 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-315/2/b/solution

Comparable hydrostatic thermal escape requires comparable Jeans escape parameter

λ = G M_{p} m / (k_{B} T R_{exo})

. For the same escaping species and

R_{exo} ≃ R_{p}

\frac{T _{J}}{T _{\oplus}} ≳ \frac{M _{J} / R _{J}}{M _{\oplus} / R _{\oplus}} ≃ \frac{318}{11.2} ≃ 28.

Thus Jupiter at 5 au needs an exobase temperature at least about thirty times Earth's at the same irradiation to have comparable Jeans escape flux.

For the inner planet, the usable EUV power is

η π R_{exo}^{2} F_{EUV}

. If the binding energy per unit escaping mass is

G M_{p} / R_{p}

\dot{M} = \frac{η π R _{exo}^{2} F _{EUV} R _{p}}{G M _{p}} .

x

of the planetary mass is therefore

t_{x} = \frac{x G M _{p}^{2}}{η π R _{exo}^{2} R _{p} F _{EUV}} .

This neglects Roche-lobe reduction, radiative cooling, changes in radius and flux, and the planet's orbital evolution. If the gas is lifted only from

R_{exo}

, replace

R_{p}

in the denominator by

R_{exo}

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