The Criterion for Chaos in Three-planet Systems

We establish the criterion for chaos in three-planet systems, for systems similar to those discovered by the Kepler spacecraft. Our main results are as follows: (i) The simplest criterion, which is based on overlapping mean motion resonances (MMRs), only agrees with numerical simulations at a very crude level. (ii) Much greater accuracy is attained by considering neighboring MMRs that do not overlap. We work out the widths of the chaotic zones around each of the neighbors, and also provide simple approximate expressions for the widths. (iii) Even greater accuracy is provided by the overlap of three-body resonances (3BRs), which accounts for the fine-grained structure seen in maps from N-body simulations, and also predicts Lyapunov times. From previous studies, it is unclear whether interplanetary chaos should be attributed to the overlap of MMRs or of 3BRs. We show that the two apparently contradictory viewpoints are in fact consistent: both predict the same criterion for chaos. (iv) We compare the predicted criterion with high-resolution maps of chaos from N-body simulations, and show that they agree at a high level of detail.