In this post, I’m adding an interesting extension to a popular physics problem that’s been circulating on the internet.
The original problem is to calculate the average power output of the Death Star when it blows up Alderaan, making the following assumptions:
- The shot takes half a second.
- Alderaan has the same mass and radius as the Earth.
- The amount of energy delivered is simply equal to the gravitational binding energy of the planet (i.e., the amount of energy required to break it up into tiny pieces and separate them by huge distances).
The extension that I’d like to add is this: Calculate the recoil velocity of the Death Star, and discuss the effect that the associated acceleration would have on the occupants.
First, a quick recap of the original problem.
Gravitational binding energy of a planet is given by the formula
where M and R are the mass and radius of the planet, and G is the gravitational constant. Plugging in the numbers for Earth gives us a total energy of 2×10^32 J. If this much energy is delivered in half a second, the power output required is 4×10^32 W. This is one million times the total power output of the Sun!
Okay, now for the extension. The amount of momentum carried by light is simply the amount of energy in the light divided by the speed of light:
By conservation of momentum, this must also be the amount of momentum carried by the Death Star as it recoils backward (ignoring certain subtle effects that won’t make a big difference in the result). To get the recoil velocity, we simply divide the momentum by the Death Star’s mass.
Crap. What is the Death Star’s mass? It’s got to be somewhere in that technical readout that R2 was carrying…
We’ll just have to get a decent approximation. According to Wikipedia, the diameter of the Death Star is 160 km, so it has a radius of 80 km. Assuming a density similar to that of our moon (3000 kg/m^3), the mass of the Death Star is 6×10^18 kg. In reality, since the Death Star isn’t solid, its mass would probably be less than this, in which case the actual recoil velocity would be greater than the number we’re about to get. But, hey, we’re just looking for an estimate anyway.
Okay, we now have everything we need to calculate the recoil velocity. Are you ready?
It turns out that the Death Star would have to recoil with a velocity of one hundred thousand meters per second. That’s about 360,000 km/h, or 200,000 mph, which is roughly eight times the Earth’s escape velocity. Accelerating to this velocity in half a second amounts to about 20,000 g’s. That’s more than enough to splatter all of the Imperial personnel on board, plumbers and all, into a runny pulp against the walls.
Just imagine the embarrassment on Admiral Tarkin’s face when he realizes, during that split second before he meets his demise against the wall of his command room, how stupid it is to stand on a battle station that’s about to put out so much energy in such a short time in only one direction!