local stability criterion for self-gravitating disks in modified gravity

we study local stability of self-gravitating fluid and stellar disk in the context of modified gravity theories which predict a yukawa-like term in the gravitational potential of a point mass. we investigate the effect of such a yukawa-like term on the dynamics of self-gravitating disks. more specifically, we investigate the consequences of the presence of this term for the local stability of the self-gravitating disks. in fact, we derive a generalized version of toomre's local stability criterion for diferentially rotating disks. this criterion is complicated than the original one in the sense that it depends on the physical properties of the disk. in the case of modified gravity theory (mog), we use the current confirmed values for the free parameters of this theory, to write the generalized toomre's criterion in a more familiar way comparable with the toomre's criterion. this generalized toomre's criterion may be used to study the global stability of stellar and  fluid disks using computer simulations.