Wednesday, October 19, 2016

How Much Does it Cost to Blow Up a Planet?

A curious reader wrote in today with an odd and ominous inquiry—how much would it cost to power the laser of the Death Star? We're by no means the first ones to turn an analytical eye to everyone's favorite space opera, but outlandish questions like this are always a good opportunity to bring a bit of fun to mathematics.

Fortunately, thanks to our legions of fellow nerds on the internet, most of the work's already been done for us: the good people at "StarDestroyer.net" went and calculated the approximate amount of energy the Death Star's laser must have imparted on to Alderaan, making the assumption that it's a planet with a roughly Earthlike mass. To do this, they found the gravitational binding energy of the planet—the amount of energy it would take to drag all the particles that comprise Earth away from one another to an infinite distance.

It might be surprising that you can pull attractive objects infinitely far apart on a finite energy budget, but it's possible thanks to the fact that the attraction between them falls off rather quickly—growing weaker as the square of the distance between them. Using this knowledge and a good bit of calculus, the StarDestroyer crew found that it would take roughly 1032 joules of energy to completely disintegrate an Earthlike planet.

We're not just talking "made uninhabitable" here, 1032 J is enough to make it look like the place never existed.
Image Credit: Still from A New Hope, poached for educational purposes.
(Please don't sue us, Disney.)
Now, that's clearly a lot. But just how much? Our reader wanted an answer in kilowatt-hours, since that's how electricity is usually billed, but a kilowatt-hour is just another unit of energy, like the joule. A watt is one joule per second (so a sixty-watt lightbulb uses sixty joules of energy/second), but a watt-hour is the amount of energy that a one-watt device would use if it ran for an hour. This is the same as the number of seconds in an hour: 3600 seconds at 1W is 3600 joules. A kilowatt-hour, then, is 1,000 times that much, or 3,600,000 joules: 3.6 megajoules (MJ). Since we can easily interconvert from joules to kWh using this factor, we can do our math in joules and then worry about converting to kWh and cost later.

Exponential numbers like 1032 are inconceivably large, though, so to get a sense for the amount of energy we're talking about here, let's look at the sun (don't literally look at the sun, please.) The sun has a luminosity of about 3.8*1026 watts, meaning it gives off that many joules per second. Already, things aren't looking good for this month's Imperial Power & Light bill—that six order-of-magnitude difference between the sun's per-second output and the Death Star's firepower means we're going to need to harness ALL the energy of a sunlike star for a while in order to get the energy necessary to blow up the planet. How long?


That looks like a lot, but it's actually only around 3 days...remember, though—we're talking about funneling all the energy of a sunlike star into some kind of battery with 100% efficiency. Although sci-fi authors have proposed ideas for structures that could do this kind of thing, the Empire has enough trouble building a planet-sized space station, let alone one that could enclose an entire star. 
How to store and unleash that much power at once is also a pretty serious problem. There are a lot of hypothetical ways to power a space station like this, fission and fusion being the most reasonable candidates (chemical energy is a no-go; you'd need about a third of the earth's mass in gasoline for each shot, at which point you might as well just throw the gas tank at your target and call it a day). But no matter what kind of alien energy storage tech the Empire has, it's not going to be able to beat the theoretical maximum, the king of energy storage efficiency: antimatter.
When a particle of antimatter meets a particle of matter, they sort of unravel one another, letting loose the energy that they contained in the form of a photon pair. This is the most direct and efficient possible means of converting matter into energy, with an output given by Einstein's most famous equation: E=m*c2. Back-calculating from that formula, we find that generating 1032 J of energy would require the annihilation energy from roughly 1.1 trillion metric tons of mass, meaning you'll need 550 billion tons of antimatter per shot. At this point, it again becomes reasonable to wonder why we're going to all the trouble of building a destruction-prone space station when you could just huck a few billion tons of antimatter at the planet and watch the fireworks—although it's possible that a sort of antimatter analogue of the Leidenfrost Effect would bounce the stuff out of the planet's atmosphere before it had a chance to touch down and do real damage.
But let's put all that aside. Here we are: you've built your space station (let's not talk about construction costs), kidnapped your princess, synthesized enough anti-lead to fill the Three Gorges Dam, and brought it into orbit around your target planet; let's talk about how much this shot will cost.
A kilowatt-hour of energy, 3.6 MJ, costs 12 cents here in the US. That number's as high as 40¢ in places like Denmark, but we'll go ahead and assume the Empire is using the ancient galactic equivalent of cheap-and-dirty fossil fuels to generate power.  (It's obviously impossible to account for things like economies of scale on a galactic level, along with inflation since "a long long time ago", so we're going to have to just sweep those under the rug. Although I'm SURE someone's gone and figured out how much a "credit" is worth in modern USD, I don't even have the patience for real economic analysis, let alone the fictional kind.)

So 1032 joules of energy at 12¢/kWh (and remember, a kWh is 3.6 MJ) gives us:

We find ourselves with another mind-bendingly large number, as expected. How much is 3*1024 dollars? Even using the most liberal definition of "money" available, by which account there's more than 1.2 quadrillion dollars in circulation and investment, this figure is still roughly 2.5 billion times the amount of money in the world.

I know that nobody wants to say it to the temperamental Sith lord with a penchant for remote asphyxiation, but with the kind of resources it took to pull off this one show of strength, the Empire could have bribed the whole galaxy into compliance a few times over.

So if you're trying to start an evil empire, maybe take a hint from a more successful one and go the Bread and Circuses route.

Stephen Skolnick

1 comment:

  1. I tend to think that in that SW galaxy that existed a long time ago and far away the laws of physics were different perhaps caused by the Force as people tended to travel among planets so quickly among other inconsistencies.

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