Anniversary: Kepler’s Third Law of Planetary Motion

I’m not sure of this, but maybe on this day, our man Johannes Kepler discovered his third law of planetary motion. Now we all know the first law, right? Planets orbit the Sun in ellipses where the Sun is at one of the foci (which I would write “focuses” if this weren’t math we were talking about). The reason that this was a big discovery is because most planets don’t have very elliptical orbits. So it was pretty subtle. The eccentricity of the earth is just 0.017 — damned close to a circle.

The second law of planetary motion is really the most interesting. Basically what it says is that planets move faster when they get closer to the Sun. Now this is exactly what we would expect, but that’s just because we all know Newton. But at that time, Newton wouldn’t even be born for another 25 years. The way the law is presented is this: “A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.” That’s the thing about Kepler: he was really a mathematician. He thought in terms of geometry. That’s probably why I’ve always felt a special kinship to him.

Now the third law of planetary motion, well, that’s an odd one. It is stated thusly: “The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.” Now that’s real analytic geometry! The semi-major axis is more or less the radius for objects like the Earth that move more or less in circles. For a highly elliptical object like Halley’s Comet, you can think of it as half the distance from its closest and furthest approaches to the Sun. So imagine that: this mathematical relationship is the same for the Earth as it is for Halley’s Comet.

Big Caveat on Planetary Motion

I’ve been slowly making my way through Small Gods and there is some discussion in there of how philosophers only raise questions. Any question that is raised only leads to dozens of new ones. So all this stuff that Kepler worked out is true in a universe that has exactly two objects. Clearly, all the planets affect each other. And that’s especially true, I would assume, of Halley’s Comet. Plus, Halley’s comet is losing material via the solar wind, which has got to screw up its momentum. So nothing I’ve discussed is exactly true. But then, what is truth? Is truth beauty? Or is beauty truth? Or should we all be weapons designers?

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