Lectures on Stellar Statistics

by C. V. L. Charlier

TABLE 6.
MEAN VALUES OF m, M AND THE PROPER MOTIONS (μ) OF PARALLAX STARS OF DIFFERENT SPECTRAL TYPES.

Sp.	Number	m	M	μ
B	15	+2.03	-1.67	0″.05
A	46	+3.40	+0.64	0.21
F	125	+5.60	+2.10	0.40
G	179	+5.77	+1.68	0.51
K	184	+6.17	+2.31	0.53
M	42	+6.02	+2.30	0.82

We shall later consider all parallax stars taken together. We find from [table 6] that the apparent magnitude, as well as the absolute magnitude, is approximately the same for all yellow and red stars and even for the stars of type F, the apparent magnitude being approximately equal to +6^m and the absolute magnitude equal to +2^m. For type B we find the mean value of M to be -1^m.7 and for type A we find M = +0^m.6. The proper motion also varies in the same way, being for F, G, K, M approximately 0″.5 and for B and A 0″.1. As to the mean values of M and μ we cannot draw distinct conclusions from this material, because the parallax stars are selected in a certain way which essentially influences these mean values, as will be more fully discussed below. The most interesting conclusion to be drawn from the parallax stars is obtained from their distribution over different values of M. In the memoir referred to, Adams has obtained the following table (somewhat differently arranged from the table of Adams),[16] which gives the number of parallax stars for different values of the absolute magnitude for different spectral types.

A glance at this table is sufficient to indicate a singular and well pronounced property in these frequency distributions. We find, indeed, that in the types G, K and M the frequency curves are evidently resolvable into two simple curves of distribution. In all these types we may distinguish between a bright group and a faint group. With a terminology proposed by Hertzsprung the former group is said to consist of giant stars, the latter group of dwarf stars. Even in the stars of type F this division may be suggested. This distinction is still more pronounced in the graphical representation given in figures ([plate IV]).

TABLE 7.
DISTRIBUTION OF THE PARALLAX STARS OF DIFFERENT SPECTRAL TYPES OVER DIFFERENT ABSOLUTE MAGNITUDES.

M	B	A	F	G	K	M		All
- 4	..	..	..	..	..	1		..
- 3	..	..	..	..	..	..		..
- 2	1	4	1	7	..	2		15
- 1	2	7	7	28	15	4		63
- 0	3	10	6	32	40	10		91
+ 0	1	11	6	7	14	11		50
+ 1	1	3	20	9	4	1		38
+ 2	..	5	48	26	..	1		80
+ 3	..	1	32	36	2	..		71
+ 4	..	1	5	25	25	..		56
+ 5	..	1	..	6	25	..		32
+ 6	..	2	..	3	10	..		15
+ 7	..	1	..	..	14	..		15
+ 8	..	..	..	..	3	7		10
+ 9	..	..	..	..	2	4		6
+10	..	..	..	..	..	..		..
+11	..	..	..	..	..	1		1
Total	8	46	125	179	154	42		554

In the distribution of all the parallax stars we once more find a similar bipartition of the stars. Arguing from these statistics some astronomers have put forward the theory that the stars in space are divided into two classes, which are not in reality closely related. The one class consists of intensely luminous stars and the other of feeble stars, with little or no transition between the two classes. If the parallax stars are arranged according to their apparent proper motion, or even according to their absolute proper motion, a similar bipartition is revealed in their frequency distribution.

Nevertheless the bipartition of the stars into two such distinct classes must be considered as vague and doubtful. Such an apparent bipartition is, indeed, necessary in all statistics as soon as individuals are selected from a given population in such a manner as the parallax stars are selected from the stars in space. Let us consider three attributes, say A, B and C, of the individuals of a population and suppose that the attribute C is positively correlated to the attributes A and B, so that to great or small values of A or B correspond respectively great or small values of C. Now if the individuals in the population are statistically selected in such a way that we choose out individuals having great values of the attributes A and small values of the attribute B, then we get a statistical series regarding the attribute C, which consists of two seemingly distinct normal frequency distributions. It is in like manner, however, that the parallax stars are selected. The reason for this selection is the following. The annual parallax can only be determined for near stars, nearer than, say, 5 siriometers. The direct picking out of these stars is not possible. The astronomers have therefore attacked the problem in the following way. The near stars must, on account of their proximity, be relatively brighter than other stars and secondly possess greater proper motions than those. Therefore parallax observations are essentially limited to (1) bright stars, (2) stars with great proper motions. Hence the selected attributes of the stars are m and μ. But m and μ are both positively correlated to M. By the selection of stars with small m and great μ we get a series of stars which regarding the attribute M seem to be divided into two distinct classes.

The distribution of the parallax stars gives us no reason to believe that the stars of the types K and M are divided into the two supposed classes. There is on the whole no reason to suppose the existence at all of classes of giant and dwarf stars, not any more than a classification of this kind can be made regarding the height of the men in a population. What may be statistically concluded from the distribution of the absolute magnitudes of the parallax stars is only that the dispersion in M is increased at the transition from blue to yellow or red stars. The filling up of the gap between the “dwarfs” and the “giants” will probably be performed according as our knowledge of the distance of the stars is extended, where, however, not the annual parallax but other methods of measuring the distance must be employed.