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### Relativistic MHD Simulations of Relativistic Jets

Yosuke Mizuno

NASA Postdoctoral Program Fellow

NASA Marshall Space Flight Center (MSFC)

National Space Science and Technology Center (NSSTC)

Context

- Introduction
- Development of 3D GRMHD code
- 2D GRMHD simulations of Jet Formation
- Stability of relativistic jets
- MHD boost mechanism of relativistic jets
- Summary

Astrophysical Jets

M87

- Astrophysical jets: outflow of highly collimated plasma
- Active Galactic Nuclei, Microquasars, Gamma-Ray Bursts, Jet velocity ～c.
- Generic systems: Compact object（White Dwarf, Neutron Star, Black Hole）+ Accretion Disk
- Key Problem of Astrophysical Jets
- Acceleration mechanism
- Collimation
- Long term stability
- Modeling of Astrophysical Jets
- Magnetohydrodynamics (MHD)+ Relativity
- MHD centrifugal acceleration
- Extraction of rotational energy from rotating black hole

Relativistic Jets in Universe

Mirabel & Rodoriguez 1998

Modeling of Astrophysical Jets

Energy conversion from accreting matter is the most efficient mechanism

- Gas pressure model
- Jet velocity ~ sound speed (maximum is ~0.58c)
- Difficult to keep collimated structure
- Radiation pressure model
- Can collimate by the geometrical structure of accretion disk (torus)
- Difficult to make relativistic speed with keeping collimated structure
- Magnetohydrodynamic (MHD) model
- Jet velocity ~ Keplerian velocity of accretion disk

make relativistic speed because the Keplerian velocity near the black hole is nearly light speed

- Can keep collimated structure by magnetic hoop-stress
- Direct extract of energy from a rotating black hole (Blandford & Znajek 1977, force-free model)

Requirment of Relativistic MHD

- Astrophysical jets seen AGNs show the relativistic speed (~0.99c)
- The central object of AGNs is supper-massive black hole (~105-1010 solar mass)
- The jet is formed near black hole

Require relativistic treatment (special or general)

- In order to understand the time evolution of jet formation, propagation and other time dependent phenomena, we need to perform relativistic magnetohydrodynamic simulations

1. Development of 3D GRMHD Code “RAISHIN”

Mizuno et al. 2006a, Astro-ph/0609004

Numerical Approach to Relativistic MHD

- RHD: reviews Marti & Muller (2003) and Fonts (2003)
- SRMHD: many authors
- Application: relativistic Riemann problems, relativistic jet propagation, jet stability, pulsar wind nebule, etc.
- GRMHD
- Fixed spacetime (Koide, Shibata & Kudoh 1998; De Villiers & Hawley 2003; Gammie, McKinney & Toth 2003; Komissarov 2004; Anton et al. 2005; Annios, Fragile & Salmonson 2005; Del Zanna et al. 2007)
- Application: The structure of accretion flows onto black hole and/or formation of jets, BZ process near rotating black hole, the formation of GRB jets in collapsars etc.
- Dynamical spacetime (Duez et al. 2005; Shibata & Sekiguchi 2005; Anderson et al. 2006; Giacomazzo & Rezzolla 2007 )

Propose to Make a New GRMHD Code

- The Koide’s GRMHD Code (Koide, Shibata & Kudoh 1999; Koide 2003) has been applied to many high-energy astrophysical phenomena and showed pioneering results.
- However, the code can not perform calculation in highly relativistic (g>5) or highly magnetized regimes.
- The critical problem of the Koide’s GRMHD code is the schemes can not guarantee to maintain divergence free magnetic field.
- In order to improve these numerical difficulties, we have developed a new 3D GRMHD code RAISHIN(RelAtIviStic magnetoHydrodynamc sImulatioN, RAISHIN is the Japanese ancient god of lightning).

4D General Relativistic MHD Equation

- General relativistic equation of conservation laws and Maxwell equations:

∇n (r Un) = 0(conservation law of particle-number)

∇nTmn= 0(conservation law of energy-momentum)

∂mFnl + ∂nFlm + ∂lF mn= 0

∇mFmn= - Jn

- Ideal MHD condition:FnmUn= 0
- metric：ds2=gmndxmdxn
- Equation of state :p=(G-1) u

(Maxwell equations)

r : rest-mass density. p : proper gas pressure. u: internal energy. c: speed of light.

h : specific enthalpy, h =1 + u +p / r.

G: specific heat ratio.

Umu : velocity four vector. Jmu : current density four vector.

∇mn : covariant derivative. gmn : 4-metric,

Tmn : energy momentum tensor, Tmn = r h Um Un+pgmn+FmsFns -gmnFlkFlk/4.

Fmn : field-strength tensor,

Conservative Form of GRMHD Equations (3+1 Form)

a: lapse function,

bi: shift vector,

gij: 3-metric

Metric:

(Particle number conservation)

(Momentum conservation)

(Energy conservation)

(Induction equation)

U (conserved variables)

Fi (numerical flux)

S (source term)

√-g : determinant of 4-metric

√g : determinant of 3-metric

Detail of derivation of GRMHD equations

Anton et al. (2005) etc.

3D GRMHD Code “RAISHIN”

Mizuno et al. (2006)

- RAISHIN utilizes conservative, high-resolution shock capturing schemes to solve the 3D general relativistic MHD equations (metric is static)
- Ability of RAISHIN code
- Multi-dimension (1D, 2D, 3D)
- Special(Minkowski spcetime) and General relativity(static metric; Schwarzschild or Kerr spacetime)
- Different coordinates (RMHD: Cartesian, Cylindrical, Spherical and GRMHD: Boyer-Lindquist of non-rotating or rotating BH)
- Use several numerical methods to solving each problems
- Use Gamma-law or variable equation of state

Relativistic MHD Shock-Tube Tests

Exact solution: Giacomazzo & Rezzolla (2006)

Relativistic MHD Shock-Tube Tests

Balsara Test1 (Balsara 2001)

- The results show good agreement of the exact solution calculated by Giacommazo & Rezzolla (2006).
- Minmod slope-limiter and CENO reconstructions are more diffusive than the MC slope-limiter and PPM reconstructions.
- Although MC slope limiter and PPM reconstructions can resolve the discontinuities sharply, some small oscillations are seen at the discontinuities.

FR

SR

SS

FR

CD

Black: exact solution, Blue: MC-limiter, Light blue: minmod-limiter, Orange: CENO, red: PPM

400 computational zones

2. 2D GRMHD Simulation of Jet Formation

Mizuno et al. 2006b, Astro-ph/0609344

Hardee, Mizuno, & Nishikawa 2007, ApSS, 311, 281

Wu et al. 2008, CJAA, submitted

2D GRMHD Simulation of Jet Formation

Initial condition

- Geometrically thin Keplerian disk (rd/rc=100) rotates around a black hole (a=0.0, 0.95)
- The back ground corona is free-falling to a black hole (Bondi solution)
- The global vertical magnetic field (Wald solution)

Numerical Region and Mesh points

- 1.1（0.75) rS < r < 20 rS, 0.03< q < p/2, with 128*128 mesh points

Schematic picture of the jet formation near a black hole

Time evolution (Density)non-rotating BH case (B0=0.05,a=0.0)

Parameter

B0=0.05

a=0.0

Color: density

White lines: magnetic field lines (contour of poloidal vector potential)

Arrows: poloidal velocity

Fast-rotating BH

Resultsr

- The matter in the disk loses its angular momentum by magnetic field and falls to a black hole.
- A centrifugal barrier decelerates the falling matter and make a shock around r=2rS.
- The matter near the shock region is accelerated by the J×B force and the gas pressure gradient and forms jets.
- These results are similar to previous work (Koide et al. 2000, Nishikawa et al. 2005).
- In the rotating black hole case, additional inner jets form by the magnetic field twisted resulting from frame-dragging effect.

b

Bf

vtot

White curves: magnetic field lines (density), toroidal magnetic field (plasma beta)

vector: poloidal velocity

Relativistic Radiation Transfer

Wu et al., 2008, CJAA, submitted

- We have calculated the thermal free-free emission and thermal synchrotron emission from a relativistic flows in black hole systems based on the results of our 2D GRMHD simulations (rotating BH cases).
- We consider a general relativistic radiation transfer formulation (Fuerst & Wu 2004, A&A, 424, 733) and solve the transfer equation using a ray-tracing algorithm.
- In this algorithm, we treat general relativistic effect (light bending, gravitational lensing, gravitational redshift, frame-dragging effect etc.).

Image of Emission, absorption & scattering

Radiation images of black hole-disk system

Radiation image seen from q=85 (optically thick)

Radiation image seen from q=85 (optically thin)

- The radiation image shows the front side of the accretion disk and the other side of the disk at the top and bottom regions because the general relativistic effects.
- We can see the formation of two-component jet based on synchrotron emission and the strong thermal radiation from hot dense gas near the BHs.
- A wired synchrotron emission (green-spark) is seen the surface of the disk (time-dependent). It becomes a origin of QPOs?

Radiation image seen from q=45 (optically thick)

3. Stability Analysis of Magnetized Spine-Sheath Relativistic Jets

Mizuno, Hardee & Nishikawa, 2007, ApJ, 662, 835

Hardee, 2007, ApJ, 664, 26

Hardee, Mizuno & Nishikawa, 2007, ApSS, 311, 281

Instability of relativistic jets

- Kelvin-Helmholtz (KH) (this talk topics)
- Presence of velocity gradients in the flow
- Important at the shearing boundary flowing jet and external medium
- Current-driven (CD) (e.g., kink instability)
- Presence of strong axial electric current
- Important in strongly twisted magnetic field
- Interaction of jets with external medium caused by such instabilities leads to the formation of shocks, turbulence, acceleration of charged particles etc.
- Used to interpret many jet phenomena
- quasi-periodic wiggles and knots, filaments, limb brightening, jet distuption etc

Spine-Sheath Relativistic Jets (observations)

M87 Jet: Spine-Sheath Configuration?

HST Optical Image (Biretta, Sparks, & Macchetto 1999)

VLA Radio Image (Biretta, Zhou, & Owen 1995)

Typical Proper Motions > c

Optical ~ inside radio emission

Jet Spine ?

Typical Proper Motions < c

Sheath wind ?

- Obsevations of OSOs show the evidence of high speed wind （~0.1-0.4c）(Pounds et al. 2003):
- Related to Sheath wind
- Spine-sheath configuration proposed to explain
- limb brightening in M87, Mrk501jets （Perlman et al. 2001; Giroletti et al. 2004）
- TeV emission in M87 (Taveccio & Ghisellini 2008)
- broadband emission in PKS 1127-145 jet (Siemiginowska et al. 2007)

Spine-Sheath Relativistic Jets (GRMHD Simulations)

- In recent general relativistic MHD simulation of jet formation (e.g., Hawley & Krolik 2006, McKinney 2006, Hardee et al. 2007),
- simulation results suggest that
- a jet spine driven by the magnetic fields threading the ergosphere
- may be surrounded by a broad sheath wind driven by the magnetic fields anchored in the accretion disk.
- This configuration might additionally be surrounded by a less highly collimated accretion disk wind from the hot corona.

Non-rotating BH Fast-rotating BH

Disk Jet

BH Jet Disk Jet

Total velocity distribution of 2D GRMHD Simulation of jet formation

(Hardee, Mizuno & Nishikawa 2007)

Key Questions of Jet Stability

- When jets propagate outward, there are several possibility to grow of instabilities
- How do jets remain sufficiently stable?
- What are the Effects & Structure of Kelvin-Helmholtz (KH) / current driven (CD) Instability in spine-sheath configuration?
- We investigate these topics by using 3D relativistic MHD simulations

3D Simulations of Spine-Sheath Jet Stability

Initial condition

Mizuno, Hardee & Nishikawa, 2007

- Cylindrical super-Alfvenic jet established across the computational domain with a parallel magnetic field (stable against CD instabilities)

- Solving 3D RMHD equations in Cartesian coordinates
- (using Minkowski spacetime)
- ujet = 0.916 c (γj=2.5), jet = 2 ext (dense, cold spine jet)
- External medium, uext = 0 (static), 0.5c (sheath wind)
- Spine Jet precessed to break the symmetry (frequency, w=0.93)
- RHD: weakly magnetized (sound velocity > Alfven velocity)
- RMHD: strongly magnetized (sound velocity < Alfven velocity)
- Numerical Resion and mesh points
- -3 rj< x,y< 3rj, 0 rj< z < 60 rj (Cartesian coordinates) with 60*60*600 computational zones, (1rj=10 computational zone)

Simulation results (nowind, weakly magnetized case)

3D isovolume of density with B-field lines show the jet is disrupted by the growing KH instability

Longitudinal cross section

y

y

z

x

Transverse cross section show the strong interaction between jet and external medium

Effect of magnetic field and sheath wind

vw=0.0

vw=0.5c

vw=0.0

vw=0.5c

- The sheath flow reduces the growth rate of KH modes and slightly increases the wave speed and wavelength as predicted from linear stability analysis.
- Substructure associated with the 1st helical body mode is eliminated by sheath wind as predicted.
- The magnetized sheath reduces growth rate relative to the fluid case and the magnetized sheath flow damped growth of KH modes.

4. MHD Boost mechanism of Relativistic Jets

Mizuno, Hardee, Hartmann, Nishikawa & Zhang, 2008, ApJ, 672, 72

A MHD boost for relativistic jets

- The acceleration mechanism boosting relativistic jets to highly-relativistic speed is not fully known.
- Recently Aloy & Rezzolla (2006) have proposed a powerful hydrodynamical acceleration mechanism of relativistic jets by the motion of two fluid between jets and external medium
- This hydrodynamical boosting mechanism is very simple and powerful
- But is likely modified by the effect of magnetic fields
- We have investigated the effect of magnetic fields on the boost mechanism by using RMHD simulations

Initial Condition (1D RMHD)

- Consider a Riemann problem consisting of two uniform initial states
- Right (external medium): colder fluid with larger rest-mass density and essentially at rest.
- Left (jet): lower density, higher temperature and pressure, relativistic velocity tangent to the discontinuity surface
- To investigate the effect of magnetic fields, put the poloidal (Bz: MHDA) or toroidal (By: MHDB) components of magnetic field in the jet region (left state).
- For comparison, HDB case is a high gas pressure, pure-hydro case (gas pressure = total pressure of MHD case)

Simulation region

-0.2 < x < 0.2 with 6400 grid

Schematic picture of simulations

Hydro Case

Solid line (exact solution), Dashed line (simulation)

In the left going rarefaction region, the tangential velocity increases due to the hydrodynamic boost mechanism.

jet is accelerated to g~12 from an initial Lorentz factor of g~7.

MHD Case

HDA case (pure hydro) : dotted line

MHDA case (poloidal)

MHDB case (toroidal)

HDB case (hydro, high-p)

- When gas pressure becomes large, the normal velocity increases and the jet is more efficiently accelerated.
- When a poloidal magnetic field is present, stronger sideways expansion is produced, and the jet can achieve higher speed due to the contribution from the normal velocity.
- When a toroidal magnetic field is present, although the shock profile is only changed slightly, the jet is more strongly accelerated in the tangential direction due to the Lorentz force.
- The geometry of the magnetic field is a very important geometric parameter.

Dependence on Magnetic Field Strength

Solid line: exact solution, Crosses: simulation

Magnetic field strength is measured in fluid flame

- When the poloidal magnetic field increases, the normal velocity increases and the tangential velocity decreases.
- When the toroidal magnetic field increases, the normal velocity decreases and the tangential velocity increases.
- Toroidal magnetic field provides the most efficient acceleration.

Summary

- We have developed a new 3D GRMHD code ``RAISHIN’’by using a conservative, high-resolution shock-capturing scheme.
- We have performed simulations of jet formation from a geometrically thin accretion disk near both non-rotating and rotating black holes. Similar to previous results (Koide et al. 2000, Nishikawa et al. 2005a) we find magnetically driven jets.
- It appears that the rotating black hole creates a second, faster, and more collimated inner outflow. Thus, kinematic jet structure could be a sensitive function of the black hole spin parameter.

Summary (cont.)

- We have investigated stability properties of magnetized spine-sheath relativistic jets by the theoretical work and 3D RMHD simulations.
- The most important result is that destructive KH modes can be stabilized even when the jet Lorentz factor exceeds the Alfven Lorentz factor. Even in the absence of stabilization, spatial growth of destructive KH modes can be reduced by the presence of magnetically sheath flow (~0.5c) around a relativistic jet spine (>0.9c)

Summary (cont.)

- We performed relativistic magnetohydrodynamic simulations of the hydrodynamic boosting mechanism for relativistic jets explored by Aloy & Rezzolla (2006) using the RAISHIN code.
- We find that magnetic fields can lead to more efficient acceleration of the jet, in comparison to the pure-hydrodynamic case.
- The presence and relative orientation of a magnetic field in relativistic jets can significant modify the hydrodynamic boost mechanism studied by Aloy & Rezzolla (2006).

Future Work

- Resistivity (extension to non-ideal MHD; e.g.,Watanabe & Yokoyama 2007; Komissarov 2007)
- Couple with radiation transfer (link to observation: collaborative works with Fuerst)
- Improve the realistic EOS
- Include Neutrino (cooling, heating)
- Include Nucleosysthesis post processing
- Couple with Einstein equation (dynamical spacetime)
- Adaptive mesh refinement
- Apply to astrophysical phenomena in which relativistic outflows and/or GR essential (AGNs, microquasars, neutron stars, and GRBs etc.)

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