Particle Acceleration by Relativistic Collisionless Shocks in ElectronPositron Plasmas

1
Particle Acceleration by Relativistic
Collisionless Shocks in Electron-Positron Plasmas

• Graduate school of science, Osaka University
  Kentaro Nagata

2
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

3
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)
4
Pulsar magnetosphere

• Pulsar radius 10km
• Magnetic field 1012G
• Gamma rays and a strong magnetic field generate a
  lot of pair plasma
• The light cylinder is defined by
• rLlight speedpulsar period
• 1500km.
• In rltrL, the magnetosphere co-rotates and
  closes.
• In rgtrL, the magnetosphere doesnt co-rotate
  and extends far away .

Pulsar magnetosphere (P.Goldreich and W.H.Julian,
1969)
5
Underluminous zone and shock
X-ray image of the Crab nebula (Chandra)

• The shock is at 0.1pc from the pulsar.
• Pair plasmas are carried along the extended
  magnetic field.
• ?pulsar wind
• magnetic field 310-4G
• Lorentz factor 3106 at the shock
• The region inside the shock is underluminous
  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
6
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)
7
A theoretical model (KC model)
shock(0.1pc)
(C.F.Kennel, and F.V.Coroniti, 1984)
2pc
underluminous zone
nebula
remnant
upstream
downstream
pulsar wind
2000km/s
expansion of nebula
adiabatic expansion
Rankine-Hugoniot relations

• upstream n0,u0(g0u0),B0,P0(0cold)
• downstream n1,u1( g11 ),B1,P1
  ?7 parameters
• Rankine-Hugoniot relations
  ?4 equations
• ?4 parameters
• in 1 equation
• One upstream parameter expresses downstream flow
  velocity and energy by ultra-relativistic
  Rankine-Hugoniot relations.

Bz0magnetic field, n0number density of
particles, g0Lorentz factor of particles
8
A theoretical model (KC model)
(C.F.Kennel, and F.V.Coroniti, 1984)
shock(0.1pc)
2pc
underluminous zone
nebula
remnant
upstream
downstream
pulsar wind
2000km/s
expansion of nebula
adiabatic expansion
Rankine-Hugoniot relations

• Adiabatic expansion is solved by 1-D
  spherically symmetrical MHD conservation laws.
• u(z) flow velocity
• zdistance/(shock radius)

9
A theoretical model (KC model)
(C.F.Kennel, and F.V.Coroniti, 1984)
shock(0.1pc)
2pc
underluminous zone
nebula
remnant
upstream
downstream
pulsar wind
2000km/s
expansion of nebula
adiabatic expansion
Rankine-Hugoniot relations

• Boundary condition
• flow velocity at nebula edge
• nebula expansion velocity
• ?s310-3
• kinetic energy dominant
• In the same model, upstream flow energy is
  decided by comparing the theoretical spectrum
  with the observation, g0 3106.

10
Spectrum of the Crab nebula
Multi-wavelength spectrum of the Crab nebula
(Aharonian et al 1998)
?106
?109
synchrotron radiation
IC

• The spectrum shows a non-thermal distribution
  from optical to low gamma range.
• There must be particle acceleration processes.

11
Acceleration mechanism

• When the magnetic field is nearly perpendicular
  to the flow, downstream particles can not go back
  to the upstream region (P.D.Hudson, 1965), and
  the Fermi acceleration doesnt work efficiently.
• Possible other acceleration mechanisms
• shock surfing acceleration, magnetic
  reconnection

parallel shock
perpendicular shock
upstream
upstream
downstream
downstream
magnetic field
magnetic field
Fermi acceleration
12
Shock surfing acceleration
z
y
Bz
Ey
? charge

• Magnetized (Bz) charged particles are injected
  from upstream. Electric field (Ey) satisfies
  perfect conductivity.
• If charged particles are trapped at the shock
  surface,
• the Ey accelerates these particles along the
  shock surface.

? charged particles
?
x
? -charge
shock surface
Ey
Ex
Bz
4Bz
shock surface
13
The simulation scheme

• 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.

relativistic equation of motion
Maxwell equations
Bz
z
Ey
y
e,e-
x
injection
reflection
M.Hoshino, et al, 1992,ApJ,390,454
14
Simulation results (s10-1)
injection
shock front
wall
upstream
0.6
downstream
0
log(N)
electrostatic field Ex/Ey0
relativistic Maxwellian
3
1
?/?0
magnetic field Bz/Bz0

• A shock surface is generated.
• The averaged of fields satisfy the R-H relations.
• The spectrum is nearly relativistic Maxwellian.

3
electron energy ?/?0
1
x/(upstream gyro radius)
0
50
15
Simulation results (s10-4)
injection
shock front
wall
relativistic Maxwellian
30
upstream
downstream
nonthermal particles
0
log(N)
electrostatic field Ex/Ey0
80
0
?/?0
magnetic field Bz/Bz0

• 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.

electron energy ?/?0
20
1
x/(upstream gyro radius)
0
20
16
Particle trajectory and time variation of energy
(electrostatic field Ex)
downstream
A
B
time ?
shock front c/2
upstream
B
A
energy (?/?0)?
1
14
space ?

• particle Anot trapped by the shock front and not
  accelerated
• particle Btrapped by the shock front and
  accelerated

The particles are accelerated at the shock front
The average of accelerating electric
fieldltEy/Ey0gt1.7 gt 0
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Past studies for the shock surfing acceleration
in electron-positron plasma

• M.Hoshino (2001) showed that
• this phenomenon is characterized by s,
• when sltlt1 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
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Trapping by a magnetic neutral sheet
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
x
y
electron
Bz
Bz
x
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Relation between s and MNS

• The reflected particles begin to gyrate and
  induce an additional magnetic field.
• By definition of , when s is
  small, the initial magnetic field is small and
  the particle kinetic energy is large. Then
• initial magnetic field
• lt
• generated magnetic field,
• and a MNS is generated.
• We focus on the shock transition region in case s
  10-4 ltlt 1.

generated magnetic field (B)
Bz
Bz0
MNS
x
y
current
electron
x
20
Enlargement of the shock transition region
injection
shock transition region
wall
velocity ux/ux0
upstream
downstream
21
Shock transition region
injection
thermalize
ux/ux0
accelerated particles
Ey/Ey0 Bz/Bz0
MNS

• 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.

22
Electromagnetic fieldsaround the trapped particle
downstream
time?
upstream
space?
?Ey ?Bz The position of
accelerated particle
23
Enlargement of the shock front
particle energy ?/?0
Ey/Ey0 Bz/Bz0
inertia length (c/?p)2
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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).

inertia length (c/?p)2
60
20
Ey/Ey0 Bz/Bz0
accelerated particles
particle energy ?/?0
0
0
9.86
9.96
x/(gyro radius)
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The structure of shock front(in shock frame)
the Lorentz transformation of the shock frame
inertia length (c/?p)2
20
50
Ey/Ey0 Bz/Bz0
accelerated particles
particle energy ?/?0
0
0
x/(gyro radius)
9.86
9.96

• Averaged Ey is constantly positive in the
  acceleration region, ltEy/Ey0gt 0-7.
• The averaged Ey which accelerate particles in
  this frame ltEy/Ey0gt2 ? consistent with the
  above estimate.

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Summary

• The acceleration phenomenon is characterized by s
  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 s is small, and then a magnetic neutral
  sheet is generated. Therefore this acceleration
  process works effectively in the Crab nebula, s
  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.

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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
  (sector structure)
• ? coupling with reconnection, etc.

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Measuring the magnetic field strength
The spectrum of the Crab nebula P.L.Marsden, et
al
IR

• The spectrum has a break frequency, n1013.
• The lifetime of the Crab is t 930 yr.

Optical
Radio