Let’s indulge in a little geek time. In “Star Wars” the planet Alderaan is (spoiler alert!) blown up by the Death Star. It’s a commonplace to notice that this represents a remarkable technological feat, given the size and mass of planets and the nature of gravity. Today, I want to address the following question:
Everyone knows that it would take a lot of energy to blow up a planet. If Alderaan were entirely composed of explosives, would the detonation of those explosives be sufficient to destroy the planet?
Yes, it’s a silly question, and we’re not going to address it rigorously. I just want to see how far we can get.
Binding Energy
You can reduce our question to one of energy: Do explosives release enough energy to destroy a planet made of them? It might seem that the obvious answer is “yes”, but difficulties arise when one considers what it means to “destroy a planet”, as opposed to simply re-arranging one.
A planet is basically a bunch of stuff held together by gravity. Changing the chemical composition of a planet doesn’t destroy it. “Destruction”, in this case, must refer to the dispersal of the planet’s mass in such a way that it doesn’t clump together again. The energy required to perform this dispersal/destroy the planet is called gravitational binding energy.
People with even more time on their hands than myself have estimated Alderaan’s binding energy to be 2.4e32 J. That’s based upon a realistic geology, however, not our all-explosive planet.
Explosive
To nail down this thought experiment a little, let’s consider Planet Semtex: a uniform sphere of pentaerythritol tetranitrate with a radius of 6400 km. Planet Semtex has a radius of 6.4e8 cm, and therefore a volume of (about) 1.1e27 cubic cm. PETN has a density of 1.77 g per cubic cm, and an explosive energy of 5810 J/g, so the chemical energy bottled up inside Planet Semtex is about 1.1e31 J. Note that this is about 5% of the energy needed to destroy Alderaan.
Calculation
Might 1.1e31 J be sufficient to destroy Planet Semtex, however? Could our choice of material drastically cut the binding energy?
Well, the gravitational binding energy of a uniform sphere is 3*G*M*M/(5*r), where G is the gravitational constant, M is the mass of the sphere in kilograms, and r is the radius of the sphere in meters. For Planet Semtex:
G = 6.673e-11 m^3/kg*s^2
M = 1.947e24 kg
r = 6.4e6 m
U = 2.37e31 J
So, no. Even though the binding energy of Planet Semtex is only about 10% that of Alderaan, Planet Semtex, if detonated, would still generate only about 50% of the energy required to blow itself up.
