A Point or Five About Lagrange

It's kind of a shame that comic book writers often do such a half-assed job of research for their pseudoscience. From Final Crisis #1:

So if you know a little bit about astronomy, the only sensible response to the Guardian's instruction is "Which Lagrange point?"

Back in the 18th century, a really smart mathematician named Joseph-Louis Lagrange was working with Newtonian mechanics on what's known in the physics biz as the "three-body problem". Basically, you use Newton's (& Leibnitz's) calculus to solve Newton's law of gravity. If there's one thing in the universe, the solution is trivial. If there are two things in the universe, the solution is straightforward and exact (what the mathematicians call "analytic"): The center-of-mass of the system has the trivial solution of the one-body problem, and the two objects move relative to the center-of-mass along paths which are one of the classic "conic sections"--circle, ellipse, parabola, and hyperbola.

However, if you add a third body to the system, now there is no analytic solution. The only way you can predict the behavior of the system is to model it numerically, solving the differential equations and propagating them forward in time. This is subject to numeric error, which is why even today astronomers can't predict the position of the objects in the solar system out to more than ten million years or so.

What Lagrange discovered was pretty neat: There is an analytic solution for a very limited case of the three-body problem. The first object has to be big and massive, and the second not so big, and the third has to be so small that it doesn't count at all. The first two objects are orbiting in the circular (or nearly circular elliptical) solution of the two-body problem, and then you can put the third object at one of five points and it'll orbit along.

Thus, you see, when the Guardian says "Seal the crime scene out to the planet's Lagrange point", the first reasonable response is "Which Lagrange point?"

The five are, in order:

• On a line between the first body and the second body, closer to the second body. The opposed gravitational forces on the third body mean that it orbits as fast as the second body around the first body, when normally being closer to the first body would make it faster.
• Beyond the second body, on a line with it and the first body. The added gravity of the first and second body speed up the third body and make it orbit as fast as the second body, when normally being further away from the first body would make it slower.
• On the other side of the first body, on a line with it and the second body. This is the "Counter-Earth" position.
• In the second body's orbit, ahead of it, so that the three bodies form an equilateral triangle.
• In the second body's orbit, behind it, so that the three bodies form an equilateral triangle.

It turns out that the first three points are only of mathematical interest, because they're unstable--if you drift away from them, you keep moving away from them. (Technically, the stability is a saddle--stable azimuthally, unstable radially.) L4 and L5, however, are stable--if you drift away from them, you're going to tend to move back toward them. There's quite a lot of cosmic debris in the form of the Trojan asteroids sitting in Jupiter's L4 and L5 points, for example.

Now, the second reasonable response to the Guardian is a totally different way to ask "Which Lagrange point?", because not only are there five Lagrange points, but the Earth is involved in two sets of them: The Sun-Earth system and the Earth-Moon system.

The Guardian also makes reference to the "gravity well", and that's another problem. Normally, the way you'd use "gravity well" is just as the gravitational field set up by an object with mass. A gravity well doesn't really have a fixed outer limit, because the gravitational field slowly drops off to, but never reaches, zero. In any case, anywhere within the solar system, you're deep within the gravity well of the sun. It's only when you're very near a planet that its gravitational field is more important than the sun's. And that's what leads us to the only way to interpret the Guardian's admonition so that it makes sense: The limit of the Earth's gravity well in this sense is the point at which an object is more strongly attracted to the Earth than it is to any other object.

But that just leads us to another problem, which is that that definition of gravity well has nothing to do with the Lagrange points. None of the Lagrange points (not even L1) are sitting at a point of equipoise, where the gravitational effects of the two bodies cancel out.

What I might have written, if I were somehow to need to write such a scene: "Seal the crime scene out to the first Lagrange point. Interdict and inspect all extraplanetary traffic."

But then I wouldn't have written the scene that way at all, because I would have the Guardians be smart enough to realize that spatial interdiction won't be effective because the perpetrator of a 1011-Deicide is unlikely to need to travel through mundane space to escape and, in particular, because of the near certainty that the assassin of a New God travels by Boom Tube.

I'm surprised I have to explain this.

Posted by Greg at June 7, 2008 10:32 PM