Pulsed gravity orbits (was Re: kinetic energy paradox?)

[John Mallinckrodt <ajmallinckro@CSUPOMONA.EDU>]

On Tue, 23 Oct 2001, Carl E. Mungan wrote:

> I could perhaps imagine making gravity turn on and off periodically
> in short bursts. (A bunch of equally spaced gravity shields
> surrounding the sun with only small open slits between them?) Then
> earth would go around the sun in a nearly equilateral figure. (With a
> smaller average radius, assuming we start with the same total
> mechanical energy, owing to the decreased duty cycle.) Yet its KE
> would still be constant.
>
> It is amusing to push this idea to its limits and consider earth
> moving around the sun in an equilateral triangle (with very slightly
> rounded corners).

To take this topic fully off on the tangent suggested by Carl, I
wonder how many of us have ever considered the shape of orbits
under the influence of a central force of the form

  F(r) = F_o(r/R_o)^N

where F_o and R_o are constants and N is a large positive number
like, say, 100.  Such a force yields orbits for virtually any
initial condition that all have apoapses near R_o.  They behave as
if gravity is essentially turned off until the body reaches R_o.
When they reach R_o they sharply turn as if spectrally reflecting
from the interior of a circle of radius R_o.  By properly tuning
the initial conditions, one can generate an orbit in the pattern
of a regular polygon with an arbitrary (within reason) number of
sides or an arbitrarily rapidly precessing polygon.  All of these
properties follow immediately from the fact that such a force
gives rise to a potential energy function that is virtually flat
out to R_o and rises sharply beyond R_o.

John Mallinckrodt      mailto:ajm@csupomona.edu
Cal Poly Pomona          http://www.csupomona.edu/~ajm