Equilibrium sequences and gravitational instability of rotating isothermal rings

Abstract

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: alpha corresponding to the thermal energy relative to gravitational potential energy, and. (R) over cap (B) measuring the ellipticity or ring thickness. Unlike in the incompressible case, not all values of. (R) over cap (B) yield an isothermal equilibrium, and the range of. (R) over cap (B) for such equilibria shrinks with decreasing a. 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 non-axisymmetric modes in rigidly rotating slender rings with angular frequency Omega(0) and central density rho(c). Rings with smaller alpha are found more unstable with a larger unstable range of the azimuthal mode number. The instability is completely suppressed by rotation when Omega(0) exceeds the critical value. The critical angular frequency is found to be almost constant at similar to 0.7(G rho(c))(1/2) for alpha greater than or similar to 0.01 and increases rapidly for smaller a. 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.