Estimates for the Set-theoretic Distance between Orbits of Celestial Bodies

K. V. Kholshevnikov

Protecting the Earth against Collisions with Asteroids and Comet Nuclei, In: A. M. Finkelstein, W. F. Huebner, V. A. Shor (Eds) Proceedings of the International Conference “Asteroid-Comet Hazard-2009”, StP: Nauka, 55–57 (2010)

Ключевые слова: Very Long Baseline Interferometry (VLBI), Small Solar System Body (SSSB), set-theoretic distance between orbits of celestial bodies, close encounters of celestial bodies with the Earth, criteria for the proximity to intersection, unperturbed orbits, semi-major axes and eccentricities, topological notion of linkage, lower bound of the distance, upper bound of the distance, a trigonometric equation of the 8-th degree

Аннотация

Prediction of close encounters of celestial bodies with the Earth or other planets requires enormous calculations. It is reasonable to use preliminary simple criteria for the proximity to intersection based on estimates of set-theoretic distance (, ) between orbits , of celestial bodies , . As a first step it is sufficient to consider orbits as unperturbed. We describe here two simple criteria and one more complicated one. The first one uses the most unfavorable orientation of orbits with given semi-major axes and eccentricities. The second one uses the topological notion of linkage. Both give us lower and upper bounds of the distance. The third one requires calculation of the distance itself and it amounts to solving a trigonometric equation of the 8-th degree