Global Time-Renormalization of the Gravitational $N$-body Problem

This work considers the gravitational N-body problem and introduces time-reparametrization functions that allow to define globally solutions of the N-body equations. First, a lower bound of the radius of convergence of the solution to the original equations is derived, which suggests an appropriate time-reparametrization. In the new fictitious time τ , it is then proved that any solution exists for all τ ∈ 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-regularization remain valid in the limit when one of the masses vanishes. Finally, numerical experiments show the efficiency of the new time-regularization functions for some N-problems with close encounters.