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Shooting for The Sun

Tom Napier

Volume 20.2, June 2010

Why the sun is a poor dumping ground for nuclear waste

Every time the question of nuclear waste disposal comes up, someone is sure to say, “Why not shoot it into the Sun?” It seems so obvious. The Sun’s a fine big target, and you’ll get there if you shoot straight up. The usual objection is that if a launch rocket failed, we’d get nuclear waste dumped on our heads. No one seems to consider the real objection: the Sun is the most inaccessible destination in our entire solar system.

Moving in Space

Let’s consider some orbital dynamics. If we launch a spacecraft at “escape velocity” (11.2 km/sec), it will never fall back to Earth. As it still shares Earth’s orbital velocity, some 30 km/sec, it remains in orbit around the Sun. It can’t fall into it any more than the Earth can.

A spacecraft launched with more than escape velocity ends up with a residual velocity relative to Earth. If this is directed in the same direction Earth is moving, the spacecraft will enter an elliptical orbit that will take it farther from the Sun. After a little more than a year it will return to Earth’s orbit, but Earth won’t be there. It will have had time to go around a little more than once.

If we direct our spacecraft’s residual velocity against the direction Earth is moving, it will enter an elliptical orbit that will take it closer to the Sun. It will take less than a year to return to Earth’s orbit and, once again, the Earth won’t be there. However, this is a step in the right direction.

The bad news is that to hit the Sun requires reducing the spacecraft’s velocity by nearly all of Earth’s orbital velocity. That is, we have to slow it by 30 km/sec. This is an enormous change in velocity; space probes to the nearer planets make velocity changes of only the order of 4 km/sec.

Since we also have to apply 11.2 km/sec just to get into space, it might seem that we need an acceleration of 41 km/sec to get to the Sun. The good news is that velocities can’t be combined; in space energies are combined. To get from the surface of Earth to the Sun we need an acceleration of “only” 32 km/sec. So how big a rocket do we need to dump one ton of waste into the Sun?

Now for Some Rocket Science

The acceleration a rocket provides depends on two things: how powerful a fuel it uses and how little the rocket’s structure and payload weigh relative to the amount of fuel it can carry. I’ll use some optimistic numbers to spare you the mathematics. For brevity, I’ll use “weighs” to mean “has a mass of.”

Assume we use the best known fuel (liquid hydrogen and liquid oxygen) and suppose that the fuel tanks, the engines, and the control system are 10 percent of the initial mass of the rocket. That is, 90 percent of its take-off mass is fuel. For the moment I’ll ignore the mass of the payload—that is, whatever useful cargo the rocket carries.

The maximum velocity this idealized rocket can achieve is just over 10 km/sec, not quite enough to escape from Earth and a long way short of the 32 km/sec we need to reach the Sun. As soon as we add a load of any kind, the final velocity will be lower. If the payload weighs the same as the structure, the final velocity will be 7.5 km/sec. Our rocket will head into space, slow to a stop, and fall back down again.

The secret of spaceflight is “staging.” Take that 7.5 km/sec rocket and put it on top of a much bigger rocket with the same proportions. The result is a rocket capable of accelerating the original payload to 15 km/sec but at the cost of a large increase in the take-off mass.

Suppose the payload weighs one ton, and assume that the structure of the topmost stage weighs the same. This structure holds nine tons of fuel, so the total mass of the top stage is eleven tons. This mass forms the payload of the bigger stage that lifts it, so the latter weighs 121 tons. If we put another stage underneath, it will weigh 1,331 tons. That’s a total of 1,463 tons, about half as much as a Saturn-V. If each stage adds 7.5 km/sec, we are up to 22.5 km/sec, a remarkable velocity but way short of the 32 km/sec we need.

To cut a long story short, the final Sun rocket not only has to have four stages, but the payload of each stage has to be cut to about 74 percent of the structure mass. To dispose of one ton of nuclear waste will require a 44,000-ton rocket. If we assume a more realistic launch mass of 3,000 tons (about Saturn-V size), the payload that finally reaches the sun will weigh about 68 kg (under 150 lbs). The trash bill comes to about $8 million per pound.

This calculation was based on rather optimistic values for fuel energy and structure mass. It also ignores the fact that a large part of the payload should consist of a steel canister. This serves three purposes: it provides radiation shielding for ground handling, it ensures that a launch failure won’t disperse the waste, and it gives the payload some chance of reaching the Sun without being vaporized and carried away in the solar wind.

Aiming for the Stars

Putting a spacecraft on a non-return flight into interstellar space is much easier than hitting the Sun. To reach the Sun you need to subtract 100 percent of Earth’s orbital velocity; to reach solar escape velocity you need only add 41 percent to it. The total velocity increment from takeoff becomes 16.73 km/sec. A three-stage rocket with a launch mass of 120 tons could send one ton of nuclear waste to the stars; a Saturn-V sized rocket could send twenty-five tons.

The Sun would be a great trash-can, but it’s too darn difficult to get to.

Tom Napier

Tom Napier is a retired physicist/engineer.  He co-founded the Philadelphia Association for Critical Thinking (PhACT) and has written extensively on pseudoscience.