Averaging on Earth-Crossing Orbits

Authors: Gronchi G.F.¹; Milani A.²

Source: Celestial Mechanics and Dynamical Astronomy, Volume 71, Number 2, 1998 , pp. 109-136(28)

Publisher: Springer

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Abstract:

The orbits of planet-crossing asteroids (and comets) can undergo close approaches and collisions with some major planet. This introduces a singularity in the N-body Hamiltonian, and the averaging of the equations of motion, traditionally used to compute secular perturbations, is undefined. We show that it is possible to define in a rigorous way some generalised averaged equations of motion, in such a way that the generalised solutions are unique and piecewise smooth. This is obtained, both in the planar and in the three-dimensional case, by means of the method of extraction of the singularities by Kantorovich. The modified distance used to approximate the singularity is the one used by Wetherill in his method to compute probability of collision. Some examples of averaged dynamics have been computed; a systematic exploration of the averaged phase space to locate the secular resonances should be the next step.

‘Alice sighed wearily. “I think you might do something better with the time” she said, “than waste it asking riddles with no answers”

(Alice in Wonderland, L. Carroll)

Keywords: asteroids; comets; averaging; semianalytical theory; secular perturbations; collisions

Language: English

Document Type: Regular paper

Affiliations: 1: Department of Mathematics, University of Pisa, Via Buonarroti 2, I-56127 Pisa, Italy (e-mail: gronchi@dm.unipi.it milani@dm.unipi.it) 2: Department of Mathematics, University of Pisa, Via Buonorroti 2, I-56127 Pisa, Italy (e-mail: gronchi@dm.unipi.it milani@dm.unipi.it)

Publication date: 1998-01-01