I noticed in my post of 02 AUG 06 that I had posted a couple of weeks
ago that I had neglected to include some further information about
the case of nearly circular orbits that I had intended to include.
Apparently I was so focused on erradicating typos that I didn't
notice the omission. So in the interest of completeness I'll,
belatedly, mention here that extra info concerning the case of
closed nearly circular orbits for a Newtonian particle subject to a
central force/potential.
If we leave things in terms of the radial coordinate for the system
as the usual polar coordinate r-variable (rather than change
variables to the inverse radial distance u = 1/r or to its
dimensionless version w = u*(L/sqrt(2*m*|E|)) ), and if we aren't
interested in the actual values of the periapsis r_p and the apoapsis
r_a which are slightly above and below the mean circular orbital
radius r_0 of the nearly circular orbit then we can write a nice
elegant expression for the closed orbit criterion. So as to not bore
the list readership too much, I'll only give the result here and skip
the intermediate mathematical steps of the derivation of the result.
If we let n be the number of radial oscillations in the orbit per
revolution, and if we let F(r) be the *magnitude* of the attractive
central force as a function of the distance from the force center,
(i.e. F(r) = dV(r)/dr so that -F(r) is the actual outward radial
force component) then the criterion for the nearly circular orbit
also being a simple closed orbit is to have n be a positive integer
where n is given nicely in terms of a log-log derivative as:
n^2 - 3 = (dln(F(r))/dln(r))_0
where the notation (...)_0 around the derivative is supposed to
signify that the derivative is to be evaluated at the value r = r_0
which is the mean radius of the nearly circular orbit about which the
actual orbital radius very slightly oscillates about as the orbit
proceeds.
The value of r_0 is given (as is usual for circular orbits by
requiring that the central force be at all times 100% centripetal
with no tangential component) as the value r which implicitly
satisfies the force balance equation:
F(r_0)*r_0^3 = L^2/m
given that m and L are the fixed values of the mass and the orbital
angular momentum magnitude respectively.
Notice that what this criterion in terms of a log-log derivative
above tells us is that n^2 - 3 is the exponent of the "effective
power-law" force function whose value and slope matches that of the
actual force law at a radial distance that is the mean radial
distance of the nearly circular orbit. This makes the application of
this criterion to the special case of the nearly circular orbits for
an *actual* power law force *especially* trivial to implement.