## Energy Limit Edit

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 Edit

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} $