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New Approach to Supernova Simulations
W. Bauer, Hirschegg 2006
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3d
• Explosion energy 3foe
• texpl = 0.1 - 0.2 s
Fryer, Warren, ApJ 02
•Very preliminary
•Similar convection
as seen in their 2d
work
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Hydro Simulations
• Tough problem for hydro
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– Length scales vary drastically in time
– Multiple fluids
– Strongly time dependent viscosity
– Very large number of time steps
Special relativity, causality, …
Huge magnetic fields
3D simulations needed
– Giant grids
Need to couple all of this to radiation transport
calculation and Boltzmann transport problem
for neutrinos
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Simulations of Nuclear Collisions
• Hydro, mean field, cascades
• Numerical solution of transport theories
– Need to work
in 6d phase space => prohibitively large
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2
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–
–
–
–
–
–
grids (20 x40 x80~10 lattice sites)
Idea: Only follow initially occupied phase space cells in
time and represent them by test particles
One-body mean-field potentials (r, p, t) via local
averaging procedures
Test particles scatter with realistic cross sections =>
(exact) solution of Boltzmann equation (+Pauli, Bose)
Very small cross sections via perturbative approach
Coupled equations for many species no problem
Typically 100-1000 test particles/nucleon
1st
Developed
@ MSU/FFM
W. Bauer, Hirschegg 2006
G.F. Bertsch, H. Kruse und S. Das Gupta, PRC (1984)
H. Kruse, B.V. Jacak und H. Stöcker, PRL (1985)
W. Bauer, G.F. Bertsch, W. Cassing und U. Mosel, PRC (1986)
H. Stöcker und W. Greiner, PhysRep (1986)
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Transport Equations
f = phase
space
density
for
baryons
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Mean field EoS
2-body scattering
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Test Particles
• Baryon phase space function, f, is Wigner
transform of density matrix
• Approximate formally by sum of delta
functions, test particles
• Insert back into integral equation to obtain
equations of motion for 6 coordinates of each
test particle
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Test Particle Equations of
Motion
Coulomb
Nuclear EoS
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Two-body scattering
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Example
• Density in
QuickTime™ and a Sorenson Video 3 decompressor are needed to see this picture.
W. Bauer, Hirschegg 2006
reaction
plane
• Integration
over
momentum
space
• Cut for
z=0+-0.5 fm
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Momentum Space
• Output quantities (not input!)
• Momentum space information on
– Thermalization & equilibration
– Flow
– Particle
production
• Shown here:
local
temperature
W. Bauer, Hirschegg 2006
QuickTime™ and a Sorenson Video 3 decompressor are needed to see this picture.
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Reproduces Experiments
 Production
Pion spectra
Disappearance
Baryon flow
of flow
Dependence on
mass
2-particle
interferometry
Pion transparency
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Try this for Supernovae!
• 2 M in iron core = 2x1057 baryons
• 107 test particles => 2x1050 baryons/test
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•
•
•
particle 
Need time-varying grid for (non-gravity)
potentials, because whole system collapses
Need to think about internal excitation of test
particles
Can create n-test particles and give them finite
mean free path => Boltzmann solution for ntransport problem
Can address angular momentum question
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Initial Conditions for Core
Collapse
Iron Core
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Woosley, Weaver 86
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Equation of
State
• Low density:
– Degenerate e-gas
• High density
– Dominated by
nuclear EoS
– Isospin term in
nuclear EoS
becomes dominant
• For now:
• High density neutron rich EoS can be explored
by GSI upgrade and/or RIA
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Electron Fraction, Ye
• Strongly density dependent
• Neutrino cooling
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Internal Heating of Test
Particles
• Test particles
represent mass
of order Mearth.
• Internal excitation
of test particles
becomes
important for
energy balance
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Neutrinos
• Neutrinos similar to pions at RHIC
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•
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•
– Not present in entrance channel
– Produced in very large numbers (RHIC: 103, here 1056)
– Essential for reaction dynamics
Different: No formation time or off -shell effects
Represent 10N neutrinos by one test particle
– Populate initial neutrino phase space uniformly
– Sample test particle momenta from a thermal distribution
Neutrino test particles represent “2nd fluid”, do NOT
escape freely (neutrino trapping), and need to be
followed in time.
Neutrinos created in center and are “light” fluid on
which “heavy” baryon fluid descends
– Inversion problem
– Rayleigh-Taylor instability
– turbulence
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Neutrino Test particles
• Move on straight lines (no mean field)
• Scattering with hadrons
– NOT negligible!
– Convolution over all sAnA2 (weak neutral current)
– Resulting test particle cross section angular distrib.:
scm(qf) = d(qf -qi)
– Center of mass picture:
Pn,i
Pn,f
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pN,i
pN,f
=> Internal excitation
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Coupled Equations
Similar to Wang, Li, Bauer, Randrup, Ann. Phys. ‘91
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Neutrino
Gain and
Loss
f a’
a’
a
fn
n
fa
f = 1 +- f
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WB, Heavy Ion Physics (2005)
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Numerical Realization
• Test particle equations of motion
• Nuclear EoS evaluated on spherical grid
• Newtonian monopole approximation for gravity
• Better: tree-evaluation of gravity
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Test Particle Scattering
• Nuclear case: test particles scatter with
(reduced) nucleon-nucleon cross sections
– Elastic and inelastic
Inelastic
Elastic
• Similar rules apply
for astro test
particles
– Scale invariance
– Shock formation
– Internal heating
cm frame
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DSMC
• Stochastic Direct Simulation Monte Carlo
– Do not use closest approach method
– Randomly pick k collision partners from given cell
– Redistribute momenta within
cell with fixed ir, iq,f
• Technical details:
– QuickSort on scattering
All particles in given cell
have same scattering index
index of each particle
makes CPU time consumption
~ k N logN
– Final state phase space
approximated by local T
Fermi-Dirac (no additional
power of N)
– Hydro limit: just generate “enough” collisions, no need
to evaluate matrix elements
WB, Acta Phys. Hung. A21, 371 (2004)
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Excluded Volume
• Collision term simulation via stochastic
scattering (Direct Simulation Monte Carlo)
– Additional advection contribution
Shadowing
– Modification to collision probability Excluded V
11 E
P' 1 8 b r (r )
2
E
3
=
; b = 3 a = 2nd Enskog
E
P 1 2b r(r )
virial coefficient
a=
n
Pauli
3
F
p
s i (1 2
) = hard

3
4 n i =1 free
( pF  pB )
sphere
1
Alexander, Garcia, Alder, PRL ‘95
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Kortemeyer,
Daffin,
Bauer, PRB ‘96
radius
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Effects of Angular Momentum
QuickTime™ and a H.263 decompressor are needed to see this picture.
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QuickTime™ and a H.263 decompressor are needed to see this picture.
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Collective Rotation
• Initial conditions
• Evolve in time while conserving global angular
momentum
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Results
• “mean field”
level
• 1 fluid: hadrons
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Max. Density vs. Angular
Momentum
• Mean field only!!!
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(a)
(b)
(c)
(d)
(e)
Initial conditions
After 2 ms
After 3 ms
Core bounce
1 ms after core bounce
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r0
120 km
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Vortex Formation
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Ratio of Densities
Bauer & Strother,
Int. J. Phys. E 14,
129 (2005)
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Some Supernovae are Not
Spherical!
• 1987A remnant shows “smoke rings”
• Cylinder symmetry, but not spherical
• Consequence of high angular momentum
collapse
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HST Wide Field Planetary Camera 2
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More Qualitative
• Neutrino focusing along poles gives preferred
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direction for neutrino flux
Neutrinos have finite mass, helicity
Parity violation on the largest scale
Net excess of neutrinos emitted along “North
Pole”
=> Strong recoil kick for neutron star
supernova remnant
=> Non-thermal contribution to neutron star
velocity distribution
– Amplifies effect of Horowitz et al., PRL 1998
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The People who did/do the Work
Tobias Bollenbach
Terrance Strother
Funding from NSF, Studienstiftung des
Deutschen Volkes, and Alexander von
Humboldt Foundation
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