Bureau des Longitudes

Abstract: Numerical study of the global stability of the spin-axis orientation (obliquity) of the planets against secular orbital perturbations shows that all of the terrestrial planets could have experienced large, chaotic variations in obliquity at some time in the past. The obliquity of Mars is still in a large chaotic region, ranging from 0° to 60°. Mercury and Venus have been stabilized by tidal dissipation, and the Earth may have been stabilized by capture of the Moon. None of the obliquities of the terrestrial planets can therefore be considered as primordial.

Abstract: LAPLACE and Lagrange made an essential contribution to the study of the stability of the Solar System by proving analytically that, to first order in the masses, inclinations and eccentricities of their orbits, the planets move quasiperiodically. Since then, many analytic quasiperiodic solutions have been sought to higher order1–10.1 have recently constructed an extensive analytic system of averaged differential equations containing the secular evolution of the orbits of the eight main planets, accurate to second order in the planetary masses and to fifth order in eccentricity and inclination, and including corrections from general relativity and the Moon8–10. Here I describe the results of a numerical integration of this system, extending backwards over 200 million years. The solution is chaotic, with a maximum Lyapunov exponent that reaches the surprisingly large value of ∼ 1/5 Myr–1. The motion of the Solar System is thus shown to be chaotic, not quasiperiodic. In particular, predictability of the orbits of the inner planets, including the Earth, is lost within a few tens of millions of years. This does not mean that after such a short timespan we will see catastrophic events such as a crossing of the orbits of Venus and Earth; but the traditional tools of quantitative celestial mechanics (numerical integrations or analytical theories), which aim at unique solutions from given initial conditions, will fail to predict such events. The problem of the stability of the Solar System will have to be set up again, and the qualitative methods initiated by Poincare definitely need to replace quantitative methods in this analysis.

Abstract: The expected changes over the past 500 million years in the principal astronomical frequencies influencing the Earth's climate may be strong enough to be detectable in the geological records, and such effects have been inferred in several cases. Calculations suggest that the shortening of the Earth-moon distance and of the length of the day back in time induced a shortening of the fundamental periods for the obliquity and climatic precession, from 54 to 35, 41 to 29, 23 to 19, and 19 to 16 thousand years over the last half-billion years. At the same time, the precessional constant increased from 50 to 61 arc seconds per year. The changes in the frequencies of the planetary system due to its chaotic motion are much smaller; their influence on the changes of the periods of climatic precession, obliquity, and eccentricity of the Earth's orbit around the sun can be neglected. Eccentricity periods used for Quaternary climate studies may therefore be considered to have been more or less constant for pre-Quaternary times.