Document Sample

Improved simulations of relativistic stellar core collapse José A. Font Departamento de Astronomía y Astrofísica Universidad de Valencia (Spain) Collaborators: • P. Cerdá-Durán, J.M. Ibáñez (UVEG) • H. Dimmelmeier, F. Siebel, E. Müller (MPA) • G. Faye (IAP), G. Schäfer (Jena) • J. Novak (LUTH-Meudon) Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 Outline of the talk • Numerical simulations of rotational stellar core collapse: gravitational waveforms • Relativistic hydrodynamics equations in conservation form (Godunov-type schemes) • Approximations for the gravitational field equations (elliptic equations – finite- difference schemes, pseudo-spectral methods) • CFC (2D/3D) • CFC+ (2D) • Axisymmetric core collapse in characteristic numerical relativity • Improved means: • Treatment of gravity: from CFC to CFC+, and Bondi-Sachs • Modified CFC equations (high-density NS, BH formation) • Dimensionality: from 2D to 3D • Collapse dynamics: inclusion of magnetic fields Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 Astrophysical motivation General relativity and relativistic hydrodynamics play a major role in the description of gravitational collapse leading to the formation of compact objects (neutron stars and black holes): Core-collapse supernovae, black hole formation (and accretion), coalescing compact binaries (NS/NS, BH/NS, BH/BH), gamma-ray bursts. Time-dependent evolutions of fluid flow coupled to the spacetime geometry only possible through accurate, large-scale numerical simulations. Some scenarios can be described in the test-fluid approximation: hydrodynamical computations in curved backgrounds (highly mature nowadays). (see e.g. Font 2003 online article: relativity.livingreviews.org/Articles/lrr-2003-4/index.html). The (GR) hydrodynamic equations constitute a nonlinear hyperbolic system. Solid mathematical foundations and accurate numerical methodology imported from CFD. A “preferred” choice: high-resolution shock-capturing schemes written in conservation form. The study of gravitational stellar collapse has traditionally been one of the primary problems in relativistic astrophysics (for about 40 years now). It is a distinctive example of a research field in astrophysics where essential progress has been accomplished through numerical modelling with gradually increasing levels of complexity in the input physics/mathematics. Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 Introduction: supernova core collapse in a nutshell The study of gravitational collapse of massive stars largely pursued numerically over the years. Main motivation in May and White’s 1967 first one-dimensional numerical relativity code. Current standard model for a core collapse (type II/Ib/Ic) supernova: (from simulations! [Wilson et al (late 1980s), MPA, Oak Ridge, University of Arizona (ongoing)]) • Nuclear burning in massive star yields shell structure. Iron core with 1.4 solar masses and 1000 km radius develops in center. EoS: relativistic degenerate fermion gas, =4/3. • Instability due to photo-disintegration and e- capture. Collapse to nuclear matter densities in ~100ms. • Stiffening of EoS, bounce, and formation of prompt shock. • Stalled shock revived by neutrinos depositing energy behind it (Wilson 1985). Delayed shock propagates out and disrupts envelope of star. • Nucleosynthesis, explosion expands into interstellar matter. Proto-neutron star cools and shrinks to neutron star. Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 Introduction (continued) May & White’s formulation and 1d code used by many groups to study core collapse. Most investigations used artificial viscosity terms in the (Newtonian) hydro equations to handle shock waves. The use of HRSC schemes started in 1989 with the Newtonian simulations of Fryxell, Müller & Arnett (Eulerian PPM code). Basic dynamics of the collapse at a glance: 1d core collapse simulations Relativistic simulations of core collapse with HRSC schemes are still scarce. Nonspherical core collapse simulations in GR very important: 1. To produce and extract gravitational waves consistently. 2. To explain rotation of newborn NS. 3. Collapse to NS is intrinsically relativistic (2M/R ~0.2-0.4) (let alone to BH!) Romero et al 1996 (radial gauge polar slicing). Purely hydrodynamical (prompt mechanism) explosion. No microphysics or -transport included! Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 Multidimensional core collapse & gravitational waves Numerical simulations of stellar core collapse are nowadays highly motivated by the prospects of direct detection of the gravitational waves (GWs) emitted. GWs, ripples in spacetime generated by aspherical concentrations of accelerating matter, were predicted by Einstein in his theory of general relativity. Their amplitude on Earth is so small (about 1/100th of the size of an atomic nucleus!) that they remain elusive to direct detection (only indirectly “detected” in the theoretical explanation of the orbital dynamics of the binary pulsar PSR 1913+16 by Hulse & Taylor (Nobel laureates in physics in 1992). International network of resonant bar detectors International network of interferometer detectors Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 Core collapse & gravitational waves (continued) • GWs are dominated by a burst associated with the bounce. If rotation is present, the GWs large amplitude oscillations associated with pulsations in the collapsed core (Mönchmeyer et al 1991; Yamada & Sato 1991; Zwerger & Müller 1997; Rampp et al 1998 (3D!)). • GWs from convection dominant on longer timescales (Müller et al 2004). • Müller (1982): first numerical evidence of the low gravitational wave efficiency of the core collapse scenario: E<10-6 Mc2 radiated as gravitational waves. (2D simulations, Newtonian, finite-difference hydro code). • Bonazzola & Marck (1993): first 3D simulations of the infall phase using pseudo-spectral methods. Still, low amount of energy is radiated in gravitational waves, with little dependence on the initial conditions. • Zwerger & Müller (1997): general relativity counteracts the stabilizing effect of rotation. A bounce caused by rotation will occur at larger densities than in the Newtonian case need for relativistic simulations: Dimmelmeier et al 2001, 2002; Siebel et al 2003; Shibata & Sekiguchi 2004, 2005; Cerdá-Durán et al 2005. Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 Supernova codes vs core collapse numerical relativity codes State-of-the art supernova codes are (mostly) based on Newtonian hydrodynamics (e.g. MPA group, Oak Ridge National Laboratory group). • Strong focus on microphysics (elaborate EoS, transport schemes for neutrinos – computationally challenging). • Often use of the most advanced initial models from stellar evolution. • Simple treatment of gravity (Newtonian, possibly relativistic corrections). However … no generic explosions yet obtained! (even with most sophisticated multi-dimensional models) Core collapse numerical relativity codes (mostly) originate from vacuum Einstein codes (e.g. Whisky (EU), Shibata’s). • No microphysics: matter often restricted to ideal fluid EoS. • Simple initial (core collapse) models (uniformly or differentially rotating polytropes). • Exact or approximate Einstein equations for spacetime metric (inherit the usual complications found in numerical relativity: formulations of the field equations, gauge freedom, long-term numerical stability, etc). Our approach: flux-conservative hyperbolic formulation for the hydrodynamics CFC, CFC+, and Bondi-Sachs for the Einstein equations Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 3+1 General Relativistic Hydrodynamics equations (1) Equations of motion: ( u ) 0 [1] local conservation laws of density current T 0 [4] (continuity equation) and stress-energy (Bianchi p p( , ) identities) [1] Perfect fluid stress-energy tensor ( g u ) 0 T hu u pg x Introducing an explicit coordinate chart: ( g T ) g T x Different formulations exist depending on: Wilson (1972) wrote the system as a set of advection equation within the 3+1 formalism. Non-conservative. 1. The choice of time-slicing: the level surfaces of can be spatial (3+1) or Conservative formulations well-adapted to numerical null (characteristic) x 0 methodology are more recent: 2. The choice of physical (primitive) • Martí, Ibáñez & Miralles (1991): 1+1, general EOS variables (, , ui …) • Eulderink & Mellema (1995): covariant, perfect fluid • Banyuls et al (1997): 3+1, general EOS • Papadopoulos & Font (2000): covariant, general EOS Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 3+1 General Relativistic Hydrodynamics equations (2) ( u ) 0 1 [1] R g R 8T [10] T 0 [4] 2 p p( , ) [1] Einstein’s equations Foliate the spacetime with t=const spatial hypersurfaces St j n ds 2 ( 2 i )dt 2 2 dxi dt dxi dx i i ij t Let n be the unit timelike 4-vector orthogonal to St such that 1 n ( t i i ) n i 1 u i Eulerian observers v vi t i nu u u: fluid’s 4-velocity, p: isotropic pressure, : rest-mass density : specific internal energy density, e=( 1+ ): energy density Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 3+1 General Relativistic Hydrodynamics equations (3) Replace the “primitive variables” in terms of the “conserved variables” : D W W 2 1 /(1 v j v j ) w , vi , S j hW 2 v j p h 1 E hW 2 p First-order flux-conservative hyperbolic system i 1 u ( w) g f ( w) Banyuls et al, ApJ, 476, s ( w) 221 (1997) g t x i Font et al, PRD, 61, 044011 (2000) u ( w) D, S j , E D where is the vector of conserved variables i i i i i i i D v f ( w) , S j v p j , E D v i pv i fluxes g ln 0 0, T j gj , T 0 s ( w) x T sources x Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 Nonlinear hyperbolic systems of conservation laws (1) For nonlinear hyperbolic systems classical solutions do not exist in general even for smooth initial data. Discontinuities develop after a finite time. For hyperbolic systems of conservation laws, schemes written in conservation form guarantee that the convergence (if it exists) is to one of the weak solutions of the original system of equations (Lax-Wendroff theorem 1960). A scheme written in conservation form reads: n 1 n t n n ˆ n n ˆ n n u j u j ( f (u j r , u j r 1 ,, u j q ) f (u j r 1 , u j r ,, u j q 1 )) x ˆ ˆ where f is a consistent numerical flux function: f (u , u ,, u ) f (u ) Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 Nonlinear hyperbolic systems of conservation laws (2) The conservation form of the scheme is ensured by starting with the integral version of the PDE in conservation form. By integrating the PDE within a spacetime computational cell [ x j 1/ 2 , x j 1/ 2 ] to n , t n1 ] the numerical flux function is an approximation [tthe time-averaged flux across the interface: ˆ 1 t n1 The flux integral depends on the solution at the f j 1/ 2 n f (u ( x j 1/ 2 , t )) dt numerical interfaces u ( x j1/ 2 , t ) time step during the t t When a Cauchy problem described by a set of Key idea: a possible procedure is to continuous PDEs is solved in a discretized form the calculate / 2 t) u ( x j1by,solving numerical solution is piecewise constant (collection of Riemann problems at every cell local Riemann problems). interface (Godunov) n n u ( x j 1/ 2 , t ) u (0; u j , u j 1 ) Riemann solution for the left and right states along the ray x/t=0. Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 Nonlinear hyperbolic systems of conservation laws (3) Any FD scheme must be able to handle discontinuities in a satisfactory way. 1. 1st order accurate schemes (Lax-Friedrich): Non-oscillatory but inaccurate across discontinuities (excessive diffusion) 2. (standard) 2nd order accurate schemes (Lax-Wendroff): Oscillatory across discontinuities 3. 2nd order accurate schemes with artificial viscosity 4. Godunov-type schemes (upwind High Resolution Shock Capturing schemes) Lax-Wendroff numerical solution of Burger’s equation at t=0.2 (left) and t=1.0 (right) 2nd order TVD numerical solution of Burger’s equation at t=0.2 (left) and t=1.0 (right) Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 Nonlinear hyperbolic systems of conservation laws (4) rarefaction wave shock front Solution at time n+1 of the two Riemann problems at the cell boundaries xj+1/2 and xj-1/2 Spacetime evolution of the two Riemann problems at the cell boundaries xj+1/2 and xj-1/2. Each problem leads to a shock wave and a rarefaction wave moving in opposite directions (Piecewise constant) Initial data at time n for the two Riemann problems at the cell boundaries xj+1/2 and xj-1/2 n 1 n t n ˆ n ˆ uj uj f j 1/ 2 f j 1/ 2 x cell boundaries where fluxes are required Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 Approximate Riemann solvers In Godunov’s method the structure of the Riemann solution is “lost” in the cell averaging process (1st order in space). The exact solution of a Riemann problem is computationally expensive, particularly in multidimensions and for complicated EoS. Relativistic multidimensional problems: coupling of all flow velocity components through the Lorentz factor. • Shocks: increase in the number of algebraic jump (RH) conditions. • Rarefactions: solving a system of ODEs. This motivated the development of approximate Roe-type SRRS (Martí et al 1991; Font et al 1994) (linearized) Riemann solvers. HLLE SRRS (Schneider et al 1993) Based on the exact solution of Riemann problems Exact SRRS (Martí & Müller 1994; Pons et al 2000) corresponding to a new system of equations Two-shock approximation (Balsara 1994) obtained by a suitable linearization of the original ENO SRRS (Dolezal & Wong 1995) one. The spectral decomposition of the Jacobian matrices is on the basis of all solvers. Roe GRRS (Eulderink & Mellema 1995) Upwind SRRS (Falle & Komissarov 1996) Approach followed by an important subset of shock- capturing schemes, the so-called Godunov-type Glimm SRRS (Wen et al 1997) methods (Harten & Lax 1983; Einfeldt 1988). Iterative SRRS (Dai & Woodward 1997) Marquina’s FF (Donat et al 1998) Martí & Müller, 2003 Living Reviews in Relativity www.livingreviews.org Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 A standard implementation of a HRSC scheme 1. Time update: Conservation form algorithm n1 n t n ˆ n ˆ u j u j f j 1/ 2 f j 1/ 2 x In practice: 2nd or 3rd order time accurate, conservative Runge-Kutta schemes (Shu & Osher 1989) 2. Cell reconstruction: Piecewise constant (Godunov), linear (MUSCL, MC, van Leer), parabolic (PPM, Colella & Woodward 1984) 3. Numerical fluxes: Approximate Riemann interpolation procedures of state-vector variables solvers (Roe, HLLE, Marquina). Explicit use from cell centers to cell interfaces. of the spectral information of the system 1 ˆ 5 ~ ~ ~ f i f i ( wR ) f i ( wL ) n n Rn 2 n 1 5 U( wR ) U( wL ) n Rn ~ ~ n 1 Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 HRSC schemes: numerical assessment Relativistic shock reflection Shock tube test • Stable and sharp discrete shock profiles • Accurate propagation speed of discontinuities • Accurate resolution of multiple nonlinear structures: discontinuities, raraefaction waves, vortices, etc V=0.99999c (W=224) Simulation of a extragalactic relativistic jet Wind accretion onto a Kerr black hole (a=0.999M) Scheck et al, MNRAS, 331, 615 (2002) Font et al, MNRAS, 305, 920 (1999) Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 Relativistic Rotational Core Collapse (CFC) Dimmelmeier, Font & Müller, ApJ, 560, L163 (2001); A&A, 388, 917 (2002a); A&A, 393, 523 (2002b) Goals extend to GR previous results on Newtonian rotational core collapse (Zwerger & Müller 1997) determine the importance of relativistic effects on the collapse dynamics (angular momentum) compute the associated gravitational radiation (waveforms) Model assumptions axisymmetry and equatorial plane symmetry (uniformly or differentially) rotating 4/3 polytropes in equilibrium as initial models (Komatsu, Eriguchi & Hachisu 1989). Central density 1010 g cm-3 and radius 1500 km. Various rotation profiles and rotation rates simplified EoS: P = Ppoly + Pth (neglect complicated microphysics and allows proper treatment of shocks) constrained system of the Einstein equations (IWM conformally flat condition) Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 CFC metric equations ij 4 ij CFC In the CFC approximation (Isenberg 1985; Wilson & Mathews 1996) the ADM 3+1 equations t ij 2K ij i j j i t K ij i j Rij KK ij 2 K im K m m m K ij K im j m K jm i m 8T ij j R K 2 K ij K ij 16 2T 00 0 i K ij ij K 8S j 0 reduce to a system of five coupled, nonlinear elliptic equations for the lapse function, conformal factor, and the shift vector: Solver 1: Newton-Raphson iteration. Discretize 5 K ij K ij 2 W P 2 equations and define root-finding strategy. 16 Solver 2: Conventional integral Poisson iteration. Exploits Poisson-like structure of metric 7 K ij K ij 5 2 h 3W 2 5 P 2 16 equations, uk=S(ul). Keep r.h.s. fixed, obtain linear Poisson equations, solve associated integrals, then iterate until nonlinear equations converge. 1 i 16 4 S i 2 K ij j 6 i k k 3 Both solvers feasible in axisymmetry but no extension to 3D possible. Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 Animation of a representative rotating core collapse simulation For movies of additional models visit: www.mpa-garching.mpg.de/rel_hydro/axi_core_collapse/movies.shtml Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 Central Density Gravitational Waveform HRSC scheme: PPM + Marquina flux-formula Type I “regular” Solid line: relativistic simulation Dashed line: Newtonian Type II “multiple bounce” Larger central densities in relativistic models Similar gravitational radiation amplitudes (or smaller in the GR case) “transition” GR effects do not improve the chances for detection (at least in axisymmetry) Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 Gravitational Wave Signals www.mpa-garching.mpg.de/Hydro/RGRAV/index.html Influence of relativistic effects on signals: Investigate amplitude-frequency diagram Spread of the 26 models does not change much Signal of a galactic supernova detectable On average: Amplitude → Frequency ↑ If close to detection threshold: Signal could fall out of the sensitivity window! Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 CFC+ metric equations Cerdá-Duran, Faye, Dimmelmeier, Font, Ibáñez, Müller, and Schäfer, A&A, in press (2005) CFC+ metric: ij ij hij , CFC CFC TT tr TT 0 h (ADM gauge) The second post-Newtonian deviation from isotropy is the solution of: 1 TTkl 1 hij TT ij (16 vk vl 4 kU lU ) 6 (Schäfer 1990) c4 c (complicated) transverse, traceless projection operator Newtonian potential Modified equations for , i and (with respect to CFC): K ij K ij 2 hW P 5 2 16 7 K ij K ij 1 TT 2 5 h 3W 2 2 5 P 16 c 2 hij ijU 1 i 16 4 S i 2 K ij j 6 i k k 3 Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 CFC+ metric equations (2) S i (4 vi v j iU jU ) x j TT We can solve the hij equations by introducing some intermediate potentials: S ij 4 vi v j iU jU 1 7 1 hij TT Sij ij S kk 3x k (i S j ) k 3 (i S j ) x j i S kk S 4 vi v j x i x j 2 4 4 1 1 1 1 1 iT j x k ij S k x k x l ij S kl ij S i R j T i (4 v j v j jU jU ) xi 4 2 4 4 4 R i i kU lU x k x l S v vk x x U x kU d x 1 M2 i 1 16 elliptic linear equations i i k i k 3 n O 2 r 2r r Linear solver: LU decomposition 1 i j ij 1 using standard LAPACK routines S v v ij x j iU d 3 x O 2 r 2 r 1 1 r Boundary conditions S vk vl x k x l d 3 x O 2 r Multipole development in compact- vk v k x i x iU d 3 x 1 2 1 M i supported integrals r Ti n O 2 2r r M2 i 1 R i n O 2 r r Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 CFC+ results: rotating neutron stars Initial models (KEH method) Mass Model Axis ratio /K (sun) RNS0 1.00 0.00 1.40 RNS1 0.95 0.42 1.44 RNS2 0.85 0.70 1.51 RNS3 0.75 0.87 1.59 RNS4 0.70 0.93 1.63 RNS5 0.65 0.98 1.67 Study the time-evolution of equilibrium models under the effect of a small amplitude perturbation. Computation of radial and quasi- radial mode-frequencies (code validation: comparison between CFC and CFC+ results, and with those of an independent full GR Equatorial profiles of the non-vanishing components of hij for the code) sequence of rigidly rotating models RNS0 to RNS5 Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 CFC+:radial modes of spherical NS quasi-radial modes of rotating NS spherical NS rotating NS No noticeable differences between CFC and CFC+ Good agreement in the mode frequencies (better than 2%), also with results from a full GR 3D code (Font et al 2002) Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 CFC+: core collapse dynamics (1) Type I (regular collapse) Type III (rapid collapse) Relative differences between CFC and CFC+ for the central density and the lapse remain of the order of 10-4 or smaller throughout the collapse and bounce. Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 CFC+: core collapse dynamics (2) Type II (multiple bounce) Extreme case (torus-like structure) Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 CFC+: core collapse waveforms Two distinct ways to extract waveforms: From the quadrupole formula: quad AE 2 t r / c h x , t 1 15 sin 2 20 8 r From the metric hij: AE2 t r / c h PN x , t r / c 1 15 2 sin 2 20 64 r 1 quad h x , t 8 Offset correction (dashed line) h PN-corrected h PN a ij hij 2 2 TT Absolute differences between CFC and CFC+ waveforms. No significant differences found. Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 “Mariage des Maillages” HRSC schemes for hydrodynamics and spectral methods for metric Reference: Dimmelmeier, Novak, Font, Müller, Ibáñez, PRD 71, 064023 (2005) The extension of our code to 3D has been possible thanks to the use of a metric solver based on integral Poisson iteration (as solver 2) but using spectral methods. MdM idea: Use spectral methods for the metric (smooth functions, no discontinuities) and HRSC schemes for the hydro (discontinuous functions). Valencia/Meudon/Garching collaboration. New metric solver uses publicly available package in C++ from Meudon group (LORENE). Communication between finite-difference grid and spectral grid necessary (high- order interpolators). It works! Spectral solver uses several (3-6) radial domains (easy with LORENE package): • Nucleus limited by rd (domain radius parameter) roughly at largest density gradient. • Several shells up to rfd. • Compactified radial vacuum domain out to spatial infinity. In contraction phase of core collapse, inner domain boundaries are allowed to move (controlled by mass fraction or sonic point). The relation between the FD grid and the spectral grid changes dynamically due to moving domains. Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 MdM code: Importance of a moving spectral grid In core collapse relevant radial scale contracts by a factor 100. Spectral grid setup with moving domains allows to put resolution where needed. Example: influence of bad spectral grid setup on collapse dynamics. rd held fixed (10% initial rse) 1. Domain radius rd must follow contraction. wrong result! 2. Domain radius rd should stay fixed at roughly rpns after core bounce. final rd too large bounce time 3. More than 3 domains needed in dynamical core collapse. only 3 radial domains 4. Compare with previous solvers in axisymmetry: 33 collocation points per domain sufficient. Gibbs-type oscillations Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 MdM code: Oscillations of rotating neutron stars Another stringent test: can code keep rotating neutron stars in equilibrium? Test criterion: preservation of rotation velocity profile (here shown after 10 ms). Compactified grid essential if rfd close to rstar (profile deteriorates only negligibly) 3d low resolution 2d high resolution Axisymmetric oscillations in rotating neutron stars 3d low resolution without artificial perturbation can be evolved as in other codes. No important differences between running the code in 2d or 3d modes. Proof of principle: code is ready for simulations of dynamical triaxial instabilities. Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 MdM code: Generic nonaxisymmetric configurations Explore nonaxisymmetric configurations in 3d. Extension from axisymmetry to 3d trivial with LORENE. Even in axisymmetry spectral solver uses coordinate with 4 collocation points (shift vector Poisson equation is calculated for Cartesian components). Setup: rotating NS with strong (unphysical) nonaxisymmetric “bar” perturbation. Rotation generates spiral arms Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 MdM code: Comparison with full GR core collapse simulations by Shibata and Sekiguchi Full GR simulations of axisymmetric core collapse available recently (Shibata & Sekiguchi 2004). Comparison between CFC and full GR possible! Shibata & Sekiguchi used rotational core collapse A3B2G4 (DFM 02) models with parameters close (but not equal) to the ones used by Dimmelmeier, Font & Müller (2002). W6 (20% gain at bounce!) Disagreements in the GW amplitude of about 20% at the peak (core bounce) and up to a factor 2 in the ringdown. Most plausible reason for discrepancy: different functional form of the density used in the wave A3B2G4 (A/rse=0.32) extraction method (W6) and the formulation (stress Shibata & Sekiguchi formulation vs first moment of momentum density (A/rse=0.25) formulation). A3B2G4 The qualitative difference found by Shibata & Sekiguchi (2004) is due to the differences in the collapse initial model, notably the small decrease of Shibata & Sekiguchi the differential rotation length scale in their model. Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 CFC metric equations: modification to allow for black hole formation Original CFC equations It turns out to be essential to rescale some K K ij of the hydro quantities with the appropriate 2 W 2 P ij 5 16 power of the conformal factor for the elliptic solvers to converge to the correct solution: 7 K ij K ij 2 5 h 3W 2 2 5 P 16 * 6 S i* 6 S i 1 i 16 4 S i 2 K ij j 6 i k k 3 P* 6 P 1 K ij K ij To obtain * , S i* , P * one 2 2 hW P 5 16 needs to first compute the conformal factor, which is 1 obtained from the evolution equation 2 h 3W 5 5 P 2 5 7 K ij K ij 16 1 k k i 16 2 S i 2 K ij j 6 i k k t 6 3 Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 Black hole formation (spherical symmetry) with rescaling without rescaling High central density TOV solution Collapse of a (perturbed) unstable neutron star to a black hole in spherical symmetry. Collapse can be followed well beyond formation of an apparent horizon. Central density grows by 6 orders of magnitude, central lapse function drops to 0.0002. Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 Rotational core collapse to high-density NS: CFC vs CFC+ Model M7C5 (Shibata and Sekiguchi 2005): • Differential rotation parameter A/R=0.1 • Baryon rest mass M*=2.464 • Angular momentum J/M2=0.664 • Polytropic EOS (=4/3, k=7x1014 (cgs)) Excellent agreement with the full BSSN simulations of Shibata & Sekiguchi (2005) max=1.4x1015 GW amplitude larger at min=0.42 bounce with CFC+ Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 Core collapse simulations using the Einstein equations for the Bondi metric Reference: Siebel, Font, Müller, and Papadopoulos, PRD 67, 124018 (2003) V ds 2 e 2 U 2 r 2 e 2 du 2 2e 2 dudr 2Ur 2 e 2 dud r 2 e 2 d 2 e 2 sin 2 d 2 r The metric functionsV ,U , and only depend on the coordinates u, r and 1 1 Gab Rab g ab R Rab h uaub g ab pgab Ricci tensor 2 2 Rrr ,r ,r r r 2 4 2 2r 2 Rr r 4 e 2( )U ,r ,r 2r 2 ,r ,r 2 ,r , , 2 ,r cot 2 r 1 r 2 e 2 g AB RAB 2V,r r 4 e 2( ) (U ,r ) 2 r 2U ,r 4rU , r 2U ,r cot 2 4rU cot 2e 2 ( ) 1 (3 , , ) cot , , ( , ) 2 2 , ( , , ) r 2 e 2 g R 2r (r ) ,ur (1 r ,r )V,r (r ,rr ,r )V r (1 r ,r )U , r 2 (cot , )U ,r e 2 ( ) 1 (3 , 2 , ) cot , 2 , ( , , ) rU 2r ,r 2 , r ,r cot 3 cot Hypersurface equations: hierarchical set for ,r ,U ,r and V,r Evolution equation for (r ),ur Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 The light-cone problem is formulated in the region of spacetime between a timelike worldtube at the origin of the radial coordinate and future null infinity. Initial data for are prescribed on an initial light cone u=0. Boundary data for , U, V and are also required on the worldtube. For the general relativistic hydrodynamics equations we use a covariant (form invariant respect to the spacetime foliation) formulation developed by Papadopoulos and Font (PRD, 61, 024015 (2000)) which casts the equations in flux-conservative, first-order form. Gravitational waves at null infinity: Null code test: time of bounce • Bondi news function (from the metric variables expansion at scri) • Approximate gravitational waves (Winicour 1983, 1984, 1987): • Quadrupole news 150 x Grid 1 : r 1 x4 • First moment of momentum formula Grid 2 : r 100 tan x 2 Time of bounce: 39.45 ms (null code 1), 38.32 ms (CFC code), 38.92 ms (null code 2). Good agreement between independent codes (less than 1% difference). Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 Gravitational waves: consistency & disagreement Good agreement in the computation of the GW strain using the quadrupole moment and the first moment of momentum formula. Equivalence valid in the Minkowskian limit and for small velocities, which explains the small differences. But Siebel et al (2002) found excellent agreement between the quadrupole news and the Bondi news when calculating GWs from pulsating relativistic stars. Quadrupole news A possible explanation: different velocities involved in both scenarios, 10-5- rescaled by a factor 10-4c for a pulsating NS and 0.2c in core collapse. 50. Functional form for the quadrupole moment established in the slow motion limit on the light cone may not be valid. Large disagreement between Bondi news and quadrupole news, both in amplitude and frequency of the signal. Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 3+1 General Relativistic (Ideal) Magnetohydrodynamics equations (1) GRMHD: Dynamics of relativistic, electrically conducting fluids in the presence of magnetic fields. Ideal GRMHD: Absence of viscosity effects and heat conduction in the limit of infinite conductivity (perfect conductor fluid). The stress-energy tensor includes the contribution from the perfect fluid and from the magnetic field measured by b observer comoving with the fluid. the T TPF TEM T hu u p g b b TPF hu u pg with the definitions: 1 1 TEM F F g F F u u g b 2 b b b 2 b b 4 2 b2 F u b p p 2 Ideal MHD condition: F u 0 b2 electric four-corrent must h h be finite. J qu F u Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 3+1 General Relativistic (Ideal) Magnetohydrodynamics equations (2) 1. Conservation of mass: ( u ) 0 Adding all up: first- order, flux- 2. Conservation of energy and momentum: T 0 conservative, hyperbolic system of 0 , F u B u B 1 3. Maxwell’s equations: F balance laws W • Induction equation: 1 t B v B + constraint (divergence-free condition) • Divergence-free constraint: B 0 i 1 u g f s Bi 0 g t x x i i 0 D ~i gj Dv T gj S j i 2 ~i h W v j v p j b b j i i x u f 2 i s k h W v p i / Dv i b 0bi ~ ~ 0 ln T 0 T B ~ ~ x v i B k v k Bi k 0 Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 Solution procedure of the GRMHD equations i 1 u g f Bi s 0 t g x i x i Constrained transport scheme (Evans & Hawley 1988, • Same HRSC schemes as for GRHD equations Tóth 2000). (HLL, Kurganov-Tadmor, Roe-type) • Wave structure information obtained Field components defined at cell interfaces. Zone- • Primitive variable recovery more involved centered vector (needed for primitive recovery and cell reconstruction & Riemann problem) obtained from Details: Antón, Zanotti, Miralles, Martí, Ibáñez, Font & Pons, in preparation (2005) staggered field components: Update of field components: Bi , j x 1 2 Bix j Bix 1, j , t B x n 1 i, j Bix j , n y i , j 1 i , j t B y n 1 i, j Biy j , n x i 1, j i , j i , j 4 1 ˆy x ˆ , ˆ ˆ f Bi 1, j f y Bix j f x Biy j 1 f x Biy j , , These equations conserve the discretization of B Bix 1, j Bix j Biy j 1 Biy j B , , , xi , j yi , j i, j Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 GRMHD equations: code tests (1) 1D Relativistic Brio-Wu shock tube test (van Putten 1993, Balsara 2001) Dashed line: wave structure in Minkowski spacetime at time t=0.4 HLL solver Open circles: nonvanishing lapse function (2), at time t=0.2 1600 zones Open squares: nonvanishing shift vector (0.4), at time t=0.16 CFL 0.5 Agreement with previous authors (Balsara 2001) regarding wave locations, maximum Lorentz factor achieved, and numerical smearing of the solution. Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 GRMHD equations: code tests (2) Magnetized spherical accretion onto a Schwarzschild BH density Test difficulty: keeping the stationarity of the solution Used in the literature (Gammie et al 2003, De Villiers & Hawley 2003) Initial data: Magnetic field of the type b b , b ,0,of the on top 0 t r hydrodynamic (Michel) accretion solution. Radial magnetic internal energy field component chosen to satisfy divergence-free condition, and its strength is parametrized by the ratio: 2p b2 radial velocity HLL solver 100 zones =1 Solid lines: analytic solution Circles: numerical solution radial magnetic field (t=350M) Increasing the grid resolution shows that code is second-order convergent irrespective of the value of Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 GRMHD equations: code tests (3) Magnetized equatorial Kerr accretion (Takahashi density et al 1990, Gammie 1999) Test difficulty: keeping the stationarity of the solution (algebraic complexity augmented, Kerr metric) Used in the literature (Gammie et al 2003, De Villiers & Hawley 2003) Inflow solution determined by specifying 4 conserved quantities: the azimuthal mass flux FM, the angular momentum flux FL, the energy flux FE, and the velocity component F of the electromagnetic tensor. a=0.5 FM=-1.0 FL=-2.815344 radial magnetic FE=-0.908382 field F=0.5 HLL solver azimuthal magnetic field Solid lines: analytic solution Circles: numerical solution (t=200M) second order convergence Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 GRMHD: spherical core collapse simulation As a first step towards relativistic magnetized core collapse simulations we employ the test (passive) field approximation for weak magnetic field. • magnetic field attached to the fluid (does not backreact into the Euler-Einstein equations). • eigenvalues (fluid + magnetic field) reduce to the fluid eigenvalues only. HLL solver + PPM, Flux-CT, 200x10 zones The divergence-free condition is fulfilled to good The amplification factor of the initial magnetic field precision during the simulation. during the collapse is 1370. Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005 Summary of the talk • Multidimensional simulations of relativistic core collapse feasible nowadays with current formulations of hydrodynamics and Einstein’s equations. • Results from CFC and CFC+ relativistic simulations of rotational core collapse to NS in axisymmetry. Comparisons with full GR simulations show that CFC is a sufficiently accurate approach. • Modification of the original CFC equations to allow for collapse to high density NS and BH formation. • Ongoing work towards extending the SQF for GW extraction (1PN quadrupole formula). • Axisymmetric core collapse simulations using characteristic numerical relativity show important disagreement in the gravitational waveforms between the Bondi news and the quadrupole news. • 3d extension of the CFC core collapse code through the MdM approach (HRSC schemes for the hydro and spectral methods for the spacetime). • First steps towards GRMHD core collapse simulations (ongoing work) Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005

DOCUMENT INFO

Shared By:

Categories:

Tags:

Stats:

views: | 7 |

posted: | 7/31/2011 |

language: | English |

pages: | 49 |

OTHER DOCS BY pengxuebo

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.