The Orbit-Orbit Distance Function: On the Maximum Number of Stationary Points, and a Fast Filter for Finding Asteroid-Asteroid Encounters

Details The Orbit-Orbit Distance Function: On the Maximum Number of Stationary Points, and a Fast Filter for Finding Asteroid-Asteroid Encounters The Orbit-Orbit Distance Function: On the Maximum Number of Stationary Points, and a Fast Filter for Finding Asteroid-Asteroid Encounters

Jul 2007

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2007dda....38.0905m&link_type=abstract

American Astronomical Society, DDA meeting #38, #9.05

Mathematics

Algebraic Geometry

Scientific paper

The function describing the distance at any given moment between two bodies in confocal Keplerian orbits has many uses in dynamical astronomy. These include asteroid mass determinations, characterization and prediction of meteor streams and showers, collisions of space debris with objects in Earth orbit, and a class of problems wherein astrometric and/or radial velocity observations of spacecraft are obtained from a platform in Earth orbit.
The distance function is a family of two-dimensional manifolds with the topology of a 2-torus (the two angles being the azimuthal coordinates of the two orbiting bodies). The stationary points (local maxima, local minima, and saddle points) on these manifolds are places of special dynamical and practical interest. Studies incorporating techniques from algebraic geometry, mainly by Gronchi and collaborators, have shown that in general the maximum possible number of stationary points is not precisely known but must be less than or equal to 16. Determining the actual upper bound is thus far an unsolved problem. Knowing this upper bound is essential for full understanding of the geometry of the orbit-orbit distance surface and hence characterization of the kinds of orbits that can suffer close approaches. This paper reports numerical calculations which strongly suggest that in the general problem the maximum number of stationary points is 12. We also characterize the regions in the six-dimensional relative orbital elements space in which exist orbit-orbit scenarios associated with 12, 10, 8, etc. stationary points.
Finally, in support of asteroid mass determination, we make note of an algorithm allowing fast filtering and determination of dates of asteroid-asteroid close approaches.

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