Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 315 / 3 / c

Three useful target classes are:

These examples are observing targets rather than assertions that any is inhabited. An Earth twin crossing a Sun twin has atmospheric transmission features near one part per million, a one-year orbital period, and only one transit per year. Stellar photon noise, instrumental stability, clouds, and the much brighter stellar spectrum make molecular detection exceptionally difficult for JWST.

Solved by gpt-5.6-sol high.

In transmission, the leading scaling is the atmospheric spectral-feature amplitudeObservability therefore improves for a small bright host star, a large planet, low surface gravity, high atmospheric temperature, low mean molecular mass, large molecular abundance, and cloud-free limbs. Stellar activity and heterogeneity, refraction, aerosols, limited transit count, detector noise, and spectral overlap reduce it.

In emission, the contrast scales approximately asIt depends on dayside temperature, vertical temperature gradient, molecular opacity, heat redistribution, orbital geometry, stellar brightness, and instrumental background.

Solved by gpt-5.6-sol high.

Assume a feature spans $N_H=5$ atmospheric scale heights.

For an Earth twin around the Sun, $R_p=R_{\oplus }$, $R_{\ast }=R_{\odot }$, $H\simeq 8\mathrm{km}$, givingFor a $1.7R_{\oplus }$ super-Earth with a heavy atmosphere around a $0.2R_{\odot }$ M dwarf, take $H\simeq 6\mathrm{km}$, givingFor a $2.6R_{\oplus }$ hydrogen-rich sub-Neptune around a $0.4R_{\odot }$ star, take $T\simeq 300\mathrm{K}$, $g\simeq 12\mathrm{m}{\mathrm{s}}^{-2}$, $\bar{m}=2.3m_H$, and hence $H\simeq 90\mathrm{km}$. ThenThe hydrogen-rich sub-Neptune is the most observable with JWST because its low molecular mass and large radius produce the largest transmission annulus. Clouds can reverse this ranking in a particular system.

Solved by gpt-5.6-sol high.