Why are planets denser as you approach the center?

Question

Gauss' law says that the net electric force inside a hollow, uniform, not rotating sphere is zero. Since gravity is also proportional to the inverse square of the distance, I assume this should apply to the force of gravity inside a hypothetical uniform shell, like a hollow planet. So what makes planets develop to become denser at the center or as you get deeper? If a planet happened to form as a shell, would gravity shape it into a sphere with a higher density as you go deeper? The centripetal force from spinning tends to make denser materials move out more in a viscous fluid, so spinning doesn't seem to create the density distributions we see.

If it's not uniform, and the center ends up denser by chance, then that non-uniformity could maintain the density distribution, but if we somehow got to a reasonably uniform, hollow shell, would there be a tendency of the shell to deform into a ball with increasing density with depth, a uniform sphere, or retain its shape as a shell?

Answer 1

If a planet formed a spherical shell (which isn't a good model for how planets actually form but we can run with it as a thought experiment), then:

• it is true that, say, a piece of dust that was floating around inside the shell wouldn't feel a gravitational force,
• but the shell itself would feel a gravitational force causing it to collapse.

It's a little easier to handle this if you suppose the shell has some finite width $a$, so the shell ranges from $R$ to $R+a$. Then the gravitational potential within the shell will change between $R$ and $R+a$, and since the gravitational force is proportional to the gradient of the potential, there will be a non-zero force on the shell. Since gravity is attractive, the direction of the force will be for the shell to collapse as opposed to expand.

As the shell collapses, it will continue to collapse until some form of pressure prevents further collapse. For planets like Earth, that pressure comes from the the fluid in the core, or from the rocky layers above it. For the outer layers, the rock will experience gravitational attraction toward the center. For inner layers, the layer will experience gravitational attraction, plus the pressure of outer layers being pulled in by gravity, so the inner layers will tend to be denser and more compressed.

Answer 2

TLDR: Every particle of matter in a planet (except for the one particle at the very center) feels net gravitational attraction toward the center. The shell theorem says that particles closer to the center feel less attraction than particles further out, but they still feel pressure due to the cumulative weight of all of the more strongly attracted particles above them. At the very center, the net gravitational force is zero, but a particle there feels the weight of the whole planet pushing in on it from every direction.