1911 Encyclopædia Britannica/Star

STAR, the general term for the luminous bodies seen in the heavens; used also by analogy for star-shaped ornaments (see Medal; Orders and Decorations) or other objects, and figuratively for persons of conspicuous brilliance. The word is common to many branches of languages: in Teutonic two forms appear, starre or sterre (cf. Du. ster), and sterne, or stern (cf. Ger. Stern, and the Scand. stjarna, stjerna, Sec). From Lat. stella, are derived Span. and Port. estrella, and Fr. étoile. The Greek is ἀστήρ, and the Sanskrit tara, for stara. The ultimate root is unknown, but may be connected with that meaning “to strew,” and the word would thus mean the points of light scattered over the heavens. The study of the stars is coeval with the birth of astronomy (see Astronomy: History); and among the earliest civilizations beneficent or malevolent influences were assigned to them (see Astrology). With the development of observational astronomy the sidereal universe was arbitrarily divided into areas characterized by special assemblages of stars; these assemblages were named asterisms or constellations, and each received a name suggested by mythological or other figures. The heavenly bodies fall into two classes: (1) the fixed stars, or stars proper, which retain the same relative position with respect to one another; and (2) the planets, which have motions of a distinctly individual character, and appear to wander among the stars proper.

Numerous counts of the number of stars visible to the naked eye have been made; it is doubtful whether more than 2000 can be seen at one time from any position on the earth. When a telescope is employed this number is enormously increased, and still more so with the introduction of photographic methods; with modern appliances more than a hundred million of these objects may be rendered perceptible.

The recognition of star’s is primarily dependent on their brightness or “magnitude”; and it is clear that stars admit of classification on this basis. This was attempted by Ptolemy, who termed the brightest stars “of the first magnitude,” and the progressively fainter stars of progressively greater magnitude. Ptolemy’s Number and
of the Stars.
classification has been adopted as the basis of the more exactly quantitative modern system. In this system one star is defined to be unit magnitude higher than another if its light is less in the ratio 1:2·512. This ratio is adopted so that a difference of five magnitudes may correspond to a light-ratio of 1:100. This subject is treated in the article Photometry, Celestial. The faintest stars visible to the naked eye on clear nights are of about the sixth magnitude; exceptionally keen, well-trained eyes and clear moonless nights are necessary for the perception of stars of the seventh magnitude. According to E. Heis the numbers and magnitudes of stars between the north pole and a circle 35º south of the equator are:—

1st mag. 2nd mag. 3rd mag. 4th mag. 5th mag. 6th mag.
14 48 152 313 854 2010

From the value of the light-ratio we can construct a table showing the number of stars of each magnitude which would together give as much light as a first magnitude star, viz.:—

1st mag. 2nd mag. 3rd mag. 4th mag. 5th mag. 6th mag.
1 2 6 16 40 100

Comparing these figures with the numbers of stars of each magnitude we notice that the total light emitted by all the stars of a given magnitude is fairly constant.

Variable Stars.—Although the majority of the stars are unchanging in magnitude, there are many exceptions. Stars whose brightness fluctuates are called variable stars. The number of known objects of this class is being added to rapidly, and now amounts to over 4000. The systematic search made at Harvard Observatory is responsible for a large proportion of the recent discoveries. Many of these stars seem to vary quite irregularly; the changes of magnitude do not recur in any orderly way. Others, however, are periodic, that is to say, the sequence of changes is repeated at regular intervals, and it is thus possible to predict when the maximum and minimum brightness will occur. Of the periodic variable stars, the lengths of the periods range from 3 hours 12 minutes, which is the shortest yet determined, to 610 days, the longest. When statistics of the lengths of the periods are collected, it is at once noticed that they fall into two fairly well-marked classes. The following table, based on S. C. Chandler’s “Third Catalogue” (Astronomical Journal, vol. xvi.), supplemented by A. W. Roberts’s list of southern variables (ibid. vol. xxi.), classifies the lengths of the periods of 330 stars.

Stars 73 8 12 22 41 45 49 50 20 6 1 2 1

It will be noticed that there are very few periods between 50 and 150 days, that a considerable number are less than 50 days (actually a large majority of these are less than 10 days), and that from 150 days upwards the number of periods increases to a maximum at about 350 days and then diminishes. We thus recognize two classes of variables, of which (1) the long-period variables have periods ranging in general from 150 to 450 days, though a few are outside these limits, and (2) the short-period variables have periods less than 50 days (in the majority pf cases less than 10 days). There is some over-lapping of these two classes as regards length of period, and it is doubtful in which class some stars, whose periods are between 10 days and 150 days, should be placed; but the two classes are quite distinct physically, and the variability depends on entirely different causes.

Long-period Variables.—The best known and typical star of this class is Mira or ο Ceti. This was the first variable star to be discovered, having been noticed in 1596 by David Fabricius, who thought it was a new star (a Nova). The varying brightness, ranging from the ninth to the second magnitude, was recognized in 1639 by John Phocylides Holwarda, and in 1667 Ismael Boulliau (1605–1694) established a periodicity of 333 days. Although the periodic outbursts of light have taken place without intermission during the two and a half centuries that the star has been Under observation, they are somewhat irregular. The different maxima differ considerably in brightness; thus in 1906 (the brightest maximum since 1779) the second magnitude was reached) but in other years (as in 1868) it has failed to reach the fifth magnitude. The minima likewise are variable, but only slightly so. Also, the period varies somewhat; the maxima occur sometimes early and sometimes late as compared with the mean period, but the difference is never more than forty days. No general law has been discovered governing these irregularities. The change of magnitude takes place gradually, but the rise to maximum brilliance is rather more rapid than the decline. Spectroscopic observation shows that the increased light accompanies an actual physical change or conflagration in the star. The spectrum is of the third type with bright hydrogen emission lines (see below, Spectra of Stars). Stars having this type of spectrum are always variable, and a large proportion of the more recently discovered long-period variables have been detected through their characteristic spectrum.

χ Cygni is another star of this class, remarkable for its range of magnitude. In its period of 406 days it fluctuates between the thirteenth and the fourth magnitudes; thus at maximum it emits 4000 times as much light as at minimum. The mean range of 75 long-period variables observed at Harvard (Harvard Annals, vol. lvii.) was five magnitudes. Another variable, R Normae[1] is of interest as having a pronounced double maximum in each period.

It is natural to compare the periodic outbursts occurring in these stars with the outbursts of activity on the sun, which have a period of about eleven years. In both cases no extraneous cause can be assigned; the period seems to be inherent in the star itself and not to be determined by the revolution of a satellite (no variability of the line-of-sight motion of Mira has been found, so that it is probably not accompanied by any large companion). In both cases the rise to a maximum is more rapid than the decline to a minimum, and in fact some of the minor peculiarities of the sunspot curve are closely imitated by the light-curves of variable stars. H. H. Turner has analysed harmonically the light-curves of a number of long-period variables, and has shown that when they are arranged in a natural series the sun takes its place in the series near, but not actually at, one end. It is necessary to suppose, if the analogy is to hold, that the sun is brightest when sunspots and faculae are most numerous; this is by no means unlikely. On the other hand, the variations in the light of the sun must be very small compared with the enormous fluctuations in the light of variable stars. Moreover, the solar period (11 years) is far outside the limits of the periods of variables. It is therefore perhaps misleading actually to class the sun with them; but it seems highly probable that whatever cause produces the periodic outbursts of spots and faculae on our sun differs only in degree from that which, in stars under a different physical condition of pressure and temperature, results in the gigantic conflagrations which we have been considering.

Short-period Variables.—Besides the shortness of the period these variables possess other characteristics which differentiate them from the long-period variables. The range of variation is much smaller, the difference between maximum and minimum rarely exceeding two magnitudes. Also the variations recur with perfect regularity. There is reason to believe that all the stars of this class are binary systems, and that the variations of brightness are determined by the different aspects presented by the two component stars during the period of revolution. There are several well- marked varieties of short-period variables; the most important are typified by the stars Algol, β Lyrae, ζ Geminorum and δ Cephei.

In the Algol variables one of the component stars is dark (that is to say, dark in comparison with the other), and once in each revolution, passing between us and the bright component, partially hides it. This class of variables is accordingly characterized by the fact that for the greater part of the period the star shines steadily with its maximum brilliancy, but fades away for a short time during each period. The variability of Algol (β Persei) was discovered in 1783 by John Goodricke (1764–1786), but, judging from its name, which signifies “the demon,” it seems possible that its peculiarity may have been known to the ancient astronomers. Algol is ordinarily of magnitude 2·3, but once in a period of 2d. 20h. 49m. it suffers partial eclipse and fades to magnitude 3·5. The duration of each eclipse is 9 1/4 hours. Ever since the variability of Algol was observed it was suspected to be due to a partial eclipse of the star by a dark body nearly as large as itself revolving round it; but the explanation remained merely a surmise until K. H. Vogel of Potsdam, by repeated measurements of the motion of Algol in the line of sight, showed that the star is always receding from us before the loss of light and approaching us afterwards. This leaves no room for doubt that an invisible companion passes between us and Algol about the time the diminution of light takes place, and so proves the correctness of the explanation. The dimensions of the Algol system have been calculated, with the result that Algol appears to have a diameter of 1,000,000 m. and its companion a diameter of 830,000 m.; the distance between their centres cannot be deduced without making certain doubtful assumptions, but may be about 3,000,000 m. When this distance is compared with those prevailing in the solar system, it seems an extraordinarily small separation between two such large bodies; we shall, however, presently come across systems in which the two components revolve almost or actually in contact. About 56 Algol variables were known in 1907; the variables of this class are the most difficult to detect, for the short period of obscuration may easily escape notice unless the star is watched continuously.

The variable star β Lyrae, which is typical of another class, was also discovered by Goodricke in 1784. It differs from the Algol type in having two unequal minima separated by two equal maxima. Thus in a period of 12d. 22h. from a maximum of magnitude 3·4 it falls to 3·9, rises again to 3·4, then falls to 4·5 and returns to magnitude 3·4. The changes take place continuously, so that there is no period of steady luminosity. The hypothesis of G. W. Myers (Astrophysical Journal, vol. vii.) affords at least a partial explanation of the phenomena. Two stars are supposed to revolve about one another nearly or actually in contact. In such a system the tidal forces must be very great, and under their influence the stars will not be spherical, but will be elongated in the direction of the line joining their centres. When the line of centres is at right angles to our line of sight, the stars present to us their greatest apparent surface, and therefore send us the maximum light. This happens twice in a revolution. As the line of centres becomes more oblique, the surface is seen more and more foreshortened and the brilliancy diminishes continuously. Supposing that the two stars are of unequal surface brilliancy, the magnitude at minimum will depend on which of the two stars is the nearer to us, accordingly there are two unequal minima in each revolution. When the two stars are of equal brilliancy the minima are equal; this is the case in variables of the ζ Geminorum type. When the orbits are eccentric, the tidal disturbance varying with the distance between the two components will probably cause changes in their absolute brilliancy; the variation due to change in the aspect of the system presented to us may thus be supplemented by a real intrinsic variation, both, however, being regulated by the orbital motion. A large eccentricity also produces an unsymmetrical light variation, the minimum occurring at a time not midway between two maxima; stars of this character are called Cepheid variables, after the typical star δ Cephei. All the best-known short-period variables have been proved to be binary systems spectroscopically, and to have periods corresponding with the period of light variation, so that to this extent the hypothesis we have described is well founded; but it is doubtful if it is the whole explanation. S. Albrecht has shown that, of the 10 members of the δ Cephei class for which both the orbits and the light-variations are thoroughly known, the maximum light always occurs approximately at the time when the brighter component is approaching us most rapidly; this relation, which seems to be well established, is a most perplexing one.

No hard and fast physical distinction can be drawn between the various classes of short-period variables; as the distance between the components diminishes the Algol variable merges insensibly into the β Lyrae type. The latter, on the other hand, is perhaps connected by insensible gradations with the ordinary simple star. Sir G. H. Darwin and H. Poincare have investigated the forms taken up by rotating masses of fluid. When the angular momentum is too great for the usual spheroidal form to persist, this gives place to an ellipsoid with three unequal axes; this is succeeded by a pear-shaped form. The subsequent sequence of events cannot be traced with certainty, but it seems likely that the pear-shaped form is succeeded by an hour-glass-shaped form, which finally separates at the neck into two masses of fluid. Ellipsoidal, pear-shaped or hour-glass-shaped stars would all give rise to the phenomena of a short-period variable, and doubtless examples of these intermediate forms exist. Certain clusters contain a remarkable number of short-period variables. Thus the cluster Messier 5 was found at Harvard to contain 185 variables-out of 900 stars examined. Solon I. Bailey, on examining 63 of them, found that with one exception their periods lay between 10h. 48m. and 14h. 59m., and the range of variation between 0·7 and 1·4 magnitudes. Moreover, the light-curves were all of a uniform type, a distinctive feature of “cluster variables” being the rapid rise to a maximum and slow decline.

Temporary Stars or Novas.—From time to time a star, hitherto too faint to be noticeable, blazes out and becomes a prominent object, and then slowly fades into obscurity. According to Miss Agnes Clerke there are records of ten such stars appearing between 134 B.C. and A.D. 1500. Since that time nine novas have appeared, which have attained naked-eye visibility; and in recent years a number of very faint objects of the same class have been detected. The brightest star of all these was the famous “Tycho’s star” in Cassiopeia. It was first observed on the 6th of November 1572 by Wolfgang Schuler. In five days its light had reached the first magnitude, and a little later it even equalled Venus in brilliancy and was observed in full daylight. After three weeks it began to decline, but the star did not finally disappear until March 1574. “Kepler’s” nova in Ophiuchus broke out in 1604 and attained a brightness greater than that of Jupiter; it likewise gradually waned, and disappeared after about fifteen months. For nearly three centuries after these two remarkable stars no nova attained a brilliancy greater than that of the ordinary stars, until in 1901 Nova Persei appeared. This star was discovered by T. D. Anderson on the 21st–22nd of February, its magnitude at that time being 2·7. In the next two days it reached zero magnitude, thus becoming the brightest star in the northern heavens, but after that it rapidly decreased. On the 15th of March it was of the fourth magnitude; during the next three months it oscillated many times between magnitudes 4 and 6, and by the end of the year it had faded to the seventh magnitude. In July 1903 it was of the twelfth magnitude, and its light has remained constant since then. In the case of this star there is evidence that the outburst must have been extremely rapid, for the region where Nova Persei appeared had been photographed repeatedly at Harvard during February, and in particular no trace of the star was found on a plate taken on the 19th of February, which showed eleventh magnitude stars. Thus a rise of at least eight magnitudes in two days must have occurred.

On the 21st of August, six months after the discovery of Nova Persei, C. Flammarion and E. M. Antoniadi discovered that a nebula surrounded it. Subsequent photographs showed that this nebula, which consisted mainly of two incomplete rings of nebulosity, was expanding outwards at the rate of from 2″ to 3″ per day. This expansion continued at the same rate until the following year. Spectroscopic examination had already suggested prodigious velocities of the order of 1000 m. per second in the gases of the atmosphere of the nova; but the velocity implied by this expansion of the nebula was unprecedented and comparable only with the velocity of light. The suggestion was made, and seems to be the true explanation, that what was actually witnessed was the wave of light due to the outburst of the nova, spreading outwards with its velocity of 186,000 m. per second, and rendering luminous as it reached them the particles of a pre-existing nebula, whose own light had been too faint to be visible.

Two possible explanations of the phenomena of temporary stars have been held. The collision theory supposes that the outburst is the result of a collision between two stars or between a star and a swarm of meteoric or nebulous matter. The explosion theory regards the outburst as similar to the outbreak of activity of a long-period variable. Probably the latter hypothesis is the one more generally accepted now. There is one unique star, which is of special interest as occupying rather an intermediate position between a nova and a long-period variable. This is the southern star η Argus (sometimes called η Carinae). From 1750 until about 1832 it seems to have varied irregularly between the second and the fourth magnitudes. For the next ten years it slowly increased (though with slight check), and in 1843 was nearly as bright as Sirius; since then it has slowly faded, but it was not till 1869 that it ceased to be visible to the naked eye. It is now about magnitude 7·5. The slowness both of the rise and decline is in great contrast with the progress of a nova. η Argus is surrounded by a nebula, the famous “Keyhole nebula”; in this respect it resembles Nova Persei.

System of Stars.—On examining the stars telescopically, many which appear single to the unaided eye are found to be composed of two or more stars very close together. In some cases the proximity is only apparent; one star may be really at a vast distance behind the other, but, Double
being in the same line of vision, they appear close together. In many cases, however, two or more stars are really connected, and their distance from one another is (from the astronomical standpoint) small. The evidence of this connexion is of two kinds. In a number of cases measures of the relative positions of the two stars, continued for many years, have shown that they are revolving about a common centre; when this is so there can be no doubt that they form a binary system, and that the two components move in elliptic orbits about the common centre of mass, controlled by their mutual gravitation. But these cases form a very small proportion of the total number of double stars. In many other double stars the two components have very nearly the same proper motion. Unless this is a mere coincidence, it implies that the two stars are nearly at the same distance from us. For otherwise, if they had from some unknown cause the same actual motion, the apparent motion in arc would be different. We can therefore infer that the two stars are really comparatively close together, and, moreover, since they have the same proper motion, that they remain close together. They may thus be fairly regarded as constituting a binary system, though the gravitational attraction between some of the wider pairs must be very weak.

Several double stars were observed during the 17th century, ζ Ursae Majoris being the first on record. In 1784 Christian Mayer published a catalogue of all the double stars then known, which contained 89 pairs. Between 1825 and 1827 F. G. W. Struve at Dorpat examined 120,000 stars, and found 31 12 double stars whose distance apart did not exceed 32″. W. S. Bumham’s General Catalogue of Double Stars (1907) contains 13,655 pairs north of declination -31°. Undoubtedly a large number of these are only optical pairs, but mere considerations of probability show that the majority must be physically connected. For only 88 of them has it been possible as yet to deduce a period, and at least half even of these periods are very doubtful. The rates of motion are so slow that many centuries’ observations are needed to determine the orbit.

The most rapid visual binary (leaving aside Capella for the moment) is δ Equulei, which completes a revolution in 5·7 years. Next to it come 13 Ceti, period 7.4 years, and κ Pegasi, period 11.4 years. From a list of systems with determined periods given by Aitken (Lick Observatory Bulletin, No. 84) there are 20 with periods less than 50 years, and 16 between 50 and 100 years. S Equulei, 13 Ceti and κ Pegasi are all extremely close pairs, and can only be resolved with the most powerful instruments. Capella, whose period is only 104 days, was discovered to be double by means of the spectroscope, but has since been measured frequently as a visual binary at Greenwich. With the best instruments a star can be distinguished as double when the separation of the two components is a little less than 0·1″. From the very few orbits that have as yet been determined one interesting result has been arrived at. Most of the orbits are remarkably eccentric ellipses, the average eccentricity being about 0·5. There is a very striking relation between the eccentricity and the period of a system; in general the binaries of longest period have the greatest eccentricities. The relation applies not only to the visual but to the spectroscopic binaries; these, having shorter periods than the visual binaries, have generally quite small eccentricities. Another interesting feature is that, where the two components differ in brightness, the fainter component is often the one possessing the greater mass.

Far within the limit to which telescopic vision can extend binary systems are now being found by the spectroscope. These systems appear as a connecting link between short-period variable stars on the one hand and telescopic double binaries stars on the other. Stars of the class to which the Algol type of variables belongs will appear to us to vary only in Spectroscopic
the exceptional case when the plane of the orbit passes so near our sun that one body appears to pass over the other and so causes an eclipse. Except when the line of sight is perpendicular to the plane of the orbit, the revolution of the two bodies will result in a periodic variation of the motion in the line of sight. Such a variation can be detected by the spectroscope. If both the bodies are luminous, especially if they do not differ much in brilliancy, the motion of revolution is shown by a periodic doubling of the lines of the spectrum; when one body is moving towards us and the other away their spectral lines are displaced (according to Doppler’s principle) in opposite directions, so that all the lines strong enough to appear in both spectra appear double; when the two bodies are in conjunction, and therefore moving transversely, their spectra are merged into one and show nothing unusual. More usually, however, only one component is sufficiently luminous for its spectrum to appear; its orbital motion is then detected by a periodic change in the absolute displacement of its spectral lines. Up to 1905, 140 spectroscopic binaries had been discovered; a list of these is given in the Lick Observatory Bulletin, no. 79. Details of the calculated orbits of 63 spectroscopic binaries are given in Publications of the Allegheny Observatory, vol. i. No. 21. According to W. W. Campbell one star in every seven examined is binary.

A continuous gradation can be traced from the most widely separated visual binaries, whose periods are many thousand years, to spectroscopic binaries, Algol and β Lyrae variables, whose periods are a few hours and whose components may even be in contact, and from these to dumb-bell shaped stars and finally to ordinary single stars. It is a legitimate speculation to suppose that these in the reverse order are the stages in the evolution of a double star. As the simple star radiates heat and contracts, it retains its angular momentum; when this is too great for the spheroidal form to persist, the star may ultimately separate into two components, which are driven farther and farther apart by their mutual tides. Tidal action also accounts for the progressively increasing eccentricities of the orbits, already referred to. This theory of the genesis of double-stars by fission is not, however, universally accepted; in particular objections have been urged by T. C. Chamberlin and F. R. Moulton. It is true that rotational instability alone is not competent to explain the separation into two components; but the existence of gravitational instability, pointed out by J. H. Jeans, enables the principal difficulties of the theory to be surmounted. Whilst there is thus no well-defined lower limit to the dimensions of systems of two stars, on the other hand we cannot set any superior limit either to the number of stars which shall form a system or to the dimensions of that system. No star is altogether removed from the attractions of its neighbours, and there are cases where some sort of connexion seems to relate stars which are widely separated in space. A curious case of this sort is that of the five stars β, γ, δ, ε and ζ of Ursa Major. These have proper motions which are almost identical in amount and in direction. The agreement is too close to be dismissed as a mere coincidence, and it is confirmed by a corresponding agreement of their radial motions determined by the spectroscope; and yet, seeing that β and ζ Ursae Majoris are 19° apart, these two stars must be distant from each other at least one-third of the distance of each from the sun; thus the members of this singular group are separated by the ordinary stellar distances, and probably each has neighbours, not belonging to the system, which are closer to it than the other four stars of the group. Further, E. Hertzsprung has shown that Sirius also belongs to this same system and shares its motion, notwithstanding that it is in a nearly opposite part of the sky. It is difficult to understand what may be the connexion between stars so Widely separated; from the equality of their motions they must have been widely separated for a very long period.

Of multiple stars the most famous is θ Orionis, situated near the densest part of the great Orion nebula. It consists of four principal stars and two faint companions. From the more complex systems of this kind, we pass to the consideration of clusters, which are systems of stars in which the components Star-
are very numerous. When examined with a telescope of power insufficient to separate the individual stars, a cluster appears like a nebula. The “beehive cluster” Praesepe in Cancer is an example of an easily resolved cluster composed of fairly bright stars. The great cluster in Hercules (Messier 13), on the other hand, requires the highest telescopic power for its complete resolution into stars. Doubtless with improved telescopes many more apparent nebulae would be shown to be clusters, but there are certainly many nebulae which are otherwise constituted. Many of the clusters are of very irregular forms, either showing no well-marked centre of condensation, or else condensed in streams along certain lines. There is, however, a well-marked type to which many of the richest clusters belong; these are the globular clusters. They have a symmetrical circular shape, the condensation increasing rapidly towards the centre. The Hercules cluster is of this form; another example is ω Centauri, in which over 6000 stars have been counted, comprised within a circle of about 40′ diameter. These clusters present many unsolved problems. Thus Perrine, from an examination of ten globular clusters (including Messier 13 and ω Centauri), has found in each case that the stars can be separated into two classes of magnitudes. About one-third of the stars are between magnitudes 11 and 13, and the remaining two-thirds are between magnitudes 15·5 and 16·5. Stars of magnitudes intermediate between these two groups are almost entirely absent. Thus each cluster seems to consist of two kinds of stars, which we may distinguish as bright and faint; the bright stars are all approximately of one standard size, and the faint stars of another standard size and brightness.

The question of the stability of these clusters is one of much interest. The mutual gravitation of a large number of stars crowded in a comparatively small space must be considerable, and the individual stars must move in irregular orbits under their mutual attractions. It does not seem probable, however, that they can escape the fate of ultimately condensing into one confused mass. If this surmise be correct, we are witnessing in clusters a counter-process of evolution to that which is taking place in double stars; the latter appear to be separating from a single original mass and the former condensing into one.

Colours and Spectra of Stars.—The brighter stars show a marked variety of colour in their light, and with the aid of a telescope a still greater diversity is noticeable. It is, however, only the red stars that form a clearly marked Colours. class by themselves. For purposes of precise scientific investigation the study of spectra is generally more suitable than the vague and unsatisfactory estimates of colour, which differ with different observers. Of the first magnitude red stars Antares is the most deeply coloured, Betelgeux, Aldebaran and Arcturus being successively less conspicuously red. Systematic study of red stars dates from the publication in 1866 of Schjellerup’s Catalogue, containing a list of 280 of them.

The two components of double stars often exhibit complementary colours. As a rule contrasted colours are shown by pairs having a bright and a faint component which are relatively wide apart; brilliant white stars frequently have a blue attendant—this is instanced in the case of Regulus and Rigel. That the effect is due to a real difference in the character of the light from the two components has been shown by spectrum analysis, but it is probably exaggerated by contrast.

The occurrence of change, either periodic or irregular, in the colour of individual stars, has been suspected by many observers; but such a colour-variability is necessarily very difficult to establish. A possible change of colour in the case of Sirius is noteworthy. In modern times Sirius has always been a typical white or bluish-white star, but a number of classical writers refer to it as red or fiery. There is perhaps room for doubt as to the precise significance of the words used; but the fact that Ptolemy classes Sirius with Antares, Alde- baran, Arcturus, Betelgeux and Procyon as “fiery red” (ὑπόκιῤῥοι) as compared with all the other bright stars which are “yellow” (ξάνθοι.) seems almost conclusive that Sirius was then a redstar.

When examined with the spectroscope the light of the stars is found to resemble generally that of the sun. The spectrum consists of a continuous band of light crossed by a greater or less number of dark absorption lines or bands. As in the case of the sun, this indicates an incandescent body Spectra of
which might be solid, liquid, or a not too rare gas, surrounded by and seen through an atmosphere of somewhat cooler gases and vapours; it is this cooler envelope whose nature the spectroscope reveals to us, and in it the presence of many terrestrial elements has been detected by identifying in the spectrum their characteristic absorption lines. Stellar spectroscopy dates from 1862, when Sir William Huggins (with a small slit-spectroscope attached to an 8-in. telescope) measured the positions of the chief lines in the spectra of about forty stars. In 1876 he successfully applied photography to the study of the ultra-violet region of stellar spectra. Various schemes of classification of spectra have been used. The earliest is that due to A. Secchi (1863–1867) who distinguished four “types”; subsequent research, whilst slightly modifying, has in the main confirmed this classification. Secchi’s Type I. or “Sirian” type includes most of the bright white stars, such as Sirius, Vega, Rigel, &c.; it is characterized by strong broad hydrogen lines, which are often the only absorption lines visible. Type II. includes the “Solar” stars, as Capella, Arcturus, Procyon, Aldebaran, their spectra are similar to that of the sun, being crossed by very numerous fine lines, mostly due to vapours of metals. The great majority of the visible stars belong to these first two types. Type III. or “Antarian” stars are of a reddish colour, such as Antares, Betelgeux, Mira, and many of the long-period variables. The spectrum, which closely resembles that of a sunspot, is marked by nutings or bands of lines sharply bounded on the violet side and fading off towards the red. It has been shown by A. Fowler that these flutings are due to titanium oxide; this probably indicates a relatively low temperature, for at a high temperature all compounds would be dissociated. Type IV. also consists of red stars with banded spectra, but the bands differ in arrangement and appearance from those in the third type, and are sharply bounded on the red side. These stars are also believed to have a comparatively low, surface temperature, and the bands are attributed to the presence of compounds of carbon. About 250 Type IV. stars are known, but none conspicuous; 19 Piscium, the brightest, is of magnitude 5·5.

Other classifications which are extensively used are those respectively of K. H. Vogel, J. N. Lockyer and the Draper Catalogue. The divergences depend mainly on the different views taken by their authors as to the order of stellar evolution. Apart from these considerations, the chief modification in the classification introduced by more recent investigators has been to separate Secchi’s Type I. into two divisions, called helium and hydrogen stars respectively. The former are often called “Orion” stars, as all the brighter stars in that constellation with the exception of Betelgeux belong to the helium type. Helium stars are generally considered to be the hottest and most luminous (in proportion to size) of all the stars. Type II. is now subdivided into “Procyon,” “Solar” and “Arcturian” stars. The “Procyon” or calcium stars form a transition between Type I. and Type II. proper, and show the lines of calcium besides those of hydrogen. An important variety of Type III. spectra has been recognized, in which, as well as the usual absorption bands, bright emission lines of hydrogen appear; stars having this particular spectrum are always variable. Finally, a fifth type has been added, the Wolf-Rayet stars; these show a spectrum crossed by the usual dark lines and bands, but showing also bright emission bands of blue and yellow light. About 100 Wolf-Rayet stars are known, of which γ Velorum is the brightest; they are confined to the region of the Milky Way and the Magellanic Clouds. (See Planet.)

Evolution of Stars.—The absence of the distinctive lines of an element in the spectrum does not by any means signify that that element is wanting or scarce in the star. The spectroscope only yields information about the thin outer envelope of the star; and even here elements may be present which do not reveal themselves, for the spectrum shown depends very greatly on the temperature and pressure. Stars of the different types are therefore not necessarily of different chemical constitution, but rather are in different physical conditions, and it is generally believed that every star in the course of its existence passes through stages corresponding to all (or most of) the different types. The stars are known to be continually losing enormous quantities of energy by radiating their heat into space. Ordinary solid or liquid masses would cool very rapidly from this cause and would soon cease to shine. But a globe of gaseous matter under similar conditions will continually contract in volume, and in so doing transforms potential energy into heat. It was shown by Homer Lane that a mass of gas held in equilibrium by the mutual gravitation of its parts actually groWs hotter through radiating heat; the heat gained by the resulting contraction more than counterbalances that lost by radiation. Thus in the first stage of a star’s history we find it gradually condensing from a highly diffused gaseous state, and growing hotter as it does so. But this cannot continue indefinitely; when the density is too great the matter ceases to behave as a true gas, and the contraction is insufficient to maintain the heat. Thus in the second stage the star is still contracting, but its temperature is decreasing. The greatest temperature attained is not the same for all stars, but depends on the mass of the star. It is, however, important to bear in mind that Lane’s theory is concerned with the temperature of the body of the star; the temperature of the photosphere and absorbing layers, with which we are chiefly concerned, does not necessarily follow the same law. It depends on the rapidity with which convection currents can supply heat from the interior to replace that radiated, and on a number of other nicely balanced circumstances which cannot well be calculated.

Conflicting opinions are held as to the various steps in the process of evolution and the order in which the various types succeed one another, but the following perhaps represents in the main the most generally accepted view. Starting from a widely diffused nebula, more or less uniform, we find that, in consequence of gravitational instability, it will tend to condense about a number of nuclei. Jeans has even estimated theoretically the average distances apart of these nuclei, and has shown that it agrees in order of magnitude with the observed distances of the stars from one another (Astrophysical Journal, vol. xxii.). As the first condensation takes place, the resulting development of heat causes the hydrogen, helium and light gases to be expelled. This may explain the existence of gaseous nebulae, which are often found intimately associated with star-clusters, a good example being the nebulosity surrounding the Pleiades. As the nuclei grow by the attraction of matter they begin to be capable of retaining the lighter gases, and atmospheres of hydrogen and helium are formed. The temperature of the photosphere at this stage has reached a maximum, and the star is now of the helium type. Then follows a gradual absorption of first the helium and then the hydrogen, the photosphere grows continually cooler, and the star passes successively through the stages exemplified by Sirius, Procyon, the Sun, Arcturus and Antares. Some authorities, however, consider the Antarian (Type III.) stars to be in a very early stage of development and to precede the helium stars in the order of evolution; in that case they are in the stage when the temperature is still rising. Type IV. (carbon) stars are placed last in the series by all authorities; they seem, however, to follow more directly the solar stars than the Antarian. If the latter are considered to be in an early state this presents no difficulty; but if both Antarian and carbon stars are held to be evolved from solar stars, we may consider them to be, not successive, but parallel stages of development, the chemical constitution of the star deciding whether it shall pass into the third or fourth type. The Wolf-Rayet stars must probably be assigned to the earliest period of evolution; they are perhaps semi-nebulous. In this connexion it may be noted that the spectrum of Nova Persei, after passing through a stage in which it resembled that of a planetary nebula, has now become of the Wolf-Rayet type.

Density of Stars.—Interesting light is thrown on the question of the physical state of the stars by some evidence which we possess as to their densities. The mean density of the sun is about 1 1/3 times that of water; but many of the stars, especially the brighter ones, have much lower densities and must be in a very diffused state. We have necessarily to turn to binary systems for our data. When the orbit and periodic time is known, and also the parallax, the masses of the stars can be found. (If only the relative orbit is known, the sum of the masses can be determined; but if absolute positions of one component have been observed, both masses can be determined separately.) But even when, as in most cases, the parallax is unknown or uncertain, the ratio of the brightness to the mass can be accurately found. Thus it is found that Procyon gives about three times as much light as the sun in proportion to its mass, Sirius about sixteen times, and ζ Orionis more than ten thousand times. In these cases evidently either the star has a greater intrinsic brilliancy per square mile of surface than the sun, or is less dense. Probably both causes contribute. The phenomena of long-period variables show that the surface brilliancy may vary very greatly, even in the same star. The Orion stars have the highest temperature of all and have admittedly the greatest surface- luminosity, but the extreme brilliancy of ζ Orionis in proportion to its mass must be mainly due to a small density. For the Algol variables it is possible to form even more direct calculations of the density, for from the duration of the eclipse an approximate estimate of the size of the star may be made. A. W. Roberts concluded in this way that the average density of the Algol variables and their eclipsing companions is about one-eighth that of the sun. For β Lyrae G. W. Myers found a density a little less than that of air; the density is certainly small, but J. H. Jeans has shown that for this type of star the argument is open to theoretical objection, so that Myers’s result cannot be accepted.

There are many stars, however, of which the brightness is less than that of the sun in proportion to the mass. Thus the faint companion of Sirius is of nearly the same mass as the sun, but gives only 1/4000 of its light. In this case the companion, being about half the mass of Sirius itself, has probably cooled more rapidly, and on that account emits much less light. T. Lewis, however, has shown that the fainter component of the binary system is often the more massive. It may be that these fainter components are stiil in the stage when the temperature is rising, and the luminosity is as yet comparatively small; but it is not impossible that the massive stars (owing to their greater gravitation) pass through the earlier stages of evolution more rapidly than the smaller stars.

Distances and Parallaxes of the Stars.—As the earth traverses annually its path around the sun, and passes from one part of its orbit to another the direction in which a fixed star is seen changes. In fact the relative positions are the same as if the earth remained fixed and the star described an orbit equal to that of the earth, but with the displacement always exactly reversed. The star thus appears to describe a small ellipse in the sky, and the nearer the star, the larger will this ellipse appear. The greatest displacement of the star from its mean position (the semi-axis major of the ellipse) is called its parallax. If π be the parallax, and R the radius of the earth’s orbit, the distance of the star is R/sin π. The determination of stellar parallaxes is a matter of great difficulty on account of the minuteness of the angle to be measured, for in no case does the parallax amount to 1″; moreover, there is always an added difficulty in determining an annual change of position, for seasonal instrumental changes are liable to give rise to a spurious effect which will also have an annual period. Very special precautions are required to eliminate instrumental error before we can compare observations, say, of a star on the meridian in winter at 6 p.m. with observations of the same star in summer on the meridian at 6 a.m. The first determination of a stellar parallax was made by F. W. Bessel in the years 1837–1840, using a heliometer. He chose for his purpose the binary star 61 Cygni, which was the star with the most rapid apparent motion then known and therefore likely to be fairly near us, although only of the sixth magnitude. He found for it a parallax of 0·35″ a value which agrees well with more modern determinations. T. Henderson at the Cape of Good Hope measured the parallax of α Centauri, but his resulting value 1″ was considerably too high. More accurate determinations have shown that this star, which is the third brightest star in the heavens, has a parallax of 0·75″, this indicates that its distance is 25,000,000,000,000 m. So far as is known α Centauri is our nearest neighbour.

Formerly attempts were made to determine parallaxes by measuring changes in the absolute right ascensions and declinations of the stars from observations with the meridian circle. The results were, however, always untrustworthy owing to annual and diurnal changes in the instrument. Nowadays the determination is more usually made by measuring the displacement of the star relatively to the stars surrounding it. Hitherto the heliometer has been most extensively used for this purpose, D. Gill, W. L. Elkin, B. E. A. Peter and others have made their important determinations with it. The photographic method; however, now appears to yield results oi equal precision, and is/ likely to be used very largely in the future. The quantity determined by these methods is the relative parallax between the star measured and the stars with which it is compared. To obtain the true parallax, the mean parallax of the comparison stars must be added to this relative parallax. It is, however, fair to assume that the comparison stars will rarely have a parallax as great as 0·01″; for it must be remembered that it is quite the exception for a star taken at random to have ab appreciable parallax; particularly if a star has an ordinarily small proper motion, it is likely to be very distant. Still exceptional cases will occur where a comparison star is even nearer than the principal star; it is one of the advantages of the photographic method that it involves the use of a considerable number of comparison stars, whereas in the heliometric method usually only two stars, chosen symmetrically one on each side of the principal star, are used. In the table are collected the parallaxes and other data of all stars for which the most probable value of the parallax exceeds 0·20″. Although much work has been done recently in measuring parallaxes, the number of stars included in such a list has not been increased, but rather has been considerably diminished; many large parallaxes, which were formerly provisionally accepted, have been reduced on revision. It cannot be too strongly emphasized that many of these determinations are subject to a large probable error, or even altogether uncertain. For one or two of the more famous stars such as α Centauri the probable error is less than ±0·01″; but for others in the list it ranges up to ±0·05″. To convert parallaxes into distance we may remember that a parallax of 1″ denotes a distance of 18 1/2 billion miles, or 206,000 times the distance of the sun from the earth. A parallax of 0·01″ denotes a distance a hundred times as great, and so on, the distance and parallax being inversely proportional. A unit of length, which is often used in measuring stellar distances, is the light year or distance that light travels in a year; it is rather less than six billion miles.

Stars with Large Parallax.
Star. Position
R.A. Dec.
Mag. Annual
Parallax. Authority
h. m. sec.
Gr. 34 0 13 +43 8·1 2·8 ·27 R, Sc, C
Ceti 1 39 -16 3·7 1·9 ·31 S
C.Z.5h243. 5 8 -45 8·5 8·7 ·31 S
Sirius 6 41 -17 -1·4 1·3 ·38 G, E
Procyon 7 34 +5 0·5 1·2 ·30 A, E
Ll. 21185 10 58 +37 7·6 4·8 ·37 R, C
Ll. 21258 11 0 +44 8·5 4·4 ·21 A, k, K, R
Ll. 25372 13 40 +15 8·5 2·3 ·20 R, E
Centauri 14 33 -60 0·2 3·7 ·76 G, E
O.A. 17415–6 17 37 +68 9·1 1·4 ·22 k
2398 18 42 +59 8·8 2·3 ·29 Sc, R
Draconis 19 32 +70 4·8 1·9 ·22 s, P
Altair 19 46 +9 0·9 0·6 ·24 E
61 Cygni 21 2 +38 4·8 5·2 ·31 many
Indi 21 56 -57 4·8 4·7 ·28 G, E
Krueger 60 22 24 +57 9·2 0·9 ·26 B, Sc, R
Lac. 9352 22 59 -36 7·4 7·0 ·28 G

Authorities.—A—A. Auwers; B—E. E. Barnard; C—F. L. Chase; E— W. L. Elkin; G—Sir David Gill; K—J. C. Kapteyn; k—K. N. A. Krüger; P—B. Peter; R—H. N. Russell and A.R. Minks; S—W. de Sitter; s—M. F. Smith; SC—F. Schlesinger.

The stars selected to be examined for parallax are usually either the brightest stars or those with an especially large proper motion. Neither criterion is a guarantee that the star shall have a measurable parallax. Brightness is particularly deceptive; thus Canopus, the second brightest star in the heavens, has probably a parallax of less than 0·01″, and so also has Rigel. These two stars must have an intrinsic brilliancy enormously greater than that of the sun for if the sun were removed to such a distance (parallax 0·01″) it would appear to be of about the tenth magnitude.

Although the parallaxes hitherto measured have added greatly to our general knowledge of stellar distances and absolute luminosities of stars, a collection of results derived by various observers choosing specially selected stars is not suitable for statistical discussion. For this reason a series of determinations of parallax of 163 stars on a uniform plan by F. L. Chase, M. F. Smith and W. L. Elkih (Yale Transactions, vol, ii., 1906) constitutes a very important addition to the available data. The stars chosen were those with centennial proper motions greater than 40″, observable at Yale, and not hitherto attacked. It is noteworthy that no parallaxes exceeding 0·20″ were found; the mean was about 0·05″. It is greatly to be desired that a general survey of the heavens, or of typical regions of the heavens, should be made with a view to determining all the stars which have an appreciable parallax. This is now made possible by photography. If three plates (or three sets of exposures on one plate) are taken at intervals of six months, when the stars in the region have their maximum parallactic displacements, the first and third plates serve to eliminate the proper motion of the star, and the detection of a parallax is easy. Some progress with this scheme has been made. But even such an attempt to systematically plumb the universe can only make us acquainted with the merest inside shell. We should learn perhaps the distribution and luminosities of the stars within a sphere of radius sixty light years (corresponding to a parallax of about 0·05″), but of the structure of the million-fold greater system of stars, lying beyond this limit, yet visible in our telescopes, we should learn nothing except by analogy. Fortunately the study of proper motions teaches us with some degree of certainty something of the general mean distances and distribution of these more distant stars, though it cannot tell us the distances of individual stars. There is another method of determining stellar distances, which is applicable to a few double stars. By means of the spectroscope it is possible to determine the relative orbital velocity of the two components, and this when compared with the period fixes the absolute dimensions of the orbit; the apparent dimensions of the orbit being known from visual observations the distance can then be found. The method is of very limited application, for in general the orbital velocity of a visual binary is far too small to be found in this way; one of its first applications has been made to α Centauri, with the result that the parallax found in the ordinary way is completely confirmed.

Proper Motions of Stars.—The work of cataloguing the stars and determining their exact positions, which is being pursued on so large a scale, naturally leads to the determination of their proper motions. The problem is greatly complicated by the fact that the equator and equinox, to which the observed positions of the stars must be referred, are not stationary in space, and in fact the movements of these planes of reference can only be determined by a discussion of the observations of stars. Halley was the first to suspect from observation the proper motions of the stars. From comparisons between the observed places of Arcturus, Aldebaran and Sirius and the places assigned to them by Alexandrian astronomers, he was led to the opinion that all three are moving towards the south (Phil. Trans. 1718). Jacques Cassini also proved that Arcturus had even since the time of Tycho Brahe shifted five minutes in latitude; for η Bootis, which would have shared in the change, if it had been due to a motion of the ecliptic, had not moved appreciably. It was early realized that the proper motions of the stars were changes of position relative to the sun, and that, if the sun had any motion, of its own as compared with the surrounding stars as a whole, this would be shown by a general tendency of the apparent motions of the stars to be directed away from the point to which the sun was moving.

To determine proper motions it is necessary to have observations separated by as long a period of time as possible. Old catalogues of precision are accordingly of great importance. By far the most valuable of these is Bradley’s catalogue of 3240 stars observed at Greenwich about 1750–1763, which has been re-reduced according to modern methods by A. Auwers. These stars include most of the brighter ones visible in the latitude of Greenwich, ranging down to about the seventh magnitude. An early catalogue which includes large numbers of stars of magnitude as low as 8·5 is that of S. Groombridge, containing 4200 stars within 52° of the north pole observed between 1806 and 1816. This has been re-reduced by F. W. Dyson and W. G. Thackeray, and proper motions derived by comparison with modern Greenwich observations. A very extensive determination of proper motions from a comparison of all the principal catalogues has been made by Lewis Boss. The results are given in his Preliminary General Catalogue (1910), which comprises the motions of 6188 stars fairly uniformly distributed over the sky, including all the stars visible to the naked eye. Of rather a different nature are J. G. Porter’s catalogue (Publications of the Cincinnati Observatory, No. 12) and J. F. Bossert’s catalogue (Paris Observations, 1890), which consist of lists of stars of large proper motion determined from a variety of sources. Recently the proper motions of faint stars have been determined by comparing photographs of the same region of the sky, taken with an interval of a number of years. At present the available intervals are too small for this method to have met with marked success. Large proper motions can however be found in this way. Their detection is especially simple when the stereo-comparator is used; this instrument enables the two eyes to combine the images of each star on two plates into one image (as in the stereoscope); when the star has moved considerably in the interval between the taking of the two plates, it appears to stand out from the rest in relief and is at once noticed.

The star with the greatest proper motion yet known was found by J. C. Kapteyn on the plates of the Cape Photographic Durchmuslerung. Its motion of 8·7″ per year would carry it over a portion of the sky equal to the diameter of the full moon in about two centuries. In the table is given a list of the stars now known to have an annual proper motion of more than 3″. The faintness of the majority of the stars appearing in this list is noteworthy.

Stars with Large Proper Motion.
Name. R.A.
Annual Proper
h. m. °
C.Z.5h243 5 8 -45·0 8·70 8·5
Gr. 1830 11 47 +38·4 7·04 6·9
Lac. 9352 22 59 -36·4 6·94 7·5
Cor.32416 0 0 -37·8 6·07 8·5
611 Cygni 21 2 +38·3 5·20 5·5
Ll. 21 185 10 58 +36·6 476 7·3
ε Indi 21 56 -57·2 4·61 5·2
Ll. 21258 11 0 +44·0 4·41 87
o2 Eridani 4 11 -7·8 4·05 4·6
μ Cassiop 1 2 +54·4 3·73 5·6
O.A. 14318 15 5 -16·0 3·68 9·1
O.A. 14320 15 5 -15·9 3·68 9·1
α Centauri 14 33 -60·4 3·60 0·2
Lac. 8760 21 11 -39·2 3·53 7·3
e Eridani 3 16 -43·4 3·12 4·4
O.A. 11677 11 15 +66·4 3·02 9·0

The majority of the stars have far smaller proper motions than these. Only 24% of the stars of Auwers-Bradley have proper motions exceeding 10″ per century, and 51% exceeding 5″ per century. With catalogues containing fainter stars the proportion of large proper motions is somewhat smaller, thus the corresponding percentages for the Groombridge stars are 12 and 31 respectively.

When the parallax of a star is known, we are able to infer from its proper motion its actual linear speed in miles per hour, in so far as the motion is transverse to the line of sight. The velocity in the line of sight can be determined by spectroscopic observation, so that in a few cases the motion of the star is completely known. Several stars appear to have speeds exceeding 100 m. per second, but of these the only one reliably determined is Groombridge 1830, whose speed is found to be about 150 m. per second. Probably the velocity of Arcturus is also over 100 m. per second; there is, however, no real evidence for the velocity of 250 m. per second which has sometimes been credited to it. The above are velocities transverse to the line of sight. The greatest radial velocities that have yet been found are about 60 m. per second; several stars (Groombridge 1830 among them) have radial speeds of this amount. The stars of the Helium type of spectrum are remarkable for the smallness of their velocities; from spectroscopic observations of over 60 stars of this class, J. C. Kapteyn and E. B. Frost have deduced that the average speed is only 8 m. per second. According to W. W. Campbell the average velocity in space of a star is 21·2 m. per second.

When the proper motions of a considerable number of stars are collected and examined, a general systematic tendency is noticed. The stars as a whole are found to be moving towards a point somewhere in or near the constellation Canis Major. The motions of individual stars, it is true, The Solar
vary widely, but if the mean motion of a number of stars is considered this tendency is always to be found. Now it is necessary to bear in mind that all observed motions are relative; and, especially in dealing with stellar motions, it is arbitrary what shall be considered at rest, and used as a standard to which to refer their movements. Accordingly this mean motion of the stars relative to the sun has been more generally regarded from another point of view as a motion (in the opposite direction—towards the constellation Lyra) of the sun relatively to the stars. In what follows we shall speak of this relative motion as a motion of the sun or of the stars indifferently, for there is no real distinction between the two conceptions. One of the problems, which has engaged a large share of the attention of astronomers in the last century, has been the determination of the direction of this “solar motion.”

The first attempt to determine the solar apex (as the point towards which the solar motion is directed is termed) was made in 1783 by Sir William Herschel. Although his data were the proper motions of only seven stars, he indicated a point near λ Herculis not very far from that found by modern researches. Again in 1805 from Maskelyne’s catalogue of the proper motions of 36 stars (published in 1790), he found the position, R.A. 245° 52′ and Dec. 49° 38′ N. The systematic tendency of the proper motions is so marked that the motions of a very few stars are quite sufficient to fix roughly the position of the solar apex; but attempts to fix its position to within a few degrees have failed, notwithstanding the many thousands of determined proper motions now available. The difficulties of the determination are twofold. There is a close interdependence between the constant of procession and the solar motion; the two determinations must generally be made simultaneously, and both depend very considerably on the systematic corrections required by the catalogues compared. But further, if these practical difficulties could be considered overcome in the best determinations, there is a vagueness in the very definition of the solar motion. The motion of the sun relative to the stars depends on what stars are selected as representative. There is no a priori reason to expect the same result from the different classes of stars, such as the brighter or fainter, northern or southern, nearer or more distant, Solar type or Sirian stars. There is for example some evidence that the declination of the solar apex is really increased when the motion is referred to fainter stars. For these reasons a really close agreement between the results of different investigators is not to be expected.

Of the various modern determinations of the apex, we give first those which depend, wholly or mainly, on the Auwers-Bradley proper motions. Setting A for the right ascension, D for the declination of the apex, these are:—

L. Boss A = 17h 48m D = +42°·8
L. Struve A = 18h 20m D = -23°·5
S. Newcomb A = 18h 10m D = +31°·3
J. C. Kapteyn A = 18h 14m D = +29°·5.

The large differences between these results, derived from the same material, depend mainly on the different systematic corrections applied by each astronomer to the declinations of Bradley. From the data of his Preliminary General Catalogue (1910), L. Boss found A = 18h 2m, D = +34°·3. Having regard to the special precautions taken to eliminate systematic error, and to the fact that the stars used were distributed nearly equally over both hemispheres, it is fair to conclude that this is the most accurate determination yet made. From the Groombridge proper motions Dyson and Thackeray found A = 18h 20m, D = +37°. Other determinations have been made by O. Stumpe (Ast. Nach. No. 3000) and J. G. Porter (Ast. Journ. xii. 91), using mainly stars of large proper motions derived from various sources; their results are of the same general character. Most of the above investigators, besides giving a general result, have determined the apex separately for bright and faint stars, for stars of greater or less proper motion, and in some cases for stars of Sirian and Solar spectra. Considerable divergences in the resulting position of the apex are found.

It will be seen that the proper motion of any star may be regarded as made up of two components. The part of the star's apparent displacement, which is due to the solar motion, is generate ally called the parallactic motion; the rest of its motion (i.e. its motion relative to the mean of all the stars, is called its peculiar motion (motus peculiaris). Regarded Speed of
as a linear velocity, the parallactic motion is the same for all stars, being exactly equal and opposite to the solar motion; but its amount, as measured by the corresponding angular displacement of the star, is inversely proportional to the distance of the star from the earth, and foreshortening causes it to vary as the sine of the angular distance from the apex. To arrive at some estimate of the speed of the solar motion, we may consider the motions of those stars whose parallaxes have been measured, and whose actual linear speed is accordingly known (disregarding motion in the line of sight). If a sufficient number of stars are considered, their peculiar motions will mutually cancel and the parallactic or solar motion can then be derived. But not much reliance can be placed on this kind of determination. A very weighty objection is that the stars whose parallaxes are determined are mainly those of large proper motion and therefore not fairly representative of the bulk of the stars; in fact their peculiar motions will not neutralize one another in the mean. A better method is to derive the speed from the radial motions observed with the spectroscope. In this way W. W. Campbell from the radial motions of 280 stars found the velocity to be 20 kilometres per second with a probable error of 1 1/2 km. per second (Astrophysical Journal, 1901, vol. xiii). This result depends on the northern stars only. By the addition of the data for southern stars, so as to obtain a distribution fairly symmetrical over the whole sphere, S. S. Hough and J. Halm deduced a velocity of 20·8 km. per second towards the apex A = 18h 5m, D = +26°. The speed is very nearly four radii of the earth's orbit per year; thus the annual parallactic motion is equal to four times the parallax, for a star lying in a direction 90 from the solar apex; for stars nearer the apex or antapex it is foreshortened. This result, while it does not afford any means of determining the parallaxes of individual stars, enables us to determine the mean parallax of a group of stars, if we may assume their peculiar motions practically to cancel one another.

In researches on the solar motion the assumption is almost always made that the motions of the stars relatively to one another —the “peculiar” motions—are at random. The correctness of this hypothesis has long been under suspicion, but it has generally been accepted as the best simple approximation to the actual distribution of the motions that could be made. Naturally exceptional regions must be recognized; for example, a connected system such as the Pleiades, whose stars have the same proper motion, must constitute an exception. There can occasionally be traced a certain community of motion over a much larger area. Thus R. A. Proctor found that between Aldebaran and the Pleiades most of the stars have a motion positive in right ascension and negative in declination, a phenomenon which he designated “star-drift.” A more precise investigation by L. Boss has shown that there is in this region a “moving cluster” of globular form. The stars composing this all have equal and parallel motions; about 40 stars brighter than the seventh magnitude are known to belong to it. The group consisting of five stars of Ursa Major together with Sirius has already been alluded to; another very marked group of 16 stars in Perseus, all of the Helium type of spectrum, form a similar association. Spectroscopic evidence has indicated that most of the stars of Orion are associated, and share nearly the same motion (or rather, in this case, absence of motion).

But, whilst recognizing the existence of local drifts and systems, and admitting the possibility of relative motion between the nearer and more distant, or other classes of stars, it is only recently that astronomers have seriously doubted the correctness of the hypothesis of random distribution of stellar motions as at least a rough representation of the truth. The hypothesis was put to the test by J. C. Kapteyn, with the result that it appears to be not even approximately accordant with the facts. His researches indicate that, instead of being haphazard, the proper motions of the star show decided preference for two “favoured” directions, apparently implying that the stars surrounding us do not constitute a simple system but a dual one. The motion of the stars in the mean towards Canis Major The Two
is thus a resultant motion, which, when examined more minutely, is found to be due to the intermingling of two great streams of stars moving in very different directions. These two streams or drifts prevail in every part of the sky examined, and contain nearly equal numbers of stars; that is to say, in whatever part of the sky we look about half the stars are found to belong to one and half to the other of the two great drifts. This hypothesis of two star-drifts does not imply that all the stars move in one or other of two directions. The stars have on this theory random peculiar motions in addition to the motion of the drift to which they belong, just as on the older theory the stars have peculiar motions in addition to the solar or parallactic motion shared by all of them. But the two theories lead to a very different statistical distribution of the stellar motions The older one—which may be called the “one-drift” hypothesis, since according to it the stars appear to form a single drift moving away from the solar apex—requires that the apparent directions of motion should be so distributed that fewest stars are moving directly towards the solar apex, and most stars along the great circle away from the solar apex, the number decreasing symmetrically, for directions inclined on either side of this great circle, according to a law which can be calculated. This is found not to agree with the facts at all. The deviation is unmistakable; in general the direction from the solar apex is not the one in which most stars are moving; and, what is even more striking, the directions, in which most and fewest stars respectively move, are not by any means opposite to one another. It seems difficult to account for the very remarkable and unsymmetrical distribution of the motions, unless we suppose that the stars form two more or less separate systems superposed; and it has been found possible by assuming two drifts with suitably assigned velocities to account very satisfactorily for the observed motions.

The phenomenon of two drifts was discovered by an examination of the Bradley proper motions (Brit. Assoc. Rep., 1905, p. 257), and has subsequently been confirmed by a discussion of the Groombridge proper motions (Mon. Not. R.A.S., 1906, 67, p. 34; 1910, 71, p. 4). By an examination of the stars of very large proper motion F. W Dyson has traced the presence of the two drifts in all parts of the sky. They have been shown to prevail among fainter stars down to magnitude 9·5, by an examination of the Greenwich-Carrington proper motions; these, however, only cover a region within 9° of the north pole. Of the behaviour of stars fainter than magnitude 9·5 there is at present no direct evidence. About 10,000 stars altogether were dealt with in the above-mentioned investigations The general results indicate that one of the drifts is moving (relatively to the sun) directly away from a point near α Ophiuchi (about R.A. 270°, Dec. +12°), and the other from a point in Lynx (R.A. 83, Dec. +60°). These two points may be called the apices of the two drifts, for they are analogues of the solar apex on the one-drift theory; they are about 110° apart. The velocities of the drifts differ considerably, the one whose apex is in Ophiuchus having about 1 1/2 times the speed of the other. We may conveniently distinguish the two drifts as the slow-moving and fast-moving drifts respectively; but it should be remembered that, since these motions are measured relatively to the sun, this distinction is not physically significant. The stars appear to be nearly equally divided between the two drifts. The magnitudes of the stars are distributed in the same way in each drift. There is also clear evidence that the mean distances of both drifts from us are very approximately the same. Thus we are led to regard the two systems as completely intermingled, a fact which adds considerably to the difficulty of explaining the phenomena otherwise than as produced by two great systems—universes they have been called—which have come together, perhaps, by their mutual attraction, and are passing through one another. The chances of individual stars of the two systems colliding are infinitesimal. Until the hypothesis has been thoroughly tested by an examination of the line-of-sight velocities of stars from the same point of view, this physical interpretation must be received with some degree of caution; but there can be no doubt of the reality of the anomalies in the statistical distribution of proper motions of the stars, and of these it offers a simple and adequate explanation.

Having determined the motions of the two drifts, and knowing also that the; stars are nearly equally divided between them, it is evidently possible to determine the mean motion of the drifts combined. This is of course that relative motion of the sun and stars which we have previously called the solar motion. The position of the solar apex calculated in this way agrees satisfactorily with that found by the usual methods. It is naturally fairly close to the apex of the faster drift, but is displaced from it in the direction of the apex of the other drift. In this connexion it may be noticed that, when the smaller and larger proper motions are discussed separately, the latter category will include an unduly great proportion of stars belonging to the fast-moving drift, and the resulting determination will lead to a solar apex too near the apex of that drift, i.e. with too low a declination. This appears to be the explanation of Stumpe’s and Porter’s results; they both divided their proper motions into groups according to theif numerical amount, and found that the declination of the solar apex progressively increased as the size of the motions used diminished. Another anomalous determination of the apex, due to H. A. Kobold (Astro. Nach., 3163, 3451, and 3491) is also explained when the two drifts are recognized. Kobold, using a peculiar and ingenious method, found for it a declination -3°, which disagrees very badly with all other determinations; but it is a peculiarity of Kobold’s method that it gives the line of symmetry of motion, which joins the apex and antapex, without indicating which end is the apex. Now the position of this line, as found by Kobpld, actually is a (properly weighted) mean between the corresponding lines of symmetry of the two drifts, but naturally it lies in the acute angle between them, whereas the line of the solar motion is also a weighted mean between the two lines of drift, but lies in the obtuse angle between them.

The Structure of the Universe.—We now arrive at the greatest of all the problems of sidereal astronomy, the structure and nature of the universe as a whole. It can by no means be taken for granted that the universe has anything that may properly be called a structure. If it is merely the aggregate of the stars, eac.h star or small group of stars may be a practically independent unit, its birth and development taking place without any relation to the evolution of the whole. But it is becoming more and more generally recognized that the stars are not unrelated; they are parts of a greater system, and we have to deal with, not merely the history of a number of independent units, but with a far vaster conception, the evolution and development of an ordered universe.

Our first inquiry is whether the universe extends indefinitely in all directions, or whether there are limits beyond which the stars are not distributed. It is not difficult to obtain at least a partial answer to this question; anything approaching a uniform distribution of the stars cannot extend Limits of the
indefinitely. It can be shown that, if the density of distribution of the stars through infinite space is nowhere less than a certain limit (which may be as small as we please), the total amount of light received from them (assuming that there is no absorption of light in space) would be infinitely great, so that the background of the sky would shine with a. dazzling brilliancy. We therefore conclude that beyond a certain distance there is a thinning out in the distribution of the stars; the stars visible in our telescopes form a universe having a more or less defined boundary; and, if there are other systems of stars unknown to us in the space beyond, they are, as it were, isolated from the universe in which we are. It is necessary however to emphasize that the foregoing argument assumes that there is no appreciable absorption of light in interstellar space. Recently, however, the trend of astronomical opinion has been rather in favour of the belief that diffused matter may exist through space in sufficient quantity to cause appreciable absorption; so that the argument has no longer the weight formerly attached to it. Another line of reasoning indicates that the boundary of the universe is not immeasurably distant, and that the thinning out of the stars is quite perceptible with our telescopes. This depends on the law of progression in the number of stars as the brightness diminishes. If the stars were all of the same intrinsic brightness it is evident that the comparison of the number of stars of successive magnitudes would show directly where the decreased density of distributibn began. Actually we know that the intrinsic brightness varies very greatly, so that each increase of telescopic power not only enables us to see stars more remote than before, but also reveals very many smaller stars within the limits previously penetrated. But notwithstanding the great variety of intrinsic brightness of the stars, the ratio of the number of stars of one magnitude to the number of the magnitude next lower (the “star-ratio”) is a guide to the uniformity of their distribution. If the uniform distribution extends indefinitely, or as far as the telescope can penetrate, the star-ratio should have the theoretical value 3·98,[2] any decrease in density or limit to the distribution of the stars will be indicated by a continual falling off in the star-ratio for the higher magnitudes. H. H. Seeliger, who investigated this ratio for the stars of the Bonn Durchmusterung and Southern Durchmusterung, came to the Conclusion (as summarized by Simon Newcomb) that for these stars the ratio ranges from 3·85 to 3·28, the former value being found for regions near the Milky Way and the latter for regions near the galactic poles. There is here evidence that even among stars of the Durchmusterung (9·5 magnitude), a limit of the universe has been reached, at least in the direction normal to the plane of the Milky Way. For the higher magnitudes J. C. Kapteyn has shown that the star-ratio diminishes still further.

In all investigations into the distribution of the stars in space one fact stands out pre-eminently, viz. the existence of a certain plane fundamental to the structure of the heavens. This is the galactic plane, well known from the fact that it is marked in the sky by the broad irregular belt of milky light called the Galaxy or Milky Way. But it The
is necessary to make a careful distinction between the galactic plane and the Galaxy itself; the totter, though it is necessarily one of the most remarkable features of the universe, is not the only peculiarity associated with the galactic plane: Its particular importance consists in the fact that the stars, bright as well as faint, crowd towards this plane. This apparent relation of the lucid stars to the Galaxy was first pointed out by Sir W. Herschel. For the stars visible to the naked eye a very thorough investigation by G. V. Schiaparelli has shown the relation in a striking manner. He indicated on planispheres the varying density of distribution of the stars over the sky. On these the belt of greatest density can be easily traced, and it follows very closely the course of the Milky Way; but, whereas the latter is a belt having rather sharply defined boundaries, the star-density decreases gradually and continuously from the galactic equator to the galactic poles. The same result for the great mass of fainter stars has been shown by Seeliger. The following table shows the density with which stars brighter than the ninth magnitude are distributed in each of nine zones into which Seeliger divided the heavens:—

Galactic latitude N. Pole
70° N.
70° N.
50° N.
50° N.
30° N.
30° N.
10° N.
10° N.
10° S.
10° S.
30° S.
50° S.
50° S.
70° S.
70° S.
S. Pole
Number of stars per square degree 2·78 3·03 3·54 5·32 8·17 6·07 3·71 3·21 3·14

The table, which is based on over 130,000 stars, shows that along the galactic circle the stars are scattered nearly three times more thickly than at the north and south poles of the Galaxy. What, however, is of particular importance is that the increase is gradual. No doubt many of the lucid stars which appear to lie in the Milky Way actually belong to it, and the presence of this unique cluster helps to swell the numbers along the galactic equator; but, for example, the increased density between latitudes 30° to 50° (both north and south) as Compared with the density at the poles cannot be attributed to the Galaxy itself, for the Galaxy passes nowhere near these zones. The star-gauges of the Herschels exhibit a similar result; the Herschels counted the number of stars visible with their powerful telescopes in different regions of the sky, and thus formed comparative estimates of the density of the stars extending to a very high magnitude. According to their results the star-density increases continuously from 109 per square degree at the poles to 2019 along the galactic equator. In general, the fainter the stars included in the discussion the more marked is their crowding towards the galactic plane. Various considerations tend to show that this apparent crowding does not imply a really greater density or clustering of the stars in space, but is due to the fact that in these directions we look through a greater depth of stars before coming to the boundary of the stellar system. Sir William Herschel and afterwards F. G. W. Struve developed the view that the stars are contained in a comparatively thin stratum bounded by two parallel planes. The shape of the universe may thus be compared to that of a grindstone or lens, the sun being situated about midway between the two surfaces. Thus the figure, represents a section of the (ideally simplified) universe cut perpendicular to the planes AB and CD between which the stars are contained, S being the sun. Imagine this stratum to be uniformly filled with stars (of course in the actual universe instead of sharply defined boundaries AB and CD, we shall have a gradual thinning out of the stars) it follows that in the two directions SP and SP′ the fewest stars will be seen; these then are the directions of the galactic poles. As we consider a direction such as SQ farther and farther from the pole the boundary of the universe in that direction becomes more and more remote so that more stars are seen, and finally in the directions SR and SR′ in the galactic plane, the boundary is perhaps beyond the limits of our telescopes. That the universe must have a boundary in the directions SR and SR′, we can hardly doubt, but nothing is known of its shape or distance except that in all directions it must be far greater than SP or SP′; in particular it is not known whether the sun is near the centre or otherwise. That the sun is nearly midway between the two boundary planes can be tested by comparing the star-densities of the northern and southern galactic hemispheres. These are zone for zone very nearly equal; the slight excess of stars in the southern hemisphere perhaps implies that the sun is a little north of the central position. This is confirmed by the fact that the Milky Way is not quite a great circle of the celestial sphere, but has a mean south galactic latitude of about 1·7°.

If, instead of considering the whole mass of stars, attention is directed to those of large proper motion, which are therefore in the mean relatively near us, the crowding to the galactic plane is much less noticeable, if not indeed entirely absent. Thus Kapteyn found that the Bradley stars having proper motions greater than 5″ per century were evenly distributed over the sky, Dyson and Thackeray’s tables show the same result for the Groombridge stars down to magnitude 6·5; but the fainter stars (with centennial proper motions greater than 5″) show a marked tendency to draw towards the galactic circle. The result is precisely what should be expected from the theory of the shape of the universe which has been set forth. If in the fig. we describe a sphere about S with radius SP so as just to touch the boundaries of the stratum of stars, then, provided a class of stars is considered wholly or mainly included within this sphere, no concentration of stars in the galactic plane is to be expected, for the shape of the universe does not enter into the question. It is only when some of the stars considered are more remote and lie outside this sphere (but of course between the two planes) that there is a galactic crowding. We infer that nearly all the stars down to magnitude 6·5, whose proper motions exceed 5″, are at a distance from the sun less than SP, whilst of the fainter stars with equally great proper motions a large proportion are at a distance greater than SP. This result enables us to form some sort of idea of the distance SP.

On considering the distribution of the stars according to their spectra, it appears that the Type II. (solar) stars show no tendency to congregate in the galactic plane. The result of course only applies to the brighter stars, for we have very little knowledge of the spectra of stars fainter than about magnitude 7·5. The explanation indicated in the last paragraph applies to this case also Type II. stars are in general much less intrinsically luminous than Type I., so that the stars known to be of this type must be comparatively near us, for otherwise they would appear too faint to have their spectra determined. They are accordingly within the sphere of radius SP (fig.), and consequently are equally numerous in every direction. The Type I. stars, being intrinsically brighter, are not so limited. According to F. McClean, of the stars brighter than magnitude 3·5, only the helium and not the hydrogen stars of Type I. show a condensation towards the galactic plane. Thus we see that the effect of limiting the magnitude to 3·5 is that the hydrogen stars are now practically all within the sphere SP, and it is only the helium stars, whose absolute luminosity is still greater, that are more widely distributed. Of the rarer types of spectra, stars of Type III. agree with those of Type II. in being evenly distributed over the sky; Types IV. and V. however, congregate towards the galactic plane. The most remarkable are the Type V. (Wolf-Rayet) stars; in their case the condensation into the galactic regions is complete, for of the 91 known stars of this type, 70 are actually in the Milky Way and the remaining 21 are in the Magellanic Clouds (two large clusters in the southern hemisphere, which resemble the Milky Way in several respects). Excluding the latter, the 70 Wolf-Rayet stars have a mean distance from the central galactic circle of only 2·6°. There can be little doubt that these stars belong to the Milky Way cluster, so that their presence is a property of the cluster rather than of the galactic plane in general. Spiral nebulae have the remarkable characteristic of avoiding the galactic plane, and it has been suggested that the space outside the limits of the stellar universe is filled with them. It does not, however, seem probable that their apparent anti-galactic tendency has such a significance; in the Magellanic Clouds spiral nebulae are very abundant, a fact which shows that there is no essential antipathy between the stars and the spiral nebulae.

As might be expected, the relative motion of the two great star-drifts is parallel to the galactic plane.

A glance at the Milky Way, with its sharply defined irregular boundaries, its clefts and diverging spur, is almost sufficient to assure us that it is a real cluster of stars, and does not merely indicate the directions in which the universe extends farthest. Barnard’s photographs of its structure leave little doubt on the matter; the numerous rifts and dark openings show that its thickness cannot be very great. To complete our representation of the universe, it is therefore necessary The
to add to the fairly uniform distribution of stars between two planes a gigantic cluster of an annular or spiral form, also lying between the planes and completely surrounding the sun. The Milky Way is not of uniform brightness, so that we are perhaps nearer to some parts of it than to others, but it is everywhere very distant from the sun. Estimates of this distance vary, but it may probably be put at more than three thousand light years (parallax less than 0·001″). Nevertheless the Milky Way contains a fair proportion of lucid stars, for these are considerably more numerous in the bright patches of the Milky Way than in the rifts and dark spaces.

It has been seen that the parallaxes afford little information as to the distribution of the main bulk of the stars and that the chief evidence on this point must be obtained indirectly Distribution
of the
from their proper motions. Our principal knowledge of this subject is due to Kapteyn (Groningen Publications, Nos. 8 and 11), and though much of his work is provisional, and perhaps liable to considerable revision when more extensive data are obtainable, it probably gives an idea of the construction of the universe sufficiently accurate in all essential respects. As has been explained the, mean distance of a group of stars can be readily determined from the parallactic motion, which, when not foreshortened, is approximately four times the parallax; but to obtain a complete knowledge of the distribution of stars it is necessary to know, not merely the mean parallax of the group, but also the frequency law, i.e. what proportion of stars have, a quarter, half, twice or three times, &c, the mean parallax. One result of Kapteyn’s investigations may be given here. Taking a sphere whose radius is 560 light years (a distance about equal to that of the average ninth magnitude star), it will contain:—

1 star giving from 100,000 to 10,000 times the light of the sun
26 stars 10,000 1,000
1,300 1,000 100
22,000 100 10
140,000 10 1
430,000 1 0.1
650,000 0.1 0.01

Whether there is an increasing number of still less luminous stars is a disputed question.

The comparative nearness of the stars, of the solar type, which we have had occasion to allude to, is confirmed by the fact, that their proper motions are on the average much larger than those of the Sirian stars. Kapteyn finds that magnitude for magnitude, the absolute brightness of the solar stars is only one-fifth of that of the Sirian stars, so that in the mean they must be at less than half the distance. As the numbers of known stars of the two types are nearly equal, it is clear that, at all events in our immediate neighbourhood, the solar stars must greatly outnumber the Sirian.

References.—Of modern semi-popular works entirely devoted to and covering the subjects treated of in this article the principal is Simon Newcomb’s The Stars, a Study of the Universe; mention must also be made of Miss A. M. Clerke’s The System of the Stars (2nd ed., 1905), which contains full references to original papers; Problems in Astrophysics, by the same author, may also be consulted. The following works of reference and catalogues deal with special branches of the subject; for variable stars, Chandler's “Third Catalogue,” Astronomical Journ. (1896), vol. xvi., is now very incomplete; Harvard Annals, vol. iv., pt. 1, and vol. lx., No. 4, together constitute a catalogue of 3734 variable stars; ephemerides of over 800 variables are given in the Vierteljahrsschrift of the Astronomische Gesellschaft. For double stars see Burnham’s General Catalogue (1907), and Lewis, Memoirs of the R.A.S., vol. lvi.; the orbits of the principal binaries are discussed in T. J. J. See, Evolution of Stellar Systems, and another list will be found in Lick Observatory Bulletin, No. 84. A list of spectroscopic binaries discovered up to 1905 is given in Lick Observatory Bulletin, No. 79. For the spectrum analysis of stars, Scheiner’s Astronomical Spectroscopy (trans. by Frost) may be consulted. The “Draper Catalogue,” Harvard Annals, vol. xxvii., gives the classification' according ito spectrum of over 10,000 stars; for the brighter stars Harvard Annals, vol. 1. forms a more complete catalogue. Of the numerous memoirs discussing stellar spectra in relation to evolution, A. Schuster, “The Evolution of Solar Stars,” Astrophysical Journ. (1903), vol. xvii., may be mentioned as giving a concise survey of the subject.  (A. S. E.) 

  1. Variable stars (except those sufficiently bright to have received special names) are denoted by the capital letters R to Z followed by the name of the constellation. The first nine variables recognized in each constellation are denoted by single letters, after which combinations RR, RS, &c, are used.
  2. This number is the 3/2th power of the ratio of the brightness of stars differing by a unit magnitude.