# Distribution function of an ideal gas

1. Nov 18, 2005

### gdumont

Hi,

I have the following problem to solve:

Consider a planet of radius $R$ and mass $M$. The plante's atmosphere is an ideal gas of $N$ particles of mass $m$ at temperature $T$. 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 $f_0(\mathbf{v})$, so the full distribution is just
$$f(\vec{v})=f_0(\vec{v})e^{-U/kT}$$
where
$$U=-\frac{GMm}{r}$$
is the gravitational potential and $r$ is the distance from the center of the planet to the molecule of velocity $\vec{v}$. I'm not sure if I should replace $r$ by $r-R$ in $U$.

Can anyone confirm if I'm right or not?

Thanks

2. Nov 18, 2005

### 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
$$U(r) = - \frac{GMm}{r} \,\,\, r > R$$
and $$U = \infty$$ for $$r < R$$.

3. Nov 18, 2005

### gdumont

OK, but is the distribution function OK?

4. Nov 18, 2005

### Physics Monkey

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