Semi-global Simulations of Star Formation in Nuclear Rings of Barred Galaxies

Nuclear rings at the centers of barred galaxies are active in star formation. To understand what determines the star formation rate (SFR) and structure of nuclear rings, we conduct semi-global, magnetohydrodynamic simulations of nuclear rings subject to various mass inflow rates with and without magnetic fields. We adopt the TIGRESS framework of Kim & Ostriker to handle radiative heating and cooling, star formation, and related supernova feedback. Our findings suggest that supernova feedback cannot destroy the nuclear ring completely or halt star formation within it, while both the mass inflow rate and supernova feedback affect the ring star formation rate. The supernova feedback is responsible for small-amplitude SFR fluctuations with a timescale of less than 40 million years, while the SFR variations over longer timescales are due to changes in the mass inflow rates. Magnetic fields seeded by the inflows are amplified in the ring due to rotational shear and supernova feedback, greatly reducing the SFR at late times. Strong magnetic tension in the ring drives radially inward accretion flows from the ring to form a circumnuclear disk in the central region, which is absent in the unmagnetized model.


Introduction
Barred galaxies are common in the universe.Including weak bars, about two thirds of disk galaxies possess a bar at their centers [1,2,3,4,5,6,7,8].A stellar bar exerts a gravitational torque to gas to produce a pair of dust lanes at the leading side of the bar, which are shocks in the gas flows [9,10,11,12,13,14,15].When gas in orbital motion hits the dust lanes, it loses angular momentum to move radially inward and fuels a star-forming nuclear ring at the location where the centrifugal force balances the gravity [14,16].By making CO observations for the gas densities and line-of-sight velocities, one can infer the mass inflow rate Ṁin along the dust lanes to the ring.In the case of NGC 1097, the estimated mass inflow rate was found to be Ṁin = 3.0 ± 2.1 M ⊙ yr −1 [17].The condition of mass conservations requires that Ṁin = ṀSF + Ṁout + Ṁring , where ṀSF is the star formation rate (SFR), Ṁout is the outflow rate from the galaxy as a wind, Ṁring is the rate of change in the ring mass.For NGC 1097, ṀSF ≃ 1.8 − 2 M ⊙ yr −1 [18,19] and Ṁout = 0.6 M ⊙ yr −1 [20].This indicates that most of the inflowing gas is currently being consumed by star formation in the ring of NGC 1097.
Our own Galaxy also possesses a nuclear ring, called the central molecular zone (CMZ), with the current SFR of ṀSF = 0.04-0.1 M ⊙ yr −1 [21,22,23], which is about 10 times smaller than the prediction of the Kennicutt-Schmidt relation for the observed CMZ mass [22,24], possibly by solenoidal turbulence driving associated with strong velocity shear [25].To explain the low SFR of the CMZ, [26] proposed a scenario of episodic star formation in which the ring becomes gradually massive as the gas driven by the large-scale bar accumulates, and undergoes gravitational instability and starburst when it becomes sufficiently massive.The associated strong supernova feedback disperses the ring material, making the ring quiescent temporarily until mass accumulates again in the ring.According to this scenario, the CMZ is currently in the quiescent phase.One-dimensional idealized simulations of [27] found that the ring SFR indeed exhibits quasi-periodic oscillations with a period of ∼ 20 Myr.However, three-dimensional global simulations with more realistic treatment of star formation and feedback show different results.For example, the simulations of [28] found that ring star formation is episodic, but the associated period is about 20 times longer than that in the onedimensional simulations of [27].[15] found that ṀSF is very similar to the mass inflow rate to the ring, while it is maintained almost constant in the simulations of [29].These diverse results from the three-dimensional global simulations imply that there is no consensus as to what controls the ring SFR.In these global simulations, the mass inflow rate to a nuclear ring is self-consistently determined by dynamical interactions between gas and a stellar bar, and thus highly time dependent.In such situations, the ring size and mass vary considerably with time, making it difficult to isolate key factors that determine the ring SFR.
Another important ingredient that governs gas dynamics and star formation in the central parts of barred galaxies is magnetic fields.Observations of radio synchrotron emission indicate that magnetic fields are mainly parallel to the dust lanes with a strength of ∼ 30 µG in the bar regions, while they penetrate the ring with a pitch angle of ∼ 40 • and have a strength of ∼ 60 µG in the ring regions [30,31].More recent SOFIA observations of dust emission suggest that magnetic fields are dominated by the m = 0 mode, while the radio polarization is dominated by the m = 2 spiral mode [32].Magnetic fields in the rings appear to suppress the ring star formation.[33] measured the magnetic field strength of 11 giant molecular clouds distributed along the nuclear ring of NGC 1097, for which the gas depletion time τ dep is obtained from [18].They found that τ dep is higher for clouds with stronger magnetic fields, suggesting that magnetic fields suppress the ring star formation.To investigate what controls star formation in nuclear rings, we have run series of threedimensional magnetohydrodynamic (MHD) simulations.Instead of running fully global simulations, we construct a semi-global model of a nuclear ring in which the bar-driven mass inflow rate Ṁin is handled by the boundary conditions.The semi-global model resolves nuclear rings better than global models and allows us to directly control the mass inflow rate and ring radius.We first run unmagnetized models with various time-fixed Ṁin [34], unmagnetized models with time-varying Ṁin [35], and then magnetized models with fixed Ṁin [36].Here we highlight the main results of [34,35,36].We refer the reader to these papers for more detailed and quantitative results and discussion.

Model and Method
As a simulation domain, we consider a cubic box with side length L = 2048 pc located at the center of a galaxy.The box is under the external gravitational potential from the central black hole and stellar bulge, corresponding to those in NGC 1097.The box rotates about the center at angular frequency Ω p = 36 km s −1 kpc −1 , equal to the bar pattern speed.Instead of modeling the bar explicitly, we incorporate the bar-driven mass inflows using gas streams through two nozzles located at the y-boundaries.To cover various galactic situations, we vary the inflow rate Ṁin and inflow speed v in of the gas in the nozzles, the latter of which is to set to give the ring size R ring = 500 or 600 pc.The box is initially filled with rarefied warm gas with temperature T = 2 × 10 4 K and number density n = 10 −5 e −|z|/(50 pc) cm −3 , which is swept by the gas streams from the boundaries.
We use the Athena code [37] to integrate the basic MHD equations expanded in the rotating frame, and adopt the TIGRESS framework [38] to handle star formation and supernova feedback self-consistently after making it suitable for non-periodic boundary conditions of our semi-global 10 -1 10 0 10 1 Star formation rate ṀSF as a function of the mass inflow rate Ṁin for unmagnetized models with R ring = 600 pc.models.A sink particle (representing a star cluster) is created in a cell that satisfies the three conditions simultaneously: (1) the cell lies at a local potential minimum; (2) the gas density is greater than the Larson-Penston density threshold that depends on the resolution and the local temperature; (3) the velocity is converging in all three directions.We follow the orbits of sink particles by using the Boris algorithm [39] which conserves the momentum and energy of the particles almost exactly.
In what follows, we present three sets of models.In the first set, we run various models with differing Ṁin = 0.125-8 M ⊙ yr −1 which is kept fixed over time.In the second set, we explore the models with a time-varying inflow rate where ∆τ in = 15, 50, 100 Myr is the oscillation period of the mass inflow rate.Finally in the third set, we make the inflow streams threaded by magnetic fields parallel to the inflow velocity by imposing the boundary condition with strength B in corresponding to the plasma parameter β in = 10, 30, 100.Table 1 lists the model parameters presented in this article.

Models with Fixed Mass Inflows
Figure 1 plots the temporal changes of the gas mass M gas in the ring and the mass M sp of newly-formed star particles, as well as the star formation rate ṀSF in the ring averaged over a 10 Myr interval.At early time when the gas streams injected from the nozzles collide each other and start to form a ring, M sp and ṀSF exhibit some fluctuations.At t ∼ 100 Myr, however, the system reaches a quasi-steady state in which the ring morphology and various statistical quantities such as the ring mass and SFR do not vary much over time.The mass of star particles keeps increasing over time due to continued star formation.In our models, star formation occurs stochastically rather than synchronously in different parts of the ring, leading to an almost constant SFR when averaged over the entire ring.Small-amplitude fluctuations in ṀSF is caused by turbulence driven by supernova feedback.We run various models for the ring radius of R ring = 600 pc with differing Ṁin .Figure 2 plots ṀSF against Ṁin , showing that ∼ 80 % of the inflowing gas turn to stars in our models.This suggests that the ring SFR is determined primarily by the mass inflow rate.Remaining ∼ 20 % escapes the simulations domain as galactic winds.
Figure 3(a) plots the SFR surface density Σ SFR = ṀSF /A ring in the ring as a function of the gas surface density Σ SFR of the ring, where A ring denotes the area covered by the ring.The best fit to the numerical results yields for our unmagnetized models.In our model, Equation ( 3) is a consequence of vertical dynamical equilibrium of the ring.That is, the total midplane pressure balances the weight of the overlying gas, as Figure 3(b) displays.Here, ∆z is the grid spacing in the vertical direction and Φ tot is the total (external and self) gravitational potential.The midplane pressure is dominated by the turbulent pressures (ρv 2 z ) due to supernova feedback, with P mid ∝ Σ 0.9 SFR .Since the vertical weight W tot is the product of the gas surface density Σ ring and the vertical gravity g z , this together with Equation (3) and Figure 3

Ṁin
, ṀSF [M yr Temporal changes in the SFR ṀSF (thick lines) and the mass inflow rate Ṁin (thin lines) for unmagnetized models with time-varying mass inflow rates dominates, our results show that both external gravity and self-gravity (including the gravity of star particles) are important in determining the vertical dynamical equilibrium.

Models with Time-varying Mass Inflows
Now we turn to models with time-varying mass inflow rate Ṁin .Figure 4 plots the mass inflow rate Ṁin (thin lines) and the ring SFR ṀSF (thick lines) for models with ∆τ in = ∞ (constant), 15 (P15), 50 (P50), 100 Myr (P100).The SFR is measured using the star particles that formed in the past 10 Myr.Note that When ∆τ in ≥ 50 Myr, an oscillating inflow rate causes largeamplitude (a factor of ∼ 5), quasi-periodic variations of the SFR.This is because the increased mass inflows first enhance the ring density to promote star formation, and the associated increase in supernova feedback then inflates the ring vertically, increasing the vertical scale height of the ring and thus reducing the midplane ring density.In models with ∆τ in ≤ 15 Myr, in contrast, the SFR shows only modest (less than a factor of 2) fluctuations with timescale < 40 Myr, due to star formation feedback.This indicates that the long-period, large-amplitude fluctuations of the SFR is caused by the temporal change in the mass rate, while the short-period, low-amplitude fluctuations of the SFR is due to supernova feedback.

Models with Magnetic Fields
We also use the same semi-global framework and introduce weak magnetic fields parallel to the velocity fields at the nozzles to explore the effect of magnetic fields on the ring star formation.We consider four models with different field strength, characterized by the plasma beta β in = ∞, 100, 30, 10.The β in = 100 model has the mean field strength of 0.76 µG at the nozzles.For all magnetized models, the mass inflow rate is fixed to Ṁin = 1 M ⊙ yr −1 .Figure 5 plots the temporal changes in the strength of the regular and turbulent components of magnetic fields in the ring.Figure 6 plots the temporal changes in the ratios of the magnetic energy E mag to the turbulent kinetic energy E trb and to the orbital kinetic energy E orb of the rings.
Early evolution of the magnetized models is similar to that of the unmagnetized counterpart.The gas streams from the nozzles at the opposite boundaries collide with each other, driving strong shocks.The streams lose their orbital kinetic energy as they pass through the shocks multiple times, and form a nuclear ring with R ring = 500 pc after about three orbital times.As the streams form a ring, magnetic fields become predominantly toroidal in the ring, although vertical expansions of superbubbles created by supernova explosions drag the toroidal fields in the vertical direction to produce poloidal fields in high-altitude regions.Driven by rotational shear and supernova feedback, the magnetic fields in the ring become stronger over time, due to actions of both small-and large-scale dynamos.Figure 5 shows that the regular fields keep increasing with time, achieving strength of ∼ 50-70 µG presumably due to large-scale α-Ω type dynamo [40,41].On the other hand, the turbulence driven by supernova feedback may amplify the turbulent magnetic fields that saturate at ∼ 30-40 µG, probably balanced by turbulent dissipation [42].Note that E mag /E trb in each model rises rapidly as the ring forms at early time, saturates over the time span when B trb dominates, and increases again at late time primarily due to the Ω effect.In all models, the saturation values of E mag /E trb exceed 0.1, which can be achieved mainly by solenoidal, low-Mach-number driving in a uniform non-rotating medium [43].It would be interesting to study how rotation and shear conspire with turbulent dynamo to set up the saturated level of E mag /E trb .As in the unmagetized cases, the magnetized ring maintains vertical dynamical equilibrium instantaneously, with the total mid-plane pressure equal to the weight of the overlying gas.Strong magnetic fields provide magnetic stress which reduces the demand for supernova feedback to produce turbulent and thermal pressures.
Figure 7(a,b) plots ṀSF and the gas depletion time t dep ≡ M gas / ṀSF for all models.At t ∼ 120 Myr, the magnetic fields in the ring become strong enough to decrease ṀSF and  7(c) denotes the observed conditions of the nuclear rings in NGC 1097 [33,44], indicating that the ring of this galaxy is in the regime where the magnetic support against gravity is dynamically important.Strong magnetic fields in the nuclear ring also exerts magnetic stress, transporting the angular momentum of rotating gas radially outward, and thus causing gas inflow.In a quasi-steady state, the mass accretion rate is given by Ṁacc ≈ ṀM + ṀR , where ṀM and ṀR are the mass accretion rate due to the maxwell stress T Rϕ = −B R B ϕ /(4π) and the Reynolds stress R Rϕ = ρv R v ϕ defined as respectively, with v c = (Ω − Ω p )R being the background circular velocity in the rotating frame.Figure 8(a) plots the numerically measured Ṁacc at R = 100 pc (solid lines) as a function of time, in comparison with the predicted ṀM (dashed lines) and ṀR (dotted lines).In the unmagetized model, Ṁacc ∼ 4 × 10 −4 M ⊙ yr −1 on average, indicating that there is almost no gas accretion without magnetic fields.Although Ṁacc remains small at early time, it increases quite rapidly to Ṁacc ∼ 0.02-0.03M ⊙ yr −1 at late time in the magnetized models.Note that Ṁacc ≃ ṀM when magnetic fields are strong, suggesting that magnetic tension is the major driver of mass accretion in our simulations.Figure 8(b) plots the radial distributions of Ṁacc for a few selected epochs for the model with β in = 100.Note that Ṁacc is not constant in radius but decreases toward small R, with a slope decreasing with time.This implies that there is a sustained mass deposition at R < R ring mediated by magnetic tension, eventually forming a nuclear disk at the galaxy center.

Figure 1 .
Figure 1.Temporal variations of (a) the gas mass M gas in the ring as well as the mass M sp of the newly formed stars and (b) the ring SFR ṀSF of the unmagnetized model with Ṁin = 2 M ⊙ yr −1 .

15thFigure 3 .
Figure 3. (a) SFR surface density Σ SFR as a function of the gas surface density Σ ring in the ring and (b) the total gas weight W against the total midplane pressure P mid for unmagnetized models with R ring = 600 pc.

Figure 5 .Figure 6 .
Figure 5. Temporal variations in the strength of (a) the regular component B reg and (b) the turbulent component B trb of magnetic fields in magnetized models with Ṁin = 1 M ⊙ yr −1 .

Table 1 .
Model Parameters (b)suggests that g z ∝ Σ 0.8 ring .Given that g z is independent of Σ ring if the external gravity dominates, and g z ∝ Σ ring if self-gravity Temporal variations of (a) the ring SFR ṀSF and (b) the gas depletion time t dep , and (c) t dep as a function of the total magnetic field strength B tot for all magnetized models with Ṁin = 1 M ⊙ yr −1 .The yellow star symbol in (c) marks the observational results of the nuclear ring in NGC 1097.
increase t dep .Figure7(c) plots t dep as a function of the total magnetic field strength B tot for all magnetized models, showing that t dep is a increasing function of B tot ≳ 30 µG.The star symbol in Figure