Radial Alignment in Simulated Clusters

0.28 MB·English

by  Maria J. Pereira

Draftversion February1,2008
PreprinttypesetusingLATEXstyleemulateapjv.10/09/06
RADIAL ALIGNMENT IN SIMULATED CLUSTERS
Maria J. Pereira, Greg L. Bryan and Stuart P. D. Gill
ColumbiaUniversity,DepartmentofAstronomy,NewYork,NY,10025
Draft versionFebruary 1, 2008
ABSTRACT
Observational evidence for the radial alignment of satellites with their dark matter host has been
accumulating steadily in the past few years. The effect is seen over a wide range of scales, from
massiveclustersofgalaxiesdownto galaxy-sizedsystems, yetthe underlying physicalmechanismhas
8 still not been established. To this end, we have carried out a detailed analysis of the shapes and
0
orientations of dark matter substructures in high-resolution N-body cosmological simulations. We
0
find a strong tendency for radial alignment of the substructure with its host halo: the distribution
2
of halo major axes is very anisotropic, with the majority pointing towards the center of mass of the

n host. Thealignmentpeaksoncethesub-halohaspassedthevirialradiusofthehostforthefirsttime,
a but is not subsequently diluted, even after the halos have gone through as many as four pericentric
J
passages. This evidence points to the existence of a very rapid dynamical mechanism acting on these

3 systems and we argue that tidal torquing throughout their orbits is the most likely candidate.
2
Subjectheadings: methods: N-bodysimulations,galaxies: kinematicsanddynamics,galaxies: clusters

]
h
1. INTRODUCTION ments between galaxies can be dealt with easily: e.g.
p
downweightingclosepairsreadilyremovescontamination
- Anisotropy in galaxy orientations has been a matter
o by alignments induced in interacting systems. However,
of debate for several decades, and many conflicting re-
r as Hirata & Seljak (2003) pointed out, if galaxy orienta-
t ports have been published. Past studies have found ev-
s tions are affected by their surrounding density field (e.g.
idence for three different types of alignment: alignment
a
a galaxy cluster), then they will also be correlated with
[ between clusters (Binggeli 1982; Plionis & Basilakos
2002), between the brightest cluster galaxy (BCG) the orientations of the background population of galax-
2 ies that is being lensed by that field. This correlation
and the satellite distribution (e.g., Yang et al. 2006)
v between widely separated redshift bins cannot trivially
and between the orientation of satellites and their
2 be removed.
host (Hawley & Peebles 1975; Pereira & Kuhn 2005;
0 Given the growing body of evidence suggesting that
Agustsson & Brainerd 2006). This last type of align-
7 galaxyorientationsareanisotropic,andthepressingneed
ment, which we will refer to as radial alignment and is
1 for an accurate quantification of intrinsic alignments for
the focus of this paper, has been the hardest to confirm
.
7 (Trevese et al.1992;Torlina et al.2007),sinceitrequires weak lensing, it seems crucial that we try and find the
0 physical cause behind these anisotropies. There are two
high quality data on small scales. In recent years, this
7 commonly proposed explanations. The first, initially
field has seen a resurgence, largely due to the arrival of
0 developed by Peebles (1969) in his tidal torque theory
the Sloan Digital Sky Survey (SDSS) (Abazajian et al.
: (TTT), explains the anisotropy as a left-over primor-
v 2005). The SDSS provides accurate measurements of
dial effect. TTT ascribes the orientation and rotation
i isophotal shapes for millions of galaxies, and this has
X of galaxies to torquing during their formation. It there-
finally allowed large statistical studies of galaxy align-
r mentsto be performed. Pereira & Kuhn(2005)targeted fore follows that the signal should be stronger on the
a outskirts of the cluster, and that it wanes with time,
galaxies in massive X-ray selected clusters and found
suchthatolder,morerelaxedclustersshouldexhibitless
a significant tendency for their radial alignment. This
tendency foralignment. The other alternative,proposed
result has since been confirmed by Faltenbacher et al.
by Pereira & Kuhn (2005), is a dynamical mechanism,
(2007) for a larger sample of groups optically selected
i.e. an interaction with the tidal field of the host clus-
fromtheSDSS.Onsmallerscales,Agustsson & Brainerd
ter that gets progressively stronger during infall and is
(2006)foundatendencyforsatellitegalaxiesintheSDSS
noterasedbysubsequentorbitalmotions. Observational
to be radiallyaligned with their hostgalaxy,whereason
studies have so far been unable to distinguish between
largescales,Mandelbaum et al.(2006a)foundaverysig-
the two, due to difficulties in constraining galaxy orbits
nificant correlation between the orientations of galaxies
and in measuring galaxy shapes accurately out to large
and the surrounding density field traced by galaxy over-
redshifts.
densities.
A different approach is needed, and a few numeri-
Initiallymotivatedbytheprospectofusinggalaxyori-
cal studies have recently been published on this sub-
entationstoprobetheirformationhistories,thesestudies
ject. Studies of simulated halo shapes and orientations
are now also driven by the need to calibrate weak lens-
havebeenperformedaroundvoids(Brunino et al.2007),
ing and cosmic shear measurements. A key assumption
along filaments and sheets (Arag´on-Calvo et al. 2007;
for lensing techniques is that the population of galaxies
Altay et al. 2006; Hahn et al. 2007) and in a Milky-Way
being lensed is randomly oriented. Some intrinsic align-
typehalo(Kuhlen et al.2007). Anisotropiesarereported
Electronicaddress: [email protected] in all three environments. The advantages of working
2
with simulated clusters are obvious - 3D spatial infor- elapsedsince their formation, whichis defined, following
mation means we do not suffer dilution from projection Lacey & Cole(1993), atthe redshift wherethe halofirst
effects. Also, with enough temporal and spatial reso- contains half of its present-day mass.
lution, we can follow the galaxies as they fall into the
cluster along filaments, and beyond, as they orbit inside Halo Rvir Mvir zform age Nsat(<rvir)
the cluster. By trackingthe effect’s evolutionwith time, #1 1.34 2.82 1.18 8.37 166
we will be able to more precisely determine its source. #2 0.97 1.05 0.87 7.17 49
Westart( 2.1)byintroducingthesimulationsusedfor #3 1.06 1.38 0.84 7.01 98
this analysis§and describing the properties of the eight #4 1.06 1.38 0.75 6.57 71
#5 1.34 2.80 0.59 5.65 168
host halos. Our methods for finding the substructure
#6 1.06 1.39 0.50 5.06 97
halos ( 2.2) and determining their shapes ( 2.3) follow,
§ § #7 1.00 1.16 0.43 4.52 54
alongwithastudyofthereliabilityofourshapemeasure-
#8 1.37 3.00 0.30 3.42 152
ments. With this information, we then show in section
3.1 that cosmologicaldark-matter simulations do indeed
TABLE 1
produce radial alignment in clusters at z 0, and we
≈ Summaryof theeighthostdarkmatterhalosatz=0.
study the correlation of this effect with various parame- Distanceis measuredin h−1 Mpc,mass in 1014h−1 M⊙,and
ters, such as host halo mass and distance from the clus- agein Gyrs. Onlysatellites with morethan 200particles
ter center ( 3.2). Having established the importance of aretalliedin thelast column.
§
the alignment effect in our simulations, we use the high
temporal resolution to study its evolution with time in
2.2. Identifying Substructure
section 3.3, and its dependence on orbital phase ( 3.4).
§
We argue in section 4.1 that tidal torquing by the host Simulation outputs merely tell us what the particle
halo tidal field is responsible for the alignment of sub- spatialand kinetic distribution is at eachredshift. They
structure, and compare our results with previous obser- giveusno informationaboutparticleassignmentorhalo
vations in 4.2. We end ( 4.3) by briefly speculating on identity- whichparticlebelongsto whichhalo? Thereis
§ §
the possible consequences of such a mechanism for the no unique answer to this question, mainly because there
morphological and orbital evolution of galaxies in clus- are many different ways in which a halo can be defined.
ters. A number of sophisticated algorithms have been de-
veloped to locate halos within simulations (Davis et al.
2. SIMULATIONS AND ANALYSIS 1985; Frenk et al. 1988; Bertschinger & Gelb
1991; Suto et al. 1992; Weinberg et al. 1997;
2.1. The Data
Klypin & Holtzman 1997). They face many chal-
The N-body simulations used in this work are pre- lenges: the dynamic environment of cosmological
sented in detail in Gill et al. (2004), and we describe simulations blurs halo boundaries, and halos are con-
them here only briefly. Using the open source adaptive tinually undergoing mergers or being stripped within a
mesh refinement code MLAPM (Knebe et al. 2001), a set host potential, making it impossible to clearly define
offour initialconditions atredshift z =45ina standard a halo edge. Furthermore, most of these do a poor
ΛCDM cosmology (Ω =0.3,Ω =0.7,Ω h2 =0.04,h= job at finding substructure in very dense background
0 λ b
0.7,σ = 0.9) were created. From an initial distribution regions, and although nearly all algorithms now use
8
of 5123 particles in a box 64h−1 Mpc wide and with a kinetic information to remove gravitationally unbound
mass resolution of m = 1.6 108h−1 M , the closest particles, they are generally not too concerned with
p ⊙
×
eight particles were iteratively collapsed, reducing the background contamination, which can be safely ignored
particle number to 1283 particles. These low resolution for most applications.
initialconditionswerethenevolveduntilz =0,atwhich Unfortunately, these issues are especially problematic
point eight clusters were selected in the mass range 1– for our analysis: once the substructure halos cross the
3 1014h−1 M . All particleswithin twotimes the virial virial radius of the host cluster, contrast is lost, and the
⊙
×
radius were then tracked back to their initial positions particle background from the host becomes very signifi-
at z = 45, where they were regenerated to their origi- cant. If we were to mistakenly assign background parti-
nalmass resolutionand positions. These highresolution clesfromtheclustertooursubstructurehalos,thiscould
pockets are surrounded by a “buffer” zone with eight mimictheradialalignmenteffectwearelookingfor,since
times the original mass resolution, which itself is nested cluster particles themselves are radially distributed. We
inparticlesthatare64timesmoremassivethanthepar- solvethisproblembyfindingthesubstructurehalosearly
ticlesatthecenterofthecluster. Theseinitialconditions on, at the formation redshift (z ) for each host. At
form
were then re-simulated to z = 0, recording 63 outputs these early times, the hosts are still starting to assemble
from z = 1.5 to z = 0 so that ∆t 0.17 Gyrs. A sum- andhalosarenotasclustered,andthereforemucheasier
≈
mary ofthe eight hosthalos is presentedin Table 1, and to identify. Once the halos have been found, their indi-
on quick inspection it should be immediately apparent vidualparticledistributionscanthenbetrackedforward
that the eight hosts have widely varying masses and as- in time through any environment, even into the dens-
semblyhistories. Wecalculatethesequantitiesasfollows: est cores of clusters, without suffering from background
the virial radius is defined as the distance at which the contamination.
average halo density drops below ρ (r ) = ∆ ρ , A more detailed description of our halo finding and
halo vir vir b
where∆ =340andρ isthecosmologicalbackground tracking methods can be found in Gill et al. (2004), so
vir b
density. The virial mass is defined to be the mass in- weprovidehereonlyabriefsummary. Wefindandtrun-
sidethisradius. Wecalculateeachhost’sageasthetime cate all the halos in our simulation volume at z
form
3
using the AMIGA Halo Finder (AHF), the successor (Frenk et al. 1988; Allgood et al. 2006). The simplest
of the MLAPM Halo Finder (MHF) (Gill et al. 2004). way to do this is to calculate the inertia tensor of the
AHF uses the adaptive grids of AMIGA to locate ha- distribution, I = m r r , which is then diago-
jk Pi i i,j i,k
loswithinthesimulation. AMIGA’sadaptiverefinement nalized to find the principal axes of the halo. However,
meshes follow the density distribution by construction. this procedure is not ideal, since it weights particles by
The grid structure naturally “surrounds” the halos, as r2, and therefore results in a shape measurement that is
the halos are simply manifestations of overdensities. As overly biased by the outlying particle distribution.
AMIGA’s gridsareadaptiveitconstructs aseriesofem- Abettermeasure(Gerhard1983)isthereduced inertia
bedded grids, the higher refinement grids being subsets tensor:
oerfagrrcihdys oofnnleoswteerdriesfiolnaetmedengtrildevsealsn.dAcHonFsttraukcetss tahi“sgrhiid- I˜jk =Xmiri,jrr2i,k. (1)
i i
tree”. Within that tree, each branch represents a halo,
thus identifying halos, sub-halos, sub-sub-halos and so which weights particles equally regardless of their dis-
on. tancetothecenterofthehalo,usingonlythedirectional
Once we have found all the halos and sub-halos in our information of halo particles to calculate their shapes.
simulations at this redshift, we can start tracking their The eigenvectors and eigenvalues of this reduced form
particle distributions throughtime. The main disadvan- of the inertia tensor give us the principal axes of the
tageofthismethodisthatanysubsequentaccretion(af- halo and a measure of their relative lengths (b/a,c/a),
ter zform) onto the halos will, by design, be ignored. although the latter are substantially overestimated, as
Thisseemsareasonablecompromise-haloswithinhalos we shall see.
travel through their environment too quickly to accrete The main source of uncertainty in determining the
significant amounts of particles and we assume that any shapes of our halos is the small number of particles we
particles acquired before the halo enters the host settle sample their potentials with. We want to characterize
into the potential well isotropically, so that we are still halo alignments over as wide a mass range as possible,
obtaining a fair sample of the shape of the halos by only so we want to know what the minimum number of par-
including the particles that were present at zform. ticles is that will still give us a reliable measure of a
At each time step we look at the distribution of parti- halo’sshape. The ability to determine the orientationof
cles for each halo and, after recalculating their center of ahalo’smajoraxisalsodepends stronglyonthe value of
mass,we checkif eachparticle is still bound to the halo. b/a: Anoblatehalowithb a,willbealmostdegenerate
This is an iterative process: starting at the center of the in its major/intermediate a≈xis orientations.
halo and moving outwards, we calculate each particle’s Inorderto addressthesequestions,wegeneratedaset
kineticandpotentialenergyinthehalo’sreferenceframe offake triaxialNFWhaloswith varyingnumbersofpar-
and remove all particles that have velocities, v > bvesc, ticles (Np) and intermediate-to-major axis ratios (b/a)
where b=1.5 is the bound factor, and the only free pa- and fed them through our pipeline. For each value of
rameter in our algorithm. We repeat the process until N and b/a we performed 100 random realizations of an
p
no further particles are removed or a minimum number NFW halo and calculated the angle between the major
(Np =200,c.f. 2.3)ofparticleshasbeenreached. Parti- axis direction measured and that which was input. The
§
clesthataredeterminedtobeunboundaresubsequently dispersioninthesevalues,θ ,isthenagoodestimateof
acc
ignored. This is a completely effective way of removing the accuracy of our measurement. The minor-to-major
theclusterbackground,asparticlesthatdonotbelongto axis ratio (c/a) does not appear to affect the determina-
the substructure halo will be quickly left behind. It also tionofthemajoraxisdirection,andtheresultspresented
allows us to track debris being stripped off the subhalos infigure1arethereforeonlyforprolatehaloswithb=c.
as they orbit inside the cluster. As expected, our accuracy depends very strongly on
When all unbound particles have been removed, we the number of particles sampled - the points on the left
fit an NFW distribution to the radial profile of the re- panelarewellfitbyarelationoftheform: θ N−0.54.
acc
maining particles. We define the halo’s radius as the Wewantacompromisebetweenindividualhalo∝accuracy
distance at which the average halo density drops below and sample size - at values of N < 200, θ increases
p acc
ρhalo(rvir) = ∆vir(z)ρb(z), where ∆vir(z) is the virial rapidly, and we pick this, somewhat arbitrarily, for our
overdensity at that redshift, and discard any particles lower limit on N . If b/a = 0.8, our measurements of
p
thatlie outsidethis limit. However,this radiusisalmost these halos would be accurate by θ 10◦. However,
acc
never reached in the case of substructure, in which case θ also depends strongly on b/a. W≈hen N = 200,
acc p
the radius of the halo is defined as the distance to the b/a< 0.8 is required to maintain the 10◦ error, with an
furthestboundparticle. Oncewehavedeterminedwhich increase in b/a leading to a rapid decrease in accuracy.
particles belong to which halo at each timestep, we are Figure 1 refers to the input values of b/a. In fact, the
ready to measure their shapes. measuredellipticities are much higher, although the two
are tightly correlated: (b/a) = (b/a)0.45 . We
2.3. Shape Measurements input measured
place an upper limit of 0.8 on the intrinsic axis ratios,
Howcanwecondense athree dimensionalparticledis- which translates to a limit on the measured values of
tribution into a few simple parameters describing its b/a<0.9.
shape? With no prior knowledge of how the particles With our limits in place for the minimum number of
are distributed this is a difficult task. However, ha- particlesandmaximumaxisratios,wearereadytostart
los produced in dark matter cosmological simulations analyzing our results. It is worth noting, however, that
seem to follow a universal density profile (Navarro et al. boththeseerrorsourceswouldbiasourshapesrandomly:
1996), and are generally well fit by triaxial ellipsoids thereis no preferreddirectionthatwill be selectedif the
4
halos are under-sampled or too spherical. This in turn be radiallyalignedwithrespectto their nearestclusters.
implies that the results on alignment presented in the If there is a primordial alignment of halos with respect
next section are, if anything, conservative. to the filaments in which they form, then even at large
distances this will appear as a radial alignment in our
3. RESULTS analysis. Thistypeofprimordialalignmentatlargeradii
3.1. Alignment at z =0 was seen by Arag´on-Calvoet al. (2007) in their study of
filamentary structures, where they found similar values
The quantity we will focus on is the angle, φ, between
for cosφ (their figure 2e).
the major axis of each halo and the vector connecting h i
The main focus of this paper, however, is what hap-
the halo to the center of the host. If halos are oriented
pens closer to the host. As the halo falls in, the am-
randomly in space, the cosine of φ will be uniformly dis-
plitude of the alignment increases dramatically, reach-
tributed between 0 and 1, with a mean value, cosφ ,
ing a peak of cosφ = 0.72 at about one-half of the
h i
of 0.5. When cosφ 1 the halo is pointing toward the h i
virialradiusofthe cluster,before decreasingagaingrad-
≈
hostcenter,whereaswhencosφ 0itisalignedtangen-
ually inside the core. The increase of the alignment
≈
tially to it, so that a value of cosφ > 0.5 implies an
withdecreasingdistancematchesthebehaviourfoundby
h i
overall tendency for radial alignment. The standard er-
Faltenbacher et al.(2007)intheirstudyofSDSSgroups,
roron cosφ isσhcosφi =σ/√N, whereN is the sample and is to be expected if the effect is caused by the tidal
h i
size and σ is its standard deviation. We show in Figure field of the host, but the dip at small radii, r < 0.3r ,
vir
2 a histogram of cosφ for all halos within 2rvir of each has not yet been observed. This is most likely due to
of the eight hosts. It is immediately apparent that our the severe projection effects that dominate the cores of
distribution is inconsistent with isotropy at a very high observed clusters.
significance level: cosφ =0.66 0.01, with most halos What causes this behavior? It appears that the align-
h i ±
pointing toward the center of the host halo. mentthatisset-upintheinfallregionsisbeingdisrupted
Whileitisclearthattheresultsoffigure2confirmpre- in the inner regions of the cluster. What causes the dis-
viousobservationalreportsofradialalignment,aprecise ruption? Is this primarily a spatial effect caused by the
quantitative comparison is rather difficult, and we de- environmentoftheclustercore,oratemporalone,given
fer this discussion to 4.2. Nevertheless, much can be thatgalaxiesclosesttothecenterhavebeeninthecluster
§
learned from a qualitative study of the effect’s behavior environment for longer? And what produced the align-
and correlation with individual (and host) halo proper- ment in the first place? The best way to answer these
ties. Figure 3 shows the same histogram as in Figure 2 questionsistotakeadvantageoftheextradimensionpro-
but now for two separate halo populations, segregated vided by simulations, and explore the evolution of this
by mass. There does not appear to be a significant dis- effect with time.
tinction between the two populations. This tells us not
only that the alignment effect is mass independent, but
3.3. Evolution with Redshift
alsoconfirmstheexperimentsin 2.3thatshowthatres-
olution effects are unimportant in§ the lowest mass halos The evolution of the alignment with redshift is plot-
considered in our analysis (N >200). ted in figure 5 for each of the eight clusters indepen-
p
We can also study how the effect depends on extrin- dently. Perhaps the most striking feature of this plot
sic characteristics of the halo, e.g. the distance to the is the self-similarity of the different curves. Every clus-
center of the host, or the host mass. We searched for ter appears to go through exactly the same evolution,
correlationswith different global host properties such as regardless of size or formation time, such that at z = 0
mass and age, and found none. The alignment mecha- theyarepracticallyindistinguishable,asdescribedinthe
nism appears to be universal, in that it is present with previous section. Figure 5 also reveals that the clusters
approximately the same strength in hosts with widely evolve monotonically, with the strength of the effect in-
varying mass, formation times and assembly histories. creasingsteadilysincetheformationtime ofeachcluster
This surprising result is also seen in observational stud- to the present day.
ies: Pereira & Kuhn (2005) found no correlation of the Whatever the source of the disruption at the cluster
alignmentstrengthwith the dynamicalstate of the clus- cores,itisseeminglynotstrongenoughtodilutetheover-
ters inferred from their x-ray morphologies. allalignmentsignal. Therearetwopossibleexplanations:
Itcouldbethat,eventhoughalignmentisdisruptedonce
3.2. Dependence on Distance to Cluster Center the halo reaches the core of the host, the constant infall
ofpristinelyalignedhalosresultsinanoverallincreaseof
Figure 4 shows the dependence of the effect on the
the average alignment per host. Alternatively, the mis-
distance from the cluster center. All halos at redshifts
alignment seen at the cores could be short-lived - a fea-
z < z are included in this analysis, in order to
form ture of each halo’s orbital motion through the potential
enhance the overall signal. The behavior appears very
of the host.
smooth: cosφ risesgraduallyasthehostisapproached,
h i Distinguishing between these two alternativesrequires
peaks slightly past its virial radius, and then decreases
adifferentapproach: weneedtotrackhalosontheirway
again toward the center.
towardthe cluster, andthen trace their orbits inside the
It is striking that already at a distance of three virial
virial radius of the host.
radii there is a small, consistent, tendency for radial
alignment. At this distance, how can the halo already
3.4. Evolution with Orbital Phase
“feel” the presence of the host? This is easily under-
stood once we consider that clusters form at the inter- Figure 6 shows the alignment evolution stacked for all
section of filaments, and hence that most filaments will halos throughout their orbits. Initially halos are tracked
5
relative to the amount of time (in Gyrs) remaining un- ter center, in a figure rotation that is co-planar with its
til they cross the virial radius of the host for the first orbital rotation and in the same direction. The radial
time. Once they cross this threshold, halo orbital times alignment is quickly reinstated, but orbital alignment is
are normalized at each passage through pericenter and lost as the halo progresses towards apocenter. Steady
apocenter. torquing throughout the orbit keeps halos oriented to-
We again detect a small alignment at large distances ward the cluster center and away from the direction of
from the cluster , which we believe is evidence for a pri- their orbits until after the apocentric passage,where or-
mordial alignment along filaments as discussed in 3.2. bitalalignmentincreasessteadilytowardspericenter,and
§
As the host is approached the signal increases signifi- a new cycle begins. Figure 8 illustrates this behaviour
cantly, peaking just before the first pericentric passage, with a sketch of a halo’s rotation as it orbits around the
and then a periodic oscillation ensues, which follows the cluster.
halo’s orbital period closely. On average, the tendency Ifhaloorbitswerecircular,haloswouldquicklybecome
foralignmentismuchlargerwithinthehostthanbefore, tidally lockedandmaintainradialalignmentthroughout
although the alignment tendency changes dramatically their orbits. In reality, their orbits are quite eccentric,
with orbitalphase. It follows that the dip observednear and their orbital speed varies significantly. Halos do not
the cluster cores in figure 4 is in fact a result of the mis- react to the tidal torquing quickly enough through the
alignmentobservedatpericenter,and,mostimportantly, pericentric passage, and the narrow dips observed are
that it is not disruptive, since the alignment tendency is the result. In fact, idealized numerical experiments in-
restored well before the next apocenter is reached. In volving a single halo in a circular orbit around a static
fact, the alignmentis quite constantthroughoutthe rest hostinvariablyleadtotidallockingofthehalo,although
of the orbit and seems to increase slightly at each pas- thetimerequiredforlockingvariessignificantlywiththe
sage. This evidence points to a stable dynamical effect original orientation of the halo (C. M. Simpson & K. V.
that is set-up as the halo orbits around the cluster. Johnston,private communication). Interestingly,for ha-
Further insights can be obtained by exploring the ori- los that start out already pointing toward the host cen-
entations of the halos with respect to their orbits. We ter, the time required is rather short, of the order of an
define a new angle, β, as the angle between each halo’s orbital period or less.
major axis and the halo’s velocity, and plot the mean Further support for this tidal torquing hypothesis is
valueofitscosineforallhalosvs. orbitalphaseinFigure shown in figure 9. Although we believe our shape mea-
7. Thesimilaritiesinbehaviorbetweenradialandorbital surements to be robust to random outliers, it is possible
alignmentatlargedistancesare simply a consequenceof thatstronglydistortedoutershells,caused,e.g.,bytidal
the radial nature of the orbits themselves - halos form stripping, could significantly bias the result. As a test,
and travelalong filaments towardthe intersecting nodes we apply four different particle cuts to each of our halos
where clusters reside. In fact, even inside the hosts, or- by varying the boundedness criteria on the particle ve-
bits are quite eccentric, with an average apocentric to locities: instead of throwing out all particles for which
pericentric distance ratio of 4:1. v >bv ,whereb=1.5,weexcludealternatelyparticles
esc
Coulditbe,then,thattheradialalignmentweobserve thathavevelocitiesgreaterthan1,0.75and0.5timesthe
within the virial radius is just a tendency for halos to escapevelocity. Forthemostconservativecriteria,which
be aligned along their orbits coupled with the fact that only retains particles that have velocities v < 0.5v ,
esc
orbits are, on average, quite radial? Figure 7 tells us more than 70% of the particles are discarded, and we
that this is not the case: once inside the cluster, we find are only probing the very bound cores of the halos. Fig-
that the orbital alignment is also correlatedwith orbital ure 9 makes clear that stripping cannot possibly be the
phase, but whereas the radial alignment is almost in- sole cause of the alignment effect, since even the most
stantly recovered after pericenter, the orbital alignment conservative cut shows significant alignment.
increases again much more slowly, and only after reach- Nevertheless, a trend is observed, in that the most
ing the next apocenter. This asymmetry around peri- boundparticlesshowslightlylesstendencyforalignment
center seems, at first, surprising, but, as will be shown overall. This could be the result of tidal stripping in the
inthefollowingsection,followsasanaturalconsequence outer layers, but more likely it is a simple statistical ef-
of tidal torquing by the cluster potential throughoutthe fect: Because we only consider halos with N > 200 in
p
halo’s orbit. this analysis, as we progressively exclude more particles
from the halo with decreasing b, some halos fall below
4. DISCUSSION thislimitandareconsequentlyignored. Henceadecrease
in b implies a smaller number of halos in each sampled
4.1. Tidal Torquing as a Mechanism for Alignment
bin,whichreducesthesignal-to-noise,andbrings cosφ
h i
Once radialand orbitalalignment information is com- closer to 0.5. Despite this trend, the conclusion remains
bined, a clearer picture emerges of what is going on in- that halo shapes are not significantly warped by tidal
side these clusters. As the halo approaches pericenter stripping, and that tidal torquing of the entire halo is a
along an eccentric orbit, it is continually torqued along better explanation for the effect.
the direction of the potential gradient, i.e. halos tend to A number of early numerical and analytical studies
point toward the host center, and, because their orbits supportthe importance oftidal torques within clusters .
are fairly eccentric, also along their orbitaldirection. At Miller & Smith (1982) performed a set of numerical ex-
pericenter, the halo is moving too fast for the torquing perimentsonarotatingbarinanexternalforcefieldand
to be completely effective, whichcausesthe dip inradial observed tidal braking of the rotation, with a rate that
alignment. It is nevertheless enough to torque the halo wasinverselyrelatedtothesquareoftheclustercrossing
awayfromitsorbitaldirectionandbacktowardtheclus- time. More recently, numerical N-body experiments by
6
Ciotti & Dutta (1994)showedthatthe time requiredfor tween light and dark matter relatively unimportant.
the alignment of a prolate galaxy with the tidal field of We therefore compare the dark matter alignment di-
a cluster is much shorter than the Hubble time, and on rectly with the galaxy observations of Pereira & Kuhn
the order of a few times the galaxy’s intrinsic dynami- (2005). We project each halo’s dark matter particles
cal time. Using a different approach, Usami & Fujimoto alongthe three spatialdimensions in our simulationand
(1997) studiedtidal effects ongaseousellipsoids orbiting computethe2Dinertiatensoroftheirprojecteddistribu-
in a centralpotential analytically, predicting that galax- tion. Theanglebetweenthehalo’s2Dmajoraxisandits
iesineccentricorbitsshouldhavetheirlong-axistrapped projectedseparationfrom the cluster center canthen be
toward the direction of the radius vector of the cluster. measured. We include in this sample all galaxies within
While this paper was being written, two studies were 2 virial radii of the cluster center - interlopers are not
published on halo alignments that describe similar re- accountedfor,sincetheSDSSgalaxieswearecomparing
sults. Kuhlen et al. (2007) studied the alignment of our results to are all spectroscopically confirmed cluster
substructure around a Milky Way type halo using the members. Figure 10 shows the results of this 2D analy-
Via Lactea simulation, and observed a radial alignment sis and compares them with the SDSS observations. We
tendency that is preserved throughout the halos’ or- plot all three independent projections and note that the
bits. Faltenbacher et al. (2007) looked at several differ- dispersionintheirvaluesshouldgiveusafairestimateof
ent types of alignment in a set of dark matter hosts at the error introduced by the projection procedure itself.
z = 0, finding similar levels of radial alignment that in- The dark matter alignment is much stronger than
crease with decreasing distance to the host. that observed: θ = 34◦.5 0◦.9 whereas θ =
h ihalo ± h igal
42◦.79 0◦.55. This mustbe in someparta reflectionof
4.2. A Comparison with Observations ±
how much harder it is to measure accurate galaxy posi-
Theresultsof 3certainlyseemtosubstantiatetheob- tion angles on an survey image than for a well-resolved
§
servationalevidenceforradialalignmentofclustergalax- halo in a cosmological simulation, where dynamical in-
ies. A quantitative comparison, however, is not easily formationallowsforamuchcleanerbackgroundremoval.
made. In order to properly “observe” these simulations, Nonetheless, the dilution caused by this measurement
semi-analyticmodelsofgalaxyformationarerequiredto noisecannotwhollyaccountforthe significantdifference
extrapolate from the dark matter halos to the luminous in radial alignment between the two components. Given
components embedded within. These then need to be the nature of the alignment mechanism established in
projected, interlopers and survey limits accounted for, 4.1,itisperhapsnottoosurprisingthatdarkmatterha-
and the resulting image fed through traditional source §los are more strongly aligned. One wouldnaively expect
extraction and isophotal analysis pipelines. This is a the dark matter halos to be more easily torqued, given
laborious procedure and it cannot yet provide accurate that they are much more extended (providing a longer
results, since galaxy formation models are still largely lever)andhavegenerallylowerspins ( andthereforeless
unconstrained in a crucial parameter: the alignment be- gyroscopicresistance)than their luminous counterparts.
tween luminous and dark matter.
4.3. Possible Consequences of Tidal Torquing in
Observationally, studies of the alignment and relative
Clusters
ellipticity of the two components are currently only pos-
sible for gravitational lens galaxies, a very rare class of Halo alignments have traditionally been studied ei-
objects. Keeton et al. (1998) analysed a sample of 17 ther as a probe of their formation history, or as a con-
lenses, mostly isolated early-types, and found that the taminant to weak lensing studies. Now that we have
luminous componentof the lens generally aligns with its established that the leading mechanism behind halo
inner halo to 10◦. In order to probe the shapes of the alignments within clusters is a dynamical effect present
≤
halos to larger radii, stackedgalaxy-galaxyweak lensing throughouttheirlifetime,itisinterestingtospeculateon
studiesareneeded. Thesearejustnowbecomingfeasible, whatpossibleevolutionaryconsequencesthismechanism
and preliminary results appear somewhat contradictory might have for the halos affected.
(Hoekstra et al. 2004; Mandelbaum et al. 2006b). Figure 8 shows us that at each point in the orbit, the
Most theoretical studies have concentratedon the for- torque acts to rotate the halo away from its orbital di-
mation of disk galaxies and their angular momentum, rection,whichnecessarilyresults in a decelerationof the
where some misalignment between baryonic and dark halo’s orbital motion, inducing orbital decay. The ha-
matter spin is commonly seen (e.g. van den Bosch et al. losanalysedin this study do indeed show a tendency for
(2002)). Ontheotherhand,Bailin et al.(2005)findthat orbitalcircularization: Gill et al.(2004)showedthatha-
the orientations of simulated halos and their embedded los with more pericentric passages have smaller orbital
disks are largely uncorrelated at large radii, and almost eccentricites. They argued that dynamical friction was
perfectly aligned at small (r <0.1r ). not a likely cause, and tentatively ascribed the effect to
vir
The current uncertainty in this parameter makes it the growth of the host halo instead. While the velocity
impossible to accurately predict the orientation of the dispersion of the satellites is seen to depend on the host
galaxies that would populate our halos. However, the halo mass, it seems possible that at least part of the or-
results presented in this paper suggest a gravitational bital decay observed is a natural result of the constant
origin for the alignment mechanism, and it is therefore torquingthroughoutthehalo’sorbit. Thisisaninterest-
reasonable to expect that the two components should ingprospect,sinceanextrasourceoforbitaldecaycould
react similarly to it. Furthermore, the tidal torquing potentiallyhelpsolvetheoutstandingproblemofcDfor-
within clusters is so effective that the halos appear to mation in massive clusters, as well as alleviate some un-
“forget” their original orientations before a single orbit resolveddiscrepanciesbetweenobservedsatellitepopula-
is completed, which renders the original alignment be- tionsandthe generallylowefficacyofdynamicalfriction
7
predicted by numerical studies (e.g. Hashimoto et al. tion of radius, with a small but significant effect
(2003); Taffoni et al. (2003)). We are currently investi- extending out to many virial radii from the clus-
gating the importance of this induced orbital decay,and ter. This signal, which has been seen in previous
this will be the subject of a future paper. work(Arag´on-Calvo et al.2007), ismostlikelyleft
Anotherpossibleconsequenceofthestrongtorquingof overfromtheprimordialimprintofthesurrounding
dark matter halos within hosts is the possibility of disk large-scale structure and can be ascribed to tidal
warping. Becauseoftheirhighangularmomentum,disks torques exerted at early times, when the cluster-
will naturally resist tidal torquing more effectively than size perturbations were just turning around (e.g.,
the surrounding dark matter halo, which will introduce Peebles 1969).
a misalignment between the halo and the disk. Even
though recent studies of (isolated) disk-halo alignments
Closer to the cluster center, within 1-2 virial radii,
show that their orientations are largely uncorrelated at •
the amplitude of the alignment increases dramati-
large radii (Bailin et al. 2005), the same is not true for
cally to a peak of cosφ = 0.72 at about one-half
theinnerhalos(r <0.1r ),wheretherotationalaxisof h i
vir of the virial radius, and then falls slowly closer to
the disk is seen to lie very close to the minor axis of the
the cluster center.
innerhalo. We haveshownthattidaltorquingaffectsall
particles in the halos, even the most bound, so it is not
Whenexaminedasafunctionoforbitalphasefora
unreasonable to expect that the inner shells should also
•
givensatellite,wefindthatthealignmentincreases
feel these torques. The question then remains whether
rapidly as the satellite falls into the cluster for the
the disk within will align itself accordingly, or whether
first time and remains high after that, except for
themisalignmentcouldbeapossiblecauseofwarpingof
a short period during pericenter passage, when it
the disk, but this will also require further study.
dips precipitously. It is this short-lived dip which
5. CONCLUSIONS gives rise to the decrease in cosφ close to the
h i
cluster center.
There is growing observational evidence that a satel-
lite’s major axis is preferentially aligned with the ra-
dial vector linking the satellite to its host. This ten- Based on these results, we conclude that the strong
dency for satellites to point at their hosts has been seen alignment seen at small radius — within two virial radii
on both cluster and group scales (Pereira & Kuhn 2005; — is due to tidal torquing by the cluster halo. The idea
Agustsson & Brainerd 2006). Motivated by this result, isverysimple–thegalaxyisonlyinastableequilibrium
we haveuseda suite of cosmologicalN-body simulations ifit is pointing atthe cluster center;otherwise there is a
to investigate the alignment between satellite and host net torque which acts to rotate the galaxy towards this
dark matter halos. equilibriumpoint. Wedemonstratethatthealignmentis
Wetakeparticularcaretoseparatesatelliteandcluster seenbothintheouterandinnerpartsofthesatellite,in-
particles using a combined halo finder plus tracker. In dicating that it is not due to some process (such as tidal
this method, an adaptive halo finder (Gill et al. 2004) is stripping) which impacts only the outer, poorly bound,
used to initially identify a set of satellite sub-halos that part of the sub-halo. We also briefly review previous lit-
we subsequently track as they enter and orbit the clus- eraturewhichhasinvestigatedtheimpactoftidaltorques
ter, removing particles as they become unbound. The oncollisionlesssystemsusinganalyticapproximationsor
advantage of this approach is that we can be sure to idealized simulations, and find that the expected ampli-
use only genuine sub-halo particles and exclude “back- tudeandtimescaleissufficienttoproducethealignments
ground”clusterparticlesthatmightbiasourshapemea- we see.
surements. We then use the reduced inertia tensor to Although we study only dark-matter halos, we expect
measure the shapes and orientations of all sub-halos this effect to extend to the luminous part of galaxies, as
which end up inside the virial radii of a set of eight observations seem to indicate. This will have an obser-
simulated clusters. We highlight here the main results vational impact on weak lensing studies and may also
obtained from this analysis: modify the distribution of stars in a satellite, as well
as the satellite’s orbital properties. We will investigate
Thesatellitesinthesimulationsshowastrongten- these possibilities in future work.
•
dency to point toward the cluster center. The
mean cosine of the angle between the major axis
of each halo and the cluster center is cosφ = The simulations presented in this paper were carried
0.66 0.01, where an isotropic distributihon woiuld outontheBeowulfclusterattheCentreforAstrophysics
have±cosφ = 0.5. This tendency for alignment &Supercomputing,SwinburneUniversity. Wewouldlike
is fouhnd fori all clusters at all redshifts analyzed, to thank Jeff Kuhn, Kathryn Johnston and Christine
and does not appear to depend on the mass of the Simpson for helpful discussions. Greg Bryan acknowl-
cluster or the satellite. edges support from NSF grants AST-05-07161,AST-05-
47823,andAST-06-06959,aswellastheNationalCenter
The amplitude of the alignment is a strong func- for Supercomputing Applications.
•
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Fig. 1.— Results for “fake” NFW halo analysis. Left: Error in major axis orientation vs. number of particles in halo. The vertical
dashedlinerepresents thelowerlimitonNforthehalos analysedinthispaper,whereas thehorizontal linesshow theminimumaccuracy
that will be tolerated. Right: The error in major axis orientation vs. the intermediate-to-major axis ratio. Errors are shown for halos
with200particles(asdots)andwith1000particles(ascrosses).
Fig. 2.—Angledistributionatz=0. Anisotropicdistributionwouldfollowthedashedline,withhcosφi=0.5
10
Fig. 3.— Angle distribution at z = 0 for high mass and low mass halos. An isotropic distribution would follow the dashed line, with
hcosφi=0.5. MasslimitsareinunitsofM⊙.
Fig. 4.—Radialalignmentvs. distancetoclustercenter forallhaloswithz<zform. Anisotropicdistributionisagainrepresentedby
thedashedline