Gravity on a toroidal planet

A couple years ago I wrote a series about how a toroidal planet would differ from a spherical one, such as how day and night in the interior would depend on latitude, how east and west would be a choice rather than a natural law, and some considerations for mapmaking. In one post I mentioned that the gravitational pull on the inner equator would be toward the center of the ring. Recently, a reader wrote to disagree and make the case that it would be toward the nearest part of the ring.

I've lost the calculations that originally led me to conclude that gravity would be toward the center of mass, but the conclusion made intuitive sense. A torus is not a gravitationally stable shape, and the overall force would have to be inward for it to collapse into hydrostatic equilibrium and become a sphere. But it also makes sense that the gravitational pull on the inner equator would be toward the nearest mass, i.e., toward the ring and away from the center of the torus.

Here's a visualization of the gravitational pull on a toroidal planet, calculated numerically (if you have suggestions for how to determine it analytically, email me):

Note the difference in gravitational force between the outer equator and inner equator. While the pull on the inner equator is indeed toward the ring, it's not as strong as the pull the opposite direction on the outer equator. In fact, for a 4:1 torus, it's only about half the magnitude. As pointed out by my reader, hydrostatic equilibrium wouldn't be achieved by matter on the inside of the ring falling in, but by matter on the outside pushing in. That's an interesting distinction, one I hadn't guessed.

Thanks, Robert, for prompting me to revisit this!