What is the pressure at the center of the Earth?

Question

In the question Does gravity increase the closer to the core you get?, it was determined that gravity reduces to zero at the center of the Earth. That is logical.

However if pressure is proportional to mass and gravity, then would pressure also reduce to zero at the center of the Earth? If all the atoms in the center of the Earth are essentially weightless (equal gravity on all sides), then how can there be any pressure?

And if there is no pressure, just weightless atoms, then could voids or bubbles form in the center of the Earth? And if a bubble did form, would its ‘skin’ not feel more gravity on the outward side of the bubble, thus inducing the void to grow?

And if such a hypothetical void did grow at the center of the Earth, at what size would it stabilize?

Answer 1

It is the pressure gradient that is proportional to the local gravitational force. When that force is integrated over a distance, the pressure gradient is integrated to accumulate a total pressure.

The maximum occurs at the point towards which gravity is directed in a spherical mass, which is the center. True, gravity at that point is zero, but it and therefore the pressure gradient have already been integrated over the whole radius (and mass) of the Earth. Instead of zero, the pressure at the center will be a maximum.

The pressure at the center of Earth is about 360 GPa [1]. Note carefully that, as expected where the gravity is locally zero, the total pressure levels off until its gradient is zero at the center; but the total pressure itself is maximized there instead of dropping to zero.

From Ref. [1].

Reference

1. Lajos Volgyesi, M Moser; "The Inner Structure of the Earth", Periodica Polytechnica Chemical Engineering (1982) 26(3).

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Addendum:

A couple additional features should be noted in the above plot. First, it was mentioned above that the pressure gradient is proportional to the local gravitational force. Actually it is proportional to force per unit volume, whereas gravity at any point is a certain amount of force per unit mass. So if there is an increase in density (more mass packed into the same volume), the gradient jumps. Thus is seen in the plot where the mantle meets the denser material in the core (rock --> iron).

Second, the maximum pressure noted above is 360 GPa. This is comparable to and maybe larger than the bulk modulus of the materials that make up the Earth; for instance Encyclopedia Britannica gives a bulk modulus of only 160 GPa for steel (an iron-based alloy) under ambient conditions. This means the pressure required to balance gravity deep inside Earth is sufficient to compress the material and make it more dense than it normally would be. In fact this compression causes Earth to become denser than Mercury, even though the latter is overwhelmingly iron. Mercury is smaller and less massive than Earth, so it cannot generate as much compression from its gravity.

Answer 2

The previous answers do a fine job already. But I'll try to add a simple thought experiment. Imagine three objects floating in space, clumping together by gravity:

###   ###
#A#|c|#B#
###   ###

Mass A, the negligible mass c and mass B, equal to A. The center is attracted to A and B and their gravity cancels out. However, A presses against c, because it is attracted by B, and B presses against c from the opposite direction. So of course, c will feel the pressure from both A and B, even though it feels no gravity.

Answer 3

Pressure at the center of the earth is non-zero. You're correct that there's no gravitational force at the center of the earth, but that doesn't mean pressure is zero - the pressure comes from the many miles of rock sitting above the center of the earth.

As an analogy, think of a balloon. The pressure inside is higher than ambient because the elastic skin of the balloon pulls in everywhere on the surface. Pressure at the center of the balloon is higher, despite the fact that no force pulls inward on the center - the increased pressure comes from the outer shell of the balloon pulling inward. You can think of the earth in much the same way - even though there's no gravity acting at the center of the earth, any arbitrary "outer shell" experiences a gravitational force pulling down which contributes to the pressure below.

Answer 4

Imagine the whole ball being separated into a handful of concentric shells, with the outermost shell being the crust with the surface, and the lower shells ever deeper, hotter, and ghastly regions of the internals of our planet.

Now, just for the thought experiment, forget that each individual shell has "stuff" going on in itself, and just imagine it as a volume of whatever material (dirt, lava, etc.) there is.

The outermost shell then is acted on by gravity and pulled inwards/down. But that's all that gravity does. It does not create pressure on its own, at all! The pressure only occurs because the next layer resists the previous layer - or Earth would implode (which, incidentally, is what actually does happen in a supernova).

The principle is the same as if you heat water on the stove, in one of those pots with a screw-on lid. To create pressure, you need hot water, but you also need the lid resisting the water vapors trying to "get out". Remove either factor, and there is an open pot without any pressure.

Back to Earth: this phenomenon continues. As you noticed, the effect of gravity on each individual shell gets less and less (the shells weigh less, the farther you go down, even if you size them such that they have the same mass as the previous ones). But the downward force from all the upper layers is still there just fine - at this point, gravity has already done its work and it does not matter why the upper shells press down. All the matter there is already compressed as much as possible under the circumstances (of course not compressed as much as physically possible - our planet is much too small/light for that), so the forces from above are transmitted further down.

Obviously, in reality there aren't any well-defined shells and the effect on gravity works on all individual atoms making up Earth, but the principle remains. All that is on top presses down, and thus there is pressure at the very center, albeit there is no or very little gravity down there.

Answer 5

Even though the Earth is mostly a solid you can treat like a viscous fluid as far as pressure calculation are performed. When you are the surface of the earth there is the weight of 14.7 pounds of air molecules above you per square inch which is 1 ATM (101 325 Pa) of pressure at sea level. The Earth is composed of various materials with rock at the surface and iron at the core with ever denser materials closer to the core.

Metals like iron are also denser at the Earth's core due to immense pressure near the Earth's core. Solid metals like iron will compress by 60 to 80 percent denser due to increased pressure as compared to the density at the surface of the Earth.

At the center of the Earth the contributed gravitational field from the materials of the Earth would be near zero and any gravitational field felt would be from the Earth's moon. The pressure derived at the center of the Earth is the integration or summation of gravitational weight of all the material from the center to the Earth's to the surface 4000 miles (6436 km) above. It is true that the weight of the materials near the center of the of Earth nears zero but there also much less material to integrate as compared to materials near closer to the Earth surface. Imagine a cone of material with the cone's vertex at the center and the earth and the base portion at the surface of the earth.

Material farthest away from the Earth center contributes more material with proportional more gravitational force since it's farther from the center of the earth. Pressure is derived from what's above and not below as referenced from the earth's center pointing outward to the earth's surface.