Choreographies in the N-body problem

Author(s)

Gabriela Iunes Depetri

• Document type: Master's Dissertation
• Press: São Paulo.
• Institution: Universidade de São Paulo (USP). Instituto de Física (IF/SBI)
• Defense date: 2011-03-03

Examining board members

Alberto Vazquez Saa; Marcus Aloizio Martinez de Aguiar; Antonio Fernando Ribeiro de Toledo Piza

• Advisor: Alberto Vazquez Saa

Abstract

The purpose of this work is the numerical computing of the periodic solutions to the N-body problem, that is, the general problem of determinig the motion of N bodies exclusively subject to gravitational forces between them. In particular, we search for solutions that were named choreographies, which have in common the property that all bodies move along the same curve. The interest in this kind of solution has recently increased due to technological advances in Gravitational Wave (GW) Physics. As the detection of Gws is foreseen for the near future, all periodic configurations of the N-body problem may be considered as possible sources of gravitational radiation. Identifying the patterns of radiation associated to these orbits is nowadays one of the pressing tasks in this field. Having this fact in mind, we calculate the GWs emitted by a system in which all bodies describe a choreographic orbit. In Chapter 1, we briefly describe the history of the search for the general solution to the N-body Problem, initially motivated by the interest in the stability analysis of the Solar System. Next, in Chapter 2, we present the main definitions and theorems to which we refer during this text. The reader may opt between following this chapter as he begins to read this thesis and consulting it only if necessary or when he is referred to. In Chapter 3, we identify the degrees of freedom of the system consisting of N bodies and determine the physical quantities it conserves, through Noethers theorem. Doing that, we establish the non-integrability of our dynamical system, in the sense of Liouville integrability, if N > 2. We also give the general solution to the 2-body problem, known as Keplers Problem, and present two particular solutions to the 3-body Problem, known as Eulers solution (1765) and Lagranges solution (1772). In Eulers solution, all three bodies are in the same line, which revolves around its center of mass, and in Lagranges solux tion they are at the vertices of an equilateral triangle, which also revolves around its center of mass. In order to describe all known periodic solutions to the N-body Problem, in Chapter 4 we study homographic orbits, that is, orbits in which the configuration at any instant can be obtained by a rotation and a dilation/contraction of the initial configuration. These solutions generalize the solutions by Euler and Lagrange mentioned above. In Chapter 5, we analyze choreographic orbits. This class of solutions was discovered by Cris Moore in 1993, who computed numerically a choreographic solution in which the bodies move along the same curve in the shape of an eight. The existence and stability of this orbit were rigorously studied by Richard Montgomery and Alain Chenciner. Here, we sketch the construction of the figure eight solution in the particular case where all masses are identical. We simulate this and other choreographic solutions, as well as some other periodic solutions described before, through the use of a fourth order Runge- Kutta method of numerical integration. Finally, in Chapter 6 we calculate the Gws emitted by the homographic and choreographic orbits simulated before. We end this work with a brief discussion comparing the GW patterns obtained to different orbits and analyzing the possibility of determining the mission source from a measurement of a GW signal. (AU)

• FAPESP's process: 08/07584-0 - N-bodies coreographies
• Grantee: Gabriela Iunes Depetri
• Support Opportunities: Scholarships in Brazil - Master