diferentes abordagens à composição e ao ambiente das estrelas … · 2013. 3. 12. ·...

Universidade de Sao Paulo

Instituto de Astronomia, Geofısica e Ciencias Atmosfericas

Departamento de Astronomia

Marcio Guilherme Bronzato de Avellar

Diferentes abordagens a Composicao e ao

Ambiente das Estrelas de Neutrons

Different approaches to the Composition of Neutron Stars and their

Environment

Sao Paulo

2012

Marcio Guilherme Bronzato de Avellar

Diferentes abordagens a Composicao e ao

Ambiente das Estrelas de Neutrons

Different approaches to the Composition of Neutron Stars and their

Environment

Tese apresentada ao Departamento de Astrono-

mia do Instituto de Astronomia, Geofısica e

Ciencias Atmosfericas da Universidade de Sao

Paulo como requisito parcial para a obtencao

do tıtulo de Doutor em Ciencias.

Area de Concentracao: Astronomia

Orientador: Prof. Dr. Jorge E. Horvath -

IAG-USP

Co-orientador: Prof. Dr. Mariano Mendez -

University of Groningen - The Netherlands

Esta e uma versao corrigida; a versao origi-

nal encontra-se disponıvel na Unidade.

Sao Paulo

2012

To Rosi and to my family: without them this Thesis would not have been possible.

Acknowledgements

There are many people I have to thank, people that have helped me during my long

journey until this moment, a journey that I have decided to take many years ago. I

apologize if I do not mention every person I should in what follows.

First, I want to thank to my family that even not understanding exactly my motivations

and dreams gave me all the support I needed to keep going in this career. Some years after

I entered in the University I met Rosi and we have been together since then. We passed

by many difficult situations together, but we had many happy moments that compensates

everything else. I have to thank her a lot, not only by her love, but also because I am a

dreamer and it is not rare I keep dreaming while awake. At these moments Rosi appears,

pulling me back to reality. “You can dream, but it is even better if you do something to

make it real”. Words of wisdom, of love, from a person that wishes the best for me. I love

her very much.

By the professional side and friendship I thank, first, my supervisor Jorge Horvath.

Besides an excellent supervisor, he is a friend. I should better call him advisor. Some

people say he is “too tough” or rigorous in tests, in talks, in classes, in Thesis etc, but I

can say, by experience, that this “toughness” have helped me a lot in locating where are

my weak points and the lacks in my knowledge. In the end he pushed me for the best of

me. And I thank for that.

Also deserve special acknowledgements prof. Marcos Diaz, the rapporteur of my Ph.D.

project and prof. Mariano Mendez, my co-supervisor for their suggestions and sharing

knowledge and experience. I thank prof. Raimundo Lopes, for the help and useful advices

specially during the COSPAR Workshop.

“Thank you very much” I say to my friends and colleagues from IAG-USP, specially

Felipe, Oscar and Pamela for the help and friendship in my beginning; to my friends and

colleagues from Molecular Sciences course, specially Maria Fernanda and Marcelo, and to

my old friends of the Takion Group of the TAPIOCA Corporation (Takion and Aggre-

gated People International Organization Committee Associated), specially Ze Henrique,

Fernando and Leo, The Originals. I cannot forget Douglas, Laura and Daniel for receiving

me so well into the Jorge’s research group and my friends and colleagues from Kapteyn

Astronomical Institute, specially Beike, Guobao, Yanpin, Andrea.

Moreover I thank all my professors for teaching and sharing their knowledge, to CAPES

for the financial support, to Instituto de Astronomia, Geofısica e Ciencias Atmosfericas

da Universidade de Sao Paulo (IAG-USP) and to Kapteyn Astronimical Institute for the

hospitality during my stage in Groningen, the Netherlands.

At long last, I want to thank the secretaries Marina, Cida, Regina and Conceicao from

the astronomy department; Marcel, Ana and Lilian from SPG; and Marco, Luis, Ulisses

and Patricia from the informatics division.

Esta tese/dissertacao foi escrita em LATEX com a classe IAGTESE, para teses e dissertacoes do IAG.

“Seeing. One could say that the whole of life lies in seeing - if not ultimately, at least

essentially. To be more is to be more united (...) But unity grows, and we will affirm this

again, only if it is supported by an increase of consciousness, of vision. That is probably

why the history of the living world can be reduced to the elaboration of ever more perfect

eyes at the heart of a cosmos where it is always possible to discern more. Are not the

perfection of an animal and the supremacy of the thinking being measured by the

penetration and power of synthesis of their glance? To try to see more and to see better is

not, therefore, just a fantasy, curiosity, or a luxury. See or perish. This is the situation

imposed on every element of the universe by the mysterious gift of existence. And thus, to

a higher degree, this is the human condition.”

“The Human Phenomenon” - Pe. Teilhard de Chardin.

“A great discovery does not issue from a scientist’s brain ready-made... it is the fruit of

an accumulation of preliminary work.”

Marie Curie

Resumo

Mesmo depois de 80 anos de pesquisas intensas, a composicao das estrelas de neutrons

permanece desconhecida, uma vez que tanto a materia densa do interior desses objetos

compactos quanto a materia se movendo em torno deles encontram-se em condicoes fısicas

extremas, irreprodutıveis em laboratorios terrestres.

Nessa Tese, seguimos quatro diferentes caminhos interconectados para abordar esses

objetos extremos. Primeiramente, exploramos a estrutura matematica das estrelas de

neutrons, construindo solucoes parametrizadas unicamente pela densidade central, apro-

priadas para estudar o comportamento estrutural dessas estrelas em diversas situacoes.

Em seguida, adotamos uma abordagem nova, a teoria da informacao, para inferir uma

hierarquia de equacoes de estado, mostrando que as estrelas de quarks seriam, por sua

conformacao, favorecidas na Natureza. Estudando a emissao em raios-X advinda do sis-

tema binario de baixa massa 4U 1608–52, que contem uma estrela de neutrons, limitamos

o tamanho fısico da fonte emissora, mostrando que nao deve estar longe da superfıcie da

estrela compacta. Para isso, empregamos uma tecnica inedita no calculo dos time lags.

Por fim, mostramos que e possıvel obter restricoes a massa do quark estranho e ao gap de

energia da CFL diretamente das observacoes.

Concluımos essa Tese com a afirmacao de que a materia estranha e, estrutural e ener-

geticamente, favorecida pela Natureza, muito embora exista uma barreira entropica a ser

superada e uma densidade central mınima a ser atingida logo apos o colapso da estrela

progenitora. Se essa barreira foi superada na Natureza, apenas observacoes futuras mais

refinadas dirao.

Abstract

Even after 80 years of intense research, the composition of neutron stars remains un-

known, since both the dense matter in the interior of these compact objects and the matter

moving around them are in extreme physical conditions, unreproducible in terrestrial lab-

oratories.

In this Thesis, we follow four different interconnected ways to approach these extreme

objects. First, we explore the mathematical structure of neutron stars, building solutions

parametrised solely by the central density, what are very appropriate to study the structural

behaviour of these stars in different situations. Then, we adopted a novel approach, the

information theory, to infer a hierarchy of equations of state, showing that quark stars

would be, by its configuration, favoured by Nature. Studying the X-ray emission arising

from the low-mass binary system 4U 1608-52, which contains a neutron star, we limit the

physical size of the emitting source, showing that it should not be far from the surface of

the compact star. For this, we employed a technique to calculate the time lags never used

before. Finally, we show that it is possible to obtain restrictions on the strange quark mass

and the energy gap of the CFL directly from observations.

We conclude this Thesis with the statement that the strange quark matter is, struc-

turally and energetically favoured by Nature, though there is an entropic barrier to be

overcame and a minimum central density to be reached just after the collapse of the pro-

genitor star. If this barrier is actually overcame in Nature, only refined observations will

tell.

List of Figures

1.1 Concept of pulsar. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.2 SED distribution of the Crab system. . . . . . . . . . . . . . . . . . . . . . 23

1.3 Bimodal distribution of pulsars. . . . . . . . . . . . . . . . . . . . . . . . . 24

1.4 Recent neutron stars mass measurements. . . . . . . . . . . . . . . . . . . 26

1.5 Mass-radius diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.6 LMXBs components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.7 Roche lobes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.8 Atoll and Z sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.9 Variability-luminosity-spectra relations. . . . . . . . . . . . . . . . . . . . . 33

1.10 Colour-colour diagram: variability components. . . . . . . . . . . . . . . . 35

1.11 kHz QPO frequencies vs spectral hardness. . . . . . . . . . . . . . . . . . . 36

1.12 “Colour-colour-colour” diagram: rms amplitude. . . . . . . . . . . . . . . . 37

1.13 “Colour-colour-colour” diagram: 2-200 keV luminosity. . . . . . . . . . . . 38

1.14 Colour coordinate representing the position along the atoll track. . . . . . 39

1.15 RMS-RMS diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1.16 The structure of the atom. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

1.17 Elementary particles predicted by the Standard Model. . . . . . . . . . . . 42

1.18 The building blocks of matter. . . . . . . . . . . . . . . . . . . . . . . . . . 42

1.19 Baryons and mesons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

1.20 Properties of the interactions. . . . . . . . . . . . . . . . . . . . . . . . . . 43

1.21 Isospin symmetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1.22 Common species inside a neutron star. . . . . . . . . . . . . . . . . . . . . 47

1.23 SLy4 equation of state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

1.24 Mass-radius relation for the SLy4 equation of state. . . . . . . . . . . . . . 51

1.25 Equation of state of MIT Bag Model. . . . . . . . . . . . . . . . . . . . . . 54

1.26 Mass-radius relation for MIT Bag Model equation of state. . . . . . . . . . 54

3.1 Intuitive definition of complexity. . . . . . . . . . . . . . . . . . . . . . . . 84

4.1 Atmospheric windows for the electromagnetic spectrum. . . . . . . . . . . 94

4.2 Mirror assembly to focus X-rays. . . . . . . . . . . . . . . . . . . . . . . . 95

4.3 Variability components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.4 Constraints on the EoS by kHz QPOs. . . . . . . . . . . . . . . . . . . . . 99

4.5 PDSs of eight data segments. . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.6 Average of the eight PDSs: the PDS for the whole observation. . . . . . . . 101

4.7 Average of the eight PDSs after the shift-and-add technique: the PDS for

the whole observation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.1 What is going on in this Thesis . . . . . . . . . . . . . . . . . . . . . . . . 20

1.2 Observation: pulsars and the neutron stars . . . . . . . . . . . . . . . . . . 22

1.2.1 The problem of the masses and the necessity of radii measurements 26

1.2.2 Low Mass X-ray Binaries . . . . . . . . . . . . . . . . . . . . . . . . 29

1.3 Nuclear and Particle Physics: theory of neutron star matter . . . . . . . . 41

1.3.1 The Equations of State used in this Thesis . . . . . . . . . . . . . . 49

1.3.1.1 SLy4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

1.3.1.2 Strange Quark Matter . . . . . . . . . . . . . . . . . . . . 52

2. Analytical solutions in the construction of strange quark stars models . . . . . . 57

2.1 Exact and quasi-exact models of strange stars . . . . . . . . . . . . . . . . 61

3. Information theory and measurements to infer a hierarchy of equations of state . 81

3.1 Entropy, complexity and disequilibrium in compact stars . . . . . . . . . . 87

4. X-ray astrophysics as a tool to study kilohertz quasi-periodic oscillations and time

lags of the X-ray emission and to probe the environment of neutron stars . . . . 93

4.1 X-ray astrophysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.2 Shift-and-add . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.3 The coherence function and the phase and time lags . . . . . . . . . . . . . 102

4.4 Time lags in the kilohertz quasi-periodic oscillations of the low-mass X-ray

binary 4U 1608–52 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Appendix 127

A. QCD Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

A.1 Self-bound models of compact stars and recent mass-radius measurements . 129

Chapter 1

Introduction

Neutron stars, one of the most exotic objects in Universe. Inside them, a sea of still

unknown particles in the densest form in Nature is supported against further gravitational

collapse by the pressure of degenerated matter (pure neutrons, at least in the simplest

form). Their very existence, though, was hypothesised by the great Russian physicist Lev

Landau in 1932, while Baade and Zwicky (1934) were the first to propose a mechanism

for neutron star formation, during their study about supernovae, all this in the two years

following the discovery of the neutron by Chadwick (1932a,b). However, in spite of being

one of the possible end points of stellar evolution, the true composition of these compact

objects remains uncertain after almost 80 years of intense studies.

The birth of neutron stars is marked by the death of a massive progenitor star with mass

between 8M⊙ and 25M⊙ in a cataclysmic explosive event that releases at least ∼ 1053erg of

energy, sufficient to overshadow the bright of an entire galaxy. After the contraction of the

iron core of the progenitor star, a neutron star with M ∼ 1, 4M⊙ and R ∼ 10km remains.

The central density is certainly above the nuclear saturation value, or ∼ 2.4× 1014g/cm3.

Neutron stars are natural laboratories to test two of the most fundamental theories of

physics: the Einstenian theory of gravitation, the General Relativity, in the strong field

regime and the physics of matter at very high densities and temperatures. Both regimes

cannot be achieved in laboratories on Earth.

However, the first evidence of their realization in Nature came from the observation

of the pulsar PSR 1919+21 (Hewish et al., 1968). A pulsar is a fast spinning object

that releases pulses of energy at a very precise and constant rate, generally seen in radio

wavelengths, but also in other wavelengths like optical, X-rays and γ-rays.

18 Chapter 1. Introduction

From the 1930’s (the Baade and Zwick’s work) to the 1960’s (the first pulsar discov-

ery) the neutron star physics remained as an exotic and speculative research field, mostly

because of the lack of methods and technology to access information about the interior

from direct observations. It was realized from the beginning that in order to theoretically

study the neutron stars one needed of General Relativity to solve for the interior structure

of these objects, since the value of the compactness factor, GM/c2R ∼ 0.2, is very high

compared to common stars: roughly 1,000 times higher than the value for a white dwarf

and 100,000 higher the same ratio for the Sun. The compactness factor is a measure for

the strength of the gravitational field around the star. In addition, the natural energy

scale is in the X-ray band, not in the visible.

The difficulty of using the General Relativistic equations comes from the fact that we

have to use the Einstein’s Field Equations for a Spherically Symmetric Perfect Fluid from

which derive the relativistic equation for the hydrostatic equilibrium. The latter is a first

order non-linear partial differential equation that must be solved coupled with the mass

equation and the equation of state of the fluid.

The first real attempts for solving the internal structure of neutron stars were the works

of Tolman (1939) and Oppenheimer and Volkoff (1939). They studied the solutions for the

following set of equations:

dp

dr= −

Gm(r)ρ(r)

r2

(

1 +p(r)

c2ρ(r)

)(

1 +4πr3p(r)

c2m(r)

)(

1−2Gm(r)

c2r

)−1

, (1.1)

dm

dr= 4πr2ρ(r), (1.2)

ρ = ρ(r) (1.3)

where the first is the relativistic hydrostatic equilibrium equation, the second is the mass

integral and the third is some functional form of the density profile (or of one of the other

quantities or some combination of them).

We will see that one of the most general ways of solving the Tolman-Oppenheimer-

Volkoff (TOV) equations is substituting the third equation by the Equation of State (EoS)

of the dense neutron star matter. It is the equation of state that provides the microphysics

or the composition of the star (see chapter 2).

Chapter 1. Introduction 19

And here we face the second major difficulty: the behaviour of matter is relatively well

known until densities of about ∼ 5 × 1014g/cm3, which is pretty high. However, neutron

stars’ densities can reach values as high as ∼ 20 × 1014g/cm3 and we cannot reproduce

such densities in laboratory. In other words we do not know the true equation of state for

neutron stars. For further discussion about this problem we refer to subsection 1.3 and to

chapter 2.

On the other hand, neutron stars do appear in Nature and we can detect their signatures

by means of astrophysical measurements, now developed and quite reliable. Astrophysics,

then, provides a window to address many of the aspects of the dense matter we cannot

address on Earth. Here we face the third major difficulty: because their small size and

the extreme physical conditions regarding their own existence, it is a real technological

challenge to our telescopes and detectors (see chapter 4 for some discussion about the

problematic).

Neutron stars appear in different astrophysical systems: as isolated neutron star, gener-

ally detected as pulsars; in binary systems with an ordinary star (high mass and low mass

X-ray binary systems, depending on the mass of the companion); and in binary compact

star systems (a neutron star and another compact object either another neutron star or

a white dwarf or a black hole). Hence, we can study neutron star in several wavelengths,

sometimes simultaneously, with different techniques.


1.1 What is going on in this Thesis

...and the supremacy of the thinking being measured by the penetration and power of

synthesis of their glance?...

Synthesis. From this word you readily perceive our intent in this Thesis. You readily

get a sense of our beliefs and soon you will see how we convey our idea of unity.

Neutron stars have not been deciphered yet, being at the limits of our theories and far

out from the scope of the direct experiments. For more than 80 years, scientists around the

world have been studying neutron stars, one of the most exotic objects in the Cosmos by

means of general relativity, particle and nuclear physics and astrophysics. Independently,

each approach bore amazing fruits toward a better understand of the physics of dense

matter, the nature of space-time and stellar evolution, with consequences that extend far

to other fields of astrophysics and physics like, for example, the chemical abundances and

the formation of the elements of periodic table, the chemical evolution of our galaxy and

even the origin of the elements necessary to life, to quote only a few.

Our technology has evolved a lot since the 1930s. We are now at the position we are

able to reach the edge of present knowledge and effectively test the limits of the theories

that mould the physical Reality, the theory of General Relativity and the Nuclear Physics.

It is now time for a new synthesis. In this Thesis, we “hedge our bets” on neutron star

composition by applying different methods of studying compact objects, some orthodox,

some completely new, forging them into a coherent whole, leading us a step further in the

comprehension of this startling neutron stars.

We first present a historic survey through the astrophysical events and hypothesis about

neutron stars (1.2), the association neutron star/pulsar/supernova and the problematic

of how we could pick up hints about the internal composition of neutron stars through

observations (1.2.1). We also give some basis to one kind of system that contains a neutron

star, showing how the astrophysical observables are linked together in an inseparable whole

(1.2.2). Then we talk about the theory of neutron star matter, necessary to understand

what we can expect about the composition, which is given by the Equation of State (1.3).

After these introductory sections we present our unified view, starting with the math-

ematical framework of neutron stars. In chapter 2 we show how to construct quark star

models with a minimum of numerical calculations, exploring exact and quasi-exact solu-

Section 1.1. What is going on in this Thesis 21

tions in order to give way to derivation of physical properties. We emphasize the role of

one of most promising exact solutions (the anisotropic solution) and of the quasi-exact

solution in allowing more profound theoretical studies. In fact, we selected the anisotropic

solution as the best for what we present in the following chapter.

Thus, in chapter 3 we examine the possibility that quark stars are preferred in Na-

ture provided an effective formation channel, the nucleation of SQM, is confirmed. The

methodology employed there, information theory, is unusual and its use in astrophysics is

in the very beginning. However, the outputs are promising, pointing towards an affirmative

answer in favour of the exotic (strange) quark stars. Strange quark stars and their siblings

hadronic stars are similar in mass and radius and, isolated, it is difficult to overrule one

in favour of the other. Then, it is necessary to study their environments (and now we

understand the title of this Thesis) to look for characteristic signals of these two types of

stars and to decide which kind of them we are really talking about. The real meaning of

what we have just said is obvious: we are deciding the internal composition.

Finally, in chapter 4 we explore how to probe the immediate environment of a class of

binary sources (the low mass X-ray binaries) to give one method to extract the radii using

the the so-called kilohertz quasi-periodic oscillations (kHz QPOs) frequencies to further

test the mass-radius relation, as done in other studies. We also study the time lags and

the coherence of the X-ray emission to find out where they are produced: if in an electron

corona or in the disc, for example. Constraints on the emission region opens a direct

window to the mass and radius of the neutron star in these systems through models of disc

reflection/thermalisation and models of scattering off in Compton clouds, for example.

An alternative form, which employs just the mass-radius of exotic models including

pairing between quarks, is included in the Appendix. There, we extract limits on the values

of the parameters of quantum chromodynamics directly from observations, discussing in

the ways some caveats of radii determinations.

Other approaches are available and being developed like the link of the thermodynamic

entropy and the leakage of information during the contraction of the iron core just after

the supernova explosion. Also, we have a larger dataset of another low mass X-ray bi-

nary system where we have much better statistics to detect and study the quasi-periodic

oscillations and time lags.


1.2 Observation: pulsars and the neutron stars

As we have already said, the first evidence that neutron stars appear in Nature came

from the observation of the pulsar PSR 1919+21. In what follows we identify pulsars

as neutron stars, but recall that not every neutron star is a pulsar. “Pulsar” denotes a

compact source that emits radiation in pulsed signals, or in well defined time intervals, like

a beacon.

It is generally accepted, as elaborated over the years (Woltjer, 1964; Pacini, 1967; Gold,

1968), that pulsars are neutron stars with high magnetic field spinning at rates that ranges

from ∼ 1ms to ∼ 2s. However, a pulsar is only theoretically characterized if there is a

tilt of a few degrees between the rotation axis and the magnetic field axis. What we see

in such situation is the electromagnetic radiation emitted by the magnetic cone swept out

by the star (although we do not have any self-consistent model for the emergence of the

pulses). Otherwise there is no pulsar at all. See figure 1.1 for a schematic view.

(a) Schematic pulsar (b) Observed pulses. The average is the period.

Figure 1.1: The concept of pulsar. We observe pulses only when the radiation cone is tilted in relation to

the rotation axis and if it passes through our sight of view. We see, in reality, what is displayed in panel

b.

They emit mostly and were discovered by their radio emission, but they can emit in

the full electromagnetic spectrum, like the well-studied example in the Crab nebula (see

figure 1.2).

Section 1.2. Observation: pulsars and the neutron stars 23

Figure 1.2: Notice that the SED distribution of the Crab system covers all the electromagnetic

spectrum.

By the conservation of magnetic flux from the progenitor to the compact object during

the stellar core contraction, the magnetic fields increases, ranging from ∼ 108G to ∼ 1014G

in the final configuration. On the other hand, the rotation periods decreases, because the

star spun-up by conservation of angular momentum, lying in the range from ∼ 1ms to

∼ 2s. A very interesting feature regarding the pulsar periods become apparent when we

plot the derivative of the rotation period versus the rotation period together with the

magnetic field strength lines: we clearly see a bimodal distribution or two populations of

pulsars (see figure 1.3).

The majority of pulsars we see has periods about 0.7s (but see below some biases and

caveats), high magnetic fields (∼ 1012G) and are usually isolated. The second popula-

tion are the so-called millisecond pulsars with periods ∼ 1ms and weak magnetic fields

(∼ 108G), appearing commonly in binaries. The first class is believed to be a younger

population or classical pulsars, formed just after the supernova explosion.

An evolutionary path to explain this bimodal distribution is certainly believed to exist.

During the pulsar’s life, angular momentum is lost to radiation and the pulsar slows down

and emits less intensely with time. The pulsar slowly approaches the so-called Death Line,


Figure 1.3: The bimodal pulsar distribution: the classical pulsars have high magnetic fields

and larger periods, while the millisecond pulsars have weak field and spin very fast. The log(P)

- log(P ) distribution of pulsars (black dots) (ATNF Pulsar Catalogue data). Binary pulsars

are marked by a circle. SGRs and AXPs are marked by stars and triangles, respectively.

The lines are: constant surface dipole magnetic field strength (dashed) and characteristic

ages (dotted). The arrows indicate a measurement of the braking index. The death line

is the pair-creation limit for generating radio pulses. Figure from F. Lamb and W. Yu,

astro-ph/0408459.

fading in the process until certain combination of the strength of the magnetic field and the

rotation period is attained. Then the pulsar turns-off. However, when there is a low mass

companion star, an accretion disc forms at a time when the pulsar has long turned-off. This

event marks a new era in the pulsar’s evolution. The accreted matter transfers angular

momentum to the neutron star, which spins up. When, again, a certain combination of the

magnetic field strength and rotation period is attained, the pulsar revives as a millisecond

pulsar (Glendenning, 2000).

The subject of the astrophysical part of this Thesis is the so-called Low Mass X-ray

Binaries (LMXBs), thought to be the ancestors of the millisecond pulsars. We will discuss


them later on this chapter.

Pacini (1967) and Gold (1968) point to the fact that highly magnetized rotating com-

pact stars must radiate an enormous amount of energy (∼ 1038erg/s) and even more must

be stored as rotation energy (∼ 1048erg). Calculating the energy input that must be ac-

celerating the Crab remnant (as observed in several wavelengths) we can derive important

relations among the quantities involved in the radiation mechanisms from the pulsar. For

instance, we get from the Crab pulsar:

P ∼1

30s, P ∼ 4× 10−13s/s (1.4)

and average density of

ρ ∼ 1.5× 1014g/cm3 (1.5)

for an estimated radius ≤ 19km and for an assumed mass of about 2M⊙.

The existence (and observation) of the quantity P eliminates vibrations as the origin

of the pulsations. Because this average density, together with the calculated compactness

and the decreasing monotonicity of the density with radius (learned from TOV equations)

and together with the fact that such compact object must be charge neutral (composed of

neutrons, therefore), we conclude that pulsars must be the so long sought neutron stars.

On the other hand, it is hard to associate pulsars/neutron stars with supernova remnant

due to the large kick velocity that the pulsar acquires just after its formation, but there

is one classical example where this association is positive (the Crab) from which we can

conclude that, in fact, pulsars are neutron stars.

It is important to notice that there are detection biases when one is trying to detect

pulsars, specially the fast ones. Because the signal intensity is diminished by an inverse-

square law of the distance and because the electrons in the interstellar medium disperse

the radio signal, there are practical limits in the detection. The dispersion is a particularly

important effect. Longer wavelengths suffer a greater delay in reaching us than the shorter.

Thus, the distribution of pulsar periods does not reflect properly the relative population

of classical and millisecond pulsars.

Another bias comes from the fact that pulsars radiate for a timescale of 10 million

years. This timescale is short compared to the age of the galaxy and the lives of massive


stars that are believe to create the pulsars. Then, from the estimated 105 active pulsars,

maybe the majority of them cannot be seen.

1.2.1 The problem of the masses and the necessity of radii measurements

Another striking characteristic of the observed neutron stars is their mass distribution

(see figure 1.4).

Figure 1.4: Recent measurements suggest a bimodal distribution in the mean mass: 1.4M⊙

and 1.6M⊙. Original reference is Lattimer and Prakash (2005). This updated figure will

appear in Annual Review of Nuclear and Particle Physics, Vol. 62, 2012.

Recent theoretical studies suggest a mass distribution at least bimodal: there would

be a peak around 1.3M⊙ and another one around 1.7M⊙, with minimum and maximum

around 1.15M⊙ and 2.1M⊙, respectively.


Now, with much more data available, the mean mass, which was in the past around

1.4M⊙, can be something around 1.68M⊙. Or even we could suppose a bimodality although

the error bars are a factor of obscurity (the situation is a bit different as discussed by

Valentim et al. (2011) with a bayesian approach). The mass distribution is intimately

related to the events that lead to the core collapse of massive stars (M & 8M⊙) and to

the subsequent supernova explosion that, then, lead to the formation of the compact star

itself.

Independently of the case, the formation of a neutron star begins when a massive star

exhausts its nuclear fuel up to the point of iron, the most bound element. The iron core

of this pre-supernova star is partially supported by the degeneracy pressure of relativistic

electrons (∝ η4/3e ) and partially by thermal pressure (while there are elements to the fusion

process to occur). The layers above the core are still fusioning elements and depositing

more iron in the core.

When the iron core exceed its effective Chandrasekhar mass1(Timmes et al., 1996),

the electron degeneracy pressure no more can support the core against the collapse and

a proto-neutron star begins to form. The reactions that occur are basically two, both of

them diminish the pressure and make the collapse even faster:

e− + 56Fe → 56Mn + νe ; e− + p→ n+ νe (1.6)

which, by electron capture, release a huge amount of neutrinos and diminish the degeneracy

pressure and

γ + 56Fe ↔ ...→ 134He + 4n ; γ + 4He→ 2p+ 2n. (1.7)

The latter removes almost all of the thermal support. Thus, the velocity of the collapse

increases and, about 100ms, the young neutron star takes its final configuration. This is

1 MCh0 = 5.83Y 2e (original Chandrasekhar mass; Ye is the electron fraction).

MCh = MCh0

[

1 − 0.0226(

Z6

)2/3]

(with electrostatic corrections due to the non-uniformity in the

electron distribution).

MCh = MCh0

[

1+(

πkBTǫf

)2]

(due to the finite entropy of the iron core; ǫf = 1.11(ρ7Ye)1/3 is the Fermi

energy).

MCh = MCh0

[

1 +(

seπYe

)2]

(written in terms of the electronic entropy per baryon; se =

0.5ρ−1/3(Ye/0.42)2/3).


the timescale that takes to the neutrons become degenerate and, because they are fermions,

their pressure halts the collapse.

The physics that leads to the formation of neutron stars is different depending on the

mass interval of the progenitor stars (Fryer et al., 2011). Theoretically there are three

cases:

1. progenitor mass between & 7.5M⊙ and ∼ 11M⊙: the main source of uncertainty

comes from the stellar evolution theory. The models do not make an accurate division

of the edge for the formation of neutron stars and white dwarfs. These stars are

moderately massive and abundant in the galaxy and probably produce reminiscent

compact stars with low average masses, ∼ 1.1M⊙ to ∼ 1.3M⊙.

2. progenitor mass between ∼ 11M⊙ and ∼ 25− 30M⊙: the main source of uncertainty

comes from the explosion mechanism itself, e.g., the mass of the reminiscent is de-

termined mainly by the fallback material onto the proto-neutron star. These stars

are massive and less abundant in the galaxy and could provide neutron stars with

masses around ∼ 1.7− 1.8M⊙.

3. progenitor mass greater than ∼ 25 − 30M⊙: the main source of uncertainty comes

from the mass loss rate. They probably form black holes, not studied here.

On the other hand, even more important than mass determinations are the radii de-

terminations, because the mass can be accurately determined when the neutron star is in

binary systems. In the end, the radii are the crucial factor and they are the most trou-

blesome to determine. Here, the major problem is the error bars. The confidence interval

always captures a big region in the mass-radius diagram, encompassing many equations

of state. Thus, it is important to combine as many methods of determining the radius

as we can. For an extensive work on masses and radius determinations combining many

techniques see Ozel (2006) and Ozel et al. (2009). Another way of determining the radius

is combining timing and spectroscopy in LMXBs (see subsection 1.2.2). The latter uses the

association of kilohertz quasi-periodic oscillations with the inner radius of the disc or with

the innermost stable circular orbit and the general relativistic distortions of the iron line

(generally seen in these systems) (Hiemstra et al., 2011; Sanna et al., 2011). This provides

a much smaller confidence range which implies smaller error bars in radius determinations.


Figure 1.5: The mass-radius diagram; original figure is from reference Demorest et al. (2010).

The horizontal strips are observational constraints. The three black dots are observed neutron

stars’ radii and their respective error bars (the black dots are from the references [20], [21]

and [22] quoted in our paper in the Appendix).

All we need is an equation of state that passes through the observed masses in the

plot. Or, in other words, the equation of state must support neutron stars’ masses at least

equal to the effective Chandrasekhar limit of the core of the progenitor star. Although the

maximum mass allowed by this equation of state may be greater than 2M⊙, it is necessarily

lower than the causal limiting mass of ∼ 3.0 − 3.5M⊙. So, it seems that we are stuck in

our search for the true equation of state of neutron stars. We need another approach to

address this question (see chapter 3).

We now move to a brief description of the astrophysical systems studied in this Thesis,

the Low Mass X-Ray Binaries.

1.2.2 Low Mass X-ray Binaries

Low mass X-ray binaries are gravitationally bound star systems in which one of them

is a compact object (a neutron star or a black hole) and the other (the companion star) is

an ordinary low mass (. 1.4M⊙) star, in contrast to high mass X-ray binary in which the

companion star mass can be as high as ∼ 20M⊙. These systems are very bright, generally

non-pulsating X-ray sources and very soon after the first observations of them (Giacconi

et al., 1962) it became clear that the energy source is provided by accretion of matter onto

the compact object. See figure 1.6 for a schematic view.


Figure 1.6: Notice all the components. We study here phenomena at the inner edge of the

disc, near the surface of the central compact star.

The accretion of matter onto the compact object in LMXBs proceeds via accretion

disc (Frank et al., 2002) which forms when the companion evolves to the giant phase

and its matter fills the Roche lobe (see figure 1.7). The angular momentum possessed by

the matter induces the formation of the disc and as matter loses angular momentum, its

gravitational energy is converted into kinetic energy that is radiated away mostly in the

X-ray band.

Figure 1.7: Roche lobes gravitational equipotentials. Here we see five Lagrangian points or

points of stable equilibrium in the gravitational potential.

The first work regarding the structure and radiative processes of the disc was the

seminal work of Shakura and Sunyaev (1973) where they derived that the disc around

the black holes is geometrically thin and optically thick, emitting energy as a collection


of black bodies (or multi-colour black body). With these assumptions we can use the

Stephan-Boltzmann law and derive the approximated temperature of the disc as function

of the radial coordinate:

T (r) ≈

[

3GMM

8πr3σ

[

1−

(

Rinner

r

)1/2]]1/4

. (1.8)

Notice that when Rinner → R⋆ and r is not far from the surface, the temperature

reaches ∼ 107K for typical values of the mass accretion rate, M ∼ 10−8M⊙/yr, and mass

of the compact object M = 1.4M⊙. That is what we detect as the X-ray radiation of a

few keV of energy.

In this Thesis we studied LMXBs with a neutron star and our concern is to address the

internal composition of this startling compact object. We have for this purpose some tools

of high energy astrophysics. These systems show up spectroscopic features, variability of

the light curve and thermonuclear bursts in the energy range from few keV up to hundreds

of keV.

The theory of accretion discs predicts the formation of a boundary layer on the surface

of the neutron star where the accreting gas has its azimutal velocity slowed down to match

the spin velocity of the star in a very abrupt process. This very thin layer has a high

density and is supposed to emit like a blackbody. Recalling that the accretion luminosity

is given by the rate of release of gravitational energy

Lacc =GMM

R⋆

(1.9)

and that the disc luminosity is

Ld = 2

∫

∞

R⋆

σT (r)42πrdr =GMM

2R⋆

, (1.10)

we see immediately that the boundary layer is responsible for at least half of the total

energy release:

LBL =GMM

2R⋆

. (1.11)

Based on equation 1.11 we calculate the blackbody temperature of the boundary layer

in ≈ 3 × 107K for typical values of the quantities involved, comfortable in the soft X-ray


band 1keV to 10keV.

Actually we observe the X-ray spectra of LMXBs in that range of energies and, based

on theoretical grounds, we represent the spectra as a sum of two components: a disc

multicolour soft blackbody plus a hard blackbody from the boundary layer/neutron star

surface, although this decomposition is often ambiguous (Gilfanov et al., 2003). Besides,

another spectroscopic feature is an emission line from iron at ∼ 7keV (Hirano et al.,

1987; Miller, 2002; Bhattacharyya et al., 2006; Hiemstra et al., 2011) that is supposed to

be distorted by general relativistic effects. In this case, the distortion of the line itself

becomes a window to access the parameters of the compact object.

Notice that the disc plays an important role on the spectra through the mass accretion

rate M . It is supposed that variations in M drive changes in spectral and temporal states,

apart from changes in luminosity.

On the other hand, LMXBs show a myriad of temporal features, e.g., variability on

scales from millihertz (mHz) to kilohertz (kHz) called quasi-periodic oscillations or QPOs,

besides some types of noise. There are models that try to explain the origin of the QPOs,

from disc instabilities and/or oscillations to orbital motion-based models like the Sonic-

point Model (Miller et al., 1998) or the Relativistic Precession Model (Stella and Vietri,

1998). So far, none of them were able to fully explain or predict the appearance of these

characteristic frequencies. Regarding the kHz QPOs, they are generally associated to

orbital motion of matter at the innermost stable circular orbit or ISCO and if it is the

case, we can derive the radius of the central neutron star (see chapter 4).

Finally, we observe in many sources the so-called thermonuclear bursts, sudden and

huge eruptions where the X-ray flux increases by several factors in a short time, ∼ 2s,

and decay time that can last several minutes. We will not enter in more details here,

but it suffices to say that thermonuclear bursts have thermal blackbody spectra and the

measurements of the bolometric fluxes and blackbody temperatures can be used to estimate

radii if the distance to the source is known (Ozel et al., 2009).

There are two major classifications of neutron star LMXBs: the atoll and Z sources

(van der Klis, 1989b,a). This division is due to the shape the sources acquire when plotted

in the colour-colour2 or colour-intensity diagrams. See the figure 1.8. They can also be

2 Colour in X-rays are defined to be the ratio of photons in a predefined high energy band relatively to


transient or persistent: while the latter have been showing detectable X-ray emission for

most of the time, the transients show long periods of inactivity.

Banana

Lower Banana

Island State

Extreme Island State

S =1

Lower Left

z

Horizontal Branch

zS =2

BananaUpper

Upper Banana

Lower Banana

Normal Branch

Flaring Branch

a b c

Figure 1.8: Colour-colour and colour-intensity diagrams. Panel a names the atoll sources

and panel c names the Z sources. This figure was extracted from van der Klis (2006).

To be more specific, we studied here the transient atoll source 4U1608–52 (see chapter

4). Let us now briefly describe some general features common to many atoll sources. For

a far more complete description see the reference Linares (2009).

To begin with, a cyclic relation between luminosity, variability and spectra is assumed

to exist, as shown in figure 1.9.

Figure 1.9: The observed quantities are supposed to relate each other.

In what follows we show a set of figures nicely provided by M. Linares that will serve

to our purpose of summarizing the general properties of atoll sources. We will state the

results but not interpret them. For interpretations we refer the reader to Linares (2009).

The author studied these relations in detail for nine atoll sources, namely, 4U 1608–52,

Aql X–1, 4U 1705–44, 4U 1636–53, 4U 0614+09, 4U 1728–34, 4U 1820–30, 4U 1735–44,

GX 3+1. Keep in mind the source 4U 1608–52, the object studied in this Thesis.

a lower energy band.


Regarding the relation variability and spectra, let us take a look in figures 1.10 and

1.11. In figure 1.10 we see which frequency component appears in which spectral state.

Important to us in this Thesis are the kHz QPOs, in blue, which appear mostly in the

intermediate and soft states. Except by the very low frequency noise complex (VLFN)

that appears in the softest states, the overall trend is an anti-correlation between frequency

and hardness, e.g., the lower the hardness, the higher the frequency. Figure 1.11 shows an

important characteristic regarding the kHz QPOs in atoll sources. Both kHz QPOs, the

lower and the upper, anti-correlate with hardness. However, they follow different paths in

the frequency vs colour diagram as can be easily seen. Because of this, when one sees only

one kHz QPO in a given observation, the identification of which one it is becomes obvious

when taking into account the typical range of frequencies of the lower and the upper kHz

QPO and the correspondent hard colour.

To end the relation between the variability and the spectra, we see in figure 1.12 that the

strength of the variability, given by the fractional rms, increases if the hardness increases.

The atoll sources display luminosities from 0.1% to 50% of the Eddington luminosity

(3 × 1035 to 1 × 1038erg/s) in the energy range 2 − 200keV . Because the hard colour is

independent of the interstellar absorption, it traces very well the changes in the spectral

state. In figure 1.13 we see that, on average, the soft states are more luminous, but there

are hard states more or equally luminous than the soft ones. Thus, the hard colour is not

a good indicative of the luminosity. From figure 1.14 we see that luminosity and spectral

hardness are anti-correlated in soft and hard states but not in intermediate states.

Finally, the authors in Linares (2009) have not found any obvious relation between

frequencies, spectra and luminosity neither in a given source nor between sources, but they

actually found a correlation between luminosity and hardness for a fixed frequency (see

figure 1.15). This means that the same temporal states can occur at different luminosities.

By studying specific spectroscopic and temporal features and their relation with the ob-

served quantities, we can find how to calculate properties of the central compact object like

the mass, radius, magnetic fields etc and determine the equation of state observationally.

With this we conclude this subsection and refer the reader to chapter 4 to our work

about the time lags in the source 4U 1608–52.


0.5

0.6

0.7

0.8

0.9

1.0

1.1

Har

d co

lor

(Cra

b)

1608 Aql X-1

Har

d co

lor

(Cra

b)

1705

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Har

d co

lor

(Cra

b)

1636 0614

Har

d co

lor

(Cra

b)

1728

0.5

0.6

0.7

0.8

0.9

1.0

1.1

0.8 1 1.2 1.4 1.6

Har

d co

lor

(Cra

b)

Soft color (Crab)

1820

0.8 1 1.2 1.4 1.6

Soft color (Crab)

1735

0.8 1 1.2 1.4 1.6

Har

d co

lor

(Cra

b)

Soft color (Crab)

GX 3+1

VLFNBREAKUPPER

Figure 1.10: Colour-colour diagrams for the nine atoll sources. Green points are measures

of the break frequency of the flat-topped broadband noise, small blue open circles mark the

detections of upper kHz QPOs, red points show those observations where VLFN is present and

grey points the cases where none of these phenomena were present. Observations combining

either VLFN or flat-topped noise with upper kHz QPO show up as blue circles with red or

green interior, respectively. The hard (Sa ≡ 1) and soft (Sa ≡ 2) vertices are marked with

a large black open circle on the upper and lower part of the atoll track, respectively. Figure

from reference Linares (2009).


Figure 1.11: kHz QPO frequencies versus spectral hardness. Black dots represent upper kHz

QPOs and grey triangles lower kHz QPOs. Squares show initially unidentified (single) kHz

QPOs. Figure from reference Linares (2009).


16080.5

0.6

0.7

0.8

0.9

1.0

1.1

Har

d co

lor

(Cra

b)

Aql X-1 1705

0

10

20

RM

S 1

-100

Hz

(%)

16360.5

0.6

0.7

0.8

0.9

1.0

1.1

Har

d co

lor

(Cra

b)

0614 1728

0

10

20

RM

S 1

-100

Hz

(%)

1820

0.8 1 1.2 1.4 1.6

Soft color (Crab)

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Har

d co

lor

(Cra

b)

1735

0.8 1 1.2 1.4 1.6

Soft color (Crab)

GX 3+1

0.8 1 1.2 1.4 1.6

Soft color (Crab)

0

10

20

RM

S 1

-100

Hz

(%)

Figure 1.12: “Colour-colour-colour” diagrams of the nine atoll sources. The colour scale

shows the 1-100 Hz fractional rms amplitude of the variability. Figure from reference Linares

(2009).


16080.5

0.6

0.7

0.8

0.9

1.0

1.1

Har

d co

lor

(Cra

b)

Aql X-1 1705

-2

-1

log 1

0 (

L / L

Edd

. )

16360.5

0.6

0.7

0.8

0.9

1.0

1.1

Har

d co

lor

(Cra

b)

0614 1728

-2

-1

log 1

0 (

L / L

Edd

. )

1820

0.8 1 1.2 1.4 1.6

Soft color (Crab)

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Har

d co

lor

(Cra

b)

1735

0.8 1 1.2 1.4 1.6

Soft color (Crab)

GX 3+1

0.8 1 1.2 1.4 1.6

Soft color (Crab)

-2

-1lo

g 10

( L

/ LE

dd. )

Figure 1.13: “Colour-colour-colour” diagrams of the nine atoll sources. The colour scale

shows the 2-200 keV luminosity, in Eddington units and logarithmic scale. The luminosity

ranges from 2.5×1035erg/s (0.1%LEdd) to 1.3×1038erg/s (50%LEdd). Figure from reference

Linares (2009).


Figure 1.14: Colour coordinate representing the position along the atoll track, Sa, versus

luminosity (in the same units as 1.13). The Sa ranges corresponding to soft, hard and

intermediate states are separated by the double dashed lines and indicated on the right-hand

axis. Figure from reference Linares (2009).


Figure 1.15: RMS-RMS diagrams of the nine atoll sources. The color scale shows the 2-

200 keV luminosity, in Eddington units and logarithmic scale. The luminosity ranges from

2.5 × 1035erg/s (0.1%LEdd) to 1.3 × 1038erg/s (50%LEdd). Figure from reference Linares

(2009).

Section 1.3. Nuclear and Particle Physics: theory of neutron star matter 41

1.3 Nuclear and Particle Physics: theory of neutron star matter

In this section we intend to show some aspects of the theory and of the experiments used

to study neutron star matter. Our goal in this section is to link what we know empirically

about nuclear matter to the matter inside the neutron stars (Krane, 1987; Glendenning,

2000, chapters 4 and 5 and references therein). Are they the same? Can we describe them

both with the same theory? As we have already stated, it is not possible to attain densities

like those reached in the interior of neutron stars in our laboratories. However, much we

have learnt about nuclear physics and the behaviour of matter and its constituents at

Earth’s laboratories conditions. With facilities like the CERN LHC, the Tevatron and

many others particle and heavy-ion accelerators around the world, we shattered the nuclei

and its constituents (see figure 1.16) and learnt about the building blocks of matter, the

elementary particles, which give rise to the Standard Model of Particle Physics.

Figure 1.16: The structure of the atom. Here we see the electron and quarks which are

elementary particles. Notice that protons and neutrons are not elementary particles since

they are formed by different combination of quarks.

The Standard Model needs the existence of 17 basic particles as shown in figure 1.17

and so far can account for all the phenomena we see in Nature. The only particle not yet

seen is the Higgs boson. Ironically, it is the most fundamental since it is (theoretically)


the responsible for the mass of all other particles.

Figure 1.17: All the 17 elementary particles predicted by the Standard Model. The Higgs

boson is the only not yet observed.

There are two classes of particles: the fermions, with half-integer spin (1/2, 3/2, 5/2...)

are the matter constituents and obey the Fermi-Dirac statistics and the Pauli Exclusion

Principle; and the bosons, with integer spin (0, 1, 2...), that are the force carriers and obey

the Einstein-Bose statistics, but not the Pauli Exclusion Principle. In figure 1.18 we show

some properties of fermions and bosons.

(a) Fermions (b) Bosons

Figure 1.18: The building blocks of matter.


Fermions by themselves form two more sets of (non elementary) particles: baryons with

half-integer spin and mesons with integer spin. We show only a few types of baryons and

mesons in figure 1.19 and we show which kind of force acts on which type of particles in

figure 1.20.

(a) Baryons (b) Mesons

Figure 1.19: All types of particles (baryons and mesons) are different combinations of quarks. Here we

show a pretty small number of baryons and mesons.

Figure 1.20: We show here the four forces of Nature and their effectiveness acting upon the

particles.

By knowing what the building blocks of matter are, we can study how they clump

and form the nuclei we see in the periodic table. Two things became apparent from the

experiments since the earliest times: that the mass of the proton and the neutron are

approximately equal and that Nature seems to prefer nuclei with approximately the same

number of protons and neutrons (see figure 1.21). We say that there is isospin symmetry in

nuclear matter or that the nuclear matter is isospin-symmetric. Isospin is a word derived


from isotopic spin and is an abstract concept mathematically resembling the ordinary spin.

By definition, protons have isospin 1/2 and neutrons have isospin −1/2. Symmetry, for

quantum systems, is related to degeneracy of energy levels. In this case, the fact that

mp = 938.28MeV/c2 ≈ mn = 939.57MeV/c2. Thus, there is an approximated symmetry

in the strong nuclear interaction (which binds the proton and neutron, for example), the

isospin symmetry meaning that the strong interaction does not depend on the charge.

Hence, the proton and the neutron are supposed to be different states of the same particle:

the nucleon. However, notice that the isospin symmetry is a theoretical idealization and

should be considered as approximate.

(a) Decay times. (b) Decay process.

Figure 1.21: The stable elements are shown in black. Notice that the isospin symmetry is an idealization:

we clearly see the deviation from the line Z = N .

For nuclear matter from which our world is composed, the importance arises from the

fact that nuclei are made by protons and neutrons which, in turn, are composed of u−

and d−quarks. Although there are many combinations of quarks, the isospin symmetry

plus the properties of the chemical potential of the particles (we will discuss thoroughly

soon) prevents the formation of stable “nuclei” with other combinations of quarks at the

actual energy density conditions of the Universe. Thus, the ground state of nuclear matter


is that of matter composed of protons and neutrons, unless some exotic process happens,

for example the formation of strange quark matter.

In short, the nuclear matter is hot non-degenerate (the thermal energy is of the same

order of the Fermi energy), symmetric (number of protons and neutrons are approximately

the same), and do not carries strangeness (because the energy needed to create strangeness

is not attained by these systems). It can be described by the nuclear mean-field theory

constrained by the charge symmetry and strangeness conservation (Glendenning, 2000).

Now, let us back to the neutron star case. The dawn of neutron star is the supernova

explosion in which the star core implodes releasing a huge amount of energy, 99% of it

carried by neutrinos. The source of this huge amount of energy is the gravitational binding

energy of the star. The neutrino emission cools down the star from ∼ 50MeV (∼ 1011K)

to ∼ 1MeV (∼ 1010K) in a few seconds. In some millions of years the temperature will

drop below to 106K. This temperature, while hot to Earth standards is quite cold in

the nuclear scale and because the Fermi energy (or the chemical potential at T = 0K)

depends on density, which is now very high, the matter becomes degenerate after the first

few seconds. In fact, the chemical potential is the quantity that we must to analyse in

order to find out which kind of particles could form, in principle, in a certain medium. The

particle will appear if its chemical potential exceeds its mass in that given medium which,

in turn, is given by its vacuum mass corrected by the interactions with other particles in

that medium. For instance, Λ-baryons will appear when µΛ = µn & mΛ ≃ 1115MeV .

The chemical potential of any particle can be written as a linear combination of the

chemical potential of any conserved charges in the medium – in neutron stars, the baryon

and electric charges:

µ = bµn − qµe, (1.12)

where b is the baryon charge and q is the electron charge and µn and µe are the chemical

potentials of neutron and electron, respectively.

As the density increases, so do the chemical potential, and the energy of the system

can be reduced by sharing the conserved baryonic number with other baryon species. In

this way we expect that the super-dense matter inside neutron stars be populated by many

baryon species or even by quarks (see subsection 1.3.1). Very important for these compact


objects is the hyperon threshold at around three times the saturation density, a value that

can be easily attained after the implosion of the iron core of the progenitor. Hyperon is a

baryon that contains a strange s-quark.

In order to know the baryon species that can be formed inside a neutron star we must

to study the evolution of that object during its hot phase until it cools down to 106K, the

point from where, for all the purposes, the star is frozen regarding the nuclear scale. For

example, when the star is still hot in that scale, reactions like 1.13 occur:

N +N → N + Λ +K, (1.13)

where N stands for nucleon, Λ is an uds-baryon (a hyperon) and K (kaon) is a su-meson.

As we can see, the strangeness begin to increase inside the neutron star, although the

K meson will rapidly decay in muons, photons and anti-neutrinos unless it condensates (a

recent exciting possibility). When the temperature is low enough, the kaon cannot form

any more because there is no available energy. However, another reaction can occur that

increases the strangeness:

n+N + µ− → N + Σ− + 2γ + ν, (1.14)

where n stands for the neutron, µ− for the muon, Σ− for the dds-hyperon, γ for the photon

and ν for the neutrino. See in figure 1.22 a schematic chart showing some common species.

In this way, processes like the above one will occur until the system reaches its ground

state, minimizing the energy since it keeps the charge neutrality for the given number of

baryons inside the star. The result is that the neutron star is not made by neutrons as at

first the scientists thought, but it is probably a rich zoo of many baryons.

In short, the neutron star matter is cold degenerate (the thermal energy much lower

than the Fermi energy), asymmetric (various baryon species), and carries strangeness (be-

cause the formation of hyperons). It can be described by the nuclear mean-field theory

for baryons plus for leptons, constrained by the charge neutrality and the general beta

equilibrium without strangeness conservation (Glendenning, 2000).

Notwithstanding what was said before, we do not know the exact behaviour at supra-

nuclear densities & ρsat. We need extrapolate the now well-established theory of nuclear

matter to the high densities of the neutron star interior. The extrapolation cannot be


Figure 1.22: Common species inside a neutron star as function of the baryon number density.

random, but must follow some strict rules (Glendenning, 2000):

1. Lorentz covariance,

2. general relativity,

3. causal equation of state (v2 = dp/dǫ ≤ 1),

4. microscopic stability known as Le Chatelier’s Principle (dp/dρ ≥ 0),

5. baryon and electric charge conservation,

6. Pauli Principle,

7. generalized beta equilibrium,

8. phase equilibrium,

9. asymptotic freedom of quarks,

10. properties of matter at saturation density.

The most used theory is the relativistic mean-field approximation because it incorpo-

rates naturally the properties of matter at the saturation point and the effective mass

corrections. To do so we write the Lagrangian of matter inside the neutron stars:


L =∑

B

ψB

(

iγµ∂µ −mB + gσBσ − gωBγµω

µ −1

2gρBγµτ · ρµ

)

ψB

+1

2

(

∂µσ∂µσ −m2

σσ2)

−1

4ωµνω

µν +1

2m2

ωωµωµ

−1

4ρµν · ρ

µν +1

2m2

ρρµρµ −

1

3bmn(gσσ)

3 −1

4c(gσσ)

4

+∑

λ

ψλ

(

iγµ∂µ −mλ

)

ψλ . (1.15)

We will not enter in details, but it is important to recall that the sum is over the baryon

species B from the baryon octet (p, n, Λ, Σ+, Σ−, Σ0, Ξ−, Ξ0) (Glendenning, 2000).

The Euler-Lagrange equations are obtained when we substitute the meson fields by

their mean values in the static uniform matter and the nucleon currents by ground-state

expectations generated in the presence of the mean meson fields (that is why we call it

mean-field approximation).

At the end of this hard work we are in the position of obtain the so-called Equation of

State (EoS) for the neutron star matter:

T µν = −gµνL+∑

φ

∂L

∂(∂µφ)∂νφ, (1.16)

where φ stands for the fields and T µν is the energy-momentum tensor given by

T µν =

ǫ 0 0 0

0 p 0 0

0 0 p 0

0 0 0 p

Solving the 7+N variables system results in ǫ(ρ) and p(ρ) which, together, provide the

equation of state. Due to the complexity of the problem, most of the equations of state is

given in tabular form, being calculated numerically, but there are a few cases in which a

closed expression for the equation of state is available (for example, the equation of state

prescription from MIT Bag Model; we will discuss this equation of state in subsection

1.3.1). Now, with the equation of state in hands we can proceed to solve for the structure

of the neutron stars by solving the TOV equations, integrating the system subject to the

initial conditions ǫ(r = 0) = ǫc and M(r = 0) = 0 (the energy density and the mass,


respectively). There are only a few analytical solutions for this problem (see the chapter

2 for further discussion on this problem).

For each value of ǫc allowed by the EoS we have a stellar model, a point in the mass-

radius diagram. Thus, varying ǫc we can construct a sequence of stars in the mass-radius

diagram correspondent to this particular EoS (see figure 1.5). And thus we close the circle.

A question that naturally arises is: what is the true equation of state of neutron star

matter? In this Thesis we studied models within two classes of equation of state. One is

the SLy4 prescription and is for a neutron star of hadronic composition. The second class

is a strange quark star following the MIT Bag Model prescription. On theoretical grounds

both classes seem to be feasible in Nature. On observational basis, it has been very hard to

access information from the interior of neutron stars, although the search for their modes

of oscillation, the evolution of the breaking index of pulsars and glitches, for example, are

very exciting promises.

We refer the reader to chapter 3 for another way to study the equation of state where

we suggest that can exist a hierarchy of equations of state to be realized in Nature.

1.3.1 The Equations of State used in this Thesis

We used here two equations of state, a hadronic equation of state (SLy4) and a strange

quark equation of state (MIT Bag Model). We will discuss a bit them both in what follows.

1.3.1.1 SLy4

Here we will briefly describe the equation of state for the hadronic neutron stars we

used in this Thesis. This equation of state seems to be very appropriate to calculate the

structure of neutron stars composed by very rich neutron matter.

This equation of state is based on Skyrme-Lyon (SLy) effective nucleon-nucleon inter-

action given as input to the calculation, once the many-body approximation is fixed. This

effective interaction is two-body in nature, but contains a term that is an average of an

original three-body problem. Besides, there is a strong dependence of a neutron excess

which makes it suitable for calculation involving neutron star matter.

The equation of state we used here is an “unified” one in the sense it accounts for the

outer and inner crust and for the liquid neutron core and it was derived by Douchin and


Haensel (2001).

For the crust, they used the Compressible Liquid Drop Model (CLDM) of nuclei

(Douchin et al., 2000). Within this model, the total energy density is a sum of the bulk

contribution of nucleons, the surface contribution of nucleons, the Coulomb interaction

contribution and the electron energy contribution. All the contributions are calculated for

the SLy forces (Douchin and Haensel, 1999, 2000).

E = EN,bulk + EN,surf + ECoul + Ee (1.17)

In their calculations for the outer crust, the authors limited the density lower limit to

ρ > 106g/cm3 and for densities below the neutron drip (4.3×1011g/cm3) they used not the

SLy equation of state, but the Haensel and Pichon equation of state (Haensel and Pichon,

1994). This equation of state is a reliable extrapolation using maximal experimental data

for nuclear masses. Then, for densities above the neutron drip, in the region of the inner

crust, the equation of state is matched with the SLy equation of state.

The transition from the (inner) crust to the liquid core occurs at constant pressure and

is accompanied by a density jump and takes place because the nuclei get closer and closer.

The liquid core is assumed to be a homogeneous plasma of neutrons, protons, electrons

and negative muons (the latter, for densities above the threshold density for the appearance

of muons). At higher densities the authors extrapolated the npeµ matter, instead of taking

into account the appearance of hyperons. The total energy density of the npeµ matter is

given by the energy density of the nucleons and of the leptons in addition to the rest energy

of the matter constituents:

E(nn, np, ne, nµ) = EN(nn, np) + nnmnc2 + npmpc

2 + Ee(ne) + Eµ(nµ) (1.18)

where the equilibrium condition with respect weak interaction is given by

µn = µp + µe, µµ = µe (1.19)

where, in turn,

µj =∂E

∂nj

, j = n, p, e, µ. (1.20)


Then, the equation of state for the npeµ is given by

ρ(nb) =E(nb)

c2, P (nb) = n2

b

d

dnb

(

E(nb)

nb

)

. (1.21)

Summarizing, this equation of state covers three main regions of the neutron star: the

outer crust, the inner crust and the liquid core. In figures 1.23 and 1.24 we show the

equation of state covering the whole neutron star structure and the resulting mass-radius

relation, respectively:

1010

1011

1012

1013

1014

1015

1028

1029

1030

1031

1032

1033

1034

1035

1036

1037

ρ [ g . cm−3 ]

P [

dyn

. cm

−2 ]

Figure 1.23: SLy4 equation of state of Douchin and Haensel. Dotted vertical line corresponds

to the neutron drip and the dashed one to the crust-liquid core interface.

0

0.5

1

1.5

2

2.5

8 10 12 14 16 18 20

M [M

sun]

R [km]

SLy4 Hadronic Stars Sequence

Figure 1.24: Mass-radius relation for the SLy4 equation of state.


Haensel and Potekhin (2004) derived an analytical representation of the SLy4 equation

of state. Analytical representations are preferred over the tabulated form because they

avoid two major problems of the latter: first, the ambiguity in the interpolation when

calculating the neutron star structure (because this leads to ambiguity in the calculated

parameters). Second, the procedure have to take into account thermodynamic relations,

which is difficult. Besides, analytical representations can have their derivatives calculated

precisely.

In this Thesis we used the analytical representation in our calculations and we talk a

bit about it in chapter 3.

1.3.1.2 Strange Quark Matter

When it was realized that the quarks are asymptotically free, the idea of a star composed

entirely or in part by quark matter seemed natural. Quark matter, in this context, is a

much larger colourless3 region than the hadronic volume through which quarks are free

to move. The asymptotic freedom of quarks leads to the possibility that the true ground

state of the strong interaction could be the strange quark matter (Bodmer, 1971; Witten,

1984) instead of the nuclei we are familiar with.

The MIT Bag Model (Chodos et al., 1974) was developed to account for the hadronic

masses in terms of their quark content. This model also explains the spatial confinement

of quarks, e.g., to explain why we do not see free quarks everywhere. In the model, quarks

are confined by a boundary, called the Bag, where even the vacuum is expelled. In reality,

the energy density of the physical vacuum exerts pressure on this boundary to impede that

quarks cross it. Inside the bag the quarks move freely and we can consider them as a Fermi

gas.

The expressions for the pressure, energy density and baryon density for a Fermi gas of

quarks in the zero temperature, T = 0, approximation (as discussed earlier, the neutron

star is frozen at strong interaction scales) read:

p = −B +∑

f

1

4π2

[

µfkf

(

µ2f −

5

2m2

f

)

+3

2m4

f ln(µf + kf

mf

)

]

(1.22)

3 Colour is the charge of the strong interaction in the same sense the electron carries the charge of

electromagnetism; a quark change its colour when it absorbs or emits a gluon.


ǫ = B +∑

f

3

4π2

[

µfkf

(

µ2f −

1

2m2

f

)

−1

2m4

f ln(µf + kf

mf

)

]

(1.23)

ρ =∑

f

k3f3π2

(1.24)

where the sum is over all flavours f of interest and kf is the Fermi momentum defined in

terms of the chemical potential µf = (m2f + k2f )

1/2. B is the energy density of the physical

vacuum.

Now, if we take the massless approximation, again justified because of the large value of

their chemical potentials at the typical densities found in neutron stars4, then the equation

of state assumes a simple form that we will use throughout this Thesis:

p = −B +∑

f

1

4π2µ3fkf (1.25)

ǫ = B +∑

f

3

4π2µ3fkf (1.26)

ρ =∑

f

k3f3π2

(1.27)

which results in the known formula 3p = ǫ+4B, with ǫ = c2ρm here where ρm is the matter

density, see figure 1.25. Notice the striking characteristic of this EoS: these are self-bound

stars, bounded not by gravity, but by the strong force. This means that when the pressure

is zero at the edge of the star, the density assumes a finite value that falls to zero in a

distance 10−13cm.

The model is controlled by three parameters: B (the vacuum constant), ms (the strange

quark mass) and αc (the strong coupling constant that accounts for the interaction among

quarks). The numerical value of B is of high interest for the stability of strange quark

matter: for αc = 0 and ms = 0, B = 57 − 91MeV/fm3; for αc = 0 and ms = 150MeV ,

B = 57 − 75MeV/fm3. In such situations, the energy per baryon of a mixture of three

quark flavours (up, down and strange) will be less than the energy per baryon of iron

(E/A(56Fe) ∼ 931MeV ) and the ground state will be the strange quark matter.

4 µs = 1.9B1/4 at the edge of the strange star and would be higher than 275MeV .


0

1e+35

2e+35

3e+35

4e+35

5e+35

6e+35

7e+35

0 5e+14 1e+15 1.5e+15 2e+15 2.5e+15

p [e

rg/c

m3 ]

ρ [g/cm3]

Figure 1.25: Equation of state of MIT Bag Model.

This equation of state produces a very different mass-radius relation than the SLy4 and

other hadronic equations of state; see Alcock et al. (1986) and figure 1.26.

0

0.5

1

1.5

2

2.5

7.5 8 8.5 9 9.5 10 10.5 11 11.5 12

M [M

sun]

R [km]

Strange Quark Stars Sequence

Figure 1.26: Mass-radius relation for strange quark equation of state from MIT Bag Model.

What was discussed above is valid for non-interacting strange quarks; if we take into

account the coupling constant of the strong interaction to the first order, the expressions

for the energy density and pressure (Farhi and Jaffe, 1984) read


ǫ = B +∑

i

(

Ωi + µini

)

(1.28)

and

p = −B +∑

i

Ωi (1.29)

where Ωi is the thermodynamic potential at T = 0 and the sum on i is over the quark

flavours and the muon and the electron. The other quantities are:

Ωf = −γf

24π2

µf

√

µ2f −m2

f

(

µ2f −

5

2m2

f

)

+3

2m4

f ln[µf +

√

µ2f −m2

f

mf

]

−2αs

π

[

3(

µf

√

µ2f −m2

f −m2f ln[µf +

√

µ2f −m2

f

µf

])2

−2(

µ2f −m2

f

)2

− 3m4f ln

2(mf

µf

)

+6ln( σ

µf

)(

µfm2f

√

µ2f −m2

f −m4f ln[µf +

√

µ2f −m2

f

mf

])

]

(1.30)

where γf is the flavour degeneracy, αc is the coupling constant and σ is the renormalization

scale. Finally, the quark number density, baryon number density and the charge density

are

nf = −∂Ωf

∂µf

(1.31)

ρ =1

3

∑

u,d,s

nf (1.32)

q =∑

i

niqi (1.33)

It is easy to show that setting αc = 0 we recover the previous expressions for T = 0.

How is it possible that strange quark matter could manifest itself in Nature? When a

compact object forms from a progenitor star with mass higher than ∼ 8M⊙, the densities

in the core reach something about ∼ 1014−15g/cm3. At such densities, phase transitions are

likely to occur because the weak interaction would convert about one third of the quarks


into strange quarks, since the mixture of the three flavours has a lower energy per baryon

than the conventional matter and also lower than the matter composed only by a mixture

of two flavours.

And how could one observationally differentiate strange stars from neutron stars? We

have seen that both, hadronic neutron stars and strange stars, produce sequences with

similar values of mass and radius. One possible way is through very fast pulsars; there

is a limit on the rotation period of gravitationally bound hadronic star. So if a pulsar is

discovered to have rotation period shorter than that limit, the conclusion that the strange

quark matter hypothesis realizes in Nature would be unavoidable.

Other possible way comes from studies of phenomenon at the surface of a strange

star. Because the matter in these stars is bound by the strong interaction, the star has a

very abrupt edge, with the density falling from ∼ 1014−15g/cm3 to 0 in a distance range

∼ 10−13cm (the strong force range). For example, the strange star can support outgoing

radiant fluxes much greater than the Eddington flux, the photon emissivity would be

around 1 and the star could be like a “silver ball” in X-rays, and at last, the magnetosphere

could be very different from hadronic stars since the electrostatic forces are not able to

remove particles from the surface (Alcock, 1991).

To end this discussion, is it possible to have these two families of neutron stars existing

simultaneously? In principle, it could be possible. However, the Universe can be too old for

this to happen. The reason is that once a strange star is formed, it is possible that an event

like a collision would expel strange matter nuggets. These nuggets of strange matter have a

net positive electric charge5 and they would be inert except in a neutron-rich environment.

Eventually, the flux of such nuggets would have contaminated the hadronic stars and

because they are neutron-rich, the nuggets would absorb the neutrons, leading to a con-

version of the whole hadronic star to a strange star. (A estimation of the strange nuggets

flux in our galaxy can be done with what is discussed in the works of Clark and Eardley

(1977) and Cappellaro and Turatto (1988)).

Thus, although all the neutron stars could be strange stars, more studies have to be

performed in order to observationally determine the EoS of neutron stars.

5 This is because the mass difference between the strange quark and the up and down quarks leads to

slight different amounts of the three in the three flavour mixture.

Chapter 2

Analytical solutions in the construction of strange

quark stars models

In this chapter we explain our motivation for seeking an analytical solution for the

structure of neutron stars. The first attempts in doing this calculations are due to Richard

Tolman (Tolman, 1934, 1939) and Oppenheimer and Volkoff (Oppenheimer and Volkoff,

1939). Since it was realized that General Relativity is the necessary framework for studying

neutron stars, physicists struggled with the equations. The reason is that for a spherically

symmetric and static perfect fluid we must solve three coupled non-linear differential equa-

tions with four variables. The equations are shown below:

Gµν =8πG

c4Tµν , (2.1)

where Gµν = Rµν − 12gµνR describes the metric of the spherically symmetric and static

space-time and T µν =(

ρ+ pc2

)

dxµ

dsdxν

ds−pgµν is the energy-momentum tensor for the perfect

fluid.

From this, we obtain the following set of equations to solve:

λ′e−λ

r+

1− e−λ(r)

r2=

8πG

c2ρ(r), (2.2)

ν ′e−λ

r−

1− e−λ(r)

r2=

8πG

c4p(r), (2.3)

e−λ[ν ′′

2+ν ′2

4−ν ′λ′

4+ν ′ − λ′

2r

]

=8πG

c4p(r). (2.4)

58 Chapter 2. Analytical solutions in the construction of strange quark stars models

with the last equation usually replaced by the contracted Bianchi Identities that express

the conservation law:

p′(r) +1

2(c2ρ(r) + p(r))ν ′(r) = 0. (2.5)

Alternatively, one can solve another form of these equations, the so-called Tolman-

Oppenheimer-Volkoff (TOV) Equations:

p′(r) = −Gm(r)ρ(r)

r2

(

1 +p(r)

c2ρ(r)

)(

1 +4πr3p(r)

c2m(r)

)(

1−2Gm(r)

c2r

)−1

, (2.6)

m′(r) = 4πr2ρ(r). (2.7)

The TOV equations describe the structure of the star and is the relativistic version of

the hydrostatic equilibrium equation. As we can readily see, we need another information

to close the system and solve for the quantities λ(r), ν(r), p(r) and ρ(r). A common

choice, specially in the past, is to give a functional profile for one of these quantities, say,

for instance, ρ(r) = a − b × r2, a and b constants (Tolman, 1939). With such choice, the

system is, in principle, solvable.

Such approach provided us about 127 solutions until 1998. Studying these solutions

Delgaty and Lake (1998) found out that only 16 satisfy the minimum criteria for being

physically acceptable. And of these 16, only 9 satisfy the additional condition that the

sound speed in the medium monotonically decreases with radius.

Being physically acceptable, although, does not mean that the equation of state is

meaningful from the point of view of nuclear and particle physics. For example, Tolman

IV solution (Tolman, 1939) provides an equation of state of the form:

ρ(p) = ρc − 5pcc2

+ 5p

c2+ 8

(pcc2− p

c2)2

pcc2+ ρc

. (2.8)

Does this equation of state represent some form of dense neutron star matter? Probably

not. Besides, a general characteristic of the equations of state coming from a choice for

one of the quantities is that the vast majority of them depends on the central density. In

other words, the equation of state would represent a new type of matter at each central

density allowed, however, maintaining the same functional ρ− p dependence.

Chapter 2. Analytical solutions in the construction of strange quark stars models 59

To avoid such consternation, physicists began to employ a functional equation of state,

for example the MIT Bag Model, p = (c2ρ− 4B)/3 (Witten, 1984), replacing the necessity

of a functional form for one of that quantities. Now we can enter the microphysics into the

problem. Now we know what kind of matter we are describing. But this time, there are no

known exact solution for the equations. The only solution is a numerical solution. Using

the MIT Bag equation of state, Alcock et al. (1986) first obtained a numeric solution for

a strange star.

The last thing we can do if we want an exact analytical solution is to overdetermine

the system, e.g., imposing a choice for one of the quantities and an equation of state

simultaneously (Ivanov, 2001). However, this procedure results in two solutions for one of

the quantities and it is necessary to force a matching of these two in order to have the

system consistently solved.

Why does a physicist want to search for an exact solution for the structure of neutron

stars? Because they allow easy calculations and, most importantly, they allow to predict

some new phenomena or behaviour where numeric solutions blur our insights.

Now we refer the reader to our paper regarding modelling a strange star, e.g., em-

ploying an exotic quark matter equation of state, with some known physically acceptable

exact solutions and one quasi-exact solution. In this contribution we have shown that the

numerical solution for the complete sequence of bare strange stars first obtained by Alcock,

Farhi e Olinto using the MIT Bag equation of state can be accurately described by some

exact analytical solutions for the Einstein Equations, all of them simply parametrized by

one single quantity, namely, the central density. In particular, the gaussian quasi-exact

solution is very useful to describe that numerical solution despite the small error in one of

the elements of the metric.

On the other hand, fully exact analytical solution that take into account the equation

of state for the strange stars were also employed and modelled. We call attention for the

anisotropic solution that also describes the strange stars equally well the gaussian solution,

although this time there is an anisotropy in pressure, which now has two components: the

radial and the tangential. The mass-radius relation thus obtained matches almost perfectly

the numerical solution. Besides the pressure anisotropy, this solution needs a slightly higher

central density to produce the same mass.


At last, if we allow the existence of a radial electric field inside the strange star, another

exact solution can be obtained, this time producing masses up to 3 solar masses.

In this contribution we also discuss the problems that results from choosing analytical

profiles to quantities other than choosing the equation of state explicitly. To illustrate our

point we discuss the Tolman IV and the Buchdahl I solutions when forcing a matching

with the Alcok, Farhi and Olinto numerical solution.

Section 2.1. Exact and quasi-exact models of strange stars 61

2.1 Exact and quasi-exact models of strange stars

WARNING: there is a misprint mistake in page 69 of this Thesis which corresponds to

page 1944 of our paper, below. The expression 5 actually reads

p(r) =c2ρ(r)

3−

4B

3.


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834

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Eat and Qua-Eat Model of Strange Star 1

s b s hv c v h sy p chs b

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Chapter 3

Information theory and measurements to infer a

hierarchy of equations of state

So far, we have discussed some difficulties to address the composition of neutron star

matter either theoretically or observationally. Our current technology does not have yet

the necessary resolution and sensibility for this task. A beautiful theory is only a beautiful

theory, if it cannot be confronted with observations. Nature does not care about theories,

the way we describe it, but our goal as scientists is to decode how Nature happens to work.

We believe that in order to circumvent some of these difficulties we need to use as many

methods of studying neutron stars as possible. Thus we could, in principle, to narrow the

window of acceptance of possible equations of state. One of these new approaches is

information theory.

Nowadays, information theory is the “flavour of the month” in physics: from cosmology

to biological systems, through condensed matter and communications and also linguistics,

information theory has been helping to elucidate many aspects of the behaviour of systems

otherwise not amenable of detailed treatment.

This theory relies on the central concept of Shannon entropy (Shannon and Weaver,

1949) (also known as information entropy or logic entropy) that is related with the infor-

mation stored in the system. It is a measure of the uncertainty associated with the value

of a quantity. In other words, how much one can say about a system with the smallest

piece of information: just one bit. Shannon entropy is defined by the quantity

S = −K∑

x

p(x)logb[

p(x)]

, (3.1)

82 Chapter 3. Information theory and measurements to infer a hierarchy of equations of state

where p(x) is the probability distribution (see below) of the quantity to be measured and

the basis b and the constant K determine the “type” of entropy being calculated. In the

specific case of Shannon entropy, K = 1 and b = 2, the resulting unit of entropy being a

bit. If instead b = e, the resulting unit is nat. It is worth to say that there is a relationship

with the thermodynamic entropy. In this case, K = kB, the Boltzmann constant, and

b = e. However, one must to take some care because the relation with thermodynamic

and statistical physics is not straightforward: the meaning of p(x) may be very different.

In the latter case, p(x) stands for the the probabilities of a given microscopic state of the

system and is related to a specific energy configuration.

Then, information is what we can get from observing the occurrence of an event (how

surprising or unexpected or what else) and, with certain reductionism, we define informa-

tion in terms of the probability of that event to occur. Another feature is that information

theory deals with any kind of probability. The definition of information, I(p), by Shannon

relies in some desired mathematical properties of this quantity:

• I(p) ≥ 0,

• I(p1 ∗ p2) = I(p1) + I(p2) (additivity),

• I(p) is monotonic and continuous in p,

• I(1) = 0,

from which we derive that information is:

I(p) = logb(1/p) = −logb(p) (3.2)

for some constant b.

Thus, flipping a fair coin once gives you −log2(1/2) = 1 bit of information.

If a source provides n symbols ai each with probability pi, then we could be inter-

ested in the average amount of information in the stream of symbols, which implies in a

weighted average. Then:

I

N=

N∑

i=1

pilogb(1/pi) = −

N∑

i=0

pilogb(pi) ≡ H(P ). (3.3)

Chapter 3. Information theory and measurements to infer a hierarchy of equations of state 83

This quantity is defined as the entropy of the probability distribution P = pi.

A good property is that the maximum of this quantity is achieved when we have

equiprobability, pi = 1/n. Example: a student and his grades in three situations. If

the grades are A, B, C, D and F, with equal probabilities, then the student will get 2.32

bits of information. However, if the probabilities are 1/10, 1/5, 2/5, 1/5, 1/10, then the

information that the student will get is 2.12 bits; if 0, 0, 0, 0, 1, then he will get 0 bit of

information.

Equation 3.1 can be generalized to the continuous case:

S = −K

∫

x

p(x)logb[

p(x)]

dx, (3.4)

and we will use it in what follows.

In describing a physical system, another statistical quantity related to information

entropy is the complexity. But what exactly is complexity? Let us state that complexity is

what does not match the requirements of being simple. It seems a tautology, but we shall

see that when dealing with physical systems it makes sense. In physics, we always begin by

studying ideal systems as a first approximation to Nature, since Nature always happens to

be much more sophisticated. We study ideal systems because they are the simplest. We

say that these systems have minimum complexity, say zero complexity, by construction.

Let us allow our definition of complexity to encode the order and the disorder (or

the self-organization in broader terms) of a system and let us analyse two ideal systems,

extremes in all aspects and opposites as well (Lopez-Ruiz et al., 1995):

• the perfect crystal has zero complexity by definition; strict symmetry rules imply

probability density centered around the prevailing state of perfect symmetry which,

in turn, gives you minimal information, e. g., a small piece of information is enough

to describe the whole system. The perfect crystal is completely ordered;

• the ideal gas also has zero complexity by definition; all the accessible states are

equiprobable which imply maximal information. The system is totally disordered.

As we see from above, the information alone is not enough to define properly the

complexity of a system. Then we define the disequilibrium as the distance to the equiprob-


ability. With this in mind we can get a “visual” intuition about what complexity would

be, as shown in figure 3.1:

Figure 3.1: Intuitive definition of complexity.

Although there is no unique definition of complexity, we shall adopt the definition given

by Lopez-Ruiz, Mancini e Calbet (1995) as modified by (Catalan et al., 2002), because it

matches the asymptotic behaviour for that two extreme ideal systems: the ideal gas and

the perfect crystal. In their definition, then:

C = H ×D, (3.5)

where H = exp(S) and S is the information entropy (or the information content of the

system) in natural logarithmic units and D is the disequilibrium (identified with the dis-

tance of the system to its state of equiprobable probability distribution). In its original

definition, the expressions for S and D are the following:

S = −

∫

ρ(r)ln[ρ(r)]dr, (3.6)

D =

∫

ρ2(r)dr. (3.7)

The quantity ρ(r) is the normalized probability distribution that describes the state of

the system. S describes the uncertainty associated to that probability distribution while D

Chapter 3. Information theory and measurements to infer a hierarchy of equations of state 85

stands for the information energy as defined by Onicescu (1966), or the quadratic distance

to the equiprobability.

These definitions have been used by a number of authors (Panos et al., 2007; Panos

and Chatzisavvas, 2009; Panos et al., 2009) to study atomic systems. With astrophysical

interest Sanudo and Pacheco (2009) applied the concepts to white dwarfs and Chatzisavvas

et al. (2009) applied them to neutron stars with a simple equation of state.

In astrophysical situations concerning the structure of compact objects, the choice for

the quantity ρ(r) may be seem as a caveat: these authors use the mass density distribution

as the quantity to enter in the integrals. This is justified by noticing that the mass density

distribution, that comes from the solution of the equations of structure for a relativistic

star (TOVs, see below), is somehow related to the probability of finding some particles

at a given location r inside the star. This is something reminiscent from the Liouville’s

theorem for the density of points in a fluid in the phase space, after the integration in the

momentum volume.

And now we face the very relevant question: how do these concepts relate to the

composition of neutron stars? Much in the same way a given equation of state determines

a particular composition and a particular sequence of stars in the mass-radius diagram,

it also yields a particular expression of information entropy through the density profile of

each star in that sequence. Hence, we can calculate the disequilibrium and the complexity

and say something about which is the state of matter inside the compact object.

In this contribution, we calculate these statistical quantities for two sequences of neu-

tron stars: one of hadronic composition (obtained with the SLy4 equation of state) and

the other of strange quark composition (obtained with MIT Bag equation of state in the

context of the anisotropic exact analytical solution).

We show that the complexity of the two sequences is very low and that there is a trend

for these stars to be at a state of minimum complexity and by comparing the outputs, we

suggest a hierarchy of equations of state to be realized in Nature, e.g., we suggest that the

quark equation of state would be preferred or would be more probable to occur after the

events that lead to the formation of the compact object. However, we also discuss that

even if that is the preferred equation of state, the central density barrier could prevent its

realization.


To make the link with the thermodynamic of neutron stars is our next goal and one

should keep in mind the G. N. Lewis statement about chemical entropy (Lewis, 1930),

“Gain in entropy always means loss of information, and nothing more”.

We now refer the reader to our paper on this subject where we discuss our contribution

to the problem.

Section 3.1. Entropy, complexity and disequilibrium in compact stars 87

3.1 Entropy, complexity and disequilibrium in compact stars


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Chapter 4

X-ray astrophysics as a tool to study kilohertz

quasi-periodic oscillations and time lags of the X-ray

emission and to probe the environment of neutron stars

4.1 X-ray astrophysics

The roots of high energy astrophysics are in the beginning of the 20th century, when

Victor Hess (Hess, 1912) found out that the average ionization of the atmosphere above

∼ 1.5km was higher than the ionization at the sea level. This ionization is due to the

cosmic rays, as named by Robert Millikan in 1925. After the World War II, the advent

of sounding rockets made possible the discovery of sources of high energy radiation from

space. In 1949, a flight carrying a Geiger counter showed that the Sun emits X-rays. But

it was only in 1962 when Riccardo Giacconi and his colleagues (Giacconi et al., 1962), after

setting up an experiment in a sounding rocket whose original purpose was to study X-rays

from the moon, that the field of high energy astrophysics was effectively born.

In that flight, the team discovered the brightest X-ray source in the sky, Sco X-1,

and a completely unexpected diffuse glow of X-rays coming from all direction – the X-ray

background. See (Melia, 2009) for a more detailed historic view.

The reason why the high energy astrophysics is so recent when compared with optical

astronomy and is mostly (but not exclusively1) space-based is that the atmosphere is

opaque to most of electromagnetic spectrum. See figure 4.1.

1 At very high energies, the atmosphere itself is a detector since its particles are collisional targets for

this very energetic radiation coming from space. When such collision happens, the outcome is a cascade of

new produced particles that produces a blue light known as Cerenkov radiation detected on the ground.

94Chapter 4. X-ray astrophysics as a tool to study kilohertz quasi-periodic oscillations and time lags of the X-ray emission

and to probe the environment of neutron stars

Figure 4.1: Atmospheric windows: how deep from the space radiation of different wavelengths

can penetrate Earth’s atmosphere.

Another feature of high energy astrophysics is that we have much less photons than in

other wavelengths since it costs more to Nature to produce them. Because all of this, the

field is a challenge from the point of view of experiment and of the theory. First, because of

the complexity of the detector (as we shall see below), but also because of the interaction

of matter and radiation under extreme conditions is at the limit of the known theories.

The energetics of the processes involved are extreme (Melia, 2009). The release of

gravitational energy of an object of mass m falling onto a typical neutron star of mass M⋆

and radius R⋆ is

Eacc =GM⋆

R⋆

≈ 1020erg/g, (4.1)

which is about 20 times greater than the energy released in the nuclear fusion of hydrogen.

As an example, during the accretion processes, the strong gravitational field acts more

on protons, whereas the radiation release from the fall and from the central object acts

more on electrons, because the latter has bigger cross-section. But because the Coulomb

interaction loosely maintains the plasma coupled, there is a balance between the gravita-

tional and radiation forces that prevents the accretion. This is the Eddington luminosity,

Section 4.1. X-ray astrophysics 95

Ledd:

Ledd ≈ 1.3× 1038(M⋆

M⊙

)

erg/s (4.2)

whose black-body effective temperature is ∼ 107K, in the X-ray band.

Because of this extreme energetics, we need special methods to focus and to properly

detect high energy photons (Melia, 2009, for a more complete description). Taking X-rays

as example: unlike the optical and UV photons, they reflect from a surface of conducting

materials only at high incidence angles (for 1keV photons, i = 87o). To focus them on the

detectors, we need a set of mirrors assembled in a shape of nested barrels (see figure 4.2).

Figure 4.2: The mirror assembly to focus X-rays is composed by a set of paraboloid and

hiperboloid mirrors nested. The satellites are usually very long.

On the other hand, the detection is not simple. The first devices for detecting photons

up to ∼ 20keV were the proportional counters. They are nothing more than a gas-filled

discharge tubes with a voltage drop across the gas. When a photon enters the tube, it

produces a high energy electron which, in turn, initiate a cascade of electron-ion pairs. This

cascade produces a current proportional to the incident photon energy and this information

is enough to reconstruct the X-ray spectrum.

Today the devices to detect photons up to ∼ 20keV evolved to the microcalorimeters.

Here, the photon energy is measured after it has been converted into heat, without wor-

rying about the characteristic charge transport properties of the detector. Generally, it is

composed by an absorber, a temperature sensor and a link to a heat sink.



For energies even greater (∼ 20keV to several MeV ), the photons are very penetrative

and we need another kind of device: the scintillation counter. Its basic element is a crystal

of CsI or NaI. The idea behind is that the energetic γ-ray Compton scatters (inelastically

at this energies) several times before the photoelectric absorption occurs. The photon loses

energy at each scatter and the ionization energy lost is converted in visible light, which is

converted into an electrical signal by a photomultiplier tube.

The late 1990s and the beginning of 2000s were the dawn of a new era in high energy

astrophysics with the launch of Rossi X-ray Timing Explorer (RXTE), the XMM-Newton

and the Chandra X-ray Telescope. Chandra and XMM are kind of complementary in

imaging and in the spectroscopic qualities: while the first has an unmatched angular

resolution, ∼ 0.′′05, eight times better than other satellites for imaging, the second has an

unprecedented large effective area, ∼ 1500cm2, providing a high energy resolution.

But our work in this Thesis is about timing in low mass X-ray binaries (LMXBs), or the

X-ray variability of these sources. For this task we used data from RXTE. Its strength is

its unparalleled temporal resolution, capable to discern variability on timescales of months

down to microseconds, in an energy range from 2 − 250keV . It was the RXTE that

discovered the kilohertz quasi-periodic oscillations in compact binaries, the subject of this

chapter.

Millisecond variability naturally occurs in processes of accretion onto a compact stellar-

mass object and these rapid variations arising from the inner accretion flow are stochastic

in nature. Therefore, statistical techniques are necessary to study them and the Fourier

analysis is the commonest used tool (van der Klis, 2006, see this reference for a more

detailed description).

The power spectrum is obtained from the Fourier transform of a X-ray lightcurve,

which is a time series, to provide the variance in terms of the Fourier frequency. The

power spectrum has many components in a frequency range from millihertz to kilohertz

(the highest frequencies so far discovered). Broad structures are called “noise” and narrow

ones quasi-periodic oscillations. In figure 4.3 we can see the some components of the power

spectra. Some components are described below (van der Klis, 2006):

• Power-law noise: follows a power law Pν ∼ να with 0 < α < 2.

• Band-limited noise: is the noise that steepens towards higher frequency (the local

Section 4.1. X-ray astrophysics 97

L

Lb

L

L

b2

VLFN

LhHz

uF

(a) Estado F

LhHz

Lu

Lb

VLFN

Lb2

LG

(b) Estado G

Lu

Lb

Lb2

VLFN

LhHz

LH

(c) Estado H

Figure 4.3: Variability components. The letters F, G and H stand for the position of the source in the

colour-colour diagram when it was in the soft state. This figure is from reference van Straaten et al. (2003)



power-law slope, −d(logPν)/d(logν), increases with ν) either abruptly or gradually.

• Quasi-periodic oscillations: is a finite-width peak in the power spectrum, generally

described by a Lorentzian, Pν ∝ λ/[(ν − ν0)2 + (λ/2)2]. ν0 is the centroid frequency

and λ is the full width at half maximum. The quality factor is the quantity Q ≡ ν0/λ

and Q > 2 denotes a QPO, by convention.

In the spirit of Fourier analysis, we describe in the next two sections the tools and

methodology used to study the kHz QPOs and the time lags in the source 4U 1608–52.

As we have already said, the upper kHz QPO is usually associated with the radius of

the compact objects through expressions for the orbital frequency and ISCO. Below we

show how it is done.

For equatorial circular orbits in a Kerr space-time (the space-time around a punctual

spinning mass)2 the orbital frequency is given by:

νφ =

√

GM/r3

2π

1

1 + j(rg/r)3/2= νk(1 + j(rg/r)

3/2)−1, (4.3)

where νk is the keplerian orbital frequency, rg = GM/c2 and j = Jc/GM2 is the angular

momentum parameter. Beside, J = Iω and ω = 2πs, where s is the spin frequency of the

compact object and I = 2MR2/5 is its moment of inertia.

To first order in j, the ISCO radius and its corresponding frequency are:

RISCO ≃ 6rg(

1− 0.54j)

(4.4)

and

νISCO ≃(

c3/2π63/2GM)(

1 + 0.75j)

(4.5)

With the equations 4.3 and 4.5 it is possible to constrain the equation of state (see

figure 4.4).

2 The Kerr space-time is a good approximation to the space-time around a spinning neutron star; more

generally, the metric depends on the EoS.

Section 4.2. Shift-and-add 99

0

0.5

1

1.5

2

2.5

3

0 5 10 15 20

M [M

sun]

R [km]

SLy4Quarks

Figure 4.4: The outermost exclusion area is the constraints to M and R from the work with 4U1608–52

for ν → νISCO = 1064Hz whose spin is 619Hz. The innermost exclusion area is for a fake source for

which we imagine νISCO = 1220Hz with spin 353Hz; the two equations of state shown in red and green

are for strange quark star following the MIT Bag Model and for hadronic stars following the SLy4. The

solid black area indicates the black hole line formation (M=R/2,964).

4.2 Shift-and-add

Now we describe a very useful technique to deal with the very fast variations in the

lightcurve, the kilohertz quasi-periodic oscillations (kHz QPOs) (Mendez, 2001). As we

have seen, we use Fourier techniques to obtain the Power Density Spectra (PDS) and to

study the temporal features of the lightcurve. This technique is called shift-and-add and

we will describe it below.

Suppose you have a lightcurve with 512s from one observation. In order to study

phenomena at small time scales, we divide the observation in contiguous segments of

uniform length to set up the required frequency resolution: for example, we can divide

the observation in 8 segments of 64s. Now, on each segment we perform a Fast Fourier

Transform (FFT):

Sj ≡

N−1∑

k=0

ske2πijk/N ≡ S(fj), (4.6)

where sk is the time series, or our counts/s, in a way that we have, in the end:



Sj,0 = Re[Sj,0] + i× Im[Sj,0],

Sj,1 = Re[Sj,1] + i× Im[Sj,1],

...

Sj,7 = Re[Sj,7] + i× Im[Sj,7],

in which each segment spans a wide frequency interval. Thus, we obtain the PDS perform-

ing the multiplication by the conjugate complex, Pk(fj) = S⋆j,kSj,k = S2

j,k:

S2j,0 = Re2[Sj,0] + Im2[Sj,0],

S2j,1 = Re2[Sj,1] + Im2[Sj,1],

...

S2j,7 = Re2[Sj,7] + Im2[Sj,7],

Thus, we have just obtained the PDS for each segment, as illustrated in the figure 4.5.

But we are interested in the PDS of the whole observation that we get by averaging over

all segments, e.g., all k from 0 to 7, in the sense that for each frequency (x-axis) in the

PDS we average the value of its correspondent power (y-axis):

P (fj) ≡ 〈S2j,k〉 =

S2j,1 + ...+ S2

j,7

N. (4.7)

We end up with figure 4.6, the PDS for the whole observation.

If the kHz QPO is strong enough to be measured in short time segments then we

can shift the frequencies of all PDSs to a reference frequency just before to perform the

operations and the average: Sj−j′ ≡∑N−1

k=0 ske2πi(j−j

′

)k/N ≡ S(fj − fj−j′ ).

Thus, Pk(fj − f′

j) = S⋆j−j

′,kSj−j

′,k = S2

j−j′,kand the average over the intervals becomes

Section 4.2. Shift-and-add 101

Figure 4.5: Power density spectrum for eight segments of 64s showing one kHz QPO each.

Figure from reference Mendez (2001).

Figure 4.6: Average of the eight PDSs of the previous figure: we get the PDS for the whole

observation. Figure from reference Mendez (2001).

P (fj − f′

j) ≡ 〈S2j−j

′,k〉 =

S2j−j

′,1+ ...+ S2

j−j′,7

N, (4.8)

where the frequency f′

j is the shift frequency for each interval. For example, if we chose

the reference frequency to be 450 Hz and the kHz QPO is in 400 Hz, f′

j = −50Hz; if,

instead, the frequency of the kHz QPO of the second interval is in 550 Hz, f′

j = 100Hz.



Applying the frequency shift we gain statistics because, since the S/N ratio for the

QPO is ∼ (T/W )1/2, where T is the total length of the observation and W is the width of

the QPO, the alignment applied to the power spectra increases the statistical significance

of this feature, making the QPO narrower. The result is striking (figure 4.7):

Figure 4.7: Average PDS of the whole observation after the shift-and-add technique. Notice

the gain in significance and the consequent appearance of the second peak of kHz QPO.

Figure from reference Mendez (2001).

Not only the stronger (lower) kHz QPO becomes even stronger, but now a “hidden”

(not significant) kHz QPO becomes apparent (significant): the upper.

In our work, we applied for the first time the shift-and-add technique to calculate the

time lags in the frequency range of the kHz QPOs, as we shall discuss soon.

4.3 The coherence function and the phase and time lags

In the same way one calculates the PDS, one can calculate quantities that are defined

for two or more concurrent processes. These quantities are related to the cross-spectrum.

Time lags are Fourier-frequency-dependent measures of the time delays between two

concurrent and correlated time series, in our case, between two X-ray light curves in

different energy bands (Nowak et al., 1999).

Given two light curves, s(t) and h(t), it is possible to find a linear transformation

Section 4.3. The coherence function and the phase and time lags 103

h(t) =

∫

∞

−∞

tr(t− τ)s(τ)dτ (4.9)

where tr is the transfer function between s and h. Taking the Fourier transform of this

convolution we go to the frequency space where H(f) = Tr(f)S(f) are, obviously, the

Fourier transform of h(t), tr(t) and s(t).

The transfer function is something reminiscent from the signal processing analysis. This

function is a linear mapping of the Laplace transform of the input signal (analogue to s(t)

in our case) to the Laplace transform of the output signal (analogue to h(t) in our case).

On the other hand, the Fourier coherence is a Fourier-frequency-dependent measure of

the linear correlation between that two time series or light curves. The coherence function

(Nowak et al., 1999) is defined as

γ2(f) =|〈S⋆(f)H(f)〉|2

〈|S(f)|2〉〈|H(f)|2〉. (4.10)

The coherence function, then, is a measure of the degree to which the transfer function

Tr(f) is constant for the data segments and frequencies over which we took the average.

If Tr(f) is constant throughout the the measurements of s(t) and h(t), then the process is

said to be perfectly coherent (unit coherence). This means that we can predict the output

signal from the input signal and vice-versa. On the other hand, small coherence implies a

net phase lag.

The Fourier phase lag is the phase of the average cross spectrum, C(f). Hence, φ(f) =

arg[C(f)] = atan(

Im[C(f)]Re[C(f)]

)

, where C(f) = 〈S⋆(f)H(f)〉.

In the same way as with the PDSs, we have the cross density spectra (CDSs) which

are the products S⋆0H0(f), S

⋆1H1(f), ..., S⋆

7H7(f) (if N = 7, or eight segments as in

our example). Taking the average of these products we get the average cross spectrum

C(f) = 〈S⋆H(f)〉. The corresponding time lag is τ(f) = φ(f)/2πf .

The importance of these statistical measurements is that they can provide strong con-

straints on models relating the energy spectra and the aperiodic variability features like the

kHz QPOs. In particular, near perfect coherence can rule out models that assume spatially

extended sources or emitting regions like thermal flares etc (Vaughan and Nowak, 1997;

Nowak et al., 1999).



4.4 Time lags in the kilohertz quasi-periodic oscillations of the low-mass

X-ray binary 4U 1608–52

This paper have not been submitted yet.

Section 4.4. Time lags in the kilohertz quasi-periodic oscillations of the low-mass X-ray binary 4U 1608–52 105

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-0.08

-0.06

-0.04

-0.02

0

0.02

4 6 8 10 12 14 16 18 20

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sec]

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0

0.2

0.4

0.6

0.8

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910-993 Hz993-1040 Hz

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Chapter 5

Conclusion

At long last we reached the end of a first stage of our long term goal of advancing in

the understanding of the superdense matter at the interior of neutron star. That is, we

have contributed to the big synthesis needed for this goal, as described so enthusiastically

in the beginning.

Admitting from the very beginning the interesting possibility of strange matter to be

the ground state of matter, which can be formed in the so high density interior of neutron

stars, we begun our study by the mathematical framework that describes these compact

objects.

We obtained exact and quasi-exact analytical solutions parametrized uniquely by the

central density with valuable predictive power for many situations physically and astro-

physically relevant. These solutions were obtained from the Einstein equations for a spher-

ically symmetric and static perfect fluid, which lead directly to the relativistic version of

the hydrostatic equilibrium equation, when supplemented with the equation of state of

dense matter (here, the strange matter equation of state from MIT).

Here, we called the attention to the exact solutions with the anisotropic pressure and

with the electric field. These two solutions, while exacts from the mathematical point of

view, still need to be physically explained. It is necessary to substantiate theoretically

the existence of the anisotropy in the pressure or the existence of a radial electric field

running through the whole extension of the star. The case of the radial electric field

seems actually unlikely and perhaps the insertion of the radial electric field degree of

freedom is only a mathematical artefact. However, several authors seriously considered

them (Ghezzi, 2005; Ray et al., 2006; Negreiros et al., 2009), since the electric field allows

118 Chapter 5. Conclusion

stars with very low masses up to the causal limiting mass of 3M⊙. Their actual occurrence

in Nature should be considered as open. The case of the anisotropy in pressure is not

strong either, but nevertheless seems to be more plausible, since it could be related with

the velocity distribution and other mechanical statistical properties of particles (Gleiser

and Dev, 2004). The quasi-exact gaussian solution is also physically relevant, but only in

a limited range of central densities.

Noticing that there are many proposals in the literature for the structure equations that

directly employ the strange matter equation of state, we move on to the analysis under the

point of view of information theory. Our intent was to find out, from the perspective of

the self-organization of the system, if we could infer something about the energetic costs to

form a strange star instead of a hadronic star. Our results indicate that strange stars should

be energetically favoured, although there is a minimum density that has to be achieved

when the contraction of the proto-neutron star happens for SQM to form (Benvenuto and

Lugones, 1999). A strange star has low complexity, like the hadronic stars, but because

its disequilibrium is lower than for hadronic stars, we conclude that strange stars are “less

ordered”, which is easier for Nature to form.

These two theoretical approaches so far discussed are complementary to each other

about the strange quark hypothesis and are in accordance with the advances arising from

the theoretical physics of strong interactions.

From the observational point of view and assuming again the strange matter hypothesis

as true (surely in the colour-flavour-locked version), we studied the mass vs radius for four

astrophysical systems for which there are such measures. From this “observational” mass-

radius relation we showed that, although the error bars are large and we indeed need

of more accurate and precise measurements, we could constrain the energy gap of CFL

and the mass of strange quark in accordance with what we could expect from independent

sources of information. This form of strange matter has a lower energy per baryon than the

previous unpaired version and can be even more “fundamental”. In spite of this, we still

cannot assure that observations favour the strange matter composition. The crucial factor

is not in the discovery of higher masses, but mainly in the detection of “neutron stars”

with very low mass and measurements of their radii: radii R⋆ ≤ 9km would definitely

indicate that we are dealing with self-bound stars (quarks), while radii R⋆ ≥ 9km would

Chapter 5. Conclusion 119

indicate hadronic stars. And so we move on to the next step: the analysis of the X-ray

emission of the low mass X-ray binaries that contain a neutron star.

At this stage very early development, we studied the variability in the X-ray lightcurve

of the system 4U 1608–52. For a long time we know that this system presents two kilohertz

quasi-periodic oscillations, known as lower and upper kHz QPOs. Generally we associate

the upper kHz QPO with the inner radius of the disc or with the innermost stable circular

orbit (ISCO). Therefore, it is possible to derive the radius (and sometimes the mass) of

the compact object or, in the worst case, some upper limit for this quantity. From the

highest upper kHz frequency detected in our dataset, νupper = 1064Hz, we put an upper

limit of 17km on the radius, for an assumed mass of 1.74M⊙ (without including relativistic

effects). We also studied the time lags in the frequencies of kHz QPOs and we found out

a strong dependence on energy, but a weak dependence (or none at all) on the Fourier

frequency. The dependence on frequency would indicate different locations to where the

lags are produced, again putting limits on the parameters of the compact object. Not only

this: we also studied the intrinsic coherence. For all the energy and frequency bands the

intrinsic coherence is of order of unit. If on the one hand the time lags impose constraints to

the size of the emitting region and to its density, the intrinsic coherence rules out models of

extended emitting sources. Models that explain the lags and the coherence have potential

to provide constraints on the neutron stars parameter and also to the geometry of these

systems.

Ultimately, from our calculations we can state that the astrophysical measures cannot

rule out the strange matter in no way, remaining this form of matter as one of the most

fantastic possibilities to be confirmed or excluded in the next years.

For the future, our goals are to study other systems LMXBs with a much bigger dataset

in order to analyse the kHz QPOs and time lags and the intrinsic coherence. Additionally

we intend to study anomalous X-ray pulsar and soft gamma repeaters, with the aid of

the exact solutions from which we intend to derive observable quantities of neutron stars,

without forgetting the interesting new approach of information theory. This is a long term

project, but we believe that will lead us directly to a revealing synthesis about the true

nature of matter inside the neutron star.

120 Chapter 5. Conclusion

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Appendix

Appendix A

QCD Parameters

A.1 Self-bound models of compact stars and recent mass-radius

measurements

Here we used the Particle and Nuclear Physics approach discussed in chapter 1 to

extract information regarding the parameters of the strange quark matter equation of

state. The study was performed with two versions of exotic equation of state, namely, the

MIT Bag Equation of State (in which we have unpaired free uds quarks) and the Colour-

Flavour Locked Equation of State (in which the three quark flavours can form Cooper

pairs which allow further lowering in the energy per baryon in the system making it more

stable).

I want to make here a special acknowledgement to prof. Laura Paulucci Marinho since

without her major effort in doing the statistical fitting procedure and without her vast

knowledge on the subject, this work would not have been possible.

In this contribution studied strange quark star with pairing (the so-called Colour-

Flavour Locked, or CFL, equation of state) and we show that the parameters of the CFL

can be effectively constrained only when our instruments are capable of producing mea-

surements of masses and radii with sufficiently small error bars.

We also discuss how the controversy in radii measurements quoted in this paper dra-

matically affects the results, not only concerning the CFL, but also in general.

130 Appendix A. QCD Parameters

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®®´ ¯ ¯®v7ae ad 76e Ùve®®´

7® a ¯®e5 6a6e5®7a ² aµ °de ® µ

ß 7aad 7a 7® ¶Ã 6a7¯5 5ad®ad7a¸

² ¶ a¶ ad °56 ° ±67® ®±µ®76e

°®5°a® ² °®® ad d75® ² a5a ®®

ºÅÀÆÅÛ¼ 6a e± ve7a¸ v7¯7e7µ ² ³76 Â5Ã

® °de® ß aeµÈ´ 7a ±765e´ v7¯7e7µ ²

6e®® ² É·L ®a¸ Â5Ã ° °de® ¬µ e7Ãeµ

¯ °® ²v¯e 6ad7d® ² ®e²Á¯5ad ®®´ 7a

®±7 ° 6°±e³ °de® 6a ¯ dv7®d ¹² ³Á

°±e 7a°d7 ±®® ®56 ® LÃ7aÁàv67aa7ÃvÁ

·5edÁ·ee ¹Là··½ °µ ¯ ±®a ºÅÜÆÅÇ¼½´ ad µ

¶5ed 6a®75 ¯a6°Ã 7a ®7a¸ ³76 6°±6

® 6°±®77a

×a ® ² 7® ¶Ã ¶ Â5a77veµ ®¶

665 ®±µ®76e °®5°a® 6a edµ 6a®7a

7°±a ±°® ² áÉâ´ e7Ã É± ±7 a¸µ

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7® 7® da ¸7va ®5¯® ² ±®a d

¢¢£ æ¤æçèé¢é

ßa 6a®7d7a¸ 5a±7d ®a¸ Â5Ã ° ¹êëì

°½ Â57a ² ® ¹Eí½ ®®5°® ®7°±e®

²° ¶a äå î ïð ñ ò óô þ õø´ ¯7a¸ ñ´ ô´ ad ø´

a¸µ da®7µ´ ±®®5´ ad ¯¸ 6a®a´ ®±67veµ

¬ aaÈ ®a¸ Â5Ã °®® ® ²²6 ² 7a6®Á

7a¸ a¸µ ² ®µ®° º¿¼ àa ad´ 7²

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a ²²6 ² e¶7a¸ a¸µ ² ®µ®°´ °Ã7a¸

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a°76 ±a7e ² É·L ®a¸ Â5Ã ° º»Ç´»Ø¼ 7a

ùif]fhúbY]Nfn=b\npYKnXNûObpNbnKbpOp<<=úpibX<n;?pnXN

rüýÿPD :RPRx D 8 T0@TT0 9QTVV^

1=211= = 1 211 A S

Section A.1. Self-bound models of compact stars and recent mass-radius measurements 131

b t oo 2, b t po og p fo t

p, v bg

C ¼ 3

222 þ B (

I t epo, t tt t m

cmp f qo mtto wtt po p tom

coop t t og f t qo ct

, t po og p t fo pom

to, t f b d fctt mo

(, fo emp, [ ] fo foto c mo

ctt m T tomgmc ptt fo t

po tt t

¼

i!u"#"$

1

42i% 2

i 5

&'2i þ

3

&')i l*+

iþ%

'i

(

wt ¼ -u þ # þ $Þ=3 t cmm om

mmtm, % ¼ & 2 þ'2$=&

T E. dg bt bg ot t og

tg / t t poo P fw

/ ¼ 306 P (7

wt t potc tg v bg 06 ¼ -%8 þ &2Þ=2,

t poo P ¼ C

c f t fct tt '$ 9 :, t E. v t b

cmpt mocg Hwvo, t E. w

omobg o bvo [;] c f t, mc

wo b oo t ctoct mp E. tt

c t ffct f po vg to qo,

p t o bvo t cot ett W tot o

g, t, bg ptt o E. fo mtto

W v pomto< t qt f tt fo

mtto

P ¼ >-?2@ 4BÞA (D

wo > pomto ot wt t tff f t E.

B cotg ot t t b ctt B p

pomto Wt t ppoemt, t g t w

tt t TmFppmoGff (TFG t E.

qt c b wott m fomJ

KLM ¼'M@M

x21þ

KM

@M1þ

4x8KM

'M1&

'M

x

NO(Q

'LM ¼ 4x2@M (R

KM ¼ >-@M 1ÞA (

wo t too qtt o t m m,

poo, m tgS x t m o,

> U O8VX ffctv pomto wc p

t qtt '$ cmpct bt vog m

ct wg

Fo m o w t v g wg t pg

t ct (YZ\^_ f t memm m t coo

pt o, , dtt cov ` ¼ `-xA >Þ T

YZ\^_ tog p, o pomto<t, >

T o t v, t o t memm m

c f t, w t ctoct vo m

o otp vog > wt ccptb o,

t c, fom aa t a7R T cmpg wt bov

t, w t oc t t &hj W t dtt

epo fo t memm m ('M"knr f c

qc to coopt o (xsy pt

>, g t ot

'M"knr ¼ :1z3>2 þ :&:1> þ :::&z (|

xsy ¼ :&z&>2 þ :3&> þ :111 (;

~to t t, w dg d t ot

hknr ¼ sy?2

'knr

xsyA (a

v tt

¼ x?)

4B' ¼ 'M

?

4B8

Ho t pt wo t pc B ppo

tc tt t o t m vg p

B bt t ot btw memm m t co

opt o (w t pgc t o oto

ZJ t p g t v f > T fto

og pot Wtt o ppo m t b

oc fo g o qt f tt

om t ffctv pomto f bv, w v

g< t omt f t mo ot b

t fo ffot t f pomto wt t fo t

fw toJ D Ra|Q [a], EF DQD| [],

D |a7a [], .~ RD7a [7] T ot

o pot t fw ct

T v f > fct f B c b cto

bg t bov m f .~ RD7a, c

o tt t memm m pov bg pcdc E.

t t q t m f t vg to T c

b bt fom Eq (|(a o bg vog t v f

> B Eq (D ctoct t coop

mo qc Wt t povg tom

o v ot t mo ot fo

t ffctv qt f tt t cto

v, w cmpo tm wt t to to to

wt 3 m poc T ffctv pom

to, t po (>A B, tt w b tb fo

ep fo t t t m tm, t bov

m o to opctv ooo bo, o t

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c t t l w t lt pl

F 1 T tpl p pl t t l w u

t pt t mppu lpt w t t

pl t pm Fu

W pll p v pw ww p Be tpt

cp v cmptlt wt pll u mpumt t

t A p cuc t mppu lpt w

t t pl F 1 p plmt cmltl

vlp

W wul l w t lpt t ctv ppmt

tm tu t ppl wt t ppmt

tm t CF mptt tpt Utuptl pltu

t E t p lp pv t lppt

E (1) t l ppl tcpll ul m mp

pampt p mp Wtut t lptt cpt

ctl lpt p Be t s p B Pvu

pttmt t uc p lpt clu t w

Al [2 ] w p m ppmt E

wp wc QC! cct t t u

t p ppmt " # 0:3 pv p

T ppmtpt l p w ap

$ t tm pmc ttpl t up p

p

%&' ¼ *3

4+,-.$

. þ3

4+,-,$

, þ Be (11)

wt -. ¼ / * " p -, p ppmt I t p

pc t tp up mp p p ppmt pct

-, pl p Be tm t t pt wc

t tpt tw up p uclp mptt p

ccu p t w tt p tp

Altu t pv pl t wt -, ¼ -,ðs; Þ

p cp pct CF up mptt ulum

(wt t pmw) ppt t tm pmc

ttpl a t .s wt l t tm

tpl t $. $, p $5 t cmcpl ttpl

tl p p lp p tvpl c pll

t ppmt p w tpl [16] W pv

c t t mp t pampt tpt w

p t pl t wt p Be p uct s B p

ppl tcpl cl m

c t t l t tp t a lpt

t ppmt E t t CF t v

t ctv u w lppt w pv tp

t ull upt tpt CF tp up mptt p

wtt E (1)7(8) p ppl t pmt t

mppu lpt tp t pm t tp p

W pv cv pcctpl p t

ppmt9 t p ctpt B < => M?@DfGH t u

cuct p 0 J K =0 M?@ p t tp up

mp s L /00 M?@ Rct mpumt v p v

c vplu t N up mp pt t cpl

2 OS9 VX:4 Y /:= M?@ [26] At p mpll cpl

t vplu ul v ( [8Z] Pptcl

Pt\up mp p c t)

w c t pt t vplu 1ZZ ^S p p lw

up

At lv t T_S upt vpl t (s

B ) p t cmp t vplu mp p pu

wt t mpumt u cpt t t p

mt u t v t mppu lpt

w F 2 T tmpl` vplu t ppmt

u p B b dV M?@DfGH b =0 M?@ p s b

/X0 M?@ (wll wt t ww tplt CF

tp up mptt tu ut t up tp

g

hijkn hijko hijkj hijkq hijkk

rx

yz|~

n

nio

niq

ni

ni

o

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rs v s am wh 3: o I shor

os b r vhoss h s h

hs s h 3 br s h scra

999. f mcbo s h v fw ras

f frm vovr h oss Thf h a

m b srr s cbo

W s h ohah h e f s f C

sa eq m oo css o bhv

h css vss a rs rcr wh

cmp f h c P ¼ ð2 BÞ s

mr h fs r B b r s

rcr fm h h S w wqa wh

mro whh srs ssm wh

ca cos m scra h rs

a so rrs c h ba s b hs

i h cr s Hwv h o f h

sa eq mss s m sbo I h oss rsssr

[ ] h hs shw h B bsrs ba or

h vm css (r h C chs os h ac

cm cs rro m whh s c

co !" wh h cw ss #cs s mr

S v smcor shm w s h B s

rcr h ba s o b shor os

c rro ms whh mc f r

ma h sffss f h ES

I hs w v hva fr vos f h cms

h op h or h r wh 3

a wh foo e f s s mcor

h sh ar am Ths qr f cbom

s os vr h ors #css mcor b

$v r Hvh [1] P ¼ %&þ' ½

2 ð ) *ÞB+

h mcor rcr f h cms " B

r (r h h o ffs h f

h M,-/ R l0567 r bos sm #

rm f h cm sc

Ah ao sc wh f a s h v sf

s f h mssrs v f cr sa eq

m I or b br h smco f h mro

sra s bw eqs h h

8 ;<=>

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U

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LYD

LYG

K

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KY?

KYD

KYG

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ny~ _

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zuk|gu ~gknj_ Ï gÎnxgj z xnjjun~j ugnkyj zu Z jknuj nug j|y zu k|g jgkj z nunxgkguj t¶gy ÐÑ

Á¿ Òg `¿ Òg ny~ Ó ¾` Òg ¿ Òg k |t|t|k k|g znk k|nk y k|j njg k|g xnjjun~j

ugnky j yjkunyg~ jgÑ ÐÑ k|g xnjj¶g jnu_

Ò_ ^_ d_ ÔÂ µÂµÇ È_ Â_ Õ¹ÇµÏÕ µÖÔ _ Åµ»»\ ÅÕ×Æ\µ ÇÂ\ÂØ Ô ÙÚÛ ¿º´¿¿º `¿

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p a oo x ( [7] fo ao

o t of t b pt wt c 0) A tp

of t a to oa aa pova

btt t wt t obva at a pp

t x owa T oa b

pott t woa b pob to

2M, fo xp, wt q

tt Eo tt woa o a to xp t v fo

a a a fo t t 4 1, E

1744, a 4 1, f t pp tt to b

t a ot

T v b , owv, tt t v of

a a fo 4 1, E 1744, a

4 1 v b t ava [] T o

ptto toa a t pvo w

ta pot ot pob tt o w

btt a t tt t a

a v fo t t Coa t t

atto v b t, Ltt, a Bow

[], w v pta t pvo to to vf

wt t CL qto of tt optb wt t

otto of t at pta b t T t of

o fo t v afft, to W

tt t v ow pt p pvo foa ow

bo xt wa o of pobt,

apta

T o b fo t t, oa to t,

Ltt, a Bow, b t pvo

oaa, a t t ta wt o

v fo t a a (M :5M a

R k! wt o vb aff fo t

w t potop a fo poto o o

aa to b t t t a, R"# $ R;

t oo oto %c at fo t bv

o) T a ow to t of pt ms, &, a

' tt oa pp xp t at, b v

t of P* J114 t qtt pova

t ot fo t qto of tt I ot woa,

o +a v t of (&, ', ms) tt tf t

oato wt t v aa t

T bvo to v ow x

wt t t q a b

ott w t v of t ot two vb a

+xa A t p pt , t x

owa o

-./ 36839<=-68=

P t fo a a of opt

t v pott fo ot t opoto

Howv, ba b p, t tt t

o b t fo opo of t a

to o fo v afft opoto wt

at

Gv t pt otov of t a at

to qota t pp, t pp to b t to

b ova fo av of t oo of ta

o >t w o oa tt o wt

o b fo bot a a atto

( t of []) t pt of t CL

qto of tt b fftv ota tw,

v t of pt a+ t CL Eo

b a to t t at I t , o ava

atto tq a p qa to

ot t a R"# x bt oa

o oa ot tt a &e?? apat

qtt pta Eo fo CL t q

tt t o popoa , pfoa

wt ptto wt ota of+t oa

b xpoa W ta to aa t b@t ft

wo

A aato tt pot t v ow of

t t o to t obva (M ¼ 0:DF K

0:0FM fo 4 1) [], t ot w xpa

b oa t voto t T owaNN

xt o pott fo t qt of xot

opoto t to I ft, o+

t woa p a of R O Q k!wt t

fboa pot, w t t of, , R S

k! woa b tot ott wt t, pot to

o opoto W t t a of + aa

t fat opoto of opt t, t t

o +t v tt v af of pobt ft ot

of b of popo

U3V86X9YZ\^Y8_=

W w to owa t ppot of t ao a

Apo Pq ao Etao a o Po a t CdPq

A a CAPE A (B) W woa o

to t g o >o fo v f ao

o ttt

hij ln ruyuz |z~n |zn n |n i n

hj n |z~n uzn |n i in

hj n uz |n un ii iin

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[2 M Beaa P R 47 20 (20

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rr ri

[ J E Hrva O G Bve a H Vei P

Rv D 44 33 (

[0 R Ra T Sar E V Ser a a M Vv

A P (Y 0 (2

[ M Ar Ra aa S R a F Wi! P

Rv D 64 33 (2

[3 Ra aa a F Wi! ar"iv#$ a

rr ri

[ Ra aa a F Wi! P Rv L 6 2

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(22

[2 T Gevr F O! A CarraLavr a P Wrwi

Ar J 71 (2

[2 F O! T Gevr a D Pai Ar J 6%& 330

(2

[22 T Gevr P Wrwi L Ca'ara a F O!

Ar J 71% 3 (2

[2 P B D'r T Pei S M Ra' M Rr

a J W T H aer (L 467 (2

[2 M Ar J A Bwr a Ra aa P Rv D

6& 3 (2

[20 J A Bwr a Ra aa P Rv D 66 02

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[2 R Caaei a G arei Rv M P 76 2

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P Rv D & (2

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M G B DE AVELLAR J E HORVATH AD L PAULUCCI PHYSICAL REVIEW D 48 (2

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diferentes abordagens à composição e ao ambiente das estrelas … · 2013. 3. 12. ·...

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