Orbit determination from one position vector and a very short arc of optical observations

In this paper we address the problem of computing a preliminary orbit of a celestial body from one topocentric position vector ${\cal P}_1$ and a very short arc (VSA) of optical observations ${\cal A}_2$. Using the conservation laws of the two-body dynamics, we write the problem as a system of 8 polynomial equations in 6 unknowns. We prove that this system is generically consistent, namely, for a generic choice of the data ${\cal P}_1, {\cal A}_2$, it always admits solutions in the complex field, even when ${\cal P}_1, {\cal A}_2$ do not correspond to the same celestial body. The consistency of the system is shown by deriving a univariate polynomial $\mathfrak{v}$ of degree 8 in the unknown topocentric distance at the mean epoch of the observations of the VSA. Through Gr\"obner bases theory, we also show that the degree of $\mathfrak{v}$ is minimum among the degrees of all the univariate polynomials solving this problem. Even though we can find solutions to our problem for a generic choice of ${\cal P}_1, {\cal A}_2$, most of these solutions are meaningless. In fact, acceptable solutions must be real and have to fulfill other constraints, including compatibility with Keplerian dynamics. We also propose a way to select or discard solutions taking into account the uncertainty in the data, if present. The proposed orbit determination method is relevant for different purposes, e.g. the computation of a preliminary orbit of an Earth satellite with radar and optical observations, the detection of maneuvres of an Earth satellite, and the recovery of asteroids which are lost due to a planetary close encounter. We conclude by showing some numerical tests in the case of asteroids undergoing a close encounter with the Earth.