{'fraction_non_alphanumeric': 0.03795189759487974, 'fraction_numerical': 0.02265113255662783, 'mean_word_length': 4.425935973955507, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Gamma-ray bursts from synchrotron self-Compton emission

The emission mechanism of the gamma-ray bursts (GRBs) is still a matter of debates. The standard synchrotron energy spectrum of cooling electrons F_E ~ E^{-1/2} is much too soft to account for the majority of the observed spectral slopes. An alternative in the form of quasi-thermal Comptonization in a high compactness source has difficulties in reproducing the peak of the observed photon distribution below a few hundred keV. We show here that for typical parameters expected in the GRB ejecta the observed spectra in the 20-1000 keV BATSE energy range can be produced by inverse Compton scattering of the synchrotron radiation in a partially self-absorbed regime. If the particles are continuously accelerated/heated over the life-time of a source rather than being instantly injected, a prominent peak develops in their distribution at a Lorentz factor gamma ~ 30-100, where synchrotron and inverse-Compton losses are balanced by acceleration and heating due to synchrotron self-absorption. The synchrotron peak should be observed at 10-100 eV, while the self-absorbed low-energy tail with F_E ~ E^2 can produce the prompt optical emission (like in the case of GRB 990123). The first Compton scattering radiation by nearly monoenergetic electrons can then be as hard as F_E ~ E^1 reproducing the hardness of most of the observed GRB spectra. The second Compton peak should be observed in the high energy gamma-ray band, possibly being responsible for the emission detected by EGRET in GRB 941017. A significant electron-positron pair production reduces the available energy per particle, moving the spectral peaks to lower energies as the burst progresses. The regime is very robust, operates in a broad range of parameter space and can explain most of the observed GRB spectra and their temporal evolution.


INTRODUCTION
Spectra of the prompt soft gamma-ray emission of gamma-ray bursts (GRBs) are still not explained and seem mysterious despite large theoretical efforts devoted to this problem. Already in the 1980-ies it was recognized that synchrotron emission from the electrons injected at high energies produces cooling spectra FE ∝ E −1/2 (described by a photon spectral index α = −3/2) which are much too soft to be consistent with that observed from GRBs (e.g. Bussard 1984;Imamura & Epstein 1987). The problem became acute when Preece et al. (2000) showed that the time-resolved spectra have the mean observed α close to −1 (i.e. FE ∝ E 0 ) and some spectra can be as hard as FE ∝ E 1 . In spite of these facts, many different versions of the synchrotron shock models were proposed recently (see e.g. Tavani 1996;Chiang & Dermer 1999;Piran 1999) to explain GRB spectra. Panaitescu & Mészáros (2000) hypothesized that inverse Compton In principle, efficient cooling of electrons can be prevented by their reacceleration (Lloyd & Petrosian 2000). In synchrotron models, this requires the fraction of particles taking part in that process to be orders of magnitudes smaller than the total number of particles (not to exceed the available energy) and they should always be the same (Ghisellini et al. 2000), conditions that are difficult to imagine.
Problems with the relativistic synchrotron (and self-Compton) models gave rise to optically thick emission models such as quasithermal Comptonization (Zdziarski & Lamb 1986;Thompson 1994;Liang 1997;Ghisellini & Celotti 1999;Stern 1999), where the energy is shared among many particles which are now mildly relativistic. If the synchrotron radiation is self-absorbed, one can achieve rather hard spectra with the peak at 10-50 keV in the co-moving frame of the ejecta. For the bulk Lorentz factor Γ ∼ 100, this peak shifts, however, to an uncomfortably high energy.
In this paper we show that optically thin synchrotron self-Compton mechanism operating at parameters expected in the GRB ejecta can naturally produce very hard spectra peaking in the BATSE energy band, if the available energy is shared among all particles and the particles are continuously accelerated/heated over the life-time of a source. In such conditions, the electron/pair distribution develops a prominent peak at a Lorentz factor γ ∼ 30−100, where synchrotron and Compton losses are balanced by particle acceleration and heating due to synchrotron self-absorption. A copious pair production reduces available energy per particle, moving the spectral peak to lower energies as the burst progresses, reproducing thus the hard-to-soft evolution observed in time-resolved spectra (Ford et al. 1995;Ryde & Svensson 2002). High energy emission observed in some GRBs (e.g. GRB 941017, González et al. 2003), and the prompt optical emission observed in GRB 990123 (Akerlof et al. 1999) also can be explained in this model simultaneously.

MAIN PARAMETERS
Let us consider the ejecta moving with Lorentz factor Γ at the distance R from the source. The main parameters determining radiation physics are the (comoving) energy dissipation rate, magnetic field strength B, size of the emission region R ′ , and the number of particles described by the Thomson optical depth τT = neR ′ σT. It is suitable to describe the available energy by the comoving compactness where U is the comoving density of a relevant kind of energy. Formally, Λ is just the optical depth τT of pairs if we would spend all available energy for their mass. In reality, most energy goes to radiation and the resulting τT ≪ Λ (e.g., τT ∼ 10 − 20 at Λ = 10 3 and τT ∼ 1 at Λ = 30). The role of magnetic fields can be described by the magnetic compactness ΛB given by Eq.
(1) with U = B 2 /(8π). The ratio ΛB/Λ is model dependent. We assume that the magnetic field is below equipartition, i.e. ΛB Λ. Even in magnetically dominating models, one does not expect necessarily ΛB ≫ Λ, since reconnection of magnetic field providing the energy dissipation can reduce its strength within the emission region to ΛB ∼ Λ. It is evident that R ′ should not exceed the size of the causally connected region, i.e. R/Γ in both transversal and radial direction (the latter in the observer's frame is R/Γ 2 ). We assume R ′ = R/Γ and measure the comoving time t ′ in units of the light crossing time of the region R ′ /c = R/(cΓ) which corresponds to the observer time R/(cΓ 2 ). The dissipation compactness (corresponding to the energy dissipation rate) is then ℓ = dΛ/dt ′ . (For a constant dissipation rate during t ′ = 1 we get ℓ = Λ.) It can be estimated from the observed isotropic energy release E rad assuming that the energy was dissipated homogeneously in a causally connected region ℓ = E rad σT mec 2 Γ4πR 2 = 7 E rad,52 Γ −1 2 R −2 15 .
(2) (Here we adopt notation Q = 10 x Qx in cgs units if not mentioned otherwise.) Figure 1 demonstrates the levels of the compactness and the observed time-scale on a R − Γ plane. The observed emission episode at the first glance should be a single pulse of time-scale R/(cΓ 2 ). Actually, most of GRBs have a complex time structure. (2). Dashed lines show the timescale in the observer frame defined as t 0 = R/(cΓ 2 ). The deceleration limit R d for different E K is given for the wind with the mass loss rate 10 −5 M ⊙ y −1 and velocity of 10 3 km s −1 as well as for the interstellar medium of n ISM = 1 (see Eq. [3]). The circle corresponds to the model run simulated in the paper.
We can then prescribe this episode to a single GRB pulse and associate E rad with its energy fluence, or to admit that the energy can be released by compact flares within the causally connected region.
In that case, E rad should be referred to the fluence of a complex emission episode and Eq. (2) then gives an average compactness in the region while the local values of ℓ can be substantially higher.
What is the typical size R where the energy dissipation takes place? For radii smaller than 10 14 cm, the compactness is large ℓ 300 (see Fig. 1), pair production is extremely efficient, and the mean energy available per particle is rather small. The main emission mechanism is then multiple Compton scattering. The spectra expected from this regime can be hard (Ghisellini & Celotti 1999;Stern 1999), but peaked at too high energies (10-50 keV in the comoving frame). At R 10 15 cm, the compactness and optical depth are smaller, the mean energy available per particle larger and the energy is radiated away by optically thin or synchrotron or synchrotron self-Compton emission.
The dissipation radius is also limited from above by deceleration of ejecta in the external environment where the ejecta sweeps up mass ∼ Mej/Γ (see solid lines in Fig. 1). Here EK = ΓMejc 2 is the isotropic kinetic energy of the ejecta, nISM cm −3 is particle concentration in the interstellar medium,Ṁ−5 is the mass loss rate by the progenitor star in units 10 −5 M⊙ y −1 , and w3 is the wind velocity in units 1000 km s −1 . We notice that the deceleration limit in the wind case restricts the low compactness (ℓ 10) regime to a relatively narrow region of parameters. If the kinetic energy is 10 53 erg, or the environment is denser, then the low-compactness regime is hardly possible at all. Thus we consider the typical dissipation radius R = 10 15 cm. The observed pulse duration of about 1 s is consistent with the fact that GRBs have very little power at time-scales below 1 s (Beloborodov, Stern & Svensson 2000).
The optical depth τT in the emission region is bounded from above by the opacity of the ejecta (assuming matter dominated ejecta and that the matter is concentrated within a causally connected shell). For external shocks, τT cannot be smaller than that determined by collected material What is the physical mechanism of the energy dissipation? In the external shock models (Rees & Mészáros 1992;Piran 1999), the dissipation occurs at R ∼ R d which can be about 10 15 cm for the wind environment if EK ∼ 10 53 erg and Γ ∼ 200 (see Fig. 1). The impulsive first-order Fermi acceleration would operate in a fast cooling regime (see Stern 2003, for a specific version of this scenario) which, as discussed in the Introduction, contradicts the data. An alternative version of shock energy dissipation is particle heating by plasma instabilities behind the shock front (see e.g. Frederiksen et al. 2003). We follow this route here. It can also operate in internal or 'refreshed' shocks which could be produced by collisions of fresh ejecta with previously ejected, partially decelerated material. In the Poynting flux dominated models (Usov 1994;Lyutikov & Blandford 2003), the magnetic field energy can be dissipated at the required distances.

Model setup
As we have seen a number of models can satisfy our requirements. Thus we do not specify the exact model for the energy dissipation, but consider a toy model where energy is injected to the emission region with the constant rate during comoving time R ′ /c. We adopt a slab geometry of the emission region which, in a zero approximation, is consistent with both internal and external shock scenarios. Indeed, we can expect that the main energy release takes place behind the shock front in a layer which is thin relative to the size of the causally connected region. This geometry can also be a satisfactory approximation for the magnetic reconnection scenario. Indeed, the magnetic field is probably predominantly transversal and the reconnection plane is again perpendicular to the direction of propagation.
We neglect the curvature of the shock front and the bulk velocity gradient thus reducing the problem to a static slab. In this approach we omit a number of effects associated with relativistic expansion of the emitting shell. These effects are important for the description of the time evolution, however they are not critical for understanding general spectral properties. The thickness of the emitting slab, ∆ < 1 (in units of R ′ ), is unknown since it depends on the relaxation process behind the shock front and is probably smaller in the case of the reconnection scenario. We take ∆ = 0.1, but the results are not very sensitive to its value.
The energy release is uniform over the slab volume and we assume that the energy is injected in a form of acceleration of electrons and pairs which obtain equal amount of energy per unit time. The optical depth can increase due to pair production. We treat the magnetic field geometry as chaotic, therefore all pairs are isotropic.

Radiative processes
Let us first consider how particles (electron and positrons) of Lorentz factor γ are heated and how do they cool. The energy gain rate of a particle is simply given by the heating rate ∝ ℓ divided by the number of particles which is proportional to the total Thomson optical depth (including pairs) across the slab τT. Particles cool by emitting synchrotron radiation and by scattering this radiation (self-Compton mechanism). The energy balance equation can be written as: Here η < 1 accounts for the reduced synchrotron cooling due to synchrotron self-absorption, and ΛT is the compactness corresponding to the energy density of soft photons in the Thomson regime (with energy ǫ ≡ E/mec 2 1/γ). The typical cooling time is then t cool ∼ 1/[(ΛT + ηΛB)γ], which, for the GRB conditions, is orders of magnitude smaller than the light-crossing time.

Simulations
The simulations were performed using a Large Particle Monte-Carlo code (LPMC) developed by Stern (1985) and Stern et al. (1995). It treats Compton scattering, photon-photon pair production and pair annihilation, synchrotron radiation and synchrotron self-absorption. The exact cross-section are used for the first three processes, while the cross-sections in the relativistic approximation are used for the synchrotron process (Ghisellini & Svensson 1991). The electron/pair and photon distributions are computed self-consistently. The code is essentially nonlinear: the simulated particles constitute at the same time a target medium for other particles. The geometry of the emission region is a pill-box of radius R ′ and thickness ∆ = 0.1. Output photons are recorded when they cross the surfaces z = ±(∆/2 + 0.1)R ′ .
As an example, we take typical parameters described in § 2. The comoving radius R ′ = 10 13 cm, the initial Thomson optical depth τ0 = 6 × 10 −4 (close to the critical τcr, but higher than τext), compactness ℓ = 3 (corresponding to Γ ≈ 130 for E rad,52 = 1, see Eq. 2), and the magnetic compactness ΛB = 0.3 (corresponding to the comoving magnetic field B ′ ∼ 1000 G, three times below the equipartition). The evolution of broad-band spectra and the electron distribution are shown in Fig. 2.
At the start of simulations, τT = τ0 < τcr and the electron heating and cooling are balanced at high γ (see dashed curves). Partially self-absorbed synchrotron radiation (solid curves, lower energy bump) peaks at ǫs ∼ 3 × 10 −7 in the comoving frame and has a hard low-energy tail FE ∝ E 2 (Ghisellini & Svensson 1991). The optical depth grows nearly linearly with time due to electronpositron pair production (see Fig. 3a) and the mean particle energy decreases as γ ∝ 1/ √ τ (Fig. 3b). The Compton parameter y = ξτT γ 2 computed from the pair distribution (see Fig. 3c) is about 8 in the beginning, reaches minimum at t ′ = 0.3 and grows to more than 10 by the end of energy injection. The 'observed' yparameter, i.e. the ratio of luminosities in the first Compton bump to the synchrotron component grows monotonically, however, from 3 at t ′ = 0.1 to 16 at t ′ = 1. This discrepancy is caused by the nonstationarity of the problem -it takes too long time to build up the spectrum and to reach a steady-state. The first Compton peak ∝ γ 2 moves to softer energies and crosses the 'BATSE window' (note that due to self-absorption, synchrotron peak energy is very stable). This spectral evolution is consistent with the observed in time-resolved pulses (e.g. Ford et al. 1995;Ryde & Svensson 2002). The photon flux in the BATSE band (Fig. 3d) shows a 'fast rise -exponential decay' behaviour often seen in GRBs. It decays much before the energy supply terminates, but has a long flat part. While the second Compton peak at ∼ 10-100 MeV, on the contrary, rises later and decays on longer timescale.
We fitted the model photon spectrum NE = FE/E in the BATSE window by a phenomenological 'GRB function' consisting of a power-law with an exponential cutoff, NE ∝ E α exp(−E(2+ α)/Ep), merging to a high-energy power-law ∝ E β (Band et al. 1993). The evolution of the fitted (observed) peak energy Ep of the EFE spectrum and the spectral slopes is shown in Fig. 3e,f. One sees softening of the spectrum as the burst progresses. The fitted α is close to −1, the most probable value in the distribution of timeresolved spectra (Preece et al. 2000). The results of spectral fitting depend somewhat on the used energy interval: α becomes larger (spectrum hardens) in a wider interval and β is softer. Often the data at lower energies (with better statistics) dominate the fitting procedure, then the fitted α can be much harder. A correlation between α and Ep is also expected. The hardest possible spectrum, α = 0, corresponds to Compton scattering by isotropic monoenergetic electrons (Blumenthal & Gould 1970;Rybicki & Lightman 1979).

SUMMARY
The radiative processes that are responsible for particle cooling depend on the optical depth in the emission region. If τT is very small, the mean γ is large and the BATSE photons would be produced by synchrotron radiation. At τT ∼ 1, quasi-thermal Comptonization operates. At intermediate τT ∼ 3×10 −4 -10 −2 , the electrons/pairs have γ ∼ 30 − 100 and the first Compton peak is produced in the BATSE band. Such optical depth seems natural for the external shock in the typical Wolf-Rayet progenitor wind as well as for the emission within the ejecta (e.g. due to magnetic reconnection or collisions of the fresh ejecta with already decelerated material). Efficient pair production at intermediate compactnesses, ℓ = 0.1-10, also can be responsible for the required τT.
Synchrotron self-Compton emission from continuously heated, nearly monoenergetic electrons can explain many observed features of GRBs. The synchrotron spectrum peaking at ǫ ∼ 3 × 10 −7 in the comoving frame will be blueshifted to the extreme UV region. The self-absorbed low-energy tail is hard, FE ∝ E 2 , and can explain the prompt optical radiation detected from GRB 990123 (Akerlof et al. 1999). The energy of the first Compton peak is expected to decrease as the burst progresses since pair production reduces the mean energy available per particle. In the case of a larger τ0 and/or a higher compactness and/or a smaller Γ, the first Compton component peaks in X-rays and possibly can be identified with the observed X-ray flashes (e.g. Heise 2003).
The GRB spectral hardness distribution (Preece et al. 2000) can also be reproduced (maybe except its hardest events, see Ghirlanda et al. 2003). Since the incident synchrotron spectrum is hard, the low energy slope of the scattered radiation, FE ∝ E 1 , is determined by kinematics of single Compton scattering by monoenergetic electrons.
The second inverse Compton peak observed at 10 MeV -10 GeV is delayed relative to the soft gamma-ray emission and lasts longer. In spite of large y-parameter, it does not necessarily dominate the total energy output because of the Klein-Nishina effect. The rather hard (α ∼ −1) spectrum at ∼ 10 MeV can match observations of GRB 941017 (González et al. 2003) and its observed slow evolution. If the mean particle energy decreases rapidly, this component can possibly produce even the second distinct pulse in the BATSE range.
Summarizing, the proposed model can explain a large fraction of GRB spectra and their time evolution. It also reproduces the high energy ∼ 10 − 100 MeV emission detected by EGRET in some bursts and the prompt optical emission. Two latter phenomena are natural within this model and do not require additional assumptions or separate emission regions.