Table Of ContentMon.Not.R.Astron.Soc.000,1–14(2011) Printed14September2011 (MNLATEXstylefilev2.2)
Fast simulations of gas sloshing and cold front formation
E. Roediger1(cid:63), J. A. ZuHone2
1Jacobs University Bremen, PO Box 750 561, 28725 Bremen, Germany
2 NASA/Goddard Space Flight Center, 8800 Greenbelt Rd., Code 662, Greenbelt, MD 20771, USA
1
1
0
Accepted1988December15.Received1988December14;inoriginalform1988October11
2
p
e
ABSTRACT
S
Wepresentasimplifiedandfastmethodforsimulatingminormergersbetweengalaxy
2 clusters. Instead of following the evolution of the dark matter halos directly by the
1 N-body method, we employ a rigid potential approximation for both clusters. The
simulations are run in the rest frame of the more massive cluster and account for
] the resulting inertial accelerations in an optimised way. We test the reliability of this
O
method for studies of minor merger induced gas sloshing by performing a one-to-one
C comparison between our simulations and hydro+N-body ones. We find that the rigid
. potential approximation reproduces the sloshing-related features well except for two
h
artefacts: the temperature just outside the cold fronts is slightly over-predicted, and
p
- the outward motion of the cold fronts is delayed by typically 200 Myr. We discuss
o reasons for both artefacts.
r
t Key words: galaxies:clusters:general-galaxies:clusters:individual:A2029 X-rays:
s
a galaxies: clusters methods: numerical - methods: N-body simulations
[
1
v
3 1 INTRODUCTION (AM06 afterwards) have shown that the sloshing scenario
9 reproduces the morphology of observed CFs.
5 During the last decade, high resolution X-ray observations
2 havefoundawealthofstructureintheintra-clustermedium
. (ICM) of galaxy clusters, among them cold fronts (see re- Inrecentyears,moreandmoresloshingCFshavebeen
9
viewbyMarkevitch&Vikhlinin2007).Thesestructuresre- reported (e.g. Owers et al. 2009; Ghizzardi et al. 2010).
0
1 veal themselves as sharp discontinuities in X-ray brightness In principle, the properties of the CFs contain informa-
1 accompanied by a jump in temperature, where the brighter tion about the merger history of the clusters, as has been
: side is the cooler one. One variety of cold fronts (CFs) was shown for the Virgo cluster (Roediger et al. 2011, R11a
v
soonunderstoodtobethecontactdiscontinuitybetweenthe hereafter),A496(Roedigeretal.2011,R11bhereafter),and
i
X gaseousatmospheresofmergingclusters(e.g.A2142:Marke- RXJ1347.5-1145 (Johnson et al. 2011). However, disentan-
vitchetal.2000;A3667:Vikhlininetal.2001;andthebullet glingthemergerhistoryofeachclusterfromtheCFproper-
r
a cluster 1E 0657-56: Markevitch et al. 2002). ties requires a set of dedicated simulations for each cluster.
A second class of CFs was found to form arcs around Doingthiswithfullhydro+N-bodysimulationsiscomputa-
the cool cores of apparently relaxed clusters (e.g. RX tionallyexpensive.Thesameistrueiftheinfluenceofmore
J1720.1+2638: Mazzotta et al. 2001; Mazzotta & Giacin- time-consuming physics like viscosity (ZuHone et al. 2010),
tucci2008;Owersetal.2009;MS1455.0+2232:Mazzotta& magnetic fields (ZuHone et al. 2011) or thermal conduction
Giacintucci2008;Owersetal.2009;2A0335+096Mazzotta (ZuHoneetal.2011,inprep.)onsloshingCFsisstudied.A
et al. 2003; Sanders et al. 2009; A2029: Clarke et al. 2004; reasonablesimplificationthatspeedsupthesimulationscon-
Million&Allen2009;A1795:Markevitchetal.2001;Bour- siderably would be very useful. The most expensive part of
din&Mazzotta2008;Perseus:Churazovetal.2003;Sanders suchsimulationsistheself-gravityofthegasanddarkmat-
et al. 2005; A496: Dupke et al. 2007; Ghizzardi et al. 2010; ter(DM)particles.Inmostcases,theself-gravityofthegas
Virgo:Simionescuetal.2010;Centaurus:Fabianetal.2005; can be neglected, because the Jeans length is about 1 Mpc.
Sanders&Fabian2006).Markevitchetal.(2001)suggested The simulations speed up substantially when the DM ha-
that this variety of CFs forms due to sloshing of the cool losofthemainclusterandthesubclusterareapproximated
central gas within the central cluster potential, where the as rigid potentials (RPs). However, AM06 have shown that
sloshingisinitiallytriggeredbyaminormergerevent.Using even in a minor merger the central part of the main cluster
hydro+N-bodysimulations,Ascasibar&Markevitch(2006) moves significantly w.r.t. the overall cluster potential, thus
this effect cannot be neglected. ZuHone et al. (2010) (Z10
hereafter) suggested that the rigid potential approximation
(cid:63) E-mail:[email protected] can be used if additionally a point-mass-like approximation
(cid:13)c 2011RAS
2 E. Roediger & J. ZuHone
ofthemotionofbothclustersistakenintoaccount,includ- 2.2.1 Orbit of the subcluster
ing the effects of inertial acceleration.
Weassumetheorbitofthesubclustertobetheorbitofatest
Here we improve this rigid potential approximation
particle free-falling through the main cluster. In the course
anddemonstratethereliabilityofthesimplifiedsimulations
of the simulation, the potential of the subcluster is shifted
for the application to gas sloshing by comparing them to
through the main cluster along this orbit. This approach
hydro+N-bodysimulations.Thisrigidpotentialapproxima-
does not include dynamical friction, which will slow down
tion has already been applied successfully in sloshing simu-
the subcluster after pericentre passage (see Sect. 3.3 for a
lations for the Virgo cluster (R11a) and for A496 (R11b).
comparison).However,thesloshingistriggeredmainlydur-
ingthepericentrepassage,andthusourresultsdonotsuffer
from this discrepancy. Given that our test particle orbit is
2 METHOD
boundtopredictincreasinglywrongsubclusterpositionsaf-
2.1 Test setup terthefirstpericentrepassage,westopoursimulationswell
before a second pericentre passage.
We consider the following scenario: The ICM in a mas-
We construct orbits that are comparable to the ones
sive spherical galaxy cluster (the main cluster) is initially
in the hydro+N-body simulations in orientation, pericentre
in hydrostatic equilibrium. A spherical, less massive, gas
distance,andvelocityhistorypriortopericentrepassage.We
freegalaxycluster(thesubcluster)passesthroughthemain
start our simulations 1Gyr prior to the pericentre passage
cluster. The gravitational impact of the subcluster initiates
of the subcluster, which we set to occur at t=0. We reran
sloshingoftheICMinthemainclustercoreandsubsequent
ourfiducialsimulationwithaninitialisationtimeof0.5Gyr
coldfrontformation(seeAM06foradetaileddescriptionof
prior to pericentre passage and found that our results are
the dynamics).
not sensitive to the choice of the initialisation time.
Hydro+N-body simulations of this scenario have been
Thesubclusterorbitisinthexy-planeofthecomputa-
performed by AM06 and Z10. Our aim here is to investi-
tionalgrid.Forthesakeofashortnotation,weidentifythe
gate to what extent simulations with a rigid potential ap-
+y-direction as ”north” (N), the −y-direction as ”south”
proximation (RPA) for the DM content of, both, the main
(S), the +x-direction as ”west” (W), and the −x-direction
cluster and the subcluster can reproduce the resulting CF
as”east”(E).ThesubclusterwillstartWofthemainclus-
structures in terms of morphology, orientation, size, tem-
tercore,hasitsclosestapproachtothemainclustercorein
perature and density distribution. As the reference, we use
the NE and moves away towards the SE.
thehydro+N-bodysimulationsofZ10.Thus,wefollowtheir
leadandtailorourinitialmodelstomatchtheirs.Themain
cluster model is based on a Hernquist potential (Hernquist
2.2.2 Basic inertial frame correction
1990) with a scale radius, a. The temperature profile is de-
scribed by the phenomenological function Astherestframeofthemainclusterisnotaninertialframe,
T c+r/a theICMinthisframeissubjecttoapseudo-accelerationdue
T(r)= 0 c, (1)
1+r/a 1+r/a to the attraction of the main cluster core towards the ap-
c
proachingsubcluster.Z10usedthemostsimpleapproxima-
whereT isameasurefortheoverallclustertemperature,c
0 tiontoaccountforthis:theyassumedthatthemaincluster
describesthedepthofthecentraldensitydrop,anda char-
c responds to the gravity of the subcluster as a whole, like a
acterisestheradiusofthisdrop(seealsoAM06).Thecorre-
rigid body. Consequently, they calculate the inertial accel-
sponding density profile resulting from hydrostatic equilib-
erationfeltbythemainclustercentreduetothesubcluster
rium is (AM06)
and add this pseudo-acceleration to all of the ICM.
(cid:18) r (cid:19)(cid:18) r (cid:19)α(cid:16) r(cid:17)β
ρ(r) = ρ 1+ 1+ 1+ (2)
0 a ca a
c c
c−1 1−a/a 2.2.3 Improved inertial frame correction
with α=−1−n and β =1−n c.
c−a/a c−a/a
c c This basic inertial frame correction is a reasonable approx-
In our setup, we initialise the main cluster with these den- imation for the central region of the main cluster, but it is
sityandtemperatureprofiles,choosingtheparameterssuch wrongfortheouterpartsofthecluster.Itwillleadtounreal-
that they fit the corresponding hydro+N-body simulation isticflowsintheouterpartsofthemaincluster.Forsmaller
(see Table 1). From these profiles, we derive the gravita- pericentre distances, during pericentre passage of the sub-
tional potential of the main cluster assuming hydrostatic cluster the inertial acceleration can be large and even pro-
equilibrium. duce supersonic motions in the cluster outskirts. These un-
As in Z10, the subcluster is initialised as a pure DM realisticflowscaninfluencetheresultingCFsatlaterstages.
structure, where the DM mass distribution is described by Hence,weproposetoapplythepseudo-accelerationonlyto
a Hernquist profile (Hernquist 1990). thecentralregionofthemainclusterinsideacharacteristic
radius, R , and dampen it outside this radius exponen-
damp
tially over a length scale, L . Thus, instead of adding
2.2 The rigid potential approximation damp
thesameinertialframeaccelerationateverypositioninthe
Oursimulationsarerunintherestframeofthemaincluster. cluster, we multiply it with a radius-dependent function,
Thegasdynamicsisdescribedbythehydrodynamicalequa-
(cid:40) 1 if r(cid:54)R
tions. Additionally, the ICM is subject to the gravitational damp
acceleration due to the main cluster and the subcluster. W(r)= exp(−r−Rdamp) else , (3)
Ldamp
(cid:13)c 2011RAS,MNRAS000,1–14
Fast gas sloshing simulations 3
we state our main finding already here: The RPA is able to
Table1.Summaryofmodelparameters.SeeSect.2.1fordetails.
reproduce the hydro+N-body simulations very well except
fortwosystematicdifferences,ofwhichonecanbefullycor-
merger characteristics
massratio 20 5 2 rected for, and the other partially:
impactparameter 200kpc 500kpc 500kpc
• After the onset of sloshing, the RP simulations lag be-
pericentredistance 62.5kpc 150kpc 135kpc
hindthehydro+N-bodyonesby200to250Myr,depending
on the cluster mass ratio. This lag can be corrected for by
main cluster
ρ0/(10−25gcm−3) 3.75 2.9 2.76 delaying the hydro+N-body results by the appropriate lag
a/kpc 615 623 615 in the comparison.1
n 5 5 5 • TheRPAgenerallyover-estimatesthetemperaturejust
T0/(107K) 15.8 13.8 11.27 outsidetheCFs.Thisdisagreementcanbeattenuatedbythe
ac 61.5 62.3 61.5 damping discussed above, but not avoided completely.
c 0.1585 0.169 0.1557
subcluster 3.1 Fiducial case: mass ratio 5, moderate damping
mass(1014M(cid:12)) 0.714 2.5 5
scaleradius(kpc) 220 350 416 Our fiducial case is the intermediate merger with a mass
ratio of 5 between the clusters. We simulate this case
damping none, none, none, with four different settings for the inertial frame correc-
(Rdamp/kpc, (500,300) (800,300), (950,400) tion as listed in Table 1: one without large-scale damp-
Ldamp/kpc) (800,100), ing, two moderate damping settings and a strong one.
(500,300)
The best results are achieved for moderate damping with
(R /kpc,L /kpc)=(800,300).Thiscasewillbede-
damp damp
scribed first. The effect of the damping will be summarised
where r is the distance to the main cluster centre. in Sect. 3.2.
The choice of R is motivated by the sphere of in-
damp
fluence of the subcluster: If the subcluster passes the main
3.1.1 Morphology and orientation of cool spiral
cluster centre at a pericentre distance larger than its own
size, it attracts the main cluster core only slightly. If the With both simulation methods, the gas sloshing evolves in
subclusterpassesthemainclustercoreatasmalldistance,it a very similar manner. We demonstrate this in Fig. 1 by
canattractatmostaregioncomparabletoitsownsize,but showing snapshots of the ICM temperature in the orbital
notbeyondthat.Hence,Rdamp shouldbecomparabletothe plane. The first and third row are for the hydro+N-body
sizeofthesubcluster,i.e.,abouttwiceitscharacteristicscale run,theremainingtworowsforourfiducialRPsimulation.
length. We test several combinations of (Rdamp,Ldamp). The top two rows focus on the onset of sloshing. We
see the subcluster pass the cluster centre (timesteps 0 and
1Gyr) from the NW over NE towards the SE. At 0.5 Gyr,
2.3 Code
sloshinghasjustsetin,andanarc-likeCFtowardstheSis
All simulations are run with the FLASH code (version 3.2, accompaniedbyacoolfantowardsN.Botharesurrounded
Dubey et al. 2009). FLASH is a modular block-structured byhotterICM.Thesepropertiesarealikeinbothmethods.
AMRcode,parallelisedusingtheMessagePassingInterface IntheRPA,thecoolfantakesamorespiral-likeappearance
(MPI) library. It solves the Riemann problem on a Carte- comparedtothehydro+N-bodycase.Alsothetemperature
siangridusingthePiecewise-ParabolicMethod(PPM).The distributionSofthesouthernCFdiffersbetweenbothmeth-
simulations are performed in 3D and all boundaries are re- ods.
flecting. We use a simulation grid of size 4×3×3Mpc3, ThebottomsixpanelsofFig.1displaythefurtherevo-
which is large enough to prevent reflected waves reaching lutionofthegassloshing.Asmentionedabove,wefindthat
the central region of interest during our simulation time. the RP simulations lag behind the hydro+N-body ones by
We resolve the inner 50 kpc with 2 kpc, the inner 130 kpc 250 Myr for this mass ratio. Further details of this lag will
with 4 kpc, the inner 260 kpc with 8 kpc and enforce de- bediscussedinSect.3.1.2.2.Hence,hereweplottheresults
creasing resolution with increasing radius from the cluster fromthehydro+N-bodysimulationswithadelayof250Myr
centretooptimisecomputationalcosts.Wehaveperformed in order to compare corresponding timesteps.
resolution tests here and also in R11a,b and found our re- Intheintermediatephase(0.7to1.5Gyr)thecoolspi-
sults to be independent of resolution. raltypicalforsloshingforms.ThemajorCFisfoundinthe
SW, and a secondary CF evolves towards the NE. At its
outside, the cool spiral is surrounded by a hot horse-shoe
shaped region, which tends to be slightly too hot in the
3 RESULTS
We present simulations of three merger scenarios which
1 Inacorrectmanner,theresultsoftheRPAshouldbebrought
cover a range in mass ratios from 2 to 20. The complete
forward instead of delaying the more accurate hydro+N-body
model parameters for each run are listed in Table 1.
ones. However, there are several RP runs for each merger case
Inthefollowingsubsectionswegiveadetailedcompar- and only one hydro+N-body run. For the sake of simplicity and
ison of the RP simulations to the hydro+N-body ones. In clarity we decide to apply the time shift to the hydro+N-body
order to aid the reader in smoothly following our analysis, runandtrustthereadertorememberthisfootnote.
(cid:13)c 2011RAS,MNRAS000,1–14
4 E. Roediger & J. ZuHone
y
d
o
b
N
+
o
r
d
y
h
al
ti
n
e
t
o
p
d
gi
ri
,
y
d
o
b
N
+
o
r
d
y
h
,
al
ti
n
e
t
o
p
d
gi
ri
,
.. , . , ,
Figure 1. Comparison of hydro+Nbody simulations (first and third row) and rigid potential simulations (second and fourth row) for
themergerwithmassratio5.ThepanelsshowtheICMtemperatureintheorbitalplane(seecolourscale).Thepanelsizeis1Mpc.The
timestepisnotedaboveeachpanel.Thehydro+NbodyresultsarefromZ10,replottedinourcolourscale.Therigidpotentialsimulation
shownhereusesthedampingsetting(R /kpc,L /kpc)=(800,300).Theuppersixpanelsshowtheonsetofsloshing,thebottom
damp damp
sixtheevolutionofthecoldfrontsandcoldspiralstructure.Ouranalysisshowsthataftertheonsettherigidpotentialsimulationslag
behindthehydro+Nbodyoneby250Myr(seeSect.3.1.2.2).Hence,inthebottomsixpanelsweplottheresultsfromthehydro+Nbody
simulationswithadelayof250Myr.
(cid:13)c 2011RAS,MNRAS000,1–14
Fast gas sloshing simulations 5
RPA.Thisevolutionisthesameinallcases,andtheresults thisreference:Atagiventimestep,theRPsimulationspro-
ofbothmethodsagreewellinmorphologyandorientationof ducesomewhattoosmallradiifortheoutermostCFs.This
thecoolspiral.Inthelatephase(2Gyr),theCFintheSW istrueforalldirectionsexceptNW.Thus,ingeneral,theRP
starts to break apart in the hydro+N-body simulation. In simulationslagbehindthehydro+N-bodyone.Plottingthe
theRPsimulation,theCFsremainintactandthemorphol- hydro+N-body result with a delay of 250 Myr (solid black
ogy remains close to spiral-like. We note that this break-up linewitherrorbars,andrememberfootnote1)compensates
of the spiral structure can be recovered by using stronger thedifferenceinallbuttheNWdirection,leadingtoagood
damping(Fig.6;seeSect.3.2foramoredetaileddiscussion agreementtotheRPsimulation.ThesecondCFsagreewell
of the effect of damping). between all runs.
ThislaginCFmotionisthemajorsystematicdifference
between two simulation methods. Consequently, we use the
3.1.2 Size of the cold front structure delay of 250 Myr for the full hydro+N-body simulation in
all other comparisons regarding the fiducial merger case.
The outwards motion of the CFs and hence growth of the
coolspiralareimportantcharacteristicsofthesloshingpro-
cess. In their studies of the Virgo and Abell 496 clusters,
3.1.3 Quantitative comparison of density and temperature
R11a,b found that the velocity of the outward motion is
distribution
largely independent of the subcluster, but is characteristic
for the potential of the main cluster. This means that the Inordertogobeyondthequalitativecomparisonofthetem-
positionsoftheCFsinagivenclusterdependmostlyonthe perature slices in Fig. 1, we derive radial temperature and
timesincethesubcluster’spericentrepassage,i.e.theageof density profiles as described above in Paragraph 3.1.2.1. In
the CFs. Therefore it is important to know to what extent Fig.3wecomparetheseprofilesfordifferentrealisationsfor
the RPA recovers this outward motion. This is the aim of the fiducial merger. Again, here we concentrate on the red
this subsection. andblacklines,whichareforthefiducialRPsimulationand
the hydro+N-body one, respectively.
At 0.5 Gyr, the sloshing is still in the onset phase and
3.1.2.1 Deriving cold front radii We study the radii
wedonotapplythedelaydiscussedabove.Still,alreadyhere
of the CFs towards the diagonal directions in the orbital
theCFintheSWisaheadinthehydro+N-bodysimulation.
plane. For this purpose, we first derive radial temperature
Here,theRPArunshowsaweakerimpactofthesubcluster
profilestowardsNW,SW,SEandNE,whereeachprofileis
averaged over an azimuthal extent of ±15o (see Fig. 3 for passage on the temperature. Also the density profile in the
northern directions are not accurately reproduced. This is
profilesandSect.3.1.3fortheirdiscussion).GiventhatCFs
the region the subcluster directly passes and the strongest
rarelyformperfectcircularconcentricarcsaroundtheclus-
differences are to be expected.
ter centre, this azimuthal averaging introduces a smearing
In all later timesteps we apply the delay of 250 Myr to
out of the intrinsically discontinuous CFs over a finite ra-
thehydro+N-bodysimulationasderivedinSect.3.1.2.2.As
dial range. This occurs even for the small azimuthal range
a result, we achieve a good agreement between both meth-
we use for averaging. Consequently, in the profiles, the CFs
ods. Especially along the SW-NE axis, which is perpendic-
donotappearasatruediscontinuity,butassteepslopesin
ulartothesubclusterorbit,theagreementisexcellent.The
temperaturethatstretchovertypically10to50kpc.Ineach
onlysystematicdifferencebetweenbothmethodsisthatthe
temperature profile, we identify CFs as regions of tempera-
RPA over-predicts the temperature just outside the CFs,
ture slopes above 0.02keV/kpc. Thus, we identify an inner
which we have already seen in Fig. 1.
and outer edge for each CF, and the nominal CF radius is
defined as the average radius between this inner and outer
edge.
3.1.4 Density and temperature across the CFs
ThefirsttimestepatwhichaCFcanbedetectedisnot
immediately after the subcluster’s pericentre passage, but Hereweaimatcomparingtheevolutionofthetemperature
typically 0.4 Gyr afterwards. In the hydro+N-body simula- and density at the CFs. For this purpose, we derive both
tion the CFs in the NW and NE direction are established quantitiesattheinnerandouteredgeofeachCFalongwith
only at t = 1Gyr. At even later times, in all but the NE its radius (see Paragraph 3.1.2.1). We note that the combi-
direction a second CF at smaller radii is detectable. nationofazimuthalandradialbinningintroducesanuncer-
taintyintemperatureofatleast±0.5keVatbothedges.In
3.1.2.2 Comparing the evolution of cold front radii Fig. 4 we plot the temperature at the inner and outer edge
Having derived the positions of the CFs at each timestep, of each CF as a function of its radius. Thus, we compare
we can now proceed to analyse their outward motion. We simulations at stages when the CF spiral has reached the
dosoinFig.2,whereweplotthetemporalevolutionofthe same size. We compare the results along the diagonal di-
CFradiitowardsthediagonaldirections(NE,NW,SE,and rections in the orbital plane. This figure is restricted to the
SW) in the orbital plane. We use error bars to indicate the outermost ring of CFs. Also here, we focus on the red and
width of each CF, i.e. from its inner to outer edge. The RP blacklines,whichareforthefiducialRPsimulationandthe
simulations with different settings are plotted by coloured hydro+N-body one, respectively.
lines.Herewefocusontheredline,whichmarksourfiducial ThetemperaturesattheinneredgesoftheCFsarere-
RP run. producedwell.TheoutertemperaturesintheSEandSWdi-
The result from the hydro+N-body simulation is the rectionsdiffersystematicallybetweentherigidpotentialand
dashedblackline.TheRPresultsdiffersystematicallyfrom the hydro+N-body simulation. In these positions, the rigid
(cid:13)c 2011RAS,MNRAS000,1–14
6 E. Roediger & J. ZuHone
250 NE NW 250
200 nodamp 200
damp 500, 300
c damp 800, 300
kp 150 hydro+Nbody, tdelay=250 Myr 150
/ F hydro+Nbody
rC 100 100
50 50
0 0
250 SE SW 250
200 200
c
kp 150 150
/ F
rC 100 100
50 50
0 0
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
t/Gyr
Figure2.Outwardmotionofthecoldfrontstowardsthediagonaldirectionsinthexy-plane(onedirectionperpanel,seelabel).Forthe
derivationofthecoldfrontradii,seeParagraph3.1.2.1.Theerrorbarsindicatethewidthofthecoldfrontasitappearsintheazimuthally
averagedprofiles.InallbuttheNEpanel,theevolutionoftheoutermostandsecondcoldfrontisshown.Thefigurecomparestheresults
for different realisations of the mass ratio 5 merger. The black dashed line without error bars represents the hydro+N-body run, the
othercoloured/brokenlinesarefortheresultsfromtherigidpotentialsimulationswithdifferentdampingsettings(seelegend).Clearly,
the rigid potential simulations lag behind the hydro+N-body one. Plotting this reference with a delay of 250Myr (and remembering
footnote1)correctsforthislag.
potential simulations first underestimate the outer temper- isreproduced,too.However,herethedetailsdependonthe
atureandoverestimateitatlatertimes.Thereasonforthis damping (see Sect. 3.2).
difference is that the rigid potential approximation leads to
a different flow field outside the central region of the clus-
ter. In all but the early times, this leads to a stronger com- 3.2 Impact of damping
pressional heating at the outer side of the CFs and thus a
The damping of the inertial frame correction described in
higher temperature. This systematic difference can be less-
Sect.2.2.3isconstructedsuchthatitgraduallyswitchesoff
ened somewhat by the damping, but is not prevented com-
the inertial frame correction in the cluster outskirts, which
pletely.
should not be applied there. Using the RPA without this
We apply the same analysis to the density inside and
”re-correction” leads to an unrealistic ICM velocity field in
outside each CF and present the result in Fig. 5. For the
theouterclusterregionsandtwoartefactscomparedtothe
fiducial RP simulation, the densities at the CFs are repro-
hydro+N-body reference run: a cooler temperature in the
duced well. Thus, even at late stages, where the RPA does
outer northern region and the hotter temperature outside
not accurately estimate the temperatures outside the CFs,
all CFs. Both effects can be seen in the temperature slices
it is still accurate for the densities.
(toppanelofFig.1)andallothercomparisonplots(Figs.3,
4 and 5).
Usingastrongdampingof(R /kpc,L /kpc)=
damp damp
(500,300)leadstoanearlycorrecttemperaturedistribution
3.1.5 Large-scale asymmetry
in the outer cluster regions as demonstrated in the temper-
In addition to the cold spiral structure in the cluster cen- ature slice in Fig. 1. However, the temperature just outside
tre, also the large-scale distribution of the ICM density at theCFsisstillslightlytoohigh(Figs.1,3,4).Moreover,the
least out to 500 kpc is reproduced well in the RPA. Both strong damping leads to slightly smaller CF radii towards
simulationmethodsfindthecharacteristicasymmetryinthe the NW and SE direction after t = 1Gyr, which is the di-
sensethatprofilesof,both,densityandtemperatureonop- rection of motion of the subcluster (Fig. 2). The CF radii
positesidesoftheclustercentrealternatearoundeachother, towards the NE and SW, i.e. along the axis perpendicular
switchingoverattheCFs.Wehaveillustratedthiseffectin to the orbit, are independent of damping at all times.
Fig. 3 in the density panels for t=1.5Gyr by overplotting In summary, the stronger the damping, the more accu-
the profiles from the NE side to the SW direction and the rate is the outer temperature distribution, but at some ex-
NW ones to the SE direction. penseoftheaccuracyintheclustercentre.Hence,weprefer
Qualitatively, the large-scale structure in temperature to use a moderate damping which ensures an accurate re-
(cid:13)c 2011RAS,MNRAS000,1–14
Fast gas sloshing simulations 7
5·10-26 0.5Gyr 1.Gyr 1.5Gyr 2.Gyr 5·10-26
3 / g/cmρ 5·1100--2276 51·01-206-27 3 / g/cmρ
hydro+Nbody
nodamp
damp 500,300
damp 800,300
10 -1207 1 100-27
9 rSE-NW / kpc 0.5Gyr rSE-NW / kpc 1.Gyr rSE-NW / kpc 1.5Gyr rSE-NW / kpc 2.Gyr 9
8 8
T / keV 67 67 T / keV
5 5
4 4
-400 -200 0 200 400 -400 -200 0 200 400 -400 -200 0 200 400 -400 -200 0 200 400
rSE-NW / kpc rSE-NW / kpc rSE-NW / kpc rSE-NW / kpc
5·10-26 0.5Gyr 1.Gyr 1.5Gyr 2.Gyr 5·10-26
3 / g/cmρ 5·1100--2276 5 1·1e0-2-267 3 / g/cmρ
10 -1207 11e0-27
9 rSW-NE / kpc 0.5Gyr rSW-NE / kpc 1.Gyr rSW-NE / kpc 1.5Gyr rSW-NE / kpc 2.Gyr 9
8 8
T / keV 67 67 T / keV
5 5
4 4
-400 -200 0 200 400 -400 -200 0 200 400 -400 -200 0 200 400 -400 -200 0 200 400
rSW-NE / kpc rSW-NE / kpc rSW-NE / kpc rSW-NE / kpc
Figure 3. Comparisonofdensityandtemperatureprofilesalongdiagonalsinthexy-plane.Thecolumnsarefordifferenttimestepsas
indicatedineachpanel.ThetwotoprowsarefortheSE-NWdirection,thetwobottomrowsfortheSW-NEdirection.Theblacksolid
linesshowthehydro+N-bodysimulation,colouredbrokenlineslinestherigidpotentialsimulationswithdifferentdamping.Profilesare
averagedover±15o aroundtheindicateddirection.Thehydro+N-bodysimulationisplottedwithadelayof250Myrinallbutthe0.5
Gyrstep,whichaccountsforthelagofrigidpotentialapproximationdescribedinSect.3.1.2.2.Forthedensityprofilesatt=1.5Gyr,
weoverplottheprofileofthepositivedirectionasthinlinesinthenegativedirectioninordertodemonstratetheasymmetry.
production of the CF radii, temperatures inside them and an algorithm to incorporate dynamical friction in a simpli-
densities at, both, the inside and outside. fied form. However, the gas sloshing we are interested in is
Thecase(800,100)isverysimilartoourfiducialsetting triggered by the pericentre passage and thus is unaffected
(800, 300) demonstrating that the results are not sensitive by this difference in subcluster position at later epochs.
to the choice of the fall-off length scale for the damping, InFig.8wedemonstratetheevolutionofthesubcluster
L . by plotting its mass within its scale radius. Clearly, during
damp
pericentre passage and up to 0.2 Gyr afterwards the sub-
cluster suffers tidal compression. It starts loosing mass sig-
3.3 Subcluster orbit and mass evolution nificantlyonly0.5Gyrafterpericentrepassage.Thisiswell
after triggering the gas sloshing and thus has no significant
InadditiontotheICMproperties,wealsocomparethesub- effectonthefurtherevolutionoftheICMinthemainclus-
cluster orbit w.r.t. the cluster centre from both methods in ter. We will discuss the differences in the evolution of the
Fig.7.ThetestparticleorbitusedintheRPsimulationsap- main cluster potential in Sect. 4.1.
proximates the true trajectory of the subcluster very well.
Wehavemarkedthesubclusterpositioninstepsof250Myr
along both orbits, demonstrating that the test particle ap- 3.4 Mergers with other mass ratios
proximation also recovers the motion of the test particle
3.4.1 Mass ratio 20
along its orbit up to a few 100 Myr after pericentre pas-
sage. After that, dynamical friction causes the subcluster We have run the same comparison for a minor merger with
to decelerate significantly, such that its new apocentre dis- alargermassratioof20.Here,thesubclusterhasasmaller
tance is only about 1.8 Mpc, and the apocentre is reached scale radius of only 220 kpc, hence the damping setting
already 1.25 Gyr after pericentre passage. The test particle (R /kpc,L /kpc)=(500,300) corresponds to mod-
damp damp
method does not capture this effect, and predicts a com- eratedampinghere.InFig.9wecomparetemperatureslices,
parablecluster-centricdistancealreadyafterabout0.6Gyr in Fig. 10 the evolution of the CF radii, and the tempera-
after pericentre passage. Taylor & Babul (2001) proposed tures inside and outside each CF in Fig. 11. Even for this
(cid:13)c 2011RAS,MNRAS000,1–14
8 E. Roediger & J. ZuHone
10 10
NE NW
9 9
8 8
V 7 7
e
T / k 6 6
5 initial T(r) 5
hydro+Nbody
4 nodamp 4
3 damp 800,300 3
damp 500,300
2 2
10 10
SE SW
9 9
8 8
V 7 7
e
T / k 6 6
5 5
4 4
3 3
2 2
50 100 150 200 250 50 100 150 200 250
r / kpc
CF
Figure 4. Temperature inside and outside of the cold front as a function of cold front radius. We show results for the four diagonal
directionsinthexy-plane.Differentlinecolours/stylescodedifferentrealisationsofthemassratio5merger,seelegend.Ineachpanel,
theupperandlowersetsoflinesdenotethetemperaturesjustoutsideandinsidethecoldfronts,respectively.Forcomparison,weplot
theinitialtemperatureprofileasthedottedline.
10-25 10-25
NE NW
-3m hydrion+itNiabl oρd(ry)
/ g cρ 10-26 ddaammpp 85n00o00d,,a33m0000p 10-26
10-25 10-25
SE SW
-3m
c
g
/ ρ 10-26 10-26
50 100 150 200 250 50 100 150 200 250
rCF / kpc
Figure 5.SameasFig.4,butfordensityinsideandoutsideofthecoldfronts(upperandlowersetoflines,respectively).
case with a much smaller subcluster, the sloshing in the exploring the limits of the RPA. Here, the subcluster has
RPA runs lags behind the hydro+N-body one by 200 Myr. a scale radius of 416 kpc and we use a damping setting of
Hence, we come to the same results and conclusions as in (R /kpc,L /kpc) = (950,400). In Fig. 12 we com-
damp damp
the fiducial case. pare temperature slices, in Fig. 13 profiles in the orbital
plane along the diagonal directions at different timesteps.
Forthiscase,wefindadelayof250MyrbetweenRPAand
3.4.2 Mass ratio 2 hydro+N-body ones.
A merger with a mass ratio of 2 is no minor merger any- Inbothmethods,thecentralICMstartssloshinginthe
more, and we perform this simulation with the purpose of clustercentre,andaclearprimaryCFisformedintheSW.
(cid:13)c 2011RAS,MNRAS000,1–14
Fast gas sloshing simulations 9
Figure12.Snapshotsofthetemperatureintheorbitalplaneforthemergerwithmassratio2.Weshowthehydro+N-bodyresults(upper
row)withadelayof250Myrcomparedtotherigidpotentialrun(bottomrow).Thishydro+N-bodysimulationhasalowresolutionof
10 kpc, which is at least partially responsible for the loss of the cool dense centre. The major cold front in the SW is still reproduced
well,butthegeneraldistortionoftheclusterbythismajormergerisnotcapturedanymorebytherigidpotentialapproximation.
5·10-26 hydro+Nbody, delay 250 Myr 5·10-26
nodamp
damp 950, 400 1.Gyr 1.5Gyr 2.Gyr
3 / g/cmρ 5·1100--2267 51·01-206-27 3 / g/cmρ
8 rSE-NW / kpc rSE-NW / kpc rSE-NW / kpc 8
7 1.Gyr 1.5Gyr 2.Gyr 7
V 6 6 V
T / ke 5 5 T / ke
4 4
3 3
-400 -300 -200 -100 0 100 200-400 -300 -200 -100 0 100 200-400 -300 -200 -100 0 100 200
rSE-NW / kpc rSE-NW / kpc rSE-NW / kpc
5·10-26 5·10-26
1.Gyr 1.5Gyr 2.Gyr
3 / g/cmρ 5·1100--2267 5 1·1e0-2-267 3 / g/cmρ
8 rSE-NW / kpc rSW-NE / kpc rSW-NE / kpc 8
7 1.Gyr 1.5Gyr 2.Gyr 7
V 6 6 V
T / ke 5 5 T / ke
4 4
3 3
-400 -300 -200 -100 0 100 200-400 -300 -200 -100 0 100 200-400 -300 -200 -100 0 100 200
rSE-NW / kpc rSW-NE / kpc rSW-NE / kpc
Figure 13.Comparisonofdensityandtemperatureprofilesalongthediagonalsinthexy-plane.SameasFig.3butforamassratioof
2betweentheclusters,whichisalreadyamajormerger.Thehydro+N-bodysimulationisplottedwithadelayof250Myr.
(cid:13)c 2011RAS,MNRAS000,1–14
10 E. Roediger & J. ZuHone
1
0.9
Msun 0.8
· 14M / 10sub, < a 0000....4567
0.3
0.2
-1 -0.5 0 0.5 1 1.5 2
t / Gyr
Figure 8. Evolution of subcluster mass within the subcluster
scale radius. During pericentre passage and up to 0.2 Gyr after-
wards the subcluster suffers tidal compression. It starts loosing
masssignificantlyonly0.5Gyrafterpericentrepassage.
,
,
Figure 6.Temperatureslicesforthefiducialrunatt=1.5Gyr,
for rigid potential simulations without damping (upper panel)
and with strong damping (lower panel). Compare to Fig. 1 for
moderatedampingandhydro+N-bodycase.
500 J
[ 21
0
c
kp -500
y /
-1000
RPA
hydro+Nbody
-1500
-2000-1500-1000-500 0 500 1000 1500 2000
x / kpc
Figure 9.Snapshotsofthetemperatureintheorbitalplanefor
Figure 7. Comparison of the subcluster orbit in the hydro+N- the merger with mass ratio 20. We show the hydro+N-body re-
bodysimulation(blacksolidline)andtherigidpotentialapprox- sults(upperrow)withadelayof200Myrcomparedtotherigid
imation (red dashed line) for the merger with mass ratio 5. We potential run (bottom row). Both methods give very similar re-
mark the subcluster positions in 250 Myr intervals. While the sults.
directionoftheorbitisreproducedwellintherigidpotentialap-
proximation,theneglectofthedynamicalfrictionleadstosignifi-
4 DISCUSSION AND SUMMARY
cantover-estimationofthevelocityafter0.5Gyrafterpericentre
passage. 4.1 Origin of the delayed evolution in the RPA
We have shown that the rigid potential approximation can
reproduce the characteristics of gas sloshing well except for
However, in the hydro+N-body code the cool central gas twoartefacts,thehighertemperaturesattheoutsideofthe
moves completely out of the cluster centre, forming a CF CFsandthetemporallaginevolution.Wehavetracedback
towards the NE of the centre only at very late stages. In the former effect to the necessarily unrealistic gas motions
the RPA, the cool gas core is never completely displaced intheouterclusterregioninthisapproximation.Inorderto
from the potential minimum. The comparison of profiles in investigatetheoriginofthelatter,wecomparetheevolution
Fig.13revealsthattheRPAstillachievesagoodagreement of radial potential profiles in the fiducial case in Fig. 14.
for the primary CF in the SW and SE, despite the nearly We show profiles towards the N, S, W and E, where we
equal masses of both clusters. plotonedirectionperpanel.Differenttimestepsarecolour-
Thus,evenforthismajormerger,theRPArecoversthe coded(codedbylinestyleinprintversion).Theresultsfrom
evolution of the major CF. However, it comes to its limit hydro+N-body and from the RPA are shown by solid lines
regardingthemorphology,theotherCFs,andthelarge-scale and dashed lines, respectively (thick and thin lines in print
distortion of the main cluster. version).
(cid:13)c 2011RAS,MNRAS000,1–14
