A generalized Buda-Lund model M. Csanád, T. Csörgő and B. Lörstad (Budapest & Lund) Buda-Lund model for ellipsoidally symmetric systems and it’s comparison to the experimental results The Buda-Lund model The elliptic flow and it’s measurements Generalization of the model Measurable quantities, limiting cases Fits to the data

The Buda-Lund model Describes a locally thermalized expanding system Successful in describing single particle spectra and correlations The original model was developed for axial symmetry only central collisions Worked for averaged data, too RHIC published position-dependent data, for example for the elliptic flow

The elliptic flow One-particle spectrum: The n-th Fourier component is the n-th flow Has (pseudo)rapidity and transverse momentum dependence These functions are measured at RHIC and SPS

Elliptic flow measurements RHIC STAR 130 GeV, nucl-ex/ RHIC PHENIX 200 GeV nucl-ex/ RHIC STAR 200 GeV, nucl-ex/ RHIC PHENIX 130 GeV nucl-ex/

Elliptic flow measurements RHIC PHOBOS, nucl-ex/ CERN NA49, nucl-ex/

Ellipsoidal symmetric case At heavy ion collisions, an expanding ellipsoid arises from the hot zone of the collision We want to describe this ellipsoidal symmetry More generalization: use of relativistic hydrodynamics (recently found solution by T. Csörgő and Y. Hama gives the emission function) The ellipsoid has an angular momentum, it may rotate, so a coordinate-transformation is necessary The moment of inertia is increasing fast, rotation stops, but a tilt angle remains (e.g Heinz, Lisa, T. Cs. with Sinyukov et al, …)

The coordinate-transformation We start from the direction of transverse momentum First, we rotate the coordinate system with  around the longitudinal axis After that, we rotate around the new y axis, with  So, we are now in the SEE System of ellipsoidal expansion (SEE): (x,y,z) System of measurable quantities: (out or transverse, side, long)

The generalized Buda-Lund model Core-halo model with In the most general hydrodynamical form: Generalized Cooper-Frye prefactor: with Boltzmann-factor for locally thermalized system:

The generalized Buda-Lund model Look for good scaling variable s and four- velocity distribution: We use From the hydro solution: So we introduce: We utilize some ansatz for the distributions: and

The saddlepoint approximation A good approximation for the product of a narrow Gaussian and a broad factor:, Exact for convolution of Gaussians, good for narrow distributions, where a parameter controls the width The saddlepoint can be computed from This method can be generalized for more dimensions

The invariant momentum distribution With the saddlepoint method, we get:, with Here, we make the coordinate transformation! This, we have to integrate, and we get the elliptic flow

Limiting cases There are some important limiting cases: 1.The saddlepoint is approx. at 0: 2.The former limit is true just for i=x,y 3.Nonrelativistic limit With these approximations, we get some formulas analytically to examine: For the distribution width (with approx. 1.):, the HBT radii are dominated by the smaller of the geometrical and the thermal radii

Limiting cases Some other formulas analytically to examine: For the effective temperatures, with approx. 3:, i=x,y,z For the slope parameter: These can be fitted not just for separate particles, but for separate momenta, too!

Limiting cases In the limiting case, where the first two saddle- point coordinates are small, we can compute the elliptic flow analytically, too:, introducing and, and so This implies, that with growing rapidity, the asymmetry vanishes, as the saddlepoint goes to the z axis, hence elliptic flow vanishes

Fit I. The parameters: YZX

Fit II. 

Results and conclusions We fitted the elliptic flow with the generalized Buda-Lund model Successful describing of the RHIC data, this means: the generalization works Another indication for deconfinement, as the temperature is above the critical in approx. 1/8 of the ellipsoid, this has a volume of Presence of a 3 dimensional Hubble flow

Next to do Try fits with the minuit fitting package, to get values for the parameters with errors Look for other data (NA49, RHIC run3, run4) Fit the correlation function and HBT radii to the data, too