structure and oscillations of neutron stars introduction neutron stars are one of the most...

Universita degli Studi di Roma “ La Sapienza”

Dottorato di Ricerca in Astronomia - XX Ciclo

STRUCTURE AND OSCILLATIONS

OF NEUTRON STARS

Thesis submitted for the degree of

Doctor Philosophiæ

PhD in Astronomy - XX cycle- October 2007

Coordinator

Prof. Paolo de Bernardis

Advisor CandidateProf.ssa Valeria Ferrari Dr. Stefania Marassi

Anno Accademico 2006-2007

“The most incomprehensible thing about the worldis that it is comprehensible”

Albert Einstein

Contents

Introduction 4

1 Neutron stars 7

1.1 Birth of a neutron star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Neutron star equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Neutron star observed properties . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Strange Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Rotating Neutron stars 21

2.1 The background model: non-rotating stars . . . . . . . . . . . . . . . . . . . . 23

2.2 The metric and the structure of rotating stars . . . . . . . . . . . . . . . . . . . 25

2.2.1 The mass-shedding limit . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.2 The algorithm used to construct a sequence of constant baryonicmass solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.2.3 The moment of inertia of rotating stars . . . . . . . . . . . . . . . . . 30

2.3 Comparison with non-perturbative results . . . . . . . . . . . . . . . . . . . . 31

2.4 The EOS’s imprint on an NS physical properties: results . . . . . . . . . . . . 39

2.4.1 Mass-shedding limit . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.4.2 Estimates of the moment of inertia . . . . . . . . . . . . . . . . . . . 42

2.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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3 Gravitational wave asteroseismology 49

3.1 Neutron star oscillations in GR . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.1.1 The perturbation equations . . . . . . . . . . . . . . . . . . . . . . . 50

3.2 Quasi-normal mode computation . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3 Neutron star quasi-normal modes and gravitational waves . . . . . . . . . . . . 54

3.4 Oscillations of quark stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4.1 The fundamental mode and its detectability . . . . . . . . . . . . . . 60


4 Oscillations of rotating neutron stars 63

4.1 Neutron stars as sources of GWs . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2 Developement of our method . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2.1 The boundary condition at radial infinity . . . . . . . . . . . . . . . . 68

4.2.1.1 Standing wave method for a spherical stars . . . . . . . . 68

4.2.1.2 The standing wave approach for rotating stars . . . . . . 69

4.2.2 Spectral methods for stellar oscillations . . . . . . . . . . . . . . . . 71

4.2.2.1 Chebyshev polynomials . . . . . . . . . . . . . . . . . . 72

4.2.2.2 Associated Legendre polynomials . . . . . . . . . . . . 73

4.2.2.3 Differential equations and boundary conditions . . . . . 74

4.2.2.4 The Regge-Wheeler equation for a spherical star as a2D-equation in r and θ . . . . . . . . . . . . . . . . . . 75


5 A test of the method: oscillations of slowly rotating stars 81

5.1 Comparison with existing results . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.2 Including the couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84


6 Conclusions 89

6

6.1 Rapidly rotating neutron stars . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.2 Oscillations of non-rotating quark stars . . . . . . . . . . . . . . . . . . . . . . 90

6.3 Oscillations of rotating neutron stars . . . . . . . . . . . . . . . . . . . . . . . 90

A Appendix 91

A.1 The first order equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

A.2 Second order equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

A.2.1 Spherical expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 93

A.2.2 Quadrupole deformation . . . . . . . . . . . . . . . . . . . . . . . . 94

A.3 Third order equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

A.3.1 The equations for w1 . . . . . . . . . . . . . . . . . . . . . . . . . . 96

A.3.2 The equations for w3 . . . . . . . . . . . . . . . . . . . . . . . . . . 98

B Appendix 101

B.1 Tensor spherical harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

B.1.1 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

C Appendix 107

C.1 The generalized Regge-Wheeler gauge: a possible gauge choice . . . . . . . . 107

C.1.1 Relations with the Regge-Wheeler gauge . . . . . . . . . . . . . . . 109

D Appendix 111

D.1 Equations for the perturbations of slowly rotating stars . . . . . . . . . . . . . 111

D.1.1 The O(ε0) equations . . . . . . . . . . . . . . . . . . . . . . . . . . 112

D.1.2 The O(ε1) equations . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Index of the figures 118

Index of the tables 123

7

Bibliography 125

8

Introduction

Neutron stars are one of the most interesting compact objects in our Universe essentially for tworeasons: first, their physical properties make them an astrophysical laboratory for the study ofthe matter at supranuclear density that is unobtainable in terrestrial experiments, second they arecandidate to be sources of gravitational waves. It is commonly believed that the experimentaldiscovery of gravitational radiation will open a new window onto the observable Universe and adifferent channel for observational astrophysics providing new information that are unlikely tobe obtained in another way.This work has the purpose to give a contribution in these two directions. In the first part of thisthesis we focus our attention to understand how the physical properties of matter in the interiorof such compact stars affects its global properties such as radius, mass, angular momentum,etc. In particular we expect to learn more on the state of matter inside a neutron star by meansthe comparison between theoretically determined values of these quantities and the observedones. A real neutron star is a very complicated object: it is rapidly rotating and very compact(a mass of about 1.4M is enclosed in a radius of the order of 10 Km) and it has to be studiedin the framework of the Einstein’s theory of gravity because general relativistic effects playing arelevant role and cannot be neglected.To model theoretically at high level of accuracy a rotating, relativistic, neutron star we use theperturbative approach, introduced by J. Hartle in 1967. Essentially, in this approach, Einstein’sequations for stellar structure are expanded perturbatively and the parameter of this expansion isthe angular velocity. We integrate the equations of stellar structure to third order in the angularvelocity and we construct models of rotating neutron stars for a set of equation of state, proposedin the literature, to model interactions in the inner core of neutron stars. We compare our results tothose that are obtained with fully non-linear codes, that directly integrate numerically Einstein’sequations. The aim of this comparison reside in establishing (determining) to what extent thirdorder corrections are needed to estimate a choosen quantity or whether the first order estimates,often used to interpret astronomical observations, are enough. We give special attention toaccurately reproduce the moment of inertia of a neutron star which rotates at rates comparable tothat of the fastest isolated pulsar observed.The second and more extended part of this thesis is devoted to the study of gravitationalwaves emission from perturbed, oscillating neutron stars (non-rotating and rotating). Thistheoretical work can play a crucial role in order to collect as much information as possibleon the gravitational waves signals that one can expect to be detectable, and on astrophysical

1

sources of gravitational radiation. This choice is also, strongly motivated, from the extraordinaryexperimental effort done, in recent years, to improve the sensitivity of the gravitational wavesdetectors like LIGO and VIRGO that now are operating at the design sensitivity and that allowsus to hope that gravitational radiation will be directly detected in the near future.When a neutron star is perturbed it can be set into non-radial oscillations and emits gravitationalwaves at the characteristic frequencies of its quasi-normal modes (QNM). These frequenciesare complex because gravitational waves acts as a damping mechanism subtracting mechanicalenergy to the system. These waves are emitted with the signature of the dynamics of the matterin the interior of the star, for this reason, as we will see in this thesis, gravitational waves providea very useful tool to extract information about the equation of state of the source and about itsnature. Moreover they are weakly interacting with the surroundings and this feature make themindipendent on the models of the atmosphere at the neutron star surface and carriers of cleaninformation. The spectrum of a pulsating relativistic star is very rich, since essentially eachfeature of the star can be directly associated with one distinct family of pulsation modes.In the first part of our study of oscillations of compact object in the framework of GeneralRelativity, we concentrate our attention on the fundamental mode (f-mode) of oscillation. Weassume that the neutron star do not rotate and we can easily compute the f-mode frequencies bysolving the equations of stellar perturbations. This perturbative approach have been formulatedin the Sixties[1, 2, 3, 4, 5], and further developed in later years[6, 7].As in Newtonian gravity, the f-mode of pulsation is related to a global oscillations of the fluidand is the most efficient mode as regards the emission of gravitational waves. In particular wewant to investigate the nature of the source with the help of the f-mode frequencies: we discussthe possibility that the detection of gravitational waves emitted by compact stars that oscillate intheir fundamental mode may allow one to discriminate between strange stars and neutron stars.Whether strange stars, i.e. compact astrophysical objects entirely made up of “strangematter1”(except, possibly, for a thin outer crust) do exist in nature is an open question, andthe astrophysical scenario in which they may form is still poorly understood. The hypothesis ofthe existence of such an object, while not being directly verifiable in terrestrial laboratories, maybe confirmed by the astrophysical observations. Furthermore, we describe strange stars with theMIT bag model of quark matter EOS2 and we will see how the detection of a signal emitted by acompact star pulsating in its fundamental mode, combined with a complementary information onthe stellar mass or the radiation radius, would allow to constrain the parameters of the MIT modelto a range much smaller than that provided by the available data from terrestrial experiments.It is important to point out that, in this thesis, we propose gravitational waves as a tool to obtaininformation about neutron stars, a less traditional tool but, in principle, very powerful becausegravitational waves have the signature of the internal composition of the star.In order to detect gravitational signals emitted from neutron stars it is fundamental to modelaccurately the source and this operation requires a deep understanding of the very extremes of

1In 1980s Bodmer and Witten suggested that the true and absolute ground state of matter consisting of degenerateup, down and strange quarks, called “strange” quark matter, might be bound and stable at zero temperature andpressure.

2Equation of State.

2

physics, including supranuclear physics, general relativity, superfluidity, strong magnetic fields,exotic particle physics, etc. Therefore to investigate the pulsation properties of any realisticneutron star is a difficult challenge mainly because it is rapidly rotating. It is important tonote, that due to rotation, some modes may grow unstable through the Chandrasekhar-Friedman-Schutz mechanism (CFS instability) [8, 9] and these instabilities may have important effects onthe subsequent evolution of the star, for example they may be associated to a further emission ofGWs, the amount of which would depend on when and whether the growing modes are saturatedby non-linear couplings or dissipative processes. If we want to have a chance to improve ourunderstanding we have to take the rotation into account. Unfortunately, at the moment, theknowledge of spectrum of rotating neutron stars is far to be complete.The perturbative approach, that we have used to compute the mode frequencies in the non-rotating case, when generalized to include rotation shows a high degree of complexity, even ifthe star is assumed to be only slowly rotating. A major difficulty arises because, when one usesthe standard spherical harmonics decomposition of the perturbed tensors, modes with differentharmonic indexes couple, giving rise to an infinite set of dynamical, coupled equations. Forthis reason, in most studies which are based on the perturbative approach further simplifyingassumptions are introduced: for example the couplings between oscillations with different valuesof the harmonic index l, that are the main feature of the rotation, are neglected, or, it is used theCowling approximation in which spacetime perturbations are neglected.For the alternative approach to the perturbative, that consists in solving the equations describinga rotating and oscillating star in full general relativity, in time domain, the situation is not reallydifferent. In fact current studies based on this approach also make use of strong simplifyingassumptions, or restrict to particular cases. For instance the Cowling approximation has beenused in several papers, or only quasi-radial modes (l = 0) have been considered. Besidesa comparison of the results for l = 0 obtained in th Cowling Approximation in [10], withthose found in full GR [11], shows that Cowling’s approximation introduces large errors in thedetermination of the f-mode frequency.In the last part of this thesis we illustrate a general method that we have developed to findthe quasi-normal mode frequencies of a rotating neutron star that overcome these difficulties.In this thesis we do not derive explicitly the perturbed equations in the general case of fastlyrotating neutron stars, we focus our attention in describing the general method. The guidelinesof our method are the following: we perturb Einstein’s equations about a stationary axisymmetricbackground describing a rotating neutron star. The perturbed quantities are expanded in circularharmonics eimφ. As we are looking for quasi-normal modes, we assume a time dependence e−iωt,with ω complex. Due to the background symmetry, perturbations with different values of ω andm are decoupled; thus, for assigned values of (m, ω), the perturbed equations to solve are a2D-system of linear differential equations in r and θ. Our method is based on two ingredients:

1. the perturbed equations are integrated using spectral methods; in particular we havedeveloped a procedure to solve the 2D-dimensional perturbation equations on a non-spherical background.

2. the boundary conditions at the center of the star and at radial infinity are defined suitably

3

generalizing, to the case of rotating stars, the boundary conditions that are used to find thequasi-normal mode frequencies of non-rotating star.

We used spectral methods because they are very powerful to solve differential equations(specially in two dimensions) and particularly useful to implement boundary conditions.To test our method, in the last part of this thesis, we applied our approach to find the frequencyof the f- mode of a slowly rotating, constant density star, as a function of the rotation rate andwe compare our results with existing results in the literature and we find a very good agreement.It is important to point out that the main advantage of our method is that it is much easier tohandle the couplings among different value of l, for this reason we are able to study the shift ofthe fundamental mode due to rotation, taking the l± 1 couplings into account, to our knowledgefor the first time in the literature.

The thesis is organized as follows.

• In Chapter 1 we present the basic concepts about neutron stars, in particular we concentrateour attention on what we know, at the moment, about neutron star observed propertiesbecause they give us a chance to obtain information about the equation of state of matterinside this compact star that, at the moment, is widely unknown;

• in Chapter 2 we construct models of rotating neutron stars using the perturbative approach,introduced by J. Hartle in 1967, and a set of equations of state proposed to modelinteractions in the inner core of neutron stars; we will show that the structure of a rapidlyrotating neutron star can be described to a high level of accuracy by using the perturbativeapproach, developed to a third order in the angular velocity Ω;

• in Chapter 3 we introduce the theory of stellar perturbations in General Relativity andwe illustrate the most relevant properties of the modes of pulsations of neutron stars;in particular we will show how the mode frequencies carry interesting information onthe inner structure of the emitting source; we discuss the possibility that the detection ofgravitational waves emitted by compact stars may allow one to put constraints on the MITbag model of quark matter EOS and also to obtain information on the nature of the source;

• in Chapter 4 we describe our new method to study the quasi-normal modes of rotatingrelativistic stars; we illustrate how the perturbed equations are integrated using spectralmethod and how it is possible to handle the boundary conditions at the center of the starand at radial infinity generalizing suitably the approach which were used to find the quasi-normal-mode frequencies of non-rotating stars;

• in Chapter 5 we apply our method and, in order to test it, we find the frequency of thef-mode of a slowly rotating star, constant density star, as a function of the rotation rate;we compare our rusults with those existing in the literature; we report our new resultsregarding the shift of the f-mode due to rotation taking the l ± 1 couplings into account;

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• finally, we summarize the main results of this work in the conclusions, Chapter 6;

• mathemathical details and relevant equations to this thesis are in the Appendices.

The present work is based on the following articles

• Omar Benhar, Valeria Ferrari, Leonardo Gualtieri, Stefania Marassi, Perturbative ap-proach to the structure of rapidly rotating neutron stars, published in Phys. Rev. D, 72,044028, (2005).

• Omar Benhar, Valeria Ferrari, Leonardo Gualtieri, Stefania Marassi, Quark matter imprinton Gravitational Waves from oscillating stars, published in Gen. Rel. Grav., 39, 1323,(2007).

• Valeria Ferrari, Leonardo Gualtieri, Stefania Marassi, A New approach to the study ofquasi-normal modes of rotating stars, accepted to Phys. Rev. D, Sep (2007).

Notations, conventions and units

In this thesis we use geometrical units in which G = c = 1, so that

1 = c = 2.9979 × 1010cm/s,

1 = G = 6.6720 × 10−8cm3g−1s−2.

The above definitions have to be considered as equations in such a way that, for example, wehave

1s = 2.9979 × 1010cm,

1g = 7.423710−29cm,

1MeV = 1.6022 × 10−6erg,

= 1.1605 × 1010K,

= 1.3234 × 10−55cm.

We often use astrophysical quantities such as

M = 1.989 × 1033g,

= 1.4766 × 105cm,

1pc = 3.26 light years = 3.09 × 1018cm

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1

Neutron stars

Neutron stars are one of the most interesting bodies in our universe: they are the smallest, denseststars known. Like all stars, neutron stars rotate, as many as a few hundred times a second. A starrotating at such a rate will experience an enormous centrifugal force that must be balanced bygravity. The balance of the two forces inform us of the lower limit regarding the stellar density.Neutron stars are 1014 times denser than Earth. Some neutron stars are in binary systems witha companion1: in this case, application of orbital mechanics allows us to calculate gravitationalmasses. This has been done, for some double systems (see, for instance, [12]) and it was foundthat the gravitational mass of neutron stars is typically (1.35 ± 0.04)M. We can therefore inferthe radius: about ten kilometers. Their central densities are of the order of 1015g/cm3 and mostof them are known to have a magnetic field of the order of 1012 Gauss.To understand the properties of matter in the interior of such extreme objects is one of themost complex tasks in physics and astrophysics. We know that the global properties, such asmasses and radii depend on the microscopic model which is used to describe nucleon-nucleoninteractions, or in other words, on the nuclear equation of state (EOS). From the comparisonbetween, theoretically determined values of these quantities and the observed ones, we expectto obtain information on the state and composition of matter at the supranuclear densities, thatprevail in the neutron star core but that are unobtainable in a laboratory.In 1934 Baade and Zwicky [13], soon after the discovery of the neutron by Chadwick and alsoLandau’s hypothesis of the existence of a star composed entirely of neutrons [14], suggestedthat neutron stars could be formed in a supernova event due to the collapse of the massive stellarcore. In 1939 Oppenheimer, Volkoff and Tolman first derived the equations of a relativistic stellarstructure from Einstein’s field equations, assuming that neutron stars were gravitationally boundstates of neutron Fermi gas.The first observational evidence of neutron star existence arrived thirty-four years later whenHewish and Bell [15] discovered the first radio pulsar and confirmed what Pacini in 1967 [16] hadpredicted: neutron stars might be observable at long radio wavelengths if they were magnetisedwith misaligned magnetic and rotation axes. More than 1100 pulsars have been found sinceHewish and Bell’s discovery. Two additional pulsars were discovered shortly afterwards, the

1It may be a white dwarf, a main-sequence star or a neutron star.

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1.1. Birth of a neutron star

Crab and Vela pulsar. Both are found within supernova remnants, the Crab and Vela nebulae,and both have very short periods. The periods of 33 ms and 89 ms provided convincing evidencethat pulsars are rotating neutron stars which form in supenova explosions.

1.1 Birth of a neutron star

Neutron stars are one of the possible final stages of stellar evolution and, in the most recentscenario, it is belived they are formed in the collapse of the core of a massive star (with massesM ≥ 8M) followed by a supernova explosion.There is a general consensus that ordinary stars are born when gaseus clouds (mostly hydrogen)contract due to the pull of gravity. As the density increases, gravitational energy is convertedinto heat by compression. If the temperature becomes high enough to ignite nuclear reactionsturning hydrogen into helium, the star has its fuel and reaches an equilibrium configuration assoon as gravitational attraction is balanced by internal pressure. When hydrogen is exhausted,the stellar core stops producing heat, the internal pressure cannot be sustained any longer and thecontraction produced by gravity resumes. If the mass of the core is high enough, its contractionassociated with a further increase of the temperature may lead to the ignition of a new fusionreaction that involve heavier nuclei. These processes can take place several times and, if the coreis sufficiently massive, as in the case of stars above eight solar masses, it is possible to reach sucha high temperature that in the central region of the star and the iron, surrounded by concentricshells of lighter materials, burns. Since nuclei heavier than iron are not stable at this stage, weare at the end point of thermonuclear reactions: the core of the star is now supported againstcollapse only by the pressure of degenerate non-relativistic electrons. Nuclear burning continuesin the surrounding shells of Silicon, Oxygen etc. overlying the then inert, central region of iron.Burning in the outer shells adds to the iron core mass. Gravity crushes the core to such a densitythat electrons become relativistic. The pressure they provide now increases less rapidly withincreasing density, this was the case at the earlier stage when the electrons were non-relativistic.Moreover, the kinetic energies of the relativistic electrons have reached the point that captureprotons, -inverse beta decay-, which produces a more favorable state. In this process, electronsinteract with protons to form neutrons and electronic neutrinos:

e− + p→ n + νe, (1.1.1)

the electrons are captured and consequently the pressure decreases. During this phase thecombined action of inverse beta decay and the process of photo-disintegration of the nuclei,

γ +56 Fe→ 134He+ 4n, (1.1.2)

that is an endothermic reaction, remove energy from the system that supports the subsequent con-traction. When the core mass goes over its maximum possible mass, called the Chandrasekharmass, gravity and the processes (1.1.1), (1.1.2), lead the core to collapse. As the core matteris crushed to high density, the core behaves like a giant atomic nucleus that resists any furthercompression and thus rebounds and sends out a schock wave whose kinetic energy is dissipated

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Chapter 1. Neutron stars

by neutrino losses and photo-disintegration of the nuclei.Through a complex and non completely understood scenario during the supernova explosion, asmall fraction of the gravitational binding energy of the star provides the kinetic energy for theejection of everything except the core: a proto-neutron star is born. In this thesis we are notinterested in the early stage of neutron star formation (proto-neutron star phase) but we assumethat the star has reached its final evolutionary stage and an old, cold, catalyzed configuration, aneutron star (NS), is established.

1.2 Neutron star equation of state

Neutron star models and equations of state (EOSs) originate from different methods that are usedto theoretically describe the nuclear force that acts between nucleons. In spite of these differentapproaches, proposed in the literature (see e.g.[17, 18] ), it is possible to formulate a generaldescription that is sufficiently independent of the model.In this thesis we assume that matter in the interior of an NS is a zero temperature: this is possiblebecause the temperature of an NS, shortly after birth, has fallen to the KeV region (an old, coldNS has a temperature of about 109 K) which is negligible on the nuclear scale (3 · 1011 − 1012)K2.The internal structure of a neutron star is believed to feature a sequence of layers of differentcomposition. The outer crust, about 300 m thick has a density ranging from ρ ∼ 107g/cm3 tothe, so-called, neutron drip density ρd = 4 · 1011g/cm3, consists of a Coulomb lattice of heavynuclei immersed in a degenerate electron gas. Proceeding from the surface towards the interiorof the star, the density increases, and so does the electron chemical potential. As a consequence,electron capture becomes more efficient and neutrons are produced in large numbers. At ρ = ρdthere are no more negative energy levels available to the neutrons that are then forced to getout of the nuclei. The inner crust sets in, about 500 m thick, consisting of neutron rich nucleiimmersed in a gas of electrons and neutrons. Moving from its outer edge towards the center,the density continues to increase, and nuclei start merging and give rise to different structures ofvariable dimensionality, changing from spheres into roads and slabs. At ρ ≤ 2 · 1014g/cm3 allstructures disappear and matter reduces to a uniform fluid of neutrons, protons, and leptons inweak equilibrium: the core starts. The density of the core ranges between ∼ ρ0, the value at theboundary with the inner crust, and a central value that can be as large as (1 ÷ 4) · 1015g/cm3. Ata density slightly larger than ρ0, the electron chemical potential exceeds the rest mass of the µmeson and the appearance of muons throught the process n→ p+µ+ νµ becomes energeticallyfavoured. All models of EOS that are based on hadronic degrees of freedom, predict that in thedensity range ρ0 ≤ ρ ≤ 2ρ0, neutron star matter consists mainly of neutrons with the mixtureof a small number of protons, electrons and muons. This picture may change at a large densitywith the appearance of strange, heavy baryons (e.g. Σ−) produced in weak interaction processes.When density in the core reaches ∼ 1015g/cm3 it is possible that matter undergoes a transition to

2MeV = 103 KeV = 1.1 · 1010 K.

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1.2. Neutron star equation of state

Figure 1-1. A representative model showing the possible internal structures of an NS.

a new exotic phase in which quarks are not longer clustered into nucleons or hadrons.In the literature there is general agreement on models of EOS for the crust because the propertiesof the matter in the crust can be obtained directly from nuclear data. In this thesis we haveemployed two well established EOSs for the outer and the inner crust: the Baym-Pethick-Sutherland (BPS) EOS [19] and the Pethick-Ravenhall-Lorenz (PRL) EOS [20], respectively.Models of the EOS at 4 · 1011 < ρ < 2 · 1014g/cm3 are somewhat based on extrapolations of theavailable empirical information, as the extremely neutron rich nuclei appearing in this densityregime are not observed at the Earth. Models of the nuclear matter EOS ρ ≥ ρ0 are mainlyobtained from two different approaches: non-relativistic nuclear many-body theory (NMBT)and relativistic mean field theory (RMFT), that are based on different dynamical assumptionsand different formalisms.The NMBT approach rests on the premise that nuclear matter can be described as a collectionof non-relativistic nucleons, interacting through phenomenological two- and three-body forces.While suffering from the obvious limitations that are inherent in its non-relativistic nature,this approach is made very attractive by its feature of being strongly constrained by the data.Calculations of the energies of the ground and low-lying excited states of nuclei with a massnumber A ≤ 10 [21], whose results are in full agreement with the experimental values, have alsoshown that NMBT has remarkable predictive power.Within RMFT, based on the formalism of relativistic quantum field theory, nucleons are describedas Dirac fermions that interact through meson exchange. In its simplest implementation, thedynamic is described by a scalar and a vector meson field [22]. This approach is very elegant,

10


but leads to a set of equations of motion that turns out to be tractable only in the mean fieldapproximation, i.e. treating the meson fields as classical fields. Both NMBT and RMFT canbe generalized to take the possible appearance of strange baryons that are produced in weakinteraction processes and that may become energetically favourable at high density into account.However, the inclusion of baryons other than protons and neutrons entails large uncertainty, aslittle is known about their interactions and the available models are only loosely constrained byfew data.In this thesis we have employed six different models of the EOS of neutron star matter in theregion of supranuclear density, as recently proposed in the literarure and that corresponds to thestar core. Those referred to as APR2, BSS1 and BSS2 are based on NMBT, while the one labelled

Symbol EOS Composition

APR2 Akmal & Pandharipande (1997) npBBS1 Baldo, Burgio &Schulze (2000) npBBS2 Baldo, Burgio &Schulze (2000) nphG240 Glendenning (2000) nph

APRB200 Akmal & Pandharipande (1997)-Benhar & Rubino npqAPRB120 Akmal & Pandharipande (1997)-Benhar & Rubino npq

Table 1-1. The composition of different EOSs refers only to the strongly interacting components(neutron, proton, hyperon and quark). For further details, see [23, 24] and references therein.

G240 has been obtained from RMFT. The APR2 and BSS1 EOS include nucleons only, andtheir differences are mainly ascribable both to the approximate treatment of the three-nucleoninteractions in the BSS1 model and to the presence of the relativistic boost corrections in theAPR2 model. The BSS2 model is a generalization of BSS1 including the hyperons Σ− andΛ0. Finally, the G240 EOS includes the full baryon octet, consisting of protons, neutrons andΣ0,±, Λ0 and Ξ±. To take the possibility that transition to quark matter may occur at sufficientlyhigh density into account, we have considered two EOSs labelled APRB200 and APRB120[24].These two models of EOS are obtained by combining a lower density phase, extending up to∼ 4ρ0 as described by the APR2 nuclear matter EOS, with a higher density phase of deconfinedquark matter described within the MIT bag model that will be illustrated in section 1.4.

1.3 Neutron star observed properties

Today the identification of pulsars as highly magnetized rotating neutron stars is very wellestablished. In these stars, the rotational and magnetic axis are misaligned, thus they emitmagneto-dipole radiation (at the expense of the rotational energy) in the form of radio waves.As observers on Earth, if we lie on the cone of the radiation that is swept out by the rotatingstar, we have a chance to observe it as a pulsed source if it is not too distant. Several hundredof the pulsars known today were discovered in large surveys conducted in the 1970s at Arecibo,

11

1.3. Neutron star observed properties

Jodrell Bank and Parkes3. All known pulsars lie within our Galaxy and in the nearby LargeMagellanic Cloud galaxy. Of course, we expect that other galaxies with populations of massivestars (M > 8M) also have pulsars in them, but they are too distant for detection of the dimsignal. Indeed, most observed pulsars lie within a third of the Galaxy radius from the Sun.The first radio binary system of pulsars was found by Hulse and Taylor [25] and they were able todetermine many of its parameters, such as masses, orbital period, separation, orbital inclinationand, the inward spiraling of the neutron stars due to the emission of gravitational waves. Thediscovery of Hulse-Taylor binary pulsars PSR 1913+16 represents the first (indirect) evidencefor the existence of gravitational waves that are one of the most important predictions of theEinstein’s theory of General Relativity (GR).Several recent astrophysical observations that include pulsars orbiting another neutron star,a white dwarf or a main-sequence star, give us more information about masses. The mostaccurately measured masses come from timing observations of the binary radio pulsars. Themass is usually determined from measurements of the parameters of the orbital motion of thebinary system, such as the orbital period P, the projection x of the binary semi-major axis onthe line of sight, the eccentricity, etc. The observed parameters are related to the masses m1 andm2 of the members of the binary through the, so-called, mass function fM = fM (m1, m2, P, x).Relativistic corrections to the orbital parameters are usually parametrized in terms of one ormore post-Keplerian parameters that describe, for instance, the advance of the periastron, theorbital decay due to gravitational wave emission, the gravitational redshift, etc. In the context ofGeneral Relativity, the measurements of the mass function fM and of one of the post-Keplerianparameters is sufficient to uniquely determine m1 and m2. A long-term, well-observed systemcan have masses that are determined to impressive accuracy, as in the textbook case of PSR1913+16, in which the masses are (1.3867±0.0002)M and (1.4414±0.0002)M, respectively[26]. As shown in Fig.1-2, in which there is an updated compilation of the measured neutronstar masses, binaries with white dwarf companions show a broader range of neutron star massesthan binary neutron star pulsars, perhaps this is due to the fact that the latest ones are formed ina rather narrow set of evolutionary circumstances4 leading to a restricted range of neutron starmasses [28]. This restriction seems to be relaxed for the other neutron star binaries: observationalevidence is that a few of the white dwarf binaries may contain neutron stars that are larger than thecanonical 1.4M value, including the interesting case of PSR J0751+1807 in which the estimatedmass (with 1σ error bars) is (2.1±0.2)M [29]. Presently, most masses are known only withinlarge errors and the error bars would be consistent with a range of masses spanning (1-1.8)M.Masses can also be estimated in binaries which contain an accreting neutron star emitting X-ray.These systems are characterized by relatively large masses but the estimated errors are also large.For the system containing white dwarfs or main-sequences stars, continued observations, which

3Arecibo is a Radio Observatory in Puerto Rico, Jodrell Bank is in Cambridge and Parkes is in Australia.4We have to note that the core collapses because of the instabilities that are triggered when the core mass

approximately attains the Chandrasekhar mass 1.4 M to 1.5M or more, depending on the lepton fraction prevalentin the core and to a lesser degree on the mass of the presupernova star. Therefore, evolutionary scenarios are relatedto the mass of the progenitor [27] but we do not expect that the mass of the core to be very different from theChandrasekhar mass value.

12


Figure 1-2. Measured and estimated masses of neutron stars in radio binary pulsars (orange,grey and blue regions) and in X-ray accreting binaries (green). For each region, simple averagesare shown as dotted lines, weighted averages are shown as dashed lines. Data and figure are from[18].

13

1.3. Neutron star observed properties

reduce the observational errors, are necessary in order to clarify this situation.Neutron star radii by direct observation is far from being so accurate like the determination ofmasses. The small size of neutron stars and distances make it very difficult to measure the radiusdirectly. However, there are different techniques and methods that can be used to estimate theactual radii of neutron stars. For example, in low-mass X-ray binaries (LMXBs)5 from thermalobservations of the neutron star’s surface, it is possible to obtain, the so-called, radiation radius:

R∞ = R/√

(1 − 2GM/Rc2), (1.3.1)

which results from a combination of flux and temperature measurements, both redshifted at theEarth from the gravitational field on the neutron star surface. A given value of R∞ implies,for an assigned star, R < R∞. This quantity defined by (1.3.1) is clearly mass-dependent. Ameasured value of R∞ sets upper limits to both R and M, but without an independent estimateof mass, limited constraints are possible on the radius. Valuable additional information maycome from recent studies that are aimed at determining the NS mass-radius ratio from redshiftmeasurements. Cottam et al. [30] have reported that the Iron and Oxygen transitions observed inthe spectra of 28 bursts of the X-ray binary EXO0748-676 correspond to a gravitational redshiftz=0.35 where

z = (1 − 2GM

Rc2)−1/2 − 1, (1.3.2)

yielding in turn a mass-radius ratio M/R=1.153 M/km.Later, in this source, 45 Hz burst oscillations in the average power spectrum of a 38 thermonu-clear X-ray burst were found and this frequency have been interpreted as the NS spin frequency[31]. The authors in [31] have shown that the widths of the lines reported in [30] are consistentwith a 45 Hz spin frequency as long as the star radius is in the range from 10 to 15 km. However,the identification of spectral lines in [30] is still controversial as shown in [32].Fig.1-3 shows the dependence of the NS mass upon its radius for the model of EOS employed

both in [23] and in this thesis, it also shows two eos models denoted SS1 and SS2 that correspondto a bare quark star (see section 1.4) and to a quark star with a crust, extending up to neutrondrip density and described by BPS EOS, respectivily. Althought the results of ref.[30] are stillcontroversial, it appears that the EOS based on a nucleonic degree of freedom, such as APR2,BBS1, that are strongly constrained by nuclear data and nucleon-nucleon scattering, fulfill therequirement of crossing the redshift line within the band that corresponds to the observationallyallowed masses. While the possible addition of quark matter in a small region in the center of thestar, as in the case of APRB200 and APRB120, does not dramatically change the picture, insteadthe occurence of a transition to hyperonic matter, at densities as low as twice the equilibriumdensity of nuclear matter, lead to a sizable softening of the EOS, as in case of BBS2 andG240, thus making the mass-radius relation incompatible with that resulting from the redshiftmeasurement in ref.[30]. The strange star model, labelled SS1 and SS2 as considered in [23],

5LMXBs are accreting binary systems in which the orbital companion of a neutron star is a low-mass main-sequence star or a white dwarf. The donor component usually fills its Roche lobe and therefore transfers mass to thecompact star.

14


0

0.5

1

1.5

2

2.5

3

3.5

2 4 6 8 10 12 14 16 18 20

M/M

o

R (km)

equations of stateAPR2

APRB200APRB120

BBS1BBS2G240

SS1SS2

Figure 1-3. NS mass versus radius for the models of EOS described in the text. The twohorizontal lines denote the boundaries of the region of observationally allowed NS masses,while the straight red line corresponds to the mass-radius relation extracted from the redshiftmeasurements of Cottam et al. [30]. Data and figure are from [23].

also appears to be compatible with the observations, irrespective of the presence of the crust, butthe corresponding radii turn out to be significantly smaller than the ones predicted by any otherEOS and that are not consistent with the range predicted in [31]. However, we have to remindyou that strange star models imply a high degree of arbitrariness in the choice of the parametersof the model, as illustrated in the next section 1.4.Obviously a simultaneous measurement of the radiation radius and the redshift for the samesource would determine both M and R. However this type of measurement in LMXBs sufferfrom large uncertainty due to the distance of the source that compromise their reliability.It is important to point out that, as long as the star’s spin frequency is less than about 600 Hz, themeasured z or R∞ in (1.3.1) and (1.3.2) refers to the equatorial radius of an NS that is modelledas a non-rotating star. The equatorial radius of a non-rotating configuration it is less than 2%different from that of the rotating star but, for higher rotation rates, the correction due to therotation to the equatorial radius must be taken into account as we will see in Chapter 2.A second possibility of obtaining a more accurate estimate of the radius is offered by thediscovery of the double-pulsar system PSR J0737-3039A & B [33, 34]. The double pulsar PSRJ0737-3039A & B has opened up a new window for testing fundamental physics under extremeconditions, in fact, for this system the general relativistic corrections to the orbital parameters arethe largest known. The orbital period is very short, Pb=2.4 h, and the eccentricity is e = 0.088.The system consisting of two pulsars with spin periods of 22.7 ms (pulsar A) and 2.77 s (pulsarB). The gravitational masses are MA=(1.338±0.001)M and MB=(1.249±0.001)M, and thesecond one is also the lowest reliably measured mass for any neutron star to date [35].In particular this double system could provide, after a few years of high precision pulsar timing,a measurement of higher order relativistic corrections to the advance of periastron ω, due to spin-

15

1.4. Strange Stars

orbit coupling, that could eventually lead to a determination of the moment of inertia IA of thestar A [34, 35, 36] through the relation:

∆ωSO ∝ I/PM2, (1.3.3)

where I is the moment of inertia, P is the orbital period and M the gravitational mass. Sincethe masses of both stars are already accurately determined, and the moment of inertia can beexpressed as a relation involving only gravitational mass and the radius and no other EOSparameters, the measurement of IA could have enormous importance in order to discriminateamong families of EOSs. In [18, 37] it is also shown how the radius could be constrainedfor moment of inertia measurement having a 10% uncertainty. There are other astrophysicaltechniques and methods that can be used to infer the NS’s radius (see e.g.[17, 18] and referencestherein for a review) that at the moment should cluster in the range (9-16) km.It is important to note that, at the moment, all the available estimates of neutron star radii areallowed by a large number of EOSs. Better constraints on the EOS of matter under extremeconditions that are present in NSs can be produced, for example, by more accurate measurementsplanned in future X-ray missions, like Constellation-X6 which has a resolution and effective areaseveral times larger than Chandra and XMM.It should be pointed out that, since the first observational evidence of their existence in latesixties, NSs have proved to be a unique astrophysical environment in which very different fieldsof physics, ranging from nuclear physics to particle physics and GR, can be tested. As we haveillustrated in this section, more accurate measurements of some properties of NSs, such as massesand radii would allow us to set more stringent constraints on the nature of nuclear forces in thefuture.In this thesis, we propose another, less traditional way in which information about neutron starscan be gleaned: gravitational waves. Let us note that, since gravitational waves iteract veryweakly with matter at the NS surface, and thus they should be independent from the particularmodel of NS atmosphere, they could represent a new and very powerful tool to obtain informationabout the EOS of the source.

1.4 Strange Stars

It is usually assumed that the ground state of hadronic matter is the state in which quarks areconfined in individual hadrons. Most of the mass of every object that we can study experimentallyin the observable universe, resides in its nuclear constituents. However, there is presently noevidence to show that the ground state of matter must be that of quarks that are confined inhadrons. The confined state may simply be a very long-lived state, but not the absolutely stable

6The Constellation-X Observatory is a combination of several X-ray telescopes working in unison to generatethe observing power of one giant telescope. With the Observatory, scientists will investigate black holes, Einstein’sTheory of General Relativity, spectral lines from NS surfaces, galaxy formation, etc. The four telescopes willcombine to provide a sensitivity 100 times greater than any past or current X-ray satellite mission.

16


one. Indeed it is not possible to prove otherwise unless a lower energy state is discovered. In the1980s Witten [38], following Bodmer’s seminal paper [39], suggested that the true and absoluteground state of matter consisting of degenerate up, down and strange quarks, called “strange”quark matter, might be bound and stable at zero temperature and pressure.This hypothesis, while not being directly verifiable in terrestrial laboratories, may be confirmedby the observation of “strange stars”, i.e. compact astrophysical objects entirely made up ofstrange matter except, possibly, for a thin outer crust. Theoretical studies of structural propertiesof quark stars, pioneered by Itho over three decades ago [40], have been booming after ROSATdiscovered the isolated pulsar RXJ1856.5-3754 [41]. The data reported in Ref. [42] seemedin fact to indicate that its radiation radius should be in the range (3.8 − 8.2) Km, so that thecorresponding stellar equatorial radius would be far too small compared to that which is typicalof neutron stars. The results of this analysis triggered speculation on the nature of RXJ1856.5-3754, and it was suggested that it may be a strange star, though this hypothesis has now beenruled out [43, 44, 45].Whether quark stars do exist in nature is an open question, and the astrophysical scenario inwhich they may form is still poorly understood. For instance, it has been argued that, besidesthe standard gravitational collapse, strange stars may form in low-mass X-ray binaries when,due to accretion, matter in a neutron star core reaches sufficiently high densities to undergo adeconfinement phase transition to quark matter [46]. If strange matter were the true ground state,then no barrier would exist for the conversion of the entire star, once the core has converted.In this way NSs would be converted to strange stars. How could one tell a strange star from aneutron star?Looking at Fig.1-3 it is evident that the mass-radius relation of strange stars is different fromthat of NSs and that for the same value of gravitational mass, strange stars could have smallerradii than NSs. We have to recall that the limiting (Kepler) spin frequency for an NS is givenapproximately by

ΩK ' α(M/R3)1/2, (1.4.1)

where α is unity in Newtonian gravity and it is empirically found to be about 0.65 in GR.Therefore, self-bound7 stars, like strange stars, at sufficiently high-energy density, having smallerradii and can rotate faster than gravitationally bound stars (such as NSs). Consequently, if apulsar with a period falling below the limit of the minimum period of NSs were discovered, itwould have to be a strange star. In Chapter 3 we will show how gravitational waves allow us todiscriminate between strange stars and NSs.Due to the complexity of the fundamental theory of strong interactions between quarks (QuantumChromo-Dynamics, or QCD), theoretical studies of strange stars are necessarily based on models.The most used is the MIT bag model [47], in which the two main elements of QCD, namelycolour confinement and asymptotic freedom, are implemented through the assumptions that:

7By self-bound we mean matter that is bound in bulk by strong interaction and does not require gravity for thestability of stellar objects made from it.

17

1.4. Strange Stars

1. quarks occur in colour neutral clusters that are confined to a finite region of space (thebag), the volume of which is limited by the pressure of the QCD vacuum (the bag constantB),

2. residual interactions between quarks are weak, and can be treated in low order perturbationtheory in the colour coupling constant αs.

The bag model parameters are the bag constant B, the quark masses mf (the index f = u, d, slabels the three active quark flavours; at the densities relevant to our work, heavier quarks donot play a role) and the running coupling constant αs, whose value at the energy scale µ can beobtained from the renormalization group relation

αs(µ) =12π

(33 − 2Nf) ln(µ2/Λ2QCD)

. (1.4.2)

In the above equation Nf = 3, denotes the number of active flavours, while ΛQCD is the QCDscale parameter whose value is constrained to the range (100 − 250) MeV by high energy data(see, e.g., [48]). At the scale typical of the quarks chemical potentials Eq. (1.4.2) yields

αs ∈ [0.4, 0.6] . (1.4.3)

As quarks are not observable as individual particles, their masses are not directly measurable.However, they can be inferred from hadron properties that are supplemented by theoreticalcalculations. According to the 2002 Edition of the Review of Particle Physics [49], the massesof up and down quarks do not exceed few MeV, and can therefore be safely neglected, while themass of the strange quark is much larger, its value being in the range

ms ∈ [80, 155] MeV . (1.4.4)

The bag constant is subject to much larger uncertainty. In early applications of the MIT bagmodel B, αs and ms were adjusted to fit the measured properties of light hadrons (spectra,magnetic moments and charge radii). This procedure leads to values of B that differ from oneanother by as much as a factor of ∼ 6, ranging from 57.5 MeV/fm3 [50] to 351.7 MeV/fm3 [51],while the corresponding values of αs turn out to be close to or even larger than unity. Moreover,the strange quark masses resulting from these analyses are typically much larger than the upperlimit given by Eq. (1.4.4). Using the parameters which are determined from fits of light hadronspectra to describe bulk quark matter is questionable, as these spectra are known to be stronglyaffected by details of the bag wave-functions, as well as by spurious contributions arising fromthe center of mass motion.The requirement that strange quark matter be absolutely stable at zero temperature and pressureimplies that B cannot exceed the maximum value Bmax ≈ 95 MeV/fm3 [52]. For values ofB exceeding Bmax, a star entirely made up of deconfined quarks is not stable. Under theseconditions quark matter can only occupy a fraction of the available volume and the star is saidto be hybrid [53]. The results of Ref. [54, 24], based on a state-of-the-art description of the

18


low-density hadronic phase, suggest that, assuming that the transition to quark matter proceedsthrough the formation of a mixed phase, hybrid stars can only exist for values of the bag constantup to ∼ 200 MeV/fm3 and contain a rather small amount of quark matter. Hybrid star modelsused in this thesis are APRB120 and APRB200. In these two models, quark matter consists ofmassless up and down quarks and massive strange quarks, with ms=150 MeV. The value of thebag constant is 200 and 120 MeV/fm3 in the APRB200 and APRB120 model, respectively andαs = 0.5. In this thesis we focus on bare strange stars and consider values of the bag constant inthe range

B ∈ [57, 95] MeV/fm3 . (1.4.5)

It should also be mentioned that, although somewhat more refined dynamic models have beenproposed, the MIT bag model appears to provide quite a reasonable description of quark matter.It is possible to use more sophisticated models to describe interaction beetween quarks, that, forexample, should include the possible occurrence of colour superconductivity [55]. However, therelative stability of the different superconducting phases discussed in the literature [55, 56] is notyet firmly established and their occurrence is expected to affect mostly transport properties andcooling, rather than the stellar structure on which we focus in this thesis.

19

1.4. Strange Stars

20

2

Rotating Neutron stars

We saw in section 1.3 that if we knew the NS’s mass, radius and moment of inertia, we wouldobtain information on the EOS. But, if we want to have a chance of putting constraints on thestate of matter inside an NS, by means of comparison with astrophysical observations, we haveto extend our investigation to other, not less important properties of an NS such as maximummass, maximum rate of rotation, baryonic mass, angular momentum, etc. In particular, it is veryuseful to know how these properties change when varing the equation of state and the star’srate of rotation. All these properties may be estimated theoretically by constructing models ofrotating neutron stars. The study of the structure of a rotating NS can essentially be handledin two different ways. The first one is based on the direct numerical integration of Einstein’sequations for stellar structure: very often, in the literature, one refers to the models as describedwith the adjective exact. The second one is the perturbative approach, introduced by Hartle in1967 [57], in which Einstein’s equations for stellar structure are expanded perturbatively andthe parameter of the expansion is the angular velocity. In the next sections we will show thatthe structure of a rapidly rotating NS can be described to a high level of accuracy by using theperturbative approach, developed to a third order in the angular velocity Ω [58].There are several reasons why the perturbative approach may be preferred to the direct numericalintegration of Einstein’s equations for stellar structure. One is that the angular behaviour of thesolution is given explicitely in terms of Legendre polynomials of the first kind, whereas the radialbehaviour can be found by integrating linear differential equations in the radial variable; thus,instead of solving the non-linear, coupled, partial differential equations of the exact theory, onesimply integrates a set of ordinary, linear differential equations. Moreover, these equations needto be integrated only inside the star, as we will illustrate in detail later, because the exteriorsolution can be found explicitely in terms of known polynomials.A further element of interest is that the perturbative approach allows us to evaluate to what extenta physical quantity changes at each order in Ω, and to check whether the nth-order contributionis needed in order to estimate a choosen quantity, or whether the first order estimates, often usedto interpret astronomical observations, are enough. For these reasons, the perturbative approachexpanded to the 2nd order in rotation rate has been used by several authors to study properties ofrotating neutron stars [59, 60, 61, 62, 63, 17, 64] (see also the review [65] and refereces therein).On the other hand, the perturbative approach has a drawback since it fails when the angular

21

velocity approaches the mass-shedding limit, i.e. when the star rotates so fast that thegravitational attraction is not sufficient to keep matter bound to the surface; numerical integrationof the exact equations shows that in these conditions, a cusp forms on the equatorial plane of thestar [66, 67, 68] which cannot be reproduced by the first few orders of the perturbative expansions(we shall discuss this point quantitatively in section 2.3). However, the rotation rate of all knownpulsars is much lower than the mass-shedding limit for most EOSs that are considered in theliterature. Thus, unless we are specifically interested in the mass-shedding problem, we canuse the perturbative approach to study the structural properties of observed neutron stars. Theproblem is to establish how many terms in the expansion we need to consider to describe, say,the fastest isolated pulsar observed so far, PSR 1937+211, whose period is T = 1.56 ms, and thisis the question we plan to answer.The equations of structure for uniformly rotating stars have been developed to a second order inthe angular velocity in [57, 59] and subsequently extended to a third order in [58]. With respectto the results from [58] we introduce two novelties:

1. the third order corrections involve two functions, w1 and w3. w1 gives a contribution to themoment of inertia, w3 affects the mass-shedding velocity. Both w1 and w3, contribute tothe dragging of inertial frames that is a pure relativistic effect (see section 2.2). In [58] thesolution for w1 was given explicitely outside the star in terms of Legendre polynomials ofa second kind; this expression contained some errors that we have corrected. In addition,we complete the third order solution giving the explicit expression of w3 outside the star.

2. In [58] a procedure was developed to construct families of constant baryonic masssolutions with varying angular velocity, this was applied to study the behaviour of themoment of inertia along such sequences for small values of angular velocities. Thisprocedure cannot be applied to fast rotating stars, and, furthermore, it was shown to beunstable when the stellar mass approaches the maximum mass for any assigned equationof state. In this Chapter we illustrate a different algorithm that we have developed andapplied and that allows us to describe a sequence of constant baryonic mass stars at anyvalue of angular velocity (smaller than the mass-shedding limit), and which is also stableat the maximum mass limit.

In section 2.2 we shall briefly explain what the corrections to the metric and to the thermody-namical functions are and that are needed to describe the structure of a rotating star to the thirdorder in Ω, and in Appendix A, we shall summarize the equations that they satisfy both insideand outside the star.In order to test the perturbative scheme, we shall compare the results we find in solving theseequations for three selected EOSs, with those of ref. [68] and [70] where the exact Einstein

1Subsequently, after the publication of our results, a new pulsar was observed in the globular cluster Terzan 5with a period T = 1.40 ms [69].

22

Chapter 2. Rotating Neutron stars

equations have been integrated for the same EOSs. Hereafter, the adjective exact used to referto Einstein’s equations will mean that the full set of equations is solved numerically and non-perturbatively. This comparison will be discussed in section 2.3 and will allow us to assess thevalidity of the third order approach.We shall then integrate the third order equations to model an NS that is made up of an outer crust,an inner crust and a core, each shell being described by a non-viscous fluid which obeys equationsof state that are appropriate in describing different density regions which we have illustated inChapter 1, section 1.2. There is a general consensus on the matter composition of the crust, forwhich we use two well established EOSs (see section 1.2): the Baym-Pethick-Sutherland (BPS)EOS [19] for the outer crust (107 ∼< ρ ∼< 4·1011 g/cm3), and the Pethick-Ravenhall-Lorenz (PRL)EOS [20] for the inner crust (4 · 1011 ∼< ρ ∼< 2 · 1014 g/cm3). For the inner core, ρ > 2 · 1014

g/cm3, we use recent EOSs which model hadronic interactions in different ways which lead toa different composition and dynamic. The main assumptions underlying these EOSs have beenoutlined in Chapter 1, section 1.2.For a given EOS there are two kinds of equilibrium configurations, ususally indicated in theliterature, as ”normal” and ”supramassive”. They both refer to rotating neutron stars withconstant baryonic mass and varying angular velocity. Each normal sequence has, as a limitingconfiguration, a non-rotating, spherical star. The supramassive sequences do not possess thislimit, and all members of a sequence have a baryonic mass larger than the maximum mass ofa non-rotating, spherical star. Our study is confined to stars belonging to the normal sequence.The results of our calculations will be discussed in section 2.4.

2.1 The background model: non-rotating stars

The simplest equilibrium model of a star is that of a non-rotating, zero temperature, sphericallysymmetric distribution of matter described by a perfect fluid. The relativistic equilibriumconfiguration can be described by using a coordinate system (t, r, θ, φ) with respect to whichthe spacetime geometry is given by the line element

ds2(0) = g(0)µνdx

µdxν = −eν(r) + eλ(r)dr2 + r2(dθ2 + sin2θdφ2) (2.1.1)

where the two functions ν(r) and λ(r) have to be determined by solving Einstein’s field equations

Gµν = 8πTµν, (2.1.2)

with the appropriate stress-energy tensor and boundary conditions. For a distribution of matterdescribed by a perfect fluid, the stress-energy tensor that acts as a source in the field, Einstein’sequations are given by

Tµν = (ε+ p)uµuν + pg(0)µν , (2.1.3)

where ε is the energy density of the fluid, p is the pressure and uµ is its covariant four-velocity.The equilibrium solution is obtained by solving the field equations together with the conservation

23

2.1. The background model: non-rotating stars

equations T µν;ν = 0. It is useful to define a function m(r) as follows

m(r) =1

2r(1 − e−λ(r)). (2.1.4)

The system of equations yields to three structure equations which relate the four unknowns ν, λ(or m), ε and p

dM(r)

dr= 4πr2ε(r), (2.1.5)

dν(r)

dr= 2

M(r) + 4πr3P (r)

r(r − 2M(r)),

dP (r)

dr= −(P (r) + ε(r))

2ν(r),r.

Equations (2.1.5) are the Tolmann-Oppenheimer-Volkoff (TOV) of hydrostatic equilibrium inGR and in spherical symmetry. The first equation clearly identifies m(r) as the gravitationalmass which is enclosed within a radius r. To determine the system of TOV equations fully, arelation between the energy density ε and the pressure p of the fluid has to be supplemented. Inthe general case, this relation, i.e. the equation of state (EOS), is given in a parametric form as

p = p(φi), ε = ε(φi), (2.1.6)

where φi are the internal variables that characterize the EOS, such as specific entropy,temperature, chemical potential, etc. However, in many applications, the EOS is assumed tohave a barotropic form

p = p(ε) (2.1.7)

in which the pressure is a function of the energy density only. Finally, the equilibrium stellarmodel is obtained numerically by integrating the TOV equations (2.1.5) from the center ofthe star r=0 up to the point r=R where the pressure vanishes, which is by definition the star’ssurface. Since from the Birkhoff theorem, the solution in the exterior of a spherically symmetricdistribution of matter is described by the Schwarzshild spacetime, the metric functions ν and λmust satisfy the following boundary condition at the surface of the star

M ≡ m(R) =1

2R(1 − eν(R)) =

1

2R(1 − e−λ(R)), (2.1.8)

where M is the total gravitational mass of the star. Hereafter, r = R and M(R) will indicate,respectively, the radius and the mass of the non-rotating star, found by solving the TOV equationsfor an assigned EOS.

24


2.2 The metric and the structure of rotating stars

In this section we introduce all the quantities that describe the star to the third order in the angularvelocity Ω. Following [57, 58, 59] we write the metric as

ds2 = −eν(r) [1 + 2h(r, θ)] dt2 (2.2.1)

+ eλ(r)

[

1 +2m(r, θ)

[r − 2M(r)]

]

dr2

+ r2 [1 + 2k(r, θ)]

dθ2 + sin2 θ [dφ− w(r, θ)dt]2

.

Here eν(r), eλ(r) and M(r) are functions of r and describe the non-rotating star solution of theTOV equations (see section 2.1 and Appendix A). The functions h(r, θ), m(r, θ), k(r, θ) andw(r, θ) are the perturbative corrections

h(r, θ) = h0(r) + h2(r)P2(θ) +O(Ω4) (2.2.2)

m(r, θ) = m0(r) +m2(r)P2(θ) +O(Ω4) (2.2.3)

k(r, θ) = k2(r)P2(θ) (2.2.4)

= [v2(r) − h2(r)] P2(θ) +O(Ω4)

w(r, θ) = ω(r) + w1(r) (2.2.5)

− w3(r)1

sin θ

dP3(θ)

dθ+O(Ω5)

P2(θ) and P3(θ) are the l = 2 and l = 3 Legendre polynomials. As described in [57, 58, 59],the function ω is of order Ω, whereas (h0, h2, m0, m2, k2, v2) are of order Ω2, and (w1, w3) are oforder Ω3. In a similar way, an element of fluid, that is located at a given (r, θ) in the non-rotatingstar, experiences a displacement due to the rotation that can be expanded in even powers of Ω:

ξ = ξ0(r) + ξ2(r)P2(θ) +O(Ω4), (2.2.6)

where both ξ0(r) and ξ2(r) are of order Ω2. We shall assume that matter in the star is describedby perfect fluid with energy momentum tensor

T µν = (ε + P )uµuν + Pgµν. (2.2.7)

Since the fluid element is displaced, its energy density and pressure change; in a reference framethat is momentarily moving with the fluid, the pressure variation is

δP (r, θ) = [ε(r) + P (r)] [δp0(r) + δp2(r)P2(θ)] (2.2.8)

and it can be expressed in terms of ξ as follows

δp0(r) = −ξ0(r)[

1

ε + P

dP

dr

]

, (2.2.9)

δp2(r) = −ξ2(r)[

1

ε + P

dP

dr

]

, (2.2.10)

25

2.2. The metric and the structure of rotating stars

where ε(r) and P (r) are the energy density and the pressure computed for the non-rotatingconfiguration. The corresponding energy density variation of the fluid element is

δε(r, θ) =dε

dP(ε+ P ) [δp0(r) + δp2(r)P2(θ)] . (2.2.11)

This last equation has been derived on the assumption that the matter satisfies a barotropic EOS.In appendix A we will write the equations that are satisfied by the corrections to the metricfunctions and to the thermodynamic variables to the various orders in Ω, and for each functionwe will specify the appropriate boundary conditions. It should be stressed that these equationshave to be numerically integrated only for r ≤ R, where R is the radius of the non-rotatingstar. For r > R the solution can be found analytically. Once the various functions have beencomputed as shown in Appendix A, we can evaluate how the star changes in shape due to rotation,the moment of inertia, the mass-shedding limit and all quantities one is interested in. Here, weshall briefly summarize what the relevant physical quantities are that we shall consider:

• Stellar deformationThe effect of rotation described by the metric (2.2.1) on the shape of the star can be dividedinto two contributions:A) A spherical expansion which changes the radius of the star and is described by thefunctions h0 and m0.B) A quadrupole deformation which is described by the functions h2, v2 and m2. Asa consequence of these two contributions, that are both of the order Ω2 and that will beindicated with the labels A and B, respectively, the radius of the star, the eccentricity andthe gravitational mass will change as follows.A fluid element on the surface of the spherical non-rotating star, i.e. at (r = R, θ), will bedisplaced by an amount δR given by

δR = δRA + δRB = ξ0(R) + ξ2(R)P2(θ) (2.2.12)

where ξ0(R) and ξ2(R) can be computed by eqs. (A.2.9) and (A.2.19) as shown inappendix A. Once we know δR we can compute, for instance the eccentricity of the star

Ec =[

(equatorial radius)2/(polar radius)2 − 1]1/2

. (2.2.13)

The change in the gravitational mass is given by

δMgrav =

[

m0(R) +J2

R3

]

. (2.2.14)

Note that δMgrav does not depend on quadrupolar perturbations (h2, v2, m2), nor on theperturbations that are first (ω) and third order (w1, w3) in Ω. The reason is that such quantityis invariant under rotations of the system, and it is even so for parity transformations (whichchange Ω → −Ω).

26


• Dragging of inertial framesThe function ω(r, θ) in the metric (2.2.1) is responsible for the dragging of inertial frames.It affects two important physical quantities, i.e. the mass-shedding limit (section 2.2.1)and the moment of inertia (section 2.2.3).The function ω(r, θ) has a very interesting and fundamental significance (see e.g [53]Ch.6): it is the angular velocity of the local inertial frames. Its acts as follows. Imagine,for example, that a particle is dropped from rest at a great distance from the star, if the starwere non-rotating, the particle would fall towards the center of the star, but if the star wererotating, the freely falling path would not be towards the center. As the particle approachedwould experience an ever increasing drag in the direction of rotation of the star, for thisreason ω(r, θ) is referred to as the angular velocity of the local inertial frames: the pluralis used because the angular velocity depends on the distance r from the star. As shown inAppendix A, outside the star ω ∼ r−3, so that spacetime is flat at infinity, and distant localinertial frames do not rotate with the star.

2.2.1 The mass-shedding limit

The mass-shedding velocity can be calculated using a procedure developed by [71]. Given a starwith assigned baryonic mass which rotates at a given Ω, let us consider an element of fluid whichbelongs to the star and is located on the surface at the equator; from the metric (2.2.1) it is easyto see that it moves with velocity

V bound = eψ(r,θ)−β(r,θ)

2

[

Ω −(

ω(r) + w1(r) −3

2w3(r)

)]

, (2.2.15)

where

eβ(r,θ) = eν(r) [1 + 2h(r, θ)] (2.2.16)

eψ(r,θ) = r2 [1 + 2k(r, θ)] sin2 θ.

In general, the fluid element will not follow a geodesic of metric (2.2.1). Using the geodesicequations we can also compute the velocity of a particle which moves on a circular orbit justoutside the equator, in the co-rotating direction

V free =(ω′ + w′

1 − 32w′

3)

ψ′eψ−β

2 (2.2.17)

+

β ′

ψ′+

[

(ω′ + w′1 − 3

2w′

3)

ψ′eψ−β

2

]21/2

,

where a prime indicates differentiation with respect to r. In general, for an assigned Ω smallerthan the mass-shedding limit, the two velocities V bound and V free are different, and in particular

V bound < V free.

27


When Ω increases, the two velocities converge to the same limit; when this limit is reached thefluid element on the surface will not be bound anymore and the star will start loosing matter fromthe equator. The value of Ω for which V bound = V free is the the mass-shedding limit and willbe indicated as Ωms. It should be stressed that V bound and V free are computed at the equatorialradius of the rotating configuration, i.e.

Rrot = R + ξ0(R) − 1

2ξ2(R). (2.2.18)

In order to apply this procedure, we need to construct sequences of stellar models with constantbaryonic mass and variable angular velocity.

2.2.2 The algorithm used to construct a sequence of constant baryonicmass solutions

The baryonic mass of the star is defined as

Mbar =

∫

t=const

√−g ut ε0 d3x, (2.2.19)

where g is the determinant of the 4-dimensional metric, d3x is the volume element on at = constant hypersurface, and ε0(r) = mNnbar(r) is the rest mass-energy density, not to beconfused with the energy density ε(r). Since all known interactions conserve the baryon number,ε0 is a conserved quantity, i.e.

(ε0uµ);µ = 0 . (2.2.20)

The expansion of Mbar in powers of Ω is

Mbar = M(0)bar + δMbar +O(Ω4) (2.2.21)

where

M(0)bar = 4π

∫ R

0

dr

(

1 − 2M(r)

r

)−1/2

r2ε0(r) (2.2.22)

and

δMbar = 4π

∫ R

0

r2dr

(

1 − 2M(r)

r

)−1/2

(2.2.23)

·[

1 +m0(r)

r − 2M(r)+

1

3r2[Ω − ω(r)]2e−ν(r)

]

ε0(r)

+dε0dP

(ε + P )δp0(r)

.

In the following, we shall briefly compare the procedure developed by Hartle in [58] to constructconstant baryonic mass sequences and the procedure we have used. The general scheme of thefirst one is:

28


• Hartle’s procedure

1. Given a central energy density ε(r = 0) and an assigned EOS, the TOV equations aresolved to find the non-rotating configuration; the model is identified by a baryonicmass Mbar = M .

2. For an assigned value of the angular velocity Ω, the equations of stellar structure aresolved to order Ω2, imposing the correction to the pressure, δp0(r = 0) is differentfrom zero. The value of δp0(r = 0) is then changed until the sought baryonic mass isreached.

Our procedure used to generate sequences with constant baryonic mass is the following:

• Our procedure

1. Is the same as point 1 of Hartle’s procedure.

2. For an assigned value of Ω we solve the equations of stellar structure up to order Ω3

for the different values of ε(r = 0) and by imposing δp0(r = 0) = 0. Among thesemodels we select the one with baryonic mass M .

If, in addition, we want to compute the mass-shedding velocity, having computed all metricfunctions for a given Ω we evaluate V bound and V free. If V bound = V free we have reached themass-shedding limit and we stop the procedure. If V bound < V free we choose a higher value ofΩ and go to point 2. This procedure can be applied to rapidly rotating stars and to models ofstars that are close to the maximum mass. It is important to note that Hartle’s procedure presentsa problem when applied to rapidly rotating stars; indeed if the rotation rate is sufficiently high,the change in the central energy density δε(r = 0) may be so large that treating this changeas a perturbation is inappropriate and produces a large error in all quantities. Instead with ourapproach, this change is treated as a background quantity (we vary ε(r = 0)), and the results aremuch more accurate (as can be seen by the comparison with the “exact” integrations of [68] and[70] as discussed in section 2.3). A second problem arises when the mass of the star is close tothe maximum mass (where we mean the maximum mass along a constant Ω sequence). SinceMbar = Mbar(Ω, εc), where

εc ≡ ε(r = 0) + δε(r = 0)

it follows that∂εc∂Ω

∣

∣

∣

Mbar

= −∂Mbar

∂Ω

∣

∣

∣

εc

/∂Mbar

∂εc

∣

∣

∣

Ω.

Since near the maximum mass∂Mbar

∂εc

∣

∣

∣

Ω−→ 0,

it follows that∂εc∂Ω

∣

∣

∣

Mbar

−→ ∞.

Consequently, since in Hartle’s procedure, ε(r = 0) is kept constant, and δε(r = 0) is treated asa perturbation, the procedure fails.

29


2.2.3 The moment of inertia of rotating stars

The angular momentum of an axisymmetric matter distribution rotating about the symmetry axiscan be expressed in terms of a conserved four vector, the angular momentum density

Jµ = T µν ξν(φ),

where ξν(φ) is the Killing vector associated to axial symmetry. The conserved total angularmomentum J tot can be found by integrating Jµ over any spacelike hypersurface. Given thecoordinate system we use, the natural choice is a t = const hypersurface, and since ξν(φ) = δν(φ)

J tot =

∫

t=const

√−g J0 d3x =

∫

t=const

√−gT tφ d3x (2.2.24)

where, as in (2.2.19), g is the determinant of the 4-dimensional metric, and d3x is the volumeelement on the chosen hypersurface. Consequently, the moment of inertia of the matterdistribution is

I =J

Ω=

1

Ω

∫

t=const

√−gT tφ d3x . (2.2.25)

Using Einstein’s equations, in (2.2.24) and (2.2.25) we can replace T tφ with 1

8πRt

φ and expandthem in powers Ωn. With this procedure it is possible to show that only terms with n oddcontribute to this expansion, and that the first terms are

J tot = J + δJ +O(Ω5) (2.2.26)

where J is the first order contribution

J =1

6

[

r4 j(r)d$

dr

]

|r=R

(2.2.27)

and δJ is the third order correction

δJ =1

6

4r3 dj

dr$(ξ0 −

1

5ξ2) + r4 j

dw1

dr

+r4 jd$

dr

[

h0 +m0

r − 2M+

4(v2 − h2)

5

−1

5

(

h2 +m2

r − 2M

)]

|r=R

. (2.2.28)

In eqs. (2.2.27) and (2.2.28) the functions $ and j(r) are defined as follows

$(r) = Ω − ω(r), (2.2.29)

j(r) = e−ν(r)/2√

1 − 2M(r)

r.

30


Thus, the corrections to the moment of inertia are

I = I (0) + δI +O(Ω4), (2.2.30)

I(0) =J

Ω, δI =

δJ

Ω. (2.2.31)

It should be noted that the angular momentum defined in (2.2.24) coincides with that definedusing the asymptotic behaviour of the metric as in [72] (Ch. 19). In appendix A, we shall showhow to find δJ from the asymptotic behaviour of the metric function w1.

2.3 Comparison with non-perturbative results

In order to verify to what extent the perturbative scheme correctly describes the physicalproperties of a rapidly rotating neutron star, we compare the results of the integration of theequations of stellar structure expanded to order Ω3 with those of ref. [68] and [70], where thefully non-linear Einstein equations have been integrated. To perform the comparison we studystellar sequences with constant baryonic mass and variable spin frequency ν = Ω/2π, and wechoose three EOSs labelled as follows: AU, which results from an approach based on nuclearmany body theory [73], APR2, (see section 1.2, APRb in [70]) which takes recent results onnuclear many body theory into account [74, 75] and L obtained from relativistic mean fieldtheory [76].AU and L have been used both in [68] and in [70]; the results of the two papers agree to betterthan 1% for both EOSs, except that for the estimate of the moment of inertia for the EOS L, forwhich the relative error is, at most, of the order of 1.3 %. APR2 has only been used in [70]. Sincewe interpolate the EOS tables with the same linear interpolation as in [68] (in [70] a 5th-orderpolynomial has been used), in the following tables our results for AU and L will be comparedwith those of [68]. For each EOS we consider two stellar sequences with constant baryonic massand varying angular velocity:

• Sequence A, such that the gravitational mass of the non-rotating configuration is M =1.4M.

• Sequence B, that corresponds to the maximum mass.

The corresponding values of the baryonic mass are given in table 2.3. The values of ν are chosenas in [68] and [70].

The results of the comparison are summarized in Tables 2-2, 2-3, 2-4 (sequence A) and2-6, 2-5, 2-7 (sequence B), where we show the stellar parameters for different values of the spinfrequency. The lines labelled CST and BS refer, respectively, to the data found by integratingthe exact equations of stellar structure in [68] and in [70]; BFGM refers to our results that werefound by integrating the equations of stellar structure, perturbed to the order Ω3 for the samemodels and EOS. The spin frequency ν is given in column 2, the central density εc in column 3,

31

2.3. Comparison with non-perturbative results

Mbar/M

AU 1.58 2.64APR2 1.55 2.69

L 1.52 3.16

Table 2-1. The baryonic masses of the stellar models used to compare the results of theperturbative approach with those found by integrating the exact Einstein equations are given forthe three considered EOSs. The data in column 2 and 3 correspond to a non-rotating star withgravitational mass M = 1.4M and Mmax, respectively.

EOS AU (Mbar = 1.58M)ν (kHz) εc/ε∗ I/I∗ Mgrav/M Req (Km) ν/νms

CST 0 1.206 1.160 1.400 10.40 0BFGM 0 1.204 1.154 1.395 10.39

∆ - 0.2% 0.5% 0.4% 0.1%CST 0.476 1.192 1.191 1.403 10.59 0.379

BFGM 0.476 1.190 1.182 1.398 10.56∆ - 0.2% 0.8% 0.4% 0.3%

CST 0.896 1.148 1.292 1.412 11.22 0.713BFGM 0.896 1.155 1.254 1.405 11.02

∆ - 0.6% 2.9% 0.5% 1.8%CST 1.082 1.111 1.386 1.412 11.87 0.861

BFGM 1.082 1.132 1.298 1.410 11.34∆ - 1.9% 6.3% 0.1% 4.5%

CST 1.257 1.048 1.566 1.432 14.44 1.000BFGM 1.257 1.108 1.344 1.414 11.73

∆ - 5.7% 14.2% 1.3% 18.8%

Table 2-2. Stellar parameters for sequence A. Data are computed along a sequence of stellarmodels with constant baryonic mass which corresponds to a non-rotating configuration of massM = 1.4M and varying spin frequency ν = Ω/2π (given in kHz in column 2). εc is the centraldensity in units of ε∗ = 1015g/cm3; the moment of inertia I is in units of I∗ = 1045g cm2, theequatorial radius Req is in km. For each value of ν and for each quantity, we give the relativeerror ∆. Mgrav is the gravitational mass in solar mass units. In the last column the ratio of thespin frequency and the mass-shedding frequency (as computed in [68] and [70]) is given.

32


EOS APR2 (Mbar = 1.55M)ν (kHz) εc/ε∗ I/I∗ Mgrav/M Req (Km) ν/νms

BS 0 0.995 - 1.403 11.55 0BFGM 0 0.988 1.310 1.389 11.58

∆ - 0.7% - 1.0% 0.3%BS 0.554 0.970 1.396 1.409 11.99 0.532

BFGM 0.554 0.964 1.373 1.394 11.97∆ - 0.6% 1.6% 1.0% 0.2%BS 0.644 0.960 1.425 1.411 12.18 0.619

BFGM 0.644 0.956 1.395 1.396 12.12∆ - 0.4% 2.1% 1.0% 0.5%BS 0.879 0.920 1.551 1.418 13.05 0.844

BFGM 0.879 0.929 1.466 1.401 12.63∆ - 1.0% 5.5% 1.2% 3.2%BS 1.041 0.870 1.737 1.428 15.04 1.000

BFGM 1.041 0.905 1.526 1.405 13.13∆ - 4.0% 12.1% 1.6% 13.0%

Table 2-3. Stellar parameters for sequence A are given as in Table 2-2. Data are computed alonga sequence of stellar models with constant baryonic mass which corresponds to a non-rotatingconfiguration of mass M = 1.4M and varying spin frequency ν = Ω/2π (given in kHz).

EOS L (Mbar = 1.52M)ν (kHz) εc/ε∗ I/I∗ Mgrav/M Req (Km) ν/νms

CST 0 0.433 2.127 1.400 14.99 0BFGM 0 0.440 2.120 1.402 14.99

∆ - 1.6% 0.3% 0.1% 0.0%CST 0.283 0.427 2.194 1.402 15.30 0.395

BFGM 0.283 0.433 2.182 1.404 15.27∆ - 1.4% 0.5% 0.1% 0.2%

CST 0.505 0.411 2.378 1.407 16.22 0.705BFGM 0.505 0.419 2.307 1.408 15.95

∆ - 1.9% 3.0% 0.1% 1.7%CST 0.609 0.398 2.548 1.412 17.17 0.851

BFGM 0.609 0.410 2.385 1.410 16.43∆ - 3.0% 6.4% 0.1% 4.3%

CST 0.716 0.377 2.881 1.419 21.25 1.000BFGM 0.716 0.400 2.473 1.414 17.06

∆ - 6.1% 14.2% 0.4% 19.7%

Table 2-4. Stellar parameters for sequence A are given as in Table 2-2. Data are computed alonga sequence of stellar models with constant baryonic mass which corresponds to a non-rotatingconfiguration of mass M = 1.4M and varying spin frequency ν = Ω/2π (given in kHz).

33


EOS AU (Mbar = 2.64M)ν (kHz) εc/ε∗ I/I∗ Mgrav/M Req (Km) ν/νms

CST 0 3.020 1.982 2.133 9.41 0BFGM 0 3.020 1.964 2.126 9.39

∆ - 0.0% 0.9% 0.3% 0.2%CST 0.602 2.547 2.077 2.141 9.74 0.357

BFGM 0.602 2.566 2.058 2.135 9.71∆ - 0.7% 0.9% 0.3% 0.3%

CST 1.174 2.101 2.281 2.170 10.40 0.697BFGM 1.174 2.159 2.218 2.158 10.19

∆ - 2.8% 2.8 % 0.6 % 2.0 %CST 1.481 1.813 2.525 2.201 11.20 0.879

BFGM 1.481 1.968 2.344 2.178 10.58∆ - 8.5% 7.2% 1.0% 6.0%

CST 1.625 1.637 2.766 2.227 12.10 0.964BFGM 1.625 1.881 2.416 2.187 10.82

∆ - 15.0% 12.7% 1.8% 10.6%CST 1.684 1.532 2.974 2.246 13.66 1.000

BFGM 1.684 1.844 2.448 2.192 10.93∆ - 20.0% 17.7% 2.4% 20.0 %

Table 2-5. The stellar parameters for sequence B, which correspond to maximum mass (seetext) are given as in table 2-2.

34


EOS APR2 (Mbar = 2.69M)ν (kHz) εc/ε∗ I/I∗ Mgrav/M Req (Km) ν/νms

BS 0 2.600 - 2.205 10.12 0BFGM 0 2.748 2.219 2.202 10.03

∆ - 5.7% - 0.1% 0.9%BS 0.609 2.300 2.352 2.217 10.45 0.408

BFGM 0.609 2.292 2.347 2.212 10.44∆ - 0.3% 0.2% 0.2% 0.1%BS 1.081 1.900 2.593 2.243 11.21 0.723

BFGM 1.081 1.958 2.532 2.234 10.99∆ - 3.0% 2.4% 0.4% 2.0%BS 1.188 1.800 2.687 2.253 11.50 0.795

BFGM 1.188 1.884 2.589 2.241 11.16∆ - 4.6% 3.6% 0.5% 3.0%BS 1.284 1.700 2.799 2.263 11.86 0.859

BFGM 1.284 1.819 2.645 2.248 11.33∆ - 12.0% 5.5% 0.7% 4.5%BS 1.494 1.400 3.337 2.307 14.17 1.000

BFGM 1.494 1.679 2.788 2.264 11.79∆ - 20.0% 18.8% 2.0% 17.0%


35


EOS L (Mbar = 3.16M)ν (kHz) εc/ε∗ I/I∗ Mgrav/M Req (Km) ν/νms

CST 0 1.470 4.676 2.700 13.70 0BFGM 0 1.486 4.530 2.662 13.63

∆ - 1.6% 3.0% 1.4% 0.5%CST 0.352 1.201 4.959 2.706 14.20 0.341

BFGM 0.352 1.220 4.786 2.668 14.09∆ - 1.6% 3.5% 1.4% 0.8%

CST 0.714 0.955 5.554 2.733 15.24 0.692BFGM 0.714 0.990 5.211 2.689 14.84

∆ - 3.6% 6.0 % 1.6% 2.6%CST 0.909 0.802 6.292 2.765 16.57 0.880

BFGM 0.909 0.896 5.545 2.707 15.46∆ - 8.4% 11.8% 2.1% 6.7%

CST 0.997 0.710 7.024 2.792 18.05 0.966BFGM 0.997 0.852 5.724 2.715 15.82

∆ - 20.0% 18.5% 2.8% 12.4%CST 1.032 0.655 7.659 2.813 20.66 1.000

BFGM 1.032 0.835 5.800 2.720 15.98∆ - 27.0% 24% 3.3% 23.0%


36


the moment of inertia I in column 4, the gravitational mass Mgrav in column 5, the equatorialradius Req in column 6, the ratio between the spin frequency and the mass-shedding frequency(as computed in [68] and [70]) in column 7. For each value of ν and for each quantity, we givethe relative error ∆.As explained in Section II, we stress that the quantities εc, Mgrav and Req are expanded in powersΩn with n even, therefore, as far as our calculations are concerned, they are computed up to theorder Ω2. Conversely, the angular momentum J tot is expanded in odd powers of Ω, and wecompute it to the order Ω3. The moment of inertia I (see eq. 2.2.25) is the angular momentumdivided by Ω, therefore, in this thesis it is the sum of two contributions: I (0), of the order Ω0

coming from J computed at order Ω, and δI , of order Ω2 coming from δJ computed at order Ω3;in this sense we refer to I as a quantity computed up to the 3rd-order in Ω.It should be noted that a difference of the order of one percent between quantities computedby integrating the exact or the perturbed equations can be attributed essentially to interpolationof the EOSs that are given in a tabulated form. The discrepancies are indeed of that order whenν = 0, only with the exception of εc of the maximum mass star for APR2. The larger discrepancyfound in this case (∆ = 5.7%) can be explained by noting that the APR2 model yields a M(εc)curve that features an extended plateau around M = Mmax. For example, a 7% change of thecentral density, from 2.65×1015g/cm3 to 2.85×1015g/cm3, only leads to a variation of the fifthdigit in the value of the mass.From Table 2-2, 2-3, 2-4 we can see that for the stars of sequence A, a critical quantity in thecomparison bewteen the exact and perturbative approach is the equatorial radius of the star;indeed, while the relative error ∆ = (Rexact − Rperturb.)/Rexact is smaller than 4.5% for ν ∼<0.85 νms, it becomes an order of 19 − 20% at mass-shedding. The reason for this discrepancyis that when the spin frequency approaches the mass-shedding limit the numerical integration ofthe exact equations shows that a cusp forms on the equatorial plane, from which, the star startsloosing matter [66]. This cusp is hardly reproduced by a perturbative expansion, and in any event,it cannot be reproduced by truncating the expansion to order Ω3. The physical quantity whichis more sensitive to the error on the radius at mass-shedding is the moment of inertia (∼ 14%).From Tables 2-2, 2-3, 2-4 it also emerges that the agreement between exact calculations and a3rd-order perturbative approach is better for the EOS APR2, which is stiffer than AU and L.For the maximum mass sequences B, Tables 2-6, 2-5, 2-7 show that the discrepancies are larger;the largest error is in the central density, and the reason is the following. In order to construct aconstant baryonic mass sequence, for a fixed value of ν, the central density is changed until therequired mass is reached; since, approaching the maximum mass

limM→Mmax

∂εc∂Mbar

= ∞, (2.3.1)

when M = Mmax the determination of εc becomes less accurate.In Table 2.3 we show the mass-shedding frequency computed by the perturbative and the exactapproach for the sequences A and B: the values we find are systematically higher than thosefound with the exact codes. The discrepancy has to be attributed to the inaccuracy of theperturbative approach near mass-shedding as explained above. However, it should be stressedthat the rotation rate of known pulsars is much lower than the mass-shedding limit as computed

37


νms in KHz

Sequence AEOS AU EOS L EOS APR2

BFGM 1.533 0.876 1.299CST 1.257 0.716 -BS 1.237 0.728 1.041

∆CST 22.0% 22.3% -∆BS 23.9% 20.3% 24.8%

Sequence BEOS AU EOS L EOS APR2

BFGM 2.057 1.277 1.842CST 1.685 1.032 -BS 1.682 1.022 1.494

∆CST 22.1% 23.7% -∆BS 22.3% 24.9% 23.3%

Table 2-8. The mass-shedding spin frequencies are given for sequences A and B (see text) andfor the considered EOSs.

for most EOSs considered in the literature; for instance, the rotation rate of the fastest isolatedpulsar observed so far is ν ∼ 641 Hz, and unless we are specifically interested in the mass-shedding problem, the results given in Tables 2-2, 2-3, 2-4 and 2-6, 2-5, 2-7 show that theperturbative approach can be used to study the structural properties of observed rapidly rotatingneutron stars. Indeed, for ν ∼ 641Hz, we find that the results of the perturbative and exactcalculations for the EOS AU and APR2 differ by, at most, ∼ 2% for all computed quantities,even for the maximum mass.For the EOS L, differences are larger, mainly because ν = 641 Hz is closer to the mass-sheddingfrequency than for other EOSs, and the perturbative approach becomes less accurate for thereasons explained above. In addition, unlike the EOS AU and APR2, the EOS L is tabulated withvery few points, expecially at high densities (only four points for εc > 1015 g/cm3), and thisintroduces further differences which depend on the interpolation scheme. The results of Hartle’sapproach have been compared to the exact results by several authors [62],[63],[64]. In all thesepapers the perturbative approach is developed up to order Ω2. [62] focuses on the calculationof the mass-shedding frequency for the maximum mass configuration Ωlim, which is computedfor a number of EOSs to see whether the scaling law defined in [77] and based on the results ofthe integration of the exact equations, is satisfied. The authors found mass-shedding frequenciessystematically larger by ∼ 10 − 15% than those obtained from the “exact” scaling law. In[63], Hartle’s procedure was applied to construct models of stars rotating at mass-shedding forthree values of the mass and for EOSs that were different from those we consider. The resultswere compared to those of [71] where the exact equations were integrated for the same EOSs,finding, that Hartle’s approach leads to results that are compatible with exact calculations downto rotational periods of the order of 0.5 ms. However, this result was subsequently disclaimed

38


in [67], where it was shown that the equatorial radius computed by the exact codes of [71] wassistematically underestimated, leading to larger values of Ωms. For this reason the agreementbetween [63] and [71] was good. The conclusion of [67] is that Hartle’s method to the order Ω2

cannot be considered very accurate in configurations that approach the mass-shedding limit, aswe also find.Finally, in [64] the second order Hartle’s approach is applied in order to construct sequencesof stellar models with a constant Mbar and varying Ω for the same five EOSs that were usedto integrate the exact codes of [70]. They used a procedure different from ours, matching theangular momentum and gravitational mass computed by the two approaches by varying ν andεc. Therefore, in their approach, J (computed to order Ω) and Mgrav are equal by construction.The comparison between the perturbative and the exact approaches is focused on the quadrupolemoment Q, which is the coefficient of the r−3P2(cos θ) term in Newtonian potential [59]. To theorder Ω2, Q is

Q =J2

M+

8

5KM3 (2.3.2)

where K is defined in eq. (A.2.15). They found a relative error which depends on the EOS andincreases with the rotation rate. For rates of the order of the fastest pulsar they found errors ofthe order of 20%. Our calculations of Q for EOS APR2, AU and L that are also used in [70]show a much better agreement with the exact results, with a relative error smaller than 2% forMgrav = 1.4 M.

2.4 The EOS’s imprint on an NS physical properties: results

In this section we will illustrate our results, and, in particular, how the EOSs and rotation affectsthe physical properties of rapidly rotating NSs.The non-rotating NS configurations which correspond to the APR2 and BSS1 EOS exhibitsimilar mass-radius relations, with a maximum gravitational mass ∼> 2 M. The appearanceof strange baryons makes the EOS softer, thus leading to sizable changes in the mass-radiusrelation. This feature is clearly visible in the case of both the BSS2 and G240 EOS [23], andleads to maximum masses of ∼ 1.22 M and ∼ 1.55 M, respectively. Comparison with theexperimental determinations of the masses of neutron stars located in binary radio pulsar systemsand yielding a narrow distribution, centered at (1.35 ±0.04) M [12], suggests that the dynamicsunderlying the BSS2 model leads to too soft an EOS. On the other hand, all other EOSs appearto be compatible with the mass of the most massive neutron star listed in Ref. [12] (∼ 1.56 M).Using the EOSs illustred in table 1-1 to describe matter in the crust and in the stellar core,and using the procedure developed in section 2.2, the equations of stellar structure perturbed tothe order Ω3 have been integrated to construct sequences of stellar configurations with constantbaryonic mass and varying angular velocity. In our study we consider six values of Mbar, plusthe maximum mass model for each EOS. The gravitational mass is

Mgrav = M + δMgrav +O(Ω4), (2.4.1)

39

2.4. The EOS’s imprint on an NS physical properties: results

where

M = 4π

∫ R

0

r2 ε(r) dr (2.4.2)

is the gravitational mass of the non-rotating configuration and δMgrav is given by eq. (2.2.14);Mgrav depends on the EOS and on angular velocity. In Table 2.4 for each EOS we give the valuesof M for the non-rotating model that corresponds to the selected Mbar

M/M

Mbar/M APR2 BBS1 BBS2 G2401.31 1.19 1.18 1.19 1.201.50 1.35 1.34 - 1.371.56 1.39 1.38 - 1.411.68 1.49 1.47 - 1.512.10 1.80 1.79 - -2.35 2.00 1.97 - -

Table 2-9. For each value of the baryonic mass in column 1, the corresponding values of thegravitational mass of the non-rotating configuration are tabulated for the selected EOSs.

1

1.2

1.4

1.6

1.8

2

2.2

2.4

5e+14 1e+15 1.5e+15 2e+15 2.5e+15 3e+15

M/M

εc (g/cm3)

EOS APR2

Mbar=1.31

Mbar=1.50Mbar=1.56

Mbar=1.68

Mbar=2.10

Mbar=2.35

ν =0ν =0.57 (kHz)ν =0.76 (kHz)ν =1.19 (kHz)

Figure 2-1. The gravitational mass for the EOS APR2 is plotted as a function of the centraldensity for different values of the spin frequency, highlighting the curves with Mbar = const.

The maximum masses (baryonic and gravitational) of the non-rotating configurations are givenin Table 2.4 for the selected EOSs. An example of how the gravitational mass changes due

40


EOS Mmaxbar /M Mmax/M

APR2 2.70 2.20BBS1 2.41 2.01BBS2 1.74 1.22G240 1.35 1.55

Table 2-10. Maximum masses of the non-rotating configurations.

to rotation is shown in Fig. 2-1, where we plot Mgrav (in solar mass units) as a function ofthe central density for different values of spin frequency for the EOS APR2 and highlight thesequences with Mbar = const.

2.4.1 Mass-shedding limit

In Fig. 2-2 we plot the mass-shedding frequency, νms, as a function of the gravitational massfor each EOS. The dashed horizontal line is the frequency of PSR 1937+21, the fastest isolatedpulsar observed so far, the mass of which is presently unknown. We see that the values of

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1.2 1.4 1.6 1.8 2 2.2

ν ms

(kH

z)

Mgrav/Mo

APR2BBS1BBS2G240

PSR 1937+21

Figure 2-2. The mass-shedding frequency νms is plotted for each EOS as a function of thegravitational mass. The horizontal line corresponds to the spin frequency of PSR 1937+21, whichis the fastest isolated pulsar observed so far and whose mass is presently unknown.

νms for G240 are sistematically lower than those of the other EOSs, while those for APR2 aresistematically higher. This behaviour can be understood from the data given in Table 2.4.1 where,for each EOS, we tabulate the ratio of the gravitational mass of the star which rotates at mass-shedding to the equatorial radius, Mms

grav/Rmseq , for three values of the baryonic mass. For a

given mass, the ordering of νms in Fig. 2-2 corresponds to the same ordering of Mmsgrav/R

mseq in

41


ν = νms ν = 0Mms

grav/Rmseq M/R

Mbar/M 1.31 1.56 2.35 1.31 1.56 2.35G240 0.106 0.127 - 0.131 0.163 -BBS1 0.114 0.136 0.210 0.143 0.169 0.267BBS2 0.117 - - 0.154 - -APR2 0.124 0.147 0.223 0.152 0.179 0.268

Table 2-11. For each EOS and for three values of the baryonic mass, we tabulate the ratio ofthe gravitational mass of the star rotating at mass-shedding to the equatorial radius, Mms

grav/Rmseq .

In the last three columns, we give the ratio M/R for the non-rotating star with the same baryonicmass.

Table 2.4.1, showing that more compact stars admit a higher mass-shedding limit. In the lastthree columns of Table 2.4.1 we also give the ratio M/R of the non-rotating star with the samebaryonic mass, and it is interesting to see that rotation affects the compactness of a star in a waywhich depends on the EOS. For instance, for Mbar = 1.31M (for which all EOSs admit a stableconfiguration), the non-rotating star with EOS BBS2 (M/R = 0.154) is more compact than thatwith EOS APR2 (M/R = 0.152), while, near mass-shedding, the compactness of the BBS2 star(Mms

grav/Rmseq = 0.117) is smaller than that of the APR2 star (Mms

grav/Rmseq = 0.124). To further

clarify this behaviour, in Fig. 2-3 we plot the energy density inside the star as a function ofthe radial distance for these two EOSs in both the non-rotating case and at mass-shedding. It isuseful to remind the reader that the main difference between the EOS APR2 and BBS2 is thatthere are hyperons in the core of the BBS2 star. From the upper panel of Fig. 2-3 we see thatwhen the BBS2 star does not rotate, hyperons are highly concentrated in the core, making thecentral density very large; however, since the EOS is soft, when the star rotates quickly, matteris allowed to distribute itself throughout the star. Conversely, due to the stiffness of the APR2EOS, in the corresponding star (Fig. 2-3 lower panel) the matter distribution does not seem tobe so affected by rotation. A similar analysis can be found in [71] where different EOSs wereconsidered. By comparing our results of order Ω3 to those of order Ω2, we find that the third ordercontributions to the mass-shedding limit are actually negligible, smaller than 1%, independentlyof the EOS.

2.4.2 Estimates of the moment of inertia

In Fig. 2-4 we plot the moment of inertia as a function of the spin frequency, for each EOS andfor given values of the baryonic mass up to the maximum mass. Since we know, as discussedin section 2.4.1, that the mass-shedding frequencies estimated by our perturbative approach areabout 20% larger than those found by integrating the exact equations, we plot the data only forν ≤ 80% νms, where νms is the mass-shedding frequency that we find. From Fig. 2-4 we see thatthe moment of inertia has a non-trivial behaviour as the baryonic mass increases. Indeed, along

42


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 2 4 6 8 10 12 14

ε/ε *

r (km)

EOS BBS2

non rotatingmass shedding

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 2 4 6 8 10 12 14

ε/ε *

r (km)

EOS APR2

non rotatingmass shedding

Figure 2-3. The energy density distribution inside the star is plotted as a function of the radialdistance for the EOS APR2 and BBS2, and for Mbar = 1.31M. ε is given in units of ε∗ =1015 g/cm3. In both cases the continuous line refers to the non-rotating configuration, the dottedline to the star rotating at mass-shedding.

43


1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

I/I*

ν (kHz)

EOS APR2

(a)

PSR 1937+21Mbar=1.31 MoMbar=1.50 MoMbar=1.56 MoMbar=1.68 MoMbar=2.10 MoMbar=2.35 MoMbar=2.60 Mo

Mbar= Mmax

,

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

I/I*

ν (kHz)

EOS BBS1

(b)

PSR 1937+21Mbar=1.31 MoMbar=1.50 MoMbar=1.56 MoMbar=1.68 MoMbar=2.10 MoMbar=2.35 Mo

Mbar= Mmax

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

I/I*

ν (kHz)

EOS BBS2

(c)

PSR 1937+21Mbar=1.00 MoMbar=1.31 Mo

Mbar= Mmax

,

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

I/I*

ν (kHz)

EOS G240

(d)

PSR 1937+21Mbar=1.31 MoMbar=1.50 MoMbar=1.56 MoMbar=1.68 Mo

Mbar= Mmax

Figure 2-4. The moment of inertia (in units of I∗ = 1045g cm2) is plotted as a function of thespin frequency, for the EOS APR2 (panel a), BBS1 (panel b), BBS2 (panel c) and G240 (paneld). For each EOS we consider a few values of the baryonic mass up to the maximum mass, andplot the data only for ν ≤ 80% νms (see text). The vertical dashed line is the spin frequency ofPSR 1937+21.

44


1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

0 0.2 0.4 0.6 0.8 1

I/I*

ν (kHz)

APR2

BBS1

G240

I(0)+δ II(0)

PSR 1937+21

Figure 2-5. The moment of inertia, measured in units of I∗ = 1045g cm2, is plotted versus thespin frequency (in kHz) for stars with baryonic mass Mbar = 1.56M. The continuous linesrefer to I that is computed by solving the stellar structure equations to the third order in theangular velocity, whereas the horizontal dashed lines refer to the first order term I(0). The verticaldashed line is the spin frequency of PSR 1937+21. For each EOS, the data are plotted only fo rν ≤ 80% νms (see text).

an Ω = constant sequence of stellar models, if Mbar Mmaxbar then I is an increasing function

of Mbar, while as Mbar approaches the maximum mass, I becomes a decreasing function ofMbar.This behaviour can be understood as follows: I ∼ MR2, and while, in general, keeping Ωconstant as the mass increases, the radius decreases, approaching the maximum mass the massremains nearly constant while the radius is continuously decreasing. This is true for any EOS,but of course it is more evident for softer EOSs like BBS2 and G240.In a recent paper [78] it has been suggested that a measurement of the moment of inertia frompulsar timing data will impose significant constraints on the nuclear EOS. The authors considerthe newly discovered binary double pulsar PSR J037-3039, and construct models of the fastestpulsar (M = 1.337 M, ν = 276.8 Hz) with different EOSs. They compute the moment ofinertia assuming the star is rigidly rotating using the fully non-linear, numerical code developedin [68], and compare the results with those obtained using Hartle’s perturbative approach thatwere developed at first order in the angular velocity, i.e. they compute the quantity given ineq. (2.2.31) which we call I (0). They show that the agreement is very good, and this was to beexpected since the spin frequency of the considered star is quite low.What we want to show now, is, that, if we consider stars that rotate faster, as for examplePSR 1937+21, the first order approach is inaccurate, and we need to consider the third ordercorrections. To this purpose, in Fig. 2-5 we plot for each EOS and for Mbar = 1.56 M the firstorder term I (0) (dashed line) and I = I (0) + δI (solid line) where δI is the correction found bysolving the perturbed equations to order Ω3. For each EOS, the data are plotted for ν ≤ 80% νms.The vertical dashed line refers to the spin frequency of PSR 1937+21. It is clear from the figure

45

2.5. Concluding remarks

that the third order correction becomes relevant at frequencies higher than ∼ 0.5 kHz. Forinstance, if ν = 641 Hz, the relative errors [δI/(I (0) + δI)] that one makes neglecting the thirdorder corrections are, 5% for APR2, 7% for BBS1 and 13% for G240.It is interesting to compare I (0), I = I (0) + δI , where δI is computed using corrections of theorder Ω3, and the value of I one gets by integrating the exact equations of stellar structure. Thisis possible for the EOS APR2 and the results are shown in Table 2.4.2 for Mgrav ∼ 1.4 M andν = 644 Hz, and for the maximum mass and ν = 609 Hz. The table shows that including thethird order is important to adequately estimate the moment of inertia of the fastest pulsars.

Iexact/I∗ I(0)/I∗ ∆(

I(0) + δI)

/I∗ ∆Mgrav ∼ 1.4 M 1.425 1.238 13% 1.395 2.1%ν = 644 Hz

Iexact/I∗ I(0)/I∗ ∆(

I(0) + δI)

/I∗ ∆Mgrav = Mmax 2.352 2.281 3% 2.347 0.2%ν = 609 Hz

Table 2-12. We compare the moment of inertia computed for the EOS APR2 by integrating theexact equations of stellar structure (column 2) [70], the perturbed equations to order Ω (column3), and to order Ω3 (column 5). The relative error bewteen the exact and the approximated resultsare given in columns 4 and 6. I is in units of I∗ = 1045g cm2.

2.5 Concluding remarks

In this Chapter we have illustrated our obtained results by solving the equations of stellarstructure developed to the third order in the angular velocity, using recent EOS which modelhadronic interactions in different ways to describe the matter in the inner core. The stellarparameters we find have been compared, when available in the literature, with those found bysolving the exact equations of stellar structure. The main conclusions we can draw from ourstudy are the following.It is known that near mass-shedding, the perturbative approach fails to reproduce the stellarproperties correctly and the reasons are well understood. We confirm this behaviour and givequantitative results in this limit. However, for lower values of the angular velocity the situationis different. Taking the rotation rate of the fastest isolated pulsar observed so far as a reference,PSR 1937+21 for which ν ∼ 641 Hz, we see that at these rates the perturbative approach allowsus to describe all stellar properties to an accuracy better than ∼ 2% even for the maximum massmodels, unless the EOS in the core is very soft, as in the case of the EOS L; indeed, for this EOS,the mass-shedding velocity is low and close to the chosen rotation rate, so that the perturbativeapproach is inaccurate.The two quantities that are affected by third order corrections are the mass-shedding velocityand the moment of inertia. For the first, we find that the third order corrections are actuallynegligible, smaller than 1%, independently of the EOS. Conversely, the moment of inertia is

46


affected by terms of order Ω3 in a significant way; for instance, for a star with M = 1.4 M

rotating at ν ∼ 641 Hz the third order correction is of the order of δI/(I (0) + δI) ∼ 5% forthe stiffer EOS we use (APR2), and as high as ∼ 13% for the softest (G240). Thus, third ordercorrections have to be included to study the moment of inertia of rapidly rotating neutron stars,whereas they are irrelevant when estimating the mass-shedding limit.The perturbative approach allow us accurate theoretical calculations of the physical properties ofrapidly rotating NSs, we hope that, in the future, with the help of missions like Constellation-X,more astronomical data on these sources will be available in order to put more tighten constraintson the EOSs.

47


48

3

Gravitational wave asteroseismology

It is commonly believed that the experimental discovery of gravitational waves (GWs) will opena new window onto our universe. GWs should permit the development of a new astronomywhich is potentially able to observe objects that are located very distant from us and phenomenathat are hardly observable through electromagnetic emission. GWs interact so weakly with thesurrounding matter that they go through it without interaction, unlike electromagnetic radiationwhich is easily absorbed and scattered, and even unlike neutrinos which, although they almostcompletely penetrate ordinary matter, should scatter thousands of times while leaving, forinstance, the core of a supernova. This difference makes it likely that, if GWs will be observed,they will generate a revolution in our view of the Universe, comparable to that which resultsfrom observations based on radio waves, X-ray, γ-ray and neutrinos. The best candidates GW’ssources are NSs.In this Chapter we will focus our attention on the non-radial pulsations of these compact objectsbecause they are one of the most promising channels of emission of GWs. Non-radial oscillationsof NSs are manifested in a variety of astrophysical situations, for instance, as a consequence ofa glitch, of a close interaction with an orbital companion, or of a phase transition to quark matterwhich occurs in the inner core or soon after the birth of an NS in a gravitational collapse.When an NS is set into non-radial oscillations, it emits GWs at the characteristic frequencies ofits quasi-normal modes (QNM). The study of these oscillations has helped, and continues to help,astrophysicists obtain relevant information about the internal structure of NSs through the use ofasteroseismological methods. Each observed mode of oscillations carries information, at least inprinciple, regarding both the structure of the star and the status of the nuclear matter in its interiorand furthermore, the amount of this information regarding the internal composition grows withthe number of modes that can be detected. As we will see in this Chapter, the spectrum ofa pulsating relativistic star is incredibly rich, since, essentially, each feature of the star can bedirectly associated with one distinct family of pulsation modes.As the emission of GWs from oscillating NSs is an important part of this thesis, in this Chapterwe briefly summarize the mathemathical theory of stellar perturbations and we illustrate some ofthe results on NS oscillation modes that have been obtained in earlier works.In section 3.4 we will show how the mode frequencies carry interesting information on the innerstructure of the emitting source: in particular we discuss the possibility that the detection of

49

3.1. Neutron star oscillations in GR

gravitational waves emitted by compact stars may allow one to put constraints on the MIT bagmodel of quark matter EOS (that we illustrated in Chapter1, section 1.4) and also discriminatebetween strange stars and NSs.

3.1 Neutron star oscillations in GR

Due to their central role in astrophysics, oscillations of stars have been studied extensively bothin the framework of the Newtonian theory of gravity and in GR. Stellar oscillations for moststars, and firstly for the Sun, can be described by Newtonian theory with a very high degreeof accuracy. However NSs are very compact objects and have to be studied in the frameworkof the GR, because, for these stars, relativistic effects must not be neglected. In particular,in accordance with GR, a compact star vibrating into non-radial modes emits GWs, whereasgravitational waves do not exist in the Newtonian theory. This difference is substantial and it isthe key point of a reformulation of the relativistic theory of stellar perturbations as proposed byThorne and collaborators in the Sixties [1, 2, 3, 4, 5], and further developed by Chandrasekharand Ferrari [79, 6, 80]. In particular, if the star is assumed to be non-rotating, the oscillationfrequencies can easily be computed by solving the equations of stellar perturbations as foundby Thorne and collaborators [1, 2, 3, 4, 5] and subsequently integrated for a large variety ofequations of state used to describe matter in an NS as, for example in [81, 23]. GWs willbe emitted with these frequencies of oscillation and with some characteristic damping timeswhich depend on the structure of the star. In particular, in [81, 23] the authors show that, inthe spectrum the identification of a detected signal of a sharp pulse which corresponds to theexcitation of a mode, would allow us to infer interesting information regarding the compositionof the inner core of an NS. These studies provide the theoretical basis for gravitational waveasteroseismology.

3.1.1 The perturbation equations

This subsection is dedicated to brief introduction of the perturbation equations for a sphericalequilibrium stellar model as they were derived by Lindblom and Detweiler [82, 83]. Thebackground unperturbed model is that described in Chapter 2, section 2.1. The motion of theperturbed stellar configuration is described by a displacement three-vector ξ i which accountsfor the motion of the fluid in respect to the coordinate system. Because of the motion of thefluids, the geometry of the spacetime inside and outside the star is no longer described by theline element (2.1.1) but by

ds2 = ds2(0) + hµνdx

µdxν, (3.1.1)

where we assume that hµν is a small perturbation of the background metric. If we assume a smallvariation in both the fluid variables and the spacetime, we must deal with perturbed Einstein’s

50

Chapter 3. Gravitational wave asteroseismology

equations, coupled with the perturbed equations of motion for the fluid

δ(Gµν − 8πT µν ) = 0, δ(T µν;µ). (3.1.2)

We shall omit the explicit derivation of these equations, which can be found in the originalpapers, here we only remark that the perturbation equations can be decomposed into tensorspherical harmonics in order to take advantage of the spherical symmetry of the backgroundand to eliminate the angular dependence (for further details see e.g. [84] and Appendix B for theexplicit expressions of tensor spherical harmonics). Furthermore, we shall work in the frequencydomain, this means we shall assume that the time dependence of all perturbations is of the formeiωt, where ω is, in general, complex because of the damping due to the emission of GWs.By making an appropriate choice of gauge, as firstly defined by Regge and Wheeler [84], theperturbed line element can be written as

ds2 = −eν(1 + rlH0Ylmeiωt)dt2 − 2iωrl+1H1Ylmdtdr + eλ(1 − rlH0Ylme

iωt)dr2

+ (1 − rlKYlmeiωt)r2(dθ2 + sin2θdφ2) − 2sinθ(h0dt+ h1dr)eiωt∂θYlmdφ

+2

sinθ(h0dt+ h1dr)eiωt∂φYlmdθ, (3.1.3)

where Ylm are the usual spherical harmonics and where the functions (H0, H1, K) are the radialpart of the polar (or even-parity) metric perturbations, while h0 and h1 represent the axial (orodd-parity) metric perturbations. The components of the displacement vector ξ i are expressed interms of the two fluid perturbation variables W and V as

ξr = e−λ/2rl−1WYlmeiωt, (3.1.4)

ξθ = −rl−2V ∂θYlmeiωt,

ξφ = rl(r sin θ)−2V ∂φYlmeiωt.

It is important to note that when one performs expansion in tensor spherical harmonics, theequations that govern the evolution of the polar perturbations are decoupled of those for the axialperturbations. The polar equations couple to the fluid perturbations, while for the axial equations,both the pressure p and the energy density ε of the fluid remain unchanged. This is due to thefact that the perturbations of p and ε are scalar fields under the rotation group, so that they areexpanded in scalar spherical harmonics, which have even-parity (polar) under rotation and do notcouple to any odd-parity (axial) perturbation. We also assume that the perturbations take placeadiabatically, i.e. that the changes in the pressure and energy density arise without dissipation.In the following we briefly describe the polar and the axial perturbation equations as they havebeen formulated in [83, 82].

• Polar perturbation equationsThe five polar perturbation functions (H0, H1, K,W, V ) are not all independent. Einstein’sequations provide relationships between the perturbation variable, and, by eliminating oneof the perturbation funcutions, we can construct a fourth-order system of linear equations

51

3.1. Neutron star oscillations in GR

for the four polar perturbation variables (H1, K,W,X):

H ′1 =

1

r

(

−[(l + 1) + eλ(2m(r)/r + r2(p− ε))]H1 + eλ[H0 +K − 4(ε + p)V ])

,

K ′ =1

r

(

H0 + (n+ 1)H1 − [(l + 1) − r1

2ν ′]K − 2(ε+ p)eλ/2W

)

,

W ′ = −(l + 1)

rW + reλ/2

(

e−ν/2X

γp− l(l + 1)V

r2+

1

2H0 +K

)

,

X ′ = − l

rX + (ε + p)eν/2

1

2

[

1

r− 1

2ν ′)H0 +

1

2r[r2ω2 + (n + 1)H1] +

1

2r(3

2rν ′ − 1)K

−ν ′ l(l + 1)V

r2− 1

r[eλ/2[(ε+ p) + ω2] − 1

2r2(r−2e−λ/2ν

′

)′]W

]

(3.1.5)

where the prime denotes the derivative with respect to r, n = (l − 1)(l + 2)/2 andω2 = ω2e−ν . V and H0 are linear combinations of (H1, K,W,X) that are defined bythe following relations:

H0 =2r2e−ν/2X − [1

2(n + 1)rν ′ − r2ω2]e−λH1 + [n− ω2r2 − 1

2ν ′(3m(r) − r + r3p)]K

3m(r)/r + n+ r2p,

V =1

ω2

[

Xe−ν/2

(ε+ p)− ν ′e−λ/2W

2r− 1

2H0

]

.

The fourth-order system of linear equations (3.1.5), for each given value of l and ω admitsfour linearly independent solutions. The system is singular at r = 0 while the physicallyrelevant solutions must be finite everywhere. By imposing the regularity of the solutionsat the center of the star, only two linearly independent solutions are selected.The regularity conditions are imposed by expanding each perturbation variable as a powerseries around r = 0. In addition, the Lagrangian1 pressure perturbation has to vanish at thesurface of the perturbed star. This requirement then selects only one physically acceptablesolution of the system (3.1.5).Outside the star, the perturbation variables associated with the fluid motion are zero andthe perturbation equations reduce to a second-order system for the metric variable H1 andK. The equations can be combined to a single one-dimensional wave equation for theZerilli function Zpol (derived in [85])

(

d2

dr2∗

+ ω2

)

Zpol = VpolZpol, (3.1.6)

Vpol ≡2(r − 2M)

r4(nr + 3M)2[n2(n+ 1)r3 + 3Mn2r2 + 9M2(M + nr)],

wherer∗ ≡ r + 2Mlog(r/2M − 1). (3.1.7)

1Lagrangian pressure perturbation is that which is measured by an observer who follows the motion of the fluid.

52


• Axial perturbation equationsThe equations for the axial perturbations [7] are simpler than the polar ones and can becombined into a single weave equation introducing the function Zax that is related to theaxial perturbation variable h0 and h1 by the following relations:

h0 = − i

ω

d

dr∗(rZax)

h1 = −e−ν(rZax). (3.1.8)

In the interior of the star the function Zax satisfies the equation:(

d2

dr2∗

+ ω2

)

Zax = VaxZax, (3.1.9)

Vax ≡eν

r3[l(l + 1)r + r3(ε− p) − 6m(r)],

which, outside the star, reduces to the Regge-Wheeler equation [84](

d2

dr2∗

+ ω2

)

Zax = VaxZax (3.1.10)

Vax ≡1 − 2M/r

r3[l(l + 1)r − 6M ].

3.2 Quasi-normal mode computation

In GR, the frequencies of oscillation of a star are complex. The presence of an imaginary partderives from the fact that the mechanical energy of vibration is exponentially damped by theemission of gravitational waves. Consequently, the corresponding modes are not called normalbut quasi-normal: the real part represents the frequency of oscillation and the imaginary part theinverse of the damping time.Usually, in a normal-mode classical analysis, one has an ordinary differential equation (or systemof equations) and imposes boundary conditions in order to cause that the wavefunction to vanishoutside a given region of space.Perturbations of stars (that are our wavefunctions) are different because they propagate throughall space and we cannot require that they should be zero outside a finite region. However, wewant to make sure that gravitational radiation that is unrelated to the initial perturbation, does notdisturb the system. Therefore, we have to impose the so-called purely outgoing wave boundarycondition to be sure that nothing is coming in from spatial infinity. We will also impose matchingconditions at the surface of the star in order to have continuous solutions.Thus a QNM is a solution of the system of the perturbation equations that satisfy the conditionsof

1. regularity at the center of the star,

53

3.3. Neutron star quasi-normal modes and gravitational waves

2. vanishing of the Lagrangian pressure perturbation at the surface of the star,

3. continuity of the solution at the surface of the star,

4. purely outgoing wave at spatial infinity.

Several numerical techniques have been developed in order to find the QNM solutions of theperturbed equations in the frequency domain, for a review on the different method see [79, 86,87]. In this thesis we explicitly compute the QNM eigenfrequencies for slowly damped modes (inparticular we concentrate our attention on the frequency of the fundamental mode), i.e. those forwhich the imaginary part of the frequency is much smaller than the real part. In this case we canuse the standing wave method for the spherical stars that have been developed in [79, 86]. Withthis approach we are able to handle conditions at infinity: the perturbed equations are integratedfor real values of the frequency from r = 0 to radial infinity where the amplitude of the Zerillifunction is computed. The frequency of a QNM can be shown to correspond to a local minimumof the Zerilli function and the damping time is given in terms of the width of the parabola whichfits the wave amplitude as a function of the frequency near the minimum. We will describe thestanding wave method, in details in Chapter 4, where we will discuss its generalization to theQNM of rotating stars. At this stage it is important to keep in mind that this method works onlyto determine the QNM frequencies of slowly damped modes but it cannot be applied to highlydamped modes (in which the imaginary part of the frequency is comparable to, or greater than,the real part) or black hole QNM.

3.3 Neutron star quasi-normal modes and gravitational waves

We know that NS non-radial oscillations in the framework of GR are associated to the emission ofGWs. The spectrum of a pulsating neutron star is enormously rich, since, essentially, each featureof the star can be directly associated with one dinstinct family of pulsation modes. The pulsationsof a non-rotating compact star, such as an NS, can be separated into two main categories: thefamily of fluid modes and the family of spacetime modes.Fluid modes of NS are associated only to even-parity (polar) perturbations and they are classifieddepending on the restoring force which prevails in bringing the perturbed element of a fluid backto the equilibrium position:

• The fundamental mode, or f-mode, as in Newtonian gravity, is related to globaloscillations of the fluid and it is nearly independent of the details of the stellar structure.For old, cold NSs, its frequency is proportional to the average density of the star andits eigenfunction has no radial modes inside the star. For a typical old NS, the f-modefrequency is about (1.5 - 3)kHz with a damping time, due to the emission of gravitationalwaves, of the order of a tenth of second [82, 88, 23].

• The pressure modes, or p-modes, are associated with acoustic waves that propagate at thevelocity of sound inside the star. The restoring force is a gradient of pressure. The radial

54


component of the fluid displacement is usually much larger than the tangential one, it isa radial nearly mode. The frequency increases with the radial order of the mode startingfrom (5 - 7)kHz for the p1-mode and the damping time is of the order of seconds[88, 23].

• The gravity modes, or g-modes, as for Newtonian stars, have a non-zero frequencyspectrum only if entropy or composition gradients are present in the interior of the star.The restoring force is mainly provided by buoyancy. Their frequency range covers a widerange, from zero frequency to 1.5 kHz.

In the spectrum of an oscillating NS, the frequencies of the g-modes are lower than those ofp-modes and the two sets are separeted by the frequency of the fundamental mode.The general relativistic spacetime has its own dynamics and it is not only the medium in whichGWs can propagate. It has its unique oscillation spectrum the, so-called, w-modes spectrum. Thefamily of spacetime modes (w-modes) it is separated into three categories that are similar bothfor (even-parity) polar and odd-parity (axial) stellar oscillations:

• The trapped modes were firstly computed by Chandrasekhar and Ferrari [7]. These modesexist only for very compact, probably unrealistic, stars with R≤3M. In fact a typical NSwith M=1.4M and R=12 km has a compactness R/M∼ 5.8. Their frequencies can be ofthe order of a few hundred Hz to a few kHz and their damping times are slower than thatof the other w-modes [7, 89].

• The curvature modes were identified by Kokkotas and Schutz [90] after the discovery ofthe trapped-modes by Chandrasekhar and Ferrari [7]. These modes, that it is customary toconsider the standard w-modes, are associated with the spacetime curvature and exist forall relativistic stars. Their main characteristics are very short damping times, of the orderof tenths of milliseconds and they don’t have any relevant fluid motion. The negligiblefluid motion is a common feature of all classes of w-modes, consequently, they exist evenif one suppress into equations 3.1.5 all the fluid perturbations2.The frequency of the first w-modes is about (6 -13 )kHz and it increases with the order ofthe mode[65].

• The interface modes, or wII modes, were discovered by Leins, Nollert and Soffel [87].They are extremely rapidly damped modes. Their frequency can vary from 2 to 15 kHzand the damping time is less then a tenth of millisecond.

In addition to these modes, other types of oscillations are be possible, as many as are theimprovements in the description of the stellar interior: for example if one considers an NS modelwith a fluid core, a solid crust and a fluid surface ocean, modes due to a non-zero shear in thecrust (s-modes) or to the solid-fluid interfaces (i-modes) do appear [91].In Table 3-1 we give the frequencies amd the damping times of the most relevant l=2 (where

2This approximation is known as the inverse Cowling Approximation ICA.

55

3.3. Neutron star quasi-normal modes and gravitational waves

1.4M

f-mode p1-mode w1-mode wII1 -mode

EOS νf (Hz) τf (ms) νp1 (Hz) τp1 (s) νw1 (Hz) τw1 (s) νIIw1(Hz) τ IIw1

(s)APR1 1818 216 5922 5.8 11092 22.4 5366 18.8APR2 1983 184 6164 4.4 11360 22.5 5906 20.2

APRB200 1997 181 6410 5.1 11407 22.6 5962 20.2APRB120 1998 181 6413 5.0 11405 22.6 5963 20.2

BBS1 1832 213 5861 4.2 11066 22.5 5394 20.6G240 1763 231 5055 1.9 10737 22.8 5100 20.7SS1 2736 109 9428 4.2 11919 26.4 8594 18.8

Table 3-1. The frequencies and damping times of the polar QNM for l=2 are tabulated for1.4M and for different EOSs. Data are from [23].

l is the harmonic index) modes of oscillations for NSs modelled with different EOSs (for adescription of the EOSs see Chapter 1, section 1.2). The values in the Table 3-1 are calculatedin [23] and are sufficiently representative of the quasi-normal mode properties of an NS. It isimportant to point out that from the point of view of the GW emission, the most important modesare the fluid, l=2, fundamental mode and the first pressure gravity mode. Higher order modes, aswell as g-,p-,s-,and i-modes do not involve large mass motion, hence they are not expected to bestrong sources of gravitational radiation[88, 91].It should be stressed that the most relevant information about the astrophysical properties ofcompact objects such an NS that GWs observation can carry is concentrated into the highfrequency band, above about 1kHz. This means that the frequency of the fundamental mode

Figure 3-1. The Virgo sensitivity curve.

lies in the region of the spectrum where GW interferometric detectors, like VIRGO and LIGO,

56


begin to be strongly limited in sensitivity by the presence of shot noise as showed in fig. 3-1. Theconstruction of GW detectors with good sensitivity in the kHz region of the spectrum is crucialin order to observe the oscillations of NSs and to enter the era of GW astronomy of compactobjects (see also section 3.4.1).

3.4 Oscillations of quark stars

In this section we discuss the possibility that detection of a gravitational signal emitted by acompact star, oscillating in its fundamental mode with frequency νf and damping time τf , willallow us to infer whether the source is a neutron star or a strange star and to constrain theoreticalmodels of the quark matter EOS described in section 1.4. The problem we shall investigate isthe following. Suppose that a gravitational signal is detected, which is emitted by a compactobject oscillating in its fundamental mode (f -mode), i.e. the mode which is known to be themost efficient as far as gravitational radiation is concerned [92]. We do not know whether thesource is a neutron star or a strange star. Since the damping time of the f -mode , τf , is known tobe of the order of a fraction of second (see e.g. Table 3-1), the mode excitation would correspondto a sharp peak in the energy spectrum of the detected signal, emitted at the mode frequency νfwhich could then be identifiable by a suitable data analysis technique. The questions we want toaddress are:

• Does the knowledge of νf (and/or of τf ) allow one to say anything about the nature of thesource?

• Assuming that we can establish that the star is a strange star, would these data allow oneto set constraints on the parameters of the MIT bag model?

To answer these questions, we compute frequency and damping time of the fundamental mode ofstrange stars, letting the parameters of the bag model vary in the range indicated by Eqs. (1.4.3),(1.4.4) and (1.4.5), which covers the parameter space that is allowed for bare strange stars. Weconsider masses in the range [0.7 M,Mmax], where Mmax is the maximum mass allowed by eachchoice of the model parameters. We consider bare stars to be without a crust, as the presenceof a crust does not affect the fundamental mode frequency in a significant way. We comparethese frequencies with those computed in [23] for neutron stars and for hybrid stars. In [23] NSswere modeled using the same set of modern EOSs that we have illustred in Chapter 1, section 1.2.These EOSs describe matter at supranuclear densites and they are obtained within non relativisticnuclear many-body theory and relativistic mean field theory, they model hadronic interactions indifferent ways, and lead to different compositions and dynamics. The hybrid stars were modeledusing the EOS APR120 and APR200 of Ref. [54], describing hybrid stars with a rather smalladmixture of quark matter. The results of this comparison are shown in Figs. 1 and 2, where weplot νf and the corresponding damping time τf , respectively, versus the gravitational mass of thestar. In both figures the shaded region refers to all values of νf and τf that are allowed for strange

57

3.4. Oscillations of quark stars

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

ν f (

kH

z )

M/M

Neutron stars

+

Strange stars

APR2APRB200APRB120

BBS1G240

Figure 3-2. The frequency of the fundamental mode is plotted as a function of the mass ofthe star, for neutron/hybrid stars described by the EOS employed in [23] and indicated with thesame labels (APR2, APRB200, APRB120, BBS1, G240) and for strange stars. The shaded regioncovers the range of parameters of the MIT bag model as considered in this thesis, i.e. αs ∈[0.4, 0.6], ms ∈ [80, 155] MeV and B ∈ [57, 95] MeV/fm3.

stars, assuming that the paramaters of the bag model vary in the selected range. Continuous linesrefer to the values of νf and τf that are computed for neutron and hybrid stars in [23]. Figure 1shows that:

• Strange stars cannot emit gravitational waves with νf . 1.7 kHz for any values of the massin the range we consider.

• For masses lower than 1.8 M, above which no stable bare strange star can exist, there isa small range of frequency where neutron/hybrid stars are indistinguishable from strangestars. However, there is a large frequency region where only strange stars can emit. Forinstance if M = 1.2 M, a signal with νf & 1.9 kHz would belong to a strange star. Notethat the fundamental mode frequency and damping time of the hybrid stars consideredin [54] and [23] and shown in Figs. 1 and 2 (EOS APR120 and APR200), are basicallyindistinguishable from that of the neutron star with the same low density EOS, except whenthe mass is close to the maximum mass. This is due to the small amount of quark matterthese hybrid stars contain.

• Even if we do not know the mass of the star (as is often the case for isolated pulsars) theknowledge of νf allows us to gain information about the nature of the source; indeed, ifνf & 2.2 kHz we can reasonably exclude that the signal is emitted by a neutron star.

Figure 2 contains complementary information: for strange stars, τf is, in general, smaller thanfor neutron/hybrid stars.Thus, the next question is: assuming that we know the signal has been emitted by a strange star,

58


0

0.2

0.4

0.6

0.8

1

1.2

0.8 1 1.2 1.4 1.6 1.8 2 2.2

τ f (

s)

M/M

Neutron stars

+

Strange stars

APR2APRB200APRB120

BBS1G240

Figure 3-3. The damping time of the fundamental mode is plotted as a function of the mass ofthe star, as in Fig. 1, for neutron/hybrid stars and for strange stars (shaded region).

can we constrain in some way the parameters of the MIT bag model ? In Fig. 3 we show to whatextent this is possible. We plot the values of νf allowed for strange stars versus the stellar mass,indicating the points that belong to the same value of the bag constant B with the same symbol.From this picture we see that for a given mass the mode frequency increases with B, and that,knowing M and νf we would be able to set constraints on B that are much more stringent thanthose provided by the available experimental data. Similar information can be derived by thesimultaneous knowledge of νf and the radiation radius

R∞ =R

√

1 − 2M/R. (3.4.1)

In Fig. 4 we plot νf as in Fig. 3, but versus R∞. In addition we plot the values of νf forthe neutron/hybrid star models as considered in [23] as continuous lines. The figure shows thatradiation radii smaller than ' 13 km should be attributed to strange stars, whereas if 13 . R∞ .

15.5 Km the star can either be a strange or a neutron/hybrid star. Higher values of R∞ can onlybelong to neutron/hybrid stars. Figure 4 further shows that the knowledge of νf constrains thevalue of B. The problem whether quark stars can be discriminated from neutron stars usinggravitational waves, which we discuss in this Chapter, has already been addressed in [93], [94]and [95]. In [93] νf and τf have been computed for strange star models that are obtained withinthe MIT bag model for two values of B, i.e. B = 56 and 67 MeV/fm3, ms = 150 MeV, andassuming αs both vanishing and different from zero. In [94] strange stars have been modeledputting αs and ms to zero and choosing B = 75 and 137 MeV/fm3. Finally, in [95] the values ofνf and τf have been plotted versus the radiation radius for the following values of the parameters:αs = 0.6,ms = 0, 150, 300 MeV, andB = 57 and 209 MeV/fm3. These values of the parametersand the star central density were chosen to fit R∞ within the range 3.8 − 8.2 Km. However,this choice implies very small values of the stellar mass (ranging from ∼ 0.05 to ∼ 0.5 M)which are hard to explain within current evolutionary scenarios for neutron star formation. As a

59

3.4. Oscillations of quark stars

1.6

1.8

2

2.2

2.4

2.6

2.8

3

0.8 1 1.2 1.4 1.6 1.8 2

ν f (

kH

z )

M/M

B=57B=80B=95

Figure 3-4. The fundamental mode frequency νf is plotted versus the gravitational mass, M ,for different values of the bag constant and αs and ms varying in the range indicated by Eqs.(1.4.3) and (1.4.4).

consequence, they obtain values of νf that are larger than ∼9 kHz, much higher than those weconsider.

3.4.1 The fundamental mode and its detectability

It is important to point out that the fundamental mode can be excited in a variety of astrophysicalprocesses, like in a glitch, in close interaction with a companion or in the aftermath of agravitational collapse. Recent simulations of gravitational collapse show that a significantfraction of the total energy emitted in gravitational waves, of the order of 10−9 − 10−8 Mc

2, isindeed emitted at the frequency of the f-mode [96, 97]. However, this energy is too low to bedetectable by current interferometric antennas like VIRGO or LIGO, unless the collapse occursin our galaxy, but we know that, unfortunately, the rate of collapse per galaxy is only of a fewper hundred years. In order to detect signals emitted by more distant sources, we would needvery sensitive, high frequency detectors, like EURO or EURO-XYLO which were considered ina preliminary assessment study some years ago (see e.g. http://www.astro.cf.ac.uk/geo/euro/).As discussed in [98], these kinds of detectors would be able to see signals emitted by oscillatingstars up to the distance of the Virgo cluster if the energy stored in the mode is of the orderof 10−7 − 10−8 Mc

2, which is not too far from present estimates. Indeed current numericalsimulations assume axisymmetric collapse, but if asymmetries are present, the emitted energymay be larger. Moreover rotation, which is certainly presents in stars, has the effect of loweringthe mode frequencies and enhancing detection chances.

60


1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

8 10 12 14 16 18 20

ν f (

kH

z )

R∞ (km)

B=57B=80B=95APR2

APRB200APRB120

BBS1G240

Figure 3-5. νf is plotted versus the radiation radius, R∞, for different values of the bag constantand αs and ms varying in the range indicated by Eqs. (1.4.3) and (1.4.4). The continuous linesrefer to the values of νf for the neutron/hybrid star models considered in [23].


The study that we have presented in this Chapter differs from the preceeding literature in so muchas it explores the entire range of allowed parameters in a systematic way. Our results show thatthe detection of a signal emitted by a compact star pulsating in its fundamental mode, combinedwith complementary information on the stellar mass or the radiation radius, would allow one todiscriminate between neutron/hybrid stars and strange stars; in addition, we show that it wouldalso be possible to constrain the bag constant to a range much smaller than that provided by theavailable data from terrestrial experiments.It has also to be mentioned that, although somewhat more refined dynamical models have beenproposed, the MIT bag model appears to provide quite a reasonable description of quark matter.The results discussed in Ref. [56] show that the EOS obtained from model (like NJL), in whichquark masses are dynamically generated through the appearance of a condensate associated withchiral symmetry breaking, is in fact similar to that one obtained from the bag model. Usingmore sophisticated models, such as the NJL model, may turn out to be required to describethe possible occurence of the pairing instability induced by the attractive one-gluon exchangeinteraction between quarks, leading to colour superconducting phases[55, 56] discussed in theliterature is not yet firmly established, and their occurrence is expected to affect mostly transportproperties and cooling, rather then the stellar structure, and consequently the f-mode frequencyon which we focus in this Chapter. As a final remark, taking the detectability of the fundamentalmode into account, we can conclude that asteroseismology will become a branch of GW researchwhen high frequency detectors operate. [99].

61


62

4

Oscillations of rotating neutron stars

Neutron stars are tremendously complicated objects: their modelling requires a detailedunderstanding of the very extremes of physics, including supranuclear physics, general relativity,superfluidity, strong magnetic fields, exotic particle physics, etc. [53].To investigate the pulsation properties of any realistic NS is therefore a serious challenge mainlybecause it is rapidly rotating. If we want to make a breakthrough in understanding the spectrumof an oscillating NS we have to take the rotation into account. As a matter of fact, thatdue to rotation, some modes may grow unstable through the Chandrasekhar-Friedman-Schutzmechanism (CFS instability) [8, 9] and these instabilities may have important effects on thesubsequent evolution of the star, for example they may be associated to a further emission ofGWs, the amount of which would depend on when and whether the growing modes are saturatedby non-linear couplings or dissipative processes. In order to detect the gravitational signalsemitted by compact stars it is therefore important to know their pulsation frequencies and tostudy under which conditions the corresponding modes become unstable. We saw in the previousChapter that if the NS is assumed to be non-rotating, the frequencies of the different families ofmodes can easily be computed by solving the equations of stellar perturbations [1, 2, 3, 4, 5].However, all neutron stars are rotating, and unfortunately our present knowledge of the QNMspectrum of rotating stars is far to be complete. The perturbative approach which works so finein the non-rotating case when generalized to include rotation shows a high degree of complexity,even if the star is only slowly rotating [100, 101, 102, 103, 104, 105, 106, 107, 108, 109].A major difficulty arises because, when using the standard spherical harmonics decompositionof the perturbed tensors, modes with different harmonic indexes couple, giving rise to an infiniteset of dynamical, coupled equations. For this reason, in most studies which are based on thisapproach further simplifying assumptions are introduced1: for example in [100, 101, 102, 103]the couplings between oscillations with different values of the harmonic index l are neglected,or, in [104, 105, 106, 107, 108] the Cowling approximation2 is used. An alternative approachconsists in solving the equations describing a rotating and oscillating star in full general relativity,

1To our knowledge, the only study of the oscillations of a slowly rotating star which does not make use of noneof these restrictive assumptions is in [109] where, however, only the r-modes have been investigated.

2In this approximation spacetime perturbations are neglected.

63

4.1. Neutron stars as sources of GWs

in time domain. However, current studies based on this approach also make use of strongsimplifying assumptions, or restrict to particular cases. For instance the Cowling approximationhas been used in several papers [110, 111, 112, 113, 114]; in [11] only quasi-radial modes (l = 0)have been considered; in [115, 116] only the neutral mode 3 has been studied; in [117] onlyaxisymmetric (m = 0) modes have been analysed, using the conformal flatness condition. In[10], the frequencies of axisymmetric modes (m = 0) with l = 0 to l = 3 have been computedfor rapidly rotating relativistic polytropes with polytropic index n = 1, using the Cowlingapproximation; a comparison of the results for l = 0 obtained in th Cowling Approximationin [10], with those found in full GR in [11], shows that Cowling’s approximation introduceslarge errors in the determination of the fundamental mode frequency.In this Chapter we illustrate a general method that we have developed to find the QNMfrequencies of a rotating neutron star.We perturb Einstein’s equations about a stationary axisymmetric background describing arotating NS. The perturbed quantities are expanded in circular harmonics eimφ. As we arelooking for quasi-normal modes, we assume a time dependence e−iωt, with ω complex. Due tothe background symmetry, perturbations with different values of ω andm are decoupled; thus, forassigned values of (m, ω), the perturbed equations to solve are a 2D-system of linear differentialequations in r and θ. In this Chapter we do not derive explicitly the perturbed equations in thegeneral case of fastly rotating neutron stars, we focus our attention in describing the generalmethod. Two are the ingredients on which our method is based:

1. the perturbed equations are integrated using spectral methods;

2. the boundary conditions at the center of the star and at radial infinity are defined suitablygeneralizing the boundary conditions that are used to find the QNM frequencies of non-rotating star.

These two points will be described in section 4.2. In the section 4.1, having in mind thepossibility that different events in the life of an NS may excite the various modes to a relevantlevel, we will concentrate our attention in describing the instabilities in rotating NS that arethought to be particularly promising gravitational-wave sources [118].

4.1 Neutron stars as sources of GWs

Gravitational waves can originate from different energetic astrophysical event such as thecoalescence and merger of massive compact object, gravitational collapse to neutron star orblack holes, rotating and oscillating neutron stars, etc. Considering that a large part of thisthesis is devoted to the study of oscillations of an NS in this section we want briefly illustrate thevarious types of instabilities that can operate in an NS. These instabilities may come in different

3In this way is called the zero-frequency mode in the rotating frame.

64

Chapter 4. Oscillations of rotating neutron stars

flavours but they have one general feature in common: they can be directly associated withunstable modes of oscillation. An unstable mode of oscillation is a mode with an exponentiallygrowing amplitude and it produces GWs whose amplitude is also exponentially growing, untilthe mode saturates due to the viscous damping or to non-linear effects. For instance such aninstability operate when a newly born NS settles down after supernova collapse and it may leadto the emission of copius amounts of GWs [119, 120, 121].The theoretical basis needed to study stellar stability in GR are due to an exstensive work ofChandrasekhar, Friedman and Schutz during the seventies [8, 9, 122, 123]. The main result ofthese studies is that the instability of a given oscillation mode sets in at the point along a sequenceof equilibrium configuration where its frequency vanishes: this point is called the neutral point.This neutral point for oscillation modes can be reached if the star is rotating.As we saw in Chapter 3, section 3.1 because of the symmetry of the non-rotating problem themodes of oscillations corresponding to different spherical harmonics indexes m are degenerate,it means that for a given harmonic index l, the modes having different value of m (with−l ≤ m ≤ l) have the same frequency. However the case of rotating stars is much morecomplicated because rotation breaks the symmetry, in such a way that, these modes becomedinstinct and thier frequency of oscillation changes, as a first approximation, according to

ωR(Ω) = ωNR(Ω = 0) −mΩ + Clm(Ω) +O(Ω2), (4.1.1)

where ωR is the mode frequency according to an inertial observer, ωNR is the frequency of themode in the non-rotating case (or in the rotating frame) and Clm is a function that depends onthe mode eigenfunction in the rotating frame and on the rotation frequency. Due to the mΩ term,prograde mode (or forwards moving) and retrograde mode (or backwards moving) are affecteddifferently by the rotation (see equation 4.1.1), the sign of their frequencies possibly changing.As we will see now this phenomenon is an indicator of instabilities. The problem is furthercomplicated by the rotationally induced change in shape of the star, which first contributes atO(Ω2) in the slow rotation expansion [118, 124]. If the star rotates at a sufficiently high frequencyΩ the mode can be reach the neutral point and hence it becomes unstable.In a simple description the onset of the instability at the neutral point works as follows:the emission of gravitational radiation associated to the star oscillations acts as a dampingmechanism and reduces the amplitude of the oscillations. Furthermore, it removes positiveangular momentum from the prograde modes and negative angular momentum from theretrograde. This is due to the fact that the two branches of modes are affected by rotation indifferent ways: a backwards moving mode is dragged forwards by stellar rotation and, if the starrotate sufficiently fast, the mode will reach its neutral point and will begin to move forwards withrespect to the inertial frame. GWs from such a mode carry away positive angular momentumfrom the star but the angular momentum of the mode is negative. Thus the GW emissionmakes the angular momentum of the mode increasingly negative and leads to an instability.This instability is called CFS instability from the acronym of the names of the authors that firststudied and explained this mechanism, Chandrasekhar, Friedman and Shutz.To reach the onset of the CFS instability the rotation rate must be high enough in order thatthe mode can reach the neutral point. However, as we saw in Chapter 2, NSs cannot rotate at

65

4.2. Developement of our method

arbitrarily high rates but the rotation frequency is limited by the mass-shedding limit.4 A goodcandidate mode to become unstable due to its short timescale for the growing of the instability isthe fundamental l = m = 2 mode. The f-mode instability is a very good source of gravitationalwaves, however several authors have been estimated that a tipycal cold NS needs to rotate veryfast, more than 90% of its mass-shedding limit, to reach the onset of the CFS instability (fora review on the subject see e.g. [120, 118, 121, 124]). This value is rather high and maybenot realistic rotation frequency for NSs. We have to point out that NSs are likely to be borndifferentially rotating. Take differential rotation (like rotation) into account is a very difficultproblem but some progress have been made, Inamura et al.[125] find that differential rotation canhave a significant effect lowering the critical point at which f-mode (l=m=2) becomes unstable.In addition rotation introduces a new classes of modes (in newtonian theory as well as in GR)the so called r-modes whose dynamics is governed by the Coriolis force and whose frequencyis proportional to the rotation frequency of the star. Thus the r-modes are always retrograde inthe rotating frame and prograde in the inertial frame and the CFS instability criterion is satisfiedat all rotation rate [126]. These modes are generically unstable in all rotating stars. The onlylimitation to the onset of the r-modes instability is provided from the competition between thegrowth time of the modes and the dissipative time-scales like viscosity, thermal conductivity, etc.that could prevent the growth of the mode. At the moment it seems plausible that an instabilitywindow exists in a given range that depends of all involved parameters like differential rotation,star’s temperature, magnetic fields and NS equation of state.At the end of this section it is clear that there are difficult issues involved in modelling NSs assources of GWs. If we want to have an accurate understanding of the spectrum of the modes ina NS we can start from taking the rotation into account and we need to improve our theoreticalmodels considerably. The next section is a first step in this direction.

4.2 Developement of our method

In this section we outline the ingredients on which our method is based. We perturb Einstein’sequations, coupled with the hydrodynamical equations, about a stationary, axially symmetricbackground. The background metric can be put in the general form [127]

(ds2)(0) = g(0)µν dx

µdxν = e−νdt2 + eψ(dφ− ωdt)2 + eµ2dx22 + eµ3dx2

3 (4.2.1)

where ν, ψ, µ2, µ3 are functions of the coordinates x2, x3. In the form (4.2.1) there is still somegauge freedom, which allows to write the metric in a simpler form, like

(ds2)(0) = e−ν(r,θ)dt2 + eψ(r,θ)(dφ− w(r, θ)dt)2 + eµ(r,θ)(dr2 + r2dθ2) (4.2.2)

as in [115], or

(ds2)(0) = e−ν(r,θ)dt2 + eλ(r,θ)dr2 + eµ(r,θ)r2(

dθ2 + sin2 θ(dφ− w(r, θ)dt)2)

(4.2.3)

4We have to remind you that when this limit is reached the fluid element on the surface will not be boundanymore and the star will start loosing matter from the equator.

66


as in [128], [58]. We assume a gauge choice for the background metric (like for instance (4.2.2)or (4.2.3)), such that the spacetime is described by the coordinates (t, r, θ, φ), and ∂

∂t, ∂∂φ

areKilling vectors. We also assume that θ, φ are polar coordinates on spheres, i.e. that the surfacest = const., r = const. are diffeomorphic (but not isomorphic) to 2-spheres, as in (4.2.2), (4.2.3).We remark that as a consequence of this, any tensor field defined in one of these surfaces canbe formally expanded in tensor spherical harmonics, even if the spacetime is not sphericallysymmetric; this will be useful later, in Section 4.2.1.2.The metric and fluid velocity perturbations can be considered as tensor field in this background.We expand them in in circular harmonics eimφ, and we perform a Fourier transform in time.Due to the stationarity and axisymmetry of the background spacetime, the linearized Einstein’sequations do not couple perturbations with different values of the index m and of the frequencyω, therefore it is consistent to study perturbations with fixed values of m and ω, i.e.

hµν(t, r, θ, φ) = hmωµν (r, θ)eimφe−iωt . (4.2.4)

The frequency ω can in principle be complex.By fixing the gauge for the perturbations, the ten components hmω

µν (r, θ) reduce to six quantitiesHmω

i (r, θ)i=1,...,6. An example of a possible gauge choice, which we called “generalizedRegge-Wheeler gauge”, is described in Appendix C.1, but other gauges can be considered aswell (see for instance [115]). For simplicity of notation, we assume hereafter that the quantitiesHmωi are scalar with respect to rotation (as it happens, for instance, in the generalized Regge-

Wheeler gauge), but our discussion remains valid for a general gauge choice.The linearized Einstein’s equations give a system of partial differential equations (PDE) in thequantities Hmω

i (and in the fluid velocity perturbations); the equations involve derivatives withrespect to r and θ.For assigned values of m, in order to find the QNM frequencies we have to solve, for differentvalues of ω, the perturbation equations in the quantities Hmω

i (r, θ), by imposing that

• all metric functions are regular near the center of the star,

• the Lagrangian perturbation of the pressure vanishes on the stellar surface,

• the solution at infinity behaves as a pure outgoing wave.

The conditions at the center and at the surface of the star can be fulfilled for every value of ω,but the outgoing wave condition at infinity is only consistent with a discrete set of (complex)frequencies ωi; such frequencies are the QNM. We will now discuss, in section 4.2.1, howto implement the boundary condition at infinity, and in section 4.2.2 the numerical approach tosolve the PDE of the perturbations.

67


4.2.1 The boundary condition at radial infinity

In what follows we shall generalize the standing wave method, commonly used to integrate theequations that describe the perturbations of a non rotating, spherical star to the rotating case.Therefore it is useful to remind how this method works.

4.2.1.1 Standing wave method for a spherical stars

This approach to handle the boundary conditions at infinity in the study of relativistic stellarperturbations has firstly been suggested by K. S. Thorne in [129], and further developed byS. Chandrashekhar and V. Ferrari in [86, 130]. It is important to stress that it works only todetermine the QNM frequencies of slowly damped modes; therefore it can be used to find thefrequencies of fluid modes, like the f -, p-, and r-modes, but it cannot be applied to highly dampedmodes, like the w-modes or black hole QNM.As we have seen in Chapter 3 (section 3.1.1) the equations describing the perturbations of a non-rotating, spherical star can be separated by expanding the perturbed metric tensor in tensorialspherical harmonics. Outside the star these equations reduce to those describing Schwarzschildperturbations, and they can be reduced to two radial wave equations, the Regge-Wheeler [84]and the Zerilli [85] equations, for two appropriately defined functions which we, now, indicateas Z lm(r, ω). These equations, when written in the frequency domain, have the following generalform

d2Z lm(r, ω)

dr2∗

+[

ω2 − V (r)]

Z lm(r, ω) = 0 l ≥ 2 , (4.2.5)

where r∗ is the usual tortoise coordinate, as defined in (3.1.7).To complex mode eigenfrequencies are those for which the solutions of eq. (4.2.5), found byimposing appropriate boundary conditions at surface of the star, behave as pure outgoing wavesat radial infinity, i.e. there is no ingoing radiation:

Z lm(r, ω) = Almout(ω)eiωr∗ r∗ −→ ∞ (4.2.6)

The standing wave approach to find these eigenfrequencies consists in the following. Let usassume that Z lm(r, ω) is an analytic function of of the complex variable ω = σ − i/τ , and beω0 = σ0 − i/τ0 the frequency of a QNM, with 1/τ0 σ0 (i.e. a slowly damped mode). Ingeneral, at radial infinity the solution of eq. (4.2.5) is a superposition of ingoing and outgoingwave,i.e.

Z lm(r, ω) = Almin (ω)e−iωr∗ + Almout(ω)eiωr∗ . (4.2.7)

If ω = ω0, by definition, Almin (ω) = 0; since Z lm is analytic and 1/τ0 σ0, we can expand

Almin (σ) near the real σ0 as follows

Almin (ω0) = Almin (σ0) −i

τ0Alm ′in (σ0) (4.2.8)

68


from which, imposing Almin (ω0) = 0 we find

Alm ′in (σ0) = −iτ0A

lmin (σ0) . (4.2.9)

Using this relation the function Almin (σ), near σ = σ0 (with σ and σ0 real), has then the form

Almin (σ) = Almin (σ0) + (σ − σ0)Alm ′in (σ0) = Almin (σ0) [1 − iτ0(σ − σ0)]

= −iτ0Almin (σ0)

[

(σ − σ0) +i

τ0

]

, (4.2.10)

and its modulus square is∣

∣Almin (σ)∣

∣

2= B2

[

(σ − σ0)2 +

1

τ 20

]

(4.2.11)

where B is a constant which does not depend on σ. Thus, to find the frequencies of the QNM itis sufficient to integrate the wave equation (4.2.5) for real values of the frequency σ, and find thevalues σi for which the amplitude of the standing wave (4.2.11) has a minimum: these are theQNM frequencies. The corresponding damping times τi can be found through a quadratic fit of∣

∣Almin (σ)∣

∣

2.

4.2.1.2 The standing wave approach for rotating stars

Let us now consider a rotating star. As discussed above, with a suitable choice of the gauge therelevant perturbed quantities reduce to a set of quantities which behave as scalars with respectto rotation: Hmω

i (r, θ)i=1,...,10 (ω complex). They must satisfy a set of PDE, obtained bylinearizing Einstein’s equation, which can be integrated once the values of these quantities areassigned at the center of the star, i.e. on a sphere of radius r0 R (hereafter R is the stellarradius)

Hmωi (r0, θ) = Hmω

0i (θ) . (4.2.12)

The Hmω0i (θ) are subject to constraints, which arise from the analytical expansion in powers of

r of the perturbed equations, from the assumption of regularity of the spacetime as r → 0, andfrom the requirement that the Lagrangian pressure perturbation must vanish on the surface of thestar. These constraints reduce the number of independent quantities from the ten Hmω

0i (θ) to asmaller number, say N , i.e. Hm

0j(θ)j=1,...,N . Being these quantities defined on a sphere r = r0,they can formally be decomposed in spherical harmonics

Hm0j(θ) =

L∑

l=|m|

H lmj Y lm(θ, 0) , (4.2.13)

where the expansion is truncated at l = L. Therefore the independent solutions of the perturbedequations correspond to the following set of N · [L− |m| + 1] constants

H lmj

with j = 1, . . . , N and l = |m|, . . . , L . (4.2.14)

69


Given these constants, the perturbed equations for the functions Hmωi (r, θ) can be integrated

for r ≥ r0. In the wave zone, far away from the star, the far field limit expansion of themetric describing a rotating star shows that the metric reduces to the Schwarzschild solution(see for instance [72], Chap. 19). This occurs because terms due to rotation decrease fasterthan the “Schwarzschild-like” components. Therefore, as when dealing with Schwarzschildperturbations, in this asymptotic region we can define the gauge invariant Zerilli and Regge-Wheeler functions, Z lm

Zer(r, ω) and Z lmRW (r, ω), in terms of the perturbed metric tensor, expanded

in tensorial spherical harmonics with l ≥ 2. This tensor is found by integrating the equationsdescribing the perturbed spacetime outside the rotating star. The well known asymptoticbehaviour of Z lm

Zer(r, ω) and Z lmRW (r, ω) is

Z lmZer(r, ω) = Alm

Zer in(ω)e−iωr∗ + AlmZer out(ω)eiωr∗

Z lmRW (r, ω) = Alm

RW in(ω)e−iωr∗ + AlmRW out(ω)eiωr∗ . (4.2.15)

A (complex) frequency ω0 belongs to a quasi-normal mode if, for an assigned value of m, thefollowing condition is satisfied for any l:

AlmZer in(ω0) = AlmRW in(ω0) = 0 ∀l (4.2.16)

i.e. if the set of 2 · [L− |m| + 1] constants

AlmZer in(ω), AlmRW in(ω)

with l = |m|, . . . , L (4.2.17)

vanishes. It should be stressed that this is a big difference with respect to the non rotating case:in that case each mode belongs to a single, assigned value of l, and there is degeneracy in m.For each assigned value of m, we define the vectors

Hm ≡

H|m|m1

H|m|+1m1

...

H|m|m2

H|m|+1m2

...

and Am ≡

A|m|mZer in(ω)

A|m|+1mZer in (ω)

...

A|m|mRW in(ω)

A|m|+1mRW in (ω)

...

(4.2.18)

the dimensionality of which is N · [L− |m| + 1] and 2 · [L− |m| + 1], respectively. Since theperturbed equations are linear, these vectors are related by the matrix equation

Am(ω) = Mm(ω)Hm ; (4.2.19)

the constants H lmj do not depend on ω. The coefficients of the complex matrix Mm(ω) have

to be evaluated by integrating the perturbed equations. Equation (4.2.16), which identifies theQNM eigenfrequencies, can be written as

Mm(ω0)Hm = 0 ∀Hm . (4.2.20)

70


A discrete set of QNM’s exists if the matrix M is square, i.e. if N = 2. Thus, equation (4.2.20)is equivalent to impose

det (Mm(ω0)) = 0 . (4.2.21)

As we will show in section 4.2.2.4 and in the Appendix D, by counting the number of independentequations in the cases of spherical stars and of slowly rotating stars we find indeed N = 2. Weexpect the same to hold also for rapidly rotating stars.Let us now restrict the frequency to the real axis. Furthermore, we normalize the constants Hm

so that the solutionsHmσi (r, θ) (and consequently Z lm

Zer(r, σ), Z lmRW (r, σ)) are real; this is always

possible, because the perturbed Einstein equations in the frequency domain have real coefficients,as long as σ is real (if we assume that the fluid is non dissipative, so that the equations in timedomain are time-symmetric). Thus, in the wave zone we have

Z lmZer(r, σ) = Alm

Zer in(σ)e−iσr∗ + AlmZer out(σ)eiσr∗ ∈ IR

Z lmRW (r, σ) = Alm

RW in(σ)e−iσr∗ + AlmRW out(σ)eiσr∗ ∈ IR . (4.2.22)

The ingoing wave amplitudes,AlmZer in andAlm

RW in, can be found, as shown in [130], by evaluatingZZer and ZRW at different values of r∗, fitting the two functions as a superposition of sin σr∗ andcos σr∗. It should be noted that although H are real, the quantities Alm

Zer in, AlmRW in are complex(they satisfy the conditions Alm

Zer out = (AlmZer in)∗, AlmRW out = (AlmRW in)

∗), thus M is complex.For σ real, the vectors Hm, Am are related by

Am(σ) = Mm(σ)Hm . (4.2.23)

By expanding equation (4.2.21) about ω0 = σ0 − i/τ0 (with |1/τ0| σ0) as we did for sphericalstars, we find that if σ ∼ σ0

detMm(σ) = detM(σ0)[1−iτ0(σ−σ0)] ⇒ |detMm(σ)| ∝√

(σ − σ0)2 +1

τ0

2

. (4.2.24)

Thus, the QNM frequencies are found by evaluating the (complex) matrix M for real valuesof the frequency σ, finding the minima of the modulus of its determinant. The standing waveapproach has thus been generalized to rotating stars.

4.2.2 Spectral methods for stellar oscillations

We have developed a procedure to solve the two-dimensional perturbation equations on a non-spherical background using spectral methods. Spectral methods, indeed, are very powerful tosolve differential equations (specially in two dimensions) and particularly useful to implementboundary conditions.

71


4.2.2.1 Chebyshev polynomials

We expand the functions of r in Chebyshev polynomials:

f(x) =

∞∑

n=0

anTn(x) with Tn(x(θ)) = cos(nθ) n = 0, 1, . . . (4.2.25)

satisfying the orthogonality relations

∫ 1

−1

Tm(x)Tn(x)dx√

1 − x2=π

2(1 + δm0) δmn . (4.2.26)

(we indicate n,m = 0, 1, . . . , i = 1, 2, . . . ). The integral on Chebyshev polynomials can beevaluated numerically using the Gaussian quadrature method [131]: truncating the polynomialexpansion at n = N ,

∫ 1

−1

g(x)dx√

1 − x2=

π

N + 1

N∑

n=0

g(xn) (4.2.27)

where the collocation points are

xn = cos

(

π(n+ 1/2)

N + 1

)

n = 0, 1, . . . , N . (4.2.28)

The matrix representation of the derivative operator is the following:

f ′(x) =∑

n,m

(Dmnan)Tm (4.2.29)

with

DNn = 0

Dk−1n = Dk+1n + 2kδkn k = 2, . . . , N

D0n =1

2[D2n + 2δ1n] . (4.2.30)

The matrix representation of the multiplication by a function is the following. Given a functionf(x) with Chebyshev expansion

f(x) =∑

n

anTn(x) , (4.2.31)

given a function V (x), the Chebyshev expansion of V (x)f(x) is

V (x)f(x) =∑

n

bnTn(x) , (4.2.32)

72


and bn = Vnmam with

Vnm =2 − δm0

N + 1

∑

k

V (xk)Tn(xk)Tm(xk) . (4.2.33)

If the integration variable is defined in a different range,

y ∈ [a, b] (4.2.34)

then we rescale it to [−1, 1] with the following transformation:

x =2y − b− a

b− a∈ [−1, 1]

y =1

2((b− a)x + (b + a))

∂y =2

b− a∂x . (4.2.35)

4.2.2.2 Associated Legendre polynomials

We expand the functions of θ in the basis of the associated Legendre polynomials

P lm(y)(−1)m(1 − y2)m/2dm

dymP l(y) (4.2.36)

withy = cos θ ∈ [−1, 1] (4.2.37)

and m fixed. They are related to the spherical harmonics by the relation

Y lm(θ, φ) =

√

2l + 1

4π

(l −m)!

(l +m)!P lm(cos θ)eimφ if m ≥ 0

Y lm(θ, φ) = (−1)l(Y l−m)∗ if m < 0 . (4.2.38)

Therefore, expanding a function in circular harmonics eimφ and in associated Legendre polyno-mials is equivalent to expand it in spherical harmonics. This choice is very convenient to expandfunctions of the angular variable θ. Indeed, they are eigenfunctions of the Laplacian operator,

(

∂2θ + cot θ∂θ −

m2

sin2 θ

)

P lm = −l(l + 1)P lm . (4.2.39)

Furthermore, they automatically satisfy the asymptotic behaviour near the axis (see (4.2.36) ):

P lm ∼ (1 − y2)m/2 = (sin θ)m if θ ' 0, π (4.2.40)

73


which is required for any angular expansion of a regular function near r = 0. The associatedLegendre polynomials P lm (with m fixed) are a complete basis of all functions of θ with theasymptotic behaviour (4.2.40), i.e. such that

limθ=0,π

P lm(cos θ)

(sin θ)m= finite . (4.2.41)

If a function fm(cos θ) has been obtained by an expansion in circular harmonics eimφ of afunction regular in the origin, it must behave near the axis like (sin θ)m. Therefore, it can beexpanded in the polynomials P lml=m,... with m fixed:

fm(y) =∑

l

almP lm(y) . (4.2.42)

In order to apply the Gaussian quadrature method to the associated Legendre polynomials, wenotice that the poynomials

P lm(y) ≡ P lm(y)

(1 − y2)m/2, (4.2.43)

are, for each m, a complete basis of polynomials, with orthogonality relation

∫ 1

−1

P lm(y)P l′m(y)(1 − y2)mdy =2

2l + 1

(l +m)!

(l −m)!δll′ . (4.2.44)

The P lm are a particular case of Jacobi polynomials J (α,β)i (y) defined by [131]

∫ 1

−1

Ji(y)Jj(y)(1 − y)α(1 + y)β ∝ δij , (4.2.45)

where α = β = m. Therefore, Gaussian integration has the form

I =

∫ 1

−1

fm(y)P lm(y)dy =∑

k

wkf(yk)P

lm(yk)

1 − y2k

(4.2.46)

where yk and wk are the collocation points and weigths for the Jacobi polynomials with α = β =m. In particular,

alm =2l + 1

2

(l −m)!

(l +m)!

∑

k

wkf(yk)P

lm(yk)

1 − y2k

. (4.2.47)

4.2.2.3 Differential equations and boundary conditions

Let us consider a one-dimensional first order differential equation

Z ′(x) + V (x)Z(x) = 0 (4.2.48)

74


for a function Z(x) with x ∈ [−1, 1]. If we expand Z(x) in a polynomial basis Tn(x), truncatedat n = N ,

Z(x) =

N∑

n=0

anTn(x) (4.2.49)

then the differential equation (4.2.48) corresponds to the algebraic equation

N∑

m=0

(Dnm + Vnm)am = 0 (n = 0, . . . , N) . (4.2.50)

Here Dnm and Vnm are the matrix forms of the derivative operator and of the multiplication by ascalar function V (x) (in the case of Chebyshev polynomials, see (4.2.30), (4.2.33) ).Equation (4.2.48), to be solved, requires the specification of a boundary condition, which weimplement using the so-called τ -method: we cut the last row of (4.2.50), getting

N∑

m=0

(Dnm + Vnm)am = 0 . (n = 0, . . . , N − 1) (4.2.51)

and we add, as a last row, a boundary condition; for instance, if Z(x0) = z0, we add, as a lastrow,

N∑

m=0

Tm(x0)am = z0 . (4.2.52)

The differential equation, with the boundary condition Z(x0) = z0, reduces to a matrix equationwhich can be solved by LU decomposition [131].This approach can be easily generalized to the case of a higher order equation, to the case of asystem of coupled differential equations, and to the case of partial differential equations in r, θ.In this case, each function is expanded in the basis Tn(x), P lm(y)

fm(x, y) =N∑

n=0

L∑

l=m

almn Tn(x)Plm(y) (4.2.53)

where N,L are the maximum orders of Chebyshev and associated Legendre polynomials,respectively.

4.2.2.4 The Regge-Wheeler equation for a spherical star as a 2D-equation in r and θ

We conclude this Chapter with a simple exercise. Choosing the “generalized Regge-Wheelergauge”, described in Appendix (C.1.2) we derive the equations which describe the axialperturbations of a non-rotating star, i.e. the equations for hmω

A (r, θ) (in this case fluidperturbations are decoupled from metric perturbations), and we show how to solve them usingspectral methods. We shall follow the lines of the well-known derivation of the Regge-Wheeler

75


equations [84], with one difference: we do not expand the perturbations in spherical harmonics.By defining the function

Zmω(r, θ) ≡ 1

rhmω

1 (r, θ)e(ν−λ)/2, (4.2.54)

from Einstein’s equations we find

hmω0 (r, θ) =

1

iσ(rZmω(r, θ))′e(ν−λ)/2, (4.2.55)

where (..)′ indicates differentiation with respect to r. Zmω(r, θ) satisfies the partial differentialequation

∂2∗Z

mω + (σ2 − V )Zmω = 0 (4.2.56)

where the coordinate r∗ is defined by

dr∗dr

= e(λ−nu)/2. (4.2.57)

In this formulation, V is a differential operator:

V ≡ eν

r2

[

−(


m2

sin2 θ

)

− 6M

r+ 4π(ρ− p)r2

]

. (4.2.58)

The usual one-dimensional Regge-Wheeler equation can be easily recovered by expandingZmω(r, θ) in scalar spherical harmonics:

Zmω(r, θ) =∑

l

Z lm(r)Y lm(θ, 0) (4.2.59)

e(ν−λ)/2(

e(ν−λ)/2(Z lm(r))′)′

+ (σ2 − V l)Z lm(r) = 0 (4.2.60)

V l ≡ eν

r2

(

l(l + 1) − 6M

r+ 4π(ρ− p)r2

)

. (4.2.61)

wherelmin ≡ max(|m|, 2), (4.2.62)

and Z lmω(r) is the standard Regge-Wheeler function.Let us briefly describe how equation 4.2.56 can be integrated using spectral methods and thestanding wave approach. We shall integrate this equation for real values of the frequency,therefore in the following we shall set ω = σ. The integration range in r∗ is r∗ = [r∗1, r∗2],i.e. given Zmσ(r∗1, θ) we want to know Zmσ(r∗2, θ). The starting point r∗1 corresponds to asmall sphere near the center of the star, with r = r0 << R, where the Regge-Wheeler functionis given by an analytical expansion (see below). The final point r∗2, corresponds to a point in thewave zone, r = r∞ >> R, where the ingoing and outgoing amplitudes can be extracted. Werescale the variable r∗ and θ as follows:

x =2r∗ − r∗1 − r∗2

r∗1 − r∗2∈ [−1, 1] (4.2.63)

y = cos θ ∈ [−1, 1],

76


so that r∗1 corresponds to x = 1, and r∗2 to x = −1. Then, we perform the double expansion(4.2.53) of the Regge-Wheeler function Zm,σ(r, θ) defined in (4.2.54), for assigned values of m,σ:

Zm,σ(x, y) =K∑

n=0

L∑

l=lmin

almσn Tn(x)Plm(y). (4.2.64)

The expansions in Chebyshev’s and associated Legendre’s polynomials are truncated at K andL, respectively (for instance, L = 10, L = 20). The boundary conditions at the center of the starare imposed by assigning Zmσ and its derivative at x = 1:

Zmσ(x = 1, y) =

L∑

l=lmin

zl0HlmP lm(y)

∂r∗Zmσ(x = 1, y) =

L∑

l=m

zl1HlmP lm(y), (4.2.65)

where the analytic expansion of the Regge-Wheeler equation gives [84]

zl0 = rl+10 +X l

in(σ)rl+30 (4.2.66)

zl1 = eν(0)[(l + 1)rl0 + (l + 3)X lin(σ)rl+2

0 ] (4.2.67)

X lin(σ) ≡ (l + 2)[1

3(2l − 1)ρ(0) − p(0)] − σ2eν(0)

2(2l + 3)(4.2.68)

(note that while in zl0,1 and X lin, the index l is a superscript, in rl0 it is an exponent).

The constants H lm form a vectorHm ≡ (H lm), (4.2.69)

which can be freely assigned; for each vector H we have one solution of the equation. Noticethat we have one constant H lm for each value of l i.e., in the language of section 4.2.1.2, N = 1for the axial parity perturbations. If, in addition to axial perturbations, polar parity perturbationsare considered, described outside the star by the Zerilli equation, then there is another constantto be assigned for each value of l. Therefore, if all metric perturbations are considered, N = 2as discussed in section 4.2.1.2.We now project equation (4.2.56) in the basis of Chebyshev’s and associate Legendre’spolynomials. The operator


m2

sin2θ(4.2.70)

is diagonal in the P lm basis, with eigenvalue −l(l + 1). Therefore, in this basis the operator Vdefined in (4.2.58) reduces to the one-dimensional Regge-Wheeler potential V l(r) (4.2.61). Weintroduce (as in section (4.2.2.1), equation (4.2.30), (4.2.33)) the derivative matrix Dnn′ and thepotential matrix Vnm obtained projecting V l(r) on Chebyshev polynomials. To write the matrixequation we define the collective index

i(l, n) = (l − |m|)(K + 1) + n+ 1 ∈ [1,N ] (4.2.71)

77


with N=(L− |m| + 1)(K + 1), and we define the N -dimensional vector of components

ami = ami(l,n)=almn. (4.2.72)

The matrix equation has the block-diagonal form

Lii′ami′ = 0 (4.2.73)

L =

(

D2nn′ + Vnn′

)

l=|m|0

(

D2nn′ + Vnn′

)

l=|m|+1

. . .

0(

D2ll′ + Vnn′

)

l=L

. (4.2.74)

The boundary conditions are implemented by replacing the last two lines of each block, i.e.each i(l, K)-th and i(l, K − 1)-th lines, with the conditions at the center in terms of the vectorHm = (H lm):

K∑

n=0

ami(l,n) = zl0Hlm

0,K∑

n,k

Dknami(l,n) = zl1H

lm . (4.2.75)

By inverting the matrix equation (4.2.73) we find the coefficients ami and then Zm,ω(−1).In this way, for each choice of (m, σ) and of the vector Hm = (Hm) of initial conditions, wehave the vector of ingoing amplitudes at infinity

Am = (Alm(σ)), (4.2.76)

This procedure is linear at any step, thus, repeating it for all Hm’s:

Hm = (1, 0, . . . , 0) , Hm = (0, 1, . . . , 0) , . . . , Hm = (0, 0, . . . , 1) (4.2.77)

we find the expression of the matrix Mm = (Mm|ll′(σ)) defined by

Alm(σ) = Mm|ll′(σ)H l′m . (4.2.78)

As explained in section 4.2.1.2, near a mode σ ∼ σ0 the determinant of this matrix behaves as

detMm(σ) ∝√

(σ − σ0)2 +1

τ0

2

(4.2.79)

and the frequency σ0 and the damping time τ0 of the mode can be found by a quadratic fit in σ.

78



In this Chapter we have illustrated a new method to find quasi-normal modes of rotatingrelativistic stars. We have described the main features of our approch, that if it is comparedwith existing literature on the subject, is innovative for two different reasons. The first one is that,until now, in the perturbative approaches, oscillations are treated as perturbations in the frequencydomain of a background describing a non-rotating star and the perturbed quantities are functionsof the radial variable r only. Instead, we have developed a procedure to solve the 2D-dimensionalperturbation equations on a non-spherical background and we have integrated the perturbedequations using spectral methods. In particular we gave explicit formulae (whose numericalimplementation is straightforward) which allow us to transform functions of r, θ in vectors,and systems of coupled differential equations (involving derivatives in r, θ) in algebraic, matrixequations. The second reason is that, once the 2D-equations describing stellar perturbationshave been solved, the frequencies and damping times of, quasi-normal modes can be found bylooking for the minima of the determinant of a properly defined matrix, evaluated as function ofreal frequency, thus generalizing the standing wave approach [129, 86, 130] to rotating stars.

79


80

5

A test of the method: oscillations ofslowly rotating stars

In this Chapter, as a test of our method we consider the oscillations of a slowly rotating star1 andwe solve the equations which describe the perturbations of a slowly rotating stars. Following[58], in this case the metric and fluid velocity of the unperturbed star are

(ds2)(0) = g(0)µν dx

µdxν = e−ν(r)dt2 + eλ(r)dr2 + r2(dθ2 + sin2 θdφ2) − 2ω(r)r2 sin2 θdtdφ

u(0) µ = (e−ν/2, 0, 0,Ωe−ν/2) (5.0.1)

where ω(r) describes the dragging of the inertial frames (as we saw in Chapter 2, section 2.2),and all quantities are expanded at first order in the angular velocity of the star, Ω. We introducea rotation parameter ε defined by

ε = Ω/√

M/R3 . (5.0.2)

In the metric (5.0.1) spherical symmetry is broken only by the term ω: when Einstein’s equationsare perturbed about (5.0.1), only terms which are linear in ω are retained, i.e. we keep terms upto order O(ε). As a consequence, perturbations with index l are coupled with perturbations withindexes l ± 1 through terms that are of order O(ε), the analytical form of which can explicitlybe derived. Therefore, when we transform the perturbed equations using the double spectraldecomposition described in Section 4.2.2, the resulting algebraic equations have a particularlysimple form: the relevant matrix is “almost-block-diagonal”, each block corresponding to onevalue of l, the off-diagonal blocks couple l ↔ l ± 1.These equations can also be obtained in a different way, i.e. by expanding in Chebyshevpolynomials the system of ordinary differential equations in r derived by Kojima in [100](hereafter, K1). This follows from the fact that Kojima’s equations are derived by expandingthe perturbed Einstein’s equations in spherical harmonics (see for details Appendix B).The general structure of Kojima’s equations is the following:

Lpol[H lm0 , K lm; σ] = mE [H lm

0 , K lm; σ] + F (±)[Z l±1mRW ; σ]

Lax[Z lmRW ; σ] = mN [Z lm

RW ; σ] + D(±)[H l±1m0 , K l±1m; σ]. (5.0.3)

1For simplicity, we do not consider differential rotation in this context.

81

5.1. Comparison with existing results

Here Lpol, Lax are operators of order O(ε0), which describe the perturbations of the star in thenon-rotating case; E , F (±), N , D(±) are O(ε1) operators, which provide the corrections due torotation.These equations have been integrated numerically in [101] (see also [102]) using a very strongsimplification: the couplings l ↔ l ± 1 were neglected (i.e. F (±) and D(±) were set to zero).Moreover, the equations were solved iteratively, finding the solution for ε = 0 first, and thenreplacing it in the terms E [H lm

0 , K lm; σ] and N [Z lmRW ; σ] of eqs. (5.0.3). In this way, the right-

hand sides of eqs. (5.0.3) become source terms.We stress that with our approach, which we described in details in Chapter 4, we do not needthese simplifications anymore. In particular, we do not need to neglect the couplings, becausewe can handle the mixing among perturbations with different l’s using the spectral methods andthe generalized standing wave approach.It should also be mentioned that, when coupling terms are included in the perturbed equations,in order to have a good numerical behaviour of the perturbations near the center of the star weneed to use a set of variables (in particular the O(ε1) terms) different from that used in [102].The equations for the new variables are given explicitly in Appendix D. In the next section wecompare our results with the existing ones and we also report our new results that include thecoupling terms.

5.1 Comparison with existing results

As mentioned above, in [101] (hereafter K2) the equations of stellar perturbations have beenintegrated for a slowly rotating star, neglecting l ↔ l ± 1 couplings, and the QNM frequencieshave been found. To reproduce these results we have used the same set of variables as in K1,the same equation of state (EOS) i.e. the polytropic EOS p = Kρ2, and we have computed thefundamental mode (f -mode) frequency, σf . We know that this frequency is complex: in K2 thereal and imaginary parts of σf are fitted as functions of the rotation parameter ε as follows:

σRf = σR0 (1 +mεσ′R) +O(ε2) ,

σIf = σI0(1 +mεσ′I) +O(ε2) . (5.1.1)

The value of σR0 we find, properly normalized, is plotted versus the stellar compactness, M/R,in Figure 5-1 a). The values are in excellent agreement with the results shown in Figure 1 ofK2, for n = 1. The correction due to rotation, σ ′

R, is plotted versus M/R in Figure 5-1 b) fordifferent values of ε. We find that for ε . 10−3 the corresponding curves are indistinguishable,and coincide with the n = 1 curve shown in Figure 1 of K2. However, for ε & 10−3 different εcorrespond to different curves, and the fit (5.1.1) becomes inaccurate: σ ′

R is no longer a constant,and further corrections to (5.1.1) are of order O(ε2), as expected theoretically.In Figure 5-2 we plot the imaginary part of the f -mode frequency, σI0 , and the correspondingrotational correction, σ′

I , as in Figure 5-1. σ′I is plotted only for ε ≥ 10−2, because for smaller

values our numerical approach becomes inaccurate. They agree with the values given in Figure2 of K2, with differences of order O(ε2).

82

Chapter 5. A test of the method: oscillations of slowly rotating stars

1.14

1.16

1.18

1.2

1.22

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24

σ 0R

R3/

2 M-1

/2

M/R

n=1

a)

0.45

0.5

0.55

0.6

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24

σ R’

M/R

n=1

a)

b)

ε=1.d-04 ε=1.d-03 ε=1.d-02 ε=2.d-02 ε=4.d-02

Figure 5-1. The real part of the f -mode frequency of a non rotating, polytropic star with n = 1,is plotted as a function of the stellar compactness M/R (a); the frequency shift due to rotation, σ ′

R

is plotted versus compactness for different values of ε (b). As in K2, couplings among differentl’s are neglected.

0.03

0.04

0.05

0.06

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24

σ 0I R

4 M-3

M/R

n=1

a)

b)

a)

2.6

2.8

3

3.2

0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24

σ’I

M/R

n=1

a)

b)

a)

b)

ε=2.d-02 ε=4.d-02

Figure 5-2. The imaginary part of the f -mode frequency (a) and the corrections due to rotation(b) are plotted as in Figure 5-1.

83

5.2. Including the couplings

5.2 Including the couplings

We now apply our method to solve eqs. (5.0.3) in full, i.e. including couplings among differentl’s. The numerical implementation of the equations presents a problem: there are terms in

the equations which depend on(

∂p∂ρ

)2

and on(

∂2p∂ρ2

)

, which strongly diverge on the stellar

surface. This divergence is particularly problematic when spectral methods are used, but it canbe cured through a regularization procedure [132]. Such regularization goes beyond the scopeof the present work, where we only want to discuss a simple implementation of our approach.Therefore, we solve the perturbed equations in the case of a constant density, slowly rotating star,such that the divergent terms vanish.We consider two background models: A, with M/R = 0.2 and B, with M/R = 0.1. Massand radius for assigned values of the central density are given in Table 5-1. The explicit form of

ρ (g/cm3) M/M R(Km) M/RA 1015 1.11 8.08 0.2B 1015 0.40 5.75 0.1

Table 5-1. Parameters of the constant density stellar models A and B we use as a background.

the equations and the boundary conditions in r = 0 and r = R are discussed in Appendix D.As mentioned above, when couplings are included the equations derived in K1 are very unstablewhen integrated near the center. For this reason we introduce a new set of variables, which satisfya new set of equations shown in Appendix D.1, where are also shown the appropriate boundaryconditions in r = 0 and r = R.Once we assign the value of the harmonic index m, these equations couple polar and axialperturbations with |m| ≤ l ≤ L. The equations have been integrated for m = ±2. We donot set |m| < 2 because in that case dipolar (l = 1) perturbations have to be taken into account,which are described by equations different from those we consider in this thesis. We would liketo stress that rotational corrections to mode frequencies withm 6= 0 are much larger than those tomode frequencies with m = 0. In K2, Kojima suggested that, at lowest order in ε, the frequencyshift is proportional to m. This is consistent with the results of our numerical integration: therelative frequency shifts found in [117], where m = 0 perturbations were studied in full generalrelativity, are an order of magnitude smaller than the relative shifts we find for m = ±2.If the star does not rotate, for any assigned value of l there is a corresponding f -mode frequency,which is the same for all m’s. If the star rotates, due to the couplings the f -mode belonging to anassigned l acquires contributions from different l’s, and its frequency and damping time change.Furthermore, the degeneracy in m is broken by rotation.The real part of the f -mode frequency, νf = σRf /(2π), is plotted as a function of the rotationparameter in Figure 5-3 for the two considered stellar models. The solid line represents thefrequency of the l = 2 f -mode of the non-rotating star. The dashed lines are the frequenciesof the lowest lying fundamental mode of the rotating star, with m = 2 and m = −2, assumingL = 4. Our calculations refer to ε ≤ 0.05, since for higher values the slow rotation approximation

84


becomes inaccurate and the results cannot be trusted anymore. Figure 5-4 shows, in a smaller

2100

2200

2300

2400

2500

0 0.01 0.02 0.03 0.04 0.05

ν f (

Hz)

ε

f-mode frequency (model A)

a)b) a)b)

non-rotating starrotating star, m=2

rotating star, m=-2 2100

2200

2300

2400

2500

2600

0 0.01 0.02 0.03 0.04 0.05

ν f (

Hz)

ε

f-mode frequency (model B)

a)b) a)b)

non-rotating starrotating star, m=2

rotating star, m=-2

Figure 5-3. The real part of the f -mode frequency, νf = σRf /(2π), is plotted versus the rotationparameter ε. The data refer to two models of constant density star (see text). We include couplingsup to l = 4. The dashed line refers to the non rotating star; the dotted lines refer to the modesm = ±2 of the slowly rotating star.

range of the rotation parameter, the frequency νf computed by truncating the expansion in l atL = 2 (i.e. without couplings), L = 3 and L = 4. It is evident that for slowly rotating stars thecontribution of the couplings is a small correction, and that there is convergence as L grows. The

2525

2530

2535

2540

2545

0.045 0.046 0.047 0.048 0.049 0.05

ν f (

Hz)

ε

f-mode frequency (m=2)

a)b) a)

b)

L=2L=3L=4

2080

2090

2100

2110

2120

0.045 0.046 0.047 0.048 0.049 0.05

ν f (

Hz)

ε

f-mode frequency (m=-2)

a)b) a)b)

L=2L=3L=4

Figure 5-4. Details of left panel of Figure 5-3: we show the contribution of the differentcouplings to the f -mode frequency, for model A. The different curves are obtained by includingin the perturbed equations couplings up to l = L.

relative frequency shift due to the couplings is well approximated by a quadratic behaviour in ε:

∆νfνf

=νL=4f (ε) − νL=2

f (ε)

νε=0f

' σ′′ε2 , (5.2.1)

as shown in Figure 5-5. Therefore, the contribution of the couplings is of order O(ε2), as argued

85

5.2. Including the couplings

0

0.0001

0.0002

0.0003

0.0004

0 0.01 0.02 0.03 0.04 0.05

∆νf /

νf

ε

Coupling‘s contributions

a)b)

a)

b)

m=2m=-2

Figure 5-5. Frequency shift as given by equation (5.2.1) for model A.

by Kojima in K2, and σRf is well described by a quadratic fit of the form

σRf = σR0 (1 +mεσ′R + ε2σ′′

R) , (5.2.2)

where we should remind that terms of order O(ε2) are of the same order of the terms whichwe are neglecting in the perturbed equations ab initio. This fit is accurate up to ε ∼ 10−3;

0.7

0.8

0.9

1

0.01 0.02 0.03 0.04 0.05

ε

σ, (m=2)

0.9

1

1.1

1.2

0.01 0.02 0.03 0.04 0.05

ε

σ, (m=-2)

0.12

0.14

0.16

0.18

0.01 0.02 0.03 0.04 0.05

ε

σ,, (m=2)

0.16

0.2

0.24

0.28

0.01 0.02 0.03 0.04 0.05

ε

σ,, (m=-2)

Figure 5-6. The coefficients σ′, σ′′, plotted as functions of the rotation rate for model A.

for larger values σ′ and σ′′ are no longer constant, as shown in Figure 5-6: as ε grows, σ ′

changes linearly, and the change is negative if m > 0, positive if m < 0, yielding in both casesa shift to lower values of the total frequency σ. This explains the small asymmetry betweenνf (m = 2) and νf (m = −2) shown in Figure 5-3. Notice that deviations from the fit (5.2.2)are always of order O(ε2), consistently with our approximation scheme. For 0.01 < ε < 0.05,the coefficients σ′′ are ∼ 0.15 (if m = 2) and ∼ 0.20 (if m = −2); for ε < 0.01 they are toosmall to be correctly extrapolated with our codes. Finally, the damping time of the f -mode isshown in Figure 5-7 as a function of the rotation rate. It is worth stressing that, as the stellar

86


0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.01 0.02 0.03 0.04 0.05

τ f (

s)

ε

f-mode damping time

m=2m=-2

Figure 5-7. Damping time of the f -mode as a function of the rotation rate, for model A.

rotation increases, the frequency of the counterrotating (i.e. m = −2) mode decreases fasterthan expected by the simple linear fit (5.1.1) . Furthermore, Figure 5-7 shows that the dampingtime of the counterrotating mode increases sharply, even for small rotation rates. This indicatesthat the CFS instability may occur for values of the rotation rate lower than expected by simplelinear estimates.In Chapter 4 we saw that r-modes satisfies the CFS instability criterion at all rotation rate and itwill be very interesting to study how the frequencies of these modes change with rotation, but theequations derived for the perturbations of a slowly rotating star in [100] and in Appendix D.1,are not appropriate to study the r-modes, because the frequency σ, which is a dimensionful scaleof the problem, is of the same order as the “small” parameter ε. It should be mentioned that theshift of the r-mode frequency due to slow rotation in a relativistic star has been studied in [109],taking the couplings between perturbations with different l’s, into account.It should be mentioned that an alternative approach to find the QNM frequencies, based ona characteristic formulation of the perturbed equations and a complexification of the radialcoordinate, which could be generalized to rotating stars, has recently been proposed [133].


In this Chapter we tested our method in the case of slow rotation. The system of partialdifferential equations from which we start is formally the same as in [100] (apart from aredefinition of some variables described in Appendix D.1). However, in our approach wetransform that differential system in a system of algebraic, coupled equations. Thus, theadvantage of our method is that it is much easier to handle the couplings among different valuesof l, which in [100] correspond to couplings between different partial differential equations. Forthis reason we are able to study the shift of the fundamental mode due to rotation, taking thel ↔ l ± 1 couplings into account, to our knowledge for the first time in the literature. In this

87


Chapter we showed that, as the rotation parameter ε increases, the frequency of counterrotatingmodes decreases at a rate higher than linear in ε. Furthermore, the corresponding dampingtime sharply increases, even for small rotation rates. This suggests that the CFS instabilityfor a generic mode should occur for values of the rotation rate lower than expected by simplelinear estimates. This result complements what found in [115], where the equations of stellarperturbations where integrated in full general relativity looking for neutral modes, and it wasshown that the CFS instability sets in for smaller rotation rates than in Newtonian gravity.

88

6

Conclusions

6.1 Rapidly rotating neutron stars

In Chapter 2 we illustrated our obtained results using the perturbative approach: we solved theequations of stellar structure developed to the third order in the angular velocity, using recentEOS which model hadronic interactions in different ways to describe the matter in the innercore. The stellar parameters, we finded, have been compared, when available in the literature,with those found by solving the exact1 equations of stellar structure. The main conclusions wecan draw from our study are the following.It is known that near mass-shedding, the perturbative approach fails to reproduce the stellarproperties correctly and the reasons are well understood. We confirm this behaviour and givequantitative results in this limit. However, for lower values of the angular velocity the situationis different. Taking the rotation rate of the fastest isolated pulsar observed so far as a reference,PSR 1937+21 for which ν ∼ 641 Hz, we see that at these rates the perturbative approach allowsus to describe all stellar properties to an accuracy better than ∼ 2% even for the maximum massmodels. The two quantities that are affected by third order corrections are the mass-sheddingvelocity and the moment of inertia. For the first, we find that the third order corrections areactually negligible, smaller than 1%, independently of the EOS. Conversely, the moment ofinertia is affected by terms of order Ω3 in a significant way; for instance, for a star with M =1.4 M rotating at ν ∼ 641 Hz the third order correction is of the order of δI/(I (0) + δI) ∼ 5%for the stiffer EOS we use (APR2), and as high as ∼ 13% for the softest (G240). Thus, thirdorder corrections have to be included to study the moment of inertia of rapidly rotating neutronstars, whereas they are irrelevant when estimating the mass-shedding limit.We can conclude that the perturbative approach allow us accurate theoretical calculations of therelevant physical properties of rapidly rotating neutron stars. We hope that, in the future, with thehelp of missions like Constellation-X, more astronomical data on these sources will be availablein order to put more tighten constraints on the EOSs.

1Solving exact equations means direct numerical integration of Einstein’s equations for stellar structure withoutapproximation.

89

6.2. Oscillations of non-rotating quark stars

6.2 Oscillations of non-rotating quark stars

In Chapter 3 we studied the f-mode frequencies of oscillation of non-rotating strange stars andneutron stars. The study that we proposed differs from the preceeding literature in so much asit explores the entire range of allowed parameters of the MIT bag model of EOS in a systematicway. Our results showed that the detection of a signal emitted by a compact star pulsating inits fundamental mode, combined with complementary information on the stellar mass or theradiation radius, would allow one to discriminate between neutron/hybrid stars and strange stars.In addition, we showed that it would also be possible to constrain the bag constant to a rangemuch smaller than that provided by the available data from terrestrial experiments.As a final remark, taking the detectability of the fundamental mode into account, it seems likelythat we will need observational data to put constraints on our theoretical models. As the newinterferometers may open the gravitational waves window to the Universe in the next few years, isis appropriate to ask to what extent the various theoretical models are falsifiable by observations.

6.3 Oscillations of rotating neutron stars

In Chapters 4, 5 we propesed a new method to study quasi-normal modes of rotating relativisticstars. Comparing with the literature this method is innovative for two different reasons. The firstone is that, until now, in the perturbative approaches oscillations are treated as perturbations inthe frequency domain of a background describing a non-rotating star and the perturbed quantitiesare functions of the radial variable r only.Instead, we have developed a procedure to solve the 2D-dimensional perturbation equations ona non-spherical background and the perturbed equations are integrated using spectral methods.The second reason resides in our choice of defining, the boundary conditions at the center of thestar and at radial infinity, generalizing, to the case of rotating stars, the boundary conditions thatare used to find the quasi-normal mode frequencies of non-rotating star. To test our method, weapplied our approach to find the frequency of the f- mode of a slowly rotating, constant densitystar, as a function of the rotation rate and we compare our results with existing results in theliterature and we find a very good agreement. It is important to point out that with our approachit is much easier to handle the couplings among different value of l, for this reason we are able tostudy the shift of the fundamental mode due to rotation, taking the l ± 1 couplings into account,to our knowledge for the first time in the literature.

90

A

Appendix

In this appendix we write the equations that govern the radial part of the metric and thermo-dynamical functions that describe the structure of a rotating star to third order in the angularvelocity Ω. For each function we specify the appropriate boundary conditions and we show howto construct the solution order by order. We give explicitely the analytic solution when available,following and completing the work of Hartle and collaborators [57, 58, 59]. In what follows weshall refer to the functions appearing in section 2.2.

We recall that the structure of a non rotating, spherical star is described by the TOV equations

dM(r)

dr= 4πr2ε(r), (A.0.1)

dν(r)

dr= 2

M(r) + 4πr3P (r)

r(r − 2M(r)),

dP (r)

dr= −(P (r) + ε(r))

2ν(r),r.

ε(r) and P (r) are the energy density and pressure distribution in the non rotating star. Since allfunctions we shall consider depend on r, we shall omit this dependency throughout.Hereafter, R and M(R) will indicate, respectively, the radius and the mass of the non rotatingstar, found by solving eqs. (A.0.1) for an assigned EOS.

A.1 The first order equations

To order Ω the shape of the star remains spherical and there is only one function, ω, to determine,which is responsible for the dragging of inertial frames (cfr. the expansion in eq. 2.2.6). Byintroducing the function $ = Ω − ω, it is easy to show that it satisfies the following equation[57]

1

r4

d

dr(r4j

d$

dr) +

4

r

dj

dr$ = 0, (A.1.1)

91

A.2. Second order equations

where j(r) = e−ν/2√

1 − 2Mr. Equation (A.1.1) can be reduced to two first order equations by

defining two auxiliary functions of r

χ = j$, u = r4jd$

dr(A.1.2)

that satisfy

r ≤ Rdχ

dr=

u

r4− 4π, r2(ε + P )χ

r − 2Mdu

dr=

16πr5(ε + P )χ

r − 2M. (A.1.3)

They have to be integrated from r = 0 to R with the following boundary conditions

χ(0) = j(0)$c, u(0) = 0. (A.1.4)

These conditions follow from the behaviour of $ near the origin

$ ∼ $c(1 +$2r2 + ....), (A.1.5)

where $c is a constant. Note that from eqs. (A.1.2) and (A.1.5) it fopllows that u goes to zerofaster than r4.

For r ≥ R, P and ε vanish, M ≡M(R), and the solution of eqs. (A.1.3) is

r ≥ R χ(r) = Ω − 2J

r3, u(r) = 6J, (A.1.6)

where J is a constant which represents the angular momentum of the star to first order in Ω. Theconstants $c and J can be found by imposing that the interior and the exterior solutions matchcontinuously at R, i.e.

$c χ(R) = Ω − 2J

R3, u(R) = 6J.

A.2 Second order equations

The second order terms affect the structure of the star in the following way: h0 and m0 producea spherical expansion, h2 and v2 a quadrupole deformation.

On the assumption that the EOS is a one parameter equation of state, Einstein’s equations admita first integral (cfr. [57]) that allows to find two equations, one relating h0 and δp0, the second h2

and δp2. These two equations are

µ = δp0 + h0 −1

3e−νr2$2 (A.2.1)

0 = δp2 + h2 +1

3e−νr2$2 , (A.2.2)

92

Appendix A. Appendix

where µ is the correction of order Ω2 to the chemical potential µc

µc = µ[1 + µ+O(Ω4)] ≡ E + Put

exp

[

−∫

dEE + P

]

(A.2.3)

(where E and P are the energy density and pressure in the rotating configuration) which can beshown to be constant with respect to r and θ throughout the star.

A.2.1 Spherical expansion

Using eq. (A.2.1), the relevant equations for the spherical deformation can be shown to be

dm0

dr= 4πr2 dε

dP[δp0(ε + P )] +

u2

12r4+

8πr5(ε+ P )χ2

3(r − 2M)

dδp0

dr=

u2

12r4(r − 2M)

−m0(1 + 8πr2P )

(r − 2M)2− 4π(ε+ P )r2δp0

r − 2M

+2r2χ

3(r − 2M)

[

u

r3+

(r − 3M − 4πr3P )χ

r − 2M

]

(A.2.4)

where u and χ are defined in (A.1.2). These equations must be integrated inside the star fromr = 0 to R with the condition that both m0 and δp0 vanish in r = 0. To start the numericalintegration it is useful to make use of the following expansion

m0(r) ∼4π

15[ε(0) + P (0)]

[(

dε

dP

)

r=0

+ 2

]

χ(0)2 r5

δp0(r) ∼1

3χ(0) r2, r → 0. (A.2.5)

For r ≥ R u and χ reduce to (A.1.6); by integrating the first eq. (A.2.4) with P and ε vanishing,one finds

m0(r) = −J2

r3+ δM (A.2.6)

where δM is a constant which represents the correction to the gravitational mass due to thespherical expansion of the star. This quantity can be determined by imposing the continuity ofthe solution at r = R, i.e.

δM = m0(R) +J2

R3, (A.2.7)

and m0(R) is found by integrating eqs. (A.2.4) for r ≤ R.

93

A.2. Second order equations

Due to the sperical expansion, the pressure of each element of fluid changes by an amount δp0(r)found by integrating eqs. (A.2.4); at the same time the element is radially displaced by an amount

r + ξ0(r), (A.2.8)

and ξ0(r) can be found in terms of δp0(r) using eq. (2.2.9); known ξ0(r) we can compute thecorresponding variation of the stellar radius (cfr. eq. 2.2.12)

δRA = ξ0(R) =δp0(R)R(R− 2M(R))

M(R). (A.2.9)

A.2.2 Quadrupole deformation

We shall now solve the equations for h2(r) e v2(r), that are responsible for the quadrupoledeformation of the star:

dv2

dr= −dν

drh2 + (

1

r+

1

2

dν

dr)

[

8πr5(ε + P )χ2

3(r − 2M)+

u2

6r4

]

(A.2.10)

dh2

dr=

[

−dνdr

+r

r − 2M(dν

dr)−1(8π(ε+ P ) − 4M

r3)

]

h2

− 4v2

r(r − 2M)(dν

dr)−1 +

u2

6r5

[

1

2

dν

drr − 1

r − 2M(dν

dr)−1

]

+8πr5(ε + P )χ2

3(r − 2M)

[

1

2

dν

drr +

1

r − 2M(dν

dr)−1

]

.

Let us consider the solution for r ≤ R first.

It is easy to check that a regular solution of eqs. (A.2.10) near the origin must behave as

r → 0 h2(r) ∼ Ar2, v2 ∼ Br4, (A.2.11)

where the constants A and B are related by the following expression

B + 2π

[

1

3ε(0) + P (0)

]

A =2

3π [ε(0) + P (0)] (j(0)$c)

2 . (A.2.12)

Following [57], we write the general solution of eqs. (A.2.10) for r ≤ R as

h2(r) = hP2 + C hH2 (A.2.13)

v2(r) = vP2 + C vH2 .

94


C is a constant to be determined (see below), (hP2 , vP2 ) are a particular solution found by

integrating eqs. (A.2.10) with the initial conditions, say, A = 1 and B given by eq. (A.2.12);(hH2 , v

H2 ) are the solution of the homogeneous system which is found by putting χ = 0 and u = 0

in eqs. (A.2.10), numerically integrated with the boundary condition

r → 0 hH2 (r) ∼ r2 (A.2.14)

vH2 (r) ∼ −2π

[

1

3ε(0) + P (0)

]

r4.

For r ≥ R eqs. (A.2.10) can be solved analytically in terms of the associated Legendre functionsof the second kind, and the solution is [57],[59]

r ≥ R h2(r) = J2(1

M(R)r3+

1

r4) +K Q2

2(ξ) (A.2.15)

v2(r) = −J2

r4+K

2M(R)

[r(r − 2M(R))]1/2Q1

2(ξ)

where K is a constant and

Q22(ξ) = [

3

2(ξ2 − 1)log(

ξ + 1

ξ − 1) − 3ξ3 − 5ξ

ξ2 − 1] (A.2.16)

Q21(ξ) = (ξ2 − 1)1/2[

3ξ2 − 2

ξ2 − 1− 3

2ξlog(

ξ + 1

ξ − 1)].

withξ =

r

M(R)− 1. (A.2.17)

We can now determine the constants C and K by imposing that the solution (A.2.13) and thesolution (A.2.15) are equal at r = R; thus, the functions h2 and v2 are completely determined.

We can now compute the quadrupole deformation of the star. Indeed, known h2 from eq. (A.2.2)we find δp2:

δp2(r) = −h2(r) −1

3e−ν(r)r2$(r)2, (A.2.18)

and from eq. (2.2.10) we find the second correction to the displacement

ξ2(r) = −δp2(r)/

[

1

ε + P

dP

dr

]

. (A.2.19)

Thus, an element of fluid located at a given r and a given θ in the non rotating star, when the starrotates moves to a new position identified by the same value of θ, and by a radial coordinate

r = r + ξ0(r) + ξ2(r)P2(θ) +O(Ω4), (A.2.20)

where ξ0(r) and ξ2(r) are found as explained above.

The remaining metric function m2 can be determined by a suitable combination of Einstein’sequations, which gives

m2 = (r − 2M)

[

−h2 +8πr5(ε+ P )χ2

3(r − 2M)+

u2

6r4

]

. (A.2.21)

95

A.3. Third order equations

A.3 Third order equations

The functions that describe the third order corrections are w1 and w3, and satisfy two secondorder differential equations with a similar structure. Since they do not couple, we shall solvethem independently.

A.3.1 The equations for w1

1

r4

d

dr(r4j

dw1(r)

dr) +

4

r

dj

drw1 = D0 −

1

5D2, (A.3.1)

where

r4D0 = −u ddr

[

m0

r − 2M+ h0

]

+ 4r3χ

[

1

j

dj

dr

]

(A.3.2)

·[

2m0

r − 2M+ (

1

(dP/dε)+ 1)δp0 +

2r3χ2

3(r − 2M)

]

,

r4D2

5=u

5

d

dr

[

4k2 − h2 −m2

r − 2M

]

+ (4r3χ

5)

[

1

j

dj

dr

]

·[

2m2

r − 2M+ (

1

(dP/dε)+ 1)δp2 −

2r3χ2

3(r − 2M)

]

.

By introducing the variables

χ1 = jw1 u1 = r4jdw1

dr(A.3.3)

eq. (A.3.1) becomes

dχ1

dr=u1

r4− 4πr2(ε+ P )χ1

r − 2M, (A.3.4)

du1

dr=

16πr5(ε + P )χ1

r − 2M+ r4D0 −

r4

5D2.

We write the general solution of this system for r ≤ R as

χ1 = χP1 + C χH1 (A.3.5)

u1 = uP1 + C uH1 .

C is a constant, (χP1 , uP1 ), are a particular solution found by integrating eqs. (A.3.4) with the

initial conditionsχP1 (0) = 0, uP1 (0) = 0;

96


(χH1 , uH1 ) is the solution of eqs. (A.3.4) in which D0 and D2 are set to zero, numerically

integrated with the asymptotic conditions

r → 0 χH1 (r) ∼ 1 +8π

5[ε(0) + P (0)] r2, (A.3.6)

uH1 (r) ∼ 16π

5[ε(0) + P (0)] r5.

For r ≥ R eqs. (A.3.4) reduce to

dχ1

dr=u1

r4, (A.3.7)

du1

dr= −u

5

d

dr[4k2 −

6J2

r4].

Note that, since j(r ≥ R) = 1, for r ≥ R χ1 and w1 coincide (cfr. eq. A.3.3).

At large distance from the source w1 must behave as

w1(r) =2δJ

r3+ 0(

1

r4), (A.3.8)

where δJ is the correction to the angular momentum of the star; this suggests to write w1, andtherefore χ1, as

w1(r) = χ1(r) =2δJ

r3+ F (r), (A.3.9)

where F (r) must go to zero at infinity faster than r3. Consequently, when r → ∞ u1 → −6δJ .Thus, by direct integration of the second eq. (A.3.7), and using eqs. (A.1.6), (2.2.5) and (A.2.15),we find

uext1 = −u5[4k2 −

6J2

r4] − 6δJ =

84J3

5r4+

24J3

5M(R)r3

+24JKQ2

2

5− 48M(R)JKQ1

2

5[r(r − 2M(R))]1/2− 6δJ. (A.3.10)

Knowing uext1 , we can now integrate the first of eqs. (A.3.7) which, using eq. (A.3.9), becomesan equation for F (r), the solution of which is

F (r) = −12J3

5r7− 4J3

5Mr6+ (A.3.11)

JK

40M3r4

[

108r4 ln(r

r − 2M) − 288r3M ln(

r

r − 2M)

+33r4 − 240r3M + 336r2M2 + 256M3r − 96M4

+192rM3 ln(r

r − 2M) + 12r4 ln(

r

r − 2M)

]

− 33JK

40M3,

97


where M ≡ M(R). This expression satisfies eq. (A.3.7), as it can be checked by directsubstitution; it may be noted that it differs from the expression of F (r) given in ref. [57] which,conversely, does not satisfy eq.(A.3.7).

In conclusion

χext1 =2δJ

r3+ F (r). (A.3.12)

At this point the situation is the following. We have integrated eqs. (A.3.4) for r ≤ R, andwe know the solution for u1 and χ1 up to the constant C (eq. A.3.5); we have found thesolution for r ≥ R for u1 (eq.A.3.10) and for χ1 (eq. A.3.12) up to the constant δJ . The twoconstants are found by imposing that the solutions for u1 and χ1 given by eqs. (A.3.5) and byeqs. (A.3.10,A.3.12) match continuously at r = R.

It should be noted that computing δJ by this procedure or by eq. (2.2.28) is equivalent; indeedwe find a relative difference smaller than 10−10. This is a further check of the correctness of theanalytic expression (A.3.11) for F (r) we derived.

A.3.2 The equations for w3

The solution for the remaining function w3 can be found along the same path. We start with theequation

1

r4

d

dr(r4j

dw3(r)

dr) +

4

r

dj

drw3 −

10jw3

r(r − 2M)=

1

5D2 (A.3.13)

and assume r ≤ R; we introduce the functions

χ3 = jw3 u3 = r4jdw3

dr(A.3.14)

and find

dχ3

dr=u3

r4− 4πr2(ε+ P )χ3

r − 2M, (A.3.15)

du3

dr=

16πr5(ε + P )χ3

r − 2M+

10χ3r3

(r − 2M)+r4

5D2.

The general solution of eqs. (A.3.15) for r ≤ R is

χ3 = χP3 + C χH3 (A.3.16)

u3 = uP3 + C uH3 .

C is a constant and (χP3 , u3) are a particular solution found by integrating eqs. (A.3.15) with theinitial conditions

χP3 (0) = 0, uP3 (0) = 0,

98


(χH3 , uH3 ) are the solution of the homogeneous system obtained by putting D2 = 0 in (A.3.15)

and integrating up to R with the boundary condition

r → 0 χH3 (r) ∼ r2, uH3 (r) ∼ 2 r5. (A.3.17)

For r ≥ R eqs. (A.3.15) reduce to

dχ3

dr=u3

r4, (A.3.18)

du3

dr=

10χ3r3

(r − 2M)+u

5

d

dr[4k2 −

6J2

r4].

The solution of this system is found as a linear combination of the solution χH3ext , uH3ext of the

homogenous system found by putting k2 = J = 0 in (A.3.18), and of a particular solution of(A.3.18), χP3ext, u

P3ext which falls to infinity faster than r−5. Both solutions have been calculated

by making use of MAPLE.

The homogeneous system admits two solutions, which, for r → ∞, behave as ∼ r−5 and ∼ r2.Since the metric must be asymptotically flat, we have excluded the latter.

Thus the general solution for r ≥ R is

χ3ext = DχH3ext + χP3ext, (A.3.19)

u3ext = DuH3ext + uP3ext,

and consequentlyw3ext = DwH3ext + wP3ext (A.3.20)

where

wH3ext(r) = (7

64r3M7)

[

−120r3M2ln(r

r − 2M) (A.3.21)

+150r4Mln(r

r − 2M) − 45r5ln(

r

r − 2M)

+20rM4 + 60r2M3 − 210r3M2 + 90r4M + 8M5]

99


and

wP3ext(r) = (J

480r7M9)[

(1152J2M9 + 700r5J2M4

+2100r6J2M3 − 7350r7J2M2 + 3150r8J2M

+280r4J2M5 + 1152Kr3M10 + 64J2rM8 − 320J2r2M7

+10080KM 5r8 + 1088KM 8r5 + 1664KM 9r4

−23520KM 6r7 + 6720KM 7r6 + 16800KM 5r8ln(r

r − 2M)

−4200r7J2M2ln(r

r − 2M) + 5250r8J2Mln(

r

r − 2M)

−5040KM 4r9ln(r

r − 2M) − 13440KM 6r7ln(

r

r − 2M)

+576KM7r6ln(r

r − 2M) − 960KM8r5ln(

r

r − 2M)

−384KM9r4ln(r

r − 2M) − 1575r9J2ln(

r

r − 2M)

]

. (A.3.22)

The constants C and D are found by matching the interior and the exterior solutions at r = R.

100

B

Appendix

B.1 Tensor spherical harmonics

Any tensor of rank n ≥ 2 can be expanded in tensor harmonics once the basis tensor harmonicsof the same rank has been defined. Since we are interested in the expansion of the perturbedmetric tensor and of the perturbed stress-energy tensor, we shall restrict our analysis to symmetrictensors, thus the independent components we need to consider are 6 in the euclidean 3-space, and10 in four the dimensional space. We shall follow the approach introduced by Regge and Wheeler(Regge, J.A. Wheeler, 1957), and subsequently developed by F. Zerilli, (F. Zerilli, 1970), and findthe basis tensor harmonics by applying to the spherical harmonics the following space-operators:

[∇∇ Y`m] , [LL Y`m] , [L∇ Y`m] ,

[erer Y`m] , [erL Y`m] , [er∇ Y`m] .

These harmonics can be used to construct 3 tensors orthogonal to the 2-sphere, and 3 tensorstangent to it

a`m = [erer Y`m] (B.1.1)

b`m = 212n(`)r [er∇ Y`m]

c`m = 212n(`) [erL Y`m]

d`m = 212m(`)r

[L∇ Y`m] +1

r[erL Y`m]

f`m = 2−12m(`) (e`m + h`m)

g`m = −2−12n(`)2 (e`m − h`m)

where

e`m = r2

[∇∇ Y`m] +2

r[er∇ Y`m]

(B.1.2)

h`m = [LL Y`m] + r [er∇ Y`m] ,

101

B.1. Tensor spherical harmonics

and

n(`) = [`(`+ 1)]−12 (B.1.3)

m(`) = [`(`+ 1)(`− 1)(`+ 2)]−12 .

These six tensors are independent and orthonormal in their inner product, and allow to expandany symmetric 2-tensor of the euclidean three-dimensional space.The extension to four dimension is accomplished as follows. We write a generic symmetrictensor as

T =

3∑

i,j=1

Tijei ⊗ ej +

3∑

i=1

T0ie0 ei + T00e

0 ⊗ e0 (B.1.4)

where indicates a symmetric tensor product, and eµ (µ = 0, 1, 2, 3) are the basis vectors inMinkowsky spacetime. Under a space rotation, the first term in the sum transforms like D(1) ⊗D(1), i.e. like the product of two vectors, i.e. like a 3 × 3 tensor, which can be expanded in thebasis defined in eqs. (B.1.2). The second term transforms as D(0) ⊗ D(1), i.e. as a vector, andtherefore can be expanded as e0 ×(vector harmonics), and finally the last term transformas D(0) ⊗ D(0), and can be expanded as e0 ⊗ e0 Yjm. Thus, in addition to the six tensorharmonics defined in eq. (B.1.2), which are extended to four dimensions by adding a zero as atime component, four further harmonics have to be introduced,

[etet Y`m] [eter Y`m][et∇ Y`m] [etL Y`m]

and the following set of tensor harmonics have to be added to (B.1.2)

a(0)`m = a

(0),Z`m = [etet Y`m] (B.1.5)

a(1)`m = −ı a(1),Z

`m = 212 [eter Y`m]

b(0)`m = −ı b(0),Z

`m = 212n(`)r [et∇ Y`m]

c(0)`m = ı c

(0),Z`m = 2

12n(`) [etL Y`m]

This means that any symmetric tensor T can be expanded in tensor harmonics:

T =∑

`m

[

A(0)`ma

(0)`m + A

(1)`ma

(1)`m + A`ma`m+

+B(0)`mb

(0)`m +B`mb`m +Q

(0)`mc

(0)`m + (B.1.6)

+Q`mc`m +G`mg`m +D`md`m + F`mf`m] ,

and the coefficients of the expansion are found by taking the inner product of the tensor and thecorresponding harmonic. If the spacetime is not flat, but a curved manifold V4, at each point

102

Appendix B. Appendix

(xα) ∈ V4 we can set a basis of four vectors, a tetrad, tangent to the manifold at P , i.e. a locallyinertial frame. In this frame the spacetime is flat, and it can be split in time+3-space and thedecomposition in harmonics continues to hold. In this case all our previous definition have alocal character.

B.1.1 Parity

The tensor harmonicsc`m, d`m and c

(0)`m = ı c

(0),Z`m (B.1.7)

have parity (−)`+1, i.e. are odd or axial. The remaining seven harmonics have parity (−)`, i.e.are even, or polar.Consequently, a tensor T can be written as

T = Tax + Tpol, (B.1.8)

whereTax =

∑

`m

[

Q(0)`mc

(0)`m + +Q`mc`m +D`md`m

]

.

Tpol =∑

`m

[

A(0)`ma

(0)`m + A

(1)`ma

(1)`m + A`ma`m+

+B(0)`mb

(0)`m +B`mb`m +G`mg`m + F`mf`m

]

, (B.1.9)

and the explicit form of the tensor harmonics is given by

a(0)`m =

(t) (ϕ) (r) (ϑ)Y`m(ϕ, ϑ) 0 0 0

0 0 0 00 0 0 00 0 0 0

(B.1.10)

a(1)`m = −ı a(1),Z

`m =1√2

(t) (ϕ) (r) (ϑ)0 0 Y`m(ϕ, ϑ) 00 0 0 0

Y`m(ϕ, ϑ) 0 0 00 0 0 0

(B.1.11)

103


a`m =

(t) (ϕ) (r) (ϑ)0 0 0 00 0 0 00 0 Y`m(ϕ, ϑ) 00 0 0 0

(B.1.12)

b(0)`m = −ı b(0),Z

`m =n(`)r√

2

(t) (ϕ) (r) (ϑ)

0 ∂Y`m∂ϕ

0 ∂Y`m∂ϑ

∂Y`m∂ϕ

0 0 0

0 0 0 0∂Y`m∂ϑ

0 0 0

(B.1.13)

b`m =n(`)r√

2

(t) (ϕ) (r) (ϑ)0 0 0 0

0 0 ∂Y`m∂ϕ

0

0 ∂Y`m∂ϕ

0 ∂Y`m∂ϑ

0 0 ∂Y`m∂ϑ

0

(B.1.14)

c(0)`m = ı c

(0),Z`m =

ın(`)r√2

(t) (ϕ) (r) (ϑ)

0 − sinϑ∂Y`m∂ϑ

0 1sinϑ

∂Y`m∂ϕ

− sin ϑ∂Y`m∂ϑ

0 0 00 0 0 0

1sinϑ

∂Y`m∂ϕ

0 0 0

(B.1.15)

c`m =ın(`)r√

2

(t) (ϕ) (r) (ϑ)0 0 0 0

0 0 − sin ϑ∂Y`m∂ϑ

0

0 − sinϑ∂Y`m∂ϑ

0 1sinϑ

∂Y`m∂ϕ

0 0 1sinϑ

∂Y`m∂ϕ

0

(B.1.16)

d`m =ım(`)r2

√2

(t) (ϕ) (r) (ϑ)0 0 0 00 − sinϑX`m 0 − sinϑW`m

0 0 0 00 − sin ϑW`m 0 1

sinϑX`m

(B.1.17)

104

Appendix B. Appendix

f`m =m(`)r2

√2

(t) (ϕ) (r) (ϑ)0 0 0 00 − sin2 ϑW`m 0 X`m

0 0 0 00 X`m 0 W`m

(B.1.18)

g`m =r2

√2

(t) (ϕ) (r) (ϑ)0 0 0 00 sin2 ϑY`m 0 00 0 0 00 0 0 Y`m

(B.1.19)

n(`) and m(`) are given in eqs. (B.1.3),and

X`m(ϑ, ϕ) = 2∂

∂ϕ

[

∂

∂ϑ− cotϑ

]

Y`m(ϑ, ϕ) (B.1.20)

W`m(ϑ, ϕ) =

[

∂2

∂2ϑ− cotϑ

∂

∂ϑ− 1

sin2 ϑ

∂2

∂2ϕ

]

Y`m(ϑ, ϕ). (B.1.21)

105


106

C

Appendix

In this Appendix we denote with A,B, . . . the coordinates t, r, and with a, b, . . . the coordinatesθ, φ; we define γab ≡ diag(1, sin2 θ).

C.1 The generalized Regge-Wheeler gauge: a possible gaugechoice

There are many possible choices for the gauge of the perturbations of a stationary, axisymmetricspacetime (see for instance [115]). Here we describe a particular gauge choice which has theproperty to reduce, when the background becomes spherically symmetric, to the well-knownRegge-Wheeler gauge [84]. As discussed in Section 4.2, we assume a background metric in thecoordinates (t, r, θ, φ) like (4.2.2) or (4.2.3), with ∂

∂t, ∂∂φ

Killing vectors. The metric perturbationshave the form

hµν(t, r, θ, φ) = hmωµν (r, θ)eimφe−iωt . (C.1.1)

We shall show that it is possible to fix the gauge in such a way that the metric perturbationshmω

µν (r, θ) takes the form

hmωµν (r, θ) =

eνHmω0 (r, θ) Hmω

1 (r, θ) − imsin θ

hmω0 (r, θ) sin θhmω

0,θ (r, θ). . . eλHmω

2 (r, θ) − imsin θ

hmω1 (r, θ) sin θhmω

1,θ (r, θ)

. . . . . . Kmω(r, θ)r2 0

. . . . . . . . . Kmω(r, θ)r2 sin2 θ

, (C.1.2)

and depends on the six quantities

[Hmω0 (r, θ), Hmω

1 (r, θ), Hmω2 (r, θ), Kmω(r, θ), hmω

0 (r, θ), hmω1 (r, θ)] (C.1.3)

These quantities behave as scalars with respect to rotations. In order to fix the gauge (C.1.2) weimpose the following conditions

hmωab (r, θ) ∝ γab (C.1.4)∫ π

0

dθ sin θ

[

Y lm,θ (θ, 0)hmω

Aθ (r, θ) − im

sin2 θY lm(θ, 0)hmω

Aφ (r, θ)

]

= 0 ∀l . (C.1.5)

107

C.1. The generalized Regge-Wheeler gauge: a possible gauge choice

These conditions correspond to setting to zero four functions of r, θ:

hmωθφ (r, θ) = 0

hmωθθ (r, θ) − 1

sin2 θhmωφφ (r, θ) = 0

∞∑

l=|m|

∫ π

0

dθ sin θ

[

Y lm,θ (θ, 0)hmω

tθ (r, θ) − im


tφ (r, θ)

]

= 0

∞∑

l=|m|

∫ π

0

dθ sin θ

[

Y lm,θ (θ, 0)hmω

rθ (r, θ) − im


rφ (r, θ)

]

= 0 , (C.1.6)

and can be impossed through a diffeomorphism generated by the vector field

ξµ(t, r, θ, φ) = ξmωµ (r, θ)eimφe−iωt (C.1.7)

which depends on four functions of r, θ.The relation of (C.1.2) with the condition (C.1.4) is trivial, but to show that (C.1.5) implies(C.1.2) is less obvious. If we integrate by parts (C.1.5), we find

∫ π

0

dθY lm(θ, 0)

[

− (sin θhmωAθ (r, θ)),θ +

im

sin θhmωAφ (r, θ)

]

= 0 . (C.1.8)

As it holds for all l’s, the square brackets is identically, i.e.

(sin θhmωAθ (r, θ)),θ +

im

sin θhmωAφ (r, θ) ≡ 0, (C.1.9)

therefore we can express the four quantities (hmωAθ (r, θ), hmω

Aφ (r, θ)) in terms of two scalarfunctions hmω

A (r, θ) such that:

hmωAθ (r, θ) = − im

sin θhmωA (r, θ)

hmωAφ (r, θ) = sin θhmω

A (r, θ),θ . (C.1.10)

This, togheter with (C.1.4), gives (C.1.2).This gauge choice is implicit in the formulation used in [100] to describe the perturbations ofslowly rotating stars. It can, in principle, also be chosen to describe perturbations of rapidlyrotating stars.We stress that the existence of the generalized Regge-Wheeler gauge, in which all perturbationsare expressed in terms of functions that are scalar with respect the rotation, is important becauseit provides a solid bases to the approach described in this thesis. Indeed, it guarantees thatexpanding the perturbations in tensorial spherical harmonics is equivalent to expand in circularharmonics eimφ firstly, and then to expand the functions appearing in the resulting equations in(r, θ), in associate Legendre polynomials, P lm(θ).

108

Appendix C. Appendix

C.1.1 Relations with the Regge-Wheeler gauge

In order to better understand how the gauge (C.1.2) is related to the Regge-Wheeler (RW)gauge, we now expand hµν(t, r, θ, φ) in tensor spherical harmonics. This is always possible (ona surface t =const., r =const.), but typically it is not useful if the background is non-spherical,since the dynamical equations couple perturbations with different l’s. Anyway for slowlyrotating stars the couplings are small, and the spherical harmonics expansion, as described inAppendix D.1 and in [100], turns out to be useful.By expanding the perturbed metric tensor in tensor spherical harmonics, before any gaugefixing, we find

hµν(t, r, θ, φ) = hmωµν (r, θ)eimφe−iωt =

∑

l

(

eνH lm0

(r)Y lm(θ, φ) H lm1

(r)Y lm(θ, φ) hlm0,pol(r)Y

lm,a (θ, φ) + hlm

0(r)Slm

a (θ, φ)

. . . eλH lm2

(r)Y lm(θ, φ) hlm1,pol(r)Y

lm,a (θ, φ) + hlm

1(r)Slm

a (θ, φ)

. . . . . . Klm(r)r2γabYlm(θ, φ) + Glm(r)Zlm

ab (θ, φ) + hlmax (r)Slm

ab (θ, φ)

)

e−iωt

(C.1.11)

where

Slma (θ, φ) = (S lmθ , Slmφ ) =

(

− 1

sin θY lm,φ , sin θY

lm,θ

)

(C.1.12)

are the axial vector harmonics, and Zab and Sab are tensor harmonics satisfying γabZab =γabSab = 0, with polar and axial parity, respectively.The RW gauge for a spherical background imposes [84]:

hlm0,pol = hlm1,pol = hlmax = Glm = 0 . (C.1.13)

If we consider (C.1.13) in the case of a non-spherical background, we see that it is equivalent tothe gauge (C.1.2). Indeed, expanding hAa(t, r, θ, φ) in vector spherical harmonics, we find

hmωAa (r, θ)eimφ =

∑

l

[

hlmApol(r)Ylm,a (θ, φ) + hlmAax(r)S

lma (θ, φ)

]

, (C.1.14)

which if hlmApol = 0, reduces to

hmωAa (r, θ)eimφ =

∑

l

hlmAax(r)Slma (θ, φ) =

∑

l

hlmAax(r)

(

− 1

sin θY lm,φ , sin θY

lm,θ

)

=

(

− im

sin θhA, sin θhA,θ

)

eimφ, (C.1.15)

where we have defined the scalar functions

hA(r, θ) ≡∑

l

hlmAax(r)Ylm(θ, 0) . (C.1.16)

109

C.1. The generalized Regge-Wheeler gauge: a possible gauge choice

110

D

Appendix

D.1 Equations for the perturbations of slowly rotating stars

This derivation of the equations describing the perturbations of a slowly rotating star is based onthe work of Kojima [100], denoted as K1. As discussed above, since we are considering slowlyrotating stars, we can first expand the perturbations in spherical harmonics, getting a system ofcoupled ODE in r, and then expand this system in Chebyshev polynomials, getting an algebraicmatrix equation. The first step of this program is equivalent to the derivation of K1, with onedifference: we are going to reformulate the equations in terms of a different set of variables,which are numerically well behaved near the center of the star. Following K1, we shall assumel ≥ 2.The background configuration, describing a slowly rotating, stationary and axially symmetric staris given by equation (5.0.1). The pressure p and the energy density ρ are found by solving theTOV equations; we assume that the equation of state of matter in the star is barotropic, p = p(ρ);

therefore, c2s ≡(

∂p∂ρ

)

= p′

ρ′. The perturbations of the background (5.0.1) can be written as

hµν =∑

lm

eνH lm0 (r)Y lm(θ,φ)H lm

1 (r)Y lm(θ,φ) hlm0 (r)Slmθ (θ,φ) hlm0 (r)Slmφ (θ,φ). . . eλH lm

2 (r)Y lm(θ,φ) hlm1 (r)Slmθ (θ,φ) hlm1 (r)Slmφ (θ,φ)

. . . . . . K lm(r)r2Y lm(θ,φ) 0

. . . . . . . . . K lm(r)r2 sin2 θY lm(θ,φ)

e−iσt

δur =eν/2−λ

4π(ρ+ p)Rlm(r)Y lm(θ,φ)e−iσt

δuθ =eν/2−λ

4π(ρ+ p)

(

V lm(r)Y lm,θ (θ,φ) + U lm(r)Slmθ (θ,φ)

)

e−iσt

δuφ =eν/2−λ

4π(ρ+ p)

(

V lm(r)Y lm,φ (θ,φ) + U lm(r)Slmφ (θ,φ)

)

e−iσt

δρ = δρlm(r)Y lm(θ,φ)e−iσt

δp = δplm(r)Y lm(θ,φ)e−iσt , (D.1.1)

111

D.1. Equations for the perturbations of slowly rotating stars

with σ real. The linearized Einstein equations for the radial part of these quantities are ordinarydifferential equations in r. As explained in Section 5, the general structure of these equations is:

Lpol[H lm0 , K lm; σ] = mE [H lm

0 , K lm; σ] + F (±)[Z l±1mRW ; σ]

Lax[Z lmRW ; σ] = mN [Z lm

RW ; σ] + D(±)[H l±1m0 , K l±1m; σ] . (D.1.2)

The quantities H lm1 , H lm

2 , Rlm, V lm, U lm, δρlm, δplm can be expressed in terms ofH lm

0 , K lm, Z lmRW , once equations (D.1.2) have been solved.

D.1.1 The O(ε0) equations

The equations at O(ε0) describe the perturbations of a non-rotating star. The equations for polarperturbations inside the star are a system of two second order ODE in K lm, H lm

0 :

L(1)lmint [H0, K] ≡ (K lm −H lm

0 )′′ − eλ

r2[2r − 10M + 4π(ρ− 5p)r3](K lm −H lm

0 )′

+eλ

r2(σ2e−νr2 − 2n)(K lm −H lm

0 )

+4eλ

r4[3Mr − 4πρr4 − eλ(M + 4πpr3)2]H lm

0 = 0 (D.1.3)

L(2)lmint [H0, K] ≡ K lm ′′ − eλ

r2[(r − 3M − 4πpr3)c−2

s − 3r + 5M + 4πρr3]K lm ′

+eλ

r2[σ2e−νr2c−2

s − n(c−2s + 1)]K lm +

c−2s − 1

rH lm ′

0

+eλ

r3[(nr + 4M + 8πpr3)c−2

s − (n+ 2)r + 8πρr3]H lm0 = 0 . (D.1.4)

Once H lm0 , K lm have been determined, H lm

1 and H lm2 can be computed through the following

relations:

H lm1 = −e

ν

iσ

(

H lm ′

0 −K lm ′

+2eλ

r2(M + 4πpr3)H lm

0

)

H lm2 = H lm

0 . (D.1.5)

Other relations (see K1) give the fluid perturbations Rlm, V lm, U lm, δρlm, δplm in terms of themetric perturbations. Outside the star, the polar perturbations reduce to a system of three first

112

Appendix D. Appendix

order ODE in K lm, H lm1 and H lm

0 :

L(1)lmext [H0, H1, K] ≡ K lm ′

+eλ(r − 3M)

r2K lm − 1

rH lm

0 − n + 1

rH lm

1 = 0 (D.1.6)

L(2)lmext [H0, H1, K] ≡ −rH lm ′

1 + eλ(K lm +H lm0 ) − 2Meλ

rH lm

1 = 0 (D.1.7)

L(3)lmext [H0, H1, K] ≡ H lm ′

0 +eλ(r − 3M)

r2K lm − eλ(r − 4M)

r2H lm

0

+

(

σ2reλ − n+ 1

r

)

H lm1 = 0 , (D.1.8)

where we have defined

H lm1 ≡ −H

lm1

iσr(D.1.9)

in order to have equations with real coefficients. Furthermore, there is an algebraic constraint:

L(4)lmext [H0, H1, K] ≡ (σ2r4eλ − nr2 −Mr +M2eλ)K lm

+(nr + 3M)rH lm0 − [σ2r4 − (n+ 1)Mr]H lm

1 = 0 .

(D.1.10)

Equations (D.1.6)-(D.1.8) are equivalent to the Zerilli equation, but they have the advantage tobe easily generalizable to rotating stars, as we will see below. The Zerilli function ZZer can becomputed in terms of K, H1:

Z lmZer =

r2

nr + 3M(K lm − eνH lm

1 ) . (D.1.11)

The axial perturbations are described by the Regge-Wheeler equation

Llm[ZRW ] ≡ d2

dr2∗

Z lmRW +

[

σ2 − eν(

l(l + 1)

r2− 6M

r3+ 4π(ρ− p)

)]

Z lmRW = 0 , (D.1.12)

where the coordinate r∗ has been defined in (3.1.7) and

hlm0 = −e(ν−λ)/2

iσ

(

Z lmRW r

)′

hlm1 = e(λ−ν)/2Z lmRWr . (D.1.13)

113


An analytical expansion of equations (D.1.3), (D.1.4), (D.1.12) near the center of the star gives,for each value of l, three independent conditions at the center:

K lm −H lm0 = rl+2 + khlm(+)rl+4

K lm = rl + klm(+)rl+2

Z lmRW = 0 (D.1.14)

K lm −H lm0 = rl+2 + khlm(−)rl+4

K lm = −rl + klm(−)rl+2

Z lmRW = 0 (D.1.15)

K lm −H lm0 = 0

K lm = 0

Z lmRW = rl+1 + zlmrl+3 (D.1.16)

where the expressions for khlm(±), klm(±), zlm can be evaluated from the analytical expansion.We notice thatK lm andH lm

0 behave, as r → 0, like rl, while the combinationK lm−H lm0 behaves

as rl+2. Consequently, when we expand K lm and H lm0 in powers of r about r = 0, we find that

the leading terms are coincident. In other words, the differential equations for the variables K lm,H lm

0 and Z lmRW are linearly dependent near the origin (see the discussion in [79]). To avoid this

problem, we use as integration variables K lm − H lm0 , K lm and Z lm

RW . Finally, we notice that atorder O(ε0), i.e. for a non rotating star, equation (D.1.5) establishes that H lm

2 = H lm0 . Therefore

it is equivalent to use either H lm0 or H lm

2 .

D.1.2 The O(ε1) equations

We definen+ ≡ l(l+3)

2n− ≡ (l−2)(l+1)

2

Ql−1m ≡√

(l−m)(l+m)(2l−1)(2l+1)

Ql+1m ≡√

(l+1−m)(l+1+m)(2l+1)(2l+3)

.

(D.1.17)

The perturbed equations inside the star have the form

L(J)lmint [H0, K] = − m

(n + 1)σE(J)lm[H0, K]

+e−(λ+ν)/2

[

iQl−1mD(J)l−1m[ZRW ]

σ(n− n−)+

iQl+1mD(J)l+1m[ZRW ]

σ(n− n+)

]

Llm[ZRW ] =m

σN lm[ZRW ] + e(λ+3ν)/2

[

iQl−1mFl−1m[H0, K]

σ(n− n−)+

iQl+1mFl+1m[H0, K]

σ(n− n+)

]

(D.1.18)

114


(J = 1, 2), where L(J)lmint [H0, K], Llm[ZRW ] are the operators defined in equations (D.1.3),

(D.1.4), (D.1.12) and E(J)lm[H0, K], D(J)l±1m[ZRW ], N lm[ZRW ], F l±1m[H0, K] are operatorsat first order in ε, whose explicit expressions are given in K1. The operators D(J)l±1m[ZRW ],F l±1m[H0, K] couple perturbations belonging to different l’s, and were neglected in thenumerical integration of K2. In order to have equations with real coefficients, we need to getrid of the factors i in (D.1.18). To this purpose, we redefine the Regge-Wheeler function by afactor −i

Z lmRW ⇒ −iZ lm

RW , (D.1.19)

thus equations (D.1.18) become


(n + 1)σE(J)lm[H0, K]

+e−(λ+ν)/2

[

Ql−1mD(J)l−1m[ZRW ]

σ(n− n−)+Ql+1mD

(J)l+1m[ZRW ]

σ(n− n+)

]

Llm[ZRW ] =m

σN lm[ZRW ] − e(λ+3ν)/2

[

Ql−1mFl−1m[H0, K]

σ(n− n−)+Ql+1mF

l+1m[H0, K]

σ(n− n+)

]

.(D.1.20)

This rescaling consistently eliminates all imaginary units from the equations. The numericalintegration of equations (D.1.20) presents a serious problem. If we perform an analyticalexpansion near the center of (D.1.20), we find that the coupling terms D(J)l−1m[Z lm

RW ] becomelarger than L(J)lm

int [H lm0 , K lm] as r → 0. The reason behind this pathological behaviour is that

near the center of the star

K lm −H lm0 ∼ rl+2 + ( O(ε) terms ) · rl

K lm ∼ rl

H lm0 ∼ rl . (D.1.21)

If we use as integration variable H lm2 instead of H lm

0 (notice that while in the non rotating caseH lm

0 = H lm2 , if the star rotates H lm

0 = H lm2 +O(ε)) this problem is overcome, since

K lm −H lm2 ∼ rl+2 . (D.1.22)

Consequently, the coupling terms in the perturbed equations are smaller than L(J)lmint [H lm

2 , K lm].For this reason, we have expressed our equations inside the star in terms of K lm − H lm

2 , K lm,Z lmRW :


(n + 1)σE(J)lm[H2, K]

+e−(λ+ν)/2

[

Ql−1mD(J)l−1m[ZRW ]

σ(n− n−)+Ql+1mD

(J)l+1m[ZRW ]

σ(n− n+)

]

Llm[ZRW ] =m

σN lm[ZRW ] − e(λ+3ν)/2

[

Ql−1mFl−1m[H2, K]

σ(n− n−)+Ql+1mF

l+1m[H2, K]

σ(n− n+)

]

.(D.1.23)

115


The operators E(J)lm, D(J)l±1m are different from E(J)lm, D(J)l±1m given in K1. Theirexpressions are the following:

E(1)lm[H2, K] =iσ

2

[

2f lm ′′(r) +

(

4ν ′ − λ′ − 6

r

)

f lm ′(r)

−(

2eλ

r2(n + 1) − (ν ′)2 +

4

r

(

2ν ′ − λ′ − 2

r

)

− 32πpeλ)

f lm(r)

]

+iσ(

2nξ(1)lm ′(r) − β(1)lm ′(r) − ζ (1)lm ′(r))

− iσ

2

(

2

r+ λ′ − 2ν ′

)

(

2nξ(1)lm(r) − β(1)lm(r) − ζ (1)lm(r))

+iσ(n+ 1)eλC(3)lm(r) − iσ(n+ 1)C(2)lm(r) (D.1.24)

E(2)lm[H2, K] = − iσ

rc−2s

[

f lm ′(r) +

(

ν ′ − 2

r+ (n+ 1)

eλ

r

)

f lm(r)

]

− iσ

rc−2s

(

2nξ(1)lm(r) − β(1)lm(r) − ζ (1)lm(r))

− iσ

2(n+ 1)c−2

s C(2)lm(r) +iσ

2(n+ 1)eλ−νC(0)lm(r) (D.1.25)

D(1)l±1m[ZRW ] = iσe(λ+ν)/2

[

2gl±1m ′′(r) +

(

4ν ′ − λ′ − 6

r

)

gl±1m ′(r)

−(

2eλ

r2(n + 1) − (ν ′)2 +

4

r

(

2ν ′ − λ′ − 2

r

)

− 32πpeλ)

gl±1m(r)

]

+2iσe(λ+ν)/2[

−2(n− 2n± − 2)χ(1)l±1m ′(r) + (n− n± − 1)α(1)l±1m ′(r) − η(1)l±1m ′(r)]

−iσe(λ+ν)/2

(

2

r+ λ′ − 2ν ′

)

[

−2(n− 2n± − 2)χ(1)l±1m(r)

+(n− n± − 1)α(1)l±1m(r) − η(1)l±1m(r)]

+iσe(3λ+ν)/2[

(n− n±)A(3)l±1m(r) + (n− n±)(n− n± − 1)B(3)l±1m(r)]

−iσe(λ+ν)/2[

(n− n±)A(2)l±1m(r) + (n− n±)(n− n± − 1)B(2)l±1m(r)]

(D.1.26)

D(2)l±1m[ZRW ] = −iσ2e(λ+ν)/2

rc−2s

[

glm ′(r) +

(

ν ′ − 2

r+ (n + 1)

eλ

r

)

glm(r)

]

−iσ2e(λ+ν)/2

rc−2s

[

−2(n− 2n± − 2)χ(1)l±1m(r)

+(n− n± − 1)α(1)l±1m(r) − η(1)l±1m(r)]

−iσe(λ+ν)/2

2c−2s

[

(n− n±)A(2)l±1m(r) + (n− n±)(n− n± − 1)B(2)l±1m(r)]

+iσe(3λ−ν)/2

2

[

(n− n±)A(0)l±1m(r) + (n− n±)(n− n± − 1)B(0)l±1m(r)]

(D.1.27)

116


where f, g, ξ(J), α(J), β(J), η(J), ζ(J), C(I), A(I), B(I) are quantities which depend on the pertur-bations H lm

2 , K lm, etc., and which are given in Appendix B of K1. At the surface of the star wecompute H lm

0 and the other perturbations in terms of H lm2 , K lm, Z lm

RW . We impose the vanishingof the Lagrangian pressure perturbation (see [102]). This reduces the number of freely assignedconstants from three (times L − |m| + 1), which correspond to the three independent solutions(D.1.14)-(D.1.16), to two (times L−|m|+1), i.e. N = 2 as discussed in Section 4.2.1.2. Finally,the equations in vacuum are, as in K1,

L(J)lmext [H0, H1, K] =

ω

σ

(

mE(J)lm[H0, H1, K] +Ql−1mD

(J)l−1m[ZRW ]

n− n−

+Ql+1mD

(J)l+1m[ZRW ]

n− n+

)

(D.1.28)

Llm[ZRW ] =ω

σ

(

mN lm[ZRW ] − Ql−1mFl−1m[H0, K]

n− n−

− Ql+1mFl+1m[H0, K]

n− n+

)

(D.1.29)

(J = 1, . . . , 4) where L(J)lmext [H0, H1, K], Llm[ZRW ] are the operators defined in (D.1.3)-(D.1.10),

(D.1.12), and the expressions of E(J)lm[H0, K], D(J)lm[ZRW ], N (J)lm[ZRW ], F (J)lm[H0, K] aregiven in K1. When r R, the background spacetime is with good approximation sphericallysymmetric, because the terms due to rotation decrease faster than the “Schwarzschild-like”components (see for instance [72], Chap. 19). Therefore, spacetime perturbations satisfy theZerilli and the Regge-Wheeler equations. In this limit, equation (D.1.29) becomes the Regge-Wheeler equation for the function Z lm

RW , whereas the Zerilli function Z lmZer is related to the

solution of equation (D.1.28) by (D.1.11). At radial infinity, the amplitude of the stationary wave(

AlmZer in(σ), AlmRW in(σ))

can be computed in terms of Z lmZer and Z lm

RW . We can then apply thestationary wave approach described in Section 4.2.1 and in Chapter 4, section 4.2.2.4. Equations(D.1.23), (D.1.28), (D.1.29) can be integrated using the spectral decomposition in Chebyshev’spolynomials as explained in Section 4.2.2. There is a main difference with respect to the exampledescribed in Chapter 4, section 4.2.2.4, which refers to the axial equation for a non-rotating star.While the matrix (4.2.74) is block-diagonal – each block corresponding to a value of l – thematrix representing equations (D.1.23), (D.1.28), (D.1.29) presents, in addition to the block-diagonal terms, components of order O(ε), which couple l ↔ l ± 1. All equations in this paperhave been checked using Maple, and we have made several cross checks in order to be sure thatthe Fortran implementation of the long expressions (D.1.24)-(D.1.27) are correct.

117

Index of the figures

118

List of Figures

1-1 A representative model showing the possible internal structures of an NS. . . . . 10

1-2 Measured and estimated masses of neutron stars in radio binary pulsars (orange,grey and blue regions) and in X-ray accreting binaries (green). For each region,simple averages are shown as dotted lines, weighted averages are shown asdashed lines. Data and figure are from [18]. . . . . . . . . . . . . . . . . . . . . 13

1-3 NS mass versus radius for the models of EOS described in the text. The twohorizontal lines denote the boundaries of the region of observationally allowedNS masses, while the straight red line corresponds to the mass-radius relationextracted from the redshift measurements of Cottam et al. [30]. Data and figureare from [23]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2-1 The gravitational mass for the EOS APR2 is plotted as a function of the centraldensity for different values of the spin frequency, highlighting the curves withMbar = const. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2-2 The mass-shedding frequency νms is plotted for each EOS as a function of thegravitational mass. The horizontal line corresponds to the spin frequency of PSR1937+21, which is the fastest isolated pulsar observed so far and whose mass ispresently unknown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2-3 The energy density distribution inside the star is plotted as a function of the radialdistance for the EOS APR2 and BBS2, and for Mbar = 1.31M. ε is given inunits of ε∗ = 1015 g/cm3. In both cases the continuous line refers to the non-rotating configuration, the dotted line to the star rotating at mass-shedding. . . . . 43

2-4 The moment of inertia (in units of I∗ = 1045g cm2) is plotted as a function ofthe spin frequency, for the EOS APR2 (panel a), BBS1 (panel b), BBS2 (panelc) and G240 (panel d). For each EOS we consider a few values of the baryonicmass up to the maximum mass, and plot the data only for ν ≤ 80% νms (seetext). The vertical dashed line is the spin frequency of PSR 1937+21. . . . . . . 44

119


2-5 The moment of inertia, measured in units of I∗ = 1045g cm2, is plotted versusthe spin frequency (in kHz) for stars with baryonic mass Mbar = 1.56M.The continuous lines refer to I that is computed by solving the stellar structureequations to the third order in the angular velocity, whereas the horizontaldashed lines refer to the first order term I (0). The vertical dashed line is thespin frequency of PSR 1937+21. For each EOS, the data are plotted only fo rν ≤ 80% νms (see text). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3-1 The Virgo sensitivity curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3-2 The frequency of the fundamental mode is plotted as a function of the mass ofthe star, for neutron/hybrid stars described by the EOS employed in [23] andindicated with the same labels (APR2, APRB200, APRB120, BBS1, G240) andfor strange stars. The shaded region covers the range of parameters of the MITbag model as considered in this thesis, i.e. αs ∈ [0.4, 0.6], ms ∈ [80, 155] MeVand B ∈ [57, 95] MeV/fm3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3-3 The damping time of the fundamental mode is plotted as a function of the massof the star, as in Fig. 1, for neutron/hybrid stars and for strange stars (shadedregion). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3-4 The fundamental mode frequency νf is plotted versus the gravitational mass,M , for different values of the bag constant and αs and ms varying in the rangeindicated by Eqs. (1.4.3) and (1.4.4). . . . . . . . . . . . . . . . . . . . . . . . 60

3-5 νf is plotted versus the radiation radius, R∞, for different values of the bagconstant and αs andms varying in the range indicated by Eqs. (1.4.3) and (1.4.4).The continuous lines refer to the values of νf for the neutron/hybrid star modelsconsidered in [23]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5-1 The real part of the f -mode frequency of a non rotating, polytropic star withn = 1, is plotted as a function of the stellar compactness M/R (a); the frequencyshift due to rotation, σ′

R is plotted versus compactness for different values of ε(b). As in K2, couplings among different l’s are neglected. . . . . . . . . . . . . 83

5-2 The imaginary part of the f -mode frequency (a) and the corrections due torotation (b) are plotted as in Figure 5-1. . . . . . . . . . . . . . . . . . . . . . . 83

5-3 The real part of the f -mode frequency, νf = σRf /(2π), is plotted versus therotation parameter ε. The data refer to two models of constant density star (seetext). We include couplings up to l = 4. The dashed line refers to the non rotatingstar; the dotted lines refer to the modes m = ±2 of the slowly rotating star. . . . 85

120


5-4 Details of left panel of Figure 5-3: we show the contribution of the differentcouplings to the f -mode frequency, for model A. The different curves areobtained by including in the perturbed equations couplings up to l = L. . . . . . 85

5-5 Frequency shift as given by equation (5.2.1) for model A. . . . . . . . . . . . . . 86

5-6 The coefficients σ′, σ′′, plotted as functions of the rotation rate for model A. . . . 86

5-7 Damping time of the f -mode as a function of the rotation rate, for model A. . . . 87

121


122

List of Tables

1-1 The composition of different EOSs refers only to the strongly interactingcomponents (neutron, proton, hyperon and quark). For further details, see[23, 24] and references therein. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2-1 The baryonic masses of the stellar models used to compare the results ofthe perturbative approach with those found by integrating the exact Einsteinequations are given for the three considered EOSs. The data in column 2 and3 correspond to a non-rotating star with gravitational mass M = 1.4M andMmax, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2-2 Stellar parameters for sequence A. Data are computed along a sequence ofstellar models with constant baryonic mass which corresponds to a non-rotatingconfiguration of massM = 1.4M and varying spin frequency ν = Ω/2π (givenin kHz in column 2). εc is the central density in units of ε∗ = 1015g/cm3; themoment of inertia I is in units of I∗ = 1045g cm2, the equatorial radius Req isin km. For each value of ν and for each quantity, we give the relative error ∆.Mgrav is the gravitational mass in solar mass units. In the last column the ratioof the spin frequency and the mass-shedding frequency (as computed in [68] and[70]) is given. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2-3 Stellar parameters for sequence A are given as in Table 2-2. Data are computedalong a sequence of stellar models with constant baryonic mass which corre-sponds to a non-rotating configuration of mass M = 1.4M and varying spinfrequency ν = Ω/2π (given in kHz). . . . . . . . . . . . . . . . . . . . . . . . . 33

2-4 Stellar parameters for sequence A are given as in Table 2-2. Data are computedalong a sequence of stellar models with constant baryonic mass which corre-sponds to a non-rotating configuration of mass M = 1.4M and varying spinfrequency ν = Ω/2π (given in kHz). . . . . . . . . . . . . . . . . . . . . . . . . 33

2-5 The stellar parameters for sequence B, which correspond to maximum mass (seetext) are given as in table 2-2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

123

Index of the tables



2-8 The mass-shedding spin frequencies are given for sequences A and B (see text)and for the considered EOSs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2-9 For each value of the baryonic mass in column 1, the corresponding values of thegravitational mass of the non-rotating configuration are tabulated for the selectedEOSs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2-10 Maximum masses of the non-rotating configurations. . . . . . . . . . . . . . . . 41

2-11 For each EOS and for three values of the baryonic mass, we tabulate the ratioof the gravitational mass of the star rotating at mass-shedding to the equatorialradius, Mms

grav/Rmseq . In the last three columns, we give the ratio M/R for the

non-rotating star with the same baryonic mass. . . . . . . . . . . . . . . . . . . . 42

2-12 We compare the moment of inertia computed for the EOS APR2 by integratingthe exact equations of stellar structure (column 2) [70], the perturbed equationsto order Ω (column 3), and to order Ω3 (column 5). The relative error bewteenthe exact and the approximated results are given in columns 4 and 6. I is in unitsof I∗ = 1045g cm2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3-1 The frequencies and damping times of the polar QNM for l=2 are tabulated for1.4M and for different EOSs. Data are from [23]. . . . . . . . . . . . . . . . . 56

5-1 Parameters of the constant density stellar models A and B we use as abackground. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

124

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