Distance Between Two Arbitrary Unperturbed Orbits

Abstract

In the paper by Kholshevnikov and Vassilie, 1999, (see also references therein) the problem of finding critical points of the distance function between two confocal Keplerian elliptic orbits (hence finding the distance between them in the sense of set theory) is reduced to the determination of all real roots of a trigonometric polynomial of degree eight. In non-degenerate cases a polynomial of lower degree with such properties does not exist. Here we extend the results to all possible cases of ordered pairs of orbits in the Two–Body–Problem. There are nine main cases corresponding to three main types of orbits: ellipse, hyperbola, and parabola. Note that the ellipse–hyperbola and hyperbola–ellipse cases are not equivalent as we exclude the variable marking the position on the second curve. For our purposes rectilinear trajectories can be treated as particular (not limiting) cases of elliptic or hyperbolic orbits.