A modified Hermite integrator for planetary dynamics

We describe a modified time-symmetric Hermite integrator specialized for the long-term integration of planetary orbits. Our time-symmetric integrators have no secular errors in the semi-major axis and the eccentricity for the integration of two-body Kepler problems as usual time-symmetric and symplectic integrators. The usual time-symmetric or symplectic integrators, however, show a secular drift in the argument of pericenter. Our new family of integrators has one free parameter, which we can adjust to reduce the error in the argument of pericenter without breaking the time-symmetry or changing the order of the integrator. We show analytically that the leading term of the error vanishes for a unique value of the parameter, which is independent of the size of the timestep and the eccentricity. It is also possible to eliminate the non-leading, higher-order terms by using a parameter value that depends on both the size of the timestep and the eccentricity. We describe the second- and the fourth-order schemes. An extension to higher order is straightforward.