On the Quasi-Periodic Oscillations of Magnetars

We study torsional Alfv\'en oscillations of magnetars, i.e., neutron stars with a strong magnetic field. We consider the poloidal and toroidal components of the magnetic field and a wide range of equilibrium stellar models. We use a new coordinate system (X,Y), where $X=\sqrt{a_1} \sin \theta$, $Y=\sqrt{a_1}\cos \theta$ and $a_1$ is the radial component of the magnetic field. In this coordinate system, the 1+2-dimensional evolution equation describing the quasi-periodic oscillations, QPOs, see Sotani et al. (2007), is reduced to a 1+1-dimensional equation, where the perturbations propagate only along the Y-axis. We solve the 1+1-dimensional equation for different boundary conditions and open magnetic field lines, i.e., magnetic field lines that reach the surface and there match up with the exterior dipole magnetic field, as well as closed magnetic lines, i.e., magnetic lines that never reach the stellar surface. For the open field lines, we find two families of QPOs frequencies; a family of "lower" QPOs frequencies which is located near the X-axis and a family of "upper" frequencies located near the Y-axis. According to Levin (2007), the fundamental frequencies of these two families can be interpreted as the turning points of a continuous spectrum. We find that the upper frequencies are constant multiples of the lower frequencies with a constant equaling 2n+1. For the closed lines, the corresponding factor is n+1 . By these relations, we can explain both the lower and the higher observed frequencies in SGR 1806-20 and SGR 1900+14.Comment: 8 pages, 7 figure

Magnetar Oscillations II: spectral method

The seismological dynamics of magnetars is largely determined by a strong hydro-magnetic coupling between the solid crust and the fluid core. In this paper we set up a "spectral" computational framework in which the magnetar's motion is decomposed into a series of basis functions which are associated with the crust and core vibrational eigenmodes. A general-relativistic formalism is presented for evaluation of the core Alfven modes in the magnetic-flux coordinates, as well for eigenmode computation of a strongly magnetized crust of finite thickness. By considering coupling of the crustal modes to the continuum of Alfven modes in the core, we construct a fully relativistic dynamical model of the magnetar which allows: i) Fast and long simulations without numerical dissipation. ii) Very fine sampling of the stellar structure. We find that the presence of strong magnetic field in the crust results in localizing of some high-frequency crustal elasto-magnetic modes with the radial number n>1 to the regions of the crust where the field is nearly horizontal. While the hydro-magnetic coupling of these localized modes to the Alfven continuum in the core is reduced, their energy is drained on a time-scale much less than 1 second. Therefore the puzzle of the observed QPOs with frequencies larger than 600 Hz still stands.Comment: 15 pages, 11 figures, submitted to MNRA

Alfv\'en Polar Oscillations of Relativistic Stars

We study polar Alfv\'en 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 evolving numerically the two-dimensional perturbation equations. Our study shows that the spectrum of the polar Alfv\'{e}n oscillations is discrete in contrast to the spectrum of axial Alfv\'{e}n 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.Comment: 10 pages, 5 figure

Magnetar Oscillations I: strongly coupled dynamics of the crust and the core

Quasi-Periodic Oscillations (QPOs) observed during Soft Gamma Repeaters giant flares are commonly interpreted as the torsional oscillations of magnetars. The oscillatory motion is influenced by the strong interaction between the shear modes of the crust and Alfven-like modes in the core. We study the dynamics which arises through this interaction, and present several new results: (1) We show that global {\it edge modes} frequently reside near the edges of the core Alfven continuum. (2) We compute the magnetar's oscillatory motion for realistic axisymmetric magnetic field configurations and core density profiles, but with a simplified model of the elastic crust. We show that one may generically get multiple gaps in the Alfven continuum. One obtains discrete global {\it gap modes} if the crustal frequencies belong to the gaps. (3) We show that field tangling in the core enhances the role of the core discrete Alfven modes and reduces the role of the core Alfven continuum in the overall oscillatory dynamics of the magnetar. (4) We demonstrate that the system displays transient and/or drifting QPOs when parts of the spectrum of the core Alfven modes contain discrete modes which are densely and regularly spaced in frequency. (5) We show that if the neutrons are coupled into the core Alfven motion, then the post-flare crustal motion is strongly damped and has a very weak amplitude. Thus magnetar QPOs give evidence that the proton and neutron components in the core are dynamically decoupled and that at least one of them is a quantum fluid. (6) We show that it is difficult to identify the high-frequency 625 Hz QPO as being due to the physical oscillatory mode of the magnetar, if the latter's fluid core consists of the standard proton-neutron-electron mixture and is magnetised to the same extent as the crust. (Abstract abridged)Comment: 22 pages, 22 figures, submitted to MNRA

Relativistic models of magnetars: structure and deformations

We find numerical solutions of the coupled system of Einstein-Maxwell's equations with a linear approach, in which the magnetic field acts as a perturbation of a spherical neutron star. In our study, magnetic fields having both poloidal and toroidal components are considered, and higher order multipoles are also included. We evaluate the deformations induced by different field configurations, paying special attention to those for which the star has a prolate shape. We also explore the dependence of the stellar deformation on the particular choice of the equation of state and on the mass of the star. Our results show that, for neutron stars with mass M = 1.4 Msun and surface magnetic fields of the order of 10^15 G, a quadrupole ellipticity of the order of 10^(-6) - 10^(-5) should be expected. Low mass neutron stars are in principle subject to larger deformations (quadrupole ellipticities up to 10^(-3) in the most extreme case). The effect of quadrupolar magnetic fields is comparable to that of dipolar components. A magnetic field permeating the whole star is normally needed to obtain negative quadrupole ellipticities, while fields confined to the crust typically produce positive quadrupole ellipticities.Comment: 25 pages, 9 figures, submitted to MNRA

Relativistic models of magnetars: the twisted-torus magnetic field configuration

We find general relativistic solutions of equilibrium magnetic field configurations in magnetars, extending previous results of Colaiuda et al. (2008). Our method is based on the solution of the relativistic Grad-Shafranov equation, to which Maxwell's equations can be reduced in some limit. We obtain equilibrium solutions with the toroidal magnetic field component confined into a finite region inside the star, and the poloidal component extending to the exterior. These so-called twisted-torus configurations have been found to be the final outcome of dynamical simulations in the framework of Newtonian gravity, and appear to be more stable than other configurations. The solutions include higher order multipoles, which are coupled to the dominant dipolar field. We use arguments of minimal energy to constrain the ratio of the toroidal to the poloidal field.Comment: 13 pages, 12 figures. Minor changes to match the version published on MNRA

Structure and deformations of strongly magnetized neutron stars with twisted torus configurations

We construct general relativistic models of stationary, strongly magnetized neutron stars. The magnetic field configuration, obtained by solving the relativistic Grad-Shafranov equation, is a generalization of the twisted torus model recently proposed in the literature; the stellar deformations induced by the magnetic field are computed by solving the perturbed Einstein's equations; stellar matter is modeled using realistic equations of state. We find that in these configurations the poloidal field dominates over the toroidal field and that, if the magnetic field is sufficiently strong during the first phases of the stellar life, it can produce large deformations.Comment: 10 pages, 5 figures. Minor changes to match the version published on MNRA

Stochastic background of gravitational waves emitted by magnetars

Two classes of high energy sources in our galaxy are believed to host magnetars, neutron stars whose emission results from the dissipation of their magnetic field. The extremely high magnetic field of magnetars distorts their shape, and causes the emission of a conspicuous gravitational waves signal if rotation is fast and takes place around a different axis than the symmetry axis of the magnetic distortion. Based on a numerical model of the cosmic star formation history, we derive the cosmological background of gravitational waves produced by magnetars, when they are very young and fast spinning. We adopt different models for the configuration and strength of the internal magnetic field (which determines the distortion) as well as different values of the external dipole field strength (which governs the spin evolution of magnetars over a wide range of parameters). We find that the expected gravitational wave background differs considerably from one model to another. The strongest signals are generated for magnetars with very intense toroidal internal fields ($\sim 10^{16}$ G range) and external dipole fields of $\sim 10^{14}$, as envisaged in models aimed at explaining the properties of the Dec 2004 giant flare from SGR 1806-20. Such signals should be easily detectable with third generation ground based interferometers such as the Einstein Telescope.Comment: 9 pages, 5 figures, accepted for publication in MNRA

Case studies of personalized learning

Deliverable 4.1, Literature review of personalised learning and the Cloud, started with an evaluation and synthesis of the definitions of personalized learning, followed by an analysis of how this is implemented in a method (e-learning vs. i-learning, m-learning and u-learning), learning approach and the appropriate didactic process, based on adapted didactic theories. From this research a list of criteria was created needed to implement personalised learning onto the learner of the future. This list of criteria is the basis for the analysis of all case studies investigated. – as well to the learning process as the learning place. In total 60 case studies (all 59 case studies mentioned in D6.4 Education on the Cloud 2015 + one extra) were analysed. The case studies were compared with the list of criteria, and a score was calculated. As a result, the best examples could be retained. On average most case studies were good on: taking different learning methods into account, interactivity and accessibility and usability of learning materials for everyone. All had a real formal education content, thus aiming at the core-curriculum, valuing previous knowledge, competences, life and work skills, also informal. Also the availability of an instructor / tutor or other network of peers, experts and teachers to guide and support the learning is common. On the other hand, most case studies lack diagnostics tests as well at the start (diagnostic entry test), during the personalized learning trajectory and at the end (assessment at the end). Also most do not include non-formal and informal learning aspects. And the ownership of personalized learning is not in the hands of the learner. Five of the 60 case studies can as a result be considered as very good examples of real personalized learning