and Cosmology

6. Clusters and Groups of Galaxies
256
Fig. 6.25. Left: Chandra
image of a 6 ′ × 6 ′ -field
with two clusters of galaxies
at high redshift. Right:
a2 ′ × 2 ′ -field centered on
one of the clusters presented
on the left (RX
J0849+4452), in B, I, and
K, overlaid with the X-ray
brightness contours
clusters for which good measurements of the brightness
profile and the X-ray temperature are available.
From the luminosity function of X-ray clusters,
a mass function can be constructed, using the relation
between L X and the cluster mass that will be
discussed in the following section. As we will explain
in more detail in Sect. 8.2, this cluster mass function is
an important probe for cosmological parameters.
6.4 Scaling Relations
for Clusters of Galaxies
Our examination of galaxies revealed the existence of
various scaling relations, for example the Tully–Fisher
relation. These have proven to be very useful not only for
the distance determination of galaxies, but also because
any successful model of galaxy evolution needs be able
to explain these empirical scaling relations. Therefore,
it is of great interest to examine whether clusters of
galaxies also fulfill any such scaling relation. As we
will see, the X-ray properties of clusters play a central
role in this.
6.4.1 Mass–Temperature Relation
It is expected that the larger the spatial extent, velocity
dispersion of galaxies, temperature of the X-ray gas,
and luminosity of a cluster are, the more massive it is.
In fact, from theoretical considerations one can deduce
the existence of relations between these parameters. The
X-ray temperature T specifies the thermal energy per
gas particle, which should be proportional to the binding
energy for a cluster in virial equilibrium,
T ∝ M r .
Since this relation is based on the virial theorem,
r should be chosen to be the radius within which the
matter of the cluster is virialized. This value for r is
called the virial radius r vir . From theoretical considerations
of cluster formation (see Chap. 7), one finds that
the virial radius is defined such that within a sphere of
radius r vir , the average mass density of the cluster is
about Δ c ≈ 200 times as high as the critical density ρ cr
of the Universe. The mass within r vir is called the virial
mass M vir which is, according to this definition,
M vir = 4π 3 Δ c ρ cr r 3 vir . (6.47)
Combining the two above relations, one obtains
T ∝ M vir
r vir
∝ r 2 vir ∝ M2/3 vir . (6.48)
This relation can now be tested on observations by using
a sample of galaxy clusters with known temperature
and with mass determined by the methods discussed in
Sect. 6.3.2. An example of this is displayed in Fig. 6.27,
in which the mass is plotted versus temperature for clusters
from the extended HIFLUGCS sample. Since it is