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

Let $R_1$ be the core radius and $R_p$ the planetary radius. For the two-layer constant-density planet,and the mantle volume givesThe enclosed mass profile is

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

At a mantle radius $R_1\le r\le R_p$, the same integration givesInside the core, $M(r)=4\pi {\rho }_1{r}^3/3$, so matching to $P_b$ givesThe central pressure is therefore

For a geometrically thin isothermal ideal-gas atmosphere, set $g=G{M}_p/R_p^2$ andHydrostatic balance gives $P(z)=P_0{e}^{-z/H}$. The observed transit photosphere at $P_{\mathrm{phot}}$ is consequently at the isothermal pressure-level transit radiusThis assumes $P_{\mathrm{phot}}<P_0$, constant $T_0$, composition, and gravity, and an opacity that selects the stated pressure.

Hydrostatic balance makes the atmospheric column mass $(P_0-P_{\mathrm{top}})/g$. Multiplying by the surface area gives the mass of a thin hydrostatic atmospherewhere the second expression neglects the top pressure and atmospheric self-gravity.