Stability of magnetic fields in non-barotropic stars: an analytic treatment

Abstract

Magnetic fields in upper main-sequence stars, white dwarfs, and neutron stars are known to persist for timescales comparable to their lifetimes. From a theoretical perspective this is problematic, as it can be shown that simple magnetic field configurations are always unstable. In non-barotropic stars, stable stratification allows for a much wider range of magnetic field structures than in barotropic stars, and helps stabilize them by making it harder to induce radial displacements. Recent simulations by Braithwaite and collaborators have shown that, in stably stratified stars, random initial magnetic fields evolve into nearly axisymmetric configurations with both poloidal and toroidal components, which then remain stable for some time. It is desirable to provide an analytic study of the stability of such fields. We write an explicit expression for a plausible equilibrium structure of an axially symmetric magnetic field with both poloidal and toroidal components of adjustable strengths, in a non-barotropic, nonrotating, fluid star, and study its stability using the energy principle. We construct a displacement field that should be a reasonable approximation to the most unstable mode of a toroidal field, and confirm Braithwaite's result that a given toroidal field can be stabilized by a poloidal field containing much less energy than the former, as given through the condition E pol /E tor 2aE tor /E grav , where E pol and E tor are the energies of the poloidal and toroidal fields, respectively, and E grav is the gravitational binding energy of the star. We find that a ≈ 7.4 for main-sequence stars, and a ∼ 200 for neutron stars. Since E pol /E grav ≪ 1, we conclude that the energy of the toroidal field can be substantially larger than that of the poloidal field, which is consistent with the speculation that the toroidal field is the main reservoir powering magnetar activity. The deformation of a neutron star caused by the hidden toroidal field can also cause emission of gravitational waves.

Many-body Forces in Magnetic Neutron Stars

“…It is important to emphasize that, because of the choice of poloidal magnetic field distributions, oblate shapes are favored for highly magnetic stars. Note, however, that several studies suggest that toroidal contributions might play an important role in the stability of magnetic stars Braithwaite & Spruit (2004), Marchant et al (2011), Lasky et al (2011), Ciolfi & Rezzolla (2013, Akgun et al (2013), Mitchell et al (2015), Armaza et al (2015), Mastrano et al (2015). Still, even in this case, we expect our qualitative results to hold.…”

Gravitational radiation from neutron stars deformed by crustal Hall drift

“…2.2 [see Sec. 4.1 of Mastrano et al (2013)]. The task now is to obtain an algorithm for finding constants κ and functions f such that the field (4) matches the output of the Alicante code within a specified tolerance at each grid point.…”

“…In this current paper, two representatives of these different initial field configurations (which return the same polar cap field structure conducive to radio pulsar emission) are taken as an input to explore their effect on the neutron star deformation. Mastrano et al (2015) recently presented a method to calculate the deformation of a neutron star caused by poloidal-toroidal magnetic fields consisting of arbitrary multipoles [see also Mastrano et al (2013)]. In order to explore whether the magnetic spots and strong toroidal fields in radio pulsars produce an ellipticity which is potentially detectable through gravitational wave (hereafter GW) emission, we do not present an exhaustive study of magnetic field structures and their resulting ellipticities; we simply aim to convince the reader that magnetic field structures arising from Hall drift in radio pulsars may induce stellar deformations, which make them potentially detectable as GW sources.…”

Order-of-magnitude physics of neutron stars

“…This effect and their mutual repulsion (which results in an effective screening of the attraction of the nucleus on the outer electrons by the inner ones) makes the electron cloud bigger, resulting in most neutral atoms 1 In fact, a0 could be obtained in an even simpler way, as the only length that can be constructed from the three relevant dimensional constants in the problem, namely e, , and me. Of course this requires to first have the judgement to eliminate other constants, such as the speed of light and the mass of the nucleus.…”