Energy equipartition in globular clusters through the eyes of dynamical models

Context. Following their birth, globular clusters (GCs) experience a very peculiar dynamical evolution. Gravitational encounters drive these systems toward energy equipartition, mass segregation, and evaporation, which alter structural, spatial, and kinematic features. Aims. We determine the dynamical state of a few GCs by means of a multi-mass King-like dynamical model. Our work focuses on the prediction of the energy equipartition degree and its relationship with model parameters. Methods. We adjusted the dynamical model parameters in order to reproduce the observed velocity dispersion – as derived from Hubble Space Telescope proper motion data – as a function of the stellar mass. By doing so, we estimated Φ0, a measure of the gravitational potential well. We repeated the same fit by means of the Bianchini relation, a function obtained by interpolating on Nbody simulation results. We studied the relationship between Φ0 and the Bianchini equipartition mass meq and discuss the structural properties, such as concentration c, the number of core relaxation timescales Ncore, and core radius rc. To obtain an independent estimate of Φ0, we also fitted observed surface brightness profiles using the predicted surface density and a mass–luminosity relation from isochrones. Results. The quality of the fits of the velocity dispersion–mass relationship obtained by means of our dynamical model is comparable to those obtained with the Bianchini function. Nonetheless, when the Bianchini function is used to fit the projected velocity dispersion, the resulting degree of equipartition is underestimated. On the contrary, our approach provides the equipartition degree at any radial or projected distance by means of Φ0. As a result, a cluster in a more advanced dynamical state shows a larger Φ0, as well as larger Ncore and c, while rc decreases. We find the estimates of Φ0 obtained by fitting surface brightness profiles to be compatible at 2σ confidence level with those from internal kinematics, although further investigation of statistical and systematic errors is required. Conclusions. Our work illustrates the predicting power of dynamical models to determine the energy equipartition degree of GCs. These models are a unique tool for determining structural and kinematic properties, and can be used where observational data are poor, as is the case for the most crowded regions of a cluster, where stars are barely resolved.