= Cluster distance from Sunyaev-Zeldovich and X-ray signals
{title2=$d_A\propto(\Delta T)^2\theta/[S_XT_e^{3/2}(1+z)^4]$}
For a uniform spherical cluster of depth $\ell\propto\theta d_A$, <thermal bremsstrahlung> gives received bolometric flux $S_X\propto n_e^2T_e^{1/2}\ell^3/d_L^2$, while the <Rayleigh-Jeans law> limit of the <thermal Sunyaev-Zeldovich effect> gives $|\Delta T|\propto n_eT_e\ell$. Eliminating $n_e$ and using $d_L=(1+z)^2d_A$ yields the displayed <angular diameter distance>. A model for the density profile fixes its numerical coefficient. Clumping, nonsphericity and <temperature> structure can bias this distance because the two observables weight the gas differently.
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