  In this paper we address the problem of computing a preliminary orbit of a
celestial 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. We
prove that this system is generically consistent, namely it admits solutions at
least in the complex field. From this system we derive a univariate polynomial
$\mathfrak{v}$ of degree 8 in the unknown topocentric distance at the mean
epoch 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 polynomials
solving this problem. The proposed 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
also show some numerical tests in the case of asteroids undergoing a close
encounter with the Earth.
