# Gamma-Ray Burst Synthetic Spectra from Collisionless Shock PIC Simulations

###### Abstract

The radiation from afterglows of gamma-ray bursts is generated in the collisionless plasma shock interface between a relativistic outflow and a quiescent circum-burst medium. The two main ingredients responsible for the radiation are high-energy, non-thermal electrons and a strong magnetic field. In this Letter we present, for the first time, synthetic spectra extracted directly from first principles particle-in-cell simulations of relativist collisionless plasma shocks. The spectra are generated by a numerical Fourier transformation of the electrical far-field from each of a large number of particles, sampled directly from the particle-in-cell simulations. Both the electromagnetic field and the non-thermal particle acceleration are self-consistent products of the Weibel two-stream instability. We find that the radiation spectrum from a shock simulation show great resemblance with observed GRB spectra—we compare specifically with that of GRB 000301C.

###### Subject headings:

radiation, acceleration of particles, gamma rays: bursts, shock waves, instabilities, magnetic fields, plasmas## 1. Introduction

Radiation from gamma-ray burst (GRB) afterglows is generally believed to be synchrotron radiation. The main ingredients in this type of radiation are a strong magnetic field and a power-law distribution of high-energy electrons. Multi-wavelength observations are more or less well fitted by hiding our ignorance about the microphysical details of these ingredients in the parameters: and describing the amount of the total shock energy that is deposited in magnetic field and electrons respectively, and ; the slope of the non-thermal electron distribution, where is the relativistic Lorentz factor. However, the true origin and nature of these two ingredients remain unexplained.

Regarding the magnetic field, observations indicate that the typical value of the equipartition parameter in afterglows is (Waxman, 1997; Wijers & Galama, 1999; Panaitescu & Kumar, 2002; Yost et al., 2003). This is many orders of magnitude larger than what can be achieved by shock compression of the interstellar medium. Even if the field is injected from the progenitor it is hard to maintain such a high level of magnetic field in the shock region. Gruzinov & Waxman (1999) and Medvedev & Loeb (1999) suggested that the Weibel two-stream instability can generate a strong turbulent magnetic field in the shock region (Weibel, 1959). This has been confirmed by numerical simulations (Silva et al., 2003; Frederiksen et al., 2004; Nishikawa et al., 2003, 2005b; Hededal et al., 2004; Hededal & Nishikawa, 2005; Hededal, 2005). It has furthermore been suggested that the radiation from non-thermal particles in this turbulent magnetic field can be described as ’jitter-’ or ’diffuse synchrotron’ radiation (Medvedev, 2000, 2005; Fleishman, 2005).

Non-thermal acceleration of particles is most commonly described as Fermi acceleration (e.g. Niemiec & Ostrowski, 2004, and references therein). Fermi acceleration relies heavily on assumptions about the nature of the magnetic field in the upstream/downstream region. The Fermi acceleration mechanism also faces several observational problems: E.g. in the Crab Nebula, the low energy electrons have a power-law distribution spectrum , much lower than the ”universal” (Weiler & Panagia, 1978). Additionally, with Fermi acceleration, one expects the presence of an X-ray halo around the shock, but such a halo is not seen in Chandra observations of SN 1006 (Long et al., 2003). Recently Baring & Braby (2004) found that particle distribution functions (PDFs) inferred from GRB observations are in contradistinction with standard acceleration mechanisms such as diffusive Fermi acceleration. We stress that the Fermi acceleration mechanism in principle may work, but that it has never been explicitly demonstrated to dominate, or even to be significant, in first principles simulations.

We are entering an era where very large simulations with three-dimensional particle-in-cell (PIC) codes are becoming computationally affordable. Such codes work from first principles and can be considered as numerical experiments. 3D PIC experiments of relativistic collisionless plasma shocks have been performed by Silva et al. (2003); Frederiksen et al. (2004); Nishikawa et al. (2003, 2005b); Hededal et al. (2004); Hededal & Nishikawa (2005); Hededal (2005). These simulations have shown that the two-stream-instability generated magnetic field is highly turbulent and varies greatly through the shock region. In the non-linear stage of the instability, filaments merge and can become as strong as . Recently, Hededal et al. (2004) have further shown that the Weibel two-stream instability also involves non-thermal acceleration of the electrons. This new acceleration mechanism differs from Fermi acceleration in the sense that it is local and instantaneous. Electrons are trapped in the potential of the generated current filaments and oscillate with strong acceleration / re-acceleration. The numerical experiments have also shown that the collisionless shock transition zone is at least of the order hundreds of ion skin-depths (Frederiksen et al., 2004; Hededal et al., 2004; Hededal, 2005). This is considerably thicker than previously thought (Gruzinov, 2001).

In this Letter we present a novel tool that allow us for the first time to extract radiation spectra directly from PIC experiments. In section 2 we present the radiation tool , in section 3 we apply the tool to investigate the 3D jitter radiation spectrum, and in section 4 we present synthetic radiation spectra from a 3D relativistic shock simulation resembling a GRB afterglow collisionless shock. Finally, in Section 5 we give a brief discussion on the results and compare with observations.

## 2. Synthetic spectra from PIC-code simulations

A particle-in-cell (PIC) code simulates plasma on a much more fundamental level than MHD. The code works from first principles, by solving the Maxwell equations with source terms for the electromagnetic fields, together with the relativistic equation of motion for a large number of charged particles. The electromagnetic fields are discretized on a three-dimensional grid whereas the particles are defined with continuous positions and momenta within the grid. The relativistic PIC code that we have used is described in Frederiksen et al. (2004) and Hededal et al. (2004). Recently, radiative cooling and synthetic spectra generation has been added to the code (Hededal, 2005).

As a particle undergoes acceleration it will emit radiation. In the PIC code we know this acceleration at each time step and hence we can reduce the energy of the particles in accordance with the Larmor radiation formula. The energy loss is highly dependent on the particles Lorentz factor and for relativistic particles, most of the radiated energy is beamed into a narrow cone with opening angle around the momentum vector. Because of this relativistic beaming, we take an approximative approach and subtract the momentum lost to radiation along the velocity vector of the particle. We have tested this implementation of radiative cooling against the analytical result for a gyro-orbiting electron in a homogenous magnetic field and find excellent agreement (Hededal, 2005).

Given the detailed information of particle positions, velocities, and accelerations it is possible to compute the emitted radiation spectrum. To do this, we start by adopting the expression from Jackson (1999) for the retarded electric field from a charged particle moving with instantaneous velocity under acceleration ,

(1) |

Here, is the unit charge, is the speed of light, is the vacuum electric permittivity, and is the distance to an observer. is a unit vector that points from the particles retarded position towards the observer. In Eq. 1 we have neglected a ”velocity term”, which is justified when the typical length scale of the emitter is negligible compared to the distance to the observer (Jackson, 1999). The radiation spectrum from an accelerated charge is then found by Fourier transforming the Poynting flux based on Eq. 1 in the retarded time frame. This gives us the energy radiated per unit solid angle and unit frequency

(2) |

This expression can only be analytically integrated under constrains on the morphology of the electromagnetic field. For example, in the case of a relativistic electron moving in a homogenous magnetic field with zero electric field, the integral reduces to the well known synchrotron case (Rybicki & Lightman, 1979). In the general electromagnetic case, no analytical solution exists. However, from the PIC experiments we readily have positions, velocities and accelerations for the particles that define the plasma. With this information, we can numerically integrate Eq. 2 to obtain a synthetic radiation spectrum from each of the particles in a plasma. Finally we can add the spectrum of each particle into a total spectrum. Adding the spectrum from each particle linearly is valid as long as the phase of each contribution is completely uncorrelated to the others.

We have implemented Eq. 2 and tested several scenarios where an analytical solution exist. This includes synchrotron radiation, bremsstrahlung, and wiggler/undulator radiation. In all tests we find that our radiation tool can effectively reproduce the theoretical expected solutions. Figure 1 shows the synthetic spectrum (solid blue) of a single relativistic electron gyrating in a homogenous magnetic field. The envelope curve (dashed red) shows the analytical solution, which shows good agreement with the simulations. The inset in Fig. 1 shows that the peaks in the spectrum fall on the relativistic gyro-frequency and its higher harmonics as expected. For more details and tests, see Hededal (2005).

## 3. Example: 3D Jitter Radiation

The PIC plasma simulation code, in combination with the radiation generation module, provides us with a powerful tool for testing various non-linear problems. In this section we examine the so-called jitter- or diffuse synchrotron radiation spectrum (Medvedev, 2000, 2005; Fleishman, 2005). This type of radiation is generated by relativistic particles that are deflected in a turbulent magnetic field, where the deflection angle is comparable to the relativistic beaming angle.

We setup PIC experiments with a periodic random magnetic field with a power spectrum that follows a power-law distribution in the Fourier domain (Fig. 2). In this field we use the PIC code to trace an isotropic momentum distributions of electrons with a Lorentz factor . The strength of the magnetic field is set such that the maximum deflection of the electrons is smaller than the relativistic beaming angle for all Fourier nodes.

Fig. 3 shows the resulting radiation spectrum for different values of the slope of the magnetic Fourier decomposition . For , the high-energy part of the spectrum follows a power-law with a slope independent of the electron energy. For , the flat spectrum continues to higher frequencies than for and the spectrum has a hard cut-off at high frequencies. In all cases, the low energy part is flat and quite similar to the case of relativistic bremsstrahlung.

We observe that as the strength of the magnetic field is increased, so that the typical deflection angle of the electrons becomes larger than the relativistic beaming angle, the resulting spectrum converges through the wiggler domain to the synchrotron case Attwood (2000).

## 4. Radiation from Collisionless Shock

Our goal is to obtain spectra from the PIC shock experiments that may be compared directly with observations. Even though it is beyond the scope of this Letter, this will eventually allow us to put constraints on the conditions and physics of GRB afterglows such as GRB environments and jet structure.

We use the numerical PIC experiment studied in Hededal et al. (2004). Two colliding plasma populations are studied in the rest frame of one of the populations (downstream, e.g. representing the shocked ISM medium at the head of a GRB jet). In this frame of reference, a less dense population (upstream, representing the unshocked interstellar medium) is continuously injected at the left hand side boundary, with a relativistic velocity corresponding to a Lorentz factor . The two populations initially differ in density by a factor of 3. We use a computational box with grid points and a total of particles. The ion rest-frame plasma frequency in the downstream medium is , rendering the box 150 ion skin depths long. The electron rest-frame plasma frequency is in order to resolve also the microphysics of the electrons. Hence, the ion-to-electron mass ratio is .

As the two plasma populations interpenetrate, we observe how the Weibel two-stream instability is excited and current filaments are formed (Weibel, 1959; Medvedev & Loeb, 1999; Silva et al., 2003; Frederiksen et al., 2004; Nishikawa et al., 2003, 2005b; Hededal et al., 2004). The current filaments induce a highly intermittent electromagnetic field with a strength of up to 10% of equipartition. The nature of the magnetic field is turbulent, with a power spectrum that follows a power-law in the Fourier domain (Frederiksen et al., 2004). We note however, that unlike the random fields in section 3 there exists phase-correlations in the generated magnetic field that give the field a more coherent structure. The instability also directly leads to acceleration of electrons through a newly discovered mechanism (Hededal et al., 2004; Nishikawa et al., 2005a). Electrons caught within the Debye cylinder surrounding the ion current filaments are strongly and repeatedly accelerated and decelerated. The acceleration is instantaneous and local, and thus differs substantially from recursive acceleration mechanisms such as Fermi acceleration. The resulting particle distribution function has a non-thermal high-energy component, correlated with the distribution of the generated electromagnetic filaments. We note in passing that a fraction of the ions are scattered back into the interstellar medium. Thus, ion Fermi acceleration may possibly take place in collisionless shocks.

To obtain the total spectrum from the accelerated particles in the Weibel-generated field, we have traced 20,000 particles from the PIC simulation described above. Placing the observer somewhere along the jet direction we have numerically integrated Eq. 2 for each of these particles.

Fig. 4 shows the radiated spectrum from the collisionless shock simulations described above. Here, we have rescaled the frequency into real space units and taken the relativistic Doppler-shift into account. We find that the spectrum peaks in the far infrared. Below the peak, we find a power-law segment with slope . This is interesting because it is steeper than the limiting 1/3 slope for synchrotron radiation, and may reconcile theory and observations on the ”line of death” issue (Preece et al., 1998) of observed GRB spectra. For frequencies above the peak frequency, the spectrum continues into the near infrared / optical band, following a power-law with slope with .

From the power-law slope in the spectrum with (), the standard procedure commonly used in the GRB community would indicate that the electrons have a power-law distribution, with slope (determined by solving ). The standard conclusion would be that this is consistent with Fermi acceleration. However, we strongly emphasize that the electrons that have produced the spectrum in Fig. 4 are not accelerated by a recursive mechanism such as Fermi acceleration. The electrons are instantaneously accelerated and decelerated in the highly complicated electric and magnetic field near the ion current channels under the emission of strong radiation. This differs substantially from the iterative acceleration in Fermi acceleration. Furthermore, the paths of the accelerated shock electrons are closer to a random walk than circular, which means that synchrotron radiation is not an adequate formalism in the current context.

## 5. Discussion and Conclusions

The main source of information we have about gamma-ray bursts (GRBs) is from multi-wavelength observations of GRB afterglows. The radiation from GRB afterglows is generated in collisionless shocks between the external plasma (ISM) and the GRB jet. This emphasizes the crucial importance of a full understanding of the physics of and the radiation mechanism in collisionless shocks. In the current standard model, the radiation is assumed to be emitted as synchrotron radiation from power-law distributed electrons in a magnetic field. Following our lack of knowledge about the plasma-physical details, it is common practice to parameterize this ignorance with the dimensionless parameters , and ( and express the fraction of the total internal energy that is deposited in magnetic field and electrons, respectively, and is the slope of the supposedly power-law electron momentum distribution function).

Since collisionless shocks are extremely non-linear and self-interacting systems, the only way to study their microphysics with any credence is through self-consistent, three-dimensional relativistic particle simulations. Using this approach we have previously explained the origin and nature of a strong electromagnetic field in the shock (Frederiksen et al., 2004) and been able to identify a new non-thermal electron acceleration mechanism (Hededal et al., 2004). These two ingredients are direct and unavoidable consequences of the Weibel two-stream instability.

In this Letter we have developed a novel tool that allows us to extract radiation spectra directly from the particle-in-cell experiments. Rather than applying the synchrotron approximation (assuming homogenous magnetic field and a single electron power-law distributed population) we trace a large number of electrons in the experiments, and calculate the exact radiation emitted from each electrons. The tool has been thoroughly tested and successfully reproduces spectra from synchrotron radiation, bremsstrahlung and undulator/wiggler radiation from small-angle deflections (Hededal, 2005).

The tool has been utilized for a parametric study of 3D jitter radiation, where a population of electrons radiate as they move in a turbulent magnetic field whose power spectrum follows a power-law in the Fourier domain . We have examined how the slope influence on the resulting radiation spectra. For , the high-energy part of the spectrum follows a power-law with a slope independent of the electron energy. For , the flat spectrum continues to higher frequencies than for and the spectrum has a hard cut-off for high frequencies.

We have furthermore examined the radiation from a typical GRB afterglow collisionless shock that propagates with through the interstellar medium. We find that the resulting radiation spectrum peaks just below Hz. Above this frequency, the spectrum follows a power-law , with . Below the peak frequency, the spectrum follows a power law with . This is steeper than the standard synchrotron value of and thus more compatible with observations. Both the slope and the peak is consistent with observations (e.g. Panaitescu (2001) who finds , a peak at Hz and for the afterglow of GRB 000301C after five days).

Finally, we would like to stress the following interesting point: If one hides all the real physical details of the magnetic field in the dimensionless parameters , , and the conclusion from standard synchrotron radiation theory would be that the slope of the electron distribution is found by solving (which is consistent with standard analysis of the GRB 000301C afterglow; Panaitescu, 2001). This would appear to be consistent with Fermi acceleration and the supposedly universal . But a closer look reveals that the acceleration mechanism is not Fermi acceleration but rather a natural consequence of the Weibel two-stream instability. The electrons are instantaneously accelerated and decelerated in the highly complicated electric and magnetic field near the ion current channels. Strong radiation is produced in this process. The electron distribution behind the radiation is a mixture of a thermal component for low energies and a power-law component with for high energies (Hededal et al., 2004). The paths of the electrons are more random than circular and the electron distribution function varies with shock depth, as does the magnetic topology and strength.

The outcome of the exercise is thus that it is indeed possible, from first principles, to produce synthetic spectra very similar to the ones that are observed in GRBs, and that the fit of such spectra with the standard model assumptions may be more or less fortuitous.

CBH acknowledges support from the Instrument Centre for Danish Astrophysics and from the University of Copenhagen for funding his PhD grant. ÅN was supported by grant number 21-04-0503 from the Danish Natural Science Research Council. Both authors thank Jacob Trier Frederiksen and Troels Haugbølle for useful discussions and comments. Computing resources were provided by the Danish Center for Scientific Computing.

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