Dynamical analysis of Maclaurin disk with velocity dispersion and its influence on bar formation

We investigate the influence of Toomre's $Q$ parameter on the bar-forming dynamics of Maclaurin disk using $N$-body simulations. According to the Toomre's criterion, local velocity dispersion parametrized by $Q\geq 1$ is required to suppress the local axisymmetric instability but, in turn, it deviates particle orbits from 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, circular orbit approximation is retrieved. The effective potential hypothesis can describe the $Q$-dependences of angular and epicyclic motions of the bar-forming processes and the established bars reasonably well provided that $Q\geq 1$. This indicates the influence of initial $Q$ that is imprinted in the entire disk dynamics, not only that $Q$ serves as the stability indicator. In addition, we perform the stability test for the disk-in-halo systems. With the presence of halo, disks are more susceptible to the bar formation 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 bar instability.