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