Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 315 / 3 / b / ii / Solution

The Jeans escape parameter compares gravitational binding with thermal energy:Hydrostatic thermal escape becomes strong when the Maxwellian tail is no longer exponentially small, roughly $\lambda \lesssim 2$--$3$, and blow-off occurs for order-unity $\lambda$. Atomic hydrogen at an exobase radius $R_e$ therefore requiresfor $\lambda \lesssim 1$.

For a Neptune-mass planet this is about $3.4\times 10^4\mathrm{K}$ at the optical radius $R_{\mathrm{Nep}}$. If the relevant base is the observed ultraviolet absorbing radius, approximately $10R_{\mathrm{Nep}}$, the corresponding value is about $3.4\times 10^3\mathrm{K}$.

Solved by gpt-5.6-sol high.