Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 315 / 3 / c

First, an Earth-like atmosphere produces signals of only order parts per million in transit and planet-star contrasts near $10^{-10}$ in reflected light, demanding extreme photon collection, calibration, and starlight suppression. Second, stellar spots, faculae, flares, and spectral variability can imitate or overwhelm atmospheric features, especially around active M dwarfs. Third, long orbital periods provide few transits, while clouds, hazes, refraction, and overlapping molecular bands make the resulting sparse spectra difficult to interpret uniquely.