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In this paper we address the problem of computing a preliminary orbit of acelestial body from one topocentric position vector and a very short arc (VSA)of optical observations. Using the conservation laws of the two-body dynamics,we write the problem as a system of 8 polynomial equations in 6 unknowns. Weprove that this system is generically consistent, namely it admits solutions atleast in the complex field. From this system we derive a univariate polynomial$\mathfrak{v}$ of degree 8 in the unknown topocentric distance at the meanepoch of the VSA. Through Gr\"obner bases theory, we show that the degree of$\mathfrak{v}$ is minimum among the degrees of all the univariate polynomialssolving this problem. The proposed method is relevant for different purposes,e.g. the computation of a preliminary orbit of an Earth satellite with radarand optical observations, the detection of maneuvres of an Earth satellite, andthe recovery of asteroids which are lost due to a planetary close encounter. Wealso show some numerical tests in the case of asteroids undergoing a closeencounter with the Earth.

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