Mass estimates from stellar proper motions: The mass of $\omega$
  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.We apply these
approaches to the enigmatic globular cluster omega Centauri, combining the
proper motion from van Leeuwen et al (2000) with improved photometric cluster
membership probabilities. We show that all mass estimates based on spherical
isotropic models yield $(4.55\pm 0.1) \times 10^6 M_{\odot} [D/5.5 \pm 0.2
kpc]^3$, 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 are
included. MLM predictions, based on an anisotropic axisymmetric Jeans model,
indicate for $\omega$ Cen that the inclusion of anisotropies is not important
for the mass estimates, but that accounting for the flattening is: flattened
models imply $(4.05\pm 0.1) \times 10^6 M_{\odot} [D/5.5 \pm 0.2 kpc]^3$, 10%
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%.
