and Cosmology

6.2 Galaxies in Clusters and Groups
rem is still justified if the main fraction of mass is not
contained in galaxies. The derivation remains valid in
this form as long as the spatial distribution of galaxies
follows the total mass distribution. The dynamical mass
determination can be affected by an anisotropic velocity
distribution of the cluster galaxies and by the possibly
non-spherical cluster mass distribution. In both cases,
projection effects, which are dealt with relatively easily
in the spherically-symmetric case, obviously become
more complicated. This is also one of the reasons for
the necessity to consider alternative methods of mass
determination.
Two-body collisions of galaxies in clusters are of
no importance dynamically, as is easily seen from the
corresponding relaxation time-scale (3.3),
N
t relax = t cross
ln N ,
which is much larger than the age of the Universe. The
motion of galaxies is therefore governed by the collective
gravitational potential of the cluster. The velocity
dispersion is approximately the same for the different
types of galaxies, and also only a weak tendency
exists for a dependence of σ v on galaxy luminosity, restricted
to the brightest ones (see below in Sect. 6.2.9).
From this, we conclude that the galaxies in a cluster
are not “thermalized” because this would mean that
they all have the same mean kinetic energy, implying
σ v ∝ m −1/2 . Furthermore, the independence of σ v from
L implies that collisions of galaxies with each other are
not dynamically relevant; rather, the velocity distribution
of galaxies is defined by collective processes during
cluster formation.
Violent Relaxation. One of the most important of the
aforementioned processes is known as violent relaxation.
This process very quickly establishes a virial
equilibrium in the course of the gravitational collapse
of a mass concentration. The reason for it are the
small-scale density inhomogeneities within the collapsing
matter distribution which generate, via Poisson’s
equation, corresponding fluctuations in the gravitational
field. These then scatter the infalling particles and, by
this, the density inhomogeneities are further amplified.
The fluctuations of the gravitational field act on the
matter like scattering centers. In addition, these field
fluctuations change over time, yielding an effective exchange
of energy between the particles. In a statistical
average, all galaxies obtain the same velocity distribution
by this process. As confirmed by numerical
simulations, this process takes place on a time-scale
of t cross , i.e., roughly as quickly as the collapse itself.
Dynamical Friction. Another important process for the
dynamics of galaxies in a cluster is dynamical friction.
The simplest picture of dynamical friction is obtained
by considering the following. If a massive particle of
mass m moves through a statistically homogeneous distribution
of massive particles, the gravitational force on
this particle vanishes due to homogeneity. But since the
particle itself has a mass, it will attract other massive
particles and thus cause the distribution to become inhomogeneous.
As the particle moves, the surrounding
“background” particles will react to its gravitational
field and slowly start moving towards the direction
of the particle trajectory. Due to the inertia of matter,
the resulting density inhomogeneity will be such
that an overdensity of mass will be established along
the track of the particle, where the density will be
higher on the side opposite to the direction of motion
(thus, behind the particle) than in the forward
direction (see Fig. 6.9). By this process, a gravitational
field will form that causes an acceleration of
the particle against the direction of motion, so that the
particle will be slowed down. Because this “polarization”
of the medium is caused by the gravity of the
particle, which is proportional to its mass, the deceleration
will also be proportional to m. Furthermore,
a fast-moving particle will cause less polarization in the
medium than a slow-moving one because each mass
element in the medium is experiencing the gravitational
attraction of the particle for a shorter time, thus
the medium becomes less polarized. In addition, the
particle is on average farther away from the density
accumulation on its backward track, and thus will experience
a smaller acceleration if it is faster. Combining
these arguments, one obtains for the dependence of this
dynamical friction
dv
dt ∝−m ρ v , (6.29)
|v| 3
where ρ is the mass density in the medium. Applied
to clusters of galaxies, this means that the most mas-
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