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.
20 Chapter 1. Introduction
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.
22 Chapter 1. Introduction
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,
24 Chapter 1. Introduction
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
Section 1.2. Observation: pulsars and the neutron stars 25
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
26 Chapter 1. Introduction
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.
Section 1.2. Observation: pulsars and the neutron stars 27
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).
28 Chapter 1. Introduction
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.
Section 1.2. Observation: pulsars and the neutron stars 29
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.
30 Chapter 1. Introduction
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
Section 1.2. Observation: pulsars and the neutron stars 31
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
32 Chapter 1. Introduction
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
Section 1.2. Observation: pulsars and the neutron stars 33
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.
34 Chapter 1. Introduction
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.
Section 1.2. Observation: pulsars and the neutron stars 35
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).
36 Chapter 1. Introduction
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).
Section 1.2. Observation: pulsars and the neutron stars 37
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).
38 Chapter 1. Introduction
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).
Section 1.2. Observation: pulsars and the neutron stars 39
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).
40 Chapter 1. Introduction
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)
42 Chapter 1. Introduction
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.
Section 1.3. Nuclear and Particle Physics: theory of neutron star matter 43
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
44 Chapter 1. Introduction
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
Section 1.3. Nuclear and Particle Physics: theory of neutron star matter 45
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
46 Chapter 1. Introduction
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
Section 1.3. Nuclear and Particle Physics: theory of neutron star matter 47
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:
48 Chapter 1. Introduction
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,
Section 1.3. Nuclear and Particle Physics: theory of neutron star matter 49
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
50 Chapter 1. Introduction
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)
Section 1.3. Nuclear and Particle Physics: theory of neutron star matter 51
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.
52 Chapter 1. Introduction
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.
Section 1.3. Nuclear and Particle Physics: theory of neutron star matter 53
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 .
54 Chapter 1. Introduction
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
Section 1.3. Nuclear and Particle Physics: theory of neutron star matter 55
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
56 Chapter 1. Introduction
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.
60 Chapter 2. Analytical solutions in the construction of strange quark stars models
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.
62 Chapter 2. Analytical solutions in the construction of strange quark stars models
Itatoan Joan o Mod Pyc D
Von. 1 No. 12 (21 1 1
c Wond Sctc Pn Copay
DI: 1.112S212 11112
EAT A QA-EAT OE OF TRAGE TAR
M!"CI #. B. $% !V&''!") ad J. &. *"V!+*
,/03i3u34 5e 603r4/47i89 ;e4<=0i>8 e ?ie/>i80 6740<eri>809
U/iver0i585e 5e @84 H8uK49
@84 H8uK4 LXXLYZL[L\@H9 ]r8^iK)78r>8veK_803r4`i8g`u0b`fr<434/_803r4`i8g`u0b`fr
"chd 2 Mac 2
"hd 1 jay 21
Cocatd y J. Pnn
W cotct ad copa t wok a haty o pn odn o ta ta
any ypottcan nlod omct ad o a cond tan ho o t qakl
no pnaa. &sact qalsact ad can odn a sad to d t ot
coocan dcpto o t omct. ! pn ad ccn paatzato o
t h t o t ctan dty ad t dxc ao t odn
a spnctny ow ad dcd. I patcna w pt a odn tat wt a
#aa 8/083^ o t dty pon tat pohd a hy accat ad anot copnt
aanytcan tato o t pon 745uK4 a ann dxc o o o t tc
pottan.
e|4r50: Sta ta~ sact onto~ copact omct.
¡¡ ¢£¡
¡ ¡ ¡ ¤¥ ¦¢ §
¨ ¢ ¡ © ª¥ ¨ ¡
¡ ¢ ¡ ¡ «¬ ®¯
¨ °© ± ²³µ¶· ¢
¡ ¤¥ ¡ ¦¢ ¡ ¨ ¸¹ ¡§
¨ ¡© º ¨ ¡¡ ¨ ¤¥ » ¡
¤ ¦ ¼© ¶§© ½¡ ¡ ¡
¢ ¾ ¿±º ½ ¿ À ¡
¡ Á¬¿Â ÷
Ä ¡¡
¢£¡ Å ²¿Æ Ų¬¥ ¸¹
¡© ± ¡ ¡ ¢©
º ¡ ¦ § ¢ ¡¡ ¢£¡
¨Ç È É ²¿Æ ¢ ¥ ¢Ç
Ê ¡ ¨ ¨
1
Section 2.1. Exact and quasi-exact models of strange stars 63
1 M. G. B. d Avll nd J. E. Hovah
t ptty s sss 0 5, t mmu w by t qutm f
stte tug t w-ss sts cy Nwtm, gmty ms mcs-
mgy stg t t mg fcs t us f g tmmstmc s w2R 0 e
s ms w-kw, t s f ts bjcts c b scmb, m st ppx-
mtm, by t sutms f t mstm qutms f pfct mstpmc, sttmc
spmcy sytmc um (ws tmc ms gm by 2 = 2!"#r)$2 !%#r)&2
&2'2 &2 sm2('*+2*/
834
26(&* =
79!:%
&;
< !:%#r)
&2> (<*
834
?@(&* =
C9!:%
&
< !:%#r)
&2> (*
@9(&* ;
<
( 26(&* ; @(&**C
9(&* = 0> (D*
w t st qutm (ctct Fmcm mtmty* xpsss t cstm f
t gyItu f t ume
ttmcy, t t st t mit sttgms t s ts
qutms/
f t pssu, t smty tmc t ms gm (f xp, by
KLOKPz*, xct umc sutm c b fu by mtgtme Qw,
tms s t gut t y ct t qutm f stt @(6*,
smm sutm fuctmse
st, mf t qutm f stt @(6* ms gm (mee t um ms cctmS f
t bgmmg*, t mtgtm c b pf (t st umcy* t
pptms f t st s fwe
f bt t qutm f stt mtm fuctm (@, 6 f t
tmc* kw, tc f t tm syst c b ttpt,
bmg m g pssmb f y ctm us f t m ptse
t ms t mTcut t msm tt t tm ut c b s py UVPWXYP
tm syst smc pms gs f f (f xp,
ctmc pssu mstpy* t t xps f mfymg t mgmy ps
pysmc pbe Z s tu t tms pmt bwe
tms wk, w pf t typs f stums/
mg f stg st by usmg s f t w-kw xct sutms
(y t s tt t pysmc qumts* f t mstm qutm
f sttmc spmcy sytmc pfct um, ege t sutms btm by
t st sttgy[
mg f stg st by t tm utm, ege tmmg t
syst, m t t y t mcpysmcs m (by s f t
64 Chapter 2. Analytical solutions in the construction of strange quark stars models
Eat and Qua-Eat Model of Strange Star 1
s b s hv c v h sy p chs b
Gss s s s b ss
A cmps h bv ss h s ps s
s H by s w m h b cs h sysm wh
h s IT b m (s bw mcy
h sysm
Wh hs w shw h h ppcby h w w yc s s
h m m hs ss h h bs c by hv
m h sysm s c sc yc sc s wh
vm sysm h p s c h my
Th w shw hw h c m s m (spcy ps
s spy h s c cc c v vm
sysm h sm m pv sc ss yc s s s
ss
2 C i!"#
2$ %&'&)*+ ,.'/03&)*40.'/
As b chm cc s w hv p h smps mc
h mssss c s ps mp by Acc
Fh O (h AFO6 Ths s m y mv by h shps h
ps hy b h pc Fs 57 I s c h y h
m whch c s msss c b b
sy (s R 8 p h pm spc h ms
9 c h m hs w s smpy h scp h s m
s s mch s pssb pssby c sm (: 5; < cs s m
=>?@ 1@ BDJJKLNPUVDX YNPJ>UZ VNXDU>[P [\ U]N JUVDP?N JUDV JN^_NPLN LDXL_XDUNY `Z jXL[Lkq =DV]>
DPY xX>PU[@z
Section 2.1. Exact and quasi-exact models of strange stars 65
1 M. G. B. d Avll nd J. E. Hovah
7 8 9
0
0
R
s]
Fg 2 u e !"# "f !$e !#ge ! e%ue#&e & &u !e by ' &"&k( F$ # O #!"6
Fg 3 De#!y p" e "f f"u e e&!e !#ge !( & &u !e by ' &"&k( F$ # O #!"6
t)* +,-t t),t /4m*5:-,; ,/< *x,-t =>5 q4,?:@*x,-tC m><*;? <> />t m,t-) *x,-t;IK :t
:? 4?*+4; t> L,5,m*t5:i* ,;; t)* +*,t45*? :/ t*5m? >+ , ?:/N;* :/L4t L,5,m*t*5K -)>?*/
t> P* t)* -*/t5,; <*/?:tI Qc w):-) m4?t ?,t:?+I Qc S TUVWX :/ >5<*5 t> P* ->/?:?t*/t
w:t) t)* <*?-5:Lt:>/ >+ t)* m,tt*5 :/ t)* YZ[ P,N m><*;K w)*5* \ ^ =WXQ TUCV_`
[> jx :<*,?K ,??4m* t),t r=zC ^ |=zC ~ :? , ?>;4t:>/ >+ t)* <*/?:tI L5>j;*`
Z+ ,/ :? N:*/ +>5 |=zCK w:t) t),t L5>j;* :t :? L>??:P;* t> >Pt,:/ ?>;4t:>/? +>5
t)* L5*??45* ,/< +>5 t)* m*t5:- *;*m*/t?` [)*/ w* jt t)* L,5,m*t*5? ^ =QcC
,/< ~ ^ ~=QcCK ?> t),t t)* m,??5,<:4? 5*;,t:>/ >+ :? 5*L5><4-*< ,? ,--45,t*;I
,? L>??:P;*` +t*5 t):? :? ,-):**<K w* -,/ L5>-**< t> <*5:* ,;; t)* +*,t45*? >+ ,/I
N:*/ m><*;`
66 Chapter 2. Analytical solutions in the construction of strange quark stars models
Eat and Qua-Eat Model of Strange Star 11
Hwv ( v q b w) b
vb s phy y pb m h h w
h wk by D y Lk8s
() =
T h q w y
k ( = !) "( = !)# Th why h mp
h h # A m ph = $ h h
2%$ pp &'$# A m = ! h h h
v by 2%&'# Hwv h p*m y h
h b h m = ! q 2%# Th =
+ ! h ( ) + ! (# R# 9)#
() , = <
A p h. by h wh v / h y
,&'/ v y bv h p*m# 0
,&' = wy m h bj / h y bv w m
h p w (## h y y)# 0 ,&'
>
h y3 h bv w m y hh h /
h # Th y h y p wh h
v h bj# Th ,&' > 4 ,&' < (# R# 9)#
() 5( ) h pv
A = ! m h b h
h m h phy h (m) y
m hv pv v h b #
(v) "( ) h pv
Th p whh ppm by (m)
y m hv pv v h pp
m#
(v)676: ; c
'
Th p h h b m h h v y h#
O pph h pb m b b w h w kw
T m IV? Bhh I?? whh y pp ()*(v) y
m #
@C@C FGJKN JPU WXJYZ[\GJKN Y]^XNZ]PY
2#2## _`ixz IV z| ~|zi I
T m IV Bhh I w w kw mh
m y mp p v ph# v hy
1 ¡ ¢£¤¥¦ § 1¨ ©
ª 1 ¡ ¢£¤¥¦ « 1 ¬ « ¨
Section 2.1. Exact and quasi-exact models of strange stars 67
1 M. G. B. d Avll nd J. E. Hovah
re pysy su fu te reuremets ste te frmer set We
e teste te ppbty f tese smpe m es t te strge str prbem
mprg tem wt te umer resuts f FO preus ttempt be
fu Ref
We se tese tw suts s beuse te esty pre be wette
r er t repr ue te FO esty pre
T IV
s sut ws fu by m s sem pper () se te ste
euts e m e !"z #$%&/' = st met tt re ere ext
tegrt f te prbem e futs tus bte re
#*=
+'2
,2
'2
-2 +
'2
,2
0 #$= Q
2 +
'2
,20
834
c25(') =
,2
+6,2
-2+6'2
-2
+'2
,2
+
,2
'2
-2
+'2
,2
2
834
c79(') =
,2
,2
-26'2
-2
+'2
,2
:
Frm tese expresss eut f stte pys r us (bu ry ';)
te stt < te tt mss f te spere re respetey
5(9) = 5> ?9>
c2+ ?
9
c2+ 8
9>
c2
9
c2
2
9>
c2+ 5>
0
'; =-
6@C2
,2
-2
@C2
0 Q2 =
'2;
-2
+'2;
,2
0
DKLNPSUXLPSU =c2';
4
'2;
-2 +
'2;
,2
+'2;
,2
0
wt
834
c25> = 6
-2 + ,2
-2,20
834
c79> =
-2 ,2
,2-2
wt , - Q tree rbtrry stts
68 Chapter 2. Analytical solutions in the construction of strange quark stars models
Eat and Qua-Eat Model of Strange Star 1
T mh m i s s m s s by AFO wih h Tm
I si, w s p h mss h is h m , h is,
mi , R s h h mss h is h m i is
p Hw , h siy b s sim sy
(h p s R, whih i mi by h is
mss A q si is wh h h si qi s is mpib
wih h i xp ssi h IT b m W siy h h Tm
I is pppi , s b s i is i , b m impy b s i
p s h R p s i h m i m s Wh R s
p y h mss s ii h AFO is, h y hpp
p h siy h m Th , h qi s i s
m m m (m s s, spi is i m mii h
sm This b hi is is i Fi 4
I is h qi s h i s m m m is s
sib mp s, si i sh h m pp i s
m sh b h i his wy
B ! "
h i h sm wy s i Tm I, w xmi h #hh I$$ si
Th xp ssis i by
%&=
2(' + C)*
2 C)*. %
/= [(' + C)
*30*
+ 5 2 C)*(6 + 2C)*7*.
89:
c*;() =
2C)
2 + 2C)*
4C)(2 C)*
(2 + 2C)**
)+
' 2 C)*
2 + 2C)*
)*.
<>?@ @ VDG>DJ>KL KN JPU UXYDJ>KL KN ZJDJU \>JP JPU ]KLZJDLJZ ^_` jGUZULJ >L kKvzDLZ zK|UvZ@~
Section 2.1. Exact and quasi-exact models of strange stars 69
1 M. G. B. d Avll nd J. E. Hovah
8
c4p(r)
2 3Cr Cr
Cr( 2Cr)2 Cr
+Cr2 Cr (2 Cr)
[( Cr)/ 2 Cr( 2Cr)(2 2Cr)r
2 Cr
2 2Cr
r,
w C it stts
Tw ts w ssib i t st w tt t mss is f
mb s s !" # !"$%&O s f W bs ttmt t t
t t mss tb sit bi' t i*t ii i' t
msss -w0 w 0tbb m t t sm bm f t biit f
t ti f stt 5'i t ti ti f stt s 0ib
titis is t sfb t mb t 6 mtt
222 Qu79:;e<7=> ?7u99:7@ DFIeK
5 isti t t sit bs st 5LN (Li' 3) s''sts simP
b b mtiRti f t sit 5s simb 7@97>z
w ssm Sssi b tt b t 0tmiti
f t sstm (si t bi UVT ti f stt ws bs ims) Wit
ts ssmtis t fbbwi' Xssis w immitb ti i bs
fms
!(r) !"
cYZ\
]/\]^
c, (+)
p(r) c!(r) +
3, ( )
_(r) r
2r` g, (j)
YZkq\x
+!"Y
Z\]/\]^r`
c
2/!"r`f
r
r`
rc
+YZ\]/\]^r`
c4
2/r`fr
r`
rc4
8r
3c4 , (y)
YZkq\x
r`(8rYZ\
]/\]^!"c 8rYZ\
]/\]^ 2+r 3c4)
3c4(r r`) (8)
T bm s i tis w s t 0i' wt t tw Xssis
f X(|(r)) s (y) (8) i0bt 5btti0b 0 if t ftis
70 Chapter 2. Analytical solutions in the construction of strange quark stars models
Eat and Qua-Eat Model of Strange Star 1
i c b s i ic isi s s
w b i c b i i . Tis is w w sk b qsi
c s is i q i s s b.
W s s bck s si s sw i AFO
ii vs ss = 99 = 9 = 4
= 5 .
T i ws ssis ! "# ( sc c si
$ s c i c si %& wic c c sss
ii.
A iv %& b i s sqc
bcs c cc ss is ic s
si ss s.
Hwv i v issib s si s wi k
i ss icsis . L i is v (i.. i ci i
c AFO s s$ b ki sss c si
is s is R ! sc "# c b b i. As
i k w s ss b i ss' w i
si c i b ci i iv ic )("$
s b s wi i vs s s s b AFO.
T s i s i w i
i ! "#. Ts i s ws 2 s (i.. 2
vs c si $ cv wi ss ii i vs. T vs
"# ! sw i Tb .
*+,/0 13 V+/607 8: ;< +>? @ 8,B+C>0? ,y CDG87C>I Bh0 7+D0 JKN D+77 P+/603
UXY1Z[\ I]^D_` MjmpYJKN` ;<xzD @
|1 |Z1Z|~Z ~1Z1Z~ 1~Z|1~
|Z3 |Z1Z1Z ~~1~1 1~|1
|Z |Z1|~~ ~Z1Z|~ 1~|~
13 |Z1~ ~~|Z| 1~Z1Z~~
1 |Z1|~ 1~~Z 1~
1 |ZZ||| 1Z 1~~Z~Z
1 1~~|1 |1||1 1~1||Z
1| 1|11 1|11
11 1~~~Z| 1Z| 1|~|1~~
1Z 11~||1 1ZZ|Z 11ZZ
1~|~ 111~~~ 1Z~~
1~|1| 1||1| Z1
~3 11Z1 1|~ ZZ|~ZZ
~ 1Z|1Z 1|| Z1|~
3 1|~~ 1Z Z~|||
1Z~Z~111 1~Z11 Z|Z~
3~ Z~||~ 1|1~| Z1|1
3| Z~|1Z| 1|||1Z1 Z||Z
3 Z||Z~ 1~Z|~1 Z|~1|
3 ZZZ 1~1~~ Z||Z11
Section 2.1. Exact and quasi-exact models of strange stars 71
1 M. G. B. d Avll nd J. E. Hovah
t u fr , w prc t r ccurt fuct
t f t fr
() 24 99e0 242e05 +92e3!5" (9)
() ## 224e050
+$%#e&!&
+94e0'
( k)" (+$)
w r *8-/67 :&;6 : * +$k qutty f t r r uty
tt b pyc ctt (<, = >?@C) t t c f t bj ct
(D +$k)F
Wt t u fr , w y w r but t ccurcy f t
trc t eIKLNF W tt buty fr t qutty OtF Pt
rt Opr w r fr t rt Qt qut (+) t c fr
t c (2)F w r t y r y R r tS Tctuy t c b c ck tt t
r t ry fr t w r t (UF #), tt t wr fr t
t t (UF ) t t r qu c F V b r c qu c
f t X wt crct r f t rtt fr t w r t , r y
cu Y cF +F
P ury, w w tt wt y t c tr ty u (qut t
y t r ccut f t r tructur tt b by p cfy t
u ), t u f c b ccut fr t ytc tF V y t rZ
tur t pr f ty pr ur wt t [u \]s\?z,
wt t t pyc ru f t ^F Uy, t c b ccut
ytcy by t rt t [u ty pr _r u t buZ
ry ct, y t u wt `+$g b cu f t t f
t tt Opr fr t Oct u rc r utF W b tt t quZ
ytc QqF (4)(i) c b u fu fr ( ry) ccurt ut f t
t r tructur r ty f tutF
Pt wrt t t t t pt tt m rk3 pr t ry
r pprc, trt wt t r ttc ct f t ryc quZ
bru t xPV b qut f tt F V y fu ry ccurt pprOt
ru fru fr tr tr, t ttc c , y,
t rtt c F V r rrr r t +g fr t fr r c %g fr
Oy rtt trF
V r fru , w r, b c cr y ccurt w ;6 | 4F r
w ur rt fr ru, t t t r fru ,3
^ ~()& %
<&;6 ( k)" (++)
# × +$&
+ e7
77
KN<&^
2=( x) (+2)
W tr tw prtt pt but t tw Opr F Urt, t y prZ
t rt u fr t rZpr ur tut, tu t f tur u
72 Chapter 2. Analytical solutions in the construction of strange quark stars models
Eat and Qua-Eat Model of Strange Star 1
4 6 8
(
F 5 Th !"# b$w" $h $w% &'!))%") %* $h m$!# +m"$ ,./02 ") $h )$3!
*%! $h m%+ w$h 7c = 59: × 1;<> ?#m@ Th !3A) %* $h )$3! ) m3!k by $h v!$#3+ +"
3$ $h " %* $h #A!v) T%'p $h $w% &'!))%") B%$$%mp $h !+3$v !!%! 3) 3 *A"#$%" %* $h
!33+ #%%!"3$
CDG HI GJKIC GDD sGLiNGOPR HINJUsI GVI IWXLIssiDCs JLI DCOP JXXLDWiYJGiDCsZ [IN\
DC]R JOGVDU^V GVIP JXXIJL qUiGI ^ICILiNR DCI sVDUO] LIYIYHIL GVJG _I UsI]
` j xzx|I~Y GD ]ILiI JC] Z ]iILICG JOUI D ` _DUO] DLNI
GVI LINJONUOJGiDC D GVI OJGGILZ
VI J]JCGJ^I D DUL JXXLDJNV is GVJG _I VJI GVI XLDOI DL R R
JC] iC NODsI] DLY DL IJNV XDssiHOI JOUI D GVI NICGLJO ]ICsiGP Z Isi]Is
GVJGR DUL DLYUOJ JLI iC NDYXOIGI J^LIIYICG _iGV GVI DLYUOJ ]ILiI] HP VIC^
JC] JLKDZ
I JOsD XDiCG DUG GVJG LINICGOPR JLJiCR [NVJCIL\iIOiNV JC] |isVUsGiCVJI
YJ]I J CUYILiNJO NJONUOJGiDC iCGI^LJGiC^ GVI ]iYICsiDCOIss IqUJGiDC _iGV J
Section 2.1. Exact and quasi-exact models of strange stars 73
1 M. G. B. d Avll nd J. E. Hovah
-
-
-
-
6 8
(
r
r rrrr
Fg T s s ! Fg 5 f"# t "$% wt &c = 1' × 1)*+ g/, 3 T $0#!,s #
bgg#2 s !t"!$ ! t t4t
79:;<> ;q?<@9i: iC D@<@; Ci> I;:;>9K C;>m9i:9K m<@@;>L NO; piDD9P979@y iC DK<79:I @O;
Di7?@9i:D iC @O; NQV p>iP7;m R<D <7>;<Sy k:iR: <:S ;ep7i9@;SU Ci> ;e<mp7;U 9:
W9@@;:XD p<p;>LY ZD9:I @O9D p>ip;>@yU [<><9:U \KO<]:;>^_9;79KO <:S`9DO?D@9: Ci?:S
< j;>y ?D;C?7 I;:;><7 DK<79:I Di7?@9i: <:S S9DK?DD;S OiR @i >;DK<7; @O;D; ;q?<@9i:D
9: i>S;> @i :S @O; Di7?@9i:D Ci> <>P9@><>y C;>m9i: m<DD;D <:S 9:@;><K@9:I D@>;:I@ODL
u: Dp9@; iC @O9D I;:;><79@yU R; pi9:@ i?@ @O<@ i?> p>iK;DD iC m<k9:I @O; ;q?<@9i:D
S9m;:D9i:7;DD 9D S9];>;:@ C>im @O; Ri>k <:SU mi>;ij;>U @O; KOi9K; iC < x<?DD9<:
Ci>m 9:S?K;D @O; p>;D;:K; iC @O; 7;:I@O z RO9KO Ki:@>i7D @O; Dp<@9<7 S;K<y iC @O;
S;:D9@yL |D < >;D?7@U 9@ 9D :i@ piDD9P7; @i Kimp<>; ;<D97y i?> q?<D9^;e<K@ miS;7D
R9@O @O;9> >;D?7@DU RO9KO >;m<9: mi>; I;:;><7 P?@ >;q?9>; @O; k:iR7;SI; iC @O;
S9m;:D9i:7;DD ~ K?>j;L
74 Chapter 2. Analytical solutions in the construction of strange quark stars models
Eat and Qua-Eat Model of Strange Star 1
223 P m
A ii h w by Li X wh !" #k
wih "yi #i $ Thi #i $ i ""y is
h h vi" "i "ik h b !" U$"y x "i
i%" bi $ h "yi Hwv hi !i" "
ii b hy " wih w !"& wih wih 'C( v!
y )*+ hw hw hy ! "i b,"ik #i $ ).
hi !i" " i R$ /0+
A " $ w !y y h h wk $ Li X hw h
"yi ii $ i ib" b i
vi y !i" $! $ h i $ !" "
i h $"" !i" wk
224 567 78 m
Th x ii !" $ .h! 9hj: i h xi,
!i $ h "" bv "y ! )h "
iy+ A hyi" !ivi i hi iy hy # Uv;
wh "i " $! i hi "y h #k $ $
b Thi i ib" h "i $ ,#k i hi i
Th x xi $ h !"
<)=+ >?)3 @ B=
D+
FG)/ @ B=D+DI )/3+
J)=+ >=K?
2)/ @ B=D+I )/4+
pN)=+ >cD?)3 @ B=
D+
24G)/ @ B=D+D
4O
3I )/0+
VWYNZ >)/ @ B=
D+cD
)cDB [?+=D @ cDI )/\+
V]YNZ > V^N_)/ @ B=D+`K)cD @ cDB=D [=D?+q zI )/+
p|)=+ > pN)=+ /
2
cD?B=
/2G)/ @ B=D+D@
cD?B=)3 @ B=
D+
\G)/ @ B=D+K
cD?)3 @ B=
D+
/2G)/ @ B=D+D@
4O
3
[=K?
2)/ @ B=D+@
4G[=KpN)=+
cD
= cD= [=K?
/ @ B=D
=I )/F+
Section 2.1. Exact and quasi-exact models of strange stars 75
1 M. G. B. d Avll nd J. E. Hovah
Fg 7 R tgt prssurs f r tw s tr pc str m s T p = 8 ×
1 gcm3; b tt m = × 1 gcm3
e e i e !""#i" e$i, e e"iy %$e, e
%e""e" (ei$ i$)& 'ee, * +-/024 5 +
69:<& >e ?e @e@ke
e i"%y i" "!$$ ("ee Ci& D) i e ie?$ I @e$ e"iie"& O@e e Iee
%!ee" KL N e e " I@i" I e @e$ e"iy PQ, e !e$"
e @!%$ee$y "%e@ie& >e "S e e"$i !""#i" e$i i Ci& U&
Ve "i!i$iy I e @?e i Ci& U e !ei@$ e"$" y WCO i" %%X
e& 'Se?e, i "%ie !""e" ii e %@i@$$y e "!e, i"%i@
!e$" e "y"e!i@$$y e"e e @ee y YZ[ "& C e\!%$e, e !\X
i!! !"" !e$ $ e "e]e@e " @e$ e"iy I ^ Y_ ` jq6x z@!4
See" i e WCO !ei@$ @$@$i e ?$e i" " ^ Yq` jq6x z@!4& Ve
e"iy %$e" e "S i Ci& |& Ve "i!i$iie" Si WCO e $" %%e&
76 Chapter 2. Analytical solutions in the construction of strange quark stars models
Eat and Qua-Eat Model of Strange Star 11
8 8 9 9
R
s]
F !"# c#!v$ %&! 'h$ ( &'!&pc m&"$) &% *h!m (" h!j+,
F D$( 'y p!&)$ &% 'h$ ( &'!&pc m&"$) &% *h!m (" h!j+,
./0234567: 53;/20 <=5< <=2 56i>?<3?@i2> A?/B0 b2 i4@?3<56< C?3 026>i<i2> G H
IJ7K ;LA4
NO T=i> i> A?6>i><26< wi<= ?/3 32>/B<>O P< >=?/B0 b2 @?i6<20 ?/<U =?w2V23U
<=5< <=2 56i>?<3?@W A?/B0 5X2A< >?42 @53542<23> BiY2 <=2 45xi4/4 45>> Z<=2 2X2A<
i> >45BB i6 ?/3 5@@3?5A=[ 560 <=2 320>=iC<U 5> 0i>A/>>20 B?6; 5;? bW \?w23> 560
^i56;O7_
`6BW <=2 565BW<iA5B >?B/<i?6 C?3 <=2 56i>?<3?@iA ><53 i> 6?< >/kAi26< C?3 5 A?4q
@B2<2 02>A3i@<i?6 ?C >/A= 56 ?bz2A<O T=i> i> b2A5/>2 i< i> i4@?3<56< <? 2x@B5i6 w=232
<=2 56i>?<3?@W A?42> C3?4 i6 ?3023 <? @3?0/A2 <=2 0iX2326A2> w2 =5V2 >=?w6O 5Y
Section 2.1. Exact and quasi-exact models of strange stars 77
1 M. G. B. d Avll nd J. E. Hovah
2 pite ut tt sure f istpy ru be istpir er-
ity istibuti f te ptires isie te st ue f empe mgetir e
tubuere rerti Peps te biggest rege f te istpir me is
te stbiity ritei C 2 swe tt i te set f istbiities ee
sm istpies migt stiry rge te stbiity f te system
We e sw tt te pessue istpy is sm f w ret esities
but bermes ge ge s te ret esity ireses Fute stuies e
eressy i u ppr t eify if f te seuere (eg i ret esity
ges is stbe
5 ! Þ" #$"
I te sme wy s befe we epe te me eepe by Kmtij
%j22 t me stge sts pmetize by sige pmete te ret
esity Tis me s s te esibe ppeties rite be wee te
eertir e is epirit furti f te psiti rite sttig fm ze
t te rete gwig up t te sufre Te e&ert f tis e te mss'
ius eti is t irese te msses tei espertie ii Te mss'ius
eti is sw i Fig )*
I tis me we e wit rge stge st It is imptt t tire
tt tis is uite i&eet fm te ssumpti me by Us sire te rge is
istibute isie te we st Cge rmprt sts s e bee stuie i
my wys by Ry 2+,2/ I tis w tey e fu tt st r e
eertir e f but 0)*2 V3m i ptiru rse f te pytpir euti f
stte It is iteestig t tire tt te ert suti by Kmtij %j
wit eertir e s pies eertir es t buy >)45 × )*2 V3m
8 9 67 66 6: 6;7<=
6
6<=
:
:<=
;
?@DLNO QSXY
Z[\\]Z^_`c
kqx 1| ~~q q ~q~ ~ ~~~
78 Chapter 2. Analytical solutions in the construction of strange quark stars models
Eat and Qua-Eat Model of Strange Star 1
3 C
W hv s wi s mhmic sis (c im i
m s qk ss A hs sis w mi by
si qiy, h c siy Bh sim c qsic ms w
ss, wih wih s m (isy cic
N sim yic m ws scib hs ss by ki h
qi s ciy ch (s c 22! is h h h
s iscss h sh b mi wih his scii i mi
I ic m h qsic si, w hv s i wih
2" is, csi 2" c siis, w hv ssis
h ms h bm h svs yicy h #isi qis
c i wih shic symmy i qi s Th sc
css his ic ch c b i Fi !!, which smmis h
mss$is is sis , s i Fi !2 wh w shw h
siy s h ss mimm mss i sm sis (Tm, Bch
h, Gssiim isic W mi h his wk s h
sim mic ms AFO s bchmk, b h hs sm
m, is bs h %IT b qi s This i bhvi,
hwv, mks h i qi sysm imssib b i yicy
ss w iv qi vmii h sysm I his is , h
mch h vmi sysm is cssy
W hv shw h i is ssib qsic si h is s
m s s i sy wy W js iv h c siy,
h qiis ik mss, is, s &') c b Wih hs yic
ssis w c ic my is h s by h c
siy Th s mi *!+ i qiis, ici h my (Fis 5
6
7 8 9 ./ .. .0 .4/
/:;
.
.:;
0
0:;
4
R<=>?@ DHJK
LPUUVLXYZ]
[\>@^_`^p>x J^=z
|zx_`>x >z= J^=z
[~
<?@@><\<pp`^>J<_z=
11
Section 2.1. Exact and quasi-exact models of strange stars 79
1 M. G. B. d Avll nd J. E. Hovah
Fg 1 Dety pre te tr mxmum m
W c s i i ssi-i s!i is "!i i
f !w c! siis# ci !s c!$ %O T isic
! is "!i f !! f c! siis is "$ i& f %O
Tis is isi ! (s i 9 c i wi i 3; b s !s
's i Sc 224) T !cic *! ! is !s "!i # b !$ i c
f c q' ss
Tis ibi!i$ b! s wi s$s f i qis
fi isic ss !cic *! Ts w s f f
's s$s ib! W " iscss w s s!s c
!i# isic# c !s w $ &c c! s!! fs
i cic
I s! b i # i si f i $sic! i&cs# i is
!$ isiss $ qi f s f s ss T ci f
is f bs"i f !-!i' issi is si!! b!ic
$ ci sbsi! icccis T' fc "!# w"# i-
is " s!! bs# ci!$ icib! wi
s !s if !# bjcs +XO 0,4/-5,567 (280 ± 082/:<=
>38/ ± 8/ ') +XO ,4?-24/6@
(84:<= ') (8,:<= 9 ') c b
scib b$ isic ! (b$ 'i i cc bs) sc-
i"!$ (280:<= 844') (839:<= 80 ') (is !s wi c!i
i cisi) W cc! is w' b$ ii si!# cic!
sciis f "i$ f s ss c b csc $ b sf!
! i is sc! b"i i $ sic $ic!
siis
80 Chapter 2. Analytical solutions in the construction of strange quark stars models
Eat and Qua-Eat Model of Strange Star 1
Rc
N I , P. Th. Ph. 44 ( 2
2 J C Cs J !""#, Ph. $%. L&&. 34 (' )')
) * W !, Ph. $%. D 3+ (/0 22
0 5 6!7" & 89., Ph. $%. D :; (' 2<
' F W!b!", P=98 8 A&>h?@89 L8B8&? G H=@98 8KO P8&?@9 Ph?@, s
! (Is U ! V #sis Ubs7, X"s !Y,
< C Ziik, * F" Z [ , A&>h. \. 3:+ (/< 2<
[ ] X!v!U J * ^"v , _K. H&. $. A&K. `@. ;4: (/ 0)
/ j m 6!7 # p qk!, wx>. Ph. wxx. ::y (/ )'
X F jiU z, A ?& w= ?K |K89 $98&?%?&, s ! (Cb"7! ~v!"s #
"!ss, Cb"7!, ~p, /'
m C 5, Ph. $%. yy () )<0
^ Z XUi, Ph. $%. :: (' 2
2 J q !" "ks, Ph. $%. L&&. 4 (2'
) p j C!7 5 ^"k, Ph. $%. D ; (2 /)
0 ] N", J ji!"X!i I sUs , Ph. $%. D 4 (2< <))
' q m U, A&>8&. Ph. 3: (2 2/
< m j" j 6 ", _K. H&. $. A&K. `@. 3y (2 2<'
~sv, Ph. $%. D + (20 <)
/ m mU!", AKK=. $%. A&K. A&>h. :+ (2 02
m q X!"s * 5 q7, A&>h. \. : (0 <'
2 p k 5 ^"k, P@. $. `@. LKOK A 4y (2) ))
2 m C, q ^!""!" N [ j s, _K. H&. $. A&K. `@. ;y () '))
22 p p " j 6 ", "v!Y" ss "Y22/, 2
(2
2) j m# & 89., 8. \. Ph. 34 (20 )
20 j m# & 89., P@O?K G &h _| _&?K, Nv!, j !"!z X!"7
m mU (!s (W" ji! i, j7Y"!, 2', Y )<
2' F [z!, H8&= 44: (2< '
2< F [z!, 5 ]Uv!" 6 s s, "v!Y" ss "Y/'2, / (2/
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-
84 Chapter 3. Information theory and measurements to infer a hierarchy of equations of state
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.
86 Chapter 3. Information theory and measurements to infer a hierarchy of equations of state
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
88 Chapter 3. Information theory and measurements to infer a hierarchy of equations of state
Pysis Lttrs A 3 ( 11
C l aalal a Sce Sc cec
www.le.c!"lca"#la
E$%&o'), *o+'p-x/%) 0$2 2/d-4u/p/5&/u+ /$ *o+'0*% d%0&d
M67686 9: ;v:<<=> ?@ J6B6 HD>v=FG
IKNOQOROT UV WNOXTKTYQZ[ \VT]^NQ_Z V `QbK_QZN WOYTN]fXQ_ZN[ gKQhVXNQUZUV UV jkT mZRnT[ qRZ UT zZOkT ||[ `QUZUV gKQhVXNQO~XQZ[ jkT mZRnT[ jm[ XZQn
WXOQ_nV QNOTXi tr i i ris r r ry A¡t r ry A i¢ ¢ ¢i r ry £i t y P¤¤ ¥¢¢
¦V§TXUN¨r ti tr¡y£¡¢©ityª¢«r¬ i ti®tr st rsªtr ¬ st rs¯° ti st t
±² ³´²µ ¶·² ´¶¸¶¹´¶¹º¸» ¼²¸´³½²¼²¾¶´ ¿À ¹¾À¿½¼¸¶¹¿¾ ²¾¶½¿ÁÂà µ¹´²Ä³¹»¹Å½¹³¼ ¸¾µ º¿¼Á»²Æ¹¶Â ¶¿ ¹¾À²½ ¸·¹²½¸½º·Â ¿À ²Ä³¸¶¹¿¾´ ¿À ´¶¸¶² À¿½ ¶Ç¿ ¶ÂÁ²´ ¿À º¿¼Á¸º¶ ´¶¸½´ À½¿¼ ¶·² Ž¿¸µ º»¸´´ ¿À ¾²³¶½¿¾ ´¶¸½´Ã¾¸¼²»Âà ǹ¶· ·¸µ½¿¾¹º º¿¼Á¿´¹¶¹¿¾ ¸¾µ ǹ¶· ´¶½¸¾È² ij¸½É º¿¼Á¿´¹¶¹¿¾Ê ˳½ ½²´³»¶´ ´·¿Ç ¶·¸¶Ã ´¹¾º²¿½µ²½ º¿´¶´ ²¾²½ÈÂà ̸¶³½² Ç¿³»µ À¸Í¿½ ¶·² ²Æ¿¶¹º ´¶½¸¾È² ´¶¸½´ ²Í²¾ ¶·¿³È· ¶·² ij²´¶¹¿¾ ¿À ·¿Ç ¶¿ À¿½¼¶·² ´¶½¸¾È² ´¶¸½´ º¸¾¾¿¶ Ų ¸¾´Ç²½²µ ǹ¶·¹¾ ¶·¹´ ¸ÁÁ½¿¸º·Ê
Î ÏÐÑÏ Ò»´²Í¹²½ ÓÊÔÊ Õ»» ½¹È·¶´ ½²´²½Í²µÊ
Ö× ØÙÚÛÜÝÞßÚàÜÙ
áâ ãäå æåçåâã èéêãë êçìåâãìêãê íæîï ðìííåæåâã éæåéê äéñå òîîóåðéã ìâíîæïéãìîâ ãäåîæô ãî çäéæéçãåæìõå èäôêìçéò éâð öìîòî÷ìçéò êôêøãåïêë ãäåìæ èéããåæâê éâð çîææåòéãìîâêù úäå ìðåé ìê ãäéã é êãéãìêãìçéòïåéêûæå îí üýþÿpx (ãî öå ðåeâåð èæåçìêåòô öåòîw åâçîðåê ãäåêåòíøîæ÷éâìõéãìîâ îí é êôêãåïë éâð òìâóê ãäå ìâíîæïéãìîâ êãîæåð ìâìã (îæ ãäå òî÷ìçlìâíîæïéãìîâ åâãæîèô ãî ìãê ðìêãéâçåa ãî ãäå êãéãåîí åûìòìöæìûï èæîöéöìòìãô ðìêãæìöûãìîâ [ù Råçåâãòôë Séûðî éâðPéçäåçî [ eæêã æåòéãåð êûçä ïåéêûæåê ãî éâ éêãæîèäôêìçéò îöjåçãë éwäìãå ðwéæí êãéæë wäìòå Cäéãõìêéññéê åã éòù [ éèèòìåð ãäåêå êéïåçîâçåèãê ãî éâîãäåæ ãôèå îí çîïèéçã êãéæêë çîòòåçãìñåòô óâîwâ éêâåûãæîâ êãéæêaë wäåæå ïéããåæ ìê ìâ ãäå ðåâêåêã íîæï óâîwâ ìâ Uâìøñåæêå éâð ìê ûâðåæ åñåâ ïîæå åãæåïå èäôêìçéò çîâðìãìîâêë âéïåòôêûèæéøâûçòåéæ ðåâêìãìåêùúäå ìïèîæãéâçå îí èåæíîæïìâ÷ ìâíîæïéãìîâ ãäåîæô êãûðìåê îâ
çîïèéçã éêãæîèäôêìçéò îöjåçãê æåêûòãê íæîï ãäå íéçã ãäéã ãäå ñåæôâéãûæå îí ãäå ïéããåæ ìâ êûçä åãæåïå èäôêìçéò çîâðìãìîâê ìê êãìòòûâçåæãéìâë éâð ãäåêå êãûðìåê çéâ êäåð é âåw òì÷äã îâ ãäìê êûöjåçãíæîï é ðìííåæåâã èîìâã îí ñìåwù áâ ãäìê Låããåæë wå éððæåêê âåûãæîâêãéæêa ïéðå îí âûçòåéæ äéðæîâìç ïéããåæ éâð ïéðå îí íæåå ûéæóê(ãäå êåòíøöîûâð êãæéâ÷å êãéæêë ïîðåòåð ìâ ãäå çîâãåã îí Máú Bé÷Mîðåòù
* £rrs¡i¬ tr¤ T¢¤ + 3 2 + 3 1 © + 3 1¤
EYZQn ZUUXVNNVN r ¢ str¤i ¬¤s¡¤r (¤¤¤ A¢¢ rt str¤i ¬¤s¡¤r (¤¯¤ ¥r t¤
úäå åãåâêìîâ îí ãäåêå ìâíîæïéãìîâ çîâçåèãê ãî éêãæîèäôêìçéòïéçæîêçîèìç îöjåçãê ìê âîã êãæéì÷äãíîæwéæðë êìâçå ãäå íîæçåê ìâøñîòñåð ìâ ãäå æåêèåçãìñå åûìòìöæìûï çîâe÷ûæéãìîâê éæå ñåæô ðìíøíåæåâã íæîï ãäéã îâåê ìâ éâ éãîïìç êôêãåïù áâ éãîïìç êôêãåïêë ãäåíéçãîæ ãäéã ðåãåæïìâåê ãäå êåòíøîæ÷éâìõéãìîâ ìê ïéìâòô ãäå Cîûòîïöìâãåæéçãìîâ éâð ãäå Fåæïì åçòûêìîâ èæìâçìèòåù Oâ ãäå îãäåæ äéâðëìâ é (îæðìâéæô âåûãæîâ êãéæë ÷æéñìãôë êãæîâ÷ éâð wåéó ìâãåæéçãìîâêéòò çîâãæìöûãåù Fìâéòòôë ìâ é ûéæó êãéæ ãäå âûçòåéæ êãæûçãûæå éâðãäå âûçòåîâê ãäåïêåòñåê äéñå öååâ öôèéêêåðë éâð ãäå ãæûòô íûâðéøïåâãéò ðå÷æååê îí íæååðîï êäîw ûè ãî íîæï é êåòíø÷æéñìãéãìâ÷ öéòòwäìçä ìê âåñåæãäåòåêê öîûâð öô êãæîâ÷ ìâãåæéçãìîâêë âîã ÷æéñìãôë éòøãäîû÷ä ãäå òéããåæ ìê êãìòò ñåæô ìïèîæãéâã íîæ ãäå îñåæéòò êãæûçãûæåùáã ìê éâ îèåâ ûåêãìîâ wäåãäåæ ãäå ìâíîæïéãìîâ ûéâãìãìåê çéâ öåûêåð íîæ é ÷æîêê ðåêçæìèãìîâ îí ãäåêå åûìòìöæìûï çîâe÷ûæéãìîâêù
Mîãìñéãåð öô ãäåêå çîâêìðåæéãìîâêë wå çîïèéæåð ãäå ìâíîæïéøãìîâ éâð çîïèòåìãô êãîæåð ìâ ãäåêå ãwî âåûãæîâa êãéæê îí ðìíøíåæåâã ïìçæîêçîèìç çîïèîêìãìîâë éâð íîûâð ãäéã ãäåêå ûéâãìãìåêéæå çîïèéæéöòå ìâ ÷åâåæéòë öûã êåâêìãìñå ãî ãäå çîïèîêìãìîâë öåøçéûêå ãäå òéããåæ ðåãåæïìâåê ãäå öåäéñìîæ îí ãäå rd îí ãäå êãéæêíîæ ãäå êéïå ïéêêù Sìâçå ãäå ñéòûå îí ãäå æéðìûê ìê éòêî éâ ìïèîæøãéâã íåéãûæå íîæ éâ îöêåæñéãìîâéò ìðåâãìeçéãìîâ [ëë ìâíîæïéãìîâãäåîæô ïéô òìâó ãäå íîæïéãìîâ éâð êãæûçãûæå éêèåçãêù
× ßÞÚàÜÙ ÙÝmÜÝo
Wå ûêåð ãäå êãéãìêãìçéò ïåéêûæå îí çîïèòåìãô éê ðåeâåð öôLèåõøRûìõë Méâçìâì éâð Céòöåã [ë éê ïîðìeåð öô Céãéòâ åã éòù[:
= H × D,
3« 0 s rt ttr Î ¯¢sir ¤!¤ A¢¢ ri¬ts rsr¤i¤ "¤¡ys¢t ¤¤¤
Section 3.1. Entropy, complexity and disequilibrium in compact stars 89
1 M d Avee, J Hova / Pyss Laas A 3 (
w = pS) S i t in!"ti! t!p# $! t i%
n!"ti! c!tt !n t # t"m i t&' '!lit"ic &it u D
i t i *&i'i+i&" $iti- wit t i tc !n t # t"
t! it tt !n *&ip!++' p!++i'it# i ti+&ti!m. I it !ili'
-iti!u t p i! n! S D t n!''!wil
0 2 4 56r7 89 56r7 :r;
< 2 5>6r7:r?
T *&tit# @B) i t !"'iC p!++i'it# i ti+&ti!
tt ci+ t tt !n t # t". S ci+ t &ctit#
!cit t! tt p!++i'it# i ti+&ti! wi' D t n! t
in!"ti! l# $ - +# Oic c& [EFmu ! t *&tic
i tc t! t *&ip!++i'it#.
I ! t! t&# !& tw! t#p !n c!"pct !+Gct i ti w#u
w '!l& t! t p!++i'it# i ti+&ti!. Kc& t
l# it# i ti+&ti!u NQ) RlUc"V]u i 't t! t p!+%
+i'it# !n -il &"+ !n ptic' i liW '!cti! i i
t tu w & t l# it# p!-' t *&tit# t! %
t i t itl' . X!wWu i t c !n t t&ct& !n !&
t u t lWitti! i !%Ywt!i w "& t !'W t
T!'"ZOppi"Z!'\!nn *&ti! u ! t *&ti! !n '%
tiWi tic #! ttic *&i'i+i&" !n t t p'& t " itl'u
+!t c!"p'"t +# t *&ti! !n tt wic ci+ t
"ic!%p# ic $c!"p! iti!m !n t t'' "ttu t! -''# -
NQ) = ^_@B)u w ^ = `×bfgh c"U i t W'!cit# !n 'ilt
@Q) RlUc"V] i t "tt it#.
I ti w!\ w & t " pp!c !n [`F t! !'W t TO
*&ti!j - t w - t + *&titi t i" i!%
' Wi+' c'il n!''!w
k6q7 2 k6q7kx z6q7 2 z6q7z;
|6q7 2 | 6q7z z 2 ~;
w Q) i t " !n t t i !' &it $mu Q) i t
p & Nh i l# it# c'u wic i t& p!Wi
& t n!''!wil n!" !n t TO *&ti! t " *&ti!
| 6q7
:q2 4?
k6q7z6q7
q>
|6q7
z6q7
?
q | 6q7
k6q74 ?
k6q7
q
;
k6q7
:q2 ?
q>z6q7?
T& u t itl' t! + W'&t
0 2 4 z6r7 89 z6r7 :r;
< 2 z>6r7:r;
w N i t i" i!' l# it# $wic i G& t ^_@UNhm
!+ti n!" t !'&ti! !n t TO *&ti!. T p"%
t h = × bf \"V
i G& t p!p'# c! *&tit# tt
"\ S D i" i!' . T itlti! i pn!" n!"
f t! t i& R\"]. !w n pt'# t! t pci-c
c !n t !ic t t $ tlm *&\ tu - +#
innt "ic!%p# ic' cipti! .
- t tt"t !n t p& !ic c + liW +#
[`Fu & il t!tic''#%"!tiWt "!' *&ti! !n tt.
i t & t !%c'' ¡¢#£ *&ti! !n tt i it '#tic
n!" [F ict'# i t +!W n!" !n t TO *&ti! u t! !+%
ti t l# it# p!-' n! c iiti' W'& !n t ct'
¤¥¦§¨ ©
ª«¬«®°®¬± ²³ °´® µ°¶
· ¸¹º»¼ · ¸¹º»¼
1 ½¾¾ 1 11½¿ÀÁ
¾ ½1¾1 11 ¾¾½ÃÃÁ
Ä ½ÁÀ¾Á 1¾ 1½ÁÃÃ
¿ ½1ľ 1Ä ¿½Ä
Á ½¿ 1¿ 1¿½
11½¿ÀÃ1 1Á ¾Ã½
à 1À½1Á 1 Â1½ÁÄ
½ÀÄ 1à 1½Á
À ½Á¿ 1 1¿½Ã
p &. '#tic' p tti! !n t *&ti! !n tt
pn !W t t+&'t ! u +c& t# W!i tw! "G!
p!+'" !n t 'ttj t "+il&it# !n t itp!'ti! i"%
p! i+i'it# !n c'c&'til t iWtiW pci '#. Å&t"!u t
'#tic' n!" i c! t&ct !+#il '' t t"!#"ic '%
ti! [F. &it+' n!" !n t *&ti! !n tt ¡¢#£ i
Æ 2Ç Ç>È ÇÈ
ÇÉÈÊ ÇË6È 4 Ç7
6ÇÌ ÇÍÈ7 Ê ÇÎ6Ç 4 È7
6Ç Ç>È7 Ê Ç6ÇÉ 4 È7
6ÇË ÇÈ7 Ê ÇÌ6ÇÍ 4 È7 ; Ï
w t c!Ðcit liW i T+' b Ñ = '!l@Ul c"V)u
Ò = '!lU# c"_).
Ti i c&t'# p!p&' c!ic n! ti' t&i !n
"tt '' &c' nt& !n it t '# +&i't%i [F.
!t ! n! c!! il t SÓÔ£ i tt it ''!w "i"&"
" !& Õu "ii"&" W'& i"i' t! t *&\ *&ti!
!n tt i c& +'!w.
T tl t "!' ' ! *&ti! !n tt ci+%
il t $ 'n%+!&m *&\ ptic' ti itcti! . Ti i
!t!i!& '# "! iW!'W t i t &c' p u ic c!%
-"t i !t #t p!p'# & t!!. T! c'c&'t t in!"%
ti! t!p#u t i *&i'i+i&" t c!"p'it# n! !& "!'
!n tl *&\ t u w & ! !n t nw '#tic' ct !%
'&ti! !n t Öi ti *&ti! $wic i !n c!& !'&ti! !n
t ttic TO *&ti!m n! pic''# #""tic !%!ttil
pnct Ø&i. Ti !'&ti! i t i !t!pic p i! - t !+%
ti +# ¡" ÙG [bfF t&i +# & i [bbF. T
Wtl !n ti W# cc&t "!' i tt i ti w# w W
'#tic' p i! n! t l# it# tt c + it%
lt i'#u "'#
z6q7 2
Ï
5ÚÛ>
z
Ï q>q
>Ü
6 q>q>Ü 7
>?
I tt p i!u @Ý i t ct' it# QÞ = QÞ@Ý) i
p"t tt c!t!' t c# !n t it# p!-'. Ti %
'#tic' !'&ti! i !+ti i"p! il t ÙIT Kl "!' n! tl
*&\ "tt *&ti! !n tt
ß 2
ÏÛ>5 4 à ; á
w â ã äEä ÙUn"V
i t l# it# !n Wc&&". Ti
i"p' p i! $`m + wi'# & +c& it i'# cp%
t& t ti' nt& !n t c!- p . å&ci' t!
!& c! iti! !n 'n%+!& $tt i u +!& t W
i t + c !n lWitti! [bÕFm i t i tc &"i%
c' W'& !n t p"tic Wc&&" l# it# âu p til
!%pt&+tiW c!-il itcti! . It i i'# !w tt n!
ti " ' *&\ c
90 Chapter 3. Information theory and measurements to infer a hierarchy of equations of state
M d Avee, J Hova / Pyss Laas A 3 ( 1 1
Fg. . I !"# $!r%r rr &'!t) T*! i!x &z!+ *! -*
r!t kr *! x % rr *! r!0%!u!) T*! +r u!r"+ S2#4
&50) &6tt + *! urr!r *! r-! 0%k ! &50) &7tt)
Fg. 8. I !"# $!r%r + %r) T*! r- u!r b%+ - r-! 0%k
! u*-! *! b!*$ !"# w * *! + %r9 u''# -w -
*! *+ u ur! &r rr $! +!'r !% - *! r''!r + t *! ":
"r ! &'!rr rr $! +!'r !% - *! r''!r + t) T*! r#b'r ! *! r!
r ; -) )
< = <>c?
@BCm = m>
cD
@BCEG KNO
QRUVU WX YZ[ \X YVU ]RU [^_UZ`^fZhU`` VY[^j` YZ[ _Y``l VU`nUpq
]^Uh|l YZ[ ^` ]RU ~VY^]Y]^fZYh pfZ`]YZ] RU jZ^]` f W YZ[ \
pf_U fj] Vf_ ]RU jZ^] `|`]U_ pRf`UZ fV ]RU pfZ`]YZ]` ^Z`^[U ]RU
`jYVU Vff]`
U ZfQ nVU`UZ] fjV VU`jh]` `RfQ^Z~ ]RU jYZ]^]^U` f ^Z]UVU`]
YZ[ RfQ ]RU| [UnUZ[ fZ ]RU _Y`` f ]RU `]YV YZ[ ]RU VU`nUp]^U
VY[^j`
U `UU ^__U[^Y]Uh| Vf_ ^~` l YZ[ ]RY] f]R ]|nU` f
`]YV` `RfQ nVU]]| _jpR ]RU `Y_U URY^fV f ]RU jYZ]^]^U` Q^]R
]RU _Y``l YZ[ Yh`f ]RY] ]R^` URY^fV ^` pfZ`^`]UZ] Q^]R ]RU fZU
f]Y^ZU[ ^Z fQUUVl ]RU URY^fV f ]RU `Y_U jYZ]^]^U` Y` Y
jZp]^fZ f ]RU VY[^j` ^~` l YZ[ [^UV UZfV_fj`h| R^hUl
fV ^Z`]YZpUl QU Z[ ]RY] ¡ ^` Y ¢£¤W£¥¦§¨© jZp]^fZ f ]RU _Y`` fV
f]R pY`U` ^Z ]RU `]YhU VU~^fZl QU pYZ `]Y]U ]RY] fV ]RU RY[VfZ^p
`]YV ]R^` ^` [jU ]f ]RU Yp] ]RY] QRUZ ]RU _Y`` ^ZpVUY`U`l ]RU VY[^j`
[UpVUY`U` `UU ^~ YZ[ ^~ YZ[ ]RU UZUV~| [UZ`^]| Upf_U
Fg. ª. « r!0% ' b % $!r%r rr) T*! r#b'r ! *! r! r ; -) )
Fg. ¬. « r!0% ' b % $!r%r + %r) !%! r ' *! ! ; -) ® r "!r!)
T*! r#b'r ! *! r! r ; -) )
Fg. ¯. °"'!x # $!r%r rr &'!t) ± u! * *! '- ru'! *! $! u' x r
u%r!r b!*$ r ' *! !"#) T*! " r %+ *! x % rr !
z!+ *! -* r!) T*! r#b'r ! *! r! r ; -) )
_fVU hfpYh^²U[ `_YhhUV VY[^j`l `_YhhUV ¡ ³Z ]RU f]RUV RYZ[l fV
]RU `UhqfjZ[ jYV´ `]YV`l ]RU URY^fV f ]RU UZ]Vfn| Q^]R VY[^j`
^` j^]U [^UVUZ] ]RU hYV~UV ]RU VY[^j`l ]RU `_YhhUV ]RU UZ]Vfn|
¡ jZ]^h Y pUV]Y^Z YhjU f ]RU fV_UVl Vf_ QRUVU QU VUpfUV Y
URY^fV j^]U YZYhf~fj` ]f ]RU RY[VfZ^p `]YV R^` ^` [jU ]f ]RU
Section 3.1. Entropy, complexity and disequilibrium in compact stars 91
1 M d Avee, J Hova / Pyss Laas A 3 (
Fg. 6. Cplx !" # Nt $! $ l% t!l f $ t!l !x t!
! b$! l! $ np# T$ bl ! $ ! ! n &%# 1#
')r* +-00)r)24 2547r) 80 49) q75rk :45r:; m5+) 80 0r)) q75rk: 52+
<872+ 48=)49)r <* 49) :4r82= -24)r5>4-82:; -2 w9->9 R ? @ w9)2
B ? @D E9)r)08r); -2 :G-4) 4954 49) )24r8G* >5rr-)+ <* 49) =r5'I
-454-825i )i+ +)>r)5:): -4: '5i7) w9)2 49) m5:: +)>r)5:):; 49)
5+m-K47r) m8r) 4952 >8mG)2:54): 49-: 52+ 49) OQOSU )24r8G* -2I
>r)5:):D
W) >52 28w +-:>7:: 87r r):7i4: -2 4)rm: 80 49) >82>)G4: 80
-208rm54-82 >824)24; +-:452>) 48 49) )q7-i-<r-7m Gr8<5<-i-4* +-:I
4r-<74-82 52+ >8mGi)K-4* 8r :)i0I8r=52-V54-82D X4 -: )KG)>4)+ 4954
49) >8mGi)K-4* '52-:9): 08r 4w8 -+)5i >5:):; -2 49) 8GG8:-4) )KI
4r)m): 80 49) >82>)G4 80 8r+)r 52+ +-:8r+)rY 49) G)r0)>4 >r*:45i Z5
G)r0)>4i* 8r+)r)+ :*:4)m[ 52+ 49) -+)5i =5: Z5 4845ii* +-:8r+)r)+
:*:4)m[ \]^D X0 w) >8mG5r) <849 :)q7)2>) 80 m8+)i: 08r >8mG5>4
:45r:; 49) >8mGi)K-4* -: ')r* i8w -2 49) 4w8 >5:):; <)-2= :m5ii)r 08r
49) 95+r82-> :45rD E9) 57498r: 80 \_^ >82>i7+)+ 4954 2)74r82 :45r:
Z98w)')r; 5: G8-24)+ 874 5<8'); 49)* 5::7m)+ 5 +-00)r)24 95+r82->
)q754-82 80 :454)[ 5r) i8wI>8mGi)K-4*; 8r+)r)+ :*:4)m: <)>57:) 49)
>8mGi)K-4* G5r5m)4)r -: i8w 52+ 49) +-:)q7-i-<r-7m -: 9-=9)r 08r
49) :45r: w-49 49) :m5ii)r r5+--; 4954 -:; 08r 49) 9-=9)r m5::):D `r8m
49):) >82:-+)r54-82: 82) >52 >82>i7+) 4954 +)2:) m544)r 95+r82->
:45r: :*:4)m: <)95') :-m-i5ri* 48 49) G)r0)>4 >r*:45iY 9-=9)r m5::
? :m5ii)r r5+--; -mGi-): -2 47r2 5 m8r) i8>5i-V)+ )2)r=* +)2:-4*;
9-=9)r +-:)q7-i-<r-7m 52+ i8w)r >8mGi)K-4*D
c2 49) 849)r 952+; w) 95') :98w2 4954 :4r52=) q75rk :45r:
+-:Gi5* 49) 8GG8:-4) <)95'-8r w-49 49) r5+-7:Y 49) i5r=)r 49) r5I
+-7: Z724-i 49) 82) >8rr):G82+-2= 48 49) m5K-m7m m5:: G8-24[;
49) 9-=9)r 49) m5::; -mGi*-2= 5 m8r) hji8>5i-V)+ )2)r=* +)2:-4*;
9-=9)r +-:)q7-i-<r-7m 52+ i8w)r >8mGi)K-4*D W) >82>i7+) 4954 49)
:4r52=) q75rk :45r: 5r) i):: 8r+)r)+ Z-D)D m8r) )24r8G->[ 4952 49)
4w-2 95+r82-> :45r:; G)2+-2= 48 49) :-+) 80 49) -+)5i =5:; <74 :4-ii
05r 0r8m -4; :-2>) 49) i544)r :98w: 5 i8w +-:)q7-i-<r-7mD
X4 -: -24)r):4-2= 48 >8mG5r) 49):) >5:): w-49 49) 849)r 4*G)
80 i8wI+)2:-4* >8mG5>4 :45r; 49) w9-4) +w5r0:; :47+-)+ -2 \u^D
E9)* 0872+ 08r 49):) q75:-Iz)w482-52 8<)>4: 4954 49) >8mGi)KI
-4* |Q~ w-49 -2>r)5:-2= m5::; r)5>9-2= 5 m5K-m7m 2-4) '5i7)
54 49) 952+r5:)k95r i-m-4D E9-: <)95'-8r -: >82:-:4)24 w-49 49)
82): 4954 95') <))2 r)G8r4)+ 08r 548m-> :*:4)m: \]_]^; -0 49)
m5:: -: r)Gi5>)+ <* 49) 548m-> 27m<)r; 52+ -4 >52 <) r)i54)+ 48
49) +)=)2)r54) )i)>4r82 =5: 0)547r):D X2 >824r5:4; <849 95+r82-> 52+
:4r52=) q75rk :45r: 5r) 49) r):7i4 80 49) -24)rGi5* <)4w))2 :4r82=
-24)r5>4-82: 52+ Z:4r82=[ =r5'-4*; 49) i544)r <)-2= Gr8=r)::-')i* i)::
-mG8r4524 08r i8w m5::): -2 49) :)>82+ >5:)D
X4 -: 5i:8 80 -24)r):4 48 :)) 49) >8mG8:-4) +)G)2+)2>) 80 49) :45I
4-:4->5i q7524-4-): :-m7i452)87:i* w-49 m5:: 52+ r5+-7:D W) :98w
-2 GG)2+-K 49) _ Gi84: Z`-=:D D; D; D[D
X2 49) Gr)>)+-2= :)>4-82 w) 95') >5i>7i54)+ 49) -208rm54-82
)24r8G*; +-:)q7-i-<r-7m 52+ >8mGi)K-4* 08r 4w8 k-2+: 80 >8mG5>4
:45r :)q7)2>):Y 95+r82-> :45r: 52+ 5 q75rk :45r:D W) 8<:)r') 5
:-m-i5r <)95'-8r 80 49):) q7524-4-): w-49 m5::; <74 5 ')r* +-00)rI
)24 82) w-49 r5+-7:D E9) q7):4-82 4954 5r-:): r)=5r+-2= 49) 4r7)
>8mG8:-4-82 80 49) 2)74r82 :45r: -:; 49)2; w9->9 :454) w87i+ <)
r)5i-V)+ -2 2547r)D c2) >87i+ =7):: 4954; :-2>) 8r+)r >8:4: )2I
)r=*; 49)2 2547r) :987i+ 05'8r )K84-> :4r52=) q75rk :45r:; <74 w)
5r) 5w5r) 4954 49) SO 08ii8w)+ 48 r)5>9 49) :454) Z95+r82-> 8r
q75rk[ -: 5i:8 ')r* -mG8r4524Y 49) G5r5m)4)r 4954 >824r8i: 49) 08rI
m54-82 80 :7>9 8<)>4: -: 4987=94 48 <) 49) >)24r5i +)2:-4* 5445-2)+
7:4 504)r 49) )')24: 4954 i)5+ 48 49) 97=) >824r5>4-82 80 49) -r82
>8r)D
)4 7: -m5=-2) 4954 5 95+r82-> :45r 95: <))2 08rm)+ 874 80 49)
-r82 >8r) 52+ :)44i): -248 5 )q7-i-<r-7m :454)D E9) :4r52=)2):: <5rI
r-)r m)24-82)+ <)08r) w87i+ Gr)>i7+) 49) 08rm54-82 80 5 :4r52=)
q75rk :45r; -2 :G-4) 80 <)-2= 49) Gr)0)rr)+ :454); 72i):: 49) >82I
+-4-82: 08r 5 G549 48w5r+: 49) i544)r >52 <) 0872+D >87Gi) 80
G8::-<-i-4-): 5r-:)Y 49) r:4 -: 49) >82')r:-82 49r87=9 7>4754-82:
\]^; G8::-<i* +)i5*-2= 49) >82')r:-82 <* 52 5:4r828m->5i 4-m)I
:>5i); <)-2= 5 :4r82= 072>4-82 80 49) m5::D E9) 849)r -: 49) 08rm5I
4-82 80 5 4w8I5'8r Z:4r52=)2):: @[; 08ii8w)+ <* 49) +)>5* 48I
w5r+: :4r52=) m544)r 82 w)5kI-24)r5>4-82: 4-m)I:>5i) Z ]@
:[D
E9) +-00)r)2>) 80 0r)) )2)r=* G)r G5r4->i); w9->9 -: <* 49) :4r52=)
m544)r 9*G849):-:; r)i)5:)+ -2 49) Gr8>)::; w87i+ >8m) 874 -2 2)7I
4r-28: \]^ 52+ 5 :4r7>47r5i 5+7:4m)24 \];]^D
c7r >5i>7i54-82: :98w 4954 49) >8mGi)K-4* 80 49):) 4w8 4*G):
>8mG5>4 :45r: w-49 +-00)r)24 >8mG8:-4-82: -: ')r* i8w; -D)D; 49)r)
-: 5 4r)2+ 08r 49):) :45r: 48 <) 54 5 :454) 80 m-2-m7m >8mGi)K-4*D
5i<)4 52+ G)VI7-V \u];u@^ 95') :98w2 4954 08r 5 :*:4)m 874 80
)q7-i-<r-7m; 8r )K4r)m7m; 49)r) -:; -2 05>4; 5 4)2+)2>* 80 49) >8mI
Gi)K-4* 48 r)5>9 52 )K4r)m7mD E97:; -0 49)r) -: 5 4r52:-4-82 0r8m
5 95+r82-> :45r 48 5 :4r52=) :45r; 49) :*:4)m 5: 5 w98i) w87i+; -2
=)2)r5i; <) 874 80 )q7-i-<r-7mD X2 49-: w5* -4 <)>8m) >i)5r 49) G5rI
5ii)i-:m <)4w))2 49) :*:4)m: 4r)54)+ 9)r) 52+ 498:) 5++r)::)+ <*
5i<)4 52+ G)VI7-V; 5 05>4 w9->9 >5ii: 08r 07r49)r :47+-): 80 49-:
:7<)>4D
`-25ii* w) m7:4 r)m-2+ 4954 49) 4r52:-4-82 w-ii QO >82:)r')
49) :45r m5:: Z49) <-2+-2= )2)r=*; w9->9 -: 2)=54-'); w-ii <) i5r=)r
504)r 49) Gr8>)::[; <74 G8::-<i* 7:4 49) 4845i <5r*82 27m<)r; 52+
49)r)08r) >5r) :987i+ <) 45k)2 -2 49) >8mG5r-:82 80 49) 95+r82->
52+ :4r52=) :45r:; -2 :G-4) 4954 49) 5>475i +-00)r)2>) -: 284 i5r=)
52+ 95: <))2 -=28r)+ 5<8')D E9)r)08r); 49) :454-> 525i*:-: 80 )q7-I
i-<r-7m >82=7r54-82: >52284 r)')5i 49) 5>475i :-4754-82 80 m544)r
-2:-+) >8mG5>4 :45r:; <74 :7==):4: 4954 :4r52=) q75rk :45r: :987i+
<) Gr)0)rr)+D
E9) q7):4-82 80 49) )24r8G* 4r)2+ 5: 5 072>4-82 80 49) >8mI
G5>42):: 08r 5 =-')2 '5i7) 80 49) m5:: 5i:8 G8-24: 48w5r+: 49)
:)i0I<872+ q75rk :45r:; <74 82) :987i+ :5* 4954 49)r) -: 52 )2I
4r8G* <5rr-)r <)4w))2 49) 4w8 :454): 4954 95: 48 <) 8')r>8m) Z5
07ii 525i*:-: 0r8m 49) :4)ii5r )'8i74-82 G8-24 80 '-)w -: -2 Gr8=r)::
52+ w-ii <) G7<i-:9)+ )i:)w9)r)[D E9-: 525i*:-: Gr8'-+): 5 >8mI
Gi)m)245r* i88k 48 49) q7):4-82 80 >8mG5>4 :45r >8mG8:-4-82 52+;
-2 G5r4->7i5r; 48 49) 5>>)::-<-i-4* 80 5 Z:4-ii 9*G849)4->[ :)i0I<872+
q75rk :454); :8m)w954 r)m-2-:>)24 80 5r549 8+8r* '-)w 80 -rr)I
')r:-<i) Gr8>)::): \uu^D
¡¢£¤¥¦¤§¤¨
W) w-:9 48 5>k28wi)+=) 49) ©ª« 52+ `©ª«© 08r 49) 252I
>-5i :7GG8r4D ¬DªDD w-:9): 48 5>k28wi)+=) 49) 252>-5i :7GG8r4 80
49) z©q =)2>* Z®r5V-i[ 49r87=9 5 r):)5r>9 :>98i5r:9-GD W) 4952k
49) r)'-)w)r: 08r 49)-r >82:4r7>4-') 52+ 9)iG07i >8mm)24:D
92 Chapter 3. Information theory and measurements to infer a hierarchy of equations of state
M d Avee, J Hova / Pyss Laas A 3 ( 1 1
. p t! " " #$%t t# &' !)up*#u$ %
c $pt&
F+g- 0-7- 24 5689 8: ;<9=85>? N89@B; 9C; mDEEG=DI@KE =;6D9@8< @< 9C; MR 56D<;? TC;
E>mO86E D=; 9C; EDm; DE @< Q@S? ?
F+g- 0-U- 24 5689 8: I@E;VK@6@O=@Km? N89@B; 9C; mDEEG=DI@KE =;6D9@8< @< 9C; MR
56D<;? TC; E>mO86E D=; 9C; EDm; DE @< Q@S? ?
F+g- 0-W- 24 5689 8: B8m56;x@9>? N89@B; 9C; mDEEG=DI@KE =;6D9@8< @< 9C; MR 56D<;?
TC; E>mO86E D=; 9C; EDm; DE @< Q@S? ?
X"#c!
[Y Z? \]5;^GZK@^_ `?\? bD<B@<@_ f? hD6O;9_ iC>E? \;99? j k lnq 2k?
[kY r? wDzKI8_ j?Q? iDBC;B8_ iC>E? \;99? j 22 lkq ?
[2Y |?hC? hCD9^@EDDE_ ~?i? iE8<@E_ h?i? iD<8E_ hC?h? b8KE9D@I@E_ iC>E? \;99? j 22
lkq 2?
[Y T? ü;=_ Q? Ö^;6_ j? \D;=EGhDO=;=D_ i? =8O6;E@_ jE9=85C>E? r? k lkq ?
[nY Q? Ö^;6_ ? D>m_ T? ü;=_ iC>E? Z;? 4 k lkq 2lZq?
[Y Z?? hD9D6<_ r? D=D>_ Z? \]5;^GZK@^_ iC>E? Z;? lkkq k?
[Y ? <@B;EBK_ h? Z? jBDI? wB@? iD=@E j lq k2?
[Y i? `D;<E;6_ j?? i89;C@<_ jE9=8<? jE9=85C>E? k lkq ?
[Y Q? 48KBC@<_ i? `D;<E;6_ jE9=8<? jE9=85C>E? 2 lkq n?
[Y Z? wCD=mD_ w?4? bDCD=D_ b8<9C6> N89@B;E Z8>D6 jE9=8<? w8B? lkq kn?
[Y b??? I; j;66D=_ r?? `8=D9C_ <9? r? b8I? iC>E? 4 lkq 2?
[kY ? @99;<_ iC>E? Z;? 4 2 lq kk?
[2Y h?i? iD<8E_ |?hC? hCD9^@EDDE_ hC?h? b8KE9D@I@E_ ?? |>=8K_ iC>E? \;99? j 22
lkq ?
[Y j? 8=S88_ Q? 4; i=8:9_ i? ;;=6@<SE_ |?4? w;<_ hC;m? iC>E? \;99? lkq ?
[nY r? wDzKI8_ Z? \]5;^GZK@^_ <9? Z;? iC>E? k lkq kk2?
[Y ? \KS8<;E_ ? 8mODB@_ iC>E? Z;? 4 k lknq nk?
[Y ?? ;<;<K98_ r?? `8=D9C_ iC>E? Z;? \;99? 2 lq ?
[Y ? 8mODB@_ ? 4D99D_ jE9=85C>E? r? n2 lk2q \?
[Y ?Q? bD==D<SC;668_ h?j?? ~DEB8<B;668E_ r?j? I; Q=;@9DE iDBC;B8_ iC>E? Z;? 4
lkkq k?
[kY f? hD6O;9_ Z? \]5;^GZK@^_ iC>E? j w9D9@E9? b;BC? j556? k lkq 2k?
[kY f? hD6O;9_ Z? \]5;^GZK@^_ iC>E? Z;? 2 lkq ?
[kkY h? hD=D9C8I8=>_ bD9C? j<<? lq 2nn?
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.
96Chapter 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
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)
98Chapter 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
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:
100Chapter 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
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.
102Chapter 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
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).
104Chapter 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
4.4 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
M N R A Sc 000, 1?? ( Pt 3 Mch 1 (MN L TEX yt t v
l i kl qd lli o
lw ! bi! 4" #$%&')*
+- G- B- ./ 25/ee6789 6a. +- +/a./:; 6a. 2- <6aa6; 6a. J- =- H>756@C8
DIFKOQOUOV WKOYVFVZQ[V\ ]^V_KQ[V ^ `^ fQ^F[QgK WOZVKj^YQ[gK\ mFQn^YKQ`g`^ `^ pgV rgUsV\ ux `V zgOgV\ ||~\ pgV rgUsV\ YgQs
gO^F WKOYVFVZQ[gs IFKOQOUO^\ ]YVFQF^F\ ^ ^O^YsgF`K
Acctt 1 tctt 1 Rtctvt 1 tctt 1 1 ct 11
¡¢£¤¥ ¦§¨ ¥© ©ª©« ¤ª¬ ¤££¥¦ ®¥¥ §¤¦® ¥© ¯°¤§£ ¦§ ¥© °«
¥¤¦ ¯¥£¤ ¡±¨ ¯ ¤ ¢¤¦ ¯¯ £ ¤ « ®¥¥ £¥ ²¦³ ¤ ¤ª ¯ ¥© £¤ ¥¦
¯´µ ¦ £ ¦¯¥¤£ ¥© ¯¶ ¦§ ¥© ¡¤ «¨ ®¥¥ª ¤ ª¦ ® £¯°¤ª ¥© ¥®
²£ª¯ · ¥³ ²¦³« £ ©ª©« ¤ª¬ ¢©¦¥¦¯ ¦§ ¥© ´²¦© ¤¥¶ ¸°£¯«¢ ¤¦
¦¯²²£¥¦¯ ¡´¹¶ º»¼¯¨½ ¾ ¯¥° ¥© ¯¬¯¥ ® ¿À 8ÁÂÃÄÅ;µ ³©© ¥©
§¤ ¸° ¬ ¦§ ¥© ´¹¶ º»¼¯ ¤£ª ¯ §¤¦® Å¿Â ¥¦ 8ÂÁ ¹¶µ ³¥© ¥© ¢ £´ ¯ ¢«
£¤£¥¦ · ª Æ ÇÂÂÈɽ ¾ Ê £ ¯ªÊ£¥ ¢ ¦§ ¥© ¥® ²£ª¯
³¥© ¤ª¬ ¡ÂËÂ8ÅÌÍÎÏ £¥ ÁÐÎÑ ¥¦ ÂËÂÒÅÌÍÎÏ £¥ 8ÃÐÎÑ ¨µ ·°¥ £ ³ £´ ¡§ £¬¨
¢ ³¥© §¤ ¸° ¬ £ ³ °¯ ¥© ¯ ¤ ¯°²¥¯ ¥¦ ¦¯¥¤£ ¥© ²¦£¥¦
£¥ ³©© ¥© ¯ º»¼¯ £¤ ¢¤¦° ½ Ó£¯ ¦ ¦°¤ ¤ ¯°²¥¯µ ³ ¦²° ¥©£¥
¥© ¯¶ ¦§ ¥© ®¥¥ª ¤ ª¦ ¯©¦°² · ¯®£²² £ Ô ¤¬ £¤ ¥© °¥¤¦ ¯¥£¤
§ ³ °¯ £ ¦³«¯£¥¥ ¤ª ®¦ ² §¦¤ ¥© ²£¬¯½
ÕÖ× ØÙÚÛÜÝ Þ«¤£¬µ ßàÞÓµ °¥¤¦ ¯¥£¤µ ¥® ²£¬¯½
á âãäåæâäã
çè éêèëìí îíîïðñî òêïó ë òðëôõí ñëöèðïê÷ðø èðùïìúè îïëì ûüýþ ëî ÿìêñëìí ëèø ë õúòwñëîî
îðsúèøëìí îïëì ïóðìð êî ëè ëssìðïêúè øêîs ïóëï ðeïðèøî vðìí sõúîð ïú ïóð îùìësð ú ïóð üý
û(ðòêè & Júîî 1þ Tóð ôêõúóðìï÷ qùëîêwÿðìêúøês úîsêõõëïêúèî ûô÷ Qîþ êè ïóêî ôêèø ú
? cvty z x
c RAS
106Chapter 4. X-ray astrophysics as a tool to study kilohertz quasi-periodic oscillations and time lags of the X-ray emission
and to probe the environment of neutron stars
2 M G B d Avl M Md A Sll ld J E Hvl
ss s a tt b p a ( cs t i o t ai is
(o a bio ri ts assis s ra Kcis 2 Kaa ac 1 a
os ti I tis pap s t sa Xa bia 4 1 !"2 si
aa o t Rssi Xa Ti i #$pc (RXT# o t 1 bs o t s
4 1 !"2 is a c ai %c c c isi (& :1Le'' asi acc ()asi
* ra Kcis 1 s i tit t )k Q+,s appa i pais a sti bss
it i i s rai o as as (-t * Rssc.p 14 Ttis
s as acs o $tibi i cas o o s o isss (/ata
ac 10
#p i 3ptas cas a o t s poc p isi a s oa
css s cs sai t Xa issi tais s a i ts ss s
(5a ac 1 /ata ac 10 Kaa ac 1 6 +a 2 I tis pap
ai i a ta aas is ca ta ta i pris s sc
aa i tacs a Q+, of si t i of o t c a
pp )k Q+,s p a o t Q+, paa s as a oi o of
Tt pap is s i t occi ay i si 2 sib t aa s a
t tc pc i s i si 7 st t scs o t aa aacsis
i si 4 isss t scs i t cit o a $isi sai c a i
si " ps s csis a pspirs
8 O9;<=V>?@OC; >CD F<?NODOPOUW
Y tar si t c ass Xa bia 4 1 !"2 si aa o t Rssi XRa
Ti #$pc (RXT# sacci o t 1 bs o 24 23731 (s abc 1
a sbs o t bsrais s i (Z[k ac 1 , aa os a pi i
tit t s as i t so3tit sa tit as ta t s ta cairc
c tass ai (ta ba3so ba a tit Xa \$ T b
pis 4 1 !"2 as i t asii pi o t ta3c t so3tit sa
(ra ]aa ac 27 Tt aa i s at it i^ %iis o
t tacs (72Z a 4Z s abc 2 o t +piac _ `a (+_`
ba RXT#
Y p gi + .si ]pa (+.] o t occ ba r 1
ss it a 5fis of o 24 )k Y i i tis acca t i
hj mnnm quwx zquw |||x ~
Section 4.4. Time lags in the kilohertz quasi-periodic oscillations of the low-mass X-ray binary 4U 1608–52 107
T l i 4 1 3
O D # s DD md
2 2! ! ! !2
" ! 2 2 222 2
$a%&' () O*s+,D-.s
2
C/D..0s 5D*s6 <E> [7e8 C/D..0s 5D*s6 <E> [7e8
9"! 9"! !9! " !92 "9 "92"! 92 2 92"2! 29 2 92! "92 22 !9
2"!! 9
$a%&' :) E.+;= *D.ds D,+D;d *= ?@A
BFGH oI tJK BowKL FMN tJK uPPKL kQR SUVH WM XY Z\]^_`b FH F IuMftWoM oI SUV ILKcuKMfgh
jM oLNKL to No tJWHn wK puHt qLHt WNKMtWIg rotJ kQR SUVH WM tJWH HouLfKh QKLK wK IoBBow
tJK HFpK PLofKNuLK NKHfLWrKN WM vxyKMNKR Kt FBh Zzz^ FMN vxyKMNKR Kt FBh Zzzzh jM HJoLtn
tJK PLofKNuLK foMHWHtKN WM tJLKK HtKPH| qLHtn to PLoNufK U~H KKLg Z\ vJKLK wK PLoNufKN
FBHo H KKLg Z\ tJKM to tLFfk tJK kQR SUV ILKcuKMfg WM tWpK WM FBB U~H FMN tJK
qMFB HtKP wFH to WNKMtWIg tJK kQR SUVH vuPPKL oL BowKL rg LKBFtWMG tJK PKFk HKPFLFtWoM to
tJK BowKL kQR SUV FMN BFtKL to tJK LFg foBoL FMN WMtKMHWtWKHn KpPWLWfFBBg WNKMtWqKN rotJ
IKFtuLKHh
K tJKM PLofKKNKN HKBKftWMG ouL NFtF WM KMKLGg FMN ILKcuKMfg rFMNH FH HJowM WM tJK
tFrBKH b FMN 3h JK fLWtKLWup IoL tJK ILKcuKMfg HKBKftWoM wFH NKqMKN to HPFM F HufWKMtBg
HpFBB LFMGK WM oLNKL to pWMWpWRK tJK KKftH oI fJFMGKH oI tJK PLoPKLtWKH oI tJK SUV wWtJ
ILKcuKMfgn FMN to JFK F HufWKMtBg BFLGK MuprKL vrFBFMfKN FpoMG tJK rFMNH oI H WM
oLNKL to LKNufK tJK HtFtWHtWfFB uMfKLtFWMtWKHh
jt WH Mot PoHHWrBK to tLFfk tJK uPPKL kQR SUV oM HJoLt tWpK HfFBKH vtFrBK 3 WM tJK HFpK
wFg wK NKHfLWrKN rKIoLKn rKfFuHK Wt WH wKFk FMN rLoFN vxyKMNKR Kt FBh Zzzzn Zzz^h Kn
tJKLKIoLKn took tJK PBot oI vxyKMNKR Kt FBh Zzz^ LKBFtWMG tJK PKFk HKPFLFtWoM to tJK BowKL
kQR SUVh K tJKM qttKN F cuFNLFtWf IuMftWoM PBuH F GFuHHWFM to tJK NFtFh jM tJWH wFg wK
wKLK FrBK to fFBfuBFtK tJK ILKcuKMfg oI tJK uPPKL kQR SUV FH F IuMftWoM oI tJK ILKcuKMfg
oI tJK BowKL kQR SUVh
108Chapter 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 M G B d Avl M Md A Sll ld J E Hvl
Fr r ( # 3 # 6
56 635 66 553 326 52 323 66 3
25 6 3 5 63 6 3536 6
Ta !" $% L&wr k Q' ())r ) p * ))r k Q' (p&wr ) p & t+ pp &o,r-t+& , + t.t m&* t&t.r
/0178 97 h:; 1h7 087f<7=>?7@ C0 1h7 DC978 :=; 1h7 <uu78 IKN OPR@U 97 :uuD?7; 1h7
@h?01s:=;s:;; 17>h=?f<7U 9h?>h >C=@?@1@ ?= @h?01?=V 1h7 >7=18C?; 087f<7=>W C0 1h7 OPR 1C :
870787=>7 087f<7=>W ?= 7:>h ;:1: @7VX7=1 b70C87 97 >:D><D:17 1h7 :Y78:V7 >8C@@ @u7>18<X C0
:DD Cb@78Y:1?C=Z [h?@ 17>h=?f<7 ?=>87:@7@ 1h7 @?V=:Ds1Cs=C?@7 8:1?C :8C<=; 1h7 OPR 07:1<87@Z
[h?@ ?@ 1h7 \8@1 1?X7 1h:1 1h7 @h?01s:=;s:;; 17>h=?f<7 ?@ :uuD?7; 1C >:D><D:17 1?X7 D:V@ :=;
1h7 >Ch787=>7 C0 IKN OPR@Z
]7 1h7= >CXb?=7; :DD 1h7 ^^[@ 9?1h?= 7:>h 087f<7=>W ?=178Y:D 1C X7:@<87 1h7 1?X7
D:V@ 0C8 1h7 DC978 :=; <uu78 IKN OPR Y?: 1h7 >8C@@s@u7>18<X C0 1h7 h?Vh78 7=78VW b:=;@
87D:1?Y7DW 1C 1h7 870787=>7 7=78VW b:=;U ;7\=7; 1C b7 1h7 :b@CD<17 >h:==7D@ _s`c C0 bC1h
XC;7@U 9?1h X7:= 7=78VW eZgi I7jZ nC1?>7 1h:1 ?1 9:@ uC@@?bD7 1C >CXb?=7 1h7 ;:1: C0 bC1h
XC;7@U ecq :=; x4qU b7>:<@7 97 ;7\=7; 1h7 @:X7 870787=>7 7=78VW b:=; 0C8 bC1hU :=;
@?X<D1:=7C<@DW ?=178>:D:17; 1h7 C1h78 >h:==7D@ 9?1hC<1 @<u78uC@?1?C= ?= 7=78VWU X<>h D?I7 :
D:;;78Z
^?V<87 ` @hC9 1h7 1?X7 D:V@ C0 1h7 :Y78:V7; >8C@@ @u7>18<X C0 1h7 Cb@78Y:1?C= :0178 1h7
@h?01s:=;s:;; h:Y7 b77= :uuD?7; ?= :DD 087f<7=>W @u:>7Z nC1?>7 1h:1 1h7 1?X7 D:V@ @hC9 D:8V7
788C8 b:8@ :=; :87 Y78W =C?@7 7y>7u1 ?= 1h7 87V?C= C0 1h7 @h?017; DC978 IKN OPRZ [h?@ 87V?C=
@u:=@ :uu8Cy?X:17DW 19?>7 1h7 0<DD 9?;1h :1 h:D0 X:y?X<X C0 1h7 IKN OPRZ
[h7 D:@1 @17u ?= >:D><D:1?=V 1h7 1?X7 D:V@ C0 1h7 IKN OPR@ 9:@ 1C :Y78:V7 1h7 1?X7 D:V@
?= 1h:1 19?>7 ^]Kq 8:=V7 0C8 1h7 V?Y7= 087f<7=>W @7D7>1?C= 0C8 7:>h 7=78VW b:=;Z
[?X7 D:V@ :87 C<8?78s087f<7=>Ws;7u7=;7=1 X7:@<87@ C0 1h7 1?X7 ;7D:W@ b71977= 19C
>C=><887=1 :=; >C887D:17; 1?X7 @78?7@U ?= C<8 >:@7U b71977= 19C zs8:W D?Vh1 ><8Y7@ ?= ;?787=1
7=78VW b:=;@Z R= 1h7 C1h78 h:=;U 1h7 C<8?78 >Ch787=>7 ?@ : C<8?78s087f<7=>Ws;7u7=;7=1
X7:@<87 C0 1h7 D?=7:8 >C887D:1?C= b71977= 1hC@7 19C D?Vh1 ><8Y7@ ?= ;?787=1 7=78VW b:=;@Z
P7807>1 >Ch787=>7 |<=?1 >Ch787=>7 X7:=@ 1h:1 97 >:= u87;?>1 1h7 @?V=:D C0 C=7 D?Vh1><8Y7
~
Section 4.4. Time lags in the kilohertz quasi-periodic oscillations of the low-mass X-ray binary 4U 1608–52 109
T l i 4 1 5
F cc f t e ! be 7"#$ k% r! 3"7& k% f t f 'ec! ce $#6 H( 776H("
)*+, -./ s0289: +) -./ +-./*o ;<// N+w9a /- 9:o =>>>? )+* 9 @+,A:/-/ /BA:989-0+8 9C+D- -./
@+./*/8@/ 98E :92s 98E .+w -+ @9:@D:9-/ -./,Go
I./ 0,A+*-98@/ +) -./s/ s-9-0s-0@9: ,/9sD*/,/8-s 0s -.9- -./J @98 A*+v0E/ s-*+82 @+8K
s-*908-s +8 ,+E/:s */:9-082 -./ /8/*2J sA/@-*9 98E -./ 9A/*0+E0@ v9*09C0:0-J )/9-D*/s :0a/ -./
aLM QOPso R8 A9*-0@D:9*? 8/9* A/*)/@- @+./*/8@/ @98 *D:/ +D- ,+E/:s -.9- 9ssD,/ sA9-09::J
/B-/8E/E s+D*@/s +* /,0--082 */20+8s ;S9D2.98 U N+w9a =>>VGo
W XYZ[\]Z
R8 ^02D*/ _ w/ A:+--/E -./ -0,/ :92s )+* -.*// +) -./ )*/`D/8@J s/:/@-0+8s )+* -./ :+w/* aLM
QOPo I./ d2D*/ s.+ws -.9- s+)- A.+-+8s :92 -./ .9*E +8/s ;S9D2.98 /- 9:o =>>Vg h99*/- /-
9:o =>>>Go I./ s+:0E @D*v/s s.+w -./ C/s-Kd--082 */sD:-s )+* -./ ,+E/: w/ Ds/E ;s// C/:+wGo
*+, -.0s ^02D*/ 0- 0s 9AA9*/8- -.9- -./*/ 0s 9 s-*+82 E/A/8E/8@/ +) -./ :92s w0-. /8/*2J?
CD- -./*/ 0s 8+ ;+* v/*J w/9aG E/A/8E/8@/ +8 -./ )*/`D/8@J +) -./ aLM QOPo
^02D*/ j s.+ws -.9- -./*/ 0s 8+ s0280d@98- E/A/8E/8@/ +) -./ :92s w0-. /8/*2J )+* -./
DAA/* aLM QOPg w/ 8+-/? .+w/v/*? -.9- -./ DAA/* aLM QOP 0s w/9a/* 98E C*+9E/* 98E
-./*/)+*/ :/ss s0280d@98- -.98 -./ :+w/* aLM QOP 08 /9@. s/:/@-0+8o L+w/v/*? 20v/8 -./ :9*2/
/**+* C9*s? w/ @988+- E0s@9*E -.9- -./ /8/*2J E/A/8E/8@/ +) -./ :92s +) -./ DAA/* aLM QOP
0s s0,0:9* -+ -.9- +) -./ :+w/* aLM QOPo
N/B-? w/ s/A9*9-/E -./ E9-9 08 -w+ 2*+DAs? E/A/8E082 +8 -./ /8/*2Jm nK=_ a/S 98E =_K_p
a/S? 98E w/ A:+--/E -./ ,/98 )*/`D/8@J 08 -./ 08-/*v9: vs -./ ,/98 E/:9J )+* /9@. 2*+DA
w0-. */sA/@- -+ -./ */)/*/8@/ /8/*2J C98E ;s// ^02D*/s n 98E 5Go R- 0s 9AA9*/8- )*+, -./ A:+-s
cq u66u xyz |xyz ~~~
110Chapter 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
6 M G B d Avl M Md A Sll ld J E Hvl
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
4 6 8 10 12 14 16 18 20
∆t [m
sec]
E [keV]
610-690 Hz690-770 Hz770-823 Hz
F 2 L [ e [ w k Q ! ie f"#e be$ %&'(%)' k *$+ %)'(,,' k *e+ e$ ,,'(7-.k *b#+/ T0 ie t0 bt 1t f t0 $eit e#b f t0 ittie ie *t0 ie$it t0 f"#e be$ t01tt$ # be t/+
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
4 6 8 10 12 14 16 18
∆(t)
[mse
c]
E [keV]
910-993 Hz993-1040 Hz
1040-1064 Hz
F 3 L [ e [ #uu k Q ! ie f"#e tie/
4584 459:9 ;< n= <;sn;>?8n4 @9C9n@9n?9 =o 459 4;D9 I8s< =n 459 o:9KN9n?O =o 459 PRU VWXY
Z59 o\49<4 o=: 459 ]\^_ P9` s;g9< C:=p a cY^c 8n@ 459 o\49<4 o=: 459 ^_\_c P9` s;g9< C:=p a
cY_6Y hn p=45 ?8<9< 459 I;n98: >4 ;< n=4 <;sn;>?8n4 p9449: 458n 8 ?=n<48n4Y
j;n8IIOm q9 ?8I?NI849@ 459 ;n4:;n<;? rn=;<9 <Np4:8?49@x j=N:;9: ?=59:9n?9 p94q99n 98?5 =o
459 9n9:sO p8n@< 4= 459 yYz P9` p8n@ r<99 j;sN:9 6x ;n 459 <8D9 q8O q9 @;@ o=: 459 4;D9
I8s<Y |9?8N<9 =o 459 C==: <484;<4;?<m 459 ?=59:9n?9 =o 459 NCC9: PRU VWX ;< Nn?=n<4:8;n9@Y
hn j;sN:9 z 459 8g9:8s9 ;n4:;n<;? ?=59:9n?9 =o 459 8g9:8s9@ ?:=<< <C9?4:ND ;n 459 o:9KN9n?O
<C8?9 ;< <5=qnY =4;?9 4584 ;n 8II pN4 9~8?4IO ;n 459 :9s;=n =o 459 I=q9: PRU VWX 459 9::=:
p8:< 8:9 p;s 8n@ 459 ?=59:9n?9 ;< n=;<OY s8;n q9 <99 5=q 459 <5;o4\8n@\8@@ 49?5n;KN9 ;<
C=q9:oNIY
-''- &
Section 4.4. Time lags in the kilohertz quasi-periodic oscillations of the low-mass X-ray binary 4U 1608–52 111
T l i 4 1 7
-0.035
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
550 600 650 700 750 800 850
<∆(
t)>
[mse
c]
<ν> [Hz]
F M [ v f , !w k"# Q$% bew & a '( k)*
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
550 600 650 700 750 800 850
<∆(
t)>
[mse
c]
<ν> [Hz]
F 5 M [ v f , !w k"# Q$% bew '( a (2 k)*
−40
−20
0
20
40
60
4 6 8 10 12 14 16 18 20
<In
tCoh
(f)>
E [keV]
579 Hz, 32M
4 6 8 10 12 14 16 18 20−3−2−10123456
<In
tCoh
(f)>
E [keV]
655 Hz, 32M655 Hz, 64M
−3−2−1
0123456
4 6 8 10 12 14 16 18 20
<In
tCoh
(f)>
E [keV]
745 Hz, 32M745 Hz, 64M
4 6 8 10 12 14 16 18 20−3−2−10123456
<In
tCoh
(f)>
E [keV]
791 Hz, 32M791 Hz, 64M
F 6 Iett +!to!- v E [k) f! - f et!s ae !a 3(M a .&M* /- ak-!t#!e t re e- rfe !-*
0 (22( R89, M:R89 ;;;, '<??
112Chapter 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
8 M G B d Avl M Md A Sll ld J E Hvl
F 7 It oo cct fo t e bc 55 k ctr to 3 5 k fo t f! to6" #$ to " #$
4 D%&'(&&%)*
+,- ./0 1-2. .9:0 ./0 2/9s.;<=>;<>> .0?/=9h@0 w<2 @20> 9= ./0 ?-,22 2C0?.-@: ., ?<K?@K<.0
./0 .9:0 K<L2 <=> ./0 ?,/0-0=?0 ,s NOP QRT2 U200 1L@-02 V <=> W s,- < =9?0 0X<:CK0YZ [/92
<KK,w0> @2 ., L<9= 2.<.92.9?2 w/0= w0 >9\9>0> ,@- ,]20-\<.9,=2 1-2. 9= 0=0-L^ <=> <s.0- 9=
s-0h@0=?^ ]<=>2Z _0 s,@=> ./<. ,= .9:0 2?<K02 ,s ./0 NOP QRT2 ./0 2,s. C/,.,=2 K<L ./0
/<-> ,=02 ]^ ` Vagijm ., ` Wagijm Un<@L/<= 0. <KZ VppWYq Uu<<-0. 0. <KZ VpppYq ?,=1-:9=L
C-0\9,@2 [email protected] ]@. w9./ < K<-L0- ><.< 20. 9= :,-0 ./<= .w9?0 0=0-L^ ]<=>2q ]029>02 ./0
>9\929,= ,s ./0 ><.< 9= s-0h@0=?^ 9=.0-\<K2q w/9?/ w<2 =,. >,=0 ]0s,-0Z [/0 2,s. K<L2 9=?-0<20
w9./ 0=0-L^ ]@. 2@-C-929=LK^ >0C0=> w0<NK^ U9s >0C0=> <. <KKY ,= ./0 NOP QRT s-0h@0=? Z
x29=L < :,>0K C-,C,20> ., 0XCK<9= ./020 >0K<^2 U+<K<=L< y [9.<-?/@N zaaWYq w0 ?<K?@K<.0
./0 29P0 ,s ./0 0:9..9=L -0L9,= <=> ?,:C<-0 w9./ ,./0- [email protected] Uu<<-0. 0. <KZ VpppYZ +9=<KK^
w0 1=> ./<. ./0 +,@-90- ?,/0-0=?0 ,= ./0 .9:0 2?<K0 ,s ./0 NOP QRT 92 ?,=292.0=. w9./
@=9. Z
[/0 1.2 L9\0 ./0 0h@9\<K0=. >0=29.^ ,s ./0 -00?.,- U./0 >92?Y| w0 ?<K?@K<.0> ./0 .^C9?<K
29P0 ,s ./0 2?<..0-9=L -0L9,= 9= ./0 >92? s,- >90-0=. ,C.9?<K >0C./2 s,- 0<?/ s-0h@0=?^ ]<=>
0X?0C. ./0 1-2. s-0h@0=?^ ]<=> U~a;Va OPY ./<. 2/,w2 =, 29L=91?<=. .-0=> ,s ./0 K<L2 w9./
0=0-L^ U200 .<]K0 YZ
_0 <22@:0>q <. 1-2.q ./<. ./020 K<L2 <-0 >@0 ., ./0 9==0- 0>L0 ,s ./0 >92? ./<. >,w=;
2?<..0-2 ./0 C/,.,=2 <=> ./@2 <>,C.0> ./0 L0,:0.-^ 9= -0sZ U+<K<=L< y [9.<-?/@N zaaWYZ _0
s,@=> 2,s. K<L2 <=> s,- ./92 -0<2,= w0 @20> ./0 <CC-,X9:<.9,= ]0K,w U\<K9> 9= ./92 ?<20Y 9=
,->0- ., 1. ,@- ><.< 200 U+<K<=L< y [9.<-?/@N zaaWYZ
""
Section 4.4. Time lags in the kilohertz quasi-periodic oscillations of the low-mass X-ray binary 4U 1608–52 113
T l i 4 1 9
Fr r ( ne [2c3] a[k] a5[k]
66 +! 0" " +! 0 6" +! 06## +! $ +! 6% +! ###%0" 0 +! & +! 0 #$ +! %
')*,- ./ S7 8 o: t; 8tt r7 r 7o 7g b t; b 8t<tt7 t; d 87t= :or > ? (a@ t;r 8;oAd o: oBt7AA t;7 d78 d> ? & (a5@ oBt7AA t;7C
DE G HI
JKLr : M
I
NOH
I
NP QIR
u UMVWXYDE
Z\ L^J_ MDE ` Z \
VWXY
L^J_P QfR
hjm pjyqsvwx zw|ssmq s| jm w~m xw wm jm xxs| DE wm jm xwq J_ sq
jm hjq| vqq qmvs| L^^ sq jm mxmv| pwsvxm m|qsy jm mmv ms Qjm
sqvR M sq jm mxvsy xsj N w| N wm jm sm|qs|xmqq m|my jm ~w|q Qjm qw
qw|q jm mmm|vm m|my ~w| w| N \ Q^M0R jmm ^M0 \ II sq jm mq
wqq jm mxmv|R hjm mws|m zw|ssmq wm u jm |~m qvwmq VWXY sq jm
psvwx mpj jm mmv w| Z sq jm qj qsm jm QmRmss| ms|
| jm jm jw| wqqs| jm sqv mx w| wqqs| wqq w| wsq \
QIP ¡H¢PI R£ Q9P¤¡HIP¢R Qh ¥¦m m wx f¢I¢R |m| qw qps| §¨X©n \ ªI9«¬
Qww| m wx f¢¢¤R w| w|msv ®mx ¯ \ QIP H IP°R ± I¢%² Qw| ³wwm| m wx f¢¢ªR
m jwm ´ Iª¢°µf
¶ ·¸¹º » I
¶ «·¸¹º » °f
¶ ¼c½·¸¹º » ¤P9¾M"
jmm ·¸¹º sq jm wsq jm s||mq qw~xm vsvxw ~s w| «·¸¹º w| ¼c½·¸¹º
wm jm qvwxm jsj w| jm m|qsy jm sqv w jm ¿³À Qjmm jm m|qsy mmq jm
vm|wx pxw|m jm sqvR
Ám | w mpm|m|vm sm xwq | m|my wq |vs| jm mzm|vy jm Â
ÃÄ Åq ®m f qjq jm ~mq ® sq v|qsqm| s| ~ms| jm qwm jm jmm mzm|vy
qmxmvs|q mqxs| s| qssxw |~m m|qssmq jm wq | jsvj jm pj|q wm qvwmm
Æ
Å| s|mmqs| ps| ~m |svm sq jw m | w pwsvxm |~m m|qsy w~
I ± I¢0M" jw sq qmjs| w~ I ± I¢Ç$¾M
" Èm| s jmm sq | ¿³À w| jm
sqv mÉm|q vxqm jm qwvm jm m|qsy x ~m ¼c » IP¾M"
Ê s m vwxvxwm jm msvwx m|qsy jm sqv Qmm~m m jwm wqqm w
Ë 00 ÌÍS= ÎÏÌÍS ÐÐÐ= ÑÒÒ
114Chapter 4. X-ray astrophysics as a tool to study kilohertz quasi-periodic oscillations and time lags of the X-ray emission
and to probe the environment of neutron stars
1 M G B d Avl M Md A Sll ld J E Hvl
s m fm t s w s tt e(z2) If w t I t t sty
t z = ! = 8" ts s ":4g#$%3 f w t r = 9:5k% t t &t sty t
z = ! = '4 ts s 1:g#$%3) m & t &t sty t t * mf t s (w
z = ! t t s t& R ms mt t t ct sty fm mc m)
I ts wy w s tt t +tm mt b ,-ty s t & ty mf t
st) . mssbty s tt t ,-s /t s tm s tc t t I fm w t
&mty ss ty t sty s s y) I ts s m t
+tm *t mc smw btw t st t * mf t s mt t
I) t mm scmc * t st)
0mw& ssc * w6 m f7c y m *t &y ; t mtm s
m w t ys mc t&y tm t * mf t st m t cb sty
&y &y y s t I)
T sty w mbt s c mw t t ( t sty ct fm t ,
s m) <ct ts s / t bcs t ss c 6y tt t /-ys wmc tt
sm t s tm b stt m;) . mssbty tt s&s m stcs s tt t
/-ys stt t s tms m scf mf t s w t m t t s
mw t m ) W mc t 6 ct &t stctc c * sm ss tm
s f w *t t tm mc sty)
.* w fmc tt t *s m mt /bt y m f7c y wt
cs tm t 6 tt t *m w t *s mc s &y s)
Ts s mmbmt by t m mf t mw 60> Q? b * m sst t wt c ty)
0mw& t y b &t mc tmmm*y@ f t s 6 tm
btw t f7c y mf t c 60> Q? t cs mf t s t ts wmc
b wy tm ct t cs m t f7c y wmc b t)
<ct t 60> Q?s ts smc s '!z) T CuDDFK = 84!z L RiN O "k%
w CuDDFK = 11"!z L RiN O 1Pk%) T ; RiN 'k% ss c ttb)
t mt t smcs ) *) 4U 1V'V-5' w t c 60> Q? s s
c w f7c y * 8!z) Xm sc ; w / t YRiN 15 "k%
w mc b tt f t s f7c y )
Z[ 2\\2 ]^_` ah]^_ jjj` nopp
Section 4.4. Time lags in the kilohertz quasi-periodic oscillations of the low-mass X-ray binary 4U 1608–52 115
T l i 4 1
5 CC A N NS
s o t dows tt!" !"o s v!e s# $$% w # s t t t d$ es !
p!od&d v!e ! t &t!o st !' ts s "!#t wt t ss!to t t t oc
sste o t Fo&!! o! wt &te p!$&ds (tdd so&!s' t s o t
dows tt!" !"o s osstt wt b" t s # o! t t! !f&e b ds)
!s&$ts ! b sd o t #od$ o dows tt!" o !d potos !o# t &t!o
st ! s&! o* t ! d" o t ds +F $ " , t !&u 2--./% d w ! ow
wo!u" to dv$op #od$ t t osd!s !p!oss" t ds)
0ot! st&de t t s $! de p!o"!ss s to st&de t t# $ "s d t o!
o! 36 787c98 &s" v #& b""! d t st)
AC:WN;<N
&to!s t u =0>?@ o! t B $ s&ppo!t d t K pte 0st!oo# $ Istc
t&t G!o"% Dt!$ ds% o! t ospt $te) 0d $so Eu H#st! % J"o
0$t #! o d G&ob o Z " o! t $p d ds&ssos)
RNLNRNCN
M) MOd% M) v d! K$s% P) QU ds% ?) =) Fo!d% V) v > ! dUs% E) 0) X &" %
YY[% 0pV% 9-9% \28% st!ocpaY[-.2[
M) MOd% st!ocpa---733v
M) MOd% M) v d! K$s% ?) =) Fo!d% P) QU ds% V) v > ! dUs% YYY% 0pV% 9 %
\3Y% st!ocpaY[ 27
E) 0) X &" % M) v d! K$s% M) MOd% V) v > ! dUs% P) 0) J) QU ds% E) 0)
X &" % Q) H) G) \w% F) K) \ #b% J) >s $ts% ?) K&&$u!s% ) ]ost!b!ou% YY.%
0pV% 3[8% \ 9% st!ocpaY.-3239% ?!! t&#^ YY[0pV)))9-Y\) 39X
M) F $ " % \) t !&u% 2--.% 0pV% 77 % -[3% st!ocpa-.-2398
>) K !t% @) >! o% ?) Fo!d% 0) @ t "$o% YYY% 0pV% 9 3% \8 % st!ocpaYY- 83Y
>) K !t% ?) =) Fo!d% YY.% @% v2.7% 98 .% 8[7c 8Y
J) E !!t% M) E tt% M) =) M$$!% 2- % 0pV% .2[^Y
M) 0) Dow u% E) 0) X &" % V) Q$#s% V) E) Jov% M) =) E"$# % _` ghj Ti
kmnl`hh qrhxy`i `z ji| g1 ~~ Ti ilj n 5 +[.3c[Y /% YYY V
-
116Chapter 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
1 M G B d Avl M Md A Sll ld J E Hvl
Vaa Nwa Xl ll Ce H C Wl
Ml! ld H e!l! Md! o GX 33"# ld C$ ! X% AJ 4&4 (')*+'),-
1../ 0a 1
0 P2a5 [6789:;<=>???@D?D]
Faa I5K5L O Q2aRTUa 0 YRa P Za\5QQa ^ 5QQT O 'T _a K5U
`QT\ b cTfaK\ g k q l!de !el! d $ h ld l l!!
o q l! XgE J%ij%#mn MpAS 4rs (,+,*t- t1t
5K5L u lx l d ld ee o k q l!de
!el! l!! Xl l! MpAS y&z (1.+1.*|- tt,
F _a F2Uaa25 _a K5U `QT\ I5K5L g A S e Sl! o # %mj~n AJ
(11+11/,- tt* 25U t
^ _5U L5Q ZaU5Ua+'a_5U\ P cUQ5w\T g !le Ml!! ld ld ! o
p Sl # %mj~n AJ &z (.,)+./*- t1t UTQ 1
0 YaU2Ra O ZaUaaU2 O aQQwa P Fa__ I5K5L F _a F2Uaa25
^ OT FaQ_ !ev o m%" H g el B ! u!el! Ml!! X
l Bl # %mj~n AAS (Y O R552T /- tt*
Z5 F a OT dv Sel Sl gl! Ml!!
Xl Bl # %mj~n AJ r (..+*t- tt, 25U 1t
Ya\T5U _a K5U `QT\ g l! o eld Xl $ ld !el
lv l!! Xl l! A A (/.+.,- 1.|. N_5R5U
_a K5U `QT\ ld Xl ll Cle Sl Xl S e! dd Wl
Me vl d ! ZaRUTK5 \2U\T\ F5UT5\ y ZaRUTK5 T_5U\T2
PU5\\
0 '5U O b\\5Q+OUI5 e e g! ld de! # %mj~n l! u
!vd EA B AJ 4y (|)t+|)/- 1..) N_5R5U 1t
c Y '5wT P Z 0\\ Aeev Sl Xl S e! dd Wl H
G ld Edld J vl d H v ZaRUTK5 T_5U\T2 PU5\\ || 1.|*
I5K5L _a K5U `QT\ 0 _a PaUaKTf\ c Y '5wT Vaa `Q5U\
c a ` 'aR aK O P\aQ2T\ 1..| 0 ).) ', a\2U+./1t|
¡¢££
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
Bibliography
Alcock C., Strange stars, Nuclear Physics B Proceedings Supplements, 1991, vol. 24, p. 93
Alcock C., Farhi E., Olinto A., Strange stars, ApJ, 1986, vol. 310, p. 261
Baade W., Zwicky F., Remarks on Super-Novae and Cosmic Rays, Physical Review, 1934,
vol. 46, p. 76
Benvenuto O. G., Lugones G., The phase transition from nuclear matter to quark matter
during proto-neutron star evolution, MNRAS, 1999, vol. 304, p. L25
Bhattacharyya S., Miller M. C., Lamb F. K., The Shapes of Atomic Lines from the Surfaces
of Weakly Magnetic Rotating Neutron Stars and Their Implications, ApJ, 2006, vol. 644,
p. 1085
Bodmer A. R., Collapsed Nuclei, Phys. Rev. D, 1971, vol. 4, p. 1601
Cappellaro E., Turatto M., A new determination of the frequency of supernovae, A&A,
1988, vol. 190, p. 10
Catalan R. G., Garay J., Lopez-Ruiz R., Features of the extension of a statistical measure
of complexity to continuous systems, Phys. Rev. E, 2002, vol. 66, p. 011102
Chadwick J., Possible Existence of a Neutron, Nature, 1932a, vol. 129, p. 312
Chadwick J., The Existence of a Neutron, Royal Society of London Proceedings Series A,
1932b, vol. 136, p. 692
Chatzisavvas K. C., Psonis V. P., Panos C. P., Moustakidis C. C., Complexity and neutron
star structure, Physics Letters A, 2009, vol. 373, p. 3901
122 Bibliography
Chodos A., Jaffe R. L., Johnson K., Thorn C. B., Baryon structure in the bag theory,
Phys. Rev. D, 1974, vol. 10, p. 2599
Clark J. P. A., Eardley D. M., Evolution of close neutron star binaries, ApJ, 1977, vol. 215,
p. 311
Delgaty M. S. R., Lake K., Physical acceptability of isolated, static, spherically symmetric,
perfect fluid solutions of Einstein’s equations, Computer Physics Communications, 1998,
vol. 115, p. 395
Demorest P. B., Pennucci T., Ransom S. M., Roberts M. S. E., Hessels J. W. T., A
two-solar-mass neutron star measured using Shapiro delay, Nature, 2010, vol. 467, p.
1081
Douchin F., Haensel P., Bounds on the existence of neutron rich nuclei in neutron star
interiors., Acta Physica Polonica B, 1999, vol. 30, p. 1205
Douchin F., Haensel P., Inner edge of neutron-star crust with SLy effective nucleon-nucleon
interactions, Physics Letters B, 2000, vol. 485, p. 107
Douchin F., Haensel P., A unified equation of state of dense matter and neutron star
structure, A&A, 2001, vol. 380, p. 151
Douchin F., Haensel P., Meyer J., Nuclear surface and curvature properties for SLy Skyrme
forces and nuclei in the inner neutron-star crust., Nuclear Physics A, 2000, vol. 665, p.
419
Farhi E., Jaffe R. L., Strange matter, Phys. Rev. D, 1984, vol. 30, p. 2379
Frank J., King A., Raine D. J., Accretion Power in Astrophysics: Third Edition, 2002
Fryer C. L., Belczynski K., Wiktorowicz G., Dominik M., Kalogera V., Holz D., Compact
Remnant Mass Function: Dependence on the Explosion Mechanism and Metallicity,
ArXiv e-prints, 2011
Ghezzi C. R., Relativistic structure, stability, and gravitational collapse of charged neutron
stars, Phys. Rev. D, 2005, vol. 72, p. 104017
Bibliography 123
Giacconi R., Gursky H., Paolini F. R., Rossi B. B., Evidence for x Rays From Sources
Outside the Solar System, Physical Review Letters, 1962, vol. 9, p. 439
Gilfanov M., Revnivtsev M., Molkov S., Boundary layer, accretion disk and X-ray vari-
ability in the luminous LMXBs, A&A, 2003, vol. 410, p. 217
Gleiser M., Dev K., Anistropic Stars:, International Journal of Modern Physics D, 2004,
vol. 13, p. 1389
Glendenning N., Compact Stars. Nuclear Physics, Particle Physics and General Relativity.-
2nd ed., 2000
Gold T., Rotating Neutron Stars as the Origin of the Pulsating Radio Sources, Nature,
1968, vol. 218, p. 731
Haensel P., Pichon B., Experimental nuclear masses and the ground state of cold dense
matter, A&A, 1994, vol. 283, p. 313
Haensel P., Potekhin A. Y., Analytical representations of unified equations of state of
neutron-star matter, A&A, 2004, vol. 428, p. 191
Hess V. F., Observation of Penetrating Radiation in Seven Balloon Flights, Physik
Zeitschrift, 1912
Hewish A., Bell S. J., Pilkington J. D. H., Scott P. F., Collins R. A., Observation of a
Rapidly Pulsating Radio Source, Nature, 1968, vol. 217, p. 709
Hiemstra B., Mendez M., Done C., Dıaz Trigo M., Altamirano D., Casella P., A strong
and broad Fe line in the XMM-Newton spectrum of the new X-ray transient and black
hole candidate XTE J1652-453, MNRAS, 2011, vol. 411, p. 137
Hirano T., Hayakawa S., Nagase F., Masai K., Mitsuda K., Iron emission line from low-
mass x-ray binaries, PASJ, 1987, vol. 39, p. 619
Ivanov B. V., Relativistic static fluid spheres with a linear equation of state, ArXiv General
Relativity and Quantum Cosmology e-prints, 2001
Krane K. S., Introductory Nuclear Physics, 1987
124 Bibliography
Lattimer J. M., Prakash M., Ultimate Energy Density of Observable Cold Baryonic Matter,
Physical Review Letters, 2005, vol. 94, p. 111101
Lewis G. N., The Symmetry of Time in Physics, Science, 1930, vol. 71, p. 569
Linares M., Accretion states and thermonuclear bursts in neutron star X-ray binaries,
Sterrenkundig Instituut ”Anton Pannekoek - University of Amsterdam, 2009, Ph.D.
Thesis
Lopez-Ruiz R., Mancini H. L., Calbet X., A statistical measure of complexity, Physics
Letters A, 1995, vol. 209, p. 321
Melia F., High-Energy Astrophysics, 2009
Mendez M., Timing the Kiloherz Quasi-periodic Oscillations in LMXR Binaries. In The
Neutron Star - Black Hole Connection , 2001, p. 313
Miller J. M., X-ray spectroscopic and timing studies of galactic black hole binaries, Mas-
sachusetts Institute of Technology, 2002, Ph.D. Thesis
Miller M. C., Lamb F. K., Psaltis D., Sonic-Point Model of Kilohertz Quasi-periodic Bright-
ness Oscillations in Low-Mass X-Ray Binaries, ApJ, 1998, vol. 508, p. 791
Negreiros R. P., Weber F., Malheiro M., Usov V., Electrically charged strange quark stars,
Phys. Rev. D, 2009, vol. 80, p. 083006
Nowak M. A., Vaughan B. A., Wilms J., Dove J. B., Begelman M. C., Rossi X-Ray Timing
Explorer Observation of Cygnus X-1. II. Timing Analysis, ApJ, 1999, vol. 510, p. 874
Onicescu O., Energie informationnelle, C. R. Acad. Sci. Paris A 263, 841-842, 1966
Oppenheimer J. R., Volkoff G. M., On Massive Neutron Cores, Physical Review, 1939,
vol. 55, p. 374
Ozel F., Soft equations of state for neutron-star matter ruled out by EXO 0748 - 676,
Nature, 2006, vol. 441, p. 1115
Ozel F., Guver T., Psaltis D., The Mass and Radius of the Neutron Star in EXO 1745-248,
ApJ, 2009, vol. 693, p. 1775
Bibliography 125
Pacini F., Energy Emission from a Neutron Star, Nature, 1967, vol. 216, p. 567
Panos C. P., Chatzisavvas K. C., Complexity classification of quantum many-body systems
according to the Pair of Order-Disorder Indices (PODI), ArXiv e-prints, 2009
Panos C. P., Chatzisavvas K. C., Moustakidis C. C., Kyrkou E. G., Comparison of SDL
and LMC measures of complexity: Atoms as a testbed, Physics Letters A, 2007, vol. 363,
p. 78
Panos C. P., Nikolaidis N. S., Chatzisavvas K. C., Tsouros C. C., A simple method for the
evaluation of the information content and complexity in atoms. A proposal for scalability,
Physics Letters A, 2009, vol. 373, p. 2343
Ray S., Malheiro M., Lemos J. P. S., Zanchin V. T., Electrically charged compact stars,
ArXiv Nuclear Theory e-prints, 2006
Sanudo J., Pacheco A. F., Complexity and white-dwarf structure, Physics Letters A, 2009,
vol. 373, p. 807
Sanna A., Hiemstra B., Mendez M., Altamirano D., Belloni T., Kilohertz QPOs and broad
iron emission lines as a probe of strong-field gravity. In The X-ray Universe 2011 , 2011,
p. 141
Shakura N. I., Sunyaev R. A., Black holes in binary systems. Observational appearance.,
A&A, 1973, vol. 24, p. 337
Shannon C. E., Weaver W., The mathematical theory of communication, 1949
Stella L., Vietri M., Lense-Thirring Precession and Quasi-periodic Oscillations in Low-Mass
X-Ray Binaries, ApJ, 1998, vol. 492, p. L59
Timmes F. X., Woosley S. E., Weaver T. A., The Neutron Star and Black Hole Initial
Mass Function, ApJ, 1996, vol. 457, p. 834
Tolman R. C., Relativity, Thermodynamics, and Cosmology, 1934
Tolman R. C., Static Solutions of Einstein’s Field Equations for Spheres of Fluid, Physical
Review, 1939, vol. 55, p. 364
126 Bibliography
Valentim R., Rangel E., Horvath J. E., On the mass distribution of neutron stars, MNRAS,
2011, vol. 414, p. 1427
van der Klis M., Quasi-periodic oscillations and noise in low-mass X-ray binaries, ARA&A,
1989a, vol. 27, p. 517
van der Klis M., The Z/atoll classification. In Two Topics in X-Ray Astronomy, Volume 1:
X Ray Binaries. Volume 2: AGN and the X Ray Background , vol. 296 of ESA Special
Publication, 1989b, p. 203
van der Klis M., , 2006 Rapid X-ray Variability. pp 39–112
van Straaten S., van der Klis M., Mendez M., The Atoll Source States of 4U 1608-52, ApJ,
2003, vol. 596, p. 1155
Vaughan B. A., Nowak M. A., X-Ray Variability Coherence: How to Compute It, What It
Means, and How It Constrains Models of GX 339-4 and Cygnus X-1, ApJ, 1997, vol. 474,
p. L43
Witten E., Cosmic separation of phases, Phys. Rev. D, 1984, vol. 30, p. 272
Woltjer L., X-Rays and Type i Supernova Remnants., ApJ, 1964, vol. 140, p. 1309
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
S m cmc s sc ms msm
M G B d Avee ad J E Hv*
I o !"#«$ Cö$ o"«$ U% &÷ P'
R (÷ 1))+ 0,,0-./00 &÷ P' &P 23'
L 45e5667
U% F' 2C R & «' 1++ 0/)10.180 & « &P 23'9:;<;=>;? @ DKN=O QTVVW KpXO=YZ;? V[ Dp\pY] QTVV^
_Z; ;`b<] <fgKfY=]=fh fi b YK;<=j< <ObYY fi <fgKb<] Y]bNYk Z=Y]fN=<bOOl N;i;NN;? ]f bY h;p]Nfh Y]bNYk =Y
Y]=OO qp=]; phuhfwhn rfYY=X=O=]=;Y Nbh\=h\ iNfg Zb?Nfh=< ]f qpbNu ?;\N;;Y fi iN;;?fgk =h<Op?=h\ Y;OitXfph?
>;NY=fhY fi ]Z; Ob]];Nk Zb>; X;;h KNfKfY;?n x; YK;<=j<bOOl b??N;YY ]Z; Yp=]bX=O=]l fi Y]Nbh\; Y]bN gf?;OY
9=h<Op?=h\ Kb=N=h\ =h];Nb<]=fhY^ =h ]Z=Y wfNuk =h ]Z; O=\Z] fi h;w g;bYpN;g;h]Y b>b=ObXO; ifN ifpN <fgKb<]
Y]bNYn _Z; bhbOlY=Y YZfwY ]Zb] ]Z;Y; ?b]b g=\Z] X; ;`KOb=h;? Xl Yp<Z bh ;`f]=< ;qpb]=fh fi Y]b];k b<]pbOOl
Y;O;<]=h\ b YgbOO w=h?fw =h KbNbg;];N YKb<;k Xp] Y]=OO h;w KN;<=Y; g;bYpN;g;h]Y bh? bOYf ipN]Z;N
]Z;fN;]=<bO ?;>;OfKg;h]Y bN; h;;?;? ]f Y;]]O; ]Z; YpXy;<]n
z| ~~~z ~ ¡
¢£ ¢¤¥¦§¨©ª¥¢§¤
¬ ® ®a¯e 7d® ¯5 6°±®77a ²
6°±6 ®® ®d ° 5ad ³° da®77®
7® 6°±®d ² da®´ 7´ °7aeµ a5a® ¶7 ®°ee
²67a® ² ±a® ad e® e6a® ·5 76e
dve±°a® ad °da ³±7°ae ®5e® ±ad
¶7ad¶ ±®®7¯7e77® ® ³7®a6 ² µ±a®
7a®7d ®56 ®® d5 7¸ da®77®´ ad ¶e a¶
6e®® ² 6ada®® ¹®´ ² ³°±e´ º»¼ ad ²a6®
7a½ Aa ±®®7¯7e7µ 7®a 7a ¾¿À® 7® ³Á
7®a6 ² d6aad Â5î 7a®7d 6°±6 ¯Ä6®´ 7
e6d aeµ 7a 7aa 6 ² ® ®® ±®a 5±
7 ®5²6® ¬ e ³° ±®®7¯7e7µ ¶® ®5¸¸®d
¯5 d6d® ¸ ºÅÆǼ´ °¸7a¸ ® a ®±µ®Á
76e e7È7a ² ®¯7e7µ ®6a7 ² ®Á6eed
®a¸ °
Éa®Â5a6® ² ®Á6eed ÊËÌÍÎÏÐ ÑÍËËÐÌ ÒÓÔÕËÒÐÊÖÊ
² 6°±6 ®® v ¯a ³a®7veµ aeµÈd ¶77a
®7°±e ²°¶Ã ² M®®65®® ×a®75 ²
¬6ae¸µ ¹M׬½ B¸ Mde º¿´Ø¼ ad Ù°¯5Á
JaÁL®7a7 °de ºÚÆ»»¼ ¹® º»Å¼ ² a vv7¶ a
®5¯Ä6½ A² a 7a77e 5ad ² ±5¯7v 6a®7dÁ
7a® º»Û´»Ü¼´ ±®®7¯7e7µ ² aa±5¯7v ±77a¸
¯¶a Â5î ºÚ´»Ýƻڼ ee¶d a¶ v7¶ a 7®
®5¯Ä6 d5 ¸ aa6°a a ¶7ad¶ ²
®¯7e7µ ² ®a¸ Â5à ° ¹v7a¸ ®a¸ Â5Ã
°®®´ ¯¸ 6a®a´ ad ave ±77a¸ a¸µ ²
Â5Ã 6ada® ® ±°® 7a 7® ±±6½ ¬7®
® ² eeµ ±7d ÛÁÞv Â5î 7® 6eed 6eÁ
Þv e6Ãd ¹É·L½ ®a¸ Â5à °´ 7a ¶76 Â5î
²° ɱ ±7®´ 5® e¶7a¸ ²5 a¸µ ²
®µ®°
M56 ¶Ã ® ¯a ±5 ²¶d 7a d v
¯ 5ad®ad7a¸ a 65e 6°±®77a ² a5a
®®´ ¯ ¯®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¸ Â5à ° °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 ºÅÜÆÅǼ½´ 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´ e7à ɱ ±7 a¸µ
¸± ¹ã½ ad ®a¸ Â5à °®® ¹äå½´ ad ®¶ ¶
7® 7® da ¸7va ®5¯® ² ±®a d
¢¢£ æ¤æçèé¢é
ßa 6a®7d7a¸ 5a±7d ®a¸ Â5à ° 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¸ Â5à °®® ® ²²6 ² 7a6®Á
7a¸ a¸µ ² ®µ®° º¿¼ àa ad´ 7²
®µ®° ±®a® ±77a¸ 7a67a® ¯¶a Â5î 7 ®
a ²²6 ² e¶7a¸ a¸µ ² ®µ®°´ °Ã7a¸
7 ° ®¯e
¬ v ¯a d7²²a ±±6® aeµÈ 7®
²²6 H´ 7a d ® ²°¶Ã ² 6e®® ²
°de® ®5d7d´ ¶ d± M׬ ¯¸ °de °dµÁ
a°76 ±a7e ² É·L ®a¸ Â5à ° 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
¡¢£¡ ¤ ¥¦¦§§¨ ¥¡©ª¨§ ¨« ¬® ¯°±¯¯° ²³¯´´µ
¶·¸¶¶·¹º
132 Appendix A. QCD Parameters
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
¡¢£ ¤£¥¡ ¦§ £¤¨ £¥¥£©¡ª«£ ¦§¬§£¡£¬® §¤¤¯°ª ± ©¯¦§¡ª²ª¤ª¡³ °ª¡¢ª ´µ °ª¡¢ ¬£©£ ¡ ¶§¡§ ·¡§¸ª ± ª ¡¯ §©©¯¹ ¡ ¡¢£ ¯²®£¬«£¶
§®®£® § ¶ ¬§¶ªª § ¶ ¡¢£ª¬ ¬£®¦£©¡ª«£ £¬¬¯¬ ²§¬®º ¡¢£ ¬ª±¢¡¨ §®®»¬§¶ª¹® £§®¹¬££ ¡® §¬£ ®¢¯° §® ©¬¯®®£® °ª¡¢ ´µ £¬¬¯¬ ²§¬® ½§¡§
¥¬¯ ¾¿À ÁÂûÄÄÅÆ ª® ®¢¯° §® § ®¢§¶£¶ ¬£±ª¯ ¨ ®ª ©£ ¡¢£¬£ ª® ¯ ¤³ ª ¥¯¬§¡ª¯ ¯ ¡¢£ §®® ¥¯¬ ¡¢ª® ¯²Ç£©¡ È¢£ ¡°¯ §®®»¬§¶ª¹®
¬£¤§¡ª¯ ® ®¢¯° ©¯¬¬£®¦¯ ¶ ¡¯ ¡¢£ ¤¯°£¬ § ¶ ¹¦¦£¬ ¤ªª¡® ¯¥ É § ¶ ÊËÌÌ ¡¢§¡ ©§ £Í¦¤§ª ¡¢£ ¶§¡§ ·®¢¯° ¯ ¡¢£ ¬ª±¢¡º
¿ÎÏ»ÐÑÒ½ Ó½ÎÏ¿ ÔÓ¾ÕÔÈ ¿ÈÕÀ¿ ÕÒ½ ÀÎÔÎÒÈ Ö Ö Ö ¾×Ø¿ÔÕÏ ÀÎÙÎÚ ½ ÛÜÝ ÆÃÅÆÆà ·Äƺ
ßàáßßàâá
Section A.1. Self-bound models of compact stars and recent mass-radius measurements 133
s N hwv h f 4 1 h mss
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 ;<=>
? @ D F G J KL KK KO KQ
U
VXV
LY?
LYD
LYG
K
KYO
KY?
KYD
KYG
O
OYO
Z\^_ _ dgjk ntuggxgyk zu k|g xnjjun~j ugnky k| uggyk
xgnjugxgykj ~nkn ugugjgykg~ nj y Zt_ yj~guyt k|g
Z gnky z jknkg_ ngj zu k|g nunxgkguj nug
ny~ _
¡ ¢£¤¥
¦ § ¨ © ª «¬ «« « «® «¯
°
±²±
¬³¯
¬³§
¬³©
«
«³
«³¯
«³§
«³©
³
¡ ¢£¤¥
¦ § ¨ © ª «¬ «« « «® «¯
°
±²±
¬³¯
¬³§
¬³©
«
«³
«³¯
«³§
«³©
³
Z\^_ _ µ jgy~ g¶nnky z k|g ugyjkukg~ ~nkn nj t¶gy y ·´`¸_ ¹y k|g gzk j~g xnjj ny~ un~j z º» ¾¿ÀÁ`` Âù ĺÁ
`ºÀ ny~ º» À`¿´¿ ny~ xnjj unytg zu ÅÆÇ È¾º``´¿ yj~guyt É Ê ÉËÌ_ ¹y k|g ut|k k|g jnxg zu É Í ÉËÌ jgg kgÎk zu
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_ Ô µÂµÇ È_ Â_ չǵÏÕ µÖÔ _ ŵ»»\ ÅÕ×Æ\µ ÇÂ\ÂØ Ô ÙÚÛ ¿º´¿¿º `¿
ÜÝßÜÜÝàÝ
134 Appendix A. QCD Parameters
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
hj n n ¡¢£ £ n ¤n ¤uz£ |n un ¥un ¦ i
in
hj §n uz¨© ª «n u¬z u¬z n in
hj ®n r¢u£ |n un ¦¯ i°n
hj ®n lz¢ £ n ¥n ±²²u |n un ¦¯ i°n
h°j ¡n ª ®n lz¢ £ n ³n ´¢£ z¬n ±n ¦µ¯
i i°n
hj ¶n ²z ¤n ·~¬ £ ln r¢¨uª n |n
¦¹ in
®¥l´« ¶´®¥ ´l ¡´¶|¡ ®¡® º º º |§» ¡¥ ®³®r ½¾ ii
¿ÀÁ¿¿ÀÂÃ
Section A.1. Self-bound models of compact stars and recent mass-radius measurements 135
[ O G Bve a G Le I J M P D 7
2 (
[ S Yi a R Se P Rv C 77 002 (2
[2 M Beaa P R 47 20 (20
[ D Baii a A Lv P R 17 20 ( a
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
(2
[ G Le a J E Hrva P Rv D 66 33
(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
(22
[2 R Caaei a G arei Rv M P 76 2
(2
[23 L Paeei E J Frrr V a Ira a J E Hrva
P Rv D & (2
[2 M Ar M Bra M Pari a S R Ar
J 6% (20
[2 C T H Davi )* +,- P Rv L 14 2
(2
[ aa'era )* +,- J P G &7 302 (2
[ O G Bve a J E Hrva M R Ar
S 41 (
[2 A W Sir J M Lai'r a E F Brw Ar
J 7 (2
[ M L Raw )* +,- Ar J 7& 20 (2
M G B DE AVELLAR J E HORVATH AD L PAULUCCI PHYSICAL REVIEW D 48 (2
./5../9: