Step-by-step explanation:
Abstract
Most planetary systems are formed within stellar clusters, and these environments can shape their properties. This paper considers scattering encounters between solar systems and passing cluster members, and calculates the corresponding interaction cross-sections. The target solar systems are generally assumed to have four giant planets, with a variety of starting states, including circular orbits with the semimajor axes of our planets, a more compact configuration, an ultracompact state with multiple mean motion resonances, and systems with massive planets. We then consider the effects of varying the cluster velocity dispersion, the relative importance of binaries versus single stars, different stellar host masses, and finite starting eccentricities of the planetary orbits. For each state of the initial system, we perform an ensemble of numerical scattering experiments and determine the cross-sections for eccentricity increase, inclination angle increase, planet ejection, and capture. This paper reports results from over 2 million individual scattering simulations. Using supporting analytic considerations, and fitting functions to the numerical results, we find a universal formula that gives the cross-sections as a function of stellar host mass, cluster velocity dispersion, starting planetary orbital radius, and final eccentricity. The resulting cross-sections can be used in a wide variety of applications. As one example, we revisit constraints on the birth aggregate of our Solar system due to dynamical scattering and find N ≲ 104 (consistent with previous estimates).