Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 45 / 3 / g / Solution

Optical cluster observations give galaxy positions and line-of-sight redshifts. Subtract the mean recession to obtain the line-of-sight velocity dispersion ${\sigma }_{\mathrm{los}}$. If the tracers are in approximate virial equilibrium with isotropic velocities, $⟨v^2⟩=3{\sigma }_{\mathrm{los}}^2$ and $K\simeq 3M{\sigma }_{\mathrm{los}}^2/2$. Write the gravitational energy as $U_G=-\alpha G{M}^2/R$, with $\alpha$ determined by the density profile and radius convention. The virial theorem givesFor a uniform sphere $\alpha =3/5$, this becomes $5R{\sigma }_{\mathrm{los}}^2/G$. Projected separations are commonly used to estimate a gravitational radius rather than literally assume a uniform sphere. Interlopers, substructure, anisotropy, tracer bias and a nonzero surface term affect the estimate. The inferred mass is total gravitating mass, not merely the galaxies' stellar mass.

For X-ray observations, the hot intracluster gas emits thermal bremsstrahlung and atomic lines. Spectroscopy determines its temperature, and the surface brightness, after deprojection, determines a gas-density profile. Under spherical hydrostatic equilibrium with thermal gas pressure, $d{P}_g/dr=-{\rho }_{g}GM(<r)/r^2$ and $P_g={\rho }_g{k}_{B}T/(\mu m_p)$. ConsequentlyThe order-of-magnitude virial version is $k_{B}T\sim \mu m_{p}GM/R$. X-ray gas temperature is therefore a second measure of the depth of the gravitational potential. Nonthermal pressure, mergers, clumping and temperature gradients must be considered before equating the thermal estimate to the true mass.

For dense clusters the vacuum force is small compared with their internal gravity. If retained in the hydrostatic equation, it changes $GM/r^2$ to $GM/r^2-\Lambda r/3$ and adds $\Lambda r^3/(3G)$ to the mass formula above. The optical virial relation likewise uses the separate vacuum contribution derived in part (c). Comparing optical and X-ray estimates tests the dynamical assumptions and exposes a gravitating mass much larger than the luminous stellar component.