Mass estimates from stellar proper motions: the mass of ω Centauri

We lay out and apply methods to use proper motions of individual kinematic tracers for estimating the dynamical mass of star clusters. We first describe a simple projected mass estimator and then develop an approach that evaluates directly the likelihood of the discrete kinematic data given the model predictions. Those predictions may come from any dynamical modelling approach, and we implement an analytic King model, a spherical isotropic Jeans equation model and an axisymmetric, anisotropic Jeans equation model. This maximum likelihood modelling (MLM) provides a framework for a model-data comparison, and a resulting mass estimate, which accounts explicitly for the discrete nature of the data for individual stars, the varying error bars for proper motions of differing signal-to-noise ratio, and for data incompleteness. Both of these two methods are evaluated for their practicality and are shown to provide an unbiased and robust estimate of the cluster mass. We apply these approaches to the enigmatic globular cluster ω Centauri, combining the proper motion from van Leeuwen et al. with improved photometric cluster membership probabilities. We show that all mass estimates based on spherical isotropic models yield (4.55 ± 0.1) × 106 M⊙D/5.5 ± 0.2 kpc3, where our modelling allows us to show how the statistical precision of this estimate improves as more proper motion data of lower signal-to-noise ratio are included. MLM predictions, based on an anisotropic axisymmetric Jeans model, indicate for ω Cen that the inclusion of anisotropies is not important for the mass estimates, but that accounting for the flattening is: flattened models imply (4.05 ± 0.1) × 106 M⊙D/5.5 ± 0.2 kpc3, 10 per cent lower than when restricting the analysis to a spherical model. The best current distance estimates imply an additional uncertainty in the mass estimate of 12 per cent.