The Dynamics of Globular Clusters and Elliptical Galaxies

A model is developed for an idealised spherical galaxy evolving from a uniform mass distribution at the epoch of galactic separation until attaining an equilibrium state through gravitational collapse. The final theoretical radial surface density is computed and shows a good fit to the observational data for two globular clusters, M15 and M80. The mean cycle time and velocity are computed, the velocity-radius curve is developed and Gaussian RMS values derived, from which half-light radius vs. mass are plotted for 544 ellipticals plus compact, massive, and intermediate-mass objects. These show a linear mean log-log $R$-$M_{vir}$ slope of ${0.604\pm0.003}$, equivalent to a Faber-Jackson slope of $\gamma=3.66{\pm}0.009$ over a mass range of 7 decades. and a slope of $0.0045\pm0.0001$ on a semi-log plot of $R_{1/2}-\sigma$. Globular clusters, dwarf elliptical and dwarf spherical galaxies show a distinct anomaly on these plots, consistent with the ellipticals containing a supermassive black hole (SMBH) whose mass increases as the velocity dispersion increases, compared with the remaining types of spherical or irregular galaxies without a massive core. Analysis of the equations of motion suggested the generalised rule that all spherical galaxies expand from their initial radius at the epoch of galactic separation to a stable maximum radius of $\simeq1.136$ times their initial radius in their relaxed state.