Close encounters of the gravitational kind Adapted from the original by Pat Murphy and Paul Doherty

We're going to start with an ordinary baseball and an ordinary basketball, and we're going to end up on a grand tour of the solar system. Really.

What do I need?
A tennis ball

You've probably noticed that if you drop even the bounciest of tennis balls from a height, it never bounces back higher than where it started. When you drop the ball, gravity pulls it down and it picks up speed. It hits the ground and squashes at the moment of impact. As the squashed ball springs back to its original shape, it pushes on the floor and the floor pushes back. The force of the floor pushing against the ball throws the ball back up into the air.

The reason it doesn't bounce higher than where it started is simple: some of the ball's energy is lost as heat when it bounces, so it doesn't have as much going up as it did coming down. Knowing that, you might figure that a ball could never bounce higher than the height from which is was dropped. And you might think that this discussion doesn't have much to do with expanding the orbit of the earth. But, as happens so often with science, you'll be surprised on both counts.

Try this!

Hold the basketball and the tennis ball at the same height, one in each hand. Drop them both to the floor at the same time. If you watch, you'll see that both balls will only bounce about three-quarters of the height from which they were dropped.

Now, hold the tennis ball on top of the basketball, and drop the two together. What happens to the tennis ball? It takes off like a rocket, bouncing back much higher than where it started.

What's going on?

The balls accelerate toward the floor and are going about 4 meters/second when they hit. The basketball hits the floor first and reverses direction, heading up at 4 meters/second. The tennis ball is still going down at 4 meters/second.

At least, that's how fast the tennis ball is going if you're watching it from the ground. But what if you were a tiny person standing on the surface of the basketball? You'd see the tennis ball traveling toward you at 8 meters/second. Its speed relative to you would be 8 meters/second.

If that sounds hard to comprehend, think of this analogy: Let's say you're driving down the road at 60 miles per hour. A car on the other side of the road is coming toward you at 60 mph. Relative to the road, you're both traveling 60 mph. But if you were to run into each other, the impact would be the sum of how fast each of you were travelling -- in other words, 120 mph.

Now, back to the bouncing balls. The tennis ball smacks into the basketball and heads in the other direction. Since little energy is lost in the collision, the tennis ball leaves the basketball at nearly the same speed at which it arrived. Because the basketball is more massive than the tennis ball, the collision doesn't slow down the basketball much. The tennis ball, on the other hand, reverses direction. From your viewpoint on the basketball, the relative speed of the balls remains constant. After the balls hit, they separate at 8 meters/second.

Ah, but here's the tricky bit. For a person standing on the ground and watching the balls bounce, the picture is different. That basketball is still moving up at 4 meters/second. The tennis ball is going up 8 meters/second faster than the basketball. So, relative to someone outside either of the balls, the tennis ball is moving up at 12 meters/second, rather than just 4 meters/second. That's triple its original speed with respect to the earth! With so much speed, the ball bounces 9 times higher than the height from which it was dropped, over your head and hopefully not into any household furnishings.

Where did it get the energy to do this? From the basketball. It takes a lot of energy to move that massive basketball. When the tennis ball bounced off the basketball, it gained just a little bit of the basketball's kinetic energy. If you watched really closely, you'd notice that the basketball dropped in tandem with the tennis ball doesn't bounce quite as high as the basketball dropped alone. That's because the tennis ball stole a bit of the basketball's energy.

The general rule is easy: when a ball bounces off a much heavier moving object and doesn't lose any energy to heat, it reverses its direction and gains twice the speed of the object it bounced off of. This means that a baseball leaves the batter at the speed the pitcher threw the ball plus twice the speed of the bat (minus some speed lost as a result of heat). It also means that a golf ball that is initially at rest leaves the tee at twice the speed of the striking club head (again minus a bit for heat).

That's nice. But what does it have to do with spacecraft and orbits?

Let's start with the gravitational assists made by NASA spacecraft. These "slingshots" accelerate the spacecraft, helping it make the distance to Jupiter, Saturn, or whatever its destination may be.

Let's say you've got a spacecraft that's orbiting the sun at the same distance as the earth. The spacecraft is traveling in the opposite direction as the earth--the earth orbits counterclockwise, and the spacecraft orbits clockwise. Both are going 30 kilometers/second. The spacecraft comes in towards the planet, swings around it in a cosmic do-se-do, and leaves moving out along the line of its approach.

Even though the spacecraft went around the Earth, rather than running into it, it's considered a planetary collision, sort of like the tennis ball and basketball hitting each other. It's called a collision because the spacecraft acts like it's just hit something: it reverses direction and heads back the other way at about the same speed. Or at least that's what you see if you're standing on the Earth. But suppose you back up and look at the collision in the frame of the distant stars. Then you see a spacecraft initially orbiting the sun at 30 kilometers/second. After the collision, you see a spacecraft going 90 kilometers/second! The spacecraft is leaving the earth at 60 kilometers/second and the earth is going 30 kilometers/second so 60 + 30 = 90! That's fast enough to give the spacecraft escape velocity from the sun, heading out toward interstellar space along a hyperbolic trajectory. The spacecraft gains kinetic energy in this encounter. Where does that energy come from? Well, just as the encounter with the tennis ball slowed the basketball down, the encounter with Galileo slowed the earth down. Not by much, of course. When the Galileo spacecraft swung by earth, it sped up by over 16,000 kilometers per hour with respect to the sun, and the earth slowed down by 10 billionths of a centimeter per year. To increase the diameter of the Earth's orbit, we'd just need to switch roles; use a slingshot to accelerate the Earth, rather than the Earth to accelerating a spacecraft. That would entail a very massive enough object -- say, 1018 kilograms (the size of a large asteroid). If the asteroid were brought in from the same direction as the Earth is orbiting, it would still swing by and be sent back the way it came into space. But because of its mass, it would exert a gravitational pull on the planet, nudging us just a wee bit wider in our orbit. According to Korycansky, who devised the Earth-moving idea, if we arranged for 6000 fly-bys of the navigational asteroid over the next billion years, we could slowly move the planet into what would be a habitable distance from the sun once it's begun its expansion. Now that's planning ahead.