An Extension of Newton's Apsidal Precession Theorem
Monthly Notices of the Royal Astronomical Society
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Newton's apsidal precession theorem in Proposition 45 of Book I of the ‘Principia’ has great mathematical, physical, astronomical and historical interest. The lunar theory and the precession of the perihelion of the planet Mercury are but two examples of the applications of this theorem. We have examined the precession of orbits under varying force laws as measured by the apsidal angle θ(N, e), where N is the index for the centripetal force law, for varying eccentricity e. The paper derives a general function for the apsidal angle, dependent only on e and N as the potential is spherically symmetric. Further, we explore approximate ways of the solution of this equation, in the neighbourhood of N= 2 which happens to be the case of greatest historical interest. Exact solutions are derived where they are possible. The first derivatives ∂θ/∂N and ∂θ/∂h[where h(N, e) is the angular momentum] are analytically expressed in the neighbourhood of N= 2 (case of the inverse square law). The value of ∂θ/∂N is computed numerically as well for 1 ≤N < 3. The resulting integrals are interesting improper integrals with singularities at both limits. Some of the integrals, especially for N= 2, can be given in closed form in terms of generalized hypergeometric functions which are reducible in terms of algebraic and logarithmic functions. No evidence was found for isolated cases of zero precession as e was increased. The N= 1 case of the logarithmic potential is also briefly discussed in view of its interest for the dynamics of eccentric orbits and its relevance to realistic galaxy models. The possibility of apsidal precession was also examined for a few cases of high-eccentricity asteroids and extrasolar planets. We find that these systems may provide interesting new laboratories for studies of gravity.