## Wednesday, September 19, 2012

### planetary atmospheres

why don't planets lose their atmospheres to outer space? one would assume that the Boltzmann distribution would always allow a certain fraction of molecules to have kinetic energies large enough to overcome the gravitational potential.

bonus question: wouldn't the same thing apply to other gravitationally bound equilibrated systems such as open clusters, globular clusters and galaxies?

the Boltzmann-factor of molecules in the Earth's atmosphere is roughly $\exp(-(330 m/s / 11.2 km/s)^2) = \exp(-1150) \simeq 10^{-500}$. each second (which is the time a molecules needs for covering the scale height of the atmosphere at the first cosmic velocity) the Earth loses a fraction $10^{-500}$ of the atmosphere. after 5 billion years the Earth would have lost a fraction $10^{-483}$, almost nothing. this calculation neglects that the mean free path of air molecules is much smaller than the scale height, so the molecules don't just rise in the gravitational potential umimpeded, which makes the leakage even smaller.
the moon is a very different case: the escape velocity is $\sqrt{6\times4}$ smaller, the Boltzmann-factor is 25 times smaller, the scale height 6 times larger. under the same approximations the moon loses in 5 billion years a fraction of $10^{-3}$ of the atmosphere, which is likely a lower limit because if we substitute a slighly higher temperature, the atmosphere is lost rapidly, likewise would solar winds affect a less tighly bound atmosphere.