📝 Original Info
- Title: Relativistic stars with purely toroidal magnetic fields with realistic equations of state
- ArXiv ID: 0904.2044
- Date: 2012-08-21
- Authors: Kenta Kiuchi, Kei Kotake, Shijun Yoshida
📝 Abstract
We investigate equilibrium sequences of relativistic stars containing purely toroidal magnetic fields with four kinds of realistic equations of state (EOSs) of SLy (Douchin et al.), FPS (Pandharipande et al.), Shen (Shen et al.), and LS (Lattimer & Swesty). We numerically construct thousands of equilibrium configurations. Particularly we pay attention to the equilibrium sequences of constant baryon mass and/or constant magnetic flux, which model evolutions of an isolated neutron star. Important properties obtained in this study are summarized as follows ; (1) The dependence of the mass-shedding angular velocity on the EOSs is determined from that of the non-magnetized case. The stars with Shen(FPS) EOS reach the mass-shedding limit at the smallest(largest) angular velocity, while the stars with SLy or Lattimer-Swesty EOSs take the moderate values. (2) For the supramassive sequences, the equilibrium configurations are found to be generally oblate for the realistic EOSs in sharp contrast to the polytropic stars. For FPS(LS) EOS, the parameter region which permits the prolately deformed stars is widest(narrowest). For SLy and Shen EOS, it is in medium. Furthermore, the angular velocities $\Omega_{\rm up}$, above which the stars start to spin up as they lose angular momentum, are found to depend sharply on the realistic EOSs. Our analysis indicates that the hierarchy of this spin up angular velocity is $\Omega_{\rm up,SLy} > \Omega_{\rm up,FPS} > \Omega_{\rm up,LS}>\Omega_{\rm up,Shen}$ and this relation holds even if the sequences have strong magnetic fields. Our results suggest the EOSs within the relativistic stars containing purely toroidal magnetic fields can be constrained by observing the angular velocity, the gravitational wave, and the signature of the spin up.
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Deep Dive into Relativistic stars with purely toroidal magnetic fields with realistic equations of state.
We investigate equilibrium sequences of relativistic stars containing purely toroidal magnetic fields with four kinds of realistic equations of state (EOSs) of SLy (Douchin et al.), FPS (Pandharipande et al.), Shen (Shen et al.), and LS (Lattimer & Swesty). We numerically construct thousands of equilibrium configurations. Particularly we pay attention to the equilibrium sequences of constant baryon mass and/or constant magnetic flux, which model evolutions of an isolated neutron star. Important properties obtained in this study are summarized as follows ; (1) The dependence of the mass-shedding angular velocity on the EOSs is determined from that of the non-magnetized case. The stars with Shen(FPS) EOS reach the mass-shedding limit at the smallest(largest) angular velocity, while the stars with SLy or Lattimer-Swesty EOSs take the moderate values. (2) For the supramassive sequences, the equilibrium configurations are found to be generally oblate for the realistic EOSs in sharp contrast
📄 Full Content
Neutron stars observed in nature are magnetized with the typical magnetic field strength ∼ 10 11 -10 13 G (Lyne & Graham-Smith 2005). For a special class of the neutron stars such as soft gamma-ray repeaters (SGRs) and anomalous X-ray pulsars (AXPs), the field strength is often much larger than the canonical value as ∼ 10 15 G, and these objects are collectively referred as magnetars (Lattimer & Prakash 2007;Woods & Thompson 2004). Although such stars are estimated to be only a subclass (∼ 10%) of the canonical neutron stars (Kouveliotou 1998), much attention has been drawn because they pose many astrophysically exciting but unresolved problems.
Giant flaring activities observed in the SGRs have given us good opportunities to study the coupling of the interior to the magnetospheric structures (Thompson & Duncan 1995, 1996), but we still know little of the relationship between the crustal fraction and the subsequent starquakes (see references in Watts (2006); Geppert & Rheinhardt (2006)). The origin of the large magnetic field is also a big problem, whether descended from the main sequence stars (Ferrario & Wickramasinghe 2006) or generated at post-collapse in the rapidly rotating neutron star (Thompson & Duncan 1993). Assuming large magnetic fields before core-collapse, extensive magnetohydrodynamic (MHD) stellar collapse simulations have been carried out recently (Kotake et al. 2004;Obergaulinger et al. 2006;Shibata et al. 2006;Livne et al. 2007;Dessart et al. 2007;Takiwaki et al. 2004;Sawai et al. 2008;Kiuchi et al. 2008;Takiwaki et al. 2009) towards the understanding of the formation mechanism of magnetars. Here it is worth mentioning that the gravitational waves could be a useful tool to supply us with the information about magnetar interiors (Bonazzola & Gourgoulhon 1996;Cutler 2002). While in a microscopic point of view, effects of magnetic fields larger than the so-called QED limit of B QED = 4.4×10 13 G, on the EOSs (e.g., Lattimer & Prakash (2007)) and the radiation processes have been also elaborately investigated (see Harding & Lai (2006) for a review). For the understanding of the formation and evolution of the magnetars, the unification of these macroscopic and microscopic studies is necessary, albeit not an easy task.
In order to investigate those fascinating issues, the construction of the equilibrium configuration of magnetars may be one of the most fundamental problems. Starting from the pioneering study by Chandrasekhar & Fermi (1953), extensive studies have been done and this research field is now experiencing “renaissance” (Boucquet et al. 1995;Ioka & Sasaki 2003, 2004;Kiuchi & Kotake 2008;Konno et al. 1999;Tomimura & Eriguchi 2005;Yoshida & Eriguchi 2006;Yoshida et al. 2006), in which different levels of sophistication in the treatment of the magnetic field structure, equations of state (EOSs), and the general relativity, have been undertaken. Among them, a more sophistication is required for the studies employing the Newtonian gravity, because the general relativity(GR) should play an important role for the equilibrium configurations of compact objects like neutron stars. As mentioned below, typical densities for neutron stars interior exceed the nuclear density ∼ 10 14 g/cm 3 . In such a high density region, the pressure P and rest mass density ρ 0 becomes comparable, namely P ∼ ρ 0 c 2 with c being the speed of light. In such a regime, the Newtonian gravity is too weak to gravitationally bind the neutron stars without the general relativity, that is GM/c 2 R becomes the orders of magnitude 10 -1 with G, M and R being gravitational constant, mass and radius of star.
It is noted here that the equilibrium configurations of relativistic stars without magnetic fields have been elaborately studied using the LORENE code (Bonazzola et al. 1998). Unfortunately, however, the method of constructing a fully general relativistic star with arbitrarily magnetic structures is still not available. In most previous studies, the weak magnetic fields and/or purely poloidal magnetic fields have been assumed. Boucquet et al. (1995) and Cardall et al. (2001) have treated relativistic stellar models containing purely poloidal magnetic fields. Konno et al. (1999) have analyzed similar models using a perturbative approach. Ioka & Sasaki (2004) have investigated structures of mixed poloidal-toroidal magnetic fields around a spherical star by using a perturbative technique.
As shown in the MHD simulations of core-collapse supernovae (see Kotake et al. (2006) for a review), the toroidal magnetic fields can be efficiently amplified due to the winding of the initial seed poloidal fields as long as the core rotates differentially. After core bounce, as a result, the toroidal fields generically dominate over the poloidal ones even if there is no toroidal field initially. Even for the weakly magnetized star prior to core-collapse, it should be mentioned that the magnetorotational instability (Balbus & Hawley 1991
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