R.A.i7 h 53 m , Dec. 427' N., having an annual proper motion of lo"-3, the largest yet known. Its parallax is o"-52, which makes it the second nearest star (a Centauri being the nearest). A faint com- panion to a Centauri (sharing the same large proper motion) was dis- covered in the same year by R. T. A. Innes; its visual magnitude is n m -o, and it has been verified that the parallax is practically the same as that of a Centauri. It appears that this companion is distant 10,000 astronomical units from the principal components, and its period of revolution round them must be a million years. It is now on the near side of its orbit so that it is actually the nearest star known; for that reason it has been named Proxima Centauri. Barnard's and Innes's stars, being both faint and close to us, must be of very low intrinsic luminosity ; with them may be grouped two other companions to stars of large parallax, forming the four in- trinsically faintest stars yet known :
Barnard star absolute visual magnitude I3 M '3
Proxima Centauri absolute visual magnitude 15 -4
Groombridge 34, comes absolute visual magnitude 13
Pi. 2 h i23, comes absolute visual magnitude 12 -3
As might be expected all four are red stars in the last stage before
extinction, so that photographically their magnitudes are even
fainter. Proxima gives less than 1/10,000 of the light of the sun.
A distant companion to Capella discovered by Furuhjelm must also
be very faint; but it is probably brighter than those above men-
tioned.
At the other end of the scale it is uncertain what is the maximum luminosity reached by the stars, because of the smallness of the parallaxes of those which are likely to be the brightest. Canopus, Rigel, and some others may approach or even surpass 5 m -o (10,000 times the sun's luminosity), but it is not possible to obtain satisfactory evidence of anything brighter. The known range of absolute stellar magnitude is thus from 5 m -o to +I5 m -o, or a hundred-million-fold ratio of luminosity, with the sun just at the middle. This range is much the same as the known range of apparent brightness (in spite of the distance factor affecting the latter) ; so that apparent brightness is practically no guide to the distance. Stars of low luminosity are far more common in space than those of high luminosity. Thus we find the four red dwarfs above men- tioned within a very small distance from the sun, and doubtless they are equally plentiful throughout the stellar system ; but we have to extend our net to very great distances to catch Canopus and Rigel representing the most brilliant stars, and they ought to be regarded as very exceptional freaks of nature. Perhaps it is unfortunate that these exceptional stars catch our attention by their brilliancy, and figure to a disproportionate extent in our catalogues.
Masses of Stars. In striking contrast to the enormous range of intrinsic brightness, is the comparative uniformity of the masses of stars. Some knowledge of their masses may be gained from a study of the orbits of visual binaries of known parallax, and also from spectroscopic binaries (in which case the parallax is not needed). In general the range of mass is surprisingly small, the result being usually between one-half and twice the sun's mass. Exceptions prob- ably appear more numerous than they really are, because of our tendency to pick out the very luminous stars, which are believed to have masses above the average. Stars of type B are found to be on the average three or four times as massive as the others, confirming the view already mentioned that only a star of large mass can attain the highest temperatures. Both components of V. Puppis (type Bi) have masses not less than 17 X sun 1 ; these are the greatest yet meas- ured, though we suspect that masses up to, say, 50 X sun may occa- sionally occur. The smallest mass known is that of the faint com- ponent of the double star Krueger 60 which is between 1/6 and 1/8 X sun. Attention to these extreme cases scarcely does justice to the uniformity of the great majority of the stars; from a theoretical relation between luminosity and mass for giant stars it is probable that 90% will have masses between J and 2 X sun.
Advantage is taken of this uniformity to determine the so-called " hypothetical parallaxes," or dynamical parallaxes, of double stars. If a is the semi-axis of the orbit in astronomical units, P the period in years, and mi+nvs the mass of the system in terms of the sun, we have
Thus a can be found if mi+m 2 is known or guessed. We may assume with fair probability that mi+m2 = 2, the possible deviations being comparatively unimportant because the cube-root is taken in determining a. But the value of a in angular measure is found from the apparent orbit in the sky; comparing the angular measure with the linear measure given by the above calculation, we at once find the distance or parallax of the star. It is possible to modify the procedure so that it can be used when only a small arc of the orbit has been observed. Dynamical parallaxes of 556 double stars have been published by J. Jackson and H. H. Furner (12) ; from these the absolute magnitudes and linear velocities (transverse to the line of sight) were calculated. The magnitudes showed clearly the bifurca- tion into giants and dwarfs. The linear velocities were combined to give a determination of the sun's motion through the stellar system, the result being a velocity of 19-1 km. per sec. towards the Apex R.A.273", Dec. +34. This agrees remarkably well with the values
'That is, 17 times the sun's mass.
generally accepted; and in particular the accordance of the speed with the value 19-5 km. per sec., obtained from the discussion of spectroscopic radial velocities, shows that the assumed mass 2-0 X sun must be almost exactly the average mass of a double star system.
Fixed Calcium Lines. In certain spectroscopic binaries, the curious phenomenon of " fixed calcium lines " is observed. Whereas the other lines of the spectrum shift to and fro as the star approaches and recedes in its orbit, the narrow K line of calcium remains stationary. It is clear that there must be, somewhere between us and the star's surface, an absorbing cloud of calcium vapour, which does not follow the star in its orbit. The phenomenon was first pointed out by Hartmann in 1904 for the star 5 Orionis; more re- cently it has been observed in other cases, and more than 20 such stars are now known. All belong to the very hottest spectral class O-B2; but this is not so significant as is often supposed, because at lower temperatures the K line appears in the spectrum of the star itself and would confuse the observation of the fixed calcium cloud. There are two possibilities, (a) that the cloud surrounds the whole binary system, the components revolving within it without appre- ciably disturbing it, (b) that the cloud has no connexion with the star, but consists of calcium vapour perhaps distributed widely in interstellar space. The hypothesis (a) was apparently contradicted by the fact that measures of the velocity of the fixed cloud did not agree with that of the centre of mass of the binary system ; but the differences are not large, and may perhaps be ascribed to errors of observation or other causes of spectral shift. Hypothesis (b) seems the simplest; it suggests that vapours in very minute quantities may be diffused through space or float in extended clouds; the rarity of detection is due to the fact that the corresponding " fixed " spectral lines would in most cases be blended with similar absorption lines occurring in the atmospheres of the stars. Miss Heger at the Lick Observatory has recently discovered that the sodium lines DI and D 2 in 5 Orionis are also " fixed."
Cepheid Variables. Many new facts have emerged with regard to the class of short-period variable stars typified by 8 Cephei. The three leading classes of variable stars are (a) long-period variables, (6) eclipsing variables, (c) Cepheids. In the first-named, the varia- tion is undoubtedly due to a physical process in the star itself, which alternately blazes up and subsides; in the second, we have to do with a double star and the change of brightness is due merely to eclipses of one component by the other; the conditions which cause the variation of the third class the Cepheids are much more puzzling. The first question is : Is the Cepheid a binary star? The spectroscope apparently answers in the affirmative, for it shows a radial velocity increasing and decreasing in the period of the light fluctuation; it has generally been taken for granted that this must represent orbital motion. But the change of light cannot be attributed to eclipses; not only is the light curve of a different character, but mini- mum brightness always occurs when the star is receding most rapidly at a time when the other component could not be between it and us. There must be an actual variation in the rate of radiation by the star, and this has been confirmed by H. Shapley (13), who showed that the spectral type (and presumably the surface tempera- ture) changes during the period. For example, 5 Cephei changes from type Fo at maximum to G2 at minimum; this periodic heating and cooling is the main cause of the change of brightness. One suggested explanation is that the orbital motion occurs in a resisting medium, so that the front side of the star is brighter than the rear side on ac- count of the impact of the medium ; this would explain why minimum brightness always occurs when the star is retreating. But opinion is now tending towards a pulsatory theory proposed by H. Shapley (14) which rejects the binary hypothesis altogether. The fact is that there is literally no room for the supposed second component re- quired by the binary hypothesis. The Cepheids are giant stars filling a large volume, and the " orbit " is always small compared with the dimensions of the star itself. When we calculate the size of the orbit of the supposed companion (which we can do, knowing the period and approximate mass of the system) we find that it would graze or even lie inside the principal star a reductio ad absurdum of the binary hypothesis. Further, a relation has been found between period and density in these stars which points to the period being determined by intrinsic conditions; such a relation is quite unintelligible if the period is provoked by an external cause, viz. the revolution of a companion. Accordingly Shapley suggests that the variable is a single star which dilates and contracts with a regular pulsation; and the observed motion of approach and recession refers, not to the star as a whole, but to the upheaval and subsidence of the part of the surface presented towards us.
The radius of 5 Cephei may be taken as about 15,000,000 km.; the semi-amplitude of the oscillation, according to the observed radial velocities, is 1,370,000 km., or about 9% of the radius. For 15 other fully observed Cepheids the semi-amplitude of the pulsation ranges from 4 to 14% of the radius; this seems an amount of com- pression and expansion suitable to produce the rather large changes of temperature required. Within narrow limits the period is inversely proportional to the square root of the star's mean density, a relation which seems significant in view of the fact that the pulsations of a gravitating sphere follow this law. Moreover the constant of proportionality is of the order of magnitude predicted by theory:
