and Cosmology

6.2 Galaxies in Clusters and Groups
density distribution is just what is needed to explain the
flat rotation curves of galaxies at large radii.
Numerical solutions of (6.13) with the initial conditions
specified above (thus, with a flat core) reveal that
the central density and the core radius are related to each
other by
9σ 2 v
ρ 0 =
4πGrc
2
. (6.15)
Hence, these physical solutions of (6.13) avoid the infinite
density of the singular isothermal sphere. However,
these solutions also decrease outwards with ρ ∝ r −2 ,
so they have a diverging mass as well. The origin
of this mass divergence is easily understood because
these isothermal distributions are based on the assumption
that the velocity distribution is isothermal,
thus Maxwellian with a spatially constant temperature.
A Maxwell distribution has wings, hence it (formally)
contains particles with arbitrarily high velocities. Since
the distribution is assumed stationary, such particles
must not escape, so their velocity must be lower than the
escape velocity from the gravitational well of the cluster.
But for a Maxwell distribution this is only achievable
for an infinite total mass.
King Models. To remove the problem of the diverging
total mass, self-gravitating dynamical models with an
upper cut-off in the velocity distribution of their constituent
particles are introduced. These are called King
models and cannot be expressed analytically. However,
an analytical approximation exists for the central region
of these mass profiles,
ρ(r) = ρ 0
[1 +
( r
r c
) 2
] −3/2
. (6.16)
Using (6.7), we obtain from this the projected surface
mass density
Σ(R) = Σ 0
[1 +
( R
r c
) 2
] −1
with Σ 0 = 2ρ 0 r c .
(6.17)
The analytical fit (6.16) of the King profile also has
a diverging total mass, but this divergence is “only”
logarithmic.
These analytical models for the density distribution
of galaxies in clusters are only approximations,
of course, because the galaxy distribution in clusters is
often heavily structured. Furthermore, these dynamical
models are applicable to a galaxy distribution only if
the galaxy number density follows the matter density.
However, one finds that the distribution of galaxies in
a cluster often depends on the galaxy type. The fraction
of early-type galaxies (Es and S0s) is often largest near
the center. Therefore, one should consider the possibility
that the distribution of galaxies in a cluster may be
different from that of the total matter. A typical value
for the core radius is about r c ∼ 0.25h −1 Mpc.
6.2.5 Dynamical Mass of Clusters
The above argument relates the velocity distribution
of cluster galaxies to the mass profile of the cluster,
and from this we obtain physical models for the density
distribution. This implies the possibility of deriving
the mass, or the mass profile, respectively, of a cluster
from the observed velocities of cluster galaxies. We
will briefly present this method of mass determination
here. For this, we consider the dynamical time-scale of
clusters, defined as the time a typical galaxy needs to
traverse the cluster once,
t cross ∼ R A
σ v
∼ 1.5h −1 × 10 9 yr , (6.18)
where a (one-dimensional) velocity dispersion σ v ∼
1000 km/s was assumed. The dynamical time-scale is
shorter than the age of the Universe. One therefore concludes
that clusters of galaxies are gravitationally bound
systems. If this were not the case they would dissolve on
a timescale t cross . Since t cross ≪ t 0 one assumes a virial
equilibrium, hence that the virial theorem applies, so
that in a time-average sense,
2E kin + E pot = 0 , (6.19)
where
E kin = 1 ∑
m i vi 2 ; E pot =− 1 ∑
2
2
i
i̸= j
Gm i m j
r ij
(6.20)
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