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This work considers the {\em gravitational} $N$-body problem and introducesglobal time-renormalization {\em functions} that allow the efficient numericalintegration with fixed time-steps. First, a lower bound of the radius ofconvergence of the solution to the original equations is derived, whichsuggests an appropriate time-renormalization. In the new fictitious time$\tau$, it is then proved that any solution exists for all $\tau \in\mathbb{R}$, and that it is uniquely extended as a holomorphic function to astrip of fixed width. As a by-product, a global power series representation ofthe solutions of the $N$-body problem is obtained. Noteworthy, our globaltime-renormalizations remain valid in the limit when one of the massesvanishes. Finally, numerical experiments show the efficiency of the newtime-renormalization functions for some $N$-body problems with closeencounters.

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