Global time-renormalization of the gravitational $N$-body problem

Abstract

This work considers the {\em gravitational} $N$-body problem and introduces global time-renormalization {\em functions} that allow the efficient numerical integration with fixed time-steps. First, a lower bound of the radius of convergence of the solution to the original equations is derived, which suggests an appropriate time-renormalization. In the new fictitious time $τ$, it is then proved that any solution exists for all $τ\in \mathbb{R}$, and that it is uniquely extended as a holomorphic function to a strip of fixed width. As a by-product, a global power series representation of the solutions of the $N$-body problem is obtained. Noteworthy, our global time-renormalizations remain valid in the limit when one of the masses vanishes. Finally, numerical experiments show the efficiency of the new time-renormalization functions for some $N$-body problems with close encounters.