Gravitational Binding Energy of a Torus

[Tom MS]

I was looking at the wikipedia page for the gravitational binding energy of a sphere, but let's say that there was a toroidal planet. What would its gravitational binding energy be?
I have attempted the solution similar to what they did on wikipedia and obtained:
$$U = -4 G \pi^5 R^2 r^4 \rho^2$$https://lh3.googleusercontent.com/fPHuQ6I0NONuFyo7tz0OBkPVGWVW6kVChq_TjWU05B-jQRFqvMmnVLfPN7Q9OMCPe3qTMw=s95
R and r are shown in the diagram to be the total radius of the torus and the radius of the tube of the torus respectively.

I believe with a substitution for $\rho$ as density, this simplifies to:
$$U = -G M^2 \pi$$

But this seems to simple. Any ideas?

 

[secur]

A torus can't form or be maintained via gravitational binding. If that were the only "force" (spacetime curvature, to be pedantic) it would have to collapse into a sphere. Even if it were rotating very fast, you wouldn't get a toroidal planet - instead, an oblate spheroid, i.e. disk.

Strong internal structure of some sort would be required to hold such a shape. Check out Larry Niven's science fiction "Ringworld" for a good example. In such case the tension would produce negative gravity (according to GR); in an extreme case it would actually repel instead of attracting! In a more normal case you could have a small amount of matter, like an asteroid, in any shape imaginable - even a statue of Elvis. But such a small object is maintained by electromagnetic (chemical) binding; gravity would be negligible.

Bottom line, it can't be analyzed as simply gravitational binding energy. By the way the same would be true for any non-sphere shape: cylinder, sheet, etc.