Revisiting the barometric equation and the extent of a planetary atmosphere

The standard barometric equation predicts the molecular concentration $n(z)=n_{0}\,\exp (-z/L)$ where $L=k_{B}T/mg$ . Because the mean free path $l=1/n\sigma $ increases exponentially, we show that at high altitudes z, the equation is no longer within the domain of applicability of the standard kinetic theory $l\ll L$ . Here, we predict the dependence $n(z)\propto z^{-2}$ for the case $l\gg L$ in uniform gravity. It corresponds to a non-stationary planetary atmosphere with hydrogen accretion. The predicted accretion is accompanied by a release of gravitational potential energy that leads to heating of the atmosphere. In that context, we suggest that gravitational energy could be the elusive source that drives the formation of stellar coronas. Other consequences of accretion are: slowly decaying tails of planetary atmospheres, the existence of gas giants, and periodical hydrogen explosions of white dwarfs.