Energy equipartition in Globular Clusters through the eyes of dynamical models

Gravitational encounters drive globular clusters toward energy equipartition, mass segregation and evaporation altering structural, spatial and kinematic features. We determine the dynamical state of a few globular clusters by means of a multi-mass King-like dynamical model, focusing on the energy equipartition degree and its relation with model parameters. We fit the observed velocity dispersion, as derived from HST proper motion data, as a function of the stellar mass $\sigma(m)$, to estimate the parameter $\Phi_0$, a measure of the gravitational potential well. The same fit is repeated by means of the Bianchini relation, providing the equipartition mass $m_\mathrm{eq}$. The relationship between $\Phi_0$ and $m_\mathrm{eq}$ has been studied and the structural properties, such as concentration $c$, number of core relaxation timescales $N_\mathrm{core}$ and core radius $r_\mathrm{c}$, are discussed. To obtain an independent estimate of $\Phi_0$, we fit observed surface brightness profiles by using the predicted surface density and a mass-luminosity relation from isochrones. The quality of the fits on $\sigma(m)$ 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 $\Phi_0$. As a result, a cluster in a more advanced dynamical state shows a larger $\Phi_0$, as well as larger $N_\mathrm{core}$ and $c$, while $r_\mathrm{c}$ decreases. The estimates of $\Phi_0$ obtained by fitting surface brightness profiles result compatible at $2\sigma$ confidence level with those from internal kinematics, although further investigation of statistical and systematic errors is required.