Regardless of the tolerance and the order of the method, the ratio of the stepsize in the perturbative approach to that in the direct one was typically an exponential function of . Thus the benefits of the near-integrable approach are clear even when the perturbation is no longer small and the equations of motion are far from analytically integrable. We found that, on average, the timestep required for the near-integrable approach can be some five times longer than in the direct method. Similar results are obtained with spatially fixed frames and multistep methods. Even though the steps take a longer time to compute in our approach due to the Keplerian transformations, the overall speed of the algorithm is usually noticeably faster than that of direct integration.
Eccentric orbits bring about the problem of close encounters.
One can, in principle, switch to a new dominating central body whenever necessary and choose a suitable near-integrable Keplerian configuration. For an asteroid approaching Jupiter, this zone is roughly between 0.02 AU and 1 AU from the planet. Proper'' close encounters are thus very rare, and even in them the time spent in the transition zone dominates the computational effort. A further problem is that the object may wander in and out of the transition zone several times during the encounter. Close encounters are thus clearly easiest to handle in conventional integration variables.