Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 65 / 4 / vi / Solution

The differential gravitational field of the Milky Way removes loosely bound outer material by tidal stripping, giving globular clusters and satellite dwarf galaxies a characteristic tidal radius. For a low-mass satellite of mass $m$ on a circular orbit of radius $R$ in a spherical host, the Jacobi tidal radius is $r_t\simeq R\{m/[(3-d\ln M_h/d\ln R)M_h(<R)]{\}}^{1/3}$; eccentric orbits often make the strongest restriction near pericentre. Tidal limitation explains the sharp outer falloff, although escaping stars can form extended tidal tails, so a fitted luminosity cutoff is not an exact boundary of all associated material.