The generation of optical emission-line filaments in galaxy clusters
aa r X i v : . [ a s t r o - ph ] F e b Mon. Not. R. Astron. Soc. , 1–15 (2007) Printed 26 November 2018 (MN L A TEX style file v2.2)
The generation of optical emission-line filaments in galaxyclusters
Edward C.D. Pope ⋆ , Julian M. Pittard , Thomas W. Hartquist , Sam A.E.G. Falle School of Physics & Astronomy, University of Leeds, Leeds, UK, LS2 9JT School of Applied Mathematics, University of Leeds, Leeds, UK, LS2 9JT
26 November 2018
ABSTRACT
Recent data support the idea that the filaments observed in H α emission near thecentres of some galaxy clusters were shaped by bulk flows within their intraclustermedia. We present numerical simulations of evaporated clump material interactingwith impinging winds to investigate this possibility. In each simulation, a clump fallsdue to gravity while the drag of a wind retards the fall of evaporated material leadingto elongation of the tail. However, we find that long filaments can only form if theoutflowing wind velocity is sufficiently large, ∼ cm s − . Otherwise, the tail ma-terial sinks almost as quickly as the cloud. For reasonable values of parameters, themorphological structure of a tail is qualitatively similar to those observed in clusters.Under certain conditions, the kinematics of the tail resemble those reported in Hatchet al.(2006). A comparison of the observations with the numerical results indicatesthat the filaments are likely to be a few tens of Myrs old. We also present argumentswhich suggest that the momentum transfer, from an outflowing wind, in the forma-tion of these filaments is probably significant. As a result, tail formation could play arole in dissipating some of the energy injected by a central AGN close to the clustercentre where it is needed most. The trapping of energy by the cold gas may providean additional feedback mechanism that helps to regulate the heating of the centralregions of galaxy clusters and couple the AGN to the ICM. Key words:
Optical emission-line nebulae commonly surround massivegalaxies in the centres of X-ray bright, cool cluster cores(Crawford et al. 1999). The origin of the H α filamentshas been attributed to a variety of processes including:condensation from an intracluster medium (ICM) evolv-ing as a cooling flow (Fabian et al. 1984; Heckman et al.1989; Donahue & Voit 1991); accretion of clouds capturedin galaxy mergers (Braine et al. 1995); expulsion from thecentral galaxy (Burbidge & Burbidge 1965).NGC 1275 at the centre of the Perseus cluster con-tains the best studied example of such a nebula (Hatch et al.2006). Its filaments are typically 50-100 pc thick and up to 30kpc long, and the majority of them are radial. Hatch et al.(2006) presented kinematic data that rules out dynamicalmodels of purely infalling filaments. The observed kinematicproperties provide strong evidence that the filaments are not ⋆ E-mail:[email protected] in gravitational free fall, because, if they were, their veloc-ities would rise sharply towards the centre of the nebula(Heckman et al. 1989). The most conclusive evidence liesin the velocity structures of the northern and northwesternfilaments. The lower half of the northern filament is red-shifted with respect to the galaxy, whilst the upper sectionis blueshifted. Thus, the upper part of the filament is flow-ing away from the galaxy while the lower part is flowing intothe galaxy.The filaments may follow streamlines of the flow of moretenuous material around them. In the Perseus cluster, somefilaments appear to have been shaped by the wakes of thebuoyant bubbles. Given that the data suggest that, at least,some parts of the filaments are outflowing, their origin maylie within the galaxy. NGC 1275 contains a large reservoirof cold molecular gas that could fuel them.The filaments have morphologies similar to those oflaminar jets, and their relatively smooth structures mayplace constraints on turbulent motions in the ICM. Alter-natively, an ordered, amplified magnetic field trailing be- c (cid:13) E.C.D. Pope, J.M. Pittard, T.W. Hartquist, S.A.E.G. Falle hind a buoyant bubble interacting with filaments may pre-vent the destruction of the filaments by turbulent motions(Ruszkowski et al. 2007).The optical nebula around NGC 1275 emits 4 . × erg s − in H α and N[II]. The nature of the power sourceremains unclear. Various excitation mechanisms for theselines have been proposed. For instance, although ionisationby the central AGN may be important for the inner re-gions, it cannot be the dominant source of power for theextended nebula because the H α intensity does not decreasewith distance from the nucleus (Johnstone & Fabian 1988).The nebula may be excited by stellar UV, but there is nospatial correlation between the filaments and the stellar clus-ters. Excitation by X-rays from the ICM seems unlikely, asthe ICM may be as much as a hundred times less luminousin UV than in the X-rays (Fabian et al. 2003). Heat con-duction from the ICM to the colder filaments has also beenproposed (Donahue et al. 2000), but it might also lead tothe evaporation of the filaments on too short a timescale(Nipoti & Binney 2004). Shocks and turbulent mixing lay-ers might play a role (Crawford & Fabian 1992). Cosmicrays, preferentially diffusing along the magnetic field linestrailing behind rising bubbles, could possibly drive the ex-citation in those filaments that are located in bubble wakes(Ruszkowski et al. 2007). The same magnetic fields linescould also prevent the filaments from evaporating due tothermal conduction.The same mechanisms that power the emission couldalso have a profound effect on the morphology of the emit-ting region. It is impossible to take all of these processes intoaccount. Consequently, we will investigate the effect of grav-ity and an impinging wind that strips material from a cloudon the morphology and kinematics of the resulting filament.The main aim of the work reported here is the calcula-tion of the density and velocity structures of material evap-orated from clumps embedded in winds from the centralgalaxies of galaxy clusters and the comparison of model re-sults with observations. Section 2 contains some preliminaryconsiderations, while the model assumptions and numericalapproach are treated in section 3. The simulation resultsappear in section 4. In section 5 we investigate the devel-opment of tails in realistic environments by including theappropriate density and gravity for a galaxy cluster. Sec-tion 6 concerns momentum tranfer between clumps and thewinds surrounding them. Section 7 discusses the possibletrends in optical emission between clusters, and section 8 isa summary. We consider a clump, or cloud, embedded in a lower-density,hotter ambient wind with a speed v w in the frame of thecentral galaxy. Material evaporates from the clump and in-teracts with the wind to form a filament.It may be possible to estimate the age of an observedfilament from its width. Previous numerical studies haveshown that for a broad range of parameters the width ofa filament is comparable to the diameter of the clump fromwhich material evaporates (e.g. Dyson et al. 2006). The ra- dius of the filament r f would have a lower bound of approx-imately: r f ∼ . c s , c ∆ t , where ∆ t is the age of the filament,and c s , c is the sound speed in the clump at the head of the fil-ament. The length of the filament l f would have a maximumgiven roughly by l f ∼ r f v w /c s , c . The largest observed l f /r f ratios would require that v w /c s , c ∼ t is p h r i = √ Dt , where D is thediffusion coefficient. If the RMS displacement due to diffu-sion equals the measured radius of a filament, the filament’slength is approximately l f ∼ v w ∆ t ∼ v w r D . (1)Substituting order of magnitude estimates: l f = 30 kpc, r f = 100 pc, v w = 10 km s − we find that D ∼ cm s − .Interestingly, this is comparable to the diffusion coefficientsfound by Roediger et al. (2007) at 30 kpc in their simula-tions of metal diffusion in the Perseus cluster. If diffusionwere contributing significantly to filament broadening, thisvalue of the diffusion coefficient would require the filamentsto be roughly ∆ t ∼ × yrs old.This diffusion coefficient greatly exceeds the Spitzerthermal and viscous diffusivities. Turbulent diffusion wouldhave to be important for such a large diffusion coefficient toobtain. The turbulent diffusion coefficient is given approx-imately by D ∼ vl/ v is the characteristic veloc-ity and l the characteristic length scale of the largest ed-dies. If the turbulence is driven by the shear of the fluidflow past the clump and along the filament, we might as-sume that the thickness of the boundary layer is roughly 3% of the length of the filament (Hartquist & Dyson 1988;Canto & Raga 1991) which exceeds the observed filamentwidths. Therefore, a diffusion picture is not appropriate forthe dynamics of the filament, but clearly viscous couplingacross the entire cross section of a filament is strong, lead-ing to a flow velocity that depends on the distance along thefilament but not on the location relative to the nearest edge. The rate at which mass is ablated from a clump, ˙ m , might beestimated by the rate at which momentum associated withthe wind is transferred to the clump, divided by a speed, c s , a ,anywhere between the sound speed of material in the clumpand the difference between the average clump speed and theaverage speed of material in the filament. The appropriatevalue of c s , a depends on how effective the viscous coupling(due to turbulence or any other mechanism) between thefilament material and the wind material is. The total masslost by a clump would then be given by m ≈ ˙ m ∆ t = ∆ t Aρ w v c s , a . (2)where A is the clump cross section and ρ w is the mass den-sity of the wind. If ∆ t is 30 Myr, the clump has a radius r c ∼
50 pc, ρ w = 10 − g cm − , v w = 10 km s − and c (cid:13) , 1–15 ptical filaments in galaxy clusters c s , a = 0 . − , the above equation gives m ∼ M ⊙ .This is comparable to the mass of a structure with a num-ber density of 10 cm − , radius of 50 pc, and length of 30kpc. A model of the clump’s motion taking into account the lossof mass and the ram pressure of the impinging wind gives,dd t ( m ˙ x ) = − ˙ m ˙ x − mg + KAρ w ( v w − ˙ x ) . (3) K is a multiplicative factor which in the simplest case wouldbe a constant of order unity. g is the gravitational fieldstrength, x is the height of the clump above the centralgalaxy’s mid-plane, and a dot indicates differentiation withrespect to time. Expansion of the left-hand-side of equation(3) demonstrates that mass loss from the clump does notaffect the motion of the clump. Therefore, the terminal ve-locity of the clump is,˙ x = v w − „ mgKAρ w « / . (4)Using the values given above for ρ w and r c and taking g ∼ − cm s − , K = 1, and m ∼ M ⊙ , we find that( mg/KAρ w ) / ∼ km s − , and so is significant for sloweroutflow velocities. For larger masses, and gravities, the ter-minal velocity is greater. The terminal velocity of the clumpis not likely to be constant, because the mass of the clumpfalls with time. Alternatively, if the drag term is negligible,then the clump will accelerate freely, and the velocity will begiven by ˙ x = gt , if g is a constant. However, this case appearsto be ruled out by observations showing slower velocities.Since material is continually stripped from the clumpas it moves relative to the surroundings, it may eventu-ally disappear completely. Clearly, if this happens duringa timescale that is less than that required for the velocityto exceed the terminal velocity, then we need not simulatethe case of a clump moving at its terminal velocity. Assum-ing constant gravity, the free-fall time from 30 kpc is about10 yr for the value of g assumed above. If the clump startedfrom that height with zero velocity with respect to the centreof the galaxy, it will have reached a speed of roughly 400 kms − in the free-fall time. Depending on the wind parametersand gravitational field, a 10 M ⊙ molecular clump will notreach terminal velocity and may not be totally evaporatedaway during free fall from 30 kpc. As mention above, the data suggest that much of the mate-rial within the filaments is outflowing. There are two scenar-ios in which filaments with the observed properties could beproduced. Firstly, clouds may form in cooling gas accretingfrom outside the galaxy. As they fall inwardly they would bestripped by an outflowing wind, and the lower regions of thefilament could be infalling while the outer portion could beoutflowing. Alternatively, such clouds may condense in gasthat has been expelled from the central galaxy, and eventu-ally begin falling back into the central galaxy as in a galacticfountain (e.g. Shapiro & Field 1976). As the clouds fall in-wardly, they would be stripped by outflowing material. In either case, an outflowing wind would ensure both infallingand outflowing portions of a filament.
In a previous subsection we discussed the idea of ram pres-sure due to an impinging wind, or outflow. However, it isunclear how precisely how fast these winds are: the densityand emission structures of transonic flows can be similar tothose of nearly hydrostatic media. So, in the absence of X-ray observations with high spectral resolution, direct obser-vational diagnosis of the kinematics is unable to distinguishfirmly between low Mach number and transonic speeds. De-spite this, X-ray maps do show evidence of weak shocks inthe hot material at radii in the range of a few kpc to atleast a few 10’s of kpc in the Virgo and Perseus clusters (e.g.Forman et al. 2007; Sanders & Fabian 2007). Thus, we knowthat there are large-scale flows with transonic speeds. Thesound speeds of 3 × K and 10 K ionised gas, for the Virgoand Perseus clusters, are 8 × cm s − and 1 . × cm s − ,respectively. In conjunction with the X-ray maps, this sug-gests that the wind speeds are likely to be ∼ cm s − .In fact, we suspect that the flows may be fountain-like andthat outwardly flowing gas eventually falls back inwards. Asa result, the wind velocity is not constant, but drops off withradius. So by the time it reaches the cold clouds the windvelocity will be much reduced from its central value.AGN activity is considered a prime candidate for pre-venting the ICM cooling from 10 K at rates of morethan 100 M ⊙ yr − . This requires an energy input rate of10 erg s − which is comparable to the kinetic energy fluxof material with a mass density of 2 × − g cm − flowingthrough a spherical surface of radius 10 kpc with a radialvelocity of 10 cm s − . Thus, the “cooling problem” and ourwind speed estimates imply comparable power inputs fromAGN.The outflows we envisage would probably be somewhat,although not hugely, collimated. X-ray images of the centreof the Perseus cluster show some asymmetry in structuresat radii approaching 10 kpc. Outside about 10 kpc, the in-teraction of AGN plasma with ambient galactic interstellarand intracluster gas appears to reduce asymmetries associ-ated with collimated AGN outflows. The filaments extendseveral times this far in radii, and are radial suggesting thatthe wind may be fairly isotropic at such radii.One would also expect the outflows to be somewhattime-dependent. However, the presence of clouds dottedthroughout the ICM may well have the effect of obstruct-ing the outflow of energy injected by the AGN. Therefore,the time-variability of the AGN-driven flow far away fromits source is likely to be fairly uncorrelated with the vari-ability at the source. Essentially this means that, althoughthe initial energy injection rate may be time-variable, atlarger radii the variations may have been smoothed-out mak-ing the flow relatively steady. Finally, intermittancy on longtimescales much greater than 3 × yr will be on timescalesgreater than the ages of the structures we are attempting toexplain, and consequently unimportant in this work. Basedon these considerations, we will assume the outflows to betransonic, isotropic and steady. c (cid:13) , 1–15 E.C.D. Pope, J.M. Pittard, T.W. Hartquist, S.A.E.G. Falle
We use hydrodynamical simulations to investigate the inter-action between a tenuous flow and a mass source. The sourceof mass is assumed to be a cold cloud from which materialis stripped due to the interaction with an incident wind.The basic approach is similar to that adopted by Falle et al.(2002), Pittard et al. (2005), and Dyson et al. (2006). Themain difference is that here we also include the effects ofgravity to make the model more appropriate for a typicalgalaxy cluster environment.Despite including the gravitational potential, we willignore self-gravity for this set of simulations, for the follow-ing reasons. Self-gravity is certainly important in the coldclump at the bottom of each filament, and would be impor-tant in our model if we were to try to calculate the mass lossrate of the clump. As we have already stated, the details ofthe injection process are not important for the question weare addressing. Self-gravity may well be important in thefilaments, but the absence of a population of bright, youngstars associated with the filaments (e.g. Hatch et al. 2006)also provides some evidence that it may not. As an illustra-tion, we give the ratio of the magnitude of the gravitationalfield to the magnitude of the thermal pressure force per unitvolume in a cylindrical filament of mass density ρ and radius R is approximately, gρRP = 2 πGρµm p R k b T ≈ . „ ρ − g cm − «„ R
30 kpc « „ T K « − , (5)where µm p is the mean mass per particle in the gas. Thus,filament parameters are in the range that may allow theratio to approach unity. However, it is clear that the valuesare sufficiently uncertain that an initial focus on cases inwhich self-gravity is neglected is reasonable. Furthermore,if self-gravity does lead to instability, shear may limit thedevelopment of clumps. We will neglect self-gravity for thepresent investigation.The simulations are split into two groups. In the first weassume constant gravity, and work in the frame of the cloud.This is to gain an understanding of the effect of the gravita-tional potential and the density contrast between the cloudand the wind. In the second group, we employ a realisticcluster atmosphere and work in the rest frame of the clus-ter, and advect the cloud. This allows us to implement whatwe have learnt from the prior simulations in the more real-istic case. The latter setup is discussed more fully in section6. In either case, the velocity of the incident wind increases,as measured in the clump frame, with time. Gravity alsoaffects the motion of material in the tail.For all models we assume a rate of mass injection perunit time and volume that may vary temporally, but that isspatially uniform within a given radius of the cloud’s centre.This mass injection prescription is simple to implement, buthas the disadvantage that the flow from the injection regionis isotropic. However, the effect of asymmetric mass loss issmall at large distances from the cloud (Pittard et al. 2005).Therefore, we emphasize that the actual details of the massinjection process are relatively unimportant. The boundary of the mass injection region is not meant to coincide with theboundary of a cloud, and the cloud could be much smallerthan the injection region.Three-dimensional calculations are required if thesources are spherical. To reduce the computational cost werestrict ourselves to two-dimensional simulations in whichthe sources are cylindrical. While we expect some differ-ences between calculations performed in two and three di-mensions, at this stage we can still gain important insightfrom less computationally demanding two-dimensional sim-ulations.In this study we treat the hot gas as adiabatic, sincein the Perseus cluster the cooling time of X-ray emittinggas at 30 kpc is ∼ yr, much longer than the flow timeof the outflow along the tail. The cooling rate of the gasin the tail itself is very high, and must obviously be offsetby a heating rate. We do not know the precise temperatureof this gas, but the optically emitting gas in the tail is atabout 10 K. An isothermal treatment of this material is rea-sonable. Thus, we investigate the simple case in which theincident wind behaves adiabatically, while the injected gasremains isothermal.To ensure isothermal behaviour, we use an advectedscalar, α , which is unity in the injected gas and zero in theambient gas. The source term in the energy equation is then καρ ( T − T ) , (6)where ρ and T are the local mass density and temperature,and where κ is large enough that the temperature alwaysremains close to the equilibrium temperature, T , in the in-jected gas. Inside the injection region we added an extraenergy source so that the gas is injected with temperature T (see Falle et al. 2002, for further details).The simulations are performed using mg gt , a second-order accurate code with adaptive mesh refinement (AMR). mg gt uses a hierarchy of grids G − G N such that the meshspacing on grid G n is ∆ x / n . Grids G and G cover thewhole domain, but the finer grids only exist where theyare needed. The solution at each position is calculated onall grids that exist there, and the difference between thesesolutions is used to control refinement. In order to ensureCourant number matching at the boundaries between coarseand fine grids, the time-step on grid G n is ∆ t / n where ∆ t is the time-step on G .In this model, we are studying the behaviour of coldclouds located ∼
10 kpc away from the centre of the cluster.At such distances, the gravitational acceleration in galaxyclusters is ∼ − cm s − . We investigate different densitycontrasts between the cloud material and the ambient gas,and compare the morphology of the results with observa-tions. The problem we are investigating is scale-free. For numer-ical reasons, we have chosen scales so that the values ofparameters are near unity. For easier comparison with theastrophysical systems of interest we will rescale the resultsto physical units. In particular, we have scaled the length bya factor L , the gravitational acceleration by G , and the den-sity by R . In this case, the hydrodynamic quantities scale c (cid:13) , 1–15 ptical filaments in galaxy clusters as, x = Lx ′ , (7) g = Gg ′ , (8) ρ = Rρ ′ , (9) t = τ t ′ (10) v = Lτ v ′ , (11) p = R „ Lτ « p ′ , (12)where the primed quantities represent the dimensionlesssimulation values, and the unprimed quantities represent thereal values. t is time, v is velocity, and p is pressure. In casesin which g is nonzero τ = r LG (13)In our simulations the values selected for parameters areroughly consistent with estimates of the properties of cloudsin the Perseus cluster: the length L = 50 pc, G = 10 − cm s − (except for one case) and R = 10 − g cm − . For cases inwhich g = 0, we leave τ = 1 .
24 Myrs, which is the samevalue as when g is nonzero. In section 5, we have employedmore accurate values of the scaling factors. To ensure that the ambient wind does not penetrate theinjection region, we take the ram pressure of the injectedmaterial at the boundary of the injection region to be largerthan that of the wind, i.e. we require, v i2 ρ i = v i r i Q > ρ w v , (14)where r i is the radius of the injection region, v i is the flowspeed of the injected material at r = r i and ρ i its densityat this point. Q is the rate per unit volume per unit time atwhich mass is injected within r = r i .If we use the pressure behind a stationary normal shockin the wind rather than the wind’s ram pressure, the abovecriterion becomes Q > γ + 1 ρ w v , v i r i . (15)In the frame of a falling clump, the wind’s ram pressureincreases with time. Thus, the minimum mass injection ratemust increase accordingly to ensure that the wind does notpenetrate the injection region. If the value of Q at t = 0 is Q and the gravity is constant, the ratio of Q to the windram pressure will remain constant if Q = Q „ gtv w , « , (16)where v w , is the relative velocity between the clump and theambient flow at t = 0. While the method that we are usingto treat mass injection requires that we take Q to increase with time, in reality the mass injection rate will increasewith ram pressure (e.g. Hartquist & Dyson 1988).The computational domain is − x − y r i = 1, v i = 1, and ρ w = 1 and initially set v w =10 and T = 1. Furthermore, we assume that the injectedmaterial is in pressure equilibrium with the ambient wind.Thus, if we assume that the injected material is a factor of10 cooler than the ambient wind, then it is also 10 timesdenser than the wind. We will use η to represent this contrastparameter.We investigate the morphology of tails for three differ-ent contrast parameters: η = 10 , 10 , and 10 . The corre-sponding initial values of the adiabatic Mach number, M ,are: 0.77, 0.24, and 0.077, respectively. These parametersdefine case 1), 2), and 3), respectively. Clearly, as the am-bient wind accelerates with respect to the cloud, the Machnumber will increase.For comparison, we also study several other cases inwhich the density contrast is 10 . Case 2i) is identical to case2) except that g ′ = 10 and G = 10 − . In cases 2ii) and 2iii)the gravity is zero, but there is a constant wind velocity of 10and 30 (in code units) respectively, corresponding to Machnumbers of 0.24 and 0.72. These parameters are summarisedin table 1. It does not make sense to specify the mass ofthe clouds, since they are likely to be composed partly ofmolecular material, and therefore considerably more massivethan an estimate of 4 πr ρ c / v ′ w = 10, corresponds to a ve-locity of v w = 3 . × cm s − , which, is comparable withthe transonic values described in the previous section. Simulations with a transonic wind pose difficulties that areabsent in those for hypersonic winds. Firstly, in order for theboundaries to have no effect on a simulation with a transonicwind, the computational domain has to be very large. Sec-ondly, the velocity shear between the injected material andthe wind is so extreme in the transonic case that the windflow separates and produces a turbulent wake downstreamof the interaction region. Calculations based on the Eulerequations cannot adequately describe such turbulence, anda turbulence model must be employed.We adopt the k − ǫ model used by Falle et al. (2002).The subgrid turbulence model allows the simulation of ahigh Reynolds number flow through the inclusion of equa-tions for the turbulent energy density and dissipation rate,from which viscous and diffusive terms are calculated. Themodel solution gives an approximation to the mean flow.The diffusive terms in the fluid equations model the turbu-lent mixing of the injected gas with the original flow due toshear instabilities. The model has been calibrated by com-paring the computed growth of shear layers with experi-ments (Dash & Wolf 1983). It is based on the assumptionsthat the real Reynolds number is very large, which is thecase in astrophysical flows, and that the turbulence is fullydeveloped. The use of the subgrid model gives more realis-tic results than those obtained from a hydrodynamical codewhere the sizes of the shear instabilities are determined by c (cid:13)000
24 Myrs, which is the samevalue as when g is nonzero. In section 5, we have employedmore accurate values of the scaling factors. To ensure that the ambient wind does not penetrate theinjection region, we take the ram pressure of the injectedmaterial at the boundary of the injection region to be largerthan that of the wind, i.e. we require, v i2 ρ i = v i r i Q > ρ w v , (14)where r i is the radius of the injection region, v i is the flowspeed of the injected material at r = r i and ρ i its densityat this point. Q is the rate per unit volume per unit time atwhich mass is injected within r = r i .If we use the pressure behind a stationary normal shockin the wind rather than the wind’s ram pressure, the abovecriterion becomes Q > γ + 1 ρ w v , v i r i . (15)In the frame of a falling clump, the wind’s ram pressureincreases with time. Thus, the minimum mass injection ratemust increase accordingly to ensure that the wind does notpenetrate the injection region. If the value of Q at t = 0 is Q and the gravity is constant, the ratio of Q to the windram pressure will remain constant if Q = Q „ gtv w , « , (16)where v w , is the relative velocity between the clump and theambient flow at t = 0. While the method that we are usingto treat mass injection requires that we take Q to increase with time, in reality the mass injection rate will increasewith ram pressure (e.g. Hartquist & Dyson 1988).The computational domain is − x − y r i = 1, v i = 1, and ρ w = 1 and initially set v w =10 and T = 1. Furthermore, we assume that the injectedmaterial is in pressure equilibrium with the ambient wind.Thus, if we assume that the injected material is a factor of10 cooler than the ambient wind, then it is also 10 timesdenser than the wind. We will use η to represent this contrastparameter.We investigate the morphology of tails for three differ-ent contrast parameters: η = 10 , 10 , and 10 . The corre-sponding initial values of the adiabatic Mach number, M ,are: 0.77, 0.24, and 0.077, respectively. These parametersdefine case 1), 2), and 3), respectively. Clearly, as the am-bient wind accelerates with respect to the cloud, the Machnumber will increase.For comparison, we also study several other cases inwhich the density contrast is 10 . Case 2i) is identical to case2) except that g ′ = 10 and G = 10 − . In cases 2ii) and 2iii)the gravity is zero, but there is a constant wind velocity of 10and 30 (in code units) respectively, corresponding to Machnumbers of 0.24 and 0.72. These parameters are summarisedin table 1. It does not make sense to specify the mass ofthe clouds, since they are likely to be composed partly ofmolecular material, and therefore considerably more massivethan an estimate of 4 πr ρ c / v ′ w = 10, corresponds to a ve-locity of v w = 3 . × cm s − , which, is comparable withthe transonic values described in the previous section. Simulations with a transonic wind pose difficulties that areabsent in those for hypersonic winds. Firstly, in order for theboundaries to have no effect on a simulation with a transonicwind, the computational domain has to be very large. Sec-ondly, the velocity shear between the injected material andthe wind is so extreme in the transonic case that the windflow separates and produces a turbulent wake downstreamof the interaction region. Calculations based on the Eulerequations cannot adequately describe such turbulence, anda turbulence model must be employed.We adopt the k − ǫ model used by Falle et al. (2002).The subgrid turbulence model allows the simulation of ahigh Reynolds number flow through the inclusion of equa-tions for the turbulent energy density and dissipation rate,from which viscous and diffusive terms are calculated. Themodel solution gives an approximation to the mean flow.The diffusive terms in the fluid equations model the turbu-lent mixing of the injected gas with the original flow due toshear instabilities. The model has been calibrated by com-paring the computed growth of shear layers with experi-ments (Dash & Wolf 1983). It is based on the assumptionsthat the real Reynolds number is very large, which is thecase in astrophysical flows, and that the turbulence is fullydeveloped. The use of the subgrid model gives more realis-tic results than those obtained from a hydrodynamical codewhere the sizes of the shear instabilities are determined by c (cid:13)000 , 1–15 E.C.D. Pope, J.M. Pittard, T.W. Hartquist, S.A.E.G. Falle
Table 1.
Simulation parameters: ambient density, cloud density, sound speed of injected material, initialMach number of the wind velocity with respect to cold material, gravitational acceleration. The impliedmass outflow rates close to the origin are 4 πr δρ amb v w ∼
10 M ⊙ yr − , where δ ∼ . ρ amb ( g cm − ) ρ c ( g cm − ) c s , c (cm s − ) M initial g (cm s − )1 10 − − . × − − − . × − − − . × −
2i 10 − − . × − − − . × − − − . × − the numerical resolution. Further details can be found inFalle (1994). In general, the morphologies of the filament density gener-ated by accelerating flows are very similar to those presentedin Pittard et al. (2005). A bow wave develops upstream ofa mass source, and the wave’s amplitude falls off as 1 /r .If the flow becomes supersonic, the bow wave becomes abow shock and a very weak tail shock may occur in thewind downstream of the mass source, when the external flowis subsonic. The injected material remains in approximatepressure equilibrium with the wind, and eventually becomesconfined to a tail with a width comparable to that of theinjection region (cf. figures 1 to 4). The opening angle of theinjected material increases with the external Mach number.It should be noted that we use a sub-grid turbulencemodel which, in effect, models a turbulent boundary layerat the surface of the filament. So even when the tail appearssmooth there is significant turbulence. For increasingly largedensity contrasts, the turbulent viscosity, which is an inte-gral part of the subgrid turbulence model, fails to stabilizethe subsonic shear layer. In this case, there is turbulence onthe grid as well as sub-grid turbulence. Large-scale insta-bilities in thin filaments demand high density contrasts andan ambient flow that remains subsonic with respect to theclump.The opening angle of a turbulent boundary layer isgiven by Canto & Raga (1991) as,d h d x = 2 σ (17)where σ is the so-called spreading parameter. Experimentsshow that at low Mach number, turbulent mixing layers havea constant opening angle, and σ ∼
11, so that d h/ d x ∼ . σ ∼
50, so d h/ d x ∼ .
3. The tailbecomes fully turbulent at a downstream distance of, x = r c M w ǫ c c w . (18)where ǫ is the entrainment efficiency, M w is the Mach num-ber of the wind with respect to the clump, c s,a is the soundspeed of the environment, c s , c is the sound speed of the tailmaterial, r c is the initial radius of the tail. To calculate this length, Canto & Raga use x =16 . M w r c , which can be rescaled as, x ≈ .
85 kpc „ M w «„ r c
50 pc « , (19)in other words, for sensible values of the wind flowing pasta clump, we should expect the tail to be turbulent on kpcscales.These figures show a clear dependence, at least at earlytimes, of the tail morphology on the density contrast be-tween the cloud and the ambient gas. For example, for η = 10 (figure 3), the tail is nearly linear, but is punc-tuated by fairly evenly spaced blobs along the length of thetail. This is because subsonic shear layers tend to be unsta-ble unless the viscosity is high. Some turbulent jets displaysimilar behaviour. The blobs are separated by increasinglylarge distances further along the tail, indicating accelera-tion. For η = 10 , (figure 1) the tail only becomes irregularat large distances away from the cloud, while for η = 10 (figure 2) most of the tail is irregular. For comparison wealso show the morphologies for a case when the gravity isweaker (figure 4) and nongravitational cases with differentambient flow velocities (figures 5 and 6).For ambient flows that remain subsonic in the frameof the clumps, the tails are irregular in appearance. In case2i) the gravity is strong enough that the initially subsonicflow becomes supersonic, and consequently the tail becomessmooth and regular. Morphological considerations by them-selves would suggest that the filaments in the Perseus clus-ter satisfy these conditions. However, an examination of thekinematic structures of the model and observed filaments isalso necessary. Figures 7 to 10 show the mean velocity of the tail materialat 100 increments along the length of the tail for each ofcases 1), 2), 3), and 2i).Except for case 2i) and to some extent case 1), eachvelocity profile shows some scatter around a mean velocitythat is roughly constant along the length of most of the fil-ament. We do not show velocity profiles for cases 2ii) and2iii) because they show behaviours similar to those for cases2) and 3). However, the velocity profiles, including thosefor the model filaments that bear the most morphologicalresemblence to the observed filaments, are not compatiblewith the observed velocity structures of the NGC 1275 fila-ments. Nevertheless, the third panel of figure 10 shows clear c (cid:13) , 1–15 ptical filaments in galaxy clusters Figure 7.
Velocity profile for case 1) for (from left to right) t ′ = 6.95, 14.5, 20.9, and 28.2 ( t = 8.3, 18, 26, and 35 Myrs), respectively.The Mach numbers of the ambient flow with respect to the cold clump are: 1.3, 1.9, 2.4, and 3, respectively. We have removed the central10 km s − to avoid any points where the tail does not exist. Figure 8.
Velocity profile for case 2) for t ′ = 3.4, 8.3, 12.1, and 16.9 ( t = 4.3, 10, 15, and 21 Myrs) respectively. The Mach numbers ofthe ambient flow with respect to the cold clump are 0.33, 0.45, 0.54 and 0.66, respectively. We have removed the central 10 km s − toavoid any points where the tail does not exist. Figure 1.
Density morphology for case 1) at t ′ ≈
28 ( t ≈ M = 3. The initial Mach number was 0.77. Eachspatial unit corresponds to 50 pc. Thus, the plotted domain, inthe x-direction, extends between ± . Note the bowshock around the clump, and also the tail shock behind this. zig-zag features, which are similar to those in the velocityprofiles given in figures 5, 7 and 14 of Hatch et al. (2006).Note that the zig-zag features in figures 7 and 10 also corre-spond to the regions of enhanced density along the tail. Thesimilarity between figures 7 and 10 suggests that the zig-zag Figure 2.
Density morphology for case 2) at t ′ ≈
17 ( t ≈ M = 0.66. The initial Mach number was 0.24. Thedensity contrast between the cold clump and the ambient gas is10 . Note the irregular structure of the tail along the entire lengthof the filament, compared to the linear tail in the previous figure. features occur when the Mach number of the ambient flow,with respect to the clump, is roughly 2-3, after ∼ few 10sMyrs.The velocities near the heads of the model filaments arelow, whereas those of the observed filaments are not. Thediscrepency could be due to the material near the heads ofthe observed filaments being warm, but neutral, and hence c (cid:13) , 1–15 E.C.D. Pope, J.M. Pittard, T.W. Hartquist, S.A.E.G. Falle
Figure 9.
Velocity profile for case 3) for t ′ = 1.93, 5.7, 12.7 and 18.1 ( t =2.4, 7.0, 12, and 22 Myrs) respectively. The Mach numbers ofthe ambient flow with respect to the cold clump are 0.09, 0.12, 0.15, 0.18 and 0.22, respectively. We have removed the central 10 km s − to avoid any points where the tail does not exist. Figure 3.
Density morphology for case 3) at t ′ ≈
18 ( t ≈ M = 0.22. The initial Mach number was 0.077. Thedensity contrast between the cold clump and the ambient gas is10 . Note the approximately linear tail, with at least 6 descreteblobs of material close to the clump. invisible in the observed optical lines. More likely, it is a re-sult of our prescription for mass injection which will producedifferent features near the injection region, but the large-scale downstream flow should be unaffected.The fact that the profile shown in figure 10 resemblesthe observed profile better than that shown in figure 7,is probably because the density contrast between the coldclump and the ambient flow is more realistic in case 2i). It islikely that if we had continued simulating cases 2) and 3) tohigh enough Mach numbers that they may also have begunto exhibit the zig-zag structure. Based on these results itappears that the velocity fluctuations (and scale-size of in-stabilities) are larger for greater density contrasts betweenthe clump and the wind. Having investigated the properties of accelerating flows pastcold clouds, we now study the more specific case of a galaxycluster atmosphere. Firstly, we assume a generic gravita-
Figure 4.
Density morphology for case 2i) at t ′ ≈
20 ( t ≈ M = 5.2. The initial Mach number is 0.24. The den-sity contrast between the cold clump and the ambient gas is 10 .The structure of the tail in this example is regular, with no obvi-ous blobs at various locations along the tail. Figure 5.
Density morphology for case 2ii) at t ′ ≈
20 ( t ≈ , and gravity is omitted from the simulation. M = 0.24. c (cid:13) , 1–15 ptical filaments in galaxy clusters Figure 10.
Velocity profile for case 2i) at t ′ = 2, 5, 10, and 20 ( t = 2.5, 6.2, 12, and 25 Myrs), respectively. The Mach numbers of theambient flow with respect to the cold clump are 0.74, 1.5, 2.7 and 5.2, respectively. We have removed the central 10 km s − to avoid anypoints where the tail does not exist. Figure 6.
Density morphology for case 2iii) at a code time of t ′ ≈
20 ( t ≈
25 Myrs). The density contrast between the coldclump and the ambient gas is 10 . Gravity is omitted from thissimulation. M = 0.72. tional potential which leads to a β -profile gas density profilein hydrostatic equilibrium (e.g. Vernaleo & Reynolds 2006), ρ = ρ [1 + ( r/r ) ] β , (20)where we have assumed spherical symmetry, r is the radialcoordinate, and r is the scale-height of the density distri-bution. The gravitational acceleration is therefore given by g = − β k b Tµm p „ rr + r « , (21)where T is the gas temperature which is assumed to be con-stant with radius, µ = 0 . x , is, v ( r ) = − „ β k b Tµm p « / » r + x r + r – / . (22)Equation (22) can be numerically integrated, using the values for specific galaxy clusters, to obtain the position ofthe cloud at each point in time, r ( t ). For example, for thePerseus cluster we use, T = 10 K, ρ = 7 × − g cm − , β = 0 .
81, and r = 28 . r ( t ) = 30 » − . „ t
75 Myr « – kpc , (23)which is roughly what one would expect for constant gravity.This correspondence occurs because the gravitational accel-eration scales as 1 /r at large radii, and r at small radii. Thecloud is initially located around the turning point where thefunction is relatively flat. This means that constant gravityis a suitable approximation and the previous set of simula-tions, in which constant gravity was assumed, retain theirvalidity.For completeness, the Mach number of the flow aroundthe clump, with respect to the ambient flow, is M = 2 βγ / „ r + x r + r « . (24)Thus, for β = 3 / r = 28 . x = 30 kpc, we find that M > r . . r ≈
26 kpc. Consequently, for higherwind velocities, the flow becomes supersonic closer to theinitial location, x . The ram pressure due to relative motionbetween the cloud and the static atmosphere is, ρ w v = ρ [ r ( t )] » d r ( t )d t – . (25)We choose an initial density contrast between the cloudand its surrounding of 10 , and set the scaling parame-ters to R = 7 × − g cm − and L = 3 . × cm. P = ρk b T /µm p ≈ . × − erg cm − in the centreof the Perseus cluster, giving G = P/ ( P ′ RL ) = 1 . × − cm s − . The temporal scaling constant, is τ = 4 . × s, while the velocities are scaled by L/τ = 7 . × cm s − . The time taken for the cloud to fall from 30 kpcto 3 kpc is 75 Myr, or 0.55 in terms of code time units. For adensity contrast of 10 , we use R = 7 × − g cm − and L =3 . × cm, giving G = 1 . × − cm s − . This meansthat the temporal scaling constant becomes τ = 1 . × s,and the velocities are scaled by L/τ = 2 . × cm s − . Thetime taken for the cloud to fall from 30 kpc to 3 kpc now,or 1.7 in terms of code time units.Finally, in the preliminary simulations we found that c (cid:13) , 1–15 E.C.D. Pope, J.M. Pittard, T.W. Hartquist, S.A.E.G. Falle the tail widths were comparable to the diameters of theclouds. Therefore, to generate filaments of the observedwidths which are constrained to be < Since the lateral velocity dispersion (zig-zag effect) appearsto increase for larger density contrasts, we will concentrateon the case η = 10 . However, the case where η = 10 is alsoof interest. The general properties are as follows: long tail-like structures only form if the wind velocity is sufficientlylarge. This wind is assumed to be the same material, at thesame temperature as the static atmosphere. For example, forlow values of v ′ w (5 and 10, corresponding to 3 . × and7 . × cm s − for η = 10 ), the tail never becomes longbecause the ram pressure due to the impinging wind cannotovercome the gravitational force acting on the tail material.As a result any tail that forms is infalling everywhere, whichcontradicts the observational data, at least for NGC 1275.It therefore appears that the length and morphology of thetails are extremely dependent on the wind velocity.The morphology of the tail also depends strongly on thedensity contrast between the cloud and its environment. Ineach figure, the cloud is shown as red and is the lowest pointalong the line defined by x=0. Material is injected (stripped)and collects behind the free-falling cloud, and some densematerial sinks either side of the cloud. Behind this cold ma-terial, a filamentary tail forms. Inhomogeneities in the tailsare dependent on the wind speed and density contrast, inagreement with the previous results. As can be seen fromfigures 11 to 14, it appears that a density contrast of η = 10 produces structures most qualitatively similar to those seenin real filaments. However, they are low density, being a few × − K. The simula-tions produce filaments of 10 − M ⊙ while the estimatesin the following section suggest that 10 M ⊙ filaments mightbe more likely. Even so, the quantity of cold gas, in the sim-ulated tail, is more than capable of producing the observedoptical luminosity (see Hatch et al. 2007, for example).The material around the head of the cloud also exhibitsdifferent behaviour in this set of simulations. Material isstripped from the cloud by the wind and builds up behind.It appears that in many cases much of this material doesnot make it into a tail but simply collects behind the cloud.Eventually this becomes unstable, probably due to Rayleigh-Taylor instabilities, and produces the structures seen in thesimulations. Similar structures have been noticed in previousnumerical simulations in the work by Murray & Lin (2004).In the simulations, initial wind velocities of ∼ cm s − are required to produce long tails which are out-flowing at their outer extent. This large velocity is expected,since the velocity of the wind, at the clump, must exceed theinfall velocity of the clump for material to be carried awayfrom it. By the time the wind has reached the clump it has Figure 11.
The density structure at 58 Myrs, for η = 10 andwind velocity = 5 . × cm s − . Panel shows data at 3 . × yrs. The length scale indicates the distance from the clustercentre in kpc. The density scale shows the density in code units( ρ ′ ). Figure 12.
The density structure at 38 Myrs, for η = 10 andwind velocity = 4 . × cm s − . The length scale indicates thedistance from the cluster centre in kpc. The density scale showsthe density in code units ( ρ ′ ). done considerable work against gravity. This leads to thevelocity dispersion along the clump of a few ×
100 km s − .Figure 15 shows the velocity profile for η = 10 and awind speed of 5 . × cm s − ( v ′ w = 80). This case proba-bly best resembles the observations. Figure 16 seems to showthat a lower wind velocity ( v ′ w = 60, v w = 4 . × cm s − )does not produce the observed zig-zag kinematics for thisparticular density contrast. These results should be com-pared with the η = 10 case.Figures 17 and 18 show the kinematics for two windvelocities: v ′ w = 80 and 60, in figures 17 and 18. These cor-respond to 1 . × cm s − and 1 . × cm s − whichare either side of the sound speed of the ambient medium(1 . × cm s − ). Figure 18 eventually shows kinematic c (cid:13) , 1–15 ptical filaments in galaxy clusters Figure 15.
Velocity profile for case η = 10 and wind velocity = 5 . × cm s − . Panels show data at 14 ,
21 and 34 Myrs. The lengthscale indicates the distance from the cluster centre. We have cut off the central 20 km s − to exclude points where there is no tail. Figure 13.
The density structure at 20 Myrs, for η = 10 andwind velocity = 1 . × cm s − . The length scale indicates thedistance from the cluster centre in kpc. The density scale showsthe density in code units ( ρ ′ ). structure that could be interpreted as a zig-zag, although itis not as clear as in figure 15. The tails extend to greaterlengths for η = 10 than η = 10 because the evaporatedmaterial is less dense and can be more easily transportedoutwards by the wind.Again, the amplitude of the zig-zags is smaller for η = 10 than η = 10 . As a result, these simulations suggestthat a density constrast between the cold material and theICM of 10 is a good description, and that a wind of ve-locity ∼ × cm s − is required to produce the observedfilamentary structures. Furthermore, the kinematic zig-zagfeatures only seem to form after about 30 Myrs, suggestingthat the Perseus filaments are approximately this old.In each case, the total mass removed from the cold cloudwas of the order of 10 M ⊙ . However, the majority of thismaterial is not in the filament, but is in the more amorphousand disrupted region behind the cloud. The material aroundthe cloud is likely to have a low ionisation and is thereforenot likely to be optically emitting. Figure 14.
The density structure at 24 Myrs, for η = 10 andwind velocity = 1 . × cm s − . The length scale indicates thedistance from the cluster centre in kpc. The density scale showsthe density in code units ( ρ ′ ). The kinematics of the filaments indicate that momentumtransfer between the cool and hot phases of the ICM occurs.If there is a large enough quantity of cool gas, this couldhave important consequences for the flow of the hot phase.Momentum P is transferred from the hot ICM phase to acloud at a rate ˙ P clump = KAρ w ( v w − ˙ x ) , (26)If the radius of the cloud is ∼ v w − ˙ x ) ∼ − cm s − then ˙ P clump ∼ − erg cm − .A rough estimate for the rate at which momentum istransferred from the external medium to the cloud’s tail is˙ P tail = m tail ¨ x tail , (27)where m tail = Aρ tail l tail , ρ tail is the density in the tail, ¨ x tail is the typical acceleration in the tail, and l tail = l f is thelength of the tail. Constant cross-sectional area, density, andacceleration along the tail are assumed. The acceleration canbe approximated by ¨ x tail = ( ˙ x − u ) / l tail where ˙ x is the c (cid:13) , 1–15 E.C.D. Pope, J.M. Pittard, T.W. Hartquist, S.A.E.G. Falle
Figure 16.
Velocity profile for case η = 10 and wind velocity = 4 . × cm s − . Panels show data at 16 ,
23 and 38 Myrs. The lengthscale indicates the distance from the cluster centre. We have cut off the central 20 km s − to exclude points where there is no tail. Figure 17.
Velocity profile for case η = 10 and wind velocity = 1 . × cm s − . Panels show data at 8 ,
12 and 20 Myrs. The lengthscale indicates the distance from the cluster centre. We have cut off the central 20 km s − to exclude points where there is no tail. velocity of the clump, and u is the maximum velocity of thetail material. If the number density in the tail is about 10cm − , the tail radius is about 50 pc, and the magnitude of | u − ˙ x | is ∼ km s − , then˙ P tail ∼ „ n
10 cm − «„ R
50 pc « „ ˙ x − u
100 km s − « erg cm − (28)The corresponding rate at which systemic kinetic energy istransferred to the tail is ˙ E tail ∼ ˙ P tail | ( u − ˙ x ) | ,˙ E tail ∼ „ n
10 cm − «„ R
50 pc « „ ˙ x − u
100 km s − « erg s − (29)If the energy dissipated as thermal energy were comparableto this and all of that thermal energy were radiated in theobserved optical lines, about 100 tails would account for theobserved optical luminosity of the NGC 1275 filaments.The estimate above suggests that the tails are extremelymassive ∼ M ⊙ , though the mass of ionised material mustbe much lower (e.g. Hatch et al. 2007). This suggests thatthe filaments may consist of molecular material surroundedby a skin of ionised material.We can compare the momentum transfer between thephases of the ICM, with that developed by an AGN jet,which may in turn drive a wind from the central galaxy. Ajet injecting material at 1 M ⊙ yr − with a velocity of 0.1 times the speed of light over a lifetime of 10 Myrs, has˙ P ∼ g cm s − . This can be compared to the total mo-mentum transferred along the length of a filament by thewind ∼ ˙ xAl tail ρ tail ∼ g cm s − . This suggests that themomentum transfer to many tails could be important fordissipating the energy injected by an AGN in the ICM. Inparticular, since much of the cold material is located near thecentre of the cluster, if this dissipation mechanism is signifi-cant, then it will increase the AGN heating efficiency in thecentral regions of the cluster. Without this obstructing coldmaterial, more of the injected energy would be deposited atlarger radii, thus failing to heat the central regions whereenergy is most needed.To summarise this point it is worth reiterating that tailsonly seem to form if there is a significant outflowing wind- they are formed by the ambient wind dragging and accel-erating material, transferring energy and momentum to thecold material. As a result they are probably a sign of activityin the central galaxy. This mechanism can provide a way ofcoupling the wind to the cold material. Given that the coldclouds are likely to proliferate near the cluster this meansthat the energy will be dissipated more centrally than inthe absence of the cold material. As such, the cold materialincreases the effective opacity of the ICM to the outflowingwind.To estimate this effect, let us assume that energy is c (cid:13) , 1–15 ptical filaments in galaxy clusters Figure 18.
Velocity profile for case η = 10 and wind velocity = 1 . × cm s − . Panels show data at 0 . , . . × yrs. Thelength scale indicates the distance from the cluster centre. We have cut off the central 20 km s − to exclude points where there is no tail. injected at a rate ˙ E in , the energy dissipation rate per tailis ˙ E tail , and there are N tails. The change in power of thewind, over a region in which there are N clouds, will be∆ ˙ E = ˙ E in − N ˙ E tail . Taking the derivative with respect toradius, and assuming that ˙ E tail is a constant we find,d ˙ E d r = − d N d r ˙ E tail , (30)where d N/ d r is obtained from the spatial distribution of theclouds, d N d r = 4 πr n ( r ) (31)and n ( r ) is the concentration of clouds per unit volume. Ifwe assume that the clouds are distributed similarly to thehot gas, in a β -profile, then n ( r ) = n (1 + ( r/r ) ) β , (32)where n is the central density and r is a scale length.Choosing β = 1 means that equation (30) can be integratedanalytically to give,˙ E ( r ) = ˙ E in − N ˙ E tail » rr − arctan „ rr «– , (33)where N = 4 πn r . Therefore, the injected energy wouldbe dissipated over a typical length-scale r diss ∼ r ˙ E in N ˙ E tail . (34)Given that r is probably a constant of the system, and˙ E tail is also probably fairly constant, the important pa-rameters are ˙ E in and N . For a moderate AGN outburst,˙ E in ∼ erg s − , with ˙ E tail ∼ erg s − and N ∼ M ⊙ ), the dissi-pation length is of the same order as the scale-height of thecloud distribution. (This may be comparable to the scale-height of the ICM, but that is not clear. It is also possiblethat the clouds are more centrally concentrated than thehot ICM, but again, this is speculation.) For more powerfuloutbursts, a significant quantity of energy may be trapped,but the majority will still escape outside r .It is worth noting that if there has been significant re-cent cooling (i.e. a deficit of heating) we may expect N tobe larger, for a given system. For a given energy injection rate, the energy will necessarily be dissipated closer to thecentre than in an otherwise identical system where N islower. As a result, the presence of cold gas may provide anadditional feedback process that helps to trap energy in thecentral regions. This effect will be greater if there has beena recent deficit of heating, with the trapping effect beingreduced in the absence of cold material.This short derivation is, of course, a drastic simplifica-tion of real life. For example, in reality, the power dissipatedin a tail is a reflection of the power of the incident wind,which will undoubtedly be a function of radius. However,it highlights the possibility that the production of filamentsmay be able to trap a fraction the energy injected by a cen-tral AGN. The current study is centred on the Perseus cluster and con-cerns the interaction of material injected from one cloud intoa hot flow unaffected by other clouds. A global considerationof the effects of a distribution of clouds on the hot flow outto ∼
10 kpc would be required to investigate the problemmore completely, and we plan to investigate such problems.However, a full answer would require the developement of adetailed theory for the spatial and mass distributions of theclouds; given the state of the theory of our own interstellarmedium this task would be very difficult.The nature and number of filaments will depend on: a)the spatial, mass and velocity distributions of cold clouds atthe onset of AGN activity; b) the rate at which the AGNdeposits mechanical and thermal energy into its outflow andthe asymmetry of these AGN outflows; c) the pressure, ve-locity, and density distributions of the hot intracluster gasat the onset of AGN activity; d) the gravitational potentialof the galaxy. Below we consider, in sequence, how each ofthese could affect the filamentary structure.a) If all other factors were the same, the total numberof filaments and their combined optical luminosity wouldbe roughly proportional to the number of clouds that couldsurvive interactions with an outflow long enough for fila-ments to evolve. If the distribution of cloud masses were c (cid:13) , 1–15 E.C.D. Pope, J.M. Pittard, T.W. Hartquist, S.A.E.G. Falle identical in all clusters, the systems with the most cold gaswould exhibit the most luminous optical emission from fila-ments. This is consistent with the findings of Edge (2001).The clouds which survive for longest are likely to be themost massive.b) The number of massive clouds will depend on AGNactivity. If the clouds are formed in a steady fountain flow,the rate at which material turns into clouds would be pro-portional to the outflow rate of hot material as measurednear its the source. Roughly speaking, this mass outflowrate may be proportional to the mechanical luminosity of theAGN wind. In addition, the destruction rate of the cloudsmay also be proportional to the outflow rate. For example,a higher outflow rate could lead to a more rapid destructionrate of the clouds. Thus, it is possible that the number ofclouds producing filaments would be fairly insensitive to theAGN mechanical luminosity. However, it would be cavalierto assume the locations at which clouds form, and the ini-tial formation of the clouds, to be independent of the outflowrate.c) The properties of the ICM could affect the initialmass function of the clouds. If significant, thermal conduc-tion might establish the minimum lengthscale on which ther-mal instability grows. In hotter clusters, the minimum cloudmass would be higher, leading to a greater fraction of theclouds being formed with high masses. This would tend toincrease the number of long filaments.High ICM densities will lead to more rapid cooling andpossibly a higher formation rate of clouds, and hence morefilaments. However, given that AGN feedback is likely tobe important, high cooling rates may cause strong AGNactivity and thus reduce the cloud formation rate. The exactbalance between heating and cooling seems to vary fromcluster to cluster.Clouds in the ICM may be so plentiful in the centre ofa cluster that a bubble blown by the AGN becomes trappednear the cluster centre, preventing the formation of longfilament. So, high ICM density need not necessarily resultin the production of many filamentary tails.d) The gravity of the central galaxy may play a role inlimiting the size of the bubble mentioned in c). For example,the greater buoyancy may cause a bubble to detach sooner.Strong gravity may prevent an extended fountain or regionof outflows. Even if it does not, it may limit the lifetimes ofclouds as it would cause them to fall more rapidly.In summary, the question about scaling concerns verycomplex issues. A very long concentrated program would berequired to address them properly, and it is unclear howreliable the results of such a program would be.
This work is intended mainly as a preliminary study into theprocesses that shape filamentary structures in the ICM, andnot an exhaustive study. It is likely that there are many otherprocesses such as magnetic fields, thermal conduction andviscosity that all play a role in the dynamics that producethe filamentary kinematics and morphology. However, wehave concentrated on the basic hydrodynamic system andhave found several interesting mechanisms that may go someway to explaining the properties of the observed filaments. The first ensemble of simulations show that the long,relatively straight filaments at the centre of the Perseus clus-ter may be formed by the interaction of a wind with clumpsof cold material. Flows of low Mach number winds generatestructure in the filaments which may be comparable withobservations. These results also suggest that the amplitudeof the velocity fluctuations along the length of the filamentgrow with increasing density contrast between the cold ma-terial and the ambient wind. The morphology of the ob-served filaments suggests that the density contrast is large( ∼ ). Interestingly, the fluctuations evolve with time pro-viding a possible diagnostic tool for estimating the ages offilaments from observations. in this set of simulations, thebest comparison with observations suggests filament ages of ∼
40 Myrs in the Perseus cluster.In a more realistic environment with spatially varyinggravity, based on the Perseus cluster, filaments only form ifthe wind velocity is sufficiently large. If there is no significantwind, then the optical emission may be more amorphous, ormore likely, filamentary on much shorter length scales. Themorphology and kinematics suggests that a density contrastof η ∼ may be more compatible with the observations,while the best agreement in the kinematic data occurs for η ∼ . As a compromise it is possible that the real condi-tions lay somewhere between these two extremes. Alterna-tively, other physical processes could be important. It shouldalso be noted that these were 2-d simulations, and a full 3-dstudy would really be required to find the model conditionsthat best match reality. Regardless of this, the results dopoint to a relatively narrow range of parameters providingpossible information about the state of the ICM, and the ageof the filaments. These results suggest that the cold materialis of the order of 10 times denser than the ambient ICM,there is an outflowing wind of velocity of ∼ km s − , andthat the filaments are approximately a few 10’s of Myrs old.The masses, and densities, of the filaments created bythese simulations seem to be low compared with values pre-sented in the previous section. The simulations produce fil-aments of 10 − M ⊙ , while the crude estimates abovesuggested that 10 M ⊙ filaments might be more likely. Itshould be noted though, that the quantity of cold gas, inthe simulated tail, is more than capable of producing theobserved optical luminosity. The simulations also show a lotof cold, dense gas in the vicinity of the cloud itself.We also highlight the possibility that momentum trans-fer between the ICM and cold clouds could be a dynamicallyimportant process in galaxy clusters. This process could ef-fectively couple the energy and momentum of an outflow-ing wind to the cold material, thereby dissipating some ofthe energy injected by a central AGN. The presence of coldgas may therefore provide an additional feedback mechanismthat traps energy in the central regions of clusters where itis most needed. We acknowledge Professor John E. Dyson and Dr NinaHatch for many useful discussions and the anonymous ref-eree for helpful comments. c (cid:13) , 1–15 ptical filaments in galaxy clusters REFERENCES
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