1. W. Bauer, Hirschegg 20061 New Approach to Supernova Simulations

2. W. Bauer, Hirschegg 20062

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6. 6 3d Fryer, Warren, ApJ 02 Very preliminaryVery preliminary Similar convection as seen in their 2d workSimilar convection as seen in their 2d work Explosion energy 3foe Explosion energy 3foe t expl = 0.1 - 0.2 s t expl = 0.1 - 0.2 s

7. W. Bauer, Hirschegg 20067 Hydro Simulations Tough problem for hydro 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, … Special relativity, causality, … Huge magnetic fields Huge magnetic fields 3D simulations needed 3D simulations needed – Giant grids Need to couple all of this to radiation transport calculation and Boltzmann transport problem for neutrinos Need to couple all of this to radiation transport calculation and Boltzmann transport problem for neutrinos

8. W. Bauer, Hirschegg 20068 Simulations of Nuclear Collisions Hydro, mean field, cascades Hydro, mean field, cascades Numerical solution of transport theories Numerical solution of transport theories – Need to work in 6d phase space => prohibitively large grids (20 3 x40 2 x80~10 9 lattice sites) – Idea: Only follow initially occupied phase space cells in time and represent them by test particles – One-body mean-field potentials ( , p,  ) 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) 1 st Developed @ MSU/FFM

9. W. Bauer, Hirschegg 20069 Transport Equations Mean field EoS 2-body scattering f = phase space density for baryons

10. W. Bauer, Hirschegg 200610 Test Particles Baryon phase space function, f, is Wigner transform of density matrix Baryon phase space function, f, is Wigner transform of density matrix Approximate formally by sum of delta functions, test particles 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 Insert back into integral equation to obtain equations of motion for 6 coordinates of each test particle

11. W. Bauer, Hirschegg 200611 Test Particle Equations of Motion Nuclear EoSTwo-body scattering Coulomb

12. W. Bauer, Hirschegg 200612 Example Density in reaction plane Density in reaction plane Integration over momentum space Integration over momentum space Cut for z=0+-0.5 fm Cut for z=0+-0.5 fm

13. W. Bauer, Hirschegg 200613 Momentum Space Output quantities (not input!) Output quantities (not input!) Momentum space information on Momentum space information on – Thermalization & equilibration – Flow – Particle production Shown here: local temperature Shown here: local temperature

14. W. Bauer, Hirschegg 200614 Reproduces Experiments  Production Baryon flow Pion transparency Disappearance of flow Dependence on mass Pion spectra 2-particle interferometry

15. W. Bauer, Hirschegg 200615 Try this for Supernovae! 2 M  in iron core = 2x10 57 baryons 2 M  in iron core = 2x10 57 baryons 10 7 test particles => 2x10 50 baryons/test particle 10 7 test particles => 2x10 50 baryons/test particle Need time-varying grid for (non-gravity) potentials, because whole system collapses Need time-varying grid for (non-gravity) potentials, because whole system collapses Need to think about internal excitation of test particles Need to think about internal excitation of test particles Can create -test particles and give them finite mean free path => Boltzmann solution for - transport problem Can create -test particles and give them finite mean free path => Boltzmann solution for - transport problem Can address angular momentum question Can address angular momentum question

16. W. Bauer, Hirschegg 200616 Initial Conditions for Core Collapse Woosley, Weaver 86 Iron Core

17. W. Bauer, Hirschegg 200617 Equation of State Low density: Low density: – Degenerate e-gas High density High density – Dominated by nuclear EoS – Isospin term in nuclear EoS becomes dominant For now: For now: High density neutron rich EoS can be explored by GSI upgrade and/or RIA High density neutron rich EoS can be explored by GSI upgrade and/or RIA

18. W. Bauer, Hirschegg 200618 Electron Fraction, Y e Strongly density dependent Strongly density dependent Neutrino cooling Neutrino cooling

19. W. Bauer, Hirschegg 200619 Internal Heating of Test Particles Test particles represent mass of order M earth. Test particles represent mass of order M earth. Internal excitation of test particles becomes important for energy balance Internal excitation of test particles becomes important for energy balance

20. W. Bauer, Hirschegg 200620 Neutrinos Neutrinos similar to pions at RHIC Neutrinos similar to pions at RHIC – Not present in entrance channel – Produced in very large numbers (RHIC: 10 3, here 10 56 ) – Essential for reaction dynamics Different: No formation time or off -shell effects Different: No formation time or off -shell effects Represent 10 N neutrinos by one test particle Represent 10 N neutrinos by one test particle – Populate initial neutrino phase space uniformly – Sample test particle momenta from a thermal distribution Neutrino test particles represent “2 nd fluid”, do NOT escape freely (neutrino trapping), and need to be followed in time. Neutrino test particles represent “2 nd 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 Neutrinos created in center and are “light” fluid on which “heavy” baryon fluid descends – Inversion problem – Rayleigh-Taylor instability – turbulence

21. W. Bauer, Hirschegg 200621 Neutrino Test particles Move on straight lines (no mean field) Move on straight lines (no mean field) Scattering with hadrons Scattering with hadrons – NOT negligible! – Convolution over all  A  A 2 (weak neutral current) – Resulting test particle cross section angular distrib.:  cm  f  f -  i  – Center of mass picture: PiPi p N,i PfPf p N,f => Internal excitation

22. W. Bauer, Hirschegg 200622 Coupled Equations Similar to Wang, Li, Bauer, Randrup, Ann. Phys. ‘91

23. W. Bauer, Hirschegg 200623 Neutrino Gain and Loss  ’’ f = 1 - f + ff f’f’ f WB, Heavy Ion Physics (2005)

24. W. Bauer, Hirschegg 200624 Numerical Realization Test particle equations of motion Test particle equations of motion Nuclear EoS evaluated on spherical grid Nuclear EoS evaluated on spherical grid Newtonian monopole approximation for gravity Newtonian monopole approximation for gravity Better: tree-evaluation of gravity Better: tree-evaluation of gravity

25. W. Bauer, Hirschegg 200625 Elastic Inelastic Test Particle Scattering Nuclear case: test particles scatter with (reduced) nucleon-nucleon cross sections Nuclear case: test particles scatter with (reduced) nucleon-nucleon cross sections – Elastic and inelastic cm frame Similar rules apply for astro test particles Similar rules apply for astro test particles – Scale invariance – Shock formation – Internal heating

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31. W. Bauer, Hirschegg 200631 DSMC Stochastic Direct Simulation Monte Carlo 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 i r, i ,  Technical details: 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)

32. W. Bauer, Hirschegg 200632 Excluded Volume Collision term simulation via stochastic scattering (Direct Simulation Monte Carlo) 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 ‘96Shadowing Excluded V = 2nd Enskog virial coefficient Pauli free = hard sphere sphere radius radius

33. W. Bauer, Hirschegg 200633 Effects of Angular Momentum

34. W. Bauer, Hirschegg 200634 Initial conditions Initial conditions Evolve in time while conserving global angular momentum Evolve in time while conserving global angular momentum Collective Rotation

35. W. Bauer, Hirschegg 200635 Results “mean field” level “mean field” level 1 fluid: hadrons 1 fluid: hadrons

36. W. Bauer, Hirschegg 200636 Max. Density vs. Angular Momentum Mean field only!!! Mean field only!!!

37. W. Bauer, Hirschegg 200637 00 (a)Initial conditions (b)After 2 ms (c)After 3 ms (d)Core bounce (e)1 ms after core bounce 120 km

38. W. Bauer, Hirschegg 200638 Vortex Formation

39. W. Bauer, Hirschegg 200639 Ratio of Densities Bauer & Strother, Int. J. Phys. E 14, 129 (2005)

40. W. Bauer, Hirschegg 200640 Some Supernovae are Not Spherical! 1987A remnant shows “smoke rings” 1987A remnant shows “smoke rings” Cylinder symmetry, but not spherical Cylinder symmetry, but not spherical Consequence of high angular momentum collapse Consequence of high angular momentum collapse HST Wide Field Planetary Camera 2

41. W. Bauer, Hirschegg 200641 More Qualitative Neutrino focusing along poles gives preferred direction for neutrino flux Neutrino focusing along poles gives preferred direction for neutrino flux Neutrinos have finite mass, helicity Neutrinos have finite mass, helicity Parity violation on the largest scale Parity violation on the largest scale Net excess of neutrinos emitted along “North Pole” Net excess of neutrinos emitted along “North Pole” => Strong recoil kick for neutron star supernova remnant => Strong recoil kick for neutron star supernova remnant => Non-thermal contribution to neutron star velocity distribution => Non-thermal contribution to neutron star velocity distribution – Amplifies effect of Horowitz et al., PRL 1998

42. W. Bauer, Hirschegg 200642 The People who did/do the Work Tobias Bollenbach Terrance Strother Funding from NSF, Studienstiftung des Deutschen Volkes, and Alexander von Humboldt Foundation