Metrics on the space of bounded Keplerian orbits and space situational awareness

A main difficulty for orbit determination of space debris is the correlation of tracks of data belonging to the same physical object. A sequence of optical measurements of an Earth orbiting object over a single track has sufficient information to determine the topocentric angles and angular rates with some degree of precision, but yields no information on the range and range-rate other than the restriction that they lie within some open set containing energetically bound nonimpact orbits. These restrictions constrain the orbit to lie on a two-dimensional submanifold, known as the admissible region, of the six-dimensional phase space that can be computed using the recorded data. Given two measurements, the orbit determination problem is tantamount to finding the ¿closest¿ point between two such submanifolds. However, distance itself can be defined in a plethora of different ways in the space of Keplerian orbits. In this paper we construct a natural metric in the four-dimensional space of Keplerian orbits of fixed energy and consider some basic topological questions of mappings from the admissible region to a natural representation of this space. Using the metric presented in this paper, the ¿shortest distance¿ between the two admissible regions can be approximated. We present this as a new approach in dealing with one aspect of the space situational awareness problem.