Orbital stability of compact three-planet systems, II: Post-instability impact behaviour

Recent observational missions have uncovered a significant number of compact multi-exoplanet systems. The tight orbital spacing of these systems has led to much effort being applied to the understanding of their stability; however, a key limitation of the majority of these studies is the termination of simulations as soon as the orbits of two planets cross. In this work we explore the stability of compact, three-planet systems and continue our simulations all the way to the first collision of planets to yield a better understanding of the lifetime of these systems. We perform over $25,000$ integrations of a Sun-like star orbited by three Earth-like secondaries for up to a billion orbits to explore a wide parameter space of initial conditions in both the co-planar and inclined cases, with a focus on the initial orbital spacing. We calculate the probability of collision over time and determine the probability of collision between specific pairs of planets. We find systems that persist for over $10^8$ orbits after an orbital crossing and show how the post-instability survival time of systems depends upon the initial orbital separation, mutual inclination, planetary radius, and the closest encounter experienced. Additionally, we examine the effects of very small changes in the initial positions of the planets upon the time to collision and show the effect that the choice of integrator can have upon simulation results. We generalise our results throughout to show both the behaviour of systems with an inner planet initially located at $1$ AU and $0.25$ AU.