The Dynamics of Merging Clusters: A Monte Carlo Solution Applied to the Bullet and Musket Ball Clusters
DDraft version June 11, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
THE DYNAMICS OF MERGING CLUSTERS:A MONTE CARLO SOLUTION APPLIED TO THE BULLET AND MUSKET BALL CLUSTERS
William A. Dawson
University of California, Davis,Physics Department, One Shields Av., Davis, CA 95616, USA
Draft version June 11, 2018
ABSTRACTMerging galaxy clusters have become one of the most important probes of dark matter, providingevidence for dark matter over modified gravity and even constraints on the dark matter self-interactioncross-section. To properly constrain the dark matter cross-section it is necessary to understand thedynamics of the merger, as the inferred cross-section is a function of both the velocity of the collisionand the observed time since collision. While the best understanding of merging system dynamicscomes from N-body simulations, these are computationally intensive and often explore only a limitedvolume of the merger phase space allowed by observed parameter uncertainty. Simple analytic modelsexist but the assumptions of these methods invalidate their results near the collision time, plus errorpropagation of the highly correlated merger parameters is unfeasible. To address these weaknesses Idevelop a Monte Carlo method to discern the properties of dissociative mergers and propagate theuncertainty of the measured cluster parameters in an accurate and Bayesian manner. I introduce thismethod, verify it against an existing hydrodynamic N-body simulation, and apply it to two knowndissociative mergers: 1ES 0657-558 (Bullet Cluster) and DLSCL J0916.2+2951 (Musket Ball Cluster).I find that this method surpasses existing analytic models — providing accurate (10% level) dynamicparameter and uncertainty estimates throughout the merger history. This coupled with minimalrequired a priori information (subcluster mass, redshift, and projected separation) and relativelyfast computation ( ∼ Subject headings: galaxies: clusters: individual (1ES 0657-558), galaxies: clusters: individual (DLSCLJ0916.2+2951), gravitation, methods: analytical, methods: statistical INTRODUCTION
Merging galaxy clusters have become important astro-physical probes providing constraints on the dark mat-ter (DM) self-interaction cross-section ( σ DM ; Markevitchet al. 2004; Randall et al. 2008; Merten et al. 2011; Daw-son et al. 2012), and the large-scale matter-antimatterratio (Steigman 2008). They are a suspected source ofextremely energetic cosmic rays (van Weeren et al. 2010),and the merger event potentially affects the evolution ofthe cluster galaxies (e.g. Poggianti et al. 2004; Hwang &Lee 2009; Chung et al. 2009). All of the respective astro-physical conclusions drawn from merging clusters dependon the specific dynamic properties of a given merger.For example, the subclass of dissociative mergers, inwhich the collisional cluster gas has become dissociatedfrom the near collisionless galaxies and dark matter, pro-vides four ways of constraining the dark matter self-interaction cross-section (Markevitch et al. 2004; Randallet al. 2008). The best constraints come from studying themass-to-light ratios ( M/L ) of the subclusters , and theoffset between the collisionless galaxies and dark matter(Markevitch et al. 2004; Randall et al. 2008). Both con-straints directly depend on the merger dynamics. Firstthe relative collision velocity will affect the expected mo-mentum transfer between each subcluster’s dark matterparticles which will in turn affect the expected dark mat- [email protected] I define subcluster as either one of the two colliding clusters, ir-respective of mass, and I define cluster as the whole two-subclustersystem. ter mass transfer from the smaller subcluster to the largersubcluster ultimately affecting the expected mass to lightratios of the clusters (Markevitch et al. 2004). Secondthe expected galaxy–dark matter offset will depend onthe observed time-since-collision ( T SC ). Initially theoffset between the galaxies and dark matter will increasewith
T SC (for σ DM >
0) as the collisionless galaxies out-run the dark matter that experienced a drag force duringthe collision, then at later
T SC the offset will decreasedue to the gravitational attraction between the galaxiesand dark matter halo. Additionally it is important toknow the velocity so that dark matter candidates withvelocity dependent cross-sections (e.g. Col´ın et al. 2002;Vogelsberger et al. 2012) can be constrained.However there is no way to directly observe the dy-namic merger parameters of principal interest: the three-dimensional relative velocity ( v ) and separation ( d )of the subclusters as a function of time, their maximumseparation ( d max ), the period between collisions ( T ), andthe time-since-collision ( T SC ). Observations are gen-erally limited to: the subcluster projected separation( d proj ), the line-of-sight (LOS) velocity of each subcluster( v i ) as inferred from their redshifts, and their mass ( M i )or projected surface mass density profile. In addition tothe obvious inability to measure a change in the mergerstate, it is difficult to constrain the dynamic parametersof interest even in the observed state. This is due to thegeneral inability to constrain the angle of the merger axis I define the time of collision to be the time of the first pericen-tric passage. a r X i v : . [ a s t r o - ph . C O ] J u l Dawsonwith respect to the plane of the sky ( α ), see Figure 1.For the Bullet Cluster it was originally thought thatestimates of the Mach number of the cluster mergerthrough X-ray observations of the gas shock feature (e.g.Markevitch 2006) could be used to estimate v , andin conjunction with measurements of the relative LOSvelocities then estimate α . Similarly, the gas pressuredifferential across cold front features seen in some merg-ing clusters have also been used to estimate the Machnumber of the cluster merger (e.g. Vikhlinin & Marke-vitch 2003). However, Springel & Farrar (2007) showedthat the Mach number only translates to an upper limiton v , and in the case of the Bullet Cluster they showedthat the Mach inferred velocity could be a factor of ∼ v . There is potential for constrain-ing α using polarization measurements of radio relics(Ensslin et al. 1998), which are associated with somecluster mergers (e.g. van Weeren et al. 2010) but not all(e.g. Russell et al. 2011). Even if for some mergers radiorelics provide constraints on α , dynamic models are stillneeded in order to ascertain the dynamic properties ofthe merger throughout time.The two most prevalent methods for ascertaining thedynamics of observed merging systems are the timing ar-gument and N-body simulations. The timing argumentis based on the solution to the equations of motion oftwo gravitating point masses, with the cosmological con-straint that as z → ∞ the separation of the two masses d → Fig. 1.—
The generic two-halo merger configuration assumed inthis work. Observable parameters are shown in dark blue, and in-clude the mass of each halo ( M i ), the projected separation ( d proj ),and the line of sight (LOS) velocity components ( v i ) as deter-mined from the halo redshifts. The generally unknown parame-ters of the mergers are shown in light blue, and include the angleof the merger axis with respect to the plane of the sky ( α ), andthe three-dimensional separation ( d ) and velocity components( v i ). Note that while just the outgoing scenario is shown in thisfigure, the method also considers the incoming scenario. the mass accretion histories of the clusters (e.g. Wechsleret al. 2002), their correction is incompatible with the tim-ing argument method as this would add a second differ-ential term to the equations of motion. Finally, the largecovariance between the merger parameters plus the com-plexity of the equations of motion makes propagation oferrors in the timing argument formalism untenable. Thishas resulted in a lack of certainty with timing argumentresults, leaving most users to run a few scenarios in aneffort to roughly bound the range of possible solutions(e.g. Boschin et al. 2012).N-body simulations provide the most accurate descrip-tion of merger dynamics, however they are computation-ally expensive which results in their application beinglimited. Despite eleven currently confirmed dissociativemergers only the Bullet Cluster has been modeled withN-body simulations, whereas most of these have beenanalyzed with the timing argument method. ExistingN-body simulation strategies to ascertain the dynamicproperties of mergers are incapable of keeping up withthe current faster than exponential rate of discovery.Even for the case of the Bullet Cluster the N-body anal-yses have been limited as far as mapping out the mergerdynamic phase space allowed by the uncertainty of theobservations, with at most 15 different scenarios beingrun (Mastropietro & Burkert 2008). G´omez et al. (2012)have come the closest to addressing this issue in their in-vestigation of potential dissociative mergers (A665 andAS1063) through the use of simplified scale-free numeri- (1) Bullet Cluster (Clowe et al. 2004); (2) A520 (Mahdaviet al. 2007); (3) MACS J0025.4-1222 (Bradaˇc et al. 2008); (4)A1240 (Barrena et al. 2009); (5) ZwCL 0008.8+5215 (van Weerenet al. 2011); (6) A2744 (Merten et al. 2011); (7) A2163 (Okabeet al. 2011); (8) A1758N (Ragozzine et al. 2012); (9) Musket BallCluster (Dawson et al. 2012); (10) ACT-CL J01024915 (Menanteauet al. 2012); (11) MACS J0717.5+3745 (Mroczkowski et al. 2012) ynamics of Merging Clusters 3cal simulations of the mergers (see G´omez et al. 2000, fordetails). However, they have still had to severely limitthe phase space probed (fixing merger parameters suchas the initial relative velocity and subcluster-subclustermass ratio); thus admittedly this approach enables con-struction of plausible models, but not a thorough ac-counting of possible or likely models.With these weaknesses in mind I present a newmethod for analyzing the dynamics of observed disso-ciative mergers. My primary objectives are to 1) obtaina solution valid near the collision state, 2) fully estimatethe covariance matrix for the merger parameters, 3) beable to analyze a dissociative merger on the order of aday using a typical desktop computer, and 4) obtain ap-proximately 10% accuracy; all assuming that only themost general merger observables and their uncertaintyare known: mass of each subcluster, redshift of each sub-cluster, and projected separation of the subclusters.In § § § § § Λ = 0 .
7, Ω m = 0 . H = 70 km s − Mpc − . METHOD
In order to obtain a valid solution of the system dynam-ics near the collision state I use a model of two sphericallysymmetric NFW halos, rather than point masses. I incor-porate this model in a standard Monte Carlo implemen-tation: draw randomly from the observables’ probabilitydensity functions (PDF’s) to generate a possible realiza-tion of the merger, use the model to calculate mergerproperties of interest, apply multiple priors, store theselikelihood weighted results as a representative randomdraw of their PDF, and repeat. The final result is a mul-tidimensional PDF for the dynamic parameters of themerger. This method agrees well with hydrodynamicsimulations, § Model
The general basis of the model is a collisionless twobody system with the mass of each body mutually con-served throughout the merger. The model requires min-imal input: the mass of each subcluster, the redshift ofeach subcluster, and the projected separation of the sub-clusters (along with associated uncertainties). It assumesconservation of energy and zero angular momentum. Themodel also assumes that the maximum relative velocityof the two bodies is the free-fall velocity of the systemassuming their observed mass. In the remainder of thissubsection I will discuss in detail these general assump-tions, their justification, and their implications. This method is similar to the one used by Dawson et al. (2012),although with several improvements (see § I model the system using two spherically symmetricNFW halos truncated at r . By default the concentra-tion of each halo is determined by the halo’s mass via themass-concentration scaling relation of Duffy et al. (2008).This is not a requirement of the model though, and mea-sured concentrations can be used, as in the case of § § = 1 .
94 and c = 7 .
12: if instead Duffy et al.(2008) inferred concentrations c = 3 .
44 and c = 2 . ∼ v ( t col ) and T SC are both less than 6%.The model assumes that the mass of each subclusteris constant and equal to the observed mass . While thisassumption is also used in the timing argument method,it is more reasonable for this method since the bulk of theresults are calculated between the observed state and thecollision state, typically lasting (cid:46) K ( t ) = 0 . µv D ( t ) , where µ is the reducedmass of the system and v D ( t ) is the relative physical ve-locity of the two subclusters at time t . The potential en-ergy of the system is assumed to be purely gravitationaland is derived in Appendix A. Since the model assumeszero impact parameter there is no rotational kinetic en-ergy term. Mastropietro & Burkert (2008) find that amoderate impact parameter of ∼ . r has less than a1% effect on the merger velocity, thus this assumptionshould have negligible effect for the case of dissociativemergers which must have had relatively small impact pa-rameters in order to dissociate the bulk of their gas.For the relative velocity of the two subclusters I apply aflat prior from zero to the free-fall velocity of the subclus-ters, assuming their observed mass. This will result in anoverestimate of the maximum possible relative velocity,due to the neglect of mass accretion. It is conceivablethat this prior could be tightened using the maximumrelative velocities observed in cosmological N-body sim-ulations as a function of subcluster masses and redshift.Another possibility for tightening the prior would be toanalytically estimate the free-fall velocity accounting formass accretion (e.g. Angus & McGaugh 2008). An ad-vantage of the Monte Carlo approach taken with thismethod is that additional priors can be applied as moreknowledge becomes available without the need to rerunthe analysis, so I opt for a conservative default approach.The model ignores the effects of surrounding large scalestructure and simply treats the two-body system. AsNusser (2008) shows, a global overdense region (10 timesdenser than the background) engulfing the system onlyaffects the dynamics substantially for extreme collisionvelocities ( ∼ − ). While global overdensities r is defined as the radius of the spherical region withinwhich the average density is 200 times the critical density at therespective redshift. For subcluster mass I refer to M , which is the mass of theindividual subcluster enclosed within a radius of r . Dawsonmay be disregarded it is not clear that the effects ofnearby structures can be disregarded, e.g. as in the casethree body systems. Thus this method should be appliedwith caution to complex cluster mergers.The model also ignores dynamical friction. Farrar &Rosen (2007) found that including dynamical friction ac-counted for an ∼
10% reduction in the inferred collisionvelocity of the Bullet Cluster in their analytic treatment.This is potentially concerning since dynamical frictionis inversely proportional to the relative velocity of themerger, thus it may become even more important formergers slower than the Bullet Cluser. However in § Monte Carlo Analysis
In this section I discuss the details of the Monte Carloanalysis workflow. I chose to implement a Monte Carloanalysis because the high degree of correlation amongthe many merger dynamic parameters made traditionalpropagation of errors unfeasible. A Monte Carlo analysishas the added advantage of easily enabling applicationof different combinations of priors ex post facto, see e.g. § M i ), redshift of each subcluster ( z i ), and projectedseparation of the subclusters ( d proj ). The potential en-ergy, V (see Appendix A), at the time of the collision isused to calculate the maximum relative velocity, v max = (cid:114) − µ V ( r = 0) . The velocity of each subcluster relative to us is esti-mated from its redshift, v i = (cid:20) (1 + z i ) − z i ) + 1 (cid:21) c, where c is the speed of light. The relative radial velocityof the subclusters is calculated from their redshifts, v rad ( t obs ) = | v − v | − v v c . Since the angle of the merger axis with respect to theplane of the sky, α , is unconstrained without prior knowl-edge of the three-dimensional relative velocity, I assumethat all merger directions are equally probable. How-ever, projection effects result in P DF ( α ) = cos( α ). Dueto symmetry it is only necessary to analyze the range0 ≤ α ≤
90 degrees. I draw randomly from this PDFfor each realization. This enables the calculation of thethree-dimensional relative velocity in the observed state, v ( t obs ) = v rad ( t obs ) / sin( α ) , (1)as well as the observed three-dimensional separation ofthe subclusters, d ( t obs ) = d proj / cos( α ) . (2) If v ( t obs ) > v max , then this realization of themerger is discarded; otherwise the relative collision ve-locity is calculated, v ( t col ) = (cid:114) v ( t obs ) + 2 µ [ V ( t obs ) − V ( t col )] . (3)Similarly if v ( t col ) > v max , then this realization isdiscarded.The change in time, ∆ t , between two separations isgiven by ∆ t = (cid:90) r r dr (cid:113) µ ( E − V ( r )) . (4)I define the time-since-collision ( T SC ) as the time ittakes the subclusters to traverse from zero separationto their physical separation in the observed state, d .Because there is a potential degeneracy in whether thesubclusters are “outgoing” (approaching the apoapsis af-ter collision) or “incoming” (on a return trajectory aftercolliding and reaching the apoapsis); I solve for both ofthese cases, T SC and T SC respectively. In determin-ing T SC it is useful to define the period, T , of the sys-tem. I define T to be the time between collisions, T = 2 (cid:90) d max dr (cid:113) µ ( E − V ( r )) , where d max is the distance from zero separation to theapoapsis, when E = V . Thus, T SC = T − T SC . During the Monte Carlo analysis any realizations with
T SC greater than the age of the Universe at the clus-ter redshift are discarded. A similar flat prior is appliedwhen calculating the statistics of T SC . To this regardsome insight into the likelihood of the system being in an“outgoing” or “incoming” state can be gained by calcu-lating the fraction of realizations with T SC less than theage of the Universe at the cluster redshift. Conceivablythese temporal priors could be strengthened, requiringthat the time to first collision ( T ) plus the respective T SC be less than the age of the Universe at the clus-ter redshift, in a fashion similar to the timing argument.However, as with the timing argument model, the modelof § T SC ) = 2 T SC T . (5)There are likely selection effects which complicate thisPDF, since it can be imagined that the X-ray luminosityis greatest near the time of the collision (see e.g. Randallet al. 2002). However this information is rarely if everknown, thus it is not included by default. In § Comparison with Hydrodynamic Simulations
For the purposes of checking the physical assump-tions of the model I reanalyze the Springel & Farrar(2007) model of the Bullet Cluster, comparing my dy-namic parameter estimates with their hydrodynamic N-body simulation based estimates. For this analysis Irun just their single case through the model (i.e. I donot perform a Monte Carlo analysis). They representthe “main” and “bullet” subclusters as NFW halos withM = 1 . × M (cid:12) , c = 1 .
94, M = 1 . × M (cid:12) ,and c = 7 .
12, respectively. They note that the gas prop-erties of their simulation most closely match the observedBullet Cluster gas properties for the time step corre-sponding to a subcluster separation of d D = 625 kpcand relative velocity of v ( t obs ) = 2630 km s − . I definethis as the “observed” state (dashed line in Figure 2)and use the model discussed in § v ) and time-since-observed state (TSO) before and after the observedstate (left and right of the dashed line in Figure 2, respec-tively). The Springel & Farrar (2007) simulation results(black circles) for these parameters are read directly fromtheir Figure 4.I compare the model results (blue boxes) with theSpringel & Farrar (2007) simulation results, and assumetheir results as truth when calculating the percent error,see Figure 2. There is better than 4% agreement be-tween v and 14% agreement between the TSO. Whilethe model results are biased, the bias appears stable andis roughly an order of magnitude smaller than the typicalrandom error in the parameter estimates (see for exam-ple Table 2). Given the stability of the bias it is con-ceivable that it could be corrected in the model results.However, to have any confidence in this bias correctionthe model results should be compared with a range ofmerger scenarios, which is beyond the scope of this cur-rent work. Note that the better agreement between thevelocity estimates than between the TSO estimates isto be expected since the velocity calculation (essentiallyEquation 3) comes from simply comparing the observedand another state of the merger whereas the TSO cal-culation (Equation 4) requires integration between thesetwo states. The results of this comparative study essen-tially validate many of the simplifying assumptions of themodel (conservation of energy, and ignoring the affectsof dynamical friction, tidal stripping of dark matter andgas during the collision).As an aside it should be noted that for this comparisonI use the Springel & Farrar (2007) NFW halo parame-ters that represent the state of the halos prior to collision.Ideally I should use the NFW parameters representativeof the state of the halos at t obs , however these proper- Fig. 2.—
Comparison of the model results (blue boxes) with thehydrodynamic simulation results of Springel & Farrar (black cir-cles; 2007) for the Bullet Cluster. The top figure is a comparisonof the velocity of the “bullet” relative to the “main” subcluster,with the subhalo separation (i.e. the three-dimensional separationof the “main” and “bullet” subclusters) as the independent vari-able. The bottom figure is a comparison of the time-since-observedstate (TSO), where the “observed” state (dashed line) is defined bySpringel & Farrar (2007) as the time step in their simulation whenthe gas properties most closely match the observed Bullet Clustergas properties. Times prior(post) to the observed state have neg-ative(positive) values. The percent error in each case is calculatedassuming the Springel & Farrar (2007) results as truth. While themodel results are biased, the bias appears stable and is roughlyan order of magnitude smaller than the typical random error inthe parameter estimates (see for example Table 2). Note that theTSO percent error calculation understandably diverges near thearbitrary choice of time equal zero. The Springel & Farrar (2007)results are read directly from their Figure 4. ties were not reported in their paper. From Figure 5of Springel & Farrar (2007) some insight into the timevariability of the halo parameters can be gained. Sincethe depth each halo’s gravitational potential at ∼ r does not change appreciably throughout the merger, itcan be inferred that M of each halo does not change.However, the gravitational potential near the center ofeach halo deepens by ∼
25% during and after the colli-sion. This can be interpreted as the concentration of eachhalo increasing. Thus for the comparison of my modelwith the Springel & Farrar (2007) hydrodynamic simu-lation to be more appropriate I should have used haloswith larger concentrations. Doing so actually brings mymodel results more in-line with the simulation results.If for example I increase the concentration of the “bul-let” halo from 1.94 to 3 and the concentration of the Dawson“main” halo from 7.12 to 8, then the percent error forthe relative velocity of the halos reduces to (cid:46)
1% andthe percent error for the TSO reduces to ∼ BULLET CLUSTER DYNAMICS
The Bullet Cluster is the prime candidate for first ap-plication of the method as it is one of the best studieddissociative mergers. It has a wealth of observationaldata necessary for input to the model, as discussed in § § Bullet Cluster Observed System Properties
I summarize the observed Bullet Cluster parametersused as input to my analysis in Table 1. The full PDF’sof these input parameters have not been published so Isimply assume Gaussian distributions. I refer to the mainsubcluster as halo 1 and the “bullet” subcluster as halo2. For the mass and concentration of each subcluster Iuse the most recently reported estimates from Springel &Farrar (2007), based upon strong and weak lensing esti-mates (Bradaˇc et al. 2006). However, they do not presenterrors for these quantities so for the mass I estimate the1– σ errors to be 10% of the mass, since this is approx-imately the magnitude of the error reported by Bradaˇcet al. (2006) for M( <
250 kpc). There is no publishedestimate for the uncertainty of the concentrations of theNFW model fits, c i , so I simply assume the concentra-tions to be known quantities (as noted in § Bullet Cluster System Dynamics Results
I first analyze the Bullet Cluster with the Monte Carloanalysis method and default priors discussed in §
2, high-lighting the complexity of merger dynamics and the inap-propriateness of analyzing a small sample of select mergerscenarios. In § § ∼ Default Priors
TABLE 1Bullet Cluster parameter input
Parameter Units µ σ
Ref.M M (cid:12)
15 1.5 a · · · b M (cid:12) a · · · b d proj kpc 720 25 3 References . — (1) Springel & Farrar 2007; (2)Barrena et al. 2002; (3) Bradaˇc et al. 2006.
Note . — A Gaussian distribution with mean, µ ,and standard deviation, σ , is assumed for all pa-rameters with quoted respective values. The mass,M , and concentration, c, are the defining proper-ties of assumed spherically symmetric NFW halos. a Estimated to be 10%, based one the error mag-nitude of M( <
250 kpc) reported in Bradaˇc et al.(2006). b No errors were presented in the reference. A singleconcentration value was used for all Monte Carlorealizations.
Fig. 3.—
The posterior of the Bullet Cluster’s time-since-collision
T SC and v ( t col ) parameters is shown in grayscale withdark and light blue contours representing 68% and 95% confidence,respectively. The green-scale triangles are from a subsample of theMonte Carlo population, which jointly satisfies the requirementof being drawn from ± . σ of the mean of each of the inputparameters, i.e. the “most likely” values for the input parame-ters. Despite representing the most probable input parameter val-ues there is considerable spread in the inferred output parameters,with the subsample clearly tracing the ridge of the distribution.The saturation of the triangles increases with increasing α , from10–86 degrees. The purple-scale circles are from a subsample nearthe bi-weight location of α = 50 ± . . While thelength of the distribution is predominantly caused by uncertaintyin α the width is predominantly caused by uncertainty in the in-put parameters. Despite the Bullet Cluster being one of the bestmeasured dissociative mergers there is still considerable and com-plex uncertainty in its merger parameters, predominantly due touncertainty in α . The main results of this analysis are that: 1) there is agreat degree of covariance between the geometry, veloc-ity, and time parameters of the merger, and 2) modelsof the system which disregard the uncertainty of α willcatastrophically fail to capture the true uncertainty inynamics of Merging Clusters 7the dynamic parameters.The two-dimensional PDF of Figure 3 exemplifies thecomplexity of the covariance between the various mergerparameters . The shape of the PDF is most easily under-stood in terms of the parameters’ dependence on α . Thisdependence is illustrated by the green-scale triangles thatrepresent a subsample of the Monte Carlo population,which jointly satisfies the requirement of being drawnfrom ± . σ of the mean of each of the input parameters,i.e. the “most likely” values for the input parameters.The saturation of the triangles increases with increasing α , from 10–86 degrees, clearly showing a monotonicallyincreasing relationship with T SC (see also Figure 15).For small α (light green triangles), Equation 1 states that v ( t obs ) must be large thus v ( t col ) must also be large,and since Equation 2 states that d ( t obs ) approaches theminimum possible observed separation, d proj , the T SC must approach a minimum. Conversely for large α (darkgreen triangles), d ( t obs ) becomes large increasing thetime required to reach the observed state, and despite v ( t obs ) approaching the minimum v rad ( t obs ) the colli-sion velocity must increase for the subclusters to havebeen able to reach the larger d ( t obs ).The bulk of the uncertainty in the geometry, velocityand time parameters is due to the uncertainty of α . Thisis exemplified by the fact that the green-scale triangles inFigure 3 closely trace the extent of the ridge line of thetwo-dimensional distribution (i.e. span the bulk of theuncertainty). Conversely the “width” of the distributionis predominantly due to uncertainty in the input param-eters. This is exemplified by the purple circles of Figure3, which are for a near constant α yet randomly sam-ple the M distribution. The saturation of the circlesincrease with increasing mass.The inability to directly measure α , coupled with itsstrong degree of correlation with the other dynamicparameters, makes it the dominant source of uncer-tainty. While it was originally believed that the three-dimensional merger velocity as inferred from the X-rayshock feature could be coupled with the redshift de-termined radial velocity to measure α , Springel & Far-rar (2007) showed that the X-ray shock inferred veloc-ity significantly overestimates the true three-dimensionalmerger velocity. So at best this information can weaklyconstrain α , and in the case of the Bullet Cluster the X-ray shock inferred velocity is significantly greater thanthe free-fall velocity, v max , thus it provides no addi-tional constraining power. In § α . Added Temporal Prior
One of the advantages of this Monte Carlo methodis that additional constraints are easily incorporated expost facto. An example of such constraints in the case ofthe Bullet Cluster is the observed X-ray shock front andfactor of 2.4 greater X-ray estimated mass to lensing es-timated mass (Markevitch 2006), due to merger relatedX-ray temperature and luminosity boost. Hydrodynamic Similar degrees of complex covariance exist for the other ge-ometry, velocity and time parameters, see e.g. the results array inAppendix B.
Fig. 4.—
The posterior of the Bullet Cluster’s
T SC and v ( t col ) parameters after application of an additional temporalprior based on X-ray observations of the Bullet Cluster (grayscale).Dark and light blue contours representing 68% and 95% confidence,respectively. The added temporal prior significantly improves theconstraint on the merger parameters (compare with Figure 3).The black diamond represents the Springel & Farrar (2007) hy-drodynamic simulation result for their defined “observed state”,whose X-ray properties best match the observed X-ray properties; d =625 kpc for this state. The green bar shows their result for d between 700 to 900 kpc, which is more in line with the observed d proj = 720 ±
25 kpc (assuming 0 < α <
35 degrees). simulations of merging clusters (e.g. Ricker & Sarazin2001; Randall et al. 2002) suggest that such transient ef-fects last of order the X-ray sound crossing time. Sincesimulations show negligible difference between the timescales of the two I chose to construct a prior based on theobserved temperature boost. Randall et al. (2002) findthat the full-width-half-max (FWHM) duration of thetemperature boost is ∼ . t sc with the entire boost du-ration being ∼ . t sc , where t sc is the sound crossing timeof the more massive of the two subclusters. The peak ofthis boost roughly coincides with the time of the colli-sion , as defined in §
1. Given the M = 15 × M (cid:12) and temperature T X = 14 keV of the “main” subcluster(Markevitch 2006), the t sc = 1 Gyr. I construct a sigmoidfunction for the T SC prior PDF based on the observedtemperature boost,PDF(
T SC ) = 12 (cid:20) − tanh (cid:18) T SC − . a . b (cid:19)(cid:21) , where a is the FWHM of the duration of the temperatureboost and b is the entire boost duration. I chose a sigmoidfunction over a simple step function since the tempera-ture boost predicted by Randall et al. (2002) does notend abruptly. This prior is coupled with the previouslydiscussed T SC prior (Equation 5).Application of this prior significantly improves the un-certainty in
T SC (180% to 67%) and v ( t col ) (28%to 19%), compare Figure 3 with Figure 4. It essen-tially removes the possibility of a T SC > . α dependence shown by the green tri-angles in Figure 3 this prior also reduces the likelihoodof α (cid:38)
50 degrees, which in turn affects both the loca-tion and uncertainty of d and v ( t obs ), see Table 2.The remaining parameter estimates are predominantlyunaffected by the prior, with only a few having theirconfidence limits affected as the result of their high end Dawson TABLE 2Bullet Cluster parameter estimates
Parameter Units Default Priors Default + Added Temporal PriorsLocation a
68% LCL–UCL b
95% LCL–UCL b Location a
68% LCL–UCL b
95% LCL–UCL b M M (cid:12) M (cid:12) z z d proj Mpc 0.72 0.69 – 0.76 0.65 – 0.80 0.72 0.68 – 0.75 0.64 – 0.79 α degree 50 27 – 73 15 – 84 24 16 – 38 11 – 53 d Mpc 1.1 0.8 – 2.6 0.7 – 7.1 0.8 0.7 – 0.9 0.7 – 1.2 d max Mpc 1.3 1.1 – 2.5 1.0 – 6.4 1.2 1.0 – 1.7 1.0 – 3.1 v ( t obs ) km s −
820 640 – 1500 550 – 2500 1600 1100 – 2500 790 – 3200 v ( t col ) km s − T SC Gyr 0.6 0.3 – 1.1 0.2 – 3.9 0.4 0.3 – 0.5 0.2 – 0.6
T SC Gyr 1.2 1.0 – 2.4 0.9 – 8.2 1.3 1.0 – 2.0 0.9 – 4.6 T Gyr 1.8 1.5 – 3.2 1.4 – 8.1 1.6 1.4 – 2.3 1.3 – 4.8 a Biweight-statistic location (see e.g. Beers et al. 1990). b Bias-corrected lower and upper confidence limits, LCL and UCL respectively (see e.g. Beers et al. 1990). c For the case of the Default + Added Temporal Prior, none of the realizations have a valid
T SC , meaning that the BulletCluster is being observed in the “outgoing” state, as discussed in § low probability tails being down weighted. Additionallythere is now essentially zero probability that the bulletsubcluster has reached the apoapsis and is on a returntrajectory, since the 95% lower confidence limit of T SC is 0.9 Gyr (see Table 2) and the prior essentially removesthe possibility of a T SC > . § d = 625 kpc;this is less than the observed d proj = 720 ±
25 kpc(Bradaˇc et al. 2006). If we instead consider their esti-mate of
T SC for d between 700 to 900 kpc (corre-sponding to d proj = 700 and 0 < α <
35 degrees), then0 . < T SC < .
33 Gyr (see green bar of Figure 4).This brings their result in line with the results of thismethod, as expected by the agreement presented in § MUSKET BALL CLUSTER DYNAMICS
I also apply the method to the Musket Ball Cluster,with the objective of updating an existing analysis andcomparing this system with the Bullet Cluster. A pre-liminary analysis of the system dynamics using a sim-ilar method (Dawson et al. 2012) suggested that theMusket Ball Cluster merger is ∼ T SC – v ( t col ) PDF with that of the single point Springel & Farrar (2007)estimate. As noted in § Musket Ball Observed System Properties
I show the observed Musket Ball Cluster parameterPDF’s in Figures 5–7, each the result of analyses pre-sented by Dawson et al. (2012). I refer to their “south”subcluster as halo 1 and “north” subcluster as halo 2.The mass PDF’s, Figure 5, are the result of an MCMCanalysis where NFW halos were simultaneously fit tothe weak lensing signal. The relative velocity distribu-tions, Figure 6, are the result of a bootstrap error anal-ysis (Beers et al. 1990) of the 38 and 35 spectroscopicmembers of the north and south subclusters, respectively.The projected subcluster separation distribution, Figure7, is the result of a bootstrap error analysis of the re-cursively estimated subclusters’ galaxy number densitycentroids (see e.g. Randall et al. 2008, for a descriptionof this method). For each Monte Carlo realization in-dividual values are drawn randomly from each of thesedistributions.
Musket Ball System Dynamics Results
This more complete analysis confirms that the MusketBall Cluster merger is both significantly slower and fur-ther progressed compared to the Bullet Cluster, see Fig-ure 8. To estimate a lower limit on how much further pro-gressed I perform an additional Monte Carlo analysis for
T SC Musket − T SC Bullet assuming the marginalized
T SC distributions (see Appendices B and C). This is a lowerlimit since the Musket Ball observations, unlike the Bul-let Clusters observations, cannot rule out the case thatits subclusters have reached the apoapsis and are on a re-turn trajectory (61% of the realizations have T SC lessthan the age of the Universe at z = 0 . . +1 . − . Gyr (3 . +3 . − . times) furtherynamics of Merging Clusters 9 Fig. 5.—
Weak lensing mass PDF’s of the Musket Ball subclus-ters (Dawson et al. 2012).
Fig. 6.—
Relative radial subcluster velocity PDF’s inferred fromspectroscopic redshifts the Musket Ball Cluster galaxies (Dawsonet al. 2012).
Fig. 7.—
Projected separation PDF of the Musket Ball subclustergalaxy density centroids (Dawson et al. 2012). progressed than the Bullet Cluster, see Figure 9. This isin line with the more approximate 3–5 times estimate ofDawson et al. (2012). The Musket Ball’s relatively large
T SC means that it has potential for providing tighterconstraints on σ DM , since the expected offset betweenthe galaxies and dark matter will initially increase withincreasing T SC . However as noted in §
1, given enoughtime the expected offset will decrease due the gravita-tional attraction between the galaxies and dark matter.Also important in determining which cluster can provide
Fig. 8.—
The posterior of the Musket Ball Cluster’s
T SC and v ( t col ) parameters is shown in grayscale with dark and light bluecontours representing 68% and 95% confidence, respectively. Forcomparison the gray dashed contours are the Bullet Cluster’s 68%and 95% confidence intervals copied from Figure 4. The MusketBall Cluster occupies a much different region of merger phase thanthe Bullet Cluster, having both a slower relative collision velocityand being observed in a much later stage of merger. Fig. 9.—
The histogram presents the
T SC Musket − T SC Bullet distribution from random draws of the respective marginalized
T SC distributions; showing that the Musket Ball Cluster mergeris at least 0 . +1 . − . Gyr (3 . +3 . − . times) further progressed than theBullet Cluster merger. The black dashed line is the biweight-statistic location (Beers et al. 1982), the dark and light blue regionsdenote the bias-corrected 68% and 95% lower and upper confidencelimits, respectively. the tightest σ DM constraints is the fact the expected off-set increases as a function of the cluster surface massdensity and collision velocity, both of which are larger inthe the case of the Bullet Cluster (compare Tables 2 & 3).Without running SIDM simulations it is difficult to knowat what T SC the offset reaches it maximum, or whichmerger parameters are most important for maximizingthe offset. The complete Musket Ball Cluster parame-ter estimates are summarized in Table 3 and plotted inAppendix C.Note that just as a temporal prior was justified forthe Bullet Cluster based on the observed shock frontand increased temperature/mass estimate, I could ap-ply a similar yet opposite prior to the Musket Ball sincethe temperature/mass estimate is consistent with theweak lensing inferred mass (additionally no shock frontis observed). According to Randall et al. (2002) if the0 Dawson TABLE 3Musket Ball Cluster parameter estimates
Parameter Units Location a
68% LCL–UCL b
95% LCL–UCL b M M (cid:12) M (cid:12) z z d proj Mpc 1.0 0.9 – 1.1 0.7 – 1.3 α degree 48 28 – 67 13 – 78 d Mpc 1.6 1.2 – 2.9 0.9 – 5.5 d max Mpc 2.1 1.5 – 3.8 1.1 – 7.3 v ( t obs ) km s −
670 390 – 1100 140 – 1500 v ( t col ) km s − T SC Gyr 1.1 0.7 – 2.4 0.5 – 5.8
T SC Gyr 3.5 2.0 – 7.2 1.4 – 12.0 T Gyr 4.8 2.9 – 10.4 2.2 – 22.7 a Biweight-statistic location (see e.g. Beers et al. 1990). b Bias-corrected lower and upper confidence limits, LCL and UCL respec-tively (see e.g. Beers et al. 1990). c
61% of the realizations with a valid
T SC (i.e. less than the age of the Uni-verse at the cluster redshift) have a valid T SC , meaning that it is possiblethat the Musket Ball Cluster is being observed in the “incoming” state. cluster mass and inferred X-ray temperature or lumi-nosity cluster mass are approximately the same then T SC (cid:38) t sc , which in the case of the Musket Ball means T SC (cid:38) .
75 Gyr. While this is consistent with my
T SC estimate for the Musket Ball, it is not entirelyappropriate to apply this prior since the X-ray observa-tions are relatively shallow and cannot confidently ruleout a temperature and luminosity boost (Dawson et al.2012). However it is conceivable that this line of reason-ing would be applicable with deeper X-ray observations,either for the Musket Ball or similar dissociative mergers. SUMMARY AND DISCUSSION
I have introduced a new method for determining thedynamic properties and associated uncertainty of dis-sociative cluster mergers given only the most generalmerger observables: mass of each subcluster, redshift ofeach subcluster, and projected separation the subclus-ters. I find that this method addresses the primary weak-nesses of existing methods, namely enabling accurate pa-rameter estimation and propagation of uncertainty nearthe collision state with a convergent solution achievedin ∼ α , the angle of the merger with respectto the plane of the sky. Analyses that fail to account forthe uncertainty in α (all existing N-body simulations ofthe Bullet Cluster) will significantly underestimate theuncertainty in their results. This highlights the need tocarefully select and model many possible realizations ofthe merger when trying to infer results from N-body sim-ulations of a real merger (I discuss this further in § . +3 . − . times further progressed than the BulletCluster, could potentially provide tighter constraints on σ DM since the offset between galaxies and dark mat-ter should initially increase with time post-merger for σ DM >
0. And the larger the expected offset, the bet-ter the dark matter constraint when applying a methodsimilar to Randall et al. (2008).
Suggested Uses of Method
While a general method for determining the dynamicsproperties of merging clusters has numerous applications,several are worth noting. As noted N-body simulations ofspecific merging clusters are computationally expensive;in particular one SIDM simulation of a single dissociativemerger requires ∼ σ DM constraining power, enabling anefficient use of limited computational resources.Additionally it is inappropriate to simply simulate onerealization of a dissociative merger due to the broadrange of merger phase space allowed by uncertainty in ob-served parameters, as discussed in detail in § v ( t col ), and T SC ); then weight the results of each simulated real-ization by the integral of the corresponding local phaseynamics of Merging Clusters 11space PDF, as determined by this method. For example,one could estimate the uncertainty distribution of the σ DM constraint inferred from SIDM simulations of theBullet Cluster by weighting the constraint from each re-alization, where a realization with v ( t col )=2800 km s − and T SC =0.4 Gyr would receive greater weight thanone with v ( t col )=4000 km s − and T SC =0.2 Gyr, seeFigure 4. Thus the results of this method will not onlyinform efficient selection of realizations to model but willreduce the number of simulations required to properlysample the posterior PDF’s. Nevertheless SIDM simula-tions of mock clusters need to be performed to determinehow much acceptable values of σ DM affect the inferredmerger dynamics properties.General merger dynamic properties are also importantfor understanding how cluster mergers relate to otherphysical phenomena, such as galaxy evolution and radiorelics. It is well established that galaxy clusters play animportant role in the evolution of their member galaxies,but it is still unclear whether cluster mergers trigger starformation (e.g. Miller & Owen 2003; Owen et al. 2005;Ferrari et al. 2005; Hwang & Lee 2009), quench it (Pog-gianti et al. 2004), or have no immediate effect (Chunget al. 2010). Studying mergers at different T SC may re-solve these seemingly conflicting results by discriminat-ing between slow-working processes (e.g. galaxy harass-ment or strangulation) and fast-acting process (e.g. rampressure stripping). Similarly, studying global mergerdynamic properties may resolve the mystery of why manymergers have associated radio relics (e.g. Barrena et al.2009; van Weeren et al. 2011) yet others don’t (e.g. Rus-sell et al. 2011).
Extensions to the Method
While this method has advantages over existing meth-ods there is room for considerable improvement. For ex-ample the method could be improved through the elim-ination of some of the simplifying assumptions of themodel (see § spin parameter , determined from linear theory of thegrowth of structure (Peebles 1993) and simulations (Bul-lock et al. 2001). However, this prior should be adjustedto account for the samount of gas dissociated during theobserved merger, since this amount will decrease as theimpact parameter increases. Without a systematic studyof various mergers in hydrodynamic simulations it is un-clear exactly what adjustment an observed large dissoci-ation of gas should infer.Another significant extension to the model could be theinclusion of SIDM physics. As mentioned in the previoussection, one of the promising uses of this method is tosuggest which mergers might provide the best σ DM con-straining power. However one could take this a step fur-ther by including an analytic treatment of SIDM physics(e.g. Markevitch et al. 2004), thereby enabling analytic estimates of σ DM relevant effects for a given merger.Then this method could be used in conjunction with ob-served dissociative mergers to place direct constraints on σ DM . Due to the increased complexity of the physics in-volved it would be necessary to verify this extension withSIDM N-body simulations. Note:
W. Dawson has made Python code implement-ing the discussed Monte Carlo method openly availableat git://github.com/MCTwo/MCMAC.git. He has alsomade all supporting work to this paper openly avail-able at git://github.com/wadawson/merging-cluster-dynamics-paper.git.I thank my adviser David Wittman who has alwaysencouraged my research-freewill, while at the same timeproviding invaluable input and correcting guidance. Ourmany fruitful discussions have touched every aspect ofthis work. I also thank the Deep Lens Survey — in par-ticular Perry Gee — for access to the 2007 Keck LRISspectra, as well as Perry Gee and Brian Lemaux for as-sistance in reduction of the 2011 Keck DEIMOS spectra.I am grateful to Jack P. Hughes for discussions on con-straining the Bullet Cluster’s TSC by the observed X-rayshock feature and boosted temperature/luminosity, andReinout J. van Weeren for discussions on the possibilitiesof constraining the dynamics and geometry of mergersusing radio relics. Finally I would like to thank MaruˇsaBradaˇc, Ami Choi, James Jee, Phil Marshall, Annika Pe-ter, Michael Schneider, Reinout van Weeren, and DavidWittman for their substantial reviews of earlier drafts.Support for this work was provided by NASA throughChandra Award Number GO1-12171X issued by CXOCenter, which is operated by the SAO for and on behalfof NASA under contract NAS8-03060. Support for pro-gram number GO-12377 was provided by NASA througha grant from STScI, which is operated by the Associationof Universities for Research in Astronomy, Inc., underNASA contract NAS5-26555. Support for this work wasalso provide by Graduate Research Fellowships throughthe University of California, Davis. I would also like toacknowledge the sultry universe who hides her secretswell.
Facilities:
CXO (ACIS-I), HST (ACS), Keck:I (LRIS),Keck:2 (DEIMOS), Mayall (MOSAIC 1 & 1.1), Subaru(Suprime-Cam), SZA.2 Dawson
APPENDIX
POTENTIAL ENERGY OF TWO TRUNCATED NFW HALOS
Generically the potential energy of a two-halo system with center to center separation r is V ( r ) = (cid:90) Φ ( r (cid:48) ) dm , (A1)where Φ ( r (cid:48) ) is the gravitational potential of halo 1 as a function of radial distance r (cid:48) from the center of the halo 1to the mass element of halo 2, dm . I derive Φ ( r ) for the case of a truncated NFW halo in § A.1. The integral ofequation A1 can be approximated as a summation over N × N mass elements, m ij , each with area dr × dθ , where i and j range from 0 → N − V ( r ) ≈ N − (cid:88) i =0 N − (cid:88) j =0 Φ ( r (cid:48) ij + (cid:15) ) m ij , where r (cid:48) ij is the distance from the center of halo 1 to the 2 nd halo’s mass element m ij , as derived in § A.2, and (cid:15) isthe softening length which reduces the effects of artificial singularities.
Truncated NFW Gravitational Potential
For an axially symmetric mass distribution the potential can be expressed as a series of Legendre PolynomialsΦ n ( r ) = − πG ( n + 1 / r n +1 (cid:90) r r (cid:48) n +2 ρ n ( r (cid:48) ) dr (cid:48) − πGr n n + 1 / (cid:90) ∞ r r (cid:48) − n ρ n ( r (cid:48) ) dr (cid:48) (A2)where ρ n ( r ) = ( n + 1 / (cid:90) π ρ ( r, θ ) P n (cos θ ) sin θ dθ. (A3)Assuming a spherical NFW halo ρ NFW ( r ) = ρ s r/r s (1 + r/r s ) only the zero th order term of Equation A3 remains ρ NFW ( r ) = ρ ( r )and Equation A2 reduces toΦ NFW ( r ) = − πGr (cid:90) r r (cid:48) ρ NFW ( r (cid:48) ) dr (cid:48) − πG (cid:90) ∞ r r (cid:48) ρ NFW ( r (cid:48) ) dr (cid:48) Φ NFW ( r ) = − πGρ s r (cid:20)(cid:90) r r (cid:48) r (cid:48) /r s (1 + r (cid:48) /r s ) dr (cid:48) + r (cid:90) ∞ r r (cid:48) r (cid:48) /r s (1 + r (cid:48) /r s ) dr (cid:48) (cid:21) . Since I truncate the NFW halo at r the ∞ in the second integral becomes r andΦ NFW T ( r ) = (cid:40) − πGr ρ s r s (cid:104) ln(1 + r/r s ) − rr s + r (cid:105) , if r ≤ r ; − GM r , if r > r . (A4) Mass Elements of a Truncated NFW Halo
Given the differential mass elements for a spherically symmetric halo dm = 2 πρ ( r, θ ) r sin( θ ) dθ dr, and discretizing the mass into elements with lengths δr = r /N and δθ = π/N the halo 2 mass elements are givenby m ij = 2 π (cid:90) ( i +1) δri δr (cid:90) ( j +1) δθj δθ ρ ( r (cid:48) ) r (cid:48) sin( θ (cid:48) ) dθ (cid:48) dr (cid:48) . For an NFW halo this becomes m ij = 2 πρ s r s [cos( j δθ ) − cos (( j + 1) δθ )] (cid:34)(cid:18) i + 1) δrr s (cid:19) − − (cid:18) i δrr s (cid:19) − + ln (cid:20) ( i + 1) δr + r s i δr + r s (cid:21)(cid:35) . ynamics of Merging Clusters 13 BULLET CLUSTER RESULT PLOTS
This section contains the parameter results array plots for the Bullet Cluster case including the added temporal priorof § Input , Geometry , and
Velocity & Time )resulting in a six subplot results array, see Figure 10. The
Input parameters consist of: M , M , z , z , and d proj ,where halo 1 refers to the “main” subcluster and halo 2 refers to the “bullet” subcluster. The Geometry parametersconsist of the randomly drawn α , and calculated d , and d max . The calculated Velocity & Time parameters consistof:
T SC , T SC , and T . Fig. 10.—
For ease of display the results array is divided into six subplots, Figures 11–15. The Input parameters consist of: M ,M , z , z , and d proj . The calculated Geometry parameters consist of: α , d , and d max . The calculated Velocity & Time parametersconsist of: v ( t obs ), v ( t col ), T SC , T SC , and T . Fig. 11.—
Bullet Cluster marginalized
Input vs. Input parameters result plots, for the case including the added temporal prior of § ynamics of Merging Clusters 15 Fig. 12.—
Bullet Cluster marginalized
Input vs. Geometry parameters result plots, for the case including the added temporal prior of §6 Dawson
Input vs. Geometry parameters result plots, for the case including the added temporal prior of §6 Dawson Fig. 13.—
Bullet Cluster marginalized
Input vs. Velocity & Time parameters result plots, for the case including the added temporal priorof § ynamics of Merging Clusters 17 Fig. 14.—
Bullet Cluster marginalized
Geometry vs. Geometry parameters result plots, for the case including the added temporal priorof § Fig. 15.—
Bullet Cluster marginalized
Geometry vs. Velocity & Time parameters result plots, for the case including the added temporalprior of § ynamics of Merging Clusters 19 Fig. 16.—
Bullet Cluster marginalized
Velocity & Time vs. Velocity & Time parameters result plots, for the case including the addedtemporal prior of § MUSKET BALL CLUSTER RESULT PLOTS
This section contains the parameter results array plots for the Musket Ball Cluster. Similar to § B the parameters aregrouped in three categories (
Input , Geometry , and
Velocity & Time ) resulting in a six subplot results array, see Figure10. The
Input parameters consist of: M , M , z , z , and d proj , where halo 1 refers to the “south” subclusterand halo 2 refers to the “north” subcluster. The calculated Geometry parameters consist of: α , d , and d max . Thecalculated Velocity & Time parameters consist of: v ( t obs ), v ( t col ), T SC , T SC , and T . Fig. 17.—
Musket Ball Cluster marginalized
Input vs. Input parameters result plots. Dark and light blue colors correspond to 68% and95% confidence intervals, respectively. The black dashed line is the biweight-statistic location (Beers et al. 1982). ynamics of Merging Clusters 21
Fig. 18.—
Musket Ball Cluster marginalized
Input vs. Geometry parameters result plots. Dark and light blue colors correspond to 68%and 95% confidence intervals, respectively.
Fig. 19.—
Musket Ball Cluster marginalized
Input vs. Velocity & Time parameters result plots. Dark and light blue colors correspondto 68% and 95% confidence intervals, respectively. ynamics of Merging Clusters 23
Fig. 20.—
Musket Ball Cluster marginalized
Geometry vs. Geometry parameters result plots. Dark and light blue colors correspond to68% and 95% confidence intervals, respectively. The black dashed line is the biweight-statistic location (Beers et al. 1982).
Fig. 21.—
Musket Ball Cluster marginalized
Geometry vs. Velocity & Time parameters result plots. Dark and light blue colors correspondto 68% and 95% confidence intervals, respectively. ynamics of Merging Clusters 25
Fig. 22.—
Musket Ball Cluster marginalized
Velocity & Time vs. Velocity & Time parameters result plots. Dark and light blue colorscorrespond to 68% and 95% confidence intervals, respectively. The black dashed line is the biweight-statistic location (Beers et al. 1982).6 Dawson