Formation of relativistic jets in Short Gamma Ray Bursts

The intense variability of the prompt emission light curves of the Gamma Ray Bursts is well known since the Compton Gamma Ray Observatory (CGRO) era. The implications associated with the intense variability provided valuable information even in this early era of observations, like the rough estimation of the emitting region dimension, the compactness problem, the required ultra-relativistic motion and the low baryon mass load of the emitting outflow. At present, some key aspects of the prompt emission phase are still ambiguous and the dissipation process that is responsible for the transformation of the outflow bulk kinetic energy to radiation is still a matter of strong debate, e.g. in Long GRBs the Internal Shock Scenario and Reconnection model are among the most prominent models for the optically thin regime. Nevertheless, all the approaches assume that the prompt variability is associated with dissipation in the body of the jet and thus require a highly inhomogeneous outflow; in its turn the outflow inhomogeneity is ascribed to the central engine’s intrinsic properties, although the interference of the outflow with the surrounding at the acceleration phase can not exclude some extra implications on the derived variability.

Our research on the central engine, accretion and jet launching, is focused on the Short GRBs, i.e. bursts believed to originate by a compact object binary. The initial state of our models assume a Black Hole – accreting torus system. Thus it is applicable for the BH-NS or for these NS-NS systems that the HMNS collapses before the main phase of the burst takes place; see for example [5] for an analysis of the previous year big star, GRB 170817A. Τhe accretion reservoir is thermally dominated, but a poloidal magnetic field also exists & β = ptherm / pmag ~ few, to few tenths in our models). The initial configuration of the magnetic field resembles the electrical field of a circular wire and leads to the formation of a magnetic barrier, [2,6] that prevents the baryon loading at the jet body and concludes to the formation of a Poynting dominated jet.

In our study we also make use of the special relativistic jet theory corollaries. The investigation of the steady state jets has shown that the acceleration of the outflow due to its Poynting context is slower than the thermal one by a spatial factor of ~2 orders of magnitude, see [8]. The extended spatial requirement poses extra difficulty to the simulations we perform. The resolution of our r-logarithmic grid resolution is optimum for the proper description of the accretion and the launch – initial acceleration of the jet, nevertheless insufficient to describe to the second phase of the Poynting dominated acceleration. For that reason, the analysis of the emerging outflow is based on the total energy flux parameter

μ = γ ξ (1+ σ)
where γ is the outflow Lorentz factor, ξ the specific enthalpy of the plasma and σ the magnetization parameter, i.e. the Poynting flux normalized to the thermal plus inertial energy flux of the flow. The relationship above demonstrates in a simple form the energetic balance between the thermal, the inertial and the Poynting flux of the plasma. In the case of a steady special relativistic jet, μ is an integral along the field line which determines also the maximum achievable Lorentz factor γinfinity = μ when all the thermal and Poynting energy is transformed to inertial. In the case of a curved spacetime the physical interpretation loses its meaning and μ is not a constant of motion, it can be seen as a purely mathematical definition.

The aim of our model is to investigate the relation of the inhomogeneity of the emerging jet with the properties of the feeding torus. One of the most prominent process for the outward transportation of the accreting material angular momentum is the magnetorotational instability (MRI, [1,4]). The integration of our model was performed by the HARM algorithm [3,7] assuming the ideal conducting limit and a polytropic equation of state. Under such conditions special care and limits on the grid resolution and the magnitude of the torus magnetic field must be set in order to fully resolve the MRI. Since the maximum growth rate ΩMRI of the instability doesn’t depend on the magnetic field strength (notice though that its characteristic wavelength does), but mostly of the rotational velocity of the torus the diversity of our models is succeeded by varying the position of the initial torus. Then the dependence of the outflow inhomogeneity δ tvar using the value of the energetic parameter μ at a point where the effect of the curved spacetime are negligible and the maximum growth rate can be established.

The results of such a simulation appears below. (Author: Kostas Sapountzis)

The formation of a low baryon and Poynting dominated jet is apparent in the panel 1 and 3 where the logarithmic of density and the magnetization parameter are exhibited. The formation of a magnetic barrier prevents matter from entering the jet funnel resulting to an outflow with 3-4 orders of magnitude lower density. The energetic content of the collimated outflow is sufficient to launch a highly relativistic jet, but as the special relativistic theory predicts the acceleration the specific spatial scales are too short to provide it (above $r>200 rg$ the resolution is less that $1 point/rg$ so we do not account this area into our considerations). But what is important for our model is the the inhomogeneity of the outflow appearing on the 2nd panel of the synthetic figure (Author: Kostas Sapountzis).

synthetic_mu

The time evolution of the energetic parameter μ at the (x,z)=(12,200) rg point. Right panel time zooming with the dashed lines representing the characteristic time of the MRI as calculated by 4 and fit at some pulsations.

 

In order to proceed to the further analysis of the inhomogeneity it is important to chose a specific point of reference. By assuming a point in the inner region of the jet, we plot the μ evolution (see Figure above). The variability is compared to the maximum growth rate in the torus, the characteristic time interval of which is described by the dashed lines of the diagram. The correlation is clear and similar results are obtained and for the rest sets of initial configurations that we have analyze so far.

We aim to extend the results obtained so far in two directions. The first path is the proper resolution of the next phase of the jet propagation, i.e the magnetic acceleration regime. For this purpose the easiest approach is the use of a modified grid that keeps sufficient points density for the next one-two orders of magnitude at the radius scale; another perhaps more robust possibility might be the use of an adaptive mesh refinement method. The second path is the extension of simulation at the 3D regime and the investigation of the validity of the derived δ tvar – ΩMRI correlation. Our group has access to the powerful Okeanos HPC of the University of Warsaw ICM, while two our owned servers also contribute to the result processing and visualization. Of course, the computational difficulties that the 3D simulations involved are always challenging, but the astrophysical value of the conclusions derived is the best reward.

 

References

[1] Balbus S. A., Hawley J. F., 1991, ApJ, 376, 214

[2] Bisnovatyi-Kogan G. S., Ruzmaikin A. A., 1976, Ap&SS, 42, 401

[3] Gammie C. F., McKinney J. C., T´oth G., 2003, ApJ, 589, 444

[4] Gammie C. F., 2004, ApJ, 614, 309

[5] Granot J., Guetta D., Gill R., 2017, ApJL, 850, L24

[6] McKinney J. C., Tchekhovskoy A., Blandford R. D., 2012, MNRAS, 423, 3083

[7] Noble S. C., Gammie C. F., McKinney J. C., Del Zanna L., 2006, ApJ, 641, 626

[8] Vlahakis & Konigl, 2003, ApJ, 596, 1080


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