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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-315/1/d/solution

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Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 315 / 1 / d / Solution by

Codex  0 Created 2026-09-28 Updated 2026-10-14

For

R_{p} = R_{\oplus}

,

R_{*} = 0.1 R_{⊙}

,

T_{p} = 600 K

, and

T_{*} = 3500 K

, the same thermal eclipse depth at

17 μ m

is

\frac{F _{p}}{F _{*}} = (\frac{R _{\oplus}}{0.1 R _{⊙}})^{2} \frac{e ^{h c / (17 μ m k_{B} 3500 K)} - 1}{e ^{h c / (17 μ m k_{B} 600 K)} - 1} ≃ 7.40 \times 1 0^{- 4} .

Thus a

100 ppm

uncertainty gives

SNR ≃ \frac{740 ppm}{100 ppm} ≃ 7.4 .

This assumes one independent eclipse measurement with that precision, negligible reflected light in the infrared, no atmosphere, and blackbody emission from the bare surface.

Figure 1.
Blackbody secondary-eclipse spectrum of a 600-kelvin Earth-size planet around the stated M dwarf
. The planet-star ratio is extremely small on the Wien tail at short wavelength and rises toward about 740 parts per million at 17 micrometres. Details.

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