Earth's interior heat vs Gravity

[Mechanic]

Earth’s interior is much hotter than the surrounding environment (i.e., empty space) so there must be a tendency of the Earth to expand - yet the Earth’s volume remains essential constant. This is so because an equilibrium has been achieved in which the heat-driven expansion is balanced by a gravity-driven contraction. My question: How is this described mathematically? How do we numerically describe the equilibrium between the amount of HEAT inside the Earth and GRAVITY?

Thanks

[Mechanic]

Maybe I should post this question to another thread??? Any suggestions? Maybe Gerneal Relativity? Or one more oriented towards thermodynamics??

Thanks

[Mechanic]

I’m curious as to the lack of responses to my question. Is this a “stupid question”? Is this simply of no interest to anybody? FWIW I spent this morning plotting (MATLAB) osculating circles per exercises in Faber’s “Differential Geometry and Relativity Theory” and read the same text while relaxing with my Saturday afternoon cigar today. Meaning – I am really looking for assitance.

Thanks

[D H]

Earth’s interior is much hotter than the surrounding environment (i.e., empty space) so there must be a tendency of the Earth to expand ...
What makes you think this is so? Asteroids have lost all of their internal heat, and do not have a tendency to expand. Solids are not gases.

[DaveC426913]

Uh, well I guess I can see his point. If the Earth were magically supercooled, it would shrink a certain amount. If it were heated, it would expand. Since it is being heated, it is larger than it would be if at ambient temperature.

But the OP seems to feel that expansion would be on-going, which is not true. As you point out, a solid is not the same as a gas.

If you have a simple sphere of steel and you heat it a certain amount, it will expand a certain amount. And then it will stop expanding. Heat it hgiher and it will expand more. And then stop. There is surely a linear relationship between the sphere's temperature and its volume.

Note that the cessation of the expansion occurs independent of gravity.

So, to answer the OP's question, the Earth will have a certain expansion due to its temperature, but it is not a continual expansion, just like the steel sphere. And all this happens before factoring gravity in.

[Mechanic]

Good point - a solid is not a gas. As to the steel sphere analogy, I understand that the sphere will stop expanding whether gravity is not present. However, it seems to me that the resultant volume will be less if gravity (or any other pressure inducing force) is present. I would be surprise to learn that the steel sphere would occupy the exact same volume when placed in the vacuum of outer space (no outside pressure) or placed at the bottom of the ocean (very high outside pressure). I would expect a small volume difference based on where equilibrium is achieved.

Thanks

[D H]

OK. Now you're getting somewhere. Solids are compressible. The bulk modulus of a solid describes how that solid reacts to pressure. One way to visualize this: Think of the individual atoms or molecules that comprise a solid as being connected via springs. Just as laying down on your bed compresses the springs in your mattress, squeezing a solid makes the atoms/molecules that form the solid squeeze a little closer together. You sink in a little bit when you lie down on your mattress. That sinking stops at the point where the spring compression balances out your weight. The same thing happens when you apply pressure to a solid. The springs in a solid ball of steel are just a whole lot stiffer than the springs in your mattress.

Another thing happens to rock when it is compressed. Some rocks change chemical composition above some critical temperature/pressure. The rocks we know on the surface simply do not exist deep beneath the Earth's surface.

[Mechanic]

"The bulk modulus of a solid describes how that solid reacts to pressure."

What is the equation governing how the bulk modulus describes how a solid reacts to pressure?

Thanks

[D H]

What is the equation governing how the bulk modulus describes how a solid reacts to pressure?
RTFI. Documented very well here (http://www.google.com). I told you the exact phrase to use in the referenced website.