# Partition function of an asteroid gas (gravity)

Consider the classical problem (Newtonian gravity) of a large number of $$N$$ identical non-interacting asteroids orbiting around a big planet. I wanted to see if the problem was solvable. I wrote my Hamiltonian:

$$H=\frac{p_r^2}{2m}+\frac{p_\theta^2}{2mr^2}+\frac{p_\varphi^2}{2mr^2\sin^2(\theta)}-\frac{GmM}{r}$$

But calculating the partition function I get: $$Z=C\int r^2e^{\beta GmM/r}\mathrm{d}r$$ (where $$C$$ is a constant dependent of $$\beta,m$$).

Sadly, this integral does not converge in any of this ranges [($$0,\infty$$) or ($$r_0,\infty$$)].

I suppose it the divergence is similar to the one in quantum case (see partition function of the hydrogen gas).

Is this a clue to some physics missing (asteroid-asteroid interactions, relativity)? Is the solution to regularize also?

Any papers, books, or references that treat this case (specially a classical gas) are welcomed.

• Perhaps this problem has some analogy to the Boltzmann statistics of a planet's atmosphere (a discussion of such a problem is present here: physics.stackexchange.com/q/436036). Feb 14, 2019 at 15:30