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 \(\mathcal{P}_1\) and a very short arc (VSA) of optical observations \(\mathcal{A}_2\). 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 \(\mathcal{P}_1, \mathcal{A}_2\), it always admits solutions in the complex field, even when \(\mathcal{P}_1, \mathcal{A}_2\) 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 Gröbner 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 \(\mathcal{P}_1, \mathcal{A}_2\), 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.