3d Fryer, Warren, ApJ 02
Very preliminary
Similar convection as seen in their 2d work Explosion energy 3foe
texpl = 0.1 - 0.2 s

Hydro Simulations:

Hydro Simulations Tough problem for hydro
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

Simulations of Nuclear Collisions:

Simulations of Nuclear Collisions Hydro, mean field, cascades
Numerical solution of transport theories
Need to work in 6d phase space => prohibitively large grids (203x402x80~109 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 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) 1st Developed
@ MSU/FFM

Transport Equations:

Transport Equations Mean field EoS
2-body scattering f = phase space density for baryons

Test Particles:

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

Test Particle Equations of Motion:

Test Particle Equations of Motion

Example:

Example Density in reaction plane
Integration over momentum space
Cut for z=0+-0.5 fm

Momentum Space:

Momentum Space Output quantities (not input!)
Momentum space information on
Thermalization & equilibration
Flow
Particle production
Shown here: local temperature

Reproduces Experiments:

Reproduces Experiments

Try this for Supernovae!:

Try this for Supernovae! 2 M in iron core = 2x1057 baryons
107 test particles => 2x1050 baryons/test 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 n-transport problem
Can address angular momentum question

Initial Conditions for Core Collapse:

Initial Conditions for Core Collapse Woosley, Weaver 86 Iron Core

Equation of State:

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

Electron Fraction, Ye:

Electron Fraction, Ye Strongly density dependent
Neutrino cooling

Internal Heating of Test Particles:

Internal Heating of Test Particles Test particles represent mass of order Mearth.
Internal excitation of test particles becomes important for energy balance

Neutrinos:

Neutrinos Neutrinos similar to pions at RHIC
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

Neutrino Test particles:

Neutrino Test particles Move on straight lines (no mean field)
Scattering with hadrons
NOT negligible!
Convolution over all sAnA2 (weak neutral current)
Resulting test particle cross section angular distrib.: scm(qf) = d(qf -qi)
Center of mass picture: Pn,i pN,i Pn,f pN,f => Internal excitation

Coupled Equations:

Coupled Equations Similar to Wang, Li, Bauer, Randrup, Ann. Phys. ‘91

Neutrino Gain and Loss:

Neutrino Gain and Loss a a’ n f = 1 - f + fa fa’ fn WB, Heavy Ion Physics (2005)

Numerical Realization:

Numerical Realization Test particle equations of motion
Nuclear EoS evaluated on spherical grid
Newtonian monopole approximation for gravity
Better: tree-evaluation of gravity

Test Particle Scattering:

Elastic Test Particle Scattering Nuclear case: test particles scatter with (reduced) nucleon-nucleon cross sections
Elastic and inelastic cm frame Similar rules apply for astro test particles
Scale invariance
Shock formation
Internal heating

DSMC:

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 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 All particles in given cell have same scattering index WB, Acta Phys. Hung. A21, 371 (2004)

Excluded Volume:

Excluded Volume Collision term simulation via stochastic scattering (Direct Simulation Monte Carlo)
Additional advection contribution
Modification to collision probability Alexander, Garcia, Alder, PRL ‘95
Kortemeyer, Daffin, Bauer, PRB ‘96 = 2nd Enskog virial coefficient

Effects of Angular Momentum:

Effects of Angular Momentum

Collective Rotation:

Initial conditions
Evolve in time while conserving global angular momentum Collective Rotation

Results:

Results “mean field” level
1 fluid: hadrons

Max. Density vs. Angular Momentum:

Max. Density vs. Angular Momentum Mean field only!!!

Slide37:

Initial conditions
After 2 ms
After 3 ms
Core bounce
1 ms after core bounce 120 km

Vortex Formation:

Vortex Formation

Ratio of Densities:

Ratio of Densities Bauer & Strother, Int. J. Phys. E 14, 129 (2005)

Some Supernovae are Not Spherical!:

Some Supernovae are Not Spherical! 1987A remnant shows “smoke rings”
Cylinder symmetry, but not spherical
Consequence of high angular momentum collapse HST Wide Field Planetary Camera 2

More Qualitative:

More Qualitative Neutrino focusing along poles gives preferred 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

The People who did/do the Work:

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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