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

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 $\gamma =-4$.

Solved by gpt-5.6-sol high.

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.

Solved by gpt-5.6-sol high.

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.

A deep ultraviolet transit and asymmetric light curve are signatures of an extended, escaping atmosphere rather than the optical planetary disc. Neutral hydrogen and ionized or neutral metals can form a comet-like tail, while interaction with the stellar wind can create an asymmetric bow shock.

For $R_{\ast }=0.5R_{\odot }$ and transit depth $D=0.5$, the absorbing radius isabout ten Neptune radii. This scale directly demonstrates that the ultraviolet absorber is gravitationally extended.

Solved by gpt-5.6-sol high.

The Jeans escape parameter compares gravitational binding with thermal energy:Hydrostatic thermal escape becomes strong when the Maxwellian tail is no longer exponentially small, roughly $\lambda \lesssim 2$--$3$, and blow-off occurs for order-unity $\lambda$. Atomic hydrogen at an exobase radius $R_e$ therefore requiresfor $\lambda \lesssim 1$.

For a Neptune-mass planet this is about $3.4\times 10^4\mathrm{K}$ at the optical radius $R_{\mathrm{Nep}}$. If the relevant base is the observed ultraviolet absorbing radius, approximately $10R_{\mathrm{Nep}}$, the corresponding value is about $3.4\times 10^3\mathrm{K}$.

Solved by gpt-5.6-sol high.

For $\lambda \gtrsim 10$--$20$, escape is well described by the dilute high-velocity tail of a nearly hydrostatic exosphere. For $\lambda \lesssim 2$--$3$, collisions couple the escaping gas into hydrodynamic atmospheric escape; the intermediate regime requires a kinetic or transonic calculation.

The hydrostatic-tail estimate follows by integrating a Maxwell distribution over outward velocities exceeding escape speed. The Jeans escape flux isThereforeWhen $\lambda \le 1$, this is of order the free thermal supply $4\pi R_e^2{\rho }_e{c}_s$; quantitatively the flow is hydrodynamic and its conserved mass-loss rate is $\dot{M}=4\pi r^2\rho u$ through the transonic wind.

Solved by gpt-5.6-sol high.

Three nonthermal atmospheric-escape mechanisms are stellar-wind ion pickup, sputtering of neutrals by energetic incident particles, and photochemical escape in exothermic reactions. Charge exchange, polar-wind acceleration, and impact erosion provide further examples.

Solved by gpt-5.6-sol high.

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.