WIGGLER RADIATION !
IN LANGMUIR TURBULENCE !
~ A POSSIBLE EMISSION MECHANISM OF GRB ~RIKEN (ABB. lab.)!

Yuto Teraki
collaborator: Fumio TakaharaRefs. [1] Teraki & Takahara, 2014, ApJ, 787, 28

Introduction

α-distribution in GRB

Number of GRBs

α-distribution

Kaneko et al. 2006

/ erg/s/Hz2
Photon Photon
spectrumspectrum 

N (E) ∝ E

αN (E) ∝ E β 

E

Inconsistent !
with!
synchrotron theory
3

Emission RegioneShock front

Debye
lengthTurbulentelectromagnetic field✏B magnetic✏E electricSironi & Spitkovsky 2009

We do not know the radiation !
perfectly for this case!

B 2 /8⇡
✏B =
nmc24

(a)

located within a thin layer of orderthe front. There is a gas Ñow throuÑux freezing, the generated magne
with the shocked material downs
replenished with a fresh one where
thus continuously
small scale magnetic
fieldproduced. Since t
lived2and does not decay in a dyna
mc
by numerical simulations (see refeB ⌧ Loeb 1999), this Ðeld will be presentTurbulent
fieldandradiationαΔθ∼1/γ

jitter radiation(b)eBFIG. 1.ÈEmission
from &various
points along
the particleÏs trajectory.
rectilinear
trajectory
perturbativeacceleration(a) a ? *h ; emission from selected parts (bold portions) of the trajectory is
seen by an observer. (b) a > *h ; emission from the entire trajectory isobserved.

3.1. Fields Produced by the EleMedvedev 2000
In this subsection we brieÑy sum
Landau& Lifshitz
1979
of the theoryof Medvedev& Loeb

ence. Since electrons are light, thethem is rapid : the typical e-folding
time times the shock speed) is much

Of course, not general!!
Strong field?!

Electric field?

Oscillation of field?
5

Photon Formation Time (Length)
for synchrotron peak
Eret (t)・P (!)

t2 eBmc2

mc=eB

Photon Formation !
Lengthspatial fluctuation scale

mceB!

Photon Formation
Time (PFT)

oscillation timescale

Langmuir (Electrostatic) waves
Dispersion relation !
of Langmuir wave

Generated by !
two stream instability!

23 2 2
2= !pe + k ve
2Dieckmann 2005

Neglect

typical scale = inertial length
crossing timeSilva 2006

✏E ⇠ 0.1! = !pec/!peT ⇠ 1/!peeEmcstpeStrong, fast oscillating electrostatic turbulence!
7

Summarizing...
We cannot ignore!
the spatial fluctuation nor!
time variability!
c!stSpatial scale!
of turbulences2⇡c⇠turb =
ktyp!st1/!st ⇠ 1/!pcrossing time ~ oscillation timescale
8

Parametrizing the turbulenceStrength parameterast

ktyp cOscillation parameterb⌘!pktyp cTypicallya⇠b⇠19Known parameter domainb

Fleishman & Toptygin 2007

DRL theory

1?
Unknown!

1?
?
aMission
Clarify the radiation spectra from!
electrons in turbulent field !
for alland ba

strength parameter!sta⌘
ktyp coscillation parameterb⌘!pktyp c
11

Numerical
Calculation

random phase, and direction

We generate 3D isotropic Langmuir turbulence by usi
which is slightly modified from
N the description for magne!
"acalone & Jokipii (1999).
It is described
E(x) =An cos by(knsuperposition· x − ωp t +randomphase, and direction
Superposition the Fouriermodes
E 2 (k)n=1µ

Description of the turbulent fieldskN!
" phase, wave v
Here,A,β,k,andωaretheamplitude,nnnpNE(x) =An cos (kn · x − ωp t +X~
kn An of each mode is defined arespectively.
The amplitude~~
n=1E(~x) =An cos (kn · ~x !p t + n )|k|$ N%−1nHere, An , βn , kn, and ωp are the amplitude, phase,n=1! wave v
ktyprespectively. The amplitude An Aof2n each
is maxdefined=σ 2modeGn k
Gn k as,$n=1
N!

Spatial scale !0 ⌘ ktyp c%−122A=
σGn of turbulenGn,
where the variable σ represents the amplitude
nn=1

for the power spectrum

Time scale

24πkofwhere the variable σ represents the amplitude
turbulentnn ∆kGn =,αfor the power spectrum
1 + (k2n Lc )4πkn ∆kn
G, ∆kn =field. Here,
where Lc is the correlation length oftheα
Lc = 2⇡/k
1 +typ(kn Lc )!pequal spacing
k-space,overfiniteinterwhere Lcinis logarithmic
the correlation
length of
the thefield.
Here,
21/23 ∆k<is E
where equal
kmin =2π/Linc , logarithmic
α = 9/2 andkmaxover
chosen10interv
kminspacingk-space,the >finite3Langstand thespectral
wherekmin =index
2π/Lcis, αfor= 3-dimensional
9/2 and kmax is isotropic
chosen
10
kmin
~
kn0
and the spectral
is for 3-dimensional
isotropic
Lang
two parameterswhichindex
characterizeradiation
spectra.
Theê =e
Mean strength ! ⌘mcz
two parameters which characterize radiation
The|knspectra.
|a≡eσ=

ω

13 st,

Calculation the radiation spectra
Inject electrons with init = 10

Solve the EOM

An example of thetrajectoryd( me v) = eEdtUse the Lienard-Wiechert potential directly

Unit vector toward observer

Retarded time!st2
a== 10!0F!

!pµ = 5/2
b== 0.1, 1, 5, 7, 10!0Hump
24!p = 101!

bdW/dω

1F!
corresponds tohigh wavenumbermodes!

ω/ω05
2

1

Time variability!
dominated
Spectral index
1F! / !

Fig. 2.— Radiation spectra for the jitter radiation to DRL regime. The parameters area = 10−2 and b = 0.1, 1, 5, 7, 10 from top down in low frequency range. The straight line
shows the power law spectrum with indexes 1 and −5/2

Typical frequency!typ =2!pa1 &Origin of the spectral shape for aF (!)

!

0!

b>1

weakly beamed!
&!
oscillation origin

1!

0

5/2! = !p&2strongly beamed !
&!
wavenumber origin

ktyp c!

Doppler boosted oscillation !
with p1(1v/c)p

0! = ktyp c cos ✓& E(ktyp )
2

p16

2

kc & power law component

Turbulenceeephoton2frequency:cv⇠cinitial frame
22E (k)2electron rest frame

c=2

k5/2

~ ⇠ |B|~
|E|

kcF

5/2

imaginary!
photona = 100, b = 20, 90, 400, 800
F!

!

13dW/dωNew!

µ = 5/2
Softer than F! / !

b1!

1

1

aWiggler Radiation ! Strength!
in Langmuir turbulemce!
dominated!
ω/ω0(WRL)
Spectral index
Typical frequency

Fig. 3.— Radiation spectra for the DRL to Wiggler-like regime. The parameters are a = 102
1and b = 20, 90, 400, 800
from
top
down.
Thestraight
lines
show
thepowerlaw
spectrum2
2
3
with indexes
1/3
and1.
typstp!

=! , (>! )F! / !

1Discussion in observer frame
peak frequency(1v1) mc2 1 ⇠c eE c

2!st

mc2eE
mc2
eE

whencircular orbit approximation should be !
!st > !papplicable in the timescale of 1/!st

We discuss the validity!
for this approximation

parallel v.s. perpendicular
d~
p=mdt

E.O.M.
d~v
~v k
dt
d~v
~v ?
dt

✓◆
3d~v
d~v
+ 2 ~v ·~v
dt
cdtv
3 d~

F~ = minertia
2times larger

dt
d~v
F~ = mdt

be accelerated almost perpendicular
2e2
P = 3
3c4

"✓dv?
dt

◆2Fm+2✓dvk
dt◆2 #Fm 3

power

2times larger

Energy change & Conclusion~E = eE · ~v t
EnergyPFT for2!stt = 1/!stelectric fieldmc
2
E . eEc ⇥= mceE

Typical frequency is written as

2Conclusion
using!stat given time

Application&Summary

Chart of spectral signatures!p=!0b=a

DRLb

F! / !

probable!
region!

1Jitter

12!pTime
F! / ! 1/2
Space2!0F! / !

01WRL

2!stF! / ! 1/3

Strengthb=a

LAE-like

nonlineara!st=!0

Gamma Ray Bursts

Number of GRBs

α-­‐distribu*on

Photon spectrumN(E)∝EHard GRBs can be!
α 

explained

N (E) ∝ E β





0 




E



α− 2 Kaneko et al. 2006
324

Polarization degree and spectral indices・Some GRBs may have high polarization degree!
e.g. observation of GRB110301A by GAP (Yonetoku et al. 2011)= 70 ± 22 % with 3.7
According to our model of GRB emission,

WRL

Soft GRBs with<2/3

High

Hard GRBs with>

2/3

Low

DRL
It is inverse of the Ito san’s prediction!

Summary
1.We studied the radiation spectra from electrons!! !
! moving in a Langmuir turbulence by using firstprinciple numerical calculation.
2.We clarify the radiation spectra including newly
1/3
found spectrum; WRL F
3.Hard spectra of GRBs can be explained bythis radiation mechanism in realistic parameter
range of turbulences in shock region in GRBs.

Thank you!

27