Does the effect of Newton's shell theorem appear only in full with perfect symmetry, or is there a gradual onset of its effects with gradually increasing symmetry?
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Posted Oct 20, 2016, 12:57 AM UTC · edited
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Posted Oct 20, 2016, 12:57 AM UTC · edited
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What do you mean by "gradually increasing symmetry"? Either ρ(r) = ρ(r), or it's not. There isn't any intermediate step, or a way to gradually go from one case to the other. If you add any angular dependence to the density, the shell theorem no longer applies.
I meant it heuristically. An egg is closer to being spherically symmetric than a garden hose is.
Or imagine deforming an egg until it is spherically symmetric. Does gravity just all of a sudden follow the shell theorem instantaneously the moment the egg becomes spherical?
But neither is spherically symmetric, so the shell theorem doesn't apply to either.
Yes.
I think we have a semantic disconnect over the meaning of "follow the shell theorem." I do not mean strictly that "no net gravitational force is exerted by the shell on any object inside," where if you obey that qualification, you follow the shell theorem, but if you deviate in the slightest, you do not.
What I want to know is if the gravitational force exerted by a shell on objects inside of it is variable in any way based on the shell's shape. Would the force gradually decrease as the egg deforms into a sphere, or suddenly vanish when it becomes a perfect sphere?
Well of course it depends on the shape/mass distribution of the object. But the problem is, how do you quantify the shape?
If you have an ellipsoidal shell, you could calculate the g-field as a function of the ellipsoid axes. I know this can be done analytically outside the shell (using Newtonian gravity); I'm not sure about the inside. It should vanish inside as the ellipsoid is made spherical. As for the outside, the field will essentially be a superposition of a monopole field and a quadrupole field. As you take the quadrupole moment to zero, you turn it into a sphere, and the quadrupole term dies. So only the monopole remains and spherical symmetry is restored.
But the inside case sounds like a nasty problem to work out, and at best you've got a solution for an ellipsoidal shell.
How do you quantify "non-sphericalness" for a general object in such a way that you can continuously make it spherical?
Question in a form that makes more sense: what is the gravitational field at some location in the interior of a uniformly dense spheroid as a function of its oblateness? What is the general form for the gravitational field inside an object whose shape is described by a multipole expansion?
A multipole expansion can be carried out inside the spheroid, but multipole expansions inside a mass distribution rather than far away is the stuff of nightmares.
This is the kind of problem I'd love to see but I never want to do. I know Landau had a sick fetish for ellipsoids, but as far as I remember, even he only treats the exterior.
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