A variety of different proxies for the stellar collision rates in globular clusters is used in the literature, depending on the quality of data available. We present comparisons between these proxies and the full integrated collision rate for different King models. The most commonly used proxy, Γ, defined to be, where ρ0 is the central cluster density, and rc is the core radius based on the 1966 King model, is an accurate representation of the collision rate from the King model to within about 25 per cent for all but the least concentrated globular clusters. By integrating over King models with a range of parameters, we show that, defined to be, where is the average density within the half-light radius, and rh 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, even with excellent data. Recent work has suggested that shallower than linear relations exist between proxies for Γ and the probability that a cluster will contain an X-ray source; we show that there is a similarly shallower than linear relationship between Γ and that can explain the relationship where is used; we also show that reasonable measurement errors are likely to produce a shallower than linear relationship even when Γ itself is used. We thus conclude that the existing evidence is all consistent with the idea that X-ray binary formation rates are linearly proportional to cluster collision rates. We also find, through comparison with Gunn-Griffin models (sometimes referred to as multimass King 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.
- Globular clusters: general