Solving the N-body problem in astrophysics

Abstract

The astrophysical N-body problem describes N point masses interacting with each other through pairwise gravitational forces. A solution of this problem is frequently necessary in dynamical astronomy. In the collisional N-body problem, the relaxation time is small compared to the timescale we are interested in studying. Collisional N-body problems include open and globular clusters and protoplanetary disks during the stage, typically lasting hundreds of Myrs, when planetary embryos collide and merge. In the first part of this Thesis, I develop new symplectic integrators which provide a solution for the N-body problem. The integrators decompose the N-body problem into a superposition of two-body problems, which are integrable. Since they are symplectic, the integrators conserve all Poincaré invariants (the evolution is Hamiltonian). We used the integrators to compute the evolution of a globular cluster through core collapse up to 20 times faster than standard techniques. In the second part of this Thesis, I apply the results from the first part of the Thesis to planetary dynamics finding that for problems with hierarchical binaries (planets with moons, planetary systems with binary stars, etc.), the integrators are far more efficient than alternatives. I show numerically that a popular code is neither symplectic nor time-symmetric, and can yield incorrect three-body dynamics. I derive symplectic integrators in various coordinate systems with different Hamiltonian splittings and compare them through backward error analysis and tests of Pluto's orbital element evolution. The final part of this Thesis is concerned with time-symmetric and time-reversible integration in astrophysics, whether we are integrating the N-body problem or other ordinary differential equations. These integrators have been proposed as an alternative to symplectic integration. I show, again using backward error analysis, that such integrators are usually useful, but can behave worse than symplectic integrators. I find time-reversibility can be eliminated in some cases while good error behavior is still maintained.