Lectures on Stellar Statistics

by C. V. L. Charlier

For two of the stars the spectrum is for the present unknown.

We find that the number of stars increases with the spectral index. The unknown stars in the siriometer sphere belong probably, in the main, to the red types.

If we now seek to form a conception of the total number in this sphere we may proceed in different ways. Eddington, in his “Stellar movements”, to which I refer the reader, has used the proper motions as a scale of calculation, and has found that we may expect to find in all 32 stars in this sphere, confining ourselves to stars apparently brighter than the magnitude 9^m.5. This makes 8 stars per cub. sir.

We may attack the problem in other ways. A very rough method which, however, is not without importance, is the following. Let us suppose that the Galaxy in the direction of the Milky Way has an extension of 1000 siriometers and in the direction of the poles of the Milky Way an extension of 50 sir. We have later to return to the fuller discussion of this extension. For the present it is sufficient to assume these values. The whole system of the Galaxy then has a volume of 200 million cubic siriometers. Suppose further that the total number of stars in the Galaxy would amount to 1000 millions, a value to which we shall also return in a following chapter. Then we conclude that the average number of stars per cubic siriometer would amount to 5. This supposes that the density of the stars in each part of the Galaxy is the same. But the sun lies rather near the centre of the system, where the density is (considerably) greater than the average density. A calculation, which will be found in the mathematical part of these lectures, shows that the density in the centre amounts to approximately 16 times the average density, giving 80 stars per cubic siriometer in the neighbourhood of the sun (and of the centre). A sphere having a radius of one siriometer has a volume of 4 cubic siriometers, so that we obtain in this way 320 stars in all, within a sphere with a radius of one siriometer. For different reasons it is probable that this number is rather too great than too small, and we may perhaps estimate the total number to be something like 200 stars, of which more than a tenth is now known to the astronomers.

We may also arrive at an evaluation of this number by proceeding from the number of stars of different apparent or absolute magnitudes. This latter way is the most simple. We shall find in a later paragraph that the absolute magnitudes which are now known differ between -8 and +13. But from mathematical statistics it is proved that the total range of a statistical series amounts upon an average to approximately 6 times the dispersion of the series. Hence we conclude that the dispersion (σ) of the absolute magnitudes of the stars has approximately the value 3 (we should obtain σ = [13 + 8] : 6 = 3.5, but for large numbers of individuals the total range may amount to more than 6 σ).

As, further, the number of stars per cubic siriometer with an absolute magnitude brighter than 6 is known (we have obtained 8 : 4 = 2 stars per cubic siriometer brighter than 6^m), we get a relation between the total number of stars per cubic siriometer (D₀) and the mean absolute magnitude (M₀) of the stars, so that D₀ can be obtained, as soon as M₀ is known. The computation of M₀ is rather difficult, and is discussed in a following chapter. Supposing, for the moment, M₀ = 10 we get for D₀ the value 22, corresponding to a number of 90 stars within a distance of one siriometer from the sun. We should then know a fifth part of these stars.

34.

Parallax stars. In [§22] I have paid attention to the now available catalogues of stars with known annual parallax. The most extensive of these catalogues is that of Walkey, containing measured parallaxes of 625 stars. For a great many of these stars the value of the parallax measured must however be considered as rather uncertain, and I have pointed out that only for such stars as have a parallax greater than 0″.04 (or a distance smaller than 5 siriometers) may the measured parallax be considered as reliable, as least generally speaking. The effective number of parallax stars is therefore essentially reduced. Indirectly it is nevertheless possible to get a relatively large catalogue of parallax stars with the help of the ingenious spectroscopic method of Adams, which permits us to determine the absolute magnitude, and therefore also the distance, of even farther stars through an examination of the relative intensity of certain lines in the stellar spectra. It may be that the method is not yet as firmly based as it should be,[15] but there is every reason to believe that the course taken is the right one and that the catalogue published by Adams of 500 parallax stars in Contrib. from Mount Wilson, 142, already gives a more complete material than the catalogues of directly measured parallaxes. I give here a short resumé of the attributes of the parallax stars in this catalogue.

The catalogue of Adams embraces stars of the spectral types F, G, K and M. In order to complete this material by parallaxes of blue stars I add from the catalogue of Walkey those stars in his catalogue that belong to the spectral types B and A, confining myself to stars for which the parallax may be considered as rather reliable. There are in all 61 such stars, so that a sum of 561 stars with known distance is to be discussed.

For all these stars we know m and M and for the great part of them also the proper motion μ. We can therefore for each spectral type compute the mean values and the dispersion of these attributes. We thus get the following table, in which I confine myself to the mean values of the attributes.