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

Let ${\tau }_{\nu }$ increase downward and let $\mu >0$ be the outward direction cosine. Assume a plane-parallel, static, nonscattering atmosphere in local thermodynamic equilibrium, so the source function is $B_{\nu }[T(P)]$. Hydrostatic balance givesThe formal solution of the radiative transfer equation between $P_1>P_2$, corresponding to ${\tau }_1>{\tau }_2$, isFor an isothermal atmosphere, $B_{\nu }(T)$ is constant andIf the lower boundary is itself thermalized, $I_{\nu ,1}=B_{\nu }(T)$ and the upward intensity remains the blackbody value.

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.

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.