An Algebraic Method to Compute the Critical Points of the Distance Function Between Two Keplerian Orbits

Abstract

We describe an efficient algorithm to compute all the critical points of the distance function between two Keplerian orbits (either bounded or unbounded) with a common focus.The critical values of this function are important for different purposes, for example to evaluate the risk of collisions of asteroids or comets with the Solar system planets. Our algorithm is based on the algebraic elimination theory: through the computation of the resultant of two bivariate polynomials, we find a 16th degree univariate polynomial whose real roots give us one component of the critical points. We discuss also some degenerate cases and show several examples, involving the orbits of the known asteroids and comets.

Asteroid (99942) Apophis: new predictions of Earth encounters for this potentially hazardous asteroid

“…The MOID (minimum orbit intersection distance) can act as a warning indicator. An asteroid brighter than an absolute magnitude H < 20 and has a MOID ≤ 0.05 AU will be considered as potentially hazardous to the Earth (Gronchi 2005;Gronchi et al 2007), and will need special attention to its orbit monitoring. The potentially hazardous asteroid (PHA) Apophis -previously designated 2004 MN 4 -is emblematic of the situation of studies of PHAs,and is one of the closest approaching asteroids to the Earth presently known.…”

Spacecraft Relative Orbital Motion

“…Gurfil and Kholshevnikov (Gurfil & Kholshevnikov (2006)) introduced a metric which helps to study the relative distance between Keplerian orbits. Gronchi (Gronchi (2006), Gronchi (2005)) also introduced a metric between two confocal Keplerian orbits and used this instrument in problems of asteroid and comet collisions.…”

Computing the critical points of the distance function between two Keplerian orbits via rigorous global optimization

“…In Gronchi (2002) an algorithm based on the resultant theory and the Fast Fourier Transform (FFT) is introduced to perform the elimination of one variable; an upper bound on the maximum number of critical points (if they are finitely many) is also obtained by using Newton's polytopes and Bernstein's theorem. Several improvements to the algorithm were later presented in Gronchi (2005) as well as the extension to unbounded Keplerian orbits.…”

“…As finding the zeros of the components of (4) is equivalent to finding the zeros of the two nonlinear functions (see Gronchi 2005)…”

“…To reduce the number of parameters involved in the computation of the objective function it is convenient to adopt the cometary mutual elements introduced in Gronchi (2005). In particular, the transformation from cometary elements to mutual cometary elements is…”