I want to derive a formula for pressure at depth in a constant density planet, which sounds pretty simple.(adsbygoogle = window.adsbygoogle || []).push({});

Setting up a force balance,

d(4∏*r^{2}P) = -4/3∏r^{3}*ρ*4∏ρ*r^{2}*dr*G/r^{2}

I'm too lazy to write down a thorough derivation, but by pretty simple calculus and integrating from the planet surface (where P~0), I get:

r^{2}P = ∏/3*ρ*ρ*(R^{4}-r^{4})

Then dividing by the surface area I get something like P = C *(R^{4}/r^{2}-r^{2})

At the center of the planet, this goes to infinity. Every peer reviewed paper etc. has an equation which is similar near R (the radius) but very much smaller and flatter approaching r=0. There's no asymptote.

Their version is easy to get assuming dP/dr=ρ*g. But this implies equal areas (a cylinder or something) on top and bottom. I can't see how they can neglect the surface area change. I tried my same derivation as a balance of forces in the z direction (z=r sin(θ)) and got the same thing.

So obviously, I am misunderstanding something basic about the problem. Is pressure just defined in a different way for some reason? Even if you posit a tiny cube at the center of the planet, it's not just the column above which presses down. It's the whole cone, and the area will not scale with volume as it will for a column.

So...what am I not getting?

Edit: Thinking about it, there is some net effect upwards from lateral forces in the shell. Is assuming equal areas just an approximation for that effect? Or is the common answer really the rigorous solution?

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# Hydrostatic equilibrium in sphere

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