Abstract
Nuclear rings at centers of barred galaxies exhibit strong star formation activities. They are thought to undergo gravitational instability when sufficiently massive. We approximate them as rigidly-rotating isothermal objects and investigate their gravitational instability. Using a self-consistent eld method, we first construct their equilibrium sequences specified by two parameters: ${\alpha}$ corresponding to the thermal energy relative to gravitational potential energy, and $R_B$ measuring the ellipticity or ring thickness. The density distributions in the meridional plane are steeper for smaller ${\alpha}$, and well approximated by those of infinite cylinders for slender rings. We also calculate the dispersion relations of nonaxisymmetric modes in rigidly-rotating slender rings with angular frequency ${\Omega}$ and central density ${\rho}_c$. Rings with smaller are found more unstable with a larger unstable range of the azimuthal mode number. The instability is completely suppressed by rotation when ${\Omega}$ exceeds the critical value. The critical angular frequency is found to be almost constant at $0.7(G{\rho}_c)^{1/2}$ for ${\alpha}$ > 0.01 and increases rapidly for smaller ${\alpha}$. We apply our results to a sample of observed star-forming rings and confirm that rings without a noticeable azimuthal age gradient of young star clusters are indeed gravitationally unstable.