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Comparing Extended System Interactions with Motions in Softened Potentials

Using an $N$-body evolution code that does not rely on softened potentials, I have created a suite of unbound interacting cluster pair simulations. The motions of the centers of mass of the clusters have been tracked and compared to the trajectories of point masses interacting via one of four different softened potential prescriptions. I find that the relationship between the impact parameter of the cluster interaction and the point-mass softening length that best approximates each cluster's center-of-mass motion depends on the adopted prescription. In general, the range of allowed softening lengths grows roughly linearly with the impact parameter, but zero softening is acceptable in the majority of situations. In an $N$-body simulation that adopts a fixed softening length, such relationships lead to the possibility of two-body effects, like dynamical friction, being either larger or smaller than the corresponding cluster situation. Further consideration of more specific $N$-body situations leads estimating that a very small fraction of point-mass encounters experience two-body effects significantly different than those of equivalent clusters.