and Cosmology

3. The World of Galaxies
94
arrives at a self-gravitating equilibrium system. In general,
it is a difficult mathematical problem to construct
such self-gravitating equilibrium systems.
( ) (
Fig. 3.9. The rotation parameter vrot
σ v
/ vrot
σ v
of elliptical
)iso
galaxies, here denoted by (V/σ) ∗ , plotted as a function of
absolute magnitude. Dots denote elliptical galaxies, crosses
the bulges of disk galaxies
The brightness profile of an elliptical galaxy is defined
by the distribution of its stellar orbits. Let us
assume that the gravitational potential is given. The stars
are then placed into this potential, with the initial positions
and velocities following a specified distribution. If
this distribution is not isotropic in velocity space, the resulting
light distribution will in general not be spherical.
For instance, one could imagine that the orbital planes
of the stars have a preferred direction, but that an equal
number of stars exists with positive and negative angular
momentum L z , so that the total stellar distribution
has no angular momentum and therefore does not rotate.
Each star moves along its orbit in the gravitational potential,
where the orbits are in general not closed. If an
initial distribution of stellar orbits is chosen such that the
statistical properties of the distribution of the orbits are
invariant in time, then one will obtain a stationary system.
If, in addition, the distribution is chosen such that
the respective mass distribution of the stars will generate
exactly the originally chosen gravitational potential, one
Relaxation Time-Scale. The question now arises
whether such an equilibrium system can also be stable
in time. One might expect that close encounters of
pairs of stars would cause a noticeable disturbance in the
distribution of orbits. These pair-wise collisions could
then lead to a “thermalization” of the stellar orbits. 3
To examine this question we need to estimate the timescale
for such collisions and the changes in direction
they cause.
For this purpose, we consider the relaxation timescale
by pair collisions in a system of N stars of mass m,
total mass M = Nm, extent R, and a mean stellar density
of n = 3N/(4πR 3 ). We define the relaxation time t relax
as the characteristic time in which a star changes its
velocity direction by ∼ 90 ◦ due to pair collisions with
other stars. By simple calculation (see below), we find
that
t relax ≈ R N
v ln N , (3.2)
or
N
t relax = t cross
ln N , (3.3)
where t cross = R/v is the crossing time-scale, i.e., the
time it takes a star to cross the stellar system. If we now
consider a typical galaxy, with t cross ∼ 10 8 yr, N ∼ 10 12
(thus ln N ∼ 30), then we find that the relaxation time is
much longer than the age of the Universe. This means
that pair collisions do not play any role in the evolution
of stellar orbits. The dynamics of the orbits are determined
solely by the large-scale gravitational field of the
galaxy. In Sect. 7.5.1, we will describe a process called
violent relaxation which most likely plays a central
role in the formation of galaxies and which is probably
also responsible for the stellar orbits establishing an
equilibrium configuration.
The stars behave like a collisionless gas: elliptical
galaxies are stabilized by (dynamical) pressure, and
they are elliptical because the stellar distribution is
3 Note that in a gas like air, scattering between molecules occurs
frequently, which drives the velocity distribution of the molecules
towards an isotropic Maxwellian, i.e., the thermal, distribution.