An almost circular orbit (i.e., < < 1) can be considered to be a circular orbit to which a small perturbation has been applied. Then

An almost circular orbit (i.e., ε << 1) can be considered to be a circular orbit to which a small perturbation has been applied. Then the frequency of the radial motion is given by Equation 8.89. Consider a case in which the force law is F(r) = – k/rn (where n is an integer), and show that the apsidal angle is π/√3 – n. Thus, show that a closed orbit generally results only for the harmonic oscillator force and the inverse-square-law force (if values of n equal to or smaller than – 6 are excluded).