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General Relativistic Magnetohydrodynamic Simulations of Binary Neutron Star Mergers

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General Relativistic Magnetohydrodynamic Simulations of Binary Neutron Star Mergers General Relativistic Magnetohydrodynamic Simulations of Binary Neutron Star Mergers Bnmo Giacomazzo Department of Astronomy, University of Maryland, Co llege Park, Maryland, USA Gravitational Astrophysics Laboratory, NA SA Goddard Space Flight Center, Greenbelt, Maryland, USA Luciano Rezzolla Max-Planck-lnstitut fur Gravitationsphysik, Alhert-Einstein-Institut, Potsdam Germany Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana, USA Luca Baiotti Institute of Laser Engineering, Osaka University, Osaka, Japan We report on our recent general-relativistic simulations of magnetized, equal-mass neutron­ star binaries and on the role that realistic magnetic fields may have in the evolution of these systems. In particular, we study the evolution of the magnetic fields and show that they can influence the survival of the hypermassive neutron star produced at the merger by accelerating; its collapse to a hlack hole. We also show how the magnetic field cau be amplified and that this can lead to the production of the relativistic jets observed in short ga1nrna-ra.y bursts. 1 Introduction The numerical investigation in full general relativity (GR) of the merger of binary neutron stars (BNSs) has produced a series of new and interesting results in the last years.1 Thanks to several numerical improvements, it has been indeed possible to start to investigate the full dynamics of 5 these systems including the formation of tori around rapidly rotating black holes (BHs) z.3A, which could not be modeled via l'\cwtonian simulations. This progress has allowed the beginning of an accurate investigation of whether Bl'\S mergers could indeed be behind the central engine of short 1-ray bursts (GRBs) . 5,7,s Fully GR simulations have shown that the end result of BNS mergers is the formation of a rapidly spinning BH surrounded by a hot torus. Driven by neutrino processes and magnetic fields, such a compact system may be capable of launching a relativistic fireball with an energy of � 1048 erg on a timescale of 0.1 - 1 s. Moreover, BNSs are also one 9 of the most powerful sources of gravitational waves (GWs) that will be detected in the next few years by advanced LICO and advanced Virgo. 10 GR simulations of BNSs have then started to provide accurate templates that can be used to infer properties of the NSs composing the binaries, such as the equation of state (EOS) of NSs, once their GWs will be detected. 11 Here we review some of the main results we published recently 5•8 and that describe the merger of magnetized equal-mass Bl'\Ss. We will show how magnetic fields can affect the dynamics of these systems and their role in powering short GRBs. Table Properties of the eight equal-mass binaries considered: baryon mass of each star; total ADM mass M, l: initial orbital angular velocity !10; mean coordinate radius along the line connecting the two stars; /\!ADM; rnaxixnurn initial magnetic field wherere* is 81 10 or 12. Bo , Binary no (rad/ms) r, (km) Bo (G) 1.445 2.680 1.78 15.0 ± 0.3 0 or 1.97 10* Ml .45-B* x 1.625 2.981 1.85 13.6 ± 0.3 0 or 1.97 10* Ml . 62-B* x 2 Numerical and Physical Setup All the details of the mathematical and numerical setup used for producing the results presented here are discussed in depth in our previous publications 5•8 and here we limit ourselves to a brief overview. We have used the general relativistic magnetohydrodynamic (GRMHD) Whisky codc 12•13•14, which solves the equations of GRMHD on dynamical curved backgrounds. In particular Whisky makes use of the Cactus framework which provides the evolution of the Einstein equations for the metric expressed in the BSSN formulation. 15 The GRMHD equations arc solved using high resolution shock capturing schemes and in particular by using the Piecewise Parabolic Method (PPM), 16 and the Harten-Lax-van Leer-Einfeldt (HLLE) approximate Riemann solver 17 to compute the fluxes. In order to guarantee the divergence-free character of the MHD equations we have employed the flux-CD approach, 18 but with one substantial difference, namely, that we use as an evolution variable the vector potential instead of the magnetic field. 5 The system of GRMHD equations is closed by an EOS and we have employed the commonly used "ideal-fluid" EOS, in which the pressure pis expressed as p= pc(r 1 , where pis the rest-mass density, - ) f is the specific internal energy and r is the adiabatic exponent . Both the Einstein and the GRMHD equations are solved using the vertex-centered AMR approach provided by the Carpet driver. 19 The results presented below refer to simulations h performed using 6 levels of mesh refinement with the finest level having a resolution of = 0.1500 221 m and covering each of the two :NSs, while the coarsest grid extends up to .!YI0 '.'.:'='. 254.4 375.7 km and has a resolution of h � 4.8 7.1 km. For all the simulations r = Af8 '.'.:'='. /'vf8 '.'.:'='. reported here we have used a reflection-symmetry condition across the z = 0 plane and a 7r­ syrnrnetry condition across the 0 plane� :i; = Finally, the initial data 2•20 were produced by Taniguchi and Gourgoulhon 21 with the multi­ b domain spectral-method code LORENE. Since no self-consistent solution is available for mag­ netized binaries yet, a poloidal magnetic field is added a-posteriori and it is initially confined inside each of the :NSs. 20•5 The main properties of the initial data are listed in Table 1 and we have considered two classes of binaries differing in the initial masses, i.e., binaries Ml . 45-B*, and binaries Ml . 62-B*. For each of these classes we have considered four different magnetiza­ tions {indicated by the asterisk) so that, for instance, Ml . 62-B12 is a high-mass binary with a maximum initial magnetic field 1.97 1012 G. Bo = x 3 Results In order to highlight some of the most salient aspects of the binary dynamics we will focus on the high-mass models Ml . 62-B* since they describe all the aspects of a typical BNS merger. The main difference with the low-mass models 1.45-B* is that the latter will collapse to a BH on a "Stated differently, we evolve only the region {x 2: 0, z 2: O} applying a 180°-rotational-syrnmetry boundary condition across the plane at x = 0. 0http://www.lorene.obspm.fr 70 Figure 1: Snapshots at representative tirnes of the evolution of the high-mass binary with initial maxi1111nn 1:.l - magnetic field of 10 G, i.e., . 1 . The first pa11el shows the initial condition, the second one the moment of the rnerger and the third oneMl 62theB HMNS2 funned after it. The rest-1nass density is visualized using volu111e rendering colors from red to yellow while the white lines are the magnetic field lines inside the stars. The la.st ( ) three panels show instead the evolution of the torus after the collapse of the HMNS to BH white sphere at the ( center . In these last three panels the white lines are the magnetic field lines in the region close to the spin axis ) of the BH and the green lines are the magnetic field lines in the torus. Figure published in Rezzolla et al 2011. 71 _ B=O G 9 B= !06 G B=l010 G B= 1012 G M162-B* 5 4 0 5 10 1 20 0 10 5 15 t [ms] log(B+ 1) [GJ Figure 2: Left Panel: Evolution of the 1naximum of the rest-n1ass density nonnalized to its initial value fur the high-mass models. Right panel: Lifetime of the Hl\1NS formed after theµ merger in the high-mass case as a function of the initial magnetic field. The error bar has been estimated from a set of sin1ulations of unmagnetized binary NS mergers at three different resolutions; in particular, we have assumed that the magnetized runs have the same relative error on the delay time of the correspondip.g unmagnetized model. Indicated with a dashed line is the continuation of the delay times to ultra-high ma!':nctic fields of 1017 G. Figure published in Giacomazzo et al 2011. much longer timescale 4 and, because of the much higher computational cost, we have followed those models only for ::::: lOms after the merger and hence before their collapse to BH. A synthetic overview of the dynamics is summarized in Fig. 1, whieh shows snapshots at representative times of the evolution of the high-mass binary with an initial maximum magnetic field of 1012 G, i.e., Ml . 62-B12. The rest-mass density pis visualized using volume rendering with colors from red to yellow, while the white lines represent the magnetic field lines. The first panel shows the initial conditions with the two NSs having a purely poloidal field contained inside each star. The second panel shows the time when the two NSs enter into contact (t ""' 7ms) and when the Kelvin-Helmholtz instability starts to curl the magnetic field lines producing a strong toroidal component. Zll.5 The third panel shows the H).l[NS formed after the merger, while the three last panels show the evolution of the formed BH and of the torus surrounding it. In the last three panel the green lines refer to the magnetic field in the torus and the white ones to the field lines near the spin axis of the BH. We will comment on these three last panels later in this section.
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