On a time-symmetric Hermite integrator for planetary N-body simulation

We describe a P(EC)(n) Hermite scheme for planetary N-body simulation. The fourth-order implicit Hermite scheme is a time-symmetric integrator that has no secular energy error for the integration of periodic orbits with time-symmetric time-steps. In general N-body problems, however, this advantage is of little practical significance, since it is difficult to achieve time-symmetry with individual variable time-steps. However, we can easily enjoy the benefit of the time-symmetric Hermite integrator in planetary N-body systems, where all bodies spend most of the time on nearly circular orbits. These orbits are integrated with almost constant time-steps even if we adopt the individual time-step scheme. The P(EC)(n) Hermite scheme and almost constant time-steps reduce the integration error greatly. For example, the energy error of the P(EC)(2) Hermite scheme is two orders of magnitude smaller than that of the standard PEC Hermite scheme in the case of an N = 100, m = 10(25) g planetesimal system with the rms eccentricity [e(2)](1/2) less than or similar to 0.03.