Appendix A

I HAVE preferred not to complicate the Story of the Companion of Sirius with details of a technical kind ; some further information may, therefore, be welcome to those readers who are curious to learn as much as possible about this remarkable star. I am also able to add a further instalment of the 'detective story' which has just come to hand, the sleuth this time being Mr. R. H. Fowler.

The star is between the eighth and ninth magnitude, so that it is not an excessively faint object. The difficulty in detecting it arises entirely from the overpowering light of its neighbour. At favourable epochs it has been seen easily with an 8-inch telescope. The period of revolution is 49 years.

The Companion is separated from Sirius by a distance nearly equal to the distance of Uranus from the sun -- or twenty times the earth's distance from the sun. It has been suggested that the light might be reflected light from Sirius. This would account for its whiteness, but would not directly account for its spectrum, which differs appreciably from that of Sirius. To reflect 1/10,000th of the light of Sirius (its actual brightness) the Companion would have to be 74 million miles in diameter. The apparent diameter of its disk would be 0".3, which, one would think, could scarcely escape notice in spite of unfavourable conditions of observation. But the strongest objection to this hypothesis of reflected light is that it applies only to this one star. The other two recognized white dwarfs have no brilliant star in their neighbourhood, so that they cannot be shining by reflected light. It is scarcely worth while to invent an elaborate explanation for one of these strange objects which does not cover the other two.

The Einstein effect, which is appealed to for confirmation of the high density, is a lengthening of the wave-length and corresponding decrease of the frequency of the light due to the intense gravitational field through which the rays have to pass. Consequently the dark lines in the spectrum appear at longer wave-lengths, i.e. displaced towards the red as compared with the corresponding terrestrial lines. The effect can be deduced either from the relativity theory of gravitation or from the quantum theory; for those who have some acquaintance with the quantum theory the following reasoning is probably the simplest. The stellar atom emits the same quantum of energy as a terrestrial atom, but this quantum has to use up some of its energy in order to escape from the attraction of the star; the energy of escape is equal to the mass multiplied by the gravitational potential at the surface of the star. Accordingly the reduced energy after escape is ; and since this must still form a quantum , the frequency has to change to a value . Thus the displacement is proportional to , i.e. to the mass divided by the radius of the star.

The effect on the spectrum resembles the Doppler effect of a velocity of recession, and can therefore only be discriminated if we know already the line-of-sight velocity. In the case of a double star the velocity is known from observation of the other component of the system, so that the part of the displacement attributable to Doppler effect is known. Owing to orbital motion there is a difference of velocity between Sirius and its Companion amounting at present to 4.3 km. per sec. and this has been duly taken into account; the observed difference in position of the spectral lines of Sirius and its Companion corresponds to a velocity of 23 km. per sec. of which 4 km. per sec. is attributable to orbital motion, and the remaining 19 km. per sec. must be interpreted as Einstein effect. The result rests mainly on measurements of one spectral line . The other favourable lines are in the bluer part of the spectrum, and since atmospheric scattering increases with blueness, the scattered light of Sirius interferes. However, they afford some useful confirmatory evidence.

Of the other white dwarfs is a double star, its companion being a red dwarf fainter than itself. The red shift of the spectrum will be smaller than in the Companion of Sirius and it will not be so easy to separate it from various possible sources of error. Nevertheless the prospect is not hopeless. The other recognized white dwarf is an unnamed star discovered by Van Maanen; it is a solitary star, and consequently there is no means of distinguishing between Einstein shift and Doppler shift. Various other stars have been suspected of being in this condition, including the Companions of Procyon, 85 Pegasi, and Mira Ceti.

If the Companion of Sirius were a perfect gas its central temperature would be about 1,000,000,000 degrees, and the central part of the star would be a million times as dense as water. It is, however, unlikely that the condition of a perfect gas continues to hold. It should be understood that in any case