We extend a general-relativistic ideal magneto-hydrodynamical code to include the effects of elasticity. Using this numerical tool we analyse the magneto-elastic oscillations of highly magnetised neutron stars (magnetars). In simulations without magnetic field we are able to recover the purely crustal shear oscillations within an accuracy of about a few per cent. For dipole magnetic fields between 5 x 10^13 and 10^15 G the Alfv\'en oscillations become modified substantially by the presence of the crust. Those quasi-periodic oscillations (QPOs) split into three families: Lower QPOs near the equator, Edge QPOs related to the last open field line and Upper QPOs at larger distance from the equator. Edge QPOs are called so because they are related to an edge in the corresponding Alfv\'en continuum. The Upper QPOs are of the same kind, while the Lower QPOs are turning-point QPOs, related to a turning point in the continuous spectrum.
The theoretical framework of relativistic elasticity has been developed recently in a series of papers by [1,2,3,4]. One natural application of this theory is the crust of a neutron star, where the structure of the matter is crystalline and it is able to support shearing motions. When calculating the oscillatory modes supported by the crust, one has to include the effects of gravitation, which influence the theoretically obtained frequencies for isolated neutron stars significantly. Candidates for observations to test theoretical models basing on relativistic elasticity can be found in the decaying tail of a giant burst of a soft-gamma repeater (SGR). In the giant flares of two such objects, SGR 1900+14 and SGR 1806-20, a number of long-lasting, quasi-periodic oscillations (QPOs) have been observed (see [5] and [6] for recent reviews).
The first attempts to explain those QPOs are based on models of purely crustal shear oscillations of an isolated neutron star (see e.g. [7,8,9,10,11,12]). However, SGRs are believed to posses ultra strong magnetic fields B ≈ 10 15 G, and it is necessary to construct self-consistent models including the interaction of the Alfvén oscillation with the crustal shear modes ([13, 14, 15, 16, 17]). A second approach to understand these QPOs is therefore based on purely Alfvén oscillations without a crust (see e.g. [18,19,20]). In those studies two families of QPOs were found. They are related to the open field lines near the pole and to the closed field lines near the equator. That Alfvén QPO model is very attractive, because it reproduces the near-integer-ratios of the observed 30, 92 and 150 Hz frequencies in SGR 1806-20. The results of the numerical simulation agree with a semi-analytic model based on standing waves in the short-wavelength limit [19].
Naturally any realistic model has to include both contributions, the crust and the magnetised core. In first simplified models, it was shown that the effect of the coupling between crust and core may lead to an absorption of shear modes into a MHD continuum of Alfvén oscillations [15]. Additionally long-lived QPOs may still appear at the turning points or edges of the continuum.
Here we extend a previous model of [19] to simulate coupled, magneto-elastic oscillations in a general-relativistic framework. We use a dipolar magnetic field and a tabulated equation of state (EOS) for dense matter. The numerical simulations are based on state-of-the-art Riemann solver methods for both the interior MHD fluid and the crust.
We use units where c = G = 1 with c and G being the speed of light and the gravitational constant, respectively. Latin (Greek) indices run from 1 to 3 (0 to 3).
Following the analysis by [19], who considered purely Alfvén oscillations of magnetars with a two-dimensional, general-relativistic, ideal magnetohydrodynamic code called MCoCoA [21], we extend this code by including the effects of the crust in general relativity. As in [22] this can be done by including an additional term T µν elas in the stress-energy tensor
where ρ is the rest-mass density, h the specific enthalpy, P the isotropic fluid pressure, u µ the 4-velocity of the fluid, b µ the magnetic field measured by a co-moving observer (with b 2 := b µ b µ ), Σ µν the shear tensor, and µ S the shear modulus, respectively. The latter is obtained according to [10]. As in the previous work we apply a number of simplifications: (i) a zero temperature EOS, (ii) axisymmetry, (iii) a purely poloidal magnetic field configuration, (iv) the Cowling approximation, (v) a spherically symmetric background and (vi) small amplitude oscillations. Assumptions (ii) and (iii) lead in the linear approximation to the decoupling of polar oscillations from axial ones. We further assume a conformally flat metric
where α is the lapse function and φ the conformal factor. Employing the induction equation for the magnetic field, the equations of the conservation of energy and momentum can be cast into a conservation law of the following form
where g and γ are the determinants of the 4-metric and 3-metric, respectively. The twocomponent state and flux vectors are given by
where B i are the magnetic field components as measured by an Eulerian observer [23], and W = αu t is the Lorentz factor. The shear tensor Σ iϕ = 1/2g ii ξ ϕ ,i contains the spatial derivatives (denoted by a comma) of the fluid displacement ξ ϕ due to the oscillations, which are related to the fluid 4-velocity according to ξ ϕ ,t = αv ϕ = u ϕ /u t , where v ϕ is the ϕ-component of the fluid 3-velocity.
The boundary conditions at the surface ξ ϕ ,r = 0 is a consequence of the continuous traction condition and vanishing currents at the surface of the star. At the crust-core interface the continuous traction condition together with condition of continuous parallel electric field (and so continuous ξ ϕ ) imply ξ ϕ core,r = (1 + δ) ξ ϕ crust,r with δ = µ S /(b r b r ) . The equilibrium models for our simulations are con
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