Dynamical evolution of star forming regions - II. Basic kinematics

We follow the dynamical evolution of young star-forming regions with a wide range of initial conditions and examine how the radial velocity dispersion, $\sigma$, evolves over time. We compare this velocity dispersion to the theoretically expected value for the velocity dispersion if a region were in virial equilibrium, $\sigma_{\rm vir}$ and thus assess the virial state ($\sigma / \sigma_{\rm vir}$) of these systems. We find that in regions that are initially subvirial, or in global virial equilibrium but subvirial on local scales, the system relaxes to virial equilibrium within several million years, or roughly 25 - 50 crossing times, according to the measured virial ratio. However, the measured velocity dispersion, $\sigma$, appears to be a bad diagnostic of the current virial state of these systems as it suggests that they become supervirial when compared to the velocity dispersion estimated from the virial mass, $\sigma_{\rm vir}$. We suggest that this discrepancy is caused by the fact that the regions are never fully relaxed, and that the early non-equilibrium evolution is imprinted in the one-dimensional velocity dispersion at these early epochs. If measured early enough ($<$2 Myr in our simulations, or $\sim$20 crossing times), the velocity dispersion can be used to determine whether a region was highly supervirial at birth without the risk of degeneracy. We show that combining $\sigma$, or the ratio of $\sigma$ to the interquartile range (IQR) dispersion, with measures of spatial structure, places stronger constraints on the dynamical history of a region than using the velocity dispersion in isolation.