Dissipative normal forms in the Lagrange problem

In the present research proposal we describe a methodology to investigate the restricted three-body problem including dissipative effects by means of normal form theory. While the use of normal forms is a well established tool to investigate the qualitative aspects of the dynamics of the problem in the conservative setting, it was not used to fully investigate dissipative systems and in particular the restricted three-body problem including dissipative effects. In a recent research study ([1], [2]) we were able to develop the mathematical theory of a new kind of normal form suitable for dissipative, nearly-Hamiltonian systems and to prove the exponential stability of the actions of the conservative system for a generic class of ordinary differential equations of motion. The present proposal describes how to adapt and implement the theory to the case of the restricted three-body problem including dissipative effects and to derive a local stability theorem in the vicinity of the Lagrangian points of the Sun-Jupiter and Earth-Moon systems. While the former is an essential model to understand the origin and nature of our Solar system, the latter is an important model for next generation space observatories. Furthermore, the results of the proposed research study will provide the tools to investigate the dynamics of protoplanetary systems by making use of normal form theory. The main goals of the present research proposal are the following: 1.) Develop a model of the restricted three-body problem valid in the vicinity of the triangular equilibrium points L4, L5 including generic dissipative effects, like the nebular gas drag or Poynting-Robertson effect. 2.) Derive the dissipative normal form in the non-resonant and resonant regime of initial conditions of the proposed problem by adapting the theory developed in [1],[2] to derive stability estimates by means of a specialized computer algebra program to perform the computations. 3.) Implement a local stability theorem valid in the vicinity of the Lagrangian equilibrium points of the Sun-Jupiter and Earth-Moon systems; provide the stability region, the stability time, the influence of the system parameters as well conditions on the form of the dissipation. The importance of the proposed research study can be seen immediately: dissipative forces are present everywhere in nature. Their influence on the long term stability of dynamical problems found in the Solar System can not be neglected. The fundamental understanding of the interplay between conservative and dissipative forces allows for both: the understanding of the physical problem to a better degree of approximation as well the improvement of industrial applications of space science, which rely on analytical results of this form. The impact on the field of dynamical systems theory can only be understood by the fact that the concept of exponential stability in dissipative systems is definitely new. This opens a number of possible applications to different other fields of mathematical physics. [1] A. Celletti, C. Lhotka: "Stability for exponential times in nearly-Hamiltonian systems: the non--resonant case", preprint 2011 [2] A. Celletti, C. Lhotka: "Stability times in nearly-Hamiltonian systems: the resonant case", preprint 2011

We determined, in the present study, the stability of motion of mass-less particles in the Lagrangeproblem by taking into account also the influence of different dissipative, non-gravitational forces. The Lagrange problem is a special case of the restricted three-body problem (e.g. Sun-Earth-Moon), in which a mass-less celestial body moves close to the equilibria of the system (e.g. Sun-Jupiter-Trojan asteroid), and where gravitational forces are balanced. This dynamical problem is of great importance for the determination of the right location of special space observatories, and it has served as a fundamental model for the mathematical treatment of the movement of all kind of natural celestial bodies. The forces that are present in this system are usually classified into gravitational forces that can be derived from a potential, and non-gravitational (dissipative) forces, where one cannot associate a potential to it. Non-gravitational forces that are found in our solar system originate in the interaction of electromagnetic radiation from the Sun (or in generic planetary systems from a central star) and a celestial body of much smaller size. In most applications these kind of special forces can safely be neglected. However, for small bodies, like dust, meteoroids, asteroids, and space-crafts, they need also be taken into acount to avoid wrong scientific predictions. It is therefore crucial to understand exactly the influence of dissipative forces on the motion of the bodies for both, fundamental research and applied space science. In the present study we deeply investigated the Poynting-Robertson drag force and drag induced by e.g. molecular clouds, sometimes called Stokes drag. The study is based on analytical and numerical methods, where it has been necessary to modify the theory of normal forms of dynamical systems to be able to cope also with non-gravitaitonal forces. The adaption of the analytical theory was part of the project and could successfully be fulfilled. Meanwhile, it was shown by numerical methods, that motion that is stable in the pure gravitational problem is rendered unstable by taking into account also dissipative forces. However, it was still possible to find conditions that lead to a phenomenon, so-called temporary capture (that is stability for finite times). The conditions for temporary capture were found by means of an analytical theory, while the theory of normal forms on the long-term stability of motion of celestial bodies could not be applied, as it was expected, leading to the conclusion, that long-term capture is not possible in the framework of the assumptions that have been made. Another highlight of the present study is the new scientific insight that in the case of Poynting-Robertson drag temporary capture is not only possible outside the orbit of the perturbing budy, but also inside of it. This statement complements previous studies, that concluded that this kind of motion is not possible. Further investigations will be necessary to see the influence and implications of this new result on other scientific fields.