Torsional oscillations of magnetized relativistic stars

Strong magnetic fields in relativistic stars can be a cause of crust fracturing, resulting in the excitation of global torsional oscillations. Such oscillations could become observable in gravitational waves or in high-energy radiation, thus becoming a tool for probing the equation of state of relativistic stars. As the eigenfrequency of torsional oscillation modes is affected by the presence of a strong magnetic field, we study torsional modes in magnetized relativistic stars. We derive the linearized perturbation equations that govern torsional oscillations coupled to the oscillations of a magnetic field, when variations in the metric are neglected (Cowling approximation). The oscillations are described by a single two-dimensional wave equation, which can be solved as a boundary-value problem to obtain eigenfrequencies. We find that, in the non-magnetized case, typical oscillation periods of the fundamental     torsional modes can be nearly a factor of 2 larger for relativistic stars than previously computed in the Newtonian limit. For magnetized stars, we show that the influence of the magnetic field is highly dependent on the assumed magnetic field configuration, and simple estimates obtained previously in the literature cannot be used for identifying normal modes observationally.     

Alfvén polar oscillations of relativistic stars

We study polar Alfvén oscillations of relativistic stars endowed with a strong global poloidal dipole magnetic field. Here, we focus only on the axisymmetric oscillations which are studied by numerically evolving the two-dimensional perturbation equations. Our study shows that the spectrum of the polar Alfvén oscillations is discrete in contrast to the spectrum of axial Alfvén oscillations which is continuous. We also show that the typical fluid modes, such as the f and p modes, are not significantly affected by the presence of the strong magnetic field.     

Quasi-toroidal oscillations in rotating relativistic stars

Quasi-toroidal oscillations in slowly rotating stars are examined within the framework of general relativity. Unlike the Newtonian case, the oscillation frequency to first order of the rotation rate is not a single value, even for uniform rotation. All the oscillation frequencies of the r -modes are purely neutral and form a continuous spectrum limited to a certain range. The allowed frequencies are determined by the resonance condition between the perturbation and the background mean flow. The resonant frequency varies with the radius according to the general relativistic dragging effect.     

Crustal oscillations of slowly rotating relativistic stars

We study low-amplitude crustal oscillations of slowly rotating relativistic stars consisting of a central fluid core and an outer thin solid crust. We estimate the effect of rotation on the torsional toroidal modes and on the interfacial and shear spheroidal modes. The results compared against the Newtonian ones for wide range of neutron star models and equations of state.     

Constraints on the magnetic field geometry of magnetars

We study the effect of the magnetic field geometry on the oscillation spectra of strongly magnetized stars. We construct a configuration of magnetic field where a toroidal component is added to the standard poloidal one. We consider a star with a type I superconductor core so that both components of the magnetic field are expelled from the core and confined in the crust. Our results show that the toroidal contribution does not influence significantly the torsional oscillations of the crust. On the contrary, the confinement of the magnetic field in the crust drastically affects the torsional oscillation spectrum. A comparison with estimations for the magnetic field strength, from observations, excludes the possibility that magnetars will have a magnetic field solely confined in the crust, that is, our results suggest that the magnetic field in whatever geometry has to permeate the whole star.     

General relativistic electromagnetic fields of a slowly rotating magnetized neutron star – I. Formulation of the equations

We present analytic solutions of Maxwell equations in the internal and external background space–time of a slowly rotating magnetized neutron star. The star is considered isolated and in vacuum, with a dipolar magnetic field not aligned with the axis of rotation. With respect to a flat space–time solution, general relativity introduces corrections related both to the monopolar and the dipolar parts of the gravitational field. In particular, we show that in the case of infinite electrical conductivity general relativistic corrections resulting from the dragging of reference frames are present, but only in the expression for the electric field. In the case of finite electrical conductivity, however, corrections resulting from both the space–time curvature and the dragging of reference frames are shown to be present in the induction equation. These corrections could be relevant for the evolution of the magnetic fields of pulsars and magnetars. The solutions found, while obtained through some simplifying assumption, reflect a rather general physical configuration and could therefore be used in a variety of astrophysical situations.     

The influence of Hall drift on the magnetic fields of neutron stars

We consider the evolution of magnetic fields under the influence of Hall drift and Ohmic decay. The governing equation is solved numerically, in a spherical shell with   r _i / r _o = 0.75  . Starting with simple free-decay modes as initial conditions, we then consider the subsequent evolution. The Hall effect induces so-called helicoidal oscillations, in which energy is redistributed among the different modes. We find that the amplitude of these oscillations can be quite substantial, with some of the higher harmonics becoming comparable with the original field. Nevertheless, this transfer of energy to the higher harmonics is not sufficient to accelerate significantly the decay of the original field, at least not at the   R _B = O (100)  parameter values accessible to us, where this Hall parameter   R _B   measures the ratio of the Ohmic time-scale to the Hall time-scale. We do find clear evidence though of increasingly fine structures developing for increasingly large   R _B   , suggesting that perhaps this Hall-induced cascade to ever-shorter length-scales is eventually sufficiently vigorous to enhance the decay of the original field. Finally, the implications for the evolution of neutron star magnetic fields are discussed.     

Axisymmetric magnetic fields in stars: relative strengths of poloidal and toroidal components

In this third paper in a series on stable magnetic equilibria in stars, I look at the stability of axisymmetric field configurations and, in particular, the relative strengths of the toroidal and poloidal components. Both toroidal and poloidal fields are unstable on their own, and stability is achieved by adding the two together in some ratio. I use Tayler's stability conditions for toroidal fields and other analytic tools to predict the range of stable ratios and then check these predictions by running numerical simulations. If the energy in the poloidal component as a fraction of the total magnetic energy is written as E_p / E , it is found that the stability condition is a ( E / U ) < E_p / E ≲ 0.8 where E /U is the ratio of magnetic to gravitational energy in the star and a is some dimensionless factor whose value is of order 10 in a main-sequence star and of order 10³ in a neutron star. In other words, whilst the poloidal component cannot be significantly stronger than the toroidal, the toroidal field can be very much stronger than the poloidal–given that in realistic stars we expect E / U < 10⁻⁶. The implications of this result are discussed in various contexts such as the emission of gravitational waves by neutron stars, free precession and a 'hidden' energy source for magnetars.     

The physics of strong magnetic fields in neutron stars

In this paper we present a new result, namely that the primal magnetic field of the collapsed core during a supernova explosion will, as a result of the conservation of magnetic flux, receive a massive boost to more than 90 times its original value by the Pauli paramagnetization of the highly degenerate relativistic electron gas just after the formation of the neutron star. Thus, the observed super-strong magnetic field of neutron stars may originate from the induced Pauli paramagnetization of the highly degenerate relativistic electron gas in the interior of the neutron star. We therefore have an apparently natural explanation for the surface magnetic field of a neutron star.     

On non-axisymmetric magnetic equilibria in stars

In previous work, stable approximately axisymmetric equilibrium configurations for magnetic stars were found by numerical simulation. Here, I investigate the conditions under which more complex, non-axisymmetric configurations can form. I present numerical simulations of the formation of stable equilibria from turbulent initial conditions and demonstrate the existence of non-axisymmetric equilibria consisting of twisted flux tubes lying horizontally below the surface of the star, meandering around the star in random patterns. Whether such a non-axisymmetric equilibrium or a simple axisymmetric equilibrium forms depends on the radial profile of the strength of the initial magnetic field. The results could explain observations of non-dipolar fields on stars such as the B0.2 main-sequence star τ Sco or the pulsar 1E 1207.4-5209. The secular evolution of these equilibria due to Ohmic and buoyancy processes is also examined.