# Presentation: 816303

## New exact solutions of perfectfluid hydrodynamics and an advanced estimate of theinitial energy density and thelife-time of the reaction

- Tamás Csörgő 1, _**Márton Nagy**_ 2 ** **and Máté Csanád 2

- 1 MTA KFKI RMKI, Budapest, Hungary
- 2 Eötvös University, Budapest, Hungary

- Accepted by Braz. J. Phys (Proceedings of ISMD 2007)

- NCRH2007, Frankfurt a. M. , Germany

- 16/04/2007

##   Introduction, motivation

**Contents**

-   The equations of relativistic hdrodynamics

-   Review of known solutions

-   Spherically symmetric flows

-   New simple solutions in 1+D dimensions

-   New simple solutions in 1+1 dimensions

-   Rapidity distribution

-   Advanced energy density and life-time estimation

-   Summary and outlook

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## Introduction

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## Introduction

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## The equations of relativistic hydrodynamics

**The equations of relativistic hydrodynamics**

**For perfect fluid:**

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## The equations of relativistic hydrodynamics

**EoS mean additional constraint**

**The equations of relativistic hydrodynamics**

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## Landau-Khalatnikov solution

- L.D. Landau, Izv. Acad. Nauk SSSR 81 (1953) 51
- I.M. Khalatnikov, Zhur. Eksp.Teor.Fiz. 27 (1954) 529
- L.D.Landau & S.Z.Belenkij, Usp. Fiz. Nauk 56 (1955) 309

- Basic relations:

- Unknown variables:

- Auxiliary function:

- Expression of                    is very complicated

**Landau-Khalatnikov solution**

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## Landau-Khalatnikov solution

**Landau-Khalatnikov solution**

- Animation (courtesy of T. Kodama)

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## Hwa-Bjorken solution

- R.C. Hwa, Phys. Rev. D10, 2260 (1974)
- J.D. Bjorken, Phys. Rev. D27, 40(1983)

**Hwa-Bjorken solution**

- _Euler equation satisfies_

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## Hwa-Bjorken solution

**Hwa-Bjorken solution**

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## Self-similar, ellipsoidal solutions

- Spherically symmetric, accelerationless velocity profile:

- // inside the lightcone //

- Let

- The A = const. surfaces: self-similarly expanding ellipsoids

- Scaling function

**Self-similar, ellipsoidal solutions**

- Heavy Ion Phys. A 21 (2004) 73

- EoS: (massive) ideal gas

- _Euler equation satisfies_

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## Self-similar, ellipsoidal solutions

- If                  , then                 is free:

- Buda-Lund type of solution

**Need for explicit simple accelerating solutions**

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## Spherically symmetric flows

**Spherically symmetric flows**

- Scaling function:

- For the continuity equations:

- Solutions:

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## Spherically symmetric flows

**Spherically symmetric flows**

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## Spherically symmetric flows

**Spherically symmetric flows**

**We need EoS to specify a solution**

- Simple choice: massless ideal gas

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## Spherically symmetric flows

**Spherically symmetric flows**

- A final equation:

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## Spherically symmetric flows / New solutions

**Spherically symmetric flows / New solutions**

- Some non-trivial solutions:

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## New simple solutions in 1+D dimensions

**New simple solutions in 1+D dimensions**

- The first new solution:

- If including temperature & conserved particle number:

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## New simple solutions in 1+D dimensions

**New simple solutions in 1+D dimensions**

- Fluid trajectories of the 1+D dimenisonal new solution

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## New simple solutions in 1+D dimensions

**New simple solutions in 1+D dimensions**

- The                       lines (black) and the                     lines (red)

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## New simple solutions in 1+1 dimensions

**New simple solutions in 1+1 dimensions**

- Possible cases (one row of the tabular is one solution):

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## New simple solutions in 1+1 dimensions

**New simple solutions in 1+1 dimensions**

- Animation of the temperature profile in the                     case

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## Rapidity distribution

**Rapidity distribution**

- Analytic formula is available using saddle-point integration:

- Realistic, finite widths (or local minimum at mid–rapidity)

- Gaussian ,,width’’:

- Specifying freeze-out conditions, numerical calculation is straightforward

- Data analysis suggests peak at

- For the       parameter it means

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## Rapidity distribution

**Rapidity distribution**

- Rapidity distribution from the 1+1 dimensional solution, for            .

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## Rapidity distribution

**Rapidity distribution**

- Rapidity distribution from the 1+1 dimensional solution, for            .

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## Rapidity distribution

**Rapidity distribution**

- BRAHMS data fitted with the analytic formula of pseudorapidity distribution

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## Rapidity distribution

**Rapidity distribution**

- BRAHMS data fitted with the analytic formula of rapidity distribution

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## Advanced energy density estimation

**Advanced energy density estimation**

- Fit result:                  in                                     collisions at RHIC

- Flow is accelerating

- Correction terms: due to work and faster expansion

- Bjorken’s initial energy density estimation needs correction

- At RHIC energies the correction can be as high as a factor of 2!

- For                 (accelerating) flows, both factor is bigger than 1

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## Advanced energy density estimation

**Advanced energy density estimation**

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## Advanced life-time estimation

**Advanced life-time estimation**

- In our treatment: correction is because of acceleration:

- With this formula the life-time is underestimated, as pointed out by many people

- Life-time estimation: for Hwa-Bjorken type of flows

- A. Makhlin & Yu. Sinyukov, Z. Phys. C 39, 69 (1988)

- At RHIC energies: correction is about +20%

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##  Explicit simple accelerating solutions of relativistic hydrodynamics

-  Explicit simple accelerating solutions of relativistic hydrodynamics

**Further aims**

-  more general EoS

-  Realistic rapidity distributions; BRAHMS data well described

-  less symmetry, ellipsoidal solutions

-  asymptotically Hubble-like flows

**Summary and outlook**

-  Analytic (approximate) calculation of observables

-  Advanced estimate of initial energy density: up to a factor of 2 at RHIC energies

-  Advanced estimate of high-energy reaction lifetime: a correction by 20% at RHIC energies

- nucl-th/0605070, submitted to PRL

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## Thank you for your attention!

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