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Towards relativistic magneto-rotational core collapse simulations

Towards relativistic magneto-rotational core collapse simulations. José A. Font. Departamento de Astronomía y Astrofísica Universidad de Valencia (Spain). Work done in collaboration with Pablo Cerdá-Durán (UVEG). Outline of the talk. Introduction and motivation Framework:

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Towards relativistic magneto-rotational core collapse simulations

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  1. Towards relativistic magneto-rotational core collapse simulations José A. Font Departamento de Astronomía y Astrofísica Universidad de Valencia (Spain) Work done in collaboration with Pablo Cerdá-Durán (UVEG)

  2. Outline of the talk • Introduction and motivation • Framework: • General relativistic magneto-hydrodynamics • Test magnetic field approximation • Gravitational field (Einstein equations in the CFC approximation) • Numerical approach • Relativistic magneto-rotational core collapse simulations: • Initial data • Code tests • Collapse dynamics and B-amplification mechanisms • Gravitational waveforms • Magneto-rotational instability • Long-term evolution of the B-field • Summary and outlook

  3. Introduction and motivation The study of gravitational stellar collapse has traditionally been one of the primary problems in relativistic astrophysics. It is a distinctive example of a research field in astrophysics where essential progress has been accomplished through numerical modeling with increasing levels of complexity in the input physics/mathematics. Basic set of equations to solve: Hydrodynamics + Gravitational field (+ B-fields + transport + …). The GR-Hydro equations (as well as the GRMHD counterpart) constitute a nonlinear hyperbolic system. There exist solid mathematical foundations and accurate numerical methodology imported from CFD. In recent years, the so-called (upwind) high-resolution shock-capturing schemes (or Godunov-type methods), based upon approximate Riemann solvers, have been successfully extended from classical to relativistic fluid dynamics(see e.g. Martí & Müller 2003 and Font 2003 articles at Living Reviews in Relativity). While such advances also hold true in the case of the MHD equations, the development still awaits here for a thorough numerical exploration. But sure enough, numerical GRMHD is a hot topic these days …

  4. Motivation: intense work in recent years on formulating/solving the MHD equations in general relativistic spacetimes (either background or dynamical). Pioneers:Wilson (1975), Sloan & Smarr (1985), Evans & Hawley (1988), Yokosawa (1993) More recently:Koide et al (1998 …), De Villiers & Hawley (2003 …), Baumgarte & Shapiro (2003), Gammie et al (2003), Komissarov (2005), Duez et al (2005 …), Shibata & Sekiguchi (2005), Antón et al (2006), Neilsen et al (2006). Both, artificial viscosity and HRSC schemes developed. Most of the applications are in the field of black hole accretion and jet formation … Development of the MRI in a magnetised torus around a Kerr black hole(Gammie, McKinney & Tóth 2003) … many others under way (you name it!) Jet formation: the twisting of magnetic field lines around a Kerr black hole. The yellow surface is the ergosphere (Koide et al 2002) The number of groups working in special relativistic MHD is even larger: Komissarov; Balsara; Koldoba et al; Del Zanna et al; Leisman et al; … Exact solution of the SRMHD Riemann problem found recently: Romero et al (2005) – particular case; Giacomazzo & Rezzolla (2005) – general case.

  5. State-of-the-art of core collapse simulations (table courtesy of Harry Dimmelmeier) Talks by Dimmelmeier and Ott will cover part of the ongoing work. Neutron stars have intense magnetic fields (1012-1013 G) or even larger at birth (1014-1015 G), inferred from studies of anomalous X-ray pulsars and soft gamma-ray repeaters (Kouveliotou et al 1998). In magnetars (extreme case) the B-field can be so strong to affect the internal structure of the star (Bocquet et al 1995). Performing GR-MHD core collapse simulations is the next natural step …

  6. Status of magneto-rotational core collapse(in a nutshell) • Proposed as a possible supernova explosion mechanism by Müller & Hillebrandt (1979); magnetohydrodynamical shock. • Initial conditions for magneto-rotational core collapse(Heger et al 2005):initial rotation rates are more than one order of magnitude smaller than • those used in past parameter studies of magneto-rotational core collapse (e.g. Dimmelmeier, Font & Müller 2002) • those predicted by previous evolutionary calculations. • Strength and distribution of the initial magnetic field in the core unknown. If initially weak, there exist several amplification mechanisms: • differential rotation (-dynamo; transforms rotational energy into magnetic energy, winding up any seed poloidal field into toroidal field). • MRI, exponential growth of the field strength. It occurs if the radial gradient of angular velocity is negative, which holds in core collapse simulations. • Magnetic fields (torques) can affect the collapse dynamics in a major way by redistributing angular momentum(Meier et al 1976; Spruit & Phinney 1998; Spruit 2002; Heger et al 2004), which can slow down the forming PNS. Slowly-rotating neutron stars at birth (10-15 ms). • Magneto-rotational core collapse may lead to jet-like explosions (distinctive GW signature; Obergaulinger et al 2006). • No relativistic magneto-rotational core collapse simulations yet available.

  7. Ideal MHD simulations, 2D, Newtonian physics + microphysics: very active field in recent years:Wheeler et al 2003; Akiyama et al 2003, Kotake et al 2004a,b, 2005; Takiwaki et al 2004; Wheeler & Akiyawa 2004; Yamada & Sawai 2004; Ardeljan et al 2005; Sawai et al 2005. • Magneto-rotational effects on the GW signature from core collapse were first studied in detail by Kotake et al (2004) and Yamada & Sawai (2004). Found differences from the signature of purely hydrodynamical models for very strong initial fields (|B|≥1012 G) • Obergaulinger, Aloy & Müller (2006): most comprehensive parameter study up-to-date of magneto-rotational core collapse. • rotating polytropes, 2D, Newtonian (modified Newtonian) • SQF to compute gravitational waves • ad-hoc initial poloidal |B| fields (no consistent solution known) Weak initial fields (|B|≤1011 G): • change neither collapse dynamics nor resulting GW signal (most relevant case, astrophysics-wise). Strong initial fields (|B|≥1011 G): • slow down core efficiently (even retrograde rotation occurs) • cause qualitatively different dynamics and GW signals • highly bipolar, jet-like outflows occur

  8. GEO LIGO VIRGO Core collapse and gravitational waves Numericalsimulationsof stellar core collapse are nowadays highly motivated by the prospects of direct detection of the gravitational wavesemitted. International network of resonant bar detectors International network of interferometers Do GRMHD effects modify the existing GW signals from hydrodynamical rotational core collapse?

  9. Early core collapse simulations: GR hydrodynamics Equations of motion: local conservation laws of density current and stress-energy • Conservative formulations well-adapted to numerical methodology: • Banyuls et al (1997); Font et al (2000): 3+1, general EOS First-order flux-conservative hyperbolic system Solved using HRSC schemes (either upwind or central)

  10. Numerical schemes: HRSC 1. Time update:Conservation form algorithm 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)interpolation proceduresof state-vector variables from cell centers to cell interfaces. 3. Numerical fluxes:Approximate Riemann solvers (Roe, HLLE, Marquina). Explicit use of the spectral information of the system High-order symmetric (central) schemes also available (Kurganov & Tadmor 2000)

  11. 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, rarefaction waves, vortices, etc V=0.99999c (W=224) Wind accretion onto a Kerr black hole (a=0.999M) Font et al, MNRAS, 305, 920 (1999) Simulation of a extragalactic relativistic jet Scheck et al, MNRAS, 331, 615 (2002)

  12. CFC metric equations 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: 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.

  13. Relativistic non-magnetized core collapse HRSC scheme: PPM + Marquina flux-formula Solid line:(CFC)relativistic simulation Dashed line:Newtonian Larger central density in relativistic models (more compact PNS) Similar gravitational radiation amplitudes Multiple bounce collapse suppressed in GR • Axisymmetric simulations: • Dimmelmeier, Font & Müller (2002) [CFC] Cerdá-Durán et al (2005) [CFC+] Shibata & Sekiguchi (2005) [BSSN] • Excellent agreement • Extensions to realistic EoS and 3D available (see talks by Dimmelmeier and Ott)

  14. CFC versus full General Relativity CFC is indeed an excellent approximation to GR for studying core collapse Dimmelmeier et al (astro-ph/0603760) N: Newtonian potential R: TOV potential A: Modified TOV potential

  15. General relativistic magneto-hydrodynamics (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. with the definitions: Ideal MHD condition: electric four-current must be finite.

  16. General relativistic magneto-hydrodynamics (2) Adding all up:first-order, flux-conservative, hyperbolic system of balance laws + constraint(divergence-free condition) • Conservationof mass: • Conservation of energy and momentum: • Maxwell’s equations: • Induction equation: • Divergence-free constraint: Antón et al. (2006)

  17. Notation and Riemann problem GRMHD equations are rewritten in terms of the conserved quantities inside a coordinate volume flat Nabla operator The associated flux-vector Jacobians in every direction are 7x7 matrices. The solution of the eigenvalue problem (Antón 2006) leads to seven types of waves: Entropic wave: Alfvén waves: Magnetosonic waves (no analytic expression): Characteristics of the GRMHD equations

  18. Test magnetic field approximation Test B-field approximation: to study the collapse of weakly magnetized stellar cores, it is a good approximation to consider that the magnetic field entering in the energy-momentum tensor is negligible when compared with the fluid part. Therefore, the magnetic field evolution (induction equation) does not affect the dynamics of the fluid, which is governed solely by the hydrodynamics equations. However, the evolution of B does depend on the fluid evolution, due to the presence of the velocity field in the induction equation. Practical numerical advantage: the seven eigenvalues of the GRMHD Riemann problem reduce to three (purely hydro) (see Antón et al (2006) for complete expressions)

  19. Solution procedure for the GRMHD equations Constrained transport scheme (Evans & Hawley 1988, Tóth 2000) Flux-CT scheme • Same HRSC schemes as for GRHD equations (HLL, Kurganov-Tadmor, Roe-type) • Wave structure information available • Primitive variable recovery more involved • See Noble et al 2006 for methods comparison • Antón et al 2006 find the roots of an 8-th order polynomial using a 2d-Newton-Raphson method. In the test B-field approximation the primitive recovery is as in the purely GRHD case (as there is no B-field components in the momentum and energy equations)

  20. Magnetic field evolution: flux-CT (1) The B-field evolution is given by the induction equation and the divergence constraint. HRSC schemes used for the hydrodynamics do not preserve the divergence-free condition. An ad-hoc scheme has to be used to update the magnetic field components. Main physical implication of divergence constraint is that the magnetic flux through a closed surface is zero: essential to the constrained transport (CT) scheme(Evans & Hawley 1988, Tóth 2000). For any given surface, the time variation of the magnetic flux across the surface is: Induction equation Stokes theorem The magnetic flux through a surface can be calculated as the line integral of the electric field along its boundary.

  21. Magnetic field evolution: flux-CT (2) Numerical implementation in axisymmetry: Assumption: B-field components constant at each cell surface E-field components constant along each cell edge Evolution equations for the B-field (CT scheme) The polodial (r and ) B-field components are defined at cell interfaces (staggered grid) (axisymmetry condition) The total magnetic flux through the cell interfaces is given by: If the initial data satisfy the divergence constraint, it will be preserved during the evolution

  22. Magnetic field evolution: flux-CT (3) Discretisation: Equations used by the code to update the B-field. The only remaining aspect is an explicit expression for the E-field. The E-field components can be calculated from the numerical fluxes of the conservation equations for the B-field. Done solving Riemann problems at cell interfaces (characteristic information of the flux-vector Jacobians incorporated in the B-field evolution). This procedure is only valid for r and  E-field components. Balsara & Spicer (1999) proposed a practical way to compute the  component of the E-field from the numerical fluxes in adjacent interfaces. Resulting scheme flux-CT

  23. Code tests in strong gravity (black holes) Magnetised spherical accretion onto a Schwarzschild BH density radial velocity Test difficulty: keep stationarity of the solution. Used in the literature (Gammie et al 2003, De Villiers & Hawley 2003) radial magnetic field internal energy 2nd order convergence density radial magnetic field Magnetised equatorial Kerr accretion (Takahashi et al 1990, Gammie 1999) Test difficulty: keep stationarity of the solution (algebraic complexity augmented, Kerr metric) Used in the literature (Gammie et al 2003, De Villiers & Hawley 2003) azimuthal velocity azimuthal magnetic field

  24. Initial data We work with a vector potential such that the associated B-field is divergence-free. Given a vector potential, the B-field components at cell interfaces can be discretized such that the numerical values of the magnetic flux over cells are zero up to round-off error. This value is preserved during the evolution using the flux-CT scheme. 1) Homogeneous “starred” B-field 3) Toroidal “starred” B-field magnetic field at r=0 2) B-field generated by a circular current loop

  25. Toroidal test Let us consider a rotating axisymmetric configuration with no meridional flows Induction equation: -dynamo amplification mechanism (Meier 1976) Equilibrium solution can be found in three cases: Purely toroidal B-field Rigid rotator Special case Useful for code validation! Toroidal test A: circular current loop + non evolving rigidly rotating fluid of constant density Global error in the toroidal B-field minmod Grid resolution: f=1 80x10 f=2 160x20 f=4 320x40 Convergence order: minmod 2.35 MC 2.16  minmod º MC TTB MC TTA + minmod x MC PPM Local convergence order

  26. Poloidal test Let us now consider a fluid with only radial velocities and a purely poloidal magnetic field From the induction equation and the continuity equation, it can be shown that the following equivalence holds in the equatorial plane: We define a Lagrangian coordinate system and check whether does not change with time (as a function of the mass enclosed in a given radius) Spherical collapse of a 4/3 polytrope: time evolution of the error during the infall phase for different reconstructions and grid resolutions t=35 ms Dotted curve: initial profile travelling shock < 1% minmod KT solver PNS boundary

  27. Summary of initial models • Hydrodynamical models follow the notation of Dimmelmeier, Font & Müller (2002). • Magnetised models follow notation of Obergaulinger, Aloy & Müller (2006). • Magnetic field configuration: • Poloidal field generated by a current loop (CL) • Toroidal field (T) • Initial magnetic field strength 1010 G (since we use the test B-field approximation all results can be rescaled to other initial field values) and rmag=400 km. • Last column indicates how stable is initially the B-field configuration. • We focus our discussion on models A1B3G5 (representative of the sample; Cerdá-Durán & Font, in preparation).

  28. A1B3G5-D3M0 dynamics: energy parameters Their evolution allows to quantify the signatures of rotation and B-field on the collapse dynamics. • B-field amplification mechanisms present in our axisymmetric simulations: • radial compression: radial velocity compress B-field lines perpendicular to the velocity; only during infall phase; acts on poloidal and toroidal components. • -dynamo: winds up poloidal field lines into toroidal field lines. The latter are generated and amplified even from purely poloidal initial configurations. The toroidal B-field dominates after core bounce. • --dynamo (Spruit 1999), toroidal  poloidal NOT possible (3D effect). • MRI (Balbus & Hawley 1991) NOT possible (test B-field approximation) total toroidal poloidal RC +   The final mag «1 (and «rot) as the B-field strength is small enough not to affect the dynamics

  29. A1B3G5-D3M0 dynamics: morphology density & velocity poloidal B-field & lines toroidal B-field & orientation time of bounce (t=30 ms) At bounce the initial poloidal configuration is highly distorted by the radial compression of the field lines. Maximum values of the poloidal B-field reached at the center. A toroidal component has formed surrounding the inner core. The travelling shock wave is visible in both the hydrodynamical variables and the magnetic field.

  30. A1B3G5-D3M0 dynamics: morphology density & velocity poloidal B-field & lines toroidal B-field & orientation final time (t=60 ms) At the final time a compact toroidal B-field is confined in the outer layers of the PNS; the inner region is dominated by the poloidal B-field. Convective motions (meridional flows) responsible of the twisting of the poloidal field, which changes direction. changes sign The -dynamo mechanism changes the sign of the toroidal B-field in a shell (B>0) surrounded by toroidal field in the opposite direction (B<0). Such configuration is maintained until the end of simulation, resulting in a linear amplification of the toroidal B-field.

  31. A1B3G5-D3M0 dynamics: field lines Equations of the magnetic field lines bounce final time The box represents the region above the equatorial plane. Axes are in km. Colours are proportional to the magnetic flux in each line (blue=low, green-red=high). • Outside the PNS the B-field is roughly poloidal. • A shell of twisted toroidal field lines formed around the inner core of the PNS. • Poloidal structure of innermost PNS region hidden by the shell. • Longer time simulations required to clarify the stability of such configuration.

  32. A1B3G5-T3M0: dynamics & B-field morphology • Purely toroidal B-fiel initially. Since no poloidal field present initially, toroidal B-field can not be amplified via -dynamo. • mag decreases during evolution as the only amplification mechanism acting is RC (much slower than -dynamo). Final PNS “less magnetised” than progenitor. • Toroidal B-field does not generate poloidal B-field due to axisymmetry (no --dynamo).Drawback of the simulation. • Mixed model (DT3M0):long-term evolution same as D3M0 model. The (initial) toroidal B-field does not amplify by RC nor transforms into poloidal, while the (initial) polodial B-field gets amplified by RC and transformed into toroidal. • If --dynamo were not important, the B-field distribution in the PNS (toroidal) would depend almost exclusively on the poloidal B-field distribution of the collapse progenitor (and barely on the toroidal one). Black: T3M0 Red: D3M0 Blue: DT3M0 Toroidal B-field PNS

  33. A4B5G5-D3M0: extreme case Faster initial rotation. Differential rotation. -dynamo stronger. Maximum density reached off-center (strong centrifugal rotation). B-field weak in the “inner hole”, since no matter flows transport field lines into this region. Strongest B-field reached outside shock location at bounce, the shock dragging the B-field lines along its path. Complicated windings and twists of the field lines by the action of -dynamo and meridional flows. Positive toroidal B-field regions formed. B-field shows large variations in small scales: finite conductivity effects such as field line reconnection could be important (magnetic diffusion proportional to B).

  34. PNS B-field evolution under -dynamo Fits of mag to obtain B-field growth and saturation timescales. rot A1B3G3-D3M0 mag We can estimate the B-field saturation timescale in the newly-formed PNS under the (unlikely) assumption that -dynamo were the only amplification mechanism at work (transforming rotational energy into magnetic energy). During the quasi-stationary evolution of the PNS, and assuming fixed angular profiles and poloidal B-field distribution, the toroidal B-field grows linearly with time. As the PNS rotates slower the B-field saturates because -dynamo cannot extract more energy from rotation: complete halting of the PNS, exchange sense of rotation, or torsion pendulum. However, MRI is likely to amplify the B-field to saturation on much shorter timescales (… in a few slides)

  35. Comparison with existing simulations • Comparisons made with models by Obergaulinger et al (2006a,b): • D3M10 models chosen from their sample, which have low magnetisation and hence our passive field approximation applies. Obergaulinger et al (2006) checked that the dynamics does nor change (MHD versus hydro) for such models. • B-field dynamics and morphology qualitatively similar in CFC models than in Newtonian gravity and TOV models (as long as the hydrodynamics is similar in both cases, which does not apply to Newtonian, multiple bounce core collapse models). • Magnetisation (mag) of final PNS smaller in CFC than in both Newton and TOV. Irrespective of numerical scheme (solver, minmod, PHM) and resolution. General trend?

  36. Gravitational waves A1B3G3 A1B3G5 A4B5G5 Black: “hydro” GW Red: “magnetic” GW, D3M0 Blue: “magnetic” GW, T3M0 Green: as blue but counter-rotating Hydro stress formula MHD The magnetic component of the GW is several orders of magnitude smaller than the hydrodynamical component. However: As the B-field amplification reaches saturation, the effects of the field on the waveforms are expected to be significant. In particular MRI may noticeably change the waveforms, as it is the most efficient amplification mechanism.

  37. Magneto-rotational instability (1) MRI is a shear instability that generates turbulence and an amplification of the magnetic field in rotating magnetized plasma, transporting angular momentum in the star (Balbus & Hawley 1991). Magnetized collapse models are indeed susceptible of developing MRI(Akiyama et al 2003; Obergaulinger et al 2005). The (Newtonian) condition for the MRI to occur (neglecting buoyancy effects and if the B-field strength is very low) is: If this condition is fulfilled and the magnetic field has a poloidal component, MRI grows exponentially in time. The timescale of the fastest growing unstable mode is: independent of B-field configuration and strength A1B3G3-D3M0 Main caveat of our simulations: no back-reaction of B-field onto the dynamics. Estimate only the potential effects of MRI.

  38. Magneto-rotational instability (2) A1B3G3-D3M0 A1B3G5-D3M0 A4B5G5-D3M0 time of bounce end of simulation Depicted in white are the regions where MRI is not fulfilled or timescale exceeds 1s. MRI does not affect the collapse dynamics (infall phase) but the evolution after core bounce is dominated by this instability. Passive B-field approximation no longer valid. Models T3M0 are not affected by MRI (no poloidal B-field). However, relaxing axisymmetry condition leads to poloidal B-fields via --dynamo, and hence to MRI.

  39. Summary of the talk • Multidimensional simulations of relativistic magneto-rotational core collapse feasible nowadays with current formulations of GRMHD and Einstein’s equations. • We have designed a method to build weakly magnetised stars in GR with either toroidal or poloidal (or both) B-field components. • Tests performed to check accuracy and convergence of our GRMHD code. • Results from CFC relativistic simulations of magneto-rotational core collapse to NS in axisymmetry presented. “Passive” B-field approximation used. • Core collapse dynamics, B-field distribution, and gravitational waveforms analysed. No major differences found with previous Newtonian studies. Strength of final B-field smaller in CFC than in Newtonian case. • B-field amplification mechanisms investigated (radial compression and -dynamo). • --dynamo and MRI not included in the modelling. Growth time of MRI shows it needs to be included as it dominates the post-bounce dynamics within a few ms. • Current GRMHD code already extended to relax passive B-field approximation. Tests in progress.

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