Fast error-controlling MOID computation for confocal elliptic orbits

Abstract

We present an algorithm to compute the minimum orbital intersection distance (MOID), or global minimum of the distance between the points lying on two Keplerian ellipses. This is achieved by finding all stationary points of the distance function, based on solving an algebraic polynomial equation of 16th degree. The algorithm tracks numerical errors appearing on the way, and treats carefully nearly degenerate cases, including practical cases with almost circular and almost coplanar orbits. Benchmarks confirm its high numeric reliability and accuracy, and that regardless of its error-controlling overheads, this algorithm pretends to be one of the fastest MOID computation methods available to date, so it may be useful in processing large catalogs.

A lower bound of the distance between two elliptic orbits

“…Namely, of all N 0 = 199 990 000 pairs in the original sample Φ, we put in each catalog those and only those pairs E and E ′ that satisfied the condition ρ(E, E ′ ) δ. The computation of the distance ρ(E, E ′ ) was carried out by means of the software described by Baluev and Mikryukov (2019) and available for download at http://sourceforge.net/projects/distlink. This software provides a numeric implementation of the algebraic method presented by Kholshevnikov and Vassiliev (1999b), similar to the one presented by Gronchi (2002Gronchi ( , 2005.…”

Fast error-safe MOID computation involving hyperbolic orbits

“…The MOID is a minimum of some distance or distance-like function ρ(u, u ) that depends on two arguments, determining positions on two orbits. Multiple methods of finding the minima of ρ(u, u ) are available Kholshevnikov and Vassiliev (1999); Gronchi (2002); Baluyev and Kholshevnikov (2005); Gronchi (2005); Armellin et al (2010); Hedo et al (2018); Baluev and Mikryukov (2019), as well as methods allowing to put useful bounds on the MOID or related quantities (Mikryukov and Baluev, 2019;Gronchi and Niederman, 2020).…”

“…Email address: r.baluev@spbu.ru (Roman V. Baluev) The fastest methods appear those in which both u and u are solved for rather than found by numeric optimization (Kholshevnikov and Vassiliev, 1999;Gronchi, 2002Gronchi, , 2005Baluyev and Kholshevnikov, 2005;Baluev and Mikryukov, 2019). The task is analytically reduced to solving a nonlinear equation with respect to u and then expressing u also analytically.…”

“…The recent work by Baluev and Mikryukov (2019) represents an efficient solution of this issue based on careful treatment of numeric errors appearing on the way. Basically, it represent a numeric implementation of the algebraic approach presented by Kholshevnikov and Vassiliev (1999), similar to the one presented by Gronchi (2002Gronchi ( , 2005.…”

“…This contrasts with (Gronchi, 2005) where this issue does not appear because they used true anomalies as free variables. However, that code revealed numeric unreliability on some orbit pairs (Hedo et al, 2018;Baluev and Mikryukov, 2019). Though the fraction of such cases is small, they often escape from being selfdiagnosed, making such code less prefered than alternatives that avoid failures.…”