and Cosmology

2.3 The Structure of the Galaxy
a component i of the velocity vector, the dispersion σ i
is defined as
σ 2
i = 〈 (v i − 〈v i 〉) 2〉 = 〈 v 2 i − 〈v i〉 2〉
= n −1 ∫
R 3 d 3 v f(v) ( v 2 i − 〈v i〉 2) . (2.33)
The larger σ i is, the broader the distribution of the
stochastic motions. We note that the same concept applies
to the velocity distribution of molecules in a gas.
The mean velocity 〈v〉 at each point defines the bulk
velocity of the gas, e.g., the wind speed in the atmosphere,
whereas the velocity dispersion is caused by
thermal motion of the molecules and is determined by
the temperature of the gas.
The random motion of the stars in the direction
perpendicular to the disk is the reason for the finite
thickness of the population; it is similar to a thermal
distribution. Accordingly, it has the effect of a pressure,
the so-called dynamical pressure of the distribution.
This pressure determines the scale-height of the distribution,
which corresponds to the law of atmospheres.
The larger the dynamical pressure, i.e., the larger the
velocity dispersion σ z perpendicular to the disk, the
larger the scale-height h will be. The analysis of stars
in the Solar neighborhood yields σ z ∼ 16 km/s for stars
younger than ∼ 3 Gyr, corresponding to a scale-height
of h ∼ 250 pc, whereas stars older than ∼ 6 Gyr have
a scale-height of ∼ 350 pc and a velocity dispersion of
σ z ∼ 25 km/s.
The density distribution of the total star population,
obtained from counts and distance determinations of
stars, is to a good approximation described by
n(R, z) = n 0
(
e
−|z|/h thin
+ 0.02e −|z|/h thick ) e −R/h R
;
(2.34)
here, R and z are the cylinder coordinates introduced
above (see Sect. 2.1), with the origin at the Galactic
center, and h thin ≈ h otd ≈ 325 pc is the scale-height of
the thin disk. The distribution in the radial direction can
also be well described by an exponential law, where
h R ≈ 3.5 kpc denotes the scale-length of the Galactic
disk. The normalization of the distribution is determined
by the density n ≈ 0.2 stars/pc 3 in the Solar neighborhood,
for stars in the range of absolute magnitudes of
4.5 ≤ M V ≤ 9.5. The distribution described by (2.34) is
not smooth at z = 0; it has a kink at this point and it is
therefore unphysical. To get a smooth distribution which
follows the exponential law for large z and is smooth in
the plane of the disk, the distribution is slightly modified.
As an example, for the luminosity density of the
old thin disk (that is proportional to the number density
of the stars), we can write:
L(R, z) =
L 0e −R/h R
cosh 2 (z/h z )
, (2.35)
with h z = 2h thin and L 0 ≈ 0.05L ⊙ /pc 3 . The Sun is
a member of the young thin disk and is located above
the plane of the disk, at z = 30 pc.
2.3.2 The Galactic Disk:
Chemical Composition and Age
Stellar Populations. The chemical composition of stars
in the thin and the thick disks differs: we observe the
clear tendency that stars in the thin disk have a higher
metallicity than those in the thick disk. In contrast, the
metallicity of stars in the Galactic halo and in the bulge
is smaller. To paraphrase these trends, one distinguishes
between stars of Population I (Pop I) which have a Solarlike
metallicity (Z ∼ 0.02) and are mainly located in
the thin disk, and stars of Population II (Pop II) that
are metal-poor (Z ∼ 0.001) and predominantly found
in the thick disk, in the halo, and in the bulge. In reality,
stars cover a wide range in Z, and the figures above
are only characteristic values. For stellar populations
a somewhat finer separation was also introduced, such
as “extreme Population I”, “intermediate Population II”,
and so on. The populations also differ in age (stars of
Pop I are younger than those of Pop II), in scale-height
(as mentioned above), and in the velocity dispersion
perpendicular to the disk (σ z is larger for Pop II stars
than for Pop I stars).
We shall now attempt to understand the origin of
these different metallicities and their relation to the
scale-height and to age. We start with a brief discussion
of the phenomenon that is the main reason for the
metal enrichment of the interstellar medium.
Metallicity and Supernovae. Supernovae (SNe) are
explosive events. Within a few days, a SN can reach
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