While working on an entirely different topic (rhymes with ‘fissile’), I got to wondering — how does the Earth’s gravity vary with depth?
Let’s start by examining what causes gravity. We have Newton’s law for gravitation:
We can use this to calculate the acceleration towards the center of the Earth at any position (which allows us to drop the unit vector and the minus sign).
This is all well and good — when we are outside the Earth. But what if we’re inside, digging down for treasure or dinosaurs? The M here is the mass pulling us down.
So what happens at the center of the Earth? We have no mass below us; in effect, all the mass around us cancels out the gravitational attraction: in the center of the Earth, you’re weightless!(And very likely compressed and on fire, but let’s ignore those inconvenient details.) So pack as many snacks as you want, you won’t gain weight, and if you do, you’ll quite literally burn off the calories (as well as everything else).
Let’s express this more rigorously. The mass below you is:
If we assume the density is constant, we can calculate the average density for the Earth at about 5495.051 kg/m³. That itself is interesting: it’s far above any form of igneous rocks (basalt, granite), which themselves are denser than sedimentary rocks (limestone, shale). This implies that whatever is down there, is definitely compressed, and our assumption about constant density is wrong from the get-go!
Let’s proceed undeterred by this litany of errors. If we assume constant density, everything is easier, and that’s what we prefer:
This is where things get interesting (I promise). If we plug this into the equation we found for gravity, the r³ cancels out against the 1/r²!
That leaves us with a linear equation for gravity below the Earth’s surface! If we combine this with the previous equation for gravity above, we get:
We can plot this, getting something that has a peak at the Earth’s surface at about 9.81 m/s² (with the maximum amount of mass below you), then dropping off as you fly off into space (hope you packed some oxygen), and dropping off linearly as you dig into the Earth (take a strong shovel).