Download shock front

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
The Crab nebula in optical three bands relatively easy (~2kpc).
(European Southern Observatory)
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
rL=light speed×pulsar period
~ 1500km.
In r<rL, the magnetosphere corotates and closes.
Pulsar magnetosphere
In r>rL, the magnetosphere
(P.Goldreich and W.H.Julian, 1969)
doesn’t co-rotate and extends
far away .
Underluminous zone and shock
X-ray image of the Crab nebula
(Chandra)
shock
• The shock is at ~0.1pc from the
pulsar.
• Pair plasmas are carried along the
extended magnetic field.
⇒pulsar wind
magnetic field ~3×10-4[G]
Lorentz factor ~ 3×106 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.
Nebula
• The thermalized pulsar
wind brights by means
of synchrotron radiation.
• Expanding filament
structures suggest that
the nebula edge
expands with a velocity
The Crab nebula in optical three bands
~2000km/s at ~2pc from
(European Southern Observatory)
the pulsar.
A theoretical model (KC model)
shock(0.1pc)
underluminous zone
nebula
upstream downstream
(C.F.Kennel, and F.V.Coroniti, 1984)
2pc
remnant
2000km/s
expansion
of nebula
pulsar wind
adiabatic expansion
Rankine-Hugoniot relations
upstream
n0,u0(g0=u0),B0,P0(=0:cold)
downstream n1,u1( g1=1 ),B1,P1
⇒7 parameters
Rankine-Hugoniot relations
⇒4 equations
 1
2
u

⇒4 parameters
1


B02
B02
1 
1
 1 



1    1 
1  2 

2
2 
2
2 
in 1 equation
u1
8n0g 0 mc  u1 
 4n0g 0 mc  u1 
One upstream parameter expresses downstream flow velocity
and energy by ultra-relativistic Rankine-Hugoniot relations.
upstream magnetic field energy density Bz 0 / 4


upstream kinetic energy density
n0g 02 mc 2
Bz0:magnetic field, n0:number density of particles, g0:Lorentz factor of particles
2
A theoretical model (KC model)
(C.F.Kennel, and F.V.Coroniti, 1984)
shock(0.1pc)
2pc
underluminous zone
nebula
upstream downstream
pulsar wind
adiabatic expansion
Rankine-Hugoniot relations
Adiabatic expansion is
solved by 1-D spherically
symmetrical MHD
conservation laws.
flow velocity
(km/s)
6000
4000
u( z )   [1  (3z 2 ) 1/ 3 ]
u(z) : flow velocity
z=distance/(shock radius)
2000
remnant
2000km/s
expansion
of nebula
A theoretical model (KC model)
(C.F.Kennel, and F.V.Coroniti, 1984)
shock(0.1pc)
2pc
underluminous zone
nebula
upstream downstream
pulsar wind
adiabatic expansion
Rankine-Hugoniot relations
• Boundary condition
flow 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, g0 ~ 3×106.
flow velocity
(km/s)
6000
4000
2000
remnant
2000km/s
expansion
of nebula
Spectrum of the Crab nebula
Multi-wavelength spectrum of the Crab nebula (Aharonian et al 1998)
γ=106
synchrotron radiation
γ=109
IC
• The spectrum shows a non-thermal distribution
from optical to low gamma range.
• There must be particle acceleration processes.
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 doesn’t work efficiently.
• Possible other acceleration mechanisms
shock surfing acceleration, magnetic reconnection
parallel shock
downstream
magnetic field
Fermi acceleration
upstream
magnetic field
upstream
perpendicular shock
downstream
Shock surfing acceleration
ｚ
Bｚ
ｙ
③
1.
+charge
Eｙ
①
charged
particles
③
-charge
②
ｘ
Eｙ
Ex
Bｚ
4Bｚ
positron
shock surface
Magnetized (Bz) charged
particles are injected
from upstream. Electric
field (Ey) satisfies perfect
conductivity.
E y  v x Bz
2. If charged particles are
trapped at the shock
surface,
3. the “Ey” accelerates
these particles along the
shock surface.
The simulation scheme
•Particle motion and electro-magnetic field are consistently solved
by the equation of motion and Maxwell equations respectively.
d (gv)
v
m
 q( E   B) : relativistic equation of motion
dt
c
1 B
1 E 4J
:　Maxwell equations
  E  4,   E  
,  B 

c t
c t
c
•Magnetized cold electron-positron plasma is uniformly injected
from the left boundary.
B
ｚ
•Initial electro-magnetic fields (Ey,Bz)
e+,esatisfy perfect conductivity.
E y  v x Bz
•Particles are reflected at the right
boundary in order to make back flow.
ｚ
Eｙ
ｙ
ｘ
injection
reflection
M.Hoshino, et al, 1992,ApJ,390,454
Simulation results
injection
shock front
0.6 upstream
-1
(=10 )
wall
log(N)
downstream
0
electrostatic field Ex/Ey0
relativistic Maxwellian
3
1
magnetic field Bz/Bz0
3 electron energy 0
1
0
x/(upstream gyro radius)
50
γ/γ0
• A shock surface is
generated.
• The averaged of fields
satisfy the R-H relations.
• The spectrum is nearly
relativistic Maxwellian.
Simulation results
injection
upstream
wall
relativistic Maxwellian
downstream
log(N)
30
shock front
0
-4
(=10 )
nonthermal particles
electrostatic field Ex/Ey0
80
0
magnetic field Bz/Bz0
20 electron energy 0
1
0
x/(upstream gyro radius)
γ/γ0
• The amplitude of the fields
is very large.
• The averaged fields still
satisfies R-H relations.
20 • Nonthermal high energy
particles appear at the
shock front.
Particle trajectory and time variation of energy
(electrostatic field Ex)
downstream
B
time ⇒
A
upstream
A
space ⇒
B
1
energy (γ/γ0)→
14
•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
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
neutral sheet
• Particles are trapped because their magnetic(MNS)
gyro motions change its direction
Bz
through the MNS.
• All that time, motional electric field
Ey accelerates the particles along
the MNS.
x
• In the shock transition region, Ey
has a finite value. On the other
y
hand, in the down stream Ey is
electron
almost zero. This is the reason why
the particle acceleration occurs only
at the shock front.
Bz
Bz
x
Relation between  and MNS
•The reflected particles begin to
gyrate and induce an additional
magnetic field.
2
2


B
/
8

N
m
g
c
•By definition of
,
z0
0 e 0
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.
generated magnetic
field (B)
Bz
～Bz0
MNS
x
y
current
electron
～Bz0
x
Enlargement of the shock transition region
injection
shock transition region
velocity ux/ux0
upstream
shock front
downstream
wall
Shock transition region
injection
shock front
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.
Electromagnetic fields
around the trapped particle
time⇒
downstream
upstream
space⇒
ーEy ーBz +The position of
accelerated 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).
steady state
Maxwell eqs
60
Bz
4

Jy,
x
c
E y
x
0
⇒evaluate Ey
inertia length (c/ωp)×２
Ey/Ey0
Bz/Bz0
20
accelerated particles
particle
energy
γ/γ0
0
9.86
x/（gyro radius）
9.96
0
The structure of shock front
(in shock frame)
the Lorentz transformation of the shock frame
E y  g s ( E y   s Bz ), Bz  g s ( Bz   s E y )
shock velocity  s  1 / 2, g s  (1   s ) 1/ 2  1.15
2
50
inertia length (c/ωp)×２
Ey’/Ey0
Bz’/Bz0
20
accelerated particles
particle
energy
γ/γ0
0
9.86
x/（gyro radius）
9.96
0
•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.
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
(sector structure)
⇒ coupling with reconnection, etc.
Measuring the magnetic field strength
The spectrum of the Crab nebula P.L.Marsden, et al
IR
Radio
Optical
• The spectrum has a
break frequency, n~1013.
• The lifetime of the Crab
is t ~ 930 yr.
E
t
, E  mc 2g ,
| dE / dt | ( synchrotron loss )
3eg 2 B 7.110 24


2mc
B 3t 2
 B  3 10  4 [G ]
dE 2e 4g 2 B 2

,
2 3
dt
3m c