Small-N collisional dynamics - III: The battle for the realm of not-so-small-N

In this paper, the third in the series, we continue our study of combinatorics in chaotic Newtonian dynamics. We study the chaotic four-body problem in Newtonian gravity assuming finite-sized particles, and we focus on interactions that produce direct collisions between any two stars. Our long-term goal is to construct an equation that gives the probability of a given collision event occurring over the course of the interaction, as a function of the total encounter energy and angular momentum as well as the numbers and properties of the particles. In previous papers, we varied the number of interacting particles and the distribution of particle radii for all equal mass particles. Here, we focus on the effects of different combinations of particle masses. We develop an analytic formalism for calculating the time-scales for different collision scenarios to occur. Our analytic time-scales reproduce the simulated time-scales when gravitational focusing is included. We present a method for calculating the relative rates for different types of collisions to occur, assuming two different limits for the particle orbits: radial and tangential. These limits yield relative collision probabilities that bracket the probabilities we obtain directly from numerical scattering experiments and are designed to reveal important information about the (time-averaged) trajectories of the particles as a function of the interaction parameters. Finally, we present a collision rate diagram (CRD), which directly compares the predictions of our analytic rates to the simulations and quantifies the quality of the agreement. The CRD will facilitate refining our analytic collision rates in future work, as we expand in to the remaining parameter space.