The Linkage Problem
Introduction
Most of the astrodynamics problems require to get the data provided by celestial bodies observations and to develop a state vector, in order to be able to predict their future positions, for example. The problem of the orbital determination is old. The first works in this field were realized by those scientists who produced astronomical tables of the positions of the planets and the stars. Among the oldest, there are the Ptolemy’s works of the 2nd century AD, such as the Almagest or the Handy Tables. Then, it is remarkable the contribution of Johann Kepler with his Rudolphine Tables in 1626 [Vallado, 1997]: he succeeded in predicting the conjunctions of Mercury and the Sun to within five hours in November 1631. The examples are numerous: Newton, Laplace, Lagrange, Halley and several other scientists studied methods to determine an initial orbit from the observations during the years. The method, which is still used today, was outlined by the German mathematician Carl Friedrich Gauss: its work is a milestone in the field, mainly thanks to the adoption of a least square method to improve the accuracy of the orbital determination. Gauss’s motivation was the discovery of the first asteroid, Ceres, in 1801: the italian astronomer Giuseppe Piazzi succeeded in following up the asteroid from the January 1 until February 11 realizing twenty-one observations.
The orbit determination and identification problems
Numerical behavior of the Keplerian Integral methods for the initial orbit determination problem
A differential algebra technique to handle observation errors
Generalization of a method by Mossotti for initial orbit determination
Patched Dynamics
Introduction
The method of patched conics is a classical technique used for the mission analysis of interplanetary trajectories [Battin, 1999]. The dynamics of a spacecraft under the gravitational influence of n bodies, i.e. the Sun and the planets, is not integrable. Thus, it is approximated by patching together basic Keplerian dynamics: when the spacecraft is sufficiently far from any planet, we assume it follows an unperturbed heliocentric orbit; on the contrary, when it is close to a planet and enters its sphere of influence, its unperturbed trajectory is considered planetocentric. A similar techinque, consisting in patching basic dynamics to model more complex problems, could be applied also in other contexts. In the following, we analyse two problems. We introduce the “Sun-shadow” dynamics, obtained by patching Kepler’s and Stark’s dynamics, to model the motion of an Earth satellite perturbed by the solar radiation pressure considering the Earth’s shadow effect. Then, we deal with the problem of close-encounters between small bodies (such as an asteroid) and the Earth. In this case, the model adopted is obtained by patching a heliocentric Keplerian dynamics and a planetocentric Keplerian dynamics as done in the classical patched-conics approximation.