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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.

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