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

In the exoplanet transmission spectrum, the constant $a$ is the wavelength-independent reference transit depth, set mainly by the opaque planetary radius and any grey cloud deck. The Gaussian term can represent a resolved atomic or molecular absorption band centered at ${\lambda }_0$, with amplitude $b$ and width controlled by $\alpha$. The power-law term represents a continuum such as Rayleigh scattering or haze extinction; for Rayleigh scattering the opacity index is approximately $\gamma =-4$.

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Where the spectral components vanish, $a=(R_p/R_{\ast }{)}^2$, soFor extinction cross-section $\sigma \propto {\lambda }^{\gamma }$, the scattering slope of a transmission spectrum obeysSince $y=(R_p/R_{\ast }{)}^2$,After subtracting the Gaussian feature, the model gives $dy/d\log \lambda =\gamma c(\lambda /{\lambda }_1{)}^{\gamma }$. At $\lambda ={\lambda }_1$,or, with $g=G{M}_p/R_p^2$, $T=\bar{m}G{M}_{p}c{R}_{\ast }^2/(2k_B{R}_p^3)$. This estimates the mean isothermal terminator temperature under hydrostatic conditions.

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For a hot Jupiter transiting a Sun-sized star,a one-percent baseline transit. Taking $T\simeq 1500\mathrm{K}$, $g\simeq 20\mathrm{m}{\mathrm{s}}^{-2}$, and an ${\mathrm{H}}_2$-dominated mean molecular mass $\bar{m}\simeq 2.3m_H$ gives an atmospheric scale height near $270\mathrm{km}$. A strong band spanning five scale heights then hasor several hundred parts per million; very strong clear-atmosphere features can approach $10^{-3}$. The spectrum is therefore a roughly one-percent baseline with a localized Gaussian-sized band near ${\lambda }_0$ and a continuum rising toward short wavelengths when $\gamma <0$. Clouds reduce both $b$ and the observable power-law slope.

Solved by gpt-5.6-sol high.