We study the chaotic orbital evolution of planetary systems, focusing on
secular (i.e., orbit-averaged) interactions, because these often dominate on
long timescales. We first focus on the evolution of a test particle that is
forced by multiple massive planets. To linear order in eccentricity and
inclination, its orbit precesses with constant frequencies. But nonlinearities
can shift the frequencies into and out of secular resonance with the planets'
eigenfrequencies, or with linear combinations of those frequencies. The overlap
of these nonlinear secular resonances drive secular chaos in planetary systems.
We quantify the resulting dynamics for the first time by calculating the
locations and widths of nonlinear secular resonances. When results from both
analytical calculations and numerical integrations are displayed together in a
newly developed "map of the mean momenta" (MMM), the agreement is excellent.
This map is particularly revealing for non-coplanar planetary systems and
demonstrates graphically that chaos emerges from overlapping secular
resonances. We then apply this newfound understanding to Mercury. Previous
numerical simulations have established that Mercury's orbit is chaotic, and
that Mercury might even collide with Venus or the Sun. We show that Mercury's
chaos is primarily caused by the overlap between resonances that are
combinations of four modes, the Jupiter-dominated eccentricity mode, the
Venus-dominated inclination mode and Mercury's free eccentricity and
inclination. Numerical integration of the Solar system confirms that a slew of
these resonant angles alternately librate and circulate. We are able to
calculate the threshold for Mercury to become chaotic: Jupiter and Venus must
have eccentricity and inclination of a few percent. Mercury appears to be
perched on the threshold for chaos.Comment: 18 pages, submitted to Ap
