According to wikipedia[1] the converse of the Shell Theorem is (nearly) true:
Suppose there is a force F between masses M and m, separated by a distance r of the form F = Mmf(r) such that any spherically symmetric body affects external bodies as if its mass were concentrated at its centre. Then what form can the function f take?
The form of f allows Newtonian gravity but not Einsteinian.
The key point there is "spherically symmetric". The Sun isn't. It bulges around its equator thanks to rotation, as does everything. That does have effects on planetary motion; an object in an inclined orbit spends a bit more time a bit farther away from a bit of the Sun's mass. Replace the Sun with a black hole of equal mass and you don't have that oblateness. The effect on planetary orbits would be very small, so macroscopically the Solar System would still be the same, just with very slight differences in orbital speeds and periods.
Interesting. However, I would have thought that planets are far out enough that classical is accurate enough. This sort of thing is why I specified planets as opposed to orbits in general.
