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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
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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)
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Galactic originGalactic origin
Extragalactic origin?Extragalactic origin?
Cosmic Ray Spectrum
Galactic—ExtragalacticTransition ~1018 eV
12 decades of energies
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A closer look at ultrahigh energy
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Alan Watson at ICRC2007
0
pp
np
CMB
CMB
Greisen-Zatsepin-Kuzmincutoff
Look for viable acceleration mechanisms
Source flux E-γ
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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
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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).
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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
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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
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2.5 5 7.5 10 12.5 15 17.5ck p
0.2
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cp12
2.5 5 7.5 10 12.5 15 17.5ckp
2
4
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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
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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
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Acceleration Gradient
Maximum wakefield (Acceleration Gradient G) excited by whistler wave in magnetized plasma is
wbc
eEa
ackG
20
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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
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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.
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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
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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
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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
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R~ 50 Rs~ 500km
θ~0.1
R
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uRs~10km
The acceleration gradient at the sweet spot
*Just need 100 km to accelerate particle to 1020 eV provided 10-4!
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at arrive weGRB, of luminosity thewith
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Does acceleration gradient really depend on surfaceB field and plasma density?
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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
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No explicit dependence on magnetic field and plasma density!
Attainable energy 1020 eV for 10-4
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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
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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