Suppose we find the optical depth \(\tau\) of CMBR photons passing through the
middle of a galaxy cluster from the Sunyaev-Zeldovich effect and we also
measure the X-ray flux \(f_{v}\) at frequency \(v\) from the hot gas in the
cluster. Assuming that the hot gas makes up a sphere of radius
\(R_{\mathrm{c}}\) with uniform electron density \(n_{\mathrm{e}}\) inside, we
obviously have \(\tau=2 \sigma_{\mathrm{T}} R_{\mathrm{c}} n_{\mathrm{e}}\) and
$$
f_{v}=\frac{\frac{4}{3} \pi R_{\mathrm{c}}^{3} \epsilon_{v}}{4 \pi D^{2}}
$$
where \(\epsilon_{v}\) is the emissivity per unit volume of the hot gas and \(D\)
is the distance of the galaxy cluster. Following (8.70), we can write
$$
\epsilon_{v}=\frac{A n_{\mathrm{e}}^{2}}{\sqrt{T}} e^{-h v /
\kappa_{\mathrm{B}} T}
$$
Now show that
$$
D=\frac{A \Delta \theta}{24 \sigma_{\mathrm{T}}^{2} \sqrt{T}} e^{-h v /
\kappa_{\mathrm{B}} T} \frac{\tau^{2}}{f_{v}}
$$
where \(\Delta \theta=2 R_{\mathrm{c}} / D\) is the observed angular size of the
X-ray emitting gas sphere. This expression for \(D\) is used to determine the
distances of galaxy clusters. [Note: If the galaxy cluster is at a large
redshift, then some corrections have to be applied to the above expression for
\(D .]\)
