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

The peak near $0.5\mu \mathrm{m}$ can be reflected starlight carrying the stellar spectral shape, while the $1\mu \mathrm{m}$ peak can be thermal emission from a very hot, young, or strongly irradiated planet. Wien displacement law then suggests $T_s\simeq 5800\mathrm{K}$ and $T_p\simeq 2900\mathrm{K}$, so $T_p/T_s\simeq 1/2$.

Assume both objects are in the far-infrared Rayleigh-Jeans law regime. Fromone obtains $R_p/R_s\simeq \sqrt{2\times 10^{-3}}\simeq 0.045$. With $R_s=0.5R_{\odot }$ and $R_J\simeq 0.10R_{\odot }$,of order a Neptune radius. This estimate is sensitive to the peak interpretation and blackbody assumptions.

If the system transits, secondary-eclipse emission spectroscopy is especially effective because it separates planetary light from starlight and directly measures the favorable $10^{-3}$ infrared contrast. For a sufficiently wide orbit, direct-imaging spectroscopy is preferable because the hot planet is self-luminous and can be spatially separated from its star.