A light-cone approach to axisymmetric stellar core collapse José A. Font Departamento de Astronomía y Astrofísica Universidad de Valencia (Spain) EU TMR Network Meeting, Trieste, 26 September 2003 Work done in collaboration with:  Florian Siebel (PhD thesis project)  Philippos Papadopoulos  Ewald Müller

 Introduction  Mathematical framework and implementation  Initial data  Core collapse dynamics  Gravitational waves  Conclusion Plan of the Talk Further information: Siebel, Font, Müller & Papadopoulos, Phys. Rev. D 67, (2003)

Introduction Supernovae have always been considered among the most important sources of gravitational radiation. GWs from core collapse supernovae were first numerically computed using Newtonian gravity (e.g. Müller 1982, Finn & Evans 1990, Mönchmeyer et al 1991, Bonazzola & Marck 1993, Zwerger & Müller 1997, Ott et al 2003). More recently, effects of general relativity have been included under the simplifying assuption of a conformally flat spatial metric (Dimmelmeier, Font & Müller, 2002) Further extensions towards full GR are currently being made, e.g. Pablo Cerda’s talk (CFC+) and Shibata & Sekiguchi axisymmetric simulations in full GR (in preparation) In all existing works GWs are not computed instantly within the numerical simulation but extracted a posteriori using the approximation of the quadrupole formalism. Motivation for our project: study the dynamics of stellar core collapse using characteristic numerical relativity, foliating the spacetime with a family of outgoing light cones emanating from a regular center. A suitable compactification of the global spacetime allows to include future null infinity on the numerical grid, where GWs can unambiguosly be extracted (Bondi news)

The Einstein equations for the Bondi metric Hypersurface equations: hierarchical set for Evolution equation for Ricci tensor

Compactification and regularity at the poles (1) Coordinate transformations and new variables: New metric: Hypersurface equations:

Compactification and regularity at the poles (2) Evolution equation: wave equation for the quantity with 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.

where fluxes source terms We use a covariant formulation developed by Papadopoulos & Font PRD, 61, (2000); gr-qc/ conserved variables General Relativistic Hydrodynamics equations First-order flux-conservative hyperbolic system (in axisymmetry):

Numerical implementation: field equations Wave equation: for the time update of we have implemented two algorithms: 1.The parallelogram algorithm based on ingoing and outgoing characteristics (Gómez, Papadopoulos & Winicour 1994) 2.A dissipative algorithm (Lehner 1999) Hypersurface equations: 1.The equations for are discretized as 2.To solve the equation for we discretize the alternative equation The derivative ensures regularity at the origin.

Numerical implementation: hydrodynamics 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) interpolation procedures of state-vector variables from cell centers to cell interfaces.

Gravitational waves at null infinity Bondi news function: The quantities K, c, H and L are read off from the metric variables expansion at scri, e.g. Approximate gravitational waves (Winicour 1983, 1984, 1987): Quadrupole moment in axisymmetry: Quadrupole news: First moment of momentum formula: Relation between quadrupole strain (i.e. gravitational wave signal) and the Bondi news:

Equation of state Hybrid EoS to include the effect of stiffening at nuclear densities and the effect of thermal heating due to the appearance of shocks (Janka et al 1993; Dimmelmeier et al 2002). change discontinuously at nuclear density from Degenerate electron gas Mixture of relativistic and non-relativistic gases

Initial data Iron core: solution of the Tolman-Oppenheimer-Volkoff equation with Non-radial quadrupolar perturbations added on top of the spherical data Different types of initial models: 1.Unperturbed spherical model 2.Perturbation of the rest-mass density 3.Perturbation of the meridional velocity component Collapse initiated by setting which mimics the softening of the EoS

Code tests (1) Shock reflection test Ultrarelativistic flow with: Nonequidistant radial grid with: MC slope limiter HLLE approximate Riemann solver 800 radial zones

Code tests (2): convergence Thermal energy during the infall phaseTime of bounce Deviations from zero converge to zero with a convergence rate of 2. Time of bounce: ms (null code 1), ms (CFC code), ms (null code 2). Good agreement between independent codes (less than 1% difference).

Core collapse dynamics (1) Time evolution of the central density Spacetime diagram for the collapse Collapse phase: central density increases by 4.5 orders of magnitude. Collapse stops at about 40 ms (stiffening in the hybrid EoS). New equilibrium supra-nuclear value. Lapse of proper time vs radius. One light cone shown after every 5ms. Mass shells trajectories. A shock forms after about 40 ms close to the origin, and then travels out.

Core collapse dynamics (2) Snapshots of radial velocity profiles Surface plot of the rest mass density around the shock front The shock front is resolved with only three radial zones. The aspherical nature of the data is most prominent at the shock front. Snapshots taken between Bondi time 30 ms and 45 ms, with a 1 ms delay between subsequent outputs.

Core collapse dynamics (3)

Fluid oscillations in the outer core (1st surprise) The meridional velocity oscillates strongly in the entire region in front of the shock. Information propagates through the metric (sound waves would need several 10 ms to cover the distance) Oscillations are created by our choice of coordinates, i.e. gauge effect. Following Bishop et al (1997) inertial coordinates can be established at future null infinity. It is possible to define an “inertial” meridional velocity for which the oscillations almost entirely disappear. Gauge effects can thus play a major role for the oscillations in the pre-shock region. Meridional velocity component vs Bondi time at the fixed location r=833 km and y=0.5 (well in front of the shock)

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 the Bondi news and the quadrupole news, both in amplitude and frequency of the signal. Quadrupole news rescaled by a factor 50.

But... agreement for pulsating neutron stars 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 can be found in the different velocities involved in both scenarios, c for a pulsating NS and 0.2c in the core collapse and bounce. The functional form for the quadrupole moment established in the slow motion limit on the light cone may not be valid. (Analogy: total mass of the spacetime and Bondi mass are equal only for vanishing fluid velocities) Siebel, Font, Müller & Papadopoulos (2002)

Evidence against the quadrupole news signal We have strong evidence that the quadrupole news signal extracted from our collapse simulations do not correspond to physical GW signals: 1. If it were the true signal, then it is difficult to understand why the Bondi signal has a significantly smaller amplitude (the various terms contribution to the Bondi news are relatively large and add up to a small signal; errors cancellation unlikely) 2. Comparisons with the CFC results by Dimmelmeier et al (2002) yield much larger quadrupole signals amplitudes in our case (CFC results are in excellent agreement with recent full GR results by Shibata & Sekiguchi) 3. Physically motivated argument given by the spatial distribution of matter in our simulations. Main contribution in the radial integral of the quadrupole moment comes from the outer, infalling layers of matter (gauge effects)

Bondi news signal: difficulties The extraction of the Bondi news signal is highly non-trivial since: 1. it involves calculating non-leading terms from the metric expansion at future null infinity 2. in the Tamburino-Winicour approach one has to take into account gauge effects. In our simulations these effects are the dominant contribution. Different contributions to the Bondi news: summing up all the signal is close to zero. When separating the third addend into angular derivatives of H and w, each single contribution has an amplitude 23 times larger than that shown in the figure. High frequency numerical noise visible.

Bondi news signal: high-frequency noise The metric quantities show high frequency noise as soon as the shock forms at bounce (at a Bondi time of about 40 ms). Location of the shock and the Bondi news signal. Noise created by the motion of the shock across the grid (discontinuous jumps between adjacent grid zones). A small, localized effect has a large impact on the gravitational wave signal, due to the radial integration of the metric variables at every time step which propagates information to future null infinity instantaneously.

Gravitational waves: adding all up Bondi news as a function of Bondi time for two representative models At bounce the Bondi news shows a spike The oscillations of the signal are associated with the oscillations of the newly- formed neutron star and the propagation of the shock to the exterior Typical oscillation frequencies are of the order kHz The gravitational wave energy is of the order of Mc 2

Summary Simulations of supernova core collapse and extraction of the associated exact gravitational waves using the characteristic formulation of general relativity is possible. Our code is accurate enough to allow for a detailed analysis of the global dynamics of the collapse. We have not found a robust method for the Bondi news GW extraction in the presence of strong shock waves (no further work reported in the literature). Possible future improvements: 1.Incorporate rotation: GW signals of larger amplitude. Numerical noise less important. 2.Rearrange the metric equations to diminish the importance of high-order derivatives. 3.Implement pseudospectral methods for the metric update. 4.Include adaptive grids and shock-fitting methods (shock location fixed during the evolution).