## Stellar Statistics

The Stellar Luminosity Function. By systematically observing all stars in the solar neighbourhood, one can find the distribution of their absolute magnitudes. This is given by the luminosity function M), which gives the relative number of main sequence stars with absolute magnitudes in the range [M — 1/2, M +1 /2]. No stars appear to be forming at present in the region of space where the luminosity function has been determined. The age of the Milky Way is 10-15 Ga, which means that all stars less massive than 0.9 Me, will still be on the main sequence. On the other hand, more massive stars, formed early in the history of the Milky Way, will have completed their evolution and disappeared. Low-mass stars have accumulated in the luminosity function for many generations of star formation, whereas bright, high-mass stars are the result of recent star formation.

By taking into account the different main sequence lifetimes of stars of different masses and hence of different magnitudes, one can determine the initial luminosity function &(M), which gives the brightness distribution at the time of star formation, the zero age main sequence luminosity function. The relation between the function & and the observed luminosity function is

where T0 is the age of the Milky Way and tE(M) is the main sequence lifetime of stars of magnitude M. Here we assume that the birth rate of stars of magnitude M has remained constant during the lifetime of the Milky Way. The initial luminosity function is shown in Fig. 17.7.

The Fundamental Equation of Stellar Statistics. The Stellar Density. A crucial problem for studies of the structure of the Milky Way is to find out how the density of stars varies in space. The number of stars per unit

Fig. 17.7. The observed luminosity function &(My) and the initial luminosity function &(My) for main sequence stars in the solar neighbourhood. The functions give the number of stars per cubic parsec in the magnitude interval [MV — 1/2, My +1/2]; they are actually the products D& and D&, where D is the stellar density function (in the solar neighbourhood)

Fig. 17.7. The observed luminosity function &(My) and the initial luminosity function &(My) for main sequence stars in the solar neighbourhood. The functions give the number of stars per cubic parsec in the magnitude interval [MV — 1/2, My +1/2]; they are actually the products D& and D&, where D is the stellar density function (in the solar neighbourhood)

volume at a distance r in the direction (l, b) from the Sun is given by the stellar density D = D(r, l, b).

The stellar density cannot be directly observed except in the immediate neighbourhood of the Sun. However, it can be calculated if one knows the luminosity function and the interstellar extinction as a function of distance in a given direction. In addition the number of stars per unit solid angle (e. g. per square arc second) can be determined as a function of limiting apparent magnitude by means of star counts (Fig. 17.8).

Let us consider the stars within the solid angle w in the direction (l, b) and in the distance range [r, r + dr ]. We let their luminosity function @(M) be the same as in the solar neighbourhood and their unknown stellar density D. The absolute magnitude M of the stars of apparent magnitude m is, as usual,

The number of stars in the apparent magnitude interval [m — 0.5, m + 0.5] in the volume element dV = wr2dr at distance r is (Fig. 17.9)

10 pc

The stars of apparent magnitude m in the given area of the sky will in reality be at many different distances.

Fig. 17.8. The stellar density is determined by means of star counts. In practice, the counting is done on photographic plates. (Cartoon S. Harris)

In order to obtain their total number N(m), one has to integrate dN(m) over all distances r:

10 pc

Equation (17.5) is called the fundamental equation of stellar statistics. Its left-hand side, the number of stars in the apparent magnitude interval [m — 0.5, m + 0.5] in the solid angle w, is obtained from the observations: one counts the stars of different magnitudes in a chosen area of a photographic plate. The luminosity function is known from the solar neighbourhood. The extinction A(r) can be determined for the chosen areas, for instance, by means of multicolour photometry. In order to solve the integral equation (17.5) for D(r, l, b), several methods have been developed, but we shall not go into them here.

Figure 17.10a shows the stellar density in the solar neighbourhood in the plane of the Milky Way, and Fig. 17.10b in the direction perpendicular to the plane. There are several individual concentrations, but e. g. spiral structure cannot be observed in such a limited region of space.

Figure 17.10a shows the stellar density in the solar neighbourhood in the plane of the Milky Way, and Fig. 17.10b in the direction perpendicular to the plane. There are several individual concentrations, but e. g. spiral structure cannot be observed in such a limited region of space.

The Distribution of Bright Objects. Using stellar statistical methods, one can only study the close neighbourhood of the Sun, out to about 1 kpc at the most. Absolutely faint objects cannot be observed very far. Since the solar neighbourhood appears to be fairly representative of the general properties of the Milky Way, its study is naturally important, giving information e. g.

on the distributions and luminosity functions of stars of various spectral types. However, in order to get an idea of the larger-scale structure of the Milky Way, one has to make use of objects that are as absolutely bright as possible, and which can be observed even at large distances.

Examples of suitable objects are stars of early spectral types, HII regions, OB associations, open star clusters, cepheids, RRLyrae stars, supergiants and giants of late spectral types, and globular clusters. Some of these objects differ greatly in age, such as the young OB associations, on the one hand, and the old globular clusters, on the other. Any differences in their space distribution tell us about changes in the general structure of the Milky Way.

The young optical objects, the H II regions, OB associations and open clusters, are strongly concentrated in the plane of the Milky Way (Table 17.1). Figure 17.11 shows that they also appear to be concentrated in three drawn-out bands, at least within the observed region. Since these types of objects in other galaxies are known to be part of a spiral structure, the observed bands in the Milky Way have been interpreted as portions of three spiral arms passing through the solar neighbourhood. Stars of later spectral types seem to be much more evenly distributed. Apart from a few special directions, interstellar dust limits observations in the galactic plane to within 3-4 kpc.

Fig. 17.10a,b. The stellar density near the Sun. (a) The stellar density of spectral classes A2-A5 in the galactic plane, according to S.W. McCuskey. The numbers next to the isodensity curves give the number of stars in 10,000 pc3. (b) The distribution of different spectral classes perpendicularly to the galactic plane according to T. Elvius. The density in the galactic plane has been normalized to one

Table 17.1. Populations of the Milky Way; z is the vertical distance from the galactic plane, and vr the velocity component perpendicular to the galactic plane

Table 17.1. Populations of the Milky Way; z is the vertical distance from the galactic plane, and vr the velocity component perpendicular to the galactic plane

 Population Typical Average age z Vr Metal objects [109 a] [pc] [k/s] abundance Halo population II Subdwarfs, globular clusters RRLyr (P > 0.4d) 14-12 2000 75 0.001 Intermediate population II Long period variables 12-10 700 25 0.005 Disc population Planetary nebulae, novae bright red giants 12-2 400 18 0.01-0.02 Old population I A stars, Me dwarfs classical cepheids 2-0.1 160 10 0.02 Young population I Gas, dust, supergiants, T Tau stars 0.1 120 0.03-0.04

Fig. 17.11. The distribution of various objects in the galactic plane. Three condensations can be discerned: the Sagittarius arm (lowest), the local arm near the Sun and (outermost) the Perseus arm

Fig. 17.11. The distribution of various objects in the galactic plane. Three condensations can be discerned: the Sagittarius arm (lowest), the local arm near the Sun and (outermost) the Perseus arm

Old objects, particularly the globular clusters, have an almost spherical distribution about the centre of the Milky Way (Fig. 17.12). The space density of old objects increases towards the galactic centre. They can be used to determine the distance of the Sun from the galactic centre; the value of this distance is about 8.5 kpc.

Stellar Populations. Studies of the motions of the stars in the Milky Way have revealed that the orbits of stars moving in the galactic plane are almost circular. These stars are also usually young, a few hundred million years at the most. They also contain a relatively large amount of heavy elements, about 2-4%. The interstel-

17.3 The Rotation of the Milky Way

Fig. 17.12. The distribution of globular clusters. (From S.R. Majewski, Stellar populations and the Milky Way, in C. Martinez Roger, I. Perez Fournon, F. Sanchez (Eds.) Globular Clusters, Cambridge University Press, 1999)

lar material similarly moves in the galactic plane in almost circular orbits. On the basis of their motions and their chemical composition, the interstellar medium and the youngest stars are collectively referred to as population I.

Outside the plane of the Milky Way, an almost spherically symmetric halo extends out to 50 kpc and beyond. The stellar density is largest near the galactic centre and decreases outwards. The halo contains very little interstellar matter, and its stars are old, perhaps up to 15 x 109 years. These stars are also very metal-poor. Their orbits may be very eccentric and show no preference for the galactic plane. On the basis of these criteria, one defines stars of population II. Typical population II objects are the globular clusters, and the RR Lyrae and W Virginis stars.

The stars of population II have large velocities with respect to the local standard of rest, up to more than 300kms—In reality their velocities at the solar distance from the galactic centre are quite small and may sometimes be opposite to the direction of motion of the LSR. The large relative velocities only reflect the motion of the LSR with a velocity of about 220kms—1 round the galactic centre.

Between these two extremes, there is a sequence of intermediate populations. In addition to populations I and II, one generally also speaks of a disc population, including the Sun, for instance. The typical motions, chemical composition and age of the various populations (Table 17.1) contain information about the evolution of our Galaxy and about the formation of its stars.

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