Resolving flows around black holes: the impact of gas angular momentum
MMon. Not. R. Astron. Soc. , 000–000 (0000) Printed 11 October 2016 (MN L A TEX style file v2.2)
Resolving flows around black holes: the impact of gasangular momentum
Michael Curtis ? , Debora Sijacki Institute of Astronomy and Kavli Institute for Cosmology, University of Cambridge, Madingley Road, Cambridge CB
HA, UK
11 October 2016
ABSTRACT
Cosmological simulations almost invariably estimate the accretion of gas on tosupermassive black holes using a Bondi-Hoyle-like prescription. Doing so ignores theeffects of the angular momentum of the gas, which may prevent or significantly de-lay accreting material falling directly on to the black hole. We outline a black holeaccretion rate prescription using a modified Bondi-Hoyle formulation that takes intoaccount the angular momentum of the surrounding gas. Meaningful implementationof this modified Bondi-Hoyle formulation is only possible when the inner vorticitydistribution is well resolved, which we achieve through the use of a super-Lagrangianrefinement technique around black holes within our simulations. We then investigatethe effects on black hole growth by performing simulations of isolated as well as merg-ing disc galaxies using the moving-mesh code
AREPO . We find that the gas angularmomentum barrier can play an important role in limiting the growth of black holes,leading also to a several Gyr delay between the starburst and the quasar phase in ma-jor merger remnants. We stress, however, that the magnitude of this effect is highlysensitive to the thermodynamical state of the accreting gas and to the nature of theblack hole feedback present.
Key words: methods: numerical - black hole physics - cosmology: theory
Gas falling on to supermassive black holes that reside in thecentres of majority galaxies may have a significant amountof angular momentum with respect to the central black hole.The accreting gas will settle into orbits that are alignedwith a plane normal to the mean angular momentum ofthe infalling gas. Once the fluid has settled on to circularorbits accretion on to the central object will be inhibited bythe centrifugal force and, as such, a disc like structure willform, with the subsequent evolution of the system governedby, amongst others, the viscous processes that are present inthe disc (Pringle 1981; Frank et al. 2002). The viscosity ofthe fluid causes inner packets of fluid to move inwards and,in doing so, they transfer angular momentum to the outerparts of the disc. This process causes most of the mass tospiral into the central object, whilst transporting angularmomentum outwards.One key problem with the predictive power of accretiondisc theory is understanding the nature and relative impor-tance of the different sources of viscosity. If we assume thatthe source of the viscosity is molecular then we can esti- ? E-mail: [email protected] mate the ratio of the inertial to viscous forces, the Reynoldsnumber, as Re = u φ Rν , (1)where u φ is the rotational velocity of the fluid, ν ∼ c s λ isthe molecular viscosity of the fluid, c s is the sound speedof the gas and λ is the mean free path length. Putting intypical values for accretion discs yields a Reynolds number ∼ , which implies an accretion timescale larger than aHubble time. Clearly, to explain the accretion rates on toobserved active galactic nuclei (AGN), we need to invokedifferent viscosity mechanisms. These are likely to includemagnetic effects, especially at small spatial scales, such asthe magnetic rotational instability, which occurs in suffi-ciently ionized flows for which u φ decreases as a functionof radius (Balbus & Hawley 1991). In such cases, the effectis to negatively torque inner fluid elements, causing themto fall on to closer orbits, and vice versa. This motion isunstable and leads to a magnetic viscosity. Another sourceof viscosity may be caused by the turbulent motion of thedisc, where the random flow and eddies of the turbulencelead to a viscosity analogous to that driven by random flowon the molecular level. The nature of such turbulence and c (cid:13) a r X i v : . [ a s t r o - ph . GA ] O c t Curtis & Sijacki at what point in the flow it sets in remains, however, a keyuncertainty.As there are still fundamental gaps in our understand-ing of gas angular momentum transport, we are limited inour ability to successfully model black hole accretion discs.Hence, much modern theory follows Shakura & Sunyaev(1973) in parameterizing the viscosity as ν = αc s H , where H is the disc scaleheight and 0 < α (cid:46) . ∼ kpc scales, theexact magnitude of these processes is affected by star for-mation efficiency, as well as by the nature of stellar andblack hole feedback processes (e.g. Springel et al. 2005; vande Voort et al. 2011; Dubois et al. 2013; Nelson et al. 2015).On sub-kpc scales, an array of physical mechanisms havebeen invoked to explain the successive gas angular momen-tum transport. These include global and local bar and spi-ral instabilities (Roberts et al. 1979), with the possibilityof a “bars-within-bars” mechanism (Shlosman et al. 1989),clumpy, turbulent discs (King & Pringle 2006; Krumholzet al. 2006; Hobbs et al. 2011) or discs supported by radia-tion pressure (Thompson et al. 2005). While the feasibility ofsome of these mechanisms has been studied by means of highresolution cosmological or isolated galaxy simulations (e.g.Mayer et al. 2007; Levine et al. 2008; Hopkins & Quataert2010; Emsellem et al. 2015), gas angular momentum trans-fer within self-gravitating discs remains an unsolved issue(Goodman 2003).It is thus unsurprising that currently, most cosmologicalsimulations estimate the growth of black holes by use of aBondi-Hoyle-Lyttleton approach (Hoyle & Lyttleton 1939;Bondi & Hoyle 1944; Bondi 1952), adopting the formula˙ M = 4 π G M ρ ∞ ( c ∞ + v ∞ ) / , (2)where M BH is the mass of the black hole, c ∞ and ρ ∞ arethe sound speed and gas density at infinity and v ∞ is therelative velocity between the black hole and the gas at in-finity. Clearly, this does not take into account the effect ofgas angular momentum. Doing so is difficult - in galaxy for-mation simulations we cannot directly resolve the relevantscales required to understand the accretion flows in regionsclose to the black hole (i.e. in the sub-kpc or sub-parsecregime) and, as discussed above, even if we could we arefundamentally limited by our understanding of the physicalprocesses on these scales. What is required to improve onthe Bondi-Hoyle-Lyttleton approach in cosmological simu- lations is a simple theoretical framework that links how theaccretion rate on small scales may be affected by the meanangular momentum of the in-falling gas on large scales.There have been several attempts to account for theeffects of angular momentum on the black hole accretionrate within galaxy formation simulation. Power et al. (2011)use an accretion-disc particle technique, whereby gas parti-cles of sufficiently low angular momentum that their orbitswould cross a specified accretion radius are added to a par-ticle that accounts for the unresolved inner accretion disc.The black hole then accretes matter from this disc on someviscous timescale. Rosas-Guevara et al. (2015) make simi-lar assumptions, and calculate the net tangential velocity ofaccreting material and compare this to the sound speed ofthe gas. For sufficiently high gas angular momentum, thetraditional Bondi rate is suppressed.In this paper we adopt an alternative method, basedon the work of Krumholz et al. (2005) (for previous relatedworks see also Abramowicz & Zurek (1981); Proga & Begel-man (2003)) who parameterise the accretion rate on to acentral object in terms of the vorticity of the gas on largescales. The effect of increased vorticity is to suppress theaccretion rate from the standard Bondi rate, as gas witha larger impact parameter than the accretion radius of theblack hole circularises before it can cross the horizon andforms a disc. In the limit of no vorticity, the prescriptionreduces to the spherically symmetric Bondi accretion case.We implement this vorticity-based accretion rate pre-scription in the moving mesh code AREPO . After validatingour implementation on a number of stringent tests, we inves-tigate how this affect the accretion rate on to supermassiveblack holes in simulations of isolated galaxies, as well as ofsome simple galaxy binary mergers. In doing so, we makeuse of the refinement technique as described by Curtis & Si-jacki (2015), which allows us to improve the resolution of thefluid flow around black holes in our simulations. Accuratelyresolving the velocity structure of the gas in this region isvital for successfully implementing the vorticity prescriptionand we show how failing to do so leads to an underestimateof the vorticity. Whilst this work focuses on galaxy-scalesimulations, we intend to present a method for use in fullycosmological simulations.This paper is organized as follows. In Section 2 we out-line our methodology and the exact nature of our new im-plementation. In Section 3 we discuss analytical expecta-tions of the vorticity-based accretion rate prescription onthe dynamical models of black hole growth. We then presentseveral numerical experiments in Section 4 to validate ournumerical implementation, before showing the main resultsof our work in Section 5 where both isolated and merginggalaxy simulations are discussed. Finally, Section 6 presentsour conclusions and plans for future work.
We use the moving-mesh code
AREPO code (Springel 2010)for all simulations in this paper.
AREPO employs the
TreePM approach (Springel et al. 2005) for handling gravitational c (cid:13) , 000–000 esolving flows around black holes: the impact of gas angular momentum interactions, and dark matter is represented by a collision-less fluid of massive particles. The fluid is modelled using aquasi-Lagrangian finite volume technique whereby the gas isdiscretized by tessellating the computational domain with aVoronoi mesh.Star formation is carried out using the model of Springel& Hernquist (2003). We stochastically form star parti-cles from cells of gas above a density threshold of ρ sfr =0 . − , with a characteristic time-scale of t sfr = 1 . q EOS = 1 in the notation ofSpringel & Hernquist (2003)), as well as a softer equation ofstate with q EOS = 0 . q EOS = 1and an isothermal medium of temperature 10 K, and whichcorresponds to a less pressurized and more clumpy medium.In addition to this, in some simulations we include metal linecooling as in Vogelsberger et al. (2013), where we assumethat the gas has a solar abundance, since we are primarilyinterested in the qualitative range of any subsequent effects.The variations of the ISM model we consider here are aimedat addressing the possible impact of the gas thermodynami-cal state on the subsequent nature of the accretion flow andhence black hole growth.
In order to improve the resolution of the simulation in theregion of black hole particles we employ the black hole re-finement scheme of Curtis & Sijacki (2015). This makes useof the ability of
AREPO to split and merge cells based onarbitrary criterion by forcing cells to decrease linearly in ra-dius as they approach the black hole. This means that forcells a distance d from a black hole particle have a cell radius R , where dR ref ( R cellmax − R cellmin ) c + R cellmin c < R , (3) dR ref ( R cellmax − R cellmin ) + R cellmin > R . (4)Here R ref is the scale over which refinement occurs, whichwe typically set to be equal to the black hole smoothinglength, c is a constant that we set to 2, whilst R cellmin and R cellmax determine the aggression of the refinement. We typically setthese to be equal to the Bondi radius and the black holesmoothing length, respectively. We adopt the model of Curtis & Sijacki (2015), based uponDi Matteo et al. (2005); Springel et al. (2005) to follow theevolution of black holes. We compare our vorticity-basedprescription outlined below to that of the standard Bondi-like formula ˙ M Bondi = 4 π G M ρ ∞ c ∞ , (5) where we do not take into account the relative velocity termin this work. Moreover in all simulations we limit the accre-tion on to the black hole to the Eddington limit˙ M Edd ≡ π GM BH m p (cid:15) r σ T c , (6)where σ T is the Thomson cross section, m p is the mass of aproton and (cid:15) r = 0 . (cid:15) f of the luminosity to the gas, leading toan energy budget of∆ E feed = (cid:15) f (cid:15) r ∆ M BH c . (7)Here, (cid:15) f = 0 .
05 is the feedback efficiency, which is set toreproduce the normalization of the locally inferred M BH − σ relation (Di Matteo et al. 2005; Sijacki et al. 2007).We allow for feedback to occur both isotropically (purethermal coupling) and in a bipolar manner. In the lattercase, we inject mass, energy and momentum within a coneof opening angle θ out = π / The benefits of the refinement method mean that in princi-ple it is possible to account for the angular momentum ofthe gas in different ways (Power et al. 2011; Rosas-Guevaraet al. 2015). In this paper we implement the unified modelof Krumholz et al. (2005), which is very well suited to ourcomputational approach. For completeness, we briefly sum-marize the method here. The authors consider an accretingpoint particle of mass M within a velocity field of finite vor-ticity at infinity, which they parameterize as v ∞ = ω ? c ∞ yr B ˆ x , (8)where r B is the Bondi radius, c ∞ is the sound speed ofthe gas at infinity, and ω ? is a dimensionless ‘vorticityparameter’, so called because the vorticity of the flow is − ω ? c ∞ /r B ˆ z , where we have assumed a Cartesian coordi-nate system centred on the black hole. The task is then toanalyse the accretion as a function of ω ? . For ω ? = 1, gaswith impact parameter equal to the Bondi radius is trav-elling at the Keplerian velocity. This suggests that there isa transition region around ω ? = 1, which leads Krumholzet al. (2005) to consider three regimes of accretion:(i) Very small ω ? , where original Bondi accretion holds.(ii) Small ω ? (iii) Large ω ? c (cid:13)000
05 is the feedback efficiency, which is set toreproduce the normalization of the locally inferred M BH − σ relation (Di Matteo et al. 2005; Sijacki et al. 2007).We allow for feedback to occur both isotropically (purethermal coupling) and in a bipolar manner. In the lattercase, we inject mass, energy and momentum within a coneof opening angle θ out = π / The benefits of the refinement method mean that in princi-ple it is possible to account for the angular momentum ofthe gas in different ways (Power et al. 2011; Rosas-Guevaraet al. 2015). In this paper we implement the unified modelof Krumholz et al. (2005), which is very well suited to ourcomputational approach. For completeness, we briefly sum-marize the method here. The authors consider an accretingpoint particle of mass M within a velocity field of finite vor-ticity at infinity, which they parameterize as v ∞ = ω ? c ∞ yr B ˆ x , (8)where r B is the Bondi radius, c ∞ is the sound speed ofthe gas at infinity, and ω ? is a dimensionless ‘vorticityparameter’, so called because the vorticity of the flow is − ω ? c ∞ /r B ˆ z , where we have assumed a Cartesian coordi-nate system centred on the black hole. The task is then toanalyse the accretion as a function of ω ? . For ω ? = 1, gaswith impact parameter equal to the Bondi radius is trav-elling at the Keplerian velocity. This suggests that there isa transition region around ω ? = 1, which leads Krumholzet al. (2005) to consider three regimes of accretion:(i) Very small ω ? , where original Bondi accretion holds.(ii) Small ω ? (iii) Large ω ? c (cid:13)000 , 000–000 Curtis & Sijacki ω ? As the work attempts to buildon the Bondi formula, it is useful to consider first the cut-off for which the angular momentum is too small to havean effect. Krumholz et al. (2005) argue that if the gas onlycircularizes at a radius that is less than that of the accret-ing object (in this case the Schwarzschild radius, r s , as weconsider non-spinning black holes) then the material will beaccreted on mostly radial orbits. This implies a conditionfor when the original Bondi rate is appropriate r circ = ω ? y / r < r s , (9)which implies ω ? < ω crit ≡ p r s / r B , (10)as the condition for the original Bondi prescription to hold. ω ? (cid:28) , ω ? > ω crit Above the critical vor-ticity derived in the previous section, the gas will circularisebefore it reaches the accretor. In these cases, the analysis ofKrumholz et al. (2005) suggests that the resulting torus thatforms will extend to r B with a scale height of ∼ r B , thusblocking accretion in the range π / < θ < π /
4, i.e. in theplane of the gaseous disc. This has the effect of decreasingthe accretion rate to ˙ M ∼ . M Bondi . ω ? (cid:62) A where the condition is satisfied, that is˙ M ≈ Z A ρ ∞ p v ∞ ( y, z ) + c ∞ d y d z , (11)which, when calculated, gives˙ M ( ω ? ) ≈ π r ρ ∞ c ∞ f ( ω ? ) , (12)where f ( ω ? ) is a numerical factor that is approximated inthe limit ω ? → ∞ by f ( ω ? ) ≈ π ω ? ln(16 ω ? ) . (13)We implement this scheme into the AREPO code. Wecalculate the vorticity parameter as ω ? = | ω | r B c ∞ , (14)by taking an exponential spline kernel-weighted (Monaghan& Lattanzio 1985) mean of the vorticities of the neighbour-ing cells to the black hole ? . We then calculate the accretion ? Note that we have explored different choices of how manyneighbouring gas cells to consider (e.g. within 0 .
1, 0 .
3, 0 . rate whenever ω ? > ω crit using the following expression˙ M Vort = 4 π G M ρ ∞ c ∞ × (cid:26) . ω ? < . . · π ω ? ln(16 ω ? ) ω ? > . . (15)since the analytic expression is well fit by a piecewise ap-proximation for ω ? greater and less than 0 .
1. The factorof 0 .
34 accounts for the same phenomenon detailed in Sec-tion 2.2.1.2, selected to match the value of 0 . M Bondi for ω crit < ω ? (cid:28)
1. It differs from exactly 0 . In order to understand the implications of our implementa-tion, we have carried out a series of simulations of an isolateddisc galaxy. This simulation setup is particularly well suitedto our topic of inquiry as, on large scales, the majority ofgas will be rotationally supported and thus the effect of gasangular momentum on the black hole accretion rate may besignificant. In fact, the observed luminosities of local AGNharboured within spiral galaxies are either rather low ormodest, and in the latter case often fall within the categoryof Seyferts, suggesting limited fuelling rates possibly drivenby bar instabilities (see e.g. Knapen et al. 2000; Laine et al.2002; Kewley et al. 2006).Our initial conditions match those of Curtis & Sijacki(2015) (for original implementation of the model and fur-ther details see Springel et al. (2005)). These consist of anisolated galactic disc in hydrostatic equilibrium, situatedwithin a dark matter halo described by a Hernquist (1990)density profile ρ dm = M dm a π r ( r + a ) , (16)where M dm is the total dark matter mass and a is a scalingparameter. We generate a self-consistent gaseous disc distri-bution in equilibrium and potential by an iterative process,based on an exponential surface density profile, with theazimuthal velocity set to v φ, gas = R (cid:18) ∂ Φ ∂R + 1 ρ gas ∂P∂R (cid:19) , (17)and the remaining velocity components v R = v z = 0. Wealso include a centrifugally supported stellar disc that alsofollows an exponential surface density profile. Our param-eters are chosen to represent a simplified Milky Way-likegalaxy, but note that we do not attempt to explicitly modelSagittarius A* in any way.In the last part of this paper we contrast our isolateddisc simulations with simulations of binary galaxy mergers.We consider only equal mass mergers on prograde parabolicorbits and initial galaxy properties that are identical to ourisolated galaxy setup. These simulations serve to explorethe difference in black hole growth reduction due to thevorticity-based accretion rate prescription in the case wherelarge scale gravitational torques affect the central gas den-sity and angular momentum distribution significantly.In Table 1 we list all simulations of isolated and merginggalaxies used in this work, where we summarize the initialnumber of gas cells, type of refinement method, black hole c (cid:13) , 000–000 esolving flows around black holes: the impact of gas angular momentum accretion and feedback prescriptions and any other varia-tions in the models, which relate to black hole seed mass,type of radiative cooling and the effective equation of statefor the ISM. The black hole accretion rate, for both the vorticity andthe standard Bondi rate prescription, is highly sensitive tothe gas sound speed, gas density and the black hole mass.Probing the full parameter space with numerical simulationsis non-viable, both because of the non linear nature of theevolution but also because of the significant computationaltime to carry out an individual simulation. To tackle this,we here explore the effects of changing the gas and blackhole properties in the context of our isolated galaxy setup(see Section 2.3) and we subsequently explore part of theparameter space with fully self-consistent simulations.It should be noted here that in both observations andsimulations the exact nature of the gas close to the blackhole’s Bondi radius is poorly understood (Frank et al. 2002;Goodman 2003). While a number of recent studies constraingas properties close to the Bondi radius in some systems,such as Sagittarius A* (Yuan et al. 2003; Xu et al. 2006),M87 (Russell et al. 2015) or typically in LINERs and Seyfertsthrough mega-masers (for a review see Lo 2005), we do notyet have a consensus on the density, temperature and veloc-ity distribution of the gas. Thus for the following analysis, wemake a very simplifying assumption and consider that thegas velocity field matches the initial (numerically derived)conditions down to the Bondi radius (see equation 17), al-though we also investigate the effect of assuming a field witha higher vorticity. The vorticity profile scales approximatelyas ω ∼ ( R . ) ˆz with radius. In Figure 1, coloured 2D histograms show the ratio of thevorticity-suppressed black hole accretion rate to the stan-dard Bondi rate as a function of gas density and soundspeed, for two fixed black hole masses of 3 . × M (cid:12) (ourdefault seed mass) and 10 M (cid:12) , in the left-hand and right-hand panels, respectively. Here, we have included the ef-fect of limiting the black hole growth by capping the ac-cretion rate at the Eddington limit, hence the suppressionis min( ˙ M Vort , ˙ M Edd ) / min( ˙ M Bondi , ˙ M Edd ). This means thatfor sufficiently low sound speeds and high densities, the ef-fective suppression is zero because the growth is Eddingtonlimited, rather than being constrained by the angular mo-mentum barrier (see yellow regions on the panels). At theother end of the spectrum, we note that for the majority ofthe parameter space the suppression ratio is at most 0 . ω ? , where: ω ? ∝ | ω | c s . (18)Whilst some of the viscous processes that transport angularmomentum might be expected to increase with increasedthermal motions of the gas, we are fundamentally limited byof our treatment of the ISM. Improving our modelling of thegas physics to self consistently resolve a realistic multiphaseISM is the subject of much ongoing research - in the futureit will certainly be interesting to see how this will affect ourunderstanding of the gas properties around the black hole. Because of the non-linear nature of black hole growth withinsimulations, looking at the ratio of the vorticity-based accre-tion rate to the standard Bondi rate alone can be a mislead-ing guide to the results of simulations. This is principallybecause of two effects. Firstly, the Bondi rate itself can varydramatically with density and (in particular) sound speedof the ambient medium. As such, if the black hole is grow-ing either very fast or very slow, then the final black holemass will be dominated by the changes in the Bondi rate c (cid:13)000
1. It differs from exactly 0 . In order to understand the implications of our implementa-tion, we have carried out a series of simulations of an isolateddisc galaxy. This simulation setup is particularly well suitedto our topic of inquiry as, on large scales, the majority ofgas will be rotationally supported and thus the effect of gasangular momentum on the black hole accretion rate may besignificant. In fact, the observed luminosities of local AGNharboured within spiral galaxies are either rather low ormodest, and in the latter case often fall within the categoryof Seyferts, suggesting limited fuelling rates possibly drivenby bar instabilities (see e.g. Knapen et al. 2000; Laine et al.2002; Kewley et al. 2006).Our initial conditions match those of Curtis & Sijacki(2015) (for original implementation of the model and fur-ther details see Springel et al. (2005)). These consist of anisolated galactic disc in hydrostatic equilibrium, situatedwithin a dark matter halo described by a Hernquist (1990)density profile ρ dm = M dm a π r ( r + a ) , (16)where M dm is the total dark matter mass and a is a scalingparameter. We generate a self-consistent gaseous disc distri-bution in equilibrium and potential by an iterative process,based on an exponential surface density profile, with theazimuthal velocity set to v φ, gas = R (cid:18) ∂ Φ ∂R + 1 ρ gas ∂P∂R (cid:19) , (17)and the remaining velocity components v R = v z = 0. Wealso include a centrifugally supported stellar disc that alsofollows an exponential surface density profile. Our param-eters are chosen to represent a simplified Milky Way-likegalaxy, but note that we do not attempt to explicitly modelSagittarius A* in any way.In the last part of this paper we contrast our isolateddisc simulations with simulations of binary galaxy mergers.We consider only equal mass mergers on prograde parabolicorbits and initial galaxy properties that are identical to ourisolated galaxy setup. These simulations serve to explorethe difference in black hole growth reduction due to thevorticity-based accretion rate prescription in the case wherelarge scale gravitational torques affect the central gas den-sity and angular momentum distribution significantly.In Table 1 we list all simulations of isolated and merginggalaxies used in this work, where we summarize the initialnumber of gas cells, type of refinement method, black hole c (cid:13) , 000–000 esolving flows around black holes: the impact of gas angular momentum accretion and feedback prescriptions and any other varia-tions in the models, which relate to black hole seed mass,type of radiative cooling and the effective equation of statefor the ISM. The black hole accretion rate, for both the vorticity andthe standard Bondi rate prescription, is highly sensitive tothe gas sound speed, gas density and the black hole mass.Probing the full parameter space with numerical simulationsis non-viable, both because of the non linear nature of theevolution but also because of the significant computationaltime to carry out an individual simulation. To tackle this,we here explore the effects of changing the gas and blackhole properties in the context of our isolated galaxy setup(see Section 2.3) and we subsequently explore part of theparameter space with fully self-consistent simulations.It should be noted here that in both observations andsimulations the exact nature of the gas close to the blackhole’s Bondi radius is poorly understood (Frank et al. 2002;Goodman 2003). While a number of recent studies constraingas properties close to the Bondi radius in some systems,such as Sagittarius A* (Yuan et al. 2003; Xu et al. 2006),M87 (Russell et al. 2015) or typically in LINERs and Seyfertsthrough mega-masers (for a review see Lo 2005), we do notyet have a consensus on the density, temperature and veloc-ity distribution of the gas. Thus for the following analysis, wemake a very simplifying assumption and consider that thegas velocity field matches the initial (numerically derived)conditions down to the Bondi radius (see equation 17), al-though we also investigate the effect of assuming a field witha higher vorticity. The vorticity profile scales approximatelyas ω ∼ ( R . ) ˆz with radius. In Figure 1, coloured 2D histograms show the ratio of thevorticity-suppressed black hole accretion rate to the stan-dard Bondi rate as a function of gas density and soundspeed, for two fixed black hole masses of 3 . × M (cid:12) (ourdefault seed mass) and 10 M (cid:12) , in the left-hand and right-hand panels, respectively. Here, we have included the ef-fect of limiting the black hole growth by capping the ac-cretion rate at the Eddington limit, hence the suppressionis min( ˙ M Vort , ˙ M Edd ) / min( ˙ M Bondi , ˙ M Edd ). This means thatfor sufficiently low sound speeds and high densities, the ef-fective suppression is zero because the growth is Eddingtonlimited, rather than being constrained by the angular mo-mentum barrier (see yellow regions on the panels). At theother end of the spectrum, we note that for the majority ofthe parameter space the suppression ratio is at most 0 . ω ? , where: ω ? ∝ | ω | c s . (18)Whilst some of the viscous processes that transport angularmomentum might be expected to increase with increasedthermal motions of the gas, we are fundamentally limited byof our treatment of the ISM. Improving our modelling of thegas physics to self consistently resolve a realistic multiphaseISM is the subject of much ongoing research - in the futureit will certainly be interesting to see how this will affect ourunderstanding of the gas properties around the black hole. Because of the non-linear nature of black hole growth withinsimulations, looking at the ratio of the vorticity-based accre-tion rate to the standard Bondi rate alone can be a mislead-ing guide to the results of simulations. This is principallybecause of two effects. Firstly, the Bondi rate itself can varydramatically with density and (in particular) sound speedof the ambient medium. As such, if the black hole is grow-ing either very fast or very slow, then the final black holemass will be dominated by the changes in the Bondi rate c (cid:13)000 , 000–000 Curtis & Sijacki ρ [ M (cid:12) / kpc ] c s [ k m / s ] . . . . . . . . . . . ˙ M v o rt / ˙ M B o nd i ρ [ M (cid:12) / kpc ] c s [ k m / s ] No FeedbackNo Feedback, No metalsNo Feedback + sEOSNo Feedback, M = 10 M (cid:12) Thermal, No metalsBipolar FeedbackMerger . . . . . . . . . . . ˙ M v o rt / ˙ M B o nd i Figure 1.
Coloured 2D histograms showing the ratio min( ˙ M Vort , ˙ M Edd ) / min( ˙ M Bondi , ˙ M Edd ) for two fixed black hole masses of 3 . × M (cid:12) (left-hand panel) and 10 M (cid:12) (right-hand panel) as a function of gas density and sound speed (see equation 15), assuming a givengas vorticity profile. Overplotted, we show the tracks from several of our simulation models, indicating the typical regimes simulatedblack holes are in (but note that simulated black hole masses grow with time along the shown tracks). In the bottom left region ofthe parameter space, for high densities and low sound speeds (coloured yellow), the Eddington rate is the dominant limitation of theaccretion rate, as the Bondi rate is very high. However, unless the sound speed is sufficiently small to allow for a large value of ω ? , thesuppressing effect of the vorticity is limited to 0.34. As such, the resulting suppression is sensitive to the presence and type of feedbackinjected, as well as the ISM physics. There is also a dependence on the mass of the central black hole, as this affects the Bondi radius.Higher mass black holes present larger regions of the parameter space with high suppression, but also a larger region in which the blackhole growth is Eddington limited. ρ [ M (cid:12) / kpc ] c s [ k m / s ] . . . . . . . . D e l a y [ G y r ] ρ [ M (cid:12) / kpc ] c s [ k m / s ] M = 3 . × M (cid:12) M = 10 M (cid:12) M = 5 × M (cid:12) M = 10 M (cid:12) . . . . . . . . . D e l a y [ G y r ] Figure 2.
2D coloured histograms showing the effective delay in Gyr for a 3 . × M (cid:12) black hole to increase in mass by an order ofmagnitude for our default gas vorticity profile (left-hand panel) and assuming a vorticity that is an order of magnitude larger than ourinitial conditions (right-hand panel). We plot the peak of this distribution for a series of different starting black hole masses with lines ofdifferent styles, as indicated on the legend. At high sound speeds, and low densities, the black hole growth is insignificant. Conversely, forlarge densities and low sound speeds, the Bondi rate is sufficiently high that the black holes grow very fast, regardless of the suppressingeffect of the vorticity prescription. In between there is a clearly defined region of the parameter space (yellow to pink shades) where thesuppressing effect has a significant impact, causing delays in black hole growth comparable with the Hubble time. For higher startingblack hole masses lower densities and higher sound speeds lead to similar delays. In practice, whether this occurs in simulations dependson the evolution of the surrounding gas, which is highly dependent on both the feedback routine and the ISM model chosen.c (cid:13) , 000–000 esolving flows around black holes: the impact of gas angular momentum Type N gas Refinement Accretion Feedback AdditionsIsolated 10 No Bondi No -Isolated 10 Yes Bondi No -Isolated 10 No Bondi Thermal -Isolated 10 No Bondi Thermal -Isolated 10 No Bondi Thermal -Isolated 10 Yes Bondi Thermal -Isolated 10 Yes Bondi Thermal -Isolated 10 Yes Bondi Thermal -Isolated 10 Yes Bondi No Metal line coolingIsolated 10 Yes Vorticity No Metal line coolingIsolated 10 Yes Bondi No sEOSIsolated 10 Yes Vorticity No sEOSIsolated 10 Yes Bondi No M = 10 M (cid:12) Isolated 10 Yes Vorticity No M = 10 M (cid:12) Isolated 10 No Vorticity Thermal -Isolated 10 Yes Vorticity Thermal -Isolated 10 Yes Bondi Bipolar sEOSIsolated 10 Yes Vorticity Bipolar sEOSMerger 10 Yes Bondi No Metal line coolingMerger 10 Yes Vorticity No Metal line coolingMerger 10 Yes Bondi No Metal line coolingMerger 10 Yes Vorticity No Metal line cooling
Table 1.
Simulation details for isolated and merger galaxy models, as indicated in the first column. The second column lists the initialnumber of gas cells employed (note that due to super-Lagrangian refinement this number can increase significantly during the simulatedtime-span). The third column indicates for which models we use our super-Lagrangian refinement around black holes (Yes) and for whichwe only use the standard (de)/-refinement present in AREPO (No). The fourth and fifth columns indicate our adopted model for blackhole accretion and feedback, while in the sixth column we list any other variations considered, such as the metal line cooling in additionto the primordial cooling (which is always switched-on), a softer equation of state (sEOS) for the ISM and a larger black hole seed massof 10 M (cid:12) , instead of our default choice of 3 . × M (cid:12) . itself, and the black hole growing according to the vorticityprescription will undergo similar growth with a slight delay.Secondly, any early growth in black holes is compoundedat later times, because of the M dependence of the Bondirate. This is a problem that will be particularly acute insimulations without strong self-regulation, which is the casefor a variety of different models of feedback and, of course,for simulations without strong black hole feedback at all.For this reason, we have carried out a series of simplifiedtime integrations of black hole growth. We make several ba-sic assumptions: in particular we do not include the effects ofblack hole feedback and we assume that the distribution ofgas properties as a function of radius remains the same overtime. We start the black holes at an initial seed mass andat each time step we calculate the updated accretion ratebased on sampling the relevant gas properties at the Bondiradius (which grows with time), and continue the integra-tions for a maximum of 14 Gyr. We will see that whilst theresults from simulated galaxies differ from these predictionsin certain key ways, we are able to reproduce the generaleffects of using the vorticity prescription over a wide rangeof parameters.In the left-hand panel of Figure 2, 2D coloured his-togram shows the delay in Gyr for a 3 . × M (cid:12) blackhole to increase its mass by an order of magnitude, as afunction of gas density and sound speed (and keeping ourdefault choice of gas vorticity distribution). The distribu-tion is clearly peaked, with a well defined maximum delayregion, which we have over plotted for different values of the initial black hole mass. The parameter space for eachis split into separate regions: in the top left (white region),for high sound speeds and low densities, the black hole isunable to grow by any significant fraction. The opposite oc-curs for high densities and low sound speeds (blue region)- the black hole grows fast regardless of suppression. Thereis, however, a tight regime in which the delay in black holegrowth can be particularly significant (yellow-pink region)and comparable to the Hubble time.On the right-hand panel of Figure 2, we show an anal-ogous plot but for an increased gas vorticity, which is now10 times higher at all radii. For a large region of parame-ter space, the resulting delay in black hole growth remainssimilar or slightly larger. The region in which we expect themost significant suppression increases in size, as the magni-tude of the suppressing effect increases, but the region of theparameter space in which we expect significant delay is stillrelatively small. Finally, we note that for larger initial blackhole masses lower gas densities and higher sound speeds leadto a similar effective delay. Whilst all simulations that attempt to implement sub-gridprescriptions face difficulty in resolving the gas propertiesaround the black hole, accurately resolving a higher orderquantity such as the velocity field is especially difficult. Inthis section we detail our numerical experiments to vali- c (cid:13)000
Simulation details for isolated and merger galaxy models, as indicated in the first column. The second column lists the initialnumber of gas cells employed (note that due to super-Lagrangian refinement this number can increase significantly during the simulatedtime-span). The third column indicates for which models we use our super-Lagrangian refinement around black holes (Yes) and for whichwe only use the standard (de)/-refinement present in AREPO (No). The fourth and fifth columns indicate our adopted model for blackhole accretion and feedback, while in the sixth column we list any other variations considered, such as the metal line cooling in additionto the primordial cooling (which is always switched-on), a softer equation of state (sEOS) for the ISM and a larger black hole seed massof 10 M (cid:12) , instead of our default choice of 3 . × M (cid:12) . itself, and the black hole growing according to the vorticityprescription will undergo similar growth with a slight delay.Secondly, any early growth in black holes is compoundedat later times, because of the M dependence of the Bondirate. This is a problem that will be particularly acute insimulations without strong self-regulation, which is the casefor a variety of different models of feedback and, of course,for simulations without strong black hole feedback at all.For this reason, we have carried out a series of simplifiedtime integrations of black hole growth. We make several ba-sic assumptions: in particular we do not include the effects ofblack hole feedback and we assume that the distribution ofgas properties as a function of radius remains the same overtime. We start the black holes at an initial seed mass andat each time step we calculate the updated accretion ratebased on sampling the relevant gas properties at the Bondiradius (which grows with time), and continue the integra-tions for a maximum of 14 Gyr. We will see that whilst theresults from simulated galaxies differ from these predictionsin certain key ways, we are able to reproduce the generaleffects of using the vorticity prescription over a wide rangeof parameters.In the left-hand panel of Figure 2, 2D coloured his-togram shows the delay in Gyr for a 3 . × M (cid:12) blackhole to increase its mass by an order of magnitude, as afunction of gas density and sound speed (and keeping ourdefault choice of gas vorticity distribution). The distribu-tion is clearly peaked, with a well defined maximum delayregion, which we have over plotted for different values of the initial black hole mass. The parameter space for eachis split into separate regions: in the top left (white region),for high sound speeds and low densities, the black hole isunable to grow by any significant fraction. The opposite oc-curs for high densities and low sound speeds (blue region)- the black hole grows fast regardless of suppression. Thereis, however, a tight regime in which the delay in black holegrowth can be particularly significant (yellow-pink region)and comparable to the Hubble time.On the right-hand panel of Figure 2, we show an anal-ogous plot but for an increased gas vorticity, which is now10 times higher at all radii. For a large region of parame-ter space, the resulting delay in black hole growth remainssimilar or slightly larger. The region in which we expect themost significant suppression increases in size, as the magni-tude of the suppressing effect increases, but the region of theparameter space in which we expect significant delay is stillrelatively small. Finally, we note that for larger initial blackhole masses lower gas densities and higher sound speeds leadto a similar effective delay. Whilst all simulations that attempt to implement sub-gridprescriptions face difficulty in resolving the gas propertiesaround the black hole, accurately resolving a higher orderquantity such as the velocity field is especially difficult. Inthis section we detail our numerical experiments to vali- c (cid:13)000 , 000–000 Curtis & Sijacki date our implementation of the vorticity prescription within
AREPO , whilst indicating limitations with the current ap-proach. Whilst much of our discussion here is specific tothe Krumholz et al. (2005) model, we note that our in-vestigations raise points that are important for alternativeapproaches of angular momentum estimation in black holeaccretion modelling.
In order to test the conservation properties of our refinementscheme, we simulate a cold, pressure free Keplerian disc.This problem, which holds modern hydrodynamics codes toa particularly stringent standard, is one that has been usedto demonstrate the effects of systematic errors in angularmomentum conservation (Cullen & Dehnen 2010; Hopkins2015; Pakmor et al. 2016). In its latest version, which in par-ticular adopts improvements to the gradient estimation andtime integration, our moving mesh code
AREPO shows verygood conservation properties - for full details see Pakmoret al. (2016). Here, we are interested in the effects that oursuper-Lagrangian refinement scheme has on the gas angularmomentum within the refinement region.We adopt the same initial setup as Pakmor et al. (2016).In particular, we initialize a disc spanning a radial range of0 . < r < . ρ = 1 .
0, and gas velocities v x = − yr − / , v y = xr − / . Here, the units used are arbi-trary. Outside of this region, we set ρ = 10 − and v = 0.In both regions we set the internal energy to u = 2 . − γ/ρ ,where we set the adiabatic index γ = 5 /
3. Our simulationuses a constant external gravitational acceleration of g = − rr ( r + (cid:15) ) , (19)where (cid:15) = 0 .
25 for r < .
25 and is zero otherwise. Sincein this case there is no black hole particle, in the simula-tion with super-Lagrangian refinement we set the refine-ment region to be R ref = 1 . R cellmax = 10 − , which represents the size required fora smooth transition at the refinement boundary. We havetested different values for R cellmin - since the disc in thiscase does not extend to the centre of the simulation wefind that our results are relatively insensitive to the exactvalue. For the plots here we show our simulation for which R cellmin = 10 − .Figure 3 illustrates the results for simulations both with(right-hand panel) and without (left-hand panel) our super-Lagrangian refinement scheme. We show the density acrossthe disc at t = 150, and note the good agreement betweenthe two simulations, which both show that the disc has re-mained stable for the entire duration of the simulation. Inparticular, the simulation with refinement shows somewhatmore stable inner edge of the cold disc. The ratio of thetotal angular momentum of the disc to the initial theoret-ical value is plotted in the bottom row. Both simulationsshow very good conservation properties throughout, indi-cating that the cell refinement and de-refinement have notintroduced neither significant angular momentum transportnor conservation issues. This is reassuring as it demonstratesthat the estimation of gas angular momentum, and hencevorticity, does not suffer from spurious numerical artefactsdue to our super-Lagrangian refinement. The Krumholz et al. (2005) model assumes a large scaledisc-like vorticity and, thus for robustness of our scheme,we need to ensure not to take into account random turbu-lent vorticity that may be present in the simulations. This isespecially problematic for radial flows (i.e flows with negli-gible vorticity) close to the centre of the simulation domain,since the discrete nature of the velocity gradient calculationmeans that the small errors this introduces are amplified.As this is a function of both the limited resolution of thesimulations, as well as the (Cartesian) coordinate system,we note here that this problem would not be avoided by us-ing angular momentum (or proxies for this, such as circularvelocity of the gas) instead of vorticity - presence of minoranisotropies in the radial flow means that, in the central re-gion, it is inevitable that gas with very large velocities onsmall but finite impact parameters will be present. In prac-tice, in our simulations with more realistic initial conditions(such as of galaxies in isolated or cosmological settings),we find that this should rarely be an issue. Nevertheless,to avoid any spurious growth suppression, we ensure thatthe mean vorticity that we measure for the fluid is repre-sentative of a sufficiently large fraction of the gas mass. Todo this, we compare the net vorticity within black hole’ssmoothing length with that of individual cells within thesame region and calculate the fraction of the mass f sort ofthe gas that has a vorticity within π / f sort ∼ .
15. By examining the test simulations, we findthat f sort = 0 .
35 represents a conservative threshold whichminimizes the vorticity effects that are likely to be drivenpurely by noise. As such, for simulations with lower values of f sort we use the standard Bondi accretion rate and not ourvorticity prescription (for more details, see Appendix A).Figure 4 shows this phenomenon in practice. We showthe velocity (first row) and vorticity (second row) fields fortwo simulations, one of a purely radial Bondi flow (left col-umn) and one with a coherent disc structure (right column).We have scaled each simulation to the region of interest -that is, to a region that is comparable to the size of theblack hole smoothing length. In each case, for context, weshow a density slice across the z = 0 plane. For the Bondisimulation, the vorticity is essentially completely random,this being just the effect of the noise inherent in floatingpoint arithmetic. The length of the arrows indicating thevorticity field have been normalized to the same value: ingeneral, the size of these is amplified by the speed of thegas, which means that they are negligible except as the gasapproaches the origin, where they blow up in size. Here, be-cause of the steep velocity gradient, these errors are rapidlymagnified, resulting in a spurious signal that we need toavoid. In the case of the isolated galaxy, we can see therotating disc demonstrates a clear vorticity signal, alignedwith the z axis.Our simple prescription based on f sort works very well indetecting spurious vorticity and recovering the correct blackhole growth rate, as shown in bottom row of Figure 4. Onthe left, we plot both the ratio of instantaneous simulatedblack hole accretion rate to the analytical Bondi rate (top)as well as the ratio of the cumulative simulated to the cumu-lative analytical Bondi rate (bottom). Both of these ratios c (cid:13) , 000–000 esolving flows around black holes: the impact of gas angular momentum − − − − t . . . . . J z / J z , t h . . . . . . . . . ρ -2 -1 0 1 2-2-10120 20 40 60 80 100 120 140 t J z / J z , t h . . . . . . . . . ρ Figure 3.
In the left-hand panel we show the density map at t = 150 of a 2D simulation of a cold Keplerian disc for a simulation withno cell refinement or de-refinement (note that the units used are arbitrary). In the right-hand panel we show an analogous plot, but for asimulation using our black hole refinement scheme, and the region within which we impose super-Lagrangian refinement is marked witha dashed circle. In both cases, we also show the ratio of the total angular momentum in the simulation to the initial theoretical value(bottom row). The simulations show very good conservation properties for the entire simulated time-span. In particular, the aggressivecell refinement and de-refinement do not cause any significant differences to the angular momentum conservation of the code. should be 1. Vorticity estimation without our f sort prescrip-tion can lead to a significant accretion rate suppression, bymore than an order of magnitude, which is purely drivenby the numerical noise. Conversely, with our f sort prescrip-tion, whilst there are occasions when (by chance) the vor-ticity alignment causes the rate to dip for a single time step,this basically makes no difference to the overall growth, asdemonstrated by the bottom plot. In the right-hand bottompanel of Figure 4 we show the mass fraction of gas which isaligned to the net vorticity direction as a function of timefor isolated disc galaxy simulation with (dashed) and with-out black hole thermal feedback (the results are very similarin the case of the bipolar feedback). This confirms that spu-rious vorticity estimate is not an issue in these simulations,as for the vast majority of simulated time the mass fractionof gas aligned is much higher than f sort = 0 . All simulations that seek to account for the angular mo-mentum of the gas in the black hole accretion rate must beable to robustly estimate the velocity field in the region ofinterest. Thus to demonstrate the validity of our approach,Figure 5 shows radial profiles of the three velocity compo-nents (left-hand panel) and the corresponding radial vortic-ity profiles (right-hand panel) in simulations with thermalblack hole feedback with super-Lagrangian refinement andwithout. This shows a problem with simulations without re-finement - because the distribution is strongly peaked in thecentral region, the lack of resolution here leads to an un- derprediction of gas vorticity. We note here that this prob-lem is just a manifestation of the different ways in whichthe simulations are capturing the gas properties around theblack hole. Previous work (Curtis & Sijacki 2015, 2016) hasshown that the increased resolution in the central regionafforded by our refinement scheme can lead to significantdifferences in the capturing of the central feedback bubbleand the subsequent transport of the feedback energy awayfrom the immediate vicinity of the black hole.More specifically, feedback has two effects: a) it causeshot gas bubbles with low angular momentum to rise buoy-antly away from the black hole on vertical (and partly ra-dial) orbits and b) it increases the sound speed of the gas,increasing turbulence that will drive viscous processes thatmay be responsible in part for transporting angular momen-tum outwards and mass inwards. This results in the gas inthe central region having a very different velocity structure.For the cold, accreting gas, cells fall on to mostly radial or-bits, implying ω z ≈
0, resulting in the drop in vorticity forthe smallest radii. In addition, however, the strong blackhole-driven outflow means that v z has an R dependence inthe innermost region. This results in a strong contributionto the vorticity from the azimuthal ω φ term.The left-hand panel of Figure 6 shows the radial profilesof total gas vorticity for simulations with and without refine-ment at three different resolutions. The thick dot-dashed lineis a simple analytic estimate of the vorticity profile whichwe calculate as follows: we assume that the dominant com-ponent in the vorticity signal is aligned with the z axis ofthe disc and is dependent on the circular velocity, whichwe approximate as p GM ( < R ) /R . We then differentiate c (cid:13)000
0, resulting in the drop in vorticity forthe smallest radii. In addition, however, the strong blackhole-driven outflow means that v z has an R dependence inthe innermost region. This results in a strong contributionto the vorticity from the azimuthal ω φ term.The left-hand panel of Figure 6 shows the radial profilesof total gas vorticity for simulations with and without refine-ment at three different resolutions. The thick dot-dashed lineis a simple analytic estimate of the vorticity profile whichwe calculate as follows: we assume that the dominant com-ponent in the vorticity signal is aligned with the z axis ofthe disc and is dependent on the circular velocity, whichwe approximate as p GM ( < R ) /R . We then differentiate c (cid:13)000 , 000–000 Curtis & Sijacki x [ p c ] y [ p c ] z [ p c ] x [ k p c ] y [ k p c ] z [ k p c ] x [ p c ] y [ p c ] z [ p c ] x [ k p c ] y [ k p c ] z [ k p c ] − − ˙ M v o rt / ˙ M B o nd i t [ yr] − − M v o rt / M B o nd i Spurious SuppressionWith Fix t [Gyr] . . . . . . . ω F r a c t i o n A li g n e d No FeedbackWith Feedback f sort = 0 . Figure 4.
The top row shows the velocity field for simulations of an idealized Bondi inflow (left column) and of an isolated disc galaxywith black hole feedback (right column). In the second row, we show the corresponding vorticity field for the same simulations. In thecase of the Bondi simulation, the arrows are normalized to a constant value. For context, we show a density slice in the z = 0 plane. Inthe Bondi case, the velocity field is entirely radial, but because of the discontinuity of the gradients in the centre of the simulation domaincoupled with the rising velocity magnitude, the simulation results are increasingly sensitive to small errors in the gradient calculation.The resulting vorticity field is entirely noise driven, but can lead to a significant spurious signal. In the case of the isolated galaxy, thevelocity structure of a disc is clearly defined, leading to a coherent, robust corresponding signal in the vorticity (aligned with the z axis).In the bottom left-hand panels we show both the instantaneous and cumulative accretion rate normalized to the analytical Bondi ratewith and without our f sort prescription used to minimize the noisy vorticity estimate. In the Bondi simulation our f sort prescriptionis needed and recovers the correct accretion rate. In the isolated disc galaxy simulation, both with and without black hole feedback,spurious vorticity is rarely significant as shown in the bottom right-hand panel, where the mass fraction of gas aligned with the netvorticity direction as a function of time is plotted. c (cid:13) , 000–000 esolving flows around black holes: the impact of gas angular momentum − R [kpc] v ( R ) [ k m / s ] v φ v R v z No Ref − R [kpc] ω [ / G y r ] ω R ω φ ω z ω t No Ref R ref Figure 5.
The left-hand panel shows radial profiles of the three velocity components in cylindrical (note that v z represents the absolutevelocity value) polar coordinates from a simulation with thermal feedback (where the time is 1 . z component of the vorticity) whilst on smaller scales, the effects of feedback create a more turbulent distribution, with a rising overallvorticity. The vertical dashed line indicates the edge of the refinement region. Our simulations without the refinement scheme (dashedlines) fail to capture much of this complexity. − R [kpc] ω [ / G y r ] No Refinement No Refinement No Refinement With Refinement With Refinement With Refinement Model t [Gyr] | ω | With Refinement . × R ref With Refinement . × R ref No Refinement . × R ref No Refinement . × R ref Figure 6.
The left-hand panel shows radial profiles of total vorticity for simulations with (continuous lines) and without (dashed lines)refinement, both performed at three different resolutions. While all simulations have comparable vorticity profiles for
R > R ref (black lines), and over a sphere with radius 0 . R ref (blue lines). The simulation with refinement not only captures asystematically higher vorticity across the whole simulated time-span, but it provides numerically meaningful results as we probe regionscloser to the black hole’s Bondi radius, while the simulation without refinement fails in this regime at the matching effective resolution.c (cid:13)000
R > R ref (black lines), and over a sphere with radius 0 . R ref (blue lines). The simulation with refinement not only captures asystematically higher vorticity across the whole simulated time-span, but it provides numerically meaningful results as we probe regionscloser to the black hole’s Bondi radius, while the simulation without refinement fails in this regime at the matching effective resolution.c (cid:13)000 , 000–000 Curtis & Sijacki this field numerically to find the curl and, hence, the vor-ticity. All three refinement simulations at increasing resolu-tions produce very similar vorticity distribution and agreevery well with our analytic estimate. Simulations withoutrefinement, however, fail to reproduce inner vorticity peaka part from the highest resolution run. This indicates thatfor a comparable effective (large-scale) resolution and CPUresources, simulations without refinement are bound to sys-tematically underestimate gas vorticity. This is illustratedexplicitly in the right-hand panel of Figure 6 where kernelaveraged vorticity measurement is plotted as a function ofsimulated time. Not only is the estimated vorticity system-atically smaller in the runs without refinement, but as weprobe regions closer to the Bondi radius the estimate is dom-inated by the numerical noise (dashed blue line), demon-strating that much higher effective (and computationallymore expensive) simulations would be needed to mitigatethis issue.
Figure 7 shows the effects of our vorticity prescription onthe evolution of the black hole mass in the isolated discgalaxy simulations. In the left-hand panel we show the re-sults for simulations with black hole feedback. Here, we cansee that for standard isotropic thermal feedback, as expectedfrom our analytic analysis, the suppression of the black holegrowth is minimal. The surrounding gas is at too high asound speed and as highlighted by Figure 1 this simulationmodel probably represents the worst case for vorticity-basedaccretion suppression to be effective.For comparison, we also show a simulation run usingbipolar feedback, which has the effect of decoupling (to acertain extent) the black hole heated gas from the cold ac-creting gas. Here, we adopt a softer equation of state for theISM as well, leading to a less pressurized and more clumpymedium. The relative suppression of the black hole growthis more significant - indeed, at its peak ˙ M Vort is around onetenth of the standard Bondi rate - and has a noticeable effectespecially at early times, with the peak growth phase of theblack hole delayed by around 1 Gyr in the vorticity-basedsimulation. Note however that the final black hole massesare broadly independent of the accretion rate prescription(and much more dependent on the feedback prescription) asblack holes enter into self-regulated growth regime in bothof these simulation models. Black hole growth is thus es-sentially determined by the amount of injected energy. Inthese cases, therefore, the suppressing effect of gas vorticityis sub-dominant compared to the feedback choice.Given the strong dependence on feedback, we now turnto investigate the effect of the suppression in simulationswith no feedback, in order to restrict our analysis to theeffects of the accretion rate alone. Whilst such simulationshave been shown to be unrealistic in many ways, they nev-ertheless provide insight into the different regimes of blackhole growth in a simplified scenario. Specifically, while itis desirable on both theoretical and numerical grounds forthe black hole growth to be self-regulated, in reality this is not necessarily the case, or it may occur on average withshort, intermittent phases of rapid, unrestricted growth inbetween. Thus, our simulations without feedback and withstrong self-regulation may each possibly book-end a morerealistic scenario.In the right-hand panel of Figure 7, we show the evo-lution of the black hole mass for simulations without feed-back, for three different setups: a fiducial simulation (whichincludes metal line cooling), a simulation with a softer equa-tion of state and finally a simulation with a larger initialblack hole mass of 10 M (cid:12) . Here, we are motivated by ex-ploring different regimes that may lead to significant sup-pression, according to our previous analysis in Section 3.Black holes in simulations with a higher initial seed massand with the softer equation of state grow rapidly, showinga very modest difference between the standard and vortic-ity simulations. This is despite the fact that in both cases,at its peak, the ratio of the vorticity-based prescription tothe standard Bondi rate is around 0 .
01. The reason for thisis that, in both cases, the fast black hole growth is drivenby changes in the Bondi-part of the accretion rate (alsodue to a strong M dependence) leading quickly to theEddington-limited growth, to the extent that even a signif-icant vorticity-based suppression has a limited effect on theoverall evolution. For the simulation with metal line cooling,however, it is possible to enter a regime where the black holegrowth is significantly suppressed by roughly three orders ofmagnitude and for the entire 10 Gyr of simulated time-span.While this highlights that gas angular momentum barriercan be extremely efficient in limiting black hole growth, westress that the relevance of this result is clearly highly de-pendent on the actual gas properties on parsec scales andbelow, which are currently not well understood. In order to study the effects of our vorticity-based prescrip-tion on a situation where the angular momentum profile ofthe gas in the galaxy is significantly disrupted, we performsimulations of galaxy binary mergers. Here, we set up twogalaxies with the same parameters as those in Section 2.3at a distance of 200 kpc, which places them outside theirrespective virial radii. The two galaxies are initially in thesame plane and collide on a prograde parabolic orbit. Welimit ourselves here to studying the most interesting setupfrom our original simulations that gives a significant sup-pressing effect, namely one with no black hole feedback andwith metal line cooling.As the merging galaxies approach each other, they slowdown due to dynamical friction and after two close passagesthey coalesce to form a spheroidal remnant † . Large scalegravitational torques during the merging process funnel gastowards the innermost regions thus increasing the gas den-sity around black holes as well as increasing the relativefraction of gas on radial versus azimuthal orbits. In the toprow of Figure 8 we show a selection of projected gas den-sity maps from different stages of the merger evolution. The † Note that due to the relative orientation of galaxies in this spe-cific merger setup, a gaseous disc reforms during the intermediatestage of the merger, which has considerable vorticity.c (cid:13) , 000–000 esolving flows around black holes: the impact of gas angular momentum t [Gyr] M B H [ M (cid:12) ] Thermal Feedback, BondiThermal Feedback, VortBipolar + sEOS, BondiBipolar + sEOS, Vort t [Gyr] M B H [ M (cid:12) ] Metals, BondiMetals, VortsEOS, BondisEOS, Vort M = 10 M (cid:12) , Bondi M = 10 M (cid:12) , Vort Figure 7.
The left-hand panel shows the growth of black holes in simulations with two feedback prescriptions: thermal and bipolarcoupled with a sEOS, taking into account the vorticity-based and standard Bondi accretion prescriptions. In the case of the thermalfeedback, the sound speed is too high to allow for significant suppression by vorticity. In the bipolar case, whilst the black holes reach asimilar final mass (which is determined in a large part by the cumulative energy injected by the feedback) there is a delay in the initialgrowth of the black hole of up to ∼ corresponding Voronoi tessellations of the computational do-main are plotted in the second row. The first galaxy passagehappens at around 0 . . . . c (cid:13)000
The left-hand panel shows the growth of black holes in simulations with two feedback prescriptions: thermal and bipolarcoupled with a sEOS, taking into account the vorticity-based and standard Bondi accretion prescriptions. In the case of the thermalfeedback, the sound speed is too high to allow for significant suppression by vorticity. In the bipolar case, whilst the black holes reach asimilar final mass (which is determined in a large part by the cumulative energy injected by the feedback) there is a delay in the initialgrowth of the black hole of up to ∼ corresponding Voronoi tessellations of the computational do-main are plotted in the second row. The first galaxy passagehappens at around 0 . . . . c (cid:13)000 , 000–000 Curtis & Sijacki − −
10 0 10 20 x [kpc] − − y [ k p c ] − . − . − . − . − . − . − . − . − . P r o j e c t e d ρ [ M (cid:12) / k p c ] − −
20 0 20 40 x [kpc] − − y [ k p c ] − . − . − . − . − . − . . P r o j e c t e d ρ [ M (cid:12) / k p c ] − − − x [kpc] − − − y [ k p c ] − . − . − . − . − . − . − . − . − . − . . P r o j e c t e d ρ [ M (cid:12) / k p c ] − . − . . . . x [kpc] − . − . . . . y [ k p c ] − . − . − . − . − . − . − . − . − . − . − . P r o j e c t e d ρ [ M (cid:12) / k p c ] − −
10 0 10 20 x [kpc] − − y [ k p c ] − −
20 0 20 40 x [kpc] − − y [ k p c ] − − − x [kpc] − − − y [ k p c ] − . − . . . . x [kpc] − . − . . . . y [ k p c ] − − − − − − ρ [ M (cid:12) / k p c ] Normal, MergerVorticity, MergerNormal, IsolatedVorticity, Isolated t [Gyr] M B H [ M (cid:12) ] NormalVorticity
Figure 8.
The top row shows the projected gas density of our simulations of a galaxy merger pair at times of 0 .
5, 0 .
7, 2 .
75 and 6 Gyr(from left to right), spanning a time sequence from the first contact to the final merger remnant. The second row shows the correspondingVoronoi mesh, which illustrates the spatial resolution of our hydro solver. Below this, the evolution of the gas density within the blackhole’s smoothing length is plotted, including the case with the standard Bondi accretion prescription (blue) and with the vorticity-basedprescription (green). For comparison we plot the same quantity from our simulations of isolated disc galaxies simulated with the samephysics (dashed lines). Finally, the bottom row shows the evolution of the black hole mass for our merger simulations with Bondi andvorticity-based prescriptions (dashed vertical lines correspond to the image sequence shown in the top row). While there is a significantdelay in black hole growth after the second passage with vorticity-based prescription, due to the large-scale torques high gas centraldensity builds up, which ultimately leads to the final black hole mass similar as in the standard Bondi case.c (cid:13) , 000–000 esolving flows around black holes: the impact of gas angular momentum smaller scales (e.g. in the “alpha-disc” regime) may domi-nate the final supply rate to the black hole. In this paper we have investigated the impact of gas angu-lar momentum on the growth of supermassive black holes.Specifically, we have limited our analysis to the model ini-tially proposed by Krumholz et al. (2005) that takes intoaccount gas vorticity as to generalize the well-known Bondiaccretion rate. While by no means exhaustive, this model iswell suited for implementation in galaxy formation simula-tions, especially in the case where the relevant gas propertiesare robustly measured close to the Bondi radius, which isthe case when we adopt our super-Lagrangian refinementtechnique (Curtis & Sijacki 2015). Note that the refine-ment is not only needed to accurately measure gas densityand sound speed, but the gas vorticity field itself, which iseven more numerically challenging. We have implementedthis vorticity-based accretion rate prescription in the mov-ing mesh code
AREPO , for general usage in (cosmological)galaxy formation simulations. We have extensively testedour implementation on a range of numerically challengingsetups obtaining robust results and have verified the con-vergence properties of the model with increasing resolution.With respect to the Bondi rate, in the vorticity-basedprescription a significant suppression of black hole growthis expected when gas angular momentum and thus vortic-ity are high. To fully investigate model predictions we havefirst developed simple analytical dynamical models, whichallowed us to establish a tight dependence of the relativesuppression magnitude on the density and sound speed ofthe gas - two properties that on the scales of interest arepoorly constrained in practice by both observations and sim-ulations. Moreover, due to the significant dependence of theaccretion rate on the black hole mass ( ∝ M ), our dynami-cal models reveal a more nuanced effect, whereby significantaccretion suppression can be outweighed by rapid growth incases where the net accretion rate is high (i.e. close to theEddington limit).We have then explored the effects that different blackhole feedback parametrizations and ISM physics choiceshave on the black hole growth with the matching set of sim-ulations of isolated disc galaxies, performed with the stan-dard Bondi and vorticity-based accretion rate prescriptions.In doing so, we have found a picture that is broadly simi-lar to our analytic predictions: black hole growth is largelygoverned by feedback but that, within this, the effect ofthe vorticity prescription is to have a mild suppression onthe accretion rate. One of the principal reasons for this isthe self-regulating nature of feedback implementations weconsidered in this study, where the final black hole mass islargely set by the amount of feedback energy injected.Conversely, in simulations without black hole feedback,we found that the black hole can be in regimes where thesuppressing effect is much more significant, even leading tono appreciable growth over a Hubble time. The contrast-ing no feedback and feedback simulation results are likelyto bracket a more realistic scenario, where black hole self-regulation is not as tight as we assumed here and occurson a natural duty cycle after longer episodes of sustained growth, when significant amounts of gas can be expelled ina large-scale outflow from the innermost region of the hostgalaxy.Finally, we have studied the case of isolated binary ma-jor mergers of two disc galaxies hosting black holes. Dueto very efficient torquing gas angular momentum is effi-ciently transported outwards and large amounts of gas arefunnelled towards the centre of the merger remnant. Withrespect to the isolated disc galaxy model, the growth ofthe black hole using the vorticity prescription is greatly in-creased as expected, but it still remains significantly sup-pressed when compared to merger remnant grown with thestandard Bondi rate. Interestingly, while the final black holemass in the merger runs with and without vorticity suppres-sion is similar, there is a several Gyr delay in reaching thismass, once the gas angular momentum barrier is taken intoaccount. This could naturally explain scarce observationalevidence of quasar triggering in galaxies with perturbedmorphologies which previously had been accounted for byobscuration effects alone. Indeed, there is observational ev-idence of a significant delay between starburst activity andthe peak of AGN activity (e.g. Davies et al. 2007; Bennertet al. 2008; Yesuf et al. 2014; Matsuoka et al. 2015).We finally caution that our work is just a first stabin the direction of understanding the interplay between theblack hole growth and the gas angular momentum from thepoint of view of galaxy formation simulations. As we havedemonstrated in this work, the black hole accretion suppres-sion within our model is highly sensitive on the gas prop-erties on parsec scales. Thus a more realistic treatment ofthe ISM physics and black hole feedback are needed to makemore progress on this front. More generally, the amount ofgas that may eventually reach the black hole will be heavilymodulated by the physics occurring on even smaller spatialspaces, where effectiveness of gas fragmentation, star forma-tion (and stellar winds) as well as (magneto-) hydrodynam-ical viscous transport processes through the accretion discneed to be considered. While numerically self-consistentlyaccounting for all these processes on such a vast range ofspatial scales will be a formidable challenge for some timeto come, simulations have now started to reach the regimewhere some of these scales can be bridged, promising toshed light on the physics that powers supermassive blackhole growth. We thank Ewald Puchwein, Martin Haehnelt and VolkerSpringel for their useful comments on our manuscript. Wethank Ruediger Pakmor for proving us with his initial con-dition setup for the Keplerian disc simulation. MC is sup-ported by the Science and Technology Facilities Council(STFC). DS acknowledges support by the STFC and theERC Starting Grant 638707 “Black holes and their hostgalaxies: co-evolution across cosmic time”. This work wasperformed on the following: the COSMOS Shared Mem-ory system at DAMTP, University of Cambridge oper-ated on behalf of the STFC DiRAC HPC Facility - thisequipment is funded by BIS National E-infrastructure cap-ital grant ST/J005673/1 and STFC grants ST/H008586/1,ST/K00333X/1; DiRAC Darwin Supercomputer hosted by c (cid:13)000
AREPO , for general usage in (cosmological)galaxy formation simulations. We have extensively testedour implementation on a range of numerically challengingsetups obtaining robust results and have verified the con-vergence properties of the model with increasing resolution.With respect to the Bondi rate, in the vorticity-basedprescription a significant suppression of black hole growthis expected when gas angular momentum and thus vortic-ity are high. To fully investigate model predictions we havefirst developed simple analytical dynamical models, whichallowed us to establish a tight dependence of the relativesuppression magnitude on the density and sound speed ofthe gas - two properties that on the scales of interest arepoorly constrained in practice by both observations and sim-ulations. Moreover, due to the significant dependence of theaccretion rate on the black hole mass ( ∝ M ), our dynami-cal models reveal a more nuanced effect, whereby significantaccretion suppression can be outweighed by rapid growth incases where the net accretion rate is high (i.e. close to theEddington limit).We have then explored the effects that different blackhole feedback parametrizations and ISM physics choiceshave on the black hole growth with the matching set of sim-ulations of isolated disc galaxies, performed with the stan-dard Bondi and vorticity-based accretion rate prescriptions.In doing so, we have found a picture that is broadly simi-lar to our analytic predictions: black hole growth is largelygoverned by feedback but that, within this, the effect ofthe vorticity prescription is to have a mild suppression onthe accretion rate. One of the principal reasons for this isthe self-regulating nature of feedback implementations weconsidered in this study, where the final black hole mass islargely set by the amount of feedback energy injected.Conversely, in simulations without black hole feedback,we found that the black hole can be in regimes where thesuppressing effect is much more significant, even leading tono appreciable growth over a Hubble time. The contrast-ing no feedback and feedback simulation results are likelyto bracket a more realistic scenario, where black hole self-regulation is not as tight as we assumed here and occurson a natural duty cycle after longer episodes of sustained growth, when significant amounts of gas can be expelled ina large-scale outflow from the innermost region of the hostgalaxy.Finally, we have studied the case of isolated binary ma-jor mergers of two disc galaxies hosting black holes. Dueto very efficient torquing gas angular momentum is effi-ciently transported outwards and large amounts of gas arefunnelled towards the centre of the merger remnant. Withrespect to the isolated disc galaxy model, the growth ofthe black hole using the vorticity prescription is greatly in-creased as expected, but it still remains significantly sup-pressed when compared to merger remnant grown with thestandard Bondi rate. Interestingly, while the final black holemass in the merger runs with and without vorticity suppres-sion is similar, there is a several Gyr delay in reaching thismass, once the gas angular momentum barrier is taken intoaccount. This could naturally explain scarce observationalevidence of quasar triggering in galaxies with perturbedmorphologies which previously had been accounted for byobscuration effects alone. Indeed, there is observational ev-idence of a significant delay between starburst activity andthe peak of AGN activity (e.g. Davies et al. 2007; Bennertet al. 2008; Yesuf et al. 2014; Matsuoka et al. 2015).We finally caution that our work is just a first stabin the direction of understanding the interplay between theblack hole growth and the gas angular momentum from thepoint of view of galaxy formation simulations. As we havedemonstrated in this work, the black hole accretion suppres-sion within our model is highly sensitive on the gas prop-erties on parsec scales. Thus a more realistic treatment ofthe ISM physics and black hole feedback are needed to makemore progress on this front. More generally, the amount ofgas that may eventually reach the black hole will be heavilymodulated by the physics occurring on even smaller spatialspaces, where effectiveness of gas fragmentation, star forma-tion (and stellar winds) as well as (magneto-) hydrodynam-ical viscous transport processes through the accretion discneed to be considered. While numerically self-consistentlyaccounting for all these processes on such a vast range ofspatial scales will be a formidable challenge for some timeto come, simulations have now started to reach the regimewhere some of these scales can be bridged, promising toshed light on the physics that powers supermassive blackhole growth. We thank Ewald Puchwein, Martin Haehnelt and VolkerSpringel for their useful comments on our manuscript. Wethank Ruediger Pakmor for proving us with his initial con-dition setup for the Keplerian disc simulation. MC is sup-ported by the Science and Technology Facilities Council(STFC). DS acknowledges support by the STFC and theERC Starting Grant 638707 “Black holes and their hostgalaxies: co-evolution across cosmic time”. This work wasperformed on the following: the COSMOS Shared Mem-ory system at DAMTP, University of Cambridge oper-ated on behalf of the STFC DiRAC HPC Facility - thisequipment is funded by BIS National E-infrastructure cap-ital grant ST/J005673/1 and STFC grants ST/H008586/1,ST/K00333X/1; DiRAC Darwin Supercomputer hosted by c (cid:13)000 , 000–000 Curtis & Sijacki
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APPENDIX A: SETTING THE F SORT
THRESHOLD
As discussed in Section 4.2, f sort is the fraction of the gasmass that has a vorticity within π / c (cid:13) , 000–000 esolving flows around black holes: the impact of gas angular momentum . . . . . . f sort . . . . . . . . . F r a c t i o n BondiDisc, No FeebackDisc, With Feedback f sort = 0 . Figure A1.
The distribution of f sort for simulations of Bondiflow, as well as simulations of isolated disc galaxies with andwithout feedback. fraction aligned with the net vorticity follows that expectedfor a random distribution. In the case of the disc galaxy,the signal is coherent, especially when there are no pertur-bations to the velocity field caused by feedback. We set thethreshold at which the vorticity suppression is active to be f sort = 0 .
35, to exclude the scenario present in the Bondisimulations. c (cid:13)000