Orbital (Planar) Motions: Exact Solution and Its Analysis

Abstract

We use conservation laws to show that the non-radial fall on the central mass is impossible, and that all infinite motions are asymptotically radial and obey the Hubble law. Its detailed form is given based on the solution in quadratures, which follows from the conservation equations. We study all the radial intervals where the motion is allowed depending on the total energy, E, and angular momentum, L. We demonstrate that finite motions are only possible for a finite interval of negative energies with the angular momentum below the critical value we found; so, unlike the classical Kepler problem, infinite motions grossly dominate the finite ones. We present and discuss all elementary function solutions existing for special values of E and L, such as circular and spiral orbits. All other cases lead to solutions in terms of Legendre elliptical integrals.