The Henryk Niewodniczański Institute of Nuclear Physics Polish Academy of Sciences Cracow, Poland Based on paper M.Ch., W. Florkowski nucl-th/ Characteristic form of 2+1 relativistic hydrodynamic equations Mikołaj Chojnacki Cracow School of Theoretical Physics May 27 - June 5, 2006, Zakopane, POLAND

2 Outline Angular asymmetry in non-central collisions  Angular asymmetry in non-central collisions Hydrodynamic equations  2+1 Hydrodynamic equations  Boundary and initial conditions  Results from hydrodynamics  Freeze-out hypersurface and v 2  Conclusions

3 Angular asymmetry in non-central collisions x y Space asymmetries transform to momentum space asymmetries Indirect proof that particle interactions take place

4 Equations of relativistic hydrodynamics Energy and momentum conservation law:  Energy and momentum conservation law: energy-momentum tensor  energy-momentum tensor  at midrapidity (y=0) for RHIC energies temperature is the only thermodynamic parameter  thermodynamic relations

5 System geometry Cylindrical coordinates ( r, ) Cylindrical coordinates ( r,  ) r vRvR vTvT v   x y z = 0 Boost – invariant symmetry Values of physical quantities at z ≠ 0 may be calculated by Lorentz transformation Lorentz factor :

6 Equations in covariant form Non-covariant notation Dyrek + Florkowski, Acta Phys. Polon. B15 (1984) 653

7 Temperature dependent sound velocity c s (T)  Relation between T and s needed to close the set of three equations. to close the set of three equations.  Potential Φ Lattice QCD model by Mohanty and Alam Phys. Rev. C68 (2003) T C = 170 [MeV]  Potential Φ dependent on T  Temperature T dependent on Φ inverse function of

8 Semifinal form of hydrodynamic equations in the transverse direction  auxiliary functions: transverse rapidity where

9 Generalization of 1+1 hydrodynamic equations by Baym, Friman, Blaizot, Soyeur, Czyz Nucl. Phys. A407 (1983) hydrodynamic equations reduce to case  angular isotropy in initial conditions  potential Φ independent of 

10 Observables as functions of a ± and   velocity  potential Φ  sound velocity  temperature  solutions

11 Boundary conditions  Automatically fulfilled boundary conditions at r = 0 r a ±,  a + (r, ,t) a - (r, ,t)  (r, ,t)  Single function a to describe a ±  Function  symmetrically extended to negative values of r extended to negative values of r a(r, ,t)  (-r, ,t)  Equal values at  = 0 and  = 2π  trtr,2,,0, 

12 Initial conditions - Temperature  Initial temperature is connected with the number of participating nucleons the number of participating nucleons Teaney,Lauret and Shuryak nucl-th/ x y AB b  Values of parameters

13 Initial conditions – velocity field  Isotropic Hubble-like flow  Final form of the a ± initial conditions

14 Results  Impact parameter b and centrality classes  hydrodynamic evolution initial timet 0 = 1 [fm] sound velocity based on Lattice QCD calculations  sound velocity based on Lattice QCD calculations  initial central temperatureT 0 = 2 T C = 340 [MeV] nitial flowH 0 = [fm -1 ]  initial flowH 0 = [fm -1 ]

15 Centrality class % b = 3.9 [fm]

16 Centrality class % b = 3.9 [fm]

17 Centrality class % b = 3.9 [fm]

18 Centrality class % b = 7.1 [fm]

19 Centrality class % b = 7.1 [fm]

20 Centrality class % b = 7.1 [fm]

21 Centrality class % b = 9.2 [fm]

22 Centrality class % b = 9.2 [fm]

23 Centrality class % b = 9.2 [fm]

24 Freeze-out  Cooper-Frye formula  Hydro initial parameters c S from Lattice QCD data c S from Lattice QCD data centrality: % centrality: % mean impact parameter mean impact parameter b = 7.6 fm b = 7.6 fm H 0 = fm -1 H 0 = fm -1 T 0 = 2.5 T C = 425 MeV T 0 = 2.5 T C = 425 MeV Freeze-out temperature  Freeze-out temperature T FO = 165 MeV T FO = 165 MeV

25 Freeze-out hypersurface

26 Azimuthal flow of Ω Data points from STAR for Ω + Ω Phys. Rev. Lett. 95 (2005) _

27 Conclusions  New and elegant approach to old problem: we have generalized the equations of 1+1 hydrodynamics to the case of angular asymmetry using the method of Baym et al. (this is possible for the crossover phase transition, recently suggested by the lattice simulations of QCD, only 2 equations in the extended r-space, automatically fulfilled boundary conditions at r=0)  Velocity field is developed that tends to transform the initial almond shape to a cylindrically symmetric shape. As expected, the magnitude of the flow is greater in the in-plane direction than in the out-of-plane direction. The direction of the flow changes in time and helps the system to restore a cylindrically symmetric shape.  For most peripheral collisions the flow changes the central hot region to a pumpkin-like form – as the system cools down this effect vanishes.  Edge of the system preserves the almond shape but the relative asymmetry is decreasing with time as the system grows.  Presented results may be used to calculate the particle spectra and the v 2 parameter when supplemented with the freeze-out model (THERMINATOR).