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

A deep ultraviolet transit and asymmetric light curve are signatures of an extended, escaping atmosphere rather than the optical planetary disc. Neutral hydrogen and ionized or neutral metals can form a comet-like tail, while interaction with the stellar wind can create an asymmetric bow shock.

For $R_{\ast }=0.5R_{\odot }$ and transit depth $D=0.5$, the absorbing radius isabout ten Neptune radii. This scale directly demonstrates that the ultraviolet absorber is gravitationally extended.

Solved by gpt-5.6-sol high.

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.

For $\lambda \gtrsim 10$--$20$, escape is well described by the dilute high-velocity tail of a nearly hydrostatic exosphere. For $\lambda \lesssim 2$--$3$, collisions couple the escaping gas into hydrodynamic atmospheric escape; the intermediate regime requires a kinetic or transonic calculation.

The hydrostatic-tail estimate follows by integrating a Maxwell distribution over outward velocities exceeding escape speed. The Jeans escape flux isThereforeWhen $\lambda \le 1$, this is of order the free thermal supply $4\pi R_e^2{\rho }_e{c}_s$; quantitatively the flow is hydrodynamic and its conserved mass-loss rate is $\dot{M}=4\pi r^2\rho u$ through the transonic wind.

Solved by gpt-5.6-sol high.

Three nonthermal atmospheric-escape mechanisms are stellar-wind ion pickup, sputtering of neutrals by energetic incident particles, and photochemical escape in exothermic reactions. Charge exchange, polar-wind acceleration, and impact erosion provide further examples.

Solved by gpt-5.6-sol high.