**5.5.4. Empirical gas distributions derived by surface brightness
deconvolution**

The gas distributions in clusters can be derived directly from
observations of the X-ray surface brightness of the cluster, if the
shape of the cluster is known and if the X-ray observations are
sufficiently detailed and accurate. This method of
analysis also leads to a very promising method for determining cluster
masses (Section 5.5.5).
The X-ray surface brightness at a photon frequency
and at a projected distance
*b* from the center of a spherical cluster is

(5.80) |

where _{} is the X-ray
emissivity. This Abel integral can be inverted to give
the emissivity as a function of radius,

(5.81) |

Because of the quantized nature of the observations of the X-ray surface brightness (photon counts per solid angle) and the sensitivity of integral deconvolutions to noise in the data, the X-ray surface brightness data are often smoothed, either by fitting a smooth function to the observations or by applying these equations to the surface brightness averaged in rings about the cluster center.

Now, the emissivity is given by equation (5.19) and depends on the elemental
abundances, the density of the gas, and the gas temperature. Thus the
distribution of these three properties in the cluster could be
determined from observations of the X-ray surface brightness
*I*_{}(*b*).
Basically, the continuum
emission is due mainly to free-free emission (equation 5.11) and is
relatively insensitive to the heavy element abundances, while the line
emission measures these abundances. For a given set of abundances, the
emissivity can be written as

(5.82) |

where _{}(*T*_{g})
is *n*_{p} / *n*_{e} times the sum on the
right side of equation (5.19).
Thus _{}(*r*) can be found from
observations of the X-ray surface brightness as a
function of photon frequency, and the frequency dependence and magnitude
of _{}
give the local gas temperature and density.

Unfortunately, observations of
*I*_{}(*b*)
are really not available for clusters.
These observations require an instrument with good spatial *and*
spectral resolution. Most of the X-ray spectra of clusters have been
taken with instruments having very poor spatial resolution (comparable
to the size of the cluster;
Section 4.3.1).
High spatial resolution observations of clusters have
primarily come from the *Einstein* X-ray observatory
(Section 4.4).
The imaging instruments on *Einstein* had only limited spectral
resolution. Moreover, the optics in *Einstein* could only focus
soft X-rays (*h*
4 keV). At
typical gas temperatures in clusters (*kT*_{g}
6 keV), most of the
X-ray emissivity is due to
thermal bremsstrahlung, and the emissivity is nearly independent of
frequency for *h*
*kT*_{g}. Thus the *Einstein* surface brightness
distributions cannot be used directly to determine the local
temperature. If the limited spectral resolution of
the *Einstein* imagers is ignored, their observations provide
<*I*_{x}(*b*)>, the surface
brightness averaged over the sensitivity of the detector as a function
of photon frequency.

From <*I*_{x}(*b*)>, the sensitivity averaged
emissivity

(5.83) |

can be found from equation (5.81). Unfortunately, even if the elemental
abundances are assumed to be known, this equation provides only one
quantity at each radius, and it is impossible to determine both
*n*_{e} and *T*_{g}.

In many analyses of X-ray cluster observations, a second equation for the density and temperature has been provided by assuming that the intracluster gas is hydrostatic and that the cluster potential is known. The hydrostatic equation in a spherical cluster (5.56) can be rewritten as

(5.84) |

where *m*(*r*) is the total cluster mass and the mean atomic
weight *µ*
0.63 is
assumed to be independent of radius. Combining this with equation (5.83)
gives

(5.85) |

This is an ordinary differential equation for
*T*_{g}(*r*), which can be integrated
given a boundary condition. This has been taken to be the central
temperature
(White and Silk, 1980)
or the intergalactic pressure
(Fabian *et al.*,
1981a).
Given *T*_{g}(*r*), equation (5.83) gives the density
profile.

Various versions of this method have been used to determine gas
distributions in a large number of clusters using data from
*Einstein*
(White and Silk, 1980;
Fabricant *et al.*,
1980;
Fabian *et al.*,
1981;
Nulsen *et al.*,
1982;
Fabricant and
Gorenstein, 1983;
Canizares *et al.*,
1983;
Stewart *et al.*,
1984a,
b).
In some cases, spectral information from *Einstein* or
low spatial resolution spectra have been used to further constrain the
temperature profiles or to determine the form of the cluster potential
necessary for a consistent fit
(Section 5.5.5). These analyses have
provided information on the mass distribution in clusters and central
galaxies (M87, in
particular), the gas distributions in clusters, and the prevalence of
cooling accretion flows in clusters. These topics will be discussed in
more detail later.

Gas temperature and density profiles could be derived more directly if high spatial and spectral resolution data at photon energies up to 10 keV were available. The proposed AXAF satellite will have these capabilities (Chapter 6).