All right, time for an actual example of this gravitational force law I’ve been ruminating on for the last two days. Today we’ll look at an alternate version of the gravitational potential that’s truly screwy:
The first thing to notice is that it’s not finite as r becomes very large. Among other things, that means there’s no such thing as an escape velocity in a universe with this kind of gravity. What goes up really must come down.
To say more, we ought to take a look at the effective potential, which takes into account the angular momentum of the orbiting object. We’ll write that effective potential down and call it V:
Here the letter “l” is the angular momentum, which is the orbiting body’s mass times its velocity times its distance from the center of the orbit. A circular orbit will occur if the orbital radius r is such that the particle is held right at the bottom of the effective potential well. If there is such a location, it’ll have to be where the derivative of the potential is zero.
In our case we can find that location pretty easily:
Which implies that the radius which produces a circular orbit is:
The condition that the orbit be a stable one is that this location is actually a local minimum of the potential. A theorem from calculus tells us that the test for this is that the second derivative of the potential is positive at that location. We can do that pretty easily too:
Substituting in the particular r that we found for the radius of the circular orbit gives:
(My notation’s a little sloppy, generally you should be sure to indicate that the derivative is being evaluated at a specific point.)
Anyway. We know that m is positive and both squared terms are positive, so the second derivative at that point is certainly positive. Therefore that point is a true minimum of the potential energy and the orbit is stable. Who’d have thunk it?
I’m not prepared to say that it’s a realistic possibility for the law of gravitation in some alternate universe. The bare fact that orbits are stable doesn’t say much about much. We don’t know if stars would be stable, and that’s certainly a bare minimum requirement for an interesting universe with life as we know it. But it is quite interesting to see that such a wild variation of gravity does in fact result in at least a few features of normal gravity in our own universe.