Computation of collision probability in Geostationary Orbit

In the present work, the probability of collision between two orbiting bodies in a geostationary (or near geostationary) orbit will be addressed. Much has been written about the topic of computing the collision probability of satellites during the past decades. Certainly, the large volume of literature addressing this issue unveils the need that exists for ensuring the safety and integrity of the constantly growing population of satellites. The density of the "cloud" seems to increase without bound and the presence of debris puts into risk the operative satellites. Moreover, there is a specific ring, inside this "cloud" in which, due to its optimal conditions for communications or meteorological purposes, the increase of the mentioned "cloud density" is even larger: the Geostationary orbit. The standard and widely accepted method for the estimation of the probability of collision is the well-known Monte Carlo simulation. Nevertheless, the large computational time required by this method may make it unfeasible to take real-time decision when there exist several collision risk situations at the same time. This is the reason why numerous approximated methods have been developed, aiming at reducing the computational time. In LEO orbits, where satellites normally have not at all similar orbits, the satellite encounters( which are bounded by 3-sigma position error ellipsoids) occur at really large relative velocity and, as a result, the encounter time is very short. This fact is of paramount importance for the calculation of collision probability, since as the encounter is very brief, one simply needs to propagate the positions of the satellites to the point of closest approach and apply the existing methods. These can be classified as linear relative motion methods, given that the relative velocity during the encounter can be considered to be linear. Now, if the orbits described by the satellites are similar, there is likelihood that their relative velocity is small. Thus, the encounter time will be large enough to prevent from assuming that the velocity does not change throughout the rendez-vous. This is the typical case observed in the region near the geostationary orbits, where all satellites try to be all as closed as possible to the mentioned orbit, which is unique. Several methods have been developed in order to solve this particular orbital rendez-vous in a relatively fast manner. They must deal with changes in relative velocity, position and position error covariance matrices, since it is not enough to analyze just the moment of closest approach. Therefore, one can divide all the methods into two main groups: linear relative motion methods and non-linear relative motion methods. In this work, we will focus on studying the latter in order to be able to estimate the collision probability of geostationary orbits .Particularly, we will analyze one method developed by Patera , which is one of the most popular in the industry currently. A few realistic cases will be studied and the validity of Patera’s method will be tested comparing its results with those obtained by means of Monte Carlo simulation, whose probability outputs cannot disagree in more than 10⁻⁴ (which is the threshold to decide whether or not performing a collision avoidance maneuver in case of collision risk). In addition, the validity of a suggested improvement of Patera’s method will be tested, using the same criterion. Prior to this study and the presentation of its results, a short background on the probability of collision calculation will be provided.Ingeniería Aeroespacia