Orbital Mechanics

Maneuvering a spacecraft in orbit requires some tactics that most of us would find counter-intuitive. This morning I got to pondering exactly how a simple maneuver would be accomplished. I set my self a simple problem: suppose I was in a perfectly circular orbit, 1 kilometer (0.6 miles) further from Earth than the space station I wanted to dock with, which was orbiting in a perfectly circular orbit at 500 kilometers (311 miles) above the Earth's surface. What maneuvers would I need to make?

Orbital mechanics doesn't get much simpler than this.

First thing you'd notice is that the space station is moving ahead of you in its orbit. That's because at its slightly lower altitude, it needs to move about a half millimeter per second (1.2 miles per hour) faster than you in order to stay in orbit. That's a leisurely walking pace, so you'd see the space station moving slowly past you – and you'd be forgiven for thinking that you had to speed up to “catch up” to it. But in fact, you need to do exactly the opposite! You need to slow down very slightly so that you will fall down to its altitude. So you aim your maneuvering rocket to slow yourself down, and light it off for a short blast, just enough to put you into an elliptical orbit whose perigee (low point) is 500 kilometers.

Now you're moving more slowly, and falling behind the space station more quickly – but as you fall, you gain a little more speed (just as you would if you jumped off a stool). Not enough to catch up, though, and when you've fallen to 500 kilometers, you're still a little bit behind the space station. But no matter – because you're in an elliptical orbit, you're actually traveling ever-so-slightly faster than the space station, so you slowly approach it. At that point, you need to slow down some more, to match the space station's speed and orbit.

You started out above the space station, moving more slowly than it was. You made two maneuvers to dock with it: both of them slowing you down! It seems weird, and somehow wrong, but that's really how it works. All of the strangeness derives from a simple fact: satellites in higher orbits have more energy than those in lower orbits. We had to slow down (twice!) to dock with the space station because we were in a higher orbit, and had more energy (for our mass) than the space station did – so we had to get rid of some...

The above analysis is based on a simple formula for the velocity of a satellite in a perfectly circular orbit around the Earth. The formula, along with other information, can be found here, here, and here.