Relmuis

2005-May-11, 07:06 PM

I wonder which different shapes a planet can have. The problem is relatively easy to describe, but extremely hard to solve.

For a planet, one can use a self-gravitating non-compressible liquid, which would have an equal density everywhere. This is not quite true, of course, but very nearly so.

For a sustainable shape, the surface of this liquid must follow an equipotential surface of the gravity field generated by that particular shape. This does't mean that the gravity vector must have the same strength at all points on this surface. But is does mean that the gravity vector must must be normal to this surface at all points on this surface (i.e. a plumbline will everywhere hang at a right angle to the surface).

We know, of course, that the sphere is a sustainable shape for a notrotating planet. If rotation is added, a "centrifugal" potential can be superposed on the gravitational potential, and an ellipsoid of revolution has now become a sustainable shape. But there might well be other sustainable shapes.

I wonder about the torus. If the torus is sustainable, either as a rotating or even as a nonrotating shape, this would not mean that torus-shaped planets would occur in nature. But it would mean that a civilization might build a torus-shaped planet and expect it to retain its shape afterwards.

My reason to expect this possibility is the following: think of a lot of planets spaced equally around the same circular orbit and having the correct speed to orbit the central star at that distance. Now slowly decrease the mass of the central star to zero, all the while increasing the mass of the several planets, in exactly the right amount to retain the same centripetal acceleration (and therefore the same orbital speed). The planets are now essentially keeping each other in the same orbit. Now subdivide the planets until you end up with a circle of contiguous small planets all around the circle. This is, of course, a self-gravitating torus, albeit a very slender one with a very big central hole.

Essentially, the question would now be how much this torus can be fattened up and by how much the central hole can be narrowed for a certain orbital speed. And also, wether there would be a ratio between the two diameters where the orbital speed is allowed to become equal to zero.

A related problem is whether one can have a toroidal star. This problem would be even harder, as in this case the fluid cannot be assumed to be noncompressible.

For a planet, one can use a self-gravitating non-compressible liquid, which would have an equal density everywhere. This is not quite true, of course, but very nearly so.

For a sustainable shape, the surface of this liquid must follow an equipotential surface of the gravity field generated by that particular shape. This does't mean that the gravity vector must have the same strength at all points on this surface. But is does mean that the gravity vector must must be normal to this surface at all points on this surface (i.e. a plumbline will everywhere hang at a right angle to the surface).

We know, of course, that the sphere is a sustainable shape for a notrotating planet. If rotation is added, a "centrifugal" potential can be superposed on the gravitational potential, and an ellipsoid of revolution has now become a sustainable shape. But there might well be other sustainable shapes.

I wonder about the torus. If the torus is sustainable, either as a rotating or even as a nonrotating shape, this would not mean that torus-shaped planets would occur in nature. But it would mean that a civilization might build a torus-shaped planet and expect it to retain its shape afterwards.

My reason to expect this possibility is the following: think of a lot of planets spaced equally around the same circular orbit and having the correct speed to orbit the central star at that distance. Now slowly decrease the mass of the central star to zero, all the while increasing the mass of the several planets, in exactly the right amount to retain the same centripetal acceleration (and therefore the same orbital speed). The planets are now essentially keeping each other in the same orbit. Now subdivide the planets until you end up with a circle of contiguous small planets all around the circle. This is, of course, a self-gravitating torus, albeit a very slender one with a very big central hole.

Essentially, the question would now be how much this torus can be fattened up and by how much the central hole can be narrowed for a certain orbital speed. And also, wether there would be a ratio between the two diameters where the orbital speed is allowed to become equal to zero.

A related problem is whether one can have a toroidal star. This problem would be even harder, as in this case the fluid cannot be assumed to be noncompressible.