Past exam of the formation of Structure in the Universe course of the University of Cambridge / 2015 / Paper 1 / 3 / b / Solution

Let $D$ be the separation of galaxy centres and $x$ a small displacement from the satellite centre along the line of centres. Expanding the central point-mass field gives the radial differential accelerationInside a uniform-density satellite, the restoring self-gravity isComparing the two at the same displacement yields the differential-gravity disruption limit of a uniform satellite:The satellite radius cancels because both accelerations are linear in $x$. This is the instantaneous tidal-gravity estimate requested by the given criterion, not a full rotating Roche calculation. Including centrifugal terms for a circular corotating orbit replaces the radial coefficient two by three, giving $D_{\min }\simeq [9M/(4\pi \rho ){]}^{1/3}$ while preserving the required scaling. An eccentric orbit requires checking the stronger tide near periapsis, and extended density profiles require a more detailed restoring-force calculation.