and Cosmology

6. Clusters and Groups of Galaxies
232
may be used. A forth parameter is a characteristic scale
of a cluster, often taken to be the core radius r c ,defined
such that at R = r c , the projected density has
decreased to half the central value, N(r c ) = N 0 /2. Finally,
one parameter is needed to describe “where the
cluster ends”; the Abell radius is a first approximation
for such a parameter.
Parametrized cluster models can be divided into those
which are physically motivated, and those which are
of a purely mathematical nature. One example for the
latter is the de Vaucouleurs profile which is not derived
from dynamical models. Next, we will consider
a class of distributions that are based on a dynamical
model.
Isothermal Distributions. These models are based on
the assumption that the velocity distribution of the
massive particles (this may be both galaxies in the
cluster or dark matter particles) of a cluster is locally
described by a Maxwell distribution, i.e., they
are thermalized. As shown from spectroscopic analyses
of the distribution of the radial velocities of cluster
galaxies, this is not a bad assumption. Assuming, in
addition, that the mass profile of the cluster follows
that of the galaxies (or vice versa), and that the temperature
(or equivalently the velocity dispersion) of the
distribution does not depend on the radius (so that one
has an isothermal distribution of galaxies), then one
obtains a one-parameter set of models, the so-called
isothermal spheres. These can be described physically
as follows.
In dynamical equilibrium, the pressure gradient must
be equal to the gravitational acceleration, so that
dP
dr
=−ρ
GM(r)
r 2 , (6.8)
where ρ(r) denotes the density of the distribution, e.g.,
the density of galaxies. By ρ(r) = 〈m〉 n(r), this mass
density is related to the number density n(r), where 〈m〉
is the average particle mass. M(r) = 4π ∫ r
0 dr′ r ′2 ρ(r ′ )
is the mass of the cluster within a radius r. By
differentiation of (6.8), we obtain
(
d r
2
dP
dr ρ dr
)
+ 4πGr 2 ρ = 0 . (6.9)
The relation between pressure and density is P = nk B T.
On the other hand, the temperature is related to the
velocity dispersion of the particles,
3
2 k BT = 〈m〉 〈
v
2 〉 , (6.10)
2
where 〈 v 2〉 is the mean squared velocity, i.e., the velocity
dispersion, provided the average velocity vector is set
to zero. The latter assumption means that the cluster
does not rotate, or contract or expand. If T (or 〈 v 2〉 )is
independent of r, then
dP
dr = k 〈
BT dρ
〉
v
2
〈m〉 dr = dρ
3 dr = σ v
2 dρ
dr , (6.11)
where σv
2 is the one-dimensional velocity dispersion,
e.g., the velocity dispersion along the line-of-sight,
which can be measured from the redshift of the cluster
galaxies. If the velocity distribution corresponds to an
isotropic (Maxwell) distribution, the one-dimensional
velocity dispersion is exactly 1/3 times the threedimensional
velocity dispersion, because of 〈 v 2〉 = σx 2 +
σy 2 + σ z 2,or
〈 〉
v
σv 2 2 = 3 . (6.12)
With (6.9), it then follows that
(
d σ
2
v r 2 )
dρ
+ 4πGr 2 ρ = 0 . (6.13)
dr ρ dr
Singular Isothermal Sphere. In general, the differential
equation (6.13) for ρ(r) cannot be solved
analytically. Physically reasonable boundary conditions
are ρ(0) = ρ 0 , the central density, and (dρ/dr) |r=0 = 0,
for the density profile to be flat at the center. One particular
analytical solution of the differential equation exists,
however: By substitution, we can easily show that
ρ(r) =
σ 2 v
2πGr 2 (6.14)
solves (6.13). This density distribution is called singular
isothermal sphere; we have encountered it before, in
the discussion of gravitational lens models in Sect. 3.8.2.
This distribution has a diverging density as r → 0and
an infinite total mass M(r) ∝ r. It is remarkable that this