Instability-driven evolution of poloidal magnetic fields in relativistic stars

DRAFT VERSION OCTOBER 24, 2018 Preprint typeset using LATEX style emulateapj v. 08/22/09 INSTABILITY-DRIVEN EVOLUTION OF POLOIDAL MAGNETIC FIELDS IN RELATIVISTIC STARS RICCARDO CIOLFI1, SAMUEL K. LANDER2,1, GIAN MARIO MANCA1, LUCIANO REZZOLLA1 Draft version October 24, 2018 ABSTRACT The problem of the stability of magnetic fields in stars has a long history and has been investigated in detail in perturbation theory. Here we consider the nonlinear evolution of a nonrotating neutron star with a purely poloidal magnetic field, in general relativity. We find that an instability develops in the region of the closed magnetic field lines and over an Alfv´en timescale, as predicted by perturbation theory. After the initial unstable growth, our evolutions show that a toroidal magnetic field component is generated, which increases until it is locally comparable in strength with the poloidal one. On longer timescales the system relaxes to a new non-axisymmetric configuration with a reorganization of the stellar structure and large-amplitude oscillations, mostly in the fundamental mode. We discuss the energies involved in the instability and the impact they may have on the phenomenology of magnetar flares and on their detectability through gravitational-wave emission. Subject headings: stars: neutron — gravitational waves — magnetohydrodynamics (MHD) — methods: nu- merical 1. INTRODUCTION During at least two points within a neutron star’s (NS) life, large-scale magnetic field rearrangement may occur. These are shortly after the formation of NSs in supernovae (Bo- nanno et al. 2003), and also during the giant flares of mag- netars (Thompson & Duncan 1996; Geppert & Rheinhardt 2006). Whilst similar rearrangements may also occur in other stars, they are likely to be particularly significant for the physics of NSs, where the fields are exceptionally strong: up to 1013 G at the surface of normal pulsars and 1015 G for mag- netars. There are a variety of instabilities in NSs, and in a proto-NS magnetic fields may actually have a stabilising ef- fect (see, e.g. Miralles et al. (2002); Bonanno et al. (2003)), but we are concerned here with the fast-acting “Tayler insta- bility” which affects purely poloidal (or purely toroidal) mag- netic fields in stars. The magnetic-field geometry of a NS is important for the star’s evolution, provides a distortion that may lead to gravi- tational radiation (Bonazzola & Gourgoulhon 1996), as well as powering the mechanisms by which these stars may be ob- served: the pulsar emission for normal NSs, and the X/γ-ray emission of magnetars. It is important therefore to determine which models of magnetised NSs are stable equilibria. The study of magnetised stellar equilibria dates back to Chandrasekhar & Fermi (1953). Since then, many possible magnetic equilibria have been studied, using both analytic and numerical techniques. These have included configurations with purely poloidal fields (Ferraro 1954; Monaghan 1965; Bocquet et al. 1995) and purely toroidal fields (Roxburgh 1963; Kiuchi & Yoshida 2008), as well as mixed poloidal- toroidal configurations (Roxburgh 1966; Haskell et al. 2008; Tomimura & Eriguchi 2005; Lander & Jones 2009; Ciolfi et al. 2009, 2010). However, constructing a configuration in equilibrium is only half the problem when modelling stellar magnetic fields; one also needs them to be stable over many dynamical timescales, since stellar magnetic fields have been observed Electronic address: ciolfir@aei.mpg.de 1 Max-Planck-Institut f¨ur Gravitationsphysik, Albert-Einstein-Institut, Potsdam, Germany 2 School of Mathematics, University of Southampton, Southampton, UK to be long-lived. This has proved to be a challenging prob- lem for analytic methods, which can only study the initial localised instability and not the resultant field configuration. With purely poloidal and purely toroidal fields known to be unstable (Markey & Tayler 1973; Wright 1973; Tayler 1973; Flowers & Ruderman 1977), only mixed-field configurations are likely to exist in stars. More recently it has become feasible to use numerical evo- lutions to study these hydromagnetic instabilities, with the benefit that the global behaviour of the instability may be studied (analytic works rely on local analyses), as well as the final outcome of the instability when the field undergoes significant rearrangement (Lander & Jones 2011; Braithwaite 2007; Geppert & Rheinhardt 2006; Kiuchi et al. 2011). De- spite this recent progress, there are still very few models of stellar magnetic-field configurations whose stability has been assessed. The instability-induced redistribution of magnetic flux is potentially a very violent event and it has been suggested as a trigger mechanism for the giant flares of magnetars (Thomp- son & Duncan 1996). This redistribution is likely to be ac- companied by a significant change to the mass quadrupole moment of a NS, making it a potentially detectable source of gravitational waves (GWs) (Kashiyama & Ioka 2011; Corsi & Owen 2011).

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