Ask a Physicist: Nuke the Sun?

Sharon from Pittsburgh, PA wants to know:

Would it be a bad thing to shoot our nuclear waste into the sun?

It’s a fun idea, and at first blush you might think it’d be a great way to get rid of something toxic—after all, what’s more “gone” than something incinerated in a giant fusion reactor, ninety million miles away? But let’s dig a little into how such a proposal could work, along with some potential pitfalls.
First off it’s important to note that, while disposing of nuclear waste is still a challenge, managing it isn’t actually as much of a problem as most people believe. XKCD did a fantastic What if? on “wet storage”, which is how most of the nuclear fuel in the US is currently stored. It turns out that keeping spent fuel rods at the bottom of a pool of ordinary water (albeit a well-maintained and heavily guarded one) is enough to ensure they don’t do any harm.  Currently, the plan involves letting the rods cool down in wet storage for a long time, until they can be transferred to a more permanent home—known as “dry storage”. While this would be preferable since it requires fewer resources and less active maintenance, there’s nothing explicitly wrong with leaving the rods in wet storage indefinitely. But let’s say once we decide to take our fuel out of wet storage, we go way beyond “dry storage” to “metal-vaporizingly-hot-storage”, and send it into the sun to be rid of it entirely.
Right out of the gate, the first issue you’d run into with this plan would be the price tag. Currently, it costs about $10,000 per pound of material that you want to send to space, and that’s just to put it in low-Earth orbit, where most satellites and the International Space Station reside. Sending something to the sun would probably be even more expensive, since you’ve got to pack the fuel necessary to get all the way out of the earth’s potential well, rather than just zooming around the rim fast enough to avoid falling back in, as in the low Earth orbit case. Only a few manmade objects have ever left Earth’s potential well—even the trip to the moon didn’t count! (Which makes sense when you consider that the moon is orbiting the earth.)
But let’s be generous here and stick with the $10,000/lb figure, assuming that economies of scale take care of the extra fuel costs and that our payload doesn’t have to reach the sun in a particularly timely manner. Since there’s no friction to slow it down in space, once it’s out of Earth’s potential well even a slight push in the exact right direction could set an object on a collision course with the sun.
So just how much space-waste are we talking about here? This GAO report indicates that spent nuclear fuel is accumulating at a rate of roughly 2,000 metric tons per year. Since a metric ton is 2205 pounds, that means it’d cost roughly $22,050,000 to send a metric ton of material to space. Multiply that by 2,000, and we find a cost estimate around $44 billion—just over twice NASA’s annual budget.
While that’s a lot, I was actually surprised to see it so affordable; I was expecting something orders of magnitude more than what we currently spend. Compared to the military’s annual budget, it’s practically a drop in the bucket!
However, there are other issues here, not the least of which is the possibility of a mishap. A program like this would involve rockets leaving every few days, to keep up with the 2000 metric tons per year we’d need to ship off and, while we’re pretty good at getting things to space in one piece, miscalculations and accidents do happen. The Challenger shuttle disaster was bad enough, and a national embarrassment to boot—the possibility of a similar occurrence scattering nuclear waste across the land is more than enough to make sure this proposal stays on the ground.
All in all, it seems like it’s probably not worth it to try, especially when there are so many other options for disposing of nuclear waste. However, out of curiosity, I thought it’d be interesting to do a rough calculation and see if this process would even be energy-positive—that is: do UNspent nuclear fuel rods contain enough energy to get themselves to space, or do the laws of physics guarantee that this would be a waste of time?

According to Wikipedia, the USA uses 4,686,400,000 Megawatt-hours of electricity per year, roughly 20% of which comes from nuclear power. This tells us that the 2,000 metric tons of spent fuel accumulating each year provide 20% of that number, or 937,300,000 Megawatt-hours.

Now, a watt is a unit of power, or energy per unit time. A lightbulb might be rated at 60 watts, meaning it uses 60 joules of energy every second. A Megawatt, then, is a million joules of energy per second, and a Megawatt-hour is the number of joules that get used up by a Megawatt-powered device running for a full hour. Since there are 3600 seconds in an hour, we can conclude that a Megawatt-hour is 3600 Megajoules. Multiplying the above number by this figure, we find that nuclear fuel annually provides the US with roughly 3.4*10^18 J of energy.
So is this enough to propel 2,000 tons of spent nuclear fuel out into the sun? It seems like it’d be a tough thing to figure out, but physics can provide us with a bit of a shortcut! We know the escape velocity of Earth, based on its gravitational mass; it’s actually equal to how fast something would be moving if it fell to Earth from an infinite distance, so to get something far enough away from Earth that it wouldn’t fall back down again, it has to start out moving at about that speed—25,000 miles per hour, almost exactly. Knowing that the kinetic energy of an object is given by the formula: K=(1/2)*M*v^2 where M is mass and v is velocity, we find that 2,000 tons of spent fuel rods moving at 25,000 mph (that’s 2E6 kg at 11,176 meters per second, for those of you calculating along) have a kinetic energy of 1.25*10^14 J.
While 1.25*10^14 might look like it’s a significant fraction of 13.4*10^18, it’s actually less than one part in twenty thousand! From that perspective, this idea isn’t a bad one, but most of the trouble with things like space travel isn’t in getting the energy, but converting it to the right kind of energy in a safe way. Now if only there were some way to use the nuclear power itself for spacecraft propulsion!
Thanks for writing in!

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