and Cosmology

6. Clusters and Groups of Galaxies
248
isothermal mass distribution – see (6.16) – as a model
for the mass distribution, we obtain
ρ g (r) = ρ g0
[1 +
( r
r c
) 2
] −3β/2
, (6.41)
where ρ g0 is the central gas density. The brightness
profile of the X-ray emission in this model is then,
according to (6.37),
I(R) ∝
[
1 +
( R
r c
) 2
] −3β+1/2
. (6.42)
The X-ray emission of many clusters is well described
by this profile, 4 yielding values for r c of 0.1 to
0.3h −1 Mpc and a value for the index β = β fit ≈ 0.65.
Alternatively, β can be measured, with the definition
given in (6.40), from the gas temperature T g and the
velocity dispersion of the galaxies σ v , which yields
typical values of β = β spec ≈ 1. Such a value would
also be expected if the mass and gas distributions were
both isothermal. In this case, they should have the same
temperature, which was presumably determined by the
formation of the cluster.
The fact that the two values for β determined above
differ from each other (the so-called β-discrepancy) is
as yet not well understood. The measured values for β fit
often depend on the angular range over which the brightness
profile is fitted; the larger this range, the larger β fit
becomes, and thus the smaller the discrepancy. Furthermore,
temperature measurements of clusters are often
not very accurate because it is the emission-weighted
temperature which is measured, which is, due to the
quadratic dependence of the emissivity on ρ g , dominated
by the regions with the highest gas density. The
fact that the innermost regions of clusters where the
gas density is highest tend to have a temperature below
the bulk temperature of the cluster may lead to
an underestimation of “the” cluster temperature. In addition,
the near independence of the spectral form of
ɛ ff
ν from T for h Pν ≪ k B T renders the measurement of
4 We point out that the pair of equations (6.41) and (6.42) is valid
independently of the validity of the assumptions from which (6.41)
was obtained. If the observed X-ray emission is very well described
by (6.42), the gas density profile (6.41) can be obtained from it,
independently of the validity of the assumptions made before.
T difficult. Only with Chandra and XMM-Newton can
the X-ray emission also be mapped at energies of up
to E 10 keV, which results in considerably improved
temperature measurements.
Such investigations have revealed that the gas is not
really isothermal. Typically, the temperature decreases
towards the center and towards the edge, while it is
rather constant over a larger range at intermediate radii.
Many clusters are found, however, in which the temperature
distribution is by no means radially symmetric, but
shows distinct substructure. Finally, as another possible
explanation for the β-discrepancy, it should be mentioned
that the velocity distribution of those galaxies
from which σ v is measured may be anisotropic.
Besides all the uncertainty as to the validity of the
β-model, we also need to mention that numerical simulations
of galaxy clusters, which take dark matter and
gas into account, have repeatedly come to the conclusion
that the mass determination of clusters, utilizing
the β-model, should achieve an accuracy of better than
∼ 20%, although different gas dynamical simulations
have arrived at distinctly different results.
Dark Matter in Clusters from X-Ray Observations.
Based on measurements of their X-ray emission, a mass
estimate can be performed for galaxy clusters. It is
found, in agreement with the dynamical method, that
clusters contain much more mass than is visible in galaxies.
The total mass of the intergalactic medium is
clearly too low to account for the missing mass; its gas
mass is only ∼ 15% of the total mass of a cluster.
The mass of clusters of galaxies consists of ∼ 3%
contribution from stars in galaxies and ∼ 15% from
intergalactic gas, whereas the remaining ∼ 80%
consists of dark matter which therefore dominates
the mass of the clusters.
6.3.3 Cooling Flows
In examining the intergalactic medium, we have assumed
hydrostatic equilibrium, but we have disregarded
the fact that the gas cools by its emission, thus it will
lose internal energy. For this reason, once established,
a hydrostatic equilibrium cannot be maintained over ar-