  We propose two algorithms to provide a full preliminary orbit of an
Earth-orbiting object with a number of observations lower than the classical
methods, such as those by Laplace and Gauss. The first one is the Virtual
debris algorithm, based upon the admissible region, that is the set of the
unknown quantities corresponding to possible orbits for objects in Earth orbit
(as opposed to both interplanetary orbits and ballistic ones). A similar method
has already been successfully used in recent years for the asteroidal case. The
second algorithm uses the integrals of the geocentric 2-body motion, which must
have the same values at the times of the different observations for a common
orbit to exist. We also discuss how to account for the perturbations of the
2-body motion, e.g., the $J_2$ effect.
