Long-term influence of asteroids on planet longitudes and chaotic dynamics of the solar system

Over timescales much longer than an orbital period, the solar system exhibits large-scale chaotic behavior and can thus be viewed as a stochastic dynamical system. The aim of the present paper is to compare different sources of stochasticity in the solar system. More precisely we studied the importance of the long term influence of asteroids on the chaotic dynamics of the solar system. We show that the effects of asteroids on planets is similar to a white noise process, when those effects are considered on a timescale much larger than the correlation time τ_Φ ≃ 10⁴ yr of asteroid trajectories. We computed the timescale τ_e after which the effects of the stochastic evolution of the asteroids lead to a loss of information for the initial conditions of the perturbed Laplace-Lagrange secular dynamics. The order of magnitude of this timescale is precisely determined by theoretical argument, and we find that τ_e ≃ 10⁴ Myr. Although comparable to the full main-sequence lifetime of the sun, this timescale is considerably longer than the Lyapunov time τ_i ≃ 10 Myr of the solar system without asteroids. This shows that the external sources of chaos arise as a small perturbation in the stochastic secular behavior of the solar system, rather due to intrinsic chaos.