Dynamical Analysis of the Maclaurin Disk with Velocity Dispersion and Its Influence on the Formation of the Bar

Abstract

We investigate the influence of Toomre’s Q parameter on the bar-forming dynamics of the Maclaurin disk using N-body simulations. According to Toomre’s criterion, the local velocity dispersion parametrized by Q ≥ 1 is required to suppress the local axisymmetric instability but, in turn, it deviates particle orbits from a nearly circular limit in which particle natural frequencies are calculated. We resolve this by including the effect of velocity dispersion, as the pressure potential, into the effective potential with the gravitational potential. With this formulation, a circular orbit approximation is retrieved. The effective potential hypothesis can describe the Q dependence of angular and epicyclic motions of the processes of the formation of a bar and the established bars reasonably well provided that Q ≥ 1. This indicates the influence of the initial Q imprinted on the entire disk dynamics and not only that Q serves as the indicator of stability. In addition, we perform a stability test for the disk-in-halo systems. With the presence of a halo, disks are more susceptible to the formation of a bar as seen by the elevated critical Q than that for the isolated disk. This is attributed to the differential rotation that builds the unstable non-axisymmetric spiral modes more efficiently, which are the ingredients of the bar instability.