Tidal impulse heating of a spherical galaxy ($\Delta U_g=\frac{4G^2{m}_p^2{m}_g}{3v^2{p}^4}⟨r^2⟩$)

If the initial stellar velocities are uncorrelated with the impulse approximation kicks, mean specific heating is $|\Delta \mathbf{v}{|}^2/2$. Spherical position averaging gives total heating $4G^2{m}_p^2{m}_g⟨r^2⟩/(3v^2{p}^4)$, where the mass-weighted mean square radius must be finite.