Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 315 / 2 / b / Solution

Assume Newtonian spherical hydrostatic equilibrium, neglect rotation and thermal density changes, and require continuous pressure at the layer boundary. Writeso the mantle mass profile is $M(r)=C+D{r}^3$. Integrating $dP/dr=-G{\rho }_{2}M(r)/r^2$ inward from $P(R_p)=P_0$ gives the boundary pressure