Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 66 / 4 / a / i

For a relaxed cluster, galaxy motions act as tracers of its gravitational potential. The virial theorem gives a dynamical mass of orderwith a profile-dependent coefficient and a projection correction because the observed velocity dispersion is usually only the line-of-sight component. For an isotropic uniform-sphere estimate using its one-dimensional dispersion, $T=3M{\sigma }_v^2/2$ and $W=-3G{M}^2/(5R)$ give $M=5R{\sigma }_v^2/G$. Measured dispersions require much more gravitating mass than the luminous galaxies provide, hence large cluster mass-to-light ratios. Interlopers, orbital anisotropy and departures from equilibrium are systematic uncertainties, so the independent gas and lensing measurements matter.