and Cosmology

6.3 X-Ray Radiation from Clusters of Galaxies
and is thus, for a cluster with T ∼ 10 8 K, significantly
shorter than the lifetime of the cluster, which can be
approximated roughly by the age of the Universe. Since
the sound-crossing time defines the time-scale on which
deviations from the pressure equilibrium are evened out,
the gas can be in hydrostatic equilibrium. In this case,
the equation
∇ P =−ρ g ∇Φ (6.34)
applies, with Φ denoting the gravitational potential.
Equation (6.34) describes how the gravitational force
is balanced by the pressure force. In the spherically
symmetric case in which all quantities depend only on
the radius r, we obtain
1 dP
ρ g dr =−dΦ dr =−GM(r) r 2 , (6.35)
where M(r) is the mass enclosed within radius r. Here,
M(r) is the total enclosed mass, i.e., not just the gas
mass, because the potential Φ is determined by the total
mass. By inserting P = nk B T = ρ g k B T/(μm p ) into
(6.35), we obtain
M(r) =− k BTr 2 ( dlnρg
Gμm p dr
+ dlnT )
dr
. (6.36)
This equation is of central importance for the X-ray astronomy
of galaxy clusters because it shows that we can
derive the mass profile M(r) from the radial profiles of
ρ g and T. Thus, if one can measure the density and temperature
profiles, the mass of the cluster, and hence the
total density, can be determined as a function of radius.
However, these measurements are not without difficulties.
ρ g (r) and T(r) need to be determined from the
X-ray luminosity and the spectral temperature, using
the bremsstrahlung emissivity (6.30). Obviously, they
can be observed only in projection in the form of the
surface brightness
I ν (R) = 2
∫ ∞
R
dr
ɛ ν (r) r
√
r2 − R 2 , (6.37)
from which the emissivity, and thus density and temperature,
need to be derived by de-projection. Furthermore,
the angular and energy resolution of X-ray telescopes
prior to XMM-Newton and Chandra were not high
enough to measure both ρ g (r) and T(r) with sufficient
accuracy, except for the nearest clusters. For this
reason, the mass determination is often performed by
employing additional, simplifying assumptions.
Isothermal Gas Distribution. From the radial profile
of I(R), ɛ(r) can be derived by inversion of (6.37).
Since the spectral bremsstrahlung emissivity depends
only weakly on T for h P ν ≪ k B T, due to (6.30), the
radial profile of the gas density ρ g can be derived
from ɛ(r). The X-ray satellite ROSAT was sensitive
to radiation of 0.1keV E 2.4 keV, so that the X-ray
photons detected by it are typically from the regime
where h P ν ≪ k B T.
Assuming that the gas temperature is spatially constant,
T(r) = T g , (6.36) simplifies, and the mass profile
of the cluster can be determined from the density profile
of the gas.
The β-Model. A commonly used method consists of fitting
the X-ray data by a so-called β-model. This model
is based on the assumption that the density profile of
the total matter (dark and luminous) is described by an
isothermal distribution, i.e., it is assumed that the temperature
of the gas is independent of radius, and at the
same time that the mass distribution in the cluster is described
by the isothermal model that has been discussed
in Sect. 6.2.4. With (6.8) and (6.11), we then obtain for
the total density ρ(r)
dlnρ
dr
=− 1
σ 2 v
GM
r 2 . (6.38)
On the other hand, in the isothermal case (6.36) reduces
to
dlnρ g
dr
=− μm p
k B T g
GM
r 2 . (6.39)
The comparison of (6.38) and (6.39) then shows that
dlnρ g /dr ∝ dlnρ/dr,or
ρ g (r) ∝ [ρ(r)] β with β := μm pσ 2 v
k B T g
(6.40)
must apply; thus the gas density follows the total density
to some power. Here, the index β depends on the ratio of
the dynamical temperature, measured by σ v , and the gas
temperature. Now, using the King approximation for an
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