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Memorie della Società Astronomica Italiana

Supplementi - Vol. 1

Computational Astrophysics in Italy: Methods and Tools

Prima Riunione Nazionale Bologna, 4-5 luglio 2002 Editor: Roberto Capuzzo Dolcetta

Foreword p. 10

People invited to the round table on the state of computational astrophysicsin Italy p. 11

List of participants

PDF file PS file p. 12

Introduction

R. Capuzzo Dolcetta Few comments about computational astrophysics in Italy p.15

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Algorithms and numerical techniques

P. Londrillo, C. Nipoti and L. Ciotti A Parallel Implementation of a New Fast Algorithm for N-body Simulations p. 18

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F. Reale High Performance Computing at INAF/OAP 12 p. 27

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D. Molteni and C. Bilello Riemann Solver in SPH p. 36

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S. Orlando, G. Peres, F. Reale, R. Rosner, T. Plewa and A. Siegel Development and Application of Numerical Modules for FLASH in Palermo:Two Astrophysical Examples p. 45

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P.F. Spinnato, S.F. Portegies Zwart, M. Fellhauer, G.D. van Albada, andP.M.A. Sloot Tools and Techniques for N-body Simulations p. 54

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R. Valdarnini Performance Characteristics of a Parallel Tree-code p. 66

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L. Tornatore and S. Borgani Few Ideas About the Energetics of ICM 61 p. 76

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Visualization and computing tools

U. Becciani, C. Gheller, V. Antonuccio, D. Ferro and M. Melotti AstroMD, a tool for Stereographic Visualization and data analysis forastrophysical data p. 80

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A. Costa, U. Becciani, V. Antonuccio, R. Capuzzo Dolcetta, P. Miocchi, P.Di Matteo, V. Rosato Astrocomp: web technologies for high performance computing on a grid ofsupercomputers p. 89

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P. Miocchi, V. Antonuccio, U. Becciani, R. Capuzzo Dolcetta, A. Costa, P.Di Matteo, V. Rosato

AstroComp: using the portal to perform astrophysical -body simulations p. 96

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A. Cacciani, P. Rapex, B. Subrizi, V. Di Martino Data reduction and analysis of Solar Dopplergram p. 103

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V. Antonuccio-Delogu, U. Becciani, D. Ferro and A. Romeo A software interface between Parallel Tree- and AMR Hydrocodes p. 109

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V. Rosato Dedicated Hardware Devices for scientific applications p. 116

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Astrophysical and cosmological supercomputing applications

M. Bottaccio, M. Montuori, L. Pietronero, P. Miocchi, and R. CapuzzoDolcetta N-body simulations for structure formation from random initial conditions p. 120

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G. Carraro, C. Dalla Vecchia, and B. Jungwiert Toward self-consistent models of Galaxy Evolution p. 130

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A. Diaferio N-body simulations, the halo model, and the distribution of galaxy clusters inSunyaev-Zeldovich effect surveys p. 140

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B. Lanzoni, A. Cappi and L. Ciotti High-resolution re-simulations of massive DM halos and the FundamentalPlane of galaxy clusters p. 145

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C. Zanni, S. Massaglia, G. Bodo, P. Rossi, A. Capetti, and A. Ferrari Propagation of extragalactic jets in stratified media p. 155

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L. Del Zanna, N. Bucciantini, P. Londrillo

A third order shock-capturing code for relativistic 3-D MHD p. 165

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E. Rasia, G. Tormen and L. Moscardini High-Resolution Simulations: Modeling Intracluster Medium and DarkMatter in Galaxy Cluster p. 176

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V. Muccione and L. Ciotti Elliptical galaxies interacting with the cluster tidal field: origin of theintracluster stellar population p. 186

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O. Zanotti and L. Rezzolla Dynamics of Oscillating Rlativistic Tori p. 192

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G. Murante, S. Bonometto, A. Curir, A.V. Maccio’ and P. Mazzei Dynamics of collisionless self-gravitating structures p. 199

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G. Tormen, L.Moscardini and E.Rasia Dynamics of the ICM in Galaxy Clusters p. 209

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L. Baiotti, I.J.Hawke, P.J.Montero and L.Rezzolla A new three-dimensional general relativistic hydrodynamics code p.210

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Two selected topics of the round table on computational astrophysics inItaly: state and perspectives

R. Rosner Issues in Advanced Computing: A US Perspective p. 220

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F. Pasian A task of advanced computing: how to process large quantities of data? p. 221

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Conclusions

G. Peres Final remarks p. 223

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[Mem. S.A.It. Homepage]

Mem. S.A.It. Suppl. Vol. 1, 18c© SAIt 2003

Memorie della

Supplementi

A parallel implementation of a new fastalgorithm for N-body simulations

P. Londrillo1, C. Nipoti2,

and L. Ciotti2

1 INAF – Osservatorio Astronomico di Bologna, via Ranzani 1, 40127 Bologna,Italy e-mail: [email protected]

2 Dipartimento di Astronomia, Universita di Bologna, via Ranzani 1, 40127Bologna, Italy e-mail: [email protected], [email protected]

Abstract.A new, momentum preserving fast Poisson solver for N-body systems sharing ef-fective O(N) computational complexity, has been recently developed by Dehnen(2000, 2002). We have implemented the proposed algorithms in a Fortran-90 code,and parallelized it by a domain decomposition using the MPI routines. The codehas been applied to intensive numerical investigations of galaxy mergers, in par-ticular focusing on the possible origin of some of the observed scaling relationsof elliptical galaxies. We found that the Fundamental Plane is preserved by anequal mass merging hierarchy, while it is not in a scenario where galaxies growby accretion of smaller stellar systems. In addition, both the Faber-Jackson andKormendy relations are not reproduced by our simulations.

Key words. methods: N-body simulations – methods: numerical – galaxies: ellip-tical and lenticular, cD – galaxies: formation – galaxies: fundamental parameters

1. Introduction

In the last few years, fast algorithms forcomputing N-body interactions relying onthe general multipole expansion techniqueshave been developed. These schemes, usu-ally referred to as Fast Multipole Methods(FMMs, see, e.g., Greengard & Rokhlin1987, 1997), are designed to have O(N)computational complexity. However, it has

Send offprint requests to: P. LondrilloCorrespondence to: Osservatorio Astronomicodi Bologna, via Ranzani 1, 40127 Bologna

been noted that only O(N logN) scaling isusually achieved in numerical implementa-tion (see, e.g., Capuzzo-Dolcetta & Miocchi1998, hereafter CM98). The O(N logN) op-eration count also characterizes the N-bodycodes based on tree algorithms, originallyproposed by Barnes & Hut (1986, hereafterBH86). In addition, it has been shown bydirect numerical tests that, for given accu-racy, tree-codes are faster than the FMMsby a factor of ∼ 4 (CM98). For this reason,most of the N-body codes currently usedfor astrophysical applications are based on

P. Londrillo et al.: N-body simulations of galaxy mergers 19

the classical BH86 tree scheme, where theparticle data are organized into a nestedoct-tree cell structure. Interactions betweendistant particles are approximated by cell–particle interactions, and the distributionof the particles on each cell is representedby a multipole expansion, usually trun-cated to the quadrupole term.

Recently, a new scheme has been intro-duced (Dehnen 2000, 2002, hereafter D02),which can be seen as an original combina-tion of the tree-based BH86 scheme andof the FMMs multipole expansion, trun-cated at a fixed low order level (the pro-posed p = 3 order results the optimalone). This scheme, implemented by theAuthor in a C++ code [named falcON,Force Algorithm with Complexity O(N)],results in a significant improvement overthe existing BH86 tree-codes, in terms ofuniform accuracy, linear momentum con-servation, and CPU time performances.Moreover, the numerical tests presented inthe D02 paper show that for an accuracylevel ε ' 10−3, for the first time an effec-tive O(N) scaling in operation count, aspredicted by the FMM theory, is in factobtained. For these reasons two of us (P.L.and C.N.) have implemented this scheme ina new Fortran-90 code (FVFPS, a FortranVersion of a Fast Poisson Solver), andhave then parallelized it by using the MPIprocedures. A completion of the code tohandle gas-dynamics using Euler equationsand the AMR (Adaptive Mesh Refinement)techniques is under development. Here webriefly describe the main characteristicsof Dehnen’s algorithm, and we presentour FVFPS implementation and its paral-lelization. The achieved accuracy level, theperformances in terms of operation countspeed-up, the effective O(N) scaling, aswell as the parallel scaling with the proces-sors number, are then documented in nu-merical tests. Finally, we describe some in-teresting astrophysical applications of thecode, in the context of the study of galaxymerging. Our version of the code is avail-able upon request.

2. Dehnen’s scheme

In the FMM approach, a system of particlesis first organized in a oct-tree structure ofnested cells, providing a hierarchical group-ing of particles. For each pair of distantcells (A, B), the interactions between par-ticles are then approximated by a Taylorexpansion of the two-point Green functionaround the geometrical center of each cellby using spherical harmonics. In these cell-cell interactions the two-body symmetry ofthe exact Green function is preserved bythe multipole approximation. In principle,a FMM-based scheme can achieve a pre-scribed accuracy ε by increasing the orderp of the Taylor polynomial, and/or by low-ering the expansion parameter η ≡ r/R(where R is the distance between the twocell centers and r gives the typical size ofa cell enclosing a particle group). Even ifa formal, asymptotic scaling O(N) is ex-pected, the proposed algorithms in the orig-inal formulation (Greengard & Rokhlin,1997) appear in fact to be less perform-ing and of difficult application. Therefore, afirst variant has been introduced in CM98,where the expansion order is kept fixed top = 2, and only the η parameter controlsthe accuracy of the Taylor expansion. Thisapproach takes advantage of the adaptivityof tree algorithms to select the pairs of cells(even belonging to different levels of refine-ments), which satisfy the opening criteriumrcrit/R ≤ θ, where θ is a preassigned con-trol parameter. To improve the efficiency ofthe scheme, a new definition of the criticalsize rcrit(A,B), has been also introduced.However, even in this improved formula-tion, the FMM scheme results to be notcompetitive with the classical BH86 tree-code.

Dehnen’s scheme basically developsalong similar lines, but with some new in-gredients, which in fact succeeded to im-prove the overall efficiency in a substantialway:

– the Taylor polynomial expansion is per-formed in Cartesian components, andthe expansion centers are the center of

20 P. Londrillo et al.: N-body simulations of galaxy mergers

mass of each cell, as in the BH86 tree-code;

– a fixed expansion order p = 3 is chosenas optimal;

– a new definition of the critical radiusrcrit is introduced for each cell, to opti-mize the adaptive selection of all pairsof well-separated cells;

– a new, mass dependent control parame-ter θ = θ(M) has been introduced, thatassures faster performances and a moreuniform error distribution. The θ(M)function is given in implicit form by

θ5

(1 − θ)2=

θ5min

(1 − θmin)2

(

M

Mtot

)

−1

3

,

(1)where θmin (the minimum value of θ, as-sociated with the total mass) enters asa new, preassigned control parameter,and where M is the total mass enclosedin a cell.

The resulting force solver is then organizedas follows:

(a) As in the BH86 code, an oct-treestructure of nested cells containing parti-cles is first constructed. A generic cell A,when acting as a gravity “source” is charac-terized by the position of its center of massXA, by its critical size rA, mass MA, andby the quadrupole tensor QA. For the samecell A, when acting as a gravity “sink” a setof (twenty) Taylor coefficients CA are usedto store the potential and acceleration val-ues at XA.

(b) To compute these coefficients, a newprocedure for the tree exploration has beendesigned: starting from the root cell, allpairs of cells (A, B) satisfying the accep-tance condition (rA+rB)/R ≤ θ, are lookedfor in a sequential way. When the condi-tion is satisfied, each cell of the selectedpair accumulates the resulting interactioncontributions in CA(B) or CB(A). In caseof nearby cells (not satisfying the accep-tance condition), the bigger cell of the pairis splitted, and the new set of pairs are thenconsidered. However, in order to avoid vis-iting the tree too deeply, some empirical

conditions are adopted to truncate the se-quence, by computing directly (exact) two-body forces for particles in the smallestcells. The Author, by using a stack struc-ture to order the visited cell-pairs, has in-troduced a powerful and efficient new al-gorithm able to perform this interactionphase in the force computations.

(c) Finally, once all the cell-cell interac-tions has been computed and stored in theCA array, a final step is needed to evaluatethe gravitational potential and the acceler-ations of the particles in the (generic) cellA. This evaluation phase can be performedby a simple and fast O(N) tree-traversal.

In our F-90 implementation of thescheme above, we followed rather closelythe algorithms as described in D02: at thisstage of development and testing, we haveneither attempted to optimize our imple-mentation, nor to look for particular tricksto save computational time.

To parallelize the force solver, we haveadopted a straightforward strategy, by firstdecomposing the physical domain into a setof non-overlapping subdomains, each con-taining a similar number of particles (a fewper cent of tolerance is allowed). The setof particles on each subdomain, Sk, is thenassigned to a specific processing unit, Pk.Each processor builds its own tree [step(a)], and computes the interaction phasefor all the particles in its subdomain [step(b)]. Steps (a) and (b) are executed in par-allel. By ordering the processors in a one-dimensional periodic chain, it is then possi-ble to exchange (particles and tree-nodes)data between them, by a “send receive′′

MPI routine, in such a way that each pro-cessor can compute the interaction phasewith the other subdomains. This phase isthen performed by the following computa-tional steps:

(i) The Pk processor sends its source-data to Pk+s and receives from the Pk−s

a copy of its source-data, where s =1, 2, ...NPE/2, and NPE is the total numberof processors.

(ii) At any given level s, each Pk com-putes, again in parallel, the interaction


phase with particles and cells of Pk−s.Mutuality of the interactions allows to eval-uate, at the same time, the results of theseinteractions for cells and particles of pro-cessor Pk−s to be stored as sink-data.

(iii) These accumulated sink-data (ac-celeration for particles and Taylor’s coeffi-cients for cells) are sent back to the Pk−s

processor, while the corresponding sink-data of the Pk processor are received fromprocessor Pk+s and added to the currentPk values. In this way, it is evident that byrepeating step (ii) NPE/2 times and step(iii) NPE/2 − 1 times, the interactions ofPk particles with all other particles in thedomain are recovered.

(iv) The final evaluation phase [step(c)], being fully local, is performed in par-allel.

Besides the force solver, the N-bodycode has been completed by a leap-frogtime integrator scheme, with a (uniform)time step size ∆t determined adaptivelyusing a local stability condition ∆t <1/√

4πGρmax, where G is the gravitationalconstant, and ρmax the maximum of par-ticle density. For typical simulations in-volving galaxy mergers, this adaptive (intime) step size keeps comparable to thevalue Tdyn/100, where Tdyn represents amacroscopic dynamical time scale, and as-sures energy conservation with relative er-rors smaller than 0.1%.

3. Performances of the FVFPS code

We ran some test simulations in order toestimate the performances of our FVFPScode, and to compare it both with fal-cON and with the BH86 tree-code. Wefocus first on the analysis of the timeefficiency in force calculation, consider-ing (one–processor) scalar simulations ofthe N-body system representing a galaxygroup, where each galaxy is initialized bya Hernquist (1990, hereafter H90) galaxymodel. Values of the total number of par-ticles N up to 2 × 106 are considered. Theopening parameter given in eq. (1) is usedby adopting θmin = 0.5, while for the BH86

code we used θ = 0.8: this choice results incomparable accuracies in the force evalua-tion. We ran our simulations on a PentiumIII/1.133 GHz PC. For falcON and for theBH86 code we refer to the data publishedby D02, who used a Pentium III/933 MHzPC.

In Fig. 1 (left panel) we plot the timeefficiency of the codes as a function of thenumber of particles; it is apparent from thediagram that FVFPS (solid line), as wellas falcON (dashed line), reaches an effec-tive O(N) scaling, for N>∼104. Our F-90implementation results to be slower thanthe C++ version by a factor of ∼ 1.5: wethink that this discrepancy may be due todifferences in the details of the implemen-tation, and also reflects a better intrinsicefficiency of the C++ with respect to the F-90 for this kind of algorithms (especially forroutines where intensive memory accessingis required). In the same diagram the datarelative to the BH86 tree-code are also plot-ted (dotted line), and the O(N logN) scal-ing is apparent. As already documented inD02, it is remarkable how, for fixed N andprescribed accuracy, the BH86 tree-code issignificantly slower than the new scheme:for example for N ' 105 the FVFPS codeis faster by a factor of ∼ 8. Therefore, evenif the present F-90 implementation doesno yet appear to be optimal, it gives fur-ther support to the efficiency of the D02scheme, independently of the adopted pro-gramming language and of the implemen-tation details.

The accuracy in force evaluation is doc-umented on the right panel of Fig. 1, wherewe plot, as a function of N , the meanrelative error on the acceleration modulus(filled circles)

⟨

∆a

a

⟩

≡ 1

N

N∑

i=1

|aiapprox − ai

direct|aidirect

, (2)

and the corresponding 99th percentile(∆a/a)99% (empty squares), for FVFPS(solid lines) and, for comparison, in fal-cON (dashed lines). As usual, the refer-ence acceleration values ai

direct are evalu-


Fig. 1. Left panel: CPU time per particle as a function of total number of particles forBH86 tree-code (dotted line), falcON (dashed line; data from D02), and FVFPS (solidline). For the BH86 code θ = 0.8, while in the last two cases θ = θ(M), with θmin = 0.5.Right panel: Mean error on the force calculation (filled circles) and the 99th percentile(empty squares), as a function of the number of particles in falcON (dashed line; datafrom D02), and in FVFPS (solid line).

Fig. 2. Left panel: number of particle processed per second by the parallel version of theFVFPS code, as a function of the number of processors. The data refer to simulations ofa group of galaxies with total number of particles N = 1048576. Right panel: CPU timeper particle as a function of the number of processors for the same simulations in leftpanel. The different contributions are plotted (see text for an explanation).


ated by direct summation of the exact two-body Green function, while the approxi-mated values are given with θ = θ(M),and θmin = 0.5. In this case, we have con-sidered only simulations having a singleH90 galaxy model. From the diagram it isapparent that, for both implementations,〈∆a/a〉 and (∆a/a)99% do not exceed afew 10−3 and a few per cent, respectively,assuring accuracies usually considered suf-ficient for astrophysical applications (see,e.g., CM98).

As outlined in the previous Section, weparallelized our code by using a domain de-composition with a load balancing essen-tially based only on particles number oneach subdomain. This load balancing cri-terium seems in fact to work well, takingadvantage of the intrinsic O(N) scaling inthe operation count of the basic scheme.In order to check the performance of theparallelized algorithm, we ran some simu-lations on the IBM Linux cluster (PentiumIII/1.133 GHz PCs) at the CINECA cen-ter, in a range NPE ≤ 16 in the num-ber of processors. In this case the initialconditions are represented by a group of8 one–component H90 galaxy models witha total number of particles N = 1048576.As can be observed in Fig. 2 (left panel),this parallelization is characterized by arather well behaved linear scaling of theCPU time per particle versus NPE. Thisscaling can be better evaluated by consid-ering the right panel of Fig. 2, where dif-ferent contributions to the total CPU con-sumption are plotted. As expected, up toNPE = 16 the main contributions comeby the tree building (“tree”) and by thethe force computation in each sub-domain(“self”), corresponding to the step (i) and(ii), respectively, of the parallel schemeoutlined in the previous Section. The con-tributions coming from the force evaluationamong particles belonging to different sub-domains (“cross”) and the cumulative timefor all communication steps among proces-sors (“trans”), which increase with NPE,keep smaller in the chosen range of parame-ters (N,NPE). Therefore, the O(N) scaling

of the base scheme, allows to estimate, forgiven N , the optimal configuration of theNPE that can be used.

4. Simulations of galaxy merging

As an astrophysical application of the de-scribed FVFPS code, we consider here thestudy of the galaxy merging process. Wechecked the reliability of the numerical sim-ulations performed with FVFPS for thiskind of application (now time evolution isalso considered), by running a few merg-ing events, between spherically symmet-ric stellar systems, both with GADGET(Springel, Yoshida & White 2001), andwith FVFPS. The results are in remarkableagreement: the remnants obtained with thetwo codes are practically indistinguishablefor three-dimensional shape, circularizedhalf–mass radius, circularized density andvelocity dispersion profiles. As an illustra-tive example, in Fig. 3 we show the densityprofiles of the end–product of the mergerobtained from the head–on parabolic en-counter of two H90 models, with a totalnumber of particles N = 65536 obtainedwith GADGET (empty symbols), and withFVFPS (filled symbols); note how also “mi-nor” details, as the slope change in the den-sity profile at r ∼ 10rM, are nicely repro-duced. In addition, in both cases the changein total energy ∆E/E is smaller than 10−4

per dynamical time, over 100 Tdyn. In thistest, when using the FVFPS code, weadopted softening parameter ε = 0.05, ini-tial time step ∆t = 0.01, and (constant)θ = 0.65; in the GADGET simulation weused ε = 0.05, αtol = 0.05, ∆tmin = 0,∆tmax = 0.03 , α = 0.02 (see Springel etal. 2001 for details), where the softeningparameters and the time step parametersare in units of the galaxy core radius anddynamical time, respectively.

On the basis of the results of these tests,we used our N-body code to investigatewith numerical simulations the effects of hi-erarchies of dissipationless galaxy mergerson the scaling relations of elliptical galax-ies. This study is described in details in


Fig. 3. Angle–average density profile of theremnant of the merging of two identicalH90 galaxy models with FVFPS (full sym-bols), and with GADGET (empty sym-bols). rM is the half–mass radius.

Nipoti, Londrillo & Ciotti (2002, hereafterNLC02). Here we report, as an example,the results relative to the effects of merg-ing on the Fundamental Plane of ellipticalgalaxies (hereafter FP; Djorgovski & Davis1987, Dressler et al. 1987). We considereda hierarchy of 5 equal mass mergers anda hierarchy of 19 accretion events, corre-sponding to an effective mass increase ofa factor of ∼ 24 and ∼ 16, respectively:in the former case the successive gener-ations are produced by merging pairs ofidentical systems obtained by duplicatingthe end–product of the previous step, whilein the latter case each end–product mergeswith a seed galaxy. Each merging is pro-duced in a head–on parabolic encounter.The seed galaxies are spherically symmetricone–component H90 galaxy models, withnumber of particles N = 16384 each. As aconsequence of the merging hierarchy, thenumber of particles in the simulations in-creases with the galaxy mass: in the lastmerging of equal mass systems and of theaccretion scenario, the total number of par-

Fig. 4. The merging end–products in the(k1, k3) plane, where the solid line rep-resents the edge–on FP relation with itsobserved 1-σ dispersion (dashed lines).Equal mass mergers are shown as trian-gles; crosses represent the accretion hier-archy. Bars show the amount of projectioneffects.

ticles involved is of the order of 5.2 × 105

and 3.2 × 105, respectively.In Fig. 4 we plot the results of equal

mass merging and accretion simulationsin the plane (k1, k3), where the observedFP is seen almost edge–on, with best–fitk3 = 0.15k1 + 0.36 (solid line) and scatterrms(k3) ' 0.05 (dashed lines; see Bender,Burstein & Faber 1992). In the diagram theend–products of equal mass mergers andaccretion are identified by triangles andcrosses, respectively. The (common) pro-genitor of the merging hierarchies (pointwithout bar) is placed on the edge–on FP.The straight lines in Fig. 4, associated withthe merger remnants, represent the rangespanned in the (k1, k3) space by each end–product, when observed over the solid an-gle, as a consequence of projection effects.It is interesting to note that for all the end–products the projection effects are of thesame order as the observed FP dispersion.In addition it is also apparent from Fig. 4


Fig. 5. Sersic best fit parameter m vs. totalstellar mass of the end–products at stage iof the merging hierarchy. Symbols are thesame as in Fig. 4

how, in case of equal mass merging, k3 andk1 increase with merging in a way full con-sistent with the observed FP tilt and thick-ness. On the other hand, in the case of ac-cretions, after few steps the end–productsare characterized by a k3 decreasing for in-creasing k1, at variance with the FP slopeand the trend shown by the end–productsof equal mass mergers. As a consequence,the last explored models (corresponding toan effective mass increase of a factor ∼ 16)are found well outside the FP scatter.

The different behavior of the end–products in the two scenarios is due mainlyto effects of structural non–homology. Thisis clearly shown by an analysis of theprojected stellar mass density profiles ofthe end–products. In fact, we fitted (overthe radial range 0.1 <∼R/〈R〉e <∼ 4) the pro-jected density profiles of the end–productswith the Sersic (1968) R1/m law, and inFig. 5 we plot the best fit parameter m asa function of the total mass of the systems,for equal mass mergers (triangles) and ac-cretion (crosses). Clearly, the best-fittingm depends on the relative orientation of

the line–of–sight and of the end–productsof the simulations: the two points for eachvalue of the mass in Fig. 5 show the rangeof values spanned by m when projecting thefinal states along the shortest and longestaxis of their inertia ellipsoids. In case ofequal mass merging, the trend is m increas-ing with galaxy mass, as observed in real el-liptical galaxies (see, e.g., Bertin, Ciotti &Del Principe 2002 and references therein).On the contrary, in case of accretion thebest Sersic m parameter decreases at in-creasing mass of the end–product, a behav-ior opposite to what is empirically found.

As it is well known, elliptical galax-ies follow, besides the FP, also other scal-ing relations, such as the Faber–Jackson(hereafter FJ, Faber & Jackson 1976), andthe Kormendy (1977) relations. It is re-markable that, in our simulations, both theequal mass merging and the accretion hier-archies fail at reproducing these two rela-tions. In fact, the end–products have cen-tral velocity dispersion lower, and effectiveradius larger than what predicted by the FJand the Kormendy relations, respectively.In case of equal mass mergers, these two ef-fects curiously compensate, thus preservingthe edge–on FP. In other words, the rem-nants of dissipationless merging like thosedescribed, are not similar to real ellipticalgalaxies, though in some cases they repro-duce the edge–on FP (see NLC02 for de-tails).

5. Conclusions

The main results of our work can be sum-marized as follows:

– Our implementation of the D02 schemein a F-90 N-body code (FVFPS) has ef-fective O(N) complexity for number ofparticles N>∼104, though its time effi-ciency is lower by a factor of ∼ 1.5 thanDehnen’s C++ implementation.

– As shown by numerical tests, the paral-lelized version of the FVFPS code hasgood scaling with the number of proces-sors (for example for numbers of parti-


cles N ' 106, at least up to 16 proces-sors).

– We found very good agreement betweenthe results of merging simulations per-formed with our FVFPS code and withthe GADGET code.

– From an astrophysical point of view,our high resolution simulations of hier-archies of dissipationless galaxy merg-ers indicate that equal mass mergingis compatible with the existence of theFP relation. On the other hand, whenan accretion scenario is considered, themerger remnants deviate significantlyfrom the observed edge–on FP (see alsoNLC02).

– The different behavior with respect tothe FP of the merger remnants in thetwo scenarios is a consequence of struc-tural non–homology, as shown by theanalysis of their projected stellar den-sity profiles.

– In any case, the end–products of oursimulations fail to reproduce both theFJ and Kormendy relations. In the caseof equal mass merging, the combinationof large effective radii and low centralvelocity dispersions maintains the rem-nants near the edge–on FP.

Acknowledgements. We would like to thankWalter Dehnen for helpful discussions and as-sistance. Computations have been performedat the CINECA center (Bologna) under theINAF financial support. L.C. was supportedby MURST CoFin2000.

References

Barnes J. E., Hut P., 1986, Nature, 324,446 (BH86)Bender R., Burstein D., Faber S. M., 1992,ApJ, 399, 462Bertin G., Ciotti L., Del Principe M.,2002, A&A, 386, 1491Capuzzo-Dolcetta R., Miocchi P., 1998,Journal of Computational Physics, 143, 29(CM98)Dehnen W., 2000, ApJ, 536, L39Dehnen W., 2002, Journal ofComputational Physics, preprint (astro-ph/0202512), (D02)Djorgovski S., Davis M., 1987, ApJ, 313,59Dressler A., Lynden-Bell D., Burstein D.,Davies R.L., Faber S.M., Terlevich R.,Wegner G., 1987, ApJ, 313, 42Faber S.M., Jackson R.E., 1976, ApJ, 204,668Greengard L., Rokhlin V., 1987, Journalof Computational Physics, 73, 325Greengard L., Rokhlin V., 1997, ActaNumerica, 6, 229Hernquist L., 1990, ApJ, 356, 359 (H90)Kormendy J., 1977, ApJ, 218, 333Nipoti C., Londrillo P., Ciotti L., 2002,MNRAS, submitted (NLC02)Sersic J.L., 1968, Atlas de galaxiasaustrales. Observatorio Astronomico,CordobaSpringel V., Yoshida N, White S.D.M.,2001, New Astronomy, 6, 79


Memorie della

Supplementi

High-resolution re-simulations of massive DMhalos and the Fundamental Plane of galaxy

clusters

B. Lanzoni1, A. Cappi1, and L. Ciotti2

1 INAF – Osservatorio Astronomico di Bologna, via Ranzani 1, 40127 Bologna,Italy e-mail: [email protected], [email protected]

2 Dipartimento di Astronomia, Universita di Bologna, via Ranzani 1, 40127Bologna, Italy e-mail: [email protected]

Abstract. Pure N-body high-resolution re-simulations of 13 massive dark mat-ter halos in a ΛCDM cosmology are presented. The resulting sample is used toinvestigate the physical origin of the Fundamental Plane, “Faber-Jackson”, and“Kormendy” relations observed for nearby galaxy clusters. In particular, we focuson the role of dissipationless hierarchical merging in establishing or modifyingthese relations. Contrarily to what found in the case of the Faber-Jackson andKormendy relations for galaxies (see Londrillo, Nipoti & Ciotti, this conference),dissipationless merging on cluster scales produces scaling relations remarkablysimilar to the observed ones. This suggests that gas dissipation plays a minor rolein the formation of galaxy clusters, and the hierarchical merger scenario is not atodd with the observed regularity of these systems.

Key words. dark matter – galaxies: clusters: general

1. Introduction

Like early-type galaxies, also galaxy clus-ters follow well defined scaling relationsamong their main observables: in partic-ular, a luminosity–velocity dispersion re-lation (i.e., a “Faber-Jackson–like” rela-tion), a luminosity–radius relation (i.e.,a “Kormendy–like” relation), and theFundamental Plane (hereafter FP; e.g.,

Send offprint requests to: B. LanzoniCorrespondence to: Osservatorio Astronomicodi Bologna,via Ranzani 1, 40127 Bologna

Schaeffer et al. 1993, hereafter S93; Adamiet al. 1998; and for ellipticals: Djorgovskiet al. 1987; Dressler et al. 1987; Faber& Jackson 1976; Kormendy 1977). Besidestheir potential importance as distance indi-cators, these relations contain informationon the cluster formation and evolution pro-cesses. Within the commonly accepted cos-mological scenario, where cold dark matter(CDM) is the dominant component of theUniverse, structure formation is driven byhierarchical dissipationless merging: smallDM halos form first, then they merge to-

146 B. Lanzoni et al.: High-resolution re-simulations and the cluster FP

gether to form larger systems, and the evo-lution of the baryonic matter follows thatof the hosting DM halos.

The importance of investigating thephysical origin of the observed scaling re-lations, and the role of dissipationless hi-erarchical merging in establishing or mod-ifying them, is therefore apparent, and ex-plains the large number of theoretical worksdevoted to this problem in the case of el-liptical galaxies (e.g. Capelato et al. 1995;Nipoti, Londrillo, Ciotti 2002a,b; Gonzales& van Albada 2002; Evstigneeva et al.2002; Dantas et al. 2002; Londrillo et al.,this conference). In particular, it has beenshown that, while the FP is reproducedby the end–products of equal mass merg-ers, the Faber–Jackson and the Kormendyrelations are not (Nipoti et al. 2002ab;Londrillo et al., this conference). In thecase of galaxy clusters however, almostno theoretical studies are devoted to theirFP (see, e.g., Pentericci, Ciotti & Renzini1995; Fujita & Takahara 1999; Beisbart,Valdarnini & Buchert 2001).

We address this problem in the presentwork, by means of high resolution re-simulations of very massive DM halos (thehosts of galaxy clusters) in a ΛCDM cos-mology. Following their hierarchical dis-sipationless evolution in detail, we verifywhether they reproduce the observed scal-ing relations, both at the present time, andat higher redshift. A full analysis of thiswork will be given elsewhere (Lanzoni etal., in preparation), while we present here aselection of the main astrophysical results,focusing on the numerical issues.

2. High-resolution re-simulations

To investigate whether the dark mat-ter counterpart of galaxy clusters, as ob-tained by numerical simulations, do de-fine the observed scaling relations, a largeenough sample of very massive DM ha-los was needed. For this purpose, we haveemployed the dissipationless “Very LargeSimulations” (VLS; see Yoshida, Sheth, &Diaferio 2001), where the simulated comov-

ing volume of the Universe is sufficientlylarge: the box side is of 479h−1Mpc, withH0 = 100 h−1 km s−1 Mpc−1, and h =0.7. The adopted cosmological model is aΛCDM Universe with Ωm = 0.3, ΩΛ = 0.7,spectral shape Γ = 0.21, and normalizationto the cluster local abundance, σ8 = 0.9.The total number of particles is 5123, of6.86 × 1010M¯/h mass each.

From these simulations, we have se-lected a sample of 13 halos with massesbetween 1014M¯/h and 2.3 × 1015M¯/h.They span a variety of shapes, from nearlyround to more elongated. The richness oftheir environment also changes from caseto case, with the less isolated halos usu-ally surrounded by pronounced filamen-tary structures, containing massive neigh-bors (up to 20% of their mass). All the se-lected halos are required to have a mas-sive progenitor at redshift z = 0.5 (theirmasses range between 6 × 1013 and 1.5 ×

1015M¯/h), and also at z = 1 (with massesbetween 4×1013 and 7×1014M¯/h). Sucha choice allows us to study galaxy clustersnot only locally, but also at higher redshift.

Given the mass resolution of the sim-ulations, less than 1500 particles composea halo of 1014M¯/h, and its properties en-tering the FP relation cannot be accuratelydetermined. We have therefore resimulatedat higher resolution the halos in our sam-ple, by means of the technique introducedby Tormen, Bouchet & White (1997), thatwe briefly summarize in the following.

The first step is to select, in a given cos-mological simulation, the halo one wantsto reproduce at higher resolution. Then,all the particles composing the selectedhalo and its immediate surroundings aredetected in the initial conditions of theparent simulation: the number of particleswithin the region thus defined is increased,and the region is therefore called “thehigh resolution (HR) region”. As a con-sequence, the mean inter-particle separa-tion decreases, and the corresponding high-frequency modes of the primordial fluctua-tion spectrum are added to those on largerscales originally used in the parent sim-

B. Lanzoni et al.: High-resolution re-simulations and the cluster FP 147

ulation. The overall displacement field isalso modified consequently. At the sametime, the distribution of surrounding par-ticles is smoothed by means of a sphericalgrid, whose spacing increases with the dis-tance from the center: in such a way, theoriginal surrounding particles are replacedwith a smaller number of macroparticles,whose mass grows with the distance fromthe HR region. Thank to this method, evenif the number of particles in the HR re-gion is increased, the total number of par-ticles to be evolved in the simulation re-mains small enough to require reasonablecomputational costs, while the tidal fieldthat the overall particle distribution exertson the HR region remains very close to theoriginal one. For the new initial configura-tion thus produced, vacuum boundary con-ditions are adopted, i.e., we assume a van-ishing density fluctuation field outside thespherical distribution of particles with di-ameter equal to the original box size L. Anew N-body simulation is then run start-ing from these new initial conditions, andallows to re-obtain the selected halo at thesuited resolution.

We have applied this technique to the13 massive DM halos selected in the VLS.For 8 out of the 13 selected halos, theresolution has been increased by a fac-tor ∼ 33, using high-resolution particlesof ∼ 2.07 × 109M¯/h each, while a fac-tor of 2 improvement has been adopted forthe 5 intermediate mass halos (the parti-cle mass is 109M¯/h in this case). Thegravitational softening used for the high-resolution region is ε = 5 kpc/h, corre-sponding to about 0.2% and 0.5% of thevirial radius of most and less massive ha-los, respectively. This scale length repre-sents the spatial resolution of the resimula-tions, to be compared with that of 30 kpc/hof the original VLS. Note that to preventlow-resolution macroparticles to “contam-inate” the final resimulated halo (i.e., toend within its virial radius), we need to de-fine the HR region not only through theparticles of the halo itself in the parentsimulation, but considering also a bound-

ary region in its immediate surroundings.This is particularly important in the caseof less massive halos, since the small num-ber of their composing particles does notallow to define their encompassing regionin the initial conditions with enough pre-cision. To avoid such a “contamination”problem in our re-simulations, a radius ofabout 3 rvir and 5 rvir for the boundary re-gion was necessary for the most and the lessmassive halos, respectively. The resultingnumber of high and low resolution particlesare listed in Table 1, together with theirsum1. To run the resimulations, the par-allel dissipationless tree-code “GADGET”(Springel, Yoshida & White 2001) has beenused on the IBM SP2 and SP3 of the CentreInformatique National de l’EnseignementSuperieur (CINES, Montpellier, France),and the CRAY T3E of the RZG ComputingCenter (Munich, Germany), that have com-parable processor speed. The number ofprocessors and the CPU time required forthe corresponding runs are also listed inTable 1.

Results have been dumped in outputfiles for 100 time steps, equally spaced inthe logarithm of the expansion factor ofthe Universe between z = 20 and z = 0.At each time step, a Spherical Overdensity(Lacey & Cole 1994) halo finder has beenused to detect the halos, and to estimatetheir virial mass and radius (see Table 1).With respect to the original virial massesand radii, those of the resimulated halosshow non-systematic and small differences(of the order of few percent), particularlyfor the most massive objects. Moreover,also the halo formation history (the wayhow mass assembly proceeds with time) isvery similar to that in the parent simula-tion, thus assuring the reliability and pre-cision of the high-resolution re-simulationtechnique we use. Visually, the noticeableimprovement provided by such a technique

1 Note that to get the same mass resolution(2.07 × 109M¯/h or 109M¯/h) in the entirevolume of the parent simulation, about 16503

of 21003 particles of would have been required.


is apparent from Figs. 1 and 2, where theimages of the 3 most massive halos, beforeand after the resimulations, are shown atthe scales of ∼ 30Mpc/h and ∼ 2 rvir, re-spectively.

3. Observed scaling relations at z = 0

Analyzing a sample of 16 nearby galaxyclusters, S93 found the following relationbetween the observed luminosity and ve-locity dispersion:

L ∝ σ1.87±0.44, (1)

where L is in units of L∗ = 1.325 ×

1010 h−2L¯ (h = 1), and σ in km/s, andbetween luminosity and radius:

L ∝ R1.34±0.17, (2)

where R is the effective radius measuredin Mpc. In both cases, the observed scat-ter is very large, while a considerable im-provement is found when linking the threeobservables together in a FP-like relation.While a bi-parametric fit was performed byS93, here we derived the FP by means ofthe Principal Component Analysis (here-after PCA; see e.g. Murtagh & Heck 1987),that searches for the most suitable combi-nations of the three observables able to de-scribe a thin plane. By performing a PCAon the data sample of S93, we obtain thenew orthogonal variables

pi ≡ αi log R + βi log L + γi log σ, (3)

with R in Mpc/h, L in 100L∗, σ in 1000km/s. In Fig. 3 we show the resulting dis-tribution of the observed clusters (blue tri-angles) in the (p1, p3) and (p1, p2) spaces,the former providing an exact edge-on viewof the FP and making apparent its smallthickness.

4. Scaling relations at z = 0 forsimulated clusters

To derive the analogous quantities for thesimulated clusters, we have constructed

their projected radial profiles by countingthe DM particles within concentric shellsaround the center of mass, for three orthog-onal directions. Thus, the projected half-mass radius Rh has been derived as the ra-dius of the shell containing half the totalnumber of particles, and the velocity dis-persion σ has been computed by averagingthe squared line of sight component of the(barycentric) velocity over all the particleswithin Rh. Since the simulated clusters, aswell as the real ones, are not spherical, sucha procedure gives different values of Rh andσ for the three considered line of sights,the maximum difference never exceeding33% and 21% in the two cases, respectively.Finally, for converting the DM mass intolight, we have adopted a mass–to–light ra-tio with a small dependence on the lumi-nosity, as suggested by observations (S93;Girardi et al. 2002): M/L ∝ Lα, withα = 0.3. In particular, we have assumed:M/L = 450(M/M0)

α/(α+1)M¯/L¯, whereM0 = 7.5 × 1014M¯.

By performing a PCA on the 13 DM ha-los for the three considered lines of sights,using Rh, σ, and L computed as just de-scribed, we obtain a very well defined andthin FP. To check whether this FP is sim-ilar to the observed one, we have also con-structed the PCA variables pi by combiningRh, σ, and L derived for the DM halos, withthe coefficients αi, βi, γi obtained for theobserved clusters. The resulting FP is prac-tically indistinguishable from the observedone, as apparent from Fig. 3. Moreover, alsothe L-σ and L-R relations of simulated andreal clusters are in remarkable agreement(Fig. 4).

Note that assuming a constant M/L ra-tio, the FP of simulated clusters is still welldefined, and only shows a mild tilt with re-spect to the observed one (see solid line inFig. 3).

5. Scaling relations at higher redshiftfor simulated clusters

While no observational data are yet avail-able for galaxy clusters at higher redshift,


Fig. 1. Images of the 3 most massive DM halos in the parent simulation (left panels)and after the high-resolution re-simulation (right panels). All panels show the projectionalong an arbitrary line of sight, of a region of ∼ 30Mpc/h around the halo center ofmass.


Fig. 2. Images of the 3 most massive DM halos in the parent simulation (left panels)and after the high-resolution re-simulation (right panels). All panels show the projectionalong an arbitrary line of sight, of a region of ∼ 2 rvir around the halo center of mass.


Table 1. Characteristics of the high-resolution resimulations of the 13 massive halos:NHR = number of high-resolution particles, NLR = number of low-resolution macropar-ticles, NT = total number of particles, Nproc = number of processors, tCPU = hours perprocessor required for the resimulations, Mvir = virial mass in 1014M¯/h, rvir = virialradius in Mpc/h.

Name NHR NLR NT Nproc tCPU Mvir rvir

g8 3700120 191733 3891853 32 80 23.42 2.75g1 2574717 202301 2777018 32 70 13.99 2.31g72 3299865 194277 3494142 32 57 11.77 2.18g696 4870197 184314 5054511 64 60 11.37 2.16g51 1677364 213477 1890841 32 44 10.78 2.12g245 3437317 215269 3652586 16 18.4 6.50 1.79g689 3252085 215809 3467894 16 18.3 6.08 1.75g564 2068981 227187 2296168 16 12.8 4.91 1.63g1777 3094123 219047 3313170 16 18.0 3.83 1.50g4478 2293433 225706 2519139 16 12.8 2.92 1.37g914 250605 247091 497696 16 7.3 1.45 1.09g3344 206140 248756 454896 16 6.3 1.09 0.99g1542 207202 248948 456150 16 6.3 0.82 0.99

our high-resolution re-simulations allow toinvestigate the scaling relations of theirDM counterparts at any epoch. For thatpurpose, the most massive progenitors atz = 0.5 and at z = 1 of our 13 DM ha-los have been considered, and their pro-jected half-mass radius, velocity dispersionand luminosity have been derived with thesame procedure used for the z = 0 objects.Also in this case, when combining Rh, σ,and L with the coefficients αi, βi, γi de-rived for the real clusters at z = 0, the re-sulting FP is very similar to the edge-on FPobserved locally, while when seen face-on,the region populated by the high-z DM ha-los is systematically shifted towards larger(smaller) values of p1 (p2) with respect tothe region occupied by the observed nearbyclusters. In addition, very well defined L-σ and L-R relations are already in placeat these redshifts. With respect to the ob-served L-σ relation at z = 0, a flatter slopeis found for increasing redshift for the sim-ulated clusters, while the opposite trend isfound in the case of the L-R relation, thatbecomes steeper for increasing z.

6. Discussion and conclusions

We have described a technique for increas-ing the (mass and spatial) resolution of ob-jects selected in a given cosmological sim-ulation. Applying it to sample of 13 mas-sive halos at z = 0 in a ΛCDM cosmology,we have investigated whether the scalingrelations observed for nearby galaxy clus-ters are also followed by their DM hosts, ina dissipationless hierarchical merging sce-nario. By considering the most massive pro-genitors at z = 0.5 and z = 1 of the 13selected halos, we have also investigatedwhether the same scaling relations were al-ready in place at those earlier epochs.

The main conclusions can be summa-rized as follows:

– the high-resolution re-simulation tech-nique is very powerful and reliable. Theprecision of the method depends on thenumber of particles composing the orig-inal object, and on the dimension of theboundary region considered to avoid the“contamination” from macroparticles.


-0.5 0 0.5 1

-0.5

0

0.5

-0.5

0

0.5

Fig. 3. Observed (blue triangles) and simulated (red hexagons) clusters in the edge-onand face-on view of the FP (upper and lower panels, respectively), at redshift z = 0. Thelinear best fit to the simulated clusters in case of a constant M/L is shown by the pinksolid line.

– The DM hosts of galaxy clusters dodefine a FP that is practically indis-tinguishable from the one observed fornearby galaxy clusters.

– A very good agreement between thescaling relations of simulated and ob-served clusters is also found in terms ofthe L-σ and the L-R relations.

– The FP of simulated clusters is alreadydefined at z = 0.5 and z = 1, and it isvery similar and as thin as the one ob-served locally; the only differences aredue to an obvious shift towards smallerradii and masses for increasing z. Alsothe L-σ and L-R relations are alreadyin place at high redshifts, with a differ-ence in the slope (with respect to the re-


Fig. 4. L-σ (left panels) and L-R (right panels) relations for the observed (blue trianglesand dashed line) and simulated (red hexagons and solid line) clusters, at redshift z = 0.Luminosities are in units of 100L∗, velocity dispersions in km/s, and radii in Mpc.

lations observed locally) due to the factthat at recent epochs, more massive ob-jects evolve more rapidly than smallersystems.

All the results about the scaling re-lations of simulated and observed clus-ters are obtained by assuming a mass-to-light ratio with mean value of the order of450hM¯/L¯ (well in agreement with whatestimated from observations of galaxy clus-ters), and with a dependence on the lumi-nosity also suggested by the observations.

Given the very different nature of thetwo components (baryonic and non bary-onic) and of the physical processes actingon them, the fact that the DM halos definethe same relations shown by real clusters isnot obvious at all. Moreover, dissipation-less hierarchical merging at galactic scalesis unable to reproduce the observed scal-ing relations of elliptical galaxies: the end-products of major mergers do lie on the ob-served FP, but substantial deviations fromthe Faber-Jackson and the Kormendy rela-tions are found in this case (see Londrillo etal., this conference). Such a result suggeststhat, while dissipation has a major role inthe formation and evolution of ellipticals, itis negligible with respect to the pure grav-

ity in settling the properties of galaxy clus-ters.

Acknowledgements. B.L. acknowledges G.Tormen, V. Springel, S. White, & G. Mamonfor their help with the N-body simulationsand for useful discussions. This work has beenpartially supported by the Italian Ministery(MIUR) grant COFIN2001 “Clusters andgroups of galaxies: the interplay betweendark and baryonic matter”, and by theItalian Space Agency grants ASI-I-R-105-00and ASI-I-I-037-01. L.C. was supported byCOFIN2000.

References

Adami C., Mazure A., Biviano A., KatgertP., & Rhee G., 1998, A&A 331, 493Beisbart C., Valdarnini R., & Buchert T.,2001, A&A 379, 412Capelato H.V., de Carvalho R.R., &Carlberg R.G., 1995, ApJ 451, 525Dantas C.C., Capelato H.V., RibeiroA.L.B., & de Carvalho R.R., 2002,MNRAS accepted (astro-ph/0211251)Djorgovski S., & Davis M. 1987, ApJ, 313,59Dressler A., Lynden-Bell D., Burstein D.,Davies R.L., Faber S.M., Terlevich R., &Wegner G. 1987, ApJ, 313, 42


Faber S.M., & Jackson R.E. 1976, ApJ,204, 668Fujita Y., & Takahara F., 1999, ApJ 519,L55Evstigneeva E.A., Reshetnikov V.P.,Sotnikova N.Ya., 2002, A&A 381, 6Girardi M., Manzato P., Mezzetti M.,Giuricin G., Limboz F., 2002, ApJ 569,720Gonzales A.C., & van Albada T.S., 2002,MNRAS submittedKormendy J., 1977, ApJ 218, 333Lacey C., & Cole S., 1994, MNRAS 271,781Lanzoni B., Cappi A., Ciotti L., TormenG., & Zamorani G., in preparationMurtagh F., Heck A., 1987, MultivariateData Analysis, D.Reidel PublishingCompany, Dordrecht, Holland

Nipoti C., Ciotti L., Londrillo P., 2002a,astro-ph/0112133 (NCL02a)Nipoti C., Ciotti L., Londrillo P., 2002b,MNRAS submitted (NCL02b)Pentericci L., Ciotti L., & RenziniA. 1995, Astrophysical Letters andCommunications, 33, 213Schaeffer R., Maurogordato S., Cappi A.,Bernardeau F., 1993, MNRAS 263, L21(S93)Springel V., Yoshida N., & White S.D.M.,2001, New Astronomy, 6, 79Tormen G., Bouchet F., & White S.D.M.,1997, MNRAS 286, 865West M.J., Dekel A., Oemler A., 1987,ApJ 316, 1Yoshida N., Sheth R.K., & Diaferio A.,MNRAS 328, 669


Memorie della

Supplementi

Elliptical galaxies interacting with the clustertidal eld: origin of the intracluster stellar

population

V. Muccione1, L. Ciotti2

1 Geneva Observatory,51 ch. des Maillettes, 1290 Sauverny, Switzerlande-mail: [email protected]

2 Dipartimento di Astronomia, Universita di Bologna, via Ranzani 1, 40127Bologna, Italy e-mail: [email protected]

Abstract.With the aid of simple numerical models, we discuss a particular aspect of theinteraction between stellar orbital periods inside elliptical galaxies (Es) and theparent cluster tidal field (CTF), i.e., the possibility that collisionless stellar evapo-ration from Es is an effective mechanism for the production of the recently discov-ered intracluster stellar populations (ISP). These very preliminary investigations,based on idealized galaxy density profiles (such as Ferrers density distributions)show that, over an Hubble time, the amount of stars lost by a representativegalaxy may sum up to the 10% of the initial galaxy mass, a fraction in interestingagreement with observational data. The effectiveness of this mechanism is dueto the fact that the galaxy oscillation periods near its equilibrium configurationsin the CTF are of the same order of stellar orbital times in the external galaxyregions.

Key words. Clusters: Galaxies – Galaxies: Ellipticals – Stellar Dynamics:Collisionless systems

1. Introduction

Observational evidences of an IntraclusterStellar Population (ISP) are mainly basedon the identification of intergalactic plan-etary nebulae and red giant branch stars(see, e.g., Theuns & Warren 1996, Mendezet al. 1998, Feldmeier et al. 1998, Arnaboldiet al. 2002, Durrell et al. 2002). Overall,the data suggest that approximately 10%(or even more) of the stellar mass of the

cluster is contributed by the ISP (see, e.g.,Ferguson & Tanvir 1998).

The usual scenario assumed to ex-plain the finding above is that gravita-tional interactions between galaxies in clus-ter and/or interactions between the galax-ies with the tidal gravitational field of theparent cluster (CTF), lead to a substantialstripping of stars from the galaxies them-selves.

V. Muccione & L. Ciotti: Origin of the ISP 187

Here, supported by a curious coinci-dence, namely by the fact that the char-acteristic times of oscillation of a galaxyaround its equilibrium position in the CTFare of the same order of magnitude of thestellar orbital periods in the external part ofthe galaxy itself, we suggest that an addi-tional effect is at work, i.e. we discuss aboutthe possible “resonant” interaction betweenstellar orbits inside the galaxies and theCTF.

In fact, based on the observationalevidence that the major axis of clusterEs seems to be preferentially oriented to-ward the cluster center, N-body simula-tions showed that galaxies tend to align re-acting to the CTF as rigid (Ciotti & Dutta1994). By assuming this idealized scenario,a stability analysis then shows that thisconfiguration is of equilibrium, and allowsto calculate the oscillation periods in thelinearized regime (Ciotti & Giampieri 1998,hereafter CG98).

Prompted by these observational andtheoretical considerations, in our numericalexplorations we assume that the galaxy isa triaxial ellipsoid with its center of massat rest at the center of a triaxial cluster.The considerably more complicate case ofa galaxy with the center of mass in rotationaround the center of a spherical cluster willbe discussed elsewhere (Ciotti & Muccione2003, hereafter CM03).

2. The physical background

Without loss of generality we assume thatin the (inertial) Cartesian coordinate sys-tem C centered on the cluster center, theCTF tensor T is in diagonal form, withcomponents Ti (i = 1, 2, 3). By using threesuccessive, counterclockwise rotations (ϕaround x axis, ϑ around y′ axis and ψaround z′′ axis), the linearized equations ofmotion for the galaxy near the equilibriumconfigurations can be written as

ϕ =∆T32∆I32

I1ϕ,

ϑ =∆T31∆I31

I2ϑ,

ψ =∆T21∆I21

I3ψ, (1)

where ∆T is the antisymmetric tensor ofcomponents ∆Tij ≡ Ti − Tj , and Ii are theprincipal components of the galaxy inertiatensor. In addition, let us also assume thatT1 ≥ T2 ≥ T3 and I1 ≤ I2 ≤ I3, i.e., that∆T32, ∆T31 and ∆T21 are all less or equalto zero (see Section 3). Thus, the equilib-rium position of 1 is linearly stable, and itssolution is

ϕ = ϕM cos(ωϕt),

ϑ = ϑM cos(ωϑt),

ψ = ψM cos(ωψt), (2)

where

ωϕ =

√

∆T23∆I32

I1,

ωϑ =

√

∆T13∆I31

I2,

ωψ =

√

∆T12∆I21

I3. (3)

For computational reasons the best ref-erence system in which calculate stellar or-bits is the (non inertial) reference systemC ′ in which the galaxy is at rest, and itsinertia tensor is in diagonal form. The equa-tion of the motion for a star in C ′ is

x′ = RTx−2Ω∧v

′−Ω∧x′−Ω∧(Ω∧x

′), (4)

where x = R(ϕ, ϑ, ψ)x′, and

Ω = (ϕ cos ϑ cos ψ + ϑ sin ψ,−ϕ cos ϑ sinψ

+ϑ cos ψ, ϕ sin ϑ + ψ). (5)

In eq. (4)

RTx = −∇x

′φg + (RTTR)x′, (6)

where φg(x′) is the galactic gravitational

potential, ∇x′ is the gradient operator in

C ′, and we used the tidal approximationto obtain the star acceleration due to thecluster gravitational field.

188 V. Muccione & L. Ciotti: Origin of the ISP

3. Period estimations

For simplicity, in the following estimates weassume that the galaxy and cluster densi-ties are stratified on homeoids. In partic-ular, we use a galaxy density profile be-longings to the ellipsoidal generalization ofthe widely used γ-models (Dehnen 1993,Tremaine et al. 1994):

ρg(m) =Mg

α1α2α3

3 − γ

4π

1

mγ(1 + m)4−γ, (7)

where Mg is the total mass of the galaxy,0 ≤ γ ≤ 3, and

m2 =

3∑

i=1

(x′i)

2

α2i

, α1 ≥ α2 ≥ α3. (8)

The inertia tensor components for thisfamily are given by

Ii =4π

3α1α2α3(α

2j + α2

k)hg, (9)

where hg =∫ ∞

0ρg(m)m4dm, and so I1 ≤

I2 ≤ I3. Note that, from eq. (3), the fre-quencies for homeoidal stratifications donot depend on the specific density distri-bution assumed, but only on the quantities(α1, α2, α3). We also introduce the two el-lipticities

α2

α1≡ 1 − ε,

α3

α1≡ 1 − η, (10)

where ε ≤ η ≤ 0.7 in order to reproducerealistic flattenings.

A rough estimate of characteristic stel-lar orbital times inside m is given byPorb(m) ' 4Pdyn(m) =

√

3π/Gρg(m),where ρg(m) is the mean galaxy density in-side m. We thus obtain

Porb(m) ' 9.35 × 106

√

α31,1(1 − ε)(1 − η)

Mg,11

×mγ/2(1 + m)(3−γ)/2 yrs, (11)

where Mg,11 is the galaxy mass normalizedto 1011M¯, α1,1 is the galaxy “core” ma-jor axis in kpc units (for the sphericallysymmetric γ = 1 Hernquist 1990 models,

Re ' 1.8α1); thus, in the outskirts of nor-mal galaxies orbital times well exceeds 108

or even 109 yrs.For the cluster density profile we as-

sume

ρc(m) =ρc,0

(1 + m2)2, (12)

where m is given by an identity similar toeq. (8), with a1 ≥ a2 ≥ a3, and, in analogywith eq. (10), we define a2/a1 ≡ 1 − µ anda3/a1 ≡ 1 − ν, with µ ≤ ν ≤ 1.

It can be shown that the CTF com-ponents at the center of a non-singularhomeoidal distribution are given by

Ti = −2πGρc,0wi(µ, ν), (13)

where the dimensionless quantities wi areindependent of the specific density profile,w1 ≤ w2 ≤ w3 for a1 ≥ a2 ≥ a3, and so theconditions for stable equilibrium in eq. (1)are fulfilled (CG98, CM03).

The quantity ρc,0 is not a well mea-sured quantity in real clusters, and for itsdetermination we use the virial theorem,Mcσ

2V = −U , where σ2

V is the virial ve-locity dispersion, that we assume to be es-timated by the observed velocity disper-sion of galaxies in the cluster. Thus, we cannow compare the galactic oscillation peri-ods (for small galaxy and cluster flatten-ings, see CM03)

Pϕ =2π

ωϕ' 8.58 × 108

√

(ν − µ)(η − ε)

a1,250

σV,1000yrs ,

Pϑ =2π

ωϑ' 8.58 × 108

√νη

a1,250

σV,1000yrs ,

Pψ =2π

ωψ' 8.58 × 108

√µε

a1,250

σV,1000yrs . (14)

(where a1,250 = a1/250kpc, and σV,1000 =σV/103km s−1) with the characteristic or-bital times in galaxies. From eqs. (11) and(14), it follows that in the outer halo ofgiant Es the stellar orbital times can beof the same order of magnitude as the os-cillatory periods of the galaxies themselvesin the CTF. For example, in a relativelysmall galaxy of Mg,11 = 0.1 and α1,1 = 1,


0.4 0.6 0.8 10

1000

2000

3000

4000

5000

Fig. 1. Histogram of m at t = 0 vs. Nesc for a galaxy of Mg = 1010 M¯ and α1 = 20kpc, with ε = 0.2 and η = 0.4. The cluster parameters are µ = 0.2, ν = 0.4, a1,250 =σV,1000 = 1, and the galaxy maximum oscillations angles in eqs. (2) are fixed to 0.2 rad.In this simulation Ntot = 105.

Porb ' 1 Gyr is at m ' 10 (i.e., at ' 5Re),while for a galaxy with Mg,11 = 1 andα1,1 = 3 the same orbital time is at m ' 7(i.e., at ' 3.5Re).

4. The numerical integration scheme

Here we present the preliminary resultsobtained by using only a very idealized

galactic density profile, while more realis-tic galaxy density distributions, such as tri-axial Hernquist models, are described else-where (CM03, Muccione & Ciotti 2003,hereafter MC03). In particular, here we ex-plore the evaporation process in the caseof a Ferrers (see, e.g., Binney & Tremaine

190 V. Muccione & L. Ciotti: Origin of the ISP

1987) models, with density profiles given by

ρg =

ρg(0)(1 − m2)n form ≤ 10 form > 1

(15)

where m is the homeoidal radius definedas in eq. (8). The case n = 0 correspondsto an anisotropic harmonic oscillator, whilefor n ≥ 0 the density distribution is radi-ally decreasing. A nice property of Ferrersmodels (for integer n) is that their gravi-tational potential can be simply expressedin algebraic form (see, e.g., Chandrasekhar1969).

To integrate the second order differ-ential equations (11) for each star, weuse a code based on an adapted Runge-Kutta routine. The initial conditions areobtained by using the Von Neumann re-jection method, and with this method wearrange Ntot initial conditions in configu-ration space which reproduce the densityprofile in eq. (15). In MC03 and CM03 thethree components of the initial velocity foreach star are assigned by considering the lo-cal velocity dispersion of the galaxy modelat rest, while here, for simplicity, each starat t = 0 is characterized by null velocity.Within t = tH (where tH = 1.5 × 1010

yrs), the code checks which of the initialconditions result in a orbit with m > 1.This escape condition is very crude, anda more sophisticated criterium is adoptedin MC03 and CM03, where untruncatedgalaxy models are used. In standard sim-ulations we use, as a rule, Ntot = 105, thus,from this point of view, we are exploring105 independent “1-body problems” in atime–dependent force field.

The simulations are per-formed on the GRAVITOR, theGeneva Observatory 132 processorsBeowulf cluster (http://obswww.unige.ch/˜pfennige/gravitor/gravitor e.html).

5. Preliminary results and conclusions

In Fig. 1 we show the results of one of ourpreliminary simulations for a galaxy model

with n = 1, Mg = 1010 M¯, semi-mayoraxis α1 = 20 kpc, flattenings ε = 0.2,η = 0.4, and maximum oscillation anglesequals to 0.2 rad. The cluster parametersare a1,250 = σV,1000 = 1, µ = 0.2, ν = 0.4,and the total number of explored orbits isNtot = 105. In the ordinate axis we plot thenumber of escapers as a function of theirhomeoidal radius at t = 0. Note how thezone of maximum escape is near m ' 0.8,and how the total number of escapers is asignificant fraction of the total number ofexplored orbits (actually, with the galaxyand cluster parameters adopted in the sim-ulations, Nesc ' 0.1Ntot). This number is asomewhat upper limit of the expected num-ber in more realistic simulations: in fact,1) as described in Section 4, we adopteda very weak escape criterium, and 2), inmore realistic galaxy models the density de-crease in the outer parts is stronger thanin Ferrers models, and so we expect that asmaller number of stars will be significantlyaffected by the CTF. Both points are ad-dressed and discussed elsewhere, in MC03in a preliminary and qualitative way for anHernquist model at the center of a clus-ter, while in CM03 we present the resultsof a systematic exploration of the parame-ter space for untruncated galaxies both atthe center and with the center of mass incircular orbit in a spherical cluster.

In any case, the simple model here ex-plored looks promising, with a number ofescapers in nice agreement with the obser-vational estimates.

Acknowledgements. We would like to thankGiuseppe Bertin and Daniel Pfenniger for use-ful discussions and the Observatory of Genevewhich has let us use the GRAVITOR for oursimulations. L.C. was supported by the grantCoFin2000.

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