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

The aim of this paper is to compare different sources of stochasticity in the solar system. More precisely we study 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 time scale much larger than the correlation time $\tau_{\varphi}\simeq10^{4}$ yr of asteroid trajectories. We compute the time scale $\tau_{e}$ 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\textendash Lagrange secular dynamics. The order of magnitude of this time scale is precisely determined by theoretical argument. This time scale should be compared with the Lyapunov time $\tau_{i}$ of the solar system without asteroids (intrinsic chaos). We conclude that $\tau_{i}\simeq10\, \text{Myr} \ll \tau_{e} \simeq10^{4}\, \text{Myr}$, showing that the external sources of chaoticity arise as a small perturbation in the stochastic secular behavior of the solar system, rather due to intrinsic chaos.