Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 321 / 3 / a / Solution

Use a local Cartesian frame rotating at $\Omega =\sqrt{GM/r_0^3}$ about a circular orbit at $r_0$, with $x=r-r_0\ll r_0$, and neglect curvature beyond leading order. The radial gravitational-plus-centrifugal potential isIts first derivative vanishes at $r_0$ and its second derivative there is $-3{\Omega }^2$. Dropping a constant gives the shearing-sheet tidal potential