Star Formation in Nuclear Rings of Barred Galaxies
AAccepted for publication in the ApJ
Preprint typeset using L A TEX style emulateapj v. 5/2/11
STAR FORMATION IN NUCLEAR RINGS OF BARRED GALAXIES
Woo-Young Seo and Woong-Tae Kim
Center for the Exploration of the Origin of the Universe (CEOU), Astronomy Program, Department of Physics & Astronomy, SeoulNational University,Seoul 151-742, Republic of Korea andFPRD, Department of Physics & Astronomy, Seoul National University, Seoul 151-742, Republic of Korea
Accepted for publication in the ApJ
ABSTRACTNuclear rings in barred galaxies are sites of active star formation. We use hydrodynamic simulationsto study temporal and spatial behavior of star formation occurring in nuclear rings of barred galaxieswhere radial gas inflows are triggered solely by a bar potential. The star formation recipes include adensity threshold, an efficiency, conversion of gas to star particles, and delayed momentum feedbackvia supernova explosions. We find that star formation rate (SFR) in a nuclear ring is roughly equal tothe mass inflow rate to the ring, while it has a weak dependence on the total gas mass in the ring. TheSFR typically exhibits a strong primary burst followed by weak secondary bursts before declining tovery small values. The primary burst is associated with the rapid gas infall to the ring due to the bargrowth, while the secondary bursts are caused by re-infall of the ejected gas from the primary burst.While star formation in observed rings persists episodically over a few Gyr, the duration of active starformation in our models lasts for only about a half of the bar growth time, suggesting that the barpotential alone is unlikely responsible for gas supply to the rings. When the SFR is low, most starformation occurs at the contact points between the ring and the dust lanes, leading to an azimuthalage gradient of young star clusters. When the SFR is large, on the other hand, star formation israndomly distributed over the whole circumference of the ring, resulting in no apparent azimuthal agegradient. Since the ring shrinks in size with time, star clusters also exhibit a radial age gradient, withyounger clusters found closer to the ring. The cluster mass function is well described by a power law,with a slope depending on the SFR. Giant gas clouds in the rings have supersonic internal velocitydispersions and are gravitationally bound.
Subject headings: galaxies: ISM — galaxies: kinematics and dynamics — galaxies: nuclei — galaxies:spiral — ISM: general — shock waves — stars: formation INTRODUCTION
Nuclear rings in barred galaxies are sites of intensestar formation (e.g., Burbidge & Burbidge 1960; Sandage1961; Phillips 1996; Buta & Combes 1996; Knapen et al.2006; Mazzuca et al. 2008; Comer´on et al. 2010; Sand-strom et al. 2010; Mazzuca et al. 2011; Hsieh et al. 2011).These rings are thought to form as a result of nonlinearinteractions of gas with a non-axisymmetric bar poten-tial (e.g., Combes & Gerin 1985; Shlosman et al. 1990;Athanassoula 1992; Heller & Shlosman 1994; Knapen etal. 1995; Buta & Combes 1996; Combes 2001; Piner etal. 1995; Regan & Teuben 2003). Due to the bar torque,the gas readily forms dust-lane shocks in the bar regionand flows inward along the dust lanes. The inflowinggas speeds up gradually in the azimuthal direction as itmoves inward, and shapes into a ring very close to thegalaxy center (e.g., Kim et al. 2012a). Consequently,nuclear rings have very large surface densities and shortdynamical time scales, capable of triggering starburst ac-tivity.There are some important observational results thatmay provide clues as to how star formation occurs in thenuclear rings. First of all, observations indicate that thestar formation rate (SFR) in the nuclear rings appears tovary with time, and differs considerably from galaxy togalaxy. Analyses of various population synthesis mod- [email protected], [email protected] els for a sample of galaxies reveal that the strength ofobserved emission lines from the nuclear rings is bestdescribed by multiple starburst activities over the last0 . . (cid:12) yr − . It appears that the SFR is largely insen-sitive to the total molecular mass in a ring, but can bestrongly affected by the bar strength. For instance, thering in a strongly-barred galaxy NGC 4314 has a rela-tively low SFR at ∼ . (cid:12) yr − (Benedict et al. 2002),which is about an order of magnitude smaller than thatin a weakly-barred galaxy NGC 1326 (Buta et al. 2000),although the total molecular mass contained in the ringis within a factor of two. In fact, the SFRs given inMazzuca et al. (2008) combined with the bar strengthpresented by Comer´on et al. (2010) show that strongly-barred galaxies tend to have a very small SFR in therings, while weakly-barred galaxies have a wide range ofthe SFRs.Second, based on the spatial distributions of star-forming regions in nuclear rings, B¨oker et al. (2008)proposed two models of star formation: “popcorn” and“pearls on a string” models (see also Sandstrom et al.2010). In the first popcorn model, star formation oc-curs in dense clumps that are randomly distributed alonga nuclear ring. This type of star formation, presum- a r X i v : . [ a s t r o - ph . C O ] A p r Seo & Kimably caused by gravitational instability of the ring itself(Elmegreen 1994), does not produce a systematic gradi-ent in the ages of young star clusters along the azimuthaldirection (see also, e.g., Benedict et al. 2002; Brandle etal. 2012). In the second pearls-on-a-string model, on theother hand, star formation takes place preferentially atthe contact points between a ring and dust lanes. Thismay happen because gas clouds with the largest densi-ties are usually placed at the contact points due to orbitcrowding (e.g., Kenney et al. 1992; Reynaud & Downes1997; Kohno et al. 1999; Hsieh et al. 2011). Since starclusters age as they orbit along the ring, this model nat-urally predicts a bipolar azimuthal age gradient of starclusters starting from the contact points (see also, e.g.,Ryder et al. 2001; Allard et al. 2006; Mazzuca et al. 2008;B¨oker et al. 2008; Ryder et al. 2010; van der Laan et al2013). Mazzuca et al. (2008) found that ∼
50% of thenuclear rings in their sample galaxies show azimuthalage gradients and that such galaxies have, on average, alarger value of the mean SFR than those without notice-able age gradients.Another interesting observational result concerns ra-dial locations of star clusters relative to the nuclear rings.In some galaxies such as NGC 1512 (Maoz et al. 2001)and NGC 4314 (Benedict et al. 2002), young star clustersare located at larger radii than the dense gas of nuclearrings. Martini et al. (2003) also reported that out of123 barred galaxies in their sample, eight galaxies havestrong nuclear spirals, all of which have star-forming re-gions outside the rings. By analyzing multi-wavebandHST archive data of NGC 1672, Jang & Lee (2013) re-cently identified hundreds of young and old star clusterswith ages in the range ∼ − Myr. They found thatthe clusters in the nuclear regions exhibit a systematicpositive radial age gradient, such that older clusters tendto be located at larger galactocentric radii, farther awayfrom the ring. Proposed mechanisms for the radial agegradient include the decrease in the ring size due to an-gular momentum loss (Regan & Teuben 2003) and mi-gration of clusters due to tidal interactions with the ring(van de Ven & Chang 2009).Numerical simulations have been a powerful tool tostudy formation and evolution of bar substructures suchas dust lanes, nuclear rings, and nuclear spirals (e.g.,Sanders & Huntley 1976; Athanassoula 1992; Piner et al.1995; Englmaier & Gerhard 1997; Patsis & Athanassoula2000; Maciejewski et al. 2002; Regan & Teuben 2003,2004; Ann & Thakur 2005; Thakur et al. 2009; Kim etal. 2012a,b). In particular, Athanassoula (1992) showedthat dust lanes are shocks formed at the downstream sidefrom the bar major axis. Dust lanes tend to be shorterand located closer to the bar major axis as the gas soundspeed increases (Englmaier & Gerhard 1997; Patsis &Athanassoula 2000; Kim et al. 2012a).Very recently, Kim et al. (2012b, hereafter Paper I) ranvarious models with differing bar strength and demon-strated that nuclear rings form not by resonant inter-actions of the gas with the bar potential, as was previ-ously thought, but instead by the centrifugal barrier thatthe inflowing gas with non-vanishing angular momentumcannot overcome. According to this idea, a more massivebar forms stronger dust-lane shocks which remove an-gular momentum more efficiently from the gas, so thatthe inflowing gas is able to move inward closer to the galaxy center, forming a smaller nuclear ring. This turnsout entirely consistent with the observational result ofComer´on et al. (2010) that “stronger bars host smallerrings”. Magnetic stress at the dust lanes takes away an-gular momentum additionally, leading to an even smallerring compared to the unmagnetized counterpart (Kim &Stone 2012). Paper I also showed that nuclear spiralsthat form inside nuclear rings unwind with time due tothe nonlinear effect (Lee & Goodman 1999), with an un-winding rate higher for a stronger bar. Thus, the prob-ability of having more tightly wound spirals is larger forgalaxies with a weaker bar, consistent with the observa-tional result of Peeples & Martini (2006).While the numerical studies mentioned above are use-ful to understand gas dynamics in the central regionsof barred galaxies, they are without self-gravity and/orprescriptions for star formation. There have been only afew numerical studies that considered star formation innuclear rings in a self-consistent way. Heller & Shlosman(1994) studied star formation in galactic disks that areunstable to bar formation. Using a smoothed particlehydrodynamics (SPH) combined with N -body method,they found that star formation in barred galaxies occursepisodically, with a time scale of ∼
10 Myr, and that theassociated SFR is well correlated with the mass accre-tion rate to the central black hole (BH). These were con-firmed by Knapen et al. (1995) who also found that tur-bulence driven by star formation tends to widen nuclearrings. Friedli & Benz (1995) used another SPH+ N -bodymethod to run various models with differing parameters,finding that star formation in the nuclear regions first ex-periences a burst phase before entering a quiescent phase(see also Martin & Friedli 1997). Since these authors em-ployed a small number ( ∼ ) of gas particles in theirmodels, however, they were unable to resolve the nuclearregions well. Kim et al. (2011) ran SPH simulations forstar formation specific to the central molecular zone inthe Milky Way. While Dobbs & Pringle (2010) studiedcluster age distributions in spiral and barred galaxies,their results were based on SPH simulations that did notconsider star formation and feedback.In this paper, we extend Paper I by including self-gravity and a prescription for star formation feedback.We focus on temporal and spatial distributions of starformation occurring in nuclear rings of strongly-barredgalaxies. Unlike the previous SPH simulations with starformation, our models use a grid-based, cylindrical codewith high spatial resolution in the central regions. Wealso allow for time delays between star formation andfeedback, which is crucial to study age gradients of starclusters that form in nuclear rings. Our main objectivesare to address important questions such as what controlsthe SFR in the nuclear rings and what are responsiblefor the presence (or absence) of the age gradients of starclusters in the rings, mentioned above.We take a simple galaxy model in which a self-gravitating gaseous disk with either uniform or expo-nential density distribution is placed under the influ-ence of a non-axisymmetric bar potential. We imple-ment a stochastic prescription for star formation thattakes allowance for a threshold density as well as a starformation efficiency. Star formation feedback is treatedonly through direct momentum injections from super-tar Formation in Barred Galaxies 3nova (SN) explosions occurring 10 Myr after star forma-tion events. By considering an isothermal equation ofstate, we do not consider gas cooling and heating, andradiative feedback, which may be important in regulat-ing star formation in disk galaxies (e.g., Ostriker et al.2010; Ostriker & Shetty 2011; Kim et al. 2011; Shetty& Ostriker 2012). In our models, the bar potential isturned on slowly over time, which not only represents asituation where the bar forms and grows but also helpsavoid abrupt gas responses. In each model, we measurethe SFR in the ring and study its dependence on variousquantities such as the gas mass in the ring, mass inflowrates to the central regions, bar growth time, etc. We alsoexplore temporal and spatial variations of star-formingregions and their connection to the SFR. In addition, westudy physical properties of star clusters and gas cloudsin the rings and compare them with observational resultsavailable.We remark on a few important limitations of our mod-els from the outset. First of all, our gaseous disks aretwo-dimensional and razor-thin. This ignores potentialdynamical consequences of vertical gas motions and re-lated mixing, which was shown important in inducingnon-steady gas motions across spiral shocks (e.g., Kim& Ostriker 2006; Kim et al. 2006, 2010). Second, weadopt an isothermal equation of state for the gas, cor-responding to the warm phase, and do not consider ra-diative cooling and heating required for production andtransitions of multiphase gas (e.g., Field 1965; Wolfireet al. 2003; McKee & Ostriker 2007). We also ignorethe effects of outflows, winds, and radiative feedbackfrom young stars, which may be of crucial importancein setting up the equilibrium pressure in galactic planes,thereby regulating star formation in disk galaxies (e.g.,Ostriker et al. 2010; Ostriker & Shetty 2011; Shetty &Ostriker 2012). Finally, we in the present work do notconsider the effects of spiral arms that may supply gas tothe bar regions. With these caveats, the numerical mod-els presented in this paper should be considered as a firststep toward more realistic modeling of star formation innuclear rings.This paper is organized as follows. In Section 2, wedescribe the numerical methods and parameters used forour time-dependent simulations. In Section 3, we presentthe temporal evolution of SFRs, and their dependence onthe model parameters. The age gradients of star clustersin the azimuthal and radial directions as well as prop-erties of star clusters and dense clouds are discussed inSection 4. In Section 5, we summarize our main resultsand discuss their astronomical implications. MODEL AND METHOD
To study star formation in nuclear rings of barredgalaxies, we extend the numerical models studied in Pa-per I by including self-gravity, conversion of gas to stars,and feedback from star formation. In this section, webriefly summarize the current models and describe ourhandling of star formation and feedback. The reader isreferred to Paper I for more detailed description of thenumerical models.
Galaxy Model
We initially consider an infinitesimally-thin, rotatingdisk. The disk is assumed to be unmagnetized and
Table 1
Model ParametersModel Σ ( M (cid:12) pc − ) τ bar /t orb f mom (1) (2) (3) (4)noSG 20 1 0.U05 5 1 0.75U10 10 1 0.75U20 20 1 0.75U30 30 1 0.75M25 20 1 0.25M50 20 1 0.50FB05 20 0.5 0.75FB20 20 2 0.75FB40 20 4 0.75E30 30 1 0.75E50 50 1 0.75E100 100 1 0.75 Note . — Gas surface density in Modelswith the prefix “E” initially have an exponen-tial distribution Σ = Σ exp ( − r/ . . isothermal with sound speed of c s = 10 km s − . Theexternal gravitational potential responsible for the diskrotation consists of four components: a stellar disk, astellar bulge, a non-axisymmetric stellar bar, and a cen-tral BH with mass M BH = 4 × M (cid:12) . This gives rise toa rotation curve that is almost flat at v c ∼
200 km s − in the bar region and its outside. The presence of theBH makes the rotation velocity rise as v c ∝ ( M BH /r ) / toward the galaxy center. The bar potential is mod-eled by a Ferrers (1887) prolate spheroid with semi-major and minor axes of 5 kpc and 2 kpc, respec-tively. The bar is rigidly rotating with a pattern speedΩ b = 33 km s − kpc − , which places the corotation res-onance radius at r = 6 kpc and the inner Lindblad res-onance (ILR) radius at r = 2 . t orb = 2 π/ Ω = 186 Myr. In our mod-els, the bar potential is turned on over the bar growthtime scale τ bar , while the central density of the spheroidalcomponent (bar plus bulge) is kept fixed. We vary τ bar to study situations where the bar grows at a differentrate. The mass of the bar, when it is fully turned on, isset to 30% of the total mass of the spheroidal componentwithin 10 kpc. All the models are run until 1 Gyr.As in Paper I, we integrate the basic equations of idealhydrodynamics in a frame corotating with the bar. Weuse the CMHOG code in cylindrical polar coordinates( r, φ ). CMHOG is third-order accurate in space and hasvery little numerical diffusion (Piner et al. 1995). To re-solve the central region with high accuracy, we set up alogarithmically-spaced cylindrical grid over r = 0 .
05 kpcto 8 kpc. The number of zones in our models is 1024 inthe radial direction and 632 in the azimuthal directioncovering the half-plane from φ = − π/ π/
2. The cor-responding spatial resolution is 0 .
25 pc, 5 pc, and 40 pcat the inner boundary, at r = 1 kpc where most starformation takes places, and at the outer radial bound-ary, respectively. We adopt the outflow and continuousboundary conditions at the inner and outer radial bound-aries, respectively, while taking the periodic boundaryconditions at φ = ± π/ To allow for dilution ofgravity due to finite thickness H of the combined disk,we take H/r = 0 . or an exponential distributionΣ exp( − r/R d ) with the scale length of R d = 3 . f mom of the radial momentumfrom SNe imparted to the disk in the in-plane directionrelative to what would be the total radial momentum ina three-dimensional uniform medium (see Section 2.2).We run a total of 13 models that differ in Σ , τ bar , and f mom . Table 1 lists the model parameters. Column (1)lists each model. Models with the prefix “E” have an ex-ponential disk, while all the others have a uniform disk.Model noSG is a control model that does not includeself-gravity and star formation. Column (2) lists Σ ofthe disk. Column (3) gives the bar growth time τ bar inunits of t orb , while Column (4) gives f mom . We takeModel U20 with Σ = 20 M (cid:12) pc − , τ bar /t orb = 1, and f mom = 0 .
75 as our fiducial model. All the models ini-tially have a Toomre Q parameter greater than unity, sothat they are gravitationally stable in the absence of abar potential. However, nuclear rings that form near thecenter achieve large density, enough to undergo runawaycollapse to form stars. Star Formation and Feedback
To model star formation and ensuing feedback, we firstidentify high-density regions whose average surface den-sity (cid:104) Σ (cid:105) within a radius R SF exceeds a critical density.The natural choice for the threshold density would beΣ th = c s GR SF = 1160 M (cid:12) pc − (cid:16) c s
10 km s − (cid:17) (cid:18) R SF
10 pc (cid:19) − , (1)from the Jeans condition. While it is desirable to choosea small value for the sizes of star-forming regions, wetake R SF = 10 pc because of numerical resolution: astar-forming cloud at r ∼ ∼ (cid:104) Σ (cid:105) ≥ Σ th immediately undergogravitational collapse and star formation since we needto consider the star formation efficiency as well as thecomputational time step (Kim et al. 2011). The SFRexpected from a cloud with mass M cloud = πR (cid:104) Σ (cid:105) ,from the Schmidt (1959) law, isSFR = (cid:15) ff M cloud t ff for (cid:104) Σ (cid:105) ≥ Σ th , (2) We ignore the gravity from the initial gas distribution in orderto make the initial rotation curve the same with that in the non-self-gravitating counterpart. where (cid:15) ff is the star formation efficiency per free-fall time, t ff , defined by t ff = (cid:18) π G (cid:104) ρ (cid:105) (cid:19) / = 3 . (cid:18) (cid:104) Σ (cid:105) (cid:12) pc − (cid:19) − / , (3)assuming a disk scale height of 100 pc. We take (cid:15) ff =0 .
01, consistent with theoretical and observational esti-mates (e.g., Krumholz & McKee 2005; Krumholz & Tan2007).The star formation probability of an eligible cloud with (cid:104) Σ (cid:105) ≥ Σ th in a time interval ∆ t is then given by p =1 − exp( − (cid:15) ff ∆ t/t ff ) ≈ (cid:15) ff ∆ t/t ff (e.g., Hopkins et al. 2011).For a given computational time step ∆ t , the probability p calculated in our models is typically ∼ − − − ,much smaller than unity. In each time step, we thusgenerate a uniform random number N ∈ [0 , N < p . When a cloud undergoesstar formation, we create a particle with mass M ∗ , andconvert 90% of the cloud mass to the particle mass. Theinitial position and velocity of the particle are set equalto the density-weighted mean values of the parent cloudwithin R SF . Each particle has a mass in the range M ∗ ∼ − M (cid:12) , which is about ∼ − times largerthan the masses of observed clusters in nuclear rings (e.g.,Maoz et al. 2001; Benedict et al. 2002; see also PortegiesZwart et al. 2010). Therefore, a massive single particle inour models can be regarded as representing an unresolvedgroup of star clusters rather than an individual cluster.We treat SN feedback using simple momentum inputto the surrounding gaseous medium. We consider onlyType II SN events since our models run only until 1 Gyr.Since we do not resolve individual stars in a cluster ortheir group, we assume that all SN explosions occur si-multaneously. Stars with mass between 8 M (cid:12) and 40 M (cid:12) explode as Type II SNe (Heger et al. 2003), which com-prise about 7% of the cluster mass under the Kroupa(2001) initial mass function. The mean mass of SN pro-genitors is then ∼
14 M (cid:12) , indicating that the number ofSNe exploding from a cluster (or their group) with mass M ∗ is N SN = M ∗ / (200 M (cid:12) ), with the total ejected mass M ejecta = 0 . M ∗ returning back to the ISM. Betweenstar formation and SN explosions, we allow a time delayof 10 Myr, corresponding to the mean life time of TypeII SN progenitors (e.g., Lejeune & Schaerer 2001). Notethat the orbital time of gas in nuclear rings is typically ∼
25 Myr in our models, so that star clusters move byabout 150 ◦ in the azimuthal angle from the formationsites before experiencing SN explosions.Each feedback, corresponding to N SN simultane-ous SNe, injects mass and radial momentum in theform of an expanding shell. In the momentum-conserving stage, a single SN with energy 10 ergwould drive radial momentum P rad , = 3 × (cid:15) / (Σ / (cid:12) pc − ) − / M (cid:12) km s − to the surround-ing gas if the background medium is uniform and in threedimensions, where (cid:15) is the SN energy in units of 10 erg(e.g., Chevalier 1974; Shull 1980; Cioffi et al. 1988). Thedependence of P rad , on Σ is due to the fact that theshell expansion in the Sedov phase is slow when the back-ground surface density is large. Since our model disks arerazor-thin by ignoring the vertical direction, the momen-tum imparted to the gas in the simulation domain wouldtar Formation in Barred Galaxies 5be smaller than P rad , . Let f mom denote the fraction ofthe total radial momentum that goes into the in-plane di-rection. If the expansion is isotropic, f mom ∼
75% (Kimet al. 2011), but f mom can be smaller in a vertically strat-ified disk since it is easier for a shell to expand along thevertical direction. The total momentum of a shell fromeach feedback is thus set to P sh = 3 × f mom N / (cid:18) Σ1 M (cid:12) pc − (cid:19) − / M (cid:12) km s − . (4)In this paper, we take f mom = 0 .
75 as a standard value,but run some models with lower f mom to study the effectof f mom on the SFR (see Table 1).As the initial radius of a shell, we take R sh = 40 pc,corresponding to the shell size at the end of the Sedovphase when N SN = 10 and the background density isΣ = 10 M (cid:12) pc − , the typical mean density of nuclearrings when star formation is active. When feedback oc-curs from a particle, we redistribute the mass and mo-mentum within a circular region with radius R sh centeredat the particle by taking their spatial averages. We thenadd the shell momentum densityΣ v sh = (cid:40) P max (cid:16) RR (cid:17) R , r ≤ R sh , , r > R sh , (5)to the gas momentum density in the in-plane direction,and M ejecta / ( πR ) to the gas surface density, while re-ducing the particle mass by M ejecta . In equation (5), R denotes the position vector relative to the particle loca-tion and P max = 2 P sh / ( πR ) is the momentum per unitarea at r = R sh . Note that v sh ( R ) ∝ R ensures aninitially divergence-free condition at the feedback center(e.g., Kim et al. 2011). STAR FORMATION IN NUCLEAR RINGS
In this section, we first describe overall evolution ofour numerical simulations. We then present the tempo-ral variations of SFRs and their dependence on the gasmass, the bar growth time, and f mom , as well as the re-lationship between the SFR surface density and the gassurface density. The properties of star clusters and gasclouds in the rings are analyzed in the next section. Overall Gas Evolution
We begin by presenting evolution of our standardmodel U20 that has a uniform density Σ = 20 M (cid:12) pc − initially. Evolution of the other models is qualitativelysimilar, although more massive disks show more activestar formation. In all models, star formation occursmostly in the nuclear rings. Figure 1 plots snapshotsof the gas distributions as well as the positions of starclusters in Model U20 at a few selected epochs. Theleft panels show the gas density in logarithmic scalesin the 5 kpc regions, with the solid ovals indicatingthe outermost x -orbit that cuts the x - and y -axes at x c = 3 . y c = 4 . In Model U30, the bar-end regions, often called ansae, formstars as well, although the associated SFR is less than 3% comparedto the ring star formation.
10 Myr, while asterisks correspond to young clusters withage <
10 Myr: the upper right colorbar represents theirages. In all panels, the bar is oriented vertically alongthe y -axis and remains stationary. The gas inside thecorotation resonance is rotating in the counterclockwisedirection.Early evolution of Model U20 before t = 0 . x -orbit responds strongly to the bar potential. Before thebar is fully turned on (i.e., t < τ bar ), no closed x - and x -orbits exist since the external gravitational potentialvaries with time. During this time, the gas streamlinesare highly transient as they try to adjust to the time-varying potential. The overdense ridges grow with timeand move slowly toward the bar major axis, develop-ing into dust-lane shocks. The gas passing through theshocks loses angular momentum and flows radially in-ward along the dust lanes. The radial velocity of theinflowing gas is so large that it is not halted at the ILR(Paper I). It gradually rotates faster due to the Coriolisforce, and eventually forms a nuclear ring after hittingthe dust lane at the opposite side. Produced by super-sonic collisions of two gas streams, the contact pointsbetween the ring and the dust lanes have the largest den-sity in the ring, and can thus be preferred sites of starformation.Over time, the nuclear ring shrinks in size and thecontact points rotate in the azimuthal direction. Toillustrate this more clearly, Figure 2 plots the temporaland radial variations of the azimuthally averaged surfacedensity in the central regions of Model U20 and its non-self-gravitating counterpart, Model noSG. The horizontaldashed line marks the location of the ILR. The ring isbeginning to form at t ∼ . r ∼ x -orbits:the major axis of the ring is inclined to the x -axis by − ◦ at t = 0 .
15 Gyr (Fig. 1a). As the ring materialcontinuously interacts with the bar potential, the contactpoints rotate in the counterclockwise direction. At t ∼ . x -orbits, and thecontact points are located close the bar minor axis.At the same time, the ring becomes smaller in size dueto the addition of low-angular momentum gas from out-side as well as by collisions of the ring material whose or-bits are perturbed by thermal pressure (Paper I). Whenself-gravity is absent, the decreasing rate of the ring ra-dius is d ln r NR /dt ∼ − . − until t < ∼ . As pointed out by the referee, Figure 1 shows that the ringstarts out fairly elongated and titled relative to the bar and be-comes rounder with time. Note that the shape of the ring inModel U20 at t = 0 .
15 Gyr is remarkably similar to that of ahighly-elongated nuclear ring in ESO 565-11 observed by Buta etal. (1999), although the latter is ∼ Seo & Kim
Figure 1.
Snapshots of logarithm of gas density (color scale) as well as the locations of star clusters in Model U20 at t =0 . , . , . , . / Σ ). The solid ovals in the left panels draw the outermost x -orbit that cuts the x - and y − axes at x c = 3 . y c = 4 . x -orbit with x c = 1 . y c = 4 . tar Formation in Barred Galaxies 7 Figure 2.
Temporal and radial variations of the azimuthally av-eraged surface density in logarithmic scale for (a) Model U20 and(b) Model noSG. The horizontal dashed line in each panel marksthe location of the inner Lindblad resonance. The colorbars labellog(Σ / Σ ). The decreasing rate of the ring size is smaller whenself-gravity and star formation are included. bits relatively intact, reducing the ring decay rate to d ln r NR /dt ∼ − . − . The decrease of the ring sizein turn causes the contact points to move radially inwardand rotate further in the counterclockwise direction (butnot more than ∼ ◦ ).As Figure 1 shows, the bar region (i.e., inside theoutermost- x orbit) becomes evacuated quite rapidly.This is because the bar potential is efficient in remov-ing angular momentum from the gas only in the bar re-gion, while its influence on the gas orbits outside theoutermost- x orbit is not significant. By t ∼ . x -orbit with x c = 1 . y c = 4 . x -orbit the inner-ring x -orbits. The formation of the innerring in our model is as follows. As mentioned before, thedust-lane shocks form first at far downstream and movestoward the bar major axis as the bar potential grows.During this time, much of the gas in the bar region in-falls to the nuclear region. Near the time when the barattains its full strength, the dust lanes find their equi-librium positions on an x -orbit, still at the leading sidefrom the bar major axis. The inner ring starts to format this time, by gathering the residual material locatedbetween the outermost and inner-ring x -orbits. Somegases located outside the outermost x -orbit experiencecollisions near the bar ends where x -orbits crowd, andare then able to lower their orbits to the inner-ring x -orbit, increasing the inner-ring mass. In Model U20, thegas added to the inner ring from outside of the outer- most x -orbit is about 70% of the total inner-ring massat t = 1 Gyr. Most of the gas inside the inner-ring x -orbit had already transited to the nuclear ring by ex-periencing the dust-lane shocks before the bar was fullyturned on. Star Formation Rate
Figure 3 plots the time evolution of the SFR, the massinflow rate to the nuclear ring ˙ M NR ≡ (cid:82) π Σ v r rdφ (mea-sured at r = 1 . M NR in Model U20. Here v r denotesthe radial velocity of the gas. In plotting these profiles,we take a boxcar average, with a window of 20 Myr. Inthis model, the bar potential grows over the time scale of τ bar = 0 .
19 Gyr, and the first star formation takes placeat the contact points at t = 0 .
12 Gyr. As the bar growsfurther, the dust-lane shocks become stronger, increasingthe amount of the infalling gas to the ring. ˙ M NR attainsa peak value ∼ (cid:12) yr − at t = 0 .
15 Gyr, which coin-cides with the time of highest density of the dust lanes(see Fig. 5 of Paper I). The decay of ˙ M NR after the peakis caused by the fact that only the gas inside the out-ermost x -orbit responds strongly to the bar potential,while the outer region is not much affected. Similarly,the SFR exhibits a strong burst with a maximum value ∼ (cid:12) yr − , which occurs ∼
30 Myr after the peak of˙ M NR . The associated SN feedback produces many holesin the gas distribution, driving a huge amount of kineticenergy to the surrounding medium. Note that the nu-clear ring, albeit somewhat patchy, is overall well main-tained despite energetic momentum injections (Fig. 1b).When the mass inflow rate to the ring is very large, starformation occurring at the contact points alone is unableto consume the whole inflowing gas. As we will show inSection 4.1, the maximum gas consumption rate affordedto the contact points is estimated to be about 1 M (cid:12) yr − for the parameters we adopt. Any surplus inflowing gaspasses by the contact points and is subsequently addedto the ring that is clumpy. Some overdense regions inthe ring are soon able to achieve the mean density largerthan the critical value and undergo star formation, in-creasing the SFR rapidly. Since the star-forming regionsare randomly distributed in the ring, there is no obviousazimuthal age gradient of star clusters in this high-SFRphase.Particles spawned from star formation orbit about thegalaxy center under the total gravity, but they do not feelgas pressure that is quite strong in the nuclear ring. Inaddition, the gaseous ring becomes smaller in size withtime. Thus, the orbits of star particles increasingly de-viate from the gaseous orbits over time. When stars inclusters explode as SNe, they are not always located inthe ring. A majority of star clusters are still in the ring,while there are some clusters ( ∼ Figure 3.
Temporal variations of (a) the SFR, (b) the mass inflow rate ˙ M NR to the nuclear ring, and (c) the total gas mass M NR in thenuclear ring of Model U20. The ordinate is in linear scale in the left panels, while it is in logarithmic scale in the right panels. In (a) and(b), the horizontal dotted lines indicate the reference rate of 1 M (cid:12) yr − . creases ˙ M NR temporarily during t = 0 . .
35 Gyr, withthe associated short bursts of star formation at t = 0 . .
32 Gyr in Model U20.At t = 0 . M NR is reducedto below 2 × M (cid:12) in Model U20 since star formationconsumes the gas in the ring, which also decreases theSFR (Fig. 3). As most of the gas in the bar region insidethe outermost x -orbit is almost lost to star formation,the galaxy evolves into a quasi-steady state where starformation is limited to small regions near the contactpoints (Fig. 1d). Parametric Dependence of SFR
Figure 4 compares the SFRs from (left) uniform-diskand (right) exponential-disk models with different Σ .In all models, the SFR displays a primary burst fol-lowed by a few secondary bursts, with time intervals of ∼ −
80 Myr, before becoming reduced to below1 M (cid:12) yr − . The primary burst is associated with therapid gas infall due to angular momentum loss at thedust-lane shocks, while the secondary bursts are causedby the re-infall of the ejected gas via SN feedback outto the bar region. Models with larger Σ start to formstars earlier and have a larger value of the maximumstar formation rate, SFR max . The duration of activestar formation, ∆ t SF , defined by the time span whenSFR ≥ SFR max /
2, is also larger for models with largerΣ , since the gas available for star formation is corre-spondingly larger. Models U20 and E50 initially havea similar gas mass inside the outermost x -orbit, butModel E50 has larger SFR max since the gas is more cen-trally concentrated and thus infalls more readily to thenuclear ring. Columns (2) and (3) of Table 2 list SFR max and ∆ t SF for all models. The phase of active star forma-tion lasts only for ∼ . τ bar in all models.Figure 5 shows how (left) the bar growth time and(right) the amount of the momentum injection affect thetemporal behavior of the SFR. In models where the bargrows more rapidly, dust-lane shocks form earlier andtar Formation in Barred Galaxies 9 Figure 4.
Temporal variations, over t = 0 − . . The horizontal dotted lines indicate SFR = 1 M (cid:12) yr − . Models with larger gas mass inside the outermost x -orbit formstars earlier and at a larger rate. Figure 5.
Temporal variations of the SFR for (a) models with differing the bar growth time τ bar and (b) models with different momentuminjection f mom . Note that the range of the abscissa is 1 Gyr in (a) and 0 . (cid:12) yr − .0 Seo & Kim
Temporal variations of the SFR for (a) models with differing the bar growth time τ bar and (b) models with different momentuminjection f mom . Note that the range of the abscissa is 1 Gyr in (a) and 0 . (cid:12) yr − .0 Seo & Kim Table 2
Simulation OutcomesModel SFR max ∆ t SF M tot ∗ M x M NR β Γ(M (cid:12) yr − ) (Myr) (10 M (cid:12) ) (10 M (cid:12) ) (10 M (cid:12) )(1) (2) (3) (4) (5) (6) (7) (8)noSG - - - 10.66 8.03 - -U05 0.53 - 1.4 2.62 0.87 13.7 3.1U10 3.55 42 3.3 5.33 1.39 14.4 2.6U20 7.46 83 9.0 10.66 1.42 12.5 2.2U30 11.8 83 13.3 15.99 2.36 12.6 2.2M25 7.77 87 8.2 10.66 1.08 9.6 2.3M50 7.78 77 8.7 10.66 1.55 10.8 2.2FB05 9.83 68 8.9 10.66 1.62 12.3 2.2FB20 4.04 163 8.7 10.66 1.73 9.7 2.2FB40 1.94 392 7.2 10.66 2.75 7.4 3.0E30 5.10 74 6.1 7.55 1.55 12.4 2.2E50 10.4 77 10.1 12.59 1.41 6.3 2.2E100 21.9 84 23.0 25.18 1.78 6.3 2.0 Note . — SFR max and ∆ t SF denote the peak rate and the duration of activestar formation with SFR ≥ SFR max /
2, respectively; M tot ∗ is the total mass instars at t = 1 Gyr; M x is the total gas mass inside the outermost x -orbit at t =0; M NR is the mass of the nuclear ring at t = 1 Gyr; β = d log( t/ yr) /d ( r/ kpc)is the radial age gradient of clusters; Γ = − d log N/d log M ∗ is the power-lawslope of the cluster mass functions. Figure 6.
Dependence of the SFR on (a) the mass inflow rate ˙ M NR to the ring and (b) the total mass M NR in the ring. The dashed linedraws SFR = ˙ M NR in (a), and SFR ∝ ˙ M . in (b). The SFR is almost equal to ˙ M NR for the whole range of the SFR, while it is not wellcorrelated with M NR when SFR ≤ (cid:12) yr − . tar Formation in Barred Galaxies 11 Figure 7. (a) Dependence of the SFR surface density Σ
SFR on the mean surface density Σ cl of dense clouds for Model U20.The colorbar indicates the epoch of star formation for each sym-bol, and the curved arrow denotes the mean evolutionary trackin the Σ SFR − Σ cl plane. (b) Σ SFR − Σ cl relationship from ourmodels compared to the observed Kennicutt-Schmidt law. Squaresand pluses represent normal and circumnuclear starburst galaxiesadopted from Kennicutt (1998), respectively, while circles are forspatially-resolved star-forming regions in NGC 1097 from Hsieh etal. (2011). initiate stronger gas inflows. This causes star formationin models with smaller τ bar to occur at a higher rate andfor a shorter period of time (see Table 2). The peak SFRis attained approximately at t ∼ (0 . − τ bar . This sug-gests that galaxies in which the bar forms more slowly arelikely to have star formation less active instantaneouslybut extended for a longer period of time. Figure 5b showsthat the SFR computed in our models is largely insensi-tive to f mom , although smaller f mom makes the secondarybursts less active.To directly address what controls star formation in nu-clear rings, we plot in Figure 6 the dependence of the SFR(left) on the mass inflow rate to the ring and (right)on the gas mass in the ring for Models U10, U20, andU30. Color indicates the star formation epoch of eachsymbol. Even though there are large scatters especiallywhen the SFR is low, the SFR is almost equal to ˙ M NR over two orders of magnitude variations in ˙ M NR . Thescatters in the SFR − ˙ M NR relation are due to the factthat star formation is stochastic in our models and thatit takes the gas some finite time ( ∼ −
30 Myr) to travelfrom r = 1 . M NR is measured) to the nu-clear ring. On the other hand, the SFR does not showa good correlation with M NR . While SFR ∝ M . forSFR > ∼ (cid:12) yr − , it is almost independent of M NR forSFR < ∼ (cid:12) yr − . Note that the change in M NR is lessthan a factor of 5 in Figure 6b, while the SFR varies bymore than two orders of magnitude. This suggests thatit is the mass inflow rate to the ring, rather than thering mass, that determines the SFR in the nuclear ring.Conversely, the SFR can be a good measure of the mass inflow rate driven by the bar potential.Column (4) of Table 2 gives the total mass in stars M tot ∗ formed until the end of the run for each model.Columns (5) and (6) list the total gas mass M x insidethe outermost x -orbit in the initial disk and the massof the nuclear ring M NR at t = 1 Gyr, respectively. Notethat M x is approximately the maximum gas mass avail-able for star formation in the ring. We find that therelation M tot ∗ = M x − M NR , with M NR = 2 × M (cid:12) fixed, explains the numerical results fairly well, indicat-ing that most of the gas inside the outermost x -orbitflows inward to form stars, with some residual gas re-maining in the nuclear ring. Compared to Model U20with τ bar /t orb = 1, Model FB40 with τ bar /t orb = 4 has M tot ∗ about 20% smaller, since the gas in the bar regionis still flowing in to the nuclear ring at the end of therun.As will be discussed in more detail in Section 5.2, theoverall temporal trend of the SFR (that is, rapid de-cline after a primary burst except for a few secondarybursts) found in our numerical models is largely similarto the numerical results of previous studies (e.g., Heller &Shlosman 1994; Knapen et al. 1995; Friedli & Benz 1995),but appears inconsistent with observations of Allard etal. (2006) and Sarzi et al. (2007) who found that star-forming nuclear rings live long, with multiple episodes ofstarburst activities. Since the ring SFR is controlled bythe mass inflows rate to the rings, this implies that ringsin real galaxies should be continually supplied with freshgas from outside for quite a long period of time. Candi-date mechanisms for additional gas inflows, over a timescale much longer than the bar growth time, include spi-ral arms and cosmic gas infalls, which are not includedin this paper. Star Formation Law
To explore the dependence of the local SFR on thelocal gas surface density, we define clouds as regionsin the simulation domain whose density is larger than300 M (cid:12) pc − . This density roughly corresponds tothe mean density of boundaries of gravitationally boundclouds in our models (see Section 4.4). While this choiceof the minimum density for clouds is somewhat arbi-trary, these clouds may represent giant molecular cloudsand their complexes including hydrogen envelopes (e.g.,Williams et al. 2000; McKee & Ostriker 2007).At each time, we calculate the mean surface densityΣ cl of, and the total area A cl occupied by, the cloudsdistributed along the ring. The mean SFR surface den-sity is then given by Σ SFR = SFR /A cl . Figure 7a plotsthe resulting Σ SFR as a function of Σ cl for Model U20.Color represents the time when each point is measured,while the curved arrow indicates the mean evolution-ary direction in the Σ SFR –Σ cl plane. When the firststar formation takes place ( t = 0 .
12 Gyr), the ring hasΣ cl ∼
560 M (cid:12) pc − and Σ SFR ∼ . (cid:12) yr − kpc − .The radial gas inflow along the dust lanes increases Σ SFR rapidly until it achieves a peak value at t = 0 .
18 Gyr.The corresponding increase of Σ cl is smaller since starformation reduces the gas content in the ring. After thepeak, Σ SFR decreases with decreasing ˙ M NR , but it has alarger value, by a factor of ∼ − cl before the peak. This is because active2 Seo & Kim Figure 8.
Histograms of the star clusters that formed in each selected time bin (with bin width of 0 .
15 Gyr) for all uniform-density modelsas a function of the azimuthal angle where they form. The arrow at the bottom of each panel indicates the mean position of a contactpoint.
SN feedback stirs the ring material vigorously, tending toincrease density contrast between clumps and the back-ground material. After the Σ
SFR peak, therefore, thetotal area covered by the gas with Σ >
300 M (cid:12) pc − becomes smaller than before, resulting in larger Σ SFR .Figure 7b plots the Σ
SFR –Σ cl relationship measuredat every 0 . M x tend to have larger Σ SFR and largerΣ cl . Note that our numerical results are overall consis-tent with the Kennicutt-Schmidt law for normal galaxies(squares) and circumnuclear starburst galaxies (pluses)adopted from Kennicutt (1998), and not much differ-ent from the observed Σ SFR –Σ cl relation for spatially-resolved star-forming regions (circles) in the nuclear ringof NGC 1097 taken from Hsieh et al. (2011). Thereare large scatters in Σ
SFR , amounting to ∼ SFR , as the Kennicutt-Schmidt law implies, while thescatters in Σ
SFR are likely due to the temporal variationsof the mass inflow rate to the ring. PROPERTIES OF STAR CLUSTERS AND GAS CLOUDS
Azimuthal Age Gradient In plotting the Σ
SFR –Σ relation, Hsieh et al. (2011) used themaximum density of a cloud, instead of the mean density, for Σ.
As mentioned in Introduction, observations indicatethat some galaxies have well-defined azimuthal age gra-dients of star clusters in nuclear rings (e.g., Ryder et al.2001; Allard et al. 2006; B¨oker et al. 2008; Ryder et al.2010; van der Laan et al 2013), while others do not (e.g.,Benedict et al. 2002; Brandle et al. 2012). Our simula-tions show that the presence or absence of the azimuthalage gradient is decided by the SFR in the ring (or, morefundamentally, on ˙ M NR ), independent of Σ and the ini-tial gas distribution.Figure 8 plots for the uniform-density models the his-tograms of star clusters formed in each selected timebin, with bin width of 0 .
15 Gyr, as a function of theangular position where they form. The arrow at thebottom of each panel marks the location of a contactpoint that is moving in the positive azimuthal directionwith time, as described in Section 3.1. Model U05 withΣ = 5 M (cid:12) yr − has SFR < ∼ . (cid:12) yr − (Fig. 4),and star-forming regions in this model are almost al-ways localized to the contact point. In Models U20 andU30, on the other hand, star-forming regions are widelydistributed along the azimuthal direction at early time( t = 0 . − .
45 Gyr) when SFR > (cid:12) yr − , whilethey are preferentially found near the contact points atlate time ( t > ∼ .
45 Gyr) when SFR < (cid:12) yr − .Star clusters age as they orbit along the ring and emitcopious UV radiations during about ∼
10 Myr afterbirth. If star-forming regions are localized to the con-tact points, therefore, clusters would appear as “pearlstar Formation in Barred Galaxies 13
Figure 9.
Positions and ages of young star clusters at selected epoches in Models U05, U20, and U30, overlaid on the gas densitydistribution in linear scale. The left panels show a clear azimuthal age gradient, while there is no age gradient in the right panels. Colorindicates the cluster ages. on a string” (e.g., B¨oker et al. 2008), with an age gra-dient along the rotational direction of the nuclear ring.This is illustrated in the left panels of Figure 9 which plotthe spatial locations of star clusters with color indicatingtheir ages ( <
10 Myr), overlaid on the density distribu-tion in linear scale, in Model U10 at t = 0 .
35 Gyr andModel U20 at t = 0 .
51 Gyr. There is clearly a positive bi-polar age gradient starting from the contact points thatare located at φ ∼ ◦ and 210 ◦ . On the other hand,when the star-forming regions are randomly distributedthroughout the ring, as in the “popcorn” model of B¨okeret al. (2008), star clusters with different ages would bemixed. In this case, there is no apparent age gradientalong the ring, as exemplified in the right panels of Fig- ure 9 for Model U30 t = 0 .
33 Gyr and Model U20 at t = 0 .
18 Gyr.Why does then the SFR (or ˙ M NR ) matter for the az-imuthal distributions of star-forming regions? The an-swer lies at the fact there is a limit on the rate of gasconsumption at the contact points that occupy very smallareas in the ring. When ˙ M NR is sufficiently small, mostof the inflowing gas to the ring can be converted intostars at the contact points, and the resulting SFR is cor-respondingly small. When ˙ M NR is very large, on theother hand, the contact points cannot transform all theinflowing gas to stars instantaneously. The excess inflow-ing gas overflows the contact points and is transferred toother regions in the ring. The ring becomes denser not4 Seo & Kimonly by the addition of the overflowing gas but also byits own self-gravity. Some clumps in the ring achieve sur-face density above the threshold value, and start to formstars. We find that the rings have the Toomre stabil-ity parameter as low as ∼ . r and∆ φ denote the radial thickness and the azimuthal extentof a contact point, respectively. Then, the maximumSFR expected from two contact points is simply˙ M ∗ , CP = 2 (cid:15) ff Σ CP r NR ∆ r ∆ φ/t ff , (6)where Σ CP is the surface density of the contactpoints. The mean density of star-forming clouds is ∼ (cid:12) pc − , which we take for Σ CP . For (cid:15) ff = 0 . r = 50 pc, ∆ φ = 30 ◦ , and r NR = 1 kpc typical in ourmodels, equation (6) yields ˙ M ∗ , CP ∼ (cid:12) yr − , consis-tent with our numerical results. Note that the specificvalue of ˙ M ∗ , CP depends on the parameters we adopt. Inparticular, ˙ M ∗ , CP ∝ c s r /R / if the ring width is pro-portional to the ring radius, suggesting that galaxies witha weak bar (to have a smaller ring) and strong turbulencewould have large ˙ M ∗ , CP . Radial Age Gradient
Although the presence of an azimuthal age gradientdepends on the SFR, we find that star clusters alwaysdisplay a radial age gradient. Figure 10 plots the spatialdistributions of (left) the formation locations and (right)the present positions of star clusters on the x - y planeat t = 1 Gyr in Model U20. Each cluster is coloredaccording to its formation time. The open circles denotethe clusters that have passed the central region with r < . t =1 Gyr. Star-forming regions at late time are concentratedon the contact points, while they are well distributed atearly time. Note that clusters that form early beforethe nuclear ring settles on an x -orbit have initial kickvelocities quite different from those on x - or x -orbits attheir formation locations. Although the ring soon takeson the x -orbit, these clusters move on eccentric orbitsand wander around the nuclear region. Figure 10b showsthat young clusters are preferentially found near the ring,while old clusters are located away from it, indicative ofa positive radial gradient of their ages.To show this more clearly, Figure 11 plots the ageof clusters as functions of their present radial positions (circles) at t = 1 Gyr as well as their formation loca-tions (plus symbols) for Models U10 and U20. Again,the open circles denote the clusters that have passed bythe galaxy center, while the filled circles are for thosethat have not. Note that the age distributions of thepresent-day and formation-epoch locations of the clus-ters are not much different from each other, althoughthe former shows a large spatial dispersion. The dis-persion is larger for older clusters. To quantify the ra-dial age gradient, we bin the clusters according to theirages, with a bin size of ∆ log( t/ yr) = 0 .
2, and calcu-late the mean age and position in each bin. Our bestfits of the ages to the formation-epoch positions are β ≡ d log( t/ yr) /d ( r/ kpc) ∼ . . β for all models. This radial age gradient results pri-marily from the decrease in the ring size with time, suchthat old clusters formed at larger galactocentric radii.Clusters diffuse out radially through gravitational inter-actions themselves and also with dense clouds in the ring,without much effect on the radial age gradient. Cluster Mass Functions
Figure 12 plots the mass functions of all the clustersthat have formed in each of the (left) uniform-disk and(right) exponential-disk models until t = 1 Gyr. Theupper panels are for the clusters formed while star for-mation is very active with SFR ≥ (cid:12) yr − , whereasthose produced when SFR < (cid:12) yr − are presentedin the lower panels. In general, the mass distribution ofclusters is described roughly by a power law, with its in-dex depending on the SFR and M x . When the SFR islarger than 1 M (cid:12) yr − , the slope of the mass function isΓ ≡ − d log N/d log M ∗ ∼ M x . When SFR ≤ (cid:12) yr − , clus-ters have a much steeper mass distribution with Γ > ∼ M x can be understood as follows. When the SFRis large, there are numerous dense regions distributedthroughout the ring. Such regions grow by accretingthe surrounding material. Since star formation occursin a stochastic manner in our models, some dense cloudshave a chance to grow as massive as, or even larger than ∼ M (cid:12) , leading to a relatively shallow mass function.On the other hand, when the SFR is small, there arenot many dense regions. Since the growth of density isquite slow in these models, they form stars at densitiesslightly above the critical value. In this case, most clus-ters have mass around ∼ − M (cid:12) , with a steepmass distribution. Giant Clouds
Finally, we present the properties of giant clouds lo-cated in nuclear rings. High-resolution radio observa-tions show that nuclear rings consist of giant molecularassociations at scale of ∼ . . ∼ M (cid:12) and are gravitationally bound (e.g., Hsieh etal. 2011).To identify giant clouds in our models, we utilize a core-finding technique developed by Gong & Ostriker (2011).This method makes use of the gravitational potential oftar Formation in Barred Galaxies 15 Figure 10.
Spatial distributions of (a) the formation locations and (b) the present positions of star clusters in Model U20 at t = 1 Gyr.The dashed lines draw the ring at this time. Open circles denote the clusters that have passed the central region with r < . Figure 11.
Ages of star clusters as functions of their currentradial locations (circles) at t = 1 Gyr and their formation positions(pluses) for Models U10 and U20. Open circles are those that havepassed by the galaxy center at a very close distance during orbitalmotions, while the filled circles are for those that have not. Thedashed lines are the fits, with slopes of β = d log( t/ yr) /d ( r/ kpc) =14 . .
5, for the initial cluster positions, for Models U10 andU20, respectively. the gas, and thus allows smoother cloud boundaries thanthe methods based on isodensity surfaces (see, e.g., Smithet al. 2009). At a given time, we search for all the localminima of the gravitational potential and find the largestclosed potential contour encompassing one and only onepotential minimum. We then define the potential min-imum and outermost contour as the center and bound-ary of a cloud, respectively. If the distance between twoneighboring minima is less than 0 . t = 0 .
36 Gyr.The left panel shows the gas surface density in linearscale, while the right panel displays boundaries of giantclouds as contours overlaid over the gravitational poten-tial of the gas. A total of 14 giant clouds are identified.The mean values of their masses M , radii R , and one-dimensional velocity dispersions σ are 10 M (cid:12) , 100 pc,and 20 km s − , respectively, corresponding to super-sonic internal motions. The average density of the cloudboundaries is found to be ∼
300 M (cid:12) pc − . The averagevalue of the virial parameter is α = 5 σ R/ ( GM ) ∼ SUMMARY AND DISCUSSION
Summary
We have presented the results of two-dimensional hy-drodynamic simulations of star formation occurring innuclear rings of barred galaxies. We initially consider aninfinitesimally thin, isothermal gas disk placed under theexternal gravitational potential. The external potentialconsists of a stellar disk, a stellar bulge, a central BH,and a non-axisymmetric stellar bar. We do not studythe effect of spiral arms in the present work. The bar6 Seo & Kim
Figure 12.
Clusters mass functions for (left) the uniform-density and (right) the exponential-density models. The upper and lower panelsplot the clusters formed when the SFR is larger or smaller than 1 M (cid:12) yr − , respectively. The mass function becomes shallower withincreasing SFR. potential is modeled by a Ferrers prolate spheroid withthe semi-major and minor axes of 5 kpc and 2 kpc, re-spectively, and rotates about the galaxy center with apatten speed of 33 km s − kpc − . The bar mass, whenit is fully turned on, is set to 30% of the total stellarmass in the spheroidal component, corresponding to astrongly barred galaxy. We fix the gas sound speed to c s = 10 km s − and the BH mass to 4 × M (cid:12) . Our sim-ulations incorporate star formation recipes that include adensity threshold corresponding to the Jeans condition,a star formation efficiency, conversion of gas to particlesrepresenting star clusters or their groups, and delayedmomentum feedback via SN explosions. To explore var-ious situations, we consider both uniform and exponen-tial density models, and vary the gas surface density, bargrowth time, and the total momentum injection in thein-plane direction. The main results of this work can besummarized as follows. The imposed bar potential readily induces a pair ofdust-lane shocks in the bar region inside the outer-most x -orbit. At early time when the bar potential isweak, the dust-lane shocks are placed at the far down-stream side from the bar major axis. As the bar poten-tial increases, the dust-lane shocks become stronger andslowly move toward the bar major axis. The gas passingthrough the shocks loses a significant amount of angu-lar momentum, infalls radially along the dust lanes, andforms a nuclear ring. The continuous gas inflows providea fuel for star formation in the ring. After the bar po-tential reaches its full strength, the dust lanes settle onan x -orbit, while the nuclear ring follows an x -orbit.The remaining gas located inside the outermost x -orbitand outside the dust lanes is gathered to form an elon-gated inner ring in the bar region, whose shape is welldescribed by an x -orbit, as well. Some of the gas locatedoutside the outermost x -orbit transits to the inner ringtar Formation in Barred Galaxies 17 Figure 13.
Distribution of giant clouds in the nuclear ring of Model U20 at t = 0 .
36 Gyr. Left: the gas surface density is shown in linearscale. Right: cloud boundaries found by the method described in the text are overlaid on the gravitational potential of the gas. near the bar ends where x -orbits crowd. Similarly, thegas in the inner ring loses angular momentum when itcollides with other gas near the bar ends, slowly infallingto the nuclear ring through the dust lanes.The contact points between the dust lanes and the nu-clear ring is a tunnel through which the inflowing gason x -orbits switches to the x -orbit of the nuclear ring.About the time when the bar potential is fully turned on,the contact points are located near the bar minor axis.Over time, the nuclear ring shrinks in size due to theaddition of low angular momentum gas from outside andby collisions of the ring material, which in turn makesthe contact points rotate slowly in the counterclockwisedirection. Since the contact points have largest densityin the nuclear ring, they are preferred sites of star for-mation, although star-forming regions can be distributedthroughout the ring when the mass inflow rate is high.The bar potential transports the gas in the bar regionto the nuclear ring very efficiently, but does not havestrong influence on the gas orbits outside the bar region.This not only makes the bar region evacuated rapidly butalso reduces the mass inflow rate ˙ M NR dramatically after ∼ . ∼ −
80 Myr.The peak SFR is attained at t ∼ (0 . − τ bar , and theduration of active star formation is roughly a half of the bar growth time. The SFR is almost equal to ˙ M NR .It has a weak dependence on the total gas mass M NR in the ring when SFR > ∼ (cid:12) yr − , and is not corre-lated with M NR when SFR < ∼ (cid:12) yr − . This suggeststhat star formation in the ring is controlled primarilyby ˙ M NR rather than M NR . The relationship betweenthe SFR surface density and the surface density of denseclumps in nuclear rings found from our numerical modelsare consistent with the usual Kennicutt-Schmidt law forcircumnuclear starburst galaxies.The presence or absence of azimuthal age gradients ofyoung star clusters in nuclear rings depends on the SFR(or ˙ M NR ) in our models. When ˙ M NR is small, most ofthe inflowing gas to the nuclear ring is consumed at thecontact points. In this case, young star clusters thatform would exhibit a well-defined azimuthal age gradi-ent along the ring. When ˙ M NR is large, on the otherhand, the contact points are unable to transform all ofthe inflowing gas to stars. The extra gas overflows thecontact points and goes into the nuclear ring. The ringbecomes massive and forms stars in clumps that becomedense enough. In this case, no apparent age gradient ofstar clusters is expected since star-forming regions arerandomly distributed over the whole length of the ring.The critical value of ˙ M NR that determines the presenceor absence of the azimuthal age gradient is estimated tobe ∼ (cid:12) yr − in our models, although it depends onvarious parameters such as the ring radius, critical den-sity, etc. (eq. [6]).Star clusters produced also exhibit a positive radialage gradient, such that young clusters are located closeto the nuclear ring, while old clusters are found awayfrom the ring. The primary reason for this is thatthe nuclear ring becomes smaller in size with time, andthus star-forming regions gradually move radially in-ward. In our models, the radial age gradient amounts8 Seo & Kimto β = d log( t/ yr) /d ( r/ kpc) ∼ −
15. Radial diffusionof clusters via mutual gravitational interactions and alsowith the gaseous ring does not affect the radial age gra-dient much.When the SFR is large ( > (cid:12) yr − ), some denseclouds are able to grow by accreting surrounding ma-terial and form massive clusters. The cluster massfunction is well described by a power law, with slopeΓ = − d log N/d log M ∗ ∼ −
3. A larger slope cor-responds to a more massive disk in which more gas isavailable for the ring star formation. When the SFR issmall, on the other hand, most clusters form near thethreshold density, leading to a steeper slope with Γ > ∼ M (cid:12) and sizes 0 . ∼
20 km s − . They are gravitationallybound with the virial parameter of α ∼ Discussion
We find that the SFR in nuclear rings shows a strongprimary burst, with its duration and peak value depen-dent on the bar growth time, and subsequently a fewweak and narrow bursts, after which the SFR becomesvery small. The peak of the primary burst is attainedroughly when the bar potential is fully turned on. Thisburst behavior of the SFR appears to be a generic fea-ture of star formation in nuclear rings of strongly-barredgalaxies found in numerical simulations. For instance, N -body+ SPH models presented by Heller & Shlosman(1994), Knapen et al. (1995), and Friedli & Benz (1995)showed that the SFR reaches its peak value, with narrowbursts superimposed, about the time when the stellar barfully develops, after which it is reduced to small values.In numerical modeling for star formation in the nuclearregion of the Milky Way, Kim et al. (2011) found thatthe SFR is maximized at t ∼ .
15 Gyr, with a peak value ∼ .
22 M (cid:12) yr − , and then drops to a relatively constantvalue ∼ . − .
07 M (cid:12) yr − . The sustained star forma-tion in Kim et al. (2011) is thought to arise because theMilky Way has a very weak bar that takes a long timeto clear out gas in the bar region. In this case, the gasinfall may proceed continuously over an extended periodof time, a situation similar to the case with a slowly-growing bar.There is observational evidence that star formation innuclear rings occurs continually over a long period oftime (a few Gyrs) with successive ∼ −
10 bursts sep-arated by a few tenths of Gyrs each (e.g., Allard et al.2006; Sarzi et al. 2007; see also van der Laan et al 2013).This is in sharp contrast to our numerical results thatshow that star formation in nuclear rings is dominatedby one primary burst before declining to small values,with ∼ . The bar poten-tial alone is unlikely responsible for gas supply needed forstar formation in real nuclear rings.
Unless bars are dy- namically young, present star formation in nuclear ringsof nearby barred galaxies requires additional gas feeding.One obvious such mechanism is spiral arms that can re-move angular momentum at spiral shocks to transportgas from outer disks to the bar regions (e.g., Lubow etal. 1986; Kim & Stone 2012), which is not consideredin the present work. Accretion of halo gas to the diskmay not only rejuvenate bars (e.g., Bournaud & Combes2002) but also enhance the SFR in the rings (e.g., Jiang &Binney 1999; Fraternali & Binney 2006, 2008). Such gasflows might actually exist, as evidenced by the presenceof an enhanced number of carbon stars in the outer spiralarms of M33 (Block et al. 2007). Temporal variations inthe bar strength (e.g., Bournaud & Combes 2002) andin the bar pattern speed (e..g, Combes & Sanders 1981)are also likely to affect the mass inflow rate to the ringand thus the SFR.Some galaxies such as IC 4933 (Ryder et al. 2010) showage gradients of star clusters along the azimuthal direc-tion in nuclear rings, while there are other galaxies suchas NGC 7552 (Brandle et al. 2012) that do not show aclear age gradient. Mazzuca et al. (2008) analyzed H α data for 22 nuclear rings and found that about half oftheir sample galaxies contain azimuthal age gradients,although most of them are not throughout the entirering. They also found that the mean SFR in galaxieswith azimuthal age gradients is 2 . ± . (cid:12) yr − , whichis slightly larger than the mean value of 3 . ± . (cid:12) yr − for galaxies with no apparent age gradient. While thisappears consistent with our numerical results, the largedispersions in the mean SFRs suggest that there is nofixed SFR that can distinguish between galaxies withand without age gradients. In addition, the criticalSFR for the absence or presence of azimuthal age gra-dients is about 1 M (cid:12) yr − in our models, while mostgalaxies in the sample of Mazzuca et al. (2008) haveSFR > (cid:12) yr − . As equation (6) suggests for themaximum SFR, ˙ M ∗ , CP , allowed at the contact points,however, the critical SFR depends on many factors thatmay vary from galaxy to galaxy. For example, NGC1343 with the most clear bi-polar age gradient in theMazzuca et al. (2008) sample has the current SFR of ∼ . (cid:12) yr − . Its ring radius is ∼ M ∗ , CP ∝ Σ / ∝ c s /R / , so that the levelof interstellar turbulence and the size of star-forming re-gions R SF may change ˙ M ∗ , CP considerably.We find that star clusters that form in nuclear ringsnaturally develop a positive radial gradient of their agesowing primarily to the decrease in the ring size over ∼ Gyr in our models. This is consistent with the re-sults of Jang & Lee (2013) who found that clusters withages < ∼ ∼ Gyrand may not apply to clusters in a small age range sincethe decay of the ring size is quite slow. Indeed, Mazzucaet al. (2008) found that two (NGC 5953 and 7570) oftheir sample galaxies show a negative radial age gradienttar Formation in Barred Galaxies 19of H II regions in the rings. Our numerical results plot-ted in Figure 11 also show that when limited to clusterswith age ∼ –10 . yr at t = 1 Gyr, younger clusterscan be found at larger radii, which is due to the stochas-tic nature of star formation and ensuing gravitationalinteractions.Our results show that the SFR in the nuclear rings istightly correlated with the mass inflow rate to the ringrather than the total gas mass in the ring (Fig. 6). Thisresult is seemingly consistent with the results of Benedictet al. (2002) who found that the SFR in the nuclear ringof a strongly-barred galaxy NGC 4314 is smaller, by afactor of 30, than that in a weakly-barred galaxy NGC1326 (Buta et al. 2000), even if the gas mass in the ringis smaller by only a factor of two. It is interesting to notethat the gas mass contained in most “gas-rich” nuclearrings of barred galaxies in the BIMA SONG sample isin a remarkably narrow range of ∼ (1 − × M (cid:12) (Sheth et al. 2005). The SFR data presented in Mazzucaet al. (2008) combined with the bar strength given inComer´on et al. (2010) show that strongly-barred galaxiesusually have very small present-day SFRs and the SFRsin weakly-barred galaxies vary in a wide range, althoughthe number of galaxies in their sample is too limited tomake a conclusive statement. It will be interesting tosee how the bar strength as well as gas inflows by spiralshocks influence the SFR in nuclear rings.While star formation is concentrated in nuclear rings inour models, observations indicate that star formation insome galaxies occurs not only in nuclear rings but also inthe bar region including dust lanes (e.g., Martin & Friedli1997; Sheth et al. 2000; Zurita & P´erez 2008; Elmegreenet al. 2009; Mart´ınez-Garc´ıa & Gonz´alez-L´opezlira 2011).While dust lanes themselves are known hostile to star for-mation due to strong velocity shear (e.g., Athanassoula1992; Kim et al. 2012a), Sheth et al. (2000) proposed thatstars form in interbar dust spurs in filamentary shapethat impact the dust lanes from the trailing side of thebar (see also Sheth et al. 2002; Zurita & P´erez 2008).Indeed, Elmegreen et al. (2009) inferred that some clus-ters in the nuclear ring of NGC 1365 actually formed inone of the dust lanes by the impact of spurs and subse-quently migrated inward to the nuclear ring. The originof these interbar spurs is yet unclear. Apparently, thereis no filamentary interbar feature in our models. Theymay originate from gas inflows due to spiral shocks fromthe region outside the bar (Elmegreen et al. 2009), frominteractions of gas with magnetic fields that are perva-sive in the bar region (Beck et al. 1999, 2005), and/orfrom other dynamical processes that involve gas cooling,self-gravity, etc., which are not considered in the presentwork. It will be an important direction of future work tostudy how spiral arms and magnetic fields affect the gasinflows and star formation in the bar and nuclear regions.We gratefully acknowledge I. S. Jang and M. G. Lee forsharing their results on the radial age gradient of clus-ters found in NGC 1672. We also thank E. C. Ostrikerand K. Sheth for helpful discussions, and are grateful Note that NGC 5953 is a non-barred galaxy. While Sheth et al. (2005) reported that the ring in NGC 6946has a mass of ∼ M (cid:12) , a higher-resolution observation of Schin-nerer et al. (2006) gives the ring mass of ∼ × M (cid:12) . to the referees for an insightful report and for the in-formation on ESO 565-11. This work was supported bythe National Research Foundation of Korea (NRF) grantfunded by the Korean government (MEST), No. 2010-0000712. 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