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, Y-V,Y-0 = 1.45 M. L.(-1) and (2) 109 regular globulars both with and without direct observations of Y-V,Y-0 show a very strong correlation between global binding energy and total luminosity, regulated by Galactocentric position: E-b = 7.2 x 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, Y-V,Y-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 Y-V,Y-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.