Core-corona separation in the UrQMD hybrid model
aa r X i v : . [ h e p - ph ] A ug Core-corona separation in the UrQMD hybrid model
J. Steinheimer
1, 2 and M. Bleicher
1, 2 Institut f¨ur Theoretische Physik, Goethe-Universit¨at,Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany Frankfurt Institute for Advanced Studies (FIAS),Ruth-Moufang-Str. 1, D-60438 Frankfurt am Main, Germany
We employ the UrQMD transport + hydrodynamics hybrid model to estimate the effects of aseparation of the hot equilibrated core and the dilute corona created in high energy heavy ioncollisions. It is shown that the fraction of the system which can be regarded as an equilibratedfireball changes over a wide range of energies. This has an impact especially on strange particleabundancies. We show that such a core corona separation allows to improve the description ofstrange particle ratios and flow as a function of beam energy as well as strange particle yields as afunction of centrality.
I. INTRODUCTION
The objective of the low energy heavy ion colliderprograms, at the RHIC facility on Long Island andthe planned projects NICA in Dubna and FAIR nearthe GSI facility, is to find evidence for the onset of adeconfined phase [1, 2]. At the highest RHIC energies,experiments [3–6] have already confirmed a collectivebehavior of the created (partonic) system, signaling achange in the fundamental degrees of freedom. Lat-tice QCD calculations indeed expect a deconfinementcrossover to occur in systems created at the RHIC. Astheoretical predictions on the thermodynamics of finitedensity QCD are quite ambiguous [7–14], one hopesto experimentally confirm a possible first order phasetransition and consequently the existence of a criticalendpoint, by mapping out the phase diagram of QCD insmall steps.Hadronic bulk observables which are usually connectedto the onset of deconfinement are the particle flow andits anisotropies as well as particle yields and ratios[15–30]. It has often been proposed, that e.g. theequilibration of strangeness would be an indication forthe onset of a deconfined phase, although this idea isstill under heavy debate [31–34].The interpretation of experimental results and theirrelation to the deconfinement phase transition is mostoften circumstantial and extensive model studies arerequired to understand the multitude of observables.Therefore dynamical models for the description ofrelativistic heavy ion collisions are needed as input forthe interpretation of the observed phenomena.Dynamical approaches to heavy ion collisions are oftenbased on two complementary theoretical concepts: thefirst being relativistic fluiddynamics [35–44]. In thisapproach one assumes that the produced system canbe described as an expanding liquid which is in localthermal equilibrium. The assumption of local equili-bration is usually disputed which led to development ofhydrodynamic models which employ viscous corrections.Apart from these complications, a general advantage ofthe hydrodynamic approach is that an equation of state, which contains the information on the active degrees offreedom of the system (potentially including a phasetransition), can be easily introduced in the model.The second type of models is based on the relativisticBoltzmann transport equation [46–52]. The applicabilityof this equation is independent of any equilibrium as-sumption which makes it superior to the hydrodynamicapproach in this respect. However the implementationof a phase transition in such a microscopic model is farmore complicated and still poses a great challenge totheorists.To obtain a more comprehensive picture of the wholedynamics of heavy ion reactions various so calledmicro+macro hybrid approaches have been developedduring the last years [53] to combine the benefits ofthe two mutually complementary approaches discussedbefore. Here, one employs initial conditions that arecalculated from a non equilibrium model followed byan ideal or viscous hydrodynamic evolution coupled tothe Boltzmann equation for the final state [44, 45, 54–62].In this paper we present a study of the effects of theassumption of local thermal and chemical equilibriumin the description of heavy ion collisions at differentbeam energies. In particular we want to discuss caseswhere only parts of a created fireball can be regarded asbeing equilibrated. Following previous explorations sucha system can be divided in a hot and dense core anda dilute corona [63, 64], where each part of the systemshould be treated on different theoretical footing. Wewill apply the UrQMD transport/hydrodynamics hybridmodel as described in [66] and modify it to allow fora consistent simultaneous description of a core-coronaseparated system. Let us remark that the present modelallows only to study the effects of local thermal andchemical equilibration, while it does not allow to pindown the actual dynamical processes which induce theearly equilibration. In fact, the processes responsiblefor fast equilibration of the produced matter (probablyinstabilities or multi particle interactions [34, 68]) stillpose a great theoretical challenge in modeling heavy ioncollisions. -8 -6 -4 -2 0 2 4 6 8-8-6-4-202468 X-Axis [fm] Y - A x i s [f m ] FIG. 1. (Color online) Contour plot of the local rest frameenergy density in the transverse plane ( z = 0) of a central( b = 0) collision of Pb+Pb at E lab = 40 A GeV. The energydensity is normalized to the ground state energy density ( ǫ ≈
145 MeV / fm ). The two green lines correspond to lines of aconstant energy density of ǫ/ǫ = 5 and 7. In the following we will first explain in short the con-cept and implementation of the UrQMD hybrid modeland how it is extend to allow for a consistent separationof the dense core and dilute corona of the fireball thatis obtained. Then we present results that are sensitiveto such a separation und discuss them in order beforemaking concluding remarks.
II. THE HYBRID MODEL
Hybrid approaches to unite hydrodynamics and trans-port equations where proposed 10 years ago [60, 62] andhave since then been employed for a wide variety of in-vestigations [56, 57, 69, 70]. The hybrid approach pre-sented here is based on the integration of a hydrody-namic evolution into the UrQMD transport model [65–67]. During the first phase of the evolution the parti-cles are described by UrQMD as a string/hadronic cas-cade. Once the two colliding nuclei have passed througheach other the hydrodynamic evolution starts at the time t start = 2 R/ p γ c.m. −
1, where γ c.m. denotes the Lorentzgamma of the colliding nuclei in their center of massframe. While the spectators continue to propagate inthe cascade, all other particles, i.e. their baryon chargedensities and energy-momentum densities, are mappedto the hydrodynamic grid. By doing so one explicitlyforces the system into a local thermal equilibrium foreach cell. In the hydrodynamic part we solve the con-servation equations for energy and momentum as well asthe net baryon number current, while for the net strangenumber we assume it to be conserved and equal to zerolocally. Solving only the equations for the net baryon -8 -6 -4 -2 0 2 4 6 8-8-6-4-202468 Y - A x i s [f m ] X-Axis [fm]
FIG. 2. (Color online) Contour plot of the local rest frameenergy density in the transverse plane ( z = 0) of a central( b = 0) collision of Pb+Pb at E lab = 160 A GeV. The energydensity is normalized to the ground state energy density ( ǫ ≈
145 MeV / fm ). The two green lines correspond to lines of aconstant energy density of ǫ/ǫ = 5 and 7. number is commonly accepted in hydrodynamical mod-els, although we have shown in earlier [71] publicationsthat net strangeness may fluctuate locally. It is plannedto also implement an explicit propagation for the netstrange density. Such an extension of the model alsorequires that the equation of state is extended in the netstrange sector which is an investigation, currently under-way and will be addressed in future publications.The hydrodynamic evolution is performed using theSHASTA algorithm [38] with an equation of state that in-corporates a chiral as well as an deconfinement crossoverand which is in agreement with thermodynamic resultsfrom lattice calculations [72]. At the end of the hydrody-namic phase the fields are mapped to particle degrees offreedom using the Cooper-Frye equation [73]. The tran-sition from the hydrodynamic prescription to the trans-port simulation is done gradually in transverse slices ofthickness 0.2 fm, once all cells in a given slice have anenergy density lower than five times the ground state en-ergy density (see also [74]). The temperature at µ B = 0which corresponds to such a switching density is roughly T = 170 MeV which is close to what is expected to be thecritical temperature. Detailed information of the transi-tion curve in the phase diagram can be found in [65].After this the final state interactions and decays of theparticles are calculated and the system freezes out dy-namically within the UrQMD framework.For an extensive description of the model the reader isreferred to [65, 74]. III. SEPARATING CORE AND CORONA
Essentially all previous hybrid model calculationshave assumed that the whole system created enters aphase of local thermal equilibrium. As the local restframe energy density varies in coordinate space, onewould expect that some portions of the created system,already in the beginning of the evolution, have densitiesthat are smaller than the transition energy density. Ingeneral, doing hydrodynamical simulations one neglectssuch portions of the system which never really enter anequilibrated phase. As a first step we want to estimatequantitatively how good such an assumption is. Asa visualization of the problem, figures 1 and 2 showcontour plots of the local rest frame energy densities(normalized to the ground state energy density) in thetransverse plane ( z = 0) of central Pb+Pb collisions attwo different energies. Figure 1 depicts the distributionfor E lab = 40 A GeV and figure 2 for E lab = 160 A GeV. One can clearly see that the energy densitiesreached in the center of the system exceed the transitioncriterion for both cases , while the energy density for E lab = 160 A GeV is roughly 5 times that for E lab = 40 A GeV. The green solid lines are lines of constant energydensity, ǫ = 5 and 7 times ǫ . For the higher beamenergy, almost all the system seems to have an energydensity larger than this criterion while for E lab = 40 A GeV this is not the case, one observes that parts ofthe system lie already outside of the hot and dense region.In the following we describe how one can separate thesystem which is produced in the heavy ion collision ina dense core part, which will be propagated using thehydrodynamic prescription, and a dilute corona part forwhich we assume the UrQMD transport approach to bethe correct model.The idea that the fireball, created in a heavy ioncollision, can be divided in a dense core which expandscollectively and a dilute corona which is dominated byhadronic scatterings is not new. The first time thisidea was adapted in a dynamical model for heavy ioncollisions was in [63]. In this approach the system wasdivided in transverse cells of a certain pseudorapidityrange. Whenever the transverse string density in sucha cell was above a certain value then it was considereda part of the equilibrated core otherwise was is con-sidered part of the corona. In a different approach thecore-corona separation was made, using a Monte CarloGlauber model and distinguishing between nucleonsthat have interacted once or more than once, while thosenucleons which have interacted only once were regardedas Corona part [75–78].For the present study we will apply a method similarto that used in [63]. At the time t start of the transitionfrom the UrQMD model to the hydrodynamic phase wecalculate the scalar constituent quark number density(Mesons count 2 for q + q and Baryons count 3 for q + q + q ) at the position of every particle. This is done by describing every hadron as a Lorentz contractedGaussian distribution of its constituent quark numberand then sum over the contributions of all particlesto a given space point. As a result one obtains thequark number density ρ q at the position of every hadronin the UrQMD model. If the density at the particlesposition is above a certain critical separation densityit is used to calculate the initial density profiles forthe hydrodynamic evolution as outlined above. If thedensity is below the separating density it will remainin the transport model and will be propagated therein parallel to the hydrodynamic evolution. After thetransition from the hydrodynamic phase, back to thetransport model occurs, all particles can then againre-interact and decouple dynamically.res, all particlescan then again re-interact and decouple dynamically.Note that this procedure is similar to tho one applied in[63], although a distinct difference is that we calculatethe scalar density at every particles position in coordi-nate space. As for higher energies, the particle densityshould be roughly independent of the pseudorapidity(boost invariance), the definition of a corona via thetransverse string density as in [63] is sufficient. For lowerenergies this relation does not hold anymore and the fullcalculation of the particle number density seems moreappropriate. On the other hand, the present approachbecomes unfeasible at some point as particles from allrapidities contribute to the local density of any otherparticle. One might therefore, optionally, apply a cutin pseudo-rapidity (of ∆ η = 0 .
5) for particles whichcontribute to the local density ρ q ( ~x ).Our procedure introduces the density ρ q as anotherparameter in the model. In the present investigationwe will constrain this parameter to lie between 4 and 5times ρ q (where ρ q = 0 . · f m − , the ground statequark density at T = 0). This choice of parameter istaken, because we try to keep the cut off density to enterthe hydrodynamic phase close to the density criterionfor the transition from the hydrodynamic phase back tothe hadronic afterburner. The values of 4 and 5 times ρ q closely correspond to the energy densities of 5 and7 times ǫ , when we consider a hadron resonance gas,which includes the same degrees of freedom as doesUrQMD. IV. ENERGY DEPENDENCE
In this section we will concentrate in the beam energydependence of effects of a core-corona separation asdescribed above. We will apply the model for mostcentral ( b < . ρ q .Figure 3 shows the fraction of the total energy ofthe colliding system (excluding spectators) which istransferred into the hydrodynamic phase. For the lowest f l u i d f r a c t i on [ % ] s [GeV] q0 q0 FIG. 3. (Color online) Fraction of the total energy of thesystem which is transferred into the hydrodynamic phase asa function of center of mass beam energy, for central ( b < . ρ q = 4 ρ q and the red dashed lineto ρ q = 5 ρ q . The error bars indicate the mean deviation ofthe fluid fraction on an event by event basis. beam energy, E lab = 2 A GeV, only a vanishing fractionof the system can be regarded as being in local thermalequilibrium. The fraction increases slowly with thebeam energy, while only at the highest SPS energiesone can regard the whole system as being equilibrated.Changing the density cut off parameter ρ q only resultsin a small shift at intermediate beam energies, whileat the highest SPS energies the density gradients ofthe produced system are so large that the result isinsensitive on the exact value of the ρ q parameter. The’error’ bars in the figure represent the mean deviation ofthe fluid fraction on an event-by-event basis.As a next step we investigate the dependence ofexperimental hadronic observables like particle yieldsand flow, on the separation procedure. Figure 4 depictsthe beam energy dependence of the ratios of protonsto pions (upper plot) and positively charged kaons andpions (lower plot) in the mid-rapidity region of central b < . | y | < . ρ q = 4 · ρ q ,circles: ρ q = 5 · ρ q ). The model results are compared todata from different experiments [79–86] which are shownas blue square symbols. K + / + at |y|<0.5 s [GeV] p/ + at |y|<0.5 FIG. 4. (Color online) Particle ratios of protons to pions, up-per plot, and positively charged kaons to pions, lower plot.The results are for the mid rapidity region ( | y | < .
5) of cen-tral ( b < . K + /π + ratio for thestandard UrQMD calculation and the data. For the non-strange protons and pions the assumptionof local thermal equilibration seems not to change theresults on the particle ratio and all different modelcalculation give a rather good description of the data.However, if one looks at the ratio of the positivelycharged kaons to pions we observe considerable dif-ferences in the results obtained from the differentapproaches. Generally, the assumption of thermal equi-librium drastically enhances the production of strangeparticles when compared to the UrQMD non-equilibriumapproach. Especially for the very low beam energies thisleads to a drastic overestimation of the ratios involvingstrange particles in the standard UrQMD hybrid model.In contrast using the core-corona separation approachthe fraction of the system, for which local equilibrium / at |y|<0.5 s [GeV] - / at |y|<0.5 FIG. 5. (Color online) Particle ratios of lambdas to pions,upper plot, and negatively charged Xi’s to pions, lower plot.The results are for the mid rapidity region ( | y | < .
5) of cen-tral ( b < . K + /π + ratio for thestandard UrQMD calculation and the data. is assumed, changes with beam energy. Therefore,for the lowest energies, one smoothly recovers thedefault UrQMD results. At intermediate energies thecore-corona result is generally in between the transportmodel and default hybrid model. It should be notedthat the position of the peak in the K + /π + ratiodepends on the core fraction and only coincides for thenew core-corona approach with the available data. Forthe default hybrid and transport model calculations,the peak appears at lower energies. As a side remarklet us state that in the usual transport simulationsthe position of the peak is solely determined by thetransition from a baryon to meson dominated system,while for the present core-corona approach the slowonset of strangeness equilibration plays the driving role. + :
5) of cen-tral ( b < . The beam energy dependence of strange baryon topion ratios is shown in figure 5, where the verticaldashed lines again indicate the positions of the peaksin the K + /π + ratio. The description of the Λ /π isagain improved, when one assumes that only a part ofthe system is fully equilibrated. Fore the Ξ − /π ratio(lower plot in figure 5) the default UrQMD calculationdrastically underpredicts the production rate of themultistrange baryon. Even the default hybrid modelresult seems to underestimate the data slightly and thisratio even decreases in our new core-corona approach.However, the effect is smaller than for the single strangehadrons. For both strange baryon ratios the peakposition is found to be independent of the applied modelparametrization.Next we turn to the investigation of average particleflow. Even more than particle yields, their momenta and N W b [fm] Pb+Pb n =30 mb FIG. 7. (Color online) The dependence of the number ofwounded nucleons N W as a function of the impact parameterfor collisions of heavy ions at E lab = 40 A GeV. The red lineshows the result from a Monte Carlo Glauber simulation, as-suming an inelastic nucleon-nucleon cross section of σ n = 30mb. The points depict the results from the hybrid modelcalculation on an event-by-event basis. collective motion could depend on the degree of equi-libration in the system. Figure 6 shows the excitationfunctions of the mean transverse masses of pions andkaons compared to data [80, 81, 85]. For the positivelycharged pions we observe almost no dependence on theparametrization of the core-corona separation, in the hy-brid model. The K + excitation function (shown in thelower part of figure 6) shows small differences at low beamenergies. Here the mean transverse mass is increased inthe non-equilibrium transport approach as compared tothe hybrid model calculation. Interestingly, the descrip-tion of the π + data at high energies is better in the defaultUrQMD approach, while for the K + it is better in thehybrid model.In [87] it has been pointed out, that the surplusof low momentum pions in the data as compared tothe hybrid model calculations, which would lead to adecrease in the K + /π + and the mean m T of pions,can be contributed to heavy resonance contributions aswell as non-equilibrium corrections to the hydrodynamicphase. V. CENTRALITY DEPENDENCE
Instead of varying the temperature and density bya beam energy scan, such a variation could also beachieved by changing the centrality of the collision. Inour calculation this can be done by changing the impactparameter b. In experiment, as the determination ofthe impact parameter is usually not straight forward,one usually gives observables as function of the number E lab = 40 A GeV 160 A GeV f l u i d f r a c t i on [ % ] b [fm] FIG. 8. (Color online) Fraction of the total energy of thesystem which is transferred into the hydrodynamic phase asa function of the impact parameter b, for collisions of Pbnuclei. The red lines with circles depict the results for E lab =40 A GeV and the black lines with squares for E lab = 160 A GeV. We compare results for both choices of the core densityparameter ρ q = 4 (solid lines) and 5 ρ q (dashed lines). Ifa cut in pseudo rapidity as applied for the calculations at E lab = 160 A GeV we obtain the green dash dotted line asa result for ρ q = 5 ρ q . The error bars indicate the meandeviation of the fluid fraction on an event by event basis. of wounded nucleons. This is the number of nucleonswhich have undergone a primary binary collision andtheir energy can therefore contribute to the fireballstotal energy.In the transport and hybrid model calculations, thedefinition of the number of wounded nucleons is byno means trivial. Late time secondary interactions,which would only excite the spectator fragment, canremove spectator nucleons from the fragment leadingto an overestimation of N W . We therefore define thenumber of wounded nucleons in our model calculations,as the number of nucleons which have not interacteduntil the time t start (see definition above). In thisway one obtains a dependence of N W on the impactparameter b which is in agreement with the estimateof a Monte-Carlo Glauber model calculation [88, 89](see figure 7). Such a Glauber model is often used toestimate N W from experimental data. Therefore wecan compare our results with experiment, using ourdefinition of N W without invoking the real experimentaltrigger conditions.Let us start with an investigation of the fluid fractionas a function of the impact parameter. Figure 8 displaysthe fluid fraction for two different energies as a functionof the impact parameter b . For each energy the resultsfor the two different cut off densities, where the solidlines correspond to ρ q = 5 ρ q and the dashed lines to ρ q = 4 ρ q . The green dash dotted line shows the effect of /N W |y|<0.5 + /N W |y|<0.5 N W FIG. 9. (Color online) Centrality dependence of the number ofΛ’s (upper plot) and pions (lower plot) per wounded nucleon,produced in heavy ion collisions at E lab = 40 A GeV, com-pared to data. Shown is the yield at mid rapidity | y | < . ρ q = 4 (triangles) and5 ρ q (circles). the cut in pseudorapidty in the definition of the densityfor E lab = 160 A GeV.For both energies one observes a dependence of the fluidfraction on the input parameter. While at E lab = 40 A GeV (red lines with circles) this dependence is ratherstrong, it is rather weak for E lab = 160 A GeV (blacklines with squares). If a cut in η for the calculation ofthe local density is applied, the impact parameter depen-dence becomes much stronger for th highest SPS energy(green dash-dotted line).In the last part of this paper we will discuss the re-sults on the centrality dependence of different hadronicobservables and compare them to data [79, 90–93](the centrality selection with regard to the numberon wounded nucleons is taken from [94]). Figure 9 /N W /N W N W FIG. 10. (Color online) Centrality dependence of the numberof Λ’s per wounded nucleon produced in heavy ion collisions at E lab = 40 A GeV (upper plot) and E lab = 160 A GeV (lowerplot), compared to data. Shown is the 4 π integrated yield.The line styles are as in the previous figure. If a cut in pseudorapidity as applied for the calculations at E lab = 160 A GeVwe obtain the green dash dotted line as a result for ρ q = 5 ρ q . displays the mid-rapidity ( | y | < .
5) yields of pions andlambdas divided by the number of wounded nucleons, asa function of N W . Here we the analysis is restricted toresults for E lab = 40 A GeV due to stronger dependenceof the fluid fraction on the centrality at this energy. Thedifferent lines depict the results for the default UrQMDmodel in its cascade mode (grey line), the defaulthybrid model (black dashed line with square symbols)and the different parametrizations of the core-coronaseparated hybrid model (red lines with symbols). Forthe most central collisions all models reproduce the dataequally well, while the centrality dependence can onlybe captured by the hybrid model, and especially thecore-corona separated versions.The pion yield on the other hand shows only a weaksensitivity on the presence of a corona and is wellreproduced with the UrQMD cascade version. v E lab =160 A GeV E lab =40 A GeV v N W FIG. 11. (Color online) Centrality dependence of the chargedparticle elliptic flow at mid rapidity | y | < . E lab = 40 (lower plot) and 160 A GeV(upper plot). The line styles are as in the previous figure.
While figure 9 only displays the results at E lab = 40 A GeV, figure 10 shows a comparison of the centrality de-pendence of the Λ multiplicity for E lab = 40 and 160 A GeV. For the lower energy the core-corona separated ver-sion again gives the best result. While at the highest SPSenergy, all hybrid model results show almost no central-ity dependence, which is in contrast to experimental data.it seems that our definition of the corona begins to failwhen the beam energy becomes so large that the pro-duced system can be regarded as being boost invariant.To obtain a rapidity independent density, we apply anadditional cut in η , as described above, for the calcula-tion of ρ q . This way one effectively reduces the systemto 2 dimensions and one obtains a result which is com-parable to that of the mere geometrical picture proposedin [63, 75–78].Figure 11 summarizes our results on the centralitydependence of the elliptic flow of charged pions, with v = h cos [2( φ − Φ R )] i , where Φ R denotes the reaction plane. Again both beam energies E lab = 40 and 160 A GeV are depicted. A general challenge for the experi-mental determination of v is the correct determinationof the reaction plane which sets the coordinate systemin which the elliptic flow is defined. The experimentalsystematic uncertainty is reflected in large error barsand is especially severe at the lower energies. Never-theless our calculations show that the value of v ishardly sensitive on the approach which is used for thecalculations. All results, from the default UrQMD andthe hybrid model, essentially give the same centralitydependence of v at 40 A GeV. At the highest SPSbeam energy the picture is already different. Here thedefault UrQMD results underestimate the elliptic flow,while the default hybrid model overestimates it. Again,the core-corona separated version of the hybrid modelimproves the description.
VI. CONCLUSION
We presented a method to explore the dynamics of thesystem produced in high energy heavy ion collisions andto effectively divide it into an equilibrated core and adilute corona. To this aim the UrQMD hybrid approachwas applied, where the dense and equilibrated core isdescribed hydrodynamically and the dilute corona bythe non-equilibrium transport approach. To distinguishbetween the two separated regions we employed a localparticle density criterion.It was found that the fraction of the system which canbe regarded as being in local thermal and chemicalequilibrium slowly increases in the energy range between E lab = 5 − A GeV. While observables of non-strangehadrons appeared to be insensitive on this separation,strange hadron properties showed considerable modifi-cations. Strange particle yields as well as their radialflow and especially the description of the ’horn’ in the K + /π + ratio were improved. From this point of viewwe explained the drastic increase in the K + /π + upto E lab = 20 − A GeV with an onset of chemicalequilibration of strangeness. In thermal models whichalso explain the steep increase in strangeness production[95–98] canonical corrections or the introduction of astrangeness saturation parameter, are usually responsi-ble for the suppression of strangeness at low energies.If the rapid equilibration of strangeness is causedby a change in the properties of the active degrees offreedom, present in the initial state of the collision,thenthe present study suggests an onset of such a new phasein reactions at beam energies of E lab ≈ − A GeV. Infact the change of the properties of the Matter involvedwould not change suddenly at some specific beam energybut rather continuously over a wide range of collisionenergies.In the second part of this paper we discussed thecentrality dependence of different hadronic observableswithin the core-corona separated approach. Again thedescription of strange hadron observables is improved inthe core-corona version, when compared to the defaulthybrid model. On the other hand the hybrid model givesno good description the excitation function of the meantransverse mass of pions and centrality dependence ofpion multiplicities, independent of the core-corona sepa-ration. For our model we apply an ideal fluid dynamicaldescription without viscous corrections. This is a possibleorigin for both observations, as viscosity leads to entropyproduction which is directly related to the pion produc-tion rate. Furthermore it can account for a decrease inthe average flow in the hydrodynamic phase. Anothercontribution to the too small pion yield, as well as thetoo large average momentum are missing contributionsfrom high mass (mesonic) resonances, decaying predom-inantly into pions, which are not explicitly included inthe Cooper-Frye transition and the transport model (aswell as in the UrQMD model).At the highest SPS energy we only obtain a very mod-erate centrality dependence of the core fraction of thesystem. This is somewhat in contrast to studies wherethe core fraction is calculated only on a 2 dimensional projection of the system on the transverse plane, indicat-ing that the system produced at E lab = 160 A GeV seemsto have a rapidity independent density. As the Lorentzcontraction of the incoming nuclei is also rather strong,a Lorentz invariant formulation of the density, as well asthe hydrodynamical equations seems more suitable. Toapproximate a rapidity independent density, we applieda cut in η for the definition of the local particle densityleading to a considerable improvement of the centralitydependence of strange particle yields which supports adefinition of the core as proposed in [63, 75], at leastat energies above E lab ≈ A GeV. However, at lowerenergies a full 3 dimensional evaluation of the createdsystem is in order.
ACKNOWLEDGMENTS
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