Multiple solutions for asteroid orbits: Computational procedure and applications

We describe the Multiple Solutions Method, a one-dimensional sampling of the six-dimensional orbital confidence region that is widely applicable in the field of asteroid orbit determination. In many situations there is one predominant direction of uncertainty in an orbit determination or orbital prediction, i.e., a ``weak'' direction. The idea is to record Multiple Solutions by following this, typically curved, weak direction, or Line Of Variations (LOV). In this paper we describe the method and give new insights into the mathematics behind this tool. We pay particular attention to the problem of how to ensure that the coordinate systems are properly scaled so that the weak direction really reflects the intrinsic direction of greatest uncertainty. We also describe how the multiple solutions can be used even in the absence of a nominal orbit solution, which substantially broadens the realm of applications. There are numerous applications for multiple solutions; we discuss a few problems in asteroid orbit determination and prediction where we have had good success with the method. In particular, we show that multiple solutions can be used effectively for potential impact monitoring, preliminary orbit determination, asteroid identification, and for the recovery of lost asteroids

Necessary conditions for classical super-integrability of a certain family of potentials in constant curvature spaces

We formulate the necessary conditions for the maximal super-integrability of a certain family of classical potentials defined in the constant curvature two-dimensional spaces. We give examples of homogeneous potentials of degree -2 on $E^2$ as well as their equivalents on $S^2$ and $H^2$ for which these necessary conditions are also sufficient. We show explicit forms of the additional first integrals which always can be chosen polynomial with respect to the momenta and which can be of an arbitrary high degree with respect to the momenta

The Asteroid Identification Problem

es. Tests on the values of these distances are used as preliminary filters, to allow more 1 extensive computations to be performed on a fraction of the total number of possible couples. The difficulties in the application of this algorithm arise from the fact that poorly observed orbits have badly conditioned covariance matrices: all the computations performed with these matrices are affected by large numerical errors. Moreover, it is clear that the larger the eigenvalues of the covariance matrix, the larger the nonlinear effects will be. For asteroids observed over a very short arc, and lost since a long time, the linear approximation fails. Nonlinear optimisation algorithms can be used but are computationally expensive [2]. Identifications based upon some of the orbital elements, e.g., excluding the mean longitude, can be effective in reducing the relevance of the nonlinear effects; the corresponding distances are defined by the marginal covariance matrices, using a formal

Orbit Determination with Very Short Arcs. II Identifications

When the observational data are not enough to compute a meaningful orbit for an asteroid/comet we can represent the data with an attributable, i.e., two angles and their time derivatives. The undetermined variables range and range rate span an admissible region of solar system orbits, which can be sampled by a set of Virtual Asteroids (VAs) selected by means of an optimal triangulation [Milani et al. 2004]. The attributable 4 coordinates are the result of a t and they have an uncertainty, represented by a covariance matrix. Two short arcs of observations, represented by two attributables, can be linked by considering for each VA (in the admissible region of the rst arc) the covariance matrix for the prediction at the time of the second arc, and by comparing it with the attributable of the second arc with its own covariance. By dening an identi cation penalty we can select the VAs allowing to t together both arcs and compute a preliminary orbit. Two attributables may not be enough to compute an orbit with convergent dierential corrections. Thus the preliminary orbit is used in a constrained dierential correction, providing solutions along the Line Of Variation which can be used as second generation VAs to further predict the observations at the time of a third arc. In general the identication with a third arc will ensure a well determined orbit, to which additional sets of observations can be at

generation asteroid/comet surveys

Unbiased orbit determination for the nex

Unbiased orbit determination for the next generation asteroid/comet surveys

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