Particle Acceleration by Relativistic Collisionless Shocks

in Electron-Positron Plasmas

Graduate school of science, Osaka University Kentaro Nagata

Contents

• Motivation

Observation and theory for the Crab nebula

Necessity of particle acceleration processes (not Fermi acceleration)

• Simulation results for relativistic collisionless shocks in electron-positron plasmas

• Discussion

Pulsar nebula• The pulsar nebula is

brightened by the energy coming from the central neutron star via pulsar wind.

• One of the best observed and studied pulsar nebula is the Crab nebula because

its age is known (951 years) and

observations are relatively easy (~2kpc).The Crab nebula in optical three bands

(European Southern Observatory)

Pulsar magnetosphere

• Pulsar radius ~ 10km• Magnetic field ~ 1012G• Gamma rays and a strong ma

gnetic field generate a lot of pair plasma

• The light cylinder is defined by

rL=light speed×pulsar period ~ 1500km.

In r<rL, the magnetosphere co-rotates and closes.

In r>rL, the magnetosphere doesn’t co-rotate and extends far away .

Pulsar magnetosphere(P.Goldreich and W.H.Julian, 1969)

Underluminous zone and shock• The shock is at ~0.1pc from the puls

ar.• Pair plasmas are carried along the e

xtended magnetic field. ⇒pulsar wind magnetic field ~3×10-4[G] Lorentz factor ~ 3×106 at the shock• The region inside the shock is under

luminous because the pulsar wind is still cold.

• The magnetic field is almost toridal because the pulsar rotates with high frequency and the wind is highly relativistic.

shock

X-ray image of the Crab nebula(Chandra)

Nebula

• The thermalized pulsar wind brights by means of synchrotron radiation.

• Expanding filament structures suggest that the nebula edge expands with a velocity ~2000km/s at ~2pc from the pulsar.

The Crab nebula in optical three bands(European Southern Observatory)

A theoretical model (KC model)

upstream n0,u0(0=u0),B0,P0(=0:cold)

downstream n1,u1( 1=1 ),B1,P1 7 parameters⇒Rankine-Hugoniot relations 4 equations⇒⇒4 parameters in 1 equationOne upstream parameter expresses downstream flow velocity

and energy by ultra-relativistic Rankine-Hugoniot relations.

2200

20 4/

mcn

B

densityenergykineticupstream

densityenergyfieldmagneticupstream z

upstream

pulsar wind

Rankine-Hugoniot relations

adiabatic expansion

underluminous zone nebula

downstream

remnant

expansionof nebula

Bz0:magnetic field, n0:number density of particles, 0:Lorentz factor of particles

(C.F.Kennel, and F.V.Coroniti, 1984)

21

2200

20

122

00

20

1

21 1

18

11

14

1

1

umcn

B

umcn

B

u

u

shock(0.1pc)2pc

2000km/s

A theoretical model (KC model)

upstream

pulsar wind

shock(0.1pc)

Rankine-Hugoniot relations

adiabatic expansion

underluminous zone nebula

downstream

remnant

expansionof nebula

(C.F.Kennel, and F.V.Coroniti, 1984)

Adiabatic expansion is solved by 1-D spherically symmetrical MHD conservation laws.

u(z) : flow velocity z=distance/(shock radius)

])3(1[)( 3/12 zzu

2pc

2000km/s

flow velocity (km/s)

6000

4000

2000

A theoretical model (KC model)

• Boundary conditionflow velocity at nebula edge = nebula expansion velocity ⇒σ~3×10-3 “kinetic energy dominant”• In the same model,

upstream flow energy is decided by comparing the theoretical spectrum with the observation, ~ 3×106.

upstream

pulsar wind

Rankine-Hugoniot relations

adiabatic expansion

underluminous zone nebula

downstream

remnant

expansionof nebula

(C.F.Kennel, and F.V.Coroniti, 1984)

flow velocity (km/s)

6000

4000

2000

shock(0.1pc) 2pc

2000km/s

Spectrum of the Crab nebula

• The spectrum shows a non-thermal distribution from optical to low gamma range.

• There must be particle acceleration processes.

Multi-wavelength spectrum of the Crab nebula (Aharonian et al 1998)

synchrotron radiationIC

γ=106

γ=109

Acceleration mechanism• When the magnetic field is nearly perpendicular to th

e flow, downstream particles can not go back to the upstream region (P.D.Hudson, 1965), and the Fermi acceleration doesn’t work efficiently.

• Possible other acceleration mechanisms shock surfing acceleration, magnetic reconnection

parallel shock perpendicular shock

magnetic field ma

gn

etic

fie

ld

upstreamdownstream

upstream downstream

Fermi acceleration

Shock surfing acceleration

1. Magnetized (Bz) charged particles are injected from upstream. Electric field (Ey) satisfies perfect conductivity.

2. If charged particles are trapped at the shock surface,

3. the “Ey” accelerates these particles along the shock surface.

zxy BvE

ｚ

ｘ

ｙB ｚ E ｙ

shock surface

①chargedparticles

③+charge

③-charge

②

E ｙ

shock surface

B ｚ 4B ｚ

Ex

positron

•Particle motion and electro-magnetic field are consistently solved by the equation of motion and Maxwell equations respectively.

•Magnetized cold electron-positron plasma is uniformly injected from the left boundary.

•Initial electro-magnetic fields (Ey,Bz) satisfy perfect conductivity.

•Particles are reflected at the right boundary in order to make back flow.

The simulation scheme

ｘ

ｙｚ B ｚ

E ｙ

e+,e-

injection

reflection

zxy BvE

M.Hoshino, et al, 1992,ApJ,390,454

)()(

Bc

vEq

dt

vdm

　c

J

t

E

cB

t

B

cEE

41,

1,4

: relativistic equation of motion

: Maxwell equations

Simulation results (=10-1)

γ/γ0

relativistic Maxwellian

log

(N)

1

0

1

3

0.6

3

injection wall

upstream downstream

magnetic field Bz/Bz0

electron energy

electrostatic field Ex/Ey0

x/(upstream gyro radius)0 50

• A shock surface is generated.

• The averaged of fields satisfy the R-H relations.

• The spectrum is nearly relativistic Maxwellian.

shock front

Simulation results (=10-4)

1

0

0

20

30injection wall

upstream downstream

magnetic field Bz/Bz0

electron energy

electrostatic field Ex/Ey0

x/(upstream gyro radius)0

shock front

80

20

γ/γ0

relativistic Maxwellian

log

(N) nonthermal particles

• The amplitude of the fields is very large.

• The averaged fields still satisfies R-H relations.

• Nonthermal high energy particles appear at the shock front.

Particle trajectory and time variation of energy

BAspace ⇒

time ⇒

energy (γ/γ0)→

shock front ~ c/2

(electrostatic field Ex)

BA

upstream

downstream

•particle A ： not trapped by the shock front and not accelerated•particle B ： trapped by the shock front and accelerated

The particles are accelerated at the shock front

The average of accelerating electric field 〈 Ey/Ey0 〉 ~1.7 > 0

1 14

Past studies for the shock surfing acceleration in electron-positron plasma

M.Hoshino (2001) showed that• this phenomenon is characterized by ,• when <<1 particle energy spectrum is nonthermal,• accelerated particles are trapped at the shock

surface, where a magnetic neutral sheet (MNS) exists.

M.Hoshino, 2001, Prog. Phys. Sup. 143, 149

Trapping by a magnetic neutral sheet

Bz

x

x

y

electron

magnetic neutral sheet(MNS)

• Particles are trapped because their gyro motions change its direction through the MNS.

• All that time, motional electric field Ey accelerates the particles along the MNS.

• In the shock transition region, Ey has a finite value. On the other hand, in the down stream Ey is almost zero. This is the reason why the particle acceleration occurs only at the shock front. Bz Bz

Relation between and MNS

x

xy

Bz

current

electron

～ Bz0

～ Bz0

generated magneticfield (B)

MNS

•The reflected particles begin to gyrate and induce an additional magnetic field. •By definition of 　　　　　　　　　 , when is small, the initial magnetic field is small and the particle kinetic energy is large. Then

initial magnetic field＜

generated magnetic field, and a MNS is generated.•We focus on the shock transition region in case = 10-4 << 1.

200

20 8/ cmNB ez

Enlargement of the shock transition regioninjection wall

upstream downstream

shock transition region

velocity ux/ux0

shock front

Shock transition region

• The injected particles go through the shock front and some of them return to the front.

• Trapping and acceleration occur at the shock front, not the whole shock transition region.

ux/ux0

Ey/Ey0 Bz/Bz0

accelerated particles

MNS

thermalize

injection shock front

ー Ey 　 ー Bz +The position of accelerated particle

space⇒

tim

e⇒

upstream

downstream

Electromagnetic fieldsaround the trapped particle

Enlargement of the shock front

Ey/Ey0

Bz/Bz0

particle energy　 γ/γ0

inertia length (c/ωp)× ２

The structure of the shock front(in downstream frame)

•The acceleration by Ey is unclear, because Ey has a large positive and negative amplitude around the MNS.•The MNS propagate upstream with velocity ~ c/2, so we should discuss in the MNS frame (= shock frame).

x/ （ gyro radius ）9.86 9.96

Ey/Ey0

Bz/Bz0

0

60

particleenergy　 γ/γ0

0

20

0,4

x

EJ

cx

B yy

z steady stateMaxwell eqs ⇒evaluate Ey

accelerated particles

inertia length (c/ωp)× ２

15.1)1(,2/1

)(),(

2/12

sss

yszszzsysy

velocityshock

EBBBEE

•Averaged Ey’ is constantly positive in the acceleration region, 〈 Ey’/Ey0 〉～ 0 － 7.•The averaged Ey’ which accelerate particles in this frame 〈 Ey’/Ey0 〉～ 2 　⇒　 consistent with the above estimate.

the Lorentz transformation of the shock frame

x/ （ gyro radius ）9.86 9.960

20Ey’/Ey0

Bz’/Bz0

0

50

particleenergy　 γ/γ0

accelerated particles

inertia length (c/ωp)× ２

The structure of shock front(in shock frame)

Summary

• The acceleration phenomenon is characterized by which is defined as the ratio of the magnetic field energy density and the particle kinetic energy density.

• The additional magnetic field induced by gyrating particles overcome the initial magnetic field when is small, and then a magnetic neutral sheet is generated. Therefore this acceleration process works effectively in the Crab nebula, ~ 10-3.

• Some particles are accelerated by motional electric field Ey while the magnetic neutral sheet traps them.

• The magnetic neutral sheet is kept stationary at the shock front. We confirmed that the averaged Ey is positive in the shock frame and that the particle acceleration occurs at the shock frame.

To go further

• The condition of trapping and detrapping for accelerated particles

⇒ Spectrum form (maximum energy, power-low or thermal)

• 2-D simulation

⇒ Weibel instability• Magnetic field consisting of opposing two regions (s

ector structure)

⇒ coupling with reconnection, etc.

Measuring the magnetic field strength

• The spectrum has a break frequency, ~1013.

• The lifetime of the Crab is ~ 930 yr.

The spectrum of the Crab nebula P.L.Marsden, et al

Radio

IR

Optical

][103101.7

2

3

,3

2,,

)(|/|

423

242

32

2242

GBBmc

Be

cm

Be

dt

dEmcE

lossnsynchrotrodtdE

E