On the gravitational stability of the Maclaurin disk

We study the global gravitational stability of a gaseous self-gravitating Maclaurin disk in the absence of a halo. Further, we replace Newtonian gravity with the specific Modified gravity theory known as MOG in the relevant literature. MOG is an alternative theory for addressing the dark matter problem without invoking exotic dark matter particles, and possesses two free parameters $\alpha$ and $\mu_0$ in the weak field limit. We derive the equilibrium gravitational potential of the Maclaurin disk in MOG and develop a semi-analytic method for studying the response of the disk to linear non-axysymmetric perturbations. The eigenvalue spectrum of the normal modes of the disk is obtained and its physical meaning has been explored. We show that Maclaurin disks are less stable in MOG than in Newtonian gravity. In fact both parameters $(\alpha,\mu_0)$ have destabilizing effects on the disk. Interestingly $\mu_0$ excites only the bar mode $m=2$ while $\alpha$ affects all the modes. More specifically, when $\alpha>1$, the bar mode is strongly unstable and unlike in Newtonian gravity can not be avoided, at least in the weak field limit, with increasing the pressure support of the disk.