Core-Collapse Supernovae and Neutrino Transport

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1 Core-Collapse Supernovae and Neutrino Transport SFB TR7 Gravitational Wave Astronomy Video Seminar B.Müller

2 Core Collapse Supernovae, General Relativity & Gravitational Waves Core collapse supernova Collapse of massive star to neutron star Shock wave launched at bounce, Shock stalls and is somehow revived (neutrino heating, hydro instabilities, MHD, acoustic mechanism, nuclear phase transition...) Gravitational field of neutron star is strong (M/R during first second), infall and postshock velocities are (mildly) relativistic GR treatment desirable Gravitational waves emitted: rotational collapse, convection, anisotropic neutrino emission

Explosion mechanism Morphology of the ejecta Nucleosynthesis yields

3 Current Issues in Supernova Physics (Besides Gravitational Waves...) Convection & SASI (standing accretion shock instability) in 2D and 3D Explosion mechanism Morphology of the ejecta Nucleosynthesis yields Detectable neutrino signals Oakridge National Laboratory Illustration adapted from Mezzacappa (2003)

4 Why Neutrino Transport Is Important Transport of energy, momentum & lepton number can be crucial for the dynamics of compact objects and their surroundings Rough criterion for importance: evolution timescale diffusion time-scale or larger Role of neutrinos in the supernova problem: Determine heating & neutron star cooling (contraction) crucial for dynamics (explosion/no explosion) Regulate nucleosynthesis conditions Can produce sizeable gravitational wave signal Neutrino transport also relevant for other scenarios involving compact objects (winds from merger disks, gamma ray bursts)

5 Neutrino Transport: Basics In general: Boltzmann equation applicable (6D problem): p M f i f p p =C [ f ] i p x Dimensionality may be reduced by introducing a hierarchy of angular moments of the distribution function (requires closure relation): f J = f d, H i = f n i d, K ik = f n i n k d, Diffusion approximation: truncate hierarchy at lowest level (only energy density J is evolved) Grey approximation is even more severe (detailed information on spectral distribution lost) J, H i, K i k, J d, H i d, K i k d,

6 Overview of Current Strategies for the Multi-D Supernova Problem 2D/3D hydro + parameterized source/sink terms (e.g. 2D: Fernandez & Thompson 2009; 3D: Iwakami et al. 2008, Nordhaus et al. 2010) 2D/3D hydro + grey transport (e.g. Fryer & Young 2007, Scheck et al. 2008) 2D (3D) hydro + multi-energy transport: Discrete ordinate (Sn) method for simplified Boltzmann equation (Livne et al. 2004) Ray-by-ray variable Eddington factor method (Buras et al. 2006) Multi-group flux-limited diffusion (full 2D: Walder et al. 2005, Swesty&Myra 2006; ray-by-ray: Bruenn et al. 2006) Isotropic diffusion source approximation (Liebendörfer et al. 2009)

7 The Garching Approach to Neutrino Transport Full 6D transport currently unfeasible More affordable: ray-by-ray transport Essential features of our approach: Energy-dependent transport (grey transport results in wrong heating & is of little value for prediction of detected neutrino signal & nucleosynthesis) r 6D phase space problem Reasonable transition from trapping to free-streaming Comprehensive set of interaction rates (including the important processes in the / sector) Correct behaviour in the optically-thick regime r ray-by-ray method in axisymmetry

8 Neutrino Transport Algorithm 1 1 K, L needed from Botzmann eq. Step 1: Advect neutrinos laterally (in θ-direction) with the fluid Step 2: Iterate moment equations (1) and modified Boltzmann equation (2) to convergence 1 2 C from solution of moment eqs.

9 Numerical Implementation Problem: Stiff source terms (Δtexplicit 10-10s) Implicit discretization of moment equations; solution with NewtonRaphson method Direct linear solver employed (condition number) Advantage of ray-by-ray method: 1D transport problems can be performed independently Easy parallelization over rays (MPI/OpenMP), scales on >2000 processors for sufficiently larger number of rays Structure of Jacobian: full block-pentadiagonal matrix

10 Moment Equations: Complications The energy-dependent moment equations for the i-th Tensor containing the neutrino flavour read: moments J, H, K, L Energy-dependent stress energy tensor, containing linear function of the moments J, H, K M i t i u =s i Two independent sets of conservation laws (for lepton number) follow from this (by integrating over energy): j j = q= Y e u e Electron neutrino and antineutrino flux four-vector e Electron number source term Electron flux four-vector T i = s i Stress-energy tensor of neutrino of flavour i Neutrino energymomentum source term

11 Maintaining Lepton Number & Energy Conservation Deriving a conservation law for lepton number from the moment equations involves the Leibniz rule for derivatives of the moment X: 1 X X = X Leibniz-rule does not hold for finite-difference representation choice between energy- or leptonnumber conserving discretization Workaround: Solve both for energy moments (J,H) and number moments (J,H) expensive & possibly inconsistent Better: Special FD representation consistent with the frequency-integrated conservation laws, good conservation properties can be obtained

Coupling to Hydrodynamics General Issues Long-time stability: explosion may occur hundreds of ms ( 10000 M) after bounce ADM energy conservation: explosion (1051erg) is only a surface phenomenon

12 Coupling to Hydrodynamics General Issues Long-time stability: explosion may occur hundreds of ms ( M) after bounce ADM energy conservation: explosion (1051erg) is only a surface phenomenon Accuracy in super- and subsonic regime (protoneutron star convection has Mach number 0.05) Control over seed perturbations, mesh artifacts A. Burrows, Conf. Talk Emerging Themes in the Theory of CC SNe explosions

13 Coupling to Hydrodnymics Hydro and metric: CoCoNuT code (Dimmelmeier et al. 2002) HRSC scheme with PPM reconstruction Tabulated nuclear EoS, baryon+radiation EoS at low densities Metric in xcfc approximation (Cordero-Carrión et al. 2009, very accurate for core collapse case), but extendible to maximally constrained formulation of the field equations (Bonazzola et al. 2004) GW extraction with quadrupole formula at the moment Improvements: Consistent Multi-Fluid Advection (CMA, Müller & Plewa) Parallelization using Message Passing Interface (MPI) Different formulations for momentum equation to reduce artefacts from spherical polar grid (e.g. optional computation of flux differences in Cartesian basis) Improved long-time energy conservation Hybrid HLLC/HLLE-Solver

are not very diffusive in this regime Also: entropy perturbations create artificial acoustic

14 Accuracy of Approximate Riemann Solvers Post-shock flow dominated by convection, pressure equilibrium reasonably well maintained Mignone & Bodo 2005 Some popular Riemann solvers (HLLE, modified Marquina) are not very diffusive in this regime Also: entropy perturbations create artificial acoustic perturbations Alternatives (HLLC, Roe, original Marquina,...) preserve small scale structures much better HLLC HLLE

15 A More Quantitative Analysis PPM MC minmod 1st order HLLE solver: damping of pure entropy wave Damping time scale in units of sound-crossing time-scale

16 Numerical Challenges Explosion may occur after several 100ms Neutron star cooling even takes several s (Kelvin-Helmholtz time-scale) What about stability and total energy (ADM energy) conservation? Careful analysis shows that spurious energy generation becomes relevant already 900ms after bounce for Nomoto progenitor Trajectories of selected mass shells 8.8 solar mass model of Nomoto '84, '87 Sound-crossing time-scale for neutron star is < 1ms No cooling! Neutron star just keeps radiating...

17 How to reduce unphysical energy generation Standard form of energy equation is non-conservative e [ e P vi ]= t (Newtonian) (GR 3+1, Banyuls et al. '97) Partially absorb source terms in fluxes e [ e P v i ]= t t i i v P v =Q i t x D t 2 v i D v i P vi = xi D t Q ' Care must be taken to avoid problems with round-off errors Zone i Zone i+1 potential mass flux (e.g. when τ D) Energy liberated=mass flux difference in potential

Relativistic Supernova Explosion Models 2D simulations of several progenitors already running Stable evolution over several 100ms is possible (record: model s15s7b2 has reached 561ms after bounce)

18 Relativistic Supernova Explosion Models 2D simulations of several progenitors already running Stable evolution over several 100ms is possible (record: model s15s7b2 has reached 561ms after bounce) Overall: qualitative agreement with the Newtonian PROMETHEUS code using a modified gravitational potential Explosion confirmed for 11.2 and 15 solar mass models (cp. Marek & Janka 2009) However: 15 solar mass model explodes earlier in GR, PROMETHEUS model was marginal 11.2 solar mass model of Woosley et al. 2002, 216ms after bounce

19 Runaway criterion Competition between advection and heating of accreted material: compare time-scales

20 2D Explosion Models

21 Movie s15.avi Please play at maximum frame rate

22 Gravitational Wave Signal progenitor s15s7b2 prompt post-shock convection and early SASI activity Signal from aspherical shock expansion during explosion phase much weaker than in previous studies: cp. Yakunin et al.

23 Gravitational Wave Spectra

24 Conclusions Review of methods for neutrino transport and hydrodynamics in first multi-dimensional GR neutrino hydrodynamics code VERTEX-CoCoNuT Strengths/important features: Stable evolution of quasi-static proto-neutron star Good conservation properties 2D explosion models of supernovae underway Qualitative agreement with Newtonian PROMETHEUS code However, GR effects (compactness of the proto-neutron star) may play a role, particularly for late explosions GW signals covering bounce, accretion and explosion phase computed Signal from late explosion of model s15s7b2 looks somewhat different than in recent Newtonian simulations

Hirschegg Supernova core collapse. dynamics of core collapse. simple and efficient parameterization of deleptonization

Hirschegg Supernova core collapse. dynamics of core collapse. simple and efficient parameterization of deleptonization Hirschegg 2006 Supernova core collapse M. Liebendörfer University of Basel U.-L. Pen & C. Thompson Canadian Institut for Theoretical Astrophysics dynamics of core collapse simple and efficient parameterization