Exact, E=0, Solutions for General Power-Law Potentials. I. Classical Orbits

TL;DR

This paper derives exact classical solutions for zero-energy orbits in all power-law potentials, revealing unique behaviors like bound orbits passing through the origin and discontinuities, with implications for orbit periods and virial theorem violations.

Contribution

It provides a comprehensive derivation of exact solutions for E=0 in all power-law potentials, including bound and unbound cases, and analyzes their properties and discontinuities.

Findings

Exact solutions for all power-law potentials at E=0.

Bound orbits pass through the origin with discontinuities.

Unbound orbits with finite travel times to infinity.

Abstract

For zero energy, $E=0$, we derive exact, classical solutions for {\em all} power-law potentials, $V(r)=-\gamma/r^\nu$, with $\gamma>0$ and $-\infty <\nu<\infty$. When the angular momentum is non-zero, these solutions lead to the orbits $\r(t)= [\cos \mu (\th(t)-\th_0(t))]^{1/\mu}$, for all $\mu \equiv \nu/2-1 \ne 0$. When $\nu>2$, the orbits are bound and go through the origin. This leads to discrete discontinuities in the functional dependence of $\th(t)$ and $\th_0(t)$, as functions of $t$, as the orbits pass through the origin. We describe a procedure to connect different analytic solutions for successive orbits at the origin. We calculate the periods and precessions of these bound orbits, and graph a number of specific examples. Also, we explain why they all must violate the virial theorem. The unbound orbits are also discussed in detail. This includes the unusual orbits which have finite travel times to infinity and also the special $\nu = 2$ case.