Lectures on Stellar Statistics Part 1

Lectures on Stellar Statistics.

by Carl Vilhelm Ludvig Charlier.

CHAPTER I.

APPARENT ATTRIBUTES OF THE STARS.

1. Our knowledge of the stars is based on their _apparent_ attributes, obtained from the astronomical observations. The object of astronomy is to deduce herefrom the real or _absolute_ attributes of the stars, which are their position in s.p.a.ce, their movement, and their physical nature.

The apparent attributes of the stars are studied by the aid of their _radiation_. The characteristics of this radiation may be described in different ways, according as the nature of the light is defined.

(Undulatory theory, Emission theory.)

From the statistical point of view it will be convenient to consider the radiation as consisting of an emanation of small particles from the radiating body (the star). These particles are characterized by certain attributes, which may differ in degree from one particle to another.

These attributes may be, for instance, the diameter and form of the particles, their mode of rotation, &c. By these attributes the optical and electrical properties of the radiation are to be explained. I shall not here attempt any such explanation, but shall confine myself to the property which the particles have of possessing a different mode of deviating from the rectilinear path as they pa.s.s from one medium to another. This deviation depends in some way on one or more attributes of the particles. Let us suppose that it depends on a single attribute, which, with a terminology derived from the undulatory theory of HUYGHENS, may be called the _wave-length_ (?) of the particle.

The statistical characteristics of the radiation are then in the first place:--

(1) the total number of particles or the _intensity_ of the radiation;

(2) the _mean wave-length_ (?_0) of the radiation, also called (or nearly identical with) the _effective_ wave-length or the colour;

(3) _the dispersion of the wave-length_. This characteristic of the radiation may be determined from the _spectrum_, which also gives the variation of the radiation with ?, and hence may also determine the mean wave-length of the radiation.

Moreover we may find from the radiation of a star its apparent place on the sky.

The intensity, the mean wave-length, and the dispersion of the wave-length are in a simple manner connected with the _temperature_ (_T_) of the star. According to the radiation laws of STEPHAN and WIEN we find, indeed (compare L. M. 41[1]) that the intensity is proportional to the fourth power of _T_, whereas the mean wave-length and the dispersion of the wave-length are both inversely proportional to _T_. It follows that with increasing temperature the mean wave-length diminishes--the colour changing into violet--and simultaneously the dispersion of the wave-length and also even the total length of the spectrum are reduced (decrease).

2. _The apparent position of a star_ is generally denoted by its right ascension (a) and its declination (d). Taking into account the apparent distribution of the stars in s.p.a.ce, it is, however, more practical to characterize the position of a star by its galactic longitude (_l_) and its galactic lat.i.tude (_b_). Before defining these coordinates, which will be generally used in the following pages, it should be pointed out that we shall also generally give the coordinates a and d of the stars in a particular manner. We shall therefore use an abridged notation, so that if for instance a = 17h 44m.7 and d = +35.84, we shall write

(ad) = (174435).

If d is negative, for instance d = -35.84, we write

(ad) = (1744{35}),

so that the last two figures are in italics.

[Transcriber's Note: In this version of the text, the last two figures are enclosed in braces to represent the italics.]

This notation has been introduced by PICKERING for variable stars and is used by him everywhere in the Annals of the Harvard Observatory, but it is also well suited to all stars. This notation gives, simultaneously, the characteristic _numero_ of the stars. It is true that two or more stars may in this manner obtain the same characteristic _numero_. They are, however, easily distinguishable from each other through other attributes.

The _galactic_ coordinates _l_ and _b_ are referred to the Milky Way (the Galaxy) as plane of reference. The pole of the Milky Way has according to HOUZEAU and GOULD the position (ad) = (124527). From the distribution of the stars of the spectral type B I have in L. M. II, 14[2] found a somewhat different position. But having ascertained later that the real position of the galactic plane requires a greater number of stars for an accurate determination of its value, I have preferred to employ the position used by PICKERING in the Harvard catalogues, namely (ad) = (124028), or

a = 12h 40m = 190, d = +28,

which position is now exclusively used in the stellar statistical investigations at the Observatory of Lund and is also used in these lectures.

The galactic longitude (_l_) is reckoned from the ascending node of the Milky Way on the equator, which is situated in the constellation _Aquila_. The galactic lat.i.tude (_b_) gives the angular distance of the star from the Galaxy. On plate I, at the end of these lectures, will be found a fairly detailed diagram from which the conversion of a and d of a star into _l_ and _b_ may be easily performed. All stars having an apparent magnitude brighter than 4m are directly drawn.

Instead of giving the galactic longitude and lat.i.tude of a star we may content ourselves with giving the galactic _square_ in which the star is situated. For this purpose we a.s.sume the sky to be divided into 48 squares, all having the same surface. Two of these squares lie at the northern pole of the Galaxy and are designated GA_1 and GA_2. Twelve lie north of the galactic plane, between 0 and 30 galactic lat.i.tude, and are designated GC_1, GC_2, ..., GC_12. The corresponding squares south of the galactic equator (the plane of the Galaxy) are called GD_1, GD_2, ..., GD_12. The two polar squares at the south pole are called GF_1 and GF_2. Finally we have 10 B-squares, between the A- and C-squares and 10 corresponding E-squares in the southern hemisphere.

The distribution of the squares in the heavens is here graphically represented in the projection of FLAMSTEED, which has the advantage of giving areas proportional to the corresponding spherical areas, an arrangement necessary, or at least highly desirable, for all stellar statistical researches. It has also the advantage of affording a continuous representation of the whole sky.

The correspondence between squares and stellar constellations is seen from plate II. Arranging the constellations according to their galactic longitude we find north of the galactic equator (in the C-squares) the constellations:--

Hercules, Cygnus, Cepheus, Ca.s.siopaea, Auriga, Gemini, Canis Minor, Pyxis, Vela, Centaurus, Scorpius, Ophiuchus,

and south of this equator (in the D-squares):--

Aquila, Cygnus, Lacerta, Andromeda, Perseus, Orion, Canis Major, Puppis, Carina, Circinus, Corona australis, Sagittarius,

mentioning only one constellation for each square.

At the north galactic pole (in the two A-squares) we have:--

Canes Venatici and Coma Berenices,

and at the south galactic pole (in the two F-squares):--

Cetus and Sculptor.

3. _Changes in the position of a star._ From the positions of a star on two or more occasions we obtain its apparent motion, also called the _proper motion_ of the star. We may distinguish between a _secular_ part of this motion and a _periodical_ part. In both cases the motion may be either a reflex of the motion of the observer, and is then called _parallactic_ motion, or it may be caused by a real motion of the star.

From the parallactic motion of the star it is possible to deduce its distance from the sun, or its parallax. The periodic parallactic proper motion is caused by the motion of the earth around the sun, and gives the _annual parallax_ (p). In order to obtain available annual parallaxes of a star it is usually necessary for the star to be nearer to us than 5 siriometers, corresponding to a parallax greater than 0?.04. More seldom we may in this manner obtain trustworthy values for a distance amounting to 10 siriometers (p = 0?.02), or even still greater values. For such large distances the _secular_ parallax, which is caused by the progressive motion of the sun in s.p.a.ce, may give better results, especially if the mean distance of a group of stars is simultaneously determined. Such a value of the secular parallax is also called, by KAPTEYN, the _systematic_ parallax of the stars.

When we speak of the proper motion of a star, without further specification, we mean always the secular proper motion.

4. Terrestrial distances are now, at least in scientific researches, universally expressed in kilometres. A kilometre is, however, an inappropriate unit for celestial distances. When dealing with distances in our planetary system, the astronomers, since the time of NEWTON, have always used the mean distance of the earth from the sun as universal unit of distance. Regarding the distances in the stellar system the astronomers have had a varying practice. German astronomers, SEELIGER and others, have long used a stellar unit of distance corresponding to an annual parallax of 0?.2, which has been called a "Siriusweite". To this name it may be justly objected that it has no international use, a great desideratum in science. Against the theoretical definition of this unit it may also be said that a distance is suitably to be defined through another distance and not through an angle--an angle which corresponds moreover, in this case, to the _harmonic_ mean distance of the star and not to its arithmetic mean distance. The same objection may be made to the unit "pa.r.s.ec." proposed in 1912 by TURNER.

For my part I have, since 1911, proposed a stellar unit which, both in name and definition, nearly coincides with the proposition of SEELIGER, and which will be exclusively used in these lectures. A _siriometer_ is put equal to 10^6 times the planetary unit of distance, corresponding to a parallax of 0?.206265 (in practice sufficiently exactly 0?.2).

In popular writings, another unit: a _light-year_, has for a very long time been employed. The relation between these units is

1 siriometer = 15.79 light-years, 1 light-year = 0.0633 siriometers.

5. In regard to _time_ also, the terrestrial units (second, day, year) are too small for stellar wants. As being consistent with the unit of distance, I have proposed for the stellar unit of time a _stellar year_ (st.), corresponding to 10^6 years. We thus obtain the same relation between the stellar and the planetary units of length and time, which has the advantage that a _velocity_ of a star expressed in siriometers per stellar year is expressed with the same numerals in planetary units of length per year.

Spectroscopic determinations of the velocities, through the DOPPLER-principle, are generally expressed in km. per second. The relation with the stellar unit is the following:

1 km./sec. = 0.2111 sir./st., = 0.2111 planetary units per year, 1 sir./st. = 4.7375 km./sec.

Thus the velocity of the sun is 20 km./sec. or 4.22 sir./st. (= 4.22 earth distances from the sun per year).

Of the numerical value of the stellar velocity we shall have opportunity to speak in the following. For the present it may suffice to mention that most stars have a velocity of the same degree as that of the sun (in the mean somewhat greater), and that the highest observed velocity of a star amounts to 72 sir./st. (= 340 km./sec.). In the next chapter I give a table containing the most speedy stars. The least value of the stellar velocity is evidently equal to zero.