Revisiting the barometric equation and the extent of a planetary atmosphere

The barometric equation predicts the molecular concentration $n(z)$ exponentially decaying with altitude $z$. Because the mean free path $l=1\mathrm{/}n\sigma$ increases exponentially, at high altitudes $z$, the equation is no longer within the domain of applicability of the standard kinetic theory. \cite{grimsditch} Here, we predict the dependence $n(z)\propto z^{-2}$ for the case $l\gg L$ in uniform gravity and $n(z)\propto [\ln (1+z\mathrm{/}{R}_0){]}^{-2}$ when $z\gg R_0$ for a planet of radius $R_0$. It corresponds to a non-stationary planetary atmosphere with hydrogen accretion. The accretion is accompanied by a release of gravitational potential energy that could be the elusive source driving the formation of stellar coronas. Other consequences include slowly decaying tails of planetary atmospheres and periodical hydrogen explosions of white dwarfs.