An RMHD study of transition between prompt and afterglow GRB phases
aa r X i v : . [ a s t r o - ph ] J a n An RMHD study of transition between prompt andafterglow GRB phases
Petar Mimica ∗ Department for Astronomy and Astrophysics, University of ValenciaE-mail: [email protected]
Dimitrios Giannios
Max-Planck Institute for AstrophysicsE-mail: [email protected]
Miguel-Angel Aloy
Department for Astronomy and Astrophysics, University of ValenciaE-mail: [email protected]
We study the afterglow phases of a GRB through relativistic magnetohydrodynamic simulations.The evolution of a relativistic shell propagating into a homogeneous external medium is followed.We focus on the effect of the magnetization of the ejecta on the initial phases of the ejecta-externalmedium interaction. In particular we are studying the condition for the existence of a reverseshock into the ejecta, the timescale for the transfer of the energy from the shell to the shockedmedium and the resulting multiwavelength light curves. To this end, we have developed a novelscheme to include non-thermal processeses which is coupled to the relativistic magnetohydrody-namic code
MRGENESIS in order to compute the non-thermal synchrotron radiation.
Supernovae: lights in the darknessOctober 3-5, 2007Maó (Menorca) ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ n RMHD study of transition between prompt and afterglow GRB phases
Petar Mimica
1. Introduction
Magnetic fields may play an important role in the relativistic flow of a gamma-ray burst (GRB),but the extent to which they are important remains uncertain. Looking at the process which pro-duces a relativistic GRB outflow, two alternatives are usually considered. On the one hand, neutrinoannihilation may be a process which leads to the formation of fireball dominated by thermal energy.Here magnetic fields are dynamically unimportant. Alternatively, powerful enough magnetic fieldscan efficiently extract the rotational energy from the central engine and launch a Poynting-fluxdominated flow.In order to achieve relativistic velocities, GRBs have to be launched with high energy-to-massratio. In the fireball model, most of the energy is assumed to be thermal [17, 30]. This can be beresult of neutrino-antineutrino annihilation in the polar region of an accreting object [35, 3, 4, 6].The acceleration of the fireball is due to the internal pressure gradient, whereby thermal energy ofthe fireball is converted into kinetic energy of the baryons. At the end of the acceleration phasefaster parts of the flow may collide with the slower ones producing the internal shocks which powerthe GRB prompt emission [31, 11, 27, 28]. Processes such as two-stream instability in the shocksmight amplify weak magnetic fields present in the flow [24]. These fields are expected to accountfor less than 1% of the energy of the flow. After the internal shock phase the flow expands andcools as it enters the afterglow phase. In this case very weakly magnetized flow is expected at theonset of the afterglow.Magnetic fields with appropriate topology can efficiently extract rotational energy from thecentral engine, be it an accretion disk [8], rotating black hole [9] or a millisecond magnetar [34]launching a Poynting-flux dominated flow (PDF). The acceleration of the PDF depends on thefield geometry and the dissipation processes. Magnetic dissipation can convert Poynting flux intokinetic energy [12, 14] and also power the GRB prompt emission [23, 13, 15]. Different studies ofMHD jet acceleration show that magnetic energy is not completely converted into kinetic energyat the end of the acceleration phase. At larger radii we expect magnetic energy of the flow to becomparable to the kinetic energy of the baryons [14] or even much larger [23, 33]. It should benoted here that the magnetization of the flow might be decreased by the pair-loading caused by the n ¯ n -annihilation near the central engine [22, 5].Fireball and PDF models respectively predict weakly and strongly magnetized flow at the onsetof the afterglow phase. The initial phases of the interaction of the GRB flow with the (circumburst)external medium depend on the strength of the magnetic fields in the flow. A particularly promisingprobe of the magnetization of the GRB flow can come from understanding the early afterglowemission [20, 36, 16].In this paper we outline the status of the ongoing study of the interaction of magnetized ejectawith external medium. In Sec. 2 we describe in more detail the current understanding of ejecta-medium interaction. Sec. 3 gives an overview of numerical methods and a plan for numericalsimulations. Summary is given in Sec. 4. 2 n RMHD study of transition between prompt and afterglow GRB phases Petar Mimica
2. Ejecta-medium interaction
We consider a homogeneous shell expanding into an external medium of constant density .At large distances from the central engine (typically R ≈ − cm) substantial interactionbegins, whereby ejecta begins to decelerate due to the accumulation of the external material. Weassume that at these distances the flow has already been accelerated and collimated. The internaldissipation mechanism, presumably responsible for the prompt emission (e.g. internal shocks,magnetic dissipation) is also expected to take place at a shorter distance from the central enginewith respect to that of the afterglow phases. After the internal dissipation is over, the flow expandsand cools. Since we are interested in the afterglow phases, the shell is assumed to be cold. Wedenote the shell Lorentz factor by g ≫ D . In a radially expanding outflowthe magnetic field component perpendicular to the direction of motion drops as r − while thecomponent in the direction of motion drops as r − , so that we expect the magnetic field to bedominated by the perpendicular component. We define the magnetization parameter as s : = E P E K = B pg r c , (2.1)where E P and E K are Poynting and kinetic energies in the shell, r and B its density and magneticfield measured in the central engine frame. With this definition a fireball corresponds to s ≪ s ≥ c is the speed of light. The total energy of the shell is E = p R D ( g r c + B / p ) = E K ( + s ) . (2.2)From Eqs. 2.1 and 2.2 we can see that s parametrizes the fraction of the total energy in the kinetic(1 / ( + s ) ) and magnetic ( s / ( + s ) ) form.We first discuss the ejecta-medium interaction for non-magnetized ejecta, and then turn to thearbitrarily magnetized case. We also focus on the conditions for the existence of a reverse shockinto ejecta of arbitrary magnetization. s ≪ forward shock propagating into the external medium, and the reverse shock propagating into the shell.Shocked shell and external medium are separated by the contact discontinuity. The forward shockis always ultra-relativistic, while the strength of the reverse shock depends on the density contrastbetween the shell and the external medium and the bulk Lorentz factor g . We distinguish between relativistic and Newtonian reverse shocks [32]. The critical parameter is x : = l / D − / g − / , (2.3)where l = ( E / p n e m p c ) / is the Sedov length, n e the external medium number density, and m p the proton mass. In the Newtonian case ( x ≫
1) the shock is non-relativistic in the shell rest Similar analysis can be performed for the wind profile where the density of the external medium scales as r − . n RMHD study of transition between prompt and afterglow GRB phases Petar Mimica frame and does not decelerate the ejecta much, rather the ejecta decelerate once they accumulatea mass g − times their own mass from the external medium. In the relativistic case ( x ≪
1) theshock crosses the ejecta quickly and slows them down considerably. After the shock crosses theejecta, there is a phase where shocks and rarefaction waves cross the ejecta, passing the energy tothe forward shock. At later stages the evolution of the ejecta only depends on their total energy andthe external medium density [7]. s The dynamics of magnetized ejecta has not been studied as thoroughly as that of unmagnetizedejecta. A qualitative difference to the unmagnetized case is that later phases of the evolution areinfluenced by the internal evolution of the magnetized shell. The initial phase of the evolution hasrecently been studied [37] by solving the ideal MHD shock conditions for arbitrarily magnetizedejecta with toroidal field. In particular, the dynamics of shock crossing has been studied assumingthat there is a reverse shock. In that case, the reverse shock crosses the shell faster the higher themagnetization is. However, as we have recently shown [16], it is not always the case that a reverseshock forms.
Cold, non-magnetized ejecta are always crossed by a reverse shock upon interacting with theexternal medium. This is the case since the sound speed of the ejecta is low and does not allowfor fast transfer of the information of the interaction with the external medium throughout theirvolume. On the other hand, in a flow that is strongly magnetized and sub-fast magnetosonic (as inthe Lyutikov & Blandford 2003 model [23]) there is no reverse shock forming. The flow adjustsgradually to the changes of the pressure in the contact discontinuity that separates the magnetizedflow from the shocked external medium. In [16] we generalize to arbitrarily magnetized ejecta andderive the condition for the formation of a reverse shock.After a detailed treatment of this problem we arrive to the following condition for the formationof a reverse shock [16] x < ( s ) / , (2.4)which can be rewritten in terms of shell parameters as s < . n / D / g . E − / . (2.5)Fig. 1 (taken from [16]) shows the division of x − s parameter space in two regions, onewhere a reverse shock forms and another where its formation is suppressed. We note that from theconditions in Eqs. 2.4 and 2.5 it follows that even for mildly magnetized shells a reverse shock canbe suppressed. This indicates that the paucity of the observed optical flashes in GRB afterglows(associated with the reverse shock emission) may be caused by the suppression of the shock inmany GRBs. We use the convention that A = A x x . n RMHD study of transition between prompt and afterglow GRB phases Petar Mimica s o x Newtonian Reverse Shock
Relativistic Reverse ShockNo Reverse Shock
Figure 1:
Existence of a reverse shock in the x − s parameter space. The dashed black line delimitsregions where a reverse shock forms from the region where there is no reverse shock, ignoring the radialshell spreading. The solid black line shows the delimitation when the shell spreading is taken into account.See [16] for details.
3. Numerical simulations
In this section we describe the reasons and motivation for performing numerical simulationsof shell-ejecta interactions:1.
Verification of the analytic approach:
Results of the analytic work described in Sec. 2,especially in Sec. 2.3, need to be verified by means of numerical simulations. We note thatthe line dividing regions of formation and suppression of the reverse shock in Fig. 1, givenby Eq. 2.4 is approximate, and it is necessary to perform numerical simulations for modelswhose initial parameters lie in the vicinity of the line.2.
Dynamics of shock propagation:
We want to use numerical models to study the influence ofthe magnetization on the propagation of the reverse shock through the shell. On Figs. 2 and3 we show snapshots of two simulations, one with the unmagnetized ( s = s =
1) shell interacting with the external medium, both shown at thesame evolutionary time. It can be seen that in the magnetized case (Fig. 3) the reverse shockhas penetrated the shell deeper than in the unmagnetized case (Fig. 2). Our goal is to performa parametric study where we study the propagation of shocks and rarefactions through theshell for different combinations of x and s .5 n RMHD study of transition between prompt and afterglow GRB phases Petar Mimica
Figure 2:
Snapshot from relativistic hydrodynamical simulation of a spherical, non-magnetized ( s = − . The total energy of the shell is E = erg, its initial width D = cm and bulk Lorentz g ≃ P ∗ (dashed line), r (solid line) standfor the gas pressure and (lab frame) density respectively (in arbitrary units). With increasing radius, one canclearly see the reverse shock, contact discontinuity and forward shock located at r ∼ . · cm. Long term evolution and energy content:
One of the important questions that simulationscan answer is the timescale of the transfer of energy from the shell to the shocked externalmedium. Long-term numerical calculations are needed to determined the dependence of theefficiency of the energy transfer on magnetization. We also want to investigate the long-termevolution of the blast wave and determine when the shell profile relaxes to the Blandford-McKee solution [7].4.
Light curves:
Finally, we wish to compute synthetic multi-wavelength afterglow light curves.Light curves are very sensitive to the magnetic field content of the shell, shock strength and,especially, detailed radial profile g ( R ) of the Lorentz factor as the shell propagates into theexternal medium. They also depend on the distribution of shock accelerated particles. Tosee why g ( R ) is crucial for the light curve, consider the difference in the arrival time to theobserver of signals emitted simultaneously from two points with radii R and R + d R , respec-tively. It turns out that the difference is d t ≈ d R g ( R ) − for a relativistic shell. As can be seen,a sudden drop in a Lorentz factor in one model as compared to the other will produce featureswhich have longer observed duration. The resulting light curves of two models can be verydifferent. High-resolution simulations are needed to resolve sufficiently short time intervalsin order to be able to study the influence of magnetization on g ( R ) and the subsequent lightcurve.We have developed a relativistic magnetohydrodynamic code MRGENESIS [29, 27, 28] which6 n RMHD study of transition between prompt and afterglow GRB phases
Petar Mimica
Figure 3:
Same as Fig. 2, but for the magnetized ( s =
1) shell. Note that the reverse shock crosses theejecta faster with respect to the non-magnetized case in agreement with analytical expectations. consists of a finite-volume, high-resolution, shock-capturing scheme
GENESIS [1, 2, 21] whichsolves for the conservation laws of relativistic magnetohydrodynamics, and a module which fol-lows the transport, evolution and radiation from non-thermal particles. For the purpose of analyzingdynamics and emission from afterglow shells, a very high resolution is needed. Numerical conver-gence tests have shown that, in order to sufficiently resolve the shell, we need to resolve the scalesof the order of D g − . This means that we need to use at least g zones within the shell. We use agrid re-mapping procedure described in [29], which can be thought of as a guided mesh refinementcentered on the shell. Even with this procedure the actual number of zones for a typical model isexpected to be several millions. We are performing simulations on supercomputers of the SpanishSupercomputing Network.
4. Conclusions
We are performing high-resolution numerical studies of the transition from prompt to the earlyafterglow phase of gamma-ray bursts. The afterglow is being modeled as the radiation from the rel-ativistic shell expanding into the homogeneous external medium. We study the difference betweenthe fireball (unmagnetized shell) and the Poynting-flux dominated (magnetized shell) models.In the context of the early optical afterglow we have shown analytically that even a moder-ate magnetization of the flow can suppress the existence of a reverse shock, and thus explain theapparent paucity of the optical flashes for a large number of early afterglows. We are currentlyperforming simulations to study the formation and suppression of relativistic shocks in detail. Tostudy the later phases of the afterglow we aim to determine the influence of the initial shell mag-7 n RMHD study of transition between prompt and afterglow GRB phases
Petar Mimica netization on the energy content, transfer of energy from the shell to the forward shock, and thelong-term flow structure.We have developed a novel scheme for treating non-thermal processes in relativistic magne-tohydrodynamic simulations. This scheme is used to compute multi-wavelength light curves fromnumerical simulations. The aim is to study the influence of the initial magnetization on the short-and long-term light curves.
Acknowledgements
PM is at the University of Valencia with a European Union Marie Curie Incoming InternationalFellowship (MEIF-CT-2005-021603). MAA is a Ramón y Cajal Fellow of the Spanish Ministry ofEducation and Science. He also acknowledges partial support from the Spanish Ministry of Edu-cation and Science (AYA2004-08067-C03-C01, AYA2007-67626-C03-01). The authors thankfullyacknowledge the computer resources, technical expertise and assistance provided by the BarcelonaSupercomputing Center - Centro Nacional de Supercomputación.
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