19/04/23 L. Del Zanna: Relativistic MHD simulations of PWNe 1

Relativistic MHD simulations of Pulsar Wind Nebulae: on the origin

of the inner jet-torus structure

Luca Del Zanna

Gruppo Plasmi Astrofisici

Dipartimento di Astronomia e Scienza dello Spazio

Università degli Studi di Firenze

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Introduction

• Active Galactic Nuclei• Compact X-ray sources• Gamma-Ray Bursts• Pulsar Wind Nebulae

Solution: shock-capturing codes for relativistic MHD

Relativistic flows and shocks in high-energy astrophysics:

Problem: 3-D dynamics of relativistic plasmas

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Relativistic MHD equations

• Equations written in covariant form:

• where:

0)( u

0])2/()[( 22 gbpbbuubw

0)( bubu

)/,(),,(),,( jii

jii

jj vBvBBvbvuxtx

1/);1/(, 2 wpcpepew s

In the following we shall consider special relativity

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Shock-capturing numerical schemes

• grid discretization (finite volumes or differences)• reconstruction of U and F at intercells (TVD, ENO)• approximate Riemann solver for fluxes (upwind step)• time integration (two-steps or higher order RK)

0)(

1

n

ii

i

x

UF

t

U

Problem: MHD and RMHD systems are not strictly hyperbolic

Any set of hyperbolic conservation laws

may be solved by high-order Godunov schemes:

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The divergence-free condition

• CT staggered discretization (Evans & Hawley, 1988) • staggered B components in fluxes (to avoid monopoles)• four-state numerical magnetic fluxes

)(,0 BvBB t

the specific operator structure (a curl) in the induction equation must be considered to preserve the constraint

We use the Upwind Constrained Transport method for MHD

(UCT: Londrillo & Del Zanna, 2000; 2004)

The equation for the B field yields, as in classical MHD:

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A central-type scheme for relativistic MHD

• A new central-type shock-capturing scheme for RMHD (Del Zanna & Bucciantini, 2002; Del Zanna, Bucciantini & Londrillo, 2003)

• RMHD code main features:• Third order accuracy in time, unsplit (Runge-Kutta)• Third order accuracy in space (Convex-ENO + MC limiter)• Simplified Riemann solver (central scheme, no characteristic waves)• Efficient inversion from conservative to primitive variables• UCT staggered method for •B=0 (preserved to machine accuracy)• Generalized orthogonal curvilinear coordinates• F90 modular structure, parallelization with MPI• Future work: AMR and GRMHD

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Pulsar Wind Nebulae

• PWNe are hot bubbles (plerions) emitting synchrotron radiation at all wavelengths (e.g. Crab)

• Originated by the interaction of the ultra-relativistic magnetized pulsar wind with the expanding SNR dense ejecta

• Theory is based on 1-D analytic (Kennel & Coroniti, 1984) and self-similar (Emmering & Chevalier, 1987) MHD models

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PWN and SNR: structure and evolution

• a: wind termination shock• b: PWN-SNR contact discontinuity• c: shell of swept up ejecta• d: reverse shock• e: SNR-ISM contact discontinuity• f: SNR blast wave

PWN

Sedov

reverberations

free expansion

CD

TS

SNR

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Jet-torus structure: Chandra X-ray images

• Crab nebula (Weisskopf et al., 2000; Hester et al., 2002)• Vela pulsar (Helfand et al., 2001; Pavlov et al., 2003)• Other objects: PSR 1509-58, G0.9+01, G54.1+0.3

Crab

Vela

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Jet-torus structure: theory

• Torus: KC84 + higher equatorial wind energy flux?• Jets: magnetic collimation not efficient for >>1:

are they launched downstream of the TS?• Bogovalov & Khangoulian, 2002• Lyubarsky, 2002

• Axisymmetric RMHD simulations of the interaction of an anisotropic relativistic magnetized wind with SN ejecta• Komissarov & Lyubarsky, 2003• Del Zanna, Amato & Bucciantini, submitted

0 BjEq

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Pulsar relativistic wind model

• Far from the pulsar light cylinder the wind is expected to be ultrarelativistic, cold, and weakly magnetized. We assume:• Isotropic mass flux, anisotropic energy flux

• Purely toroidal magnetic field (residual field in a striped wind):

• Parameters of the wind model:

)](sin)1([)(),(/)/(),( 200

200 rrr

)sin()/(),( 00 rrBrB

14

,1)2/,(

)0,(,1

200

2

20

0

00

c

B

rF

rF

:)( 2F

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Simulation setup

• Central-type conservative RMHD code (here second order)

• Spherical geometry, axial symmetry (r,)• Poloidal velocity and purely toroidal magnetic field• Computational grid: 400 points in r, 100 in • Boundaries: injection for r=0.05 ly, extrapolation for r=20 ly• Long time simulations (beginning of reverberation phase)• High accuracy near the center: extremely small timesteps!• Initial conditions:

• Pulsar ultrarelativistic wind

• Spherical shell of expanding dense ejecta

• Static unmagnetized ISM

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PWN self-similar evolution and TS shape

• Expected TS profile:

)(sin)1()( 20 RR

003.0,1.0,1000

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PWN elongation

• Magnetic pinching effect (Begelman & Li, 1992):

r

BP

B

rP

B

z 48,0

8

222

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TS structure and flow pattern

=0.003• The wind anisotropy shapes the TS structure. A complex flow pattern arise:

• A: ultrarelativistic pulsar wind• B: subsonic equatorial outflow• C: supersonic equatorial funnel• D: super-fastmagnetosonic flow• a: termination shock front• b: rim shock• c: fastmagnetosonic surface

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Formation of polar jets by hoop stresses

=0.003 =0.03=0.01

• The flow pattern changes drastically with increasing • For high magnetization (>0.01) a supersonic jet is formed

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PWN crunching by instabilities at CD

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Dependence on the field shape

• Initial magnetic field with a narrow equatorial neutral sheet

)]2/(tanh[)sin(),( 00

br

rBrB

b=10

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Summary and conclusions

• Many PWNe show a jet-torus structure (Crab, Vela, …)• The torus is explained with a higher equatorial energy flux• The jet poses severe theoretical difficulties, as collimation

is not effective in the wind: hoop stresses inside PWN?• RMHD axisymmetric simulations confirm this scenario:

• The TS has a toroidal shape, a strong equatorial flow is produced• For >0.01 hoop stresses divert the flow toward the axis• Plasma is compressed and a polar jet with v=0.5-0.7c is launched• CD instabilities driven by large vortexes may crunch the PWN• A neutral equatorial sheet is enough to prevent the crunching

Thank you