On the reliability of proxies for globular cluster collision rates

Abstract

(Abridged) Proxies for the stellar collision rates in globular clusters are often used. We present comparisons between these proxies and the full integrated collision rate for King models. Gamma, defined to be rho_0^3/2 r_c^2$, where $\rho_0$ is the central cluster density, and r_c is the core radius, is an accurate representation of the collision rate from the King model to within about 25% for all but the least concentrated clusters. Gamma_h, defined to be rho_h^3/2 r_h^2, where rho_h is the average density within the half-light radius, and $r_h$ is the half-light radius, is only marginally better correlated with the full King model collision rate than is the cluster luminosity. The two galaxies where results of King model fitting have been reported in detail show a dearth of core-collapsed clusters relative to that seen in the Milky Way, indicating that the core radii of the most concentrated clusters are probably slightly overestimated. Recent work has suggested that shallower than linear relations exist between proxies for Gamma and the probability that a cluster will contain an X-ray source; we show that there is a similarly shallow relationship between Gamma and Gamma_h; we also show that measurement errors are likely to produce a shallower than linear relationship even when Gamma itself is used. The evidence is thus consistent with the idea that X-ray binary formation rates are linearly proportional to cluster collision rates. We also find, through comparison with multimass models, suggestive evidence that the retention fractions of neutron stars in globular clusters may be related to the present day concentration parameters, which would imply that the most concentrated clusters today were the most concentrated clusters at the time of their supernovae.