Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 315 / 1 / a / Solution

Treat the morning and evening halves of the day-night terminator as independent, isothermal, hydrostatic atmospheres with the same reference radius $R_0$, reference pressure $P_0$, composition, gravity $g$, and extinction coefficient ${\kappa }_{\lambda }$. Their atmospheric scale heights areFor $H_i\ll R_0$, the slant optical depth of half $i$ at tangent altitude $z$ is approximatelyThe standard isothermal effective altitude is thereforeup to a wavelength-independent choice of reference radius. The two semicircular limbs add in projected area, so the exoplanet transmission spectrum isEquivalently, $R_{\mathrm{tr}}^2=[(R_0+z_m{)}^2+(R_0+z_e{)}^2]/2$. Differentiating with respect to $\log {\kappa }_{\lambda }$ givesThus a homogeneous retrieval measures, to leading order,provided the opacity and composition do not themselves differ between the two limbs.