  Symplectic integrators have made it possible to study the long-term evolution
of planetary systems with direct N-body simulations. In this paper we reassess
the accuracy of such simulations by running a convergence test on 20Myr
integrations of the Solar System using various symplectic integrators.
  We find that the specific choice of metric for determining a simulation's
accuracy is important. Only looking at metrics related to integrals of motions
such as the energy error can overestimate the accuracy of a method. As one
specific example, we show that symplectic correctors do not improve the
accuracy of secular frequencies compared to the standard Wisdom-Holman method
without symplectic correctors, despite the fact that the energy error is three
orders of magnitudes smaller. We present a framework to trace the origin of
this apparent paradox to one term in the shadow Hamiltonian. Specifically, we
find a term that leads to negligible contributions to the energy error but
introduces non-oscillatory errors that result in artificial periastron
precession. This term is the dominant error when determining secular
frequencies of the system. We show that higher order symplectic methods such as
the Wisdom-Holman method with a modified kernel or the SABAC family of
integrators perform significantly better in secularly evolving systems because
they remove this specific term.
