On Broucke's velocity-related series expansions in the two-body problem

Abstract

This short article supplements a recent paper by Dr R. Broucke on velocity-related series expansions in the two-body problem. The derivations of the Fourier and Legendre expansions of the functionsF(v),$\sqrt {F(\upsilon )}$ and$\sqrt {{1 \mathord{\left/ {\vphantom {1 {F(\upsilon )}}} \right. \kern-\nulldelimiterspace} {F(\upsilon )}}}$ are given, where

$$F(\upsilon ) = (1 - e^2 )/(1 + 2e\cos \upsilon + e^2 ), e< 1$$

In the two-body problem,v is identified with the true anomaly,e the eccentricity andF(v) equals (an/V)².

Some interesting relations involving Legendre polynomials are also noted.