Mercury's Chaotic Secular Evolution as a Subdiffusive Process

Mercury's orbit can destabilize, generally resulting in a collision with either Venus or the Sun. Chaotic evolution can cause g₁ to decrease to the approximately constant value of g₅ and create a resonance. Previous work has approximated the variation in g₁ as stochastic diffusion, which leads to a phenomological model that can reproduce the Mercury instability statistics of secular and N-body models on timescales longer than 10 Gyr. Here we show that the diffusive model significantly underpredicts the Mercury instability probability on timescales less than 5 Gyr, the remaining lifespan of the solar system. This is because g₁ exhibits larger variations on short timescales than the diffusive model would suggest. To better model the variations on short timescales, we build a new subdiffusive phenomological model for g₁. Subdiffusion is similar to diffusion but exhibits larger displacements on short timescales and smaller displacements on long timescales. We choose model parameters based on the behavior of the g₁ trajectories in the N-body simulations, leading to a tuned model that can reproduce Mercury instability statistics from 1–40 Gyr. This work motivates fundamental questions in solar system dynamics: why does subdiffusion better approximate the variation in g₁ than standard diffusion? Why is there an upper bound on g₁, but not a lower bound that would prevent it from reaching g₅?