The Extended Tisserand-Poincaré graph for multi-body trajectory design

Abstract

The decision on the dynamical model used to design a spacecraft’s trajectory is a fundamental choice influenced by many factors. The complexity of such a model affects the accuracy of the designed trajectory but also the difficulty of the design process. One approach to simplify this task consists of splitting the trajectory design process into two phases. In the first phase a first-guess reference trajectory is generated through the usage of a simple dynamical model considering the most important forces in the system. In the second phase this reference trajectory is taken and refined with the addition of the perturbing forces that have been neglected in the first phase. The success of this approach depends on the proper sharing of the perturbing forces between the two phases. In this sense the backbone of the trajectory is designed in the first phase and it is fundamental that in this phase the main forces are taken into account to not avoid to consider them as negative perturbing effects in the second phase, that often is performed using an optimization technique.

The research covered in this report is focused on the first phase of this process, in particular for what concerns the gravitational influences generated by multiple bodies. In this sense the simplest dynamical model that could be used considers only the gravitational influence of the main attracting body. Such model however is not able to capture interesting and useful phenomena existing in a more realistic multi-body environment that allow to save a considerable amount of ∆V . This aspect is of crucial importance for missions such as EQUULEUS and DESTINY, whose limited ∆V capabilities need to be compensated by a clever design process actively exploiting both lunar and solar perturbation effects.

This is achieved in this research through the usage of a patched Circular Restricted Three-Body Problem model describing the motion of the spacecraft under the influence of two primaries at a time. Such a patched model is an approximated model that represents the trajectory of a spacecraft under the influence of three bodies. A graphical technique called Tisserand-Poincaré graph exploits this dynamical model to design trajectories in several multi-body environments. The applicability of this technique however is strongly tied to the definition of the so-called Tisserand parameter. The latter however exists only about the primary, limiting the patching between different Circular Restricted Three-Body Problems to those systems sharing the most attracting body such as the Jupiter-Europa-Ganymede or Sun-Earth-Jupiter systems. To overcome this limitation, that would make it impossible to apply the same technique in the Sun-Earth-Moon system for the EQUULEUS and DESTINY missions, the theoretical framework of the technique is extended in this research. In particular a modified version of the Tisserand parameter defined about the secondary that possesses the same constancy property of the classical Tisserand parameter but on a specifically defined family of Poincaré sections is derived. The concept of the modified Tisserand parameter is based on a relationship between the parameter and the Jacobi constant of the system that can be expressed by the fundamental equation J = T +P .The new parameter and associated Poincaré sections are used to characterize the Earth-Moon and Sun-Earth systems. Through the usage of a graphical tool, the definition of the modified parameter and the definition of a specific family of Poincaré sections, both solar and lunar perturbations are characterized and exploited in the trajectory design process in the multi-body environment. A database approach to be used for the EQUULEUS trajectory design in a quasi-four body problem is developed. Databases of flyby trajectories about the Moon and Jupiter in the Earth-Moon and Sun-Jupiter CR3BP models are also developed. A standard patching procedure is tested in the Sun-Jupiter-Europa environment to demonstrate a capture trajectory about Jupiter.