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

[David Bowman <David_Bowman@GEORGETOWNCOLLEGE.EDU>]

Regarding John Mallinckrodt's tangent:

> 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.

I've considered the problem, but not necessarily for a particularly
large value of N.  In general, the problem is *analytically* quite
difficult (but not necessarily for a numerical solution) in that the
necessary integral doesn't yield a nice simple analytic formula for
the orbit for arbitrary N values with arbitrary energy and angular
momentum.

I have only considered the two simplifying cases when either: a) N
takes on only the particular values -2, and 1 for all possible
energies and all possible angular momenta, and b) The orbits only
differ from circles by an infinitesimal amount making the angular
momentum as nearly as large as possible for a given amount of energy
for a continuum of all possible N values.  It never occurred to me to
think about the c) N --> infinity limit special case.  This latter
case is effectively a free particle in a hard-walled spherical box
(of radius R_0).  In this latter case the particle repeatedly
rebounds off the inner wall of the box obeying the rule that the
angle of incidence equals the angle of reflection, and when the
particle is not colliding with the wall it is just traveling in a
straight line chord across the inside of the box.

> 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.

If we consider case b) above for nearly circular orbits we find that
the orbit tends to close on itself in a plane figure with the
symmetry of a regular m-gon (but nearly circular in actual shape)
when the value of N obeys the formula: N = m^2 - 3 .  Here m is the
number of radial oscillations (for a bound state) per orbit.  When m
is an integer the orbit tends to be a closed figure.  The interesting
cases are N = -2, (Kepler problem with the force center at a focus of
an elliptical orbit, N = 1 (3-d SHO problem) where the force center
is at the center of an elliptical orbit, N = 6 where the orbit is
slightly distorted from a circle with the symmetry of an equilateral
triangle, N = 13 where the orbit is slightly distorted from a circle
with the symmetry of a square, N = 22 where the orbit is slightly
distorted from a circle with the symmetry of a regular pentagon, etc.

David Bowman
David_Bowman@georgetowncollege.edu