  As a follow-up of a previous work of the authors, this work considers {\em
uniform global time-renormalization functions} for the {\em gravitational}
$N$-body problem. It improves on the estimates of the radii of convergence
obtained therein by using a completely different technique, both for the
solution to the original equations and for the solution of the renormalized
ones. The aforementioned technique which the new estimates are built upon is
known as {\em majorants} and allows for an easy application of simple
operations on power series. The new radii of convergence so-obtained are
approximately doubled with respect to our previous estimates. In addition, we
show that {\em majorants} may also be constructed to estimate the local error
of the {\em implicit midpoint rule} (and similarly for Runge-Kutta methods)
when applied to the time-renormalized $N$-body equations and illustrate the
interest of our results for numerical simulations of the solar system.
