Large Thick but Hollow Planet with small inner planet
Earlier today I came across this question, asking as to the effects of having a planet with a shell around it acting as a planet itself, so there was a planet within a planet with neither in contact with the other but held in place with gravity/atmospheric pressure or somesuch, which piqued my interest, however all the answers to that question seem to assume a thinner shell with the majority of the mass being the inner planet, I'm interested in the idea of the outer planet being far far thicker with a rather small inner planet and a smaller but still reasonably large atmosphere for the inner planet, perhaps along the lines of a large marble within a bowling ball.
Part of my idea behind this is that rather than the shell planet being affected by the inner planets gravity, the inner planet is affected by the shell planets gravity.
I also wondered about the possibility of three or more planets rather than just the two, though assumably this would require either the inner planets to be too small to be of any real use for life to survive, or for the outer planets to be far larger than would make sense for them to exist in a solar system similar to our own.
Edit: As for how this would have formed, I hadn't really made any decisions as to whether it was manufactured or naturally occurring, I suppose if I was going to, in the world I was pondering implementing this in, it would have to be naturally occurring, or manufactured in the sense that it was mined out, not built that way.
As for how it would be mined out I doubt it would be hugely feasible for it to just be mined by conventional means, but if a group had that as their specific intention and the required resources and ability to plan I assume they would be able to, or some form of magic, such as an explosion that wrapped around spherically and completely destroyed the area leaving no debris, or simply teleported the area somewhere else.
This post was sourced from https://worldbuilding.stackexchange.com/q/57059. It is licensed under CC BY-SA 3.0.
1 answer
Sefa's answer is correct. Due to Newton's shell theorem, the force on any object inside (and due to) a spherically symmetric object of uniform mass density of exactly zero. The exact integration (see e.g. Wikipedia or these notes) is messy, but it indeed relies on the density of the shell, $\rho$.
Now, if we give the shell non-uniform density, then there will indeed be a net gravitational pull on the "planet" inside it. We can compute the force on this planet by computing the potential in any location inside the shell. Basically, if we assume that $\rho=\rho(r,\theta,\phi)$ (in spherical coordinates), we can solve the equation using spherical harmonics (where the factor of $\epsilon_0^{-1}$, used in electrostatics problems, is replaced with the appropriate set of gravitational constants).
This is not easy to solve analytically; numerical solutions are often better. Also - and I don't normally say this - there's a point at which you have to consider if this much effort is worth it to solve a worldbuilding problem. Odds are good that the mass density will be roughly spherically symmetric, as will the shell, and so the "planet" inside will feel very little force.
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