Lectures on Stellar Statistics

C.) by MuLLER and KEMPF, which was published simultaneously with H. 50.

It contains the magnitude of 14199 stars and embraces all stars on the northern hemisphere brighter than 7m.5 (according to B. D.). We have already seen that the zero-point of H. 50 and P. G. C. is somewhat different and that the magnitudes in P. G. C. must be increased by -0m.16 if they are to be reduced to the Harvard scale. The difference between the two catalogues however is due to some extent to the colour of the stars, as has been shown by Messrs. MuLLER and KEMPF.

25. _Photographic magnitudes._ Our knowledge of this subject is still rather incomplete. The most comprehensive catalogue is the "Actinometrie" by SCHWARZSCHILD (1912), containing the photographic magnitudes of all stars in B. D. down to the magnitude 7m.5 between the equator and a declination of +20. In all, 3522 stars. The photographic magnitudes are however not reduced for the zero-point (compare --6).

These is also a photometric photographic catalogue of the stars nearest to the pole in PARKHURST's "Yerkes actinometry" (1912),[11] which contains all stars in B. D. brighter than 7m.5 between the pole and 73 northern declination. The total number of stars is 672.

During the last few years the astronomers of Harvard and Mount Wilson have produced a collection of "standard photographic magnitudes" for faint stars. These stars, which are called the _polar sequence_,[12] all lie in the immediate neighbourhood of the pole. The list is extended down to the 20th magnitude. Moreover similar standard photographic magnitudes are given in H. A. 71, 85 and 101.

A discussion of the _colour-index_ (_i.e._, the difference between the photographic and the visual magnitudes) will be found in L. M. II, 19.

When the visual magnitude and the type of spectrum are known, the photographic magnitude may be obtained, with a generally sufficient accuracy, by adding the colour-index according to the table 1 in --15 above.

26. _Stellar spectra._ Here too we find the Harvard Observatory to be the leading one. The same volume of the Annals of the Harvard Observatory (H. 50) that contains the most complete catalogue of visual magnitudes, also gives the spectral types for all the stars there included, _i.e._, for all stars to 6m.5. Miss CANNON, at the Harvard Observatory, deserves the princ.i.p.al credit for this great work. Not content with this result she is now publishing a still greater work embracing more than 200000 stars. The first four volumes of this work are now published and contain the first twelve hours of right ascension, so that half the work is now printed.[13]

27. _Radial velocity._ In this matter, again, we find America to be the leading nation, though, this time, it is not the Harvard or the Mount Wilson but the Lick Observatory to which we have to give the honour. The eminent director of this observatory, W. W. CAMPBELL, has in a high degree developed the accuracy in the determination of radial velocities and has moreover carried out such determinations in a large scale. The "Bulletin" No. 229 (1913) of the Lick Observatory contains the radial velocity of 915 stars. At the observatory of Lund, where as far as possible card catalogues of the attributes of the stars are collected, GYLLENBERG has made a catalogue of this kind for the radial velocities.

The total number of stars in this catalogue now amounts to 1640.[14]

28. Finally I shall briefly mention some comprehensive works on more special questions regarding the stellar system.

On _variable stars_ there is published every year by HARTWIG in the "Vierteljahrschrift der astronomischen Gesellschaft" a catalogue of all known variable stars with needful information about their minima &c.

This is the completest and most reliable of such catalogues, and is always up to date. A complete historical catalogue of the variables is given in "Geschichte und Literatur des Lichtwechsels der bis Ende 1915 als sicher veranderlich anerkannten Sterne nebst einem Katalog der Elemente ihres Lichtwechsels" von G. MuLLER und E. HARTWIG. Leipzig 1918, 1920.

On _nebulae_ we have the excellent catalogues of DREYER, the "New General Catalogue" (N. G. C.) of 1890 in the "Memoirs of the Astronomical Society" vol. 49, the "Index catalogues" (I. C.) in the same memoirs, vols. 51 and 59 (1895 and 1908). These catalogues contain all together 13226 objects.

Regarding other special attributes I refer in the first place to the important Annals of the Harvard Observatory. Other references will be given in the following, as need arises.

FOOTNOTES:

[Footnote 7: "Bonner Sternverzeichnis" in den Astronomischen Beobachtungen auf der Sternwarte zu Bonn, Dritter bis Funfter Band. Bonn 1859-62.]

[Footnote 8: "Bonner Durchmusterung", Vierte Sektion. Achter Band der Astronomischen Beobachtungen zu Bonn, 1886.]

[Footnote 9: "The Cape Photographic Durchmusterung" by DAVID GILL and J.

C. KAPTEYN, Annals of the Cape Observatory, vol. III-V (1896-1900).]

[Footnote 10: "Cordoba Durchmusterung" by J. THOME. Results of the National Argentine Observatory, vol. 16, 17, 18, 21 (1894-1914).]

[Footnote 11: Aph. J., vol. 36.]

[Footnote 12: H. A., vol. 71.]

[Footnote 13: H. A., vol. 91, 92, 93, 94.]

[Footnote 14: A catalogue of radial velocities has this year been published by J. VOUTE, embracing 2071 stars. "First catalogue of radial velocities", by J. VOUTE. Weltevreden, 1920.]

CHAPTER III.

SOME GROUPS OF KNOWN STARS.

29. 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 princ.i.p.ally to such stars as are characterized either by great brilliancy or by a great proper motion. The princ.i.p.al reason why these stars are better known than others is that they lie rather near our solar system. Before pa.s.sing 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 1m.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 lat.i.tude. 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 13th 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._

+--+---------------------+----------+--------+-----+-------+-------+-------+ | 1| 2 | 3 | 4 | 5 | 6 | 7 | 8 | +--+---------------------+----------+--------+-----+-------+-------+-------+ | | | Position | Distance | | | _Name_ |----------+--------+-----+-------+-------+-------+ | | | (ad) | Square | _l_ | _b_ | p | _r_ | +--+---------------------+----------+--------+-----+-------+-------+-------+ | | | | | | | | sir. | | 1|Sirius |(0640{16})| GD_7 | 195| - 8 |0?.876 | 0.5 | | 2|Canopus |(0621{52})| GD_8 | 229 | -24 | 0.007 | 29.5 | | 3|Vega |(183338) | GC_2 | 30 | +17 | 0.094 | 2.2 | | 4|Capella |(050945) | GC_5 | 131 | + 5 | 0.066 | 3.1 | | 5|Arcturus |(141119) | GA_2 | 344 | +68 | 0.075 | 2.7 | | 6|a Centauri |(1432{60})| GD_10 | 284 | - 2 | 0.759 | 0.3 | | 7|Rigel |(0509{08})| GD_6 | 176 | -24 | 0.007 | 29.5 | | 8|Procyon |(073405) | GC_7 | 182 | +14 | 0.324 | 0.6 | | 9|Achernar |(0134{57})| GE_8 | 256 | -59 | 0.051 | 4.0 | |10| Centauri |(1356{59})| GC_10 | 280 | + 2 | 0.037 | 5.6 | |11|Altair |(194508) | GD_1 | 15 | -10 | 0.238 | 0.9 | |12|Betelgeuze |(054907) | GD_6 | 168 | - 8 | 0.030 | 6.9 | |13|Aldebaran |(043016) | GD_5 | 149 | -19 | 0.078 | 2.8 | |14|Pollux |(073928) | GC_6 | 160 | +25 | 0.064 | 3.2 | |15|Spica |(1319{10})| GB_8 | 286 | +51 | .. | .. | |16|Antares |(1623{26})| GC_11 | 320 | +14 | 0.029 | 7.1 | |17|Fomalhaut |(2252{30})| GE_10 | 348 | -66 | 0.138 | 1.5 | |18|Deneb |(203844) | GC_2 | 51 | + 1 | .. | .. | |19|Regulus |(100312) | GB_6 | 196 | +50 | 0.033 | 6.3 | |20| Crucis |(1241{59})| GC_10 | 270 | + 3 | 0.008 | 25.8 | +--+---------------------+----------+--------+-----+-------+-------+-------+ | | | | | | | | sir. | | | Mean | .. | .. | .. | 23.5|0?.134 | 7.3 | +--+---------------------+----------+--------+-----+-------+-------+-------+

+--+---------------------+------+--------+---------+---------+----+------+ | 1| 2 | 9 | 10 | 11 | 12 | 13 | 14 | +--+---------------------+------+--------+---------+---------+----+------+ | | | Motion | Magnitude | Spectrum | | | _Name_ +------+--------+---------+---------+----+------+ | | | | _W_ | _m_ | _M_ |_Sp_| _m'_ | +--+---------------------+------+--------+---------+---------+----+------+ | | | |sir./st.| | | | _m'_ | | 1|Sirius | 1?.32| - 1.56| -1m.58 | -0m.3 |A |-1.58 | | 2|Canopus | 0.02| + 4.39| -0.86 | -8.2 |F |-0.40 | | 3|Vega | 0.35| - 2.91| 0.14 | -1.6 |A | 0.14 | | 4|Capella | 0.44| + 6.38| 0.21 | -2.8 |G | 1.13 | | 5|Arcturus | 2.28| - 0.82| 0.24 | -1.9 |K | 1.62 | | 6|a Centauri | 3.68| - 4.69| 0.33 | +3.2 |G | 1.25 | | 7|Rigel | 0.00| + 4.77| 0.34 | -7.0 |B8p | 0.25 | | 8|Procyon | 1.24| - 0.74| 0.48 | +1.5 |F5 | 1.17 | | 9|Achernar | 0.09| .. | 0.60 | -2.4 |B5 | 0.87 | |10| Centauri | 0.04| + 2.53| 0.86 | -2.9 |B1 | 0.45 | |11|Altair | 0.66| - 6.97| 0.89 | +1.2 |A5 | 1.12 | |12|Betelgeuze | 0.03| + 4.43| 0.92 | -3.3 |Ma | 2.76 | |13|Aldebaran | 0.20| +11.63| 1.06 | -1.2 |K5 | 2.67 | |14|Pollux | 0.07| + 0.82| 1.21 | -1.3 |K | 2.59 | |15|Spica | 0.06| + 0.34| 1.21 | .. |B2 | 0.84 | |16|Antares | 0.03| - 0.63| 1.22 | -3.0 |Map | 3.06 | |17|Fomalhaut | 0.37| + 1.41| 1.29 | +0.4 |A3 | 1.43 | |18|Deneb | 0.00| - 0.84| 1.33 | .. |A2 | 1.42 | |19|Regulus | 0.25| .. | 1.34 | -2.7 |B8 | 1.25 | |20| Crucis | 0.06| + 2.74| 1.50 | -5.6 |B1 | 1.09 | +--+---------------------+------+--------+---------+---------+----+------+ | | | | | | | | _m'_ | | | Mean | 0?.56| 3.26| +0m.64 | -2m.1 |F1 |+1.13 | +--+---------------------+------+--------+---------+---------+----+------+

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 1m.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 -1m.6, nearly the same as that of Mars in his opposition. Its absolute magnitude is -0m.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 11th 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 6h 40m and its declination -16. It lies in the square GD_7 and its galactic coordinates are seen in the 5th and 6th columns.

The next brightest star is _Canopus_ or a Carinae at the south sky. If we might place absolute confidence in the value of _M_ (= -8.2) in the 12th 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 l.u.s.tre 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.

s.p.a.ce 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 princ.i.p.ally acc.u.mulated 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 part.i.tion 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 s.p.a.ce 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 part.i.tion 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, -2m.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 6th 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 ma.s.ses 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 princ.i.p.al 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. 5h.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 10th 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 11m.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.