Preliminary orbits with overdetermined systems of Keplerian conservation laws

We consider different choices of Keplerian conservation laws to compute preliminary orbits with two very short arcs (VSAs) of astrometric observations. In total we have 7 equations in 4 unknowns that are the radial distances and velocities at the epochs of the two VSAs. Adding two auxiliary variables we can embed the full set of conservation laws into a polynomial system of 9 equations. This complete system is generically inconsistent, i.e., it admits no solutions. However, by combining these equations, Gronchi et al. (2015) showed that an overdetermined polynomial system can be obtained that is consistent and, through variable elimination, leads to a univariate polynomial p9 of degree 9 in one radial distance. This corresponds to taking a subsystem with 7 equations of the complete system, see Gronchi et al. (2017). In this paper, we consider all the other possibilities and we find two additional overdetermined cases which are consistent and lead to a univariate polynomial (say p18 and p~18, respectively) of degree 18 in the same variable as p9. In the other overdetermined cases, the corresponding systems are inconsistent. We also present a method to compute an approximate gcd of p9 and p18 that can allow us to find preliminary orbits that approximately satisfy inconsistent systems of conservation laws. We conclude with some numerical tests with real asteroid data.