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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)
Björn Bäuchle, ITP Ffm & UiB MACE 07-04-17: NCRH2007 3/17
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
Björn Bäuchle, ITP Ffm & UiB MACE 07-04-17: NCRH2007 4/17
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
Björn Bäuchle, ITP Ffm & UiB MACE 07-04-17: NCRH2007 5/17
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
Björn Bäuchle, ITP Ffm & UiB MACE 07-04-17: NCRH2007 6/17
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
Björn Bäuchle, ITP Ffm & UiB MACE 07-04-17: NCRH2007 7/17
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.
Björn Bäuchle, ITP Ffm & UiB MACE 07-04-17: NCRH2007 8/17
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
Björn Bäuchle, ITP Ffm & UiB MACE 07-04-17: NCRH2007 9/17
-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
Björn Bäuchle, ITP Ffm & UiB MACE 07-04-17: NCRH2007 10/17
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?
Björn Bäuchle, ITP Ffm & UiB MACE 07-04-17: NCRH2007 11/17
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
Björn Bäuchle, ITP Ffm & UiB MACE 07-04-17: NCRH2007 12/17
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.
Björn Bäuchle, ITP Ffm & UiB MACE 07-04-17: NCRH2007 13/17
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!
Björn Bäuchle, ITP Ffm & UiB MACE 07-04-17: NCRH2007 14/17
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
!
Björn Bäuchle, ITP Ffm & UiB MACE 07-04-17: NCRH2007 15/17
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?
Björn Bäuchle, ITP Ffm & UiB MACE 07-04-17: NCRH2007 16/17
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
Björn Bäuchle, ITP Ffm & UiB MACE 07-04-17: NCRH2007 17/17
Thank you very much
�for your attention�
�
Vielen Dank für Ihre Aufmerksamkeit
Björn Bäuchle, ITP Ffm & UiB MACE 07-04-17: NCRH2007 19/17
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
Björn Bäuchle, ITP Ffm & UiB MACE 07-04-17: NCRH2007 4/17
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
This is the hidden picture:
Pic: Österreichische Nationalbibliothek (via en.wikipedia.org)