What the heck is specific heat capacity? This is a term that tells you how much energy per mass per degree Celsius an object has. It depends only on the type of material. If you have a gram of water and a gram of styrofoam at the same temperature, the water will have more energy because it has a higher specific heat capacity. This means you need to know the kind of material you are getting energy from; for geothermal, it’s mostly rock near the surface and iron in the core.
For the interior of the Earth, this thermal energy comes from two sources: gravity and radioactivity. The gravity part has to do with the formation of the planet. Stuff in the early solar system had a gravitational attraction to other stuff such that it “fell” together. As chunks of matter moved together, they increased in speed and collided, getting hotter.
So, you go through this process of changing from gravitational potential energy to an increase in kinetic energy and then finally an increase in thermal energy. The same thing happens when you drop something on the floor. The object might have started off with gravitational potential energy, but then it ended up on the ground with a slightly higher temperature. That’s what happened with the Earth.
OK, but that was a long time ago. Why is it still hot? It’s true that the Earth has been cooling off for like 5 billions years, radiating energy out into space. But the reason it’s still hot inside has to do with the physics of scale. In short, big things are not like small things. The thermal energy in the interior of the Earth is proportional to its volume, which scales as the cube of the planet’s radius (r3). The radiant loss of energy goes through the surface of the Earth, which is proportional to the square of the radius (r2).
What that means: If you double the radius, the thermal energy increases by a factor of 8 (= 23), but the surface area increases only by a factor of 4 (= 22). So the larger object, the longer it takes to cool off. That’s why the moon’s interior is much cooler than the Earth’s.
The gravitational formation of the Earth is not enough to account for its current interior temperature, though. The other source of energy is the radioactive decay of some heavier elements like uranium, thorium, and potassium.
So how long would it take to use up all of our planet’s thermal energy? That depends on how much there is and how fast we deplete it.
How Much Is There?
Let’s start by estimating the total thermal energy in the Earth. Just to be clear, estimations are like onions—no, not because they make you cry. It’s because estimations have layers. (Parfaits have layers too, and they don’t make you cry.)
At the outermost layer of this estimation problem, I can just use some rough assumptions. I like to start simple and see how far I get; you can always drill down and complicate things later if it seems necessary. So let’s just start with the following data: