2016ApJ...829...45K

Nuclear rings at the centers of barred galaxies exhibit strong star formation activities. They are thought to undergo gravitational instability when they are sufficiently massive. We approximate them as rigidly rotating isothermal objects and investigate their gravitational instability. Using a self-consistent field method, we first construct their equilibrium sequences specified by two parameters: α corresponding to the thermal energy relative to gravitational potential energy, and R_{B} measuring the ellipticity or ring thickness. Unlike in the incompressible case, not all values of R_{B} yield an isothermal equilibrium, and the range of R_{B} for such equilibria shrinks with decreasing α. The density distributions in the meridional plane are steeper for smaller α, and well approximated by those of infinite cylinders for slender rings. We also calculate the dispersion relations of non-axisymmetric modes in rigidly rotating slender rings with angular frequency Ω_0 and central density ρ_
c. Rings with smaller α are found more unstable with a larger unstable range of the azimuthal mode number. The instability is completely suppressed by rotation when Ω_0 exceeds the critical value. The critical angular frequency is found to be almost constant at ∼0.7{(Gρ_
c)}^{1/2} for α >= 0.01 and increases rapidly for smaller α. 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.