Atmospheric Mass Loss

Energy Limit

The mass effluence from a star is limited by the incoming energy from a nearby star. Suppose that the nearby star emits energy at a rate $L$. If the planet is at a distance $r$ from the star, and its radius is $R$, then total incoming energy per unit time is

${\frac {L}{4\pi r^{2}}}=\pi R^{2}{\dot {E}}\Rightarrow {\dot {E}}={\frac {1}{4}}\left({\frac {R}{r}}\right)^{2}$

Matter is ejected from the star at the escape velocity

$v_e = \sqrt{\frac{G M_p}{R}} \approx \sqrt{G \rho} R$

where $\rho$ is the average density of the planet. If the planet has no other way of dissipating this energy, it must expel wind, and the mass loss will be limited by

$\dot{M} \approx \frac{\dot{E}}{v_e^2} \approx \frac{L}{r^2 G \rho}$

This calculation assumes that the scale height of the atmosphere is much smaller than the radius of the planet.

Recombination Limited Photo - evaporation

Above a certain threshold, most of the incident radiation on a planet will be re - radiated back to space by recombination. The net absorbed flux (incident minus the re - radiated by recombination) is proprtional to the number density of ionised particles $n_i$. We assume that in the absence of the incident radiation all of the planetary atmosphere would be neutral. The recombination rate is a binary process, and so it must be quadratic in the density of ions. We therefore express the cooling rate per unit volume per unit time as $\Lambda n_i^2$. In radiative equilibrium,

$F R^2 \approx \Lambda n_i^2 R^2 \frac{1}{\sigma n_0} \Rightarrow n_i \approx \sqrt{\frac{F n_0 \sigma}{\Lambda}}$

where $F=L/r^2$ is the incident flux, $n_0 = \frac{M}{\mu R^3}$ is the number density of the neutral particles, $\mu$ is the mass of a single neutral particle and $\sigma$ is the cross section for the absorption of a photon by a neutral particle. The mass flux is therefore given by $\dot{M} \approx \sqrt{\frac{G M_p}{R}} n_i \mu R^2 \approx \sqrt{\frac{G M_p}{R} \frac{F n_0 \sigma}{\Lambda}} \mu R^2$. The critical flux where the transition between the two regime occurs when the two estimates for the mass accretion rates are the same

$F \approx \frac{G^3 M_p^3 n_0 \sigma \mu^2}{R^3 \Lambda}$