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MACE

�Mach-Cones-Evolution

�Numerical and Conceptual Issues in Relativistic Hydrodynamics� NCRH2007, ITP Frankfurt am Main, 2007-04-17.

Bjørn Bä[email protected]

Björn Bäuchle, ITP Ffm & UiB MACE 07-04-17: NCRH2007 2/17

Collaborators/Supporters � This work is supported by the German National

Academic Foundation and the International Office @ Uni Frankfurt.

� It is the major part of my diploma thesis which I write under supervision of�Horst Stöcker (Frankfurt)�Laszlo P. Csernai (Bergen)

� I use models by Csernai, Strottman and Magas (PIC � Hydro Code) and Miklos Zétényi (Freeze Out)


Outline� What are Mach-Cones?� What is their connection to HI-Physics?� MACE-model:

�How we model the jet�How we model sound waves�How we see the cone in the model�How we (try to) get particle spectra

� Summary


IntroductionWhat are Mach-Cones?

� Named after Austrian physicist Ernst Mach (1838-1916)

� Describes the wave front occuring behind a supersonic projectile

� Commonly known from supersonic airplanes

Pic: Österreichische Nationalbibliothek (via en.wikipedia.org)

αParticle emission

Machfrontcos�=

cs

v

U.S. Navy photo by Ensign John Gay


Introduction

� First predicted by Stöcker in 1975

� Developed further by Shuryak

� More works by Stöcker and Satarov, Stöcker & Mishustin in 2005 and many more

� Claimed to be seen at RHIC!

Mach-Cones in nuclear matter

STAR:PRL 93 (2004) 252301


MACE

� �Jet� is created:� random position within the extent of matter� random direction (as long as it ends up in the

detector)� Jet is considered

� to cause sound like perturbations� to be massless (i.e., moves with speed of light)� to have infinitely high energy� to move straight and with constant velocity

Setting up the reaction


MACEModell ing the perturbation

� Sound waves are modelled as perturbation on top of hydro

� Elementary sound-waves are represented by a (big) set of �wave markers� with random directions

� Only carry information about position and direction

� Move with speed of sound in the local rest frame of the matter.


MACE

� Sound has always the same speed w.r.t. the medium

� Wave markers must therefore have speed of sound Å flow velocity u

� Relativistic addition by adding rapidity vectors

� Of course, outside the medium there is no sound propagation.

Propagating the perturbations

�u �v

�yu= �u�12ln

1��u�1�u�

�yv= �v�12ln1�cS

1cS

�v '=�u�v


-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

-0.15-0.1-0.05 0 0.05 0.1 0.15

y

JetWave markers (odd)

Wave markers (even)

zx

y

-1.05-1

-0.95-0.9 0.75 0.8 0.85 0.9 0.95 1 1.05

-4.2

-4.15

-4.1

-4.05

-4

-3.95

-3.9

-3.85

-3.8

yJet

Wave markers (odd)Wave markers (even)

z

x

y

-1.05-1

-0.95-0.9 0.75 0.8 0.85 0.9 0.95 1 1.05

-4.2

-4.15

-4.1

-4.05

-4

-3.95

-3.9

-3.85

-3.8

yJet

Wave markers (odd)Wave markers (even)

z

x

y

PropagationStatic medium,jet started in origo

�real� medium (RHIC 65+65 AGeV), jet started at arbitrary point

t=6 dt t=6 dt

t=10 dt

x

z

y

x

zy

y

xz

JetWave markers started at odd timestepsWave markers started at even timesteps


Wave markers after 10 timesteps (static medium)

Mach-Cones!?

� Soundwave-propagation works fine BUT

� Soundwaves are spherical�no direction is

emphasized (or only through hydro flow)

� But we can see there is a cone!

So, how do we get Mach-Cones now?


Mach-Cones!?So, how do we get Mach-Cones now? (cont.)� Interference does not work:

�during (isocronic) FO only directions survive, not positions (no contradiction to experience with planes)

� taking the amplitude neglects directions� Only source of information is the shape of the

affected region


Finding the Surface� Surface is created by taking a

number of lines, �wave lines� that go along the outside.

� Wave lines all start at the position of the jet and go in different plains around the jet (cmp. longitudes on a sphere wrt to the axis)

� The wave lines finally represent the surface. Particle emission is perpendicular to them.


Wavelines in detail� For each wave line, only

a subset (an azimuthal slice) of all wave markers are relevant

� Consider only this slice:� Draw line along this slice� Do not draw along the

last elementary wave!� Easy in static medium� not easy in less-uniform

distributions of the markers!


Wavelines in more detail

� They should never go backwards

� They should not be too jagged�Artefact of too few

elementary waves�Artefact of too few

wave markers� Should not conceal

physical effects

What the wavelines should NOT look l ike:

Maximize angle between jet-axis and waveline

Allow for single elementary waves to be ignored

!


Particle flow

� Perpendicular to wave lines� Distance between sucessive

nodes is arbitrary � no good criterion for insertion of momentum

� Momentum inserted at regular intervals along the lines

� Perturbations taken into account when calculating measurables

How do we get spectra?


Summary� Elementary waves and jet are not propagated

together with the medium but as pertubations� Jet goes straight� Mach-cone is treated as collective phenomenon

that appears through the interplay of many elementary waves�not the cone is propagated

� Model is not complete yet (FO and Initial State),� sanity checks have yet to been done


Thank you very much

�for your attention�

�

Vielen Dank für Ihre Aufmerksamkeit


Backup Slides


Experience with PlanesHow do we experience supersonic booms?� Visually:

�See the shape of the machfront�use spatial information�NO information on direction!

� Audible:��FO� at constant place(s)�Evaluation of pressure

information time-dependent

back


IntroductionWhat are Mach-Cones?

� Named after Austrian physicist Ernst Mach (1838-1916)

� Describes the wave front occuring behind a supersonic projectile

� Commonly known from supersonic airplanes


αParticle emission

Machfrontcos�=

cs

v

U.S. Navy photo by Ensign John Gay

This is the hidden picture:


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