RELATIVISTIC MODELS OF MAGNETARS
PHD Research Project
Sapienza - Universit`adi Roma, Dipartimento di Fisica
Candidate: Riccardo Ciolﬁ (XXIII ciclo) Supervisor: Prof. Valeria Ferrari
September 25, 2009
During the last decades, observational evidences have shown that two classes of astrophysical sources, the soft-gamma repeaters (SGRs) and the anomalous X-ray pulsars (AXPs), are associated with neutron stars (NSs) having extremely strong magnetic ﬁelds, as large as 1014 - 1015 G at the surface and even stronger in the interior. These highly magnetized NSs have been called magnetars. Magnetars are very interesting objects, for several reasons. Such a strong magnetic ﬁeld has a dramatic eﬀect on both the equilibrium structure of the star, inducing deformations in the mass- energy distribution, and the stellar dynamics, aﬀecting the oscillations of the star, its thermal evolution, and so on, with remarkable consequences on the electromagnetic and gravitational wave emission properties. The magnetic ﬁeld is thought to be liable for the “magnetar activity” period- ically observed in SGRs and AXPs , and for the so-called giant ﬂares, in the aftermath of which quasi-periodic oscillations (QPOs) have recently been detected . These detections are crucial, being the only direct measurements of NS oscillations. It has also been suggested that magnetars could be the central engine of gamma-ray bursts. Furthermore, magnetars may be interesting sources of gravitational waves, with both transient emission associated with stellar oscillations and persistent emission due to the misalignment between the magnetic (and approximate symmetry) axis and the rotation axis. It should be stressed that the observational properties of magnetars might provide important hints concerning the internal composition of NSs and thus the behaviour of matter at supranuclear densities (not reproducible in lab experiments). At present, the number of magnetars discovered is 16 (including 3 candidates), but is widely believed that an important fraction of NSs, about 10%, would possibly become magnetars at some stage of their evolution. In this PHD Thesis our intent is to give a contribution to the understanding of magnetars’ structure and dynamics. The research project is divided into diﬀerent parts, outlined below. 2
1. Equilibrium properties of magnetars
A necessary step towards the understanding of the observed magnetar dynamical processes, is to have an accurate description of the equilibrium conﬁguration of the star. An equilibrium model must be built in the framework of general relativity, since we are dealing with compact objects and strong gravity, and must include a toroidal (φ) component of the magnetic ﬁeld in addition to the poloidal one (r, θ) usually considered; indeed, it is now widely believed that a mixed ﬁeld is needed for a stable conﬁguration. In the last years, models of magnetars have been constructed which take into account only the dipolar (l = 1) component of the ﬁeld, thus neglecting the contribution from higher multipoles and their couplings, and assume for simplicity particular ﬁeld geometries (magnetic ﬁeld entirely conﬁned in the interior [3, 4], or with a constant ratio between toroidal and poloidal components inside the star and null ratio outside, with a surface discontinuity ). Recently, numerical magnetohydrodynamic simulations  have shown that the magnetic ﬁeld is likely to rapidly evolve into a particular shape, called twisted torus, in which the toroidal ﬁeld is conﬁned in a torus-shaped region inside the star and tangent to the surface, while the poloidal ﬁeld extends throughout the entire star and the exterior. Our ﬁrst aim is to build a consistent relativistic equilibrium model, in which a mixed toroidal- poloidal magnetic ﬁeld organized in a twisted torus geometry is present, and the contribution from higher multipoles and couplings is taken into account. In building such a model we can safely neglect the eﬀects of rotation (which is relatively slow in magnetars), and treat the magnetic ﬁeld as a perturbation of the well known spherically symmetric TOV background. In addition, we shall consider axisymmetric conﬁgurations and work within the assumption of ideal magnetohy- drodynamics (MHD), which consists in neglecting the eﬀects of ﬁnite electrical conductivity. The above assumptions are standard in the present literature. Furthermore, we shall consider ‘realistic’ equations of state. One of the uncertainties of previous models concerns the ratio between toroidal and poloidal ﬁelds. On the other hand, the possibility to give a prediction of this ratio is very important: while direct observations give a measure of the surface poloidal ﬁelds, through the spin down formula, we have no direct information about how much energy associated with the toroidal ﬁeld is hidden inside the star, and this amount of energy strongly aﬀects both equilibrium properties and dynamical processes of magnetars. In our model we shall ﬁx this ratio using arguments of minimal energy. The same arguments can also be used to set the relative contribution of higher multipoles (which is not otherwise ﬁxed in the model). 3
Once the magnetic ﬁeld conﬁguration is known, many applications are possible. In general, any dynamical process involving magnetars depends on the equilibrium magnetic ﬁeld conﬁguration, thus we can study the eﬀects of our twisted torus conﬁgurations on diﬀerent processes already described in the literature (an example, discussed below, is given by magnetar oscillations). A signiﬁcant application concerning the equilibrium structure of the star is the computation of the quadrupolar deformations induced by magnetic ﬁeld on the star’s mass-energy distribution. These deformations, liable for a deviation from the spherical shape, have important consequences on the persistent gravitational wave emission of a magnetar, which is associated with its rotation around an axis misaligned with the magnetic (and approximate symmetry) axis. The amplitude of the emitted gravitational waves in this case is indeed proportional to the quadrupole ellipticity of the star, i.e. the quantity which measures the quadrupolar deformations. Furthermore, the sign of the deformation (positive or negative if the star’s shape is oblate or prolate respectively) is crucial: in the case of negative deformation, a “spin ﬂip” mechanism may arise, giving a ﬁnal conﬁguration with the symmetry axis orthogonal to the rotation axis. This conﬁguration is associated with a much stronger gravitational wave emission. As our ﬁrst application, we plan to compute the quadrupolar deformations induced by the twisted torus magnetic ﬁeld conﬁgurations we have previously obtained.
2. Magnetar oscillations
In a following step, we shall focus on the study of magnetar oscillations. Presently, neutron star oscillations have been detected in the electromagnetic spectra only in connection with the giant ﬂares of SGR 1806-20 and SGR 1900+14. The neutron stars involved in these processes are magnetars, and their magnetic ﬁeld is now believed to have a crucial role in the oscillation mechanism (some attempts to explain the QPO frequencies with pure elastic crust oscillations have been carried out, but with unsatisfactory results ). Therefore, in order to have a useful test against these observations, we need a model of NS oscillations which includes the eﬀects of a strong magnetic ﬁeld. In recent years some eﬀort has been dedicated to the development of models which describe the coupled oscillation of the ﬂuid and the magnetic ﬁeld in magnetars [8, 9]. In presence of magnetic ﬁelds the treatment of NS oscillations is much more complicated, thus strong simplifying assump- tions are used. Furthermore, the crust elasticity is not taken into account; as it is widely believed, the oscillation mechanism is likely to depend crucially on both crust elasticity and magnetic ﬁeld, 4 thus such models are to be considered as a useful step towards a complete model including both the eﬀects. In the more realistic of the above models  the background magnetic ﬁeld is assumed to be purely dipolar, and has both poloidal and toroidal components with the toroidal ﬁeld added in the same fashion as in . A possible improvement of this model, that we have planned to accomplish in collaboration with the authors of , is to consider background magnetic ﬁeld conﬁgurations which include the higher multipoles in addition to the dipole and a mixed toroidal-poloidal ﬁeld organized in a twisted torus geometry, as in our equilibrium model. We shall then study the eﬀects of the new background magnetic ﬁeld on the predicted oscillation spectrum. The main features of the model are the following:
• oscillations are treated within a perturbative approach, and using the Cowling approxima- tion, in which the perturbations of the spacetime metric are neglected;
• simulations are performed in the time domain and in 2 space dimensions (even if with an appropriate coordinate choice the equations can be rewritten into unidimensional form), thus the perturbations depend on both the r and θ coordinates and no harmonic expansion is used for them (while the background ﬁeld is still expanded in multipoles);
• the other assumptions are the same as in our equilibrium model.
An alternative way to tackle the topic of magnetar oscillations is to solve the perturbation equations in the frequency domain. This approach is in some sense complementary to the time do- main approach, and we can get interesting information from the comparison of the results obtained within the two. A model based on the frequency domain approach useful for such a comparison is still missing, and we intend to devote some eﬀorts to explore its feasibility.
3. Magnetic ﬁeld evolution
A possible further extension of our work on equilibrium magnetic ﬁeld conﬁgurations, in a diﬀerent direction, concerns the evolution of the magnetic ﬁeld associated with dissipative processes in the crust. The assumption of ideal MHD we use is very good in the ﬁrst days of the magnetar’s life, before the formation of a solid crust or the occurrence of the phase transition to superﬂuid states inside the star, but after that non-negligible ﬁnite resistivity eﬀects arise. Thus, a more realistic description of the magnetic ﬁeld conﬁguration has to take into account these eﬀects. 5
In the last years some progresses have been recorded in modelling the dissipative evolution of magnetic ﬁelds. In particular, in a recent work  the coupled magnetic and thermal evolution of the magnetar’s crust is described, including the state-of-the-art microphysics. In the above model superﬂuidity is not taken into account, but it is planned as a future improvement. A relevant limitation of such model is that the magnetic ﬁeld is assumed to vanish in the core, extending only in the crust and in the exterior. Our basic idea is to perform the evolution in the more realistic case of a magnetic ﬁeld perme- ating the whole star; such work is to be accomplished in collaboration with the authors of , by merging their evolution code with our model of magnetic ﬁeld conﬁgurations. To begin with, we should limit ourselves to the magnetic evolution alone, but in a following step the model can be extended to include the eﬀects of thermal evolution coupled to the magnetic ﬁeld. The magnetic ﬁeld conﬁgurations obtained as a result of such evolution are more realistic from the physical point of view and could serve to conﬁrm or update our previous results.
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