 Solution

For an isotropic velocity dispersion, $⟨v^2⟩=3{\sigma }_{\mathrm{los}}^2$. To specify the order-unity structure factor, approximate the gravitating system by a uniform sphere, whose gravitational potential energy is $W=-3G{M}^2/(5R)$. The virial theorem givesThe numerical value uses $R=1\mathrm{Mpc}$ and ${\sigma }_{\mathrm{los}}=900\mathrm{km}{\mathrm{s}}^{-1}$. Different mass profiles change the prefactor, rather than the scaling $R{\sigma }^2/G$.

For fully ionized intracluster gas, take mean particle mass $\mu m_p$ with $\mu \simeq 0.6$. In the same uniform-sphere virial estimate, replace ${\sigma }^2$ by $k_{B}T/(\mu m_p)$ to obtainAlternatively, hydrostatic equilibrium gives $M(<R)=-k_{B}TR(G\mu m_p{)}^{-1}(d\log {\rho }_g/d\log r+d\log T/d\log r)$. An isothermal gas with density proportional to $r^{-2}$ gives $1.3\times 10^{15}{M}_{\odot }$. The supplied radius and temperature do not specify the profile, so the meaningful inference is a mass of order $10^{15}{M}_{\odot }$, with the displayed model choices making the coefficient explicit. The high supplied temperature need not yield the same estimate as the galaxy dispersion in a crude single-profile calculation.

The stellar estimate is much smaller:It counts luminous stellar matter, rather than all matter in the cluster.