Black hole fueling in galaxy mergers: A high-resolution analysis
DDraft version January 26, 2021
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BLACK HOLE FUELING IN GALAXY MERGERS: A HIGH-RESOLUTION ANALYSIS
Joaquin Prieto , Andr´es Escala , George C. Privon & Juan d’Etigny Departamento de Astronom´ıa, Universidad de Chile, Casilla 36-D, Santiago, Chile. Department of Astronomy, University of Florida, 211 Bryant Space Sciences Center, Gainesville, FL 32611, USA.
Draft version January 26, 2021
ABSTRACTUsing parsec scale resolution hydrodynamical adaptive mesh refinement simulations we have studiedthe mass transport process throughout a galactic merger. The aim of such study is to connect boththe peaks of mass accretion rate onto the BHs and star formation bursts with both gravitational andhydrodynamic torques acting on the galactic gaseous component. Our merger initial conditions werechosen to mimic a realistic system.The simulations include gas cooling, star formation, supernovaefeedback, and AGN feedback. Gravitational and hydrodynamic torques near pericenter passes triggergas funneling to the nuclei which is associated with bursts of star formation and black hole growth.Such episodes are intimately related with both kinds of torques acting on the galactic gas. Pericenterstrigger both star formation and mass accretion rates of ∼ few (1 − M (cid:12) /yr. Such episodes last ∼ (50 −
75) Myrs. Close passes also can produce black hole accretion that approaches and reaches theEddington rate, lasting ∼ few Myrs. Our simulation shows that both gravitational and hydrodynamictorques are enhanced at pericenter passes with gravitational torques tending to have higher valuesthan the hydrodynamic torques throughout the merger. We also find that in the closest encounters,hydrodynamic and gravitational torques can be comparable in their effect on the gas, the two helpingin the redistribution of both angular momentum and mass in the galactic disc. Such phenomena allowinward mass transport onto the BH influence radius, fueling the compact object and lighting up thegalactic nuclei. Subject headings: galaxies: formation — large-scale structure of the universe — stars: formation —turbulence. INTRODUCTIONCosmological N-body numerical simulations of struc-ture formation shows that dark matter (DM) haloes wereformed by mergers between smaller haloes in a hierar-chical way (e.g. Angulo et al. 2012). Similar kinds ofsimulations including baryonic physics have shown thatgalaxies were formed inside DM haloes in this hierarchi-cal model (e.g. Dubois et al. 2014; Genel et al. 2014).In this sense, mergers and interaction between galaxiesare a fundamental piece of the galaxy formation processand certainly they influence the galaxies evolution. Infact, observation of irregular and disrupted systems areconsistent with mergers and interactions between galax-ies (e.g. Toomre &Toomre 1972; Schweizer 1982; Engelet al. 2010; Bussmann et al. 2012).The infrared and optical properties of interactinggalaxies are different compared with isolated systems(e.g. Sanders & Mirabel 1996). Such differences can be aconsequence of star formation burst associated to galac-tic interactions (e.g. Sanders et al. 1998; Duc et al. 1997;Jogee et al. 2009). Beside the radiative signatures of starformation, some interacting galaxies also show nuclearactivity which can be associated with black hole (BH)fueling (e.g. Petric 2011; Stierwalt et al. 2013) These twofeatures, i.e. SF bursts and active galactic nuclei (AGN),suggest that galactic encounters are able to redistributegas inside galaxies, moving material toward their cen-tral regions to feed massive BHs and trigger SF bursts(e.g. Barnes & Hernquist 1991; Mihos & Hernquist 1996;Springel et al. 2005).Smoothed Particle Hydrodynamic (SPH) numericalsimulations of galactic mergers with ∼ few (10 − ∼ ∼ pc-resolution simulations have not beenperformed.When choosing to deal with idealized mergers usingAMR codes over a Lagrangian code, problems with theadvection of material and grid alignment issues mayarise, which could result in a loss of angular momen-tum conservation (Wadsley et al. 2008; Hahn et al. 2010;Hopkins 2015). This issues are minimized as high spatialresolution is imposed at central galaxy regions, minimiz-ing spurious field misalignments, and also, since peri-center passes have short durations (few orbital times atmost) there are no significant orbital angular momentum a r X i v : . [ a s t r o - ph . GA ] J a n Prieto et al.deviations with respect to the ideal case. Furthermore,the galaxies are maintained at resolutions that are highenough for the AMR technique to be effective at resolv-ing shocks throughout the merger process and thereforecontact discontinuities are captured.As the objective of our simulation is to properly andfully characterize a generic galaxy merger that exhibitsrealistic dynamics, we have to choose appropriate or-bital initial conditions that have been proven capableof nearly-reproducing such observed controlled environ-ments. Due to the degeneracy of the problem and thelarge parameter space of galaxy interactions, constrain-ing the initial conditions with hydrodynamic simulationswould be prohibitively time-consuming. Privon et al.(2013) used the Identikit code to find the orbital parame-ters capable to reproduce the morphology and kinematicsof tidal features of four known observed galaxy mergers(NGC 5257/8, The Mice, Antennae and NGC 2623). Inthis work we will adopt their orbital parameters (as anansatz) for NGC 2623. Whilst the objective of this workwill not be to reproduce neither the morphology of thissystem, we cite its characteristics as order of magnitudecontrol values.NGC 2623 is a low-redshift, luminous infrared galaxy(LIRG) with an infrared luminosity of L IR = 3 . × L (cid:12) from Armus et al. (2009). The system has beenclassified as an M4 merger (Larson et al. 2016), i.e. theyare galaxies with apparent single nucleus and evidenttidal tails. The merger shows two tidal tails of ∼ − M (cid:63) = 2 . × M (cid:12) witha molecular hydrogen mass of M H = 6 . × M (cid:12) .This values do not stray afar from typical LIRG val-ues found in samples like the GOALS survey (Armus etal. 2009). Haan et al. (2011) show that although thereis some spread in the central BH mass values found inthe GOALS survey, masses are generally found in the10 − M (cid:12) range.In this paper, for the first time we will study the evo-lution of a merger system from its early stages, up tothe point where their BHs coalesce, using a ∼ § § § METHODOLOGY AND NUMERICALSIMULATION DETAILS2.1.
Initial conditions
As already mentioned, in this work we use the param-eters found in Privon et al. (2013) for NGC 2623, asinitial conditions for a high resolution hydrodynamic nu-merical simulation. Table 2.1 shows the initial orbitalparameters of the simulated merger used in this workand table 2.1 shows the initial position and velocity forboth galactic centers. In addition to the orbital ICs, it is necessary to spec-ify both the mass content and the mass distribution foreach component of the galaxies, including the gaseousdisc, stellar disc, and stellar bulge. In order to createICs for the DM haloes, gas and stars for each galaxy wehave used the DICE code (Perret et al. 2014). For oursetup the gaseous disc follows an exponential profile witha characteristic radius of 1 kpc. The stellar disc is mod-elled with a Myamoto-Nagai profile with a characteristicradius of 0.677 kpc. The stellar bulge follows a Einastoprofile with a characteristic radius of 0.6 kpc. Finally forthe DM profile we employ a Navarro Frenk and Whiteprofile (Navarro, Frenk & White 1996) with a concen-tration parameter equal to 10. The SFR in the stellardisc follows Bouch´e (2010). Table 2.1 shows a completesummary of the galactic parameters of the system.
TABLE 1Initial orbital parameters. D ini [kpc] e p [kpc] µ ( i ; ω ) ( i ; ω )50.0 1.0 0.6 1.0 (30 ◦ ; 330 ◦ ) (25 ◦ ; 110 ◦ )the orbit e , pericentral distance of the orbit p , mass galaxiesratio µ and disk orientation for both galaxies ( i, ω ) with respectthe orbital plane. TABLE 2Initial position and velocity.
Coordinates Gal Gal (x, y, z) [kpc] (-25, 0, 0) (25, 0, 0)(v x , v y , v z ) [km/s] (25, 4.1, 0.0) (-25, -4.1, 0.0)The orbital plane has been rotated 45 ◦ in the polar directionand 45 ◦ in the azimuthal direction. Those data are in the ref-erence frame of the simulated box. Gas physics
The simulation was performed with the cosmologicalN-body hydrodynamical code RAMSES (Teyssier 2002).The code uses adaptive mesh refinement, and solves theEuler equations with a second-order Godunov methodand MUSCL scheme using a MinMod total variation di-minishing scheme to reconstruct the cell centered valuesat cell interfaces.The galaxies were set inside a computational box ofL box = 400 kpc. The coarse level of the simulation cor-responds to (cid:96) min = 7 and ∆ x coarse = 3 .
125 kpc. Weallowed 10 levels of refinement to get a maximum reso-lution at (cid:96) max = 17 of ∆ x min = 3 .
05 pc. The refinementis allowed inside a cell if i) its total mass is more than8 times that of the initial mass resolution, and ii) theJeans length is resolved by less than 4 cells (Trueloveet al. 1997). If we take into account grid regions wherenumber density is above 0.01 cm − , the worst cell refine-ment we find is at level 12 with ∆ x ≈
97 pc, and cells atrefinement level 14 and 15 account for almost ∼
60% ofthe total number of cells throughout such areas. Abovethese, at level 16 we account for the ∼
16% of cells andat level 17 we account for ∼ TABLE 3Initial isolated galaxy setup.
Gaseous disc(Exponential profile)Mass [10 M (cid:12) ] 1.0Characteristic radius [kpc] 1.0Truncation radius [kpc] 5.0Stellar disk(Myamoto-Nagai profile)Mass [10 M (cid:12) ] 2.975Number of particles 11900000Characteristic radius [kpc] 0.677Truncation radius [kpc] 5.0Stellar bulge(Einasto profile)Mass [10 M (cid:12) ] 0.975Number of particles 390000Characteristic radius [kpc] 0.6Truncation radius [kpc] 1.0Dark mater halo(NFW profile)Mass [10 M (cid:12) ] 20Number of particles 200000Concentration parameter 10Truncation radius [kpc] 60 (1993) model down to temperature T = 10 K with acontribution from metals, assuming a primordial compo-sition of the various heavy elements. Below this temper-ature, the gas can cool down to T = 10 K due to metalline cooling (Dalgarno & MacCray 1972).We adopted a star formation number density thresh-old of n = 250 H cm − with a star formation effi-ciency (cid:15) (cid:63) = 0 .
03 (e.g. Rasera & Teyssier 2006; Dubois& Teyssier 2008). When a cell reaches the conditionsfor star formation, stellar (population) particles can bespawned following a Poisson distribution with a massresolution of m (cid:63), res ≈ × M (cid:12) . In order to ensurenumerical stability we do not allow cells to convert morethan 50% of the gas into stars within a single time step.After 10 Myr the most massive stars explode as SNreleasing a specific energy of E SN = 10 erg/10 M (cid:12) , re-turning 20 per cent of the stellar particle mass back intothe gas with a yield (fraction of metals) of 0 . r SN = 2∆ x min . In order to capture the delayof stellar feedback energy release from non-thermal pro-cesses, we used the delayed cooling implementation ofSNe feedback (Teyssier et al. 2013). In this work we use t diss ≈ .
25 Myr and the energy threshold e NT is the oneassociated to a turbulent velocity dispersion σ NT ≈ M Bondi .In such accretion implementation the gas density is com-puted as an average weighted value taken from the sink’scloud particles using a kernel following Krumholz et al.(2004) as presented in Dubois et al. (2012). Through-out the simulation we cap the accretion rate at the Ed-dington rate. The initial BH mass for both galaxies is M BH = 10 M (cid:12) , approximately lying on the M BH − σ (cid:63) Fig. 1.—
Gas number density projection (left column) and stellarmass projection (right column) at different times. From top tobottom: central BHs are at 10 kpc, the first pericenter, the firstapocenter and the point of coalescence relation. Assuming a sound speed of c s ≈ −
30 km/s(for a gas temperature T gas ≈ − K, note thatmost of the time the gas is at 10 K, the 10 K are asso-ciated to AGN activity when the gas is expelled from theBH vicinity), the BH influence radius at the beginningof the simulation is R BH = G M BH /c s ≈ −
42 pc ≈ −
14 ∆ x min . In other words, we can resolve the BHinfluence radius with several cells (note that such radiusincreases throughout the simulation). In addition to theforementioned grid refinement criteria, we impose that a20-sided cubic volume surrounding sink particles, staysfixed at maximum spatial resolution, helping to resolvethe BH influence radius and to account for any potentialnon trivial physical processes occurring nearby. Finally,the BHs particles merge if their separation is lower than d merge = 2∆ x min and if they are gravitationally bound.We have also included AGN feedback from the central Prieto et al. Fig. 2.—
BH particles distance as a function of time. The dif-ferent gray vertical lines mark the first, second and third pericen-ter passes of the orbit (from left to right). After the third peri-center the BHs approach each other until they finally merge att merger = 1 .
25 Gyr, ∼
800 Myr after the first pericenter pass.
BHs. AGN feedback is modeled with thermal energyinput (Teyssier et al. 2011; Dubois et al. 2012). The rateof energy deposited by the BH inside the injection radius r inj ≡ x min is ˙ E AGN = (cid:15) c (cid:15) r ˙ M BH c . (1)In the above expression, (cid:15) r = 0 . (cid:15) c = 0 .
15 is the fraction of this en-ergy coupled to the gas in order to reproduce the localBH-galaxy mass relation (Dubois et al. 2012). As ex-plained in Booth & Schaye (2009), in order to avoid gasover-cooling the AGN energy is not released instanta-neously every time step ∆ t but it is accumulated un-til the surrounding gas temperature can be increased by∆ T min = 10 K. In order to reduce the heating effect ofthe AGN we have included an extra multiplicative factorof 0.1 to ˙ E AGN , which is done as otherwise the feedbackis too effective at preventing accretion onto the centralBHs, and is consistent with the scaling of radiative effi-ciency and BH mass found in Davis & Laor (2011). Suchfactor can be interpreted as a lower radiative efficiency,a lower energy coupling or a combination of both effects.This lowering of factor feedback is consistent with howNGC 2623’s energetics are dominated by star formationover AGN feedback Privon et al. (2013). RESULTSFigure 1 shows four different snapshots throughout themerger, namely (from top to bottom) when the BHs areat a distance of 10 kpc, the first pericenter, the firstapocenter and time where systems coalesce (marked bythe time in which the SMBHs merge). After the firstpericenter pass the system develops two prominent tidaltails producing a “double-tailed” object, which can beappreciated in how the system looks like, at the point ofits first apocenter in figure 1.Before we show results related with SF properties, BHgrowth and gas dynamics throughout the merger, it is il-lustrative to look at the BH separation evolution shown
Fig. 3.—
Enclosed SFR at different distances from BH particle forboth galaxies as a function of time: Inside 5 kpc in black, 1 kpc inviolet, 0.5 kpc in blue, 0.2 kpc in green, 0.1 kpc in orange and 0.05kpc in red. The gray solid vertical lines mark the pericenters of theorbit. It is clear that after each pericenter pass there is a SF burst.Both the second and the third pericenter passes trigger nuclear SFon scales below few 100 pc whereas the first one produces moreextended SF at ∼ kpc scales. in figure 2. This quantity is a good proxy for the galacticcenter separation. The figure shows the time for peri-center passes (in solid gray vertical lines). After thethird pericenter the BHs start to orbit around each other,rapidly decreasing their separation until they merge att merger = 1 .
275 Gyr. This time corresponds to ∼ Star formation rate
A number of works have shown how galactic mergers-interactions trigger bursts of SF (e.g. Barnes & Hernquist1991; Cox et al. 2006; Di Matteo et al. 2007; Hopkins etal. 2008; Moreno et al. 2019). The SF in mergers is notrestricted to the galactic nuclear regions (the inner ∼ kpc) but it can also be triggered in gaseous tails of thesystem (e.g. Soifer et al. 1984; Keel et al. 1985; Lawrenceet al. 1989). In this analysis we will focus on the nu-clear (not tail) SF burst produced by enhancement ofgas density due the galactic interactions and we will notigh resolution BH fueling in a merger system 5 Fig. 4.—
Kennicutt-Schmidt relation for both galaxies in smallcircles. The different colors mark different times. In black solidline the Kennicutt (1998) relation, in green solid line the Daddiet al. (2010) relation for normal galaxies and in blue solid line theDaddi et al. (2010) relation for star bursts. The thin lines markthe error of the corresponding relation. study the extended, in-tail SF (e.g. Barnes 2004; Chien& Barnes 2010; Renaud et al. 2014).Figure 3 shows the enclosed SFR inside a given radiusof both galaxies throughout the evolution. The SFR iscomputed as the ratio between the total stellar mass pro-duced in the last ∼ t (cid:63), avg = 1 (cid:80) i m (cid:63),i (cid:88) i t (cid:63),i m (cid:63),i . (2)Then, SFR = 1 t (cid:63), avg (cid:88) i m (cid:63),i , (3)where m (cid:63),i and t (cid:63),i are the new stellar population massand its age, respectively. We computed the SFR inside asphere centered at the BH position for different radius,namely R SFR = 5 , , . , . , . , .
05 kpc.The SFR in figure 3 shows intermittent behavior dueto both stellar and BH feedback. Before any pericenterpassages between the galaxies, at scales of ∼ few kilo-parsec, formation rates generally fluctuate around ∼ M (cid:12) / yr. We see SF episodes lasting ∼ −
40 Myr afterthe pericenter passes. These relatively short periods areexplained by the response of the medium to feedbackfrom the massive stellar particles. We also note that starformation evolves differently for both galaxies after theirfirst encounter. At ∼
70 Myr before the first pericenterpass (376 Myr after the start of the simulation, see thefirst image of figure 1) the spiral arms of the galaxiesstart to collide (at this time galactic center distance is10 kpc). This working interface progressively increasesthe gas density, translating to a burst of SF.
Fig. 5.—
Mass accretion rate onto the BHs in black solid line.The solid vertical gray lines mark the pericenters of the orbit. TheEddington mass accretion rate limit is in blue solid line. Boththe second and the third pericenter produce a clear increase in theBHAR.
Once the galaxies reach the first pericenter a clear SFburst appears across all distance scales in Galaxy 1, withthe SFR reaching ∼ few M (cid:12) / yr at distances over 0 . (cid:38)
30 Myr (longer for bigger scales), after which the systemis left with a relatively high but slowly decreasing nuclearSFR. In the case of Galaxy 2 there is a not too drasticincrease in nuclear star formation at 0 . − . − ∼ kpc, due to spiral arms collision) SF whereasthe second and third pericenter produce new stars at thenuclear region (inside ∼ − M (cid:12) / yr, below theSFRs measured by Evans et al. (2008) corresponding to Prieto et al. ∼ − M (cid:12) / yr, and also below the ∼ M (cid:12) / yr foundby Howell et al. (2010) for a system with similar initialconditions as ours (NGC 2623). The simulated valuesare closer to the 8 M (cid:12) / yr rate found for the system’s re-cent past in in Cortijo-Ferrero et al. (2017). This SFRsvalues are realistic for a merger system (Pearson et al.2019), albeit it would put our simulation below typicalstarburst galaxy rates.Figure 4 shows the Kennicutt-Schmidt relation(Schmidt 1959; Kennicutt 1998, here after K98) for bothgalaxies as a function of time. In order to compute thegalactic disc surface density Σ gas and the surface SFRΣ SFR we have defined a radius R disc and a height h disc inside a box with 12 kpc of side centered at the BH po-sition; the SFR is computed within this cylinder. Theequatorial plane of the cylinder is constructed with apoint and a normal vector, namely the BH position andthe gas angular momentum vector computed inside 2 kpcfrom the BH position (see appendix A for a discussionabout rotational center). Inside the 12 kpc side box wecompute the enclosed mass in both the positive and thenegative ˆ z direction as a function of height z . The discheight h disc corresponds to the altitude z where the cylin-der contains 90% of the baryonic mass. Following ananalogous method we computed the radius R disc as theradius where for a cylinder height h disc the disc contains90% of the baryonic mass. After this procedure we defineΣ gas = M gas πR , (4)where M gas is the gas mass inside the cylinder andΣ SFR = SFR πR , (5)with the SFR computed each ∼ ∼
600 Myr (green to orange tran-sition) where we found a steady decline in SFRs at figure3, the galaxy 2 system starts going below both the K98and D10 relations, exhibiting how star formation is notable to keep up with the amount of gas stripped fromthe galaxy by the merger interaction. Later pericenterpassages bring both galaxies above the star burst D10 re-lation, and after the violent episodes of both SF and AGNfeedback that the systems are subjected to, the eventu-ally merged galaxy evolves progressively to the regionbelow the D10 normal galaxies, showing a clear decreasein SFR (see Renaud et al. 2014).3.2.
Black hole evolution
Observational evidence suggests that galactic encoun-ters can trigger AGN activity (e.g. Veilleux et al. 2002;Giavalisco et al. 2004; Treister et al. 2012). In order tofeed the BHs the galactic gas should reach the sphere ofinfluence of the central massive objects. To accrete onto the BH, gas orbiting around the BH must lose an-gular momentum, resulting in an inward gas mass flowin the galaxy. This can be triggered by gravitationaltorques acting on the gas due to the galaxy-galaxy in-teraction (e.g. Barnes 1988; Barnes & Hernquist 1991;
Fig. 6.—
BH mass evolution. In solid black line the BH massassociated to galaxy 1 and in solid blue line the BH mass associatedto galaxy 2. The solid vertical gray lines mark the pericenters ofthe orbit.
Di Matteo et al. 2005; Cox et al. 2008) or in general byany type of torque (gravitational, pressure gradients/ hy-drodynamic, magnetic, or viscous) capable of changingthe angular momentum of the gaseous component of thegalaxy.Figure 5 shows the BH mass accretion rate as a func-tion of time for both galaxies. The black hole accre-tion rates (BHAR) oscillate in the range of ∼ − − − M (cid:12) / yr over the first ∼
600 Myr, where we seegalaxy 1 approaching (and eventually peaking at) theEddington limit on different occasions. After the firstpericenter passage, AGN feedback strongly regulates ac-cretion rates, lowering them for at least an order of mag-nitude.Following the low BHAR after the first pericenter pas-sage, in the two following passes, both systems exhibitclear peaks due to the funnelling of gas towards the cen-tral galaxy regions. These peaks are more pronouncedon galaxy 1 than in galaxy 2, the first reaching BHARvalues of a few 10 − M (cid:12) / yr, whilst the second only haslow (albeit pronounced) peaks of a 10 − M (cid:12) / yr. Afterthese two last encounters the BHs merge (where just be-fore this, galaxy 1 accreted at the Eddington rate for ashort period of time).Figure 6 shows the BH masses as a function of time.Because the differences in their mass accretion rate theBH masses are also different for both objects. The BH mass shows a clearer increase with the first pericentercompared with BH , as can be seen from its mass accre-tion rate (figure 4). We confirm that after undergoinga strong mass gaining episode, BH ’s mass stays nearlyconstant for ∼ ∼ . × M (cid:12) .During this time interval the galaxies have reached theirfirst pericenter producing an enhancement in the BH mass accretion rate and consequently in its mass. ThisBH growth is not associated with galactic bulge coales-cence and show that BHs can grow in stages before thegalactic bulge merge (Medling et al. 2013).In contrast to the BH evolution, the second compactobject does show a clear increase in the first pericenterbut it is substancially lower, as can be seen from thelow mass accretion rate shown in figure 4 (which caps atigh resolution BH fueling in a merger system 7 Fig. 7.—
BH mass-bulge stellar velocity dispersion relation. Dif-ferent colors mark different times. Filled circles mark the relationtaking into account all stars inside 1 kpc around the central BHand empty squares mark the relation for stars inside 0.5 kpc aroundthe central BH. The broad black solid line shows the McConnell etal. (2011) relation, the broad blue solid line shows the McConnell& Ma (2013) relation and the broad green solid line shows theG¨ultekin et al. (2009) relation. The thin green, blue and black lineare the corresponding relation 0.4 dex above and below the centralone. ∼ − M (cid:12) / yr, which is not necessarily low, but there isa big amount of perceivable variation in rates). The evo-lution becomes nearly flat, with a low amount of growthuntil the merger. At the time of coalescence, BH hadgrown nearly twice as much as what BH had grownthrough accretion, and after merger, the remnant BHends up at ∼ . × M (cid:12) . This final value would putthe final BH mass well below the LIRG galaxy mergers(like NGC 2623) found in the GOALS sample (Haan etal. (2011)), and although initially this is neither an in-dication that the black holes are not accreting enoughgas through the merger evolution, nor that the BH ini-tial masses are wrong, we can further the analysis bychecking how the M- σ relation evolves throughout thesimulation.Given the BH mass at each point and the stellar veloc-ities, it is possible to compute the M BH − σ (cid:63) (“M-sigma”)relation. (McConnell et al. 2011; McConnell & Ma 2013;G¨ultekin et al. 2009) Figure 7 shows such relation as afunction of time. We have initialized the simulation witha BH mass M BH = 10 M (cid:12) and a bulge velocity dis-persion σ b ≈
110 km / s, which means our setup is insidethe empirical relation of McConnell & Ma (2013) whenusing the velocity dispersion from the < . − σ values should partially stray to the right ofthe relation), we see in the figure both galaxies quicklymoving far away from the empirical relation. This means Fig. 8.—
Inward gas mass accretion rate as a function of time atdifferent distances from the galactic stellar center of mass: 2 kpc inblack, 1 kpc in violet, 0.5 kpc in blue, 0.1 kpc in green, 0.05 kpc inorange and 0.01 kpc in red. The solid vertical gray lines mark thepericenters of the galactic orbit. The figure shows that pericenterscorrelate with peaks of mass accretion rate. that the feeding of the BHs is not being able to catch upwith the growth of velocity dispersion (it should be notedthat after the stellar bulges merge, velocity dispersionshould not increase, and BH feeding could slowly bringthe system back to the empirical relation as the galaxystabilizes).To further support how BHs are growing less than whatis expected of them, we see for instance, that BH growsfrom 10 M (cid:12) to ∼ × M (cid:12) in 1 . M Edd ( t ) = M BH ( t ) t Sal (with theSalpeter timescale being t Sal = 4 . ≈ , × − ˙ M Edd , well under typical feeding expectedfor radiative AGN feedback to be relevant.The apparent culprit of these overall lower than ex-pected average BHAR, would be the amount of thermalfeedback being put back into the grid, which heats thegas surrounding the vicinity of our BHs too effectivelyfor accretion to be steadily maintained. This is furtherevidenced by how even though torques at the hill radiusare sustained all through the simulation (see section 3.4),this does not translate into a feeding of the black holes, aswe see instead that the only important feeding episodesoccur in the initial stages of the simulation and at closepassages (where material is too efficiently transported to-wards the center, allowing gas dynamics to overcome theheating effect of feedback). Prieto et al.The straightforward AGN feedback approach that weare using from Dubois et al. (2012) was developed forcosmological simulations, and even though it has seensuccessful use in that context, the main difference hereis that at the high resolutions we achieve, such simplerecipe may result in the failure to capture the correctsmall scale physics that model the heating of the cen-tral bulges by the BHs. It has been shown that differentmethods for dealing with BH feedback may yield quitedifferent results, and that direct injection of thermal en-ergy to galactic cores may produce strong and persistentoutflows or cavities in central regions that suppress ac-cretion (Wurster & Thacker 2013). It is then imperativethat we try to capture the more detailed heating struc-ture that is produced by the radiative transfer of thesoft X-ray photons that produce quasar-mode feedback.There have been successful efforts at capturing the heat-ing rate from the expected X-ray emission of the centralAGN from Choi et al. (2012), but this recipe is still atits heart a direct injection of energy back to the galac-tic core, and does not offer any accounting of radiativetransfer effects.A more consistent option for improving our feedbackrecipe, would be to include radiation coupling to ourhydrodynamics through RAMSES-RT, in the code pre-sented by Rosdahl et al. (2013) and Rosdahl & Teyssier(2015). Quasar feedback has already been modelled inthis way (Bieri et al. 2017), and it relies in the coupling ofhydrodynamics with the radiative transfer of photons be-ing introduced by the sink particle into the grid throughan AGN template spectral distribution, allowing for a de-tailed accounting of the production and reprocessing ofX-ray radiation (and therefore the overall heating mech-anisms) produced by the innermost regions. The intro-duction of the RT module would also allow for a moreconsistent modelling of SN feedback, and presents theopportunity for future work.3.3.
Gas accretion rate
In the last section we showed that peaks of BH mass ac-cretion rate correlate with the pericentric passages, sug-gesting a connection between close encounters and en-hancement of gas inflows in galaxies. Under this sce-nario it is useful to look at the inward gas mass accre-tion rate at different radii. Figure 8 shows the inwardgas mass accretion rate at different distances from thestellar center of mass. As with the KS computation, wehave constructed a disc perpendicular to the gas angularmomentum vector. After that, in order to compute thegas accretion rate we look for the stellar center of massinside 2 kpc around the BH for each galaxy. Given theposition (cid:126)r CM and bulk velocity (cid:126)v CM of the center of masswe have computed the inflowing gas mass accretion rateas ˙ M g = (cid:88) i ρ i ( (cid:126)v i − (cid:126)v CM ) · ∆ (cid:126)A i . (6)where (cid:126)v i is the gas cell velocity and (cid:126)A i is the surface el-ement crossed by the gas in a direction parallel to theradial vector (cid:126)r i − (cid:126)r CM , with (cid:126)r i the gas cell position. Thesum is computed inside an annulus of width ∆ x min for r ≤
200 pc, corresponding to the level of refinement 17and 32 ∆ x min for r ≥
500 pc, corresponding to the re-finement level 12. Figure 8 shows a clear correlation between peaks of gasmass accretion rate on scales (cid:38) . ∼ M (cid:12) / yr. A few Myr before thesecond pericenter pass mass accretion rates reach ∼ few5 − M (cid:12) / yr at large scales. Such episodes of inflowingmass on large scales are consistent with enhancement ofSF in pericenters as shown in figure 3. The causal rela-tion between these two phenomena can be seen by com-paring the SFR and the mass accretion rate: the closepasses produce mass inflows which are followed after afew Myr by bursts of SF.At small scales (below ∼
100 pc), the enhancementin mass accretion rate at the first pericenter is not assignificant as it is in large scales, except for very shortbursts of inflow at 0 . − . ∼ M (cid:12) / yr in galaxy 1 in the nucleus at 10 pcin this brief burst (which happens at distances below theorder of the BH sphere of influence), but rates are gen-erally around the ∼ M (cid:12) / yr value, and are sustained ina somewhat irregular fashion before the first encounter.Galaxy 2 shows a slightly more consistent mass accre-tion rate in the same period of time at similar scales, butrates are not perceivably higher. The second and thirdpericenter passes show an enhancement in mass accretionat small scales. the amount of inflowing mass is able totrigger (after few Myr) SF bursts and feed the BHs ashas been shown in the previous sections. In particular, atthe third pericenter pass the gas inflow rate approaches ∼ M (cid:12) / yr due to the gas bulges coalescence.We conclude that there is a correlation between peaksof gas mass accretion rate, SFR, and BHAR associatedwith pericenter passages. In other words, close galacticencounters trigger mass inflows crossing the BH influ-ence radius, producing SF bursts and lighting up AGNactivity in galactic centers.3.4. Torques on the gas
At this point we have shown that throughout themerger process there are episodes of efficient gas inflowstoward the galaxy centers. In order to fully understandthe origin of mass transport into the galactic center it isnecessary to quantify the torques acting on the gas, in or-der to link mass inflow episodes with angular momentumloses (see appendix C).Figure 9 shows the torques acting on the galactic discat different radius as a function of time. Before comput-ing the torques, we have defined the galactic disc in thesame way we did it to compute the KS relation and thegas mass transport.We have computed the torques with respect the stellarcenter of mass (cid:126)r CM as a proxy for the rotational center ofeach spiral galaxy (see appendix A for a discussion aboutrotational centers). In order to do that, it is necessary toset a non-rotating coordinate system free falling with thestars. In such a frame, the acceleration of a particle be-comes (cid:126)a (cid:48) i = (cid:126)a i − (cid:126)a CM , where (cid:126)a i is the particle accelerationwith respect an inertial reference frame (the center of thefixed simulation box in our case) and (cid:126)a CM is the acceler-ation of the stellar center of mass with respect the sameinertial frame (see appendix B). Then, in the co-movingigh resolution BH fueling in a merger system 9 Fig. 9.—
Left: Total gravitational torque on the gas associated to inward mass transport at different distances from the stellar centerof mass: 2 kpc in black, 1 kpc in violet, 0.5 kpc in blue, 0.1 kpc in red, 0.05 kpc in orange and 0.01 kpc in green. The solid vertical graylines mark the pericenters of the orbit. Right: Same as left column but for the hydrodynamic torque. The figure shows that pericentersare associated with increases in torques. reference frame the torques can be computed as (cid:126)τ (cid:48) = (cid:88) i m i ( (cid:126)r i − (cid:126)r CM ) × ( (cid:126)a i − (cid:126)a CM ) . (7)In the previous expression (cid:126)r i is the cell position, m i is thegas cell mass. The acceleration (cid:126)a i is the combination ofthe gravitational acceleration −∇ φ i and the accelerationassociated to hydrodynamic on the gas ∇ P i /ρ i , where φ i is the gravitational potential at a given cell and P i is thepressure in the same cell.Because the galactic disc is defined in terms of the discangular momentum, negative torques imply a loss of an-gular momentum and a resulting inward mass transport.Figure 9 shows − (cid:126)τ (cid:48) , i.e. torques producing net inwardmass transport inside an annulus at a given distance fromthe galactic centers (the regions without data are dom-inated by outward mass transport torques). The sumis computed inside an annulus of width ∆ x min for r ≤
200 pc and 32 ∆ x min for r ≥
500 pc, as in the gas massaccretion rate computation.The left column of figure 9 shows that at large scales( (cid:38) . ∼ few Myr after the first passage. On the other hand, ingalaxy 2 no gravitational torque peak can be identifiedat large scales.In galaxy 2 the late pericenter passages are associ-ated with increased gravitational torques on larger scales(mainly at 0.5 kpc scales for the second passage). Ingalaxy 1 it is more difficult to recognize a torque increase(only having measured one relevant torque spike at 1 kpcbetween passages). We also see some torque presence at0.5 kpc after the merger of the systems.Gravitational torques acting on the galactic central re-gion, i.e. less than 100 pc from the stellar center of mass,show a clear enhancement associated with the secondand third pericenter passes but an almost imperceptiblechange during the first pericenter pass for both galaxies.Figure 9 shows that the inner galactic region, besides avery short increase of torque at 10 pc in galaxy 1 af-ter the first passage, feels the maximum gravitationaltorque around the third pericenter pass (with a very bigspike at the smallest scales for galaxy 2 when the systemsare about to merge). Such strong torques acting on thegalactic gas produces gas inflows and feeds the centralmassive objects, lighting up the AGN. It is also of note,that gravitational torques feature most importantly, at100 pc scales, which aligns with how the BH influencesphere helps with gas transport at this distance.The right column of figure 9 shows the hydrodynamic0 Prieto et al.torques associated with inward mass transport. At largespatial scales hydrodynamic torques are lower and moresporadic than gravitational torques. This hydrodynami-cal torque values become closer to gravitational ones atpericenter passages, where especially for galaxy 2, we sefeatures at every passage.Within the galactic nuclei the hydrodynamic torquesmatch quite well with gravitational torques at the small-est scales, showing peaks in the same places where theircounterparts do. Such enhancements are at the samelevel as the gravitational torques showing that bothmechanisms are working to redistribute matter in thelater stages of the merger. In other words, hydrody-namic torques work in tandem with gravitational torquein order to redistribute mass and angular momentum inthe galactic disc. DISCUSSION AND CONCLUSIONSWith the aim of studying the connection betweentorques and mass transport in galactic discs, we havesimulated a galaxy merger employing realistic initial con-ditions based in Privon et al. (2013). The SFR reachesvalues of ∼ − M (cid:12) / yr, below the observational mea-surements from (Evans et al. 2008; Howell et al. 2010)for NGC 2623 specifically, but closer to the values pre-sented in Cortijo-Ferrero et al. (2017), this puts our sys-tem below the star forming capabilities of a starburstingsystem, but inside expected rates for generic merger sys-tems (Pearson et al. 2019). The final BH mass of thesystem is M BH ≈ . × M (cid:12) , around one or two orderof magnitudes below the usual values presented in Haanet al. (2011) and below the dispersion of the “M-sigma”relation (G¨ultekin et al. 2009). This low BH mass is dueto low amounts of accretion stemming from the effective-ness feedback has at heating the immediate environmentaround our sink particles, calling for an improvement ofthe feedback model at our resolution, one option beingthe inclusion of a fully coupled radiation hydrodynamicalfeedback (see Bieri et al. (2017)).Our results confirm that galactic encounters can trig-ger bursts of SF (e.g. Barnes & Hernquist 1991; Mihos &Hernquist 1996; Springel et al. 2005; Gabor et al. 2016).The first pericenter pass clearly increase the SF of bothgalaxies but those increases are more evident beyond ∼
500 pc from the galactic center, when it reaches ∼ few M (cid:12) / yr. At these higher scales the SFR enhancementis due to the gas density increase triggered by the colli-sion of the gaseous galactic spiral arms. Because the firstpericenter pass has a nuclear separation of ∼ ∼ few M (cid:12) / yr. At this stage the gas density hasincreased due to mass transport, resulting in a prominentnuclear SF burst.Besides the SFR, the BHAR peaks also show corre-lations with pericenter encounters. Whereas one of theBHs has a growth rate correlated with its three pericen-ter passess the other one correlates better with its secondand third pericenter passes. In both cases it is evidentthat the second and third pericenter passes increase the BHAR, reaching values of ∼ − ∼
25% of thecorresponding Eddington limits for the BHs (correspond-ing to a few ∼ − M (cid:12) / yr and a few ∼ − M (cid:12) / yr).Such high mass accretion rate onto the compact objectswill trigger the AGN activity.Both phenomena described above, i.e., star formationactivity and BH accretion, are driven by the amount ofgas available to form stars and to feed the BHs. Oursimulation shows that pericenter passes correlate withpeaks of gas mass accretion rates driving gas mass den-sity variations in the BH vicinity, i.e. inside its influenceradius. The first encounter produces a direct mass in-flow of ∼ M (cid:12) / yr outside of ∼
500 pc, associated withthe galaxy-galaxy crossing. This encounter triggers ∼ kpc scale SF in both galaxies. On the other hand, atsmaller scales ( r (cid:46)
100 pc) the first pericenter producesa big increase in the mass accretion rate for one of thegalaxies (reaching a short peak of ∼ M (cid:12) / yr), and asmaller increase for the second one, but still enough toproduce SF and to feed one of the BHs. The second andthird pericenter passes produce a clear enhancement inmass accretion rate onto the nuclear galactic region. Infact, at the third closest passage the gas mass inflow atinner scales is simultaneously high for both systems andas such, the galactic gas entering the BH sphere of in-fluence efficiently feeds the BHs and triggers nuclear SFbursts.Neglecting magnetic fields and viscosity, any variationon the gas angular momentum will be due to torques fromboth gravitational and pressure gradient forces (see ap-pendix C). In other words, the merger triggers changesin the gas angular momentum due to variations in thegravitational potential and gas pressure. The former areproduced due to the dynamics of the merger which ischaracterized by strong gravitational interactions, andthe latter is produced by gas layers with strong differ-ences in density and/or temperature. Such conditionsnaturally arise when both galaxies cross each other andfinally merge. We have shown that pericenter passes cor-relate with both gravitational and hydrodynamic torquepeaks. In general, gravitational torques dominate overhydrodynamic torques but at inner scales pressure gra-dient torques can reach values approaching that of thegravitational ones helping to radially transport gas ingalactic disc. These torques redistribute angular mo-mentum allowing inward mass transport onto the galac-tic center. The high resolution of our simulation showedthat such gas inflows can cross the BH influence radiusproducing peaks in the BHAR and triggering SF burst.ACKNOWLEDGEMENTSPowered@NLHPC: This research was partially sup-ported by the supercomputing infrastructure of theNLHPC (ECM-02). The Geryon cluster at the Centrode AstroIngenieria UC was extensively used for the anal-ysis calculations performed in this paper. JP is fundedby ESO-Chile Comite Mixto grant ORP 79/16. AE ac-knowledges partial support from the Center for Astro-physics and Associated Technologies CATA (PFB06) andProyecto Regular Fondecyt (grant 1181663). G.C.P. ac-knowledges support from the University of Florida. REFERENCESAngulo, R. E., Springel, V., White, S. D. M., Jenkins, A., Baugh,C. M., & Frenk, C. S. 2012, MNRAS, 426, 2046 Armus, L., et al. 2009, PASP, 121, 559 igh resolution BH fueling in a merger system 11
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APPENDIX: ROTATION CENTERThe rotational center of a system composed by particles of mass m i at position (cid:126)r i and acceleration (cid:126)a i , and where theamount of particles well-represent the phase space near such center, can be defined as the point (cid:126)r rot where the torque (cid:126)τ rot = (cid:88) i m i ( (cid:126)r i − (cid:126)r rot ) × ( (cid:126)a i − (cid:126)a rot ) (8)inside a given volume is null, with (cid:126)a rot the rotational center acceleration. In a well approximated system whichis supported by ideal rotation, all the accelerations will point to a common center, the rotational center, then thecross product position-acceleration will be null. In systems with a given degree of turbulence and strong noise in itsacceleration field, such null point does not necessarily exist, here the task reduces to searching for minima in the torquefield to define our rotational center, which necessarily introduces degeneracy in its estimation.A kinematic approach to identify the rotational center of a system can be based in the previous dynamical definition.In this case, instead of focusing on the particle accelerations it is useful to look at the particle velocities (cid:126)v i . Then therotational center will be the point where the angular momentum (cid:126)L rot = (cid:88) i m i ( (cid:126)r i − (cid:126)r rot ) × ( (cid:126)v i − (cid:126)v rot ) (9)inside a given volume is maximized. Here (cid:126)v rot is the velocity of the rotational center. Note that in this case the crossproduct position-velocity should be a maximum. As with the dynamical definition, if the system has a given degree ofturbulence, it would be possible to find more than one center of rotation. We note that if our context was understoodas a generic dynamical system, our search criterion reduces to finding the best candidate fulfilling the characteristicsof a non-stationary irrational vortex, where (cid:126)L rot is the local circulation field maxima.The process of identifying a rotational center is computationally expensive as it requires computing the angularmomentum (or torque) inside a given volume for each point in the space. Given the 3D map for the modulus ofthe angular momentum it is necessary to look for peaks in the angular momentum distribution. In other words it isnecessary to look for “clumps” of angular momentum. Given the “clumps” of angular momentum it is possible tocompute the centroid of such objects to define rotational centers. Thus, identifying the stellar center of mass given anansatz for the rotational center (the BH positions for instance) is faster, computationally. B. APPENDIX: NON-INERTIAL FRAMESInside an accelerating reference frame the Newtonian dynamical equations are modified. In such moving frame anobserver will describe the movement of any object as influenced by “fictitious forces”. Quantitatively, from a movingsystem with position (cid:126)R with respect an inertial reference frame the force described by an observer at (cid:126)R acting on aparticle at position (cid:126)r i is m i d (cid:126)r (cid:48) i d t = (cid:126)F i − m i d (cid:126)Rd t − m i (cid:126)ω × ( (cid:126)ω × (cid:126)r (cid:48) i ) − m i (cid:126)ω × (cid:126)v (cid:48) i − m i d (cid:126)ωd t , (10)where m i is the mass particle (cid:126)r (cid:48) i = (cid:126)r i − (cid:126)R is the particle position with respect the moving system position (cid:126)R and (cid:126)r i the particle position with respect an inertial reference frame. (cid:126)F i is the net force acting on the particle i (due to themagnetic, gravitational, viscous or hydrodynamic contribution), (cid:126)ω is the angular velocity of the moving system and (cid:126)v (cid:48) i = d(cid:126)r (cid:48) i /dt .In the simple case when (cid:126)ω = (cid:126)
0, i.e. a moving reference frame without rotation, with (cid:126)a (cid:48) i the particle accelerationwith respect the moving system, (cid:126)a i the particle acceleration with respect an inertial frame and (cid:126)A the moving systemacceleration with respect the same inertial reference frame it is possible to write (cid:126)a (cid:48) i = (cid:126)a i − (cid:126)A, (11) C. APPENDIX: TORQUES-MASS TRANSPORT RELATIONThe momentum conservation equation in its conservative form in Cartesian coordinates x i can be written as ∂ ( ρv k ) ∂t + ∂∂x l ( R kl + P kl + B kl − G kl − S kl ) = 0 , (12)where ρ is the gas mass density and v i is the Cartesian component of the gas velocity. R kl , P kl , B kl , G kl and S kl are the hydrodynamical stress, the pressure stress, the magnetic stress, the gravitational stress and the viscous stress,respectively. The stresses are defined by: R kl = ρv k v l , (13) P kl = δ kl P, (14) B kl = 14 π (cid:18) B k B l − B δ kl (cid:19) (15)igh resolution BH fueling in a merger system 13 G kl = 14 πG (cid:20) ∂φ∂x k ∂φ∂x l −
12 ( ∇ φ ) δ kl (cid:21) , (16) S kl = ρν (cid:18) ∂v k ∂x l + ∂v l ∂x k − δ kl ∇ · (cid:126)v (cid:19) , (17)where P is the gas pressure, B k the cartesian component of the magnetic field, B the modulus of the magnetic field, φ the gravitational potential, ν is the kinematic viscosity and δ kl is the Kronecker delta symbol.Neglecting the magnetic term and the dissipative-viscous term (Balbus 2003) the momentum conservation equationcan be written as ∂∂t ( ρv k ) + ∂∂x l ( ρv k v l ) + ∂P∂x k − ρ ∂φ∂x k = 0 , (18)and taking the cross product between the Cartesian position (cid:126)x and eq. 18 (applying (cid:15) ijk x j , with (cid:15) ijk the Levi-Civitasymbol) it is possible to derive the angular momentum conservation equation and after some algebra it is possible towrite the ˆ z component of this equation as ∂∂t ( ρ(cid:96) z ) = − (cid:2) (cid:96) z ρ ∇ · (cid:126)v + (cid:126)v · ∇ ( ρ (cid:96) z ) + τ Pz + τ Gz (cid:3) , (19)from where it is possible to get the gas mass density variation ∂ρ∂t = − (cid:96) z (cid:20) ρ ∂(cid:96) z ∂t + (cid:96) z ρ ∇ · (cid:126)v + (cid:126)v · ∇ ( ρ(cid:96) z ) + τ Pz + τ Gz (cid:21) , (20)where (cid:96) z = ( (cid:126)x × (cid:126)v ) · ˆ z is the z component of the gas specific angular momentum, τ Gz = ρ ( (cid:126)x × ∇ φ ) · ˆ z is the z componentof the gravitational torque and τ Pz = ( (cid:126)x × ∇ P ) · ˆ z is the z component of the hydrodynamic torque.Equation 20 relates the changes in gas density ρ with torques τ P,Gz acting on the gas. For a system starting froman axisymmetric stationary state with (cid:126)v = v ( r ) ˆ θ and ρ = ρ ( rr