Fast simulations of gas sloshing and cold front formation
MMon. Not. R. Astron. Soc. , 1–14 (2011) Printed 15 November 2018 (MN L A TEX style file v2.2)
Fast simulations of gas sloshing and cold front formation
E. Roediger (cid:63) , J. A. ZuHone Jacobs University Bremen, PO Box 750 561, 28725 Bremen, Germany NASA/Goddard Space Flight Center, 8800 Greenbelt Rd., Code 662, Greenbelt, MD 20771, USA
Accepted 1988 December 15. Received 1988 December 14; in original form 1988 October 11
ABSTRACT
We present a simplified and fast method for simulating minor mergers between galaxyclusters. Instead of following the evolution of the dark matter halos directly by theN-body method, we employ a rigid potential approximation for both clusters. Thesimulations are run in the rest frame of the more massive cluster and account forthe resulting inertial accelerations in an optimised way. We test the reliability of thismethod for studies of minor merger induced gas sloshing by performing a one-to-onecomparison between our simulations and hydro+N-body ones. We find that the rigidpotential approximation reproduces the sloshing-related features well except for twoartefacts: the temperature just outside the cold fronts is slightly over-predicted, andthe outward motion of the cold fronts is delayed by typically 200 Myr. We discussreasons for both artefacts.
Key words: galaxies: clusters: general - galaxies: clusters: individual: A2029 X-rays:galaxies: clusters methods: numerical - methods: N-body simulations
During the last decade, high resolution X-ray observationshave found a wealth of structure in the intra-cluster medium(ICM) of galaxy clusters, among them cold fronts (see re-view by Markevitch & Vikhlinin 2007). These structures re-veal themselves as sharp discontinuities in X-ray brightnessaccompanied by a jump in temperature, where the brighterside is the cooler one. One variety of cold fronts (CFs) wassoon understood to be the contact discontinuity between thegaseous atmospheres of merging clusters (e.g. A2142: Marke-vitch et al. 2000; A3667: Vikhlinin et al. 2001; and the bulletcluster 1E 0657-56: Markevitch et al. 2002).A second class of CFs was found to form arcs aroundthe cool cores of apparently relaxed clusters (e.g. RXJ1720.1+2638: Mazzotta et al. 2001; Mazzotta & Giacin-tucci 2008; Owers et al. 2009; MS1455.0+2232: Mazzotta &Giacintucci 2008; Owers et al. 2009; 2A0335+096 Mazzottaet al. 2003; Sanders et al. 2009; A2029: Clarke et al. 2004;Million & Allen 2009; A1795: Markevitch et al. 2001; Bour-din & Mazzotta 2008; Perseus: Churazov et al. 2003; Sanderset al. 2005; A496: Dupke et al. 2007; Ghizzardi et al. 2010;Virgo: Simionescu et al. 2010; Centaurus: Fabian et al. 2005;Sanders & Fabian 2006). Markevitch et al. (2001) suggestedthat this variety of CFs forms due to sloshing of the coolcentral gas within the central cluster potential, where thesloshing is initially triggered by a minor merger event. Usinghydro+N-body simulations, Ascasibar & Markevitch (2006) (cid:63)
E-mail: [email protected] (AM06 afterwards) have shown that the sloshing scenarioreproduces the morphology of observed CFs.In recent years, more and more sloshing CFs have beenreported (e.g. Owers et al. 2009; Ghizzardi et al. 2010).In principle, the properties of the CFs contain informa-tion about the merger history of the clusters, as has beenshown for the Virgo cluster (Roediger et al. 2011, R11ahereafter), A496 (Roediger et al. 2011, R11b hereafter), andRXJ1347.5-1145 (Johnson et al. 2011). However, disentan-gling the merger history of each cluster from the CF proper-ties requires a set of dedicated simulations for each cluster.Doing this with full hydro+N-body simulations is computa-tionally expensive. The same is true if the influence of moretime-consuming physics like viscosity (ZuHone et al. 2010),magnetic fields (ZuHone et al. 2011) or thermal conduction(ZuHone et al. 2011, in prep.) on sloshing CFs is studied. Areasonable simplification that speeds up the simulations con-siderably would be very useful. The most expensive part ofsuch simulations is the self-gravity of the gas and dark mat-ter (DM) particles. In most cases, the self-gravity of the gascan be neglected, because the Jeans length is about 1 Mpc.The simulations speed up substantially when the DM ha-los of the main cluster and the subcluster are approximatedas rigid potentials (RPs). However, AM06 have shown thateven in a minor merger the central part of the main clustermoves significantly w.r.t. the overall cluster potential, thusthis effect cannot be neglected. ZuHone et al. (2010) (Z10hereafter) suggested that the rigid potential approximationcan be used if additionally a point-mass-like approximation c (cid:13) a r X i v : . [ a s t r o - ph . C O ] S e p E. Roediger & J. ZuHone of the motion of both clusters is taken into account, includ-ing the effects of inertial acceleration.Here we improve this rigid potential approximationand demonstrate the reliability of the simplified simulationsfor the application to gas sloshing by comparing them tohydro+N-body simulations. This rigid potential approxima-tion has already been applied successfully in sloshing simu-lations for the Virgo cluster (R11a) and for A496 (R11b).
We consider the following scenario: The ICM in a mas-sive spherical galaxy cluster (the main cluster) is initiallyin hydrostatic equilibrium. A spherical, less massive, gasfree galaxy cluster (the subcluster) passes through the maincluster. The gravitational impact of the subcluster initiatessloshing of the ICM in the main cluster core and subsequentcold front formation (see AM06 for a detailed description ofthe dynamics).Hydro+N-body simulations of this scenario have beenperformed by AM06 and Z10. Our aim here is to investi-gate to what extent simulations with a rigid potential ap-proximation (RPA) for the DM content of, both, the maincluster and the subcluster can reproduce the resulting CFstructures in terms of morphology, orientation, size, tem-perature and density distribution. As the reference, we usethe hydro+N-body simulations of Z10. Thus, we follow theirlead and tailor our initial models to match theirs. The maincluster model is based on a Hernquist potential (Hernquist1990) with a scale radius, a . The temperature profile is de-scribed by the phenomenological function T ( r ) = T r/a c + r/a c r/a c , (1)where T is a measure for the overall cluster temperature, c describes the depth of the central density drop, and a c char-acterises the radius of this drop (see also AM06). The corre-sponding density profile resulting from hydrostatic equilib-rium is (AM06) ρ ( r ) = ρ (cid:18) ra c (cid:19) (cid:18) rca c (cid:19) α (cid:16) ra (cid:17) β (2)with α = − − n c − c − a/a c and β = 1 − n − a/a c c − a/a c . In our setup, we initialise the main cluster with these den-sity and temperature profiles, choosing the parameters suchthat they fit the corresponding hydro+N-body simulation(see Table 1). From these profiles, we derive the gravita-tional potential of the main cluster assuming hydrostaticequilibrium.As in Z10, the subcluster is initialised as a pure DMstructure, where the DM mass distribution is described bya Hernquist profile (Hernquist 1990).
Our simulations are run in the rest frame of the main cluster.The gas dynamics is described by the hydrodynamical equa-tions. Additionally, the ICM is subject to the gravitationalacceleration due to the main cluster and the subcluster.
We assume the orbit of the subcluster to be the orbit of a testparticle free-falling through the main cluster. In the courseof the simulation, the potential of the subcluster is shiftedthrough the main cluster along this orbit. This approachdoes not include dynamical friction, which will slow downthe subcluster after pericentre passage (see Sect. 3.3 for acomparison). However, the sloshing is triggered mainly dur-ing the pericentre passage, and thus our results do not sufferfrom this discrepancy. Given that our test particle orbit isbound to predict increasingly wrong subcluster positions af-ter the first pericentre passage, we stop our simulations wellbefore a second pericentre passage.We construct orbits that are comparable to the onesin the hydro+N-body simulations in orientation, pericentredistance, and velocity history prior to pericentre passage. Westart our simulations 1 Gyr prior to the pericentre passageof the subcluster, which we set to occur at t = 0. We reranour fiducial simulation with an initialisation time of 0 . xy -plane of the computa-tional grid. For the sake of a short notation, we identify the+ y -direction as ”north” (N), the − y -direction as ”south”(S), the + x -direction as ”west” (W), and the − x -directionas ”east” (E). The subcluster will start W of the main clus-ter core, has its closest approach to the main cluster core inthe NE and moves away towards the SE. As the rest frame of the main cluster is not an inertial frame,the ICM in this frame is subject to a pseudo-acceleration dueto the attraction of the main cluster core towards the ap-proaching subcluster. Z10 used the most simple approxima-tion to account for this: they assumed that the main clusterresponds to the gravity of the subcluster as a whole, like arigid body. Consequently, they calculate the inertial accel-eration felt by the main cluster centre due to the subclusterand add this pseudo-acceleration to all of the ICM.
This basic inertial frame correction is a reasonable approx-imation for the central region of the main cluster, but it iswrong for the outer parts of the cluster. It will lead to unreal-istic flows in the outer parts of the main cluster. For smallerpericentre distances, during pericentre passage of the sub-cluster the inertial acceleration can be large and even pro-duce supersonic motions in the cluster outskirts. These un-realistic flows can influence the resulting CFs at later stages.Hence, we propose to apply the pseudo-acceleration only tothe central region of the main cluster inside a characteristicradius, R damp , and dampen it outside this radius exponen-tially over a length scale, L damp . Thus, instead of addingthe same inertial frame acceleration at every position in thecluster, we multiply it with a radius-dependent function, W ( r ) = (cid:40) r (cid:54) R damp exp( − r − R damp L damp ) else , (3) c (cid:13)000
This basic inertial frame correction is a reasonable approx-imation for the central region of the main cluster, but it iswrong for the outer parts of the cluster. It will lead to unreal-istic flows in the outer parts of the main cluster. For smallerpericentre distances, during pericentre passage of the sub-cluster the inertial acceleration can be large and even pro-duce supersonic motions in the cluster outskirts. These un-realistic flows can influence the resulting CFs at later stages.Hence, we propose to apply the pseudo-acceleration only tothe central region of the main cluster inside a characteristicradius, R damp , and dampen it outside this radius exponen-tially over a length scale, L damp . Thus, instead of addingthe same inertial frame acceleration at every position in thecluster, we multiply it with a radius-dependent function, W ( r ) = (cid:40) r (cid:54) R damp exp( − r − R damp L damp ) else , (3) c (cid:13)000 , 1–14 ast gas sloshing simulations Table 1.
Summary of model parameters. See Sect. 2.1 for details. merger characteristics mass ratio 20 5 2impact parameter 200 kpc 500 kpc 500 kpcpericentre distance 62.5 kpc 150 kpc 135 kpc main cluster ρ / (10 − g cm − ) 3 .
75 2 . . a/ kpc 615 623 615 n T / (10 K) 15 . . a c c subcluster mass (10 M (cid:12) ) 0 .
714 2 . damping none, none, none,( R damp / kpc , (500, 300) (800, 300), (950, 400) L damp / kpc) (800, 100),(500, 300) where r is the distance to the main cluster centre.The choice of R damp is motivated by the sphere of in-fluence of the subcluster: If the subcluster passes the maincluster centre at a pericentre distance larger than its ownsize, it attracts the main cluster core only slightly. If thesubcluster passes the main cluster core at a small distance, itcan attract at most a region comparable to its own size, butnot beyond that. Hence, R damp should be comparable to thesize of the subcluster, i.e., about twice its characteristic scalelength. We test several combinations of ( R damp , L damp ). All simulations are run with the FLASH code (version 3.2,Dubey et al. 2009). FLASH is a modular block-structuredAMR code, parallelised using the Message Passing Interface(MPI) library. It solves the Riemann problem on a Carte-sian grid using the Piecewise-Parabolic Method (PPM). Thesimulations are performed in 3D and all boundaries are re-flecting. We use a simulation grid of size 4 × × ,which is large enough to prevent reflected waves reachingthe central region of interest during our simulation time.We resolve the inner 50 kpc with 2 kpc, the inner 130 kpcwith 4 kpc, the inner 260 kpc with 8 kpc and enforce de-creasing resolution with increasing radius from the clustercentre to optimise computational costs. We have performedresolution tests here and also in R11a,b and found our re-sults to be independent of resolution. We present simulations of three merger scenarios whichcover a range in mass ratios from 2 to 20. The completemodel parameters for each run are listed in Table 1.In the following subsections we give a detailed compar-ison of the RP simulations to the hydro+N-body ones. Inorder to aid the reader in smoothly following our analysis, we state our main finding already here: The RPA is able toreproduce the hydro+N-body simulations very well exceptfor two systematic differences, of which one can be fully cor-rected for, and the other partially: • After the onset of sloshing, the RP simulations lag be-hind the hydro+N-body ones by 200 to 250 Myr, dependingon the cluster mass ratio. This lag can be corrected for bydelaying the hydro+N-body results by the appropriate lagin the comparison. • The RPA generally over-estimates the temperature justoutside the CFs. This disagreement can be attenuated by thedamping discussed above, but not avoided completely.
Our fiducial case is the intermediate merger with a massratio of 5 between the clusters. We simulate this casewith four different settings for the inertial frame correc-tion as listed in Table 1: one without large-scale damp-ing, two moderate damping settings and a strong one.The best results are achieved for moderate damping with( R damp / kpc , L damp / kpc) = (800 , With both simulation methods, the gas sloshing evolves ina very similar manner. We demonstrate this in Fig. 1 byshowing snapshots of the ICM temperature in the orbitalplane. The first and third row are for the hydro+N-bodyrun, the remaining two rows for our fiducial RP simulation.The top two rows focus on the onset of sloshing. Wesee the subcluster pass the cluster centre (timesteps 0 and1 Gyr) from the NW over NE towards the SE. At 0.5 Gyr,sloshing has just set in, and an arc-like CF towards the S isaccompanied by a cool fan towards N. Both are surroundedby hotter ICM. These properties are alike in both methods.In the RPA, the cool fan takes a more spiral-like appearancecompared to the hydro+N-body case. Also the temperaturedistribution S of the southern CF differs between both meth-ods. The bottom six panels of Fig. 1 display the further evo-lution of the gas sloshing. As mentioned above, we find thatthe RP simulations lag behind the hydro+N-body ones by250 Myr for this mass ratio. Further details of this lag willbe discussed in Sect. 3.1.2.2. Hence, here we plot the resultsfrom the hydro+N-body simulations with a delay of 250 Myrin order to compare corresponding timesteps.In the intermediate phase (0.7 to 1.5 Gyr) the cool spi-ral typical for sloshing forms. The major CF is found in theSW, and a secondary CF evolves towards the NE. At itsoutside, the cool spiral is surrounded by a hot horse-shoeshaped region, which tends to be slightly too hot in the In a correct manner, the results of the RPA should be broughtforward instead of delaying the more accurate hydro+N-bodyones. However, there are several RP runs for each merger caseand only one hydro+N-body run. For the sake of simplicity andclarity we decide to apply the time shift to the hydro+N-bodyrun and trust the reader to remember this footnote.c (cid:13) , 1–14
E. Roediger & J. ZuHone h y d r o + N b o d y r i g i dp o t e n t i a l h y d r o + N b o d y r i g i dp o t e n t i a l Figure 1.
Comparison of hydro+Nbody simulations (first and third row) and rigid potential simulations (second and fourth row) forthe merger with mass ratio 5. The panels show the ICM temperature in the orbital plane (see colour scale). The panel size is 1 Mpc. Thetimestep is noted above each panel. The hydro+Nbody results are from Z10, replotted in our colour scale. The rigid potential simulationshown here uses the damping setting ( R damp / kpc , L damp / kpc) = (800 , (cid:13)000
Comparison of hydro+Nbody simulations (first and third row) and rigid potential simulations (second and fourth row) forthe merger with mass ratio 5. The panels show the ICM temperature in the orbital plane (see colour scale). The panel size is 1 Mpc. Thetimestep is noted above each panel. The hydro+Nbody results are from Z10, replotted in our colour scale. The rigid potential simulationshown here uses the damping setting ( R damp / kpc , L damp / kpc) = (800 , (cid:13)000 , 1–14 ast gas sloshing simulations RPA. This evolution is the same in all cases, and the resultsof both methods agree well in morphology and orientation ofthe cool spiral. In the late phase (2 Gyr), the CF in the SWstarts to break apart in the hydro+N-body simulation. Inthe RP simulation, the CFs remain intact and the morphol-ogy remains close to spiral-like. We note that this break-upof the spiral structure can be recovered by using strongerdamping (Fig. 6; see Sect. 3.2 for a more detailed discussionof the effect of damping).
The outwards motion of the CFs and hence growth of thecool spiral are important characteristics of the sloshing pro-cess. In their studies of the Virgo and Abell 496 clusters,R11a,b found that the velocity of the outward motion islargely independent of the subcluster, but is characteristicfor the potential of the main cluster. This means that thepositions of the CFs in a given cluster depend mostly on thetime since the subcluster’s pericentre passage, i.e. the age ofthe CFs. Therefore it is important to know to what extentthe RPA recovers this outward motion. This is the aim ofthis subsection.
We study the radiiof the CFs towards the diagonal directions in the orbitalplane. For this purpose, we first derive radial temperatureprofiles towards NW, SW, SE and NE, where each profile isaveraged over an azimuthal extent of ± o (see Fig. 3 forprofiles and Sect. 3.1.3 for their discussion). Given that CFsrarely form perfect circular concentric arcs around the clus-ter centre, this azimuthal averaging introduces a smearingout of the intrinsically discontinuous CFs over a finite ra-dial range. This occurs even for the small azimuthal rangewe use for averaging. Consequently, in the profiles, the CFsdo not appear as a true discontinuity, but as steep slopes intemperature that stretch over typically 10 to 50 kpc. In eachtemperature profile, we identify CFs as regions of tempera-ture slopes above 0 .
02 keV / kpc. Thus, we identify an innerand outer edge for each CF, and the nominal CF radius isdefined as the average radius between this inner and outeredge.The first time step at which a CF can be detected is notimmediately after the subcluster’s pericentre passage, buttypically 0.4 Gyr afterwards. In the hydro+N-body simula-tion the CFs in the NW and NE direction are establishedonly at t = 1 Gyr. At even later times, in all but the NEdirection a second CF at smaller radii is detectable. Having derived the positions of the CFs at each timestep,we can now proceed to analyse their outward motion. Wedo so in Fig. 2, where we plot the temporal evolution of theCF radii towards the diagonal directions (NE, NW, SE, andSW) in the orbital plane. We use error bars to indicate thewidth of each CF, i.e. from its inner to outer edge. The RPsimulations with different settings are plotted by colouredlines. Here we focus on the red line, which marks our fiducialRP run.The result from the hydro+N-body simulation is thedashed black line. The RP results differ systematically from this reference: At a given timestep, the RP simulations pro-duce somewhat too small radii for the outermost CFs. Thisis true for all directions except NW. Thus, in general, the RPsimulations lag behind the hydro+N-body one. Plotting thehydro+N-body result with a delay of 250 Myr (solid blackline with error bars, and remember footnote 1) compensatesthe difference in all but the NW direction, leading to a goodagreement to the RP simulation. The second CFs agree wellbetween all runs.This lag in CF motion is the major systematic differencebetween two simulation methods. Consequently, we use thedelay of 250 Myr for the full hydro+N-body simulation inall other comparisons regarding the fiducial merger case.
In order to go beyond the qualitative comparison of the tem-perature slices in Fig. 1, we derive radial temperature anddensity profiles as described above in Paragraph 3.1.2.1. InFig. 3 we compare these profiles for different realisations forthe fiducial merger. Again, here we concentrate on the redand black lines, which are for the fiducial RP simulation andthe hydro+N-body one, respectively.At 0.5 Gyr, the sloshing is still in the onset phase andwe do not apply the delay discussed above. Still, already herethe CF in the SW is ahead in the hydro+N-body simulation.Here, the RPA run shows a weaker impact of the subclusterpassage on the temperature. Also the density profile in thenorthern directions are not accurately reproduced. This isthe region the subcluster directly passes and the strongestdifferences are to be expected.In all later timesteps we apply the delay of 250 Myr tothe hydro+N-body simulation as derived in Sect. 3.1.2.2. Asa result, we achieve a good agreement between both meth-ods. Especially along the SW-NE axis, which is perpendic-ular to the subcluster orbit, the agreement is excellent. Theonly systematic difference between both methods is that theRPA over-predicts the temperature just outside the CFs,which we have already seen in Fig. 1.
Here we aim at comparing the evolution of the temperatureand density at the CFs. For this purpose, we derive bothquantities at the inner and outer edge of each CF along withits radius (see Paragraph 3.1.2.1). We note that the combi-nation of azimuthal and radial binning introduces an uncer-tainty in temperature of at least ± . c (cid:13) , 1–14 E. Roediger & J. ZuHone r C F / k p c t/GyrSE 0 50 100 150 200 250 r C F / k p c NE nodampdamp 500, 300damp 800, 300hydro+Nbody, t delay =250 Myrhydro+Nbody
Figure 2.
Outward motion of the cold fronts towards the diagonal directions in the xy -plane (one direction per panel, see label). For thederivation of the cold front radii, see Paragraph 3.1.2.1. The error bars indicate the width of the cold front as it appears in the azimuthallyaveraged profiles. In all but the NE panel, the evolution of the outermost and second cold front is shown. The figure compares the resultsfor different realisations of the mass ratio 5 merger. The black dashed line without error bars represents the hydro+N-body run, theother coloured/broken lines are for the results from the rigid potential simulations with different damping settings (see legend). Clearly,the rigid potential simulations lag behind the hydro+N-body one. Plotting this reference with a delay of 250 Myr (and rememberingfootnote 1) corrects for this lag. potential simulations first underestimate the outer temper-ature and overestimate it at later times. The reason for thisdifference is that the rigid potential approximation leads toa different flow field outside the central region of the clus-ter. In all but the early times, this leads to a stronger com-pressional heating at the outer side of the CFs and thus ahigher temperature. This systematic difference can be less-ened somewhat by the damping, but is not prevented com-pletely.We apply the same analysis to the density inside andoutside each CF and present the result in Fig. 5. For thefiducial RP simulation, the densities at the CFs are repro-duced well. Thus, even at late stages, where the RPA doesnot accurately estimate the temperatures outside the CFs,it is still accurate for the densities. In addition to the cold spiral structure in the cluster cen-tre, also the large-scale distribution of the ICM density atleast out to 500 kpc is reproduced well in the RPA. Bothsimulation methods find the characteristic asymmetry in thesense that profiles of, both, density and temperature on op-posite sides of the cluster centre alternate around each other,switching over at the CFs. We have illustrated this effect inFig. 3 in the density panels for t = 1 . The damping of the inertial frame correction described inSect. 2.2.3 is constructed such that it gradually switches offthe inertial frame correction in the cluster outskirts, whichshould not be applied there. Using the RPA without this”re-correction” leads to an unrealistic ICM velocity field inthe outer cluster regions and two artefacts compared to thehydro+N-body reference run: a cooler temperature in theouter northern region and the hotter temperature outsideall CFs. Both effects can be seen in the temperature slices(top panel of Fig. 1) and all other comparison plots (Figs. 3,4 and 5).Using a strong damping of ( R damp / kpc , L damp / kpc) =(500 , t = 1 Gyr, which is the di-rection of motion of the subcluster (Fig. 2). The CF radiitowards the NE and SW, i.e. along the axis perpendicularto the orbit, are independent of damping at all times.In summary, the stronger the damping, the more accu-rate is the outer temperature distribution, but at some ex-pense of the accuracy in the cluster centre. Hence, we preferto use a moderate damping which ensures an accurate re- c (cid:13) , 1–14 ast gas sloshing simulations -27 -26 -27 -26 ρ / g / c m r SE-NW / kpc 0.5Gyrhydro+Nbodynodampdamp 500,300damp 800,300 4 5 6 7 8 9 10 -400 -200 0 200 400 T / k e V r SE-NW / kpc 0.5Gyr5·10 -27 -26 -27 -26 ρ / g / c m r SW-NE / kpc 0.5Gyr 4 5 6 7 8 9 10 -400 -200 0 200 400 T / k e V r SW-NE / kpc 0.5Gyr r
SE-NW / kpc 1.Gyr-400 -200 0 200 400r
SE-NW / kpc 1.Gyrr
SW-NE / kpc 1.Gyr-400 -200 0 200 400r
SW-NE / kpc 1.Gyr r
SE-NW / kpc 1.5Gyr-400 -200 0 200 400r
SE-NW / kpc 1.5Gyrr
SW-NE / kpc 1.5Gyr-400 -200 0 200 400r
SW-NE / kpc 1.5Gyr 5·10 -27 -26 -27 -26 ρ / g / c m r SE-NW / kpc 2.Gyr-400 -200 0 200 400 4 5 6 7 8 9 10 T / k e V r SE-NW / kpc 2.Gyr 5·10 -27 -26 ρ / g / c m r SW-NE / kpc 2.Gyr-400 -200 0 200 400 4 5 6 7 8 9 10 T / k e V r SW-NE / kpc 2.Gyr
Figure 3.
Comparison of density and temperature profiles along diagonals in the xy -plane. The columns are for different timesteps asindicated in each panel. The two top rows are for the SE-NW direction, the two bottom rows for the SW-NE direction. The black solidlines show the hydro+N-body simulation, coloured broken lines lines the rigid potential simulations with different damping. Profiles areaveraged over ± o around the indicated direction. The hydro+N-body simulation is plotted with a delay of 250 Myr in all but the 0 . t = 1 . production of the CF radii, temperatures inside them anddensities at, both, the inside and outside.The case (800, 100) is very similar to our fiducial setting(800, 300) demonstrating that the results are not sensitiveto the choice of the fall-off length scale for the damping, L damp . In addition to the ICM properties, we also compare the sub-cluster orbit w.r.t. the cluster centre from both methods inFig. 7. The test particle orbit used in the RP simulations ap-proximates the true trajectory of the subcluster very well.We have marked the subcluster position in steps of 250 Myralong both orbits, demonstrating that the test particle ap-proximation also recovers the motion of the test particlealong its orbit up to a few 100 Myr after pericentre pas-sage. After that, dynamical friction causes the subclusterto decelerate significantly, such that its new apocentre dis-tance is only about 1.8 Mpc, and the apocentre is reachedalready 1.25 Gyr after pericentre passage. The test particlemethod does not capture this effect, and predicts a com-parable cluster-centric distance already after about 0.6 Gyrafter pericentre passage. Taylor & Babul (2001) proposed an algorithm to incorporate dynamical friction in a simpli-fied form. However, the gas sloshing we are interested in istriggered by the pericentre passage and thus is unaffectedby this difference in subcluster position at later epochs.In Fig. 8 we demonstrate the evolution of the subclusterby plotting its mass within its scale radius. Clearly, duringpericentre passage and up to 0.2 Gyr afterwards the sub-cluster suffers tidal compression. It starts loosing mass sig-nificantly only 0.5 Gyr after pericentre passage. This is wellafter triggering the gas sloshing and thus has no significanteffect on the further evolution of the ICM in the main clus-ter. We will discuss the differences in the evolution of themain cluster potential in Sect. 4.1.
We have run the same comparison for a minor merger witha larger mass ratio of 20. Here, the subcluster has a smallerscale radius of only 220 kpc, hence the damping setting( R damp / kpc , L damp / kpc) = (500 , c (cid:13) , 1–14 E. Roediger & J. ZuHone T / k e V r CF / kpcSE 2 3 4 5 6 7 8 9 10 T / k e V NEinitial T(r)hydro+Nbodynodampdamp 800,300damp 500,300
Figure 4.
Temperature inside and outside of the cold front as a function of cold front radius. We show results for the four diagonaldirections in the xy -plane. Different line colours/styles code different realisations of the mass ratio 5 merger, see legend. In each panel,the upper and lower sets of lines denote the temperatures just outside and inside the cold fronts, respectively. For comparison, we plotthe initial temperature profile as the dotted line. -26 -25 NW 50 100 150 200 250 10 -26 -25 SW10 -26 -25
50 100 150 200 250 ρ / g c m - r CF / kpcSE10 -26 -25 ρ / g c m - NEinitial ρ (r)hydro+Nbodynodampdamp 800,300damp 500,300 Figure 5.
Same as Fig. 4, but for density inside and outside of the cold fronts (upper and lower set of lines, respectively). case with a much smaller subcluster, the sloshing in theRPA runs lags behind the hydro+N-body one by 200 Myr.Hence, we come to the same results and conclusions as inthe fiducial case.
A merger with a mass ratio of 2 is no minor merger any-more, and we perform this simulation with the purpose of exploring the limits of the RPA. Here, the subcluster hasa scale radius of 416 kpc and we use a damping setting of( R damp / kpc , L damp / kpc) = (950 , c (cid:13)000
A merger with a mass ratio of 2 is no minor merger any-more, and we perform this simulation with the purpose of exploring the limits of the RPA. Here, the subcluster hasa scale radius of 416 kpc and we use a damping setting of( R damp / kpc , L damp / kpc) = (950 , c (cid:13)000 , 1–14 ast gas sloshing simulations Figure 12.
Snapshots of the temperature in the orbital plane for the merger with mass ratio 2. We show the hydro+N-body results (upperrow) with a delay of 250 Myr compared to the rigid potential run (bottom row). This hydro+N-body simulation has a low resolution of10 kpc, which is at least partially responsible for the loss of the cool dense centre. The major cold front in the SW is still reproducedwell, but the general distortion of the cluster by this major merger is not captured anymore by the rigid potential approximation. -27 -26 -26 ρ / g / c m r SE-NW / kpc 1.Gyrhydro+Nbody, delay 250 Myrnodampdamp 950, 400 3 4 5 6 7 8-400 -300 -200 -100 0 100 200 T / k e V r SE-NW / kpc 1.Gyr5·10 -27 -26 -26 ρ / g / c m r SE-NW / kpc 1.Gyr 3 4 5 6 7 8-400 -300 -200 -100 0 100 200 T / k e V r SE-NW / kpc 1.Gyr r
SE-NW / kpc 1.5Gyr-400 -300 -200 -100 0 100 200r
SE-NW / kpc 1.5Gyrr
SW-NE / kpc 1.5Gyr-400 -300 -200 -100 0 100 200r
SW-NE / kpc 1.5Gyr 5·10 -27 -26 -26 ρ / g / c m r SE-NW / kpc 2.Gyr-400 -300 -200 -100 0 100 200 3 4 5 6 7 8 T / k e V r SE-NW / kpc 2.Gyr 5·10 -27 -26 ρ / g / c m r SW-NE / kpc 2.Gyr-400 -300 -200 -100 0 100 200 3 4 5 6 7 8 T / k e V r SW-NE / kpc 2.Gyr
Figure 13.
Comparison of density and temperature profiles along the diagonals in the xy -plane. Same as Fig. 3 but for a mass ratio of2 between the clusters, which is already a major merger. The hydro+N-body simulation is plotted with a delay of 250 Myr.c (cid:13) , 1–14 E. Roediger & J. ZuHone
Figure 6.
Temperature slices for the fiducial run at t = 1 . -1500-1000-500 0 500-2000-1500-1000 -500 0 500 1000 1500 2000 y / k p c x / kpc RPAhydro+Nbody Figure 7.
Comparison of the subcluster orbit in the hydro+N-body simulation (black solid line) and the rigid potential approx-imation (red dashed line) for the merger with mass ratio 5. Wemark the subcluster positions in 250 Myr intervals. While thedirection of the orbit is reproduced well in the rigid potential ap-proximation, the neglect of the dynamical friction leads to signifi-cant over-estimation of the velocity after 0.5 Gyr after pericentrepassage.
However, in the hydro+N-body code the cool central gasmoves completely out of the cluster centre, forming a CFtowards the NE of the centre only at very late stages. Inthe RPA, the cool gas core is never completely displacedfrom the potential minimum. The comparison of profiles inFig. 13 reveals that the RPA still achieves a good agreementfor the primary CF in the SW and SE, despite the nearlyequal masses of both clusters.Thus, even for this major merger, the RPA recovers theevolution of the major CF. However, it comes to its limitregarding the morphology, the other CFs, and the large-scaledistortion of the main cluster. M s ub , < a / M s un t / Gyr Figure 8.
Evolution of subcluster mass within the subclusterscale radius. During pericentre passage and up to 0.2 Gyr after-wards the subcluster suffers tidal compression. It starts loosingmass significantly only 0.5 Gyr after pericentre passage.
Figure 9.
Snapshots of the temperature in the orbital plane forthe merger with mass ratio 20. We show the hydro+N-body re-sults (upper row) with a delay of 200 Myr compared to the rigidpotential run (bottom row). Both methods give very similar re-sults.
We have shown that the rigid potential approximation canreproduce the characteristics of gas sloshing well except fortwo artefacts, the higher temperatures at the outside of theCFs and the temporal lag in evolution. We have traced backthe former effect to the necessarily unrealistic gas motionsin the outer cluster region in this approximation. In order toinvestigate the origin of the latter, we compare the evolutionof radial potential profiles in the fiducial case in Fig. 14.We show profiles towards the N, S, W and E, where weplot one direction per panel. Different timesteps are colour-coded (coded by line style in print version). The results fromhydro+N-body and from the RPA are shown by solid linesand dashed lines, respectively (thick and thin lines in printversion). c (cid:13)000
We have shown that the rigid potential approximation canreproduce the characteristics of gas sloshing well except fortwo artefacts, the higher temperatures at the outside of theCFs and the temporal lag in evolution. We have traced backthe former effect to the necessarily unrealistic gas motionsin the outer cluster region in this approximation. In order toinvestigate the origin of the latter, we compare the evolutionof radial potential profiles in the fiducial case in Fig. 14.We show profiles towards the N, S, W and E, where weplot one direction per panel. Different timesteps are colour-coded (coded by line style in print version). The results fromhydro+N-body and from the RPA are shown by solid linesand dashed lines, respectively (thick and thin lines in printversion). c (cid:13)000 , 1–14 ast gas sloshing simulations -12-11-10-9-8-7 0 50 100 150 200 250 300 350 400 450N -1Gyr0Gyr0.1Gyr0.2Gyr1Gyr 0 50 100 150 200 250 300 350 400 450-12-11-10-9-8-7W-12-11-10-9-8-7 0 50 100 150 200 250 300 350 400 450 Φ i n k m / s r / kpcS-12-11-10-9-8-7 0 50 100 150 200 250 300 350 400 450E Figure 14.
Evolution of radial potential profiles in the main cluster towards N, W, S, E. The profiles are averaged azimuthally over ± o .Hydro+N-body simulations are shown by solid lines (thick lines in print version), rigid potential simulations by dashed lines (thin linesin print version). The colours (line styles in print version) code different timesteps, see legend. In both methods, the subcluster passageleads to a temporary deepening of the central potential. Additionally, the hydro+N-body method captures the tidal compression of themain cluster centre, leading to a stronger deepening and steepening of the potential. Both effects wear off with time and 1 Gyr afterlater the initial potential is nearly recovered. However, already after about 0.25 Gyr, the slopes of the potential are very similar in bothmethods. r C F / k p c t/GyrSE 0 20 40 60 80 100 120 140 160 r C F / k p c NE hydro+Nbodynodampdamp 500,300
Figure 10.
Same as Fig. 2, but for the merger with a mass ratioof 20. The result from the hydro+N-body simulation is plottedwith a delay of 200 Myr.
Generally, the central potential deepens and steepensduring the passage of the subcluster. The direct overlap withthe subcluster potential can also lead to a temporary localflattening of the potential, e.g. at t = 0 . T / k e V r CF / kpcSE 3 4 5 6 7 8 9 T / k e V NE Figure 11.
Same as Fig. 4, but for the merger with a mass ratioof 20. passage evident from Fig. 14. This leads to a temporarilydifferent evolution of the central potentials between bothmethods. At 1 Gyr after pericentre passage, in both methodsthe potential is nearly back to its initial state, where theclosest ”recovery” is achieved in the S. This explains whythe major, southern CF evolves so similar in both methods.In order to access the timescales of the potential evolu-tion more directly, we study the evolution of the radial gravi-tational acceleration in different positions. Along each direc- c (cid:13) , 1–14 E. Roediger & J. ZuHone - a / k m s - M y r - t / Gyr S 6 8 10 12 14 16 -1 -0.5 0 0.5 1 1.5E Figure 15.
Evolution of radial gravitational acceleration at different radii towards N, W, S, E. Comparison between hydro+N-bodysimulations (solid lines) and rigid potential simulations (dashed lines). The colours code different radii, see labels. The effect of thesubcluster passage is clearly seen in both methods. The effect is stronger in the hydro+N-body method because it captures the tidalcompression of the cluster core. After about 0.25 to 0.5 Gyr, the acceleration is again similar in both methods. Only in the inner 100kpc, the acceleration is slightly lower in the hydro+N-body method. Thus, after a slightly different onset, the sloshing proceeds in a verysimilar way afterwards. tion, we calculate the gravitational acceleration at four radiiand plot their temporal evolution in Fig. 15. Again, there isone panel per direction, and the radii are colour-coded. Solidlines are for the hydro+N-body code, dashed for the RPA.Here we see more clearly the modification of the gravita-tional field described above. Prior to pericentre passage, theaccelerations in both methods agree well. During pericentrepassage, the tidal compression leads to stronger accelera-tions in the hydro+N-body method. After about 0.25 to 0.5Gyr, the accelerations have reached the final levels at whichthey remain until the next pericentre passage. Outside 200kpc, the initial acceleration is nearly recovered, whereas in-side 100 kpc the final acceleration is slightly lower than theoriginal one. The similarity of the accelerations, i.e. the po-tential slopes, after the onset of sloshing accounts for thesimilar evolution of the CFs in both methods. The period ofdifferent potential evolution at and shortly after the pericen-tre passage is responsible for the different evolution duringthis phase and sends the hydro+N-body method ahead, be-cause in a steeper potential the sloshing oscillations tend tobe faster.
In Sect. 2.2.3 we suggested that the damping radius, R damp ,outside which the inertial frame correction will be switched off, should be comparable to the diameter of the subclus-ter, i.e. twice the subcluster scale radius. We investigatedthe impact of the damping parameters (Sect. 3.2) for allcases studied here and found that this choice leads indeedto the best possible reproduction of all features. Moreover,we find that the evolution of CF radii is the same with thismild damping and in the undamped case, but CF radii aresomewhat lower if the damping is chosen too strong. Thisdifference in behaviour can be used to confirm the choice of R damp for clusters where no hydro+Nbody simulation is yetavailable. As long as the CF radii in the damped simulationagree with the ones in the undamped case, the choice of R damp is reasonable. Regarding the damping scale length, L damp , we found that our results are not sensitive to thisparameter, and we recommend using L damp ≈ . R damp , aswe have done here. We also followed this strategy for choos-ing R damp and L damp in our simulations of Virgo and A496(R11a,b). We investigated the reliability of the rigid potential approxi-mation described in Sect. 2.2 for simulations of minor mergerinduced gas sloshing. We use the hydro+N-body simulationsof Z10 of the same scenario as the reference. Those capture c (cid:13)000
In Sect. 2.2.3 we suggested that the damping radius, R damp ,outside which the inertial frame correction will be switched off, should be comparable to the diameter of the subclus-ter, i.e. twice the subcluster scale radius. We investigatedthe impact of the damping parameters (Sect. 3.2) for allcases studied here and found that this choice leads indeedto the best possible reproduction of all features. Moreover,we find that the evolution of CF radii is the same with thismild damping and in the undamped case, but CF radii aresomewhat lower if the damping is chosen too strong. Thisdifference in behaviour can be used to confirm the choice of R damp for clusters where no hydro+Nbody simulation is yetavailable. As long as the CF radii in the damped simulationagree with the ones in the undamped case, the choice of R damp is reasonable. Regarding the damping scale length, L damp , we found that our results are not sensitive to thisparameter, and we recommend using L damp ≈ . R damp , aswe have done here. We also followed this strategy for choos-ing R damp and L damp in our simulations of Virgo and A496(R11a,b). We investigated the reliability of the rigid potential approxi-mation described in Sect. 2.2 for simulations of minor mergerinduced gas sloshing. We use the hydro+N-body simulationsof Z10 of the same scenario as the reference. Those capture c (cid:13)000 , 1–14 ast gas sloshing simulations the full evolution of the ICM and DM components includingdynamical friction, tidal compression and tidal stripping. Incontrast, the rigid potential approximation treats the poten-tials of the individual clusters as static and models only theirrelative motion. This simplification makes the rigid potentialsimulations faster by about a factor of 5 for the resolutionused here, and more for higher resolution due to the im-perfect scaling of the Poisson solver. This speed-up is veryuseful in several circumstances. For example, constrainingthe merger history of a given cluster by reproducing the ob-served sloshing signatures usually requires a large set of sim-ulations (see R11a for Virgo cluster and R11b for Abell 496).Also investigations of the impact of more time-consumingphysics like viscosity (Z10) and thermal conduction benefitfrom a fast method for the basic process.While we expect and find temporal differences in theevolution of the gravitational potentials, regarding the gassloshing, our main interest lies in the evolution of the ICM.Hence, we have simulated three representative merger sce-narios using both methods and compared them in detail. Wehave shown that, except for two (correctable) artefacts, therigid potential approximation reproduces the results of thehydro+N-body runs very well:(i) The rigid potential approximation reproduces the typ-ical sloshing cold fronts and central cold spiral structure in,both, morphology and orientation (Figs. 1, 9, 12).(ii) The minor merger induces a characteristic large-scaleasymmetry in the main cluster’s ICM beyond the centralcool spiral structure, which is reproduced at least qualita-tively in, both, density and temperature (Fig. 3).(iii) The outward motion of the cold fronts with time isdelayed in the rigid potential approximation by typically200 Myr compared to the hydro+N-body one. This origi-nates from a temporarily different potential shape duringpericentre passage because the rigid potential approxima-tion does not capture tidal compression. The significantlydifferent evolution is, however, short-lived, and the evolutionproceeds very similar afterwards. Therefore, if this delay iscorrected for, also the outward motion of the cold fronts isreproduced very well by the rigid potential approximation(Figs. 2, 10).(iv) The ICM density on both sides of the cold fronts isreproduced well, and so is the temperature inside the coldfronts (Figs. 4, 5, 11). The temperature at the outside of thecold fronts tends to be slightly too high compared to the ref-erence simulations due to a necessarily unrealistic velocityfield in the cluster outskirts in the rigid potential approxi-mation (Sect. 3.2). This effect is strongest if only the basicinertial frame correction is used (Sect. 2.2.2), and is milderfor the improved version, where the inertial frame correctionis only applied to the central part of the galaxy cluster andgradually dampened towards large radii (Sect. 2.2.3). Toostrong dampening, however, causes slightly too small coldfront radii towards the directions aligned with the subclus-ter orbit. Our tests recommend dampening outside cluster-centric radii of about twice the subcluster scale radius, witha characteristic fall-off scale length comparable to the sub-cluster scale radius.Thus, the rigid potential approximation method can be em-ployed e.g. in order to disentangle the merger history for anobserved cluster (R11a,b). The agreements in items (i) and (ii) guarantee that the orientation of the subcluster orbitcan be inferred correctly. Item (iii) ensures that the age ofthe cold front can be estimated reasonably, although here theuncorrected rigid potential approximation will over-estimatethe age by about 200 Myr. If the age needs to be determinedmore accurately, especially during the onset of sloshing orthe early evolution of the cold fronts, hydro+N-body sim-ulations are required. The mass of the subcluster can inprinciple be decoded from the contrast of density and tem-perature across the cold fronts. However, this attempt is in-trinsically difficult, because usually the cold fronts are foundwithin the cool cores of their host clusters, where the gen-eral gradient of, both, the density and temperature is in thesame direction as the cold front discontinuity itself. Giventhat azimuthal and radial averaging in deriving radial pro-files smears out the cold fronts over a certain radial extent,the contrast of all quantities between the inner and outeredge of a given cold front includes the general variation ofthe quantity across in this radial range. The contributionsof the general profile and the cold front discontinuity to thiscontrast are hard to separate, no matter which simulationmethod has been used. ACKNOWLEDGMENTS
ER is supported by the Priority Programme ”Witnesses ofCosmic History” of the DFG (German Research Founda-tion) and the supercomputing grants NIC 3711 and 4368at the John-Neumann Institut at the ForschungszentrumJ¨ulich. JAZ is supported under the NASA postdoctoral pro-gram. We thank Marcus Br¨uggen for helpful discussions,and the referee Max Ruffert for his clarifying comments.The results presented were produced using the FLASH code,a product of the DOE ASC/Alliances-funded Center forAstrophysical Thermonuclear Flashes at the University ofChicago.
REFERENCES
Ascasibar Y., Markevitch M., 2006, ApJ, 650, 102Bourdin H., Mazzotta P., 2008, A&A, 479, 307Churazov E., Forman W. R., Jones C., Bohringer H., 2003,ApJ, 590, 225Clarke T. E., Blanton E. L., Sarazin C. L., 2004, ApJ, 616,178Dubey A., Antypas K., Ganapathy M. K., Reid L. B., RileyK., Sheeler D., Siegel A., Weide K., 2009, Parallel Com-puting, 35, 512Dupke R., White III R. E., Bregman J. N., 2007, ApJ, 671,181Fabian A. C., Sanders J. S., Taylor G. B., Allen S. W.,2005, MNRAS, 360, L20Ghizzardi S., Rossetti M., Molendi S., 2010, A&A, 516,A32Hernquist L., 1990, ApJ, 356, 359Johnson R. E., ZuHone J. A., Jones C., Forman W., Marke-vitch M., 2011, eprint arXiv:1106.3489Markevitch M., Gonzalez A. H., David L., Vikhlinin A.,Murray S., Forman W. R., Jones C., Tucker W., 2002,ApJ, 567, L27 c (cid:13) , 1–14 E. Roediger & J. ZuHone
Markevitch M., Ponman T. J., Nulsen P. E. J., BautzM. W., Burke D. J., David L. P., Davis D., Donnelly R. H.,Forman W. R., Jones C., Kaastra J., Kellogg E., Kim D.,Kolodziejczak J., Mazzotta P., Pagliaro A., Patel S., VanSpeybroeck L., Vikhlinin A., Vrtilek J., Wise M., Zhao P.,2000, ApJ, 541, 542Markevitch M., Vikhlinin A., 2007, Physics Reports, 443,1Markevitch M., Vikhlinin A., Mazzotta P., 2001, ApJ, 562,L153Mazzotta P., Edge A. C., Markevitch M., 2003, ApJ, 596,190Mazzotta P., Giacintucci S., 2008, ApJ, 675, L9Mazzotta P., Markevitch M., Vikhlinin A., Forman W. R.,David L. P., VanSpeybroeck L., 2001, ApJ, 555, 205Million E. T., Allen S. W., 2009, MNRAS, 399, 1307Owers M. S., Nulsen P. E. J., Couch W. J., Markevitch M.,2009, ApJ, 704, 1349Roediger E., Br¨uggen M., Simionescu A., B¨ohringer H.,Churazov E., Forman W. R., 2011, MNRAS, 413, 2057Roediger E., Lovisari L., Dupke R. A., Ghizzardi S.,Br¨uggen M., 2011, MNRAS, submittedSanders J. S., Fabian A. C., 2006, MNRAS, 371, 1483Sanders J. S., Fabian A. C., Dunn R. J. H., 2005, MNRAS,360, 133Sanders J. S., Fabian A. C., Taylor G. B., 2009, MNRAS,396, 1449Simionescu A., Werner N., Forman W. R., Miller E. D.,Takei Y., B¨ohringer H., Churazov E., Nulsen P. E. J.,2010, MNRAS, 405, 91Taylor J. E., Babul A., 2001, ApJ, 559, 716Vikhlinin A., Markevitch M., Murray S. S., 2001, ApJ, 551,160ZuHone J. A., Markevitch M., Johnson R. E., 2010, ApJ,717, 908ZuHone J. A., Markevitch M., Lee D., 2011, eprintarXiv:1108.4427This paper has been typeset from a TEX/ L A TEX file preparedby the author. c (cid:13)000