Formation of runaway stars in a star-cluster potential

We study the formation of runaway stars due to binary-binary (2+2) interactions in young star-forming clusters and/or associations. This is done using a combination of analytic methods and numerical simulations of 2+2 scattering interactions, both in isolation and in a homogeneous background potential. We focus on interactions that produce two single stars and a binary, and study the outcomes as a function of the depth of the background potential, within a range typical of cluster cores. As reference parameters for the observational properties, we use those observed for the system of runaway stars AE Aur and $\mu$ Col and binary $\iota$ Ori. We find that the outcome fractions have no appreciable dependence on the depth of the potential, and neither do the velocities of the ejected single stars. However, as the potential gets deeper and a larger fraction of binaries remain trapped, two binary populations emerge, with the escaped component having higher speeds and shorter semi-major axes than the trapped one. Additionally, we find that the relative angles between the ejected products are generally large. In particular, the angle between the ejected fastest star and the escaped binary is typically $\gtrsim 120-135^{\circ}$, with a peak at around $160^{\circ}$. However, as the potential gets deeper, the angle distribution becomes broader. Finally, we discuss the implications of our results for the interpretation of the properties of the runaway stars AE Aur and $\mu$ Col.