Particle Acceleration by Relativistic Collisionless Shocks in Electron-Positron Plasmas Graduate...
-
Upload
sydney-clarke -
Category
Documents
-
view
223 -
download
0
Transcript of Particle Acceleration by Relativistic Collisionless Shocks in Electron-Positron Plasmas Graduate...
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