Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2015 / S2 / 3

The virial theorem for a stationary isolated system with inverse-square gravity iswhere $K$ is total kinetic energy in the system's centre-of-mass frame and $W$ is its negative gravitational potential energy. The brackets denote a suitable time or ensemble average. Surface pressure, rapid evolution or mass fluxes add terms, so equilibrium and an appropriate boundary are substantive assumptions.

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.

Nonbaryonic dark matter can provide the additional gravitating mass. Gravitational lensing independently maps cluster mass, while the cosmic microwave background and Big Bang nucleosynthesis constrain the cosmic matter and baryon budgets. The separation of lensing mass from strongly interacting X-ray gas in merging clusters is evidence that much of the gravitating component is not that hot gas.

Uncounted baryonic matter, including the observed hot gas, faint stars and compact remnants, certainly adds to the stellar estimate. X-ray emission directly establishes a large gas reservoir. However, baryon-budget constraints and microlensing constraints on compact objects prevent arbitrarily filling the entire discrepancy with ordinary hidden objects.

Modified gravity could change the inference from a measured acceleration to a Newtonian mass. Such a theory must also reproduce the observed lensing and cosmological evidence; explaining some rotation curves alone is insufficient, and cluster mass discrepancies remain a demanding test. Departures from equilibrium and projection can bias individual mass estimates, but would have to explain the independent observations systematically rather than merely modify one virial prefactor.