Simple orbital mechanics

The Moon is slowly siphoning off the Earth’s rotational energy, in a phenomenon called tidal acceleration. This moves the Moon’s orbit farther from the Earth. According to Kepler’s laws of planetary motion, the Moon must orbit more slowly as it gets farther away. Well, which is it? Is the Moon speeding up, or slowing down?

Here’s how I think about it. Suppose we’re in a spaceship in a circular orbit around a massive body. We’re orbiting at, say, 10 km/sec. (I’m just making up these numbers, so the diagrams are not to scale. And it doesn’t matter whether we’re moving clockwise or counterclockwise, so just pick one.)

At some point A in our orbit, we fire our engines, quickly accelerating from 10 km/sec to 12 km/sec. Assuming 12 km/sec is less than the escape velocity at point A, this puts us in an elliptical orbit: the purple orbit shown below.

Notice that the new orbit necessarily returns to point A, the point at which we last used our engines. Whenever our orbit brings us back to point A, we’ll again be moving at 12 km/sec. But at the farthest point of our orbit, point B, we’ll only be only moving at, let’s say, 6 km/sec.

We wait until we are at point B, and fire our engines again, quickly accelerating from 6 km/sec to 8 km/sec. Let’s suppose this makes our orbit circular (the green circle below), so we are now moving at 8 km/sec all the way around.

We’ve accelerated twice, adding a total of 4 km/sec to our velocity. And yet we’ve somehow slowed down from 10 km/sec to 8 km/sec. It’s counter-intuitive, but that’s how it works.

The idealized form of the orbital maneuver we just performed is called a Hohmann transfer orbit. A Hohmann transfer is usually the most energy-efficient way to get from one circular orbit, to another circular orbit in the same plane.

Instead of firing our engines in short bursts, we could have fired them continuously at low power, in the right direction to keep our (would-be) orbit circular at all times. We would then be continuously accelerating, yet continuously slowing down. This is similar to what happens with the Moon.

Imagine the Earth as being at the bottom of a depression: its gravity well. As the Moon moves farther away, up and out of Earth’s gravity well, it loses kinetic energy, while gaining gravitational potential energy. The gain in potential energy is larger than the loss in kinetic energy, so the Moon gains energy overall, even as it slows down.

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