I had an email from Graeme Waddington pointing out that the periodic wiggles due to Jupiter and the other planets disappear if you use a barycentric coordinate system rather than heliocentric. This is because, when the comet is far out, it is orbiting around the centre of mass of the Solar System (the barycentre) which is not exactly at the centre of the Sun. Graeme’s plots are attached. The y axis is 1/a which, if you remember from my first post, is equal to (1-e)/q so it has the big advantage that it is well conditioned for orbits where e is near to 1 (i.e. it doesn’t tend to infinity as e tends to 1). Graeme notes “At the ends of the time frame plotted the comet is at 244 au and 250 au from the sun –> P(orig.) = 4674 yr, P(future) = 7200 yr. For a similar integration using the JPL15 orbit I get 4434 and 6718 years respectively.”
