When a celestial body, e.g., an asteroid, has been observed only over a short time, its orbit is not well determined but may be anywhere in a confidence region where the astrometric residuals are acceptable. This region can be sampled by a swarm of Virtual Asteroids (VA) sharing the reality of the asteroid: one of them is real, but we do not know which one. The problem is how to sample the confidence region with a small number of VA, still being able to solve the main problems of asteroid recovery/identification and impact monitoring.
One class of methods uses random sampling of the confidence region to mimic with the VA population the probabilistic distributions of the orbits. This class includes the Monte Carlo and the Statistical Ranging methods. When it is critical to detect a very small probability (e.g., of a catastrophic impact) by computing a small number of VA orbits, and also when a large catalog of asteroids has to be handled, it is more efficient to sample the confidence region with a geometric object, such as a smooth manifold: it can be sampled uniformly, taking into account its dimension. Our group has developed in the last 6-7 years 1-dimensional sampling methods based upon a differentiable curve, the Line Of Variations (LOV), which can represent, in suitable cases, the spine of the confidence region. The LOV is sampled by uniformly spaced VA, thus interpolation between consecutive VA is possible. This is the basis for the current algorithms of Impact Monitoring, used in Pisa and at JPL. The LOV method is also used for recovery of lost asteroids and for identification of independent discoveries of the same object.
When the asteroid has moved on the sky while being observed by $<1^\circ$, the confidence region is wide in two directions and the LOV may be an inappropriate way of sampling it. We have recently developed 2-dimensional sampling methods based upon the concept of Admissible Region, a 2-dimensional manifold parameterized by a compact subset of the range/range-rate plane. This region is then sampled by triangulation, with each node used as a VA. This allows to define methods for asteroid identification/recovery and for impact monitoring starting from very poor data, such as the ones collected during a single night of observations.
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