Effects of orbital parameter uncertainties

A number of important space flight mechanics problems require accurate uncertainty estimates of the position and velocity of an orbiting object. The lack of information might simply derive from a tracking error or from a propagation error, and in many cases it can be of a significant magnitude to the extent that many objects orbiting around our planet are on orbits from which very little can be inferred, in particular, space debris. To deal with these kinds of problems, both astrodynamicists and engineers have developed many different techniques and models, many of which date back to the 1970s. E. Opik and N. G. Dennis laid the basis of the primary theories that are used to deal with these problems. Both the theories, though, require restrictive hypotheses that limit their use to particular cases. In this work, an original and general approach to the problem of dealing with uncertainties in some or in all of the six orbital parameters is proposed. No restrictive hypothesis is introduced. The case of osculating orbital parameters is then considered, and a set of six new formulas is derived. One of these six expressions is shown to represent a generalization of the Dennis statistical theory of satellite motion.