Magnetowave Induced Plasma Wakefield Acceleration for UHECR
Guey-Lin LinNational Chiao-Tung University
and Leung Center for Cosmology and Particle astrophysics, National Taiwan University
Blois 2008
Work done withF.-Y. Chang (KIPAC/Stanford & NCTU), P. Chen (KIPAC/Stanford & NTU)K. Reil (KIPAC/Stanford) and R. Sydora (U. of Alberta)
axXi:v: 0709.1177 (astro-ph)
Galactic originGalactic origin
Extragalactic origin?Extragalactic origin?
Cosmic Ray Spectrum
Galactic—ExtragalacticTransition ~1018 eV
12 decades of energies
A closer look at ultrahigh energy
Alan Watson at ICRC2007
0
pp
np
CMB
CMB
Greisen-Zatsepin-Kuzmincutoff
Look for viable acceleration mechanisms
Source flux E-γ
Cosmic Particle Acceleration Models
• Conventional models
Fermi Acceleration (1949) (= stochastic accel. bouncing off magnetic domains)
Diffusive Shock Acceleration (1970s) (a variant of Fermi mechanism)
( Krymsky, Axford et al, Bell, Blandford&Ostriker)
Limited by the shock size, acceleration time, synchrotron radiation losses, etc.
• Examples of new ideas Unipolar Induction Acceleration (R. Blandford, astro-ph/9906026, June 1999)
Plasma Wakefield Acceleration
(Chen, Tajima, Takahashi, Phys. Rev. Lett. 89 , 161101 (2002))
Many others
We shall focus on the plasma wakefield acceleration
plasma wakefield acceleration • Idea originated by Chen, Tajima and Takahashi in 2002
• Plasma wakefield generated in relativistic astrophysical outflows.
Good features of plasma wake field acceleration: —The energy gain per unit distance does not depend (inversely) on the particle's instantaneous energy.
—The acceleration is linear.
•The resulting spectral index
Stochastic encounters of accelerating-decelerating phase
results in the power-law spectrum: f(E) ~ E-2.
Energy loss (not coupled to the acceleration process) steepens the energy spectrum to f(E) ~ E-(2+β).
B
• Laser Plasma Wakefield Accelerator (LPWA)
A Single short laser pulse
T. Tajima and J. Dawson, Phys. Rev. Lett. (1979)
• Plasma Wakefield Accelerator (PWFA)
A High energy electron bunch
P. Chen, et al., Phys. Rev. Lett. (1985)
• Magnetowave Plasma Wakefield Accelerator (MPWA)
A single short magneto-pulse in magnetized plasma
P. Chen, T. Tajima, Y. Takahashi, Phys. Rev. Lett. (2002)
Three Ways of Driving Plasma Wakefield
A magneto-pulse can be excited in a magnetized plasma
more relevant to astrophysical application
But high intensity lasers or e-beams may be hard to find in astrophysical settings
Waves in Magnetized Plasma
• If k║B, the dispersion relation of wave in magnetized plasma
ce
pe
ci
pick11
22222
+ – right-handed , – + left-handed
and 4 possible modes exist
ω=kc
We call the branches below the light curve (=kc) “Magneto-waves” because of their phase velocities are lower than the speed of light.
E/B = vph/c <1
One can always find a reference frame where the wave has only B component.
pi ,pe : plasma frequency for ion& e-
ci,ce :cyclotron frequency for ion & e-
ω=kc
2.5 5 7.5 10 12.5 15 17.5ck p
0.2
0.4
0.6
0.8
1
vhpc
c p1
c p6
cp12
2.5 5 7.5 10 12.5 15 17.5ckp
2
4
6
8
10
12
14
p
cp1
cp6
c p12
Whistler Mode Dispersion Relation v.s. Magnetic Field B
We aim for the large B case.
As B increases, the relation approaches to a linear curve and the slope is closed to c.
The range of k in simulation
Take k and B to be along +z direction, the whistlerwave packet induces the ponderomotive force
Amplitude of whistler pulse
Perpendicularto k and B
This leads to the plasma wakefield
Simulation results
whistler pulseplasma wakefield
Acceleration Gradient
Maximum wakefield (Acceleration Gradient G) excited by whistler wave in magnetized plasma is
wbc
eEa
ackG
20
20
2
22
1)(
mceEmceAa
emcE
w
pwb
//
/
20
whereχ~O(1): Form factor of pulse shape
Vg ~ c
Cold wavebreaking limit
Lorentz-invariant normalized vector potential
“strength parameter”
a0 <<1 linear
a0 >>1 nonlinearif
wb
wb
Ea
EaG
0
20
The wakefield acceleration is efficient only when p < < c
Verified for a0 <<1 by simulation
Applications to UHECR acceleration
• The astrophysical environment is extremely nonlinear, while our simulations are performed in the linear regime
• In view of successful validation of linear regime, we have confidence to extend the theory to the nonlinear regime.
Strength parameter a0=eEw/mc
G
Varying Ew while fixing k and The dependence of G on the strength parameter a0 verified!
G a0 for a0>>1
Numerical result
Fitted curve
Arbitrary
unit
Extension to a0>>1 is done analytically
Acceleration in GRB Assume NS-NS merger as short burst GRB progenitor, where trains of magneto-pulses were excited along with the out-burst
R
Typical neutron star radius ~ 10 km
Surface magnetic field B ~ 1013 G
Jet opening angle θ ~ 0.1
Total luminosity L~ 1050 erg/s
Initial plasma density n0~1026 cm-3
θ
Due to the conservation of magnetic flux, B decreases as 1/r2. The plasma density also decrease as 1/r2. Therefore
while 21
rBc rnp
1
Wakefield excitation most effective when p~~c.
Where is the sweet spot (choose c/p=6)?
Location for the sweet spot: R ~ 50 RNS ~500 km
2.5 5 7.5 10 12.5 15 17.5ck p
0.2
0.4
0.6
0.8
1
vhpc
c p1
c p6
cp12
2.5 5 7.5 10 12.5 15 17.5ckp
2
4
6
8
10
12
14
p
cp1
cp6
c p12
Whistler Mode Dispersion Relation v.s. Magnetic Field B
We aim for the large B case.
As B increases, the relation approaches to a linear curve and the slope is closed to c.
The range of k in simulation
R~ 50 Rs~ 500km
θ~0.1
R
.10
eV/cm10125.025.0
3/~for 10
240
.4
4
modes in whister cksmagnetosho
theinto goingenergy outburst offraction :
,10~4
413
00
40
2
222
2220
223262
pwb
c
GRBw
GRBw
GRB
GRB
mcaeEaG
a
E
cm
e
mc
eEa
EEcmergE
uRs~10km
The acceleration gradient at the sweet spot
*Just need 100 km to accelerate particle to 1020 eV provided 10-4!
at arrive weGRB, of luminosity thewith
)(314
applying and 3
Taking
.4
parameterstrength The
.12
1
conditionspot sweet theApplying
.44
, Write
.6spot sweet the takesLet'
2
2c
222
220
0
0
0222
0
p
c
L
RR
cR
LEu
ucm
ea
mnc
B
R
R
R
R
m
ne
m
ne
R
R
cm
eB
cm
eB
GRB
GRB
GRBe
ens
ns
ee
pp
ns
eec
Rns=10 kmθ~0.1
R
Does acceleration gradient really depend on surfaceB field and plasma density?
LncmRB
cmaeEaG
cmn
L
Ra
ens
pewb
ens
00
00
30
0
4
3325.025.0
,1
4
3
Let us take the range of the sweet spot of order 0.1R.Then, within the 0.1R range, a proton can be accelerated to the energy
./10 and 1.0with
1075.040
31.0
50
222
sergL
eVc
LeRG
No explicit dependence on magnetic field and plasma density!
Attainable energy 1020 eV for 10-4
Acceleration in AGN
Take nAGN 1010 cm-3, B104 G at the core of AGNL1046 erg/s
eV/cm)10( ),1010(For
eV/m 1025.0 ,10243
200
OG
eEaGa wb
Acceleration distance for achieving 1021 eV is about 10 pc, much smaller than typical AGN jet size
** is the fraction of total energy imparted into the magnetowave modes.** Frequency of magnetowave in this case is in the radio wave region. can be inferred from the observed AGN radio wave luminosity
Summary
• The plasma wakefield acceleration is a possible mechanism to explain the UHECR production.
• Our simulations confirm, for the first time, the generation of the plasma wakefield by a whistler wave packet in a magnetized plasma. We have studied k||B case, simulation for a general angle is in progress. Simulations for production of whistler wave packet is also in progress.
• When connecting it to relativistic GRB outflow, we suggest that super-GZK energy can be naturally produced by MPWA with a 1/E2 spectrum.
•Same mechanism is also applicable to AGN
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