The center of the Earth has microgravity and is thought to be solid and almost 6000 degrees hot. What is the pressure like there? The confusing thing, for the common man I suppose, is that there's no gravity there, but lots of matter around it which does gravitate towards the center. How does that add up? Is the center just a gas cloud or is it highly compressed?
And I could add the aside question about what it is like in the microgravity center of a neutron star or a black hole, but I don't expect anyone to know about that.
Answer
The pressure at any point is the total weight of everything above an area of a square metre. So for example the atmospheric pressure at Earth's surface, 101325 Pa, means the total weight of atmosphere above a square metre at the surface is 101325 N.
So when you ask for the pressure at the centre of the Earth, the way to calculate this is to work out the total weight of all the iron and rock above a square metre. The problem is that the weight is equal to the mass multiplied by the acceleration due to gravity, and as you say in the question the acceleration due to gravity changes as we go down from the surface to the centre.
Doing the calculation requires a knowledge of the density change with depth, and actually doing the calculation is somewhat involved so i'm going to chicken out and just show you the results. The pressure depth curve looks like:
So the answer to your question is that the pressure at the centre of the Earth is about 365 GPa. For comparison, this is about 3.6 million atmospheres.
The centre is very highly compressed. The gravity may be zero at the centre, but it still has all the matter above it pressing down. The core is solid iron-nickel with a density of about 13000 kg per cubic metre. For comparison, the density at the surface of iron is about 8000 kg/m$^3$ and nickel is 9000 kg/m$^3$, so you can se how compressed the core is.
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