Polarization in Pulsar Wind Nebulae Delia Volpi In collaboration with: L. Del Zanna - E. Amato - N. Bucciantini Dipartimento di Astronomia e Scienza dello.

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Polarization in Pulsar Wind Nebulae

Delia Volpi

In collaboration with: L. Del Zanna - E. Amato - N. Bucciantini

Dipartimento di Astronomia e Scienza dello Spazio-Università degli Studi di Firenze-Italy

Observations: optical and X-ray Continnum emission

Jet-torus structure

Crab Nebula

PWN

Optical (Hubble)

X-ray (CHANDRA)

Vela PSR 1509-58 G 54.1+0.3

RELATIVISTIC MHD Models

Analytical 1-D models (Kennel & Coroniti ,1984 -- Emmering and Chevalier,1987): no jet-torus structure.

Analytical 2-D models (Bogovalov & Khangoulian, 2002 -- Lyubarsky, 2002): anisotropic Poynting+kinetic energy flux torus and oblate TS. Jets collimated by hoop stresses downwards TS.

RMHD simulations (Komissarov and Lyubarsky, 2003 – Del Zanna et al., 2004) solve 2-D hyperbolic equations and confirm jet-torus morphology theories

Lyubarsky, 2001

IDEAL RMHD EQUATIONS

€

∂ ργ( )∂t

+∇ • ργr v ( ) = 0 continuity equation

∂r S

∂t+∇ •

r T = 0 equation of the momentum conservation

∂U∂t

+ c2∇ •r S = 0 equation of the energy conservation

∂r B

∂t−∇×

r v ×

r B ( ) = 0 Faraday induction equation

r S =

w

c2γ 2r

v +1

4πc

r E ×

r B

r T =

w

c2γ 2 r

v r v −

14π

r E

r E +

r B

r B ( ) + P + uem( )I U = wγ 2 − P + uem

w = ρc2 +Γ

Γ −1P Γ =

43

uem =E2 + B2

8π

r E = -

r v ×

r B

c Ohm law

Adiabatic equation of state

ECHO: GRMHD 3-D VERSION (Del Zanna et al., 2007, A&A, 473, 11)

Numerical model• 2-D (axysimmetric) RMHD shock capturing code in spherical coordinates (Del

Zanna et al., 2002, 2003, 2004) • Initial cold ultrarelativistic Pulsar Wind with v radial and r-2 +SNR+ISM Lorentz factor (conservation of energy along streamlines) :

Anisotropic energy flux:

Toroidal field:

• Space and time evolution of RMHD equations+ maximum particle energy (Lorentz factor) equation with adiabatic and synchrotron losses averaged on the pitch angle:

€

γ θ( )=γ0α

0+ 1−α

0( )sin 2 θ( ) γ 0= 100 α

0= 0.1 anisotropy parameter

€

F0 r,θ( ) = F0r0r

⎛ ⎝ ⎜

⎞ ⎠ ⎟2

α 0 + 1− α 0 +σ( )sin2 θ[ ] σ =B2

4πwγ 2= magnetization parameter

€

B r,θ( ) = B0r0r

⎛ ⎝ ⎜

⎞ ⎠ ⎟sinθ tanh b

π2

−θ ⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎡

⎣ ⎢ ⎤

⎦ ⎥ b = width of striped wind region

(b =/2)

w=c2+4p

€

d

dt'lnε

∞=

d

dt'ln n1/3 +

1

ε∞

dε∞

dt'

⎛

⎝ ⎜

⎞

⎠ ⎟sync

dε

∞dt'

⎛

⎝ ⎜

⎞

⎠ ⎟sync

= −4e4

9m3c5 B'2ε∞

2

Synchrotron emission recipes

• Emitting particles’ isotropic distribution function at termination shock (TS = 0)

(Kennel & Coroniti, 1984b):

• Post-shock distribution function (obtained from conservation of particles’number along streamlines and under condition of 0.5):

€

f0(ε0 ) =A

4πε0

−(2α +1) ε 0min≤ ε0 ≤ ε0

max A = Knn0 = Kpp0

mc2

€

f (ε ) =K

p

4π

p

mc2ε −(2α +1) ε <ε

∞ K

p= constant

spectral index

Synchrotron emission recipes

• Emission coefficient in observer’s fixed frame:

Relativistic corrections:

• Cut-off frequency for synchrotron burn-off (evolved in the code from the maximum particle energy):

€

jν(ν , ˆ n )∝Cp

r B ' × ˆ n '

α +1

Dα + 2ν −α ν∞

≥ ν

jν(ν , ˆ n ) = 0 ν

∞< ν

D =1

γ (1−r β • ˆ n )

= Doppler boosting factor r β =

r v

c

ˆ n ' = D ˆ n +γ 2

γ +1

r β • ˆ n −γ

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟r β

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

r B ' =

1

γ

r B +

γ 2

γ +1

r β •

r B ( )

r β

⎡

⎣ ⎢

⎤

⎦ ⎥

€

ν ∞ =D3e

4πmc

r B ' × ˆ n ' ε

∞

2

observer direction versor

optical or X-ray frequency of observation

Synchrotron emission recipes• Surface brightness:

• Stokes parameters (linear polarization):

• Polarization fraction () and polarization direction (P):

• Spectral index () for two frequencies (1, 2) and integrated spectra (F):

€

Iν (ν ,Y ,Z) = jν−∞

+∞

∫ (ν , X,Y ,Z)dX X = line of sight Y, Z = plane of sky

€

Qν (ν ,Y ,Z) =α +1

α + 5/3jν

−∞

+∞

∫ (ν , X,Y ,Z) cos 2χ( )dX

Uν ν ,Y ,Z( ) =α +1

α + 5/3jν

−∞

+∞

∫ (ν , X,Y ,Z) sin 2χ( )dX local polarization position anglebetween emitted electric field and Z

€

Πν =Q

ν

2+U

ν

2

Iν

r P

ν= Π

νsin χ( ) ˆ x + cos χ( ) ˆ y ( )

€

αν ν1,ν

2,Y ,Z( ) = −

log Iν

ν2,Y ,Z( ) / I

νν

1,Y ,Z( )[ ]

log ν2/ν

1( )

Fν

ν( ) =1

d 2I

νν ,Y ,Z( )∫∫ dYdZ d=distance of emitting object

Wind magnetization Runs with effective=0.02

(averaged over ) A (=0.025, b=10) B (=0.1,

b=1)

Results: flow structure maps

• RunA: a) Stronger pinching forces smaller wind zone; b) Equipartition near TS; c) Larger magnetized region • particles loose most of their energy nearer to TS; d) Less complex magnetization map.• Supersonic jets and equatorial outflow: v 0.5-0.7c (as in Crab Nebula-Hester-2002, Vela-Pavlov 2003).

=0.025, b=10 narrow striped wind region

=0.1, b=1 large striped wind region

Results: surface brightness maps

• Optical and X emitting particles: = 0.6 (Veron-Cetty & Woltjer,1993)

• Cut-off frequencies: =5364 Å in optical maps (V.C.1993) h=1keV in X maps (Chandra)

• Angles (Weisskopf, 2000): inclination of symmetry axis:300

rotation respect to North: 480

• I normalized respect to maximum value, logarithmic scales

• Larger emitting regions in optical than in X bandsynchrotron burn-off• Internal regions: system of rings (connected to external vortices), brighter arch (inner ring), a central knot (connected to polar cusp region) due to Doppler boosting (very strong near TS, vc)• Stronger emission near TS where magnetization and velocity are higher



HIGH RESOLUTION POLARIZATION MAPS TOY MODEL

UNIFORM EMITTING TORUS WITH B TOROIDAL AND V RADIAL : ANGLE SWING INCREASES WITH V AND BIGGER IN THE FRONTv=0.2C θ=90° - - v=0.5c θ=75° -- v=0.8c θ=60°=>INFORMATION ABOUTFLOW VELOCITYPOLARIZATION FRACTION=>MAXIMUM EVERYWHERE

SAME EFFECT WITH A KC FLOW

Results: optical polarization maps Synchrotron emission

linear polarization

Polarization fraction: Bpoloidal

• Normalization against (+1)/(+5/3)70%• Along polar axis: higher polarized fraction (projected B line of sight)• Outer regions: depolarization (opposite signs of projected B along line of sight)

Polarization direction: vflow

• Ticks: ortogonal to B, lenght proportional to Π (Schmidt, 1979)• Polarization angle swing (deviation of vector direction) in brighter arcs, v c, strong Doppler boost Bigger effect in the front side Origin of the knot????? RunB: more complex structure slow velocity



Results: X-ray polarization maps=0.025, b=10 narrow striped wind region


Results: spectral index maps

• Values of Crab Nebula• Optical maps obtained with: 1=5364Å, 2=9241Å (Veron-Cetty & Woltjer, 1993)• X-ray maps obtained with: h1=0.5keV, h2=8keV (Mori et al., 2004)• Spectral index grows from inner to outer regions• RunA: X-ray simulated spectral index maps similar to ones of Crab Nebula (Mori et al, 2004)

= + 1



Observations: gamma-ray• MAGIC Telescope (J.Albert et al., Arxiv:0705.3244v1, 2007)

• Crab Nebula: gamma-ray standard candle → target of new instruments

• Emission: accelerated electrons+target photons (CMB+FIR+sync)

• HESS: TeV frequencies; GLAST: 20MeV-300GeV

• Disantangle magnetic field and distribution function+adronic component

Synchrotron and IC emission recipes

• Primordial isotropic radio-emitting population (A&A,1996):

• Accelerated wind population at TS (A&A,1996):

• Evolved distribution function (Del Zanna et al., 2006):

• Integration between spectral power per unit of frequency and distribution function:

synchrotron=> monochromatic frequency

IC => respect to ε and ν, total differential cross section, 3 targets (FIR, CMB, SYNC) (Blumenthal and Gould, 1970)

€

fr(ε) =Ar4π

ε−(2αr +1)

e(−ε /ε*r)

€

fTS

(εTS

) =A

TS4π

(ε +εTS

)−(2α

w+1)

e(−ε

TS/ε*

w ) ε

TS= ε

ρ TSρ

⎛

⎝ ⎜

⎞

⎠ ⎟

1/31

1 -ε

ε∞

€

fw(ε) = ρρTS

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

4/3εTS

ε

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2fTS(εTS)

€

νc = 0.293

4πe

mcB⊥

'ε 2 critical frequency

Results: IC

Multislopes Disconnetted areas in maximum particle energyIC emission in excessEnergy map: Compression around TS of B.Parameter? Distribution function?


Results: IC

Size reduction with increasing frequency: along y-axisJet and torus visible for radio electron distribution, no observational counterpart


Results: IC

Time-variability: gamma-rays similar to X-rays.


Conclusions• Spectra: well reproduced from radio to X-ray. Excess in gamma-ray due to

compression of B around TS (flux vortices).

• Brightness maps: jet-torus structure in gamma-rays as in X-rays (high resolution). Observed dimensions.

• Gamma-ray (as X-ray) time-variability is well reproduced by MHD motions.

COMPLETE SET OF DIAGNOSTIC TOOLS FOR PWNe AND OTHER EMITTING SOURCES (AGN, GRB)

• Future work: direct evolution of the distribution function; investigation of the parameter space; applications to other PWNe (different evolution stages) and other non-thermal emitting sources (AGN, GRB).

• Paper: Simulated synchrotron emission from Pulsar Wind Nebulae (L.Del Zanna, D.Volpi, E.Amato, N.Bucciantini, A&A, 453, 621-633, 2006)

• Paper: Non-thermal emission from relativistic MHD simulations of pulsar wind nebulae: from synchrotron to inverse Compton, D.Volpi, L. Del Zanna, E. Amato, N. Bucciantini, A&A, 2008, 485, 337

WHICH KIND OF CONTINUUM EMISSION FROM RADIO TO SOFT-GAMMA ?

CRAB NEBULA: SAME FRACTION AND POSITION ANGLE OF POLARIZATIONFROM RADIO TO X-RAYS=> SIGNATURE OF SYNCHROTRON EMISSIONOPTICAL (SHKLOVSKY, 1952- DOMBROVSKY, 1954)

SYNCHROTRON EMISSION=> LINEAR POLARIZATION WITH A MAXIMUMOF ≈ 80%

IMPORTANCE OF POLARIZATION: 1) GEOMETRY OF THE SOURCE (PULSAR WIND)2) PROPERTIES OF THE SOURCE=>MAGNETIC FIELD STRENGHT AND DIRECTION3) ACCELERATION OF PARTICLES

OBTAIN SYNTHETIC MAPS FROM NUMERICAL SIMULATIONS AND COMPAREWITH OBSERVATIONS: STUDY OF POLARIZATION

IC emission recipes

Integration between distribution function (primordial and wind) and power per unit of frequency respect to ε and ν(Blumenthal&Gould, 1970)

Incident photon density per unit of frequency: IC from CMB target

IC from FIR target

IC from SYN target

€

nνIC−CMB = 8π

c3νt

2

ehνt /kBT

−1 blackbody formula → T= 2.7K

€

nνIC−FIR = 1

chνt

LνFIR

4πR2 Lν → uniformly emitting region of blackbody radiation

€

nνIC−SYN r

r ,νt( ) = 1chνt

jνsyn r

r ',νt ⎛ ⎝ ⎜

⎞ ⎠ ⎟

r r ' −

r r

∫ dV ' integration over the entire nebula

€

Qν (ν ,Y ,Z ) =α +1

α + 5/3jν (ν ,X ,Y ,Z )cos2χdX

−∞

∞

∫

Uν (ν ,Y ,Z ) =α +1

α + 5/3jν

−∞

∞

∫ (ν ,X ,Y ,Z )sin2χdX

r ′ e ∝

r ′ n ×

r ′ B emitted electric field in the comoving frame

r ′ b =

r ′ n ×

r ′ e radiated magnetic field

r e = γ

r ′ e -

γγ +1

r ′ e •

r β ( )

r β −

r β ×

r ′ b

⎡

⎣ ⎢

⎤

⎦ ⎥ Lorentz transformation

r e ∝

r n ×

r q in the plane of the sky

r q =

r B +

r n ×

r β ×

r B ( )

qY = 1− βX( )BY + βY BX qZ = 1− βX( )BZ + βZBX

cos2χ =qY

2 − qZ2

qY2 + qZ

2 sin2χ = -

2qYqZ

qY2 + qZ

2

Polarization in Pulsar Wind Nebulae Delia Volpi In collaboration with: L. Del Zanna - E. Amato - N. Bucciantini Dipartimento di Astronomia e Scienza dello.

Documents

narrow striped wind

b toroidal

xray slide

large striped wind region

p slide

spectral index slide

wind magnetization

xray polarization maps