Rapid Xray Declines and Plateaus in Swift GRB Light Curves Explained by A Highly Radiative Blast Wave
Abstract
GRB Xray light curves display rapid declines followed by a gradual steepening or plateau phase in % of GRBs in the Swift sample. Treating the standard relativistic blastwave model in a uniform circumburst medium, it is shown that if GRBs accelerate ultrahigh energy cosmic rays through a Fermi mechanism, then the hadronic component can be rapidly depleted by means of photopion processes on time scales – s after the GRB explosion. While discharging the hadronic energy in the form of ultrahigh energy cosmic ray neutrals and escaping cosmicray ions, the blast wave goes through a strongly radiative phase, causing the steep declines observed with Swift. Following the discharge, the blast wave recovers its adiabatic behavior, forming the observed plateaus or slow declines. These effects are illustrated by calculations of model bolometric light curves. The results show that steep Xray declines and plateau features occur when GRB sources take place in rather dense media, with cm out to cm.
Subject headings:
gamma rays: bursts — stars: winds, outflows — nonthermal radiation physics — cosmic rays1. Introduction
The Swift telescope is providing a new database of GRB light curves consisting of a BAT light curve in the 15 – 150 keV range followed, after slewing within s, by a detailed 0.3 – 10 keV XRT Xray light curve and UVOT monitoring (Gehrels et al., 2004). Extrapolating the BAT light curve to the XRT range gives keV Xray light curves since the trigger. O’Brien et al. (2006) present a catalog of the combined 0.3 – 10 keV light curves of 40 GRBs, of which or so have measured redshifts, % display rapid Xray declines, and an additional % display features unlike simple blast wave model predictions. In some GRBs, the 0.3 – 10 keV flux can decrease by 4 or 5 orders of magnitude over a period of seconds within several minutes after the GRB trigger (e.g., GRB 050915B, GRB 050422, GRB 050819). About onehalf the XRT sample shows Xray flares or short timescale () structure at s after the GRB trigger, and in some cases out to s (e.g., GRB 050904 at and GRB 050730 at ).
Following the rapid Xray declines, a more gradual steepening commences in most XRT light curves, as in the cases of GRB 050219A or GRB 050607. In several GRBs, e.g., GRB050315 or GRB050822, a plateau phase with rising and decaying features within – day after the GRB trigger are observed; this phase may also be found in other GRBs (GRB 050724 or GRB 050819), but could be contaminated or mimicked by a latetime Xray flare. In a few GRBs (e.g., GRB 050401, GRB 050717), the Xray decline is gentle and monotonic, but in general the XRT light curves, whether displaying overall convexity or concavity, reveal temporal structure and, oftentimes, Xray flares (e.g., GRB 050712, GRB 050716, GRB 050726). A phenomenological model with two distinct components can fit the rapid decays and hardenings (Willingale et al., 2006), but lacks a physical basis. An acceptable physical model for GRB afterglows must be able to explain this diversity of behaviors.
A combined leptonichadronic GRB model is proposed in this paper as the cause of the rapid Xray declines and plateaus discovered with Swift. The analysis presented here follows the standard blastwave model (e.g., Mészáros, 2006). We show that if GRBs accelerate cosmic rays to ultrahigh energies, then for certain classes of GRBs, the blastwave will become strongly radiative during the early afterglow, and consequently will exhibit rapid Xray declines. This class of GRBs is defined by a range of blast wave and environmental parameters. One set of parameters that produces such declines has initial Lorentz factors – 300, apparent total energy releases ergs (and absolute energy releases ergs), and surrounding medium density cm, which is assumed to be proton dominated. The blast wave microphysical parameters must both be .
For these GRBs, Fermi processes in the blast wave are assumed to accelerate proton and ions, like they do electrons, to ultrarelativistic energies. By making reasonable approximations for the acceleration rate as a fraction of the Larmor rate, and particle escape through Bohm diffusion, we find that photohadronic losses and particle escape significantly deplete the internal energy of the blast wave, causing the blast wave dynamics to be strongly affected. Photopion interactions by the ultrarelativistic protons and secondary neutrons with the internal synchrotron photons make a source of escaping neutrons, neutrinos and cascade rays, in addition to a generally weaker proton synchrotron component.
In Section 2, a standard blastwave physics analysis for emissions from a GRB external shock in the prompt and early afterglow phase is presented, and timescales for the various processes are calculated for an adiabatic blast wave that decelerates by sweeping up material from a uniform surrounding medium. Parameter sets that allow a large fraction of the internal energy to be radiatively discharged through hadronic processes are graphically examined in Section 3. In Section 4, the equations for blastwave evolution are solved in the case of internal energy that is promptly radiated or is exponentially depleted with time. Synthetic bolometric light curves are calculated. Although these cannot be directly compared to the Swift Xray light curves without taking into account spectral effects and more complicated radiation physics, the bolometric light curves exhibit many of the features observed with Swift.
Discussion of multiwavelength and multichannel ray, cosmicray, and neutrino predictions for this model is found in Section 5, including a comparison with other models for the Xray declines. The study is summarized in Section 6.
2. Analysis
The energy flux where cm is the luminosity distance, the source luminosity , is the distance of the blast wave from the explosion center, and is the blast wave Lorentz factor at . The measured dimensionless photon energy is related to the emitted photon energy through the relation , and the differential distance traveled by the blast wave during reception time is , where is the source redshift, and primes denote quantities in the comoving fluid frame. Thus the comoving energy density
(1) 
where gives the photon energy density that produces a received spectrum with peak flux (Dermer, 2004). Relating to the measured variability time gives the proper spectral density of the radiating fluid as a function of the Doppler factor. The GRB spectrum is approximated by the broken powerlaw form
(2) 
where , ergs cm s is the peak flux at the peak photon energy , and are the indices, and the Heaviside function for and otherwise restricts the lower and upper branches of the spectrum to their respective ranges.
2.1. Adiabatic Blast Wave
Consider a blast wave with coasting Lorentz factor and apparent total isotropic energy release ergs, so that its absolute energy release, due to collimation of the relativistic outflows, is ergs. If the blast wave sweeps through a uniform surrounding medium with proton density cm, it will slow down on the deceleration length scale
(3) 
(Mészáros & Rees, 1993). A relativistic adiabatic blast wave decelerates according to the relation
(4) 
(Böttcher & Dermer, 2000), from which can be derived the asymptotes
(5) 
and
(6) 
Here the dimensionless time , where the deceleration timescale is
(7) 
and the inverse of the comoving deceleration timescale is
(8) 
The available time in the comoving frame is
(9) 
2.2. Blast Wave Physics
We treat the photopion process in the fast cooling regime (Sari et al., 1998). The minimum Lorentz factor , where and . The emission detected by Swift is assumed to be predominately nonthermal synchrotron radiation. The mean magnetic field in the fluid frame is G, and the minimum mean observed synchrotron photon energy from electrons with is , where
(10) 
The cooling Lorentz factor , and the dimensionless cooling frequency (in units of ) is given by
(11) 
Comparing eqs. (10) and (11) shows that we are in the strong cooling regime, , when
(12) 
The photon energy at the peak of the synchrotron spectrum for the fastcooling blast wave is given by
(13) 
and the peak flux is given by
(14) 
In this expression, is the magneticfield energy density in the comoving frame. For the fastcooling regime, , and . Thus
(15) 
2.3. Photopion Losses
The energyloss rate due to photopion production on the GRB synchrotron radiation field is
(16) 
after substituting eq. (2) into eq. (16), with . Here we use the approximation of Atoyan & Dermer (2003), where the product of the photopion cross section and inelasticity is b , and the threshold dimensionless photon energy for photopion production is (i.e., MeV).
The asymptotes of eq. (16) are
(17) 
where defines the Lorentz factor of protons that interact primarily with internal synchrotron photons at the peak frequency . We call the peak cosmicray proton energy, as it is the characteristic energy of protons with Lorentz factor that would escape from the blast wave with Lorentz factor as measured by a local observer. Hence the peak cosmicray proton energy is
(18) 
and
(19) 
where cm.
2.4. Rates and Limits for Ultrarelativistic Protons
2.4.1 Adiabatic Loss Rate
Adiabatic expansion losses operate on the same timescale as the available time, so for comparison of the adiabatic loss rate to other rates, we write , using eq. (9).
2.4.2 Photopion Loss Rate
2.4.3 Acceleration Rate
Because a significant energy gain by a particle can take place through Fermi acceleration mechanisms on times not shorter than the Larmor time (Rachen & Mészáros, 1998), the acceleration rate in the proper frame can be written as with the acceleration parameter . Hence the acceleration rate at the peak cosmicray proton energy is given by
(22) 
The acceleration rate for eV cosmic ray protons is
(23) 
Here we consider a standard acceleration parameter . Values of require unreasonably efficient particle acceleration, with particles gaining a large fraction of their energy in a single Larmor timescale. If , then GRBs would not accelerate cosmic rays sufficiently rapidly to make ultrahigh energy cosmic rays.
2.4.4 Escape Rate
The mean escape rate using the Bohm diffusion approximation is given by , where is the diffusion coefficient. For particle acceleration in GRB blast waves, the characteristic dimension is the shell width , and for the width of the shocked fluid shell swept up from the circumburst medium by an adiabatic relativistic blast waves (e.g., Panaitescu & Mészáros, 1999). Thus the escape rate is
(24) 
where is a parameter that allows particle escape on timescales shorter () or longer () than the escape timescale set by Bohm diffusion. If the GRB blastwave shell entrains a randomly oriented, tangled magnetic field, then depending on the coherence length of the disordered magnetic field, the particles could diffuse more rapidly than given by Bohm diffusion, so that . In contrast, if the GRB blast wave is assumed to entrain an ordered field, for example, a toroidal geometry in a jetted fireball, then escape could be impeded compared to the Bohm timescale, so that . Here we consider the Bohm limit for the rate at which particles escape, keeping in mind that the actual escape rate could be quite different.
The escape rate for protons with characteristic energy , eq. (21), is
(25) 
The escape rate for eV cosmic ray protons is
(26) 
2.4.5 Size Scale Limitation
We also have the Hillas (1984) condition that the Larmor radius be smaller than the characteristic size scale of the system, which is the shell width . Requiring implies a limit to maximum proton energy, given by
(27) 
using the asymptotes, eqs. (5) and (6). Note the slow latetime decline when (Vietri, 1998; Böttcher & Dermer, 1998). Thus we see that standard parameter values allow Fermi acceleration of protons to ultrahigh energies in GRB blast waves when and , making GRBs a viable candidate for UHECR production.
2.4.6 Proton Synchrotron Energy Loss Rate
The inverse of the synchrotron energyloss timescale for an escaping proton with Lorentz factor is
(28) 
The mean proton synchrotron photon energy, in units of , from protons with energies eV () as measured in the stationary frame, is given by
(29) 
independent of time in the relativistic deceleration phase—provided of course that is timeindependent and
3. Results
For initial parameter estimation, we adopt the Standard Parameter Set given in Table 1, with and . This set is motivated by values that reproduce typical peak fluxes and durations for BATSE GRBs at (Chiang & Dermer, 1999), except that here rather than .
Stnd. Set  Set 1  Set 2  

1  1  1  
1  1  0.5  
1  1  10  
1  10  10  
1  3  1  
1  3  3  
0.1  0.1  0.1  
1  1  1  
(s)  19.3  8.9  122 
( cm)  2.6  1.2  4.1 
Fig. 1(a) shows the Standard Parameter rates of acceleration, photopion losses, proton synchrotron losses and escape for cosmic rays with energy , the peak photon energy , the cosmicray peak energy , and the mean proton synchrotron photon energy radiated by protons with energy . Here and throughout we use an acceleration factor , an escape factor , and kinematic factor .
From the top panel in Fig. 1, one sees that the acceleration rate exceeds the inverse of available time throughout the early afterglow phase, so cosmic rays with energies are in principle easily accelerated through Fermi processes to energies . Only at several hours into the afterglow do photopion losses limit acceleration to , which by then is eV. At these late times, proton synchrotron emissions make a 1% contribution to the total loss rate. The diffusive escape rate of protons can appear as a 1% effect on the total rate, but is generally insignificant in the early afterglow. The external shock emission from this GRB is brightest s after first being detected, though shell collisions could make brighter features during the prompt and afterglow phases (see Discussion).
The acceleration, escape, and loss rates for a cosmic ray proton with eV are plotted in Fig. 1b. As can be seen, there isn’t enough time to accelerate cosmic rays to eV energies for these parameters, so there cannot be significant eV (superGZK) cosmic ray production or photopion losses from such GRBs (unless ). Protons would also escape before a significant fraction could be accelerated to such energies.
A set of parameters that overcomes these limitations is easily found. Consider Parameter Set 1 in Table 1, with giving the rates, fluxes, and energies shown in Fig. 2. For eV cosmic rays shown in the lower panel, an interesting conjunction occurs when , which happens here at s. Protons accelerated to eV energies are converted, of the time^{1}^{1}1The chargechanging fraction is not , as expected for resonance excitation and decay, due to the inclusion of direct pion channels above threshold, and multipion production at energies far above threshold., to ultrarelativistic neutrons that escape from the blast wave to form one component of a neutral beam (Atoyan & Dermer, 2003), in addition to neutrinos and rays.
The GRB formed in Parameter Set 1, Fig. 2, has a rather high MeV—which may be irrelevant in an internal/external scenario—but values of will lower during the prompt phase and lengthen the prompt phase duration. Fig. 3 shows the results for Parameter Set 2 with This GRB peaks s after the trigger, has a lower , and reaches a slightly lower peak flux than in Fig. 2.
Parameter Sets 1 and 2 model fastcooling GRBs with and , respectively, that exhibit a radiative photopion phase. By letting = 30% for Set 1, it is understood that a large fraction of the sweptup power is found in nonthermal electrons rather than in baryons or fields. A large body of parameter values clustering around the Parameter Set 1 values predict strong photopion losses in the early afterglow phase, especially if is allowed. If is assumed, as in Parameter Set 2, then agreement with GRB energetics for GRBs with apparent bolometric ray energy ergs, which represents a significant fraction of preSwift GRBs (Friedman & Bloom, 2005), means that a large fraction of GRBs also have apparent total energy , due to the unknown efficiency of converting the total energy into rays. Even for small models,the necessary large values of total apparent total energy can result in strong photopion losses. Thus if GRBs accelerate UHECRs, then photohadronic losses efficiently deplete GRB internal energy and affect the blastwave dynamics over a wide range of GRB parameters.
4. Blast Wave Evolution with Radiative Discharge
The previous section showed that with reasonable parameter values, photopion production and escape can rapidly deplete internal blastwave energy in the early afterglow when hadronic processes become important. We now calculate model light curves resulting from the radiative discharge of internal energy through photohadronic processes. The dynamics of the blast wave is numerically solving using the equation of relativistic blastwave evolution, including adiabatic losses, given by
(30) 
(Dermer & Humi, 2001). Here the blast wave momentum , is the initial baryon loading, that is, the baryonic mass mixed into the explosion, and is the sweptup mass (all masses are now in energy units). The internal energy, excluding rest mass energy, is given by
(31) 
where is the proton’s dimensionless momentum in the comoving frame, its Lorentz factor, and
(32) 
solves the equation for momentum evolution of a nonthermal proton in a blast wave with thickness (Panaitescu & Mészáros, 1999; Dermer & Humi, 2001). Here only adiabatic losses are considered for the protons, and is assumed to remain constant with and take the value . The differential sweptup mass
(33) 
where is the circumburst medium density, assumed to be radially symmetric about the GRB source. The internal energy changes due to volume expansion and adiabatic losses according to the relation
(34) 
We consider two case scenarios to simulate the change in internal energy due to a photohadronic discharge:

The internal energy is radiated on an exponential dissipation timescale , followed by a recovery of the blast wave to adiabatic behavior at . Thus
(37) and
(38) for all .
We consider first Case 1 with Parameter Set 1,^{2}^{2}2The microphysical parameters are irrelevant in this treatment of blastwave dynamics, though this may not be true in general, particularly for radiative GRBs with in the fast cooling regime. with a sudden loss of radiative energy taking place at cm, corresponding to an observer time of s after the start of the GRB—essentially the same timescale when photopion losses and escape become important (Fig. 2b). In Fig. 4(a), the numerical solutions to the equations for the evolution of are plotted for . Also shown are some approximations to the dynamics of the blast wave when (see Dermer & Humi, 2001); the curve labeled “momentum conservation solution” has the term set equal to zero; the curve labeled “analytic blast wave” is eq. (4); the curve labeled “momentum conservation solution” shows the analytic form for a fully radiative blast wave (Blandford & McKee, 1976; Chiang & Dermer, 1999). The inset gives as a function of observer time using the numerical solution to eqs. (30), (35), and (38), for different values of .
Fig. 4b shows the terms and in the numerator of eq. (30) for Parameter Set 1 with , and 0.99, and Fig. 4c shows the terms , and in the denominator of eq. (30). A reasonable approximation to blastwave dynamics can be obtained, when the blast wave is relativistic, by neglecting the term because this term is a small, constant fraction of the sweepup term . When the blast wave becomes nonrelativistic, at cm for these parameters, the term must be retained. In Fig. 4c, the range of where the , amd terms dominate the value of the denominator of eq. (30) define the coasting, relativistic selfsimilar, and nonrelativistic selfsimilar regimes, respectively. In the Case 1 scenario, after the instantaneous discharge at , the blast wave rapidly decelerates, but then recovers its adiabatic behavior as it sweeps up additional material from the circumburst medium.
An instantaneous discharge may be oversimplified, however, because a discharge of any sort must take place over a finite time exceeding at least , the light travel timescale across the blastwave width. For this reason we now consider the Case 2 Scenario, again with Parameter Set 1. Here the blastwave is assumed to recover its adiabatic behavior at cm after suffering an exponential discharge with comoving (exponential)loss timescales , 20, 40, and 80 ks. Although this is sufficient to solve the blastwave dynamics, at least in the limit , it still is not clear what is the shape of the light curve.
We can derive some idealized light curves by weighting internal energies and rates by the beaming factor that boosts energy and rate in a spherical blastwave. The following weightings are employed, keeping in mind that this sort of approach neglects particle cooling and spectral effects. The weightings considered are

, so that the comoving power is assumed to be proportional to the total internal energy, as might correspond to a single species emitter that radiates in the adiabatic limit;

, where the comoving power is set equal to the sweptup power , so this case would represent the bolometric leptonic luminosity of a GRB radiating in the strong cooling regime;

, so that the comoving power is proportional to the square of the internal energy, as might hold in a scenario involving waveparticle coupling; and

, where the comoving power is assumed proportional to the product of the total internal energy and energy density, for example, of the magnetic field.
Fig. 5 shows some numerical solutions for the Case 2, Parameter Set 1 Scenario with the different weightings just described. The blast wave is assumed to recover its adiabatic character at . The notation here is that ergs and ergs s. Overlooking the shape of the light curves at early times—a point to which we return soon—these synthetic light curves resemble the Swift BAT/XRT Xray light curves of GRBs. On top of these generic light curve shapes are the Xray flares made by some as yet unspecified mechanism.
The various weightings reflect different underlying assumptions of the physical model and yield a variety of temporal behaviors that could explain the range of Xray light curves observed with Swift. The sensitivity of the model light curves to the location where the blast wave begins to evolve adiabatically and the comoving discharge timescales is illustrated in Fig. 6, showing model light curves with a range of Xray declines and plateaulike features.
5. Discussion
The central idea proposed here is that the rapid Xray declines and plateaus discovered in keV light curves of GRBs by the Swift team (Barthelmy et al., 2005; Tagliaferri et al., 2005; O’Brien et al., 2006) are signatures of UHECR acceleration in GRBs. The solution to the problem of UHECR origin may involve a variety of source classes, but must involve at least one source class, and GRBs offer a very attractive solution: They are found outside our Galaxy (and possibly in our Galaxy; see Wick et al. (2004); Melott et al. (2004); Dermer & Holmes (2005)), which must be the case to make UHECRs with Larmor radii larger than can be contained in the Milky Way; they are energetic, with each explosion releasing as much as ergs kinetic energy (Friedman & Bloom, 2005; Le & Dermer, 2007), much of it in very clean, highly relativistic bulk particle or fielddominated outflows; and they are impulsive, at least on recurrence times day. This last is important by leading to a prediction of no clustering or angular correlations on the sky of UHECRs originating from GRBs (Waxman & MiraldaEscude, 1996).
Simple (preBeppoSAX) BATSE estimates for the required local emissivity in superGZK eV) UHECRs compared with the emissivity in Xrays and rays from GRBs, assumed to be at a mean redshift , agree to within order of magnitude, suggesting a connection between the two (Waxman, 1995; Vietri, 1995; Dermer, 2002). If GRBs are the progenitors of UHECRs, then the superposed cosmicray intensity spectrum from all the UHECR GRB sources over cosmic time display features of propagation. Most pronounced are the GZK feature and the ankle feature, the latter of which is likely due to pair production (Berezinskii & Grigor’eva, 1988; Wick et al., 2004; Berezinsky et al., 2005). If Auger measures a spectrum similar to the HighRes spectrum at energies eV, then these spectral features of cosmic rays, accumulated during transport from the source to the detector, are reasonably wellexplained within a scenario where UHECRs originate from cosmic rays, with clearcut predictions for GZK neutrino fluxes.
Here we extend the scenario that UHECRs originate from GRBs by entertaining the hypothesis that the Xray light curves of GRBs show a spectral feature associated with UHECR acceleration. The synthetic bolometric light curves in Figs. 5 and 6 are idealized, but contain the basic features of more realistic calculations that depend on a wide range of underlying assumptions about particle acceleration and the fraction of energy going into magnetic fields and waves. Depending on the radiation model, the internal and external shock contribution, and the setting of the zero of time, a wide range of Xray decays can be understood within this picture.
An important feature of this scenario is that a large fraction of the sweptup internal energy resides in protons accelerated to ultrahigh energies. Photopion processes effectively discharge the particle energy in the form of a neutral beam (Atoyan & Dermer, 2003; Dermer & Atoyan, 2004). The decay of the internal particle energy density weakens the magnetic field through feedback of the particle energy into field energy, halting further acceleration while hastening escape of the remaining UHECRs found in the blast wave, until a dominant fraction, % (depending on the initial amount of energy contained in the nonthermal protons), is lost from the blast wave as a neutral beam.
Developing detailed physical models to simulate leptonic and hadronic acceleration and loss is beyond the scope of a single paper, but the various physics issues that enter into a combined leptonic/hadronic GRB model can be described in somewhat more detail. We break the discussion into a consideration of

the acceleration mechanism;

light curves in an internal/external shock scenario;

light curves in an external shock scenario;

explanation for the rapid Xray declines;

explanation for the plateau phase; and

predictions of this scenario.
5.1. Acceleration
Acceleration to superGZK energies is possible via stochastic processes in the blastwave shell (Dermer & Humi, 2001) and through acceleration by the internal shocks in a colliding shell model (e.g., Waxman, 1995). Acceleration to such energies through shock Fermi processes is not possible through relativistic external shocks formed in a surrounding medium with magnetic field G (Gallant and Achterberg, 1999), but might be possible if the GRB takes place in a highly magnetized environment, as might be expected if the surroundings are formed by the stellar winds of highmass stars (e.g., Völk and Biermann, 1988, noting limitations imposed by total energy contained in the magnetized wind), or if the upstream field is amplified by streaming cosmic rays, for example, by the Bell & Lucek (2001) mechanism (O’C. Drury et al., 2003).
Here, acceleration of sweptup protons to ultrahigh energies is assumed to take place by secondorder gyroresonant processes in the blast wave shell. This is reflected in the derivation of the acceleration rate, eq. (23), which contains only shocked fluid quantities. A turbulent, stochastic Type 2 Fermi mechanism in the blastwave shell formed by a relativistic (internal or external) shock can make a highly efficient accelerator (Dermer & Humi, 2001), which is a sort of turbulent boiler discussed decades ago (Ochelkov & Prilutskii, 1975, and references therein), though here found in the shocked fluid shell of a relativistic blast wave. The basic coupling involves gyroresonant acceleration of ions and electrons interacting with MHD wave turbulence (in the context of Solar flares, see, e.g., Steinacker and Miller, 1992; Miller and Ramaty, 1989; Miller and Roberts, 1995), the form of which is modeldependent and may furthermore involve anisotropic coupling depending on wave type (Goldreich, 2001). But for isotropic powerlaw wave turbulence, acceleration through the stochastic Fermi mechanism makes hard number spectra (), with and most of the energy content consequently in the highest particle energies (Schlickeiser, 1984, 1989; Dermer et al., 1996; Dermer & Humi, 2001). Combined first and secondorder processes (e.g., Schlickeiser, 1984; Krülls, 1992; Ostrowski & Schlickeiser, 1993) suggest that steeper spectra, with , would be formed in this sort of scenario. But if , most of the energy resides in the highest energy particles, which can be accelerated to energies eV (Dermer, 2006).
The gyroresonant waveparticle interactions accelerate particles dynamically, so no steady state is reached (analytic solutions to timedependent particle evolution through secondorder processes are given by, e.g., Park & Petrosian, 1995; Becker, Le, & Dermer, 2006), and energy flows between waves, particles, and fields until it is discharged from the system. The bulk kinetic energy of the blast wave, initially dissipated in the shell as highly nonthermal internal particle kinetic energy as well as field and wave energy, is transformed into a component of ultrarelativistic protons carrying a large fraction of the total energy. This internal energy, including the wave energy that continues to be fed into the particles, is discharged when photopion losses become sufficiently great. Growth of the target photon field, for example, from proton synchrotron and photohadronic secondaries followed by electromagnetic cascading, can lead to an photohadronic or protonsynchrotron loop instability of the type considered by Kirk & Mastichiadis (1992) to extract increasingly more energy of the UHECRs until the internal energy content is effective discharged. Obviously, more model studies are required to quantify the total fraction of internal energy that can be discharged in this way.
Under the given circumstances, eV photopion neutrinos are created from the decay of photomeson secondaries, ultrahigh energy neutrons escape if they avoid further photopion interactions, and an electromagnetic channel consisting of rays and e and e initiates a cascade, ultimately leading to a flux of escaping rays (Atoyan & Dermer, 2003). The cosmicray neutron discharge from GRB blast waves, as considered here, produces a component of the UHECRs, causing the ionic composition of UHECRs to be protondominated. The decaying UHE neutrons and, indeed, photoprocesses by the highest energy protons and ions make an UHE synchrotron decay halo (Dermer, 2002; Ioka et al., 2004) around host galaxies. The mean injection spectrum per GRB by this process has not been calculated from first principles, but could range from a flat over a narrow energy range – eV, as in the model of Waxman & Bahcall (1999), to a powerlaw injection , as in the model by Wick et al. (2004). In principle, this process could also operate in radio/loud AGNs, and produce spectra as steep as found in the models of Berezinsky et al. (2006).
The acceleration mechanism in a colliding shell scenario is much different than stochastic acceleration in the fluid shell, but the photohadronic discharge mechanisom could certainly operate in such a system, though the details still need to be worked out. Incidentally, the potential importance of secondorder processes in GRB blast waves may negate the concerns of Ghisellini et al. (2000) that the GRB must display a strongcooling spectrum. Secondorder acceleration competing with leptonic radiative and adiabatic energy losses can form a lowenergy pileup in the electron distribution, so that no cooling tail appears. Indeed, the shape of broadband timeaveraged spectra of GRBs (Schaefer et al., 1998), interpreted as nonthermal leptonic synchrotron radiation, seems to require an abrupt lowenergy cutoff and may be consistent with an electron pileup at some lower energy due to combined first and secondorder processes with cooling. This question will take greater urgency when GLAST provides combined GLAST GBM, GLAST LAT, and Swift BAT and XRT GRB light curves, permitting detailed spectral analysis of bright GRBs from the prompt to afterglow phases. This will reveal the transition from the internal shocks (in the internal/external shock scenario) to external shocks, as now considered.
5.2. Internal/External Model
In the internal/external scenario (Piran, 1999, 2005; Mészáros, 2006), the transition from the prompt phase formed by colliding shells of jet plasma to the afterglow phase made by external shocks takes place no later than the time of BeppoSAX reorientation, hrs, and likely much sooner. In this picture, the internal shocks make a distinct radiation signature from the external shock emission received later. The zero of time for the external shock emissions from the GRB can, in the internal/external scenario, be set to various times, but is determined observationally when the GRB detector is triggered, e.g., by precursor emission. In the case of a GRB exploding in a uniform circumburst medium, the latest the zero of time can be set would be about when the external shock signature reaches its maximum brightness and triggers the GRB detector, which is generally near the deceleration time . The freedom to set the zero of time (Lazzati & Begelman, 2006; Kobayashi & Zhang, 2007) when modeling data with the external shock component considered here, as shown in Fig. 7, adds an additional distortion and increased variety to the model light curves, similar to various behaviors of the Xray light curves found with Swift.
An alternate view of the internal/external model is to have all the prompt emissions up to the time of the rapid Xray declines made by internal shock processes (see §5.5). Here it is surprising that the Swift BAT light curves are rather smooth and apparently not a superposition of generic kinematic curvature light pulses (Yamazaki et al., 2006; Dermer, 2004), although the last pulse in some GRBs, for example, Swift XRT data of GRB 061202 s after burst trigger (GCN Report 19, 2006), do appear possibly to represent the last major accretion events from a hypothetical collapsar torus. In either variant of the internal/external scenario, a complete model GRB light curve would combine the model light curves calculated in an external shock treatment with a second component consisting of ray pulses and Xray flares from colliding shells made by central engine activity.
Placing these calculations in the context of the internal/external approach, the tran