Orbit determination with very short arcs

Abstract

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, A., Gronchi, G.F., de' Michieli Vitturi, M., Knežević, Z., 2004. Celest. Mech. Dyn. Astron. 90, 59–87]. The attributable 4 coordinates are the result of a fit 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 first 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 defining an identification penalty we can select the VAs allowing to fit together both arcs and compute a preliminary orbit. Two attributables may not be enough to compute an orbit with convergent differential corrections. Thus the preliminary orbit is used in a constrained differential 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 identification with a third arc will ensure a well determined orbit, to which additional sets of observations can be attributed. To test these algorithms we use a large scale simulation and measure the completeness, the reliability and the efficiency of the overall procedure to build up orbits by accumulating identifications. Under the conditions expected for the next generation asteroid surveys, the methods developed in this and in the preceding papers are efficient enough to be used as primary identification methods, with very good results. One important property is that the completeness in finding the possible identifications is as good for comparatively rare orbits, such as the ones of Near-Earth Objects, as for main belt orbits.

Introduction

Astrometric observations of asteroids/comets are reported by the observers as Very Short Arcs, that is sequences of observations closely spaced in time and assumed to belong to the same physical object. When, as in most cases, the information contained in such a data set is not enough to compute a full (6 parameters) set of orbital elements, we refer to them as Too Short Arcs (TSAs). In such a case, the problem of orbit determination must begin with the task of linkage, that is identification of two TSAs belonging to the same physical object. Such a 2-identification is, in most cases, enough to allow for an orbit, although it will be of very poor accuracy. Next we need to find 3-identifications, that is to attribute another TSA to the 2-identification orbit, and so on. This way of thinking of the orbit determination as a procedure inextricably connected to the identification problem is a significant change with respect to the classical paradigm, going back to (Gauss, 1809). The procedure used by modern surveys to discover asteroids/comets (and other small bodies) is very different from the one of ancient times, thus the classical methods solve a problem different from the one we are facing today (Milani and Knežević, 2005).

The present paper continues research meant to establish a new paradigm of population orbit determination, suitable to handle the observational data of the current and next generation surveys. In (Milani et al., 2004), hereafter referred to as Paper I, we have found the following properties of the TSAs. First, the essential information contained in most TSAs can be summarized by an attributable: two angles and their time derivatives. Second, for each attributable we can define an admissible region, a subset of the half plane of the undetermined coordinates range and range–rate where the orbits of Solar System objects can be found, thus excluding satellites of the Earth, heliocentric hyperbolic orbits and tiny meteoroids. Third, we have found an efficient algorithm to sample the admissible region by means of a Delaunay triangulation.

This paper is organized as follows. In Section 2 we describe the procedure to compute the attributable with its uncertainty and discuss whether the TSA contains information beside the one expressed by the attributable. In Section 3 we define the attributable orbital elements with their uncertainties, a set of values defining the initial conditions of one orbit with the two angles and the two angular rates of the attributable plus the range and range rate (with respect to the observer). Then we give a generalized definition of covariance matrix applicable to an orbit determined by using one TSA and one node of the Delaunay triangulation. In this way we define a set of Virtual Asteroids (VAs) sampling the space of orbits compatible with the available observations.

In Section 4 we show how, given a VA with a generalized covariance, to compute a prediction for future/past observations with a formal uncertainty like the one of a full least squares orbit. In Section 5 we define a criterion, based on an identification penalty, to assess the likelihood that another attributable, computed from an independently detected TSA, actually belongs to the same object. We scan the swarm of VAs associated with the first TSA and select the ones for which the identification penalty is low enough, if any. We discuss different possibilities to compute a preliminary orbit which can fit two TSAs.

In Section 6 we show how to apply a constrained differential correction algorithm to find a set of orbits fitting two TSAs, following (Milani et al., 2005a). A constrained solution is essentially a five parameter solution, with one additional parameter taking an arbitrary value. In this way we extract, from the 2-dimensional swarm of triangulation nodes, a 1-dimensional swarm of solutions. The procedure can be repeated to attribute to some of these second generation VAs a third TSA: in this case it is possible, in most cases, to compute a full 6-parameter vector of orbital elements according to the classical least squares principle. To further attribute other TSAs to the 3-identification orbit we can use methods already established and well tested. In principle, we have thus defined a new paradigm for orbit determination (Milani and Knežević, 2005).

In Section 7 we test the new algorithms on a simulated next generation survey. The results are very encouraging, and in Section 8 we conclude that our method is suitable as primary orbit determination method, entirely replacing the classical paradigm for the processing of the data of the present and future surveys. We also outline the work needed to apply these methods to realistic full-scale simulations of future surveys and to real data, when available.