Lectures on Stellar Statistics

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._

+--+---------------------+----------+--------+-----+-------+-------+-------+ | 1| 2 | 3 | 4 | 5 | 6 | 7 | 8 | +--+---------------------+----------+--------+-----+-------+-------+-------+ | | | Position | Distance | | | _Name_ |----------+--------+-----+-------+-------+-------+ | | | (ad) | Square | _l_ | _b_ | p | _r_ | +--+---------------------+----------+--------+-----+-------+-------+-------+ | | | | | | | | sir. | | 1|Barnards star |(175204) | GC_12 | 358| +12 |0?.515 | 0.40 | | 2|C. Z. 5h.243 |(0507{44})| GE_7 | 218 | -35 | 0.319 | 0.65 | | 3|Groom. 1830 |(114738) | GA_1 | 135 | +75 | 0.102 | 2.02 | | 4|Lac. 9352 |(2259{36})| GE_10 | 333 | -66 | 0.292 | 0.71 | | 5|C. G. A. 32416 |(2359{37})| GF_2 | 308 | -75 | 0.230 | 0.89 | | 6|61 Cygni |(210238) | GD_2 | 50 | - 7 | 0.311 | 0.66 | | 7|Lal. 21185 |(105736) | GB_5 | 153 | +66 | 0.403 | 0.51 | | 8|e Indi |(2155{57})| GE_9 | 304 | -47 | 0.284 | 0.73 | | 9|Lal. 21258 |(110044) | GB_4 | 135 | +64 | 0.203 | 1.02 | |10|O^2 Eridani |(0410{07})| GE_5 | 168 | -36 | 0.174 | 1.19 | |11|Proxima Centauri |(1422{62})| GD_10 | 281 | - 2 | 0.780 | 0.26 | |12|Oe. A. 14320 |(1504{15})| GB_9 | 314 | +35 | 0.035 | 5.90 | |13| Ca.s.siopeiae |(010154) | GD_4 | 93 | - 8 | 0.112 | 1.84 | |14|a Centauri |(1432{60})| GD_10 | 284 | - 2 | 0.759 | 0.27 | |15|Lac. 8760 |(2111{39})| GE_10 | 332 | -44 | 0.248 | 0.83 | |16|Lac. 1060 |(0315{43})| GE_7 | 216 | -55 | 0.162 | 1.27 | |17|Oe. A. 11677 |(111466) | GB_8 | 103 | +50 | 0.198 | 1.04 | |18|Van Maanens star |(004304) | GD_8 | 92 | -58 | 0.246 | 0.84 | +--+---------------------+----------+--------+-----+-------+-------+-------+ | | | | | | | | sir. | | | Mean | .. | .. | | 41 |0?.298 | 0.87 | +--+---------------------+----------+--------+-----+-------+-------+-------+

+--+---------------------+------+--------+---------+---------+----+------+ | 1| 2 | 9 | 10 | 11 | 12 | 13 | 14 | +--+---------------------+------+--------+---------+---------+----+------+ | | | Motion | Magnitude | Spectrum | | | _Name_ +------+--------+---------+---------+----+------+ | | | | _W_ | _m_ | _M_ |_Sp_| _m'_ | +--+---------------------+------+--------+---------+---------+----+------+ | | | |sir./st.| | | | _m'_ | | 1|Barnards star |10?.29| -19 | 9m.7 | +11m.7|Mb |11.5 | | 2|C. Z. 5h.243 | 8.75| +51 | 9.2 | +10.1 |K2 |10.6 | | 3|Groom. 1830 | 7.06| -20 | 6.5 | +5.0 |G5 | 7.6 | | 4|Lac. 9352 | 6.90| +2 | 7.5 | +8.2 |K | 8.9 | | 5|C. G. A. 32416 | 6.11| +5 | 8.2 | +8.5 |G | 9.1 | | 6|61 Cygni | 5.27| -13 | 5.6 | +6.5 |K5 | 7.2 | | 7|Lal. 21185 | 4.77| -18 | 7.6 | +9.1 |Mb | 8.9 | | 8|e Indi | 4.70| -8 | 4.7 | +5.4 |K5 | 6.3 | | 9|Lal. 21258 | 4.47| +14 | 8.5 | +8.5 |Ma |10.3 | |10|O^2 Eridani | 4.11| -9 | 4.7 | +4.3 |G5 | 5.8 | |11|Proxima Centauri | 3.85| .. | 11.0 | +13.9 |.. |13.5 | |12|Oe. A. 14320 | 3.75| +61 | 9.0 | +5.1 |G0 | 9.9 | |13| Ca.s.siopeiae | 3.73| -21 | 5.7 | +4.4 |G3 | 6.8 | |14|a Centauri | 3.68| -5 | 0.3 | +3.2 |G | 1.2 | |15|Lac. 8760 | 3.53| +3 | 6.6 | +7.0 |G | 7.5 | |16|Lac. 1060 | 3.05| +18 | 5.6 | +5.1 |G5 | 6.7 | |17|Oe. A. 11677 | 3.03| .. | 9.2 | +9.1 |Ma |11.0 | |18|Van Maanens star | 3.01| .. | 12.3 | +12.7 |F0 |12.9 | +--+---------------------+------+--------+---------+---------+----+------+ | | | |sir./st.| | | | _m'_ | | | Mean | 5?.00| 17.8 | 7m.3 | +7m.6|G8 | 8.7 | +--+---------------------+------+--------+---------+---------+----+------+

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 7m. 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

_Type_ _Number_ _Mean value of M_ G 8 5.3 K 4 7.5 M 4 9.6

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. 5h.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 10th 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._

+--+---------------------+----------+--------+-----+-------+-------+-------+ | 1| 2 | 3 | 4 | 5 | 6 | 7 | 8 | +--+---------------------+----------+--------+-----+-------+-------+-------+ | | | Position | Distance | | | _Name_ |----------+--------+-----+-------+-------+-------+ | | | (ad) | Square | _l_ | _b_ | p | _r_ | +--+---------------------+----------+--------+-----+-------+-------+-------+ | | | | | | | | sir. | | 1|A. G. Berlin 1366 |(040822) | GD_5 | 141| -20 |0?.007 | 30.8 | | 2|Lal. 1966 |(010361) | GD_4 | 93 | - 2 | 0.016 | 12.9 | | 3|A. Oe. 14320 |(1504{15})| GB_9 | 314 | +35 | 0.035 | 5.9 | | 4|C. Z. 5h.243 |(0507{44})| GE_7 | 218 | -35 | 0.319 | 0.6 | | 5|Lal. 15290 |(074730) | GC_6 | 158 | +26 | 0.023 | 9.0 | | 6|53 Ca.s.siop. |(015563) | GC_4 | 98 | + 2 | .. | .. | | 7|A. G. Berlin 1866 |(055719) | GD_6 | 159 | - 2 | 0.021 | 9.8 | | 8|W Lyrae |(181136) | GC_2 | 31 | +21 | .. | .. | | 9|Boss 1511 |(0559{26})| GD_7 | 200 | -20 | 0.012 | 17.0 | |10|? Pavonis |(1849{60})| GD_11 | 304 | -24 | .. | .. | |11|A. Oe. 20452 |(2017{21})| GE_10 | 351 | -31 | 0.015 | 13.5 | |12|Lal. 28607 |(1537{10})| GB_10 | 325 | +34 | 0.033 | 6.2 | |13|A. G. Leiden 5734 |(161132) | GB_1 | 21 | +45 | 0.002 | 89.2 | |14|Lal. 37120 |(192932) | GC_2 | 33 | + 6 | 0.050 | 4.1 | |15|Lal. 27274 |(1454{21})| GB_9 | 308 | +34 | 0.013 | 16.2 | |16|Lal. 5761 |(030225) | GD_5 | 126 | -28 | 0.039 | 5.1 | |17|W. B. 17h.517 |(172906) | GC_12 | 358 | +20 | 0.014 | 14.1 | |18|Lal. 23995 |(1247{17})| GB_8 | 271 | +46 | 0.012 | 17.0 | +--+---------------------+----------+--------+-----+-------+-------+-------+ | | | | | | | | sir. | | | Mean | .. | .. | | 23.9|0?.041 | 16.7 | +--+---------------------+----------+--------+-----+-------+-------+-------+

+--+---------------------+------+--------+---------+---------+----+------+ | 1| 2 | 9 | 10 | 11 | 12 | 13 | 14 | +--+---------------------+------+--------+---------+---------+----+------+ | | | Motion | Magnitude | Spectrum | | | _Name_ +------+--------+---------+---------+----+------+ | | | | _W_ | _m_ | _M_ |_Sp_| _m'_ | +--+---------------------+------+--------+---------+---------+----+------+ | | | |sir./st.| | | | _m'_ | | 1|A. G. Berlin 1366 | 0?.54| +72 | 8m.9 | +1m.4 |F0 | 9.4 | | 2|Lal. 1966 | 0.64| -69 | 7.9 | +2.3 |F3 | 8.5 | | 3|A. Oe. 14320 | 3.75| +61 | 9.0 | +5.1 |G0 | 9.9 | | 4|C. Z. 5h.243 | 8.75| +51 | 9.2 | +10.1 |K2 |10.6 | | 5|Lal. 15290 | 1.96| -51 | 8.2 | +3.4 |G0 | 9.1 | | 6|53 Ca.s.siop. | 0.01| -44 | 5.6 | .. |B8 | 5.5 | | 7|A. G. Berlin 1866 | 0.76| -40 | 9.0 | +4.0 |F0 | 9.9 | | 8|W Lyrae | .. | -39 | var. | .. |Md | var. | | 9|Boss 1511 | 0.10| +39 | 5.2 | -1.0 |G5 | 6.4 | |10|? Pavonis | 0.14| +38 | 5.1 | .. |K | 6.5 | |11|A. Oe. 20452 | 1.18| -38 | 8.1 | +2.4 |G8p | 9.4 | |12|Lal. 28607 | 1.18| -36 | 7.3 | +3.3 |A2p | 7.4 | |13|A. G. Leiden 5734 | 0.04| -35 | 8.3 | -1.5 |K4 | 9.9 | |14|Lal. 37120 | 0.52| -34 | 6.6 | +3.5 |G2 | 7.6 | |15|Lal. 27274 | 0.79| +34 | 8.3 | +2.2 |F4 | 8.9 | |16|Lal. 5761 | 0.86| -32 | 8.0 | +4.4 |A3p | 8.1 | |17|W. B. 17h.517 | 0.63| -31 | 8.6 | +2.8 |F1 | 9.1 | |18|Lal. 23995 | 0.88| +30 | 8.2 | +2.0 |F3 | 8.8 | +--+---------------------+------+--------+---------+---------+----+------+ | | | |sir./st.| | | | _m'_ | | | Mean | 1?.34| 16.7 | 7m.7 | +3m.0 |F9 | 8.5 | +--+---------------------+------+--------+---------+---------+----+------+

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 10th 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 a.s.sociated with the generally small ma.s.s 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 6m (the table has 8 such stars), and probably also all near stars brighter than 7m (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._

+--+----------------------+----------+--------+-----+-------+-------+-------+ | 1| 2 | 3 | 4 | 5 | 6 | 7 | 8 | +--+----------------------+----------+--------+-----+-------+-------+-------+ | | | Position | Distance | | | _Name_ |----------+--------+-----+-------+-------+-------+ | | | (ad) | Square | _l_ | _b_ | p | _r_ | +--+----------------------+----------+--------+-----+-------+-------+-------+ | | | | | | | | sir. | | 1|Proxima Centauri |(1422{62})| GD_10 | 281| - 2 |0?.780 | 0.26 | | 2|a Centauri |(1432{60})| GD_10 | 284 | - 2 | 0.759 | 0.27 | | 3|Barnards p. m. star |(175204) | GC_12 | 358 | +12 | 0.515 | 0.40 | | 4|Lal. 21185 |(105736) | GB_5 | 153 | +66 | 0.403 | 0.51 | | 5|Sirius |(0640{16})| GD_7 | 195 | - 8 | 0.376 | 0.55 | | 6| .. |(1113{57})| GC_6 | 158 | + 3 | 0.337 | 0.60 | | 7|t Ceti |(0139{16})| GF_1 | 144 | -74 | 0.334 | 0.62 | | 8|Procyon |(073405) | GC_7 | 182 | +14 | 0.324 | 0.64 | | 9|C. Z. 5h.243 |(0507{44})| GE_7 | 218 | -35 | 0.319 | 0.65 | |10|61 Cygni |(210238) | GD_2 | 50 | - 7 | 0.311 | 0.66 | |11|Lal. 26481 |(1425{15})| GB_9 | 124 | -40 | 0.311 | 0.66 | |12|e Eridani |(0328{09})| GE_5 | 153 | -42 | 0.295 | 0.70 | |13|Lac. 9352 |(2259{36})| GE_10 | 333 | -66 | 0.292 | 0.71 | |14|Pos. Med. 2164 |(184159) | GC_2 | 56 | +24 | 0.292 | 0.71 | |15|e Indi |(215557) | GE_9 | 304 | -47 | 0.284 | 0.73 | |16|Groom. 34 |(001243) | GD_3 | 84 | -20 | 0.281 | 0.73 | |17|Oe. A. 17415 |(173768) | GC_8 | 65 | +32 | 0.268 | 0.77 | |18|Kruger 60 |(222457) | GC_3 | 72 | 0 | 0.256 | 0.81 | |19|Lac. 8760 |(2111{39})| GE_10 | 332 | -44 | 0.248 | 0.88 | |20|van Maanens p. m. star|(004304) | GE_3 | 92 | -58 | 0.246 | 0.84 | |21|Altair |(194508) | GD_1 | 15 | -10 | 0.238 | 0.87 | |22|C. G. A. 32416 |(2359{37})| GF_2 | 308 | -75 | 0.230 | 0.89 | |23|Bradley 1584 |(1129{32})| GC_6 | 252 | +28 | 0.216 | 0.95 | +--+----------------------+----------+--------+-----+-------+-------+-------+ | | | | | | | | sir. | | | Mean | .. | .. | .. | 30.8|0?.344 | 0.67 | +--+----------------------+----------+--------+-----+-------+-------+-------+

+--+-----------------------+------+--------+---------+---------+----+------+ | 1| 2 | 9 | 10 | 11 | 12 | 13 | 14 | +--+-----------------------+------+--------+---------+---------+----+------+ | | | Motion | Magnitude | Spectrum | | | _Name_ +------+--------+---------+---------+----+------+ | | | | _W_ | _m_ | _M_ |_Sp_| _m'_ | +--+-----------------------+------+--------+---------+---------+----+------+ | | | |sir./st.| | | | _m'_ | | 1|Proxima Centauri | 3?.85| .. | 11m.0 | +13m.9 |.. |13.5 | | 2|a Centauri | 3.68| - 5 | 0.33 | + 3.2 |G | 1.25 | | 3|Barnards p. m. star | 10.29| -19 | 9.7 | +11.7 |Mb |11.5 | | 4|Lal. 21185 | 4.77| -18 | 7.6 | + 9.1 |Mb | 8.9 | | 5|Sirius | 1.32| - 2 | -1.58 | - 0.3 |A |-1.58 | | 6| .. | 2.72| .. | .. | .. |.. |12.5 | | 7|t Ceti | 1.92| - 3 | 3.6 | + 4.6 |K0 | 4.6 | | 8|Procyon | 1.24| - 1 | 0.48 | + 1.5 |F5 | 0.90 | | 9|C. Z. 5h.243 | 8.75| +51 | 9.2 | +10.1 |K2 |10.6 | |10|61 Cygni | 5.27| -13 | 5.6 | + 6.5 |K5 | 7.2 | |11|Lal. 26481 | 0.47| .. | 7.8 | + 8.7 |G5 | 8.9 | |12|e Eridani | 0.97| + 3 | 3.8 | + 4.6 |K0 | 4.8 | |13|Lac. 9352 | 6.90| + 2 | 7.5 | + 8.2 |K | 8.9 | |14|Pos. Med. 2164 | 2.28| .. | 8.9 | + 9.6 |K |10.3 | |15|e Indi | 4.70| - 8 | 4.7 | + 5.4 |K5 | 6.3 | |16|Groom. 34 | 2.89| + 1 | 8.1 | + 8.8 |Ma | 9.5 | |17|Oe. A. 17415 | 1.30| .. | 9.1 | + 9.7 |K |10.5 | |18|Kruger 60 | 0.94| .. | 9.2 | + 9.6 |K5 |10.8 | |19|Lac. 8760 | 3.53| + 3 | 6.6 | + 7.0 |G | 7.5 | |20|van Maanens p. m. star | 3.01| .. | 12.3 | +12.7 |F0 |12.9 | |21|Altair | 0.66| - 7 | 0.9 | + 1.2 |A5 | 1.12 | |22|C. G. A. 32416 | 6.11| + 5 | 8.2 | + 8.5 |G | 9.1 | |23|Bradley 1584 | 1.06| - 5 | 6.1 | + 6.2 |G | 6.9 | +--+-----------------------+------+--------+---------+---------+----+------+ | | | |sir./st.| | | | _m'_ | | | Mean | 3?.42| 9.1 | 6m.3 | +7m.3 |G6 | 7.5 | +--+-----------------------+------+--------+---------+---------+----+------+

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

_M_ 0 1 3 4 5 6 7 8 9 10 11 12 13 Number 1 2 1 2 1 2 1 4 4 1 1 1 1.

What is the absolute magnitude of the near stars that are not contained in table? Evidently they must princ.i.p.ally be faint stars. We may go further and answer that _all stars with an absolute magnitude brighter than 6m_ must be contained in this list. For if _M_ is equal to 6 or brighter, _m_ must be brighter than 6m, if the star is nearer than one siriometer. But we have a.s.sumed that all stars apparently brighter than 6m are known and are contained in the list. Hence also all stars _absolutely_ brighter than 6m must be found in table 5. We conclude that the number of stars having an absolute magnitude brighter than 6m 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:--

Spectral type B A F G K M Number 0 2 2 5 9 3.

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 9m.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 a.s.sume 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 6m), we get a relation between the total number of stars per cubic siriometer (_D_0_) and the mean absolute magnitude (_M_0_) of the stars, so that _D_0_ can be obtained, as soon as _M_0_ is known. The computation of _M_0_ is rather difficult, and is discussed in a following chapter. Supposing, for the moment, _M_0_ = 10 we get for _D_0_ 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 resume 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._

+---+-------+-------+-------+-------+ |Sp.|Number | _m_ | _M_ | | +---+-------+-------+-------+-------+ | 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 +6m and the absolute magnitude equal to +2m. For type B we find the mean value of M to be -1m.7 and for type A we find M = +0m.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 p.r.o.nounced 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 p.r.o.nounced in the graphical representation given in figures (plate IV).