A Combined Method to Compute the Proximities of Asteroids

Abstract

We describe a simple and efficient numerical-analytical method to find all of the proximities and critical points of the distance function in the case of two elliptical orbits with a common focus. Our method is based on the solutions of Simovljević's (1974) graphical method and on the transcendent equations developed by Lazović (1993). The method is tested on 2 997 576 pairs of asteroid orbits and compared with the algebraic and polynomial solutions of Gronchi (2005). The model with four proximities was obtained by Gronchi (2002) only by applying the method of random samples, i.e., after many simulations and trials with various values of elliptical elements. We found real pairs with four proximities.