Is a Torus More Stable than a Sphere? A Mathematical Evaluation

[wedge]

Hi all
I'd like to know your opinion about this: how to evaluate if an autogravitating body is stable? I'd like to know which analytic consideration should we do.
Example: experience tells us that a sphere is pretty stable. But... something else? A torus is stable? Or should it evolve into something different, maybe a thin ring? How to evaluate this, in a mathematical view?
(Obviously I suppose there are differences if the body is rigid or not.)
Thank you
Wedge!

PS I hope it's not too complicated (I'm at the first year of University...)

 

[Andrew Mason]

Hi all
I'd like to know your opinion about this: how to evaluate if an autogravitating body is stable? I'd like to know which analytic consideration should we do.
Example: experience tells us that a sphere is pretty stable. But... something else? A torus is stable? Or should it evolve into something different, maybe a thin ring? How to evaluate this, in a mathematical view?
(Obviously I suppose there are differences if the body is rigid or not.)
Thank you
Wedge!

PS I hope it's not too complicated (I'm at the first year of University...)

I think the principle is that a fluid will occupy a shape that minimizes energy. So for each element of mass dm, you want to minimise r. Conceptually, you can do that by building the object by starting with the first element of mass and then adding. You will see that in order to minimize potential energy, you have to add the mass in concentric shells of growing radius. If a mass is added that is not a concentric shell, you could reduce the energy of a mass element by moving it closer to the surface.

AM

 

[vanesch]

I think the principle is that a fluid will occupy a shape that minimizes energy. So for each element of mass dm, you want to minimise r. Conceptually, you can do that by building the object by starting with the first element of mass and then adding. You will see that in order to minimize potential energy, you have to add the mass in concentric shells of growing radius. If a mass is added that is not a concentric shell, you could reduce the energy of a mass element by moving it closer to the surface.

Well, statically this is correct. But if you introduce dynamics, many more solutions are possible. If you want to know which ones, look around you

A planetary system is possible (ok, the basic units are still obloid spheres of course) ; but when looking at Saturn, rings with a big central mass are also possible. And maybe many more.