Binding Energy and the Fundamental Plane of Globular Clusters

This paper develops a physical description of the fundamental plane of Galactic globular clusters that explains all empirical trends and correlations in a large number of cluster observables and provides a small but complete set of truly independent constraints on theories of cluster formation and evolution in the Milky Way. In a very good first approximation, the internal structures and dynamics of Galactic globulars are described by single-mass, isotropic King models and are thus characterized by four nominally independent physical parameters. Within this theoretical framework, it is shown that (1) 39 regular (non-core-collapsed) globulars with measured central velocity dispersions all share a common core mass-to-light ratio, ϒ V ,0 = 1.45 M ☉ L , and (2) 109 regular globulars both with and without direct observations of ϒ V ,0 show a very strong correlation between global binding energy and total luminosity, regulated by Galactocentric position: E b = 7.2 × 10 39 ergs( L / L ☉ ) 2.05 ( r gc /8 kpc) -0.4 . The observational scatter about either of these two constraints can be attributed fully to random measurement errors, making them the defining equations of a fundamental plane (FP) to which real clusters are confined in the larger, four-dimensional parameter space of general King models. They are shown to underlie a pair of bivariate correlations first found, and used to argue for the existence of a globular cluster FP, by Djorgovski. A third, weaker correlation, between clusters' total luminosities and King-model concentration parameters, is related to the (nonrandom) distribution of globulars on the FP. With L , ϒ V ,0 , E b , and the central concentration c thus chosen as the four physical quantities that define any single globular cluster, the FP equations for ϒ V ,0 and E b ( L , r gc ) are used to derive expressions for any other observable in terms of only L , r gc , and c . Results are obtained for generic King models and applied specifically to the globular cluster system of the Milky Way.