NASA Technical Reports Server (NTRS) 20090007486: e(sup +/-) Pair Loading and the Origin of the Upstream Magnetic Field in GRB Shocks

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NASA Marshall Space Flight Center
DRAWV ERSION OCTOBER 27,2006
Reprint m e tw ing Uijjs tyle emulateapj v. 10/09/06
e* PAIR LOADING AND THE ORIGIN OF THE UPSTREAM MAGNETIC HELD IN GRB SHOCKS
ENRICORA MIREZ-RUIZ'~, KEN-ICHNII SHIKAWA~A ND CHRISTIABN. HEDEDAL~
Drafr version October 27,2006
ABSTRACT
We investigate.h ere the effects of plasma instabilities driven by rapid ef 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 ef 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 ef 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
1. INTRODUCTION at which the relativistic fireball has become optically thin to -
yy collisions (Piran 1999). However, the observed spectra
More than three decades ago, it was pointed out that y-rays
produced in suficiently luminous and compact astrophysical are hard, with a significant fraction of the energy above the -
yy e+e- formation energy threshold, and a high compact-
sources would create ef pairs by collisions with lower energy -+
ness parameter can result in new pairs being formed outside
photons: yy e+e- (Jelley 1966). This mCchanism both de-
4
the originally optically thin shocks responsible for the primary
pletes the escaping radiation at high energies and also changes
the composition and properties of the radiating gas through radiation (Thompson & Madau 2000). Radiation scattered by
the external medium, as the collimated y-ray front propagates
the injection of new particles. An approximate condition for
through the ambient medium, would be decollimated, and, as
such pair creation to become significant is that a sizable frac-
2
long as 1 1, absorbed by the primary beam. An ef pair cas-
tion of the radiation from the object be emitted aboye the elec-
cade can then be produced as photons are back-scattered by
tron mass energy, me2 = 51 1 keV, and that the compactness
the newly formed ef pairs and interact with other incoming
of the source.
seed photons (Thompson & Madau 2000; Beloborodov 2002,
2005; M6szt'k-os et al. 2001; Ramirez-Ruiz et al. 2002; Li et
- al. 2003; Kumar & Panaitescu 2004).
exceeds 1, where L is the total luminosity, and rl is the char- In this Letter, we consider the plasma instabilities gener-
acteristic source dimension. When 1 1, a photon of energy
ated by rapid ef pair creation in GRBs. The injection of
E = hu/m2 1 has an optical depth of unity for creating an ef pairs induces strong streaming motions in the ambient
e* pair.
medium ahead of the forward shock. This sheared flow.wil1
As an illustration consider a spherical source with a spec-
be Weibel unstable (Medvedev & Loeb 1999), and if there is
trum emitting a power L in each decade of frequencies. 1
time before the shock hits, the resulting plasma instabilities
would be > 1f or y-rays of energy E if the radius of the source will generate sub equipartition quasi-static long-lived mag-
satisfies'. -
netic fields on the collisionless temporal and spatial scales
across the ef pair-enriched region. This is studied in $3 us-
-
ing three-dimensional kinetic simulations of monoenergetic
The relevant values of L range from 160 - 16er~g s -l for and broadband pair plasma shells interpenetrating an unmag-
GRBs. In addition, the short spikes observed in the high en- netized medium. The importance of the electromagnetic fil-
er& light curires suggest that GRBs dissipate a significant amentation instability in providing an effective coupling be-
fraction of their energy at r << rl, so that the source is indeed tween ef pairs and ions is investigated in $2. The implica-
so small that equation (2) is easily satisfied. This argument tions for the origin of the upstream magnetic field, in partic-
is,. however, only 'applicable to y-riy photons emitted isotrop- ular in the context of constraints imposed by observations of
jcallx and is alleviated when the rad-ia ting source itself expand GRB afterglows, are discussed in $4.
at a relativistic speed (Piran 1999). In this case, the photons
are bearned'hfto a harrow angle 8 l/I' along the direction 2. PAIR LOADING AND MAGNEI'IC FIIEU)G ENERATION
of motion,,and, as2 result, the threshold energy for pair pro- Given a certain external baryon density np at- a radius r out-
ductionkrithin the beamed is increased to E w I?. side the shocks producing the GRB primordial spectrum, the
The non-thermal spectrum of GRB sources is therefore initial Thomson scattering optical depth is T n,oi.r and a
thought to arise in shocks which develop beyond the radius fraction r of the rimordial photons will be scattered back,
initiating a pair ee cascade. Since the photon flux drops as
Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA , r-2, for a uniform (or decreasing) external ion density most of
Department of Astronomy and Astrophysics, University of California, the scattering occurs between r and r/2, and the scattering and
Santa Cruz, CA 95064, USA
pair formation may be approximated as a local phenomenon.
National Space Science and Technology Center, Huntsville, AL 35805 -
Consider an initial input GRB radiation spectrum of the
Dark Cosmology Center, NieIs Bohr Institute, Jnliane Maries Vej 30,
2100 Copenhagen, Denmark form F(E)= F~(E/E~fo)r- e~ > eb, where eb 0.2 - 1.0 is the
break energy above which the spectral index cr 1 -2 (for the
N
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)
where AT = l/(nyuT) M c/(FbflT) 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 ef 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 w/AT and a fraction w/X,, of these
N
photons are absorbed Beloborodov (2002). One ef pair
per ambient electron is injected when w = (XTA,)lI2 M
XTE~'~/~I(2~ A)TI, ~wh/e~re n(a) = 2-"(7/12)(1+ cr)-'I3
(Svensson 1987) and 6th is the photon threshold energy for ef
6 5 6 5
formation. For 1.5 a 2, one has 15XT WI 25AT, so
that pair ereation substantially lags behindkelectrons cattering
(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- FIG.1 . - ef_.pair momentum distribution functions att = 20.8~;~(s olid
ally moving) pairs. A photon of energy E, << 1 that is side- curves) and t = 59.8q1 (dotted curves). The blue (red) curves are for a
monoenergetic (broadband)i nitial momentum distribution fhction. The top
scattered through an angle 8, creates a pair if it collides with
N (bottom) panel show the distribution functions for particles moving parallel
apother photon with energy exceeding E* N 4(8;~,)-~. The (perpendicular) to the injected ef pair plasma
injected pair will be relativistic with y* w E* 2 1. The dis-
tribution of injected pairs is therefore directly determined by (Beloborodov 2002). In what follows we assume the scatter-
the high energy spectral index a. This motivates our study in ing medium to be composed purely of ef pairs since it greatly
93 of radially streaming, relativistic pair plasma shells with a simplifies the calculations.
broadband kinetic energy distribution interpenetrating an un-
magnetized medium. 3. THE WEIBEL INSTABILITY IN ef PAIR CASCADES
5
' This pair-dominated plasma, as long as its density nf 3.1. Simulation Model . .
nq(mi/2me), is initially held back by the inertia of its con-
Here we illustrate the main features of the collision of an
sbtuent ions, provided that the pairs remain coupled to the
ek pair plasma shell into an magnetized medium, initially
baryons (Thompson & Madau 2000; Beloborodov 2002).
at rest, using a modified version of the PIC code TRISTAN
The latter is likely to be the case in the presence of weak
- first developed by (Buneman 1993) ,and most recently up-
magnetic fields' (Thompson & Madau '2000). In the absence
dated by (Nishikawa et al. 2005,2006). The simulations were
of coupling,' the pair density would not exponentiate, mainly
- performed on a 85 x 85 x 640 grid (the axes are labeled as
due to the (1 p) term in the scattering cross section, where
x, y, z) with a total of 3.8 x 10' particles for 4600 time steps,
p = V/C. Instabilities caused by pair streaming relative to the with periodic ([x,y] plane) and radiative (z direction) bound-
medium at rest, as we argued in 93, are able to generate long-
ary conditions. In physical units, the box size is-8.9 x 8.9 x
lived magnetic fields on the collisionless temporal and spatial
66.7 (c/w,)~, and the-simulations ran for 60(we)-f . .
scales, which are modest multiples of the electron plasma fre-
In the simuIations, acharge-neutral plasma shell, consisting
q~ency,~i=ct (~4 ~e'n,/m~)*/a~h,d the collisionless skin depth, of e: psjirs and moving with a bulk mbmenpm u, = jov,/c
along't@ez direction, penetrates an ambient pIasma ieajly at
rest, Here yo is &e initial Lorentz factor of the pairs and v, is
the bulk velocity of the shell along z. Pairs are continuously
Here ne is the electron number density. Plasma instabilities injected at z = 2.6Ae = 256, where A = Xe/9.6 is the g~@ size.
therefore evolve faster than the time between successive scat- The ef pairs in the ambient medium (i.e., a mass w om e/mp
terings a dm uch before the scattered photons are absorbed of 1) have a thermal spread with an rms velocity v*/c = 0.1.
by the primary radiation (Beloborodov 2002). In the pres- The shell and the ambient plasma have a pair density ratio of
ence of transverse magnetic field B the pairs gyrate around 0.75. Two different bulk Lorentz factor configurations for the
the field lines frozen into the medium on the Larrnor time, injected (cold) pairs are considered: a monoenergetic (u, =
wil = mec/(Be). The net momentum of the eh pairs is thus 15.0, V*/C = 0.01) and a broadband distribution (4.0 5 u, 5
efficiently communicated to the medium. Magnetic coupling 100.O,vh/c = 0.01). Both distributions have similar kinetic
may dominate if wg > we, which requires B2/47r > nemec2 energy contents and plasma temperatures (Fig 1).