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 Incoming links: Planet-star radius estimate in the Rayleigh-Jeans limit

Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 315 / 1 / c / Solution Created 2026-10-03 Updated 2026-10-06

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A natural interpretation of the short-wavelength peak is reflected starlight, while the longer-wavelength peak is planetary thermal radiation. A reflected spectrum approximately follows the stellar spectrum multiplied by the wavelength-dependent geometric albedo; thermal radiative flux approximately follows the planet's Planck function, modulated by molecular opacity. Thus two peaks need not represent two planetary surface temperatures.

For a quantitative estimate assume both are broad

F_{λ}

peaks, reflection has a slowly varying geometric albedo, and star and planet have approximately blackbody spectral envelopes. Wien's displacement law then gives

T_{*} ≃ \frac{2898 μ m K}{0.7 μ m} ≃ 4140 K, T_{p} ≃ \frac{2898 μ m K}{1.5 μ m} ≃ 1930 K .

(1)

The stellar estimate is compatible at order of magnitude with an old, relatively small main-sequence star; equal age does not mean equal temperature to the Sun. Assume the far-infrared signal is thermal and sufficiently long-wavelength that the Rayleigh-Jeans law applies to both bodies. The planet-star radius estimate in the Rayleigh-Jeans limit gives

\frac{F _{p}}{F _{*}} = (\frac{R _{p}}{R _{*}})^{2} \frac{T _{p}}{T _{*}}, R_{p} = \frac{R _{⊙}}{2} 1 0^{- 3} \frac{4140}{1930},

(2)

and therefore

R_{p} ≃ 0.023 R_{⊙} ≃ 1.6 \times 1 0^{7} m ≃ 2.5 R_{\oplus} .

(3)

This is a small volatile-rich-planet-sized estimate, not a Jupiter-sized one. It is conditional: peaks caused by molecular windows, strongly chromatic reflection, or a spectrum expressed as

F_{ν}

rather than

F_{λ}

do not support those two Wien estimates. Without

T_{p} / T_{*}

, the far-infrared ratio only fixes

R_{p}^{2} T_{p} / T_{*}

For a close-in hot planet around a small star, transit-based atmospheric observations are the natural route if the orbital geometry allows them. Exoplanet transmission spectra gain from the small stellar radius, with a limb signal scaling as

2 R_{p} H / R_{*}^{2}

; they probe composition at the day-night terminator. The stated

1 0^{- 3}

thermal contrast also makes exoplanet secondary eclipse measurements a particularly useful route to the dayside exoplanet emission spectrum. Resolving such a close-in small planet by exoplanet direct imaging is much harder. Transit and secondary-eclipse spectroscopy are favoured for a transiting close-in interpretation; the supplied spectrum alone does not determine the orbital geometry or a unique best technique.

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