Science Fiction?

Question

As you move away from the earth its gravity gets weaker. But suppose it did not? Suppose it got stronger? If that fictitious law were so, would it be possible for things, like the moon, to orbit the earth?

a) Yes, just as they presently do.
b) Yes, but unlike they presently do.
c) No, orbital motion could not occur.

Answer 1

b) If the force was directly proportional to distance (a Hookian response), for example, it would be like the moon was attached to a spring on a bearing. There would be no preference for circular orbits, but there could be one. You'd even get elliptical orbits, but the earth would be at the center of the ellipse instead of at a focus, and the period would be independent of distance vs. Kepler's Laws.

Answer 2

Good tough question.

Dr. R is right. And they would work -- at least for a while. But Meg might be right about the orbits being unstable. Have to figure this one out.
.................

Thanks to Meg for citing Bertrand's Theorem. The force would have to vary with r for the orbit to be stable, otherwise positive feedback caused by minor variations would make it unstable.

Go to your neighborhood skateboard park or snowboard half pipes to see such slopes with projectiles (i.e., boarders) in action. Congrats to Dr. R..

Answer 3

c) in fact stable orbits can only occur with a 1/r^2 force law. Even 1/r or 1/r^3 and it would not be possible for the moon to orbit the earth.

Edit: I looked it up and a force law that varies like r^2 , as Dr R suggested, will also produce stable orbits. see
http://en.wikipedia.org/wiki/Bertrand's_theorem

Answer 4

c. You'd have the Big Crunch, where everything would get sucked into each other.

Answer 5

csnarf?