Nick Camus Niccolo BucciantiniPhilip Hughes Maxim Lyutikov

Serguei Komissarov (University of Leeds)

RMHD simulations of the Crab Nebula

Plan of the talk

1. Crab Nebula and its wisps;2. Theoretical MHD models;3. High-resolution 2D simulations – strong variability;4. Modelling synchrotron emission – moving wisps; 5. Statistical analysis of the variability; 6. Gamma-ray emission;7. Summary.

Thermal filaments(supernova remnant)

Non-thermal diffuse emission (plerion)

Optical image

The total mass,MN + MNS ~ 6M3 < 9M3, too low for core-collapsesupernova ?

I. The Crab Nebula and its wisps

X-ray map of of the inner Crab Nebula

Chandra image ( Weisskopf et al. 2000)

“torus”

jet

pulsar

X-rays

Visible light

Hester et al.(1995)

Wisps of The Crab Nebula

HST movieQuasi-periodic (?) emission of wisps.

Knot 1 is the most compactpermanent feature on the map, 0.5 arcsec or 6 light days

B

Bv

II. MHD model

v

1D relativistic MHD model;

Particle dominatedrelativistic pulsar wind with purely azimuthalmagnetic filed;

-problem: conversionof the magnetic energy into the kinetic energy of the wind. Dissipation of magneticenergy.

Termination of spherical wind

B

B

vv

Termination of equatorial wind

Lyubarsky (2002);Bogovalov & Khangoulian (2002)

“jet”

“torus”

unshocked pulsar wind with ram pressure

Michel (1973)

termination shock

pr am / sin2 µ

“torus”

“jet”

shocked pulsarwind

v

Magnetic hoop stressredirects the flow towards the poles

The same setup as in Komissarov & Lyubarsky (2004) but 1) higher resolution; 2) improved model of synchrotron emission;3) no equatorial symmetry imposed.

III. High-resolution 2D simulations

RMHD equations:

Numerical scheme: improved version of Komissarov (1999);conservative upwind scheme; second order in time and third order in space; spherical coordinates; hierarchical time stepping; dynamical grid size (not AMR).

Simulations setup:

outer boundary: supersonic outflow

inner boundary: supersonic inflow(stationary pulsar wind)

Initial solution and boundary conditions

Supernova ejecta(cold Hubble flow)

Unshocked pulsar wind

Duration of runs ~ the Crab’s age (1000 yr)

Number of grid points in nµ = 100;200;400;800

2D; axisymmetry.

Supernova remnant (shell):

½= const; P = 0; v = verre

: scaled to fit the observed expansion rate of the Crab Nebula

Total energy flux:(scaled to fit the spindown power of the Crab pulsar)

Kinetic energy flux:

- bulk Lorentz factor;

Model of pulsar wind:

Stot =f 0

r2 (sin2 µ+ a0)

Skin = ½c3° = Stot ¡B2

Á

4¼c:

Michel (1973)

= (Poynting flux)/(kinetic flux) ~ 0.01 (too low?)

a0 = 0:1 (?)

Dissipative current sheet (magnetic dissipation, flow acceleration)

37

3

Coroniti (1990), Michel(1994), Lyubarsky & Kirk (1991) etc.

B2Á =

4¼f 0»2

cr2 sin2 µ(1¡ 2µ=¼)2;

Azimuthal magnetic field:

- magnetization parameter

(particle dominated flow).

RESULTS:

Strong variability of the plerion flow

Animation 1:

Total pressure ( CGS units ) near the end of the run with the highestresolution.

Only the inner part of the computational domain is shown.

RESULTS:

Strong variability of the plerion flow

Animation 2:

Magnetic field ( Gauss ) near the end of the run with the highestresolution

- Doppler factor, - normal to the line of sight component of comoving magnetic field, - radiation frequency.

f (²) = A³ n0

n

´ ¡ (2+¡ )=3µ

1¡²

²1

¶ ¡ ¡ 2

²¡ ¡ ;

f (²) = A²¡ ¡ (¡ = 2:2; ² < ²max);

Initial electron spectrum at the termination shock:

Downstream spectrum (synchrotron + adiabatic losses):

n – density of advective tracer; n0 ~ r -2 – its value at the shock; - cut-off energy. ²1

IV. Modelling synchrotron emission

Synchrotron emissivity:

j º =C8¼

D (¡ +2)=2B0?

³ n0

n

´ ¡ (¡ +2)=3²(º)1¡ ¡

µ1¡

²(º)²1

¶ ¡ ¡ 2

B0?

²2(º) = º=c1B0?

D

º

r ¹ (nu¹ ) = 0

r ¹ (n0nu¹ ) = 0

r ¹ (²1 n23 u¹ ) = ¡ ec2B02²2

1 n23

Evolution equations for the spectrum parameters:

- suspension equation for n ;

- advective scalar equation for n0 ;

²1 - reaction-advection eq. for

²1 evolves due to synchrotron and adiabatic losses.

These equations are integrated simultaneously with the main system (of RMHD).

Doppler beaming switched off

Role of the Doppler beamingin the appearance of the nebula. OFF

Synthetic optical image.

Doppler beaming switched on

Role of the Doppler beamingin the appearance of the nebula. ON

HST

knot 1

Synthetic optical image.

Lorentz factor of the post-shock flow

Emissivity in the frame ofthe observer

observer

HST

Origin of the knot 1

pulsarknot 1

Geometry of the knot 1

°2 ' 1=±1 (¾1 · 1)

±1 = Ãp(dn=dk)

ÁD ' 1=°2 ' ±1

Ã? ' ÁD(dk=dn) ' Ãp

- oblique shock equations

- Doppler beam angle

- flow towards the observer

Transverse size of the knot:

Animation 3:

Synchrotron emission near the end of the run with the highestresolution

• Initial wisp speed ~ 0.5c;

• wisps slow down and pile-up further out;

• sometimes wisps contract;

• proper motion in the jet.

V. Statistical analysis of data

Time series to study numerical convergence and to quantify variability

Measure the magnetic field at a point near equator downstream of the termination shock.

Four runs with increasingresolution.

nµ=400 nµ=800

nµ=200nµ=100

Auto-correlation function,

numerical noise

turbulence

< (B(t) ¡ B(t + ¿))2 >

characteristictime scale ofvariability

nµ=100nµ=200nµ=400nµ=800

Convergence not reached.Getting close?

The characteristic time scaledecreases with resolution; around 1 year for . nµ=800

Two quasi-periods: ~1.5yr and ~3yr. Observations suggest: ~ few months.

Search for quasi-periods (Wavelet transform). Morlet

In simulations a) the shock radius is ~3 times higher, b) the period decreases with resolution.

Unexplained feature – the “inner ring” ( its bright knots ).

VI. Gamma-rays from the termination shock ?

tcool ' 3:7D1=2µ

B103G

¶ ¡ 3=2 µEph;ob

100MeV

¶ ¡ 1=2

days;

Termination shock size ~ 120 light days.

Gamma-rays come from the very vicinity of the termination shock and hence must be subject to strong Doppler beaming.

Extrapolation of the knot 1 optical emission using the power law

Fº / º ¡ 0:64 (Tziamtzis et al., 2009)

gives the observed total flux at 100 MeV !

knot 1

wisps

Emaxph =

2716¼

´mhc3

e2 = 236´ MeV ;

E = ´B ( ´ < 1)

- maximum energy of electrons accelerated by the electric field

( Vittorini et al. 2011 )

Eobph = DEph;

Dmax » 2°2 » 10;

Evidence of Doppler boosting ?

Variability of the knot 1 at 100 MeV

time in years

( No data for shorter time scales )

Mechanism of the gamma-ray variability ?

j º;ob = D2+®j º - Doppler boosting

Variable Doppler boosting ?

When the viewing angle decreases from 1/ to 0 the Doppler factor increases from to 2For = 3 this yields 30-fold increase of the observed emissivity.

What can result is such a variability of the flow pattern on the timescale of gamma-ray flares ???

VII. Summary

• New high-resolution axisymmetric MHD numerical simulations reveal highly unsteady flow dynamics in pulsar wind nebulae; The most dynamic region is inside the TS cusp – the jet base. Statistical analysis indicates turbulent cascade in the main body of the nebula.

• Synthetic synchrotron maps are remarkably similar to the HST and Chandra maps of the Crab Nebula: Jet, torus, knots, wisps, fine fibrous structure of emission. The inner ring is still a puzzle; • Wisps move with relativistic speeds similar to the observed; The predominant motion is expansion, though contraction is also seen from time to time;

• For the highest resolution, the characteristic time-scale of the flow variability and wisp production is around 1 year; Wavelet transform reveals quasi-periods of ~1.5yr and ~3.0 yr (only slightly longer compared to the observations).

• Inner knot (knot 1) is a highly Doppler-boosted patch of the termination shock. It could be the main contributor to the observed gamma-ray emission from the Crab nebula. A correlation of the optical emission from the knot and the gamma-ray emission is expected.

• How different is the 3D dynamics ???

The End

Crab movie

Hester et al.(2002)

Fine fibrous structure ofthe synchrotron emission,similar to that of the Crab Nebula(Scargle1969, Hester et al. 1995)

other knots

snapshot at ~the Crab’s age

HST knot 1

bright wisps

fine wisps

north-south asymmetry (Doppler beaming)

Wavelet transform (Morlet)

nµ=200 nµ=400 nµ=800

Search for quasi-periods.

There is a quasi-periodic behaviour (more than one period?) ! The period decreases with resolution … .