Long-term changes in the semimajor axes of the outer planets

One of the oldest problems of celestial mechanics is that of the long-term behaviour of the semimajor axes a of the planetary orbits. Analytical theories^1,2 predict periodic variations in a, some of which may have very long periods, but these terms have never been computed. We have now performed a 9.3-Myr numerical integration of the orbits of the outer planets, using a pure newtonian point mass model. An accurate integrator and an effective low-pass filtering of the output allow us to detect high-order variations in the energies, and hence also in a, with periods ranging from tens of thousand to millions of years. The most interesting feature is an energy exchange between Uranus and Neptune with a period of 1,119,000 years, the same as the period of the libration between the perihelia of Jupiter and Uranus^3,4. The mechanism involves Jupiter and also Saturn; moreover, their energy shows puzzling longer-term trends. The energy of Pluto changes mostly with periods close to that of the 3:2 libration in mean motion with Neptune. Its spectrum in this region shows a very complicated structure; however, we have found no indication of chaotic behaviour.