Topocentric orbit determination: Algorithms for the next generation surveys

Abstract

The process of calculating a good orbit from astrometric observations of the same object involves three main steps: preliminary orbit determination, least squares orbit fitting, and quality control assessing the orbit's uncertainty and reliability. For the next generation sky surveys, with much larger number density of observations, new algorithms, or at least substantial revisions of the classical ones, are needed. The classical theory of preliminary orbit algorithms was incomplete in that the consequences of the topocentric correction had not been fully studied. We show that it is possible to rigorously account for topocentric observations and that this correction may increase the number of alternate preliminary orbits without impairing the overall performance. We have developed modified least squares algorithms including the capability of fitting the orbit to a reduced number of parameters. The restricted fitting techniques can be used to improve the reliability of the orbit computing procedure when the observed arcs have small curvature. False identification (where observations of different objects are incorrectly linked together) can be discarded with a quality control on the residuals and a ‘normalization’ procedure removing duplications and contradictions. We have tested our algorithms on two simulations based on the expected performance of Pan-STARRS—one of the next generation all-sky surveys. The results confirm that large sets of discoveries can be handled very efficiently resulting in good quality orbits. In these tests we lost only 0.6 to 1.3% of the possible objects, with a false identification rate in the range 0.02 to 0.06%.

Introduction

The problem of preliminary orbit determination is old, with very effective solutions developed by Laplace (1780) and Gauss (1809). However, the methods of observing Solar System bodies have changed radically since classical times and have been changing even faster recently due to advances in digital astrometry. The question is, what needs to be improved in the classical orbit determination algorithms to handle the expected rate of data from the next generation all-sky surveys? Alternatively, what can we now use in place of the classical algorithms?

The issue is not one of computational resources because these grow at the same rate as the capability of generating astrometric data. Reliability is the main problem when handling tens of millions of detections of Solar System objects. An algorithm failing once in 1000 usages may have been considered reliable a few years ago, but now we must demand better performance.

This is particularly important because of the strong correlation between difficulties in the orbit computation and the scientific value of the discovered object. Main Belt Asteroids (MBAs) are commonplace and their orbits are easily computed. In modern asteroid surveys only a few in a 1000 of the objects (to a given limiting magnitude) are the more interesting Near Earth Objects (NEOs) while a few in 100 are the equally interesting Trans Neptunian Objects (TNOs); in both cases the computation may be much more difficult for reasons explained later. Thus an algorithm that successfully computes orbits for 99% of the discoveries may still fail on a large fraction of the most interesting objects like NEOs and TNOs.

To determine if there are problems with the classical orbit determination techniques we must re-examine whether the assumptions invoked in the derivations of the classical techniques are still appropriate; e.g., neglecting multiple preliminary orbits and dismissing the topocentric corrections is wrong when searching for NEOs; neglecting the case in which the observed track on the sky has insignificant curvature is wrong for TNOs. Another consideration is the immensely superior computing power available to us: in the trade off between simpler and more reliable computations we almost always select the latter, while the classical authors were forced to do the opposite.

It has long been known that Gauss' (Gauss, 1809) and Laplace's (Laplace, 1780) preliminary orbit determination methods are equivalent to some level of approximation but it is important here to understand the approximations under which this is true and to check whether they are still applicable for contemporary observations. If the observations are geocentric the two methods are equivalent up to the algebraic equation of degree 8 corresponding to a quadratic approximation in time. We will show below that with topocentric observations they are not equivalent. Crawford et al (1930, p. 99) and Marsden (1985) have shown that Gauss' method, at least in the version of Merton (1925), can be used with topocentric observations employing the same formulae but Laplace's method cannot.

The difference between topocentric and geocentric observations is not negligible apart from some special cases. Laplace's method can account for topocentric observations but only within an iterative procedure whose convergence cannot be guaranteed. Following a suggestion by Poincaré (1906) Laplace's method can be modified to account for topocentric observations in the degree 8 equation.

The very useful qualitative theory of Charlier (1910) that allows us to compute the number of preliminary orbit solutions for Laplace's method does not apply to the modified method. It also does not apply to Gauss' and cannot describe iterative methods. This is a problem if we are concerned with the reliability of our orbit determination algorithm. Although it is always possible to improve the orbit with iterative methods, if the first approximation provides a wrong number of solutions it is possible to completely miss the correct one!

In Section 4 we develop a new qualitative theory, fully accounting for topocentric observations, for the solutions of the equation of degree 8 for both Gauss' and the modified Laplace–Poincaré methods. We show that the number of solutions can be larger than in Charlier's theory, e.g., there may be double solutions near opposition and triple solutions at low solar elongations. Some examples are given to show that the existence of additional solutions can affect the reliability of the orbit determination for real asteroids. This progress tips the balance in favor of applying Gauss' method to preliminary orbit determination because it can handle topocentric observations in a natural way.

While these mathematical arguments are interesting, an abstract qualitative theory is useful only if it is exploited by software which can compute and keep track of all the solutions of the algebraic equations. This has not been the case to date. A tricky problem arises when, after the solution of the degree 8 equation, an iterative method is used to improve the preliminary orbit: the number of limit points of the iteration may be different from the number of starting solutions. As a compromise we use a two step procedure in which the two versions, with and without iterative improvement, are used in sequence.

Preliminary orbit methods fail when the components of curvature in the observations are so poorly determined that even their sign is uncertain. This happens when either the observed arc is too short or the object is very distant. In Section 5 we present how to detect when these conditions occur and how to estimate the corresponding orbit uncertainty.

To deal with these low curvature cases we use a Virtual Asteroids (VA) method. A VA is a complete orbit compatible with the available observations but by no means determined by them. A number of VA are selected, either at random (Virtanen et al., 2003) or by some geometric construction (Milani et al., 2004), among the orbits compatible with the observations (many VA methods are described in the literature; for a recent review see Milani, 2005). A very effective method for the problem at hand uses just one VA that is derived from the theory of the Admissible Region (Milani et al., 2004).

The set of preliminary orbits consistent with the observations is used as a first guess for the nonlinear optimization procedure (differential corrections) that computes the nominal orbit fulfilling the principle of least squares of the observation residuals. If the preliminary orbits are well defined then at least one of them is likely to belong to the convergence domain of some least squares orbit in the differential corrections. In this case it is enough to use all the preliminary orbits as input to the differential correction routine. If the preliminary orbits are poorly defined they may be so far from the nominal solution that the differential corrections may not converge at all or may converge to the wrong solution.

Thus, it is essential to increase the size of the convergence domain by using modified differential corrections methods. A number of these methods exist and most of them have one feature in common: the number of orbital parameters determined is less than 6. There are 4-fit methods in which 2 variables are kept fixed (e.g., Milani et al., 2006) and 5-fit or constrained least squares solutions in which one parameter is fixed (the fixed parameter need not be one of the orbital elements but may be defined in some intrinsic way; Milani et al., 2005a).

In Section 6 we describe two tests: one based on small, focused simulations with only the most difficult orbits (NEOs and TNOs), another including all classes of Solar System objects in their correct ratio (dominated by MBAs). Both tests were performed with synthetic realizations of the Pan-STARRS (Jedicke et al., 2007) next generation survey. Moreover, we will show that all the innovative solutions we have proposed in this paper are essential for the accurate and reliable determination of the most difficult orbits.