An analytic technique for constraining the dynamical origins of multiple star systems containing merger products

We present a technique to identify the most probable dynamical formation scenario for an observed binary or triple system containing one or more merger products or, alternatively, to rule out the possibility of a dynamical origin. Our method relies on an analytic prescription for energy conservation during stellar encounters. With this, observations of the multiple star system containing the merger product(s) can be used to work backwards in order to constrain the initial orbital energies of any single, binary or triple systems that went into the encounter. The initial semimajor axes of the orbits provide an estimate for the collisional cross-section and therefore the time-scale for the encounter to occur in its host cluster. We have applied our analytic prescription to observed binary and triple systems containing blue stragglers (BSs), in particular the triple system S1082 in M67, and the period distribution of the BS binaries in NGC 188. We have shown that both S1082 and most of the BS binaries in NGC 188 could have a dynamical origin and that encounters involving triples could be a significant contributor to BS populations in old open clusters. In general, our results suggest that encounters involving triples could make up a significant fraction of those dynamical interactions that result in stellar mergers, in particular, encounters that produce multiple star systems containing one or more BSs.