Improved simulations of relativistic stellar core collapse

Institute for Pure and Applied Mathematics, University of California, Los Angeles, May 2005

Improved simulations of relativistic stellar core collapse

José A. FontDepartamento 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)


• 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

Outline of the talk


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.

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.


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.

Introduction: supernova core collapse in a nutshell


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).

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!)

Introduction (continued)

Basic dynamics of the collapse at a glance: 1d core collapse simulations

Romero et al 1996 (radial gauge polar slicing). Purely hydrodynamical (prompt mechanism) explosion. No microphysics or -transport included!


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


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.


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)

Supernova codes vs core collapse numerical relativity codes

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


3+1 General Relativistic Hydrodynamics equations (1)

pguhuT

[1] ),( pp

[1] 0)( u

[4] 0 T

Perfect fluid stress-energy tensor

Equations of motion:

local conservation laws of density current (continuity equation) and stress-energy (Bianchi identities)

Introducing an explicit coordinate chart:

TgTgx

ugx

)(

0)(

Different formulations exist depending on:

1. The choice of time-slicing: the level surfaces of can be spatial (3+1) or null (characteristic)

2. The choice of physical (primitive) variables (, , ui …)

Wilson (1972) wrote the system as a set of advection equation within the 3+1 formalism. Non-conservative.

Conservative formulations well-adapted to numerical methodology are more recent:

• Martí, Ibáñez & Miralles (1991): 1+1, general EOS

• Eulderink & Mellema (1995): covariant, perfect fluid

• Banyuls et al (1997): 3+1, general EOS

• Papadopoulos & Font (2000): covariant, general EOS

0x



[1] ),( pp

[1] 0)( u

jdxidxij

dtidxi

dtii

ds 2)( 222

Foliate the spacetime with t=const

spatial hypersurfaces t

Let n be the unit timelike 4-vector orthogonal to St such that

u: fluid’s 4-velocity, p: isotropic pressure, : rest-mass density : specific internal energy density, e=( 1+): energy density

)(1

n ii

t

[10] 82

1 TRgR

[4] 0 T

Einstein’s equations

Eulerian observersun

nv

i

i

t

ii

u

uv

1

n

t


phWE

vhWS

WD

vw jji

2

2 ,,

)1/(12

jjvvW

)( )()(1

wsx

wfg

t

wu

g i

i

DESDwu j ,,)(

i

iii

j

ii

j

iii pvvDEpvSvDwf

,,)(

Replace the “primitive variables” in terms of the “conserved variables” :

where

00 ln,,0)(

Tx

Tgx

gTws j

j

p

h 1

fluxes

sources

Banyuls et al, ApJ, 476, 221 (1997)

Font et al, PRD, 61, 044011 (2000)

is the vector of conserved variables


First-order flux-conservative hyperbolic system


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:

)),,,(ˆ

),,,(ˆ

( 1111 n

qjn

rjn

rjn

qjn

rjn

rjnj

nj uuufuuuf

x

tuu

where is a consistent numerical flux function: )(),,,(ˆ

ufuuuf

f̂



1

)),((1ˆ

2/12/1

n

n

t

t jj dttxuft

f

],[],[ 12/12/1

nn

jj ttxx

),( 2/1 txu j

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 the numerical flux function is an approximation to the time-averaged flux across the interface:

),( 2/1 txu j

),;0(),( 12/1nj

njj uuutxu

The flux integral depends on the solution at the numerical interfaces during the time step

Key idea: a possible procedure is to calculate by solving Riemann problems at every cell interface (Godunov)

Riemann solution for the left and right states along the ray x/t=0.

When a Cauchy problem described by a set of continuous PDEs is solved in a discretized form the numerical solution is piecewise constant (collection of local Riemann problems).


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)


cell boundaries where fluxes are required

shock frontrarefaction wave

Solution at time n+1 of the two Riemann problems at the cell boundaries xj+1/2 and xj-1/2

(Piecewise constant) Initial data at time n for 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

n

jnj

nj

nj ff

x

tuu 2/12/1

1 ˆˆ



Approximate Riemann solversIn 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 (linearized) Riemann solvers.

Based on the exact solution of Riemann problems corresponding to a new system of equations obtained by a suitable linearization of the original one. The spectral decomposition of the Jacobian matrices is on the basis of all solvers.

Approach followed by an important subset of shock-capturing schemes, the so-called Godunov-type methods (Harten & Lax 1983; Einfeldt 1988).

Roe-type SRRS (Martí et al 1991; Font et al 1994)

HLLE SRRS (Schneider et al 1993)

Exact SRRS (Martí & Müller 1994; Pons et al 2000)

Two-shock approximation (Balsara 1994)

ENO SRRS (Dolezal & Wong 1995)

Roe GRRS (Eulderink & Mellema 1995)

Upwind SRRS (Falle & Komissarov 1996)

Glimm SRRS (Wen et al 1997)

Iterative SRRS (Dai & Woodward 1997)

Marquina’s FF (Donat et al 1998)

Martí & Müller, 2003

Living Reviews in Relativity www.livingreviews.org


A standard implementation of a HRSC scheme

n

jnj

nj

nj ff

x

tuu 2/12/1

1 ˆˆ

nn

nnLiRii Rwfwff

~~~)()(

2

1ˆ 5

1

5

1

~~)U()U(n

nnLR Rww

1. Time update: Conservation form algorithm

In practice: 2nd or 3rd order time accurate, conservative Runge-Kutta schemes (Shu & Osher 1989)

3. Numerical fluxes: Approximate Riemann solvers (Roe, HLLE, Marquina). Explicit use of the spectral information of the system

2. Cell reconstruction: Piecewise constant (Godunov), linear (MUSCL, MC, van Leer), parabolic (PPM, Colella & Woodward 1984) interpolation procedures of state-vector variables from cell centers to cell interfaces.


HRSC schemes: numerical assessment

Shock tube testRelativistic shock reflection

V=0.99999c (W=224)

• Stable and sharp discrete shock profiles

• Accurate propagation speed of discontinuities

• Accurate resolution of multiple nonlinear structures: discontinuities, raraefaction waves, vortices, etc

Simulation of a extragalactic relativistic jet

Scheck et al, MNRAS, 331, 615 (2002)

Wind accretion onto a Kerr black hole (a=0.999M)

Font et al, MNRAS, 305, 920 (1999)


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)


CFC metric equations

kk

ij

ijii

ijij

ijij

KS

KKPWh

KKPW

3

1216

16

75232

162

64

25

25

In the CFC approximation (Isenberg 1985; Wilson & Mathews 1996) the ADM 3+1 equations

reduce to a system of five coupled, nonlinear elliptic equations for the lapse function, conformal factor, and the shift vector:

08

016

82

2

0022

jijiji

ijij

ijm

ijmm

jimijmmm

jimijijjiijt

ijjiijijt

SKK

TKKKR

TKKKKKKKRK

K

Solver 1: Newton-Raphson iteration. Discretize equations and define root-finding strategy.

Solver 2: Conventional integral Poisson iteration. Exploits Poisson-like structure of metric equations, uk=S(ul). Keep r.h.s. fixed, obtain linear Poisson equations, solve associated integrals, then iterate until nonlinear equations converge.

Both solvers feasible in axisymmetry but no extension to 3D possible.

ijij 4CFC


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


“

transi

tion”

Type II “m

ult

iple

bounce

” T

ype I “

reg

ula

r”

Central Density Gravitational WaveformHRSC scheme:

PPM + Marquina flux-formula

Solid line: relativistic simulation

Dashed line: Newtonian

Larger central densities in relativistic models

Similar gravitational radiation amplitudes (or smaller in the GR case)

GR effects do not improve the chances for detection (at least in axisymmetry)


Gravitational Wave Signalswww.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!


CFC+ metric equations

0 tr, TTTTij

CFCij

CFCij hh

kk

ij

ijii

ijTTij

ijij

ijij

KS

Uhc

KKPWh

KKPhW

3

1216

1

16

75232

162

64

225

25

CFC+ metric:

The second post-Newtonian deviation from isotropy is the solution of:

64

1)416(

1

cUUvv

ch lklk

TTklij

TTij

Modified equations for , i and (with respect to CFC):

Cerdá-Duran, Faye, Dimmelmeier, Font, Ibáñez, Müller, and Schäfer, A&A, in press (2005)

(complicated) transverse, traceless projection operator Newtonian potential

(ADM gauge)

(Schäfer 1990)


CFC+ metric equations (2)

TTijh

lk

lkii

ij

jj

ji

jiji

jijiij

jjijii

xxUUR

xUUvvT

xxvvS

UUvvS

xUUvvS

)4(

4

4

)4(

jiijklijlk

kijk

ji

kki

jjikji

kkkijij

TTij

RSSxxSxT

SxSSxSSh

4

1

4

1

4

1

2

1

4

1

4

133

4

7

2

1)()(

We can solve the equations by introducing some intermediate potentials:

16 elliptic linear equations

Linear solver: LU decomposition using standard LAPACK routines

2

2

2

23

23

23

2

23

1

1

2

1

11

1

2

1

1

2

1

rOn

r

MR

rOn

r

MxdUxxvv

rT

rOxdxxvv

rS

rOxdUxvv

rS

rOn

r

MxdUxUxxvv

rS

ii

iiikk

i

lklk

ij

ijjiij

ik

kikk

ii

Boundary conditions

Multipole development in compact-supported integrals


CFC+ results: rotating neutron stars

Model Axis ratio K

Mass

(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

Initial models (KEH method)

Equatorial profiles of the non-vanishing components of hij for the sequence of rigidly rotating models RNS0 to RNS5

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 code)


CFC+:radial modes of spherical NS quasi-radial modes of 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)

spheri

cal N

S

rota

ting N

S


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.


CFC+: core collapse dynamics (2)

Type II (multiple bounce) Extreme case (torus-like structure)


CFC+: core collapse waveforms

r

crtAtxh

/sin

15

8

1,

2E202quad

From the quadrupole formula:

From the metric hij:

txh

r

crtAcrtxh

,8

1

/sin

15

64

1/,

quad

E2202PN2

Two distinct ways to extract waveforms:

TTPN2corrected-PN2ij

ijhahh

Offset correction (dashed line)

Absolute differences between CFC and CFC+ waveforms.

No significant differences found.


“Mariage des Maillages”HRSC schemes for hydrodynamics and spectral methods for metric

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!

Reference: Dimmelmeier, Novak, Font, Müller, Ibáñez, PRD 71, 064023 (2005)

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.


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.

1. Domain radius rd must follow contraction.

2. Domain radius rd should stay fixed at roughly rpns after core bounce.

3. More than 3 domains needed in dynamical core collapse.

4. Compare with previous solvers in axisymmetry: 33 collocation points per domain sufficient.

rd held fixed (10% initial rse)

final rd too largebounce time

only 3 radial domains

Gibbs-type oscillations

wrong result!


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 resolution3d low resolution without artificial perturbation

Axisymmetric oscillations in rotating neutron stars 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.


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


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 models with parameters close (but not equal) to the ones used by Dimmelmeier, Font & Müller (2002). 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 extraction method (W6) and the formulation (stress formulation vs first moment of momentum density formulation).

A3B2G4 (DFM 02)

W6 (20% gain at bounce!)

The qualitative difference found by Shibata & Sekiguchi (2004) is due to the differences in the collapse initial model, notably the small decrease of the differential rotation length scale in their model.

A3B2G4 (A/rse=0.32)

Shibata & Sekiguchi (A/rse=0.25)

A3B2G4

Shibata & Sekiguchi


CFC metric equations: modification to allow for black hole formation

kk

ij

ijii

ijij

ijij

KS

KKPWh

KKPW

3

1216

16

75232

162

64

25

25

kk

ij

ijii

ijij

ijij

KS

KKPWh

KKPhW

3

1216

16

75532

162

62

521

521

Original CFC equations

PP

SS ii

6*

6*

6*

It turns out to be essential to rescale some of the hydro quantities with the appropriate power of the conformal factor for the elliptic solvers to converge to the correct solution:

kkt

6

*** , , PS iTo obtain one needs to first compute the conformal factor, which is obtained from the evolution equation


Black hole formation (spherical symmetry)

High central density TOV solution

with rescaling

without rescaling

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.


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)

min=0.42

max=1.4x1015

GW amplitude larger at bounce with CFC+


Core collapse simulations using the Einstein equations for the Bondi metric

222222222222222 sin22 dederdudeUrdudreduerUer

Vds

and , scoordinate on the dependonly and ,, functions metric The ruUV

cot3cot22

)(2cot)23(1)(cot

)1()()1()(2

)(2)(cot)3(12cot4

cot4)(2

12

cot22

222

24

,,,

,,,,,,)(2

,,2

,,,,,,,22

,,,2

,,,,,)(2

,2

,,22

,)(24

,22

,,,,,,2

,,)(242

2,,

rr

r

rrrrrrur

rrrrABAB

rrrrrrr

rrrr

rrrU

eUr

UrrVrVrrrRger

erU

UrrUUrUerVRger

rrUerRr

rR

r

ababbaabababab pgguuhRRgRG

2

1

2

1

Hypersurface equations: hierarchical set for rrr VU ,,, and ,urr ,)( Evolution equation for

Ricci tensor

Reference: Siebel, Font, Müller, and Papadopoulos, PRD 67, 124018 (2003)


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:

• Bondi news function (from the metric variables expansion at scri)

• Approximate gravitational waves (Winicour 1983, 1984, 1987):

• Quadrupole news

• First moment of momentum formula

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).

xr

x

xr

2tan100 :2 Grid

1

150 :1 Grid

4

Null code test: time of bounce


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.

Large disagreement between Bondi news and quadrupole news, both in amplitude and frequency of the signal.

Quadrupole news rescaled by a factor 50.

But Siebel et al (2002) found excellent agreement between the quadrupole news and the Bondi news when calculating GWs from pulsating relativistic stars.

A possible explanation: different velocities involved in both scenarios, 10-5-10-4c for a pulsating NS and 0.2c in core collapse.

Functional form for the quadrupole moment established in the slow motion limit on the light cone may not be valid.


3+1 General Relativistic (Ideal) Magnetohydrodynamics equations (1)

bbbguuFFgFFT

pguhuT

TTT

2EM

PF

EMPF

2

1

4

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 the observer comoving with the fluid.

0

uF

buF

bbgpuuhT

2

2

2

2

bhh

bpp

bbb

b

with the definitions:

Ideal MHD condition: electric four-corrent must be finite.

q uFuJ


3+1 General Relativistic (Ideal) Magnetohydrodynamics equations (2)

1. Conservation of mass:

2. Conservation of energy and momentum:

3. Maxwell’s equations:

• Induction equation:

• Divergence-free constraint:

BvBt

1

0)( u

0 T

BuBu

WFF 1

, 0

0 B

0 1

i

i

i

i

x

Bs

x

fg

t

u

g

k

j

B

S

D

u

ikki

iiiij

iij

ij

i

i

BvBv

bbvDpvWh

bbpvvWh

vD

f

~~

~/~

~

~

02

2

k

jj

Tx

T

gx

gT

s

0

ln

0

00

Adding all up: first-order, flux-conservative, hyperbolic system of balance laws

+ constraint (divergence-free condition)


Solution procedure of the GRMHD equations

0

i

i

x

B

Constrained transport scheme (Evans & Hawley 1988, Tóth 2000).

Field components defined at cell interfaces. Zone-centered vector (needed for primitive recovery and cell reconstruction & Riemann problem) obtained from staggered field components:

xji

xji

xji BBB ,1,, 2

1Update of field components:

yji

xyji

xxji

yxji

yji

jiji

nyji

nyji

jiji

nxji

nxji

BfBfBfBf

x

tBB

y

tBB

,1,,,1,

,,1,

1

,

,1,,

1

,

ˆˆˆˆ4

1

sx

fg

t

u

g i

i

1

These equations conserve the discretization of B

ji

yji

yji

ji

xji

xji

ji y

BB

x

BBB

,

,1,

,

,,1,

• Same HRSC schemes as for GRHD equations (HLL, Kurganov-Tadmor, Roe-type)

• Wave structure information obtained• Primitive variable recovery more involved

Details: Antón, Zanotti, Miralles, Martí, Ibáñez, Font & Pons, in preparation (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=0.4Open circles: nonvanishing lapse function (2), at time t=0.2Open squares: nonvanishing shift vector (0.4), at time t=0.16

HLL solver1600 zonesCFL 0.5

Agreement with previous authors (Balsara 2001) regarding wave locations, maximum Lorentz factor achieved, and numerical smearing of the solution.



Magnetized spherical accretion onto a Schwarzschild BH

Test difficulty: keeping the stationarity of the solutionUsed in the literature (Gammie et al 2003, De Villiers & Hawley 2003)

Initial data: Magnetic field of the type on top of the hydrodynamic (Michel) accretion solution. Radial magnetic field component chosen to satisfy divergence-free condition, and its strength is parametrized by the ratio:

0,0,, rt bbb

2

2

b

p

HLL solver100 zones=1

Solid lines: analytic solutionCircles: numerical solution (t=350M)

Increasing the grid resolution shows that code is second-order convergent irrespective of the value of

density

internal energy

radial velocity

radial magnetic field



Magnetized equatorial Kerr accretion (Takahashi 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 mass flux FM, the angular momentum flux FL, the energy flux FE, and the component F of the electromagnetic tensor.

a=0.5FM=-1.0FL=-2.815344FE=-0.908382F=0.5

HLL solver

Solid lines: analytic solutionCircles: numerical solution (t=200M)

density

radial magnetic field

azimuthal velocity

azimuthal magnetic field

second order convergence


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 precision during the simulation.

The amplification factor of the initial magnetic field during the collapse is 1370.


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)

Improved simulations of relativistic stellar core collapse

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