A significant fraction of nearby elliptical galaxies are known to have high-density gas disks in their circumnuclear (CN) region (0.1 to a few kpc). Yet ellipticals, especially luminous ones, show little sign of recent star formation. To investigate the possible cause of the dearth of star formation in these systems, we study the gravitational stability of CN gas disks embedded within the gravitational potentials of both the stellar bulge component and the central massive black hole (BH) in elliptical galaxies. We find that CN disks in higher mass galaxies are generally more stable than those in lower mass galaxies, because higher mass galaxies tend to have more massive BHs and more centrally concentrated stellar density profiles. We also consider the case in which the central stellar density profile has a core, which is often observed for ellipticals whose total stellar mass is higher than about 10~(11) M_☉. Such a cored stellar density profile leads to more unstable CN disks than the power-law stellar density profile characteristic of less massive galaxies. However, the more massive BHs in high-mass galaxies act to stabilize the CN disk. Our results demonstrate that the gravitational potentials of both the central BH and the stellar component should be taken into account when studying the properties of CN disks, as their stability is sensitive to both the BH mass and the stellar density profile. Our results could explain the observed trend that less luminous elliptical galaxies have a greater tendency to exhibit ongoing star formation than giant ellipticals.
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