Lectures on Stellar Statistics

What is the absolute magnitude of the near stars that are not contained in table? Evidently they must principally be faint stars. We may go further and answer that all stars with an absolute magnitude brighter than 6^m must be contained in this list. For if M is equal to 6 or brighter, m must be brighter than 6^m, if the star is nearer than one siriometer. But we have assumed that all stars apparently brighter than 6^m are known and are contained in the list. Hence also all stars absolutely brighter than 6^m must be found in table 5. We conclude that the number of stars having an absolute magnitude brighter than 6^m amounts to 8.

If, finally, the spectral type of the near stars is considered, we find from the last column of the table that these stars are distributed in the following way:—

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.

TABLE 6.

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

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.

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.

TABLE 8.

THE ABSOLUTELY FAINTEST STARS.

Regarding the absolute brightness of the stars we may draw some conclusions of interest. We find from table 7 that the absolute magnitude of the parallax stars varies between -4 and +11, the extreme stars being of type M. The absolutely brightest stars have a rather great distance from us and their absolute magnitude is badly determined. The brightest star in the table is Antares with M = -4.6, which value is based on the parallax 0″.014 found by Adams. So small a parallax value is of little reliability when it is directly computed from annual parallax observations, but is more trustworthy when derived with the spectroscopic method of Adams. It is probable from a discussion of the B-stars, to which we return in a later chapter, that the absolutely brightest stars have a magnitude of the order -5^m or -6^m. If the parallaxes smaller than 0″.01 were taken into account we should find that Canopus would represent the absolutely brightest star, having M = -8.17, and next to it we should find Rigel, having M = -6.97, but both these values are based on an annual parallax equal to 0″.007, which is too small to allow of an estimation of the real value of the absolute magnitude.

If on the contrary the absolutely faintest stars be considered, the parallax stars give more trustworthy results. Here we have only to do with near stars for which the annual parallax is well determined. In table 8 I give a list of those parallax stars that have an absolute magnitude greater than 9^m.

There are in all 19 such stars. The faintest of all known stars is Innes' star “Proxima Centauri” with M = 13.9. The third star is Barnard's star with M = 11.7, both being, together with α Centauri, also the nearest of all known stars. The mean distance of all the faint stars is 1.0 sir.

There is no reason to believe that the limit of the absolute magnitude of the faint stars is found from these faint parallax stars:—Certainly there are many stars in space with M > 13^m and the mean value of M, for all stars in the Galaxy, is probably not far from the absolute value of the faint parallax stars in this table. This problem will be discussed in a later part of these lectures.

Scientia Publisher

K. G. MALMQUIST:

A study of the stars of spectral type A. L. M. II, 22.

Price: 6 Kr.

C. V. L. CHARLIER:

Vorlesungen über die Grundzüge der mathematischen Statistik.

Price: 10 Kr.

F. ESSCHER:

Über die Sterblichkeit in Schweden. 1886-1914. Eine Stabilitäts- und Korrelationsuntersuchung. L. M. II, 23.

Price: 6 Kr.

B. FÄNGE:

On the distances and luminosities of stars of spectral type G as derived from their proper motions. L. M. II, 25.

Price: 5 Kr.

C. V. L. CHARLIER:

Introduction to Stellar Statistics.

Price: 4 Kr.

NILS H. RASMUSON:

A Research on moving clusters. L. M. II, 26.

Price: 5 Kr.

SIGFRID RAAB:

A Research on open clusters. L. M. II, 28.

Price: 8 Kr.

CARL WIRTZ:

Flächenhelligkeiten von 566 Nebelflecken und Sternhaufen. Nach photometrischen Beobachtungen am 49 cm.-Refraktor der Universitäts-Sternwarte Straßburg (Elsaß) 1911-1916. L. M. II, 29.

Price: 5 Kr.

C. V. L. CHARLIER:

Stargauges by William Herschel (H) and John Herschel (h). L. M. II, 30.

Price: 5 Kr.

PRINTED BY LÜTCKE & WULFF, HAMBURG

Transcriber's Note: The following corrections have been made to the original text.

Page 4: "Terrestial distances" changed to "Terrestrial distances"

Page 9: "we must chose," changed to "we must choose,"

Page 12: "acromasie" changed to "achromatism"

Page 15: "inparticular" changed to "in particular"

"supposing, that" changed to "supposing that"

Page 16: "393.4 mm" changed to "393.4 μμ"

Page 20: "for which stars if" changed to "for which stars it"

Page 22: "sphaerical" changed to "spherical"

Page 23: "principal scource" changed to "principal source"

Page 25: "lense with 15 cm" changed to "lens with 15 cm"

Page 27: "Through the spectroscopie method" changed to "Through the spectroscopic method"

"made to its principal" changed to "made its principal"

"american manner" changed to "American manner"

Page 35: "many tenths of a year" changed to "many tens of years"

Page 38: "same appearence" changed to "same appearance"

Page 47: "red stears" changed to "red stars"

"dispersion in M" changed to "dispersion in M"

Page 49: "smaller the" changed to "smaller than"