Orbit Determination with the two-body Integrals

TL;DR

This paper presents a novel method for preliminary orbit determination of solar system bodies using two-body integrals, applicable to limited observational data, with algorithms tested on simulated Pan-STARRS observations.

Contribution

The paper introduces two algorithms leveraging two-body integrals for orbit determination from minimal data and includes a method to test data linkage and estimate uncertainties.

Findings

Algorithms successfully compute preliminary orbits from limited data.

Method effectively tests if two observations belong to the same object.

Performance validated with extensive simulated Pan-STARRS data.

Abstract

We investigate a method to compute a finite set of preliminary orbits for solar system bodies using the first integrals of the Kepler problem. This method is thought for the applications to the modern sets of astrometric observations, where often the information contained in the observations allows only to compute, by interpolation, two angular positions of the observed body and their time derivatives at a given epoch; we call this set of data attributable. Given two attributables of the same body at two different epochs we can use the energy and angular momentum integrals of the two-body problem to write a system of polynomial equations for the topocentric distance and the radial velocity at the two epochs. We define two different algorithms for the computation of the solutions, based on different ways to perform elimination of variables and obtain a univariate polynomial. Moreover we use the redundancy of the data to test the hypothesis that two attributables belong to the same body (linkage problem). It is also possible to compute a covariance matrix, describing the uncertainty of the preliminary orbits which results from the observation error statistics. The performance of this method has been investigated by using a large set of simulated observations of the Pan-STARRS project.