Standing on the edge of Niagra Falls you can watch the water pour over. Falling down the gravity of the earth, it exchanges its potential energy for kinetic energy by picking up speed. Some of that energy is extracted by turbines and lights the homes and businesses of Yankees and Canucks alike. Some of that energy is used to pump water up into water towers to maintain the water pressure which those same people use to cook and clean.
Move with a gravitational field, get energy. Move against it, lose energy.
Now let’s say you wanted to take apart the earth. Yes, the whole thing. You want to grab each bit of dirt and pull it out of the earth’s gravity and move it out to deep space. How much energy in total would it take? There’s some practical reasons for wanting to know. The engineers who built the Death Star would need for their superlaser to deliver at least that much energy. And what could be more practical than that?
But figuring out exactly how much energy requires some finesse – figure out how much energy it takes to remove a kilogram worth of earth, and suddenly the earth has one fewer kilogram worth of gravity. So the next kilogram won’t take quite as much energy to remove, and so on. We’re going to have to do some thinking about how to get around this. We’ll start with the equation that tells you the potential energy of an object in the gravitational field of a uniform spherical mass. The potential energy is just the energy required to pull the mass entirely out of the gravity of the sphere. Here G is the gravitational constant, M is the mass of the uniform sphere, m is the mass of the object in the sphere’s field, and r is the distance between the mass and the center of the sphere:
Now here’s my plan to get figure out how to take the changing mass of the sphere into account. Imagine you scrape off a very thin but uniform layer off of the entire earth, like removing the peel from an apple. The surface area of a sphere is 4 pi times its radius, so the volume of that layer is going to be 4 pi times the radius times the thickness of that layer. Its mass is its density times its volume. In the laconic language of math, that means:
Where the greek letter rho is the density of the layer, and dr is the thickness. I choose to call the thickness “dr” so that the calculus-fluent can at this point nod their heads sagely and recognize my line of attack. Now we’ll do the same thing for M, the mass of the sphere. Clearly it’s just going to be the volume of the sphere (which is 4/3 pi r^3) times its density:
Now that’s the energy required to remove one layer. All that remains to do is add up the successive energy required to remove all those infinite numbers of successive infinitely thin layers. Sounds difficult. But that’s why God created Newton, who created calculus. We can add up the total energy to remove all the layers like this:
That’s just the energy equation with M and m replaced, and an integral sign in front of it noting that we’re adding all the layers from the center of the earth at r = 0 to the normal radius of the earth r = R. Perform the integral and get:
But the density rho is itself just the mass of earth divided by its volume, which is just M/(4/3 pi R^3). Plugging that in and reducing:
And that’s our final answer, which is also called the gravitational binding energy of a sphere. For the earth, it works out to around 2.2 x 1032 joules.
That’s a preposterously huge amount of energy! It’s a solid week of the sun’s entire power output. Dumping it in about a single second, as required to blow up Alderaan, is a very, very impressive feat. Doubly so when you take into account the fact that the binding energy is just enough to dissociate the planet into a diffuse cloud. If you want to actually blow the thing up into pieces flying out at many times escape velocity, you need much more energy.
Darth Vader told us that the power to destroy a planet is insignificant next to the power of the Force. Either he’s being a typically mystical ex-Jedi offering wisdom along the lines of the pen outmatching the sword, or the Force must be pretty dang powerful.