Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 315 / 4

The day-night contrast is controlled mainly by the ratio of radiative cooling time to horizontal advection and wave-adjustment times. A useful scaling isStronger irradiation raises $T$ and sharply shortens $t_{\mathrm{rad}}$, allowing the dayside to reradiate before circulation reaches the nightside; the contrast therefore generally increases with irradiation. In ultra-hot atmospheres, hydrogen dissociation and recombination can transport latent heat and partly reduce it, while magnetic drag can weaken winds and increase it.

At low pressure, small atmospheric mass and short $t_{\mathrm{rad}}$ produce a large contrast. At greater pressure, the radiative time grows, waves and winds redistribute heat more effectively, and the contrast decreases. The observed contrast is wavelength dependent because each wavelength probes a different pressure.

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Without scattering and in local thermodynamic equilibrium, the radiative transfer equation has the emergent solutionThe Eddington-Barbier relation gives the useful approximationA molecular band has larger opacity than its neighboring continuum and therefore reaches optical depth unity at lower pressure.

If temperature decreases outward, the band samples cooler gas and appears in absorption. If the atmosphere is isothermal, both levels have the same source function and the feature disappears. If an atmospheric thermal inversion makes the upper layer hotter, the band appears in emission. For a weak separation of formation pressures,which explicitly shows that feature sign and amplitude measure the vertical temperature gradient.

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Three important atmospheric chemical processes are:

Condensation and rainout provide another major process, removing species such as silicates or water from the gas phase where their saturation curves are crossed.

A solid or liquid surface supplies reservoirs and sinks through weathering, dissolution, volcanism, deposition, and possible biological cycling; it also caps the atmospheric mass. A surface-free sub-Neptune instead has a deep envelope merging continuously into high-pressure volatile or hydrogen-rich layers, with composition governed more by bulk elemental inventory, mixing, and deep thermochemistry.

At Earth-like equilibrium temperature, a sub-Neptune can retain methane, ammonia, and water in a hydrogen-rich atmosphere, subject to photochemistry and condensation. A hot Jupiter is hotter, usually hydrogen dominated, and more strongly driven toward CO, water, and nitrogen at depth; at extreme irradiation molecules dissociate and atomic, ionic, and negative-hydrogen opacity become important.

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The mass fractions of the $10M_{\oplus }$ planet areOn an exoplanet bulk-composition ternary diagram with iron, silicate, and hydrogen vertices, it lies in the interior on the line of equal iron and silicate fractions, one fifth of the way from the iron-silicate edge toward the hydrogen vertex.

Neglecting its atmosphere and minor volatile reservoirs, Earth lies on the iron-silicate edge at approximately

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Consider a ray bundle from projected source area $d{A}_s\cos {\theta }_s$ into solid angle $d{\Omega }_s$. In empty space its power is conserved, while geometric propagation preserves the étendueSince specific intensity is power divided by this étendue and by frequency interval,In a static medium with no redshift, frequency is unchanged and $I_{\nu }$ itself is independent of distance. The apparent solid angle shrinks as distance squared while the physical beam area grows by the same factor.

For a full plane-parallel angular field linear in direction cosine,the radiation-field moments areand $F=4\pi H=4\pi I_1/3$. At a surface with no incoming intensity and the same law only for $0<\mu <1$,

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