Abstract
$N$-body integrations are used to model a wide range of astrophysical
dynamics, but they suffer from errors which make their orbits diverge
exponentially in time from the correct orbits. Over long time-scales, their
reliability needs to be established. We address this reliability by running a
three-body planetary system over about $200$ e-folding times. Using nearby
initial conditions, we can construct statistics of the long-term phase-space
structure and compare to rough estimates of resonant widths of the system. Our
statistics are approximately consistent for a wide range of numerical methods,
including a Runge--Kutta method, Wisdom--Holman method, symplectic corrector
methods, and a method by Laskar & Robutel. "Improving" an integrator did not
affect the phase space accuracy, but simply increasing the number of initial
conditions did.
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