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Einstein-GRMHD simulations of GRB Progenitors

Einstein-GRMHD simulations of GRB Progenitors. Ken Nishikawa National Space Science & Technology Center/ University of Alabama in Huntsville (NSSTC/UAH). Cactus Retreat, April 28 – May 1, 2004. G. Richardson ( NSSTC ) M. Aloy ( MPA ) S. Koide ( Toyama University )

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Einstein-GRMHD simulations of GRB Progenitors

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  1. Einstein-GRMHD simulations of GRB Progenitors Ken Nishikawa National Space Science & Technology Center/ University of Alabama in Huntsville (NSSTC/UAH) Cactus Retreat, April 28 – May 1, 2004

  2. G. Richardson(NSSTC) M. Aloy(MPA) S. Koide (Toyama University) K. Shibata (Kyoto University) T. Kudoh (Univ, of W. Ontario) E. Ramizez-Ruiz (IAS) H. Sol (Obs. de Paris-Meudon) P. Hardee (Univ. of Alabama, Tuscaloosa) G.J. Fishman (NSSTC/MSFC) R. Preece (NSSTC/UAH) Collaborators

  3. Contents • Brief summary of gamma-ray bursts • Gamma-ray burst progenitors • Recent 3-D GRMHD simulations of jet formation • Collapsar model Hydro simulations Magnetohydodynamic simulations Einstein-GRhydro simulations • Merger model Hydro simulations Einstein-GRhydro simulations • Development of Einstein-GRMHD models of collapsar and mergers • Future plans

  4. Brief History of GRBs • First discovered in late 1960s. • 1991 BATSE launched onboard CGRO. (Meagan et al. 1992) • 1997 BeppoSax pinpointed afterglow. (GRB 970228) • 1999 Evidence of collimation.(Kulkarni & Harrison 1999) • 2003 Supernova found in optic afterglow(Galama et al. 1998, Kulkarni et. Al, 1998, Supernova 1998b vs. GRB 980425 Hjorth et al. 2003, Stanek et al. 2003)

  5. Cosmological origins

  6. Schematic GRB from a massive stellar progenitor (Meszaros, Science 2001) Prompt emission Polarization ? Accelerated particles emit waves at shocks

  7. Routes to a BH-Disk-Jet Death of massive stars a) Collapsars (Woosley) b) Hypernovae (Pacynski) ⇒ Sources of long GRBs Mergers of compact binaries (BH,NS) (Pacynski, Goodman, Eichler et al., Narayana et al., Mochkovitch et al.) ⇒ Sources of short GRBs & strong GWs

  8. Motivations • Jet formation from black holes • Accretion disk dynamics including azimuthal instabilities such as magnetorotational instability (MRI) and accretion-ejection instabilities with various initial and magnetic field geometries • Variabilities of relativistic jets due to these instabilities in the accretion disk and their effects on jet propagations • Modeling high/soft and low/hard states with AGNs related to mass accretion rates and angular momentum of black holes. • Examination of Blandford-Znajek model with a Kerr black hole as a possible energy source for Gamma-ray Bursts?

  9. Jets from binary stars (Schematic figure) BH or NS Accretion stream Mass donor star Accretion disk Jets

  10. Theoretical models of jet formation Lovelace (1976), Blandford (1976) Blandford & Znajek (1997): Kerr black hole Blandford and Payne (1982): Magneto-centrifugal force-driven jet Begeleman, Blandford, & Rees (1984): “Theory of extragalactic radio sources” Uchida & Shibata (1985): Magnetically driven Koide, Shibata & Kudoh (1998): (2-D GRMHD) “Gas pressure” & Magnetically driven

  11. Fundamental equations • ∇γ(ρUγ) = 0 (Conservation of mass) • ∇μTμν = 0 (Conservation of momentum) • ∂μFαβ + ∂αFβμ+ ∂βFμα = 0 (Conservation of energy for single component conductive fluid) • ∇αFαβ=  Jβ(Maxwell’s equations) Fαβ(electromagnetic field-strength tensor) Fαβ=∂αAβ ∂βAα FαβU μ= 0 (Frozen-in condition)

  12. Uγ : velocity 4-vector Jγ:current 4-vector ρ : proper mass density p : proper pressure ein = ρc2 + p/(Γ- 1): energy density Γ: specific-heat ratio (5/3) ∇γ: covariant derivative Tμν=pgμν+(ein+p)UμUν+FσμFνσ–g μνFλκFλκ/4: general relativistic energy momentum tensor Aμ: potential 4-vector

  13. 3+1 Formalism of General Relativistic MHD Equations ∂D/∂t = –·(Dv) ∂P/∂t = –·[pI+γ2(e+p)vv/c2–BB–EE/c2+0.5*(B2+E2/c2)I] ∂ε/∂t = –·[{γ2(e+p) –D2c2}v+E  B] ∂B/∂t = –  E (1/c2) ∂E/∂t +J= –  B (1/c2) ·E = ρc ·B = 0 E =– v  B (Frozen-in condition) γ [1 –(v/c)2]–1/2, D = γρ, P = γ2(e+p)v/c2 + EB/c2 ε = γ2(e+p) –p –Dc2+0.5*(B2+E2/c2)

  14. 3. Simulation models • General relativistic MHD codes axisymmetric (2-D) and full 3-D models Schwarzschild and Kerr black holes with simplified Total Variation Diminishing (TVD) method (Davis 1984) (Lax-Wendroff’s method with additonal diffusion term) needs better methods with Div B = 0 constraints

  15. Metric and Coordinates Schwarzschild metric:ds2 = gμνdxμdxν Boyer-Lindquist set (ct, r,θ,φ) Off-diagonal elements of metric are zero g μν= 0 (μν) g00 = –h02, g11= h12, g22 = h22, g33 = h32 h0= α, h1 = 1/α, h2 =r, h3 = r cos θ α (1– rS/r)1/2 (lapse function)

  16. Tortoise Coordinates d/dr*(r – rS)d/dr r* = ln(r –rS) Schwarzschild radius: rS 2GMBH/c2 Time Constant: τS rS/c Boundary conditions at r = 1.1 rS, 20 rS:radiating CFL numerical stability condition is severe at r =1.5 rS Polytropic equation of state: p = ρΓ Γ =5/3 and H = 1.3

  17. Initial conditions • Free-falling corona (these simulations) accretion disk relativistic Keplerian velocityvθ = vK c/[2(r/rS–1)]1/2 ρ = ρffc + ρdis rD  3 rS ρdis =  100ρffcif r > rD and |cotθ| < δ (δ= 0.125)  0 if rrD or |cot θ| δδ: thickness of disk (vr, vθ, vφ) =  (0, 0, vK) if r > rD and |cot θ| < δ  (–vffc, 0, 0) if rrD or |cot θ| δ

  18. 3-D simulation mass density(log10ρ) t = 39.2τS t = 0.0 τS Shocks t = 80.0τS t = 128.9τS Wind Black hole Jet Accretion disk Nishikawa et al (2004)

  19. Plasma pressure to magnetic pressure 39.2τS t = 0.0τS 128.9τS jet 60.0τS

  20. Twisted magnetic fields by accretion disk t = 0.0τS 39.2τS r = 2.67 rS 60.0τS 128.9τS r = 5.33 rS Black hole Nishikawa et al (2004)

  21. t = 60τS Snapshot at z = 5.6 rS ρ WEM B2z/2 Wgp p Bz vz vx vy By Bx

  22. Velocities at Z = 5.6rS t = 0.0τS t = 39.2τS t = 60.0τS t = 128.9τS vx vy vz

  23. X-ray binary system States of BHCs (Fender 2001) • How does the mass accretion rate determine the state? • How does the angular momentum, j affect the state? • How is instability involved with the state transition?

  24. Summary • Comparing the axisymmetric (2-D) simulations 3-D simulation show slower growth of jet formation • The additional freedom in the azimuthal direction without the mirror symmetry at the equatorial plane slows down the pile-up due to shocks near the black hole • In order to see effects of instabilities we need to seed initial perturbations with accretion disks

  25. Future Plans for jet formation study • Investigation of jet generation from Schwarzschild and Kerr black holes using full 3-D GRMHD simulations with better resolutions and for a long time with a new code with non-ideal fluid using finite element method (FEM) • In order to investigate different states of black holes, we will examine how the accretion disk dynamics and associated jet formation depend on initial conditions including magnetic field geometries and accreting stream from mass donor stars • Improve 3-D displays in order to understand physics involved in simulations using AVS • Implement a better boundary condition at the horizon • Investigate dynamics of the inner accretion disk near black holes in comparison with observations by Chandra, BATSE, XMM, INTEGRAL, ASTRO E2, GLAST, and Constellation-X • Investigate the dynamics of Kerr black hole as an energy source for Gamma-ray bursts

  26. Requirements on the Central Engine and its Immediate Surroundings (long-soft bursts)• Provide adequate energy at high Lorentz factor (Γ> 200; KE ~ 5 x 1051 erg)• Collimate the emergent beam to approximately 0.1 radians (6º)• Last approximately 20 s, but much longer in some cases • Produce a (Type Ib/c) supernova in some cases• In the internal shock model, provide a beam with rapidly variable Lorentz factor• Make bursts in star forming regions • Allow for the observed diversity seen in GRB light curves• Explain diverse events like GRB 980425• Make x-ray lines

  27. CactusEinstein for GRB progenitorcoupled with GHRMD CactusEinstein Progenitor GRMHD GRHydro MHD Hydro (Magnetic fields, Maxwell eq.) Equation of State (EOS) (polytropes, Neutrino annihilation to e+e-, etc.)

  28. Collapsar simulations • 2-D Hydrodynamic simulations MacFadyen and Woosley, 1999 MacFadyen, Woosley and Hegar, 2001 • 2-D Relativistic hydrodynamic simulations Aloy et al. 2000 • 2-D Magnetohydrodynamic simulations Proga et al. 2003 • 2-D GRMHD simulations Mizuno et al. 2003 (Schwarzschild) Mizuno et al. 2004 (Kerr) • 3-D Einstein-Hydro simulations of collapsing stars Biaotti et al. 2004 (Schwarzschild & Kerr)

  29. 2-D Relativistic hydrodynamic simulations with equation of state (EOS): nonrelativistic nucleons treated as a mixture of Boltzmann gases, radiation, and an approximate correction due to e+e pairs v > 0.3c E > 51019 ergs g-1 Stellar atmosphere Aloy et al. 2000, ApJ, 531, L119 Stellar surface

  30. 2-D MHD simulations with EOS (postnewtonian) t = 0.2735 s a small, latitude-dependent angular momentum and a weak radial magnetic Field (≫1) jet a realistic equation of state, neutrino cooling, photo-disintegration of helium, and resistive heating (artificial viscosity, no self-gravity and no nuclear burning) Φpw = GM/(r RS) Schwarzschild radius, RS =2GM/c2 Proga et al. 2003 ApJ Black hole

  31. 2-D GRMHD simulations by Mizuno et al. 2004 (Kerr BH) Rigid-like rotation Schematic picture of our simulation Gray: rotation The distribution on equatorial plane Uniform magnetic field （Bruenn, 1992） Distribution of mesh point

  32. color： density line：magnetic field lines Density in 2-D GRMHS simulations (Kerr BH) Jet-like ejection near the central BH τS = rS/c Disk-like structure

  33. Summary（Kerr BH case） • The formation mechanism of the jet in the rotating BH case is the same as that of the non-rotating BH case (mainly by the magnetic field) • The jet velocity in the co-rotating case is comparable to that of the non-rotating case, vjet～0.3c。 • In the co-rotating case, the kinetic energy flux is comparable to the Poynting flux • As the rotation parameter of BH increases, the poloidal velocity of the jet and magnetic twist increase gradually and toroidal velocity of the jet decreases. Because the magnetic field is twisted strongly by the frame dragging effect, it can store much magnetic energy and converts to kinetic energy of the jet directly

  34. t = 3-D Einstein-GRHydro simulation 0 Collapse sequence for the rapidly rotating model D4. 0.49 Accretion disk is formed no jet is formed Y Z for collapsar model needs EOS, magnetic fields, etc. (relativistic jet) 0.67 excised region x x Apparent horizon Biaotti et al. 2004

  35. Mergers of a system of compact binaries • Hydrodynamic simulations with realistic EOS Rosswog & Ramirez-Ruiz • Relativistic hydrodynamic simulations with realistic EOS Aloy, Janka & Müller (2003), in preparation • Einstein-GRhydrodynamic simulations Shibata, et al.

  36. Neutrino emission above the merged remnant of model i1.4 (2 x 1.4M⊙, no spin) Equatorial plane Meridian plane Et log (η) log (η) Rosswog & Ramirez-Ruiz 2003, MNRAS

  37. The first steps in thelife of a short GRB Aloy’s group goals: The viability of the scenario of merging SCBs for producing ultrarelativistic outflows (winds, jets, radial outflows?). Mechanism of collimation (if any) of the outflowing plasma. Expected durations of the GRB events generated in this framework and their relation to the time during which the source of energy is active (Ta). Howto: GRHD simulations -Build up a likely initial model Schwarzschild BH + thick accretion torus -Release energy in a baryon clean environment -EoS: ideal gas of neutrons + e+e- + radiation (Witti, Janka & Takahashi 1994) Aloy, Janka & Müller (2003), in preparation

  38. Initial model Two approaches: Type-B: Follow Font & Daigne (2003) prescription to build up equilibrium tori around a BH. Outside use Michel (1972) spherical accretion solution. Mtorus ~ 0.17 Msun MBH ~ 2.44 Msun Menv ~ 10- 7 Msun Ruffert & Janka (2001), A&A, 380, 544 free-falling corona Aloy, Janka & Müller (2003), in preparation

  39. Modeling the energy release Guided by previous results of Janka, Ruffert et al. showing that both in NS-NS mergers (Ruffert & Janka 1999) and in BH-NS mergers (Janka et al. 1999), can be released up to 1051ergs above the poles of the black hole in a region that contains less than 10-5 Msun of baryonic matter. The dependence in z-distance is: q(z) = q0 / zn; z = r sin; n ~ 5; 0~[30o, 75o] z Janka et al. (1999), ApJ, 527, L39 Ruffert & Janka (1999), A&A, 344, 573

  40. 1049 erg/s 2x1050 erg/s 1051 erg/s Limit of the deposition region Results Type B Morphology: For P> Pthr ~ 1048 erg/s the outflows are always conical, wide angle, ultrarelativistic jets. Outflow opening half-angle: ~20oto 25o. It is determined by the inclination angle of the side walls of the torus (large P). Propagation speed: larger than ~0.9999c Log 10Γ

  41. Recent Einstein-Hydrodynamic simulation • Stellar-mass BH + disk + jet ⇒ GRB central engine • BS-NS merger, NS-BH merger, stellar collapse • Best theoretical approach is simulation in NR

  42. EOS: Initial; P = K ρΓ, Γ = 2; K = 1.535e5 cgs, During the evolution: P = (Γ-1)ρε Typical grid size : 633 * 633 * 317

  43. Development of Einstein-GRMHD code for GRB central engines Present status • Einstein-GRMHD has been formulated (Baumgarte & Shapiro, 2003, ApJ) • Needs coupling between Einstein & GRMHD equations • Possibly needs further modification with methods

  44. Basic equations (Baumgarte & Shapiro, 2003, ApJ, 585, 921) ADM form Einstein eq  Stress-energy tensor Tμν= pgμν+(ein+p)UμUν+FσμFνσ –g μνFλκFλκ/4

  45. Fundamental equations • ∇γ(ρUγ) = 0 (Conservation of mass) • ∇μTμν = 0 (Conservation of momentum) • ∂μFαβ + ∂αFβμ+ ∂βFμα = 0 (Conservation of energy for single component conductive fluid) • ∇αFαβ=  Jβ(Maxwell’s equations) Fαβ(electromagnetic field-strength tensor) Fαβ=∂αAβ ∂βAα FαβU μ= 0 (Frozen-in condition)

  46. Equation of state (EOS) • Photodisintegration • Resistivity • Neutrino annihilation (pair creation) • Gas of nuclei • radiation

  47. Applications of this code • Investigation of GRB central engines • NS-NS mergers to BH • BH-BH mergers • Dynamics of Magnetars • Jet formations (energy extraction) from BHs • Investigation of generation of primordial magnetic fields • Star collapse

  48. Necessity of 3-D full particle simulation for particle acceleration • MHD simulations provide global dynamics of relativistic jets including hot spots • MHD simulations include heating due to shocks, however do not create high energy particles (MHD simulation + test particle (Tom Jones)) • In order to take account of acceleration the kinetic effects need to be included • Test particle (Monte Carlo) simulations can include kinetic effects, but not self-consistently • Particle simulations provide particle acceleration and emission self-consistently, however due to the computational limitations, the size of jet is small comparing with MHD simulations

  49. Electron density(arrows: Bz, Bx) ωpet = 23.4 jet X/Δ Z/Δ Weibel instability

  50. Collisionless shock ∂B/∂t = –E ∂E/∂t = B – J dm0γv/dt = q(E +vB) ∂ρ/∂t +•J = 0 Electric and magnetic fields created self-consistently by particle dynamics randomize particles (Buneman 1993) jet jet electron ambient electron ambient ion jet ion

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