from any one member the radial velocity of which has been measured. The result is found to be 45*6 km. per sec.

We next find the transverse velocity for each star, which is equal to

45 '6 sin AOS km. per sec.

And the stars' distances are given by

transverse velocity = distance x observed proper motion

when these are expressed in consistent units.

It will be seen that the distance of every star is found by this method, not merely those of which the radial velocity is known. The distance is found with a percentage accuracy

* The motions considered here are all measured relatively to the Sun.

60 STELLAR MOVEMENTS CHAP.

about equal to that of the observed proper motion ; for the other quantities which enter into the formulae are very well-determined. As the proper motions of these stars are large and of fair accuracy, the resulting distances are among the most exactly known of any in the heavens. The parallaxes range from 0"*021 to 0"*031, with a mean of 0"*025. A direct determination by photography, the result of the cooperation of A. S. Donner, F. Kiistner, J. C. Kapteyn, and W. de Sitter,2 has yielded the mean value 0"'023:±:0"'0025, which, though presumably less accurate than the result obtained indirectly, is a satisfactory confirmation of the legitimacy of Boss's- argument.

From these researches the Taurus-cluster appears to be a globular cluster with a slight central condensation ; its. whole diameter is rather more than ten parsecs. The question arises whether this system can be regarded as- similar to the recognised globular clusters revealed by the telescope. If there were no more members than the thirty- nine at present known, the closeness of arrangement of the stars would not be greater than that which we have found in the immediate neighbourhood of the Sun. But it appears- that the 39 are all much brighter bodies than the Sun, and it is not fair to make a comparison with the feebly luminous stars discussed in the last chapter. According to a rough calculation the members of the Taurus-cluster may be classified as follows :

5 stars with luminosity 5 to 10 times that of the Sun

18 „ „ 10 „ 20

11 „ „ 20 „ 50

5 „ „ 50 „ 100

In the vicinity of the Sun we have nothing to compare with this collection of magnificent orbs. These stars, it is true, are separated by distances of the usual order of magnitude ; but their exceptional brilliancy marks out this portion of space from an ordinary region. Whether

iv MOVING CLUSTERS 61

there are or are not other fainter members accompanying them, the term cluster is appropriate enough. There can be 110 doubt that, viewed from a sufficient distance, this assemblage would have the general appearance of a globular star-cluster.

The known motion of the Taurus-cluster permits us to trace its past and future history. It was in perihelion 800,000 years ago ; the distance was then about half what it is now. Boss has computed that in 65,000,000 years it will (if the motion is undisturbed) appear as an ordin- ary globular cluster 20' in diameter, consisting largely of stars from the ninth to the twelfth magnitude.

It is interesting to note that a cluster of the size of this Taurus group must contain many interloping stars not belonging to it. Even if we omit the outlying members, the system fills a space equal to a sphere of at least 5 parsecs radius. Now such a sphere in the neighbourhood of the Sun contains about 30 stars. We cannot suppose that a vacant lane among the stars has been specially left for the passage of the cluster. Presumably then the stars that would ordinarily occupy that space are actually there —non-cluster stars interspersed among the actual members of the moving cluster. It is a significant fact that the penetration of the cluster by unassociated stars has not disturbed the parallelism of the motions or dispersed the members.

The Ursa Major system is another moving cluster of which detailed knowledge has been ascertained. It has long been known that five stars of the Plough, viz., @, 7, S, e and { Ursae Majoris, form a connected system. By the work of Ejnar Hertzsprung it has been shown that a number of other stars, scattered over a great part of the sky, belong to the same association. The most interest- ing of these scattered members is Sirius ; and for it the evidence of the association is very strong. Its parallax and radial velocity are both well -determined, and agree

62 STELLAR MOVEMENTS CHAP.

with the values calculated from • the motion of the whole cluster. The method by which the common velocity is found, and the individual stars are located in space, is the same as that employed for the Taurus-cluster. The velocity is 18'4 km. per sec. towards the convergent point R.A. 127°'8, Dec. 4- 4 0°* 2, when measured relatively to the Sun. When the solar motion is allowed for, the " absolute " motion is 28*8 km. per sec. towards R.A. 285°, Dec. -2°. As this point is only 5° from the galactic plane, the motion is approximately parallel to the galaxy.

In Table 6 particulars of the individual stars are set down, including the parallaxes and radial velocities deduced by Hertzsprung from the known motion of the system. In most cases the calculated radial velocities have been confirmed by observation ; 3 but for nearly all the parallaxes, it has not been possible as yet to test the values given. It is quite likely that one or more stars have been wrongly included ; but there can be little doubt that the majority are genuine members of the group. The rectangular co-ordinates are given in the usual unit (the parsec), the Sun being at the origin, Oz directed towards the convergent point R.A. 127°'8, Dec. +40°'2, and Ox towards R.A. 307°'8, Dec. + 49°'8, so that the plane zOx contains the Pole. If a model of the system is made from these data, it is found, as H. H. Turner4 has shown, that the cluster is in the form of a disk ; its plane being nearly perpendicular to the galactic plane. The flatness is very remarkable, the average deviation of the individual stars above or below the plane being 2'0 parsecs, a distance small in comparison with the lateral extent of the cluster, viz., 30*to 50 parsecs. In the last column are given the absolute luminosities in terms of the Sun as unit ; it is interesting to note that the three stars of Type F are the faintest, with luminosities 10, !>. ;md 7 respectively.

IV

MOVING CLUSTERS

TABLE C>. The Ursa Major System.

1
			Computed.
Star.		Spec-- trum.	"i. v		Rectangular Co-ordinates.	Lumin- osity (8un=l).
Paral-	Radial
			lax.	Velocity:
			//	km./Bec.	x y z
3 Eridani . . . 2'92	A2 0-034	-7-5	-13-8 -23-3 12-1	96
.d Auriga- . . . 2 '07	Ap 0-024	-iti-u	7-8 -18-8 36-3	410
Shins .... —1*58	A 0-387	-8'5	-2-0 -I'l 1-2	46
37 Tis;.- Maj. . 5-16	F 0-045	-16-6	7'6 5-8 19-9	7
0 Urs« Maj. . 2 '44	A	0-047	-16-1 7'6 6-9 18-7	76
8 Leonis ... 2 '58	A 2	0-084	-14-4 -2-3 7-1 9-3	21
y Urste Ma]. . 2'.~»4	A 0-042	-15-0	8-9 10-5 19-3	87
8 Urste Maj. . . 3 '44	A2	0-045	-14-4	9-8 97 17-2	32
Groom. 1930 .	5-87	F 0-028	-13-4	18-6 15-1 257	9
€ Urs» Mai. - 1'68	Ap	0-042	-13-2	11-4 117 16-9	190
78 Ursjt Maj. . 4 '89	F 0-042	-13-0	12-0 11-8 16-8	10
£ rrsa? Maj. . .	/2-40 \3-96	~v?| °'043	-12-2	11-9 12-5 15-3	J93 ) 22
a Coronse . . .	2-31	A	0-041	-2-2	12-0 20-9 2-9	110

The stars of the Orion type of spectrum present several examples of moving clusters. In the Pleiades we have an evident cluster, in the ordinary sense of the term, and, as might be expected, the motions of the principal stars and at least fifty fainter stars are equal and parallel.^ The bright stars of the constellation Orion itself (with the exception of Betelgeuse, the spectrum of which is not of TypeB) also appear to form a system of this sort, the evidence in this case being mainly derived from their radial velocities, since the transverse motions are all exceedingly small. In Orion a faint nebulosity forming an extension of the Great Orion Nebula, has been discovered, which appears to fill the whole region occupied by the stars ; it probably consists of the lighter gases and other materials not yet absorbed by the stars which are developing. The velocity of the nebula in the line of sight agrees with that of the

* This is the case as regards the proper motions. The radial velocities of the six brightest stars show some surprising differences (Adams, Astro- phijsical Journal, Vol. 19, p. 338), but owing to the difficult nature of the spectra the determinations are not very trustworthy.

STELLAR MOVEMENTS

CHAP.

stars of the constellation. A similar nebulosity is found in the Pleiades.

In the case of these, the youngest of the stars, the argument by which we deduced the accurate equality of motion in the Taurus-cluster scarcely applies; particularly in Orion, the dimensions of which must be at least a hundred times greater than those of the Taurus-cluster, it is just possible that the associated stars may be dispersing rather rapidly.

4V

I

FIG. 4. — Moving Cluster of " Orion " Stars in Perseus.

A group to which the name moving cluster may be applied more legitimately is to be found in the constellation Perseus ; it was detected simultaneously by J. C. Kapteyn, B. Boss, and the writer. If we examine all the stars of the Orion type (Type B) in the region of the sky between K. A. 2h and 6h and Dec. + 36° and + 70 (about one-thirtieth of the whole sphere), we shall find that their motions fall into two groups. In Fig. 4 the motion of each star is

iv MOVING CLUSTERS 65

denoted by a cross, the star having a proper motion which would carry it from the origin 0 to the cross in a century. If all the stars were to start from the origin at the same instant with their actual observed proper motions, then after the lapse of a century they would be distributed as shown in the diagram. Only one star has travelled beyond the limits of the figure and is not shown; with this exception the figure includes all the Type B stars in the region for which data are available.

The upper group of crosses, which is close to the origin, consists of stars with very minute proper motions, all less than 1"*5 per century, and scarcely exceeding the probable error of the determinations. These are clearly the very remote stars, and there is not the slightest evidence that they are really associated with one another ; they appear to cling together because the great distance renders their diverse motions inappreciable. The lower group consists of seventeen stars sharing very nearly the same motion both as regards direction and magnitude. They evidently form a moving cluster similar in character to those we

o

have considered. Their association is further confirmed by the fact that'they are not scattered over the whole area investigated, but occupy a limited region of it.

Table 7 shows the stars which constitute this group. It has been pointed out by T. W. Backhouse 5 that Nos. 742 to 838 form part of a very striking cluster visible to the naked eye. The stars a Persei and o- Persei, which are not of the Orion Type, are included in the visual cluster ; the motion of the latter shows that it has no connection with this system, but a Persei appears to belong to it and may therefore be added to the group. The other stars in this part of the sky have also been examined so far as possible, but none of them show any evidence of connection with the moving cluster. All but three of the stars are arranged in a sort of chain, which may indicate a flat cluster (on the plan of the Ursa Major system) seen edge-

66

STELLAR MOVEMENTS

CHAP.

ways. It is always likely that some spurious members may be included through an accidental coincidence of motion, and it may be suspected that the three outlying stars are not really associated with the rest ; on the other hand, they may well be regarded as original members, which have been more disturbed by extraneous causes than the others.

Owing to the small proper motion, the convergent point of this group cannot well be determined. The motion deviates appreciably from the direction of the solar antapex ; so that this cluster possesses some velocity of its own apart from that attributable to the Sun's own motion.

TABLE 7. Mooing Cluster in Perseus.

Boss's No.	Name of Star.	Type.	Mag.	R.A.	Dec.	Centen- nial Motion.	Direc- tion.
				h. m.	0	/>	, o
678	Pi. 220 ....	Bo	5-6	2 54	+ 52	4-3	51
740	30Persei . . .	B5	5-5	3 11	4-44	3-8	55
742	29 Persei . . .	B3	5-3	3 12	+ 50	4-5	52
744	31 Persei . . .	B3	5-2	3 12	+ 50	4-2	51
767	Pi. 37 ...	Bo	5-4	3 16	+ 49	3-6	45
780	Brad. 476 ...	B8	5-1	3 21	+ 49	3-2	47
783	Pi. 56	B5	5-8	3 22	+ 50	4*9	37
790	34 Persei . . .	B3	4-8	3 22	+ 49	4-4	«-*• 57
796	Brad. 480 ...	B8	6-1	3 24	+ 48	4-7	54
817	•v/f Persei . . .	B5	4-4	3 29	+ 48	4-3	42
838	8 Persei ....	B5	3-0	3 36	+ 47	4-6	51
898	Pi. 186 ....	B5	5-5	3 49	+ 48	3-9	40
910	t Persei ....	BO	2-9	3 51	+ 40	3-9	49
947	c Persei ....	B3	4-2	4 1	+ 47	4-4	43
1003	d Persei ....	B3	4-9	4 14	+ 46	4-5	55
1253	15 Camelopardi .	B3	6'4	5 11	+ 58	3-5	37
1274	p Aurigae . . .	B3	5'3	5 15	+ 42	4-5	40
772	a Persei ....	F5	17	3 17	+ 50	3-8	55

In the last column the "direction" is the angle between the direction of motion and the declination circle at 4 hours R.A.

The tracing of these connections between stars widely separated from one another is an important branch of

iv MOVING CLUSTERS 67

modern stellar investigation ; and, as the proper motions of more stars become determined, it is likely that further interesting discoveries will be made. It seems worth while at this stage to consider what are the exact criteria by which we may determine whether a group of stars possesses that close mutual relation which is denoted by the term " moving cluster." Since some thousands of proper motions are available, it must be possible, if we take almost any star, to select a number of others the motions of which agree with its motion approximately. This is especially the case if, the parallax and radial velocity being unknown, the direction of motion is alone considered ; but, even if the velocities were known in all three co-ordinates, we could pick out groups which would agree approximately ; j ust as in a small volume of gas there must be many molecules having approximately identical velocities. Clearly the agreement of the motions is no proof of association, unless there is some further condition which indicates that the coincidence is in some way remarkable. There is a further difficulty that, as we have already seen in Chapter II. , stars, scattered through the whole region of the universe that has been studied, show common tendencies of motion, so that they have been divided into the two great star-streams. We must be careful not to mistake an agreement of motion arising from this general cosmical condition for the much more intimate association which is seen in the Taurus and Ursa Major systems.

In the case of the Taurus and Perseus clusters the discrimi- nation is comparatively simple. These are compact groups of stars, so that only a small region of the sky and a small volume of space are considered, and the extraneous stars which might yield chance coincidences are not numerous. In the Taurus-cluster the large amount of the motion makes the group remarkable ; and, although in the Perseus-cluster the proper motion is not so great, we have

F 2

68 STELLAR MOVEMENTS CHAP.

been careful to show by the diagram that it is very dis- tinctive. In the latter cluster, moreover, the resemblance of its members in type of spectrum helped to render the detection possible.

The Ursa Major system, which is spread over a large part of the sky, presents greater difficulties. The dis- crimination of its more scattered members was only possible owing to the fact that its motion is in a very unusual direction. Its convergent point is a long way from the apex of either star-stream and from the solar apex, and stars moving in or near that direction are rare. In the writer's investigation of the two star-streams based on Boss's Preliminary General Catalogue, a striking peculiarity was presented in one region, which on enquiry proved to be due to five stars of this system ; that five stars moving in this way should attract attention, sufficiently illustrates the fact that motion in this par- ticular direction is exceptional. We are thus to a large extent safeguarded from chance coincidences ; never- theless, our ground is none too certain, and it may reasonably be suspected that one or two of the members at present assigned to the group will prove to be spurious.

When the supposed cluster is not confined to one part of the sky and to one particular distance from the Sun, when there is nothing remarkable in its assigned motion, and when the choice of stars is not sufficiently limited, by the consideration of a particular spectral type or otherwise, not much weight can be attached to an approximate agreement of motion. A careful statistical study of groups in these adverse circumstances may eventually lead to important results ; but for the present we cannot be satis- fied to admit clusters the credentials of which do not reach the standard that has been laid down.

In concluding this chapter we may try to sum up the importance of the discovery of moving clusters in stellar a-tromony. An immediate result is that in the Taurus

iv MOVING CLUSTERS 69

and Ursa Hajor stream we have been able to arrive at precise knowledge of the distance, relative distribution, and luminosity of stars which are far too remote for the ordinary methods of measurement to be successful. An im- portant extension of this knowledge may be expected when the proper motions of fainter stars have been accurately determined. Further, the possibility of stars widely sundered in space preserving, through their whole life-time up to now, motions which are equal and parallel to an astonishingly close approximation, is a fact which must be reckoned with when we come to consider the origin and vicissitudes of stellar motions. Generally the stars which show these associations are of early types of spectrum ; but in the Taurus cluster there are many members as far advanced in evolution as our Sun, some even of type K, whilst in the more widely diffused Ursa Major system there are three stars of type F. Some of these systems would thus appear to have existed for a time comparable with the life-time of an average star. They are wandering through a part of space in which are scattered stars not belonging to their system — interlopers penetrating right among the cluster stars. Nevertheless, the equality of motion has not been seriously disturbed. It is scarcely possible to avoid the conclusion that the chance attractions of stars passing in the vicinity have no appreciable effect on stellar motions ; and that if the motions change in course of time (as it appears they must do) this change is due, not to the passage of individual stars, but to the central attraction of the whole stellar universe, which is sensibly constant over the volume of space occupied by a moving cluster.

REFERENCES. —CHAPTER IV.

1. 'rYo/i//<</<'/i Publication*, No. 14, p. 87.

-. (rfaninfien P>il>lii-<ifii>n*, No. 23.

3. Plummer, Monthly X»f »'••••.«, Vol. 73. p. 4<>r>, Table X. (last two columns).

4. Turner, The Observatory, Vol. 34, p. -J4r,.

5. Backhouse, Monthly y<>(i<-<>.-<. Vol. 71, p. 523.

70 STELLAR MOVEMENTS CH. iv

BIBLIOGRAPHY.

Taurus Cluster L. Boss, Astron. Joum., No. 604.

Ursa Major Cluster .... Ludendorff, A sir. Nar,h., No. 4313-14;

Hertzsprung, Astrophytu'id Joiim., Vol.

30, p. 135 (Erratum, p. 320). Perseus Cluster Kapteyn, Internat. Solar Union, Vol. 3,

p. 215 ; Benjamin Boss, Astron. Journ.,

No. 620 ; Eddington, Monthly Notices,

Vol. 71, p. 43.

There are in addition two less defined clusters, which have attracted some attention, viz., a large group of Type B stars in Scorpius and Centaurus, and the 61 Cygni group, which consists of stars scattered over most of the sky having a linear motion of the large amount 80 km. per sec. 5

Scorpius-Centaurus Cluster . Kapteyn, loc. cit., p. 215; Eddington, loc.

cit., p. 39. 61 Cygni Cluster B. Boss, Astwn. Joum., Nos. 629, 633.

CHAPTER V

THE SOLAR MOTION

IT was early recognised that the observed motions of the stars were changes of position relative to the Sun, and that part of the observed displacements might be attributed to the Sun itself being in motion. The question " What is the motion of the Sun ? " raises at once the philosophical difficulty that all motion is necessarily relative. In reality the manner in which the observed motion is to be divided between the Sun and the star is indeterminate ; these bodies are moving in a space absolutely devoid of fixed reference marks, and the choice and definition of a framework of reference that shall be considered at rest is a matter of convention. Probably philosophers of the last century believed that the undisturbed aether provided a standard of rest which might suitably be called absolute ; even if at the time it could not be apprehended in practice, it was an ultimate ideal which could be used to give theoretical precision to their statements and arguments. But according to modern views of the sether this is no longer allowable. Even if we do not go so far as to discard the aether-medium altogether, it is generally considered that no meaning can be attached to the idea of measuring motion relative to it ; it cannot be used even theoretically as a standard of rest.

In practice the standard of rest has been the " mean of

71

72 STELLAR MOVEMENTS CHAP.

the stars," a conception which may be difficult to define rigorously, but of which the general meaning is sufficiently obvious. Comparing the stars to a flock of birds, we can distinguish between the general motion of the flock and the motions of particular individuals. The convention is that the flock of stars as a whole is to be considered at rest. It is not necessary now to consider the reasons that may have suggested that the mean of the stars was an absolute standard of rest ; it is sufficient to regard it as a conventional standard, which has considerable usefulness. If there is any real unity in the stellar system, we may expect to obtain a simpler and clearer view of the pheno- mena by referring them to the centroid of the whole rather than to an arbitrary star like the Sun. By the centroid is meant in practice the centre of mass (or rather the centre of mean position) of those stars which occur in the catalogues of proper motions that are being discussed. As it is only the motion of this point that is being considered, its actual situation in space is not of conse- quence. If the motion of the centroid varied considerably according to the magnitude of the stars used or the particular region of the sky covered by the catalogue, it would be a very inconvenient standard. It is not yet certain what may be the extent of the variations arising from a particular selection of stars ; but, as the data of observation have improved, the wide variations shown in the earlier investigations have been much reduced or satisfactorily explained. At the present day, whilst few would assert that the " mean of the stars " is at all a precise standard, the indeterminateness does not seem sufficiently serious to cause much inconvenience.

The determination of the motion of the stars in the mean relative to the Sun, and the determination of the solar motion (relative to the mean of the stars) are two aspects of the same problem. The relative motion, which- ever way it is regarded, is shown in our observations by a

v THE SOLAR MOTION 73

strong tendency of the stars to move towards a point in the sky, which according to the best determinations is near ft Columbae. Although individual stars may move in widely divergent or even opposite directions, the tendency is so marked that the mean of a very few stars is generally sufficient to exhibit it. Sir William Herschel's l first determination in 1783 was made from seven stars only, vt't lie was able to indicate a direction which was a good first approximation. From his time up till recent years the determination of the solar motion was the principal problem in all statistical investigations of the series of proper motions which were measured from time to time. This investigation was usually associated with a determina- tion of the constant of precession — a fundamental quantity which is closely bound up with the solar motion in the analysis. In fact, both quantities are required to define our framework of reference ; the solar motion defines what is to be regarded as a fixed position, and the precession- constant defines fixed directions among the continually shifting stars. The numerous older determinations of the solar motion are now practically superseded by two results published in 1910-11, which rest on the best material yet available.

The determination by Lewis Boss 2 from the proper motions of his Preliminary General Catalogue of 6188 Stars gives,—

fR.A. 270-5 ± 1-5 SolarAPex \Dec. +34-3 ± 1-3

The determination by W. W. Campbell 3 from the radial velocities (measured spectroscopically) of 1193 stars gives,—

fR.A. 268-5 ± 2-0 ^olarApex \Dec. +25-3 ± 1-8

Speed of the solar motion 19'5±0'6 kilometres per second.

The probable errors are not given by Campbell ; but the

74 STELLAR MOVEMENTS CHAP.

foregoing approximate values are easily deduced from the data in his paper.

The discordance in declination between these two results, derived respectively from the transverse and the radial motions, is considerably greater than can be attributed to the accidental errors of the determinations. Possibly the discordance may be attributed to the different classes of stars used in the two investigations. Campbell's result depends almost wholly on stars brighter than 5m*0, whereas Boss included all stars to the sixth magnitude and many fainter stars. Moreover, Boss's result depends more especially on the stars nearest to the Sun ; for in forming the mean proper motion in any region the near stars (having the largest angular motions) have most effect, whereas in forming the mean radial motion the stars contribute equally irrespective of distance. So far as can be judged, however, these differences will not explain the discordance. Boss made an additional determination of the solar apex, rejecting stars fainter than 6m>0 ; the resulting position RA. 269°'9, Dec. + 34°'6 is almost identical with his main result. The writer,4 examining the same proper motions on the two star-drift theory, by a method which gives equal weight to the near and distant stars, arrived at the position R.A. 267°'3,' Dec. + 36°'4, again a scarcely appreciable change. The cause of the difference between the results from the proper motions and the radial motions thus remains obscure.

One of the most satisfactory features of Boss's deter- mination of the solar apex is the accordance shown by the stars of different galactic latitudes. If there is any relative motion between stars in different parts of the sky, it would be expected to appear in a division according to galactic latitude. The following comparison of the results derived from regions of high and low galactic latitudes is given by Boss.

v THE SOLAR MOTION 75

Solar Apex.

Galactic Latitude of Zones. R.A. Dec.

- 7Dto +7D 269° 40' +33° 17'

-19° to - 73, and +19° to + 7° 270 55 29 52

-42° to -19°, and +42° to +19° 269 51 34 18

S. Gal. Pole to -42°, and N. Gal. Pole to +42° 270 32 36 27

The differences are quite as small as could be expected from the accidental errors.

Another comparison of different areas of the sky can be made from the results of an analysis by the two-drift theory, which has the advantage that the position depends equally on all the stars used, instead of (as in the ordinary method) the nearest stars having a preponderating share.

We find,-

Solar Apex. Region. R.A. Dec.

P*rA» g*t5Sig} 265-5 + 37-0

Equatorial Area Dec. -36° to + 363 269° '4 + 36°'4

There is thus a very satisfactory stability in the position of the apex determined from different parts of the sky.* The evidence is less certain as to its dependence on the magnitude and spectral type of the stars. There is some indication that the declination of the apex tends to increase for the fainter stars ; but it is not entirely conclusive. The range of magnitude in Boss's catalogue is scarcely grea't enough to provide much information ; so far as it goes, it is opposed to the view that there is any alteration in the apex for stars of different magnitudes, for, as already mentioned, the stars brighter than 6m'0 give a result almost identical with that derived from the whole catalogue. From the Groombridge stars (Dec. +38° to N. Pole), F. \V. Dyson and W. G. Thackeray 5 found-

* In the foregoing comparisons antipodal regions have always been taken together. There remains a possibility of a discordance between opposite hemispheres.

76 STELLAR MOVEMENTS CHAP.

Solar Apex.

Magnitude. , • s

m. in. R.A. Dec. No. of Stars.

1-0-4-9 245 3 +16-0 200

5-0-5-9 268° + 27='0 454

6-0—6-9 278° + 33-0 1,003

7-0-7-9 280° +38^5 1,239

8-0—8-9 2723 +43°-0 811

This shows a steady increase in declination with diminishing' magnitude. It must, however, be noted that the area covered by the Groombridge Catalogue is particularly unfavourable for a determination of the declination of the apex.

Other evidence pointing in the same direction has been found by G. C. Comstock,0 who made a determination of the solar motion from 149 stars of the ninth to twelfth magnitudes. He was able to obtain proper motions of these stars, because they had been measured micro- metrically as the fainter companions of double stars, but had been found to have no physical connection with the principal stars. The resulting position of the apex is

R.A. 300° Dec. +54°

In a more recent investigation 7 the same writer has used 479 faint stars, with the results

Magnitude 7m>0 to 10m'0 Apex R.A. 280° Dec. +58°

10m-0 „ 13m-0 „ R.A. 288: Dec. +71°

The weight of these determinations cannot be great, but they tend to confirm the increase of declination with the faintness of the stars.

Earlier investigations in which the stars were classified by magnitude are those of Stumpe and Newcomb. Tint former, using stars of large proper motion only, found a considerable progression in declination with faintness. Newcomb, on the other hand, who used stars of small proper motion only, found that the declination is steady. We now know that, owing to the phenomenon of star- streaming, the exclusion of stars above or below certain limits of motion is not legitimate, so that the contradictory character of these two results is not surprising.

v THE SOLAR MOTION 77

The comparative uncertainty of the proper motions of the fainter stars requires that results based on them should be received with caution. In particular, since the mean distance of the stars increases with faintness, the average parallactic motion becomes smaller, and a sys- tematic error in the declinations of any zone has a greater effect on its apparent direction. This is particularly serious, because these investigations have been usually based on northern stars only or on an even less extensive

region.

R.A.	Dec.	No. of Stars.
274= -4	+ 34° -9	490
270-0	28-3	1,647
2653'9	28-7	656
259-3	42-3	444
275-4	40-3	1,227
273-6	38-8	222

There is fairly consistent evidence that the declination of the solar apex depends to some extent on the spectral type of the stars, being more northerly for the later types. In Boss's investigation 8 the following results were found,—

Solar Apex.

Type.

Oe5— B5

B8— A4

A5-F9

G

K

M

The later types G, K, M thus yield a declination differing markedly from the earlier types ; or, if we prefer to set aside the results for the groups containing few stars, which may be subject to large accidental errors, and confine attention to types A (B8 — A4) and K, the difference of 12° between the results of these two classes is evidently significant.

The results of Dyson and Thackeray from the Groom- bridge stars show the same kind of progression.

Solar Apex.

Type. R'A. ~~De7. No. of Stars.

B, A 269° +23^ 1100

F, G, K 27M :;; 866

Other investigations of this relation depend mainly on stars now included in Boss's catalogue, and used in his

78 STELLAR MOVEMENTS CHAP.

discussion. It therefore does not seem necessary to quote them.

To sum up the results we have arrived at, it appears that we can assign a point in the sky at about R.A. 270°, Dec. +34° towards which the motion of the Sun relative to the stellar system is directed. For some reason at present unknown, the determinations of this point by means of the spectroscopic radial velocities differ appreciably from those based on the transverse motions, giving a declination nearly 10° lower than the point mentioned. When different parts of the sky are examined the results are generally in good agreement, so that there can be little relative motion of the stars as a whole in different regions. There is some evidence that the solar apex increases in declination as successively fainter stars are considered, and it seems certain that for the later types of spectrum the declination is higher than for the earlier types. From all causes the solar apex from a special group of stars may (apart from accidental error) range from about +25° to + 40° in declination ; variations in right ascension appear to be small and accidental.

The speed with which the Sun moves in the direction thus found can only be measured from the radial motions. The result derived from the greatest amount of data is 19 '5 km. per sec.

Attention has been lavished on the investigation of the solar motion, not only on account of its intrinsic interest, but also because it is a unit of much importance in many investigations of the distribution of the more distant stars. The annual or centennial motion of the Sun is a natural unit of comparison in dealing with the stellar system, generally superseding the radius of the earth's orbit, which is too small to be employed except for a few of the nearest stars. It provides a far longer base-line than can be obtained in parallax-observations ; for the annual motion of the Sun amounts to four times the radius of the earth's

v THE SOLAR MOTION 79

orbit, and the motion of fifty or a hundred years, or even longer, may be used. The apparent displacement of the star attributable to the solar motion is called the parallactic motion. By determining the parallactic motion (in arc) of any class of stars their average distance can be found, just as the distance of an indi- vidual star is found from its annual parallax. It is not possible to find by observation the parallactic motion of an individual star, because it is combined with the star's individual motion ; but for a group of stars which has no systematic motion relative to the other stars, these indi- vidual motions will cancel in the mean.

It may be appropriate to add some remarks on the theory of the determination of the solar motion from the observations. The method usually adopted for discussing a series of proper motions is that known as Airy's.

Take rectangular axes, Ox being directed to the vernal equinox, Oy to R.A. 90°, and Oz to the north pole. The parallactic motion (opposite to the solar motion) may be represented by a vector, with components X, Y, Z, directed to the solar antapex. X, Y, Z are supposed to be expressed in arc, so as to give the parallactic motion of a star at a distance corresponding to the mean parallax of all the stars considered.

Taking a small area of the sky let the mean proper motion of the stars in the area be /*a, ^ in right ascension and declination respectively. Then considering the pro- jection of (X, F, Z) on the area considered, we have

- X sin a + Y cos a = /ua

— X cos a sin 8 — Y sin a sin d + Z cos 6 = pi

where it is assumed that these stars are at the same mean distance as the rest, and that their individual motions cancel out. If these assumptions are not exactly satisfied, the deviations are likely to be mainly of an accidental

:••

8o STELLAR MOVEMENTS CHAP.

character. Taking the above equations for each region, a least-squares solution may then be made to determine X, Y, Z. The right ascension and declination A, D of the solar antapex are given by

tan A = Y/X tanD = Z;(X*+Y*)l

Additional terms in the equations of condition involving the correction to the preoessional constant and to the motion of the equinox are often inserted, but these need not concern us here. When stars distributed uniformly over the whole sky are considered, the additional terms have no effect on the result.

Although the argument is clearer when we use the mean proper motion over an area for forming equations of condition, it is quite legitimate to use each star separately. For it is easily seen that the resulting normal equations are practically identical in the two procedures. In using the mean proper motion, it is easier and more natural to give equal weights to equal areas of the sky instead of weighting according to the number of stars ; this is generally an advantage. Further the numerical work is shortened.

There are two weak points in Airy's method. First the mean proper motion (which, if not formed separately for each area, is virtually formed in the least-squares solution) is generally made up of a few large motions and a great number of extremely small ones. It is therefore a very fluctuating quantity, the presence or omission of one or two of the largest motions making a big difference in the mean. In a determination based nominally on 6000 stars, the majority may play only a passive part in the result, and the accuracy of the result is scarcely proportionate to the great amount of material used. The second point is more serious, since it leads to systematic error. We have , i— umed that the mean parallax of the stars in each area differs only by accidental fluctuations from the mean of

v THE SOLAR MOTION 81

the whole sky ; but this is not the case. The stars near the galactic plane have a systematically smaller parallax than those near the galactic poles.

It has often been recognised that this property of the galactic plane may cause a systematic error in the apex derived from the discussion of a limited part of the sky. Perhaps it is not so generally known that it will also cause error even when the whole sky is used. It seems worth while to examine this point at length. Happily it turns out that the error is not very large, but this could scarcely have been foreseen.

If the mean parallax in any area is p times the average parallax for the whole sky, we may take account of the variation with galactic latitude by setting

p = l + f P2 (cos 6)

where 6 is the distance from the galactic pole, e is a coefficient and

We shall consider the case when the observations extend uniformly over the whole sky.

The equations of condition should then read

— Xp sin a + Yp cos a = fia

— Xp cos a sin 8 — Yp sin a sin 8 + Zp cos 8 = /ig.

We wish to re- interpret the results of an investigator who has not taken p into account. We therefore form normal equations, just as he would do, viz., from the right ascensions :

X'S.p sin -a - Y^p sin a cos a = - S^ia sin a

— X"S,p sin a cos a + YZp cos -a = S/ja cos a.

and from the declinations :

X^p cos -'a sin -8 4- YZp sin a cos a sin -8 — Z'S.p cos a sin 5 cos 8 =

- 2^5 cos a sin 8 X2p sin a cos a sin 28 + Y^.p sin 2a sin -8 — Z2p sin a sin 8 cos 8=

— 2^s sin a sin 8

- A'2p cos a sin 8 cos 8 - Y2p sin a sin 8 cos 8 + ZXp cos -8 = i>6 cos 8

G

82 STELLAR MOVEMENTS CHAP.

giving the combined equations :

X'Zp (sin 2a + cos -a sin 28) — Y2p sin a cos a cos 28 — Z'S.p cos a sin 8 cos 8 =

- 2(/ua sin a + ^5 cos a sin 8)

— X"2.p sin a cos a cos 28 + Y2p (cos 2a + sin -a cos 28) — Z'S.p sin a sin 8 cos 8 =

2(/ia cos a — ps sin a sin 8) - X'S.p cos a sin 8 cos 8 - YZp sin a sin 8 cos 8 + Z^.p cos -8 = 2/xg cos 8.

Now it clearly can make no difference in a least-squares solution whether we resolve our proper motions in right ascension and declination or in galactic latitude and longitude. The value of the solar motion, which makes the sum of the squares of the residuals in R.A. and Dec. a minimum, must be the same as that which makes the sum of the squares of the residuals in Gal. Lat. and Long. a minimum. We may therefore treat a solution as though it had been made in galactic co-ordinates, although the actual work was done in equatorial co-ordinates.

Let then a, S now stand for galactic longitude and latitude, so that A", F, Z is the paral lactic motion vector referred to rectangular galactic co-ordinates. We shall have

Taken over a whole sphere the mean value of

2 1

p(sin -a + cos 2a sin 28) = - + ~e

3 15

3 lo

»cos28 = - -_?e

3 15

The other coefficients vanish when integrated over a sphere. Thus the normal equations become (setting N for the total number of stars used)

2 / 1 \

x JT ( 1 + J~Q€ J = - 2 (pa sin a + (i& cos a sin 8)-r- N

2 r / 1 \

o Y ( 1 -f jTj€ J = 2 (/*a cos a — ps sin u sin 8) -r- N

v THE SOLAR MOTION 83

And if X0, yo, ZQ are the solutions obtained when the P2 term is neglected,

The original and corrected galactic latitudes of the antapex bein \0, X, we have

tan X =

l-t<

whilst the galactic longitude is unaltered.

The effect of the decrease of parallax towards the galactic plane is thus to make X0 numerically less than \. The uncorrected position of the solar apex is too near the galactic plane.

Inserting numerical values, X0 = 20°, and e may perhaps be ^ (i.e., mean parallax at the pole / mean parallax in the plane =8/5), we find X = 21°57r. The correction is just under 2°. Reverting to equatorial co-ordinates, the correction is mainly in right ascension, the right ascension given by the ordinary solution being about 2°'4 too great.

It is quite practicable to work out the corresponding corrections, when the proper motions cover only a zone of the sky limited by declination circles. In this case we have to retain equatorial co-ordinates throughout, and express P2 (cos 6) in terms of a and S. The mean values of the functions of sin a, cos a, sin S and cos S that occur are readily evaluated for the portion of the sphere used. As the numerical work depends on. the particular zone chosen, we shall not pursue this matter further.

A second method of finding the solar apex from the proper motions, known as Besse.l's method, has been used by H. Kobold.9 Each star is observed to be moving along a great circle on the celestial sphere. Consider the poles of these great circles. If the stellar motions all converged to a point on the sphere, the poles would all lie along the

G 2

84 STELLAR MOVEMENTS CHAP.

great circle equatorial to that point. Thus a tendency of the stars to move towards the solar antapex should be indicated by a crowding of the poles towards the great circle equatorial to the antapex. This affords a means of finding the direction of the solar motion by determining the plane of greatest concentration of the poles. . It is to be noted, however, that this method makes no discrimina- tion between the two ways in which a star may move along its great circle. Two stars moving in exactly opposite directions will have the same pole. A paradoxer might argue that, as the effect of the solar motion is to cause a minimum number of stars to move towards the solar apex, the solar motion will be indicated by a tendency of the poles to avoid the plane equatorial to its direction. How- ever, if the individual motions are distributed according to the law of errors, the crowding to the plane will be found to outweigh the avoidance, so that the method is legitimate

o o

though perhaps a little insensitive. But if the individual motions follow some other law, the result may be altogether incorrect. In the light of modern knowledge of the presence of two star-streams, Bessel's method can no longer be regarded as an admissible way of finding the solar apex ; but it is interesting historically, for in Kobold's hands it first foreshadowed the existence of the peculiar distribution of stellar motions which is the subject of the next chapter.

The determination of the solar motion from the radial velocities presents no difficulty. If (X, Y, Z] is the vector representing the parallactic motion in linear measure, each Mar yields an equation of condition :

X cos a cos d + Fsinacos5-H^sin^ = radial velocity.

A least-squares solution is then made, the individual motions of the star> IM-MI- treated as though they were accidental errors. The numerical work can be shortened by using in the equations of Condition tin- nu-an radial velocity for a

v THE SOLAR MOTION 85

sniiill area of the sky, instead of the individual results. The resulting normal equations are practically unaltered, and there is no theoretical advantage.

REFERENCES. — CHAPTER V.

1. Sir W. Herschel, Collected Papers, Vol. 1, p. 108.

2. Boss, Ash-on. Journ., Nos. 612, 614.

3. Campbell, Lick Bulletin, No. 196.

4. Ecldingtoii, Monthly AW/<v.s, Vol. 71, p. 4.

5. Dyson and Thackeray, Monthly Notices, Vol. 65, p. 428.

6. Comstock, Astron. Journ., No. 591.

7. Comstock, Astron. Journ., No. 655.

8. Boss, Astron. Journ., Nos. 623-4.

9. Kobold, Nova Ada der Kais. Leop. Carol. Deutschen Akad., Vol. 64 ; A«tr. Nach., Nos. 3163, 3435, 3591.

BIBLIOGRAPHY.

The following references are additional to those quoted in the Chapter. Owing to the recent improvement in the data of observation, and the change of theoretical views due to the recognition of star-streaming, the interest of these papers is now, perhaps, mainly historical.

Argelander, Memoires presenter a VAcad. des. Sci., Paris, Vol. 3, p. 590

(1837).

Bravais, Liouvilles's Journal, Vol. 8 (1843). Airy, Memoirs R.A.S., Vol. 28, p. 143 (1859). Stumpe, Astr. Nach., No. 3000. Porter, Cincinatti Trans., No. 12. L. Struve, Memoires St. Petersboury, Vol. 35, No. 3 ; Astr. Xach. Nos. 3729,

3816.

Newcomb, Astron. Papers of the American Ephemeri*. Vol. 8, Pt. 1. Kapteyn, Astr. Nach., Nos. 3721, 3800, 3859. Boss, Astron. Journal, No. 501. Weersma. Qroningen Publications, No. 21.

CHAPTER VI

THE TWO STAR STREAMS

THE observed motion of any star can be regarded as compounded of two parts ; one part, which is attributable to the motion of the Sun as point of- reference, is the parallactic motion ; and the other residual part is the star's motus peculiaris or individual motion. It must be borne in mind that this division cannot generally be effected in practice for the proper motion of a star ; because, although the parallactic motion in linear measure is known, we cannot tell how much it will amount to in angular measure, unless we know the star's distance, and this is very rarely the case. On the other hand, the spectroscopic radial velocities, being in linear measure, can always be freed from the parallactic motion, if desired. As the greater part of our knowledge of stellar movements is derived from the proper motions, we cannot study the mottis peculiares directly, but must deduce the phenomena respecting them from a statistical study of the whole motions.

In researches on the solar motion, it has usually, though not always, been assumed that the mottis peculiares of the stars are at random. This was the natural hypothesis to make, when nothing was known as to the distribution of these residual motions ; and certainly, when we consider

N

CH. vi THE TWO STAR STREAMS 87

how vast are the spaces which isolate one star from its neighbour, and how feeble must be any gravitational forces exerted across such distances, it might well seem improbable that any general tendency or relation could connect the individual motion of one star with another. Yet many years ago the phenomenon of local drifts of stars, or, as they are now called, Moving Clusters, was known. But although such instances of departure from the strict law of random distribution of motions must have been recognised as occurring exceptionally, probably few astronomers doubted that the hypothesis was substantially correct. In 1904, however, Prof. J. C. Kapteyn l showed that there is a fundamental peculiarity in the stellar motions, and that they are not even approximately haphazard. This deviation is not confined to certain localities, but prevails throughout the heavens, wherever statistics of motions are available to test it.

Instead of moving indiscriminately in all directions, as a random distribution implies, the stars tend to move in two favoured directions. It does not matter whether the parallactic motion is eliminated or not. A tendency to move in one favoured direction would disappear, when the parallactic motion was removed ; but a tendency in two directions can only be an intrinsic property of the individual motions of the stars. It may seem strange that this striking phenomenon was so long overlooked by those who were working on proper motions ; but usually investigators, having the solar motion mainly in their minds, as a first step towards gathering their data into a manageable form grouped together the stars in small regions of the sky, and used the mean motion. This unfortunately tends to conceal any peculiarity in the individual motions. In order to exhibit the phenomenon it is necessary to find some means of showing the statistics of the separate stellar motions ; this may be done conveni- ently in the following way.

88

STELLAR MOVEMENTS

CHAP.

ANALYSIS OF PROPER MOTIONS

Confining our attention to a limited area of the sky, so that the apparent motions are seen projected on what is practically a plane, we count up the number of stars observed to be moving in the different directions. If in classifying directions we proceed by steps of 10°, we shall then form a table of the number of stars moving in 36 directions, towards position angles 0°, 10°, 20° . . . 350°.

Velocity 0-3 Unit

Velocity 0-6 Unit

Velocity 1-5 Unit

FIG. 5. — Simple Drift Curves.

The result can be conveniently shown on a polar diagram, i.e., a curve is drawn so that the radius is proportional to the number of stars moving in the corresponding direction. Before considering the diagrams actually derived from observation, let us examine what form of curve would be obtained if the hypothesis of random motions were correct. The curve would not be a circle because of the parallactic motion ; for, as these observed motions are referred to the Sun, there would IM« superposed on the random individual motions the motion of the star-swarm as a whole. If, for example, this latter motion were towards the north, then

vi THE TWO STAR STREAMS 89

clearly there would be a maximum number of stars moving north and fewest south, the number falling off symmetri- cally on either side from north to south. The exact form can be calculated on the hypothesis of random distribution ; it varies with the magnitude of the parallactic motion compared with the average motus peculiaris, being more elongated the greater the parallactic motion. In Fig. 5

270° 240° 300°

ISO-

ISO0 30

12tf I 60°

90

FIG. 6. — Observed Distribution of Proper Motions. (Groombridge Catalogue— II. A. 14h to 18h, Dec. +38' to +70'.)

examples of this curve are given ; it may be noted how sensitive is the form of the curve to a small change in the parallactic velocity. It is convenient to have a name for a system such as is represented in these figures, in which the individual motions are haphazard, but the system as a whole is in motion relative to the Sun ; we call such a system a drift.

As an example of a curve representing the observed distribution of proper motions we take Fig. 6.2 This

9o STELLAR MOVEMENTS CHAP.

corresponds to a region of the sky between K.A. 14h and 18h, Dec. 4- 38° and 4- 70°, the proper motions being taken from Dyson and Thackeray's " New Reduction of Groom- bridge's Catalogue." The motions of 425 stars are here summarised. It is quite clear that none of the single drift curves of Fig. 5 can be made to fit this curve derived from observation. Its form (having regard to the position of the origin) is altogether different. No one of the theoretical curves corresponds to it in even the roughest manner. It will be noticed that there are two favoured directions of motion ; the stars are streaming in directions 80°aud 225°, the latter being the more pronounced elongation. The actual number of stars moving in each direction is given in the fifth column of Table 8 below. Neither of the favoured directions coincides with that towards the solar antapex, viz., 205°, for this part of the sky. It is true the mean motion of all these stars is towards the antapex, but we see that that is merely a mathematical average between the two partially opposed streams that are revealed in the diagram.

It is possible to obtain a theoretical figure that will correspond approximately with Fig. 6 in the following way. Suppose, instead of the single drift we have hitherto considered, there are two star-drifts. Let one, consisting of 202 stars, be moving in the direction 225° with velocity* 1*20, and the other, consisting of 232 stars, be moving in the direction 80° with the much smaller velocity 0*45. The corresponding curves are P and Q, Fig. 7. If these were seen mingled together in the sky, the resulting distribution would be represented by the curve R. Each radius of R is, of course, formed by adding together the corresponding radii of P and Q. If R is carefully compared with the observed curve, it will be seen that the resemblance is close. The numerical

* The unit velocity l/7i, which is related to the mean individual motion, is denned in the mathematical theory in the next chapter.

VI

THE TWO STAR STREAMS

comparisons, which these diagrams illustrate, are given in Table 8 ; it is there shown that by adding together two

TABLE 8.

Analysis of Proper Mat inns in the Region R.A. 14* to 18A, Dec. +38° to +70°.

	Calculated.
Direction.			Observed.	Difference. Obs. — Calc.
	Drift I.	Drift II.	Total.
0 5	066	4	-2
15	077	5 -2
25	088	6	-2
35	0 10 10	9 -1
45	0 11 11	10 -1
55	0 12 12	14 +2
65	0 12 12	14 +2
75	0 13 13	14 +1
85	0 13 13	13 0
95	0 12 12 12 0
105	1 12 13	10 -3
115	1 11 12	11 -1
125	1	10 11	10 -1
135	1	8	9	10 +1
145	2	7	9	7 -2
155	3	6	990
165	5	6	11	9 -2
175	7	5	12	14 +2
185	11	4	15 14 - 1
195	15	4	19 16 -3
205	19	3	22 21	-1
215	23	3	26 27 +1
225	24	3	27 29	+ 2
235	23	3	26 26	0
245	19	3	22 19	-3
255	15	3	18 17	-1
265	11 3	14	12	-2
275	7 3	10 11	+ 1
285	5 3 8 11	+ 3
295	3 3	6 8	+ 2
305	2 3	5 7	+ 2
315	1346	+ 2
325	1 4	5 6	+ 1
335	1 4	5 5	0
345	1	5	6 5	-1
355	0 6	6	4	-2
Totals .	. 202	232	434 425	—

92 STELLAR MOVEMENTS CHAP.

theoretical drifts, the observed distribution of motions is approximately obtained. Without pressing the conclusion that a combination of two simple star-drifts will represent the actual distribution in all its detail, we may at least assert that it represents its main features, whereas not even the roughest approximation can be obtained with

Q

FIG. 7. — Calculated Distributions of Proper Motions.

a single drift, i.e., with the hypothesis of random- motions.

As another illustration, we may take Fig. 8, which refers to a different part of the sky, the proper motions this time being taken from Boss's Preliminary General Catalogue. The uppermost curve, which has so interesting an appear- ance, is derived from the observed proper motions. Curve B is the best approximation that can be found on the assumption of a random distribution of motions plus the parallactic motion. It may be remarked that since the solar apex is a rather well determined point, the direction of elongation of the curve B is not arbitrary ; it is necessary to draw it pointing in the known direction of the parallactic motion. The curve C is an approximation by a combination of two star-drifts ; these again were not taken as arbitrary in direction, but were made to point towards appropriate apices deduced from a general discus- sion of the whole sky. It is quite probaldr that there are differences between A and C which are not purely

VI

THE TWO STAR STREAMS

93

accidental, but it will at least be admitted that, whereas the curve B bears scarcely any resemblance to the ob- served curve, the curve C - reproduces all the main features of the distribution, and from it we can if we choose proceed to investigate the irregularities of de- tail.

The foregoing examples illustrate a method of analysis which has been applied successfully to a great many parts of the sky. It consists in find- ing, generally by trial and error, a combination of two drifts which will give a distribution of motions agreeing closely

with that actually observed. In comparing the results obtained from different parts of the sky, it must be remembered that we are studying the two- dimensional projections of a three-dimensional phenom- enon, and the diagrams will vary in appearance as the circumstances of projection vary. The most accu- rate series of proper motions at present available is contained in Lewis Boss's " Preliminary General Catalogue," and it is of special interest to examine fully the results derivable from it.3 The catalogue contains 6188 stars well-distributed over the whole sky; practically

• Double-drift approximation

FIG. 8.— Observed and Calculated Distributions of Proper Motions. (Boss, Region VIII.)

94 STELLAR MOVEMENTS CHAP.

all stars down to the sixth magnitude are included, and the fainter stars appear to be fairly representative and have not been selected on account of the size of their proper motions. A very high standard has been attained in the elimination of systematic error — the main cause of trouble in these researches — though no doubt there is still a possibility of improvement in this respect. There can be no doubt that the catalogue represents the best data that it is at present possible to use.

After excluding certain classes* of stars for various reasons, 5322 remained for investigation. These were divided between seventeen regions of the sky, each region consisting of a compact patch in the northern hemis- phere together with an antipodal patch in the southern hemisphere. By taking opposite areas together in this way we double the number of stars without unduly extending the region, for the circumstances of projection in opposite regions are identical. The regions were numbered from I. to XV IT., I. being the circular area Dec. 4- 70° to the Pole; II. to VII. formed the belt between + 36 = and +70° with centres at O1', 4h, 8h, 12h, 16h, and 20h respectively; and VIII. to XVII. formed the belt 0° to +36° with centres at lh 12ra, 3h 36'", etc. (These are the positions of the northern portions ; the antipodal part of the sky is also to be included in each case.)

The diagrams for 11 of the 17 regions are given in Fig. 9. The arrows marked Antapex point to the antapex of the solar motion (R.A. 90°*5, Dec. — 34 *3) ; the arrows I. and II. point to the apices of the two drifts, found from the collected results of this discussion. It will be seen that the evidence for the existence of two star-streams is very strong. The tendency to move in the directions of the arrows I. and II. is plainly visible, and it is scarcely

* Viz., stars of the Orion type, members of moving clusters, and the fainter components of binary systems.

VI

THE TWO STAR STREAMS

95

necessary again to emphasise that there is no resemblance to a symmetrical single-drift curve pointing along the antapex arrow. In certain cases, notably Regions XIV.

270"

i. Region I. ; centre, North Pole. 270°

180

ii. Region II. ; centre, R.A. Oh, Dec. +503.

270°

180

II

in. Region V. ; centre, R.A. 12h, Dec. + 50°.

FIG. 9. — Diagrams for the Proper Motions of Boss's " Preliminary General Catalogue."

STELLAR MOVEMENTS

270° 270°

180

0° 180

iv. Region VI. ; centre R.A. 16h, Dec. +50°.

270

\

180 -,

v. Region VII. ; centre R.A. 201', Dec. +50°.

II

A

Antsipex vi. Region VIII. ; centre, R.A. lh 12m, Dec. +17°.

270

180

II

90

vii. Region XII. ; centre, R.A. 10h 48m, Dec. +17 .

Fit;. 9 (continued).— Diagrams for the Proper Motions of Boss' "Preliminary

THE TWO STAR STREAMS

II

viii. Region XIII. ; centre R.A. 13h 12m, Dec. +17°.

270°

180

90 II

ix. Region XIV. ; centre R.A. 15h 36m, Dec. +17°.

180

>-— 0

x. Region XVI. ; centre R.A. 20h 24m, Dec. +17

FIG. 9 (continued) . — Diagrams for the Proper Motions of Boss's "Preliminary General Catalogue."

H

98

STELLAR MOVEMENTS

270°

CHAP,

180

xi. Region XVII. ; centre R.A. 22h 48m, Dec. +17°.

FIG. 9 (continued). — Diagrams for the Proper Motions of Boss's " Preliminary General Catalogue/'

and XVI. , there appears to be a streaming towards the antapex in addition to the streaming in the directions of the two drifts, so that the -curve appears three-lobed — like a clover leaf. This is an important qualification of our conclusion, but for the present we shall not discuss it ; later it will be considered fully. The eleven regions chosen for representation are those in which the separ- ation into two drifts ought to be most plainly indicated. It will be understood that there are parts of the sky in which the projection is such that they are not very plainly separated. In fact there must be one plane of projection on which the drifts have identical transverse motions, and would therefore become indistinguishable, except by having recourse to the radial velocities. The fact, then, that in the regions which are not here represented the phenomenon is shown less plainly, in no way weakens the argument, but rather confirms it.

Let us suppose now that wre have succeeded in analysing the stellar motions in each of the seventeen regions into their constituent drifts, and have thus determined the directions and velocities of the two drifts at seventeen points of the celestial sphere. If the drift motion in each region is really the same motion seen in varying projec- tions, we must find that <>n plotting the directions on a

VI

THE TWO STAR STREAMS

99

globe they will all converge to one point. This will be true for each drift separately. The convergence actually found is shown in Figs. 10 and 11. Imagine the great circles traced

+ 30

+20-

+ 10-

o° 5° 10° 15° 20'

Approximate Scale

FIG. 10. — Convergence of the Directions of Drift I. from the 17 Regions.

+65°

+50

+55

+45°

5 1° 15

Approximate Scale

FIG. 11.— Convergence of the Directions of Drift II. from the 17 Regions.

on the sky and a photograph taken of the part of the sky where they converge ; the great circles on such a photograph will appear as straight lines. These are shown on the

H 2

ioo STELLAR MOVEMENTS CHAP.

two diagrams, and the Roman numeral attached to each line indicates the region from which it comes. Each diagram represents an area of the sky measuring about 60° by 30° ; this would correspond on the terrestrial globe to a map of Northern Africa from the Congo to the Mediter- ranean. The apex marked on each diagram is the defini- tive apex of the drift, determined by a mathematical solution. For one of the drifts, called Drift L, the con- vergence of the directions is so evident as to need no comment. Owing to the smaller velocity of Drift II., its direction in any region cannot be determined so accu- rately, and a greater deviation of the great circles must be expected. Having regard to this, the agreement must be considered good, Region VII. being the only one showing important discordance. To appreciate the evidence of this diagram we may make a terrestrial comparison ; — if from seventeen points distributed uniformly over the earth tracks (great circles) were drawn, every one of which passed across the Sahara, they might fairly be considered to show strong evidence of convergence ; the distribu- tion of the Drift II. directions is quite analogous.

The analysis of the regions gives not only the directions of the two drifts, but also their speeds in terms of a certain unit ; and both sets of results may be used in finding definitive positions of the apices towards which the two drifts are moving. The results of a solution by least squares are as follows :

Drift I. Drift II.

Apex.	Apex.
R.A.	Dec.	Speed.	R.A	Dec.	Speed.
10 Equatorial Regions	(.>2	•4	-14°	1	1	•507	286°	•5	-63°	•6	0-869
7 Polar Regions . .	HU-	•3	- 1<>	•7	1	•536	289°	•1	83	•r>	O'§16
Whole Sphere . .	GO'	•8	-14C	•6	1	•516	287°	•8	-64°	•1*	1 0-855

* The fact that the declination derived from tho whole sphere does not lie between the declinations from the two portions looks paradoxical, but is due to the unequal weights of the determinations of the Z component from the two portions.

vi THE TWO STAR STREAMS >oi

The speeds are measured in terms of the usual theo- retical unit 1 //.

The drift-speed in an}7 region should (owing to fore- shortening) vary as the sine of the angular distance from the drift-apex, being greatest 90° away from the apex and diminishing to zero at the apex and antapex. This progressive decrease as the regions get nearer the apex is well shown in the observed values, and the sine-law is followed with very fair accuracy.

Another fact derived from the analysis is the proportion in which the stars are divided between the two streams ; this seems to vary somewhat from region to region, but the mean result is that 59*6 per cent, belong to Drift I. and 40*4 per cent, to Drift II. ; that is a proportion of practically 3:2.

To sum up, the result of this analysis of the Boss proper motions is to show that the motions can be closely represented if there are two drifts. That which we have called Drift I. moves with a speed of 1'52 units, the other, Drift II., with a speed of 0'86 unit. The first drift contains f of the stars and the second drift f . Their direc- tions are inclined at an angle of 100°.

It will be remembered that these motions are measured relative to the Sun. In Fig. 12, let SA and SB repre-

B c A

FIG. 12.

sent the drift-velocities, making an angle of 100°. Divide AB at C so that A C : CB = 2:3 corresponding to the proportion of stars in the two drifts. Then SO represents

STELLAR MOVEMENTS CHAP.

the motion of the centroid of all the stars relative to the Sun, and accordingly CS represents the solar motion and points towards the solar apex. AB and BA repre- sent the motion of one drift relative to the other ; the points in the sky towards which this line is directed are called the Vertices. The positions found from the num- bers above are

v fR.A. 94>-2 Dec.

3 • ' • ' \R.A. 274=-2 Dec. -11° '9.

The relative velocity of the two drifts is 1*87 units.

It is a remarkable fact that the vertices fall exactly in the galactic plane, so that the relative motion of the two drifts is exactly parallel to the galactic plane.

The solar motion CS found from the same numbers is 0*91 towards the

Solar Apex. . . R.A. 267C'3 Dec. +36° '4

This may be compared with Prof. Boss's determination from the same catalogue by the ordinary method 4 :

Solar Apex. . . R.A. 270°'5 Dec. + 34°-3

The agreement is interesting because the principles of the two determinations are very different ; moreover, in Boss's result the magnitudes of the proper motions as well as their directions were used, whereas the analysis on the two-drift theory depends solely on the directions.

Since the speed of the solar motion has also been measured in kilometres per second, this provides an equation for converting our theoretical unit into linear measure. We have 0*91 unit =19*5 km./sec., whence the theoretical unit l/h is 21 kilometres per second. We can thus, if we wish, convert any of the velocities previously given into kilometres per second.

It will b(- seen from Fig. 12 that the direction of motion of Drift I., SA, is inclined at a comparatively small angle to the parallactic motion, SC. But that the directions

vr THE TWO STAR STREAMS 103

are clearly distinct may be appreciated by referring back to Fig. 10. The solar an tapex actually falls just outside that diagram, so it is clear that the convergence is not towards the solar antapex but towards a different apex at the point indicated.

When referred to the centroid of the stars instead of to the Sun, the motions CA and CB of the two drifts are seen to be opposite to one another. It is perhaps not easy to realise that the inclination of the two stream- motions is a purely relative phenomenon depending on the point of reference chosen ; but this is the case. If we divest our minds of all standards of rest and contemplate simply two objects in space — two star-systems — all that can be said is that they are moving towards or away from or through one another along a certain line. The distinction between meeting directly or obliquely disappears. It is clear that this line joining the vertices must be a very important and fundamental axis in the distribution of stellar motions. It is an axis of symmetry, along which there is a strong tendency for the stars to stream in one direction or the other. It is this point of viewT that has led to an alternative mode of representing the phenomenon of star-streaming, the ellipsoidal theory of K. Schwarz- schild.5

We have, hitherto, analysed the stars into two separate systems, which move, one in one direction, the other in the opposite direction, along the line of symmetry ; but Schwarzschild has pointed out that this separation is not essential in accounting for the observed motions. It is sufficient to suppose that there is a greater mobility of the stars in directions parallel to this axis than in the perpendicular directions. The distinction is a little elusive, when it is looked into closely. It may be illus- trated by an analogy. Consider the ships on a river. One observer states that there are two systems of ships moving in opposite directions, namely, those homeward

io4 STELLAR MOVEMENTS CHAP.

bound and those outward bound ; another observer makes the non-committal statement that the ships move generally along the stream (up or down) rather than across it. This is a not unfair parallel to the points of view of the two-drift and ellipsoidal hypotheses. The distinction is a small one and it is found that the two hypotheses express very nearly the same law of stellar velocities ; but by the aid of different mathematical functions. This will be shown more fully in the mathe- matical discussion in the next chapter. Meanwhile we may sum up in the words of F. W. Dyson 6 : " The dual character of Kapteyn's system should not be unduly emphasised. Division of the stars into two groups was incidental to the analysis employed, but the essential result was the increase in the peculiar velocities of stars towards one special direction and its opposite. It is this same feature, and not the spheroidal character of the distribution, which is the essential of Schwarzschild's representation."

The phenomenon of star-streams (by which we mean the tendency to stream in two favoured directions, which both the two-drift and ellipsoidal theories agree in admitting) is shown very definitely in all the collections of proper motions that are available for discussion. Very careful attention has been given to the question whether it could possibly be spurious, and due to unsuspected systematic errors in the measured motions.7 It is not difficult for the investigate* to satisfy himself that such an explanation is quite out of the question, but it is not so easy to give the evidence in a compact form. Happily we are able to give one piece of evidence which seems absolutely con- clusive. F. W. Dyson 8 has made an investigation of the stars (1924 in number) with proper motions exceeding 20" a century. In this case we are not dealing with small quantities just perceptible with refined measurements, but

VI

THE TWO STAR STREAMS

105

with large movements easily distinguished and cheeked. These large motions show the same phenomenon that has been described for the smaller motions of Boss's catalogue. In fact, the two streams are shown more prominently when we leave out the smaller motions. This does not mean that the more distant stars are less affected by star- streaming than the near stars ; it is easy to show that for stars at a constant distance the small proper motions must

N

Scale 20" 30" 40"

FIG. 13.— Distribution of Large Proper Motions (Dyson).

necessarily be distributed more uniformly in position angle than the large motions, and the enhancement of the stream- ing when the small motions are removed is due to this cause. The diagram Fig. 13 is taken from Dyson's paper ; it refers to the area R. A. 10h to 14h,Dec. -30° to +30°. The motion of each star is represented by a dot, the displace- ment of the dot from the origin representing a century's motion on the scale indicated. The blank space round the origin is, of course, due to the omission of all motions less than 20" ; we can imagine it to be filled with an extremely

io6 STELLAR MOVEMENTS CHAP.

dense distribution of dots. It is clear that the distribution shown in the diagram represents a double streaming approximately along the axes towards 6h and S. No single stream from the origin could scatter the dots as

o o

they actually are. Although the general displacement is towards the solar antapex (i.e., towards the lower right corner), this is accompanied by an extreme elongation of the distribution in a direction almost at right angles. Thus the phenomenon of two streams is well shown in the largest, and proportionately most trustworthy, motions that are known, so that it requires no particular .delicacy of observation to detect it.

RADIAL MOTIONS

Doubtless the most satisfactory confirmation of this phenomenon found in the transverse motions of the stars would be an independent detection of the same phenomenon in the spectroscopically measured radial velocities. Although great progress has been made in the determination and publication of radial velocities, the stage has scarcely yet been reached when a satisfactory discussion of this question is possible. We shall see that the results at present available are quite in agreement with the two-stream hypothesis, and afford a valuable confirmation of it in a general way ; but a larger amount of data is required before we can see how precise is the agreement between the two kinds of observations.

In the transverse motions the two streams were detected by considering the stars in a limited area of the sky ; there was no need to go outside a single area, except at a later stage when it was desired to show that the different parts of the sky were concordant. But with the radial veloci- ties, we can learn nothing of star-streaming from a single region ; that is the drawback of a one-dimensional projection compared with a two-dimensional. To pass from one area to another involves questions of stellar dis-

VI

THE TWO STAR STREAMS

107

tribution, which complicate the problem. In particular it is necessary to pay attention to spectral type. It is well known that the early type stars are more numerous near the galaxy than elsewhere ; as these have on the average smaller residual motions than the later types, there will be a tendency for the radial motions near the galactic plane to be smaller than near the poles. But evidence of star- streaming should be looked for in a tendency for the residual radial velocities to be greater near the vertices (which are in the galactic plane) than elsewhere. The two effects are opposed, and there is a danger that they will mask one another.

By treating the different types separately the difficulty is avoided, but in that case the data become rather meagre. For Type A the results have been worked out by Campbell 9 who gives the following table