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: 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 r 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 ( ) r G 0.7 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.

The centres of M83 and the Milky Way: opposite extremes of a common star formation cycle

“…Given the circumference of M83's circumnuclear gas ring, the clumps should be separated by ∼60 pc. Considering the idealised nature of the Kim & Moon (2016) model (e.g. uniform density, circular orbits) the similarity with the predicted clump spacing suggests gravitational instabilities are a plausible explanation for the observed quasi-regular gas spacing.…”

“…We now investigate what might cause this quasi-regular spacing in the gas properties. Kim & Moon (2016) model the gravitational instability of rotating isothermal rings at the centres of barred galaxies, like M83, to understand their star formation potential. Using the observed circumnuclear ring radius (∼100 pc), circular velocity (∼75 kms −1 ), mass (∼ 5 × 10 8 M ) and velocity dispersion (17 kms −1 , see below) we calculate the Kim & Moon (2016) (virial parameter) andΩ 0 (critical angular frequency) parameters for M83's circumnuclear gas ring through their Eq.…”

A Fast Poisson Solver of Second-order Accuracy for Isolated Systems in Three-dimensional Cartesian and Cylindrical Coordinates

“…together with Equation (1), where Ω s ≡ (Ω 2 0 − Ω 2 e ) 1/2 is the angular velocity due to self-gravity alone (Kim & Moon 2016). One can show that the equilibrium configurations are completely specified by two dimensionless parameters: α ≡ c 2 s /(GR 2 A ρ c ) and Ω s ≡ Ω s /(Gρ c ) 1/2 , where ρ c and R A denote the maximum density and the maximum radial extent of an equilibrium object, respectively.…”

“…By performing a linear-stability analysis, Kim & Moon (2016) found that the above equilibrium is gravitationally unstable to non-axisymmetric perturbations for a range of the azimuthal mode number m. The most unstable modes were found to have m = 9 and 10, with an almost equal growth rate Im(ω) ≈ 0.8(Gρ c ) 1/2 ≈ 17 T −1 orb and a phase speed 5 The weak scaling test shown in Figure 6 hints some performance degradation from Ncore = 1 to 64 relative to t wall ∝ ln Ncore expected for the theoretical FFT. Our parallel FFT utilizes a "transpose algorithm" known to be efficient when a data size for communication is larger than the critical size that depends on the latency/bandwidth of the interconnecting network device and the network topology (e.g., Foster & Worley 1997).…”

The equilibrium of over-pressurised polytropes

“…The situation is similar when the fluid is embedded in a ambient medium (Umemura & Ikeuchi 1986;Horedt 2000;Kim & Moon 2016), but there is no background matter here (over-pressure is due to photons). Note that Ĥe can vary in space.…”