Majorant series for the $N$-body problem

TL;DR

This paper improves the estimates of convergence radii for the gravitational N-body problem using majorant series techniques, enhancing the understanding of solution behavior and numerical method accuracy in celestial simulations.

Contribution

It introduces a new approach using majorants to obtain better convergence estimates and error bounds for time-renormalized N-body equations.

Findings

Radii of convergence approximately doubled compared to previous estimates.

Majorants effectively estimate local errors of implicit midpoint and Runge-Kutta methods.

Results are relevant for improving numerical simulations of the solar system.

Abstract

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.