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Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 315 / 4 / g / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-06

One approach uses exoplanet transmission spectra. Three aerosol signatures are:

Muted molecular-band amplitudes in the near-infrared, for example weak water structure around $1.4 μ m$ : an optically thick exoplanet cloud deck raises the continuum and hides part of the gas column.
A nearly grey transit-radius continuum across optical/near-infrared wavelengths, roughly $0.5$ -- $2 μ m$ , when particles are large relative to the wavelengths. Suppressed optical atomic-line wings provide a related indication that deep high-pressure layers are hidden.
A radius increasing toward blue/ultraviolet wavelengths, roughly $0.3$ -- $1 μ m$ , for small-particle scattering of light. In a simple Rayleigh scattering limit, $σ \propto λ^{- 4}$ , and the scattering slope of a transmission spectrum is $d R_{p} / d lo g λ = - 4 H$ .

A second approach uses the dayside spectrum and brightness obtained with exoplanet secondary eclipses, optionally supported by orbital phase variations. Its three useful aerosol signatures are:

Enhanced reflected-light geometric albedo or characteristic colour in the visible, roughly $0.4$ -- $0.9 μ m$ , for a scattering exoplanet cloud. Absorbing atmospheric haze can instead lower the reflected flux, so high albedo is not a universal aerosol rule.
Weak near-/mid-infrared gas features and a cloud-set continuum brightness temperature, roughly $1$ -- $10 μ m$ , when an opaque cloud moves the thermal photosphere to a different pressure. Changes in the exoplanet thermal phase curve can support a spatially patchy cloud interpretation, though circulation also affects it.
Broad condensate-dependent infrared spectral structure, for example a silicate resonance around $10 μ m$ , if particles are sufficiently small and the optical-depth/temperature geometry permits the feature. Its presence and sign depend on particle composition, size and vertical thermal structure.

Transmission measures aerosol extinction along the limb; eclipse/reflection measurements test dayside scattering and emission. No single flattened feature proves clouds: small atmospheric scale height, low molecular abundances, stellar heterogeneity and thermal-gradient degeneracies must be considered. Joint wavelength coverage and geometries make the inference stronger.

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