Isaac Newton famously proved the shell theorem and stated that:
"A spherically symmetric body affects external objects gravitationally as though all of its mass were concentrated at a point at its center.
I would like help to come up with a similar theorem for an oblate spheroid. I have a hunch (or a hope) that the oblate shell equation will be something like $$F = \frac{G M m}{2} \left( \frac 1 {R_1^2} + \frac{1}{R_2^2} \right)$$
where $R_1$ and $R_2$ are the distances of the test object of mass m from the two foci $F_1$ and $F_2$ of the elliptical cross section of the oblate spheroid as in the diagram below:
I can just about follow the proof in Wikipedia for the spherical case, but I don't think my calculus is good enough to extend this to the oblate spheroid case. Is there a know solution or has it even been attempted?
Going by the diagram I guess a similar method can be used as is used for the spherical case, integrating $\theta$ from $\theta = 0$ to $\theta = \pi$, but using implicit differentiation and using using $S$ as the integration variable instead of $\theta$.
There is the additional difficulty that in this case $R$ varies with $\theta$ or $S$. From the Wikipedia page on the ellipse, I have gathered from the standard parametric presentation of the ellipse that $$R = \sqrt{(a \ cos \theta)^2 + (b \ cos \theta)^2 } $$ where a is the usual semi-major axis and b is the semi-minor axis. After that, I am out of my depth. Ideally, although it might be asking too much, I would like the more general case for m at any location outside or on the oblate spheroid surface.
Shell method fails in spheroids and triaxial ellipsoid because of symmetry breaking.
Ellipsoidal coordinates should be used instead.
Please refer to the following links:
Oblate spheroid in Science World here.
Prolate spheroid in Science World here.
Please refer to my older post for triaxial ellipsoid here.