Analytical insight into the motion of the inner planets requires an appropriate dynamical modelling of the long-term evolution of their orbits. On the one hand, such a model must be consistent with the predictions of the reference numerical integrations available in literature (Laskar 1990c, 2008; Laskar et al. 2004; Laskar & Gastineau 2009) to ensure that it reproduces the dynamical features of the inner system with sufficient precision. On the other hand, the corresponding Hamiltonian should be set in a form suitable for the systematic application of canonical perturbation techniques (Hori 1966; Deprit 1969), which is essential to an unbiased analysis of such a high-dimensional dynamics. Moreover, the possibility of numerically integrating the equations of motion in an efficient way is fundamental for studying the chaotic evolution of the orbital solutions in a statistical way. Unfortunately, the construction of such a model is a delicate task. In principle, we might just consider the full N-body Hamiltonian of the Newtonian gravitational interactions among the Solar System planets, with the addition of the leading corrections coming from general relativity and the Earth–Moon interaction. This already reproduces the precession frequencies of the inner orbits with a precision better than 0.01′′ yr−1 (Laskar 1999). However, this Hamiltonian is unnecessarily complicated because it includes short-time harmonics, with periods shorter than 5000 yr (e.g. Carpino et al. 1987), which are known to generate small quasi-periodic oscillations in the inner orbits, without being implied in chaos generation. The inner planets are not, indeed, involved in any relevant mean-motion resonance. At the same time, the long-term numerical integration of the corresponding equations of motion is very time consuming, the resulting solutions needing to be filtered to extract the secular trend of the orbits (Carpino et al. 1987; Nobili et al. 1989). These facts suggest considering a secular Hamiltonian to directly describe the slow movement of the planet perihelia and nodes after proper averaging over the short-time orbital motion (Laskar 1984, 1985). Secular dynamics includes the essential planet interactions responsible for chaos in the inner Solar System and allows performing the fastest long-term numerical integrations (Laskar 1988, 1989, 1994, 2008). This enabled the discovery of chaos in the inner system before the use of symplectic integration schemes (Sussman & Wisdom 1992). Unfortunately, an effective secular model for the entire Solar System has to be of high order in planet masses, principally because of the 5:2 near mean-motion resonance between Jupiter and Saturn, the so-called great inequality (Laplace 1785; Laskar 1996). A simple averaging of the N-body Hamiltonian over the planet mean longitudes, resulting in a first-order secular dynamics in planet masses, would reproduce the fundamental frequencies g5 and g6 (which dominate the perihelion precession of Jupiter and Saturn, respectively) very poorly1 (Laskar 1988). The construction of higher-order models requires the manipulation of large Poisson series and the use of sophisticated computer algebra systems. This is probably the reason why they are still not widely used, at least as a basis of extensive research.
Citation: Laskar 1999
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