Radiation from Kinetic Poynting Flux Acceleration
Radiation from Kinetic Poynting Flux Acceleration
Edison Liang1 and Koichi Noguchi1
ABSTRACT
We derive analytic formulas for the power output and critical frequency of radiation by
electrons accelerated by relativistic kinetic Poynting flux, and validate these results with Particle-
In-Cell plasma simulations.  We find that the in-situ radiation power output and critical
frequency are much below those predicted by the classical synchrotron formulae.  We discuss
potential astrophysical applications of these results.
Subject Headings: Acceleration of particles– Radiation mechanisms: non-thermal - Gamma-
rays:bursts
Online Material: color figures
1. INTRODUCTION
In popular paradigms of radiation from blazars, pulsar wind nebulae (PWN), gamma-ray
bursters (GRB) and other gamma-ray sources, relativistic outflow energy (hydrodynamic or
electromagnetic) from the central compact object (black hole or neutron star) is first converted
into relativistic nonthermal electrons via some collisionless dissipation mechanisms (e.g. shocks,
Dermer 2003, Meszaros 2002, Lyubarski 2005).  These nonthermal electrons are then
hypothesized to radiate synchrotron-like radiation, including small-pitch-angle synchrotron
(Epstein and Petrosian 1973, Lloyd and Petrosian 2000), or “jitter” radiation if the magnetic field
is too chaotic (e.g. due to Weibel instability, Weibel 1958, Medvedev 2000, Medvedev et al
2005).  In addition, inverse Comptonization of the synchrotron photons (SSC) or external soft
photons (EC) may account for the high-energy (e.g. MeV-TeV) gamma-rays (Dermer et al 2000,
2003).  Most popular astrophysical models invoke the classical synchrotron formulas (Rybicki
and Lightman 1979).  However, two outstanding questions remain unsolved: (a) exactly how is
the outflow energy converted into nonthermal electron energy via collisionless shocks (CS,
Hoshino et al 1992, Gallant et al 1992, Silva et al 2003, Nishikawa et al 2003, Spitkovski 2006),
or electromagnetic Poynting flux (PF, Smolsky & Usov 2000, Lyutikov & Blackman 2002, Van
Putten & Levinson 2003, Lyutikov and Blanford 2003)? (b) do the accelerated electrons always
radiate synchrotron radiation, since the synchrotron models do not agree with observations in
many cases (Fenimore 2002, Dermer & Chang 1999, Preece et al 2000)?   In this paper we
present concrete examples of acceleration mechanisms whose radiation process is drastically
different from classical synchrotron radiation.
Over the past few years we have used sophisticated Particle-in-Cell (PIC) codes for
relativistic collisionless plasmas (Langdon and Lasinksi 1976, Birdsall & Langdon 1991,
Langdon 1992) to study nonthermal electron acceleration and radiation processes (Liang et al
2003, Liang & Nishimura 2004, Nishimura et al 2003, Liang & Noguchi 2005, 2006).  A unique
feature of our PIC simulations is that the power radiated in-situ by each superparticle
(=numerical representation of a charged particle) can be computed simultaneously as the
superparticles are accelerated by the Lorentz force (Noguchi et al 2005, Liang and Noguchi
2005, 2006).  This approach provides a fully self-consistent treatment of the intrinsic radiation
output during the acceleration process.  In this paper we focus on the radiation of plasmas
accelerated directly by intense electromagnetic pulses or Poynting flux (PF), and derive analytic
formulas for this radiation from first-principles.   Section 2 reviews the basic physics of PF
acceleration.  Section 3 briefly summarizes the key result of the numerical radiation power.  In
Section 4 we derive the critical frequency of PF radiation.  In Section 5 we derive an analytic
formula for the radiation power output.  In Section 6 we speculate on the astrophysics scenarios
of PF acceleration.  In Section 7 we apply the analytic formulas to a sample PF model of long
GRBs.  Section 8 is devoted to discussion and summary.
A common misconception about PIC simulations is that such simulations are too small in
physical scale (measured in units of plasma skin depths and electron gyroradii) to be relevant to
macroscopic astrophysical phenomena.  However, unlike MHD simulations, the purpose of PIC
simulations is not to try to reproduce macroscopic phenomena, but to discover and quantify
microphysical laws governing particle energization, radiation mechanisms, wave-particle
interaction and dissipation processes, which operate at the level of plasma skin depths and
gyroradii.  Once discovered via numerical simulations, such physical laws should be rederived
analytically from first principles.  These validated laws are then applicable to macroscopic
phenomena irrespective of the space and time scales.   This is the approach we will adopt in this
paper.
2. ACCELERATION BY KINETIC POYNTING FLUX
In this paper we are interested in relativistic collisionless plasmas whose Coulomb mean
free paths are much larger than the relevant plasma scale sizes (see Sec.7 for sample numbers).
The interaction of such plasmas with intense EM fields lies outside the classical MHD regime
and can only be treated correctly with kinetic theory.  Here we define “Poynting flux” as a
directed plasma outflow dominated and driven kinetically by transverse electromagnetic (EM)
fields with Ωe/ωpe=B/(4πnm)1/2 >1, (Ωe = eB/m =electron gyrofrequency, ωpe =(4πne2/m)1/2 =
electron plasma frequency, m=electron mass, n=imbedded electron density, c = 1 throughout this
paper except in Sec.7), in the absence of a flow-aligned longitudinal magnetic field.  Hence we
will not consider diffusive particle acceleration by scattering with classical Alfven and whistler
waves (see discussions in Sec.8) in a background magnetic field, or electrostatic acceleration by
longitudinal plasma (Langmuir) waves (Boyd and Sanderson 1969).  Instead we focus on particle
acceleration by the ponderomotive (JxB) force.  Astrophysical examples of such kinetic
electromagnetic outflows include the equatorial stripe wind of pulsars and magnetars (Lyubarsky
2005, Skjaeraasen et al 2005), and the front end of a low-density magnetic tower jet driven by
magnetized accretion disks around black holes (Koide et al 2004).
There are two opposite situations in which a kinetic PF can efficiently transfer its EM
energy into the nonthermal kinetic energy of a plasma: pushing or pulling the plasma with the
JxB force (Fig.1).  The common “pushing” mechanism involves an intense EM pulse impinging
on an overdense plasma (ωpe>2π/λ, λ=characteristic wavelength of the EM field) from the
outside (Fig.1a).  This phenomenon is well known from the interaction of intense radio waves
with the ionosphere, or from laser-plasma interactions (Kruer et al 1975).  The intense EM field
induces a ponderomotive (jxB) force near the critical surface (where ωpe=2π/λ, Fig.1a), which
accelerates the surface electrons inward to Lorentz factors characterized by the local
dimensionless vector potential ao (=eBλ/2πme, Wilks et al 1992).  When ao>>1, the EM field
“snowplows” all upstream electrons.  The reflection front moves forward relativistically so that
the EM pulse suffers little reflection (Kruer et al 1975).  In extreme cases (e.g. in an e+e-
plasma) the bulk of the EM energy is transferred to relativistic particles instead of being
reflected.  However, this type of “pushing” PF acceleration mainly produces quasi-Maxwellian
“superthermal” electron spectra with kT ~ ao (Fig.2a, Wilks et al 1992), instead of the power-law
spectra (Preece et al 2000) commonly observed in astrophysics.  We call this “pushing”
acceleration by an intense EM pulse “leading Poynting or ponderomotive acceleration” or LPA
(Fig.1a).  PIC simulations and analytic theory (Harteman and Kerman 1996) suggest that the
maximum Lorentz factor achievable by LPA is limited to max(Ωe2/ωpe2, ao2/2) due to energy and
momentum conservation, since all upstream electrons must share the PF momentum and energy.
However, because the snowplow is moving at almost light speed and the electrons are highly
collisionless, in most cases no forward shock (in the conventional sense) is observed to form in
the upstream plasma.  Astrophysically, LPA is potentially relevant to the interaction of pulsar or
magnetar winds with a dense environment in the kinetic limit.
In contrast to the LPA, the “trailing Poynting or ponderomotive acceleration” or TPA,
occurs when an intense EM pulse pulls, instead of pushes, an overdense plasma (Fig.1b, TPA
replaces the acronym DRPA used in our early publications, Liang et al 2003, Liang & Nishimura
2004).  Consider for example a situation in which a strongly magnetized, overdense collisionless
plasma with B/(4πnm)1/2 >1 suddenly expands due to force imbalance. The expansion disrupts
the sustaining current, so that 4πJ < Curl B.  In the absence of an external EMF regenerating the
current, the excess displacement current (∂E/∂t) then generates a transverse EM pulse, which
tries to escape from the embedding plasma (Fig.1b).  As the EM pulse emerges from the plasma,
it “pulls” out the surface electrons via the jxB force, where j is the self-induced polarization
current (Boyd and Sanderson 1969).  When the jxB force is ultra-intense, the accelerated
electrons can stay comoving with the group velocity of the EM pulse which is < c due to plasma
loading, and the acceleration can be sustained.  Unlike the LPA case in which the maximum
Lorentz factor is limited by momentum conservation, TPA transfers the PF energy and
momentum to a decreasing number of fast electrons, as slower electrons continuously fall behind
the pulse (Liang & Nishimura 2004).  We find that the maximum Lorentz factor achieved by
TPA is unlimited until radiation damping or dephasing (e.g. due to wave-front curvature)
become important.  PIC simulations show that TPA always accelerates the high-energy electrons
into a simple power law independent of the initial conditions or the pulse width (Fig.2b).  The
TPA mechanism is exceedingly robust and efficient, typically converting > 50% of the EM
energy into accelerated particle energy over a short distance(see Sec.7).
Physically both LPA and TPA are caused by a relativistic E x B drift in which the
transverse EM field comoves with the particle drift velocity.  The key difference is that in LPA
the plasma load snowplowed by the EM pulse increases or stays constant with time, thereby
limiting the Lorentz factor, whereas the TPA Lorentz factor increases indefinitely due to
decreasing plasma loading (Liang et al 2003, Liang and Nishimura 2004). Density-wise, LPA
involves collisionless compression of the plasma, while TPA involves rarefaction of the plasma.
In contrast to shocks, in which bulk flow energy is converted into EM energy, either via
compression of upstream fields or via Weibel (1959) and other instabilities, LPA and TPA
converts EM energy into accelerated particle energy.  We emphasize that both LPA and TPA are
strictly kinetic phenomena with no analog in the MHD limit.  The detailed physics of LPA and
TPA has been reviewed extensively elsewhere (Liang and Noguchi 2005, 2006), so they will not
be repeated here.
TPA may be relevant to radiation in astrophysics at two different levels: global and local.
Globally, EM pulses with large-scale ordered fields may be generated due to reconnection of a
magnetic tower jet or magnetar wind (Koide et al 2004), or from the merger of strongly
magnetized neutron stars into a black hole.  For example, TPA can take place when the front end
of a disconnected magnetic tower jet propagates down the steep density gradient of the collapsar
envelope and turns into an unconfined kinetic EM pulse (see Sec.7).  Similarly, when a
millisecond magnetar or a merging strongly magnetized neutron star binary collapses to form a
black hole, part of its energy may be emitted in the form of an intense EM pulse.
However, TPA may also occur at the local level even in the absence of large scale
ordered EM fields.  For example, relativistic EM turbulence generated by shocks and shear
layers may dissipate via the TPA mechanism as the nonlinear waves propagate into low density
regions with Ωe/ωpe>1.   In this case sustained comoving particle acceleration can last locally
until dephasing occurs.  We emphasize that the only piece of physics invoked in the radiation
calculation below is that the particle is accelerated locally by a nearly comoving transverse EM
wave.  No assumption about the global geometry, topology or size scale of the EM field is
required.  Hence the radiation formulas derived in this paper have more general validity and
much broader applications than the small scale PIC simulations performed so far based on the
simplistic LPA and TPA scenarios (Liang and Noguchi 2005, 2006).
3. NUMERICAL RADIATION POWER OUTPUT
In this paper we focus on the radiation output of electrons (and positrons) accelerated by
a comoving kinetic Poynting flux.  Numerically, we compute the radiation power output by
incorporating the relativistic dipole formula (Rybicki & Lightman 1979) into our PIC code:
                     Prad  = 2e2 (F||2 + γ2F+2)/3m2                 (1)
where γ = Lorentz factor, F|| = force component along velocity v, and F+ = force component
orthogonal to v. We compute the total power loss of each superparticle in the PIC simulation, by
interpolating the field data from the cell boundaries to the instantaneous superparticle position,
so that F  and v refer to the same time and space point.  We have carefully calibrated this
numerical procedure with known analytic results.  Fig.3 compares the numerical radiation output
for an isotropic thermal plasma in a static uniform B field with that computed using the analytic
synchrotron formula (Rybicki and Lightman 1979).  Their excellent agreement, especially for the
high-energy electrons, validates our numerical algorithm.   However, PIC simulation cannot be
used to compute the radiation spectrum numerically because the PIC simulation time step
(typically = 0.25 gyroperiod) is too large to accommodate the high frequencies.
The upper panels of Figs.4 & 5 illustrate the evolution of the Prad distribution of
superparticles in sample LPA and TPA runs.  In both cases a plane, linearly polarized EM pulse
accelerates the same slab of overdense e+e- plasma, one from the outside and one from the
inside.  While the energies of the pairs increase monotonically due to the acceleration, the power
radiated by the electrons initially rises to a maximum, but then declines monotonically.  We find
that in both cases Prad << the classical synchrotron power Psyn = 2e4B2p+2/3m2 (mp+=momentum
orthogonal to B).  This suppression of radiative power can be understood as follows.  As
particles are accelerated to higher and higher γ, v aligns increasingly with the Lorentz force F.
So the F+ term in Eq.(1) decreases relative to the F|| term.   However, Psyn comes only from the F+
term (Rybicki & Lightman 1979).  Hence Psyn >> Prad for high γ particles.   In Sec.5 we will
rigorously derive a general analytic formula for Prad.  Here we will only state that, for electrons
almost comoving with the EM pulse, Prad can be approximated by:
          Panalytic = Psyn sin4α                (2)
where α is the angle between p+ and the Poynting vector k (Fig.6).  Fig.7 compares the
numerical Prad with Panalytic for the runs of Figures 4 & 5.  It shows good correlation for the high-γ
particles. Since sinα <<1 for high-γ particles (c.f. Fig.8), Eq.(2) explains why Prad << Psyn.  The
rise and decline of Prad in Figs.4 & 5 are caused by the competition between increasing γ (and
Psyn) and decreasing α.
4. RADIATION CRITICAL FREQUENCY
A prominent feature of GRB and blazar spectra is the presence of a low energy spectral
break Epk (hundreds of keV for classical GRBs, radio-IR for blazars).  This spectral break is an
indicator of the overall spectral hardness, and is usually interpreted as the critical frequency of
synchrotron radiation ωcrsyn ~ 1.5Ω eγop+o (Rybicki & Lightman 1979) by electrons with low
energy cutoff γo.  This interpretation of the spectral break, together with some assumptions about
energy equipartition, is often used to constrain the Lorentz factor and magnetic field of the
source.  However, as we show below, for radiation emitted by TPA and LPA electrons, the
asymptotic critical frequency ωcr is << ωcrsyn.
To derive the formula for ωcr, we follow the approach of Landau and Lifshitz (1980): ωcr
is determined by the time (measured in detector frame) it takes the radiation beam of opening
angle 1/γ to sweep past the detector due to the curvature of the particle trajectory. For electrons
comoving or almost comoving with the PF, the parallel momentum px (x is the direction along k,
Fig.6) increases monotonically while pz (momentum along E) asymptotes to a constant (Liang &
Nishimura 2004, note that py along B  is constant to first order).  Hence the change in the
radiation beam direction due to bending of particle trajectory is dominated by the change in px:
Δθ ~ pzΔpx/px2.  From the Lorentz force equation we have dγ/dt=eEzpz/mγ.  Hence the time in the
laboratory frame for the radiation beam to change by an angle Δθ~2/γ is Δt=2γ2m/(eEzpz2) where
we have used the approximation γ~px(>>pz, py). This translates into a duration in the detector
frame Δtob=Δt/2γ2=m/eEzpz2.   Thus the critical frequency (Rybicki and Lightman 1979):
ωcr=1.5/Δtob=1.5eEzpz2/m=1.5Ωepz2=1.5Ωe p+2sin2α ∼ ωcrsyn sin2α (3).
Since sinα<<1 at high γ (Fig.8), ωcr << ωcrsyn   In Sec.6 we will discuss the major implications of
this result for modeling GRB and blazar data. .  In the lower panels of Figs.4 & 5 we plot the
evolution of ωcr of the same LPA and TPA runs.   These snapshots highlight the evolution of the
spectral hardness.  Again we see that ωcr first rises to a maximum before declining monotonically
due to the competition between increasing γ and decreasing α.  However the decline of Prad is
more rapid than ωcr due to the extra factors of sinα.
5.  RADIATION POWER FORMULA FOR KINETIC PF ACCELERATION
In this section we derive a general analytic approximation for the power radiated by an
electron accelerated locally by a comoving kinetic PF.  While the following derivation assumes
linearly polarized plane waves for simplicity, the results should be valid in general 3D geometry
as long as the wave front curvature and transverse gradients are << 1/(acceleration distance)
(distance for e-folding increase in γ).  We emphasize that this radiation formula should be
applicable to any acceleration by transverse EM fields almost comoving with the local ExB drift
velocity.   Hence its potential applications in astrophysics are much broader than the specific
LPA or TPA scenarios discussed above.
For particle motion in a linearly polarized plane wave with (E,B) = (Ez, By) (Fig.6), we
have Fx=-evzBy; Fy=0; Fz=e(Ez+vxBy).  Here x is the direction of Poynting vector k.  After a little
algebra we find:
 F||=eEzvz/v; F+2=e2By2[sin2α(v2-vw2)+(vx-vw)2] (4)
where vw=-Ez/By is the local profile speed of the EM field (vw<1 due to plasma loading) and
sinα=vz/v.  Substituting Eq.(4) into Eq.(1) we obtain:
Panalytic = 2e4By2[sin2α(γ2-1)(1-vw2)+γ2(vx-vw)2]/3m2  (5).
Hence the power radiated by a PF-accelerated electron depends in general on two key
parameters: the local EM field profile speed vw and the angle α between velocity v and Poynting
vector k.  Eq.(5) simplifies in various special limits:
A. Comoving particles (vx=vw):
In this case Eq.(5) simplifies to Panalytic = 2e4By2(pz2+py2)sin2α/3m2 when γ>>1.  In all of
our runs, pz>>py at late times.  So this reduces to
Panalytic = 2e4By2p+2sin4α/3m2
which is Eq.(2).  As Fig.7 shows, Eq.(2) is a good approximation for electrons comoving with
vw.   However, Eq.(2) is not a good approximation for electrons out of phase with vw (note that
electrons can have vx > vw or vx < vw).  Liang and Nishimura (2004) suggested that vw
corresponds roughly to the peak γ of the particle distribution function f(γ).
B. Vacuum pulse limit (vw=1):
In the limit vw=1, the PF propagates as a vacuum EM pulse, Eq.(5) becomes for γ >>1:
Panalytic =2e4By2γ2 (1-vx)2/3m2~e4By2γ2sin4α/6m2    (6)
Since p+ ~ γ, Eq.(6) has the same form as Eq.(2) but is a factor of 4 less in magnitude.  It defines
the lower limit to the radiative power loss of a PF accelerated electron since in reality vw<1.
C. Slightly subluminal PF (1-vw=ε<<1)
For most astrophysics applications, the PF will be slightly subluminal.  We can simplify
Eq.(5) by Taylor expanding 1-vw=ε<<1 to lowest order.  Eq.(5) then reduces to
Panalytic ~ 2e4By2γ2(ε + sin2α/2)2/3m2 (7).
This formula fully explains the physical origin of the result Prad << Psyn.  In relativistic PF
acceleration, both ε and sinα are <<1.   Fig.9 shows an example for which the best-fit ε = 0.03.
Eq.(7) shows that Panalytic behaves differently depending on whether ε >> or << sin2α/2.  In the
former case Panalytic depends only on the EM field profile speed vw and not on α:
  Panalytic ~2e4By2γ2ε2/3m2 (8)
In the second case we regain Eq.(6) which depends only on α and not on vw. Therefore when we
model astrophysical data using these formulas, we obtain different physical information
depending on the ratio ε:sin2α/2, which depends on the PF initial condition, such as To, Ωe/ωpe
.etc.  Eqs.(2), (6) & (8), which contain only 3 unknowns: (By, γ, α) or (By, γ, ε), are much easier
to use for modeling astrophysical data than Eq.(5) or Eq.(7), which contain 4 unknowns.  In
practice, ε fluctuates rapidly in both time and space so its extraction from astrophysical data
would be more difficult, whereas PIC simulation results suggest that sinα has a narrower range
for high-γ particles (Fig.8).  So the above analytic approximation is most useful for data
modeling in the regime ε << sin2α/2, which seems to be the case for GRB’s (cf. Sec.7) and may
also be the case for blazars.
We emphasize that the above analytic radiation formulas are derived from first principles
independent of any PIC simulation results.  Hence their validity is completely independent of
any numerical simulation size scale.  In fact we have carefully validated these analytic formulas
using simulations spanning a dynamic range of >105 (i.e. PF pulse widths ranging from 102 to107
gyroradii).
One may argue that the above result comes about only because we work in the lab frame,
and that the classical synchrotron formula must apply if we transform to a (primed) Lorentz
frame in which E’ = 0 and B’ is static.  This is indeed true, and one can rederive the above
formulas using appropriate Lorentz transformations of the classical synchrotron power formula
(Rybicki and Lightman 1979).  However, finding the Lorentz frame with E’ = 0 is impractical,
since E/B varies rapidly in both space and time due to modulation by self-generated currents and
current instabilities (Liang and Nishimura 2004).  There is no single Lorentz transformation that
can lead to E’ = 0 for any meaningful fraction of the EM pulse.  So in practice it is more
convenient to work in a global lab frame, measure E, B and p in this frame and use the above
analytic formulas.  Such lab-frame quantities are more relevant to astrophysical observations
anyway.
6. IMPLICATIONS FOR ASTROPHYSICAL PF MODELS
Using PIC simulations and analytic methods, we have demonstrated above that electrons
accelerated by Poynting flux which comoves with the local ExB drift velocity, radiate at a rate
(in the lab-frame) much below the classical synchrotron power, and the critical frequency of their
radiation spectrum is also much below the classical synchrotron critical frequency. These results
have major implications for the interpretation of astrophysical data from GRBs and blazars, if
their energy supply is coming from Poynting flux.  One scenario would be that the classical
synchrotron model applies only to the radiation zone, which is separate from the particle
acceleration zone, and the synchrotron model (B, γ) values refer only to the radiation zone but
not the acceleration zone, which likely has higher B and γ.  An alternative scenario is that the
observed radiation is intrinsic to the acceleration process.  Then we have to use Eq.(5) or its
various limits (Eq.(2)-(8)) to model the astrophysics data.  This will lead to much higher (B, γ)
values for the source than in conventional synchrotron models since ε and α are <<1.  Clearly the
overall energetics and parameters of the two models will be very different and such differences
may be testable.  As an example of the application of the results of Secs. 4 & 5, we will consider
a simplistic model of long GRBs in Sec.7.  Another important consequence of the suppression of
synchrotron radiation in PF scenarios is that inverse Comptonization of external soft photons
(EC) may dominate even when the (lab-frame) magnetic energy density greatly exceeds the
external soft photon energy density (Rybicki & Lightman 1979), and the conventional SSC + EC
model of blazars (Dermer & Boettcher 2002) may need to be revamped.  These scenarios will be
reconsidered in future papers.
7. APPLICATION TO A PF MODEL OF LONG GRB’s
Currently there is no universally accepted model of GRB energization and radiation.
Two popular paradigms are hydrodynamic versus electromagnetic outflows from a central
engine (e.g. a newly formed black hole), dissipating at a distance of 1014-15 cm via nonthermal
electrons and gamma-rays (Meszaros 2002, Piran 2000).  If GRBs are indeed the manifestations
of intense PF outflow, a dissipation mechanism such as TPA is attractive due to its high energy
conversion efficiency and power-law spectra (cf. Sec.2).  Liang and Nishimura (2004) also
pointed out several tantalizing similarities between the observable properties of TPA in e+e-
plasmas and GRB phenomenology.  To demonstrate the utility of the results of Secs.4&5, here
we apply the analytic radiation formulas to a simplistic “toy” model of classical long GRBs,
assuming that the PF contains only e+e- pairs with no ions (e-ion models will be considered in
subsequent papers).  We will derive the value of the spectral break energy Epk from these
formulas.  The underlying astrophysical framework is that some central engine activity lasting
10’s of seconds launches an intense EM pulse of width ~ 1012 cm and energy ~1051 ergs, loaded
with only low-density e+e- plasma so that Ωe/ωpe >1.  This intense EM pulse initially propagates
through the collapsar envelope as a non-dissipative subluminal MHD pulse as long as the
ambient density is high enough so that the formal Alfven speed vA=B/(4πρp)1/2 <  c (ρp= ambient
proton mass density).  But the pulse will eventually reach a point where the envelope ion density
is so low that vA > c, and the MHD pulse turns into a freely-expanding kinetic EM pulse. This
triggers the TPA dissipation and rapid conversion of EM energy into e+e- kinetic energy.   We
have performed simulations of relativistic magnetosonic pulses propagating down steep density
gradients.  The preliminary results seem to support the above picture.
For long GRBs it is useful to scale the burst parameters with the following benchmark
values (Fishman & Meegan 1998, Preece et al 2000): total energy E51=Etot/1051erg, burst duration
T30=T90/30sec, prompt-γ emission distance R14=R/1014cm.  We assume that the EM pulse is a
quasi-spherical shell with thickness ΔR=cT90=1012cmT30 (in this section we write out c explicitly)
and solid angle Ω4π=Ω/4π.  To simplify the model we assume that the shell is uniform with mean
field B and mean lepton (e- + e+) density n.  All physical quantities are measured in the “lab-
frame”, which we assume to be the rest frame of the GRB central engine or host galaxy. In
reality the field, density and momentum profiles are highly structured due to current instabilities
(Liang and Nishimura 2004), and the following parameters refer primarily to those leptons at the
peak of the momentum distribution function.  All our simulations with pair plasmas suggest that
at late times, particle energy Eparticle ≥ 0.6Etot, EM energy (= 2EB)≤ 0.4Etot (Fig.10, see also Liang
et al 2003).  Let N = total number of leptons (e+ + e-) in the pulse and Γ = average Lorentz factor
of the lepton distribution = <γf(γ)>.  (Γ ~ the group veocity Lorentz factor of the EM pulse
Γw=(1-vw2)-1/2, Liang and Nishimura 2004).  We thus have dimensionally in cgs units:
NΓmc2 ~ 6x1050 E51        (9)
B2ΔRR2Ω~16πx1050 E51  (10)
Eq.(10) gives:
B ~2x105 G (R14-1 Ω4π-1/2 E511/2T30-1/2) (11)
Next we estimate Γ for the radiation epoch by invoking the condition: radiative cooling rate =
particle acceleration rate.  Liang and Nishimura (2004) derived from the Lorentz force equation
the particle acceleration rate dΓ/dt = fΩe/Γ, where f is a fudge parameter of O(1) that depends on
the initial conditions.  We emphasize that this formula depends solely on the comoving nature of
the EM pulse and is independent of other global assumption of the TPA. Using Eq.(7) and
assuming ε << sinα  (and check for consistency below), we obtain:
f ecB/Γ = e4By2Γ2sin4α/6m2c3 (12).
The peak value of the late-time sinα  distribution for high-γ electrons (Fig.8) lies in the range
0.01 to 0.2 in the simulations performed so far.  Hence we scale sinα with 0.1 below:
α.1=sinα/0.1.   Solving Eq.(12) for Γ we obtain:
Γ ~1.2x105 (f 1/3 R141/3 Ω4π1/6 E51-1/6T301/6α.1-4/3) (13)
Hence ε~1/Γ << sinα and our assumption is justified.  Note that this Γ is measured in the lab-
frame so it is actually smaller than the composite Lorentz factor of internal shock models (with a
bulk Lorentz factor of 102 times internal Lorentz factors of 103-104).  Using this in Eq.(9) we
find:
N~ 6x1051 (f -1/3 R14-1/3 Ω4π-1/6 E517/6T301/6 α.1 4/3)        (14)
From Eqs.(3), (11) and (13) we obtain the value of the spectral break energy, taken as the critical
frequency  corresponding to Γ:
Epk = hωcr/2π ~ 490 keV(f 2/3 R14-1/3 Ω4π-1/6 E511/6T30-1/6 α.1 -2/3) (15)
This value agrees with the observed spectral breaks of typical long GRBs: Epk ~ 250 keV (1+z) ~
500 keV for z ~1 (Preece et al 2000) in the host-Galaxy frame.  Note that Epk in Eq.(15) depends
only weakly on the various uncertainty factors. Eq.(14) gives the mean lepton density:
n = N/(ΩΔRR2)~ 5x1010(f -1/3 R14-7/3 Ω4π-1/6 E517/6T30-7/6 α.14/3) (16)
and the frequency ratio:
Ωe/ωpe ~ 250 (f 1/6 R141/6 Ω4π-5/12 E51-1/12T301/12 α.1-2/3) >>1 (17)
which justifies our EM-domination assumption.  At this density the pairs are completely
collisionless (Coulomb mean free path > 1020cm).  We note that the local acceleration (=cooling)
time of an individual electron with the above B, Γ and sinα values  is very short: tcool =tcool ~10-2
sec.  However, this should not conflict with the 30 second observed GRB duration.  The
radiation duration of the shell is determined by the conversion time of overall EM energy into
particle energy, which is proportional to the light crossing time of the shell thickness ΔR/c, since
the EM pulse takes at least that long to emerge and energize the embedded plasma (Fig.10).
Moreover, radiation emitted by the front and back of the plasma arrive at the detector with a time
delay of ΔR/c.  These two effects combine to make the GRB duration measured by the detector ~
ΔR/c = 30 sec, irrespective of the short acceleration/cooling time of individual leptons.  An
analogy with internal shock models is in order here.  Even though the internal shock thickness
and particle acceleration length are << 1012 cm, the GRB radiation duration is governed by the
shock crossing time of the colliding shells, which have thicknesses of ~ 1012 cm.   We note that
1012cm corresponds to ~1014 gyroradii.  Even the acceleration/cooling distance of ~108cm equals
1010 gyroradii.   Both scales are much larger than the largest PIC simulation we have performed
(~107 gyroradii).  Hence it is tempting to question the applicability of our results to the GRB
regimes.  But we emphasize that the only physics used to derive the radiation rate Eq.(7) and the
acceleration rate of Liang and Nishimura (2004) is the comoving assumption of the local EM
wave with the local ExB drift speed.  This assumption is completely independent of the global
geometry, structure and size scale of the fields and plasmas, which affect only the duratiion and
longevity of the acceleration process.  In addition, we have validated the analytic radiation and
acceleration rates with simulations spanning over 4 decades in physical size (103 to107
gyroradii).  This gives us added confidence in their general validity.
However one puzzle remains.  What makes the EM pulse dissipation to occur at R~1014
cm from the central engine, two orders of magnitude larger than the EM pulse width and six
orders of magnitudes larger than the lepton acceleration/cooling length?  We speculate that it
may be the environment which determines this dissipation distance.  Here we venture a
somewhat speculative but plausible scenario that gives rise to such a large dissipation distance.
In reality, the TPA action takes place not at a sharp boundary but in an external density gradient
whose scale height is much larger than the acceleration length.  In such cases we believe that the
Liang-Nishimura (2004) acceleration rate and the radiation cooling rate of Sec.5 are applicable
only when the ambient ion mass density drops below the internal pair mass density.  Otherwise
the EM expansion and particle acceleration would be inhibited by the external ion inertia (cf.
discussion at the beginning of Sec.7).  In the collapsar model the GRB progenitor is likely
surrounded by a Wolf-Rayet wind whose mass density ~ A 5x1011 r-2 g.cm-1 (Chevalier and Li
2000) where the parameter A depends on the mass loss rate.  Hence the PF “breakout” or
dissipation distance, using the pair density of Eq.(16), becomes  rbreakout ~ A1/21014 cm.  In other
words, the TPA action is inhibited until the PF reaches an ambient ion mass density of ≤ 5x10-17
g.cm-3, and this can only occur at a distance ≥1014 cm in a Wolf-Rayet wind.  Fig.11 illustrates
the relevant length scales of this scenario.
8. DISCUSSION AND SUMMARY
Using PIC simulations and analytic theory, we have shown in this paper that when
electrons are accelerated by a comoving Poynting flux with Ωe/ωpe >1, the in-situ radiation power
output and critical frequency are much lower than those given by the classical synchrotron
formulas.  This is because the most energetic electrons have their momentum closely aligned
with the local Poynting vector or ExB drift direction. We apply our analytic formulas for the
radiation power output and critical frequency to a simple PF model of classical long GRBs, and
find that the predicted spectral break energy lies in the range of observed data.
Besides the LPA and TPA mechanisms which involve ponderomotive acceleration  of
overdense plasmas, there are many other Poynting flux scenarios that may result in nonthermal
particle acceleration.  For example, electron acceleration by comoving longitudinal wakefields
generated by PF in an underdense plasma (similar to laser accelerators in the laboratory, Tajima
and Dawson 1979) may occur in special astrophysical situations.   We have also not considered
Poynting flux propagating along flow-aligned guide fields such as Alfven and whistler waves.
Preliminary PIC simulation results suggest that linear Alfven waves (ΔB << Bo where Bo is the
longitudinal guide field) cannot accelerate nonthermal particles efficiently via the ponderomotive
force, since the net E x B drift direction is misaligned from the (strong) guide field.  On the other
hand, if the Alfven wave is highly nonlinear (ΔB > Bo), it behaves like transverse EM waves.
Then the TPA results may apply to first order.  Nonlinear Alfven waves also couple to
longitudinal modes via parametric decay, and the Langmuir waves can then accelerate the
electrons.  In general, waves can transfer energy to electrons via a large variety of resonant
interactions (Boyd and Sanderson 1969).   But such resonant interactions act on only a small
population of the electrons infrequently, whereas the ponderomotive force can accelerate the
bulk of the plasma.  PF acceleration in e-ion plasmas is more complex than in e+e- plasmas due
to charge separation (Nishimura et al 2003).   Their radiation will be treated in a separate paper.
This work was partially supported by NSF AST0406882 and NASA NNG06GH06G.
1. Rice University, Houston TX 77005-1892.
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Figure  Captions
Fig.1 Sample PIC simulation outputs illustrating the two different mechanisms of kinetic
Poynting flux acceleration of an e+e- plasma slab: (a) in LPA, an intense EM pulse incident on
an overdense plasma interface induces a ponderomotive force (JxB is along Poynting vector k)
that snowplows relativistically all upstream electrons which must share the Poynting flux
momentum, thus limiting their Lorentz factor; (b) in TPA, an intense EM pulse escaping from an
overdense plasma induces a ponderomotive force which pulls out the surface electrons
relativistically.  Only the fastest electrons can keep up with the EM pulse, so that the plasma
loading of the EM pulse decreases with time.  TPA leads to the sustained comoving acceleration
of a decreasing number of fast electrons, with no limit to their Lorentz factor.  In all figures of
this paper, x is expressed in units of 3c/ωpe.
Fig.2 (a) Electron energy spectrum accelerated by LPA resembles a superthermal quasi-
Maxwellian distribution; (b) electron energy spectra accelerated by TPA for different initial PF
thicknesses (103 and 104 c/ωpe) both show a robust power-law of index ~ -3 to -4.  The low-
energy spectral breaks correspond roughly to the Lorentz factor of the EM pulse group velocity .
Fig.3  Calibration of the numerical radiation power Prad computed from the PIC simulation
(Eq.(1)) against the analytic synchrotron formula Psyn for a 5 MeV thermal plasma in a static
uniform B field shows excellent agreement.  The scatter at low energies is due to small errors
from interpolating the field values to the particle position.  In all figures of this paper, Prad, Psyn
and Panalytic are expressed in units of 2e2Ωe2/2700.
Fig.4  Upper panel: Snapshots of Prad distribution (dots) and By profile (solid, in units of Bo/15)
vs. x for an e+e- plasma slab initially located at x=180 with n=16ncr, thickness = 12c/ωpe and
snowplowed by a vacuum EM pulse with Ωe/ωpe=10 from left to right.  Prad of the accelerated
electrons reaches a maximum at ~2 light crossing times after the EM pulse hits the plasma
surface, followed by rapid monotonic decay.  This behavior is caused by the decrease in angle α
competing with the increase in electron energy.  Lower panel: Snapshots of the critical frequency
(Eq.3) distribution show that the evolution of the spectral hardness of radiation follows that of
Prad. ωcr is expressed in units of10Ωe.
Fig.5 Upper panel: Snapshots of Prad distribution (dots) and By profile (solid, in units of Bo/150)
vs. x for a e+e- plasma slab accelerated by TPA with initial plasma temperature kTo=0.005m,
thickness Lo=12c/ωpe, Ωe/ωpe=10 and initially located at x=180.  Prad of the accelerated electrons
reaches a maximum at ~5 light crossing times after the emergence of the EM pulse, followed by
monotonic decay which is slower than in the LPA case.  Lower panel: Snapshots of the critical
frequency (Eq.3) distribution shows that the evolution of the spectral hardness of radiation
follows that of Prad. ωcr is expressed in units of10Ωe.  Note that ωcr of Fig.5 is a factor of 10 larger
than that of Fig.4.  The radiation at t=0 is thermal cyclotron radiation due to the finite initial
temperature.  But it has very low ωcr
Fig.6 Diagram showing the angle α between the Poynting vector k and p+, the momentum
component orthogonal to B.
Fig.7  Scatter plot of Prad compared with Panalytic  of Eq.(2) for the runs of (a) Fig.4  and (b) Fig.5.
At these times most of the high–γ particles are comoving with the EM pulse.
Fig.8  Scatter plot of the distribution of Lorentz factor γ vs. sinα=pz/γ for a Ωe/ωpe=10 e+e- slab
accelerated by TPA shows that the highest–γ particles have their sin α distribution peaking in the
range ~ 0.01- 0.2.
Fig.9 Scatter plot of Prad compared with Panalytic of Eq.(7) for an Ωe/ωpe=10 e+e- slab accelerated
by TPA,  At this time the best correlation for high-γ electrons is obtained when ε = 0.03.
Fig.10  Decay curves of EM energy for an Ωe/ωpe=10, kTo=5MeV e+e- slab TPA’s with different
initial thicknesses: (A) Lo=10800c/ωpe; (B) Lo=90c/ωpe; (C) Lo=12c/ωpe.  This confirms that the
conversion time of EM energy into particle energy is directly proportional to the light crossing
time Lo/c.
Fig.11  Diagram illustrating the different size scales in the “breakout” of a PF from a Wolf-Rayet
wind model of long GRBs.  The wavy arrow denotes the (lab-frame) PF thickness (ΔR=1012cm)
along the observer line of sight.  The PF breakout distance (~1014 cm) is determined by the radius
at which the wind mass density drops below the PF pair mass density (~5x10-17 g.cm-3).  Despite
the short acceleration/cooling length (~3x108cm) of individual leptons accelerated by the PF, the
detector-measured GRB duration at infinity is ~ΔR/c=30 sec due to the transit time of the PF
crossing rbreakout and the light path difference between the front and back of the PF (upper-right
space-time diagram).
                             
Plasma
Plasma
Entering
Exiting
EM pulse
Fig.1
Fig.2
Fig.3
Fig.6
Fig.7
Panalytic
       
sin α
Fig.8
