The number of cases in which all the eight attributes of the stars discussed in the first chapter are well known for one star is very small, and certainly does not exceed one hundred. These cases refer principally to such stars as are characterized either by great brilliancy or by a great proper motion. The principal reason why these stars are better known than others is that they lie rather near our solar system. Before passing on to consider the stars from more general statistical points of view, it may therefore be of interest first to make ourselves familiar with these well-known stars, strongly emphasizing, however, the exceptional character of these stars, and carefully avoiding any generalization from the attributes we shall here find.

30.

The apparently brightest stars. We begin with these objects so well known to every lover of the stellar sky. The following table contains all stars the apparent visual magnitude of which is brighter than 1^m.5.

The first column gives the current number, the second the name, the third the equatorial designation (ad). It should be remembered that the first four figures give the hour and minutes in right ascension, the last two the declination, italics showing negative declination. The fourth column gives the galactic square, the fifth and sixth columns the galactic longitude and latitude. The seventh and eighth columns give the annual parallax and the corresponding distance expressed in siriometers. The ninth column gives the proper motion (), the tenth the radial velocity W expressed in sir/st. (To get km/sec. we may multiply by 4.7375). The eleventh column gives the apparent visual magnitude, the twelfth column the absolute magnitude (M), computed from m with the help of r. The 13^th column gives the type of spectrum (Sp), and the last column the photographic magnitude (m'). The difference between m' and m gives the colour-index (c).

TABLE 2.
THE APPARENTLY BRIGHTEST STARS.

The values of (ad), m, Sp are taken from H. 50. The values of l, b are computed from (ad) with the help of tables in preparation at the Lund Observatory, or from the original to plate I at the end, allowing the conversion of the equatorial coordinates into galactic ones. The values of p are generally taken from the table of Kapteyn and Weersma mentioned in the previous chapter. The values of are obtained from B. P. C., those of the radial velocity (W) from the card catalogue in Lund already described.

There are in all, in the sky, 20 stars having an apparent magnitude brighter than 1^m.5. The brightest of them is Sirius, which, owing to its brilliancy and position, is visible to the whole civilized world. It has a spectrum of the type A0 and hence a colour-index nearly equal to 0.0 (observations in Harvard give c = +0.06). Its apparent magnitude is -1^m.6, nearly the same as that of Mars in his opposition. Its absolute magnitude is -0^m.3, i.e., fainter than the apparent magnitude, from which we may conclude that it has a distance from us smaller than one siriometer. We find, indeed, from the eighth column that r = 0.5 sir. The proper motion of Sirius is 1.32 per year, which is rather large but still not among the largest proper motions as will be seen below. From the 11^th column we find that Sirius is moving towards us with a velocity of 1.6 sir/st. (= 7.6 km/sec.), a rather small velocity. The third column shows that its right ascension is 6^h 40^m and its declination -16°. It lies in the square GD₇ and its galactic coordinates are seen in the 5^th and 6^th columns.

The next brightest star is Canopus or a CarinÆ at the south sky. If we might place absolute confidence in the value of M (= -8.2) in the 12^th column this star would be, in reality, a much more imposing apparition than Sirius itself. Remembering that the apparent magnitude of the moon, according to §6, amounts to -11.6, we should find that Canopus, if placed at a distance from us equal to that of Sirius (r = 0.5 sir.), would shine with a lustre equal to no less than a quarter of that of the moon. It is not altogether astonishing that a fanciful astronomer should have thought Canopus to be actually the central star in the whole stellar system. We find, however, from column 8 that its supposed distance is not less that 30 sir. We have already pointed out that distances greater than 4 sir., when computed from annual parallaxes, must generally be considered as rather uncertain. As the value of M is intimately dependent on that of r we must consider speculations based on this value to be very vague. Another reason for a doubt about a great value for the real luminosity of this star is found from its type of spectrum which, according to the last column, is F0, a type which, as will be seen, is seldom found among giant stars. A better support for a large distance could on the other hand be found from the small proper motion of this star. Sirius and Canopus are the only stars in the sky having a negative value of the apparent visual magnitude.

Space will not permit us to go through this list star for star. We may be satisfied with some general remarks.

In the fourth column is the galactic square. We call to mind that all these squares have the same area, and that there is therefore the same probability a priori of finding a star in one of the squares as in another. The squares GC and GD lie along the galactic equator (the Milky Way). We find now from column 4 that of the 20 stars here considered there are no less than 15 in the galactic equator squares and only 5 outside, instead of 10 in the galactic squares and 10 outside, as would have been expected. The number of objects is, indeed, too small to allow us to draw any cosmological conclusions from this distribution, but we shall find in the following many similar instances regarding objects that are principally accumulated along the Milky Way and are scanty at the galactic poles. We shall find that in these cases we may generally conclude from such a partition that we then have to do with objects situated far from the sun, while objects that are uniformly distributed on the sky lie relatively near us. It is easy to understand that this conclusion is a consequence of the supposition, confirmed by all star counts, that the stellar system extends much farther into space along the Milky Way than in the direction of its poles.

If we could permit ourselves to draw conclusions from the small material here under consideration, we should hence have reason to believe that the bright stars lie relatively far from us. In other words we should conclude that the bright stars seem to be bright to us not because of their proximity but because of their large intrinsic luminosity. Column 8 really tends in this direction. Certainly the distances are not in this case colossal, but they are nevertheless sufficient to show, in some degree, this uneven partition of the bright stars on the sky. The mean distance of these stars is as large as 7.5 sir. Only a Centauri, Sirius, Procyon and Altair lie at a distance smaller than one siriometer. Of the other stars there are two that lie as far as 30 siriometers from our system. These are the two giants Canopus and Rigel. Even if, as has already been said, the distances of these stars may be considered as rather uncertain, we must regard them as being rather large.

As column 8 shows that these stars are rather far from us, so we find from column 12, that their absolute luminosity is rather large. The mean absolute magnitude is, indeed, -2^m.1. We shall find that only the greatest and most luminous stars in the stellar system have a negative value of the absolute magnitude.

The mean value of the proper motions of the bright stars amounts to 0.56 per year and may be considered as rather great. We shall, indeed, find that the mean proper motion of the stars down to the 6^th magnitude scarcely amounts to a tenth part of this value. On the other hand we find from the table that the high value of this mean is chiefly due to the influence of four of the stars which have a large proper motion, namely Sirius, Arcturus, a Centauri and Procyon. The other stars have a proper motion smaller than 1 per year and for half the number of stars the proper motion amounts to approximately 0.05, indicating their relatively great distance.

That the absolute velocity of these stars is, indeed, rather small may be found from column 10, giving their radial velocity, which in the mean amounts to only three siriometers per stellar year. From the discussion below of the radial velocities of the stars we shall find that this is a rather small figure. This fact is intimately bound up with the general law in statistical mechanics, to which we return later, that stars with large masses generally have a small velocity. We thus find in the radial velocities fresh evidence, independent of the distance, that these bright stars are giants among the stars in our stellar system.

We find all the principal spectral types represented among the bright stars. To the helium stars (B) belong Rigel, Achernar, Centauri, Spica, Regulus and Crucis. To the Sirius type (A) belong Sirius, Vega, Altair, Fomalhaut and Deneb. To the Calcium type (F) Canopus and Procyon. To the sun type (G) Capella and a Centauri. To the K-type belong Arcturus, Aldebaran and Pollux and to the M-type the two red stars Betelgeuze and Antares. Using the spectral indices as an expression for the spectral types we find that the mean spectral index of these stars is +1.1 corresponding to the spectral type F1.

31.

Stars with the greatest proper motion. In table 3 I have collected the stars having a proper motion greater than 3 per year. The designations are the same as in the preceding table, except that the names of the stars are here taken from different catalogues.

In the astronomical literature of the last century we find the star 1830 Groombridge designed as that which possesses the greatest known proper motion. It is now distanced by two other stars C. P. D. 5^h.243 discovered in the year 1897 by Kapteyn and Innes on the plates taken for the Cape Photographic Durchmusterung, and Barnard's star in Ophiuchus, discovered 1916. The last-mentioned star, which possesses the greatest proper motion now known, is very faint, being only of the 10^th magnitude, and lies at a distance of 0.40 sir. from our sun and is hence, as will be found from table 5 the third nearest star for which we know the distance. Its linear velocity is also very great, as we find from column 10, and amounts to 19 sir/st. (= 90 km/sec.) in the direction towards the sun. The absolute magnitude of this star is 11^m.7 and it is, with the exception of one other, the very faintest star now known. Its spectral type is Mb, a fact worth fixing in our memory, as different reasons favour the belief that it is precisely the M-type that contains the very faintest stars. Its apparent velocity (i.e., the proper motion) is so great that the star in 1000 years moves 3°, or as much as 6 times the diameter of the moon. For this star, as well as for its nearest neighbours in the table, observations differing only by a year are sufficient for an approximate determination of the value of the proper motion, for which in other cases many tens of years are required.

Regarding the distribution of these stars in the sky we find that, unlike the brightest stars, they are not concentrated along the Milky Way. On the contrary we find only 6 in the galactic equator squares and 12 in the other squares. We shall not build up any conclusion on this irregularity in the distribution, but supported by the general thesis of the preceding paragraph we conclude only that these stars must be relatively near us. This follows, indeed, directly from column 8, as not less than eleven of these stars lie within one siriometer from our sun. Their mean distance is 0.87 sir.

TABLE 3.
STARS WITH THE GREATEST PROPER MOTION.

That the great proper motion does not depend alone on the proximity of these stars is seen from column 10, giving the radial velocities. For some of the stars (4) the radial velocity is for the present unknown, but the others have, with few exceptions, a rather great velocity amounting in the mean to 18 sir/st. (= 85 km/sec.), if no regard is taken to the sign, a value nearly five times as great as the absolute velocity of the sun. As this is only the component along the line of sight, the absolute velocity is still greater, approximately equal to the component velocity multiplied by v2. We conclude that the great proper motions depend partly on the proximity, partly on the great linear velocities of the stars. That both these attributes here really cooperate may be seen from the absolute magnitudes (M).

The apparent and the absolute magnitudes are for these stars nearly equal, the means for both been approximately 7^m. This is a consequence of the fact that the mean distance of these stars is equal to one siriometer, at which distance m and M, indeed, do coincide. We find that these stars have a small luminosity and may be considered as dwarf stars. According to the general law of statistical mechanics already mentioned small bodies upon an average have a great absolute velocity, as we have, indeed, already found from the observed radial velocities of these stars.

As to the spectral type, the stars with great proper motions are all yellow or red stars. The mean spectral index is +2.8, corresponding to the type G8. If the stars of different types are put together we get the table

We conclude that, at least for these stars, the mean value of the absolute magnitude increases with the spectral index. This conclusion, however, is not generally valid.

32.

Stars with the greatest radial velocities. There are some kinds of nebulae for which very large values of the radial velocities have been found. With these we shall not for the present deal, but shall confine ourselves to the stars. The greatest radial velocity hitherto found is possessed by the star (040822) of the eighth magnitude in the constellation Perseus, which retires from us with a velocity of 72 sir/st. or 341 km/sec. The nearest velocity is that of the star (010361) which approaches us with approximately the same velocity. The following table contains all stars with a radial velocity greater than 20 sir/st. (= 94.8 km/sec.). It is based on the catalogue of Voute mentioned above.

Regarding their distribution in the sky we find 11 in the galactic equator squares and 7 outside. A large radial velocity seems therefore to be a galactic phenomenon and to be correlated to a great distance from us. Of the 18 stars in consideration there is only one at a distance smaller than one siriometer and 2 at a distance smaller than 4 siriometers. Among the nearer ones we find the star (050744), identical with C. P. D. 5^h.243, which was the “second” star with great proper motion. These stars have simultaneously the greatest proper motion and very great linear velocity. Generally we find from column 9 that these stars with large radial velocity possess also a large proper motion. The mean value of the proper motions amounts to 1.34, a very high value.

In the table we find no star with great apparent luminosity. The brightest is the 10^th star in the table which has the magnitude 5.1. The mean apparent magnitude is 7.7. As to the absolute magnitude (M) we see that most of these speedy stars, as well as the stars with great proper motions in table 3, have a rather great positive magnitude and thus are absolutely faint stars, though they perhaps may not be directly considered as dwarf stars. Their mean absolute magnitude is +3.0.

Regarding the spectrum we find that these stars generally belong to the yellow or red types (G, K, M), but there are 6 F-stars and, curiously enough, two A-stars. After the designation of their type (A2 and A3) is the letter p (= peculiar), indicating that the spectrum in some respect differs from the usual appearance of the spectrum of this type. In the present case the peculiarity consists in the fact that a line of the wave-length 448.1, which emanates from magnesium and which we may find on plate III in the spectrum of Sirius, does not occur in the spectrum of these stars, though the spectrum has otherwise the same appearance as in the case of the Sirius stars. There is reason to suppose that the absence of this line indicates a low power of radiation (low temperature) in these stars (compare Adams).

TABLE 4.
STARS WITH THE GREATEST RADIAL VELOCITY.

33.

The nearest stars. The star a in Centaurus was long considered as the nearest of all stars. It has a parallax of 0.75, corresponding to a distance of 0.27 siriometers (= 4.26 light years). This distance is obtained from the annual parallax with great accuracy, and the result is moreover confirmed in another way (from the study of the orbit of the companion of a Centauri). In the year 1916 Innes discovered at the observatory of Johannesburg in the Transvaal a star of the 10^th magnitude, which seems to follow a Centauri in its path in the heavens, and which, in any case, lies at the same distance from the earth, or somewhat nearer. It is not possible at present to decide with accuracy whether Proxima Centauri—as the star is called by Innes—or a Centauri is our nearest neighbour. Then comes Barnard's star (175204), whose large proper motion we have already mentioned. As No. 5 we find Sirius, as No. 8 Procyon, as No. 21 Altair. The others are of the third magnitude or fainter. No. 10—61 Cygni—is especially interesting, being the first star for which the astronomers, after long and painful endeavours in vain, have succeeded in determining the distance with the help of the annual parallax (Bessel 1841).

From column 4 we find that the distribution of these stars on the sky is tolerably uniform, as might have been predicted. All these stars have a large proper motion, this being in the mean 3.42 per year. This was a priori to be expected from their great proximity. The radial velocity is, numerically, greater than could have been supposed. This fact is probably associated with the generally small mass of these stars.

Their apparent magnitude is upon an average 6.3. The brightest of the near stars is Sirius (m = -1.6), the faintest Proxima Centauri (m = 11). Through the systematic researches of the astronomers we may be sure that no bright stars exist at a distance smaller than one siriometer, for which the distance is not already known and well determined. The following table contains without doubt—we may call them briefly all near stars—all stars within one siriometer from us with an apparent magnitude brighter than 6^m (the table has 8 such stars), and probably also all near stars brighter than 7^m (10 stars), or even all brighter than the eighth magnitude (the table has 13 such stars and two near the limit). Regarding the stars of the eighth magnitude or fainter no systematic investigations of the annual parallax have been made and among these stars we may get from time to time a new star belonging to the siriometer sphere in the neighbourhood of the sun. To determine the total number of stars within this sphere is one of the fundamental problems in stellar statistics, and to this question I shall return immediately.

TABLE 5.
THE NEAREST STARS.

The mean absolute magnitude of the near stars is distributed in the following way:—

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 (s) of the absolute magnitudes of the stars has approximately the value 3 (we should obtain s = [13 + 8] : 6 = 3.5, but for large numbers of individuals the total range may amount to more than 6 s).

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 . 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 () 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 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.

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 . But m and are both positively correlated to M. By the selection of stars with small m and great 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 a 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.

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