2011MNRAS.413.2509G

The evolution of globular clusters due to two-body relaxation results in an outward flow of energy and at some stage all clusters need a central energy source to sustain their evolution. Hénon provided the insight that we do not need to know the details of the energy production in order to understand the relaxation-driven evolution of the cluster, at least outside the core. He provided two self-similar solutions for the evolution of clusters based on the view that the cluster as a whole determines the amount of energy that is produced in the core: steady expansion for isolated clusters, and homologous contraction for clusters evaporating in a tidal field. The amount of expansion or evaporation per relaxation time-scale is set by the instantaneous radius or number of stars, respectively. We combine these two approximate models and propose a pair of Unified Equations of Evolution (UEE) for the life cycle of initially compact clusters in a tidal field. The half-mass radius increases during the first part (roughly half) of the evolution, and decreases in the second half, while the escape rate approaches a constant value set by the tidal field. We refer to these phases as `expansion dominated' and `evaporation dominated'. These simple analytical solutions of the UEE immediately allow us to construct evolutionary tracks and isochrones in terms of cluster half-mass density, cluster mass and galactocentric radius. From a comparison to the Milky Way globular clusters we find that roughly one-third of them are in the second, evaporation-dominated phase and for these clusters the density inside the half-mass radius varies with the galactocentric distance R_G as ρ_h∝R^–2_G. The remaining two-thirds are still in the first, expansion-dominated phase and their isochrones follow the environment-independent scaling ρ_h∝M², where M is the cluster mass; that is, a constant relaxation time-scale. We find substantial agreement between Milky Way globular cluster parameters and the isochrones, which suggests that there is, as Hénon suggested, a balance between the flow of energy and the central energy production for almost all globular clusters.