Past exam of the mathematics course of the University of Cambridge 2017 iii Paper 315 4 j Solution Created 2026-10-03 Updated 2026-10-06
Three research directions natural in the 2017 setting are:
- Atmospheric characterization of small planets, extending beyond bright hot Jupiters to super-Earth-sized and terrestrial bodies. The aim is to determine whether they retain light envelopes or heavier secondary atmospheres and to understand exoplanet habitability. Potential exoplanet biosignatures require a chemical and stellar context because abiotic processes can mimic individual molecules; future infrared observatories such as the then-planned James Webb Space Telescope offered a route to better spectra, without guaranteeing a biological interpretation.
- Connecting measured composition to planet formation and planetary migration. Atmospheric metallicity of a giant planet, atmospheric carbon-to-oxygen ratio, orbital architecture and bulk mass-radius measurements can constrain accretion of gas and solids. Atmospheric condensate rainout, disequilibrium chemistry in an exoplanet atmosphere and exoplanet interior-composition degeneracy make a one-to-one mapping from a measured ratio to a birth location unreliable; comparative samples and coupled models are needed.
- Resolving atmospheric structure in space and time. Exoplanet thermal phase curves, repeated spectra, high-resolution planetary spectroscopy and exoplanet direct imaging can test day-night heat redistribution, winds, patchy exoplanet clouds, atmospheric escape and chemical heterogeneity. The challenge is to distinguish actual variability or circulation from changes in stellar light and instrumental calibration.
Past exam of the mathematics course of the University of Cambridge 2019 iii Paper 315 4 e Solution Created 2026-10-03 Updated 2026-10-05
First refine the ephemeris, planetary mass, stellar radius, and stellar variability using exoplanet transit photometry and the radial-velocity method. Then combine observations that probe different regions rather than relying on one spectrum. A present-day programme could use the following complementary measurements; in the 2019 setting of the paper, James Webb Space Telescope observations would have been a future capability.
- Exoplanet transmission spectrum at roughly – with the Hubble Space Telescope or optical ground-based spectroscopy: constrain exoplanet clouds, atmospheric haze, the scattering slope of a transmission spectrum, and sodium or potassium absorption.
- Near-infrared exoplanet transmission spectrum with NIRISS at – and NIRSpec modes covering roughly –: measure water, carbon monoxide, carbon dioxide, and methane bands, then constrain atmospheric metallicity of a giant planet and atmospheric carbon-to-oxygen ratio through a joint atmosphere model.
- Exoplanet secondary eclipses at near- and mid-infrared wavelengths with NIRSpec and MIRI, especially about – for the latter's time-series low-resolution mode: infer brightness temperatures, the vertical pressure-temperature structure, and whether an atmospheric thermal inversion turns bands into emission.
- A full-orbit exoplanet thermal phase curve in one or more infrared bands with the James Webb Space Telescope: constrain day-night heat redistribution, nightside emission, and the offset of the hottest region, with different bands probing different pressures.
- High-resolution near-infrared spectroscopy around molecular bands such as carbon monoxide near using CRIRES on the Very Large Telescope: resolve the planetary Doppler shift and seek wind velocities or rotation broadening after accounting for the orbital velocity.
- Ultraviolet transit spectroscopy with the Hubble Space Telescope, or ground-based near-infrared helium spectroscopy at : search for atmospheric escape and an extended upper atmosphere.
Together these address aerosols, molecular composition, elemental enrichment, vertical thermal structure, horizontal heat transport, winds, and escape. Repeat key events and monitor stellar activity, since stellar contamination and instrumental trends can imitate atmospheric signals. Use actual brightness, saturation limits, and predicted feature amplitudes to choose observing modes and exposure times.
The wavelength ranges and time-series capabilities are documented in the NIRISS SOSS guide, NIRSpec overview, MIRI spectroscopy guide, and ESO's CRIRES description.
Past exam of the mathematics course of the University of Cambridge 2022 iii Paper 315 4 b Solution 2026-09-28
In an isothermal hydrostatic atmosphere, pressure falls as . If a water-band line core becomes optically thick at pressure and an opaque cloud fixes the nearby continuum at , the exoplanet transmission spectrum feature spansA two-scale-height feature therefore requiresTaking a representative near-infrared water-band pressure gives , so the appropriate estimate is an exoplanet cloud deck top near . The numerical value scales directly with the assumed line-core pressure.
Other explanations include a high mean molecular weight, subsolar water abundance, a colder terminator, atmospheric haze, patchy two-limb clouds, stellar contamination, or instrumental systematics. Optical scattering slopes, broader James Webb Space Telescope molecular coverage, repeated transits, secondary-eclipse spectra, phase curves, and precise mass and radius measurements can distinguish these possibilities.
Past exam of the mathematics course of the University of Cambridge 2025 iii Paper 315 3 e Solution Created 2026-09-24 Updated 2026-09-25
An Earth analogue has an iron-silicate interior and a thin secondary nitrogen-dominated atmosphere with water and carbon dioxide. Contextual coexistence of oxygen or ozone with methane and surface-water indicators would be a target biosignature. Its small scale height and extreme reflected-light contrast put a true Sun-Earth analogue beyond routine current atmospheric work, but future large direct-imaging missions and the Extremely Large Telescope target nearby terrestrial planets.
A temperate M-dwarf rocky planet has a similar solid interior but may retain carbon-dioxide, nitrogen, or water atmospheres under synchronous rotation. Water, carbon dioxide, methane, and contextual oxygen chemistry are observable targets. Its small host gives deeper transits, so the James Webb Space Telescope and large ground telescopes can test nearby systems, although stellar activity and atmospheric erosion complicate interpretation.
A Hycean planet or water-rich sub-Neptune has a water-rich interior or ocean below a hydrogen-rich atmosphere. Methane, carbon dioxide, ammonia, water, and proposed sulfur-bearing biosignatures must be interpreted against abiotic photochemistry. Its large radius and hydrogen scale height make transmission spectroscopy comparatively favorable for JWST and future Ariel space telescope surveys, but whether a clement ocean exists beneath the atmosphere remains model-dependent.
Past exam of the mathematics course of the University of Cambridge 2025 iii Paper 315 4 e Solution Created 2026-09-24 Updated 2026-09-25
The Hubble Space Telescope uses ultraviolet through near-infrared transit and eclipse spectroscopy. It established detections of alkali metals, water, aerosols, and escaping hydrogen but has a small aperture and restricted continuous infrared coverage. High-resolution ground facilities such as the Very Large Telescope separate Doppler-shifted planetary lines from telluric and stellar spectra and directly image young giants; the atmosphere and thermal background limit broad-band precision. The James Webb Space Telescope uses NIRISS, NIRSpec, NIRCam, and MIRI for roughly visible-to-mid-infrared transit, eclipse, phase-curve, and direct-imaging spectroscopy. It has measured broad molecular inventories, including the clear carbon-dioxide feature and photochemical sulfur dioxide in WASP-39 b.
JWST improves on earlier facilities through its 6.5-metre collecting area, cold space environment, stable time-series spectroscopy, and broad infrared wavelength coverage spanning several bands of water, carbon dioxide, methane, carbon monoxide, ammonia, and sulfur compounds. The Ariel space telescope is designed for a uniform atmospheric census of about one thousand planets, while the Extremely Large Telescope will add spatial resolution and high-dispersion spectroscopy for nearby giant and terrestrial planets.
Three directions enabled or sharply advanced by JWST are chemically complete retrievals across multiple molecular bands rather than isolated detections; atmospheric tests of smaller and cooler sub-Neptunes and rocky planets; and measurements of photochemistry, clouds, heat redistribution, and vertical thermal structure through repeated transit, eclipse, and phase-resolved spectra.