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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.