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MHD cellular automata simulations: Application to GRB X-ray afterglows Bogdan D˘ anil˘ a1 , Tiberiu Harko2 and Gabriela Raluca Mocanu3∗ 1 Astronomical
Institute, Astronomical Observatory Cluj-Napoca, Str. Cire¸silor 19, RO-400487 Cluj-Napoca, Romania E-mail:
[email protected]
The Fourteenth Marcel Grossmann Meeting Downloaded from www.worldscientific.com by 154.16.42.68 on 01/07/18. For personal use only.
2 Department
of Mathematics, University College London, Gower Street London WC1E 6BT, UK E-mail:
[email protected]
3∗ Astronomical
Institute, Astronomical Observatory Cluj-Napoca, Str. Cire¸silor 19, RO-400487 Cluj-Napoca, Romania ∗ E-mail:
[email protected]
We perform two dimensional MHD Cellular Automata simulations for a two-dimensional model of a magnetized flux tube with a background flow. Self Organized Criticality is reached by the system. The lifetime distribution and energy emission as a function of time are discussed regarding an application of the model to X-Ray emission of Gamma Ray Bursts. Keywords: Magnetic reconnection MHD; gamma-ray burst: general.
1. Introduction Many important natural dynamical systems show the presence of long-range spatial and temporal correlations. The possibility that SOC appears in astrophysical phenomena has been intensively investigated recently, and the study of SOC has become a major field of research in astrophysics 1,2,6 . It was found observationally that a class of Gamma-Ray Bursts (GRBs) called Soft Gamma Ray Repeaters (SGR) do exhibit SOC characteristics 2,8 . From the statistics of the SGR 1806-20 bursts it was shown that the fluence distribution of bursts observed with different instruments is well described by power laws with indices 1.43, 1.76 and 1.67, respectively. A very similar result was obtained for the case of the X-ray flares of GRBs with known redshifts by 12 , who have shown that X-ray flares and solar flares share in common three statistical properties: power-law frequency distributions for energies, durations, and waiting times. All these distributions are specific for the physical framework of a SOC system 7 . As suggested in 12 , both types of flares may be driven by a one dimensional SOC magnetic reconnection process. The possibility of appearance of Self-Organized Criticality in an one dimensional 9 and two dimensional 5 magnetized flow was carefully investigated. Diffusion laws similar to those used to model magnetic reconnection with Cellular Automata in various astrophysical phenomena were implemented in the model, as well as a background flow. Under the assumption that the parameter relevant for X-ray afterglows is the magnetic field, the magnetic energy released by one volume during one
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individual relaxation event was computed. The obtained results show that indeed in this system SOC is established. In order to describe the temporal evolution of GRBs we develop a twodimensional CA simulation algorithm, which implements the magnetic induction equation in the MagnetoHydrodynamic (MHD) approximation framework. The bulk advection motion is, as a novelty with respect to other models, taken explicitly into account for various velocity profiles. The end purpose is to find a model which is simple to implement but which still captures the important macroscopic characteristic of observed X-Ray afterglows in GRBs. The present paper is organized as follows. The simulation procedure is described in Section 2, and the discretized MHD equation for the magnetic induction are written down. The simulation results, and some of their astrophysical implications, are presented in Section 3. We discuss and conclude our findings in Section 4. 2. Setup of the simulation The mathematical model is based on the induction equation 10,11 ∂B + η∇2 B, = ∇ × v × B ∂t
(1)
= B(x, where B y, z, t) is the magnetic field, v = v (x, y, z, t) is the plasma velocity, and η is the total magnetic diffusivity coefficient. We define the control parameter = −∇2 B/4, to be computed for the four neighbours in a in the plasma flow as G two dimensional rectangular grid (left, right, up, down). With a configuration given = (Bx (y, z, t), 0, 0), v = (0, 0, vz (z, t)), the magnetic field evolution equation by B becomes
∂ 2 Bx 1 ∂ 2 Bx ∂ 2 Bx ∂(vz Bx ) ∂ 2 Bx ∂Bx . (2) + η , G = − + =− +η x ∂t ∂z ∂y 2 ∂z 2 4 ∂z 2 ∂y 2 The advective regime is the main framework in which we develop our model. If the control parameter becomes critical, the magnetic field evolution is given, for a brief period of time, by a diffusive behaviour. Once this local criticality is relaxed, the control is given back to the advective evolution ∂(Bx vz ) ∂Bx 2 − 2∂z , high Rm >> 1, = (3) ∂t η ∂∂yB2x + ∂∂zB2x , low Rm
