= Solution
The hot intracluster gas emits X-rays, which constrain its temperature and density profile. If its pressure is predominantly thermal and it is in <hydrostatic equilibrium>,
$$
\frac{dP}{dr}=-\rho_g\frac{GM(<r)}{r^2},\qquad P=\frac{\rho_g k_BT}{\mu m_p}.
$$
Eliminating the pressure gives
$$
\boxed{M(<r)=-\frac{k_BT(r)r}{G\mu m_p}
\left(\frac{d\log\rho_g}{d\log r}+\frac{d\log T}{d\log r}\right).}
$$
Here $\mu m_p$ is the mean mass per gas particle. A gas at the observed cluster temperatures would not remain confined without a deep <gravitational potential>; the inferred total mass exceeds the gas and stellar mass together. Nonthermal pressure, mergers, gas clumping and temperature-profile errors can bias this estimate, but do not turn X-ray luminosity itself into a direct measurement of total mass.
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