[L'angolo del PhD] Riccardo Ciolfi - XXIII Ciclo - 2010

Relativistic Models of Magnetars

Scuola di Dottorato in Scienze Astronomiche, Chimiche, Fisiche e Matematiche “Vito Volterra” Dottorato di Ricerca in Fisica – XXIII Ciclo

Candidate Riccardo Ciolfi ID number 695787

Thesis Advisor Prof. Valeria Ferrari

A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics December 2010

Riccardo Ciolfi. Relativistic Models of Magnetars.

Ph.D. thesis. Sapienza – University of Rome © December 2010 ISBN: 000000000-0

version: 15 December 2010 email address: riccardo.ciolfi@roma1.infn.it

Ringraziamenti

Il mio primo ringraziamento è per la prof. Valeria Ferrari. Per tutto il mio percorso universitario Valeria è stata la mia guida, dal giorno che ho messo piede nel suo studio la prima volta, quasi 6 anni fa, ad oggi. Ho avuto lei come relatore delle mie tesi (laurea triennale, laurea specialistica, dottorato) ed ho potuto godere della sua esperienza e del suo costante supporto. Per tutto questo mi ritengo fortunato.

In questi anni è stata capace di trasmettere la sua passione per questo lavoro, al punto che, nonostante mi si siano aperte altre strade, non ho potuto fare a meno di continuare a seguire lei. Per tutto ciò che mi ha insegnato, per le possibilità che mi ha offerto, le sono infinitamente grato.

Un ringraziamento speciale va a Leonardo Gualtieri, al quale devo moltissimo.

Leonardo è stato un maestro ed un collabratore essenziale per tutto il lavoro che ho svolto in questi anni. Senza di lui questa tesi non sarebbe stata possibile.

Con affetto ringrazio Stefania Marassi. Oltre ad essere stata un’eccellente collaboratrice, ha sempre dimostrato grande stima nei miei confronti e mi ha offerto un supporto morale per me fondamentale. È per me un piacere ringraziare gli altri membri del gruppo passati e presenti, in particolare Francesco Pannarale, Giovanni Corvino e Andrea Maselli, cosí come i colleghi dottorandi e gli altri amici della ‘saletta’. Con tutti loro ho condiviso gli affanni e le gioie del dottorato.

Desidero inoltre ringraziare il prof. Fulvio Ricci, che mi ha sempre offerto il suo supporto e non ha mai mancato di dimostrare affetto e stima. È stato ed è per me un punto di riferimento.

Meritevole di un sincero grazie è il prof. Josè Pons, con il quale ho avuto il piacere di collaborare. È stato anche lui importante fonte di supporto ed incoraggiamento.

Sono poi estremamente grato al prof. Luciano Rezzolla, per avermi invitato a collaborare con lui e per avermi infine accolto nel suo gruppo.

Mi piace concludere la serie dei ringraziamenti con il prof. Kostas Kokkotas, che è stato il referee di questa tesi. Lo ringrazio per la sua attenta lettura del mio lavoro e per i suoi apprezzamenti. Non potevo chiedere un referee migliore.

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Contents Introduction

1

1 Neutron stars: a brief overview 1.1 Formation and basic structure . . . . . . . . . . . . . . . . . . . . . .

1.2 Observational properties . . . . . . . . . . . . . . . . . . . . . . . . .

5 6 9

2 Strongly magnetized neutron stars 2.1 The magnetar model . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2 Magnetar phenomenology . . . . . . . . . . . . . . . . . . . . . . . .

13 13 15

3 Stationary axisymmetric magnetized neutron stars: a relativistic model 3.1 The role of equilibrium models . . . . . . . . . . . . . . . .

3.2 Model set up: basic assumptions and equations . . . . . . .

3.2.1 Equations of ideal MHD in General Relativity . . .

3.2.2 Perturbative approach and electromagnetic potential 3.2.3 The relativistic Grad-Shafranov equation . . . . . .

3.2.4 Boundary conditions . . . . . . . . . . . . . . . . . .

3.3 Equations of state . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

17 17 19 20 21 22 25 26

. . . . . . . . . . . . . .

27 27 30 32 34 36 37 39 39 39 43 43 45 49 50

. . . . . . .

. . . . . . .

. . . . . . .

4 Twisted-torus magnetic field configurations 4.1 Internal magnetic field geometry . . . . . . . . . . . . . . . . . .

4.2 The purely dipolar field case . . . . . . . . . . . . . . . . . . . . .

4.3 The case with l = 1 and l = 2 multipoles . . . . . . . . . . . . . .

4.4 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.1 The case with multipoles l = 1, 3 . . . . . . . . . . . . . .

4.4.2 The case with multipoles l = 1, 3, 5 . . . . . . . . . . . . .

4.4.3 Higher order multipoles . . . . . . . . . . . . . . . . . . .

4.4.4 An example of antisymmetric solution . . . . . . . . . . .

4.5 Magnetic helicity and energy . . . . . . . . . . . . . . . . . . . .

4.6 Model extensions . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.6.1 Relative strength of different multipoles . . . . . . . . . .

4.6.2 A more general choice of the function β(ψ) . . . . . . . .

4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.8 Outlook: emergence and stability of twisted-torus configurations v

. . . . . . .

. . . . . . . . . . . . . .

5 Strongly magnetized neutron stars as gravitational wave sources 5.1 Quadrupolar deformations and gravitational waves . . . . . . . . . .

5.2 Single source emission . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3 Gravitational wave background produced by magnetars . . . . . . .

5.3.1 Birth rate evolution . . . . . . . . . . . . . . . . . . . . . . .

5.3.2 Background computation . . . . . . . . . . . . . . . . . . . .

5.3.3 Detectability . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3.4 Wobble angle effects . . . . . . . . . . . . . . . . . . . . . . .

53 53 55 61 62 63 65 66

Conclusions

69

A The Tolman-Oppenheimer-Volkoff (TOV) solution

71

B The GS system in the l = 1, 3, 5 case B.1 β(ψ) chosen according to Eq. (4.1) . . . . . . . . . . . . . . . . . . .

B.2 β(ψ) chosen according to Eqns. (4.22) and (4.23) . . . . . . . . . . .

73 73 74

C The energy of the system C.1 β(ψ) chosen according to Eq. (4.1) . . . . . . . . . . . . . . . . . . .

C.2 β(ψ) chosen according to Eqns. (4.22) and (4.23) . . . . . . . . . . .

77 77 79

D Quadrupolar deformations

81

Bibliography

85

vi

Introduction Neutron stars are the protagonists of the present Thesis. These extremely dense stars, whose compactness is higher than any other object in our Universe with the exception of black holes, are of great interest in both Astrophysics and fundamental Physics. They represent natural “laboratories” to study Physics under extreme conditions, impossible to realize in terrestrial experiments; in particular, they offer a unique chance to understand the behaviour of matter at supranuclear densities, as their observable properties, reflecting the internal microphysics, can be used to test our theoretical predictions on hadron interactions in superdense systems. The gravitational field of a neutron star is so strong that any accurate description of its structure requires General Relativity; furthermore, neutron stars are involved in highly dynamical astrophysical processes and this creates the ideal conditions for the emission of gravitational waves, an important additional element which makes them appealing from the theoretical and observational point of view.

Neutron stars manifest theirselves as different classes of astrophysical sources, associated to a wide phenomenology. Among them, we will focus on magnetars or strongly magnetized neutron stars, associated to Soft Gamma Repeaters and Anomalous X-ray Pulsars. The magnetic field of these objects, reaching surface strenghts of ∼ 1015 G, is so intense that it significantly affects the star’s structure and evolution. Moreover, it is liable for peculiar emission processes (already observed or potentially observable) which offer exciting prospects in discerning neutron star properties. An example is given by giant flares, so far the only events in which neutron star oscillations have been directly detected.

From the point of view of theoretical modelling, our capability to explain the observations involving magnetars is strongly limited by our scarce knowledge about their internal magnetic field configuration. Motivated by the relevance of such information, in the last years several efforts have been devoted to build equilibrium models of strongly magnetized neutron stars endowed with magnetic fields of different geometries. A crucial element is to establish how the toroidal and poloidal field components are distributed and how the magnetic energy is partitioned between the two1 . We only have direct observations of the poloidal magnetic field component, as it extends outside the star, while toroidal fields are confined in the star’s interior; however, we have many indirect evidences suggesting the existence of a significant toroidal component. Assessing the amount of energy associated to toroidal fields and then “hidden” inside the star is essential to improve the predictive power of models describing magnetars’ dynamical processes.

1

In polar coordinates (r, θ, φ) the φ-component of the magnetic field is the toroidal ot azimuthal one, while the components (r, θ) define the poloidal field.

1

2

Introduction

Equilibrium magnetar models have been developed in both newtonian and relativistic frameworks and with different approaches (we discuss them in Chap. 3).

The first models proposed include only poloidal fields [19, 58], while more recently some attempts have been made to describe mixed poloidal-toroidal configurations [52, 48, 28]; nevertheless, these attempts rely on simplifying (and quite resticting) assumptions on the distribution of toroidal fields.

Even if the actual magnetic field geometry realized in magnetars’ interior is unknown, important hints come from recent results obtained by performing 3D numerical magnetohydrodynamics simulation of magnetizd stars in newtonian framework, which suggested that the so-called twisted-torus configurations could be favoured as they appear more stable than others [13, 15, 14]. In twisted-torus-like configurations the toroidal component of the magnetic field is confined to a torus-shaped region inside the star, and the poloidal component extends throughout the entire star and in the exterior. These results provide strong motivation for studying magnetized neutron stars with such magnetic field geometry.

In this Thesis we present the equilibrium magnetar model we have developed.

The model is built in the framework of General Relativity, and follows a perturbative approach. It describes a non-rotating strongly magnetized neutron star surrounded by vacuum, with the assumption that the magnetic field acts as a stationary axisymmetric perturbation of a static and spherically symmetric unmagnetized star.

Inside the star we adopt ideal magnetohydrodynamics, i.e. we neglect finite electrical conductivity effects. The above assumptions are quite standard in the literature.

The magnetic field configurations we find reproduce the twisted-torus geometry (similar geometries have also been considered in recent newtonian studies [123, 56, 133, 134, 66]). Moreover, we include the contribution from the higher (l > 1) multipolar components of the magnetic field and their couplings, in addition to the dipolar (l = 1) component usually considered. We use an argument of minimal energy to find, among the possible solutions, the energetically favoured configuration. This allows us to evaluate the ratio of toroidal and poloidal fields in terms of magnetic field energy. All these elements constitute an improvement with respect to previous models.

The equilibrium configurations we obtain can be used as input for studies on dynamical processes involving magnetars. In this Thesis we also consider one of the possible applications: the emission of gravitational waves from magneticallydeformed rotating neutron stars. We compute the quadrupolar deformation induced by the magnetic field on the star’s structure and the resulting gravitational wave emission spectrum. For completeness, we extend the analysis to other models. Finally, we estimate the stochastic gravitational wave background produced by the entire magnetar population, which results from the superposition of the single source emissions; we also evaluate the detectability of such background by third generation detectors such as the Einstein Telescope [138].

The Thesis is organized as follows.

• In Chapter 1 we provide a basic introduction to neutron stars. We sketch their basic properties and discuss their relevance for Astrophysics and fundamental Physiscs. We describe how neutron stars are formed and their overall structure.

3

Then, we depict their most important observational properties.

• In Chapter 2 we focus on strongly magnetized neutron stars, discussing the magnetar model and the observational evidences supporting it. Next, we give an insight into magnetar phenomenology, providing further motivation to the study of magnetars.

• The original part of the Thesis starts with Chapter 3. Here we illustrate our equilibrium model describing the structure and magnetic field configuration of a strongly magnetized neutron star. We discuss the basic equations and formalism, as well as the physical inputs (such as mass, equation of state and magnetic field strenght) and the boundary conditions employed.

• In Chapter 4 we focus on the specific twisted-torus-like magnetic field geometries we consider and present the results of our computations. We discuss in detail the configurations we have found, emphasizing the improvements with respect to the previous literature.

• Chapter 5 is devoted to gravitational wave emission from strongly magnetized neutron stars.

• Finally, we draw our conclusions.

The content of Chapters 3, 4 and 5 is based on the following papers: 1. Ciolfi R., Ferrari V., Gualtieri L., Pons J.A., Relativistic models of magnetars: the twisted-torus magnetic field configuration, MNRAS 397, 913 (2009) ; 2. Ciolfi R., Ferrari V., Gualtieri L., Structure and deformations of strongly magnetized neutron stars with twisted-torus configurations, MNRAS 406, 2540 (2010) ; 3. Marassi S., Ciolfi R., Schneider R., Stella L., Ferrari V., Stochastic background of gravitational waves emitted by magnetars, accepted for publication in MNRAS (2010) .

Notation and units We employ the 4-metric signature (−, +, +, +).

In Chapters 3, 4 and 5, unless otherwise specified, we will adopt the geometrized unit system, in which the fundamental constants c = 2.99792 · 1010 cm/s ,

G = 6.67428 · 10−8 cm3 /g s2 are set to unity, c = G = 1. As a consequence, we have for example 1 s = 2.99792 · 1010 cm ,

1 g = 7.4237 · 10−29 cm .

4

Introduction

In addition, throughout this Thesis we will use the gaussian system of units for electromagnetic fields.

We will often use the Astrophysical quantity M⊙ (solar mass), defined as M⊙ = 1.989 · 1033 g .

Chapter 1

Neutron stars: a brief overview Neutron stars (NSs) are extremely compact stars, born in supernova explosions at the end of a massive star’s life. They have a typical mass of ∼ 1.4 M⊙ and a typical radius of ∼ 10 km; the resulting compactness M/R is then ∼ 105 times higher than the Sun and makes them the most compact objects endowed with a structure in our Universe (a compactness 3-4 times higher would lead to an event horizon, i.e. to a black hole).

A nice feeling of the unique properties of a NS is given in [46] (Sec. 1.5), where the sentence “neutron stars are superstars” (attributed to David Pines) is jokingly stated as a theorem by means of a number of superproperties. They are indeed superdense (mean density is of the order of the nuclear density ρ0 ∼ 2.8 · 1014 g/cm3 ), endowed with superstrong gravity, such that any good description of a NS requires the general theory of Relativity. They are superfast rotators (fastest known NS has a spin frequency of ∼716 Hz) and superprecise timers (up to 10 significant digits, more than atomic clocks), but also superglitching objects (see Sec. 1.2). NS matter is partially superconducting or superfluid. NSs possess superstrong magnetic fields (surface magnetic fields up to 1013 G for ordinary NSs and up to 1015 G for magnetars). In addition, NSs are superrich in the Physics involved: all the four fundamental forces play a crucial role in determining their structure and dynamical processes, and the fields of Physics concerned include Nuclear and Particle Physics, Condensed Matter Physics, Plasma Physics and Magnetohydrodynamics, General Relativity, Radio, Optical, X, Gamma, Neutrino and Gravitational Wave Astronomy, Physics of stellar structure and evolution, Seismology, and so on..

NSs are very interesting as they represent natural “laboratories” to test Physics under extreme conditions, impossible to reproduce on Earth. In particular, they offer a unique chance to understand the behaviour of matter at supranuclear densities through the interplay between theory and experiments. Indeed, different theoretical equations of state (EOS) reflecting our models of hadron interactions in superdense systems predict different global and observable properties of NSs (such as the compactness). A second major element of fundamental Physics justifying the growing interest for NSs is the emission of gravitational waves. Still unobserved, these spacetime perturbations are predicted by General Relativity and their existence has been confirmed indirectly by the Hulse and Taylor observations of the binary pulsar PSR 1913+16 (nobel prize in 1993). A significant and potentially observable amount of gravitational waves can only be produced in association with high-mass 5

6

1. Neutron stars: a brief overview

and strongly dynamical astrophysical systems, and NSs represent ideal candidates as gravitational wave emitters (see Sec. 1.2 and Chapter 5).

The existence of NSs was predicted in 1934 by Baade and Zwicky [7], just after the discovery of the neutron by Chadwick [21]. They suggested that a supernova explosion would lead to a very high-density star which “consists of closely packed neutrons”. It is worth mentioning that one year before Chadwick’s discovery Landau speculated on the existence of stars of nuclear density [65], even if without the neutron he reached the wrong conclusion that such a star would form by virtue of quantum mechanics violations. An important step was made by Tolman, Oppenheimer and Volkoff [122, 84], who derived the general relativistic equations of hydrostatic equilibrium for a spherically symmetric star, a fundamental element for any NS model. The first observation of a NS as a pulsar came in 1967 by Bell and Hewish [49], giving confirmation to Pacini’s suggestion [85] that a NS could manifest itself as a pulsating radio source if its magnetic and spin axes are misaligned. Later, two new pulsars were observed, the famous Crab and Vela, which further confirmed the association with rotating NS. Nowadays, about two thousand of NSs are known, all lying in our Galaxy and in the nearby Large Magellanic Cloud (most of them are within a third of the Galaxy radius from the Sun). They are divided in different classes of astrophysical sources. The most common are the ordinary radio pulsars (PSRs), for which the identification with magnetized rotating NSs is well established; several hundreds of PSRs were indeed discovered early, in the 70ies, in the large surveys conducted by the Arecibo, Jodrell Bank and Parkes observatories. Among the isolated NSs, in addition to PSRs, we now observe central compact objects in supernova remnants (CCOs), rotating radio transients (RRATs), radio-quiet thermally emitting radio sources (XDINSs or magnificent seven), and finally soft gamma repeaters (SGRs) and anomalous X-ray pulsars (AXPs), also called magnetars and interpreted as strongly magnetized neutron stars (the most interesting to the purpose of this Thesis). Then, NSs are observed in binary systems with an ordinary companion, or a compact one (a white dwarf, (potentially) a black hole or again a NS). These binaries include low and high-mass X-ray binaries (LMXBs, HMXBs), X-ray pulsars, soft (SXTs) and hard X-ray transients. All these manifestations of NSs provide us with an extremely rich phenomenology, with persistent and transient electromagnetic emission covering the whole spectrum, neutrino emission and gravitational wave emission. In Chapter 2 we shall focus on strongly magnetized neutron stars and describe in some detatil the main steps leading from source discovery to theoretical interpretation.

In what follows we briefly describe the evolutionary path leading to a NS and its basic structure (Sec. 1.1), and discuss the most interesting observed and potentially observable NS features (Sec. 1.2).

1.1

Formation and basic structure

In a star, the hydrostatic equilibrium is assured by the balance between self-gravity and radiation pressure due to thermonuclear reactions taking place in the stellar interior. The basic reaction, in which Hydrogen is converted into Helium, is ignited

1.1 Formation and basic structure

7

for first when the gravitational contraction of the proto-star cloud produces sufficiently high temperatures in its center and continues until the Hydrogen reserve is over. At this point self-gravity is no longer supported: a new contraction leads to higher temperatures and could ignite the nuclear burning of Helium and heavier elements. The following evolution depends on the star’s mass: a small mass star (e.g. our Sun) continues its contraction, but its internal burning stops before involving elements as heavy as Iron, and in the final stage it becomes a white dwarf; higher mass stars (> ∼ 8 M⊙ ) reach central temperatures high enough to burn elements up to Iron, and experience a very different final stage. Iron has the greatest binding energy, thus producing heavier elements requires energy insted of releasing it; as a consequence, when a star has formed an Iron core it suddenly happens that nuclear burning stops and this core contracts on free fall timescales due to self-gravity. Extremely high densities are achieved in such contraction, until quantum degeneracy pressure is able to contrast gravity: the superdense core formed is a proto-neutron star; the external layers fall on it and bounce off at high speed producing a supenova explosion, a spectacular event whose luminosity is comparable to that of our Galaxy.

This ejected mass will form the supernova remnant (SNR). If the progenitor’s mass is very high (> ∼ 20 M⊙ ) the same event, instead of a NS, produces a black hole. Supernova explosions are classified depending on their light curves: type Ib, Ic and II are produced by the mechanism described above (core-collapse supernovae), while type Ia supernovae are the result of the collapse of a white dwarf in a binary system, when it exceeds the Chandrasekhar mass limit due to accretion of matter from the companion star (accretion-induced collapse). This different type of supernova leads again to a NS or a black hole.

During the collapse leading to a proto-neutron star, the inverse β-decay is extremely efficient in merging together protons and electrons into neutrons, which grow in number. While in white dwarfs the degeneracy pressure which contrast selfgravity is due to electrons, in the case of a NS the degeneracy pressure is given by neutrons. The neutronization is accompanied by the emission of a huge number of neutrinos, with a consequent energy loss and significant cooling of the proto-neutron star.

Once the NS is formed its structure is given by a sequence of layers (see Fig. 1.1). A solid crust formed within the first day of the NS’s life and ∼ 1 km thick surrounds the NS core. Both crust and core are separated in inner and outer regions.

Matter density increases proceeding inwards.

In the outer crust density ranges between 107 and 1011 g/cm3 . It is composed by heavy nucleons organized in a Coulomb lattice and embedded in a degenerate electron gas. Approaching the inner boundary the inverse beta decay is more efficient and the fraction of neutrons to protons in nuclei grows; when all bound states of neutrons are occupied they start leaving nuclei, forming a neutron gas. This so-called neutron drip point marks the transition to the inner crust. Here density increases up to the nuclear density, 2.67 · 1014 g/cm3 . In this region we have two coexisting phases: proton rich matter, composed of protons and neutrons, and neutron gas. An electron gas is also present and assures the matter neutrality. Proton rich matter is clustered in nuclei, chains of nuclei and higher dimensional structures as density increases (so-called pasta phases), until we have a uniform gas of protons, neutrons and leptons (electrons and eventually muons) in beta equilibrium, where only ∼ 10% of nucleons are protons. Such unform gas composes the outer core, up

8

1. Neutron stars: a brief overview

Figure 1.1. Overall structure of a neutron star.

to densities of 2-3 times the nuclear density. The composition of the inner core, with densities up to ∼ 1015 g/cm3 , is unknown. This constitutes the main NS mistery.

We may expect the same composition of the outer core, but more exotic states are possible: the different hypotheses include the appearence of hyperons (strange heavier baryons, e.g. Σ or Λ) or meson condensates, and the transition to a deconfined phase in which quarks are no longer clustered into nucleons or hyperons.

Crust and core have radically different characteristics and they are described with different equations of state (EOS). While the EOS for the inner and outer crust receive a wide consensus, because the models can be tested against data produced in Nuclear Physics experiments, for supranuclear densities we only have theoretical models poorly constrained by extrapolations from lab measurements.

Such models are based on two different approaches: the non-relativistic many-body theory (NMBT) and the relativistic mean field theory (RMFT). In the NMBT approach nuclear matter is described as a collection of non-relativistic nucleons, whose interactions are modeled through phenomenological 2- and 3-body potentials.

It has a good predictive power, being easier to constrain with experimental data (in the density regime in which they are available) [89], while its main limit is the inconsistence with relativistic effects (e.g. it may allow causality violations).

The RMFT is more elegant and consistent as its formalism is that of relativistic quantum field theory. The simplest formulation describe the interactions through a scalar and a vector meson field [127]. On the other hand, the dynamics of the system can be solved only within a mean field approximation (i.e. a classical treatment of the meson fields). Both the approaches can be generalized to include hyperons, at the expense of a larger uncertainty because in this case the interactions are much less constrained by data [74]. Concerning the hypothesis of deconfined quark matter, a first principle description is prohibitive due to the complexity of QCD. An alternative approach widely used is based on the MIT bag model [25], which assumes that (i) quarks are confined to a region of space (the “bag”) whose volume is limited by the pressure B (the bag constant) and (ii) quark interactions are weak

1.2 Observational properties

9

and can be treated in lowest order perturbation theory.

In our work we shall model the NS core using two different EOS called APR2 and GNH3, which belong respectively to the NMBT and RMFT approaches and define a wide range of compactness. They are briefly described in Section 3.3. We also use two standard EOS for the outer and inner crust, the Baym-Pethick-Sutherland EOS (BPS) and the Pethick-Ravenhall-Lorentz EOS (PRL) [9, 88].

1.2

Observational properties

Here we briefly discuss the most interesting properties of NSs inferred from observations and some main processes which have the potential to enlarge our understanding of NS Physics.

We start discussing what is probably the most important property of a NS: its mass. The most accurate measures of NS masses available come from timing observations of pulsars in binary systems, both with a main-sequence or a compact companion star (i.e. a white dwarf or a second NS). Usually the doppler effect allows to determine the orbital size as well as the total mass of the binary; then, in many cases the detection of relativistic effects such as shapiro delay or orbit shrinkage due to gravitational wave emission permits a constraint on the inclination angle and a measure of the two masses. In some cases the masses are determined with impressive accuracy (one example is the Hulse-Taylor binary PSR 1913+16). A second possibility is to infer the mass of an accreting NS in X-ray binaries, but with less accurate results. For each EOS employed to model the NS interior we have a value for its maximum mass, the NS analogous of the Chandrasekhar mass limit for white dwarfs. Therefore, the mass measurement alone can rule out all the EOS with a mass limit lower than the highest value observed. Presently, the mass range 1 − 2 M⊙ is compatible with the data, even if some low accuracy observations suggest the possibility of larger masses.

Measuring the radius of a NS is a much harder task. On the other hand, the combined knowledge of a NS mass and radius is of crucial importance as it allows to put a direct constraint on the EOS. A possible way to infer the NS radius is from the measurement of both X-ray flux and surface temperature in low-mass X-ray binaries (LMXB). Once combined, these quantities allow to compute thepradiation radius (R∞ ), related to the star’s radius (R) by the expression R∞ = R/ 1 − 2GM/Rc2 ; the contemporary knowledge of the mass (M ) then gives the radius. An alternative way is to determine the mass-radius ratiopby means of gravitational redshift (z) measurements, through the relation z = 1 − 2GM/Rc2 − 1 . It is worth mentioning the result reported in [30], where a gravitational redshift of z = 0.35 is obtained from the observation of Iron and Oxygen transitions in the burst spectrum of the X-ray binary EXO0748-676. This measure is quite controversial [106], but it would be compatible with most of the proposed EOS, except for the softer ones including hyperons. More accurate estimates can be obtained using moment of inertia measurements, as in the promising case of the double-pulsar system PSR J0737-3039 A&B. This system has indeed the largest relativistic corrections known to the orbital parameters and few years of high precision pulsar timing can lead to an accurate determination of the moment of inertia of the NS with the highest mass; in combination with the already known value of the mass we would have a

10

1. Neutron stars: a brief overview

precise measure of the radius [72, 61, 101]. There exist other methods based on astrophysical observations, for which we refer to [94, 67] and references therein. The present observations altogether suggest NS radii in the range 9 − 16 km.

The spin period (P ) and its derivative (P˙ ) are fundamental quantities for an isolated NS. Their simultaneous knowledge is important for estimating the age (the so-called spindown age τSD = P/2P˙ ) and the surface magnetic field strenght (see below). The observed spin periods range between 1.56 ms (fastest spinning NS) and ∼ 1 − 10 s (SGRs and AXPs), while the spin derivatives vary in the wide range 10−21 − 10−9 (adimesional). As already mentioned spin periods can be extremely precise, but many sources (over 100 pulsars) has also shown occational jumps in the spin evolution, known as glitches, consisting in a sudden increase of the spin frequency (with a frequency variation in the range 10−9 − 10−6 ). Their origin is still matter of controversy. The leading model envisages a fast transfer of angular momentum from a superfluid component to the rest of the star, which includes the crust [6]. A superfluid rotates by forming a dense array of vortices which can be pinned to the other component, until the spindown of the crust causes the unpinning.

At this point the vortices are free to move and transfer angular momentum to the crust. Once explained, this phenomenon has the potential to shed light on NS matter properties.

Another process which carries crucial information about the NS internal properties, is NS cooling [87, 132]. Present models predict the time evolution of NS temperature distribution depending on the internal composition (EOS, superfluidity, and so on); in particular, the different theoretical surface temperatures (Ts ) as function of time can be compared with the observations, when a simultaneous measure of the age and Ts of a source is available. Estimating the surface temperature of a NS is very hard because most of the electromagnetic emission of NSs is usually non-thermal, and we can get useful information only from a small number of objects.

Despite their great potential, the predictive power of cooling theories is presently limited by (i) a strong model-dependence and (ii) lack of accuracy in the age and temperature measurements. Standing on the current status of this research field, NS cooling data support the idea that superfluid/superconductive states occur in NS matter, as suggested by our understanding of pulsar glitches.

For the purposes of this Thesis, an essential NS property is its magnetic field.

The surface magnetic field strenght at the pole is usually inferred according to Bp =

s

3Ic3 P P˙ , 2π 2 R6 sin2 α

(1.1)

where α is the misalignment angle between spin and magnetic axes. The formula is obtained from the assumption that the external field is dipolar and the observed spindown is entirely due to magnetic braking. There exist other methods based on estimating the effects of a given magnetic field on the electromagnetic emission (e.g. cyclotron resonant absorption) to be compared with specific X-ray spectral features observed in some isolated NSs, but they are limited by the strong model dependence of spectral fitting. Typical dipole magnetic field strenghts for ordinary pulsars are in the range 1011 − 1013 G, even if some millisecond pulsars have Bp values down to 108 G; on the other hand, magnetars (SGRs and AXPs) have dipole magnetic fields of the order of 1014 − 1015 G. In our work we focus on such strongly

1.2 Observational properties

11

magnetized NSs: in the next Chapter we discuss the observational evidences supporting the existence of magnetars and the crucial role played by magnetic ďŹ elds in determining their evolution and their observational features. Then, the following Chapters describe the original research carried out during the PhD work, and some aspects of magnetars structure and phenomenology will be further discussed.

Concerning the future prospects, the observation of gravitational waves will represent an additional formidable channel to study the nature of NSs. They are indeed the most promising sources of detectable gravitational signals (together with black holes) through a large number of processes, from the merger of binary systems to the continuous emission of rotating NSs, the emission associated with rotational and other instabilities, or related to NS oscillations, and so on. In particular, we will address the gravitational wave emission from strongly magnetized NSs in Chapter 5.

Ground based detectors such as VIRGO [135] and LIGO [136] already inaugurated the era of gravitational wave astronomy by putting the ďŹ rst physical constraints on some known sources; nevertheless, the direct detection is still missing and will hopefully come soon when the advanced versions fo such detectors will be in operation.

Larger prospects are then entrusted in future space detectors such as LISA [137] and third-generation detectors on earth such as the Einstein Telescope [138].

12

1. Neutron stars: a brief overview

Chapter 2

Strongly magnetized neutron stars In the previous Chapter we have introduced NSs on general grounds, and we have listed a number of different astrophysical sources identified as NS manifestations.

Among them, SGRs and AXPs (commonly referred to as magnetars) are the most interesting for the purposes of this Thesis and they deserve a separate introduction.

In the present Chapter we sketch the basic elements supporting the magnetar hypothesis and discuss the theoretical relevance of the observable processes involving strongly magnetized NSs.

2.1

The magnetar model

The history of magnetars starts in 1979, with the detection of an extraordinary event on March 5. A 0.2 seconds pulse of gamma rays a hundred times more intense than any previous gamma ray emission detected flooded many space detectors, followed by a fainter and softer tail signal fading out in about three minutes [79]. This tail interestingly revealed a periodic modulation with a period of about 8 seconds.

Surprisingly, another fainter burst were observed the day after from the same spot on the sky and other sixteen in the following four years. The burst repetition, together with the relatively soft gamma ray emission, allowed to exclude that the event belonged to the ordinary gamma ray bursts (GRBs) already observed. The source was indeed named “Soft Gamma Repeater”.

By comparing the time of detection of different instruments the source was located in the Large Magellanic Cloud (a small galaxy 160 thousand light-years away), and that was again a surprise. At first, the exceptional intensity of the burst led to the idea that the source was in the galactic neighborhood, while the actual measured position was 1000 times farther, implying an enormous energy release. Moreover, the position matched that of a young (∼ 104 years old) supernova remnant. These evidences in combination with the 8 s periodicity strongly suggested that the source was a rotating NS. Such discovery raised a number of questions. The burst mechanism, with such strong energy emission and the repetition feature, as well as the spin period of 8 s (a slow rotation compared to ordinary pulsars) were hard to explain. Moreover, the NS had a persistent X-ray emission (in addition to the bursts) which could 13

14

2. Strongly magnetized neutron stars

not be powered by rotational energy, nor accretion (there were no evidence for a companion star). Finally, the position of the source with respect to the supernova remnant showed an unusually high recoil velocity. In the meanwhile, two other similar sources were discovered and it was realized that periods of burst activity were alternating with periods of quiescence.

A theoretical explanation came in 1992, as a result of a study originally aimed to understand the orgin of pulsar’s magnetic fields. In the famous paper by Duncan and Thompson [34] a scenario is proposed which would lead to the formation of unusually strong magnetic fields, of the order of 1014 − 1015 G, as a consequence of the interplay between rotation and convective dynamo action during the first seconds of life of a very fast spinning NS. Convection in a newly born NS can be driven by the huge neutrino emission accompanying its early phases, with a convective overturn time of about 1 ms. If a NS is born with a period as small as 1 ms differential rotation can support a very efficient large scale dynamo, in which a dipole field up to 1015 G can be generated; if instead the spin period exceeds ∼ 30 ms, differential rotation does not play a significant role in the magnetic field amplification, which acts on smaller scales and can produce a 1012 − 1013 G field resulting from the incoherent sum of many smaller dipoles. This second path is what should happen in the case of ordinary pulsars. In the second (and most interesting) part of that work, they suggest that the peculiar properties of the 1979 March burst SGR can be explained by identifying the source with a magnetar, i.e. a NS endowed with an ultra-strong magnetic field (1014 − 1015 G).

First of all, by equating the estimated age of the supernova remnant (∼ 104 yr) with the spindown age τSD = P/2P˙ and using the dipole radiation formula (1.1) with a spin period of 8 s, they obtain a surface dipole field of ∼ 6 · 1014 G, matching the magnetar range. Magnetar-like magnetic fields result in a fast electromagnetic spindown (high P˙ ), which also justifies the relatively slow spin frequency of the SGR. Moreover, they propose different mechanisms which would favour a magnetar to have an unsually large recoil velocity, explaining another surprising feature of the source. Finally, they argue that the alternation of periods of burst activity and quiescence reveals an evolution compatible with the effects of a dissipating strong magnetic field. According to the above evidences the SGR involved in the 1979 March event represents the “smoking gun” for such strong magnetic fields.

The magnetar hypotesis has been further explored in a series of papers in the 90ies [116, 117, 118], where the origin of magnetar-like magnetic fields, as well as the radiative mechanism for bursting and for quiescent emission are discussed in detail. Concerning the burst emission, a distinction is made between soft repeated bursts and so-called giant flares such as the 1979 March 5 event. The basic idea is that during its evolution, the magnetic field accumulates stress in the NS solid crust eventually leading to cracking. In this respect the bursts would result from starquakes occurring on the NS surface. A giant flare is instead likely triggered by a sudden large-scale rearrangement of the magnetic field, occurring when it becomes unstable to field lines reconnection (note that the growth time of such instability is compatible with the 0.2 s width of the initial hard spike observed in the March 5 event). The persistent X-ray emission detected at that time by Einstein and ROSAT detectors (in addition to the transient burst signals) has not an easy explanation if not assumed to be powered by the surface magnetic field. In this case, a rough estimate implies a surface magnetic field as strong as ∼ 1015 G. Moreover, in [118]

2.2 Magnetar phenomenology

15

the authors suggest that a second class of sources, the AXPs, have much in common with the SGRs and they are probably magnetars as well, even if in a more quiescent phase. This idea has been later supported by the observation of bursts from some AXPs. Since those years, a number of new observations further supported the magnetar model. Particularly remarkable are the results obtained by Kouveliotou and collaborators [60]: they measured a pulsation period of 7.47 s for the SGR 1806-20, very close to the 8 s period of SGR 0526-66 (the one first discovered in the 1979 March event); then, from the observed spindown rate, they could estimate a dipole magnetic field of ∼ 1015 G, giving confirmation to the magnetar nature of SGRs. Soon after that, a second giant flare was observed from SGR 1900+14. The same team measured a period of 5.16 s, and obtained similar results for the spindown rate and the magnetic field strenght [50].

Nowadays, It is widely accepted that SGRs and AXPs are magnetars, protagonists of an evolutionary scenario dominated by their strong magnetic fields, and thus essentially different from the case of rotation- or accretion-powered NSs. We know 21 magnetars [80]: 9 SGRs (7 confirmed, 2 candidates), and 12 AXPs (9 confirmed, 3 candidates). They have periods in the range ∼ 2 − 12 s and inferred dipole magnetic fields from ∼ 0.3 · 1014 G to ∼ 2.1 · 1015 G. Three SGRs have shown giant flares (a third one has been observed in 2004 [86]), all the SGRs and many AXPs have a burst activity, and glitches have been observed in three AXPs. It is worth mentioning that despite the relatively small number of magnetars observed, there is a significant consensus about the possibility that a relevant fraction (of the order of 10%) of all NSs are magnetars.

2.2

Magnetar phenomenology

Magnetar-like magnetic field strenghts are liable for a number of interesting features which distinguish a magnetar from the other NS manifestations. As we have seen, a strong magnetic field leads to a distinct evolutionary path and it can power, in addition to a peculiar persistent emission, a complex burst activity which includes extraordinary events like giant flares, but it has also relevant effects on the global structure of a NS as well as on its microphysics, transport properties, and so on. In order to further justify the theoretical interest for magnetars and give the feeling of the potential they represent, we discuss here an example of physical process among the most interesting from the point of view of theoretical modelling, which has been only observed in magnetars: magnetar quasi-periodic oscillations (QPOs). Another example is represented by gravitational wave emission from magnetically-induced structure deformations. This topic has been addressed in the PhD research activity described in this Thesis and will be the subject of Chapter 5.

The observation of giant flares from three SGRs not only played a fundamental role in the discovery of magnetars, but also offers very exciting prospects on NS Physics. Analysis of the X-ray data taken in the aftermath of these events, corresponding to the decaying emission tail lasting few minutes, has indeed revealed a number of periodicities in addition to the expected spin frequency modulation, with the following frequencies: 43 Hz for SGR 0526-66 [8]; 28, 53, 84, and 155 Hz for SGR 1900+14 [112]; 18, 26, 30, 92, 150, 625 and 1840 Hz for SGR 1806-20

16

2. Strongly magnetized neutron stars

[53, 128, 113]. These features of the emission has been soon interpreted as NS torsional oscillations1 ; being the oscillating behaviour superimposed to an exponential damping, they are referred to as quasi-periodic oscillations (QPOs). If the interpretation is correct, such observations represent the only direct detection of NS oscillations, and give us a unique chance to carry out asteroseismology studies, i.e. to infer NS properties by analyzing the oscillation modes. With this precise purpose in the last decades many efforts have been devoted to the topic of NS oscillations, even if without the above observations there were no available data to be compared with theoretical models.

The first attempt to explain the observed frequencies regards them as elastic normal modes of the magnetar’s crust excited by the flare, consistently with theoretical expectations [35] (see [114] for a review and references). Many frequencies match indeed the expected range for crustal seismic oscillation modes [47]. Unluckily, the lower frequencies are difficult to reconcile with the seismic mode interpretation, indicating that the real physical scenario is more complicated. As first suggested in [70, 43] (see also [71, 68, 69]), it has been realized that the crust motions are likely coupled to the magnetar’s core magnetic field and that a predictive model should account for the coupled crust-core system, involving both matter and magnetic fields.

In this respect magnetic fields are of crucial importance not only in producing the giant flare, but also in determining the resulting oscillation spectrum. The background magnetic field geometry itself could have important effects on the process2 .

Many recent papers addressed the problem by modelling elastic oscillations of the crust [90, 107, 100] (and references therein) or Alfvèn (magnetic field) oscillations [108, 29, 20], but the first studies of the full crust-core system including the coupling between elastic and Alfvèn modes did not appear until 2010 [57, 126, 40]. The topic deserves further investigation.

1

“Torsional” indicates that matter oscillates along the φ-direction.

This possibility has been considered in this PhD work (in collaboration with the Tuebingen group headed by prof. Kokkotas); the results are not published.

2

Chapter 3

Stationary axisymmetric magnetized neutron stars: a relativistic model In the present Chapter we focus on equilibrium models of strongly magnetized neutron stars. We introduce our relativistic model, discussing the basic assumptions and the formalism. Results from simulations will be presented in the following Chapter (4), where we go deeper into the specific magnetic field configurations we have studied.

The content of the present and the next Chapter is based on [26, 27]. Here and in the following, unless otherwise stated, we adopt the geometrized unit system, in which c = G = 1.

3.1

The role of equilibrium models

In the previous Chapter we have discussed the wide phenomenology associated to magnetars, which makes them astrophysical objects of growing interest. The present-day observations provide strong motivations from the theoretical point of view, as an accurate modelling of strongly magnetized neutron stars, through the comparison with the experimental data, has the potential to shed light on unknown properties of magnetars and neutron stars in general, leading to a deeper understanding of their internal structure and the Physics involved in their dynamical processes and evolution.

Any attempt to explain the observed features of magnetars, such as quasiperiodic oscillations or (potentially) gravitational wave emission, relies on a good description of their equilibrium configurations. With the above motivation, in the last years some efforts have been devoted to the description of magnetic field equilibrium configurations of strongly magnetized neutron stars, with both newtonian and relativistic models.

The present models are affected by our lack of knowledge on the geometry and strenght of internal magnetic fields. A crucial aspect is whether the field has a strong toroidal component hidden inside the star, in addition to the poloidal component 17

18

3. Stationary axisymmetric magnetized neutron stars: a relativistic model

already observed in the exterior1 . Toroidal fields are associated with force free currents (see Sec. 3.2.3) and they can exist where such currents are supported; as a consequence, they are allowed only inside the star or, eventually, within a magnetosphere surrounding it. This is why we only have direct observations of the poloidal component of the field. Nevertheless, the existence of a toroidal component is suggested by a number of indirect evidences: (i) toroidal fields are likely formed in the first phases of a magnetar’s life [34, 116]; (ii) they are required for stability: in the case of non-rotating stars, it has been established that neither a poloidal field alone nor a purely toroidal field are stable, and that they both evolve on dynamical timescales towards a mixed field configuration (see Sec. 4.1); (iii) they help explaining the amount of magnetic energy released in processes like bursts and giant flares [34, 117]. It is very important to have a prediction of the relative strenght of toroidal and poloidal fields inside the star, as it strongly affects most of the magnetar dynamical processes. In particular, we shall discuss the effect on magnetar gravitational wave emission, which is the subject of Chaper 5. Other examples are the already cited burst activity and the thermal evolution ([92] and references therein).

The problem of modelling equilibrium configurations of strongly magnetized neutron stars has been faced with different approaches. A full general relativistic model has been presented in [19], where the coupled Einstein-Maxwell equations are solved assuming space-time circularity. Such requirement, which corresponds to the existence of two hypersurface-orthogonal Killing vectors, makes the implementation of numerical schemes much simpler, but represents an important limitation of the model as it escludes the possibility of a mixed poloidal-toroidal magnetic field (which would break circularity). This model describes only purely poloidal configurations.

A different approach has been used in [58], where the magnetic field is treated as a pertubation of a spherically symmetric unmagnetized star. The relativistic model presented in these works still describes purely poloidal configurations. In [51, 52] this perturbative approach has been generalized to include toroidal fields; mixed equilibrium magnetic field configurations have been found under the quite restrictive assumption that the magnetic field is vanishing outside the star, in contrast with the observations. An analogue assumption is adopted in the newtonian work [48].

A further improvement has been achieved in [28], where the authors consider mixed field configurations as in [52], but with a different treatment of the surface boundary conditions which allows the magnetic field to extend outside the star. Both the models include a toroidal field permeating the whole star and vanishing outside, where vacuum is assumed. In the case of [28] the presence of a purely poloidal field in the exterior implies a surface discontinuity in the toroidal field, which can only be justified with surface currents whose existence is problematic. This point will be further discussed in Sec. 4.1.

These first attempts to describe mixed poloidal-toroidal fields inside the star rely on the assumption that the function which controls the ratio of toroidal and poloidal fields is constant, the simplest possible choice (see Sec. 4.1). Apart from problems with the surface boundary conditions, this simple guess on the distribution of the toroidal component may not be a good approximation of the actual 1

We recall that the toroidal or azimuthal component is the φ-component in spherical coordinates, while the poloidal field is given by the (r, θ)-components.

3.2 Model set up: basic assumptions and equations

19

internal magnetic field geometry, which is unknown. In recent studies Braithwaite and collaborators [13, 15, 14] performed numerical magnetohydrodynamics (MHD) simulations in the framework of newtonian gravity, following the time evolution of magnetic fields in stars. Starting from a number of different random configurations, they found that a particular mixed field geometry, called twisted-torus, is a quite generic outcome of the evolution, and it appears to be stable on dynamical timescales (see Chapter 4). This result holds for ordinary stars as well as for neutron stars.

Our relativistic model, which is the subject of the present Chapter and Chapter 4, follows the approach of the cited perturbative works. We build configurations in which the distribution of poloidal and toroidal fields reproduces a twisted-torus geometry, as suggested by the above results in numerical MHD simulations. At the same time, this choice allows us to overcome the problems associated with the surface boundary conditions (see Sec. 4.1).

In the next Section (3.2) we introduce our equilibrium model on general grounds, we discuss the assumptions adopted and we present the basic equations and formalism. The following Section (3.3) is devoted to the description of the equations of state employed. In the present Chapter we still consider a generic distribution of poloidal and toroidal fields, while we specialize to the case of twisted-torus geometries in the next Chapter (4), where we present our results.

3.2

Model set up: basic assumptions and equations

We consider a non-rotating strongly magnetized neutron star surrounded by vacuum with the assumption that the system is stationary and axisymmetric.

The magnetized fluid composing the star is described within the framework of ideal MHD, in which the effects of finite electrical conductivity are neglected. Rigorously speaking, this approximation is only justified while the crust is still completely liquid and while the core matter has not yet performed the phase transition to the superfluid state, which is expected to occur at most a few hours after birth (see e.g. Section 5.1 in [1] and references therein). The onset of superfluidity and/or crystallization limits the period during which magnetostatic equilibrium can be established. Both the melting temperature and the critical temperature of transition to the superfluid state, are between 109 and 1010 K, and a typical neutron star quickly cools down below these temperatures in a few hours. However, since the characteristic Alfvén time is of the order of τA ≈ 0.01 − 10 s, depending on the background field strength, there is ample time for the magnetized fluid to reach a stable state while the state of matter is still liquid (as shown, for example, in [15]). We remark that even in presence of a stable stratification of the chemical composition, a magnetic field as strong as B ∼ 1015 G is still allowed to evolve throughout the star on a dynamical timescale [120]. After the crust is formed, the magnetic field is frozen in, and it only evolves on a much longer timescale (of the order of 103 − 105 years) due to dissipative effects like ohmic decay, ambipolar diffusion and Hall drift [45, 130, 91]. Therefore, it is reasonable to expect that the MHD equilibrium configuration set within the first day after formation, will fix the magnetic field geometry for a long time.

In what follows we first summarize the basic equations of ideal MHD in the

20

3. Stationary axisymmetric magnetized neutron stars: a relativistic model

framework of General Relativity and then we introduce our perturbative approach.

Next, we obtain the form of the electromagnetic potential in the general case (not yet specialized to twisted-torus configurations), and we derive the relativistic GradShafranov equation, which can be integrated to give the magnetic field configuration.

We use spherical coordinates, xµ = (t, xa , φ), where xa = (r, θ). A stationary axisymmetric space-time admits two killing vectors, η = ∂/∂t and ξ = ∂/∂φ, and with our coordinate choice all the quantities (including the components of the metric tensor gµν ) are independent on t and φ.

3.2.1

Equations of ideal MHD in General Relativity

The electromagnetic field is governed by the Maxwell’s equations F µν;ν = 4πJ µ ,

(3.1)

F[µν; α] = 0 ,

(3.2)

where F µν is the electromagnetic tensor, J µ is the four-current and the square brackets indicate the antisymmetric combination of indexes µ, ν, α. The Equations (3.2) are automatically satisfied by an antisymmetric electromagnetic tensor of the form Fµν = ∂ν Aµ − ∂µ Aν , (3.3)

where Aµ is the electromagnetic potential.

According to a comoving observer with four-velocity uµ , the electric and magnetic fields are defined as Eµ ≡ Fµν uν , 1 Bα ≡ ǫαβγδ uβ F γδ , 2

(3.4) (3.5)

√ where ǫαβγδ = −g [αβγδ] is the Levi-Civita antisymmetric tensor ( [t r θ φ] = 1 ), and g is the metric determinant. From (3.4) and (3.5) we have Eµ uµ = 0 ,

Bµ uµ = 0 .

(3.6)

In ideal MHD the fluid resistivity is assumed to be negligible: matter behaves as a perfect conductor and there is no electric charge density in the reference frame comoving with the fluid. This gives the additional condition Eµ = Fµν uν = 0 .

(3.7)

The complete set of MHD equations include also the continuity equation and the equations of motion T µν;ν = 0 . The continuity equation expresses the baryon conservation in a perfect fluid in thermodynamical equilibrium: (nuµ );µ = 0 ,

(3.8)

where n is the baryon number density. The stress-energy tensor of a perfect fluid with an electromagnetic field in ideal MHD is µν T µν = Tfµν luid + Tem ,

(3.9)

3.2 Model set up: basic assumptions and equations

21

where µ ν µν Tfµν , luid = (ρ + P )u u + P g

1 1 µν µν 2 µ ν µ ν Tem = B −B B , u u + g 4π 2

(3.10) (3.11)

with ρ and P mass-energy density and fluid pressure respectively, and B 2 = B µ Bµ .

Euler’s equations, which determine the system dynamics, are found by projecting the equation T µν;ν = 0 orthogonally to uµ : (ρ + P )aµ + P,µ + uµ uν P,ν − fµ = 0 ,

(3.12)

where fµ ≡ Fµν J ν is the Lorentz force and aµ = uν uµ;ν is the four-acceleration.

Here we summarize the full set of ideal, general relativistic MHD equations.

GENERAL RELATIVISTIC MHD

3.2.2

(i)

Continuity equation

(nuµ );µ = 0

(ii)

Maxwell’s equations

F µν;ν = 4πJ µ

(iii)

Ideal MHD condition

Eµ = Fµν uν = 0

(iv)

Euler’s equations

(ρ + P )aµ + P,µ + uµ uν P,ν − fµ = 0

Perturbative approach and electromagnetic potential

We assume that the magnetic field acts as a stationary axisymmetric perturbation of a static and spherically symmetric background star described by the well known Tolman-Oppenheimer-Volkoff (TOV) solution of the Einstein equations (see Appendix A).

The background metric is ds2 = −eν(r) dt2 + eλ(r) dr 2 + r 2 (dθ 2 + sin2 θdφ2 ) ,

(3.13)

where ν(r), λ(r) are given by the unperturbed Einstein equations (TOV) for assigned equations of state (EOS). The unperturbed 4-velocity writes uµ = (e−ν/2 , 0, 0, 0) .

(3.14)

We model the background star with two different “realistic” EOS proposed in the literature called APR2 [5] and GNH3 [44], which differ significantly for compactness.

They are completed by standard EOS for the stellar crust (see Sec. 1.1 and [11]).

The last Section of this Chapter (Sec. 3.3) provides an essential description of these EOS. It is worth stressing that the EOS we employ are barotropic, i.e. the pressure is only a function of the mass-energy density, P = P (ρ) .

In addition, we fix the mass of the neutron star as M = 1.4 M⊙ ; for the two EOS (APR2, GNH3) this results in a star radius of R = 11.58 km and 14.19 km respectively.

22

3. Stationary axisymmetric magnetized neutron stars: a relativistic model

If the magnetic field represents a perturbation of order O(B), it can be shown that (Fµν , Aµ , J µ ) are of order O(B), while the perturbations (δuµ , δρ, δP , δn, δgµν , δGµν , δTµν ) are of order O(B 2 ) (see for instance [28]); furthermore, (B t , At , J t , Ftν )= O(B 3 ) and (ft , fφ ) = O(B 4 ). Therefore, at first order the magnetic field is only coupled to the unperturbed background metric (3.13), whereas the deformation of the stellar structure induced by the magnetic field appears at order O(B 2 ). Hence, the approach allows us to split the problem to obtain the equilibrium configuration of a magnetized star into two parts: (i) solving the electromagnetic field equations in the given unperturbed spacetime geometry and for the unperturbed matter distribution; (ii) solving the Einstein equations which determine the variations of spacetime geometry and matter distribution under the magnetic field configuration found in the previous step. Here we only consider the first step; in Chapters 4 and 5 we will solve some components of the perturbed Einstein equations, in order to evaluate the total energy of the system and to compute the quadrupolar deformation of the star induced by the magnetic field.

The potential Aµ , at O(B), has the form Aµ (r, θ) = (0, Ar , Aθ , Aφ ). With an appropriate gauge choice we can impose Aθ = 0 and write the potential as Aµ = (0, e

λ−ν 2

Σ, 0, ψ) ,

(3.15)

where the components of Aµ are expressed in terms of two unknown functions, Σ(r, θ) and ψ(r, θ). A further simplification of Aµ is possible by exploiting the fact that fφ = −ψ,r J r − ψ,θ J θ = O(B 4 ). Using Maxwell’s equations and neglecting higher order terms, we find ψ˜,θ ψ,r = ψ˜,r ψ,θ , (3.16) ˜ where ψ˜ ≡ sin θ Σ,θ . This result implies ψ˜ = ψ(ψ) and allows us to write sin θ Σ,θ = β(ψ) ,

(3.17)

where β(ψ) is a function of ψ of order O(B).

We can also write β(ψ) = ζ(ψ)ψ, where ζ(ψ) is of order O(1). Toroidal and poloidal components of the magnetic field come respectively from derivatives of the r− and φ−component of the electromagnetic potential, thus the function ζ represents the ratio between the two; ζ = 0 gives the special case of purely poloidal field (no toroidal field). Different choices of the function β (or ζ) lead to qualitatively different field configurations; in particular, a proper choice lead to a twisted-toruslike configuration (Sec. 4.1).

3.2.3

The relativistic Grad-Shafranov equation

The Grad-Shafranov (GS) equation, which allows to determine the magnetic field configuration, can be derived by equating Jφ computed from the φ-component of Maxwell’s equations, Jφ = −

1 ν,r − λ,r e−λ ψ,r − [ψ,θθ − cot θψ,θ ] , ψ,rr + 4π 2 4πr 2

(3.18)

3.2 Model set up: basic assumptions and equations

23

and Jφ obtained from the a-components of Euler’s equations (3.12), as follows.

Euler’s equations give (a = r, θ) fa = (ρ + P )aa + P,a + ua uν P,ν

ν ν t − e 2 δu + P,a + O(B 4 ) . = (ρ + P ) 2 ,a

(3.19)

For barotropic equations of state P = P (ρ), the first principle of thermodynamics allows to write

ρ+P P,a = (ρ + P ) ln , (3.20) n ,a then (3.19) yields fa = (ρ + P )χ,a ,

(3.21)

where χ = χ(r, θ). On the other hand, the a-components of the Lorentz force fµ = Fµν J ν can be written as (see [28]) fa = with

ψ,a ˜ Jφ 2 r sin2 θ

(3.22)

,

e−ν dβ β .

J˜φ = Jφ − 4π dψ

(3.23)

Therefore, χ,a =

ψ,a J˜φ . (ρ + P )r 2 sin2 θ

(3.24)

From χ,rθ − χ,θr = 0 it follows that ψ,r

J˜φ (ρ + P )r 2 sin2 θ

which implies

!

,θ

J˜φ (ρ + P )r 2 sin2 θ

− ψ,θ

J˜φ (ρ + P )r 2 sin2 θ

!

!

=0 , ,r

= F (ψ) .

(3.25)

In addition to β(ψ), we have a second arbitrary function F (ψ), which controls the component J˜φ of the current Jφ . The conclusion that the total azimuthal current writes as the sum of two terms, each containing an arbitrary function of ψ, Jφ =

dβ(ψ) e−ν β(ψ) + (ρ + P )r 2 sin2 θF (ψ) , 4π dψ

(3.26)

is the relativistic version of a general result for stationary axisymmetric magnetized stars with barotropic EOS [22, 23, 95]. The total current Jµ is itself given by the sum of two terms, Jµpol and JµF F , where Jµpol = (0, 0, 0, J˜φ ) is purely azimuthal and −ν

β(ψ) generates a poloidal field, while JµF F = − e 4πψ Bµ is a force-free current (proportional to the magnetic fied) generating a mixed poloidal-toroidal field. Therefore, toroidal fields are generated together with a poloidal component, and they only exist where currents are present, i.e. inside the star.

24

3. Stationary axisymmetric magnetized neutron stars: a relativistic model

We assume the following simple form for F (ψ): F (ψ) = c0 + c1 ψ ,

(3.27)

with c0 , c1 constants of order O(B), O(1) respectively. Hence, Jφ turns out to be Jφ =

e−ν dβ β + (ρ + P )r 2 sin2 θ[c0 + c1 ψ] .

4π dψ

(3.28)

The usual choice adopted in the literature is even simpler: F = constant .

In all the cited relativistic works ([58, 52, 28]) the magnetic field is expanded in multipoles and then the dipolar (l = 1) contribution is the only one considered, neglecting couplings with higher (l > 1) multipoles2 . In our model we perform a multipolar expansion as well (see below), but we consider the full coupled multipole problem. The choice F = constant adopted in the literature is such to avoid these couplings; our choice is more general and allows for a consistent treatment of the system when other multipolar components are included in addition to the dipole.

From Eqns. (3.18), (3.28) the relativistic GS equation at first order in B is finally obtained: 1 ν,r − λ,r e−λ ψ,r − [ψ,θθ − cot θψ,θ ] ψ,rr + 4π 2 4πr 2 e−ν dβ β = (ρ + P )r 2 sin2 θ [c0 + c1 ψ] . − 4π dψ −

(3.29)

If we now define ψ(r, θ) ≡ sin θ a(r, θ),θ and expand the function a(r, θ) in Legendre polynomials a(r, θ) =

∞ X

al (r)Pl (cos θ) ,

(3.30)

l=1

the GS equation rewrites as

∞ ′ ′ l(l + 1) sin θ X −λ ′′ −λ ν − λ ′ Pl,θ e al + e al − al − 4π l=1 2 r2

−

e−ν dβ(ψ) β(ψ) 4π dψ

2

"

2

ψ=

P∞

l=1

= (ρ + P )r sin θ c0 + c1

al Pl,θ sin θ

∞ X l=1

al Pl,θ sin θ

#

.

(3.31)

Here and in the following we denote with primes the differentiation with respect to r.

Finally, projecting Eq. (3.31) onto the different harmonic components, we obtain a system of coupled ordinary differential equations for the functions al (r). The projection is performed using the property 2l′ + 1 2l′ (l′ + 1)

Z

0

π

Pl,θ Pl′ ,θ sin θ dθ = δll′ .

(3.32)

If we consider the contribution of n different harmonics, we need to solve a system of n coupled radial equations, obtained from Eq. (3.31), for the n functions al (r).

2

In [28] the case of a purely quadrupolar (l = 2) field is also considered.

3.2 Model set up: basic assumptions and equations

3.2.4

25

Boundary conditions

Here we discuss the boundary conditions we impose to integrate the system of radial equations obtained from the harmonic projection of the GS equation.

The functions al (r) must have a regular behaviour at the origin; by taking the limit r → 0 of the GS equation one can find al (r → 0) = αl r l+1 ,

(3.33)

where αl are arbitrary constants to be fixed. Outside the star, where there is vacuum and the field is purely poloidal, Equations (3.31) decouple, and can be solved analytically for each value of l. The solution can be expressed in terms of the generalized hypergeometric functions (F ([l1 , l2 ], [l3 ], z)), also known as Barnes’ extended hypergeometric functions, as follows: al = A1 r −l F ([l, l + 2], [2 + 2l], z) +A2 r l+1 F ([1 − l, −1 − l], [−2l], z) ,

(3.34)

where z = 2M/r and A1 and A2 are arbitrary integration constants, which must be fixed according to the values of the magnetic multipole moments. Regularity of the external solution at r = ∞ requires A2 = 0 for all multipoles. For example, for l = 1, 2, 3 we have a1 ∝ r

2

"

z2 ln(1 − z) + z + 2

#

a2 ∝ r 3 (4 − 3z) ln(1 − z) + 4z − z 2 −

z3 6

a3 ∝ r 4 (15 − 20z + 6z 2 ) ln(1 − z) + 15z −

z4 25z 2 + z3 + 2 12

.

(3.35)

At the stellar surface we require the field to be continuous. This condition is satisfied if al and a′l are continuous. For practical purposes, the boundary conditions at r = R can be written as l (3.36) a′l = − fl al R where fl is a relativistic factor which only depends on the star compactness 2M/R (in the Newtonian limit all fl = 1), and can be numerically evaluated with the help of any algebraic manipulator. For our APR2 model (2M/R = 0.357), the values of fl for the first five multipoles are 1.338, 1.339, 1.315, 1.301, and 1.292 respectively.

In general, there are n + 2 arbitrary constants to be fixed: the n constants αl , associated to the condition (3.33), plus c0 and c1 . Thus, we need to impose n + 2 constraints, of which n + 1 are determined by the boundary conditions: n conditions are provided by Eq. (3.36), i.e. by imposing continuity in r = R of the ratios a′l /al ; the overall normalization of the field gives another condition, which is fixed by imposing that the value of the l = 1 contribution at the pole is Bpole = 1015 G (this corresponds to set a1 (R) = 1.93 · 10−3 km). The reason for this choice is that the surface value of the magnetic field is usually inferred from observations by applying the spin-down formula, and assuming a purely dipolar external field; for magnetars, the value of Bpole estimated in this way is ∼ 1014 − 1015 G.

26

3. Stationary axisymmetric magnetized neutron stars: a relativistic model

The remaining condition is imposed as follows. The last freedom concerns the relative strenght of the higher order multipoles (l > 1) with respect to the dipolar component of the field (l = 1). We shall fix the constraint in two different ways: (i) by imposing that the external contribution of all the l > 1 harmonics, i.e.

P 2 l>1 al (R) , is minimum; (ii) by looking at the contribution of higher multipoles which minimizes energy in a normalization independent way (thus the energetically favoured one).

3.3

Equations of state

Here we brefly discuss about the two different EOS we employ to model the NS core, called APR2 [5] and GNH3 [44]. They are barotropic (i.e. of the form P = P (ρ)) and belong to the so-called “realistic” EOS, which are tabulated from Nuclear Physics calculations (as opposite to the analytic ones such as the polytropic EOS, very common in the literature).

The Akmal-Pandharipande-Ravenhall EOS (APR2) relies on the assumption that matter consists of protons, neutrons, electrons and muons in weak equilibrium.

It is based on the NMBT approach (Sec. 1.1) and employs the Argonne v18 twonucleon potential [129], modified in order to include relativistic corrections (necessary to use the nucleon-nucleon potential in a locally inertial frame associated to the star), and the Urbana IX three-nucleon potential [96] (also modified consistently).

The ground state energy is computed using variational techniques [2, 5]. The GNH3 EOS proposed by Glendenning belongs to the RMFT approach (Sec. 1.1). Its most relevant feature is that it accounts for the appearence of hyperons over a density threshold ρH ∼ 2ρ0 , where ρ0 is the nuclear density. The hyperons start replacing the highest energy nucleons when the nucleon Fermi level overcomes the hyperon rest mass, making such replacement convenient. This causes a significant softening of the EOS as compared to the pure nucleon case, because the hyperons have much lower kinetic energies than the replaced nucleons. For a star of 1.4 M⊙ (the value we assume), the resulting compactness is much lower in the GNH3 case than in the APR2 case (the radii are 14.19 km and 11.58 km respectively). With these two EOS we can span a wide range of compactness, which is useful to account for the EOS dependence of our results (see Chapters 4 and 5).

Chapter 4

Twisted-torus magnetic field configurations In this Chapter we specialize to twisted-torus magnetic field geometries. We discuss the details of our relativistic model and the equilibrium configurations found in numerical simulations. Our studies on gravitational wave emission from strongly magnetized neutron stars, to which the next Chapter (5) is devoted, are based on the results presented here.

Unless otherwise stated, the results refer to the APR2 EOS (see Sec. 3.3); in Sec. 4.6 we will also consider the GNH3 EOS (and similarly, both the EOS will be employed in Chapter 5).

4.1

Internal magnetic field geometry

The modelling of equilibrium magnetic field configurations in stars has always been accompanied by the study of stability of such fields. In principle, it is reasonable to expect the existence of some equilibrium: the two magnetic field or Lorentz force degrees of freedom (three degrees of freedom from the spatial components ~ or f~L diminished by one because of the zero-divergence constraint) of the vector B could be balanced by pressure and temperature gradients. Once a given equilibrium configuration is found by combining Maxwell’s and Euler equations, the following step should be to assess its stability to an arbitrary perturbation.

The stability of simple magnetic field configurations in non-rotating axisymmetric stars has been studied using analytic tools: in [115] necessary and sufficient conditions for the stability of a purely toroidal field are established, with the result that since such conditions appear to be impossible to satisfy in the whole star a purely toroidal field will evolve towards a mixed toroidal-poloidal configuration on Alfvèn timescales; similar adiabatic stability analyses have been performed for a purely poloidal field [77, 78, 131], and again it results to be unstable on Alfvèn timescales. From these studies it has been concluded that a stable magnetic field should have a mixed nature, at least in non-rotating stars1 .

The final aim of such stablity studies is to establish which magnetic field configurations are actually realized in ordinary stars or in neutron stars during the different 1

It has been argued that in some cases rotation should be able to stabilize a purely poloidal field, but this point is still controversial [41, 17].

27

28

4. Twisted-torus magnetic field configurations

Figure 4.1. Meridional view of a star endowed with a twisted-torus magnetic field (from [15]). A poloidal field extends throughout the entire star and in the exterior; a toroidal component is also present within the shaded region, defined by the magnetic field lines closed inside the star.

phases of their life. Recent progress in numerical MHD simulations, besides enabling numerical studies which have confirmed the instability of purely poloidal and purely toroidal fields in the non-rotating case [17, 16], has allowed to start facing the problem with a different approach, which consists in taking different arbitrary magnetic field configurations (not necessarily equilibrium configurations) and following their time evolution in order to see if the system tends towards any kind of stable equilibrium.

In [13, 14, 15] Braithwaite and collaborators performed 3D MHD simulations of magnetic fields in non-rotating stars (both ordinary and neutron stars) in the framework of newtonian gravity, evolving in time a number of different random initial configurations. They found as a generic outcome of the evolution a particular configuration which appears to be in stable equilibrium on dynamical timescales.

Moreover, the results suggest that this could be the only possible stable equilibrium for non-rotating stars. This so-called twisted-torus configuration is axisymmetric and consists of a poloidal field extending throughout the entire star and in the exterior, and a toroidal field confined in a torus-shaped region defined by the poloidal field lines which are closed inside the star (an illustrative example is given in Fig. 4.1).

On physical grounds, such configuration is the reasonable result of the simultaneous (i) damping of the toroidal field along the poloidal field lines reaching the vacuum exterior (where the toroidal component vanishes) and (ii) toroidal field persistence due to magnetic helicity conservation in the interior, where electric conductivity is very high. The combined effect causes the migration and confinement of the toroidal component in the region where the field lines are closed inside the star.

These hints about the geometry of internal magnetic fields provide strong motivation to study equilibrium models of magnetized neutron stars with a twistedtorus-like configuration. Our relativistic model joins recent newtonian studies in

4.1 Internal magnetic field geometry

29

which this kind of configuration is considered as well [123, 56, 133, 134, 66].

In our equilibrium model the distribution of poloidal and toroidal fields is dictated by the choice adopted for the arbitrary funcion β(ψ) or, equivalently, ζ(ψ) (see Sec. 3.2.2). The simplest choice is to take ζ = constant. This case has been studied in [52, 28, 48]. If the space outside the star is assumed to be vacuum, the toroidal field, and consequently ζ, must vanish for r > R, where R is the neutron star radius; therefore, the choice ζ = constant yields an inconsistency, unless one assumes that surface currents cancel the toroidal field outside the star resulting in a surface discontonuity of the toroidal field (as in [28]), or imposes that the constant ζ assumes very particular values (eigenvalue problem) such that all the components of the field vanish outside the star (as in [52, 48]).

In our work, in order to reproduce a twisted-torus-like configuration, we start by choosing the following form for the function ζ(ψ) ¯ − 1 · Θ(|ψ/ψ| ¯ − 1) . ζ(ψ) = ζ0 |ψ/ψ|

(4.1)

ζ0 is a constant of order O(1); ψ¯ is a constant of order O(B): it is the value of ψ at the boundary of the toroidal region where the toroidal field is confined (this boundary is ¯ − 1) is the usual Heaviside function. tangent to the stellar surface); finally, Θ(|ψ/ψ| With this choice, the function ζ vanishes at the stellar surface, where r = R, and the magnetic field λ

λ

Bµ =

0 ,

e− 2 e− 2 ψ , − ψ,r , ,θ r 2 sin θ r 2 sin θ

¯ −1 e− 2 ζ0 ψ |ψ/ψ| ν

−

r 2 sin2 θ

¯ − 1) Θ(|ψ/ψ|

!

(4.2)

has no discontinuities: the magnetic field for r > R becomes purely poloidal, consistently with the assumption of vacuum outside the star. In Sec. 4.6.2 we shall extend the model to different and more general choices for the ζ-function.

In the following we discuss the details of the twisted-torus configurations we have found, starting from the simple case of a purely dipolar field and proceeding with the inclusion of higher multipoles towards the general case (Sections 4.2-4.4).

In Sec. 4.5 we describe an argument of minimal energy which allows us to find a favoured value for the constant ζ0 and then fix the exact ratio of magnetic energy in toroidal and poloidal fields. In this analysis we set the relative strenght of the higher multipoles with respect to the dipole by taking their minimal possible contribution (see Sec. 3.2.4). A different way to set such relative strenght, which is based on the same minimal energy argument used for ζ0 , is then employed in Sec. 4.6. All the results discussed are then summarized in Sec. 4.7.

It is worth stressing that the emergence of a twisted-torus configuration in stars needs to be supported by further magnetic field evolution studies, possibly in a relativistic context. Sec. 4.8 presents a brief introduction to our work in progress on this subject.

30

4.2

4. Twisted-torus magnetic field configurations

The purely dipolar field case

We begin discussing the simplest case of a purely dipolar configuration, in which all couplings with higher order multipoles are neglected in Eq. (3.31) (al>1 = 0).

In this case, for any assigned value of ζ0 there exists an infinite set of solutions, each corresponding to a value of c1 ; these solutions describe qualitatively similar magnetic field configurations.

However, when higher order harmonics are taken into account, as we will see in the following Sections, the picture changes. For instance, when ζ0 = 0 and the l = 1, 2 harmonic components are included, the equations for a1 and a2 decouple: the equation for a1 is the same as in the purely dipolar case, but a solution for a2 satisfying the appropriate boundary conditions exists only for a unique value of c1 (see Equations (4.7)). This is true also for ζ0 6= 0 and for all the higher multipoles; therefore, in the general case c1 is not a truly free parameter, and the fact that in the purely dipolar case it looks as such, is an artifact of the truncation of the l > 1 multipoles. In order to provide a mathematically simple example, which is useful to understand the structure of the twisted-torus configurations, in this Section we consider the simplest case c1 = 0. The same choice is widely adopted in the literature, where the dipolar component is usually the only one considered.

By projecting Eq. (3.31) onto the l = 1 harmonic, and neglecting all contributions from l > 1 terms we find e−ν 2 ν ′ − λ′ ′ 1 a1 − 2 a1 − e−λ a′′1 + e−λ 4π 2 r 4π

× ζ02

"

Z

0

π

!

−a sin2 θ

1

(3/4) Θ

−1

ψ¯

#

−a sin2 θ

1 3 4 2 ¯ − a1 + 3a1

− 2a1 sin θ/ψ sin3 θ dθ

ψ¯ Z π

= (3/4)

0

c0 (ρ + P )r 2 sin3 θ dθ = c0 (ρ + P )r 2 .

(4.3)

The tetrad components of the magnetic field (i.e. the components measured by a locally inertial observer) are: B(r) =

ψ,θ 2 r sin θ

,

λ

B(θ) B(φ)

e− 2 = − ψ,r , r sin θ

ν ¯ −1 e− 2 ζ0 ψ |ψ/ψ| ¯ − 1) , · Θ(|ψ/ψ| = − r sin θ

(4.4)

where ψ = −a1 sin2 θ.

The profiles of the tetrad components of the field inside the star, are plotted in Fig. 4.2 for increasing values of ζ0 ; B(r) is evaluated in (θ = 0) and B(θ) , B(φ) in (θ = π/2). In Fig. 4.3 we show the projection of the field lines in the meridional plane, for ζ0 = 0.40 km−1 . Figs. 4.2 and 4.3 show that the toroidal field B(φ) is confined within a torus-shaped region; its amplitude ranges from zero, at the border of the region, to a maximum, close to its center. At the stellar surface and in the exterior B φ vanishes, and there is no discontinuity in the toroidal field. The panels of Fig. 4.2 show the field profiles for different values of ζ0 : larger values of ζ0

4.2 The purely dipolar field case

31

-1

( ζ0 = 0 km ) B(r)

2

B(µ) / (10

15

G)

4

B(φ)

0 -2

B(θ) -4 0

0.2

0.4

0.6

0.8

1

r/R -1

( ζ0 = 0.40 km ) B(r)

2

B(µ) / (10

15

G)

4

B(φ)

0 -2

B(θ) -4 0

0.2

0.4

0.6

0.8

1

r/R

-1

B(µ) / (10

( ζ0 = 0.80 km )

B(r)

2

15

G)

4

B(φ)

0 -2

B(θ)

-4 0

0.2

0.4

0.6

0.8

1

r/R

Figure 4.2. The profiles of the tetrad components of the magnetic field (B(r) (θ = 0), B(θ) (θ = π/2), B(φ) (θ = π/2)) are shown for the purely dipolar case with ζ0 = 0 km−1 , ζ0 = 0.40 km−1 and ζ0 = 0.80 km−1 .

32

4. Twisted-torus magnetic field configurations

Figure 4.3. The projection of the field lines in the meridional plane is shown for the purely dipolar case with ζ0 = 0.40 km−1 . The toroidal field is confined within the marked region.

correspond to a toroidal field with increasing amplitude. Interestingly, the toroidal field is confined in an increasingly narrow region close to the stellar surface, while the amplitude of the poloidal components (B(r) , B(θ) ) decreases.

The above feature of our twisted-torus configurations is remarkable, as it implies that inside the star we cannot have a magnetic field geometry where the toroidal component dominates with respect to the poloidal one: if |B(φ) | becomes larger with respect to |B(r) | and |B(θ) |, the domain where it is non vanishing shrinks. The same holds in the general case, when other multipoles are included in addition to the dipole.

4.3

The case with l = 1 and l = 2 multipoles

We now proceed with our investigation considering the l = 1 and l = 2 contributions, and setting al>2 = 0. The projection of the GS equation (3.31) onto the harmonics l = 1 and l = 2 gives the following coupled equations: 1 2 ν ′ − λ′ ′ a1 − 2 a1 e−λ a′′1 + e−λ 4π 2 r

Z π −ν

−a1 − 3a2 cos θ e sin2 θ

− 1 − (3/4) Θ

4π 0 ψ¯

×

ζ02

"

−a1 − 3a2 cos θ

2

− a1 − 3a2 cos θ + 3(a1 + 3a2 cos θ)

sin θ

¯

ψ

4

+2 sin θ −

a31

−

9a21 a2 cos θ

4 = (ρ + P )r 2 c0 − c1 a1 5

−

,

27a1 a22 cos2 θ

−

27a32

#

cos θ /ψ¯2 sin3 θ dθ 3

(4.5)

4.3 The case with l = 1 and l = 2 multipoles

33

6 ν ′ − λ′ ′ 1 a2 − 2 a2 e−λ a′′2 + e−λ 4π 2 r

Z π −ν

−a1 − 3a2 cos θ e (5/12) Θ

+ sin2 θ

− 1 4π 0 ψ¯

"

−a1 − 3a2 cos θ 2

sin θ

− a1 − 3a2 cos θ + 3(a1 + 3a2 cos θ)

¯

×ζ02

ψ

4

+2 sin θ −

a31

−

9a21 a2 cos θ

− 27a1 a22 cos2 θ

−

27a32

4 ×(−3 cos θ sin3 θ) dθ = − (ρ + P )r 2 c1 a2 .

7

cos θ /ψ¯2 3

#

(4.6)

We integrate this system by imposing the boundary conditions discussed in Sec. 3.2.4, i.e. (i) a regular behaviour at the origin (Eq. (3.33)), (ii) continuity at the stellar surface of a1 , a′1 , a2 , a′2 with the analytical external solutions given by Equations (3.35), and (iii) the requirement that the surface (or exterior) contribution of higher multipoles is minimum (in this case this corresponds to minimizing |a2 (R)|).

Let us first consider the simple case ζ0 = 0. Eqns. (4.5), (4.6) decouple, and become ν ′ − λ′ ′ a1 − 2 ν ′ − λ′ ′ e−λ a′′2 + e−λ a2 − 2 e−λ a′′1 + e−λ

2 4 a1 = 4π(ρ + p)r 2 c0 − c1 a1 , 2 r 5 6 16π a2 = − (ρ + p)r 2 c1 a2 . r2 7

(4.7)

There are four constants to fix (α1 , α2 , c0 , c1 ) and three conditions: a1 (R) = 1.93 · 10−3 km (normalization) and the ratios a′1 (R)/a1 (R) and a′2 (R)/a2 (R) from the matching with the exterior solutions; thus, we need an additional requirement. We remark that we cannot impose c1 = 0 as in the purely dipolar case, because the ratio a′2 (R)/a2 (R) depends only on c1 , and the matching with the exterior solution is possible only for a particular value of c1 , i.e. c1 = 0.84 km−2 .

If we impose that |a2 (R)| is minimum, we find that this condition yields the trivial solution a2 (r) ≡ 0 (with non-vanishing a1 ). Indeed, from Eqns. (4.7) it is straightforward to see that a2 (r) ≡ 0 is a solution of the system. When ζ0 6= 0, equations (4.5), (4.6) are coupled, but they still allow the trivial solution a2 (r) ≡ 0, which minimizes |a2 (R)| with non-vanishing a1 . The existence of this solution is a remarkable property of this system, and it is due to the fact that the integral in θ on the left-hand side of Eq. (4.6) vanishes for a2 = 0 (the integrand becomes odd for parity transformations θ → π − θ). Hence, if we look for a solution which minimizes the contributions from the l > 1 components at the stellar surface, we have to choose the trivial solution a2 (r) ≡ 0.

If, instead, we do not require that a2 (R) is minimum, and we assign a finite value to the ratio a2 (R)/a1 (R), we find a non-trivial field configuration which is non symmetric with respect to the equatorial plane. This feature is shown in Fig. 4.4, where the projection of the field lines in the meridional plane is plotted for ζ0 = 0 and a2 (R)/a1 (R) equal to 1, 1/2 and 1/4 respectively.

34

4. Twisted-torus magnetic field configurations

Figure 4.4. The projections of the field lines in the meridional plane are shown for ζ0 = 0 km−1 and a2 (R)/a1 (R) = 1, 1/2, 1/4 respectively, and for al>2 = 0. The dashed line corresponds to ψ = 0.

As discussed in the following, the existence of the trivial solution a2 (r) ≡ 0 comes from a general property of the system holding in presence of more than one multipolar component. The present case with the l = 1 and l = 2 multipoles is an helpful example to introduce such property.

4.4

The general case

When all harmonics are taken into account, there exist two distinct classes of solutions: those symmetric (with respect to the equatorial plane), with vanishing even order components (a2l ≡ 0), and the antisymmetric solutions, with vanishing odd order components (a2l+1 ≡ 0). Both solutions satisfy the GS equation (3.31). Let us consider the symmetric class: if a2l = 0 for a given radial coordinate, the integrals arising when Equation (3.31) is projected onto the even harmonics, which couple odd and even terms, vanish since the integrands change sign under parity transformations, leading to a′′2l , a′2l = 0 for that radial coordinate, and then for all r, i.e. a2l ≡ 0. Therefore, the symmetric solutions can be found by setting a2l ≡ 0, projecting Eq. (3.31) onto the odd harmonics and solving the resulting equations for a2l+1 . Similarly, the integrals in Equation (3.31) projected onto the odd harmonics vanish when a2l+1 = 0; thus, we can consistently set a2l+1 ≡ 0, and find the antisymmetric solutions using the same procedure.

In Section 4.3, we set the value of a1 at the stellar surface to be 1.93 · 10−3 km and we minimized the l = 2 contribution. It is clear that, since the l = 1 and l = 2 multipoles belong to different families, any attempt to minimize the relative weight of one with respect to the other leads to the trivial solution. The properties of Equation (3.31) discussed above, tell us that if a1 (R) 6= 0 we cannot consistently set to zero the remaining odd order components a2l+1 . However, we have the freedom of setting to zero all even terms a2l . Therefore, since we have chosen to minimize the contributions of the l > 1 harmonics outside the star, we shall focus on the symmetric family of solutions (a2l ≡ 0); we will briefly discuss an example belonging to the antisymmetric family in Section 4.4.4.

4.4 The general case

35

3

-1

( ζ0 = 0 km )

B(µ) / (10

15

G)

2 1

B(r)

B(φ)

0 -1 -2

B(θ)

-3 -4 0

0.2

0.4

0.6

0.8

1

r/R 3

-1

( ζ0 = 0.40 km )

B(µ) / (10

15

G)

2

B(r)

1 0 -1

B(φ)

B(θ)

-2 -3 -4 0

0.2

0.4

0.6

0.8

1

r/R 3

-1

( ζ0 = 0.80 km )

B(µ) / (10

15

G)

2 1 0

B(r)

-1 -2

B(θ)

B(φ)

-3 -4 0

0.2

0.4

0.6

0.8

1

r/R

Figure 4.5. The profiles of the tetrad components of the magnetic field (B(r) (θ = 0), B(θ) (θ = π/2), B(φ) (θ = π/2)) are shown for ζ0 = 0 km−1 , ζ0 = 0.40 km−1 and ζ0 = 0.80 km−1 , and l = 1, 3.

36

4. Twisted-torus magnetic field configurations

Figure 4.6. The projection of the field lines in the meridional plane is shown for ζ0 = 0, 0.40, 0.80 km−1 respectively, and l = 1, 3. The dashed lines correspond to the ψ = 0 surfaces, and the toroidal field is confined within the marked region.

4.4.1

The case with multipoles l = 1, 3

We now consider the system of equations including only the l = 1 and l = 3 components. The projected system is 1 2 ν ′ − λ′ ′ a1 − 2 a1 e−λ a′′1 + e−λ 4π 2 r Z

e−ν π ¯ + 2ψ 3 /ψ¯2 (3/4) ζ02 ψ − 3ψ|ψ/ψ| − 4π 0

3 4 ¯ ×Θ(|ψ/ψ| − 1) sin θ dθ = c0 − c1 a1 − a3 (ρ + P )r 2 , 5 7

(4.8)

1 12 ν ′ − λ′ ′ a3 − 2 a3 e−λ a′′3 + e−λ 4π 2 r Z π −ν

e ¯ + 2ψ 3 /ψ¯2 (7/48) ζ02 ψ − 3ψ|ψ/ψ| + 4π 0 ¯ − 1)(3 − 15 cos2 θ) sin θ dθ = 2 c1 (ρ + P )r 2 (a1 − 4a3 ) , (4.9) ×Θ(|ψ/ψ| 15

where

"

#

a3 (3 − 15 cos2 θ) ψ = −a1 + sin2 θ .

2

(4.10)

We again impose regularity at the origin (Eq. (3.33)), continuity in r = R of a1 , a′1 , a3 , a′3 with the vacuum solutions for a1 (r), a3 (r) given by Eq. (3.35), and we fix a1 (R) = 1.93 · 10−3 km by normalization. For the remaining constraint we choose the solution that minimizes the absolute value of a3 (R). We find that there is a discrete series of local minima of |a3 (R)|, and we select among them the absolute minimum.

Fig. 4.5 shows the profiles of the tetrad field components (see Eq. (4.4)) obtained by numerically integrating Eqns. (4.8), (4.9), for different values of ζ0 . B(r) is evaluated at θ = 0, while B(θ) , B(φ) are evaluated at θ = π/2. As ζ0 increases, the magnitude of the toroidal field becomes larger, but the region where it is confined shrinks, as already found in Section 4.2. The projection of the field lines in the meridional plane is shown in Fig. 4.6 for the same values of ζ0 . It shows that, for −1 ζ0 > ∼ 0.40 km , the magnetic field lines lie in disconnected regions, separated by

4.4 The general case

37

Figure 4.7. The projection of the field lines in the meridional plane is shown for ζ0 = 0 km−1 and l = 1, 3. The left panel refers to the solution corresponding to the absolute minimum of |a3 (R)/a1 (R)|; in this solution ψ has no nodes. The center and right panels refer to solutions corresponding to relative minima of |a3 (R)/a1 (R)|; in these cases ψ has one and two nodes, respectively. The dashed lines corresponds to the ψ = 0 surfaces.

dashed lines in the figure. Inside these regions, the function ψ has opposite sign and no toroidal field is present. A similar phenomenon has been discussed in [28]. As we will see in the next Section, the occurrence of these regions is likely an artifact of the truncation in the harmonic expansion, and disappears as higher order harmonics are included.

For completeness we also mention that the solutions corresponding to the local minima of |a3 (R)| different from the absolute minimum, correspond to very peculiar field configurations (see Fig. 4.7). The function ψ has nodes on the equatorial plane, therefore the field lines lie in disconnected regions; for a fixed value of ζ0 , the number of nodes increases as |a3 (R)| increases. These peculiar solutions exist for any value of ζ0 , and appear also when higher order harmonic components are considered. Thus, they are not artifacts of the l-truncation.

4.4.2

The case with multipoles l = 1, 3, 5

We now include the l = 5 contribution. The three equations obtained by projecting the GS equation (3.31) onto l = 1, 3, 5 are given in Appendix B (Sec. B.1). The boundary conditions are essentially the same as in the previous Section; in particular, we look for the absolute minimum of a3 (R)2 + a5 (R)2 , with fixed a1 (R) = 1.93 · 10−3 km.

In Fig. 4.8 the profiles of the tetrad components of the magnetic field are plotted for values of ζ0 in the range 0 ≤ ζ0 ≤ 3.00 km−1 . Fig. 4.9 shows the projections of the field lines in the meridional plane corresponding to the same values of ζ0 . Comparing the results with the case l = 1, 3 we see that the presence of the harmonic l = 5 modifies the magnetic field shape, but most of the features discussed in the previous Section are still present.

An interesting difference is the following. While in the case l = 1, 3 for ζ0 > ∼ 0.40 km−1 we find field configurations which exhibit two disconnected regions where the function ψ has opposite sign and the magnetic field lines are confined (regions within dashed lines in Fig. 4.6), this does not occur when the l = 5 component is taken into account. This suggests that the above feature is an artifact of the truncation in the harmonic expansion.

38

4. Twisted-torus magnetic field configurations

3

-1

B(µ) / (10

15

G)

2

( ζ0 = 0 km )

B(r)

1 0

B(φ)

-1 -2

B(θ)

-3 -4 0

0.2

0.4

0.6

0.8

1

r/R 3

-1

B(µ) / (10

15

G)

2

B(r)

( ζ0 = 0.61 km )

1 0

B(φ)

-1 -2

B(θ)

-3 -4 0

0.2

0.4

0.6

0.8

1

r/R 3

-1

( ζ0 = 3.00 km )

B(µ) / (10

15

G)

2

B(r)

1 0

B(φ)

-1 B(θ)

-2 -3 -4 0

0.2

0.4

0.6

0.8

1

r/R

Figure 4.8. The profiles of the tetrad components of the magnetic field (B(r) (θ = 0), B(θ) (θ = π/2), B(φ) (θ = π/2)) for the case including l = 1, 3, 5, with ζ0 = 0 km−1 , ζ0 = 0.61 km−1 and ζ0 = 3.00 km−1 .

4.5 Magnetic helicity and energy

39

Figure 4.9. The projection of the field lines in the meridional plane is shown for ζ0 = 0 km−1 , ζ0 = 0.61 km−1 and ζ0 = 3.00 km−1 respectively, and for l = 1, 3, 5. The toroidal field is confined within the marked region.

4.4.3

Higher order multipoles

Up to now we have included components with l < 7, neglecting the contribution from l ≥ 7. In order to test the accuracy of this approximation, we have studied the convergence of the harmonic expansion. To this purpose, we have solved the GS equation (3.31) including odd harmonic components up to l = 7, for ζ0 = 0 and ζ0 = 0.61 km−1 , and we have computed the quantities

ψl≤5 (r, θ) − ψl≤3 (r, θ)

,

ψ¯

ψl≤7 (r, θ) − ψl≤5 (r, θ)

.

¯

∆(5) (r, θ) =

∆(7) (r, θ) =

ψ

(4.11)

These functions are shown in Fig. 4.10. They are plotted only inside the star since outside they are much smaller. Fig. 4.10 shows that the error in neglecting l ≥ 7, −1 quantified by the function ∆(7) , is < ∼ 2% for ζ0 = 0 and < ∼ 4% for ζ0 = 0.61 km .

Furthermore, a comparison of ∆(5) and ∆(7) shows that the harmonic expansion converges.

From the above results we are confident that a truncation of the harmonic expansion at l = 5 corresponds to a good approximation. Therefore, in what follows we shall consider configurations including the l = 1, 3, 5 multipoles.

4.4.4

An example of antisymmetric solution

Here we show an example of a solution belonging to the antisymmetric family, corresponding to l = 2, 4. In Fig. 4.11 we plot the field lines projected on the meridional plane, for ζ0 = 0 km−1 and ζ0 = 0.30 km−1 . We remark that the field lines are antisymmetric with respect to the equatorial plane; as a consequence, the total magnetic helicity is zero (see Section 4.5). Similar zero-helicity configurations have been considered in Braithwaite [18].

4.5

Magnetic helicity and energy

The stationary configurations of magnetized neutron stars which we have found depend on the value of the free parameter ζ0 , i.e. on the ratio between the toroidal

40

4. Twisted-torus magnetic field configurations

(5)

(7)

∆

0.05 0.04 0.03 0.02 0.01 0 -1

∆

0.05 0.04 0.03 0.02 0.01 0 -1

1 0.5 0

y/R

-0.5

-0.5

0 x/R

0.5

0.5 0

y/R

-0.5

-0.5

-1

0

0.5

x/R

-1

(7)

(5)

∆

∆

0.08

0.08

0.06

0.06

1

0.04

1

0.04

0.5

0.02 0 -1

1

0.5

0.02 0 -0.5

-0.5

0 x/R

0.5

0

0 -1

y/R

-1

y/R

-0.5

-0.5

0

0.5

x/R

-1

Figure 4.10. The functions ∆(5) (left panels) and ∆(7) (right panels) are shown for ζ0 = 0 (upper panels) and ζ0 = 0.61 km−1 (lower panels) in the meridional plane for 0 ≤ r ≤ R.

and the poloidal components of the magnetic field. In this Section, we provide an argument to assign a value to ζ0 . Furthermore, we compute the magnetic energy of the system to compare the contributions from poloidal and toroidal fields.

The total energy of the system (the star, the magnetic field and the gravitational field) can be determined by looking at the far field limit (r → ∞) of the spacetime metric [82, 121]. Following [52, 28], we write the perturbed metric as h

i

ds2 = −eν 1 + 2h(r, θ) dt2 + eλ 1 + 2

i

h

+r 1 + 2k(r, θ)

2

2

2eλ m(r, θ) dr 2 r

2

dθ + sin θ dφ

+2i(r, θ)dtdr + 2v(r, θ)dtdφ + 2w(r, θ)drdφ where, in particular, m(r, θ) = system is

P

l

(4.12)

ml (r)Pl (cos θ). The total mass-energy of the

E = M + δM ,

(4.13)

where M is the gravitational mass of the unperturbed star and δM = lim m0 (r) . r→∞

(4.14)

In Appendix C (Sec. C.1), we discuss the equations which allow to determine E. We remark that δM includes different contributions, due to magnetic energy, deformation energy, and so on.

4.5 Magnetic helicity and energy

41

Figure 4.11. The projection of the field lines in the meridional plane is shown for ζ0 = 0 km−1 and ζ0 = 0.30 km−1 respectively, and for l = 2, 4. The dashed line corresponds to ψ = 0; the toroidal field is confined within the marked region.

In order to evaluate the magnetic contribution to E, it is convenient to use the Komar-Tolman formula for the total energy (see for instance Chapter 4 in [111]): E=2

Z

Tµν

V

1 − T gµν η µ nν dV 2

(4.15)

(where V is the 3-surface at constant time, η µ is the timelike Killing vector, nµ is the normalized, future-directed normal to V ); the magnetic contribution comes µν (see Eq. (3.11)), i.e. from the stress-energy tensor of the electromagnetic field Tem Em = 2 =

1 2

Z ZV ∞

em Tµν

r2e

1 − T em gµν η µ nν dV 2

λ+ν 2

0

dr

Z

π

sin θ B 2 dθ .

(4.16)

0

The total (integrated) magnetic helicity Hm of the field configuration is Hm =

Z

√ 0 , d3 x −gHm

(4.17)

0 is the t-component of the magnetic helicity 4-current, defined as where Hm

1 α Hm = ǫαβγδ Fγδ Aβ .

2

(4.18)

Explicitly, we have Hm = −2π

Z

R

dr 0

Z

0

π

[Ar ψ,θ − ψAr,θ ]dθ ,

(4.19)

where λ−ν

e 2 2 ¯ − 1 · Θ(|ψ/ψ| ¯ − 1) , ψ ζ0 |ψ/ψ| sin θ Z θ

λ−ν ψ ¯ − 1 · Θ(|ψ/ψ| ¯ − 1)dθ ′ . = ψ,θ e 2 ζ0 |ψ/ ψ| ′ sin θ 0

ψAr,θ = ψ,θ Ar

(4.20)

The functional dependence of Hm on the potential of the toroidal field, Ar (see Equation (4.19)), shows that regions of space where the toroidal field vanishes do not contribute to the magnetic helicity.

42

4. Twisted-torus magnetic field configurations

7

δM [10-6 km]

6

Em [10-6 km]

5 4 3 2 1 0 0

0.5

1

1.5 ζ0 [km-1]

2

2.5

3

Figure 4.12. The functions δM and Em are plotted as functions of ζ0 , for l = 1, 3, 5 and Hm = 1.75 · 10−6 km2 .

In ideal MHD, the magnetic helicity is a conserved quantity [10, 15]. Thus, if we consider magnetic field configurations having the same value of the magnetic helicity and different energies, the lowest energy configuration is energetically favoured.

In Fig. 4.12 we plot δM and Em as functions of ζ0 , for a fixed helicity Hm = 1.75 · 10−6 km2 . δM , and consequently the total energy M + δM , has a minimum at ζ0 = 0.61 km−1 . A fixed value of Hm corresponds, for any assigned value of ζ0 , to a different normalization constant Bpole . Since δM , Hm and Em have the same quadratic dependence on the magnetic field normalization, this means that if we change Hm the plots of δM and of Em as functions of ζ0 are simply rescaled with respect to that shown in Fig. 4.12. Consequently, for any fixed value of Hm the position of the minimum of the total energy is the same as that shown in Fig. 4.12.

We conclude that the configuration with ζ0 ≃ 0.61 km−1 is energetically favoured.

This configuration is shown, among others, in Figs. 4.8, 4.9. From Fig. 4.12 we also see that the contribution of the magnetic energy to δM is ∼ 50-70%.

In Fig. 4.13 we show the ratio of poloidal to total magnetic field energy, Ep /Em , as a function of ζ0 , for the configurations (l = 1, 3, 5) studied. This plot is interesting because, as already discussed, the relative weight of the poloidal and the toroidal components of the field significantly affects many astrophysical processes involving magnetars, like magnetar activity [130], their thermal evolution [92], their gravitational wave emission [31]. It should be stressed that the surface poloidal field is inferred from spin-down measurements which, however, provide no hint about the toroidal field hidden inside the star. We find that for ζ0 = 0.61 km−1 , Ep /Em ≃ 0.93; if we only consider the poloidal contribution inside the star, the resulting ratio is Ep /Em ≃ 0.91. It is interesting to note the approximate correspondence between the minimal energy configuration and the configuration with smaller ratio Ep /Em : a larger toroidal component is energetically favoured. Since the toroidal contribution is never higher than ∼ 10%, all our twisted-torus configurations are dominated by poloidal fields.

In the following (Sec. 4.6) we shall see how the above results change for a different (non-minimal) contribution of the higher (l > 1) multipoles and for a different and more general form of the ζ−function, which controls the internal geometry of toroidal and poloidal magnetic field components.

4.6 Model extensions

43

1 0.99

Ep / Em

0.98 0.97 0.96 0.95 0.94 0.93 0.92 0

0.5

1

1.5 ζ0 [km-1]

2

2.5

3

Figure 4.13. The ratio Ep /Em is shown as a function of ζ0 , for l = 1, 3, 5.

4.6

Model extensions

In the present Section we discuss two subsequent extentions of our model.

The first one (Sec. 4.6.1) concerns the way we fix the strength of higher (l > 1) multipoles of the magnetic field with respect to the dipolar (l = 1) component.

As explained in Sec. 3.2.4, in our coupled system of equations in order to have a complete set of boundary conditions we require a constraint on the relative contribution of higher multipoles. Here we remove the condition of minimal contribution from higher multipoles adopted so far, thus considering more general configurations.

Then, we fix the constraint by choosing among the configurations obtained the energetically favoured one, by means of the minimal energy argument already used to fix the parameter ζ0 (Sec. 4.5).

In the second extention (Sec. 4.6.2) we reconsider our choice of the arbitrary function which determines the ratio of toroidal and poloidal fields, ζ(ψ) . Different choices will help understanding how the magnetic field geometries obtained depend on the form adopted for this function.

4.6.1

Relative strength of different multipoles

If we remove the condition of minimal contribution from higher order multipoles, i.e. q a23 + a25 minimum for r ≥ R ,

(to hereafter, this will be named the Minimum High Multipole (MHM) condition), the boundary conditions are not sufficient to fix all the parameters of the problem and we are left with a free arbitrary constant. We choose c1 as a “free” parameter and we proceed as follows. For an assigned value of ζ0 : • we solve the GS equations for the al ’s for different values of c1 • we compute δM/Hm for the corresponding configurations • we compute the surface contribution of the l > 1 multipoles,

q

a23 (R) + a25 (R) .

At this point we look for the configuration having the lower q value of δM/Hm ; this represents the configuration having the favoured value of

a23 (R) + a25 (R) for the

44

4. Twisted-torus magnetic field configurations

0.095

0.086

ζ0 = 0.65 km-1 0.61 km-1 0.59 km-1 0.58 km-1 0.52 km-1

( ζ0 = 0.61 km-1 )

A

δM / Hm [km-1]

δM / Hm [km-1]

0.085 0.09

0.085 B

0.084 0.083 0.082

0.08

0.081 3

3.2 3.4 3.6 3.8 4 2 1/2 2 [10-4 km] ( a3 (R)+a5 (R) )

4.2

3.3

3.4 3.5 3.6 3.7 2 1/2 2 [10-4 km] ( a3 (R)+a5 (R) )

3.8

p Figure 4.14. The function δM/Hm is plotted as a function of a23 (R) + a25 (R); on the left ζ0 = 0.61 km−1 , on the right the cases ζ0 = 0.65, 0.61, 0.59, 0.58 and 0.52 km−1 are shown together for comparison.

assigned ζ0 . The following step is to extend the analysis to different values of ζ0 and minimize energy q with respect to the couple of parameters (ζ0 , c1 ) (equivalent to the couple ζ0 ,

a23 (R) + a25 (R) ). This gives the final favoured configuration. q

In Fig. 4.14 we plot the ratio δM/Hm as a function of a23 (R) + a25 (R) . In the left panel we fix ζ0 = 0.61 km−1 . In Sec. 4.5 we have shown that, under the MHM assumption, the quantity δM/Hm is minimum for this value of ζ0 . This MHM configuration corresponds to the point A on the curve plotted in Fig. 4.14.

Since we now q drop the MHM condition, the minimum of δM/Hm occurs for a

different value of a23 (R) + a25 (R) (point B in Fig. 4.14), which corresponds to the energetically favoured configuration with ζ0 = 0.61 km−1 . For an assigned value of Hm , the relative variation of the total energy of the configuration B with respect to A is of the order of 13%. Fig. 4.14 refers to a star with EOS APR2. Similar results are obtained for the GNH3 star.

In the right panel of Fig. 4.14 we plot δM/Hm for selected values q of ζ0 , and com-

pare the different profiles. We have explored the parameter space ( a23 (R) + a25 (R), ζ0 ), finding that theqfunction δM /Hm has a minimum (δM/Hm = 0.0817) for

ζ0 = 0.59 km−1 and a23 (R) + a25 (R) = 3.6 · 10−4 km. It is worth reminding that the l = 1 contribution is a1 (R) = 1.93 · 10−3 km. We shall refer to this configuration as the Minimal Energy 1 (ME1) configuration.

In Fig. 4.15 we compare the profiles of the tetrad components of the magnetic field for the MHM and the ME1 configurations. We see that, whereas for the MHM configuration B(θ) and B(r) are significantly different from zero throughout the star, for the ME1 configuration, obtained with no assumption on the relative strengths of the different multipoles for r > R, these field components are strongly reduced near the axis. Conversely, the toroidal component B(φ) has a similar behaviour in both configurations. The two panels of Fig. 4.15 illustrate how the magnetic field rearranges inside the star when the MHM condition is removed. The situation can be explained as follows. The magnetic helicity Hm can be written as Hm = −2π

Z

0

R

dr

Z

π 0

(Ar ψ,θ − ψAr,θ )dθ ;

(4.21)

4.6 Model extensions

45

3

3 B(r)

1 0

B(φ)

-1 -2

B(θ)

-3

ME1

2 B(µ) / (1015 G)

B(µ) / (1015 G)

2

MHM

1

B(r)

0

B(φ)

-1 B(θ)

-2 -3

-4

-4 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

r/R

0.6

0.8

1

r/R

Figure 4.15. The profiles of the tetrad components of the magnetic field B(r) (θ = 0), B(θ) (θ = π/2), B(φ) (θ = π/2) are plotted as functions of the radial distance normalized to the stellar radius. The left panel refers to the MHM configuration (energy is minimized assuming that the contribution of the multipoles higher than l = 1 is minimum for r > R); in this case ζ0 = 0.61 km−1 . The right panel refers to the minimal energy configuration ME1, obtained with no assumption on the relative strengths of the different multipoles, and for ζ0 = 0.59 km−1 .

therefore, Hm vanishes if either ψ = 0, i.e. the poloidal field vanishes, or Ar = 0, i.e. the toroidal field vanishes. In the twisted-torus model the toroidal field is zero in the inner part of the star, thus Hm receives contributions only from the magnetic field in the region where B(φ) 6= 0. Since in that region the field components of the MHM and ME1 configurations are similar, these configurations have nearly the same magnetic helicity Hm . On the other hand, the energy δM receives contributions from the field components throughout the entire star, and these contributions are not vanishing in the region where B(φ) = 0. When we minimize the function δM/Hm in the ME1 configuration, the l > 1 multipoles, which were kept minimum in the MHM configuration, do not change Hm significantly, but they change δM , and since we require δM/Hm to be minimum, they combine as to reduce the field in the inner region of the star.

4.6.2

A more general choice of the function β(ψ)

In this section we construct twisted-torus configurations choosing two different forms of the function β(ψ) (remember β(ψ) = ψζ(ψ)), namely σ

¯ − 1) , Θ(|ψ/ψ|

σ

¯ − 1) , Θ(|ψ/ψ|

¯ −1 β(ψ) = ψζ0 |ψ/ψ|

(4.22)

(note that σ = 1 corresponds to Eq. (4.1)), and ¯ −1 β(ψ) = −β0 |ψ/ψ|

(4.23)

where β0 is a constant of order O(B). A choice similar to (4.22) has been considered in [66] by Lander and Jones, who have studied the field configurations in a newtonian framework. Although Eqns. (4.22), (4.23) do not exhaust all possible choices of the function β(ψ), they are general enough to capture the main features of the stationary twisted-torus configurations.

46

4. Twisted-torus magnetic field configurations

3

3 ME1

1

B(r)

0

B(φ)

-1 B(θ)

-2

ME2

2 B(µ) / (1015 G)

B(µ) / (1015 G)

2

-3

B(r)

1 0

B(φ)

-1 -2

B(θ)

-3

-4

-4 0

0.2

0.4

0.6 r/R

0.8

1

0

0.2

0.4

0.6

0.8

1

r/R

Figure 4.16. The profiles of the tetrad components of the magnetic field [B(r) (θ = 0), B(θ) (θ = π/2), B(φ) (θ = π/2)] are shown (upper panels). In the lower panels we show the projection of the field lines in the meridional plane. Left and right panels refer, respectively, to the configuration ME1 and ME2.

4.6 Model extensions

47

For β given by Eq. (4.22) the magnetic field components and the GS equation are λ

B

µ

λ

e− 2 e− 2 0 , 2 ψ,θ , − 2 ψ,r , r sin θ r sin θ

=

ν

−

σ

¯ −1 e− 2 ζ0 ψ |ψ/ψ|

r 2 sin2 θ

¯ − 1) Θ(|ψ/ψ|

!

(4.24)

and e−λ ′′ ν ′ − λ′ ′ 1 ψ − [ψ,θθ − cot θψ,θ ] − ψ + 4π 2 4πr 2

2σ

2σ−1 e−ν ζ02 ¯ ¯ ¯ − + σ|ψ/ψ| |ψ/ψ| − 1 ψ |ψ/ψ| − 1 4π ¯ − 1) = (ρ + P )r 2 sin2 θ[c0 + c1 ψ] . ×Θ(|ψ/ψ|

(4.25)

For β given by Eq. (4.23) they are:

λ

λ

B

µ

e− 2 e− 2 ψ,θ , − 2 ψ,r , 0 , 2 r sin θ r sin θ

=

σ

¯ −1 e− 2 β0 |ψ/ψ| ν

¯ − 1) Θ(|ψ/ψ|

!

,

(4.26)

e−λ ′′ ν ′ − λ′ ′ 1 ψ − [ψ,θθ − cot θψ,θ ] ψ + 4π 2 4πr 2

2σ−1 e−ν β02 ¯ |ψ/ψ| ¯ −1 ¯ − 1) σ|ψ/ψ| Θ(|ψ/ψ| − 4πψ = (ρ + P )r 2 sin2 θ[c0 + c1 ψ] .

(4.27)

r 2 sin2 θ

and −

The field configurations are now identified by three parameters: (σ, c1 , ζ0 ) for the choice (4.22), and (σ, c1 , β0 ) for the choice (4.23). As in the previous Section, we look for the minimal energy configuration at fixed magnetic helicity; furthermore, we compute the ratio of the poloidal magnetic energy to the total magnetic energy.

We solve the system of GS equations (they are given in Appendix B, Sec. B.2 for both cases) with the boundary conditions discussed in Section 3.2.4; the relative strenght of higher (l > 1) multipoles is fixed by energy minimization as explained in the previous Section (4.6.1). For each configuration we compute the magnetic helicity Hm , the correction to the total energy δM , and the poloidal and toroidal contributions to the magnetic energy Em . The equations to determine δM are given in Appendix C (Sec. C.2). The energetically favoured configurations are found by minimizing δM/Hm with respect to the three parameters.

Let us firstly consider the case in which the relation between toroidal and poloidal fields is given by Eq. (4.22). We find that the minimal energy configuration (for the APR2 EOS) corresponds to σ = 0.18, q

ζ0 = 0.20 km−1 ,

a23 (R) + a25 (R) = 3.4 · 10−4 km .

(4.28)

48

4. Twisted-torus magnetic field configurations

3 ME3

B(µ) / (1015 G)

2 B(r)

1 0

B(φ)

-1 -2

B(θ)

-3 -4 0

0.2

0.4

0.6

0.8

1

r/R

Figure 4.17. The profiles of the tetrad components of the magnetic field are shown for the configuration ME3, corresponding to β given by Eq. (4.23); the values of the parameters are given in Eq. (4.29).

We shall refer to this configuration as the ME2 configuration. In Fig. 4.16 the configurations with σ = 1 (ME1) and σ = 0.18 (ME2) are compared. In ME2 the magnetic field has a slightly different shape: in particular, the toroidal component is larger near the surface of the star, and the extension of the toroidal field region along the y axis is smaller. The ratio of the poloidal magnetic energy to the total magnetic energy inside the star is Ep /Em = 0.91

for ME1

Ep /Em = 0.87

for

ME2 .

Furthermore, we find that the minimal energy configuration is nearly the configuration with smaller ratio Ep /Em , i.e. with larger toroidal component, as already found in Sec. 4.5; thus, the σ = 0.18 case also corresponds to the minimum value of Ep /Em which can be obtained with the choice (4.22). We can conclude that if σ is not assumed to be 1, we can obtain configurations with a larger toroidal contribution, but only by a small amount.

We now consider the choice (4.23) for the function β. In this case the minimal energy configuration (for the APR2 EOS) corresponds to σ = 0.42, q

β0 = 9 · 10−4 ,

a23 (R) + a25 (R) = 3.7 · 10−4 km .

(4.29)

This configuration, which is shown in Fig. 4.17, will be referred to as the ME3 configuration. A comparison with the right panel of Fig. 4.16 shows that the ME2 and ME3 configurations are very similar. Inside the star Ep /Em = 0.88 and, as in the previous case, it is the minimum value which can be obtained for this choice of the function β.

For the GNH3 EOS, we obtain similar results. The minimal energy configuration is obtained with the choice (4.22), and σ = 0.30, q

ζ0 = 0.13 km−1 ,

a23 (R) + a25 (R) = 5.1 · 10−4 km .

(4.30)

4.7 Summary

49

The ratio of the poloidal energy to the total magnetic energy inside the star for this configuration is Ep /Em = 0.93.

We conclude that when we allow for a non-minimal contribution of the l > 1 multipoles and for a more general parametrization of the function β(ψ), the magnetic field changes with respect to the MHM configuration found in Sec. 4.5 as follows: the poloidal field near the axis of the star is smaller, and the toroidal field near the stellar surface is larger. In all cases the toroidal field never contributes to more than ∼ 13% of the total magnetic energy stored inside the star. This means that in our configurations magnetic fields are always dominated by the poloidal component.

4.7

Summary

We have found numerical solutions of equilibrium magnetic field configurations of strongly magnetized NSs in General Relativity. The family of solutions we have found reproduce the so-called twisted-torus magnetic field geometry, in which the toroidal component is confined to a torus-shaped region inside the star, and the poloidal component extends throughout the entire star and in the exterior. Twistedtorus-like configurations have been found as a final outcome of Newtonian MHD simulations with generic initial conditions [13, 14, 15], and appear to be more stable than others. In these configurations, the existence of a toroidal component of the magnetic field in the star’s interior is allowed, while vanishing in the exterior, without assuming neither discontinuities (associated to surface currents) nor the vanishing of all magnetic field components outside the star. This is an improvement with respect to previous works [52, 28].

A further improvement is that we consider the contribution from higher (l > 1) multipoles of the magnetic field in addition to the dipole (l = 1), as well as their couplings. We have included multipoles up to l = 5, showing that this choice corresponds to a good approximation. We have fixed the relative contribution of higher multipoles with respect to the dipole at first by taking the minimum, then using an argument of minimal energy, based on magnetic helicity conservation in ideal MHD systems.

Our configurations are determined by one or more parameters, depending on the choice of the function relating toroidal and poloidal fields. In the different cases considered, we have found the set of parameters minimizing the energy in a normalization independent way, by means of the same argument mentioned above.

This also allowed to determine the favoured ratio of toroidal and poloidal magnetic field energies (the relevance of this quantity for magnetar models has been widely discussed). As a result, we have found that all our configurations are dominated by the poloidal field, with a maximum toroidal contribution of ∼ 10%.

The equilibrium configurations we have found can be used as input for studies on magnetar dynamical processes. The next Chapter (5) is devoted to one of such applications: gravitational wave emission from strongly magnetized NSs. We stress that the (newtonian) numerical studies leading to the idea that twistedtorus magnetic fields could be favoured in real stars do not represent the final step.

This result needs to be confirmed in a relativistic context and with more general assumptions. In the last months of the PhD we started working on a project aimed to provide further hints on the subject. This is sketched in the next Section.

50

4.8

4. Twisted-torus magnetic field configurations

Outlook: emergence and stability of twisted-torus configurations

The important step accomplished by Braithwiate and collaborators [13, 14, 15], allowed by recent progress in numerical MHD, has given relevant indications about the possible magnetic field configuration realized in magnetized stars (see Sec. 4.1).

Nevertheless, the topic deserves further investigation.

The results obtained only refer to a limited set of initial configurations, in which the star is non rotating and magnetic fields are random. A useful extension should account for a more general sample of configurations. On the other hand, a fundamental improvement would consist in obtaining more reliable predictions at a quantitative level, to be used in models concerning magnetars’ structure and dynamics; the results obtained up to know have supplied inputs to magnetar models (such as the one presented in this Thesis), but only at a qualitative level. In this respect, as already pointed out, one of the main limitations so far is that the simulations are performed in a newtonian (non relativistic) framework, while accurate computations involving NSs require General Relativity.

Following this research line, we recently started a project in collaboration with the Numerical Relativity group at the Albert Einstein Institute (AEI, Potsdam, Berlin), headed by prof. Luciano Rezzolla. The basic idea of this project is to perform 3D MHD simulations in General Relativity, with the purpose of (i) confirming the twisted-torus geometry as favoured, (ii) extending the analysis to different and more general cases, exploring the universality of this result, and (iii) providing quantitative predictions to be used as input for the different models involving strongly magnetized NSs. More specifically, we aim to define the conditions in which a stable equilibrium can be reached, to establish if any other kind of equilibrium geometry is possible, and to collect a representative sample of configurations by exploring the physical parameter space. An additional advance with respect to the present results on the subject will be the extension to the case of a rotating star, planned for future developments of the project. The final goal is to improve the present knowledge of the magnetic field configuration realized in strongly magnetized NSs, which is of crucial importance in order to have a fruitful comparison between models and observations.

This research is only at the preliminary stage and here we only give some additional detail. The physical system of our interest consists in an isolated NS surrounded by vacuum and endowed with a magnetic field permeating the star and extending to the exterior. At first, we consider a non rotating star. We want to follow the MHD evolution of the system prepared according to a given set of initial conditions, in order to establish if an equilibrium magnetic field configuration is reached within the timescale we consider. We are interested in the existence of such equilibrium and its features, depending on the initial condition chosen and on a set of physical parameters which define the general properties of the star (such as mass and EOS) and of its magnetic field (such as magnetic helicity). Simulations are carried out by means of numerical codes available (as developer) to the Numerical Relativity group at the AEI: the ‘Cactus’ infrastructure [139] with all its basic components and in particular the hydrodynamical evolution code ‘Whisky’ [140] in its MHD version [42]. The main features of ‘WhiskyMHD’ are the following:

4.8 Outlook: emergence and stability of twisted-torus configurations

51

(i) it employs a flux-conservative formulation of the general relativistic MHD equations and high-resolution shock-capturing schemes; (ii) fluxes are computed using the HLLE approximate Riemann solver; (iii) the divergence-free condition for the magnetic field is guaranteed using the constrained-transport approach (see [42] and references therein). Starting from the existing codes we are developing an opportunely modified version of them, optimized for the purposes of the project. To run the simulations we employ the computational resources of the institute.

We adopt some physical assumptions. The most relevant, on which WhiskyMHD is based, consists in treating the system in ideal MHD, i.e. in the limit of infinite electrical conductivity. As discussed in Sec. 3.2, this assumption is valid while the star is still completely liquid (i.e. the solid crust has not yet formed) and the core matter has not yet performed the phase transition into the superfluid state, which is expected to happen at most a few hours after birth. On the other hand, the Alfvèn timescale for the evolution of the magnetic field is orders of magnitude shorter, thus there is ample time for the magnetized fluid to reach an equilibrium configuration before the crust formation or the onset of superfluidity; the following evolution proceeds on much longer timescales, hence the field geometry stays unchanged for a long time. In conclusion, the configurations we find describe the magnetic field of a NS at the time of crust formation, but they are also representative of the configurations realized in following times. As an additional assumption we work in the so-called ‘Cowling approximation’, in which the spacetime evolution is neglected.

This approximation is fully justified for the system we consider and allows a significant reduction of the simulation time; the code can work without this assumption, thus we are able to give a comparative demonstration that the approximation is valid.

At the moment our work is focused on developing and testing the code. In addition to a number of standard numerical tests (e.g. convergence tests), we are employing and comparing two different methods to assure the divergence-free condition for the magnetic field: the constrained-transport approach already implemented in WhiskyMHD, and a vector potential formulation of the induction equation, which leads naturally to a null magnetic field divergence. This will allow us to choose the more suitable method. We are also running a number of physical tests, in order to assure the correct behaviour of the system (e.g. tests on the evolution timescales).

A tricky element is the description outside the star, where, given that the present code cannot simulate vacuum, a low-density atmosphere in which the fluid velocity is set to zero is necessarlily present. If we want that the magnetic field evolution in the exterior behaves with good approximation as prescribed by Maxwell’s equations in vacuum, the assumption of ideal MHD outside the star results inappropriate. A better approximation can be obtained by introducing a dissipative term, whose effect is to decouple the magnetic field from the fluid in the atmosphere, allowing its evolution, and to damp the non-zero laplacian components of the field, the same components that would propagate to infinity according to Maxwell’s equations. The evolution in the atmosphere has to be object of accurate tests in order to guarantee the soundness of results.

At the end of this preliminary phase of the project we will be ready to start physically relevant simulations. The initial conditions will range from random distributions of poloidal and toroidal fields to different configurations, intended to reproduce the possible field geometries present in the first phases of the star’s life.

52

4. Twisted-torus magnetic field configurations

By adjusting the initial magnetic helicity we will be able to change the ratio of toroidal and poloidal fields, to study its effect on the final equilibrium and to establish the range of values for which reaching an equilibrium is possible. Since purely poloidal or purely toroidal configurations appear to be unstable in non rotating stars (see Sec. 4.1), we expect that a stable ratio between the two components is limited within a finite range of values. We stress (again) that the assessment of this ratio is of great importance, because observations can only give us direct information about the external poloidal fields, which in turn can provide us with an indication of the internal poloidal field strenghts; on the other hand, we have no direct way to evaluate the internal toroidal fields and how much magnetic energy they hide inside the star. We will consider a number of polytropic or ideal fluid EOS (in future developments, it will be possible to employ also ‘realistic’ EOS) and we will vary the star’s mass and the magnetic field strenght. At last, we will have a sample of configurations representative of the possible field geometries realized in strongly magnetized NSs.

Chapter 5

Strongly magnetized neutron stars as gravitational wave sources The present Chapter is devoted to the emission of gravitational waves (GWs) from magnetically-deformed rotating NSs. We discuss the quadrupolar deformations of a strongly magnetized NS and the related GW emission mechanism. Then, we compute, according to our equilibrium model and other models, both the emission spectrum for a single magnetar and the stochastic background produced by the entire magnetar population. Finally, we discuss the detectability of such background by future GW detectors.

The content of the Chapter is based on [27, 76].

5.1

Quadrupolar deformations and gravitational waves

As we have already discussed in the previous Chapters (3, 4), the NS magnetic field induces perturbations on the static and spherically symmetric spacetime metric of the unperturbed star. If the system is assumed to be stationary and axisymmetric the perturbed metric can be written in the form [52, 28]

ds2 = −eν 1 + 2[h0 (r) + h2 (r)P2 (cos θ)] dt2 h

i

+2 i1 (r)P1 (cos θ) + i2 (r)P2 (cos θ) + i3 (r)P3 (cos θ) dtdr ∂ ∂ ∂ P1 (cos θ) + v2 P2 (cos θ) + v3 P3 (cos θ) dtdφ ∂θ ∂θ ∂θ

∂ ∂ +2 sin θ w2 P2 (cos θ) + w3 P3 (cos θ) drdφ ∂θ ∂θ

λ

2e λ +e 1 + m0 (r) + m2 (r)P2 (cos θ) dr 2 r

+2 sin θ v1

2

h

i

+r 1 + 2k2 (r)P2 (cos θ)

2

2

2

dθ + sin θdφ

.

(5.1)

The above expression is analogous to Eq. (4.12), but here the perturbations h, m, k, i, v, w, functions of r and θ, have been expanded into the relevant multipolar 53

54

5. Strongly magnetized neutron stars as gravitational wave sources

components. The off-diagonal metric corrections i, v and w are associated with frame dragging effects, while the functions h, m and k determine structure deformations. In particular, the structure deformations consist in (i) mass, radius and energy variations, alredy considered in Sec. 4.5 and related to the radial functions h0 and m0 , and (ii) quadrupolar deformations, given by h2 , m2 and k2 , which are relevant for GW emission.

In this Section we compute the quadrupolar deformations induced by the magnetic field for the twisted-torus configurations previously obtained. To this purpose, we have to solve some perturbed Einstein’s equations; in particular, the components involved are [rr], [θθ], [φφ] and [rθ]. In Appendix D we discuss in detail the resulting coupled system of equations and its integration.

Quadrupolar deformations are usually quantified through the star’s ellipticity, which measures deviations from spherical shape resulting from ‘compression’ of the star along the symmetry axis. According to its basic definition, the ellipticity of the star is given by (equatorial radius) − (polar radius) , (5.2) ε= (polar radius) and can be written as a function of the radial coordinate r as [24] 3 2δp2 ε(r) = − + k2 2 rν ′ (ρ + P )

,

(5.3)

where δp2 is the l = 2 component of the pressure perturbation. The surface ellipticity is then εsurf = ε(R). However, from the point of view of GW emission, the relevant quantity is the quadrupole ellipticity εQ , which accounts for the massenergy distribution; εQ has a different definition and has to be distinguished from ε.

The quadrupole ellipticity is defined as Q , (5.4) I where Q is the mass-energy quadrupole moment, and I is the mean value of the moment of inertia of the star. The quadrupole moment Q is given by the far field limit of the spacetime metric εQ =

Q , (5.5) r3 and then it is obtained by solving the relevant Einstein’s equations and computing the metric correction h2 (r). The value of I can be estimated from the limit ω → 0 of the ratio J/ω in a slowly rotating star model, where ω is the angular velocity and J the angular momentum. For 1.4 M⊙ we have I = 98.39 km3 (APR2 EOS) and I = 98.39 km3 (GNH3 EOS). Note that εQ is of order O(B 2 ) and then scales 2 as ∝ Bpole with the magnetic field normalization.

The GW emission mechanism of our interst is the following. If an axisymmetric deformed star rotates about an axis misaligned with the symmetry axis, we have a time-varying mass quadrupole moment, which implies a continuous emission of GWs. The GW amplitude of such signal, detected from a source at a distance r, is [12] 4Iω 2 |εQ | sin α , (5.6) h0 ≃ r h2 (r → ∞) ∼

5.2 Single source emission

55

where ω is the star’s angular velocity, α is the misalignment (or wobble) angle, and εQ is here the total quadrupole ellipticity, which measures deformations induced by magnetic fields as well as other possible causes. Eq. (5.6) shows the close relation between GW emission and quadrupole ellipticity.

In strongly magnetized NSs the quadrupolar deformation is determined essentially by the magnetic field configuration and strength, thus the εQ that we compute from magnetic deformation can be safely used in determining the GW emission.

Fast rotation will also induce a non-negligible deformation; however, being symmetric with respect to the spin axis, such deformation does not contribute to the GW emission and it is not included in our analysis.

We can now come to the results of our computations. It is well known that while the poloidal field tends to make the star oblate (εQ > 0), the toroidal field tends to make it prolate (εQ < 0). In a mixed toroidal-poloidal field configuration the actual deformation results from the balance of the two opposite effects. Since in our configurations the poloidal field dominates over the toroidal one, εQ is always positive and the deformation is larger for configurations in which the toroidal contribution is smaller. We have found that, for the APR2 equation of state and Bpole = 1015 G (see Sec. 3.2.4), εQ = 3.5 · 10−6 and εQ = 3.7 · 10−6 respectively for the energetically favoured configurations ME2 and the ME3 (see Sec. 4.6.2). It is also interesting to consider twisted-torus configurations which do not correspond to minimal energy.

We have determined the entire range of possible ellipticities for the twisted-torus −6 configurations analyzed in our model; we have found 3.5 · 10−6 < ∼ εQ < ∼ 4.8 · 10 −6 −6 for the APR2 EOS, and 8.1 · 10 < ∼ εQ < ∼ 9.6 · 10 for the GNH3 EOS. The larger ellipticities are obtained in the purely poloidal limit, whereas the smaller refer to the minimal energy configurations, being also the configurations having the highest contribution of toroidal fields (see Sections 4.5, 4.6). We note that, as expected, given the mass (1.4 M⊙ in our case), less compact stars (GNH3) have larger ellipticities. We also note that the values of εQ we find for the purely poloidal case are comparable to the maximal ellipticity found in [66].

Summarizing, the quadrupole ellipticity εQ corresponding to Bpole = 1015 G would lie in the quite narrow ranges (3.5, 4.8)·10−6 for the APR2 EOS and (8.1, 9.6)· 10−6 for the GNH3 EOS, i.e.

Bpole εQ ≃ k 1015 G

2

· 10−6 ,

(5.7)

with k ≃ 4 for the APR2 EOS and k ≃ 9 for the GNH3 EOS. As an example of the behaviour of εQ when the toroidal field contribution changes, in Fig. 5.1 we plot εQ versus the parameter β0 for configurations obtained choosing β(ψ) as in Eq. (4.23), assuming that σ = 1 and that the contribution of the l > 1 multipoles is fixed by energy minimization; the EOS employed is APR2.

In particular, we see that the maximal deformation is given by the purely poloidal configuration.

5.2

Single source emission

In this Section we evaluate the GW emission of a single magnetar as predicted by our equilibrium model; for completeness, we will also consider other models. In

56

5. Strongly magnetized neutron stars as gravitational wave sources

4.9

4.5

-6

εQ [10 ]

4.7

4.3 4.1 3.9 3.7 0

10

20

30

40

50

-4

β0 [10 ]

Figure 5.1. Ellipticities versus β0 for σ = 1, with β(ψ) given by (4.23). Here we assume Bpole = 1015 G , M = 1.4 M⊙ and the EOS employed is APR2.

Section 5.3 we will compute the GW stochastic background produced by the entire magnetar population, for which the single source emission constitutes the basic ingredient; then, we will evaluate the detectability of such background by third generation detectors such as the Einstein Telescope [138].

Following the literature on the subject, here and in the next Section we abandon the unit system c = G = 1 previously adopted, and we conform to a notation in which the factors c and G are explicit.

The GW energy spectrum emitted by a single source writes as

dfe −1 dEGW = E˙ GW

, dfe dt

(5.8)

where fe is the emission frequency. The GW luminosity of a rotating NS with spin axis forming a wobble angle α with the magnetic axis is composed of two contributions, one at the spin frequency νR , one at its double 2νR ; it can be written as 2G E˙ GW = 5 I 2 ε2Q ω 6 sin2 α(cos2 α + 16 sin2 α) , (5.9) 5c where εQ is the star quadrupole ellipticity induced by the magnetic field, ω is the angular velocity and I is the moment of inertia. The term sin2 α cos2 α is from the emission component at the spin frequency, the term 16 sin4 α from the component at twice the spin frequency.

The star loses rotational energy mainly due to electromagnetic radiation and GW emission (we shall neglect other effects, e.g. relativistic winds [32]). According to the well known vacuum dipole radiation model, the energy loss rate due to dipole radiation is given by 2 R6 1 Bpole dip ˙ ω 4 sin2 α . |EROT | = 6 c3

(5.10)

5.2 Single source emission

57

The total spin-down rate obtained from Eqs. (5.9) and (5.10) is |ω| ˙ = |ω˙ dip | + |ω˙ GW | 2 R6 2G 1 Bpole ω 3 sin2 α + 5 Iε2Q ω 5 sin2 α(1 + 15 sin2 α) . = 3 6 Ic 5c

(5.11)

Using the above quantity and remembering that the first term of the GW luminosity given in Eq. (5.9) is emitted at fe = νR = ω/2π, while the second at fe = 2νR , we can compute the single source emission spectrum according to Eq. (5.8).

The choice of the initial spin period P0 for the NS sets an upper limit on the frequency ranges where the two components of the GW emission contribute: the emission at νR contributes to frequencies below 1/P0 , that at 2νR to frequencies below 2/P0 . Therefore, for fe < 1/P0 the overall emission has both contributions, while for 1/P0 < fe < 2/P0 the only contribution comes from the emission at 2νR .

In conclusion, the terms to be considered when computing the GW energy spectrum are: for fe < P10 , 32π 4 G 2 2 3 dEGW = I εQ fe dfe 5c5 #−1

2 R6 Bpole 8π 2 G 2 2 × cos α + IεQ fe (1 + 15 sin2 α) 6Ic3 5c5

(

"

2

#−1 )

2 R6 Bpole 2π 2 G 2 2 + sin α + IεQ fe (1 + 15 sin2 α) 6Ic3 5c5 2

for

1 P0

< fe <

2 P0

"

;

(5.12)

, dEGW 32π 4 G 2 2 3 = I εQ fe dfe 5c5 #−1

2 R6 Bpole 2π 2 G 2 2 + IεQ fe (1 + 15 sin2 α) × sin α 6Ic3 5c5 2

for fe >

2 P0

"

;

(5.13)

, dEGW =0 . dfe

If we put reasonable numbers in the equations, assuming Bpole = 1014 − 1015 G, R ∼ 10 km, I ∼ 1045 g cm2 and fe < ∼ 61 kHz, we see that even for quadrupole 2 R B 2 is much larger than 8π5c5G Iε2Q fe2 ; in this ellipticities as large as 10−4 the term pole 6Ic3 case the contribution of GW emission to the spin-down is negligible. As shown in the following, this holds in most of the cases we consider. It is worth noting that, when α 6= 0 and (5.12) is

2 Bpole R6 3 6Ic

≫

8π 2 G 2 2 5c5 IεQ fe ,

for fe < 1/P0 the dominant term in Eq.

dEGW 32π 4 G 2 2 3 = I εQ fe dfe 5c5

2 R6 Bpole 6Ic3

!−1

,

(5.14)

58

5. Strongly magnetized neutron stars as gravitational wave sources

which does not depend on the wobble angle α. However, for dominant term is 2 R6 Bpole dEGW 32π 4 G 2 2 3 2 = I ε f sin α Q e dfe 5c5 6Ic3

!−1

1 P0

,

< fe <

2 P0 ,

the

(5.15)

and it depends on α.

It should be stressed that in general the wobble angle depends on time. The misalignment of magnetic and rotation axes causes, in the NS frame, the free precession of the angular velocity around the magnetic axis with period Pprec ≃ P/|εQ |, where P is the spin period [63, 64]. The star internal viscosity damps such precessional motion and reduces the wobble angle towards the aligned configuration (α = 0), if the star has an oblate shape (εQ > 0), whereas it increases α towards the orthogonal configuration α = π/2 (“spin-flip”), if the shape is prolate (εQ < 0) [62, 31]. The second case is more favourable for GW emission.

The timescale of the process is given by τα = nP0 /εQ , where P0 /εQ is the initial precession period and n is the expected number of precession cycles in which the process takes place; estimates for slowly rotating NSs indicate that n ∼ 102 − 104 [4], however the value of n is actually unknown [31]. The evolution of the misalignment angle is relevant for our analysis only if the associated timescale, τα , is short compared to the spin-down timescale, τsd ; conversely, if τα ≫ τsd the process takes place when the source is no longer an efficient GW emitter.

In principle, an accurate estimate of the GW emission from a single magnetar should account for (i) its initial wobble angle (to be chosen according to a proper population distribution), and (ii) the evolution of such misalignment with time. This kind of analysis would, however, be affected by the wide uncertainties on both τα and the initial angle distribution. As we shall show in Section 5.3.4, from the point of view of the GW background produced by the entire magnetar population and its detectability, the value of α assigned to the single magnetar and its eventual change with time do not significantly affect the results. Since our main interest is focused on such detection prospects, we proceed here with the simplifying assumption that each single magnetar is born with α = π/2 and that the misalignment evolution is ineffective. Then, in Section 5.3.4, we will consider the effects of a generic wobble angle. Note that α = π/2 implies that the emission is at frequency 2νR only.

An essential input for dEGW /dfe is the magnetic field strength at the pole, Bpole : it determines the electromagnetic spin-down rate and it affects the stellar deformations. If we want the value of Bpole to be representative of the magnetar population, it should be chosen as a suitable average. Such an average is uncertain at present; however the values of Bpole inferred from AXPs and SGRs lie in the 1014 − 1015 G range [81]. Our choice here is to span this range by studying its two extremes, Bpole = 1014 and 1015 G. As we have seen, the overall GW emission depends on P0 , the initial spin period.

Following [98] we set P0 = 0.8 ms. For α = π/2 this gives femax = 2/P0 = 2500 Hz. If α < π/2, part of the GW emission is at the spin frequency and the corresponding contribution has a frequency cutoff femax = 1/P0 = 1250 Hz. The chosen value of P0 implies a very fast spinning newborn NS, but still consistent with the believed range of NS spin rates at birth. We remark that, according to current scenarios of magnetar formation, strongly magnetized NSs are those that are born with periods

5.2 Single source emission

59

of the order of ms, much faster than ordinary pulsars [34]. At the end of Section 5.3.3 we will sketch the effect of assuming lower initial spin frequencies.

In the following, we compute the single source GW emission spectrum according to our model and its predictions on εQ discussed in the previous Section, but the same analysis is also extended to other models. In particular, these models (described below) account for purely poloidal fields or mixed fields dominated by the toroidal component. The different models considered are representative of the possible magnetic field configurations realized in magnetars, according to our present knowledge.

Purely poloidal magnetic field Here we consider two purely poloidal magnetic field configurations which have already been used in [98] to evaluate the magnetar GW emission and the corresponding GW background. It will be useful to include these configurations in our analysis, in order to have a comparison with the previous literature on the subject.

Since the toroidal field is absent the star’s ellipticity predicted in these cases is always positive. In addition to α = π/2 and M = 1.4 M⊙ , the numerical inputs to compute dEGW /dfe are Bpole , εQ , I and R. As previously discussed, we adopt two different values of Bpole : 1014 and 1015 G. Following [59], we write the quadrupole ellipticity as 2 R4 Bpole , (5.16) εQ = g GM 2 where the value of the dimensionless parameter g accounts for the magnetic field geometry and the EOS. As in [98], we consider two models with g = 13 (Model A) and g = 520 (Model B), respectively. The first model refers to an incompressible fluid star and a dipolar magnetic field [39]; similar values are obtained in relativistic models based on polytropic EOS [59]. Model B describes a scenario in which the NS core is a superconductor of type I, implying that the internal magnetic field is confined to the crustal layers [12]. This scenario gives much stronger deformations (see also [28]). The other parameters are fixed as follows: R = 10 km and I = 1045 g cm2 .

Toroidal-dominated magnetic field An additional model we consider is based on the hypotesis of very strong toroidal fields inside the star. It assumes a magnetic field configuration with an internal toroidal field of ∼ 2 · 1016 G (core-averaged value), in addition to a poloidal field of ordinary strength (1014 −1015 G). In [110] it is suggested that toroidal field strengths of this order are needed to explain the time-integrated emission of magnetars as inferred from the extremely bright giant flare that took place on December 27, 2004 from SGR 1806-20 (which liberated an energy of 5 · 1046 erg). Giant flares of this magnitude could result from large-scale rearrangements of the core magnetic field or instabilities in the magnetosphere [119, 73]. Such a huge toroidal field would induce prolate deformations as strong as εQ ≃ −6.4 · 10−4 [31]. The deformation associated to poloidal fields, whose strength is fixed by the choice of Bpole , tends to oppose the above deformation; however in the cases considered here (Bpole = 1014 and 1015 G) the corresponding correction is negligible (of the order of 10−4 − 10−2 ,

60

5. Strongly magnetized neutron stars as gravitational wave sources 1e+50

1e+50

Bp=10

1e+48

14

G

15

G

1e+46

dE/dfe [erg/Hz]

1e+46

dE/dfe [erg/Hz]

Bp=10

1e+48

1e+44

1e+42

1e+40

TT-APR2 TT-GNH3 P-A P-B TD

1e+38

1e+36

1e+34

1e+44

1e+42

1e+40

TT-APR2 TT-GNH3 P-A P-B TD

1e+38

1e+36

1e+34 10

100

1000

fe [Hz]

10

100

1000

fe [Hz]

Figure 5.2. Spectral gravitational energy emitted by a single magnetar as a function of the emitted frequency, for the different models we consider: P-A and P-B stand for the purely poloidal models A and B, TT are the predictions of our twisted-torus model for the two EOS considered (APR2 and GNH3), TD indicates the toroidal-dominated model. Left panel: Bpole = 1014 G; right panel: Bpole = 1015 G.

respectively). We assume again α = π/2 and M = 1.4 M⊙ . The other physical inputs are R = 10 km and I = 1045 g cm2 .

The GW emission predicted by the present model with Bpole = 1014 G could be regarded as an upper limit among the different magnetar models (excluding exotic scenarios), as in this case there occurs the most favorable combination of magnetic fields: an extremely strong toroidal field dominates the deformation while the lower value of the poloidal field strength results in a slower spin-down. In addition, the shape of the star is prolate and the evolution of the wobble angle α, if effective, leads the axis of the magnetically-induced deformation towards the orthogonal configuration, resulting in a stronger GW emission. According to this model, the GW signal emitted by a newly-born fast spinning magnetar could be observable with the Advanced Virgo/LIGO class detectors up the distance of the Virgo cluster [32].

Gravitational wave emission spectrum according to the different models We can now present the GW spectrum emitted by a single magnetar obtained according to the different models considered: the two purely poloidal models A and B (hereafter P-A and P-B), our twisted-torus model (hereafter TT) for the two equations of state considered (APR2, GNH3), and the toroidal-dominated model (hereafter TD). In Fig. 5.2 we plot dEGW /dfe as a function of the emitted frequency fe . In the left (right) panel we assume Bpole = 1014 G (Bpole = 1015 G); as already discussed, these values define a likely range for Bpole .

The first important indication which emerges from Fig. 5.2 is that the uncertainty related to the different magnetar models is always much higher (3-5 orders of magnitude) than the spread associated to the adopted range of Bpole . This means that our predictive power is mainly limited by the present lack of knowledge on the internal magnetic field configuration.

Let us now focus on the Bpole = 1014 G case (left panel). The TD model is by far the most favorable for GW emission, having the optimal combination of strong

5.3 Gravitational wave background produced by magnetars

61

deformation and slow electromagnetic spin-down. The second strongest emission is obtained with the P-B model, where large deformations are achieved even for this lower field strength. The emission predicted by the P-A model is lower by more than three orders of magnitude, due to the difference in the g2 factor appearing in the expression of dEGW /dfe . The two TT models are expected to give even weaker signals; they differ for the assumed EOS and the one which gives less (more) compact stars, GNH3 (APR2), is associated to stronger (weaker) deformations and GW emission.

If we consider higher external poloidal fields (Bpole = 1015 G, right panel) the picture changes. For the P-A, P-B and TT models the value of Bpole controls both 4 2 ), and the spin-down rate, the GW luminosity, which scales as Bpole (εQ ∝ Bpole 2 which has the electromagnetic contribution proportional to Bpole plus a very small 2 correction due to GW emission. As a result, dEGW /dfe ∝ Bpole , which translates in a factor 100 increase from Bpole = 1014 to 1015 G. Conversely, in the TD model the deformation is determined by the dominant toroidal field in the stellar interior (with poloidal field corrections up to ∼ 1% for 1015 G), and an increase in Bpole only results in a higher electromagnetic spin-down and in a smaller overall GW emission. As long as the electromagnetic spin-down dominates over the GW spindown, dEGW /dfe is reduced by a factor of 100 from Bpole = 1014 to 1015 G. The final result is that when Bpole = 1015 G, the prediction of TD and P-B models are comparable.

It is worth noting that in all the considered models the contribution given by the GW emission to the spin-down is negligible, with the exception of the early time evolution in the TD model with Bpole = 1014 G (hereafter TD14 ). This is shown in Fig. 5.2, where the energy spectra are linear in logarithmic scale, reflecting the behaviour dEGW /dfe ∝ fe3 , while the TD14 model is characterized by a lower emission level at high frequency, due to a non-negligible GW spin-down. This effect is even more evident in Fig. 5.3, where we compare the GW spectrum for the same TD model shown in the left panel of Fig. 5.2 with a TD model where GW spin-down is neglected (dashed line). It is clear that the GW contribution starts to be relevant at fe ∼ 300 Hz. For all the other models we have discussed, this contribution becomes relevant at much higher emission frequencies.

5.3

Gravitational wave background produced by magnetars

It is well known that a variety of astrophysical processes are able to generate a stochastic GW background, resulting from the superposition of a large number of unresolved sources and with distinct spectral properties and features [36, 37, 102, 103, 99, 75]. The detection of these astrophysical GW backgrounds can provide insights into the cosmic star formation history and constrain some of the physical properties of compact objects (white dwarfs, NSs and black holes). Moreover, these signals may act as foreground noise for the detection of cosmological GW backgrounds over much of the accessible frequency spectrum (in particular, we will consider the primordial background generated during the Inflationary Epoch). Here we compute the GW background produced by magnetars.

Using population synthesis methods to evaluate the initial period and the mag-

62

5. Strongly magnetized neutron stars as gravitational wave sources 1e+51

dE/dfe [erg/Hz]

1e+50

Bp = 1014 G

1e+49

1e+48

1e+47

TD-II TD-I 1e+46 100

1000

fe [Hz]

Figure 5.3. Spectral gravitational energy emitted by a single source according to the toroidal-dominated model as a function of the emitted frequency. TD-I is the same as TD in the left panel of Fig. 5.2; in TD-II (dashed line) the contribution of GW emission to the spin-down is neglected.

netic field distributions of magnetars, in [98] the GW background due to the magnetar population has been computed assuming the two different purely poloidal magnetic field configurations we have discussed in the previous Section (5.2). We now reconsider and extend the above analysis; we take into account the same purely poloidal models, our twisted-torus model and a toroidal-dominated model (also discussed in Sec. 5.2). These different models span the possible magnetic field geometries realized in magnetars.

Fundamental elements to compute the GW background, in addition to the single source emission spectrum evaluated in Sec. 5.2, are the cosmic star formation rate evolution and the corresponding magnetar birthrate. This is discussed in the following Section. These inputs depend on the cosmological model assumed: in our analysis we adopt a ΛCDM cosmological model with parameters ΩM = 0.26, ΩΛ = 0.74, h = 0.73, Ωb = 0.041, in agreement with the three-year WMAP results [109].

5.3.1

Birth rate evolution

Following [75], we use the cosmic star formation rate density evolution predicted by the numerical simulation of [124]. We only consider the formation rate of Population II stars, adopting a Salpeter Initial Mass Function (IMF) Φ(M ) ∝ M −(1+x) with x = 1.35 (normalized between 0.1 − 100 M⊙ ), in regions of the Universe which have been already polluted by the first metals and dust grains [104, 105, 83]. In [97, 93] the statistical properties of highly magnetized NSs (B≥ 1014 G) have been derived using population synthesis methods; it is shown that NSs born as magnetars represent 8-10% of the total simulated population of NSs. Here we assume a fraction fMNS = 10%; we further assume that magnetar progenitors have masses in the 8 M⊙ - 40 M⊙ range. It should be noted that the mass range of magnetar progenitors is still debated (see for instance [38, 33]). However, the range we consider is sufficiently large to include the proposed evolutionary models.

5.3 Gravitational wave background produced by magnetars

63

1

.

-1 -3 ρ⋆ [M⊙ yr Mpc ]

0.1

0.01

0.001

1e-04

1e-05

1e-06 0

2

4

6

8

10

12

14

10

12

14

z 1e+09

-1

MNS rate [yr ]

1e+08

1e+07

1e+06

1e+05

1e+04 0

2

4

6

8

z

Figure 5.4. Top panel: redshift evolution of the comoving star formation rate density.

Bottom panel: redshift evolution of the number of magnetars formed per unit time.

The top panel of Fig. 5.4 shows the redshift evolution of the cosmic star formation rate density inferred from the simulation1 . The number of magnetars formed per unit time out to a given redshift z can be computed by integrating the cosmic star formation rate density, ρ˙⋆ (z), over the comoving volume element, while restricting the integral over the stellar IMF in the proper range of progenitor masses; that is Z z Z 40 M⊙ ′ ′ dV ρ˙ ⋆ (z ) RMNS (z) = fMNS dz ′ dM Φ(M ) , (5.17) dz (1 + z ′ ) 8 M⊙ 0 where the factor (1 + z) at the denominator accounts for the time-dilation effect, and the comoving volume element can be expressed as dV = 4πr 2

h

c ǫ(z)dz H0

ǫ(z) = ΩM (1 + z)3 + ΩΛ The result is shown in the bottom panel of Fig. 5.4.

5.3.2

(5.18) i− 1

2

.

Background computation

We are now ready to compute the GW backgrounds produced by the different magnetar models we consider. Following [75], the spectral energy density of the 1

The results shown in Fig. 5.4 refer to the fiducial run in [124] with a box of comoving size L = 10h−1 Mpc and Np = 2 · 2563 (dark+baryonic) particles.

64

5. Strongly magnetized neutron stars as gravitational wave sources Bp = 10

1e-06

14

G

Bp = 10

1e-06

15

S/N = 40 1e-08

1e-12

1e-14

CGWB

S/N = 1.1

1e-10

ΩGW(f)

ΩGW(f)

S/N = 2.5

1e-08

S/N = 0.026

1e-10

1e-16

G

1e-12

1e-14

CGWB 1e-16

TT-APR2 TT-GNH3 P-A P-B TD Einstein Telescope

1e-18

1e-20

1e-22 1

10

100

TT-APR2 TT-GNH3 P-A P-B TD Einstein Telescope

1e-18

1e-20

1e-22

1000

1

10

100

f [Hz]

1000

f [Hz]

Figure 5.5. The predicted closure energy density (ΩGW ) as a function of the observational frequency, for the different magnetar models discussed: P-A and P-B stand for purely poloidal models A and B, TT are the twisted torus model predictions for the two EOS considered (APR2 and GNH3), TD is the toroidal-dominated model. Left panel: Bpole = 1014 G; right panel: Bpole = 1015 G. In both panels the shaded region indicates the foreseen sensitivity of the Einstein Telescope, and the horizontal dotted line (CGWB) is the upper limit on primordial backgrounds generated during the Inflationary Epoch. A given background is detectable by the Einstein Telescope if the corresponding signal-to-noise ratio is larger than the detection threshold S/N = 2.56 (see text).

GW background can be written as dE = dSdf dt

Z

0

zf

Z

Mf

dE

dR(M, z)

Mi

dSdf

,

(5.19)

where dR(M, z) is the differential source formation rate dR(M, z) =

ρ˙⋆ (z) dV Φ(M )dM dz , (1 + z) dz

(5.20)

dE and dSdf is the locally measured average energy flux emitted by a source at distance r. For sources at redshift z it becomes

dE

dSdf

=

(1 + z)2 dEGW [f (1 + z)] , 4πdL (z)2 dfe

(5.21)

where f = fe (1 + z)−1 is the redshifted emission frequency fe , and dL (z) is the luminosity distance to the source.

It is customary to describe the GW background by a dimensionless quantity, the closure energy density ΩGW (f ) ≡ ρcr −1 (dρgw /dlogf ), which is related to the spectral energy density by the equation ΩGW (f ) =

f dE 3 c ρcr dSdf dt

,

(5.22)

where ρcr = 3H02 /8πG is the cosmic critical density.

In Fig. 5.5, we show ΩGW as a function of the observational frequency (f ) for the different magnetars models. Differences in the predicted stochastic backgrounds reflect differences in the corresponding single source emission spectrum. The maximum amplitude is always achieved around 1 kHz: in the left panel, it ranges from

5.3 Gravitational wave background produced by magnetars

65

∼ 4 · 10−16 to ∼ 2 · 10−8 , while in the right panel the range is ∼ 4 · 10−14 − 2 · 10−9 .

The higher value is obtained with the TD model in the first case, and with the P-B model in the second case; the lower value is given in both cases by the TTAPR2 model. In both panels, model predictions are compared with the foreseen sensitivity of the Einstein Telescope (shaded region) and with the upper limit to primordial backgrounds generated during the Inflationary Epoch (horizontal dotted line labelled CGWB). The latter contribution is estimated from Eq. (6) of [125] assuming a tensor/scalar ratio of r = 0.3 and no running spectral index of tensor perturbations [54, 55].

A comparison between the estimated magnetar GW background and the upper limit to the primordial background predicted by Inflationary scenarios (the horizontal dotted line in Fig. 5.5 labelled CGWB) shows that, for the models of magnetar we consider, the magnetar GWB is always larger than the primordial background in some region of frequency (the only exception is the twisted-torus model TT-APR2 with Bp = 1015 G). For instance, for the toroidal dominated models TD15 and TD14 this is true, respectively, for f > ∼ 13 Hz and f > ∼ 4 Hz. Thus, the GWB generated by magnetars may act as a limiting foreground for the future detection of the primordial background even at frequencies as low as few tens of Hz. In addition, specific magnetar models lead to a cumulative signal which is potentially detectable by the Einstein Telescope. A more quantitative assessment of the detectability is reported in the following Section.

5.3.3

Detectability

The gravitational wave signal produced by the magnetar population can be treated as continuous. Indeed, if ∆τgw is the average time duration of a signal produced by a single magnetar, and dR(z) is the number of sources formed per unit time at redshift z, the duty cycle D out to redshift z, defined as D(z) =

Z

0

z

dR(z)∆τgw (1 + z) ,

(5.23)

satisfies the condition2 D ≫ 1. Consequently, the stochastic signal appears in the detector outputs as a time-series noise which, by the central limit theorem, is expected to have a Gaussian-normal distribution function. In this case, as suggested by [3, 99], the optimal detection strategy is to cross-correlate the output of two (or more) detectors, assumed to have independent spectral noises.

The optimized S/N for an integration time T is given by [3],

S N

2

=

9H04 T 50π 4

Z

0

∞

df

γ 2 (f )Ω2GW (f ) , f 6 P1 (f )P2 (f )

(5.24)

where P1 (f ) and P2 (f ) are the power spectral noise densities of the two detectors, and γ is the normalized overlap reduction function, characterizing the loss of sensitivity due to the separation and the relative orientation of the detectors.

The sensitivity of detector pairs is given in terms of the minimum detectable 2

If we take ∆τgw = τsd we have always D higher than 103 .

66

5. Strongly magnetized neutron stars as gravitational wave sources

amplitude for a flat spectrum ΩMIN (ΩMIN = constant) defined as ΩMIN =

1 10π 2 √ T 3H02

"Z

0

∞

γ 2 (f ) df 6 f P1 (f )P2 (f )

·(erfc−1 (2α) − erfc−1 (2γ)) ,

#−1/2

(5.25)

where T is the observation time, α the false alarm rate, γ the detection rate and erfc−1 the complementary error function (for more details see [3]).

If we consider the cross-correlation of two detectors with the sensitivity of the Einstein Telescope, we get ΩMIN = 1.13 · 10−11 for an integration time T of one year, a false alarm rate α = 10% and a detection rate γ = 90% (T. Regimbau, private communication); these values, inserted in Eq. (5.24), lead to a detection threshold S/N of 2.56.

A given background is detectable by the Einstein Telescope if the corresponding signal-to-noise ratio given by Eq. (5.24) is larger than the detection threshold. For instance, the predicted ΩGW for the purely poloidal model P-B with Bp = 1015 G gives S/N = 2.49, that is slightly smaller than such threshold; consequently, there is no chance to detect this signal. Conversely, in the most optimistic magnetar model T D14 (toroidal-dominated model with Bp = 1014 G) we obtain S/N = 40, a very promising value. This result leads to the conclusion that third-generation gravitational wave detectors, such as the Einstein Telescope, hold the potential to reveal the cumulative GW signal from magnetars in the Universe.

It is worth noting that the above results refer to the assumed initial spin period of P0 = 0.8 ms. A higher value would lead to a lower frequency cutoff and, consequently, to a weaker GWB. For the T D14 model, for example, the detection threshold S/N = 2.56 corresponds to P0 = 5.2 ms. Hence the GW background would still be detectable up to this value.

5.3.4

Wobble angle effects

So far we have assumed a constant misalignment α = π/2 between the spin and the magnetic axis, in which case the GW signal is emitted only at twice the spin frequency fe = 2νR = ω/π. For a generic misalignment, we have also the emission at the spin frequency. We now focus on the model TD14 and explore the consequences of α < π/2. In this model the stellar deformation induced by the magnetic field are larger; being the most optimistic model for gravitational wave emission, this case allows to clearly show the effects of the wobble angle on detectability.

In Fig. 5.6 we compare the sensitivity of the Einstein Telescope with the background generated by model TD14 for different (constant) values of the wobble angle.

The plot clearly shows that when α > ∼ π/4 there is a single dominant contribution with the frequency cutoff at 2500 Hz, while for smaller angles there is a dominant contribution with f max = 1250 Hz and a secondary contribution with lower amplitude extending up to f max = 2500 Hz. Similar effects hold for the alternative magnetar models which have been presented in the previous Sections. A difference between the models potentially arises if the timescale for the evolution of the wobble angle is short compared to the spin-down timescale (see Section 5.2): in this case, the star rapidly tends (i) to the orthogonal configuration for the TD model,

5.3 Gravitational wave background produced by magnetars

67

1e-06

Bp= 1014 G

1e-07

ΩGW(f)

1e-08

1e-09

1e-10

α = π/2 α = π/4 α = π/18 α = π/60 Einstein Telescope

1e-11

1e-12 10

100

1000

f [Hz]

Figure 5.6. The TD14 background (see text) is plotted for different wobble angles, spanning the range π/60 − π/2, and compared with the Einstein Telescope sensitivity.

thus increasing gravitational wave emission, and (ii) to the aligned configuration, for models P-A, P-B and TT models, thus decreasing the emission.

As shown in Fig. 5.6, for the TD14 model (as well as for the other models) the gravitational wave backgrounds corresponding to different wobble angles exhibit significant differences at large frequencies, approximately above ∼ 800 Hz, where the Einstein Telescope sensitivity is too low even for this model; therefore, the signal detectability is only marginally affected. Variations in the signal-to-noise ratio are at most 2-3% in the TD14 case, and if the gravitational wave background is weaker (e.g. for Bpole > 1014 G) the effects on the S/N are even smaller.

We can conclude that the initial value of α and its evolution in time do not have significant effects on the GW background detectability with the Einstein Telescope.

68

5. Strongly magnetized neutron stars as gravitational wave sources

Conclusions In the present PhD Thesis we have studied the stucture and magnetic field configuration of strongly magnetized neutron stars, or magnetars. In the first Chapters (1 and 2) we have introduced neutron stars on general grounds and we have discussed their relevance for Astrophysics and fundamental Physics; then, we have focused our attention on magnetars, providing motivations for our work.

The central part of the Thesis (Chapters 3 and 4) has been devoted to the magnetar model we have developed. Such model, built in the framework of General Relativity, describes a strongly magnetized neutron star surrounded by vacuum under the assumption of stationary and axisymmetric spacetime and in the limit of high conductivity in the star’s interior. Here we summarize the main improvements with respect to the current literature.

• The family of solutions we have found reproduce the so-called twisted-torus magnetic field geometry; in twisted-torus-like configurations the toroidal component is confined to a torus-shaped region inside the star, and the poloidal component extends throughout the entire star and in the exterior. The main motivation for studying magnetized neutron stars with such magnetic field geometry comes from recent results obtained in newtonian numerical magnetohydrodynamics studies of magnetizd stars, which suggested that twisted-torus configurations could be favoured as they appear more stable than others.

• We have considered the contribution from higher (l > 1) multipoles of the magnetic field in addition to the dipole (l = 1), as well as their couplings.

• We have found the set of parameters of the model minimizing the energy in a normalization independent way, thus identifying the energetically favoured configuration among the possible solutions. This also allowed to determine the favoured ratio of toroidal and poloidal magnetic field energies, which is of great relevance for the interpretation of magnetar observations. As a result, we have found that all our configurations are dominated by the poloidal field, with a maximum toroidal contribution of ∼ 10%.

The equilibrium configurations we have found can be used as input for studies aimed to understand the peculiar observational properties of magnetars, improving the predictive power of present dynamical models.

In the last Chapter (5) we have discussed a direct application of the above results: gravitational wave emission from strongly magnetized neutron stars. We have computed, according to our twisted-torus model, the quadrupolar deformation induced by the magnetic field on the magnetar’s structure and we have illustrated the 69

70

Conclusions

gravitational wave emission mechanism related to such deformation. Then, we have estimated both the emission spectrum for a single magnetar and the gravitational wave background resulting from the superposition of signals produced by the entire magnetar population. For completeness, this analysis has also been extended to other magnetar models. Finally, we have discussed about the detectabilty of this gravitational wave background by third generation detectors such as the Einstein Telescope. We have found that: • different magnetar models produce a spread in the resulting gravitational wave emission, revealing that the main uncertainties on present predictions come from our lack of knowledge on the internal magnetic field configuration; • in some cases the gravitational wave background produced by magnetars results detectable by the Einstein Telescope, with signal-to-noise ratios up to 40; • the misalignment angle between magnetic and spin axes, whose distribution over the magnetar population is unknown, and its eventual evolution in time have poor effects on the background detectability; • for all the models considered, the signal produced by magnetars acts as a limiting foreground for the future detection of the primordial gravitational wave background of cosmologic origin produced during the Inflationary Epoch.

Future developments of the research work presented in this Thesis include (i) further improvements of our magnetar equilibrium model, to be achieved by adopting more general assumptions, (ii) additional applications (e.g. magnetar quasiperiodic oscillations or cooling of magnetized neutron stars), and (iii) a study on the emergence and stability of twisted-torus configurations to be accomplished in a relativistic framework (this is sketched in Sec. 4.8).

Appendix A

The Tolman-Oppenheimer-Volkoff (TOV) solution In this Appendix we discuss the equations describing the hydrostatic equilibrium of a static and spherically symmetric star; they were first proposed in [122, 84] and named TOV after the authors. In our model we regard such a static and spherically symmetric system as the unperturbed condition, in which the star has no magnetic field. In this case the spacetime metric is given by Eq. (3.13).

The TOV equations are obtained form the combination of Einstein equations Gµν = 8πTµν

(A.1)

and stress-energy tensor law T µν;ν = 0, explicitly written as

1 ∂ √ T µν;ν = √ −g T µν + Γµλν T λν = 0 , ν −g ∂x

(A.2)

where Γµλν are the Christoffel symbols and g is the metric determinant. We recall that the stress-energy tensor (we are assuming a perfect fluid) is given by T µν = (ρ + P )uµ uν + P gµν .

(A.3)

We introduce the function m(r), which measures the gravitational mass contained within a sphere of radius r from the star’s centre: m(r) =

Z

r

4πr 2 ρ dr .

(A.4)

0

It is related to λ(r) through

2m .

(A.5) r The only non-trivial component of Eqns. (A.2) is given by µ = r, from which we have 2P,r ν,r = − , (A.6) ρ+P e−λ = 1 −

71

72

A. The Tolman-Oppenheimer-Volkoff (TOV) solution

while the relevant Einstein equations are [tt] [rr]

1 d [r(1 − e−λ )] = 8πρ , r 2 dr ν,r 1 = 8πP eλ . − 2 (eλ − 1) + r r

(A.7) (A.8)

Eq. (A.7), rewritten in terms of m(r), becomes dm = 4πr 2 ρ , dr

(A.9)

which is one of the TOV equations, while from Eq. (A.8), combined with Eq. (A.9), we have # " m + 4πr 3 P .

(A.10) ν,r = 2 r(r − 2m) Finally, by substituting Eq. (A.10) in Eq. (A.6) we obtain the second TOV equation: "

m + 4πr 3 P P,r = −(ρ + P ) r(r − 2m)

#

.

(A.11)

The TOV equations are summrized below. dm = 4πr 2 ρ dr "

m + 4πr 3 P P,r = −(ρ + P ) r(r − 2m)

(A.12) #

(A.13)

If we provide the EOS, the two equations give a unique solution for an assigned value of the central density.

Appendix B

The GS system in the l = 1, 3, 5 case Here we write explicitly the system of equations obtained from the GS equation (3.31) by taking into account the l = 1, 3, 5 multipolar components and projecting the equation onto the respective harmonics. In this case we have

ψ =

"

#

a3 (3 − 15 cos2 θ) a5 (−315 cos4 θ + 210 cos2 θ − 15) + sin2 θ . −a1 + 2 8

We consider three different cases, in which the function β(ψ) = ψζ(ψ) is chosen according to Eqns. (4.1), (4.22), (4.23) respectively.

B.1

β(ψ) chosen according to Eq. (4.1)

2 ν ′ − λ′ ′ 1 a1 − 2 a1 e−λ a′′1 + e−λ 4π 2 r Z

e−ν π ¯ + 2ψ 3 /ψ¯2 − (3/4) ζ02 ψ − 3ψ|ψ/ψ| 4π 0

3 4 ¯ ×Θ(|ψ/ψ| − 1) sin θ dθ = c0 − c1 a1 − a3 (ρ + P )r 2 , 5 7

1 12 ν ′ − λ′ ′ a3 − 2 a3 e−λ a′′3 + e−λ 4π 2 r Z

e−ν π ¯ + 2ψ 3 /ψ¯2 + (7/48) ζ02 ψ − 3ψ|ψ/ψ| 4π 0 ¯ − 1)(3 − 15 cos2 θ) sin θ dθ ×Θ(|ψ/ψ|

8 10 2 a1 − a3 + a5 , = c1 (ρ + P )r 2 15 15 33

(B.1)

73

(B.2)

B. The GS system in the l = 1, 3, 5 case

74

30 ν ′ − λ′ ′ 1 a5 − 2 a5 e−λ a′′5 + e−λ 4π 2 r Z π

−ν e ¯ + 2ψ 3 /ψ¯2 (11/60) ζ02 ψ − 3ψ|ψ/ψ| + 4π 0 4 2 ¯ − 1) (−315 cos θ + 210 cos θ − 15) sin θ dθ ×Θ(|ψ/ψ| 8

20 4 2 a3 − a5 . = c1 (ρ + P )r 21 39

B.2

(B.3)

β(ψ) chosen according to Eqns. (4.22) and (4.23)

Choice (4.22) of the function β(ψ) gives

1 e−ν 2 ν ′ − λ′ ′ a1 − 2 a1 − e−λ a′′1 + e−λ 4π 2 r 4π

2σ−1

¯ |ψ/ψ| ¯ −1 +σ|ψ/ψ|

3 4 = c0 − c1 a1 − a3 5 7

Z

0

π

(3/4) ζ02 ψ

¯ − 1) sin θ dθ Θ(|ψ/ψ|

(ρ + P )r 2 ,

12 ν ′ − λ′ ′ 1 a3 − 2 a3 e−λ a′′3 + e−λ 4π 2 r

Z π

2σ−1 2σ −ν e 2 ¯ ¯ ¯ + σ|ψ/ψ| |ψ/ψ| − 1 + (7/48) ζ0 ψ |ψ/ψ| − 1 4π 0 ¯ − 1)(3 − 15 cos2 θ) sin θ dθ ×Θ(|ψ/ψ|

2 8 10 2 = c1 (ρ + P )r a1 − a3 + a5 , 15 15 33

(B.4)

30 ν ′ − λ′ ′ 1 a5 − 2 a5 e−λ a′′5 + e−λ 4π 2 r

Z

2σ−1 2σ e−ν π 2 ¯ ¯ ¯ + σ|ψ/ψ| |ψ/ψ| − 1 + (11/60) ζ0 ψ |ψ/ψ| − 1 4π 0 4 2 ¯ − 1) (−315 cos θ + 210 cos θ − 15) sin θ dθ ×Θ(|ψ/ψ| 8

20 4 a3 − a5 . = c1 (ρ + P )r 2 21 39

2σ

¯ −1 |ψ/ψ|

(B.5)

(B.6)

B.2 β(ψ) chosen according to Eqns. (4.22) and (4.23)

75

Choice (4.23) of the function β(ψ) gives 2 ν ′ − λ′ ′ 1 a1 − 2 a1 e−λ a′′1 + e−λ 4π 2 r Z

2σ−1 e−ν π β2 ¯ |ψ/ψ| ¯ −1 ¯ − 1) sin θ dθ − (3/4) 0 σ|ψ/ψ| Θ(|ψ/ψ| 4π 0 ψ

4 3 = c0 − c1 a1 − a3 (ρ + P )r 2 , (B.7) 5 7

12 ν ′ − λ′ ′ 1 a3 − 2 a3 e−λ a′′3 + e−λ 4π 2 r Z π −ν

2σ−1 2 e β ¯ |ψ/ψ| ¯ −1 + (7/48) 0 σ|ψ/ψ| 4π 0 ψ ¯ − 1)(3 − 15 cos2 θ) sin θ dθ ×Θ(|ψ/ψ|

8 10 2 2 a1 − a3 + a5 , = c1 (ρ + P )r 15 15 33

1 30 ν ′ − λ′ ′ a5 − 2 a5 e−λ a′′5 + e−λ 4π 2 r Z

2σ−1 β02 e−ν π ¯ |ψ/ψ| ¯ −1 (11/60) σ|ψ/ψ| + 4π 0 ψ 4 2 ¯ − 1) (−315 cos θ + 210 cos θ − 15) sin θ dθ ×Θ(|ψ/ψ| 8

20 4 2 a3 − a5 . = c1 (ρ + P )r 21 39

(B.8)

(B.9)

76

B. The GS system in the l = 1, 3, 5 case

Appendix C

The energy of the system In the present Appendix we discuss the equations and integration procedure to compute the total energy of the system E = M + δM (where M is the known unperturbed mass). We consider three different cases, in which the function β(ψ) = ψζ(ψ) is chosen according to Eqns. (4.1), (4.22), (4.23) respectively.

C.1

β(ψ) chosen according to Eq. (4.1)

The perturbation of the total energy of the system can be determined from the far field limit of the spacetime metric [82, 121]: δM = lim m0 (r) , r→∞

(C.1)

where the perturbed metric is given by equation (4.12). The functions h(r, θ) and m(r, θ) are expanded as h(r, θ) =

X

hl (r)Pl (cos θ) ,

l

m(r, θ) =

X

ml (r)Pl (cos θ) .

(C.2)

l

The perturbed Einstein equations ([tt] and [rr] components), projected onto l = 0 , allow to determine the quantity m0 (r): ρ′ δp0 = P′ 2 1 ′ 2 −λ 6 ′ 2 −λ 15 ′ 2 −λ (a ) e + (a3 ) e + (a5 ) e + 2 a21 3 1 7 11 3r 450 2 72 a + 2 a23 + 7r " 11r 2 5 # Z π 2 2 ψ e−ν 2 ¯ − 1) ¯ − 1 Θ(|ψ/ψ| dθ , ζ0 |ψ/ψ| + 4 sin θ 0

m′0 − 4πr 2

77

(C.3)

78

C. The energy of the system

1 − e m0 2 + 8πP − 4πreλ δp0 = r 1 ′ 2 6 ′ 2 15 ′ 2 2eλ 2 (a1 ) + (a3 ) + (a ) − 3 a1 3r 7r 11r 5 3r λ λ 72e 450e 2 − 3 a23 − a 7r " 11r 3 5 # Z π 2 2 ψ eλ−ν ¯ − 1) ¯ − 1 Θ(|ψ/ψ| ζ02 |ψ/ψ| dθ . + 4r sin θ 0 2λ

h′0

(C.4)

δp0 is the l = 0 component of the pressure perturbation (and vanishes outside the P star), and ψ = sin θ l=1,3,5 al Pl,θ . Using the relation (arising from T rν;ν = 0) ν ′ ρ′ = − + 1 δp0 − (ρ + P )h′0 2 P′

3 4 2 ′ − a1 (ρ + P ) c0 − c1 a1 − a3 3 5 7

12 ′ 2 8 10 − a3 (ρ + P )c1 a1 − a3 + a5 7 15 15 33

4 20 10 ′ a3 − a5 , − a5 (ρ + P )c1 11 21 39

δp′0

(C.5)

Eqns. (C.3), (C.4) can be rearranged in the form ρ′ 6 15 1 δp0 = (a′1 )2 e−λ + (a′3 )2 e−λ + (a′5 )2 e−λ ′ P 3 7 11 72 2 450 2 2 2 a + 2 a1 + 2 a3 + 3r " 7r 11r 2 5 # Z π 2 2 e−ν ψ 2 ¯ − 1) ¯ − 1 Θ(|ψ/ψ| + dθ , ζ0 |ψ/ψ| 4 sin θ 0 m′0 − 4πr 2

ν′ 2

(C.6)

ρ′ + 1 + 4πreλ (ρ + P ) δp0 P′

1 +e2λ m0 (ρ + P ) 2 + 8πP = r δp′0 +

2 3 4 (ρ + P ) − a′1 c0 − c1 a1 − a3 3 5 7

2 8 10 12 a1 − a3 + a5 − a′3 c1 7 15 15 33

4 1 20 6 10 ′ a3 − a5 − (a′1 )2 − (a′3 )2 − a5 c1 11 21 39 3r 7r 15 ′ 2 2eλ 2 72eλ 2 450eλ 2 − (a ) + 3 a1 + a + a 11r 5 3r 7r 3 3 11r 3 5 #!

"Z

2 2 π eλ−ν ψ ¯ − 1) ¯ − 1 Θ(|ψ/ψ| − ζ02 |ψ/ψ| dθ .

4r sin θ 0

(C.7)

By imposing a regular behaviour at r ≃ 0 we find m0 (r → 0) = Ar 3

,

δp0 (r → 0) = Cr 2 ,

(C.8)

C.2 β(ψ) chosen according to Eqns. (4.22) and (4.23)

79

where C = − A =

2(Pc + ρc ) · (α21 + α1 c0 )

dρ 3 + 4π r 2 dP

c

Pc

1 dρ 2α21 + 4πC r 2 3 dP

(C.9)

.

c

The subscript c means that the quantity is evaluated at r → 0. We remark that the solution of Eqns. (C.6), (C.7) does not depend on new arbitrary constants. Outside the star, the equation for m0 reduces to m′0 =

1 ′ 2 6 ′ 2 15 ′ 2 (a ) + (a3 ) + (a5 ) 3 1 7 11 72 2 450 2 2 2 a . + 2 a1 + 2 a3 + 3r 7r 11r 2 5

1−

2M r

(C.10)

Solving Eqns. (C.6), (C.7), (C.10) we find δM from (C.1).

C.2

β(ψ) chosen according to Eqns. (4.22) and (4.23)

If we adopt the form of the function β(ψ) given by Eq. (4.22), the final system becomes 6 1 ρ′ δp0 = (a′1 )2 e−λ + (a′3 )2 e−λ ′ P 3 7 2 2 72 2 450 2 15 ′ 2 −λ a + (a5 ) e + 2 a1 + 2 a3 + 11 " 3r 7r 11r 2 5 # Z

2σ π ψ2 e−ν 2 ¯ ¯ ζ0 |ψ/ψ| − 1 dθ , Θ(|ψ/ψ| − 1) + 4 sin θ 0 m′0 − 4πr 2

ν′ + 2

(C.11)

ρ′ + 1 + 4πreλ (ρ + P ) δp0 P′

1 2λ +e m0 (ρ + P ) 2 + 8πP r δp′0

(

2 3 4 = (ρ + P ) − a′1 c0 − c1 a1 − a3 3 5 7

2 8 10 12 a1 − a3 + a5 − a′3 c1 7 15 15 33

4 1 10 ′ 20 6 − a5 c1 a3 − a5 − (a′1 )2 − (a′3 )2 11 21 39 3r 7r λ λ λ 15 ′ 2 2e 2 72e 2 450e 2 (a ) + 3 a1 + a + a − 11r 5 3r 7r 3 3 11r 3 5 #) "Z

2σ π ψ2 eλ−ν 2 ¯ ¯ ζ0 |ψ/ψ| − 1 dθ . Θ(|ψ/ψ| − 1) − 4r sin θ 0

(C.12)

The system obtained by choosing the function β(ψ) according to Eq. (4.23) is given by the above Equations (C.11), (C.12) with the following substitution: ψ 2 ζ02 → β02 .

80

C. The energy of the system

Appendix D

Quadrupolar deformations As discussed in Sec. 5.1, in order to compute the mass-energy quadrupole moment Q and then the quadrupole ellipticity εQ , we need to solve a system of coupled linearized Einstein’s equations involving the metric corrections h2 (r), m2 (r) and k2 (r). In particular, these equations are obtained from the l = 2 projection (see Eq. (3.32)) of the following components of the Einstein’s equations: [rθ], the combination [θθ] − [φφ]/ sin2 θ and [rr]. If we adopt the choice (4.22) for the function which determines the ratio of toroidal and poloidal fields, the system writes ν ′ 1 eλ 1 ν′ − + − m2 + − h2 r 2 2 r r Z π 2 (3 cos θ − 1) cot θ 5 ψ,θ ψ,r dθ , = 2 4r 0 sin θ k2′

h′2

π eλ 5 ¯ − 1)2σ h2 + m2 = 2 − (ψ,r )2 r 2 e−λ + e−ν ζ02 ψ 2 r 2 (|ψ/ψ| r 4r 0

(3 cos2 θ − 1) ¯ ×Θ(|ψ/ψ| − 1) dθ , sin θ

Z

2 ′ 4 6 2 ν + k2 + h′2 − 2 eλ k2 − 2 eλ h2 r r r r

2λ 1 2e − 2 + 8πP m2 − 8πeλ δp2 r r Z 5 λ π − (ψ,θ )2 + (ψ,r )2 r 2 e−λ = 4e 4r 0

′

¯ − 1)2σ Θ(|ψ/ψ| ¯ − 1) +e−ν ζ02 ψ 2 r 2 (|ψ/ψ| ×

(3 cos2 θ − 1) dθ , sin θ

(D.1)

where ψ =

#

"

a3 (3 − 15 cos2 θ) a5 (−315 cos 4 θ + 210 cos 2 θ − 15) + sin2 θ , −a1 + 2 8 81

82

D. Quadrupolar deformations

and δp2 is the l = 2 component of the pressure perturbation δP = δp0 +δp2 P2 (cos θ) .

The integration can be simplified by introducing the auxiliary function y2 = k2 + h2 + W (r, θ) ,

(D.2)

where 5e−λ π − eλ (ψ,θ )2 + r 2 (ψ,r )2 − 2rψ,θ ψ,r cot θ 16r 2 0 (3 cos2 θ − 1) × dθ . sin θ

Z

W (r, θ) =

(D.3)

This generalizes the variable change adopted in [52, 28]. With the above substitution we are left with two coupled equations 5 4r 2

y2′ + ν ′ h2 = W ′ +

2 2 −λ

× − (ψ,r ) r e

+

Z

π

0

ψ,θ ψ,r cot θ +

¯ − e−ν ζ02 ψ 2 r 2 (|ψ/ψ|

ν′ 1 + 2 r

¯ − 1) 1) Θ(|ψ/ψ| 2σ

(3 cos2 θ − 1) dθ , sin θ " # 8πeλ 2 4 λ ′ λ ′ (ρ + P ) + ′ 2 (e − 1) h2 h2 + ′ 2 e y2 + ν − νr ν′ νr

(D.4)

×

5 = 2 8r

Z

π

0

− ν ′ e−λ r 2 (ψ,r )2 + 2ψ,r ψ,θ cot θ

2 λ ¯ − 1)2σ Θ(|ψ/ψ| ¯ − 1) e (|ψ/ψ| ν′ Z π (3 cos2 θ − 1) 10π × [c0 + c1 ψ]ψ,θ sin2 θ cos θdθ , (D.5) dθ + ′ eλ (ρ + P ) sin θ ν 0

+e−ν ζ02 ψ 2 ν ′ r 2 −

where we have used the following relation (arising from T θν;ν = 0):

δp2 = −(ρ + P ) h2 +

5 4

π

Z

0

[c0 + c1 ψ]ψ,θ sin2 θ cos θdθ

.

(D.6)

To solve the system we impose a regular behaviour at the origin for h2 and y2 , getting h2 (r → 0) = Ar 2 , y2 (r → 0) = Br 4 , (D.7)

where

4π 16π 2 α1 (Pc + ρc /3) − α1 c0 (Pc + ρc ) , (D.8) B = −2πA + 3 3 with Pc and ρc indicating respectively the pressure and mass-energy density at the star’s centre. Inside the star we can write h2 and y2 as

h2 = b1 hh2 + hp2

,

y2 = b1 y2h + y2p ,

(D.9)

where the superscripts h and p denote the homogeneous and one particular solution respectively, while b1 is a constant. We can determine hp2 and y2p by integrating the system with A = 1, while hh2 and y2h are found by solving the homogeneous

83

system. The constant b1 is then obtained by imposing the continuity with the exterior solution, which is written again as (ext)

h2

h (ext)

= Kh2

p (ext)

+ h2

(ext)

,

h (ext)

= Ky2

y2

p (ext)

h (ext)

h (ext)

p (ext)

+ y2

,

(D.10)

p (ext)

are determined by , y2 and h2 , y2 where K is a constant. h2 integrating the system from r → ∞ to r = R. For the particular solution we use the following initial conditions p (ext)

h2

(r → ∞) = −

6µ20 5M r 3

p (ext)

,

y2

(r → ∞) = −

3µ20 , 5r 4

where "

8M 3 2M µ0 = − a1 (R) ln 1 − 3R2 R

2M 2 2M + + R R2

#−1

(D.11)

,

(D.12)

while for the homogeneous case we have a known analytic solution [52, 28] h (ext)

h2

= Q22 (z)

h (ext)

,

y2

2 = −√ Q12 (z) , 2 z −1

(D.13)

where z = r/M − 1 and Qnm are the associated Legendre functions of the second (ext) (ext) kind. By imposing h2 = h2 and y2 = y2 in r = R we can fix both K and b1 .

At this point we have k2 (r), h2 (r), δp2 (r) and m2 (r), and we can compute εQ . If we adopt the other choice for the relation between toroidal and polidal fields, given by (4.23), we proceed in the same way. In this case the final system of equations writes y2′

5 + ν h2 = W + 2 4r ′

′

Z

0

π

ψ,θ ψ,r cot θ +

ν′ 1 + 2 r

¯ − 1)2σ Θ(|ψ/ψ| ¯ − 1) × − (ψ,r )2 r 2 e−λ + e−ν β02 r 2 (|ψ/ψ| (3 cos2 θ − 1) dθ , sin θ " # 4 λ 8πeλ 2 ′ ′ λ h2 + ′ 2 e y2 + ν − (ρ + P ) + ′ 2 (e − 1) h2 νr ν′ νr

×

=

5 8r 2

Z

π

0

(D.14)

− ν ′ e−λ r 2 (ψ,r )2 + 2ψ,r ψ,θ cot θ

2 ¯ − 1)2σ Θ(|ψ/ψ| ¯ − 1) ν r − ′ eλ (|ψ/ψ| ν Z π (3 cos2 θ − 1) 10π × [c0 + c1 ψ]ψ,θ sin2 θ cos θdθ . (D.15) dθ + ′ eλ (ρ + P ) sin θ ν 0 +e−ν β02

′ 2

84

D. Quadrupolar deformations

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