Stability of disks in cusped potentials

We confirm that a high rate of shear at the center is able to stabilize an entire stellar disk against bar-forming (m = 2) modes, irrespective of the dark halo density. Our simulations of unstable power-law disks also yield the growth rate and pattern speed of the dominant lopsided (m = 1) mode, in close agreement with that predicted by linear theory in our cleanest case. The one-armed modes, which dominate models with extensive disks, can be controlled by tapering the outer disk. Two-armed instabilities of hard-centered disks are more difficult to identify because they are easily overwhelmed by particle noise. Nevertheless, we have detected the predicted m = 2 modes in simulations with very large numbers of particles. Such bisymmetric instabilities in these disks are provoked only by sharp edges and are therefore easily eliminated. We have constructed a cool disk model with an almost flat rotation curve and a quasi-exponential density profile that is unambiguously stable. The halo in this stable model has a large core radius, with the disk and bulge providing almost all the rotational support in the inner parts.