Distribution function of an ideal gas

[gdumont]

Hi,

I have the following problem to solve:

Consider a planet of radius [itex]R[/itex] and mass [itex]M[/itex]. The plante's atmosphere is an ideal gas of [itex]N[/itex] particles of mass [itex]m[/itex] at temperature [itex]T[/itex]. Find the equilibrium distribution function of the gas accounting for the gas itself and the gravitationnal potential of the planet.

Here are my thoughts

The equilibrium function of the gas alone is simply the Maxwell-Boltzmann distribution function [itex]f_0(\mathbf{v})[/itex], so the full distribution is just
[tex]
f(\vec{v})=f_0(\vec{v})e^{-U/kT}
[/tex]
where
[tex]
U=-\frac{GMm}{r}
[/tex]
is the gravitational potential and [itex]r[/itex] is the distance from the center of the planet to the molecule of velocity [itex]\vec{v}[/itex]. I'm not sure if I should replace [itex]r[/itex] by [itex]r-R[/itex] in [itex]U[/itex].

Can anyone confirm if I'm right or not?

Thanks

 

[Physics Monkey]

The gas molecules can't penetrate into the surface of the planet, so it makes sense for your model to be described by a potential which is
[tex]
U(r) = - \frac{GMm}{r} \,\,\, r > R
[/tex]
and [tex] U = \infty [/tex] for [tex] r < R [/tex].

 

[gdumont]

OK, but is the distribution function OK?

 

[Physics Monkey]

Yes, the Maxwell-Boltzmann distribution is the correct one.