[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: [email protected]

mailto:[email protected]

Ringraziamenti

Il mio primo ringraziamento è per la prof. Valeria Ferrari. Per tutto il miopercorso universitario Valeria è stata la mia guida, dal giorno che ho messo piede nelsuo studio la prima volta, quasi 6 anni fa, ad oggi. Ho avuto lei come relatore dellemie tesi (laurea triennale, laurea specialistica, dottorato) ed ho potuto godere dellasua 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, alpunto che, nonostante mi si siano aperte altre strade, non ho potuto fare a meno dicontinuare a seguire lei. Per tutto ciò che mi ha insegnato, per le possibilità che miha 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 hosvolto in questi anni. Senza di lui questa tesi non sarebbe stata possibile.

Con affetto ringrazio Stefania Marassi. Oltre ad essere stata un’eccellente col-laboratrice, ha sempre dimostrato grande stima nei miei confronti e mi ha offertoun 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 comei colleghi dottorandi e gli altri amici della ‘saletta’. Con tutti loro ho condiviso gliaffanni e le gioie del dottorato.

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

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

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

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

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Contents

Introduction 1

1 Neutron stars: a brief overview 5

1.1 Formation and basic structure . . . . . . . . . . . . . . . . . . . . . . 61.2 Observational properties . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Strongly magnetized neutron stars 13

2.1 The magnetar model . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Magnetar phenomenology . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Stationary axisymmetric magnetized neutron stars:

a relativistic model 17

3.1 The role of equilibrium models . . . . . . . . . . . . . . . . . . . . . 173.2 Model set up: basic assumptions and equations . . . . . . . . . . . . 19

3.2.1 Equations of ideal MHD in General Relativity . . . . . . . . 203.2.2 Perturbative approach and electromagnetic potential . . . . . 213.2.3 The relativistic Grad-Shafranov equation . . . . . . . . . . . 223.2.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Equations of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Twisted-torus magnetic field configurations 27

4.1 Internal magnetic field geometry . . . . . . . . . . . . . . . . . . . . 274.2 The purely dipolar field case . . . . . . . . . . . . . . . . . . . . . . . 304.3 The case with l = 1 and l = 2 multipoles . . . . . . . . . . . . . . . . 324.4 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.4.1 The case with multipoles l = 1, 3 . . . . . . . . . . . . . . . . 364.4.2 The case with multipoles l = 1, 3, 5 . . . . . . . . . . . . . . . 374.4.3 Higher order multipoles . . . . . . . . . . . . . . . . . . . . . 394.4.4 An example of antisymmetric solution . . . . . . . . . . . . . 39

4.5 Magnetic helicity and energy . . . . . . . . . . . . . . . . . . . . . . 394.6 Model extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.6.1 Relative strength of different multipoles . . . . . . . . . . . . 434.6.2 A more general choice of the function β(ψ) . . . . . . . . . . 45

4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.8 Outlook: emergence and stability of twisted-torus configurations . . 50

v

5 Strongly magnetized neutron stars as gravitational wave sources 535.1 Quadrupolar deformations and gravitational waves . . . . . . . . . . 535.2 Single source emission . . . . . . . . . . . . . . . . . . . . . . . . . . 555.3 Gravitational wave background produced by magnetars . . . . . . . 61

5.3.1 Birth rate evolution . . . . . . . . . . . . . . . . . . . . . . . 625.3.2 Background computation . . . . . . . . . . . . . . . . . . . . 635.3.3 Detectability . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.3.4 Wobble angle effects . . . . . . . . . . . . . . . . . . . . . . . 66

Conclusions 69

A The Tolman-Oppenheimer-Volkoff (TOV) solution 71

B The GS system in the l = 1, 3, 5 case 73B.1 β(ψ) chosen according to Eq. (4.1) . . . . . . . . . . . . . . . . . . . 73B.2 β(ψ) chosen according to Eqns. (4.22) and (4.23) . . . . . . . . . . . 74

C The energy of the system 77C.1 β(ψ) chosen according to Eq. (4.1) . . . . . . . . . . . . . . . . . . . 77C.2 β(ψ) chosen according to Eqns. (4.22) and (4.23) . . . . . . . . . . . 79

D Quadrupolar deformations 81

Bibliography 85

vi

Introduction

Neutron stars are the protagonists of the present Thesis. These extremely densestars, whose compactness is higher than any other object in our Universe with theexception of black holes, are of great interest in both Astrophysics and fundamentalPhysics. They represent natural “laboratories” to study Physics under extremeconditions, impossible to realize in terrestrial experiments; in particular, they offera unique chance to understand the behaviour of matter at supranuclear densities,as their observable properties, reflecting the internal microphysics, can be used totest our theoretical predictions on hadron interactions in superdense systems. Thegravitational field of a neutron star is so strong that any accurate description ofits structure requires General Relativity; furthermore, neutron stars are involved inhighly dynamical astrophysical processes and this creates the ideal conditions forthe emission of gravitational waves, an important additional element which makesthem 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 magnetarsor strongly magnetized neutron stars, associated to Soft Gamma Repeaters andAnomalous X-ray Pulsars. The magnetic field of these objects, reaching surfacestrenghts of ∼ 1015 G, is so intense that it significantly affects the star’s struc-ture and evolution. Moreover, it is liable for peculiar emission processes (alreadyobserved or potentially observable) which offer exciting prospects in discerning neu-tron star properties. An example is given by giant flares, so far the only events inwhich neutron star oscillations have been directly detected.

From the point of view of theoretical modelling, our capability to explain theobservations involving magnetars is strongly limited by our scarce knowledge abouttheir internal magnetic field configuration. Motivated by the relevance of such in-formation, in the last years several efforts have been devoted to build equilibriummodels of strongly magnetized neutron stars endowed with magnetic fields of differ-ent geometries. A crucial element is to establish how the toroidal and poloidal fieldcomponents are distributed and how the magnetic energy is partitioned between thetwo1. 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 significanttoroidal component. Assessing the amount of energy associated to toroidal fieldsand then “hidden” inside the star is essential to improve the predictive power ofmodels describing magnetars’ dynamical processes.

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

1

2 Introduction

Equilibrium magnetar models have been developed in both newtonian and rel-ativistic frameworks and with different approaches (we discuss them in Chap. 3).The first models proposed include only poloidal fields [19, 58], while more recentlysome 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 un-known, important hints come from recent results obtained by performing 3D numer-ical magnetohydrodynamics simulation of magnetizd stars in newtonian framework,which suggested that the so-called twisted-torus configurations could be favoured asthey appear more stable than others [13, 15, 14]. In twisted-torus-like configurationsthe toroidal component of the magnetic field is confined to a torus-shaped regioninside the star, and the poloidal component extends throughout the entire star andin the exterior. These results provide strong motivation for studying magnetizedneutron 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 perturbativeapproach. It describes a non-rotating strongly magnetized neutron star surroundedby vacuum, with the assumption that the magnetic field acts as a stationary ax-isymmetric perturbation of a static and spherically symmetric unmagnetized star.Inside the star we adopt ideal magnetohydrodynamics, i.e. we neglect finite electricalconductivity effects. The above assumptions are quite standard in the literature.

The magnetic field configurations we find reproduce the twisted-torus geom-etry (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 addi-tion to the dipolar (l = 1) component usually considered. We use an argumentof minimal energy to find, among the possible solutions, the energetically favouredconfiguration. This allows us to evaluate the ratio of toroidal and poloidal fields interms of magnetic field energy. All these elements constitute an improvement withrespect to previous models.

The equilibrium configurations we obtain can be used as input for studies ondynamical processes involving magnetars. In this Thesis we also consider one ofthe possible applications: the emission of gravitational waves from magnetically-deformed rotating neutron stars. We compute the quadrupolar deformation inducedby the magnetic field on the star’s structure and the resulting gravitational waveemission spectrum. For completeness, we extend the analysis to other models. Fi-nally, we estimate the stochastic gravitational wave background produced by theentire magnetar population, which results from the superposition of the single sourceemissions; we also evaluate the detectability of such background by third generationdetectors 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 theirbasic properties and discuss their relevance for Astrophysics and fundamentalPhysiscs. 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 themagnetar model and the observational evidences supporting it. Next, we givean insight into magnetar phenomenology, providing further motivation to thestudy of magnetars.

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

• In Chapter 4 we focus on the specific twisted-torus-like magnetic field geome-tries we consider and present the results of our computations. We discuss indetail the configurations we have found, emphasizing the improvements withrespect to the previous literature.

• Chapter 5 is devoted to gravitational wave emission from strongly magnetizedneutron 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 mag-netized neutron stars with twisted-torus configurations, MNRAS 406, 2540(2010) ;

3. Marassi S., Ciolfi R., Schneider R., Stella L., Ferrari V., Stochastic backgroundof gravitational waves emitted by magnetars, accepted for publication in MN-RAS (2010) .

Notation and units

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

In Chapters 3, 4 and 5, unless otherwise specified, we will adopt the geometrizedunit 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 forelectromagnetic 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 atthe end of a massive star’s life. They have a typical mass of ∼ 1.4 M⊙ and a typicalradius of ∼ 10 km; the resulting compactness M/R is then ∼ 105 times higher thanthe Sun and makes them the most compact objects endowed with a structure in ourUniverse (a compactness 3-4 times higher would lead to an event horizon, i.e. to ablack hole).

A nice feeling of the unique properties of a NS is given in [46] (Sec. 1.5), wherethe sentence “neutron stars are superstars” (attributed to David Pines) is jokinglystated as a theorem by means of a number of superproperties. They are indeed su-perdense (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 requiresthe general theory of Relativity. They are superfast rotators (fastest known NS hasa 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 mat-ter is partially superconducting or superfluid. NSs possess superstrong magneticfields (surface magnetic fields up to 1013 G for ordinary NSs and up to 1015 G formagnetars). In addition, NSs are superrich in the Physics involved: all the fourfundamental forces play a crucial role in determining their structure and dynamicalprocesses, and the fields of Physics concerned include Nuclear and Particle Physics,Condensed Matter Physics, Plasma Physics and Magnetohydrodynamics, GeneralRelativity, Radio, Optical, X, Gamma, Neutrino and Gravitational Wave Astron-omy, Physics of stellar structure and evolution, Seismology, and so on..

NSs are very interesting as they represent natural “laboratories” to test Physicsunder extreme conditions, impossible to reproduce on Earth. In particular, they of-fer a unique chance to understand the behaviour of matter at supranuclear densitiesthrough the interplay between theory and experiments. Indeed, different theoreticalequations of state (EOS) reflecting our models of hadron interactions in superdensesystems predict different global and observable properties of NSs (such as the com-pactness). A second major element of fundamental Physics justifying the growinginterest for NSs is the emission of gravitational waves. Still unobserved, thesespacetime perturbations are predicted by General Relativity and their existencehas been confirmed indirectly by the Hulse and Taylor observations of the binarypulsar PSR 1913+16 (nobel prize in 1993). A significant and potentially observableamount 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 candidatesas gravitational wave emitters (see Sec. 1.2 and Chapter 5).

The existence of NSs was predicted in 1934 by Baade and Zwicky [7], just afterthe discovery of the neutron by Chadwick [21]. They suggested that a supernovaexplosion would lead to a very high-density star which “consists of closely packedneutrons”. It is worth mentioning that one year before Chadwick’s discovery Lan-dau speculated on the existence of stars of nuclear density [65], even if without theneutron he reached the wrong conclusion that such a star would form by virtueof quantum mechanics violations. An important step was made by Tolman, Op-penheimer and Volkoff [122, 84], who derived the general relativistic equations ofhydrostatic equilibrium for a spherically symmetric star, a fundamental element forany NS model. The first observation of a NS as a pulsar came in 1967 by Bell andHewish [49], giving confirmation to Pacini’s suggestion [85] that a NS could manifestitself 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 confirmedthe association with rotating NS.

Nowadays, about two thousand of NSs are known, all lying in our Galaxy andin the nearby Large Magellanic Cloud (most of them are within a third of theGalaxy radius from the Sun). They are divided in different classes of astrophysicalsources. The most common are the ordinary radio pulsars (PSRs), for which theidentification with magnetized rotating NSs is well established; several hundreds ofPSRs were indeed discovered early, in the 70ies, in the large surveys conducted bythe Arecibo, Jodrell Bank and Parkes observatories. Among the isolated NSs, inaddition to PSRs, we now observe central compact objects in supernova remnants(CCOs), rotating radio transients (RRATs), radio-quiet thermally emitting radiosources (XDINSs or magnificent seven), and finally soft gamma repeaters (SGRs)and anomalous X-ray pulsars (AXPs), also called magnetars and interpreted asstrongly magnetized neutron stars (the most interesting to the purpose of this The-sis). Then, NSs are observed in binary systems with an ordinary companion, or acompact one (a white dwarf, (potentially) a black hole or again a NS). These binariesinclude 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 withan extremely rich phenomenology, with persistent and transient electromagneticemission covering the whole spectrum, neutrino emission and gravitational waveemission. In Chapter 2 we shall focus on strongly magnetized neutron stars anddescribe in some detatil the main steps leading from source discovery to theoreticalinterpretation.

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

1.1 Formation and basic structure

In a star, the hydrostatic equilibrium is assured by the balance between self-gravityand radiation pressure due to thermonuclear reactions taking place in the stellarinterior. 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 suffi-ciently high temperatures in its center and continues until the Hydrogen reserve isover. At this point self-gravity is no longer supported: a new contraction leads tohigher temperatures and could ignite the nuclear burning of Helium and heavier el-ements. 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 involvingelements as heavy as Iron, and in the final stage it becomes a white dwarf; highermass stars (∼> 8 M⊙) reach central temperatures high enough to burn elements upto Iron, and experience a very different final stage. Iron has the greatest bindingenergy, thus producing heavier elements requires energy insted of releasing it; as aconsequence, when a star has formed an Iron core it suddenly happens that nuclearburning stops and this core contracts on free fall timescales due to self-gravity. Ex-tremely high densities are achieved in such contraction, until quantum degeneracypressure is able to contrast gravity: the superdense core formed is a proto-neutronstar; the external layers fall on it and bounce off at high speed producing a supenovaexplosion, 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 massis very high (∼> 20 M⊙) the same event, instead of a NS, produces a black hole. Su-pernova explosions are classified depending on their light curves: type Ib, Ic and IIare produced by the mechanism described above (core-collapse supernovae), whiletype 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 thecompanion star (accretion-induced collapse). This different type of supernova leadsagain to a NS or a black hole.

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

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 thicksurrounds 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 composedby heavy nucleons organized in a Coulomb lattice and embedded in a degenerateelectron gas. Approaching the inner boundary the inverse beta decay is more effi-cient and the fraction of neutrons to protons in nuclei grows; when all bound statesof neutrons are occupied they start leaving nuclei, forming a neutron gas. Thisso-called neutron drip point marks the transition to the inner crust. Here densityincreases up to the nuclear density, 2.67 · 1014 g/cm3. In this region we have twocoexisting phases: proton rich matter, composed of protons and neutrons, and neu-tron gas. An electron gas is also present and assures the matter neutrality. Protonrich matter is clustered in nuclei, chains of nuclei and higher dimensional structuresas density increases (so-called pasta phases), until we have a uniform gas of protons,neutrons and leptons (electrons and eventually muons) in beta equilibrium, whereonly ∼ 10% of nucleons are protons. Such unform gas composes the outer core, up


Figure 1.1. Overall structure of a neutron star.

to densities of 2-3 times the nuclear density. The composition of the inner core, withdensities 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 arepossible: the different hypotheses include the appearence of hyperons (strange heav-ier baryons, e.g. Σ or Λ) or meson condensates, and the transition to a deconfinedphase in which quarks are no longer clustered into nucleons or hyperons.

Crust and core have radically different characteristics and they are describedwith different equations of state (EOS). While the EOS for the inner and outercrust receive a wide consensus, because the models can be tested against dataproduced in Nuclear Physics experiments, for supranuclear densities we only havetheoretical models poorly constrained by extrapolations from lab measurements.Such models are based on two different approaches: the non-relativistic many-bodytheory (NMBT) and the relativistic mean field theory (RMFT). In the NMBTapproach 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 isthe inconsistence with relativistic effects (e.g. it may allow causality violations).The RMFT is more elegant and consistent as its formalism is that of relativisticquantum field theory. The simplest formulation describe the interactions through ascalar and a vector meson field [127]. On the other hand, the dynamics of the systemcan be solved only within a mean field approximation (i.e. a classical treatment ofthe meson fields). Both the approaches can be generalized to include hyperons,at the expense of a larger uncertainty because in this case the interactions aremuch less constrained by data [74]. Concerning the hypothesis of deconfined quarkmatter, 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], whichassumes that (i) quarks are confined to a region of space (the “bag”) whose volumeis 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 definea wide range of compactness. They are briefly described in Section 3.3. We alsouse two standard EOS for the outer and inner crust, the Baym-Pethick-SutherlandEOS (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 observa-tions and some main processes which have the potential to enlarge our understand-ing 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 timingobservations of pulsars in binary systems, both with a main-sequence or a compactcompanion star (i.e. a white dwarf or a second NS). Usually the doppler effect allowsto determine the orbital size as well as the total mass of the binary; then, in manycases the detection of relativistic effects such as shapiro delay or orbit shrinkagedue to gravitational wave emission permits a constraint on the inclination angleand a measure of the two masses. In some cases the masses are determined withimpressive accuracy (one example is the Hulse-Taylor binary PSR 1913+16). Asecond possibility is to infer the mass of an accreting NS in X-ray binaries, but withless accurate results. For each EOS employed to model the NS interior we have avalue for its maximum mass, the NS analogous of the Chandrasekhar mass limit forwhite dwarfs. Therefore, the mass measurement alone can rule out all the EOS witha mass limit lower than the highest value observed. Presently, the mass range 1− 2M⊙ is compatible with the data, even if some low accuracy observations suggest thepossibility of larger masses.

Measuring the radius of a NS is a much harder task. On the other hand, thecombined knowledge of a NS mass and radius is of crucial importance as it allows toput a direct constraint on the EOS. A possible way to infer the NS radius is from themeasurement of both X-ray flux and surface temperature in low-mass X-ray binaries(LMXB). Once combined, these quantities allow to compute the radiation 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 alternativeway is to determine the mass-radius ratio by means of gravitational redshift (z)measurements, through the relation z =

√

1− 2GM/Rc2 − 1 . It is worth men-tioning the result reported in [30], where a gravitational redshift of z = 0.35 isobtained from the observation of Iron and Oxygen transitions in the burst spec-trum 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 softerones including hyperons. More accurate estimates can be obtained using momentof inertia measurements, as in the promising case of the double-pulsar system PSRJ0737-3039 A&B. This system has indeed the largest relativistic corrections knownto the orbital parameters and few years of high precision pulsar timing can leadto an accurate determination of the moment of inertia of the NS with the highestmass; in combination with the already known value of the mass we would have a


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

The spin period (P ) and its derivative (P ) are fundamental quantities for anisolated NS. Their simultaneous knowledge is important for estimating the age (theso-called spindown age τSD = P/2P ) and the surface magnetic field strenght (seebelow). 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 range10−21 − 10−9 (adimesional). As already mentioned spin periods can be extremelyprecise, but many sources (over 100 pulsars) has also shown occational jumps inthe spin evolution, known as glitches, consisting in a sudden increase of the spinfrequency (with a frequency variation in the range 10−9 − 10−6). Their origin isstill matter of controversy. The leading model envisages a fast transfer of angularmomentum from a superfluid component to the rest of the star, which includes thecrust [6]. A superfluid rotates by forming a dense array of vortices which can bepinned 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 thecrust. Once explained, this phenomenon has the potential to shed light on NSmatter properties.

Another process which carries crucial information about the NS internal prop-erties, is NS cooling [87, 132]. Present models predict the time evolution of NStemperature distribution depending on the internal composition (EOS, superfluid-ity, and so on); in particular, the different theoretical surface temperatures (Ts) asfunction of time can be compared with the observations, when a simultaneous mea-sure of the age and Ts of a source is available. Estimating the surface temperatureof a NS is very hard because most of the electromagnetic emission of NSs is usuallynon-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 presentlylimited by (i) a strong model-dependence and (ii) lack of accuracy in the age andtemperature measurements. Standing on the current status of this research field,NS cooling data support the idea that superfluid/superconductive states occur inNS 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 =

√

3Ic3PP

2π2R6 sin2 α, (1.1)

where α is the misalignment angle between spin and magnetic axes. The formulais obtained from the assumption that the external field is dipolar and the observedspindown is entirely due to magnetic braking. There exist other methods basedon estimating the effects of a given magnetic field on the electromagnetic emission(e.g. cyclotron resonant absorption) to be compared with specific X-ray spectralfeatures observed in some isolated NSs, but they are limited by the strong modeldependence of spectral fitting. Typical dipole magnetic field strenghts for ordinarypulsars are in the range 1011 − 1013 G, even if some millisecond pulsars have Bpvalues down to 108 G; on the other hand, magnetars (SGRs and AXPs) have dipolemagnetic 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 sup-porting the existence of magnetars and the crucial role played by magnetic fieldsin determining their evolution and their observational features. Then, the followingChapters describe the original research carried out during the PhD work, and someaspects of magnetars structure and phenomenology will be further discussed.

Concerning the future prospects, the observation of gravitational waves will rep-resent an additional formidable channel to study the nature of NSs. They are indeedthe most promising sources of detectable gravitational signals (together with blackholes) through a large number of processes, from the merger of binary systems tothe continuous emission of rotating NSs, the emission associated with rotational andother instabilities, or related to NS oscillations, and so on. In particular, we willaddress the gravitational wave emission from strongly magnetized NSs in Chapter 5.Ground based detectors such as VIRGO [135] and LIGO [136] already inauguratedthe era of gravitational wave astronomy by putting the first physical constraints onsome known sources; nevertheless, the direct detection is still missing and will hope-fully 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].

Chapter 2

Strongly magnetized neutronstars

In the previous Chapter we have introduced NSs on general grounds, and we havelisted a number of different astrophysical sources identified as NS manifestations.Among them, SGRs and AXPs (commonly referred to as magnetars) are the mostinteresting for the purposes of this Thesis and they deserve a separate introduction.In the present Chapter we sketch the basic elements supporting the magnetar hy-pothesis and discuss the theoretical relevance of the observable processes involvingstrongly magnetized NSs.

2.1 The magnetar model

The history of magnetars starts in 1979, with the detection of an extraordinary eventon March 5. A 0.2 seconds pulse of gamma rays a hundred times more intense thanany previous gamma ray emission detected flooded many space detectors, followedby a fainter and softer tail signal fading out in about three minutes [79]. Thistail interestingly revealed a periodic modulation with a period of about 8 seconds.Surprisingly, another fainter burst were observed the day after from the same spoton 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 theevent belonged to the ordinary gamma ray bursts (GRBs) already observed. Thesource was indeed named “Soft Gamma Repeater”.

By comparing the time of detection of different instruments the source waslocated in the Large Magellanic Cloud (a small galaxy 160 thousand light-yearsaway), and that was again a surprise. At first, the exceptional intensity of theburst led to the idea that the source was in the galactic neighborhood, while theactual measured position was 1000 times farther, implying an enormous energyrelease. Moreover, the position matched that of a young (∼ 104 years old) supernovaremnant. These evidences in combination with the 8 s periodicity strongly suggestedthat the source was a rotating NS.

Such discovery raised a number of questions. The burst mechanism, with suchstrong energy emission and the repetition feature, as well as the spin period of 8s (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 acompanion star). Finally, the position of the source with respect to the supernovaremnant showed an unusually high recoil velocity. In the meanwhile, two othersimilar sources were discovered and it was realized that periods of burst activitywere alternating with periods of quiescence.

A theoretical explanation came in 1992, as a result of a study originally aimedto understand the orgin of pulsar’s magnetic fields. In the famous paper by Duncanand Thompson [34] a scenario is proposed which would lead to the formation ofunusually strong magnetic fields, of the order of 1014 − 1015 G, as a consequenceof the interplay between rotation and convective dynamo action during the firstseconds of life of a very fast spinning NS. Convection in a newly born NS canbe driven by the huge neutrino emission accompanying its early phases, with aconvective overturn time of about 1 ms. If a NS is born with a period as small as1 ms differential rotation can support a very efficient large scale dynamo, in whicha 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 fieldamplification, which acts on smaller scales and can produce a 1012 − 1013 G fieldresulting from the incoherent sum of many smaller dipoles. This second path is whatshould 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 burstSGR can be explained by identifying the source with a magnetar, i.e. a NS endowedwith 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, matchingthe magnetar range. Magnetar-like magnetic fields result in a fast electromagneticspindown (high P ), which also justifies the relatively slow spin frequency of theSGR. Moreover, they propose different mechanisms which would favour a magnetarto have an unsually large recoil velocity, explaining another surprising feature ofthe source. Finally, they argue that the alternation of periods of burst activity andquiescence reveals an evolution compatible with the effects of a dissipating strongmagnetic field. According to the above evidences the SGR involved in the 1979March event represents the “smoking gun” for such strong magnetic fields.

The magnetar hypotesis has been further explored in a series of papers in the90ies [116, 117, 118], where the origin of magnetar-like magnetic fields, as well asthe radiative mechanism for bursting and for quiescent emission are discussed indetail. Concerning the burst emission, a distinction is made between soft repeatedbursts and so-called giant flares such as the 1979 March 5 event. The basic ideais that during its evolution, the magnetic field accumulates stress in the NS solidcrust eventually leading to cracking. In this respect the bursts would result fromstarquakes occurring on the NS surface. A giant flare is instead likely triggered by asudden large-scale rearrangement of the magnetic field, occurring when it becomesunstable to field lines reconnection (note that the growth time of such instability iscompatible with the 0.2 s width of the initial hard spike observed in the March 5event). The persistent X-ray emission detected at that time by Einstein and ROSATdetectors (in addition to the transient burst signals) has not an easy explanationif not assumed to be powered by the surface magnetic field. In this case, a roughestimate 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 commonwith the SGRs and they are probably magnetars as well, even if in a more quiescentphase. This idea has been later supported by the observation of bursts from someAXPs.

Since those years, a number of new observations further supported the magne-tar model. Particularly remarkable are the results obtained by Kouveliotou andcollaborators [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 1979March event); then, from the observed spindown rate, they could estimate a dipolemagnetic 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 sameteam measured a period of 5.16 s, and obtained similar results for the spindownrate and the magnetic field strenght [50].

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

2.2 Magnetar phenomenology

Magnetar-like magnetic field strenghts are liable for a number of interesting featureswhich 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, inaddition to a peculiar persistent emission, a complex burst activity which includesextraordinary events like giant flares, but it has also relevant effects on the globalstructure of a NS as well as on its microphysics, transport properties, and so on. Inorder to further justify the theoretical interest for magnetars and give the feeling ofthe potential they represent, we discuss here an example of physical process amongthe most interesting from the point of view of theoretical modelling, which has beenonly observed in magnetars: magnetar quasi-periodic oscillations (QPOs). Anotherexample is represented by gravitational wave emission from magnetically-inducedstructure deformations. This topic has been addressed in the PhD research activitydescribed in this Thesis and will be the subject of Chapter 5.

The observation of giant flares from three SGRs not only played a fundamentalrole in the discovery of magnetars, but also offers very exciting prospects on NSPhysics. Analysis of the X-ray data taken in the aftermath of these events, cor-responding to the decaying emission tail lasting few minutes, has indeed revealeda 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 Hzfor 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 tor-sional oscillations1; being the oscillating behaviour superimposed to an exponentialdamping, they are referred to as quasi-periodic oscillations (QPOs). If the inter-pretation is correct, such observations represent the only direct detection of NSoscillations, 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 purposein 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 comparedwith theoretical models.

The first attempt to explain the observed frequencies regards them as elastic nor-mal modes of the magnetar’s crust excited by the flare, consistently with theoreticalexpectations [35] (see [114] for a review and references). Many frequencies matchindeed the expected range for crustal seismic oscillation modes [47]. Unluckily, thelower frequencies are difficult to reconcile with the seismic mode interpretation, in-dicating 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 likelycoupled to the magnetar’s core magnetic field and that a predictive model should ac-count 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 thegiant flare, but also in determining the resulting oscillation spectrum. The back-ground magnetic field geometry itself could have important effects on the process2.Many recent papers addressed the problem by modelling elastic oscillations of thecrust [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 couplingbetween elastic and Alfvèn modes did not appear until 2010 [57, 126, 40]. The topicdeserves further investigation.

1 “Torsional” indicates that matter oscillates along the φ-direction.2 This possibility has been considered in this PhD work (in collaboration with the Tuebingen

group headed by prof. Kokkotas); the results are not published.

Chapter 3

Stationary axisymmetricmagnetized neutron stars: arelativistic model

In the present Chapter we focus on equilibrium models of strongly magnetizedneutron stars. We introduce our relativistic model, discussing the basic assumptionsand the formalism. Results from simulations will be presented in the followingChapter (4), where we go deeper into the specific magnetic field configurations wehave studied.

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

3.1 The role of equilibrium models

In the previous Chapter we have discussed the wide phenomenology associatedto magnetars, which makes them astrophysical objects of growing interest. Thepresent-day observations provide strong motivations from the theoretical point ofview, as an accurate modelling of strongly magnetized neutron stars, through thecomparison with the experimental data, has the potential to shed light on unknownproperties of magnetars and neutron stars in general, leading to a deeper under-standing of their internal structure and the Physics involved in their dynamicalprocesses and evolution.

Any attempt to explain the observed features of magnetars, such as quasi-periodic oscillations or (potentially) gravitational wave emission, relies on a gooddescription of their equilibrium configurations. With the above motivation, in thelast years some efforts have been devoted to the description of magnetic field equi-librium configurations of strongly magnetized neutron stars, with both newtonianand relativistic models.

The present models are affected by our lack of knowledge on the geometry andstrenght of internal magnetic fields. A crucial aspect is whether the field has a strongtoroidal 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 freecurrents (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 amagnetosphere surrounding it. This is why we only have direct observations of thepoloidal component of the field. Nevertheless, the existence of a toroidal componentis suggested by a number of indirect evidences: (i) toroidal fields are likely formedin 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 poloidalfield alone nor a purely toroidal field are stable, and that they both evolve ondynamical timescales towards a mixed field configuration (see Sec. 4.1); (iii) theyhelp explaining the amount of magnetic energy released in processes like burstsand giant flares [34, 117]. It is very important to have a prediction of the relativestrenght of toroidal and poloidal fields inside the star, as it strongly affects mostof the magnetar dynamical processes. In particular, we shall discuss the effect onmagnetar gravitational wave emission, which is the subject of Chaper 5. Otherexamples are the already cited burst activity and the thermal evolution ([92] andreferences therein).

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

A different approach has been used in [58], where the magnetic field is treated asa pertubation of a spherically symmetric unmagnetized star. The relativistic modelpresented in these works still describes purely poloidal configurations. In [51, 52]this perturbative approach has been generalized to include toroidal fields; mixedequilibrium magnetic field configurations have been found under the quite restrictiveassumption that the magnetic field is vanishing outside the star, in contrast withthe observations. An analogue assumption is adopted in the newtonian work [48].A further improvement has been achieved in [28], where the authors consider mixedfield configurations as in [52], but with a different treatment of the surface boundaryconditions which allows the magnetic field to extend outside the star. Both themodels 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 fieldin the exterior implies a surface discontinuity in the toroidal field, which can onlybe justified with surface currents whose existence is problematic. This point will befurther discussed in Sec. 4.1.

These first attempts to describe mixed poloidal-toroidal fields inside the starrely on the assumption that the function which controls the ratio of toroidal andpoloidal fields is constant, the simplest possible choice (see Sec. 4.1). Apart fromproblems with the surface boundary conditions, this simple guess on the distribu-tion 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 Braithwaiteand collaborators [13, 15, 14] performed numerical magnetohydrodynamics (MHD)simulations in the framework of newtonian gravity, following the time evolutionof magnetic fields in stars. Starting from a number of different random configura-tions, they found that a particular mixed field geometry, called twisted-torus, is aquite generic outcome of the evolution, and it appears to be stable on dynamicaltimescales (see Chapter 4). This result holds for ordinary stars as well as for neutronstars.

Our relativistic model, which is the subject of the present Chapter and Chapter4, follows the approach of the cited perturbative works. We build configurationsin which the distribution of poloidal and toroidal fields reproduces a twisted-torusgeometry, as suggested by the above results in numerical MHD simulations. Atthe same time, this choice allows us to overcome the problems associated with thesurface 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 formal-ism. The following Section (3.3) is devoted to the description of the equations ofstate employed. In the present Chapter we still consider a generic distribution ofpoloidal and toroidal fields, while we specialize to the case of twisted-torus geome-tries 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 vacuumwith the assumption that the system is stationary and axisymmetric.

The magnetized fluid composing the star is described within the framework ofideal MHD, in which the effects of finite electrical conductivity are neglected. Rigor-ously speaking, this approximation is only justified while the crust is still completelyliquid and while the core matter has not yet performed the phase transition to thesuperfluid state, which is expected to occur at most a few hours after birth (seee.g. Section 5.1 in [1] and references therein). The onset of superfluidity and/orcrystallization limits the period during which magnetostatic equilibrium can be es-tablished. Both the melting temperature and the critical temperature of transitionto the superfluid state, are between 109 and 1010 K, and a typical neutron starquickly cools down below these temperatures in a few hours. However, since thecharacteristic Alfvén time is of the order of τA ≈ 0.01−10 s, depending on the back-ground field strength, there is ample time for the magnetized fluid to reach a stablestate while the state of matter is still liquid (as shown, for example, in [15]). Weremark 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 thestar on a dynamical timescale [120]. After the crust is formed, the magnetic field isfrozen 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 config-uration set within the first day after formation, will fix the magnetic field geometryfor a long time.

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


framework of General Relativity and then we introduce our perturbative approach.Next, we obtain the form of the electromagnetic potential in the general case (notyet specialized to twisted-torus configurations), and we derive the relativistic Grad-Shafranov equation, which can be integrated to give the magnetic field configuration.We use spherical coordinates, xµ = (t, xa, φ), where xa = (r, θ). A stationaryaxisymmetric space-time admits two killing vectors, η = ∂/∂t and ξ = ∂/∂φ, andwith our coordinate choice all the quantities (including the components of the metrictensor 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

Fµν;ν = 4πJµ , (3.1)

F[µν; α] = 0 , (3.2)

where Fµν is the electromagnetic tensor, Jµ is the four-current and the squarebrackets indicate the antisymmetric combination of indexes µ, ν, α. The Equations(3.2) are automatically satisfied by an antisymmetric electromagnetic tensor of theform

Fµν = ∂νAµ − ∂µAν , (3.3)

where Aµ is the electromagnetic potential.According to a comoving observer with four-velocity uµ, the electric and mag-

netic fields are defined as

Eµ ≡ Fµνuν , (3.4)

Bα ≡12ǫαβγδ u

βF γδ , (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

Eµuµ = 0 , Bµu

µ = 0 . (3.6)

In ideal MHD the fluid resistivity is assumed to be negligible: matter behavesas a perfect conductor and there is no electric charge density in the reference framecomoving with the fluid. This gives the additional condition

Eµ = Fµνuν = 0 . (3.7)

The complete set of MHD equations include also the continuity equation andthe equations of motion T µν;ν = 0 . The continuity equation expresses the baryonconservation 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 fluidwith an electromagnetic field in ideal MHD is

T µν = T µνfluid + T µνem , (3.9)


where

T µνfluid = (ρ+ P )uµuν + Pgµν , (3.10)

T µνem =1

4π

[(

uµuν +12gµν

)

B2 −BµBν]

, (3.11)

with ρ and P mass-energy density and fluid pressure respectively, and B2 = BµBµ.Euler’s equations, which determine the system dynamics, are found by projectingthe equation T µν;ν = 0 orthogonally to uµ:

(ρ+ P )aµ + P,µ + uµuνP,ν − fµ = 0 , (3.12)

where fµ ≡ FµνJν is the Lorentz force and aµ = uνuµ;ν is the four-acceleration.Here we summarize the full set of ideal, general relativistic MHD equations.

GENERAL RELATIVISTIC MHD

(i) Continuity equation (nuµ);µ = 0

(ii) Maxwell’s equations Fµν;ν = 4πJµ

(iii) Ideal MHD condition Eµ = Fµνuν = 0

(iv) Euler’s equations (ρ+ P )aµ + P,µ + uµuνP,ν − fµ = 0

3.2.2 Perturbative approach and electromagnetic potential

We assume that the magnetic field acts as a stationary axisymmetric perturba-tion of a static and spherically symmetric background star described by the wellknown Tolman-Oppenheimer-Volkoff (TOV) solution of the Einstein equations (seeAppendix A).

The background metric is

ds2 = −eν(r)dt2 + eλ(r)dr2 + r2(dθ2 + sin2 θdφ2) , (3.13)

where ν(r), λ(r) are given by the unperturbed Einstein equations (TOV) for as-signed 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 theliterature 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 theseEOS. It is worth stressing that the EOS we employ are barotropic, i.e. the pressureis 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 twoEOS (APR2, GNH3) this results in a star radius of R = 11.58 km and 14.19 kmrespectively.


If the magnetic field represents a perturbation of order O(B), it can be shownthat (Fµν , Aµ, Jµ) are of order O(B), while the perturbations (δuµ, δρ, δP , δn, δgµν ,δGµν , δTµν) are of orderO(B2) (see for instance [28]); furthermore, (Bt, At, J

t, Ftν)=O(B3) and (ft, fφ) = O(B4). Therefore, at first order the magnetic field is only cou-pled to the unperturbed background metric (3.13), whereas the deformation of thestellar structure induced by the magnetic field appears at order O(B2). Hence, theapproach allows us to split the problem to obtain the equilibrium configuration ofa magnetized star into two parts: (i) solving the electromagnetic field equations inthe given unperturbed spacetime geometry and for the unperturbed matter distribu-tion; (ii) solving the Einstein equations which determine the variations of spacetimegeometry and matter distribution under the magnetic field configuration found inthe previous step. Here we only consider the first step; in Chapters 4 and 5 we willsolve some components of the perturbed Einstein equations, in order to evaluatethe total energy of the system and to compute the quadrupolar deformation of thestar induced by the magnetic field.

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

Aµ = (0, eλ−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 factthat fφ = −ψ,rJr − ψ,θJθ = O(B4). Using Maxwell’s equations and neglectinghigher order terms, we find

ψ,θψ,r = ψ,rψ,θ , (3.16)

where ψ ≡ sin θΣ,θ. This result implies ψ = ψ(ψ) and allows us to write

sin θΣ,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 ofthe r− and φ−component of the electromagnetic potential, thus the function ζrepresents the ratio between the two; ζ = 0 gives the special case of purely poloidalfield (no toroidal field). Different choices of the function β (or ζ) lead to qualitativelydifferent field configurations; in particular, a proper choice lead to a twisted-torus-like configuration (Sec. 4.1).

3.2.3 The relativistic Grad-Shafranov equation

The Grad-Shafranov (GS) equation, which allows to determine the magnetic fieldconfiguration, can be derived by equating Jφ computed from the φ-component ofMaxwell’s equations,

Jφ = −e−λ

4π

[

ψ,rr +ν,r − λ,r

2ψ,r

]

− 14πr2

[ψ,θθ − cot θψ,θ] , (3.18)


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 + uauνP,ν

= (ρ+ P )(

ν

2− e ν2 δut

)

,a+ P,a +O(B4) . (3.19)

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

P,a = (ρ+ P )(

lnρ+ P

n

)

,a, (3.20)

then (3.19) yieldsfa = (ρ+ P )χ,a , (3.21)

where χ = χ(r, θ). On the other hand, the a-components of the Lorentz forcefµ = FµνJ

ν can be written as (see [28])

fa =ψ,a

r2 sin2 θJφ , (3.22)

with

Jφ = Jφ −e−ν

4πβdβ

dψ. (3.23)

Therefore,

χ,a =ψ,a

(ρ+ P )r2 sin2 θJφ . (3.24)

From χ,rθ − χ,θr = 0 it follows that

ψ,r

(

Jφ(ρ+ P )r2 sin2 θ

)

,θ

− ψ,θ(


)

,r

= 0 ,

which implies(


)

= F (ψ) . (3.25)

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

Jφ =e−ν

4πβ(ψ)

dβ(ψ)dψ

+ (ρ+ P )r2 sin2 θF (ψ) , (3.26)

is the relativistic version of a general result for stationary axisymmetric magnetizedstars with barotropic EOS [22, 23, 95]. The total current Jµ is itself given by thesum of two terms, Jpolµ and JFFµ , where Jpolµ = (0, 0, 0, Jφ) is purely azimuthal and

generates a poloidal field, while JFFµ = − e−νβ(ψ)4πψ Bµ is a force-free current (propor-

tional to the magnetic fied) generating a mixed poloidal-toroidal field. Therefore,toroidal fields are generated together with a poloidal component, and they onlyexist where currents are present, i.e. inside the star.


We assume the following simple form for F (ψ):

F (ψ) = c0 + c1ψ , (3.27)

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

Jφ =e−ν

4πβdβ

dψ+ (ρ+ P )r2 sin2 θ[c0 + c1ψ] . (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 amultipolar expansion as well (see below), but we consider the full coupled multipoleproblem. The choice F = constant adopted in the literature is such to avoid thesecouplings; our choice is more general and allows for a consistent treatment of thesystem 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 isfinally obtained:

−e−λ

4π

[

ψ,rr +ν,r − λ,r

2ψ,r

]

− 14πr2

[ψ,θθ − cot θψ,θ]

−e−ν

4πβdβ

dψ= (ρ+ P )r2 sin2 θ [c0 + c1ψ] . (3.29)

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

a(r, θ) =∞∑

l=1

al(r)Pl(cos θ) , (3.30)

the GS equation rewrites as

−sin θ4π

∞∑

l=1

Pl,θ

(

e−λa′′l + e−λν ′ − λ′

2a′l −

l(l + 1)r2

al

)

−e−ν

4π

[

β(ψ)dβ(ψ)dψ

]

ψ=∑

∞

l=1alPl,θ sin θ

= (ρ+ P )r2 sin2 θ

[

c0 + c1

∞∑

l=1

alPl,θ sin θ

]

. (3.31)

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

Finally, projecting Eq. (3.31) onto the different harmonic components, we obtaina system of coupled ordinary differential equations for the functions al(r). Theprojection is performed using the property

2l′ + 12l′(l′ + 1)

∫ π

0Pl,θPl′,θ sin θ dθ = δll′ . (3.32)

If we consider the contribution of n different harmonics, we need to solve a systemof 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.


3.2.4 Boundary conditions

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

The functions al(r) must have a regular behaviour at the origin; by taking thelimit r→ 0 of the GS equation one can find

al(r → 0) = αlrl+1 , (3.33)

where αl are arbitrary constants to be fixed. Outside the star, where there isvacuum and the field is purely poloidal, Equations (3.31) decouple, and can besolved analytically for each value of l. The solution can be expressed in terms ofthe 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 rl+1 F ([1 − l,−1− l], [−2l], z) , (3.34)

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

a1 ∝ r2

[

ln(1− z) + z +z2

2

]

a2 ∝ r3[

(4− 3z) ln(1− z) + 4z − z2 − z3

6

]

a3 ∝ r4[

(15 − 20z + 6z2) ln(1− z) + 15z − 25z2

2+ z3 +

z4

12

]

. (3.35)

At the stellar surface we require the field to be continuous. This condition is satisfiedif al and a′l are continuous. For practical purposes, the boundary conditions at r = Rcan be written as

a′l = − l

Rflal (3.36)

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 helpof any algebraic manipulator. For our APR2 model (2M/R = 0.357), the values offl 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 constantsαl, associated to the condition (3.33), plus c0 and c1. Thus, we need to imposen + 2 constraints, of which n + 1 are determined by the boundary conditions: nconditions are provided by Eq. (3.36), i.e. by imposing continuity in r = R of theratios a′l/al; the overall normalization of the field gives another condition, which isfixed 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 isthat the surface value of the magnetic field is usually inferred from observations byapplying the spin-down formula, and assuming a purely dipolar external field; formagnetars, the value of Bpole estimated in this way is ∼ 1014 − 1015 G.


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

l>1 al(R)2, is minimum; (ii) by looking at the contribution of higher multipoleswhich minimizes energy in a normalization independent way (thus the energeticallyfavoured 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)) andbelong to the so-called “realistic” EOS, which are tabulated from Nuclear Physicscalculations (as opposite to the analytic ones such as the polytropic EOS, verycommon in the literature).

The Akmal-Pandharipande-Ravenhall EOS (APR2) relies on the assumptionthat 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 two-nucleon potential [129], modified in order to include relativistic corrections (neces-sary to use the nucleon-nucleon potential in a locally inertial frame associated to thestar), and the Urbana IX three-nucleon potential [96] (also modified consistently).The ground state energy is computed using variational techniques [2, 5]. The GNH3EOS proposed by Glendenning belongs to the RMFT approach (Sec. 1.1). Its mostrelevant feature is that it accounts for the appearence of hyperons over a densitythreshold ρH ∼ 2ρ0, where ρ0 is the nuclear density. The hyperons start replacingthe highest energy nucleons when the nucleon Fermi level overcomes the hyperonrest mass, making such replacement convenient. This causes a significant softeningof the EOS as compared to the pure nucleon case, because the hyperons have muchlower kinetic energies than the replaced nucleons. For a star of 1.4 M⊙ (the valuewe assume), the resulting compactness is much lower in the GNH3 case than in theAPR2 case (the radii are 14.19 km and 11.58 km respectively). With these two EOSwe can span a wide range of compactness, which is useful to account for the EOSdependence of our results (see Chapters 4 and 5).

Chapter 4

Twisted-torus magnetic fieldconfigurations

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

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

4.1 Internal magnetic field geometry

The modelling of equilibrium magnetic field configurations in stars has always beenaccompanied by the study of stability of such fields. In principle, it is reasonableto expect the existence of some equilibrium: the two magnetic field or Lorentzforce degrees of freedom (three degrees of freedom from the spatial componentsof the vector ~B or ~fL diminished by one because of the zero-divergence constraint)could be balanced by pressure and temperature gradients. Once a given equilibriumconfiguration is found by combining Maxwell’s and Euler equations, the followingstep should be to assess its stability to an arbitrary perturbation.

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

The final aim of such stablity studies is to establish which magnetic field configu-rations 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 poloidalfield, 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 toroidalcomponent is also present within the shaded region, defined by the magnetic field linesclosed inside the star.

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

In [13, 14, 15] Braithwaite and collaborators performed 3D MHD simulationsof magnetic fields in non-rotating stars (both ordinary and neutron stars) in theframework of newtonian gravity, evolving in time a number of different randominitial configurations. They found as a generic outcome of the evolution a particularconfiguration which appears to be in stable equilibrium on dynamical timescales.Moreover, the results suggest that this could be the only possible stable equilibriumfor non-rotating stars. This so-called twisted-torus configuration is axisymmetricand consists of a poloidal field extending throughout the entire star and in theexterior, and a toroidal field confined in a torus-shaped region defined by the poloidalfield 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 vacuumexterior (where the toroidal component vanishes) and (ii) toroidal field persistencedue to magnetic helicity conservation in the interior, where electric conductivity isvery high. The combined effect causes the migration and confinement of the toroidalcomponent in the region where the field lines are closed inside the star.

These hints about the geometry of internal magnetic fields provide strong mo-tivation to study equilibrium models of magnetized neutron stars with a twisted-torus-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 dic-tated 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 beenstudied in [52, 28, 48]. If the space outside the star is assumed to be vacuum, thetoroidal field, and consequently ζ, must vanish for r > R, where R is the neutronstar radius; therefore, the choice ζ = constant yields an inconsistency, unless oneassumes that surface currents cancel the toroidal field outside the star resulting in asurface discontonuity of the toroidal field (as in [28]), or imposes that the constantζ assumes very particular values (eigenvalue problem) such that all the componentsof the field vanish outside the star (as in [52, 48]).

In our work, in order to reproduce a twisted-torus-like configuration, we startby choosing the following form for the function ζ(ψ)

ζ(ψ) = ζ0

(

|ψ/ψ| − 1)

·Θ(|ψ/ψ| − 1) . (4.1)

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

Bµ =

(

0 ,e−λ2

r2 sin θψ,θ , − e−

λ2

r2 sin θψ,r ,

−e−ν2 ζ0ψ

(

|ψ/ψ| − 1)

r2 sin2 θΘ(|ψ/ψ| − 1)

)

(4.2)

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

In the following we discuss the details of the twisted-torus configurations wehave found, starting from the simple case of a purely dipolar field and proceedingwith 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 afavoured value for the constant ζ0 and then fix the exact ratio of magnetic energy intoroidal and poloidal fields. In this analysis we set the relative strenght of the highermultipoles 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 thesame minimal energy argument used for ζ0, is then employed in Sec. 4.6. All theresults discussed are then summarized in Sec. 4.7.

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


4.2 The purely dipolar field case

We begin discussing the simplest case of a purely dipolar configuration, in whichall 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 similarmagnetic field configurations.

However, when higher order harmonics are taken into account, as we will seein the following Sections, the picture changes. For instance, when ζ0 = 0 and thel = 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 thatin the purely dipolar case it looks as such, is an artifact of the truncation of thel > 1 multipoles. In order to provide a mathematically simple example, whichis useful to understand the structure of the twisted-torus configurations, in thisSection we consider the simplest case c1 = 0. The same choice is widely adopted inthe literature, where the dipolar component is usually the only one considered.

By projecting Eq. (3.31) onto the l = 1 harmonic, and neglecting all contribu-tions from l > 1 terms we find

14π

(

e−λa′′1 + e−λν ′ − λ′

2a′1 −

2r2a1

)

− e−ν

4π

∫ π

0(3/4) Θ

(∣

∣

∣

∣

∣

−a1 sin2 θ

ψ

∣

∣

∣

∣

∣

− 1

)

× ζ20

[

− a1 + 3a1

∣

∣

∣

∣

∣

−a1 sin2 θ

ψ

∣

∣

∣

∣

∣

− 2a31 sin4 θ/ψ2

]

sin3 θ dθ

= (3/4)∫ π

0c0(ρ+ P )r2 sin3 θ dθ = c0(ρ+ P )r2 . (4.3)

The tetrad components of the magnetic field (i.e. the components measured by alocally inertial observer) are:

B(r) =ψ,θ

r2 sin θ,

B(θ) = − e−λ2

r sin θψ,r ,

B(φ) = −e−ν2 ζ0ψ

(

|ψ/ψ| − 1)

r sin θ·Θ(|ψ/ψ| − 1) , (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 meridionalplane, 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 theborder of the region, to a maximum, close to its center. At the stellar surface andin the exterior Bφ vanishes, and there is no discontinuity in the toroidal field. Thepanels 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

-4

-2

0

2

4

0 0.2 0.4 0.6 0.8 1

B(µ

) / (

1015

G)

r/R

B(r)

B(φ)

B(θ)

( ζ0 = 0 km-1 )

-4

-2

0

2

4

0 0.2 0.4 0.6 0.8 1

B(µ

) / (

1015

G)

r/R

B(r)

B(φ)

B(θ)

( ζ0 = 0.40 km-1 )

-4

-2

0

2

4

0 0.2 0.4 0.6 0.8 1

B(µ

) / (

1015

G)

r/R

B(r)

B(φ)

B(θ)

( ζ0 = 0.80 km-1 )

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.


Figure 4.3. The projection of the field lines in the meridional plane is shown for the purelydipolar 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 toroidalfield is confined in an increasingly narrow region close to the stellar surface, whilethe amplitude of the poloidal components (B(r), B(θ)) decreases.

The above feature of our twisted-torus configurations is remarkable, as it impliesthat inside the star we cannot have a magnetic field geometry where the toroidalcomponent dominates with respect to the poloidal one: if |B(φ)| becomes largerwith respect to |B(r)| and |B(θ)|, the domain where it is non vanishing shrinks. Thesame holds in the general case, when other multipoles are included in addition tothe 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 harmonicsl = 1 and l = 2 gives the following coupled equations:

14π

(

e−λa′′1 + e−λν ′ − λ′

2a′1 −

2r2a1

)

−e−ν

4π

∫ π

0(3/4) Θ

(∣

∣

∣

∣

−a1 − 3a2 cos θ

ψsin2 θ

∣

∣

∣

∣

− 1)

× ζ20

[

− a1 − 3a2 cos θ + 3(a1 + 3a2 cos θ)∣

∣

∣

∣


ψsin2 θ

∣

∣

∣

∣

+2 sin4 θ(

− a31 − 9a2

1a2 cos θ − 27a1a22 cos2 θ − 27a3

2 cos3 θ)

/ψ2

]

sin3 θ dθ

= (ρ+ P )r2(

c0 −45c1a1

)

, (4.5)

4.3 The case with l = 1 and l = 2 multipoles 33

14π

(

e−λa′′2 + e−λν ′ − λ′

2a′2 −

6r2a2

)

+e−ν

4π

∫ π

0(5/12) Θ

(∣

∣

∣

∣


ψsin2 θ

∣

∣

∣

∣

− 1)

×ζ20

[

− a1 − 3a2 cos θ + 3(a1 + 3a2 cos θ)∣

∣

∣

∣


ψsin2 θ

∣

∣

∣

∣

+2 sin4 θ(

− a31 − 9a2

1a2 cos θ − 27a1a22 cos2 θ − 27a3

2 cos3 θ)

/ψ2

]

×(−3 cos θ sin3 θ) dθ = −47

(ρ+ P )r2c1a2 . (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 thestellar surface of a1, a

′1, a2, a

′2 with the analytical external solutions given by Equa-

tions (3.35), and (iii) the requirement that the surface (or exterior) contribution ofhigher 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, andbecome

e−λa′′1 + e−λν ′ − λ′

2a′1 −

2r2a1 = 4π(ρ+ p)r2

[

c0 −45c1a1

]

,

e−λa′′2 + e−λν ′ − λ′

2a′2 −

6r2a2 = −16π

7(ρ+ p)r2c1a2 . (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 thematching with the exterior solutions; thus, we need an additional requirement. Weremark that we cannot impose c1 = 0 as in the purely dipolar case, because theratio a′2(R)/a2(R) depends only on c1, and the matching with the exterior solutionis 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 thetrivial solution a2(r) ≡ 0 (with non-vanishing a1). Indeed, from Eqns. (4.7) it isstraightforward 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 isa remarkable property of this system, and it is due to the fact that the integralin θ on the left-hand side of Eq. (4.6) vanishes for a2 = 0 (the integrand becomesodd for parity transformations θ → π − θ). Hence, if we look for a solution whichminimizes the contributions from the l > 1 components at the stellar surface, wehave to choose the trivial solution a2(r) ≡ 0.

If, instead, we do not require that a2(R) is minimum, and we assign a finitevalue to the ratio a2(R)/a1(R), we find a non-trivial field configuration which isnon 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 forζ0 = 0 and a2(R)/a1(R) equal to 1, 1/2 and 1/4 respectively.


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 linecorresponds to ψ = 0.

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

4.4 The general case

When all harmonics are taken into account, there exist two distinct classes of solu-tions: those symmetric (with respect to the equatorial plane), with vanishing evenorder components (a2l ≡ 0), and the antisymmetric solutions, with vanishing oddorder components (a2l+1 ≡ 0). Both solutions satisfy the GS equation (3.31). Letus consider the symmetric class: if a2l = 0 for a given radial coordinate, the in-tegrals arising when Equation (3.31) is projected onto the even harmonics, whichcouple odd and even terms, vanish since the integrands change sign under paritytransformations, 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 equationsfor a2l+1. Similarly, the integrals in Equation (3.31) projected onto the odd har-monics vanish when a2l+1 = 0; thus, we can consistently set a2l+1 ≡ 0, and find theantisymmetric 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 = 1and l = 2 multipoles belong to different families, any attempt to minimize therelative weight of one with respect to the other leads to the trivial solution. Theproperties of Equation (3.31) discussed above, tell us that if a1(R) 6= 0 we cannotconsistently set to zero the remaining odd order components a2l+1. However, wehave the freedom of setting to zero all even terms a2l. Therefore, since we havechosen to minimize the contributions of the l > 1 harmonics outside the star, weshall focus on the symmetric family of solutions (a2l ≡ 0); we will briefly discuss anexample belonging to the antisymmetric family in Section 4.4.4.

4.4 The general case 35

-4

-3

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1

B(µ

) / (

1015

G)

r/R

B(r)B(φ)

B(θ)

( ζ0 = 0 km-1 )

-4

-3

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1

B(µ

) / (

1015

G)

r/R

B(r)

B(φ)B(θ)

( ζ0 = 0.40 km-1 )

-4

-3

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1

B(µ

) / (

1015

G)

r/R

B(r)

B(φ)B(θ)

( ζ0 = 0.80 km-1 )

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 = 0.80km−1, and l = 1, 3.


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 = 0surfaces, 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 = 3components. The projected system is

14π

(

e−λa′′1 + e−λν ′ − λ′

2a′1 −

2r2a1

)

−e−ν

4π

∫ π

0(3/4) ζ2

0

(

ψ − 3ψ|ψ/ψ|+ 2ψ3/ψ2)

×Θ(|ψ/ψ| − 1) sin θ dθ =[

c0 −45c1

(

a1 −37a3

)]

(ρ+ P )r2 , (4.8)

14π

(

e−λa′′3 + e−λν ′ − λ′

2a′3 −

12r2a3

)

+e−ν

4π

∫ π

0(7/48) ζ2

0

(

ψ − 3ψ|ψ/ψ|+ 2ψ3/ψ2)

×Θ(|ψ/ψ| − 1)(3 − 15 cos2 θ) sin θ dθ =215c1(ρ+ P )r2(a1 − 4a3) , (4.9)

where

ψ =

[

−a1 +a3(3− 15 cos2 θ)

2

]

sin2 θ . (4.10)

We again impose regularity at the origin (Eq. (3.33)), continuity in r = R ofa1, 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 wechoose the solution that minimizes the absolute value of a3(R). We find that thereis a discrete series of local minima of |a3(R)|, and we select among them the absoluteminimum.

Fig. 4.5 shows the profiles of the tetrad field components (see Eq. (4.4)) obtainedby numerically integrating Eqns. (4.8), (4.9), for different values of ζ0. B(r) isevaluated at θ = 0, while B(θ), B(φ) are evaluated at θ = π/2. As ζ0 increases, themagnitude of the toroidal field becomes larger, but the region where it is confinedshrinks, as already found in Section 4.2. The projection of the field lines in themeridional plane is shown in Fig. 4.6 for the same values of ζ0. It shows that, forζ0 ∼> 0.40 km−1, 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 = 0km−1 and l = 1, 3. The left panel refers to the solution corresponding to the absoluteminimum of |a3(R)/a1(R)|; in this solution ψ has no nodes. The center and right panelsrefer to solutions corresponding to relative minima of |a3(R)/a1(R)|; in these cases ψ hasone 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 andno toroidal field is present. A similar phenomenon has been discussed in [28]. As wewill see in the next Section, the occurrence of these regions is likely an artifact of thetruncation in the harmonic expansion, and disappears as higher order harmonicsare included.

For completeness we also mention that the solutions corresponding to the localminima of |a3(R)| different from the absolute minimum, correspond to very peculiarfield 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 numberof nodes increases as |a3(R)| increases. These peculiar solutions exist for any value ofζ0, 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 projectingthe GS equation (3.31) onto l = 1, 3, 5 are given in Appendix B (Sec. B.1). Theboundary 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 theprojections of the field lines in the meridional plane corresponding to the samevalues of ζ0. Comparing the results with the case l = 1, 3 we see that the presenceof the harmonic l = 5 modifies the magnetic field shape, but most of the featuresdiscussed in the previous Section are still present.

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


-4

-3

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1

B(µ

) / (

1015

G)

r/R

B(r)

B(φ)

B(θ)

( ζ0 = 0 km-1 )

-4

-3

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1

B(µ

) / (

1015

G)

r/R

B(r)

B(φ)

B(θ)

( ζ0 = 0.61 km-1 )

-4

-3

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1

B(µ

) / (

1015

G)

r/R

B(r)

B(φ)

B(θ)

( ζ0 = 3.00 km-1 )

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 = 0.61km−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 = 0km−1, ζ0 = 0.61 km−1 and ζ0 = 3.00 km−1 respectively, and for l = 1, 3, 5. The toroidalfield is confined within the marked region.

4.4.3 Higher order multipoles

Up to now we have included components with l < 7, neglecting the contributionfrom l ≥ 7. In order to test the accuracy of this approximation, we have studiedthe convergence of the harmonic expansion. To this purpose, we have solved theGS equation (3.31) including odd harmonic components up to l = 7, for ζ0 = 0 andζ0 = 0.61 km−1, and we have computed the quantities

∆(5)(r, θ) =∣

∣

∣

∣

ψl≤5(r, θ)− ψl≤3(r, θ)

ψ

∣

∣

∣

∣

,

∆(7)(r, θ) =∣

∣

∣

∣

ψl≤7(r, θ)− ψl≤5(r, θ)

ψ

∣

∣

∣

∣

. (4.11)

These functions are shown in Fig. 4.10. They are plotted only inside the star sinceoutside they are much smaller. Fig. 4.10 shows that the error in neglecting l ≥ 7,quantified by the function ∆(7), is ∼< 2% for ζ0 = 0 and ∼< 4% for ζ0 = 0.61 km−1.Furthermore, a comparison of ∆(5) and ∆(7) shows that the harmonic expansionconverges.

From the above results we are confident that a truncation of the harmonicexpansion at l = 5 corresponds to a good approximation. Therefore, in what followswe 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 themeridional plane, for ζ0 = 0 km−1 and ζ0 = 0.30 km−1. We remark that the fieldlines are antisymmetric with respect to the equatorial plane; as a consequence, thetotal magnetic helicity is zero (see Section 4.5). Similar zero-helicity configurationshave been considered in Braithwaite [18].

4.5 Magnetic helicity and energy

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


∆(5)

-1-0.5

0 0.5x/R -1

-0.5

0

0.5

1

y/R 0 0.01 0.02 0.03 0.04 0.05

∆(7)

-1-0.5

0 0.5x/R -1

-0.5

0

0.5

1

y/R 0 0.01 0.02 0.03 0.04 0.05

∆(5)

-1-0.5

0 0.5x/R -1

-0.5

0

0.5

1

y/R 0

0.02

0.04

0.06

0.08

∆(7)

-1-0.5

0 0.5x/R -1

-0.5

0

0.5

1

y/R 0

0.02

0.04

0.06

0.08

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 anargument to assign a value to ζ0. Furthermore, we compute the magnetic energy ofthe system to compare the contributions from poloidal and toroidal fields.

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

ds2 = −eν[

1 + 2h(r, θ)]

dt2 + eλ[

1 +2eλ

rm(r, θ)

]

dr2

+r2[

1 + 2k(r, θ)]

(

dθ2 + sin2θ dφ2)

+2i(r, θ)dtdr + 2v(r, θ)dtdφ + 2w(r, θ)drdφ (4.12)

where, in particular, m(r, θ) =∑

lml(r)Pl(cos θ). The total mass-energy of thesystem is

E = M + δM , (4.13)

where M is the gravitational mass of the unperturbed star and

δM = limr→∞

m0(r) . (4.14)

In Appendix C (Sec. C.1), we discuss the equations which allow to determineE. 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 = 0km−1 and ζ0 = 0.30 km−1 respectively, and for l = 2, 4. The dashed line corresponds toψ = 0; the toroidal field is confined within the marked region.

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

E = 2∫

V

(

Tµν −12Tgµν

)

ηµnνdV (4.15)

(where V is the 3-surface at constant time, ηµ is the timelike Killing vector, nµ

is the normalized, future-directed normal to V ); the magnetic contribution comesfrom the stress-energy tensor of the electromagnetic field T µνem (see Eq. (3.11)), i.e.

Em = 2∫

V

(

T emµν −12T emgµν

)

ηµnνdV

=12

∫ ∞

0r2e

λ+ν2 dr

∫ π

0sin θ B2dθ . (4.16)

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

Hm =∫

d3x√−gH0

m , (4.17)

where H0m is the t-component of the magnetic helicity 4-current, defined as

Hαm =

12ǫαβγδFγδAβ . (4.18)

Explicitly, we have

Hm = −2π∫ R

0dr

∫ π

0[Arψ,θ − ψAr,θ]dθ , (4.19)

where

ψAr,θ =eλ−ν

2

sin θψ2ζ0

(

|ψ/ψ| − 1)

·Θ(|ψ/ψ| − 1) ,

ψ,θAr = ψ,θeλ−ν

2 ζ0

∫ θ

0

ψ

sin θ′

(

|ψ/ψ| − 1)

·Θ(|ψ/ψ| − 1)dθ′ . (4.20)

The functional dependence of Hm on the potential of the toroidal field, Ar (seeEquation (4.19)), shows that regions of space where the toroidal field vanishes donot contribute to the magnetic helicity.


0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3

ζ0 [km-1]

δM [10-6 km]

Em [10-6 km]

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

In ideal MHD, the magnetic helicity is a conserved quantity [10, 15]. Thus, if weconsider magnetic field configurations having the same value of the magnetic helicityand 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 minimumat 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 samequadratic dependence on the magnetic field normalization, this means that if wechange Hm the plots of δM and of Em as functions of ζ0 are simply rescaled withrespect to that shown in Fig. 4.12. Consequently, for any fixed value of Hm theposition 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 wealso 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 inter-esting because, as already discussed, the relative weight of the poloidal and thetoroidal components of the field significantly affects many astrophysical processesinvolving magnetars, like magnetar activity [130], their thermal evolution [92], theirgravitational wave emission [31]. It should be stressed that the surface poloidal fieldis inferred from spin-down measurements which, however, provide no hint about thetoroidal 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 isEp/Em ≃ 0.91. It is interesting to note the approximate correspondence betweenthe minimal energy configuration and the configuration with smaller ratio Ep/Em:a larger toroidal component is energetically favoured. Since the toroidal contribu-tion is never higher than ∼ 10%, all our twisted-torus configurations are dominatedby poloidal fields.

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

4.6 Model extensions 43

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

0 0.5 1 1.5 2 2.5 3

Ep

/ Em

ζ0 [km-1]

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 acomplete set of boundary conditions we require a constraint on the relative contri-bution of higher multipoles. Here we remove the condition of minimal contributionfrom higher multipoles adopted so far, thus considering more general configurations.Then, we fix the constraint by choosing among the configurations obtained the en-ergetically favoured one, by means of the minimal energy argument already used tofix the parameter ζ0 (Sec. 4.5).

In the second extention (Sec. 4.6.2) we reconsider our choice of the arbitraryfunction which determines the ratio of toroidal and poloidal fields, ζ(ψ) . Differentchoices will help understanding how the magnetic field geometries obtained dependon 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.

√

a23 + a2

5 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 problemand we are left with a free arbitrary constant. We choose c1 as a “free” parameterand 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,√

a23(R) + a2

5(R) .

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

represents the configuration having the favoured value of√

a23(R) + a2

5(R) for the


0.08

0.085

0.09

0.095

3 3.2 3.4 3.6 3.8 4 4.2

δM /

Hm

[km

-1]

( a32(R)+a5

2(R) )1/2 [10-4 km]

( ζ0 = 0.61 km-1 )A

B

0.081

0.082

0.083

0.084

0.085

0.086

3.3 3.4 3.5 3.6 3.7 3.8

δM /

Hm

[km

-1]

( a32(R)+a5

2(R) )1/2 [10-4 km]

ζ0 = 0.65 km-1

0.61 km-1

0.59 km-1

0.58 km-1

0.52 km-1

Figure 4.14. The function δM/Hm is plotted as a function of√

a23(R) + a2

5(R); on the leftζ0 = 0.61 km−1, on the right the cases ζ0 = 0.65, 0.61, 0.59, 0.58 and 0.52 km−1 are showntogether for comparison.

assigned ζ0. The following step is to extend the analysis to different values of ζ0

and minimize energy with respect to the couple of parameters (ζ0, c1) (equivalent

to the couple ζ0,√

a23(R) + a2

5(R) ). This gives the final favoured configuration.

In Fig. 4.14 we plot the ratio δM/Hm as a function of√

a23(R) + a2

5(R) . In the

left panel we fix ζ0 = 0.61 km−1. In Sec. 4.5 we have shown that, under the MHMassumption, the quantity δM/Hm is minimum for this value of ζ0. This MHMconfiguration corresponds to the point A on the curve plotted in Fig. 4.14.

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

different value of√

a23(R) + a2

5(R) (point B in Fig. 4.14), which corresponds to the

energetically favoured configuration with ζ0 = 0.61 km−1. For an assigned value ofHm, the relative variation of the total energy of the configuration B with respect toA is of the order of 13%. Fig. 4.14 refers to a star with EOS APR2. Similar resultsare obtained for the GNH3 star.

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

pare the different profiles. We have explored the parameter space (√

a23(R) + a2

5(R),ζ0), finding that the function δM/Hm has a minimum (δM/Hm = 0.0817) for

ζ0 = 0.59 km−1 and√

a23(R) + a2

5(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 configurationas the Minimal Energy 1 (ME1) configuration.

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

Hm = −2π∫ R

0dr

∫ π

0(Arψ,θ − ψAr,θ)dθ ; (4.21)


-4

-3

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1

B(µ

) / (

1015

G)

r/R

MHMB(r)

B(φ)

B(θ)

-4

-3

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1

B(µ

) / (

1015

G)

r/R

ME1

B(r)

B(φ)

B(θ)

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 normalizedto the stellar radius. The left panel refers to the MHM configuration (energy is minimizedassuming 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 configurationME1, obtained with no assumption on the relative strengths of the different multipoles, andfor 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 zeroin the inner part of the star, thus Hm receives contributions only from the magneticfield in the region where B(φ) 6= 0. Since in that region the field components of theMHM and ME1 configurations are similar, these configurations have nearly the samemagnetic helicity Hm. On the other hand, the energy δM receives contributionsfrom the field components throughout the entire star, and these contributions arenot vanishing in the region whereB(φ) = 0. When we minimize the function δM/Hm

in the ME1 configuration, the l > 1 multipoles, which were kept minimum in theMHM configuration, do not change Hm significantly, but they change δM , and sincewe require δM/Hm to be minimum, they combine as to reduce the field in the innerregion 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 formsof the function β(ψ) (remember β(ψ) = ψζ(ψ)), namely

β(ψ) = ψζ0

(

|ψ/ψ| − 1)σ

Θ(|ψ/ψ| − 1) , (4.22)

(note that σ = 1 corresponds to Eq. (4.1)), and

β(ψ) = −β0

(

|ψ/ψ| − 1)σ

Θ(|ψ/ψ| − 1) , (4.23)

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


-4

-3

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1

B(µ

) / (

1015

G)

r/R

ME1

B(r)

B(φ)

B(θ)

-4

-3

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1

B(µ

) / (

1015

G)

r/R

ME2

B(r)

B(φ)

B(θ)

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 theprojection of the field lines in the meridional plane. Left and right panels refer, respectively,to the configuration ME1 and ME2.


For β given by Eq. (4.22) the magnetic field components and the GS equationare

Bµ =

(

0 ,e−λ2


λ2

r2 sin θψ,r ,

−e−ν2 ζ0ψ

(

|ψ/ψ| − 1)σ

r2 sin2 θΘ(|ψ/ψ| − 1)

)

(4.24)

and

−e−λ

4π

[

ψ′′ +ν ′ − λ′

2ψ′]

− 14πr2


−e−νζ2

0

4πψ

[

(

|ψ/ψ| − 1)2σ

+ σ|ψ/ψ|(

|ψ/ψ| − 1)2σ−1

]

×Θ(|ψ/ψ| − 1) = (ρ+ P )r2 sin2 θ[c0 + c1ψ] . (4.25)

For β given by Eq. (4.23) they are:

Bµ =

(

0 ,e−λ2


λ2

r2 sin θψ,r ,

e−ν2 β0

(

|ψ/ψ| − 1)σ

r2 sin2 θΘ(|ψ/ψ| − 1)

)

, (4.26)

and

−e−λ

4π

[

ψ′′ +ν ′ − λ′

2ψ′]

− 14πr2


−e−νβ2

0

4πψσ|ψ/ψ|

(

|ψ/ψ| − 1)2σ−1

Θ(|ψ/ψ| − 1)

= (ρ+ P )r2 sin2 θ[c0 + c1ψ] . (4.27)

The field configurations are now identified by three parameters: (σ, c1, ζ0) forthe choice (4.22), and (σ, c1, β0) for the choice (4.23). As in the previous Section, welook 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 forboth cases) with the boundary conditions discussed in Section 3.2.4; the relativestrenght of higher (l > 1) multipoles is fixed by energy minimization as explainedin the previous Section (4.6.1). For each configuration we compute the magnetichelicity Hm, the correction to the total energy δM , and the poloidal and toroidalcontributions to the magnetic energy Em. The equations to determine δM are givenin Appendix C (Sec. C.2). The energetically favoured configurations are found byminimizing δM/Hm with respect to the three parameters.

Let us firstly consider the case in which the relation between toroidal andpoloidal fields is given by Eq. (4.22). We find that the minimal energy configu-ration (for the APR2 EOS) corresponds to

σ = 0.18, ζ0 = 0.20 km−1,√

a23(R) + a2

5(R) = 3.4 · 10−4 km . (4.28)


-4

-3

-2

-1

0

1

2

3

0 0.2 0.4 0.6 0.8 1

B(µ

) / (

1015

G)

r/R

ME3

B(r)

B(φ)

B(θ)

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

We shall refer to this configuration as the ME2 configuration. In Fig. 4.16 theconfigurations with σ = 1 (ME1) and σ = 0.18 (ME2) are compared. In ME2 themagnetic field has a slightly different shape: in particular, the toroidal componentis larger near the surface of the star, and the extension of the toroidal field regionalong the y axis is smaller. The ratio of the poloidal magnetic energy to the totalmagnetic 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 configu-ration with smaller ratio Ep/Em, i.e. with larger toroidal component, as alreadyfound in Sec. 4.5; thus, the σ = 0.18 case also corresponds to the minimum valueof Ep/Em which can be obtained with the choice (4.22). We can conclude thatif σ is not assumed to be 1, we can obtain configurations with a larger toroidalcontribution, but only by a small amount.

We now consider the choice (4.23) for the function β. In this case the minimalenergy configuration (for the APR2 EOS) corresponds to

σ = 0.42, β0 = 9 · 10−4,√

a23(R) + a2

5(R) = 3.7 · 10−4 km . (4.29)

This configuration, which is shown in Fig. 4.17, will be referred to as the ME3configuration. A comparison with the right panel of Fig. 4.16 shows that the ME2and ME3 configurations are very similar. Inside the star Ep/Em = 0.88 and, as inthe previous case, it is the minimum value which can be obtained for this choice ofthe function β.

For the GNH3 EOS, we obtain similar results. The minimal energy configurationis obtained with the choice (4.22), and

σ = 0.30, ζ0 = 0.13 km−1,√

a23(R) + a2

5(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 thisconfiguration is Ep/Em = 0.93.

We conclude that when we allow for a non-minimal contribution of the l > 1 mul-tipoles and for a more general parametrization of the function β(ψ), the magneticfield 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 thestellar 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 ourconfigurations magnetic fields are always dominated by the poloidal component.

4.7 Summary

We have found numerical solutions of equilibrium magnetic field configurations ofstrongly magnetized NSs in General Relativity. The family of solutions we havefound reproduce the so-called twisted-torus magnetic field geometry, in which thetoroidal component is confined to a torus-shaped region inside the star, and thepoloidal component extends throughout the entire star and in the exterior. Twisted-torus-like configurations have been found as a final outcome of Newtonian MHDsimulations with generic initial conditions [13, 14, 15], and appear to be more sta-ble than others. In these configurations, the existence of a toroidal component ofthe magnetic field in the star’s interior is allowed, while vanishing in the exterior,without assuming neither discontinuities (associated to surface currents) nor thevanishing of all magnetic field components outside the star. This is an improvementwith 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 theircouplings. We have included multipoles up to l = 5, showing that this choicecorresponds to a good approximation. We have fixed the relative contribution ofhigher multipoles with respect to the dipole at first by taking the minimum, thenusing an argument of minimal energy, based on magnetic helicity conservation inideal MHD systems.

Our configurations are determined by one or more parameters, depending onthe choice of the function relating toroidal and poloidal fields. In the differentcases considered, we have found the set of parameters minimizing the energy in anormalization independent way, by means of the same argument mentioned above.This also allowed to determine the favoured ratio of toroidal and poloidal magneticfield energies (the relevance of this quantity for magnetar models has been widelydiscussed). As a result, we have found that all our configurations are dominated bythe poloidal field, with a maximum toroidal contribution of ∼ 10%.

The equilibrium configurations we have found can be used as input for studieson magnetar dynamical processes. The next Chapter (5) is devoted to one of suchapplications: gravitational wave emission from strongly magnetized NSs.

We stress that the (newtonian) numerical studies leading to the idea that twisted-torus 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 generalassumptions. In the last months of the PhD we started working on a project aimedto provide further hints on the subject. This is sketched in the next Section.


4.8 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 aboutthe 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 whichthe star is non rotating and magnetic fields are random. A useful extension shouldaccount for a more general sample of configurations. On the other hand, a fun-damental improvement would consist in obtaining more reliable predictions at aquantitative level, to be used in models concerning magnetars’ structure and dy-namics; 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 thisrespect, as already pointed out, one of the main limitations so far is that the sim-ulations are performed in a newtonian (non relativistic) framework, while accuratecomputations involving NSs require General Relativity.

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

This research is only at the preliminary stage and here we only give some ad-ditional detail. The physical system of our interest consists in an isolated NS sur-rounded by vacuum and endowed with a magnetic field permeating the star andextending to the exterior. At first, we consider a non rotating star. We want tofollow the MHD evolution of the system prepared according to a given set of ini-tial conditions, in order to establish if an equilibrium magnetic field configurationis reached within the timescale we consider. We are interested in the existence ofsuch equilibrium and its features, depending on the initial condition chosen and ona set of physical parameters which define the general properties of the star (suchas mass and EOS) and of its magnetic field (such as magnetic helicity). Simula-tions are carried out by means of numerical codes available (as developer) to theNumerical Relativity group at the AEI: the ‘Cactus’ infrastructure [139] with allits 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 equa-tions and high-resolution shock-capturing schemes; (ii) fluxes are computed usingthe HLLE approximate Riemann solver; (iii) the divergence-free condition for themagnetic field is guaranteed using the constrained-transport approach (see [42] andreferences therein). Starting from the existing codes we are developing an oppor-tunely modified version of them, optimized for the purposes of the project. To runthe simulations we employ the computational resources of the institute.

We adopt some physical assumptions. The most relevant, on which WhiskyMHDis based, consists in treating the system in ideal MHD, i.e. in the limit of infiniteelectrical conductivity. As discussed in Sec. 3.2, this assumption is valid while thestar is still completely liquid (i.e. the solid crust has not yet formed) and the corematter has not yet performed the phase transition into the superfluid state, which isexpected to happen at most a few hours after birth. On the other hand, the Alfvèntimescale for the evolution of the magnetic field is orders of magnitude shorter, thusthere is ample time for the magnetized fluid to reach an equilibrium configurationbefore the crust formation or the onset of superfluidity; the following evolution pro-ceeds on much longer timescales, hence the field geometry stays unchanged for along time. In conclusion, the configurations we find describe the magnetic field ofa NS at the time of crust formation, but they are also representative of the config-urations realized in following times. As an additional assumption we work in theso-called ‘Cowling approximation’, in which the spacetime evolution is neglected.This approximation is fully justified for the system we consider and allows a signif-icant reduction of the simulation time; the code can work without this assumption,thus we are able to give a comparative demonstration that the approximation isvalid.

At the moment our work is focused on developing and testing the code. Inaddition to a number of standard numerical tests (e.g. convergence tests), we areemploying and comparing two different methods to assure the divergence-free condi-tion for the magnetic field: the constrained-transport approach already implementedin WhiskyMHD, and a vector potential formulation of the induction equation, whichleads naturally to a null magnetic field divergence. This will allow us to choose themore suitable method. We are also running a number of physical tests, in order toassure 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 presentcode cannot simulate vacuum, a low-density atmosphere in which the fluid velocityis set to zero is necessarlily present. If we want that the magnetic field evolution inthe exterior behaves with good approximation as prescribed by Maxwell’s equationsin vacuum, the assumption of ideal MHD outside the star results inappropriate. Abetter approximation can be obtained by introducing a dissipative term, whose ef-fect is to decouple the magnetic field from the fluid in the atmosphere, allowing itsevolution, and to damp the non-zero laplacian components of the field, the samecomponents that would propagate to infinity according to Maxwell’s equations. Theevolution in the atmosphere has to be object of accurate tests in order to guaranteethe soundness of results.

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


By adjusting the initial magnetic helicity we will be able to change the ratio oftoroidal and poloidal fields, to study its effect on the final equilibrium and to estab-lish the range of values for which reaching an equilibrium is possible. Since purelypoloidal or purely toroidal configurations appear to be unstable in non rotatingstars (see Sec. 4.1), we expect that a stable ratio between the two componentsis limited within a finite range of values. We stress (again) that the assessmentof this ratio is of great importance, because observations can only give us directinformation about the external poloidal fields, which in turn can provide us withan indication of the internal poloidal field strenghts; on the other hand, we have nodirect way to evaluate the internal toroidal fields and how much magnetic energythey hide inside the star. We will consider a number of polytropic or ideal fluidEOS (in future developments, it will be possible to employ also ‘realistic’ EOS) andwe will vary the star’s mass and the magnetic field strenght. At last, we will havea sample of configurations representative of the possible field geometries realized instrongly magnetized NSs.

Chapter 5

Strongly magnetized neutronstars as gravitational wavesources

The present Chapter is devoted to the emission of gravitational waves (GWs) frommagnetically-deformed rotating NSs. We discuss the quadrupolar deformations ofa strongly magnetized NS and the related GW emission mechanism. Then, wecompute, according to our equilibrium model and other models, both the emissionspectrum for a single magnetar and the stochastic background produced by theentire magnetar population. Finally, we discuss the detectability of such backgroundby 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 fieldinduces perturbations on the static and spherically symmetric spacetime metric ofthe unperturbed star. If the system is assumed to be stationary and axisymmetricthe perturbed metric can be written in the form [52, 28]

ds2 = −eν(

1 + 2[h0(r) + h2(r)P2(cos θ)])

dt2

+2[

i1(r)P1(cos θ) + i2(r)P2(cos θ) + i3(r)P3(cos θ)]

dtdr

+2 sin θ(

v1∂

∂θP1(cos θ) + v2

∂

∂θP2(cos θ) + v3

∂

∂θP3(cos θ)

)

dtdφ

+2 sin θ(

w2∂

∂θP2(cos θ) + w3

∂

∂θP3(cos θ)

)

drdφ

+eλ[

1 +2eλ

r

(

m0(r) +m2(r)P2(cos θ))

]

dr2

+r2[

1 + 2k2(r)P2(cos θ)]

(

dθ2 + sin2θdφ2)

. (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 withframe dragging effects, while the functions h, m and k determine structure defor-mations. In particular, the structure deformations consist in (i) mass, radius andenergy variations, alredy considered in Sec. 4.5 and related to the radial functionsh0 and m0, and (ii) quadrupolar deformations, given by h2, m2 and k2, which arerelevant for GW emission.

In this Section we compute the quadrupolar deformations induced by the mag-netic field for the twisted-torus configurations previously obtained. To this purpose,we have to solve some perturbed Einstein’s equations; in particular, the componentsinvolved are [rr], [θθ], [φφ] and [rθ]. In Appendix D we discuss in detail the resultingcoupled 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 thestar along the symmetry axis. According to its basic definition, the ellipticity of thestar is given by

ε =(equatorial radius)− (polar radius)

(polar radius), (5.2)

and can be written as a function of the radial coordinate r as [24]

ε(r) = −32

[

2δp2

rν ′(ρ+ P )+ k2

]

, (5.3)

where δp2 is the l = 2 component of the pressure perturbation. The surface el-lipticity 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 mass-energy distribution; εQ has a different definition and has to be distinguished fromε.

The quadrupole ellipticity is defined as

εQ =Q

I, (5.4)

where Q is the mass-energy quadrupole moment, and I is the mean value of themoment of inertia of the star. The quadrupole moment Q is given by the far fieldlimit of the spacetime metric

h2(r →∞) ∼ Q

r3, (5.5)

and then it is obtained by solving the relevant Einstein’s equations and computingthe metric correction h2(r). The value of I can be estimated from the limit ω → 0of the ratio J/ω in a slowly rotating star model, where ω is the angular velocityand 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(B2) and then scalesas ∝ B2

pole 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 havea time-varying mass quadrupole moment, which implies a continuous emission ofGWs. The GW amplitude of such signal, detected from a source at a distance r, is[12]

h0 ≃4Iω2

r|εQ| sinα , (5.6)

5.2 Single source emission 55

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

In strongly magnetized NSs the quadrupolar deformation is determined essen-tially by the magnetic field configuration and strength, thus the εQ that we computefrom magnetic deformation can be safely used in determining the GW emission.Fast rotation will also induce a non-negligible deformation; however, being symmet-ric with respect to the spin axis, such deformation does not contribute to the GWemission and it is not included in our analysis.

We can now come to the results of our computations. It is well known that whilethe poloidal field tends to make the star oblate (εQ > 0), the toroidal field tends tomake it prolate (εQ < 0). In a mixed toroidal-poloidal field configuration the actualdeformation results from the balance of the two opposite effects. Since in our con-figurations the poloidal field dominates over the toroidal one, εQ is always positiveand the deformation is larger for configurations in which the toroidal contributionis 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 energeticallyfavoured configurations ME2 and the ME3 (see Sec. 4.6.2). It is also interesting toconsider twisted-torus configurations which do not correspond to minimal energy.We have determined the entire range of possible ellipticities for the twisted-torusconfigurations analyzed in our model; we have found 3.5 · 10−6 ∼< εQ ∼< 4.8 · 10−6

for the APR2 EOS, and 8.1 · 10−6 ∼< εQ ∼< 9.6 · 10−6 for the GNH3 EOS. The largerellipticities are obtained in the purely poloidal limit, whereas the smaller refer tothe minimal energy configurations, being also the configurations having the highestcontribution 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 ellip-ticities. We also note that the values of εQ we find for the purely poloidal case arecomparable to the maximal ellipticity found in [66].

Summarizing, the quadrupole ellipticity εQ corresponding to Bpole = 1015 Gwould 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.

εQ ≃ k(

Bpole1015 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 obtainedchoosing β(ψ) as in Eq. (4.23), assuming that σ = 1 and that the contribution ofthe 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 poloidalconfiguration.

5.2 Single source emission

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


3.7

3.9

4.1

4.3

4.5

4.7

4.9

0 10 20 30 40 50

ε Q [1

0-6]

β0 [10-4]

Figure 5.1. Ellipticities versus β0 for σ = 1, with β(ψ) given by (4.23). Here we assumeBpole = 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 entiremagnetar population, for which the single source emission constitutes the basicingredient; then, we will evaluate the detectability of such background by thirdgeneration detectors such as the Einstein Telescope [138].

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

The GW energy spectrum emitted by a single source writes as

dEGWdfe

= EGW

∣

∣

∣

∣

dfedt

∣

∣

∣

∣

−1

, (5.8)

where fe is the emission frequency. The GW luminosity of a rotating NS withspin axis forming a wobble angle α with the magnetic axis is composed of twocontributions, one at the spin frequency νR, one at its double 2νR; it can be writtenas

EGW =2G5c5

I2ε2Qω

6 sin2 α(cos2 α+ 16 sin2 α) , (5.9)

where εQ is the star quadrupole ellipticity induced by the magnetic field, ω is theangular velocity and I is the moment of inertia. The term sin2 α cos2 α is from theemission component at the spin frequency, the term 16 sin4 α from the componentat twice the spin frequency.

The star loses rotational energy mainly due to electromagnetic radiation andGW emission (we shall neglect other effects, e.g. relativistic winds [32]). Accordingto the well known vacuum dipole radiation model, the energy loss rate due to dipoleradiation is given by

|EdipROT | =16

B2poleR

6

c3ω4 sin2 α . (5.10)


The total spin-down rate obtained from Eqs. (5.9) and (5.10) is

|ω| = |ωdip|+ |ωGW | (5.11)

=16

B2poleR

6

Ic3ω3 sin2 α+

2G5c5

Iε2Qω

5 sin2 α(1 + 15 sin2 α) .

Using the above quantity and remembering that the first term of the GW luminositygiven in Eq. (5.9) is emitted at fe = νR = ω/2π, while the second at fe = 2νR, wecan 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 thefrequency ranges where the two components of the GW emission contribute: theemission at νR contributes to frequencies below 1/P0, that at 2νR to frequenciesbelow 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 spectrumare:for fe < 1

P0,

dEGWdfe

=32π4G

5c5I2ε2

Qf3e

×

cos2 α

[

B2poleR

6

6Ic3+

8π2G

5c5Iε2Qf

2e (1 + 15 sin2 α)

]−1

+ sin2 α

[

B2poleR

6

6Ic3+

2π2G

5c5Iε2Qf

2e (1 + 15 sin2 α)

]−1

; (5.12)

for 1P0< fe <

2P0

,

dEGWdfe

=32π4G

5c5I2ε2

Qf3e

× sin2 α

[

B2poleR

6

6Ic3+

2π2G

5c5Iε2Qf

2e (1 + 15 sin2 α)

]−1

; (5.13)

for fe > 2P0

,

dEGWdfe

= 0 .

If we put reasonable numbers in the equations, assuming Bpole = 1014 − 1015

G, R ∼ 10 km, I ∼ 1045 g cm2 and fe ∼< 1 kHz, we see that even for quadrupole

ellipticities as large as 10−4 the termB2pole

R6

6Ic3 is much larger than 8π2G5c5 Iε2

Qf2e ; in this

case the contribution of GW emission to the spin-down is negligible. As shown inthe following, this holds in most of the cases we consider. It is worth noting that,

when α 6= 0 andB2pole

R6

6Ic3 ≫ 8π2G5c5 Iε2

Qf2e , for fe < 1/P0 the dominant term in Eq.

(5.12) is

dEGWdfe

=32π4G

5c5I2ε2

Qf3e

(

B2poleR

6

6Ic3

)−1

, (5.14)


which does not depend on the wobble angle α. However, for 1P0

< fe <2P0

, thedominant term is

dEGWdfe

=32π4G

5c5I2ε2

Qf3e sin2 α

(

B2poleR

6

6Ic3

)−1

, (5.15)

and it depends on α.It should be stressed that in general the wobble angle depends on time. The mis-

alignment of magnetic and rotation axes causes, in the NS frame, the free precessionof the angular velocity around the magnetic axis with period Pprec ≃ P/|εQ|, whereP is the spin period [63, 64]. The star internal viscosity damps such precessionalmotion and reduces the wobble angle towards the aligned configuration (α = 0), ifthe star has an oblate shape (εQ > 0), whereas it increases α towards the orthogonalconfiguration α = π/2 (“spin-flip”), if the shape is prolate (εQ < 0) [62, 31]. Thesecond case is more favourable for GW emission.

The timescale of the process is given by τα = nP0/εQ, where P0/εQ is theinitial precession period and n is the expected number of precession cycles in whichthe 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 themisalignment 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 processtakes place when the source is no longer an efficient GW emitter.

In principle, an accurate estimate of the GW emission from a single magnetarshould account for (i) its initial wobble angle (to be chosen according to a properpopulation distribution), and (ii) the evolution of such misalignment with time. Thiskind of analysis would, however, be affected by the wide uncertainties on both ταand the initial angle distribution. As we shall show in Section 5.3.4, from the pointof view of the GW background produced by the entire magnetar population and itsdetectability, the value of α assigned to the single magnetar and its eventual changewith time do not significantly affect the results. Since our main interest is focusedon such detection prospects, we proceed here with the simplifying assumption thateach single magnetar is born with α = π/2 and that the misalignment evolution isineffective. Then, in Section 5.3.4, we will consider the effects of a generic wobbleangle. 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 stellardeformations. If we want the value of Bpole to be representative of the magnetarpopulation, it should be chosen as a suitable average. Such an average is uncertainat present; however the values of Bpole inferred from AXPs and SGRs lie in the1014 − 1015 G range [81]. Our choice here is to span this range by studying its twoextremes, 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 fmaxe = 2/P0 = 2500 Hz.If α < π/2, part of the GW emission is at the spin frequency and the correspondingcontribution has a frequency cutoff fmaxe = 1/P0 = 1250 Hz. The chosen value ofP0 implies a very fast spinning newborn NS, but still consistent with the believedrange of NS spin rates at birth. We remark that, according to current scenarios ofmagnetar formation, strongly magnetized NSs are those that are born with periods


of the order of ms, much faster than ordinary pulsars [34]. At the end of Section5.3.3 we will sketch the effect of assuming lower initial spin frequencies.

In the following, we compute the single source GW emission spectrum accordingto our model and its predictions on εQ discussed in the previous Section, but thesame analysis is also extended to other models. In particular, these models (de-scribed below) account for purely poloidal fields or mixed fields dominated by thetoroidal component. The different models considered are representative of the pos-sible magnetic field configurations realized in magnetars, according to our presentknowledge.

Purely poloidal magnetic field

Here we consider two purely poloidal magnetic field configurations which have al-ready been used in [98] to evaluate the magnetar GW emission and the correspond-ing 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 isalways positive. In addition to α = π/2 and M = 1.4 M⊙, the numerical inputs tocompute dEGW/dfe are Bpole, εQ, I and R. As previously discussed, we adopt twodifferent values of Bpole: 1014 and 1015 G. Following [59], we write the quadrupoleellipticity as

εQ = gB2poleR

4

GM2, (5.16)

where the value of the dimensionless parameter g accounts for the magnetic fieldgeometry 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 incompressiblefluid star and a dipolar magnetic field [39]; similar values are obtained in relativisticmodels based on polytropic EOS [59]. Model B describes a scenario in which theNS core is a superconductor of type I, implying that the internal magnetic field isconfined 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 toroidalfields inside the star. It assumes a magnetic field configuration with an internaltoroidal field of ∼ 2 · 1016 G (core-averaged value), in addition to a poloidal field ofordinary strength (1014−1015 G). In [110] it is suggested that toroidal field strengthsof this order are needed to explain the time-integrated emission of magnetars asinferred from the extremely bright giant flare that took place on December 27, 2004from SGR 1806-20 (which liberated an energy of 5 · 1046 erg). Giant flares of thismagnitude could result from large-scale rearrangements of the core magnetic fieldor instabilities in the magnetosphere [119, 73]. Such a huge toroidal field wouldinduce prolate deformations as strong as εQ ≃ −6.4 · 10−4 [31]. The deformationassociated to poloidal fields, whose strength is fixed by the choice of Bpole, tends tooppose 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,


1e+34

1e+36

1e+38

1e+40

1e+42

1e+44

1e+46

1e+48

1e+50

10 100 1000

dE/d

f e [e

rg/H

z]

fe [Hz]

Bp=1014 G

TT-APR2TT-GNH3

P-AP-BTD

1e+34

1e+36

1e+38

1e+40

1e+42

1e+44

1e+46

1e+48

1e+50

10 100 1000

dE/d

f e [e

rg/H

z]

fe [Hz]

Bp=1015 G

TT-APR2TT-GNH3

P-AP-BTD

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

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

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

Gravitational wave emission spectrum according to the different models

We can now present the GW spectrum emitted by a single magnetar obtainedaccording to the different models considered: the two purely poloidal models A andB (hereafter P-A and P-B), our twisted-torus model (hereafter TT) for the twoequations 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 frequencyfe. In the left (right) panel we assume Bpole = 1014 G (Bpole = 1015 G); as alreadydiscussed, these values define a likely range for Bpole.

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

Let us now focus on the Bpole = 1014 G case (left panel). The TD model is byfar 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 isobtained with the P-B model, where large deformations are achieved even for thislower field strength. The emission predicted by the P-A model is lower by morethan three orders of magnitude, due to the difference in the g2 factor appearingin the expression of dEGW /dfe. The two TT models are expected to give evenweaker signals; they differ for the assumed EOS and the one which gives less (more)compact stars, GNH3 (APR2), is associated to stronger (weaker) deformations andGW emission.

If we consider higher external poloidal fields (Bpole = 1015 G, right panel) thepicture changes. For the P-A, P-B and TT models the value of Bpole controls boththe GW luminosity, which scales as B4

pole (εQ ∝ B2pole), and the spin-down rate,

which has the electromagnetic contribution proportional to B2pole plus a very small

correction due to GW emission. As a result, dEGW /dfe ∝ B2pole, which translates

in a factor 100 increase from Bpole = 1014 to 1015 G. Conversely, in the TD modelthe 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 Bpoleonly results in a higher electromagnetic spin-down and in a smaller overall GWemission. As long as the electromagnetic spin-down dominates over the GW spin-down, dEGW /dfe is reduced by a factor of 100 from Bpole = 1014 to 1015 G. Thefinal result is that when Bpole = 1015 G, the prediction of TD and P-B models arecomparable.

It is worth noting that in all the considered models the contribution given bythe GW emission to the spin-down is negligible, with the exception of the early timeevolution in the TD model with Bpole = 1014 G (hereafter TD14). This is shownin Fig. 5.2, where the energy spectra are linear in logarithmic scale, reflectingthe behaviour dEGW /dfe ∝ f3

e , while the TD14 model is characterized by a loweremission level at high frequency, due to a non-negligible GW spin-down. This effectis even more evident in Fig. 5.3, where we compare the GW spectrum for the sameTD model shown in the left panel of Fig. 5.2 with a TD model where GW spin-downis neglected (dashed line). It is clear that the GW contribution starts to be relevantat fe ∼ 300 Hz. For all the other models we have discussed, this contributionbecomes relevant at much higher emission frequencies.

5.3 Gravitational wave background produced by mag-

netars

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

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


1e+46

1e+47

1e+48

1e+49

1e+50

1e+51

100 1000

dE/d

f e [e

rg/H

z]

fe [Hz]

Bp = 1014 G

TD-IITD-I

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

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

Fundamental elements to compute the GW background, in addition to the sin-gle source emission spectrum evaluated in Sec. 5.2, are the cosmic star formationrate evolution and the corresponding magnetar birthrate. This is discussed in thefollowing Section. These inputs depend on the cosmological model assumed: inour 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 bythe numerical simulation of [124]. We only consider the formation rate of PopulationII stars, adopting a Salpeter Initial Mass Function (IMF) Φ(M) ∝ M−(1+x) withx = 1.35 (normalized between 0.1 − 100 M⊙), in regions of the Universe whichhave 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 beenderived using population synthesis methods; it is shown that NSs born as magnetarsrepresent 8-10% of the total simulated population of NSs. Here we assume a fractionfMNS = 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 isstill debated (see for instance [38, 33]). However, the range we consider is sufficientlylarge to include the proposed evolutionary models.


1e-06

1e-05

1e-04

0.001

0.01

0.1

1

0 2 4 6 8 10 12 14

ρ. ⋆ [M

⊙ y

r-1 M

pc-3

]

z

1e+04

1e+05

1e+06

1e+07

1e+08

1e+09

0 2 4 6 8 10 12 14

MN

S r

ate

[yr-1

]

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 forma-tion rate density inferred from the simulation1. The number of magnetars formedper unit time out to a given redshift z can be computed by integrating the cosmicstar formation rate density, ρ⋆(z), over the comoving volume element, while restrict-ing the integral over the stellar IMF in the proper range of progenitor masses; thatis

RMNS(z) = fMNS

∫ z

0dz′

dV

dz′ρ⋆(z′)

(1 + z′)

∫ 40 M⊙

8 M⊙dMΦ(M) , (5.17)

where the factor (1 + z) at the denominator accounts for the time-dilation effect,and the comoving volume element can be expressed as

dV = 4πr2(

c

H0

)

ǫ(z)dz (5.18)

ǫ(z) =[

ΩM (1 + z)3 + ΩΛ

]− 12 .

The result is shown in the bottom panel of Fig. 5.4.

5.3.2 Background computation

We are now ready to compute the GW backgrounds produced by the differentmagnetar 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 sizeL = 10h−1 Mpc and Np = 2 · 2563 (dark+baryonic) particles.


1e-22

1e-20

1e-18

1e-16

1e-14

1e-12

1e-10

1e-08

1e-06

1 10 100 1000

ΩG

W(f

)

f [Hz]

Bp = 1014 G

S/N = 40

S/N = 0.026

CGWB

TT-APR2TT-GNH3

P-AP-BTD

Einstein Telescope 1e-22

1e-20

1e-18

1e-16

1e-14

1e-12

1e-10

1e-08

1e-06

1 10 100 1000

ΩG

W(f

)

f [Hz]

Bp = 1015 G

S/N = 2.5

S/N = 1.1

CGWB

TT-APR2TT-GNH3

P-AP-BTD

Einstein Telescope

Figure 5.5. The predicted closure energy density (ΩGW) as a function of the observa-tional frequency, for the different magnetar models discussed: P-A and P-B stand for purelypoloidal models A and B, TT are the twisted torus model predictions for the two EOS con-sidered (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 sen-sitivity of the Einstein Telescope, and the horizontal dotted line (CGWB) is the upper limiton primordial backgrounds generated during the Inflationary Epoch. A given background isdetectable by the Einstein Telescope if the corresponding signal-to-noise ratio is larger thanthe detection threshold S/N = 2.56 (see text).

GW background can be written as

dE

dSdfdt=∫ zf

0

∫ Mf

MidR(M,z)

⟨ dE

dSdf

⟩

, (5.19)

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

dR(M,z) =ρ⋆(z)

(1 + z)dV

dzΦ(M)dMdz , (5.20)

and⟨

dEdSdf

⟩

is the locally measured average energy flux emitted by a source at dis-tance r. For sources at redshift z it becomes

⟨ dE

dSdf

⟩

=(1 + z)2

4πdL(z)2

dEGWdfe

[f(1 + z)] , (5.21)

where f = fe(1 + z)−1 is the redshifted emission frequency fe, and dL(z) is theluminosity 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 thespectral energy density by the equation

ΩGW(f) =f

c3ρcr

[

dE

dSdfdt

]

, (5.22)

where ρcr = 3H20/8πG is the cosmic critical density.

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


∼ 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 theP-B model in the second case; the lower value is given in both cases by the TT-APR2 model. In both panels, model predictions are compared with the foreseensensitivity of the Einstein Telescope (shaded region) and with the upper limit toprimordial backgrounds generated during the Inflationary Epoch (horizontal dottedline 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 tensorperturbations [54, 55].

A comparison between the estimated magnetar GW background and the upperlimit to the primordial background predicted by Inflationary scenarios (the horizon-tal dotted line in Fig. 5.5 labelled CGWB) shows that, for the models of magnetarwe consider, the magnetar GWB is always larger than the primordial background insome region of frequency (the only exception is the twisted-torus model TT-APR2with Bp = 1015 G). For instance, for the toroidal dominated models TD15 andTD14 this is true, respectively, for f ∼> 13 Hz and f ∼> 4 Hz. Thus, the GWBgenerated by magnetars may act as a limiting foreground for the future detectionof 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 po-tentially detectable by the Einstein Telescope. A more quantitative assessment ofthe detectability is reported in the following Section.

5.3.3 Detectability

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

D(z) =∫ z

0dR(z)∆τgw(1 + z) , (5.23)

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

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

(

S

N

)2

=9H4

0

50π4T

∫ ∞

0dfγ2(f)Ω2

GW(f)f6P1(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 sensi-tivity 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.


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

ΩMIN =1√T

10π2

3H20

[

∫ ∞

0df

γ2(f)f6P1(f)P2(f)

]−1/2

·(erfc−1(2α) − erfc−1(2γ)) , (5.25)

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

If we consider the cross-correlation of two detectors with the sensitivity of theEinstein 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, privatecommunication); these values, inserted in Eq. (5.24), lead to a detection thresholdS/N of 2.56.

A given background is detectable by the Einstein Telescope if the correspondingsignal-to-noise ratio given by Eq. (5.24) is larger than the detection threshold. Forinstance, the predicted ΩGW for the purely poloidal model P-B with Bp = 1015 Ggives S/N = 2.49, that is slightly smaller than such threshold; consequently, thereis no chance to detect this signal. Conversely, in the most optimistic magnetarmodel TD14 (toroidal-dominated model with Bp = 1014 G) we obtain S/N = 40,a very promising value. This result leads to the conclusion that third-generationgravitational wave detectors, such as the Einstein Telescope, hold the potential toreveal the cumulative GW signal from magnetars in the Universe.

It is worth noting that the above results refer to the assumed initial spin pe-riod of P0 = 0.8 ms. A higher value would lead to a lower frequency cutoff and,consequently, to a weaker GWB. For the TD14 model, for example, the detectionthreshold S/N = 2.56 corresponds to P0 = 5.2 ms. Hence the GW backgroundwould 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 andthe magnetic axis, in which case the GW signal is emitted only at twice the spinfrequency fe = 2νR = ω/π. For a generic misalignment, we have also the emission atthe spin frequency. We now focus on the model TD14 and explore the consequencesof α < π/2. In this model the stellar deformation induced by the magnetic field arelarger; being the most optimistic model for gravitational wave emission, this caseallows 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 back-ground 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 contributionwith the frequency cutoff at 2500 Hz, while for smaller angles there is a dominantcontribution with fmax = 1250 Hz and a secondary contribution with lower am-plitude extending up to fmax = 2500 Hz. Similar effects hold for the alternativemagnetar models which have been presented in the previous Sections. A differencebetween the models potentially arises if the timescale for the evolution of the wob-ble angle is short compared to the spin-down timescale (see Section 5.2): in thiscase, the star rapidly tends (i) to the orthogonal configuration for the TD model,


1e-12

1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

10 100 1000

ΩG

W(f

)

f [Hz]

Bp= 1014 G

α = π/2α = π/4

α = π/18α = π/60

Einstein Telescope

Figure 5.6. The TD14 background (see text) is plotted for different wobble angles, span-ning 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) thegravitational wave backgrounds corresponding to different wobble angles exhibitsignificant differences at large frequencies, approximately above ∼ 800 Hz, wherethe Einstein Telescope sensitivity is too low even for this model; therefore, the signaldetectability is only marginally affected. Variations in the signal-to-noise ratio areat 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 havesignificant effects on the GW background detectability with the Einstein Telescope.

Conclusions

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

The central part of the Thesis (Chapters 3 and 4) has been devoted to themagnetar model we have developed. Such model, built in the framework of GeneralRelativity, describes a strongly magnetized neutron star surrounded by vacuumunder the assumption of stationary and axisymmetric spacetime and in the limit ofhigh conductivity in the star’s interior. Here we summarize the main improvementswith respect to the current literature.

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

• We have considered the contribution from higher (l > 1) multipoles of themagnetic 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 ina normalization independent way, thus identifying the energetically favouredconfiguration among the possible solutions. This also allowed to determinethe favoured ratio of toroidal and poloidal magnetic field energies, which is ofgreat 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 studiesaimed to understand the peculiar observational properties of magnetars, improvingthe predictive power of present dynamical models.

In the last Chapter (5) we have discussed a direct application of the above re-sults: gravitational wave emission from strongly magnetized neutron stars. We havecomputed, according to our twisted-torus model, the quadrupolar deformation in-duced 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 haveestimated both the emission spectrum for a single magnetar and the gravitationalwave background resulting from the superposition of signals produced by the entiremagnetar population. For completeness, this analysis has also been extended toother magnetar models. Finally, we have discussed about the detectabilty of thisgravitational wave background by third generation detectors such as the EinsteinTelescope. We have found that:

• different magnetar models produce a spread in the resulting gravitational waveemission, revealing that the main uncertainties on present predictions comefrom our lack of knowledge on the internal magnetic field configuration;

• in some cases the gravitational wave background produced by magnetars re-sults detectable by the Einstein Telescope, with signal-to-noise ratios up to40;

• the misalignment angle between magnetic and spin axes, whose distributionover the magnetar population is unknown, and its eventual evolution in timehave poor effects on the background detectability;

• for all the models considered, the signal produced by magnetars acts as alimiting foreground for the future detection of the primordial gravitationalwave 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 adopt-ing more general assumptions, (ii) additional applications (e.g. magnetar quasi-periodic oscillations or cooling of magnetized neutron stars), and (iii) a study onthe emergence and stability of twisted-torus configurations to be accomplished in arelativistic framework (this is sketched in Sec. 4.8).

Appendix A

TheTolman-Oppenheimer-Volkoff(TOV) solution

In this Appendix we discuss the equations describing the hydrostatic equilibriumof a static and spherically symmetric star; they were first proposed in [122, 84] andnamed TOV after the authors. In our model we regard such a static and sphericallysymmetric system as the unperturbed condition, in which the star has no magneticfield. 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

T µν;ν =1√−g

∂

∂xν(√−g T µν

)

+ ΓµλνTλν = 0 , (A.2)

where Γµλν are the Christoffel symbols and g is the metric determinant. We recallthat the stress-energy tensor (we are assuming a perfect fluid) is given by

T µν = (ρ+ P )uµuν + Pgµν . (A.3)

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

m(r) =∫ r

04πr2ρ dr . (A.4)

It is related to λ(r) through

e−λ = 1− 2mr

. (A.5)

The only non-trivial component of Eqns. (A.2) is given by µ = r, from whichwe have

ν,r = − 2P,rρ+ P

, (A.6)

71

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

while the relevant Einstein equations are

[tt]1r2

d

dr[r(1− e−λ)] = 8πρ , (A.7)

[rr] − 1r2

(eλ − 1) +ν,rr

= 8πPeλ . (A.8)

Eq. (A.7), rewritten in terms of m(r), becomes

dm

dr= 4πr2ρ , (A.9)

which is one of the TOV equations, while from Eq. (A.8), combined with Eq. (A.9),we have

ν,r = 2

[

m+ 4πr3P

r(r − 2m)

]

. (A.10)

Finally, by substituting Eq. (A.10) in Eq. (A.6) we obtain the second TOV equation:

P,r = −(ρ+ P )

[

m+ 4πr3P

r(r − 2m)

]

. (A.11)

The TOV equations are summrized below.

dm

dr= 4πr2ρ (A.12)

P,r = −(ρ+ P )

[

m+ 4πr3P

r(r − 2m)

]

(A.13)

If we provide the EOS, the two equations give a unique solution for an assignedvalue 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 projectingthe equation onto the respective harmonics. In this case we have

ψ =

[

−a1 +a3(3− 15 cos2 θ)

2+a5(−315 cos4 θ + 210 cos2 θ − 15)

8

]

sin2 θ .

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

B.1 β(ψ) chosen according to Eq. (4.1)

14π

(

e−λa′′1 + e−λν ′ − λ′

2a′1 −

2r2a1

)

−e−ν

4π

∫ π

0(3/4) ζ2

0

(

ψ − 3ψ|ψ/ψ|+ 2ψ3/ψ2)

×Θ(|ψ/ψ| − 1) sin θ dθ =[

c0 −45c1

(

a1 −37a3

)]

(ρ+ P )r2 , (B.1)

14π

(

e−λa′′3 + e−λν ′ − λ′

2a′3 −

12r2a3

)

+e−ν

4π

∫ π

0(7/48) ζ2

0

(

ψ − 3ψ|ψ/ψ|+ 2ψ3/ψ2)

×Θ(|ψ/ψ| − 1)(3 − 15 cos2 θ) sin θ dθ

= c1(ρ+ P )r2(

215a1 −

815a3 +

1033a5

)

, (B.2)

73

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

14π

(

e−λa′′5 + e−λν ′ − λ′

2a′5 −

30r2a5

)

+e−ν

4π

∫ π

0(11/60) ζ2

0

(

ψ − 3ψ|ψ/ψ|+ 2ψ3/ψ2)

×Θ(|ψ/ψ| − 1)(−315 cos4 θ + 210 cos2 θ − 15)

8sin θ dθ

= c1(ρ+ P )r2(

421a3 −

2039a5

)

. (B.3)

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

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

14π

(

e−λa′′1 + e−λν ′ − λ′

2a′1 −

2r2a1

)

− e−ν

4π

∫ π

0(3/4) ζ2

0ψ

[

(

|ψ/ψ| − 1)2σ

+σ|ψ/ψ|(

|ψ/ψ| − 1)2σ−1

]

Θ(|ψ/ψ| − 1) sin θ dθ

=[

c0 −45c1

(

a1 −37a3

)]

(ρ+ P )r2 , (B.4)

14π

(

e−λa′′3 + e−λν ′ − λ′

2a′3 −

12r2a3

)

+e−ν

4π

∫ π

0(7/48) ζ2

0ψ

[

(

|ψ/ψ| − 1)2σ

+ σ|ψ/ψ|(

|ψ/ψ| − 1)2σ−1

]

×Θ(|ψ/ψ| − 1)(3− 15 cos2 θ) sin θ dθ

= c1(ρ+ P )r2(

215a1 −

815a3 +

1033a5

)

, (B.5)

14π

(

e−λa′′5 + e−λν ′ − λ′

2a′5 −

30r2a5

)

+e−ν

4π

∫ π

0(11/60) ζ2

0ψ

[

(

|ψ/ψ| − 1)2σ

+ σ|ψ/ψ|(

|ψ/ψ| − 1)2σ−1

]

×Θ(|ψ/ψ| − 1)(−315 cos4 θ + 210 cos2 θ − 15)

8sin θ dθ

= c1(ρ+ P )r2(

421a3 −

2039a5

)

. (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

14π

(

e−λa′′1 + e−λν ′ − λ′

2a′1 −

2r2a1

)

−e−ν

4π

∫ π

0(3/4)

β20

ψσ|ψ/ψ|

(

|ψ/ψ| − 1)2σ−1

Θ(|ψ/ψ| − 1) sin θ dθ

=[

c0 −45c1

(

a1 −37a3

)]

(ρ+ P )r2 , (B.7)

14π

(

e−λa′′3 + e−λν ′ − λ′

2a′3 −

12r2a3

)

+e−ν

4π

∫ π

0(7/48)

β20

ψσ|ψ/ψ|

(

|ψ/ψ| − 1)2σ−1

×Θ(|ψ/ψ| − 1)(3 − 15 cos2 θ) sin θ dθ

= c1(ρ+ P )r2(

215a1 −

815a3 +

1033a5

)

, (B.8)

14π

(

e−λa′′5 + e−λν ′ − λ′

2a′5 −

30r2a5

)

+e−ν

4π

∫ π

0(11/60)

β20

ψσ|ψ/ψ|

(

|ψ/ψ| − 1)2σ−1

×Θ(|ψ/ψ| − 1)(−315 cos4 θ + 210 cos2 θ − 15)

8sin θ dθ

= c1(ρ+ P )r2(

421a3 −

2039a5

)

. (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 tocompute the total energy of the system E = M + δM (where M is the knownunperturbed 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 farfield limit of the spacetime metric [82, 121]:

δM = limr→∞

m0(r) , (C.1)

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

h(r, θ) =∑

l

hl(r)Pl(cos θ) ,

m(r, θ) =∑

l

ml(r)Pl(cos θ) . (C.2)

The perturbed Einstein equations ([tt] and [rr] components), projected onto l = 0 ,allow to determine the quantity m0(r):

m′0 − 4πr2 ρ′

P ′δp0 =

13

(a′1)2e−λ +67

(a′3)2e−λ +1511

(a′5)2e−λ +2

3r2a2

1

+727r2

a23 +

45011r2

a25

+e−ν

4

[

∫ π

0ζ2

0

(

|ψ/ψ| − 1)2

Θ(|ψ/ψ| − 1)ψ2

sin θdθ

]

, (C.3)

77

78 C. The energy of the system

h′0 − e2λm0

(

1r2

+ 8πP)

− 4πreλδp0 =

13r

(a′1)2 +67r

(a′3)2 +1511r

(a′5)2 − 2eλ

3r3a2

1

−72eλ

7r3a2

3 −450eλ

11r3a2

5

+eλ−ν

4r

[

∫ π

0ζ2

0

(

|ψ/ψ| − 1)2

Θ(|ψ/ψ| − 1)ψ2

sin θdθ

]

. (C.4)

δp0 is the l = 0 component of the pressure perturbation (and vanishes outside thestar), and ψ = sin θ

∑

l=1,3,5 alPl,θ . Using the relation (arising from T rν;ν = 0)

δp′0 = −ν′

2

(

ρ′

P ′+ 1

)

δp0 − (ρ+ P )h′0

−23a′1(ρ+ P )

[

c0 −45c1

(

a1 −37a3

)]

−127a′3(ρ+ P )c1

(

215a1 −

815a3 +

1033a5

)

−1011a′5(ρ+ P )c1

(

421a3 −

2039a5

)

, (C.5)

Eqns. (C.3), (C.4) can be rearranged in the form

m′0 − 4πr2 ρ′

P ′δp0 =

13

(a′1)2e−λ +67

(a′3)2e−λ +1511

(a′5)2e−λ

+2

3r2a2

1 +727r2

a23 +

45011r2

a25

+e−ν

4

[

∫ π

0ζ2

0

(

|ψ/ψ| − 1)2

Θ(|ψ/ψ| − 1)ψ2

sin θdθ

]

, (C.6)

δp′0 +[

ν ′

2

(

ρ′

P ′+ 1

)

+ 4πreλ(ρ+ P )]

δp0

+e2λm0(ρ+ P )(

1r2

+ 8πP)

=

(ρ+ P )

(

− 23a′1

[

c0 −45c1

(

a1 −37a3

)]

−127a′3c1

(

215a1 −

815a3 +

1033a5

)

−1011a′5c1

(

421a3 −

2039a5

)

− 13r

(a′1)2 − 67r

(a′3)2

− 1511r

(a′5)2 +2eλ

3r3a2

1 +72eλ

7r3a2

3 +450eλ

11r3a2

5

−eλ−ν

4r

[

∫ π

0ζ2

0

(

|ψ/ψ| − 1)2

Θ(|ψ/ψ| − 1)ψ2

sin θdθ

])

. (C.7)

By imposing a regular behaviour at r ≃ 0 we find

m0(r → 0) = Ar3 , δp0(r → 0) = Cr2 , (C.8)

C.2 β(ψ) chosen according to Eqns. (4.22) and (4.23) 79

where

C = −2(Pc + ρc) · (α21 + α1c0)

3 + 4π(

r2 dρdP

)

cPc

A =13

[

2α21 + 4πC

(

r2 dρ

dP

)

c

]

. (C.9)

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

m′0 =(

13

(a′1)2 +67

(a′3)2 +1511

(a′5)2) (

1− 2Mr

)

+2

3r2a2

1 +727r2

a23 +

45011r2

a25 . (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 systembecomes

m′0 − 4πr2 ρ′

P ′δp0 =

13

(a′1)2e−λ +67

(a′3)2e−λ

+1511

(a′5)2e−λ +2

3r2a2

1 +727r2

a23 +

45011r2

a25

+e−ν

4

[

∫ π

0ζ2

0

(

|ψ/ψ| − 1)2σ

Θ(|ψ/ψ| − 1)ψ2

sin θdθ

]

, (C.11)

δp′0 +[

ν ′

2

(

ρ′

P ′+ 1

)

+ 4πreλ(ρ+ P )]

δp0

+e2λm0(ρ+ P )(

1r2

+ 8πP)

= (ρ+ P )

− 23a′1

[

c0 −45c1

(

a1 −37a3

)]

−127a′3c1

(

215a1 −

815a3 +

1033a5

)

−1011a′5c1

(

421a3 −

2039a5

)

− 13r

(a′1)2 − 67r

(a′3)2

− 1511r

(a′5)2 +2eλ

3r3a2

1 +72eλ

7r3a2

3 +450eλ

11r3a2

5

−eλ−ν

4r

[

∫ π

0ζ2

0

(

|ψ/ψ| − 1)2σ

Θ(|ψ/ψ| − 1)ψ2

sin θdθ

]

. (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:ψ2ζ2

0 → β20 .

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 momentQ and then the quadrupole ellipticity εQ, we need to solve a system of coupledlinearized Einstein’s equations involving the metric corrections h2(r), m2(r) andk2(r). In particular, these equations are obtained from the l = 2 projection (seeEq. (3.32)) of the following components of the Einstein’s equations: [rθ], the com-bination [θθ]− [φφ]/ sin2 θ and [rr]. If we adopt the choice (4.22) for the functionwhich determines the ratio of toroidal and poloidal fields, the system writes

k′2 + h′2 − h2

(

1r− ν ′

2

)

−m2

(

ν ′

2+

1r

)

eλ

r

=5

4r2

∫ π

0ψ,θψ,r

(3 cos2 θ − 1) cot θsin θ

dθ ,

h2 +eλ

rm2 =

54r2

∫ π

0

(

− (ψ,r)2r2e−λ + e−νζ20ψ

2r2(|ψ/ψ| − 1)2σ

×Θ(|ψ/ψ| − 1))

(3 cos2 θ − 1)sin θ

dθ ,

(

ν ′ +2r

)

k′2 +2rh′2 −

4r2eλk2 −

6r2eλh2

−(

1r2

+ 8πP)

2e2λ

rm2 − 8πeλδp2

=5

4r4eλ∫ π

0

(

− (ψ,θ)2 + (ψ,r)

2r2e−λ

+e−νζ20ψ

2r2(|ψ/ψ| − 1)2σΘ(|ψ/ψ| − 1))

×(3 cos2 θ − 1)sin θ

dθ , (D.1)

where

ψ =

[

−a1 +a3(3− 15 cos2 θ)

2+a5(−315 cos4 θ + 210 cos2 θ − 15)

8

]

sin2 θ ,

81

82 D. Quadrupolar deformations

and δp2 is the l = 2 component of the pressure perturbation δP = δp0+δp2P2(cos θ) .The integration can be simplified by introducing the auxiliary function

y2 = k2 + h2 +W (r, θ) , (D.2)

where

W (r, θ) =5e−λ

16r2

∫ π

0

(

− eλ(ψ,θ)2 + r2(ψ,r)2 − 2rψ,θψ,r cot θ

)


dθ . (D.3)

This generalizes the variable change adopted in [52, 28]. With the above substitutionwe are left with two coupled equations

y′2 + ν ′h2 = W ′ +5

4r2

∫ π

0

[

ψ,θψ,r cot θ +(

ν ′

2+

1r

)

×(

− (ψ,r)2r2e−λ + e−νζ2

0ψ2r2(|ψ/ψ| − 1)2σΘ(|ψ/ψ| − 1)

)]


dθ , (D.4)

h′2 +4ν ′r2

eλy2 +

[

ν ′ − 8πeλ

ν ′(ρ+ P ) +

2ν ′r2

(eλ − 1)

]

h2

=5

8r2

∫ π

0

[

− ν ′e−λr2(ψ,r)2 + 2ψ,rψ,θ cot θ

+e−νζ20ψ

2(

ν ′r2 − 2ν ′eλ)

(|ψ/ψ| − 1)2σΘ(|ψ/ψ| − 1)]


dθ +10πν ′

eλ(ρ+ P )∫ π

0[c0 + c1ψ]ψ,θ sin2 θ cos θdθ , (D.5)

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

δp2 = −(ρ+ P )(

h2 +54

∫ π

0[c0 + c1ψ]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) = Ar2 , y2(r → 0) = Br4 , (D.7)

where

B =(

−2πA+16π

3α2

1

)

(Pc + ρc/3)− 4π3α1c0(Pc + ρc) , (D.8)

with Pc and ρc indicating respectively the pressure and mass-energy density at thestar’s centre. Inside the star we can write h2 and y2 as

h2 = b1hh2 + hp2 , y2 = b1y

h2 + yp2 , (D.9)

where the superscripts h and p denote the homogeneous and one particular solutionrespectively, while b1 is a constant. We can determine hp2 and yp2 by integratingthe system with A = 1, while hh2 and yh2 are found by solving the homogeneous

83

system. The constant b1 is then obtained by imposing the continuity with theexterior solution, which is written again as

h(ext)2 = Kh

h (ext)2 + h

p (ext)2 , y

(ext)2 = Ky

h (ext)2 + y

p (ext)2 , (D.10)

where K is a constant. hh (ext)2 , yh (ext)

2 and hp (ext)2 , yp (ext)

2 are determined byintegrating the system from r → ∞ to r = R. For the particular solution we usethe following initial conditions

hp (ext)2 (r →∞) = − 6µ2

0

5Mr3, y

p (ext)2 (r →∞) = −3µ2

0

5r4, (D.11)

where

µ0 = −8M3

3R2a1(R)

[

ln(

1− 2MR

)

+2MR

+2M2

R2

]−1

, (D.12)

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

hh (ext)2 = Q2

2(z) , yh (ext)2 = − 2√

z2 − 1Q1

2(z) , (D.13)

where z = r/M − 1 and Qnm are the associated Legendre functions of the secondkind. By imposing h2 = h

(ext)2 and y2 = y

(ext)2 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. Ifwe adopt the other choice for the relation between toroidal and polidal fields, givenby (4.23), we proceed in the same way. In this case the final system of equationswrites

y′2 + ν ′h2 = W ′ +5

4r2

∫ π

0

[

ψ,θψ,r cot θ +(

ν ′

2+

1r

)

×(

− (ψ,r)2r2e−λ + e−νβ20r

2(|ψ/ψ| − 1)2σΘ(|ψ/ψ| − 1))]


dθ , (D.14)

h′2 +4ν ′r2

eλy2 +

[

ν ′ − 8πeλ

ν ′(ρ+ P ) +

2ν ′r2

(eλ − 1)

]

h2

=5

8r2

∫ π

0

[

− ν ′e−λr2(ψ,r)2 + 2ψ,rψ,θ cot θ

+e−νβ20

(

ν ′r2 − 2ν ′eλ)

(|ψ/ψ| − 1)2σΘ(|ψ/ψ| − 1)]


dθ +10πν ′

eλ(ρ+ P )∫ π

0[c0 + c1ψ]ψ,θ sin2 θ cos θdθ . (D.15)

84 D. Quadrupolar deformations

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[l'angolo del phd] riccardo ciolfi - xxiii ciclo - 2010

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