The life cycle of star clusters in a tidal field

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.