Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 315 / 1 / b / Solution

At 10 parsecs, $0.01$ arcseconds corresponds to $a=0.10$ astronomical units. Take a solar-radius star with $T_{\star }=5772\mathrm{K}$, zero Bond albedo, full day-night heat redistribution, negligible internal heating, and a hydrogen-helium atmosphere with mean particle mass $2.3m_H$. The planetary equilibrium temperature isFor Jupiter mass and radius, $g\simeq 24.8\mathrm{m}{\mathrm{s}}^{-2}$. Its atmospheric scale height isAssume a strong band spans $N_H=5$ atmospheric scale heights and saturates in the annulus model for transmission spectroscopy. The atmospheric spectral-feature amplitude isFor a five-standard-deviation detection, the uncertainty of the measured differential contrast must satisfyThis estimate scales linearly with the assumed feature height; one atmospheric scale height would require about $7.5\mathrm{ppm}$. The question gives no opacity or abundance from which to fix $N_H$, so an atmospheric detection threshold is necessarily assumption-dependent.

For thermal emission at $10\mu \mathrm{m}$, assume the planet and star emit as blackbodies. The thermal eclipse depth from the Planck law isThus the uncertainty required for a five-standard-deviation exoplanet secondary eclipse detection isThese are uncertainties of the final transit or eclipse contrasts, including the uncertainty of their reference levels. The distance affects photon counts and observing time but cancels from the flux ratios. An opaque exactly isothermal atmosphere emits a featureless blackbody spectrum: an eclipse detects its thermal light, while identifying atmospheric composition requires spectral features and a nonisothermal structure or other diagnostics.