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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-315/4/a/solution

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

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

Three complementary methods available in the setting of this 2017 paper are:

Exoplanet transmission spectra during primary transit. The planet's atmosphere filters stellar light at the day-night terminator. Its advantages are a bright background source and sensitivity to atomic, molecular and aerosol opacity; its limitations are the need for a transit, a small annular signal, exoplanet cloud masking, and degeneracies among abundance, reference pressure, temperature and mean molecular weight. The useful physical range is ultraviolet through near- and mid-infrared, subject to instrumentation. The Hubble Space Telescope is an appropriate example, with ultraviolet/optical instruments and near-infrared transit spectroscopy, particularly around $1.1$ -- $1.7 μ m$ with WFC3.
Exoplanet secondary eclipse spectroscopy or photometry. Subtracting the stellar-only light in eclipse from the combined light outside eclipse isolates the dayside thermal and reflected spectrum. It measures dayside brightness temperature and composition and can reveal an atmospheric thermal inversion. Limitations include eclipse geometry, a weak planet-star contrast, and the joint dependence of features on opacity and the atmospheric pressure-temperature profile. Reflection is most useful in the optical, and thermal emission in the near- to mid-infrared. The Spitzer Space Telescope is a major example: earlier cryogenic observations included infrared spectroscopy, while in 2017 the warm mission provided 3.6- and 4.5-micrometre photometry.
Exoplanet direct imaging. Spatial separation isolates the planet's own spectrum without requiring a transit, and favours young hot giants on wide orbits. Limitations are small angular separation, overwhelming stellar leakage, restricted inner working angle and instrument contrast. Near-infrared spectra of young giants are especially useful, with optical reflected light and thermal infrared also possible for suitable targets. The Very Large Telescope, for example with its SPHERE imager, provides optical/near-infrared high-contrast observations using adaptive optics and stellar-light suppression.

Transmission probes the limb, eclipses probe the dayside, and direct imaging isolates spatially resolved planetary light. Radial velocities alone measure orbital motion rather than atmospheric composition; high-resolution planetary Doppler spectroscopy can instead detect atmospheric spectral lines if planetary light is isolated statistically.

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