Orbit determination with very short arcs. I admissible regions

Abstract

Most asteroid discoveries consist of a few astrometric observations over a short time span, and in many cases the amount of information is too limited to compute a full orbit according to the least squares principle. We investigate whether such a Very Short Arc may nonetheless contain significant orbit information, with predictive value, e.g., allowing to compute useful ephemerides with a well defined uncertainty for some time in the future.For short enough arcs, all the significant information is contained in an attributable, consisting of two angles and two angular velocities for a given time; an apparent magnitude is also often available. In this case, no information on the geocentric range r and range-rate _ r is available from the observations themselves. However, the values of ðr; _ rÞ are constrained to a compact subset, the admissible region, if we can assume that the discovered object belongs to the Solar System, is not a satellite of the Earth and is not a shooting star (very small and very close). We give a full algebraic description of the admissible region, including geometric properties like the presence of either one or two connected components.The admissible region can be sampled by selecting a finite number of points in the ðr; _ rÞ plane, each corresponding to a full set of six initial conditions (given the four component attributable) for the asteroid orbit. Because the admissible region is a region in the plane, it can be described by a triangulation with the selected points as nodes. We show that triangulations with optimal properties, such as the Delaunay triangulations, can be generated by an effective algorithm; however, the optimal triangulation depends upon the choice of a metric in the ðr; _ rÞ plane. Each node of the triangulation is a Virtual Asteroid, for which it is possible to propagate the orbit and predict ephemerides. Thus for each time there is an image triangulation on the celestial sphere, and it can be used in a way similar to the use of the nominal ephemerides (with their confidence regions) in the classical case of a full least square orbit.

Associating optical measurements of MEO and GEO objects using Population-Based Meta-Heuristic methods

“…This is done by applying the so-called Admissible Region (Milani et al, 2004(Milani et al, , 2011Tommei et al, 2007;Schumacher et al, 2013). This method allows for the restriction of the search space by placing constraints on e.g.…”

From Astrometry to Celestial Mechanics: Orbit Determination with Very Short Arcs

“…The procedures and all the results are described in full detail in Milani et al (2004a). The admissible region cannot have more than two connected components.…”

“…The admissible region cannot have more than two connected components. More precisely, the degree six polynomial resulting from condition (C) cannot have more than three real positive roots: when there is only one such root, the admissible region has only one component, when there are three, it has two components (see Figures 1 and 2 in Milani et al 2004a).…”

“…If this is the case, we need a preliminary orbit to be used as first guess to start the differential correction procedure. One possibility is to use X i as the preliminary orbit, but there are ways to define a better one, taking into account that an optimal "compromise" attributable between the predicted A i and the observed A 1 is provided by the algorithms of Milani et al (2001Milani et al ( , 2004c:…”

“…Our research plan consists of several steps, of which three are completed (Milani et al, 2004a(Milani et al, , 2004b(Milani et al, , 2004c; these are described in the present paper. The basic idea is as follows: a TSA comprises a number of observed positions with deviations from alignment compatible with a random observational error.…”