We use a hydrodynamic model to study the space-time evolution transverse to
the beam direction in ultrarelativistic heavy-ion collisions with nonzero
impact parameters. We focus on the influence of early pressure on the
development of radial and elliptic flow. We show that at high energies elliptic
flow is generated only during the initial stages of the expansion while radial
flow continues to grow until freeze-out. Quantitative comparisons with SPS data
from semiperipheral Pb+Pb collisions suggest the applicability of
hydrodynamical concepts already $\approx$ 1 fm/c after impact.Comment: 4 pages, 5 figures, proceedings for Quark Matter 9

Heinz, U.

Kolb, P. F.

Ruuskanen, P. V.

Sollfrank, J.

English

arXiv

We use a hydrodynamic model to study the space-time evolution transverse to the beam direction in ultrarelativistic heavy-ion collisions with nonzero impact parameters. We focus on the influence of early pressure on the development of radial and elliptic flow. We show that at high energies elliptic flow is generated only during the initial stages of the expansion while radial flow continues to grow until freeze-out. Quantitative comparisons with SPS data from semiperipheral Pb+Pb collisions suggest the applicability of hydrodynamical concepts already $\approx$ 1 fm/c after impact.We use a hydrodynamic model to study the space-time evolution transverse to the beam direction in ultrarelativistic heavy-ion collisions with nonzero impact parameters. We focus on the influence of early pressure on the development of radial and elliptic flow. We show that at high energies elliptic flow is generated only during the initial stages of the expansion while radial flow continues to grow until freeze-out. Quantitative comparisons with SPS data from semiperipheral Pb+Pb collisions suggest the applicability of hydrodynamical concepts already $\approx$ 1 fm/c after impact.We use a hydrodynamic model to study the space-time evolution transverse to the beam direction in ultrarelativistic heavy-ion collisions with nonzero impact parameters. We focus on the influence of early pressure on the development of radial and elliptic flow. We show that at high energies elliptic flow is generated only during the initial stages of the expansion while radial flow continues to grow until freeze-out. Quantitative comparisons with SPS data from semiperipheral Pb+Pb collisions suggest the applicability of hydrodynamical concepts already ≈ 1 fm/c after impact

Kolb, P.F.

Ruuskanen, P.V.

Heinz, Ulrich W.

CERN Document Server

1Hydrodynamic simulation of elliptic flow∗P.F. Kolba,b, J. Sollfrankb, P.V. Ruuskanenc, and U. HeinzaaCERN/TH, CH-1211 Geneva 23bInstitut fu¨r Theoretische Physik, Universita¨t Regensburg, D-93040 RegensburgcDepartment of Physics, University of Jyva¨skyla¨, FIN-40351 Jyva¨skyla¨We use a hydrodynamic model to study the space-time evolution transverse to thebeam direction in ultrarelativistic heavy-ion collisions with nonzero impact parameters.We focus on the influence of early pressure on the development of radial and elliptic flow.We show that at high energies elliptic flow is generated only during the initial stages ofthe expansion while radial flow continues to grow until freeze-out. Quantitative compar-isons with SPS data from semiperipheral Pb+Pb collisions suggest the applicability ofhydrodynamical concepts already ≈ 1 fm/c after impact.1. Hydrodynamic model with longitudinal boost invarianceThe transverse expansion dynamics in non-central heavy-ion collisions at SPS energieshas recently attracted much attention [1{7]. We here study it within the hydrodynamicmodel. In order to reduce the complexity of the numerical task we follow [4] and implementanalytically Bjorken scaling flow with vz = z=t in the longitudinal direction and onlysolve the transverse dynamics numerically. The Bjorken ansatz holds exactly at innitebeam energy, but properly restricted to a nite rapidity interval it is phenomenologicallysuccessful also at SPS and AGS energies [8]. It breaks down, however, near target andprojectile rapidities; using it we can therefore reliably compute the transverse expansiononly near midrapidity.The system of hydrodynamic equations is closed by an equation of state (EOS) p(e; n)giving the pressure as a function of energy and baryon density. Hydrodynamics thusprovides a direct relation between the EOS and the dynamical evolution of the system.To study the dynamical eects of a softening of the EOS in the neighborhood of a phasetransition to quark-gluon plasma we use three dierent equations of state. EOS I is thehard equation of an ideal ultrarelativistic gas, p = e=3. EOS H is the much softer EOSfor a gas of interacting hadron resonances; for n ≈ 0 it satises p ≈ 0:15 e . A Maxwellconstruction between these two EOS, adding a bag pressure B1/4 = 230 MeV, results inEOS Q which has a phase transition at Tcr(n = 0) = 164 MeV with a latent heat of 1.15GeV/fm3 [9]. The system is frozen out at a xed decoupling temperature Tdec, and allunstable resonances are allowed to decay before we compare with experimental data.∗This work was supported by BMBF, DFG and GSI.22. Space time evolution of the reaction zoneWe initialize the reaction zone with transverse energy and baryon density proles whichare taken to be proportional to the transverse density of wounded nucleons calculatedfrom the Glauber model [4,10]. The initial conguration is thus parametrized by theequilibration time 0 and the maximum energy and baryon densities e0 and n0 in central0 5 10 15−0.100.10.20.3τ (fm/c)ε xEOS IEOS HEOS Qb = 8 fm0 5 10 1500.050.10.150.2τ (fm/c)ε pEOS IEOS HEOS Qb = 8 fm0 5 10 1500.20.40.60.81〈 〈 v ⊥ 〉 〉EOS IEOS HEOS Qb = 8 fmτ (fm/c)Figure 1. Time evolution of thespatial eccentricity x (top), themomentum anisotropy p (middle),and the radial flow 〈〈v?〉〉 (bottom).collisions. For each EOS these parameters and thedecoupling temperature are xed by a t [10] tothe negative hadron and net proton mt-spectra atmidrapidity from central Pb + Pb collisions at 158AGeV [11]. The spectra for non-central collisionsare then predicted without extra parameters.The hydrodynamic evolution provides the time-dependence of the matter in coordinate and mo-mentum space. In non-central collisions, the ini-tial spatial deformation of the reaction zone inthe transverse plane, characterized by its spatialeccentricity x=hhy2−x2iihhy2+x2ii , leads to anisotropic pres-sure gradients and a preferred buildup of trans-verse flow in the shorter x-direction [4]. (x liesinside, y points orthogonal to the collision plane.〈〈: : :〉〉 denotes the energy density weighted spatialaverage at xed time.) This leads to a growingflow anisotropy, characterized by p=hhTxx−T yyiihhTxx+T yyii . Atfreeze-out this hydrodynamic quantity is directlyrelated to the elliptic flow v2 = 〈cos(2’)〉, denedby an average over the nal particle momentumspectrum; for pions p≈2 v2 [7]. In contrast to v2,p can be studied as a function of time and givesaccess to the buildup of elliptic flow.The developing stronger flow into the collisionplane leads to a decrease of x with time; thebuildup of elliptic flow thus slows down and even-tually shuts itself off. This is clearly seen in theupper two panels of Fig. 1: p saturates when xpasses through zero. For a hard EOS this happensfaster than for a soft one; also, the total amountof elliptic flow which can be generated by a givenEOS increases with its hardness c2s =∂e∂p.Fig. 1 was computed for 158 A GeV Pb+Pb collisions at b = 8 fm. For EOS Q thevertical lines indicate when the center of the reaction zone goes from plasma to mixedphase and from mixed to hadron phase, respectively. One sees that 1/6 of the nal ellipticflow is generated before the pure QGP phase disappears, 1/2 in the mixed phase, andabout 1/3 in the hadronic phase. The stars indicate the freeze-out point (Tdec = 120 MeV).3For EOS I the system freezes out before the elliptic flow is fully developed; for EOS H andEOS Q the opposite is true. The radial flow, characterized by 〈〈v?〉〉=〈〈γ√v2x + v2y 〉〉/〈〈γ〉〉(where γ is the Lorentz factor), does not saturate: Fig. 1 shows that it continues togrow monotonously until freeze-out, even after the elliptic flow has saturated, due to thecontinued presence of essentially azimuthally symmetric radial pressure gradients.Note the important role of the EOS: a softer EOS, especially its softening near a phasetransition, delays the buildup of both radial and elliptic flow. It also reduces the maximallyachievable value of the latter. At low energies, freeze-out (driven by cooling and expansiondue to radial flow) happens before the elliptic flow has fully developed. To achieve fullydeveloped elliptic flow lower beam energies are required for softer equations of state [7,10].This reflects both the lower saturation value of p for the softer EOS and the slower buildupof radial flow, resulting in more available time until freeze-out.3. Transverse mass spectra and applicability of hydrodynamicsCollective flow aects the measurable momentum spectra of the produced particles. Weshowed in [7] that our model is able to describe the measured asymmetries of the parti-cle spectra; the calculated elliptic flow v2=〈cos(2’)〉 at midrapidity agrees well with thepublished data [1]. Here we discuss the impact parameter dependence of the azimuthally0 0.5 1 1.5 2 2.5 310−1210−1010−810−610−410−2100102104mT−m0 in GeV1/mT d2N/dydmT in GeV−20−1%1−5%5−10%10−20%20−40%40−60%60−80%80−100%σ/σgeopi02.35 < y < 2.95WA98, preliminary dataFigure 2. Preliminary 0 transverse massspectra from Pb+Pb collisions of varyingcentrality measured by WA98 [2]. The linesshow our hydro results.0 0.5 1 1.5 210−810−710−610−510−410−310−210−1100101102mT−m0 in GeV1/mT d2N/dydmT in GeV−20−2.3%2.3−5%5−7.7%7.7−11%11−15%15−22%22−35%σ/σgeo2.0 < y < 2.1net protonsCERES dataFigure 3. Same as Fig. 2, but for net pro-tons measured by the CERES Collaboration[3]. Note the dierent impact parameterbinning in Figs. 2 and 3.integrated transverse mass spectra and show that, once tuned to central collision data,hydrodynamics successfully reproduces the magnitude and shape of the spectra up toimpact parameters of about 8-10 fm. Figs. 2 and 3 show preliminary data on transversemass spectra of neutral pions (WA98 Collaboration [2]) and net protons (CERES Collab-oration [3]) from 158 A GeV Pb+Pb collisions of varying centrality. The lines indicate4our hydrodynamical results at midrapidity, obtained with EOS Q and initial conditionstuned to central collisions as described above. The WA98 data extend to very peripheralcollisions: the lowest spectrum in Fig. 2 corresponds to b=13 fm where hydrodynamicscertainly looses its applicability. At such large impact parameters one does not observethe collision of two nuclei, but rather two dilute nucleon clouds penetrating each other.At smaller impact parameters the model fails in the high-mt region; here hard scatteringprocesses begin to dominate which cannot be modeled hydrodynamically. Up to b ≈ 10fm (bin 6 represents impact parameters up to 11 fm [2]) and transverse masses of about 2GeV, however, hydrodynamics works very well, both for the pion and net proton spectra.(For the latter the CERES data in Fig. 3 do not extend to very peripheral collisions, thelargest measured impact parameters corresponding to about 8.4 fm [3].)4. SummaryWe have demonstrated the interplay between spatial eccentricity as the driving forcefor generating momentum space asymmetries and the back-reaction of the latter on theformer. Comparison with measured spectra from Pb+Pb collisions with varying impactparameter showed that the hydrodynamical model successfully reproduces the data up tob=8-10 fm. The good quantitative agreement between data and model suggests ratherrapid thermalization in the reaction zone. If nal data conrm that the elliptic flow isessentially saturated in Pb+Pb collisions at the SPS, this would provide strong evidencefor very early pressure in the system. In our calculations a large fraction of the nally ob-served elliptic flow is generated while the energy density exceeds the critical value ecr = 1GeV/fm3 for deconnement. This conrms the suggestion [5] that elliptic flow is a probefor the early collision stage.We thank Th. Peitzmann (WA98) and F. Ceretto (CERES) for sending us their pre-liminary data prior to publication. P.K. wishes to express his gratitude to the CERNSummer Student Programme and thanks T. Peeter’s group for their warm hospitality.REFERENCES1. H. Appelsha¨user et al. (NA49 Coll.), Phys. Rev. Lett. 80 (1998) 41362. C. Blume, PhD thesis, Universita¨t Mu¨nster, 1998; M.M. Aggarwal et al. (WA98 Col-laboration), Phys. Rev. Lett 81 (1998) 40873. F. Ceretto, PhD Thesis, Universita¨t Heidelberg, 1998; P. Braun-Munzinger, J. Stachel,Nucl. Phys. A 638 (1998) 3c4. J.Y. Ollitrault, Phys. Rev. D 46 (1992) 2295. H. Sorge, Phys. Lett. B 402 (1997) 251; Phys. Rev. Lett. 82 (1999) 20486. D. Teaney, E. Shuryak, nucl-th/99040067. P.F. Kolb, J. Sollfrank, U. Heinz, nucl-th/9906003, Phys. Lett. B, in press8. H. Dobler, J. Sollfrank, U. Heinz, nucl-th/9904018, Phys. Lett. B, in press9. J. Sollfrank et al., Phys. Rev. C 55 (1997) 39210. P.F. Kolb, Diploma Thesis, Universita¨t Regensburg, 1999; P.F. Kolb, J. Sollfrank,U. Heinz, in preparation11. H. Appelsha¨user et al. (NA49 Coll.), Phys. Rev. Lett. 82 (1999) 2471

Hydrodynamic simulation of elliptic flow

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