Electromagnetic Catastrophe in Ultrarelativistic Shocks and the Prompt Emission of Gamma-Ray Bursts Boris E

Mon. Not. R. Astron. Soc. 345, 590–600 (2003)

Electromagnetic catastrophe in ultrarelativistic shocks and the prompt emission of gamma-ray bursts Boris E. Stern1,2 1Institute for Nuclear Research, Russian Academy of Sciences, Moscow 117312, Russia 2Astro Space Centre of Lebedev Physical Institute, Moscow, Profsoyuznaya 84/32, 117997, Russia

Accepted 2003 June 30. Received 2003 June 23; in original form 2003 January 20 Downloaded from https://academic.oup.com/mnras/article/345/2/590/1747042 by guest on 25 September 2021

ABSTRACT It is shown that an ultrarelativistic shock with the Lorentz factor of the order of tens or higher, propagating in a moderately dense interstellar medium (density above ∼1000 cm−3), undergoes a fast dramatic transformation into a highly radiative state. The process leading to this phenomenon resembles the first-order Fermi acceleration with the difference that the energy is transported across the shock front by photons rather than protons. The reflection of the energy flux crossing the shock front in both directions is due to photon–photon pair production and Compton scattering. Such a mechanism initiates a runaway non-linear pair cascade fed directly by the kinetic energy of the shock. Eventually, the cascade feeds back the fluid dynamics, converting the sharp shock front into a smooth velocity gradient and the runaway evolution changes to a quasi-steady-state regime. This effect has been studied numerically using the non- linear large particle Monte Carlo code for the electromagnetic component and a simplified hydrodynamic description of the fluid. The most interesting application of the effect is the phenomenon of gamma-ray bursts (GRBs) where it explains a high radiative efficiency and gives a perspective to explain spectra of GRBs and their time variability. The results predict a phenomenon of ‘GeV bursts’ which arise if the density of the external medium is not sufficiently high to provide a large compactness. Key words: shock waves – methods: numerical – gamma-rays: bursts.

other hand, Fermi acceleration has a number of limiting factors in 1 INTRODUCTION available power and radiative efficiency (see Achterberg et al. 2001; Astrophysical objects where we observe – active galactic nuclei Bednarz & Ostrowski 1999). For this reason, Fermi acceleration is (AGN) – or infer – gamma-ray bursts (GRBs), soft gamma-ray re- usually considered as a source of high-energy cosmic rays while the peaters – a relativistic bulk motion demonstrate surprisingly high intensive radiation is traditionally explained in terms of the internal gamma-ray luminosity. In some cases, the energy release in gamma- energy dissipation. rays can be comparable to the total kinetic energy of the bulk motion In this paper, we suggest a highly radiative analogue of Fermi in the source. This implies a very efficient mechanism of the energy acceleration where the main role is played by photons rather than conversion of the bulk motion into hard radiation. It is clear that, by charged particles. A similar idea about the important role of in the case of explosive phenomena such as GRBs, the radiation neutral particles in shocks has been suggested independently in a originates from ultrarelativistic shocks (Cavallo & Rees 1978; Rees very recent paper by Derishev et al. (2003). For a simple illustration, &Mészáros1992). This is less certain in the case of AGN jets; the shock and the external medium can be represented as two mirrors however, there exist a large number of works considering shocks as moving with an ultrarelativistic velocity respectively to each other. the main source of AGN jet radiation (for a review, see Kirk 1997). Let ξ s and ξ e be their reflection coefficients (depending on the type We can describe the energy stored in the shock in two ways: as a and the energy of a particle) and let be the Lorentz factor of the total kinetic energy of ejecta and shocked matter or as their internal shock. A photon being elastically reflected head-on from a moving energy. Two different ways of shock energy release are known: the mirror gains a factor of ∼2 in its energy. Thus, the total energy of first-order Fermi acceleration and different mechanisms of internal photons participating in back and forth motion between the mirrors energy dissipation. Fermi acceleration does not require dissipation can be roughly described by the equation as it is fed directly by the kinetic energy of the bulk motion. On the 2 dE/dt = ξsξe − S E/tc (1)

where t c is a characteristic time of reﬂection cycle and S is the E-mail: [email protected] probability of the particle escape. Actually, equation (1) represents

C 2003 RAS Ultrarelativistic shocks and GRBs 591 no more than an instructive toy model because a real situation is into a smoother velocity gradient. This transformation removes the essentially non-local (t c is uncertain) and requires an integration radiative divergence; now an upstream photon most probably in- 2 through particle spectra. If ξ sξ e > S, the total energy of particles teracts in the pre-accelerated medium and the energy gain in the will grow exponentially until mirrors decelerate. interaction becomes much less than 2. We should expect the for- In the case of the first-order Fermi acceleration the reflection re- mation of some steady-state regime where the fluid velocity pattern sults from the diffusion of supra-thermal charged particles in the is adjusted in a way to satisfy the energy conservation law. The magnetic field. How will this two-mirror approach look when ap- luminosity at this stage should be close to its ultimate value L ∼ plied to photons? If a soft photon (e.g. of a synchrotron nature) 2 dm/dt where m is the mass of the external matter swept up by interacts only with electrons, the reflection coefficients ξ s and ξ e the shock. are of the order of the Thomson optical depth of each medium (ξ s ∼ Summarizing, we can expect the following evolution: (i) a seed ξ e ∼ τ T, see Section 2). This case does not differ essentially from the radiation due to dissipation of the internal energy; (ii) Compton well-known thermal Comptonization in relativistic regime at elec- boosting and the cascade boosting of the electromagnetic component tron temperature − 1. The photon boosting terminates as soon as in linear exponential regime; (iii) a sharper non-linear evolution Downloaded from https://academic.oup.com/mnras/article/345/2/590/1747042 by guest on 25 September 2021 the photon gains a sufficient energy to interact in the Klein–Nishina when the boosted high-energy component contributes to the soft regime when reflection coefficients decrease. The Compton boost- target photon field; (vi) transformation of the shock front and setting ing is probably not very important by itself, however it could provide up a highly luminous quasi-steady-state regime. The entire event is photons above pair production cut-off and give rise to a new much so dramatic that it deserves the name ‘electromagnetic catastrophe’. more powerful effect. This qualitative illustration is confirmed below at a quantitative The importance of pair production in GRB shocks and AGN jets level. The main tool in this study is the large particle non-linear has been emphasized in many works, for example, by Thompson Monte Carlo (LPMC) code described in Stern et al. (1995). (1997), Ramirez-Ruiz et al. (2001) and Mesz´ aros,´ Ramirez-Ruiz While the mechanism can work in different astrophysical phe- and Rees (2001). Thompson (1997) and Ramirez-Ruiz et al. (2001) nomena, this study is focused on its application to GRBs. Corre- stated that the pair production can enhance the radiation efficiency spondingly, the explored region of parameters represents the pre- of the shock, providing new supra-thermal particles as seeds for vailing view on GRBs (large Lorentz factors, moderately high Fermi acceleration. In this paper, we show that the pair production densities). plays a crucial role in initiating a runaway energy release. In Section 2 we describe the general formulation of the problem, A necessary condition to trigger this process is the existence of a the accepted assumptions and the range of parameters. In Section 3 seed radiation field in the vicinity of the shock. This radiation should we briefly describe the approach to the numerical simulations of the be dense in terms of the photon number, np Rσ T 1, where R is effect. In Section 4 we present the results of simulation runs demon- a characteristic scale. Both np and R correspond to the comoving strating the evolution of the system from a low luminosity initial frame of the shock where photons are approximately isotropic. Such state through the catastrophe to a highly radiative post-catastrophe photons can appear, for example, as the result of an internal energy regime. In Section 5 we extend the range of the phenomenon to lower dissipation and undergo Compton boosting. densities and outline the threshold where the catastrophe is possible Now let a photon of energy ε 1 (hereafter ε is the photon en- at certain assumptions. In Section 6 we describe a phenomenon of ergy in electron mass units) moves upstream across the shock front. GeV bursts, which should arise when the compactness is insufficient Then, interacting with a soft photon, it produces an e+e− pair in to provide optically thick pair loading. Possible associations with unshocked matter (ξ e ∼ 1 in this case). The pair is bound to the existing data are discussed. In Section 7 we discuss various GRB rest frame of the external matter if the e+ and e− Larmor radii do scenarios associated with electromagnetic catastrophe and the issue not exceed the scale characterizing the problem. Particles are ad- of the GRB time variability. vected downstream across the shock front, losing a fraction of their energy to Comptonization. Some of the downstream Comptonized 2 FORMULATION OF THE PROBLEM photons can produce new pairs behind the shock front which in turn upscatter soft photons upstream. So we deal with multiple electro- 2.1 The simplified description of the shock magnetic interactions and the whole process can be characterized as an electromagnetic cascade fed by kinetic energy of two converging For convenience, we consider the shock in the comoving system media or cascade boosting. The energy involved in the cascade can which is preferable for its higher symmetry; the density contrast grow exponentially and eventually the process can reach a non-linear is moderate (a factor of 4 in an idealized case), and upstream and stage when the boosting rate increases. downstream energy fluxes can be comparable. More explicitly, we All the above conclusions have been made assuming that the have bound our reference system to the contact discontinuity, and its shock front separating unshocked and shocked media is very sharp Lorentz factor hereafter is . The shock front in this frame moves and can be characterized as a viscous jump. It is known from studies upstream with semirelativistic velocity βs (hereafter, β is the ve- of non-relativistic shocks that the viscous jump can disappear, trans- locity in units of the velocity of light). We assume that the shocked forming into a smother transition due to radiation pressure ahead the external matter between the shock front and the contact disconti- shock. Thompson & Madau (2000) have studied pre-acceleration of nuity is of a constant density and in rest relatively to the contact external matter for the case of a GRB external shock and demon- discontinuity. Actually this region can be subject to a semirelativis- strated that the pair production (pair loading) is extremely important tic bulk motion (perhaps with a turbulent component) depending on in this respect and feeds back the radiation efficiency. Beloborodov the distribution of external matter and shock dynamics. Neglecting (2002) has considered a more extreme case when the radiation of such motion we remove an additional source of the free energy and GRB ejecta sweeps up the external medium far ahead of the shock. because we are interested in an explosive energy release this is a In this case, the energy should be supplied from internal shocks. conservative assumption. Our case is closer to that of Thompson & Madau (2000). The pre- We do not know and do not consider the state of the ejecta behind acceleration of the external medium transforms the viscous jump the contact discontinuity, prescribing to it the role of a heavy piston,

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ρ carrying the main fraction of the kinetic energy. It is assumed that magnetic field depends on the external density√ and the reasonable −5 −3 the ejecta does not emit or reflect radiation and absorbs all pho- assumption in this case is He ∼ 10 ρ G, where ρ is in cm tons. Actually the ejecta can contribute to the seed radiation and the units. Hereafter, we treat H e as the transversal component. downstream–upstream reflectivity. The lower limit for the magnetic field in the shock H follows We assume a spherical symmetry at least within a cone of open- from Lorentz transformation and the compression by a factor of 4: ing angle 1/ respectively to the line of sight. Due to relativistic H > H g = 4 H e. However, this field can be much stronger, being effects we almost do not observe the emission of the shock beyond generated at the shock. At this step, we can rely on the numerical this cone. Events in the shock separated by an angle of more than study of the two-stream instability by Medvedev & Loeb (1999). 1/ are not causally connected since the acceleration stage. Most According to their simulations, the ensured value of energy density probably, the ejecta is strongly beamed within a solid angle 1. of generated field, U B,is∼0.01–0.1 of electron energy density in We exclude from the consideration using the isotropic equivalents the shock, U e, and, tentatively, the field can be amplified up to U B ∼ for global energy parameters: the kinetic energy of ejecta, E, and 0.01–0.1 of proton energy density. the luminosity, L. The actual energetics is then described by E The parameters of the problem are initial Lorentz factor , to- Downloaded from https://academic.oup.com/mnras/article/345/2/590/1747042 by guest on 25 September 2021 and L. tal kinetic energy E, external number density ρ, characteristic rest The spherical shock viewed from the comoving system is parabol- frame size R, seed luminosity of soft photons Ls, their spectrum, ically curved and the matter moves outward from the line of sight, values of magnetic field in both regions, H (shock) and H e.Asa relativistically at the angular displacement ∼1/ from the line of reasonable hypothesis for the seed spectrum, we accept a power law α sight. For simplicity, we adopt one-dimensional slab geometry with dN/dε ∝ ε , in the range ε1 <ε<ε2 (see Section 2.3). no bulk motion in the axial direction. This is not a conservative sim- In the case of GRBs, the isotropic kinetic energy of ejecta can plification because in this way we violate kinematic and causality vary within two orders of magnitude or more. The maximal observed constraints on interactions of photons emitted from distant regions hard X-ray–gamma-ray GRB energy release is ∼5 × 1054 erg. We of the shock. In order to avoid this problem, the consideration was accept E = 1054 erg as a baseline. In this paper, we study the case restricted to a comparatively small time (less than a half deceleration of a constant external density which is a reasonable starting point. time) when the above constraints are still not important. The scale of the problem can be characterized by the deceleration The unshocked interstellar medium (ISM) in the comoving sys- radius (Rees & Mesz´ aros´ 1992): tem is a cold fluid with bulk Lorentz factor . The ordered particle 1/3 2 4 2 motion dissipates into chaotic motion with the same average ther- Rd = E mpc πρ mal Lorentz factor when the fluid crosses the shock front. The 3 energy density contrast across the front and the front velocity in the = . × 16 1/3ρ−1/3−2/3 . (4) 0 24 10 E54 6 2 cm comoving system, βs, result from the energy conservation: In the spherical geometry, the scale changes as the shock propa- Usβs = Uo(β + βs). (2) gates, therefore we adopt R = 0.5Rd which approximately describes Here, U s is the energy density of the shocked matter, β is the external the shock propagation between 0.5Rd and Rd. Using the rest-frame 2 2 fluid velocity, U o = ρmpc , ρ is the rest frame ISM number scale R we can define dimensionless distance, x, and time, t,towork density, and from the pressure–momentum balance within the comoving system, x = X/(R/), t = T /(R/c), where 1 X and T are comoving distance and time. Note that the allowed P = U = U β(β + β ). (3) working range is x < 1 and t < 1. The time in the observer frame s 3 s o s is T = tR/(22c). β = β = obs In the ultrarelativistic limit ( 1) the equations yield s It is convenient to describe the energy budget in terms of di- / = 1 3U s 4U o. For a more detailed description of a relativistic shock, mensionless compactness. The latter defines the importance of pair see Blandford & McKee (1976). production and, more generally, the level of non-linearity of the sys- The thickness of the shock (between the shock front and the tem. The simplest way to introduce this parameter is via the energy contact discontinuity) at the constant external density and spherical column density through the system normalized to the electron mass = / / geometry is X 1 12R in the comoving system. and multiplied by the Thomson cross-section U R ω = σ , (5) 2 T 2.2 Main parameters and dimensionless units mec

The external magnetic field H e is different for cases of the stel- where U is an energy density in the comoving system. Particularly, lar wind dominated and the ISM dominated environments. In the the magnetic compactness is first case, the field is predominantly radial and processes leading H 2 R ω = σ . (6) to the catastrophe are inhibited (ξ e is very low) unless the field is B 2 T 8πmec affected by a radiation front ahead of the shock. Most probably, it must be affected by the two-stream instability generating a chaotic This is the compactness describing the energy content of the system. component. Moreover, a spherical asymmetry of the environment Traditionally, the compactness is expressed through the luminosity and ejecta together with the charge asymmetry (Compton scattering of the system L: on electrons) can induce a large-scale transversal field similar to that Lσ = T . (7) induced in an atmospheric nuclear explosion. 3 Rmec In the ISM dominated case, the field originally should have a large-scale geometry and a substantial transversal component. In The rest(observer)-frame compactness for the ultrarelativistic this paper, only the ISM case is studied quantitatively. Note that the shock is meaningless as the radiation is strongly collimated. The interaction of the shock with a slow shell of matter ejected by a GRB parameter has the above-stated physical meaning only in the shock progenitor essentially does not differ from the ISM case. The ISM comoving system where particles have a wide angular distribution.

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Assuming the comoving size of the system R/ (with the transver- the shock, it is sufficient if it just changes its direction upstream. sal size of the same order, see Section 2.1) and substituting L by the Then photons Comptonized by the particle moving upstream can luminosity flux, L ∼ F(R/)2, with an accuracy of the order of π, cross the shock. The deflection of a particle from downstream to we obtain upstream hemispheres is a much more probable event than its diffu- F R sion upstream of the shock, required for Fermi acceleration. Indeed, = σ . (8) 3 T in the comoving frame of the shocked matter the shock front moves mec ahead semirelativistically, with β = 1/3, and a charge particle has The total energy budget in the comoving frame constitutes the kinetic s to diffuse upstream faster. energy flux of the fluid crossing the shock front. The corresponding The downstream–upstream reflection of a particle can proceed in dimensionless expression, kinetic compactness, is the following ways. σ R m = ρ2m c3 T = τ p . (9) o p 3 T (i) If the Larmor radius is less than the correlation length of the mec me magnetic field, the particle turns at a half Larmor orbit. This can be Hereafter we denote energy compactness as ω and the luminosity Downloaded from https://academic.oup.com/mnras/article/345/2/590/1747042 by guest on 25 September 2021 the case for particles of a moderate energy. (power) compactness as , and = dω/dt. (ii) If the Larmor radius exceeds the correlation length, the particle diffuses in the chaotic field. 2.3 Seed radiation (iii) Except for the chaotic field H, there could exist a large-scale component, resulting from a large-scale external magnetic field due The seed radiation in the shock can be contributed by: (i) the to the flux conservation: H ∼ 4 H . Then a high-energy particle synchrotron–self-Compton radiation of shocked electrons which g e can turn around at a half Larmor orbit in the field H . have the thermal Lorentz factor ; (ii) other channels of inter- g nal energy dissipation (e.g. dissipation of magnetic field, plasma Processes (ii) and (iii) are competing and the leading one depends waves, turbulence); (iii) external (partially side-scattered) photons on concrete parameters. In this paper, only the effect of the global from the fireball and a progenitor star; (iv) high-energy seed photons component has been taken into account as the latter should exist in associated, for example, with Fermi acceleration and photo-meson the adopted model of the external environment. The effect of the production. diffusion is a matter for future studies. The most confident source is (i) unless synchrotron radiation of The reflection efficiency depends on the fraction of energy which shocked electrons is strongly self-absorbed. Its fraction in the total the particle loses before it turns around upstream. The energy loss / energy budget is limited by the factor me mp. At some parameters, rate is described by this seed radiation is sufficient to start the evolution leading to the dγ H 2 σ catastrophe (see Section 3). In such a case, we deal with the most − = + U T γ 2 = Cγ 2, (10) ph 2 conservative ‘minimal hypothesis’. At other parameters, the mini- dX 8π mec mal hypothesis is insufficient for the catastrophic evolution. Then where X is the comoving distance. The equation yields we have to consider additional channels of internal energy dissipa- γ tion (ii) or external photons (iii) and (iv). We would like to avoid γ X = o , ( ) + γ (11) a serious consideration of corresponding mechanisms in this paper 1 o XC and use a very moderate ad hoc hypothesis of the seed soft photon where field. ∼ 2 Uph −20 −1 As a baseline, we accept the compactness of such emission s C = H 1 + 3.3 × 10 cm . (12) UB τ T which is only a me/mp fraction of the total energy budget. The model spectrum of this seed radiation has been taken as a power- Here, U B and U ph are energy densities of the magnetic field and soft law fragment with −2. <α<−1.5 (a free parameter) in the range radiation field, respectively. The term Uph/UB describes the relative ε ε ∼ −8 between 1 (typically 1 10 which is close to the self-absorption contribution of Compton losses. From equation (11) we estimate ∼ ε cut-off at H 100 G) and 2 (a free parameter). Such weak seed the maximum energy of the downstream–upstream reflected particle radiation is sufficient to give rise to the catastrophe in most trials γ max ∼ 1/XC. Substituting X by the half Larmor orbit described in this paper. In Section 7 we discuss the situations when γ 3 max the effect implies a stronger seed radiation. πRL = 5.3 × 10 cm, (13) Hg

2.4 Reflection of the energy flux we obtain the final expression for the maximal energy of a reflected particle: Reflection of a particle flux from the external environment and from − / H 1/2 1 2 the shock is of primary importance for the cascade boosting (see 8 g Uph γmax = 0.75 × 10 1 + . (14) equation 1). Here we consider the reflection of a high-energy com- H UB ponent of the electromagnetic cascade where the direct Compton reflection of photons is inefficient. In this case, the reflectivity of The downstream–upstream reflection is inefficient if U ph/U B the medium is due to photon–photon generated pairs gyrating in the 1. Then the main energy of an electron is emitted as a relatively magnetic field. The requirement that an electron can change its di- soft synchrotron radiation rather than hard Comptonized photons. rection to the opposite without a substantial energy loss is a serious The limiting energy of synchrotron photons emitted by an electron limiting factor for the boosting rate. after a half Larmor orbit is ε ∼ 200 at H g ∼ H. This energy is In the case of the first-order Fermi acceleration, the main process still sufficient for pair production, however the cascade boosting in responsible for the particle scattering upstream across the shock this case will be substantially slower and the compactness threshold is the diffusion in a chaotic magnetic field (for a recent review, see for the electromagnetic catastrophe will be higher. In this paper, Gallant 2002). In our case the charged particle does not need to cross we study the case of low H and weak dissipation. If the generated

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field is much stronger but its dissipation into the electromagnetic The version of the LPMC code used here treats Compton scatter- component is fast, then we have U ph/U B ∼ 1 again. The boosting ing, synchrotron radiation, photon–photon pair production and pair is inhibited if the field is strong and its dissipation is slow (U ph annihilation. Synchrotron self-absorption was neglected as it con- U B). sumes too much computing power and is not very important in this In the case of the external environment (upstream–downstream application. All these processes are reproduced without any simpli- reflection) the electron synchrotron losses√ are small. Indeed, in the fications at the microphysics level. On the other hand, a number of −5 rest frame, equation (14) with He ∼ 10 ρ yields serious macroscopic simplifications have been performed.

10 −1/2 −1/2 Because the number of LPs which is limited by the computing γ , = 1.3 × 10 ρ (1 + U /U ) (15) r max 6 ph B power available at the moment did not allow full three-dimensional where the subscript ‘r’ refers to the rest frame and γ max ∼ γ r,max. simulation, the problem was reduced to one dimension, x, along the The term U ph/U B is large in the external environment, (i.e. Comp- shock propagation. While the locations and momenta of LPs were ton losses are much larger). Therefore, the main limiting factor three-dimensional, the target density was averaged over slab layers in upstream–downstream reflection is Compton cooling of high- and the fluid representation was one-dimensional. Downloaded from https://academic.oup.com/mnras/article/345/2/590/1747042 by guest on 25 September 2021 energy pairs at the first Larmor orbit. A charged particle produced The trajectories of electrons and positrons in the magnetic field exactly upstream in the external environment turns around very fast were simulated directly assuming transversal geometry of the field when viewed from the shock comoving√ frame. Indeed, this occurs H g in the shock and H e in the external matter. Thus, the problem when it gyrates by the angle φ ∼ 2/ where φ is defined in the of the reflection qualitatively discussed in Section 2.4 was imple- rest frame. At that moment, the particle gains a factor of 4 in its mented in the numerical simulation. Protons are assumed to be cold comoving energy (while the maximal possible energy gain is 4 2). in the fluid frame unless they have crossed the dissipative shock Then the comoving energy increases as front. The hydrodynamic part of the problem becomes very difficult as γ = γ (1 − β cos φ), (16) r the energy and momentum transferred by LPs to the fluid fluctuate. where γ r . Taking into account that the time interval corre- Attempts to simulate the fluid with internal pressure have led to sponding to dφ transforms to the comoving frame as rising instabilities at semirelativistic velocities. Therefore, a dust approximation (where the internal fluid pressure is neglected) has dT = (1 − β cos φ)dφ RL/c, (17) been adopted and the fluid velocity behind the shock front was φ for 1 equations (16) and (17) yield artificially fixed to zero. The dust approximation works satisfactorily until the fluid has dγ γrc cHe = 2 = 2 . (18) a moderate temperature and energy density in its comoving frame dT R φ φ1.7 × 103 cm Gs L (i.e. when it is ultrarelativistic in the shock comoving frame). This We can see that the rate of the energy gain decreases as the angle circumstance was used by Beloborodov (2002) and Thompson & φ increases. On the other hand, the Compton cooling rate increases Madau (2000) who also described the pre-acceleration of the ex- as γ 2. Therefore, the final energy gain can be substantially less than ternal medium by the radiation front in terms of dust approxima- 2. On the other hand, the particle can be advected back across the tion. The approximation does not work at all in the shock where shock front before it deflects by the angle φ ∼ 1. In this case, it has the pressure at a relativistic temperature is comparable to the ra- less energy gain than 2 again. Both effects were accounted for in diation pressure. The adopted hydrodynamic ansatz allows us to the numerical simulation. The upper energy limit for an upstream– describe the effect in general and fails to account for possible 6 8 downstream reflected particle is γ max ∼ 3 × 10 − 10 for conditions interesting effects associated with semirelativistic motion behind probed in Sections 4 and 5. the radiation front. The approach can be considered as a ‘zero Note that the limits for downstream–upstream (equation 14) and approximation’. upstream–downstream reflection have an opposite dependence on the magnetic field. A faster boosting takes place in the case of a 4 EVOLUTION OF THE SYSTEM THROUGH stronger external field and a weaker field in the shock. The above THE CATASTROPHE estimates are given for the orientation while the numerical simulation reproduces directly the particle trajectory and the energy loss To illustrate qualitative arguments discussed in Section 1 with nu- along it. The quantitative effect of these parameters on the boosting merical simulations, we present results of two runs with different rate and examples of upstream and downstream particle spectra are parameters: first, for Lorentz factor = 100 and the density of ex- demonstrated in Section 5. ternal environment ρ = 5 × 105 cm−3; secondly, for = 30, ρ = 106 cm−3. Other parameters for these cases are R = 2 × 1015 cm, = 120, T = 3.1 s and R = 3.5 × 1015 cm, = 130, T = 3 MONTE CARLO SIMULATIONS o o o o 133 s. The first case is closer to the traditionally assumed value of The entire problem outlined above is too difficult for an analytical for GRBs, however it is more difficult for numerical treatment treatment. In this paper we study it numerically using a LPMC because statistical fluctuations are associated with a finite number code developed by Stern (1985) and Stern et al. (1995). The code of LPs being amplified by a factor of 2. For this reason, the first is essentially non-linear; the simulated particles constitute at the run tracks the evolution of the system only until the beginning of same time a target medium for other particles. The LP method in the post-catastrophe stage. this application means that each simulated particle represents w The initial state is empty of photons and consists only of fluid of real particles (for GRBs w>1050). The most reasonable weighting cold protons and electrons with bulk Lorentz factor , representing scheme, except for some special cases, is to set wε = const, where the external medium viewed from the comoving system. Seed pho- ε is the energy of the particle and the constant can vary as the total tons appear as the (partially self-absorbed) synchrotron radiation energy involved in the simulation can change. The number of LPs of shocked electrons. In this case, the self-absorption is not very was 217 = 131 072. strong and can be ignored. The Comptonization of the synchrotron

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0.51 0.46 Downloaded from https://academic.oup.com/mnras/article/345/2/590/1747042 by guest on 25 September 2021 0.4

0.31

0.17

Figure 1. Instantaneous spectra of photons for = 100, ρ = 5105 cm−3 during pre-catastrophe and catastrophe stages. Upper panel shows the up- Figure 2. The evolution of the electromagnetic component with time. Upper ω / stream and lower – the downstream photon spectra. Labels in the lower panel panel shows the time derivative of the total dimensionless energy d u dt ω / indicate the dimensionless time from the start of the simulation. Curves in the of upstream (lower curve) and downstream d d dt (upper curve) photons = ρ = 5 −3 = ρ = upper panel correspond to the same time sequence. Time unit corresponds for 100, 510 cm . Middle panel: the same for 30, 6 −3 to 3.1 s in the observer frame. 10 cm . Lower Panel shows the evolution of pair Thomson optical depth for = 30, ρ = 106 cm−3. peak gives second and third peaks at higher energies (further Comptonization is inefficient because it proceeds in the Klein– Fig. 3 shows post-catastrophe photon spectra which can be treated Nishina regime). These peaks are visible in Fig. 1 at t = 0.17. The as a continuation of the spectra set presented in Fig. 1 to later ages. Comptonized photons are produced both by shocked and external Fig. 4 shows the spatial distribution of the main quantities: the en- electrons. ergy density of upstream photons, the pair number distribution and At t = 0.31, a high-energy tail appears due to pair production by the bulk Lorentz factor. At the moment of the catastrophe, the main photons of the third peak in the external medium (which provides fraction of photon energy is tightly concentrated to the fluid velocity comoving energy gain by factor 2). Around t = 0.4, the evolution accelerates (see also Fig. 2) reaching the non-linear phase. The effect is more prominent in the upstream photon flux. Finally, the total energy of photons rises by more than three orders of magnitude (the power supplied to photons rises by a factor of ∼mp/me), and the spectrum transforms from a harder to a softer one. The evolution from t = 0.4 to t = 0.5 will take about 0.3 s from the point of view of an external observer. The simulation has been terminated at t = 0.52 because the numerical problems associated with the high Lorentz factor become more serious. These problems are not principal and can be overcome just with an increase in the computing power. The evolution of the system with time is shown in Fig. 2. This example does not require any additional source of energy except the bulk kinetic energy of the fluid and any external photons. In this sense, the case can be interpreted as a ‘minimal model’; however we still have to assume a non-radial external magnetic field. The case of = 30, ρ = 106 cm−3 does require additional assumptions. First of all, the synchrotron radiation of electrons with ε = 30 is deeply self-absorbed. Therefore, an internal energy source ac- celerating a fraction of shocked electrons is required. We adopted for this source a me/mp share in the total energy budget: s ∼ me/mp × . The acceleration was simulated as an injection of a num- − o Figure 3. Instantaneous spectra of photons for = 30, ρ = 106cm 3 during γ = × 3 ber of LPs at 3 10 . This injection energy is an ad hoc catastrophe and post-catastrophe stages. Upper panel shows the upstream and value which is not very important as the primary synchrotron lower shows the downstream photon spectra. The dimensionless time, from photons emitted by injected electrons then undergo a multiple lower to upper curves, is: 0.49, 0.53, 0.60, 0.74, 0.98. Time unit corresponds Comptonization. to 52 s in observer frame.

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emission of the downstream flux, E u ∼ η E d, where η is the average opacity of the system for the downstream photon flux. After the transformation of the upstream component to the rest (observer) frame we have 2 2 Eu,o ∼ ηmc . (19) The opacity rises during the evolution and approaches unity in the case of high compactness. Indeed, all soft photons scatter in an optically thick layer of pairs, while hard photons are absorbed due to pair production. The deceleration of the shock can be roughly described as 2 d =−η (20) Downloaded from https://academic.oup.com/mnras/article/345/2/590/1747042 by guest on 25 September 2021 dm M where M is the invariant mass (comoving energy) of the shock. Note that at η<1 the deceleration is slower than in the adiabatic case which is described in the same way as equation (20) at η = 1. It is remarkable that the upstream spectrum evolves towards a GRB-like spectral shape, however still does not reach it. The peak energy of the latest spectrum in Fig. 2 is around 70 keV. When blueshifting the spectrum to the observer frame and assuming a cosmological redshift z = 1, the peak energy comes out around 1 MeV. There are GRBs with such peak energy, however the typical peak energy is less – between 200 and 300 keV. The low-energy Figure 4. The distributions of the upstream photon energy density, the bulk slope in this example is α ∼−1.5 while for a typical GRB α ∼−1. Lorentz factor and the pair number density across the shock (upper, medium The values of the peak energy and α resulting from this run are quite and lower panels respectively). x = 0 corresponds to contact discontinuity. robust; the peak energy comes out from transition from the Thomson Labels indicate the dimensionless time since the start of the simulation. to Klein–Nishina regime in Compton scattering on non-relativistic or semirelativistic electrons, and α =−1.5 corresponds to cool- jump which has a step-like character (see Fig. 4, upper and middle ing spectrum of relativistic electrons (see, for example, Ghisellini, panels, curve at t = 0.49). Then the photons disperse and a leading Celotti & Lazzati 2000). fraction of them form a radiation front moving with the speed of Summarizing, we should conclude that at the present level the light. The distribution of the bulk Lorentz factor smoothes and ad- model does not reproduce GRB spectra. Nevertheless, the phe- justs to the radiation front (Fig. 4, middle panel) moving ahead with nomenon provides a perspective to address the problem due to ef- a near-luminal velocity. The absence of a viscous jump at the post- ficient pair production and generation of an optically thick layer catastrophe stage means that new protons join the shock are cold; of pairs behind the radiation front (see Figs 3 and 4, lower pan- they are smoothly decelerated (or accelerated in the rest frame) by els). Once we have an optically thick pair plasma, we can expect radiative reaction via pairs and magnetic field. that it will produce a thermal Comptonization peak which may de- The fluid behind the main radiation front (where it has a semirela- scribe GRB spectra (Thompson 1997; Ghisellini & Celotti 1999; tivistic velocity) is not pressure balanced in this approximation. The Liang 1999). In this simulation such a peak cannot be reproduced thermal pressure of protons at x < 0.2 swept up before the catas- because cooled pairs are not Maxwellized; their energy distribution trophe exceeds the radiation pressure of downstream photons. This is in Compton equilibrium with the photon spectrum. This is a con- imbalance should affect the velocity distribution and a real pattern sequence of the simplified approach omitting various processes of can be more complicated. We can expect a semirelativistic forward potential importance. Particularly, the spectra could be modified in bulk motion at the left from the broad peak of the photon distribution, a proper direction by thermal bremsstrahlung and double Compton which will compress and push the pair-photon gas ahead. scattering (for a review, see Thompson 1997). The model is very simplified and works progressively worse in Actually, cool pairs should be subject to a variety of collective attempts to trace the evolution further. At the present level without a phenomena, i.e. dissipation of plasma waves or turbulence. As a careful hydrodynamic treatment we are not able to study the shock result, a prominent Comptonization peak can appear. Note that the deceleration stage and cannot reproduce the final time profile of the asymptotic case of the Comptonization peak is Wien distribution radiation. We also cannot say when and how the highly radiative which is much harder in the low-energy part (α =+2) than GRB post-catastrophe state terminates. spectra. A partially developed Wien peak could be a promising so- In order to represent the results in terms of observed gamma-ray lution of the problem. energy release, we have to consider different photon components in The contradiction in the location of the spectral break εp can be the comoving system, as follows. relaxed if the initial Lorentz factor of the shock is 20–40 rather than 100–300 and the peak luminosity is emitted by a partially (a) The downstream component, which takes the main fraction of decelerated shock. the momentum and the energy of the decelerating external medium, after the viscous shock front disappears. The energy of downstream 5 EXPONENTIAL BOOSTING OF A SEED photons and pairs is related to the mass of the external medium m 2 HIGH-ENERGY COMPONENT swept up after the catastrophe as E d ∼ mc ( − 1). (b) The upstream component (which is actually isotropic in the The first example in Section 4 presents a full non-linear simulation comoving frame) results from the absorption and quasi-isotropic re- of the system without any initial seed photons. Here we show that

C 2003 RAS, MNRAS 345, 590–600 Ultrarelativistic shocks and GRBs 597 the existence of a very small amount of high-energy photons at the Table 1. The boosting rate of the high-energy component for = 50, ρ = 4 = start extends the possible range of the catastrophic evolution down 10 ( o 7.3) depending on other parameters. Second column gives those to a much lower external density than was probed in Section 4. parameters which differ from the standard set (see text). Third column shows The most suitable source of such photons is Fermi acceleration of the boosting rate in terms of the distance of the shock propagation S where protons with photo-meson production and possible p–n conversion the high-energy component grows by an order of magnitude. Fourth column gives the total growth of the high-energy component (orders of magnitude) suggested by Derishev et al. (2003). 16 when the shock propagates the distance R = 0.5 Rd = 1.17 × 10 cm, where Because of the high computing cost of the full non-linear simula- 54 the deceleration distance Rd implies the initial isotropic kinetic energy 10 tion, in this section we explore the parameter space simulating the = × −3 o = ωo = erg. The standard parameters are: e 3.5 10 , H 14.7 G, B 1.4 × −3 o = −3 ε = −4 α =− cascade boosting in a linear regime using a linearized version of the 10 , H e 10 G, 2 10 , 1.5. LPMC code. This means that the target LP field is not affected by the simulated high-energy component. A high boosting rate ensures ‘Non-standard’ parameters S14 R/S that the system evolution eventually reaches the catastrophe stage 1 Standard set 8.9 12.5 Downloaded from https://academic.oup.com/mnras/article/345/2/590/1747042 by guest on 25 September 2021 even at a small seed amount of high-energy photons. = o 2 H e 0.2 H e 14.5 8.0 During the boosting, the spectrum of the high-energy component = o 3 H e 5 H e 6.5 17.8 evolves until it reaches some steady-state shape that we do not know ω = ωo 4 B 10 B 37. 3.1 a priori. Because the boosting rate depends on the spectrum, we have ω = ωo 5 B 0.1 B 7.3 16.0 to track the evolution for a sufficiently long time until the steady- 6 α =−1.75 15. 7.51 state exponential regime is evident. Because the number of cascade 7 α =−2. 40.0 2.9 particles grows exponentially we have to use statistical weights; to 910−7 <ε<10−3 15. 7.7 − − cut some cascade branches compensating this by an increase of the 10 10 8 <ε<10 5 27.3 4.3 statistical weight of remaining branches. The magnetic field and the seed (target) radiation field were taken according to Sections 2.2 and 2.3. The parameters of the seed spectrum were frozen at α =−1.5 (which corresponds to the slope of the −4 fast cooling spectrum) and ε2 = 10 . With these assumptions, we reduce the parameter space to two dimensions: and ρ. Because the assumptions are not obvious, it would be reasonable to accept the following strategy: to explore , ρ space with the standard set of assumptions and, at one point, to defreeze other parameters and to estimate the effect of their variation. Fig. 5 shows the average breeding time profiles for = 50, ρ = 104 cm−3 corresponding to the first three lines in Table 1. We can see a confident exponential rise in the number of hard photons crossing the shock upstream. Table 1 summarizes the results on the breeding rate for = 50, ρ = 104 with different sets of other parameters. First of all, it should be noticed that the breeding rate at moderate parameters = 50, ρ = 104 cm−3 is greater than that required to trigger the catastrophe in some of the trials. We can see a strong dependence on magnetic field. The effects of H and H e variation are of opposite signs, as was qualitatively shown in Section 2.4. Fig. 6 demonstrates spectra of particles crossing the shock (the energy of all particles is in the comoving system). We can see that the

Figure 6. Spectra of particles crossing the shock for = 50, ρ = 104 cm−3 (upper panel) and = 50, ρ = 103 cm−3 (lower panel) and standard set of other parameters. Solid lines: upstream photons; dotted lines: downstream photons; dashed lines: downstream pairs. Spikes at ε = 105 represents monochromatic photons injected at t = 0.

fractions of energy transferred downstream across the shock by high- energy Comptonized photons and by advected pairs are comparable. The turnover in spectra of upstream photons qualitatively agrees 6 4 with equation (14) yielding γ max ∼ 3 × 10 for ρ = 10 and γ max ∼ 0.7 × 107 for ρ = 103. Fig. 7 demonstrates the map of –ρ space when other parameters are frozen as described above. Levels of the constant compactness Figure 5. Average breeding time histories for = 50, ρ = 104 cm and and the constant time-scale are given for total isotropic kinetic en- different external magnetic ﬁeld. Lower curve corresponds to the line 2, ergy E = 1054 erg. Both the compactness and the time-scale vary as upper curve to the line 3 in Table 1. E 1/3 in the spherical geometry and at constant density. The solid and

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0.01

0.001

0.0001

-5 0 5

Figure 8. Late spectrum of upstream photons (t = 0.71) for = 50, ρ = Downloaded from https://academic.oup.com/mnras/article/345/2/590/1747042 by guest on 25 September 2021 103 cm−3.

−8 −4 s = 3ome/mp,10 <ε<10 , α =−1.5. The catastrophe Figure 7. The explored area in − ρ coordinates (other parameters are at such parameters develops due to the breeding of external high- frozen as described in the text). The solid (standard assumptions) and dot- energy photons. It was assumed that the high-energy component due −5 ted (external field is by factor 5 higher) lines shows the threshold for the to preceding breeding has gained the luminosity ωh =∼×10 o. catastrophe estimated with the conventional condition that the seed high- The corresponding amount of upstream high-energy photons with energy component grows at least by 10 orders of magnitude as the shock ε = 105 has been injected at the start of the simulation. = propagates the distance R 0.5Rd. Dashed lines represent the levels of The resulting upstream photon spectrum is shown in Fig. 8. Gen- 2 the constant observer time-scale R/( c) and dotted lines correspond to the erally, at a low compactness we can expect a maximum of ε2dN/d = 54 constant comoving compactness o assuming E 10 erg. Circles repre- ε distribution in high energies and an approximate power-law spec- sent parameters for full non-linear simulation runs described in Sections 4 trum in a wide range with the index between α =−1.5 (fast cooling and 6. spectrum) and α =−2 (saturated pair cascade; Svensson 1987). In this case, the maximum energy release is at the rest-frame en- dotted lines show the catastrophe threshold which is conventionally ergy ∼5 GeV (the spectrum in Fig. 8 is not final, however, because defined as the condition that the cascade is boosted at least by 10 some of the hardest photons will be absorbed on the way to the orders of magnitude when the shock propagates the distance R = observer). 0.5Rd. Perhaps we could use a weaker requirement; we prefer to According to Fig. 7, the catastrophe threshold at ∼ 50 is o ∼ use this conservative criterion because the real intensity of the seed 1 when the main energy is still released in hard gamma-rays. This high-energy radiation is unknown. Note that the boosting rate di- value is model-dependent; the threshold can be higher for a less rectly depends only on the density of the soft seed radiation field. favourable combination of parameters, e.g. for a higher ωB/ωs ra- = / The dependence on density appears in assumption s ome mp. tio. Nevertheless, if the environment of GRB progenitors is diverse, If the soft seed radiation is denser, then the threshold is correspond- we can expect that GeV bursts should be as usual a phenomenon = ρ ∼ ingly lower. For example, if s 0.01 0 then the threshold is as classic GRBs. On the other hand, they are more difficult for ob- − 200 cm 3 at = 50. servations. Such events should be weaker in the soft gamma range The threshold line in Fig. 7 depends on the frozen parameters, (soft gamma fluence is one to two orders of magnitude less than the i.e. it is model-dependent. As mentioned in Section 2.4, the strong high-energy fluence). Also, more importantly, they should be longer slowly dissipating magnetic field in the shock would substantially and smoother (see Figs 7 and 9). The example of this section cor- raise the threshold. On the other hand, if the field dissipates very fast responds to the characteristic time-scale of 133 s (the total duration leaving a dominating large-scale component, the threshold would can be longer). Such events are more difficult for a triggering and an be lower. The range of the catastrophe would be extended if an off-line search than classic GRBs, unless an event is strong enough. external field stronger than assumed is induced due to spherically The long duration also makes difficult a time-location cluster anal- asymmetric Compton drag as discussed in Section 2.2. ysis for high-energy photons in existing data. Has something like this been observed? A candidate for such a GeV burst is GRB 970417a detected simultaneously by the Burst 6 GEV BURSTS and Transient Source Experiment (BATSE) and Milagrito (Atkins The two runs described in Section 4 correspond to a comparatively et al. 2000) in the range above 50 GeV. The inferred fluence above high compactness parameter when the main energy release occurs 50 GeV is at least an order of magnitude greater than the fluence in the soft gamma range. In Section 5 it is shown that the catastrophe in the BATSE range which corresponds to the type of the spectrum is possible at much less compactness (see Fig. 7). In such a case, shown in Fig. 8 if the maximum is at an order of magnitude higher we can expect to observe a much harder radiation, which would energy. However, it was a comparatively short burst with T 90 = approximately correspond to early spectra in Fig. 1. In order to 7.9 s. Then, following Fig. 7 we have to assume a high Lorenz demonstrate this directly, we performed a run for = 50, ρ = factor ( ∼ 200), or less kinetic energy at a lower . Note that there 3 −3 10 cm . The compactness is o = 1.1, the ‘standard’ magnetic should be a strong selection effect in favour of short GeV bursts. field in the shock H = 4.4 G, the characteristic radius R = 2.0 × 1016 Another hint at GeV bursts is provided by EGRET data. The cm, and the observer time-scale T o = 133 s. The ‘minimal model’ famous GRB 940217 (Hurley 1994) where EGRET has detected does not work at such parameters, therefore the similar standard ad a delayed GeV emission during an hour after the BATSE trigger hoc hypothesis of the seed radiation as in Section 5 has been used: can hardly match the concept of a GeV burst. It rather looks like a

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2002). This issue requires a separate study. Anyway, the density range assumed in this work is not extreme. Is it possible that the threshold conditions are still never satisfied despite a sufficient density of the environment? Formally, it could be if the external magnetic field is always purely radial. However, as discussed in Section 2.2, the radial field structure can hardly survive the early stage of a GRB in any scenario. Another inhibiting factor could be a strong domination of magnetic energy density over radiation energy density. However, we do observe GRBs implying a high radiation density in the source. Whatever produces this radiation at the start should end up with the electromagnetic catastrophe. The effect is associated mainly with the external shock scenario

as internal shocks have too low a relative Lorentz factor. On the Downloaded from https://academic.oup.com/mnras/article/345/2/590/1747042 by guest on 25 September 2021 other hand, internal shocks can radiate only a relatively small fraction of the total kinetic energy because of kinematic constraints. Competition with a much more efficient mechanism presents a new difficulty for the internal shock scenario. For example, let us consider the typical internal shock scenario: prompt GRB emission from R ∼ 1013−1014 cm due to collisions of multiple shocks and Figure 9. The evolution of the electromagnetic component with time for the following afterglow due to an external shock at larger radii. An = ρ = 3 −3 50, 10 cm . The time derivative of the total energy of upstream important modification to this scenario introduces the effect dis- (lower curve) and downstream (upper curve) photons is shown. covered by Beloborodov (2002): the radiation emitted by internal shocks sweeps out the external matter up to radius R ∼ 1015−1016 superposition of a classic GRB and a GeV burst. In principle, this is cm leaving an empty cavity. Now let us consider the situation when possible when the shock meets localized dense clouds while propa- the shock reaches the edge of the cavity. It meets the environment gating in a less dense environment. Better candidates for GeV bursts with enhanced density and optical depth (as the swept-out matter is can be several EGRET detected GRBs reported by Catelli et al. compressed and loaded by a large amount of pairs), with a non-radial (1998) and Dingus et al. (1998). Some of these have high-energy magnetic field (the radial field structure can hardly survive such spectral indices α ∼−2 up to GeV range. Eventually the issue of event), with intense seed radiation (side scattered photons). These GeV bursts will be clarified by the Gamma-ray Large Area Space are perfect conditions to give rise to the electromagnetic catastrophe Telescope (GLAST) mission. with a proper GRB time-scale. If so, the radiative energy release in such a collision would outshine a burst from internal shocks by 1.5 to two orders of magnitude (the main fraction of total kinetic energy 7 DISCUSSION of the ejecta at the full radiative efficiency versus a few per cent of the kinetic energy at unknown radiative efficiency). This would While the effect of the electromagnetic catastrophe can, in principle, look like the main event, rather than the afterglow. Even if the main be applied to different astrophysical phenomena, its application to energy release is in the GeV range (low compactness, see Section 6), GRBs is of the highest interest. Below we discuss the main issues such events can hardly escape systematic observations. where the effect can be relevant.

7.2 The issue of the time variability 7.1 The radiative efficiency and the GRB scenario The internal shock model is motivated mainly by difficulties with ∼ The considered effect straightforwardly provides the ultimate ( 100 description of the GRB time variability arising in the external shock per cent) radiation efficiency and, in this respect, it probably has no scenario (Fenimore et al. 1996; Dermer 2000). On the other hand, competing mechanisms. The phenomenon of GRBs does require a it suggests a number of possible solutions for external shocks. The high radiation efficiency. The total isotropic soft gamma-ray ener- effect of the electromagnetic catastrophe may extend the list of pos- 52 × getics of GRBs with measured redshift varies between 10 and 5 sible solutions. 1054 erg. A low efficiency would imply much higher values for the isotropic kinetic energy of the ejecta which probably contradicts the (1) The catastrophe is a non-linear effect which can amplify any afterglow data. fluctuations in external density and seed radiation. In addition, it is The catastrophe is a model-independent effect which, however, sensitive to the geometry of magnetic field. The one-dimensional has a set of threshold conditions. If GRBs were associated with treatment performed in this work loses many important features coalescence of neuron star binaries, the effect would never work which could appear in a three-dimensional simulation with realistic because the expected density of the environment is too low: 0.1– hydrodynamic treatment. The catastrophe most probably will de- 1cm−3. However, a wealth of recent data support the collapsar velop locally in many spots within the 1/ cone at different times. scenario (see Mesz´ aros´ 2002 for a review) where we can expect a The spherical symmetry of the shock should be distorted due to a much higher density and the phenomenon should occur. In principle, strong feedback between the radiation and the fluid pattern. As a if the progenitor is a Wolf–Rayet star then we can expect a density result, we can observe a complex time structure even at an approx- as high as 1010−1011 cm−3 near the star surface (assuming the mass- imate symmetry in the initial state. loss rate 10−5 M). However, it is not obvious whether the external (2) The post-catastrophe evolution was interpreted here as a shock can start near the start surface; the wind can be swept out by quasi-steady state. Actually, especially in the case of decreasing photons preceding the ejecta to a larger distance (see Beloborodov density, it could be unstable and recurrent. This could be, for

C 2003 RAS, MNRAS 345, 590–600 600 B. E. Stern example, due to a version of the Beloborodov effect: the radiation variability should certainly rely on a detailed three-dimensional hy- of external shock passing through the dense matter sweeps out less drodynamics. The understanding of the spectra requires a more dense matter ahead, and radiation decays until the shock catches up complete description of the shock structure and physics of parti- with the swept matter. Then the shock regenerates producing a new cle interactions for non-relativistic and semirelativistic pairs. emission episode. In this way, we can explain episodes separated by long quiescent intervals. ACKNOWLEDGMENTS (3) The spherical symmetry of ejecta can be completely broken The author thanks K. Hurley, A. Illarionov, P. Ivanov, J. Poutanen prior to the emission stage up to fragmentation into a bunch of for useful discussions and Y. Tikhomirova for assistance. This work separate droplets of different transversal size X < R/ (this implies T was supported by the Russian Foundation for Basic Research (grant a hydrodynamic transversal confinement of droplets which needs a 00-02-16135). Part of this work was carried out during a visit to the separate study). This requires a strong instability (most probably of University of Oulu with the support of the grant from the Jenny and Rayleigh–Taylor type) at some stage. A model representing a GRB Antti Wihuri foundation. ejecta as a shower of blobs was proposed by Heinz & Begelman Downloaded from https://academic.oup.com/mnras/article/345/2/590/1747042 by guest on 25 September 2021 (1999). This is the most attractive possibility because, in addition REFERENCES to the complex time structure, it can produce a chain reaction; the catastrophe in one droplet supplies seed photons to a neighbouring Achterberg A., Gallant Y. A., Kirk J. G., Guthmann A. 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