Abstract
The Space Surveillance Network detects and registers Earth-orbiting man-made objects of a size larger than 10 cm, currently accounting for over 24, 000 objects, including nearly 2, 000 active satellites and 3, 000 non-operational satellites, with these numbers going up as new sensing capabilities become available in the coming years. The knowledge of their orbits is of paramount importance for two basic types of problems, namely the propagation and the determination of their orbits. In orbit propagation, the dynamical state of the satellite is known a priori at a given instant, and the aim is to compute the dynamical state at later instants. On the contrary, orbit determination is about estimating the dynamical state of the satellite at a given instant, making use of observation data from one or more tracking stations. Propagation and orbit determination are thus inverse processes and complementary to each other, since the determination process requires the propagation of the orbit, and vice versa. The application of these techniques naturally arises as a fundamental element in the field of space situational awareness, where automated computer systems routinely perform the precise orbit determination of both operational and inactive artificial satellites. Also, operational satellites require the precise knowledge of their orbits to predict the moment they fly over tracking stations, as well as to safely operate active satellites and plan collision avoidance maneuvers. Consequently, these automated systems not only need to be robust and reliable, but due the large amounts of data that need to be processed daily, they also need to be computationally efficient. The aim of this thesis is to develop robust and efficient numerical methods to be applied in both orbit determination and orbit propagation problems. The primary input for orbit determination are observations. For the determination of a Keplerian orbit six independent observations are required, although often many more are available. Thus, preliminary orbit determination is devoted to the estimation of a Keplerian orbit using only six observations. Observational data can be of a varied nature (angles only, range and range-rate, ...), leading to a wealth of wellknown classical techniques, e.g. Gibbs, Herrick-Gibbs, Laplace and Gauss methods. Solving Kepler’s equation is a commonplace to all of them. As a consequence, fast, accurate, robust and reliable methods for solving Kepler’s equation are of a vital importance in preliminary orbit determination. This dissertation proposes one such algorithm that, based on an improved initial guess, succeeds to solve the elliptic and hyperbolic Kepler’s equation more efficiently. Another recurring element in many preliminary orbit determination methods is solving the two-body Lambert’s problem. This dissertation revisits this problem from a new viewpoint, providing an innovative, insightful approach that enables a robust and efficient solution to the problem. Real satellite orbits, however, are not Keplerian due to the many perturbing forces acting upon them, and thus six observations are not enough to realistically capture the dynamical state of the satellite. If more observations are available, a better estimate is possible; this is the focus of statistical orbit determination, that aims to determine the initial state of a satellite from multiple observations and under a realistic, non-Keplerian dynamical model. The corresponding techniques are usually based on a linearization around the outcomes of preliminary orbit determination, leading to a succession of values converging to the sought solution, so that the propagation with the complete dynamical model yields an orbital solution that provides the best possible fit to the actual observations. Consequently, these methods rely on robust, accurate and efficient orbit propagation methods, so the choice of an appropriate propagation technique is fundamental. In this regard, classical methods (e.g. Cowell’s method) are commonly used; these methods, although effective and simple to implement, have the drawback of an exponential error growth and singular equations of motion. Conversely, regularized orbital formulations, such as DROMO, Kustaanheimo-Stiefel, or Sperling-Bürdet, allow to reformulate the equations of motion so that the orbital dynamics is described by a system of non-singular equations, which facilitates the numerical integration and reduces the negative effects of the Lyapunov instability associated to Keplerian motion, thus providing far more accurate and efficient propagation routines. With this vision in mind, the second part of this dissertation is devoted to orbit propagation with regularized formulations, and makes new contributions to this field by developing a new reformulation of DROMO, which is specifically tailored for highly-perturbed dynamical environments. This dissertation is organized as follows: • The first part covers the research contributions to orbit determination. Chapter 1 introduces the basics on orbit determination and describes the classical methods; Chapters 2 and 3 present novel methods for solving the elliptic and hyperbolic Kepler’s equation, respectively; Chapter 4 presents a new formulation, valid for any type of orbit, to solve the two-body Lambert’s problem. • The second part of the dissertation focuses on orbit propagation with regularized formulations. Chapter 5 introduces the basics on regularized special perturbation methods; Chapters 6 and 7 describe in detail the DROMO and ElliDROMO regularized formulations, respectively; Chapter 8 extends the two formulations to enhance their performance in strongly perturbed environments, leading to new methods referred to as DROMO-SPE and ElliDROMOSPE; these propagators are tested in different scenarios and their performance is shown; Chapter 9 explores the application of DROMO and DROMO-SPE to the propagation of meteor trajectories and presents a sensitivity analysis of the parameters considered in the physical modeling. Finally, Chapter 10 summarizes the main conclusion of the thesis and traces potential avenues for future work.