Introduction

Sooner or later during the integration of an N-body system close encounters create configurations that lead to difficulties or at best become very time-consuming if studied by direct methods. On further investigation one usually finds a binary of short period slowing down the calculation and introducing unacceptable systematic errors. Moreover, the eccentricity may attain a large value that necessitates small time-steps in the pericentre region unless special features are introduced. It can be seen that a relative criterion of the type (ηR/|F|)½ for a binary yields an approximate time-step Δt ∝ R3/2, where R is the two-body separation. From this it follows that eccentric orbits require more steps for the same period. Even a relatively isolated binary may therefore become quite expensive to integrate as well as cause a significant drift in the total system energy. It is convenient to characterize the systematic error of a binary integration by the relative change per Kepler orbit, α = Δa/a. For example, using the basic Hermite method we find α = –1.3×10–6 with 270 steps per orbit and an eccentricity e = 0.9. At this rate of inward spiralling, the binary energy would be significantly affected after 104 – 105 periods. Although better behaved, less eccentric systems are also time-consuming, giving α = –4×10–8 for e = 0.2 and 135 steps per orbit.