Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 315 / 1 / b / Solution

At secondary eclipse the removed monochromatic planetary flux is $\pi B_{\lambda }(T_p)R_p^2/d^2$, while the stellar flux is $\pi B_{\lambda }(T_s)R_s^2/d^2$. The thermal exoplanet secondary eclipse depth is thereforeA hot Jupiter has tiny visible thermal contrast, rapidly improving contrast toward the infrared, and molecular absorption or emission features superposed on its continuum. Reflected light may add a visible component. In the Rayleigh-Jeans law limit, $B_{\lambda }(T)\propto T/{\lambda }^4$, soa wavelength-independent asymptote for ideal blackbodies.