We investigate here the effects of plasma instabilities driven by rapid e(sup plus or minus) pair cascades, which arise in the environment of GRB sources as a result of back-scattering of a seed fraction of their original spectrum. The injection of e(sup plus or minus) pairs induces strong streaming motions in the ambient medium. One therefore expects the pair-enriched medium ahead of the forward shock to be strongly sheared on length scales comparable to the radiation front thickness. Using three-dimensional particle-in-cell simulations, we show that plasma instabilities driven by these streaming e(sup plus or minus) pairs are responsible for the excitation of near-equipartition, turbulent magnetic fields. Our results reveal the importance of the electromagnetic filamentation instability in ensuring an effective coupling between e(sup plus or minus) pairs and ions, and may help explain the origin of large upstream fields in GRB shocks

Hededal, Christian B.

Nishikawa, Ken-Ichi

Ramirez-Ruiz, Enrico

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DRAFT VERSION OCTOBER 27,2006 
Preprint typeset using style emulateapj v. 10/09/06 
ek PAIR LOADING AND THE ORIGIN OF THE UPSTREAM MAGNETIC FIELD IN GRB SHOCKS 
ENRICO RAMIREZ-RUIZ"~, KEN-ICHI NISHIKAWA~ AND CHRISTIAN B. HEDEDAL~ 
Draft version October 27, 2006 
ABSTRACT 
We investigate here the effects of plasma instabilities driven by rapid e* pair cascades, which arise in the 
environment of GRB sources as a result of back-scattering of a seed fraction of their original spectrum. The 
injection of e* pairs induces strong streaming motions in the ambient medium. One therefore expects the 
pair-enriched medium ahead of the forward shock to be strongly sheared on length scales comparable to the ra- 
diation front thickness. Using three-dimensional particle-in-cell simulations, we show that plasma instabilities 
driven by these streaming es pairs are responsible for the excitation of near-equipartition, turbulent magnetic 
fields. Our results reveal the importance of the electromagnetic filamentation instability in ensuring an effective 
coupling between e* pairs and ions, and may help explain the origin of large upstream fields in GRB shocks. 
Subject headings: gamma rays: bursts - instabilities - magnetic fields - plasmas - shock waves 
More than three decades ago, it was pointed out that y-rays 
produced in sufficiently luminous and compact astrophysical 
sources would create e* pairs by collisions with lower energy 
photons: ?-y t e+e- (Jelley 1966). This mechanism both de- 
pletes the escaping radiation at high energies and also changes 
the composition and properties of the radiating gas through 
the injection of new particles. An approximate condition for 
such pair creation to become significant is that a sizable frac- 
tion of the radiation from the object be emitted above the elec- 
tron mass energy, mec2 = 51 1 keV, and that the compactness 
of the source, 
exceeds 1, where L is the total luminosity, and r, is the char- 
acteristic source dimension. When 1 N 1, a photon of energy 
E = hv/mec2 N 1 has an optical depth of unity for creating an 
e* pair. 
As an illustration consider a spherical source with a spec- 
trum emitting a power L in each decade of frequencies. 1 
would be > 1 for y-rays of energy E if the radius of the source 
satisfies 
L 
o 5 1017e ( ) cm. 1049erg s-I 
The relevant values of L range from lo5' - 10" erg s-' for 
GRBs. In addition, the short spikes observed in the high en- 
ergy light curves suggest that GRBs dissipate a significant 
fraction of their energy at r << rr, so that the source is indeed 
so small that equation (2) is easily satisfied. This argument 
is, however, only applicable to y-ray photons emitted isotrop- 
ically and is alleviated when the radiating source itself expand 
at a relativistic speed (Piran 1999). In this case, the photons 
are beamed into a narrow angle 0 -- l/r along the direction 
of motion, and, as a result, the threshold energy for pair pro- 
duction within the beamed is increased to € ,., r. 
The non-thermal spectrum of GRB sources is therefore 
thought to arise in shocks which develop beyond the radius 
Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA , 
' Department of Astronomy and Astrophysics, University of California, 
Santa Cruz, CA 95064, USA 
National Space Science and Technology Center, Huntsville, AL 35805 
Dark Cosmology Center, Niels Bohr Institute, Juliane Maries Vej 30, 
2100 Copenhagen, Denmark 
at which the relativistic fireball has become o~ticallv thin to 
yy  collisions (Piran 1999). However, the odserved spectra 
are hard, with a significant fraction of the energy above the 
yy  t e+e- formation energy threshold, and a high compact- 
ness parameter can result in new pairs being formed outside 
the originally optically thin shocks responsible for the primary 
radiation (Thompson & Madau 2000). Radiation scattered by 
the external medium, as the collimated y-ray front propagates 
through the ambient medium, would be decollimated, and, as 
long as 1 2 1 ,  absorbed by the primary beam. An e* pair cas- 
cade can then be produced as photons are back-scattered by 
the newly formed e* pairs and interact with other incoming 
seed photons (Thompson & Madau 2000; Beloborodov 2002, 
2005; MCsz6ros et al. 2001; Ramirez-Ruiz et al. 2002; Li et 
al. 2003; Kumar & Panaitescu 2004). 
In this Letter, we consider the plasma instabilities gener- 
ated by rapid es pair creation in GRBs. The injection of 
e A  pairs induces strong streaming motions in the ambient 
medium ahead of the forward shock. This sheared flow will 
be Weibel unstable (Medvedev & Loeb 1999), and if there is 
time before the shock hits, the resulting plasma instabilities 
will generate sub equipartition quasi-static long-lived mag- 
netic fields on the collisionless temporal and spatial scales 
across the es pair-enriched region. This is studied in $3 us- 
ing three-dimensional kinetic simulations of monoenergetic 
and broadband pair plasma shells interpenetrating an unmag- 
netized medium. The importance of the electromagnetic fil- 
amentation instability in providing an effective coupling be- 
tween es pairs and ions is investigated in 132. The implica- 
tions for the origin of the upstream magnetic field, in partic- 
ular in the context of constraints imposed by observations of 
GRB afterglows, are discussed in $4. 
2. PAIR LOADING AND MAGNETIC HELD GENERATION 
Given a certain external baryon density n,, at a radius r out- 
side the shocks producing the GRB primordial spectrum, the 
initial Thomson scattering optical depth is T N n,aTr and a 
fraction T of the rimordial photons will be scattered back, 
initiating a pair eP cascade. Since the photon flux drops as 
rV2, for a uniform (or decreasing) external ion density most of 
the scattering occurs between r and r/2, and the scattering and 
pair formation may be approximated as a local phenomenon. 
Consider an initial input GRB radiation spectrum of the 
form F(E) = Fb(e/eb)-a for E > ~ b ,  where ~b N 0.2- 1.0 is the 
https://ntrs.nasa.gov/search.jsp?R=20070017972 2019-08-30T00:45:59+00:00Z
break energy above which the spectral index a N 1 -2 (for the 
present purposes the exact low energy slope is unimportant). 
Radiation scattered by the external medium is therefore decol- 
limated, and then absorbed by the primary beam.The impact 
of such process strongly depends on how many photons each 
electron is able to scatter (Beloborodov 2002) 
ATY4x lo8(+)' ( cm < A, (3) 10 cm lo5] erg s-l 
where AT = l/(n,cT) M c / (Fbc~)  is the electron mean free 
path and n, is the photon density. The photons forced out 
from the beam by the electrons can then produce e* pairs as 
they interact with incoming photons, so that a large number 
of scatterings implies a large number of pairs created per am- 
bient electron. 
At some distance w < A from the leading edge of the ra- 
diation front, the number of photons scattered by one am- 
bient electron is N w/AT and a fraction ,,, w/A,, of these 
photons are absorbed Beloborodov (2002). One e* pair 
per ambient electron is injected when w = ( x ~ x , , ) ~ ~ ~  M 
~Tf:'~/[n(Cu)]'/~ 2 AT, where n(a)  = 2-"(7/12)(1+ a)-513 
(Svensson 1987) and eth is the photon threshold energy for ef 
formation. For 1.5 5 a 5 2, one has 15AT 6 WI 5 25AT, so 
that pair creation substantially lags behind electron scattering 
(Beloborodov 2002). 
Most of the momentum deposited through this process in- 
volves the side-scattering of very soft photons, which col- 
lide with hard y-rays to produce energetic (and almost radi- 
ally moving) pairs. A photon of energy e, << 1 that is side- 
scattered through an angle w 8, creates a pair if it collides with 
another photon with energy exceeding eth N 4(8:e,.)-'. The 
injected pair will be relativistic with y* eth 2 1. The dis- 
tribution of injected pairs is therefore directly determined by 
the high energy spectral index a. This motivates our study in 
$3 of radially streaming, relativistic pair plasma shells with a 
broadband kinetic energy distribution interpenetrating an un- 
magnetized medium. 
This pair-dominated plasma, as long as its density n* 5 
np(mp/2me), is initially held back by the inertia of its con- 
stltuent ions, provided that the pairs remain coupled to the 
baryons (Thompson & Madau 2000; Beloborodov 2002). 
The latter is likely to be the case in the presence of weak 
magnetic fields (Thompson & Madau 2000). In the absence 
of coupling, the pair density would not exponentiate, mainly 
due to the (1 - 0) term in the scattering cross section, where 
/3 = V/C. Instabilities caused by pair streaming relative to the 
medium at rest, as we argued in $3, are able to generate long- 
lived magnetic fields on the collisionless temporal and spatial 
scales, which are modest multiples of the electron plasma fre- 
quency, we = ( 4 ~ e ~ n ~ / m ~ ) ' / ~ ,  and the collisionless skin depth, 
C 
x , = - - - - ~ x ~ o ~  - 
we 
( c2-3) -'I2 AT < A, . (4) 
Here ne is the electron number density. Plasma instabilities 
therefore evolve faster than the time between successive scat- 
terings and much before the scattered photons are absorbed 
by the primary radiation (Beloborodov 2002). In the pres- 
ence of transverse magnetic field B the pairs gyrate around 
the field lines frozen into the medium on the Larmor time, 
wjl = mec/(Be). The net momentum of the e* pairs is thus 
efficiently communicated to the medium. Magnetic coupling 
may dominate if wg > we, which requires B2/47r > nemec2 
F I G .  1 .  - e* pair momentum distribution functions atr = 20.8~;' (solid 
curves) and t = 5 9 . 8 4  (dotred curves). The blue (red) curves are for a 
monoenergetic (broadband) initial momentum distribution function. The top 
(botroin) panel show the distribution functions for particles moving parallel 
(perpendicular) to the injected e* pair plasma. 
(Beloborodov 2002). In what follows we assume the scatter- 
ing medium to be composed purely of e* pairs since it greatly 
simplifies the calculations. 
3. THE WEIBEL INSTABILITY IN e* PATR CASCADES 
3.1. Simulation Model 
Here we illustrate the main features of the collision of an 
e* pair plasma shell into an unmagnetized medium, initially 
at rest, using a modified version of the PIC code TRISTAN 
- first developed by (Buneman 1993) and most recently up- 
dated by (Nishikawa et al. 2005,2006). The simulations were 
performed on a 85 x 85 x 640 grid (the axes are labeled as 
x, y, z) with a total of 3.8 x lo8 particles for 4600 time steps, 
with periodic ([x,y] plane) and radiative (z direction) bound- 
ary conditions. In physical units, the box size is 8.9 x 8.9 x 
66.7 (~ /w , )~ ,  and the simulations ran for 60(we)-I. 
In the simulations, a charge-neutral plasma shell, consisting 
of ef pairs and moving with a bulk momentum u, = yovz/c 
along the z direction, penetrates an ambient plasma initially at 
rest. Here yo is the initial Lorentz factor of the pairs and v, is 
the bulk velocity of the shell along z. Pairs are continuously 
injected at z = 2.6Ae = 25A, where A = Ae/9.6 is the grid size. 
The e* pairs in the ambient medium (i.e., a mass ratio me/m,, 
of 1) have a thermal spread with an rms velocity vth/c = 0.1. 
The shell and the ambient plasma have a pair density ratio of 
0.75. Two different bulk Lorentz factor configurations for the 
injected (cold) pairs are considered: a monoenergetic (u, = 
15.0, vth/c = 0.01) and a broadband distribution (4.0 < u, < 
1 0 0 . 0 , ~ ~ ~ / c  = 0.01). Both distributions have similar kinetic 
energy contents and plasma temperatures (Fig I). 


e+/- Pair Loading and the Origin of the Upstream Field in GRB Shocks

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