The chaotic motion of the solar system: A numerical estimate of the size of the chaotic zones

Abstract

In a previous paper (J. Laskar, Nature 338, (237–238)), the chaotic nature of the Solar System excluding Pluto was established by the numerical computation of the maximum Lyapunov exponent of its secular system over 200 myr. In the present paper an explanation is given for the exponential divergence of the orbits: it is due to the transition from libration to circulation of the critical argument of the secular resonance 2 ( g 4 − g 3) − ( s 4 − s 3) related to the motions of perihelions and nodes of Earth and Mars. Another important secular resonance is identified: ( g 1 − g 5) − ( s 1 − s 2). Its critical argument stays in libration over 200 myr with a period of about 10 myr and amplitudes from 85 to 135°. The main features of the solutions of the inner planets are now identified when taking these resonances into account. Estimates of the size of the chaotic regions are determined by a new numerical method using the evolution with time of the fundamental frequencies. The chaotic regions in the inner Solar System are large and correspond to variations of about 0.2 arcsec/year in the fundamental frequencies. The chaotic nature of the inner Solar System can thus be considered as robust against small variations in the initial conditions or in the model. The chaotic regions related to the outer planet frequencies are very thin except for those of g 6 which present variations sufficiently large to be significant over the age of the Solar System.