Download Equations of state and structure of neutron stars

Martin Urbanec
Equations of state and structure of neutron stars
DISSERTATION
Silesian University in Opava, Faculty of Philosophy and Science
Institute of Physics
Equations of state and structure
of neutron stars
DISSERTATION
Supervisor: Prof. RNDr. Zdeněk Stuchlík, CSc.
Opava 2010
Martin Urbanec
Annotation
The work will be focused on the equations of state of neutron star matter and
on its impact on neutron star properties. The author will also focus particular
interest on the impact of rotation on the neutron star models.
Acknowledgements
I am very grateful to my parents Evženie and Zdeněk for their support and
patience during the whole studies. This work is dedicated to them. I would like
to thank my supervisor Prof. RNDr. Zdeněk Stuchlı́k, CSc. for giving me the
opportunity to work in the subject of neutron stars and for his leadership, help
and encouragement. The special thanks go to collaborators and close friends
Eva, Gabriel, Jirka, Pavel and Petr for lots of discussions and for their advices.
Great thanks belong also to Gabriela.
I have been honored by meeting several people who helped a lot to finish this
work. Among others the greatest thanks belongs to Prof. Jiřina Řı́kovská Stone
and Prof. John Miller who both played a key role in the introduction to the
field of neutron stars. I would like to thank particulary Jiřina for providing me
the set of her collected data of equations of state and also John deserves my
gratitude for his help with the code for slowly rotating neutron stars and for all
the discussion connected to this subject.
Finally, I would like to acknowledge the financial support from Czech grant
LC 06014.
Declaration
I declare that I have written this thesis by myself under the supervision of Prof.
RNDr. Zdeněk Stuchlı́k, CSc. and that I have cited all the sources that I used.
In Opava July 2010
...........................................
Martin Urbanec
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
I.
Overview
Chapter 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.
1
5
First ideas and observations . . . . . . . . . . . . . . . . . . . . . . .
6
Chapter 2. Equation of State of Neutron Star Matter . . . . . . . . . . . . .
9
2.1.
Nucleon - nucleon interaction and matter in the cores of neutron stars
2.1.1. Skyrme . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2. Variational theory . . . . . . . . . . . . . . . . . . . . . . .
2.1.3. Relativistic mean field theory . . . . . . . . . . . . . . . . .
2.1.4. Equation of state of neutron star matter . . . . . . . . . . . .
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14
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17
Chapter 3. Neutron Star Models . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.1.
Static, spherically symmetric neutron stars . . . . .
3.1.1. Equations of Structure . . . . . . . . . . . .
3.1.2. Gross properties of neutron stars . . . . . .
3.1.3. Surface properties . . . . . . . . . . . . . .
3.1.4. Numerical solution . . . . . . . . . . . . . .
3.1.5. Maximum mass and stability of neutron stars
3.2. Rotating Neutron Stars . . . . . . . . . . . . . . . .
3.2.1. Hartle - Thorne approximation . . . . . . .
3.3. Results . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1. Moment of Inertia . . . . . . . . . . . . . .
3.3.2. Gravitational radius, Compactness . . . . .
3.3.3. Quadrupole moment . . . . . . . . . . . . .
3.3.4. Angular momentum . . . . . . . . . . . . .
3.3.5. Mass change . . . . . . . . . . . . . . . . .
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Chapter 4. Strange Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
4.1.
4.2.
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MIT Bag Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Strange star models . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1. Neutron to strange star transition . . . . . . . . . . . . . . . .
37
38
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Chapter 5. Theory Versus Observations . . . . . . . . . . . . . . . . . . . . .
43
5.1.
Astrophysical observations testing equations of state . . . . . . . . . .
5.1.1. Maximum mass . . . . . . . . . . . . . . . . . . . . . . . . .
vii
43
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5.1.2. Double pulsar J0737–3039 . . . . . . . . . . . . . . . . . . .
5.1.3. Isolated neutron star RX J1856.5–3754 . . . . . . . . . . . . .
5.2. ISCO frequency in the field of rotating neutron stars . . . . . . . . . .
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Chapter 6. Summary and Discussion . . . . . . . . . . . . . . . . . . . . . .
51
6.1.
New results . . . . . . . . . . . . . . . . . . . . . . . .
6.1.1. Test of equation of state . . . . . . . . . . . . .
6.1.2. Analytical representation of external HT metric
6.2. Future research . . . . . . . . . . . . . . . . . . . . . .
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Appendix A. Nuclear matter at β -equilibrium . . . . . . . . . . . . . . . . .
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Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
II. Individual Papers
viii
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Preface
This thesis represents the results of work that was done during Ph.D. studies at
the Institute of physics at Silesian university in Opava. New results were found
and are concentrated in two connected fields related to structure of the neutron
(strange) stars and their external gravitational field.
The first one is oriented on testing of equation of state for dense nuclear matter
by astrophysical observations. We used set of parameterizations that has been
found to be a good description of matter at supranuclear densities to model static
neutron stars and compared resulting neutron star properties with observations
(published in paper [1]).
The second field is focused on impact of rotation on neutron star structure
with particular interest to relation of quadrupole moment of rotating neutron
star to the properties of static non-rotating neutron star. This investigation
was motivated mostly by the existence of analytic, equation of state almost
independent, dependency of moment of inertia on the mass and radius of static
neutron star. We have found that the quadrupole moments could be also expressed
in terms of mass and radius of static neutron stars and its rotational frequency. All
calculations are done in the slow rotation approximation introduced by Hartle
and Thorne [2] and extended by Chandrasekhar and Miller [3] that uses all
the quantities expanded up to second order in rotation frequency. This order
of expansion was however found to be satisfactory for all the astrophysical
situations known in the time of finishing this thesis.
The results obtained in modelling structure of the rotating neutron stars were
applied to improve estimates of mass of particular source exhibiting high–frequency
quasiperiodic oscillations in the X-ray observations (published in [4]), and
vice versa is used to restrict applicability of some orbital resonance model of
high–frequency quasiperiodic oscillations (published in [5]). As an independent
part of this thesis can be considered an introductory study of trapping of neutrinos
in extremely compact stars represented by simplifying models of uniform–density
stars reflecting to some reasonable extend realistic neutron stars (published in
[6]).
1
2
Preface
Thesis consists of two parts. Part I presents an overview of the modelling
the neutron star structure and its exterior and brings new results that have not
been published yet, while part II presents published papers. Part I of the Thesis
starts with short introduction on neutron stars in the Chapter 1. Equations of state
are presented in Chapter 2 with particular focus on several models. Both static
and rotating neutron stars models are calculated and presented in Chapter 3 and
compared with models of strange stars in Chapter 4. A numerical code, based on
the Hartle–Thorne model of rotating neutron star, was developed in the scope of
this Thesis and is used to model both the properties of neutron (strange) stars and
their external gravitational field that is given by three parameters: mass, angular
momentum and quadrupole moment. Results that are compared to observations
in Chapter 5 extend those that has been published (or accepted for publishing).
Part I ends by Chapter 6 with discussion and plans for future research.
Part I
Overview
Chapter 1
Introduction
Stars spend their lives burning their nuclear fuel and producing the light that we
see by our eyes or telescopes. When the life of the stars is approaching end, i.e.,
when the pressure gradient can not balance the gravity any more, the interior of
the stars starts to collapse and different products can be formed. The star can
collapse to a white dwarf, neutron star or black hole∗ . It is necessary to note
that neutron stars, being aim of this thesis, could be born also by the collapse of
white dwarf if it accretes enough matter from its binary companion.
White dwarfs are objects with masses close to solar mass and with radii around
hundred times smaller than the radius of Sun (i.e. with radii comparable with the
radius of Earth). Unlike stars, where the pressure originates in thermal energy
of their gas content, the pressure that balance gravitation attraction of white
dwarfs comes from degeneracy of electrons. Since electrons are Fermi particles,
their behavior is governed by the Fermi–Dirac statistics. For white dwarfs the
volume per electron is so small that quantum effects start to play a key role and
the kinetic energy of particles is substantially affected by Heisenberg relations
and additional pressure coming from temperature is negligible if compared with
the degeneracy pressure. On the other hand, white dwarfs are objects where
general relativistic effects play only minor role. Chandrasekhar [7] has found
the maximal mass limit of white dwarfs that is called after him Chandrasekhar
limit. The maximum allowed mass of white dwarfs is ≈ 1.44M¯ (see e.g. [8]
for more details).
Neutron stars can have masses up to ≈ 3M¯ , however the observed masses
for different objects are close to 1.5M¯ (see e.g. [9] for review on masses of
neutron stars in binary systems). The typical radius of neutron stars is close to
ten kilometers, i.e., it is around thousand times smaller than the radius of white
dwarfs. Because of this compactness, the general relativity starts to play a key
∗ The star can also transform to more hypothetical strange stars or another, nowadays unknown,
form of matter. However, in this introductory part we focus on general principles and we postpone
the discussion of differences between the neutron and strange stars on later.
5
6
Chapter 1. Introduction
role for equations governing the structure of neutron stars. Neutron stars could
exist and be observed as isolated or in binary systems. For neutron stars in
binaries the estimation of masses of both members of the system could be done
from measurements of post-Keplerian parameters of the orbit (see, e.g., [10, 11]).
Isolated neutron stars are usually observed as pulsars that are neutron star with
magnetic field having the axis oriented towards observer (and misaligned with the
axis of rotation). Therefore, we can observe magnetic dipole radiation generated
by the periodic variation of magnetic field due to star rotation. These flashes are
coming with frequency corresponding to the rotational frequency of the neutron
star and are observed with high accuracy. There is also the small number of
isolated neutron star candidates, that are not observed as pulsars but their thermal
radiation is observed. The probably most discussed source of this kind is RX
J1856–3754 discovered by Walter, Wolk and Neuhäuser in 1996 [12].
1.1. First ideas and observations
First theoretical idea of neutron stars comes from Baade and Zwicky [13], being
based on the analysis of supernova explosions. They stated that supernova explosion could be transition a from star to a neutron star. They also predicted that
neutron stars consist of closely packed neutrons in the object of small radius. It is
amazing that just two years after Chadwick discovered the neutron [14], the idea
of neutron stars were so close to the neutron stars as we know them today. In 1939
the issue of Physical Review contained two papers [15, 16] providing derivation of full general relativistic equation of hydrostatic equilibrium for spherically
symmetric objects (called up to these days Tolman–Oppenheiemer–Volkoff equation). Oppenheimer and Volkoff [15] solved this equation assuming the matter
consists of noninteracting neutrons and obtained the maximal allowed mass to
be 0.71M¯ . It was shown latter that inclusion of nuclear energy coming from
interaction of neutrons increases this value substantially.
The attempts of observational proof of neutron star existence started with the
active search for neutron stars using X-ray telescopes. Several X-ray sources
were discovered (including the one in the Crab Nebula and the first non-solar
source of the X-rays in the Scorpius constellation named Sco X-1) but it was
not proofed that these sources could be connected with neutron stars (see e.g.
[8, 10, 17]). The discovery came in another field of electromagnetic spectra. In
1967 Jocelyn Bell discovered the sources of pulsating radio beams. It has been
shown latter that sources of these pulsations, named pulsars, are rotating neutron
stars with strong magnetic field (see e.g. [8, 10, 17] for overview and references
1.1. First ideas and observations
7
Figure 1.1. Crab nebula and pulsar. Left: Hubble Space Telescope figure. Right:
Composite picture of three NASA telescopes: Hubble Space Telescope (yellow and
red), Chandra X-ray Telescope (blue) and infrared Spitzer Space Telescope (purple).
See NASA web page for original figures and more details.
therein). Figure 1.1 shows Crab pulsar and surrounding nebula as viewed by
different NASA instruments (visit NASA web page http://www.nasa.gov for
picture gallery of pulsars, supernovae and others).
At these days the neutron stars are commonly accepted objects. We know
hundreds of pulsars, some of them are in binaries some of them are isolated.
The pulse periods range from hundreds of seconds to milliseconds (see e.g.
[10] or Pulsar Catalogue on the web page of Australia Telescope National Facility (ATNF) http://www.atnf.csiro.au/research/pulsar/psrcat that
contains almost two thousand pulsars).
Of crucial interest can recently be considered low mass X-ray binary systems
(LMXBs) containing neutron stars. Such systems exhibit a variety of (relativistic)
astrophysical phenomena in the observed X-ray fluxes and its time variability.
The most crucial seem to be the high frequency quasiperiodic oscillations (HF
QPOs) with frequencies comparable to the orbital motion of matter in the inner
parts of accretion discs around the neutron stars. Such phenomena give a substantial information on the part of the gravitational fields close to the neutron
stars and enable us to restrict, at least in principle, the validity of the equations
of state describing the interior of neutron stars. On the other hand, testing all the
variety of realistic equations of stat in the modelling rotating neutron stars could,
in some situations, bring a restriction on validity of various orbital resonance
models of HF QPOs in the observed LMXBs [5].
Chapter 2
Equation of State of Neutron Star
Matter
Behavior of matter at the neutron star interiors is governed by the acting pressure.
At the surface matter consist of iron being the stable form of matter at zero
pressure. As one goes deeper, matter starts to be formed by nuclei that are more
and more neutron rich. At densities ≈ 4.3 × 1011 g.cm−3 the neutrons start to
drip of the nuclei, since electrons are pushed towards nuclei and they react with
protons and form neutrons via inverse β -decay∗ . The residual nuclei coexist
with the free neutrons up to densities ≈ 1014 g.cm−3 . Exact value of density
(pressure) at which no nuclei could be found is not known, and it depends on
details of the theory that has to be found.
The inner part of neutron stars, where all the neutrons are in β -equilibrium
with electrons and protons is called the core. Outside the core, where nuclei are
in equilibrium with free neutrons, the layer called inner crust could be found.
At this layer nuclei could form different shapes called nuclear pasta for its
appearance [18]. Region that lies on the top of the inner crust and contains nuclei
of different proton to neutron ratio is called, as one will guess, the outer crust.
At the surface of neutron star the atmosphere and ocean could be present (see
e.g. [17] for details). The schematic picture of neutron star structure created by
Dany N. Page (http://www.astroscu.unam.mx/neutrones/home.html) is
given on Fig. 2.1.
For outer layers of neutron stars we use the following standard equations of
state for all models calculated and presented in this Thesis.
Outer crust
• Feynman-Metropolis-Teller EoS for 7.9 g.cm−3 ≤ ρ ≤ 104 g.cm−3 where
matter consists of e− and 56
26 Fe, [19];
∗ Condition of inverse β
decay gives the density of neutron drip, however this density depends
on the description of the matter that consists of nuclei, electrons and eventually free neutrons.
9
10
Chapter 2. Equation of State of Neutron Star Matter
Figure 2.1. Illustration of neutron star structure. See the original picture by Dany P.
Page at the web page of Neutron star group at UNAN.
• Baym-Pethick-Sutherland EoS for 104 g.cm−3 ≤ ρ ≤ 4.3 × 1011 g.cm−3 with
Coulomb lattice energy corrections [20];
Inner crust
• Baym-Bethe-Pethick EoS for 4.3 g.cm−3 × 1011 ≤ ρ ≤ 1014 g.cm−3 : here,
e− , neutrons and equilibrated nuclei calculated using the compressible liquid
drop model [21].
2.1. Nucleon - nucleon interaction and matter in the cores of neutron stars
11
Outer envelopes as the atmosphere and ocean play a key role for spectral
feature of thermal radiation coming from neutron star and its model also depends
whether or not the neutron star accretes matter from its binary companion and
on the presence and strength of magnetic field (see e.g. [17]). On the other hand
influence of the atmosphere and the ocean on the structure of the neutron star is
negligible and thus we do not consider them in our models.
For neutron star cores, where matter consists of free neutrons in β -equilibrium
with protons and electrons and for higher densities also with muons, we use
different equations of state of nuclear matter to model neutron stars. The inner
crust–core boundary we set to be given by continuous dependency of energy
density on pressure, if possible.
2.1. Nucleon - nucleon interaction and matter in the
cores of neutron stars
A wide spectrum of different equations of state (EoS) of nuclear matter has been
investigated and used for modelling the neutron star structure (see, e.g., [8, 17, 22]
for reviews and references therein for details). All EoS have to give almost the
same properties close to the standard nuclear (saturation) density nS ≈ 0.16 fm−3 ,
but they could lead to substantially different behavior at high densities that could
be found at the center of neutron stars. The saturation density corresponds to
minimum of energy per baryon for symmetric nuclear matter (matter where the
density of protons is equal to the density of neutrons).
We focus our attention on different ways how to describe the nuclear matter at
the cores of neutron stars, namely the Skyrme model (see, e.g., [23] and references
there in), the Akmal Pandharipande Ravenhall EoS [24] and the relativistic mean
fieald theory (RMF) [25]. The matter in the core of neutron stars has to be
electrically neutral and in β -equilibrium. We now focus in more detail to general
properties of n-p-e-(µ ) matter, before we start to look closely on different ways
how to treat nucleon–nucleon interaction.
The total energy density of n-p-e matter is given as
E = nB EB (nB , xp ) + Ee (ne ),
(2.1)
where EB (nB , xp ) is binding energy per particle for asymmetric nuclear matter
including kinetic energies of baryons, ni is number density of i-th kind of particle,
12
Chapter 2. Equation of State of Neutron Star Matter
nB = np +nn is baryon number density and xp = np /nB is proton fraction. Because
the electrons are degenerate the energy densities could be written as
2
Ee = 3
h
pF(e)
Z ¡
¢1/2
4π p2 dp,
m2e c4 + p2 c2
(2.2)
0
where pF(e) is the Fermi momentum of electrons that could be expressed in terms
of number density by the relation
pF(e) = h̄(3π 2 ne )1/3 .
(2.3)
Binding energy per particle of asymmetric nuclear matter with proton fraction
xp = np /nB is fully given by binding energy of symmetric nuclear matter E0 (nB )
and symmetry energy S0
EB (nB , xp ) = E0 (nB ) + (1 − 2xp )2 S0 (nB ).
(2.4)
The symmetry energy S0 is the factor corresponding to second order term in expansion of binding energy in terms of asymmetry parameter δ = (nn − np )/(nn +
np ) = 1 − 2xp
¯
1 ∂ 2 EB (nB , δ ) ¯¯
.
S0 =
¯
2
∂δ2
δ =0
(2.5)
From equation (2.4) one can see that symmetry energy is the difference of binding
energy per particle between pure neutron matter and symmetric nuclear matter.
S0 = EB (nB , δ = 1) − EB (nB , δ = 0).
(2.6)
The matter in neutron star cores is in β -equilibrium, i.e., in equilibrium with
respect to reactions
n ↔ p + e− .
(2.7)
The condition of equilibrium could be written using chemical potentials as
µn = µp + µe ,
(2.8)
where chemical potential of each kind of particle is given by
µi =
∂E
.
∂ ni
(2.9)
2.1. Nucleon - nucleon interaction and matter in the cores of neutron stars
13
For matter consisting of neutrons, protons and electrons the final equation for
xp reads
¡
¢
4S0 (nB ) 1 − 2xp = ch̄(3π 2 xp nB )1/3 .
(2.10)
See Appendix A for derivation of this formula and how this formula is changed
assuming presence of muons. Dependency of energy density on baryon number
density for pure neutron matter and symmetric nuclear matter are used to obtain
symmetry energy and then the equations governing β -equilibrium (2.10, A.23)
are solved to find densities of all considered particles. Fact that we need only
energy density for pure neutron and symmetric nuclear matter and not general formula, following directly from quadratic dependency of energy density of matter
on asymmetry, simplifies practical calculation of β -equilibrium substantially.
Let us focus now on description of asymmetric nuclear matter that we use in
our calculations. Overview of different methods could be found, e.g., in [17, 22].
2.1.1. Skyrme
Skyrme equation of state represents the non-relativistic version of mean field
method. There exists a huge number of different versions of Skyrme equation
of state since it contains nine empirical parameters (t0 ,t1 ,t2 ,t3 , x0 , x1 , x2 , x3 and
α ). All of these give similar agreement with experimentally established nuclear
ground states at the saturation density nS , but they imply varying behavior of
both symmetric and asymmetric nuclear matter when density grows (up to 3nS
or even higher).
Energy density functional for Skyrme interaction is given by
H = K + H0 + H3 + Heff ,
(2.11)
14
Chapter 2. Equation of State of Neutron Star Matter
where K is the kinetic energy term, H0 zero-range term, H3 density depending
term and Heff effective-mass-dependent term and they read
K
=
H0 =
H3 =
Heff =
+
h̄2
τ,
2m
¤
1 £
t0 (2 + x0 )n2 − (2x0 + 1)(n2p + n2n ) ,
4
£
¤
1
t3 nα (2 + x3 )n2 − (2x3 + 1)(n2p + n2n ) ,
24
18 [t1 (2 + x1 ) + t2 (2 + x2 )] τn
1
[t2 (2x2 + 1) − t1 (2x1 + 1)] (τ p n p + τn nn ).
8
(2.12)
(2.13)
(2.14)
(2.15)
Rikovska Stone et al. [23] tested 87 different parameterizations of Skyrme
potentials calculating neutron star models. They found that only 27 of these
parameterizations satisfies the requirements given by neutron star observations.
We decided to use three of these parameterizations. Namely, SkT5 giving the
lowest maximal mass (Mmax = 1.82M¯ ) of all the Skyrme equations of state that
passed the test, SLy4 that is the probably most frequently used and SV that gives
highest the maximal mass (Mmax = 1.82M¯ of all parameterizations tested by
Řikovská Stone et al. [23].
2.1.2. Variational theory
The typical representant of variational theory is the very popular equation of
state developed by Akmal, Pandharipande and Ravenhall [24]. The variational
theory is an alternative way to (relativistic) mean field theory for calculations
of asymmetric nuclear matter properties. Akmal, Pandharipande and Ravenhall
[24] found the effective Hamiltonian for their models to be
¶
h̄2
−p4 nB
+ (p3 + p5 (1 − xp ))nB e
=
τn
2m
¶
µ 2
h̄
−p4 nB
+
+ (p3 + p5 xp )nB e
τp
2m
¡
¢
¡
¢2
+ g(xp = 0.5) 1 − (1 − 2xp )2 + g(xp = 0) 1 − 2xp ,
µ
Heff
(2.16)
where
τp =
¢5/3
¢5/3
1 ¡ 2
1 ¡
3π nB xp
, τm rmn = 2 3π 2 nB (1 − xp )
2
5π
5π
(2.17)
15
2.1. Nucleon - nucleon interaction and matter in the cores of neutron stars
Table 2.1.
Labels of models considered in this work and their labels in the original
paper [24].
A18 + δ v + UIX*
A18 + UIX
A18 + δ v
A18
APR
APR2
APR3
APR4
0.2
300
APR
APR2
APR3
APR4
APR
APR2
APR3
APR4
0.18
0.16
APR 1
APR 2
APR 3
APR 4
0.14
0.12
150
xp
[MeV]
APR
APR2
250 APR3
APR4
200
0.1
100
0.08
50
0.06
0.04
0
0.02
-50
0
0.3
0.6
nB [fm-3]
0.9
0.3
0.6
nB [fm-3]
0.9
0.3
0.6
nB [fm-3]
0.9
0
0.1
0.3
0.5
0.7
nB [fm-3]
0.9
1.1
1.3
Figure 2.2. Left: Binding energy per particle for pure neutron, symmetric nuclear and
neutron star matter (from left to right) for all considered APR models. Right: Proton
fraction of matter at β -equilibrium as it depends on baryon number density.
and functions g(x p ) are also functions of baryon number density nB and they have
different forms for low and high densities (see Appendix A of original paper [24]
for more details and for sets of 21 parameters p1 –p21 ).
We have used all four parameterizations from original paper and Table 2.1
indicates our notation of models. Note that UIX(∗) stand for model of three
body force and δ v indicates the inclusion of relativistic boost correction.
Unlike in the case of Skyrme equations of state, where we had tables relating
pressure and energy density together with baryon number density, we have
used the effective hamiltonian to calculate the dependency of energy density
on number density for pure neutron matter and for symmetric nuclear matter.
These were used to calculate the proton fraction of matter at β -equilibrium and
obtained results were used as a test of the code by comparison with results in the
original paper [24] (see Fig. 2.2).
2.1.3. Relativistic mean field theory
Relativistic mean field theory is standard method how to calculate the energy
of nuclear matter and was used also for models of neutron stars [17, 22]. In
particular we use here the parameterizations obtained by Kotulič Bunta and
Gmuca [26]. They have used relativistic mean field theory to fit full relativis-
16
Chapter 2. Equation of State of Neutron Star Matter
tic Brueckner-Hartree-Fock calculations by different authors [27–29]. Fitted
parameterizations have not been previously used for models of neutron stars.
The Lagrangian density includes the nucleon field ψ , isoscalar scalar meson
field σ , isoscalar vector meson field ω , isovector vector meson field ρ ,and
isovector scalar meson field δ , including also the vector cross-interaction. The
Lagrangian density in the form used by Kotulič Bunta and Gmuca [26] reads
L (ψ , σ , ω , ρ , δ ) = ψ̄ [γµ (i∂ µ − gω ω µ ) − (mN − gσ σ )]ψ
1
1
1
+ (∂µ σ ∂ µ σ − mσ 2 σ 2 ) − ω µν ω µν + mω 2 ω µ ω µ
2
4
2
1
1
1
− bσ mN (gσ σ )3 − cσ (gσ σ )4 + cω (gω 2 ω µ ω µ )2
3
4
4
1
1
1
+ (∂µ δ ∂ µ δ − mδ 2 δ 2 ) + mρ 2 ρ µ ρ µ − ρ µν ρ µν
2
2
4
1
+ ΛV (gρ 2 ρ µ ρ µ )(gω 2 ω µ ω µ ) − gρ ρ µ ψ̄γ µ τψ + gδ δ ψ̄τψ ,
2
(2.18)
where the antisymmetric tensors are
ω µν ≡ ∂ν ω µ − ∂µ ω ν ,
ρ µν ≡ ∂ν ρ µ − ∂µ ρ ν ;
(2.19)
the strength of the interactions of isoscalar and isovector mesons with nucleons is
given by (dimensionless) coupling constants g’s and the self-coupling constants
(also dimensionless) are bσ (cubic), cσ (quartic scalar) and cω (quartic vector).
The second and the fourth lines represent non-interacting Hamiltonian for all
mesons, ΛV is the cross-coupling constant of the interaction between ω and
ρ mesons. These constants represents the parameters that are obtained during
fitting of Dirac-Brueckner-Hartree-Fock∗∗ calculations of [27–29] to get best χ 2 .
Furthermore, mN is the nucleon mass, ∂ µ ≡ ∂∂xµ and γ ’s are the Dirac matrices
([22, 26]).
We have used three parameterizations (see Tab 2.2 for details) that represent
the best fits to results of previous calculations.
The advantage of relativistic mean field theory is also in relatively simple
inclusion of other particles. At higher densities in the core hyperons, of various
kind could condensate. Namely, appearance of Λ, Σ− , Ξ− could start at densities
around 2 − 3 × nS . At higher densities other hyperons like Σ0+ , Ξ0 could be
∗∗ Dirac-Brueckner-Hartee-Fock
Brueckner-Hartree-Fock.
is
the
alternative
name
for
the
relativistic
17
2.1. Nucleon - nucleon interaction and matter in the cores of neutron stars
Table 2.2. Models considered in this work and their original names in [26].
Here Original paper
H
HA
L
LA
M
MA
Fitted results
Huber, Weber and Weigel [27]
Lee et.al. [28]
Li, Machleidt, and Brockmann [29]
0.18
300
H
L
250 M
H
L
M
H
L
M
0.16
0.14
0.12
150
xp
[MeV]
200
0.1
0.08
100
0.06
50
0.04
0
H
L
M
0.02
-50
0
0.3
0.6
nB [fm-3]
0.9
0.3
0.6
nB [fm-3]
0.9
0.3
0.6
nB [fm-3]
0.9
0
0
0.2
0.4
0.6
nB [fm-3]
0.8
1
1.2
Figure 2.3. Left: Binding energy per particle for pure neutron, symmetric nuclear and
neutron star matter (from left to right) for all considered models and relativistic mean
field equations of state. Right: Proton fraction of matter at β -equilibrium for selected
parameterizations.
also present. Possibilities of hyperon existence in the core of neutron stars has
been discussed in the literature as well as its impact on neutron star models (see
e.g. [17] and references therein). Kotulič Bunta and Gmuca have also included
hyperons in their parameterizations [30], however we use only their previous
calculations [26] here. The resulting energies per particle for various kind of
matter as well as proton fraction of matter at β -equilibrium are illustrated at
Fig. 2.3.
2.1.4. Equation of state of neutron star matter
We have shown that we need to know energy density of pure neutron matter and
symmetric nuclear matter to calculate the composition and corresponding energy
density E of neutron star matter for given baryon number density. The pressure
P is then given by the law of thermodynamics
P = n2B
∂
∂ nB
µ
¶
E (nB , xp )
.
nB
(2.20)
The fact that only energy densities of pure neutron matter and symmetric nuclear
matter are enough input to calculate the equation of state extremely simplifies
18
Chapter 2. Equation of State of Neutron Star Matter
1038
1037
P [dyne.cm-2]
1036
1035
1034
SkT5
SLy4
SV
H
L
M
APR
APR2
APR3
APR4
1033
1032
1013
1014
1015
E [g.cm-3]
1016
Figure 2.4. Energy density E and pressure P for all models considered in this Thesis.
the practical calculation. In our models the equation of states come as a table
relating energy density E , pressure P and baryon number density nB . Calculated
equations of state are depicted on Fig. 2.4.
Chapter 3
Neutron Star Models
In this chapter we will focus on the neutron star models and how can they
be calculated. The important input in calculation of neutron star model are
the equation of state of matter inside the neutron stars and general relativistic
effects that play a key role for neutron stars.We will calculate models of static
spherically symmetric neutron star at first place and then, the step to rotating
neutron stars will be done. Models of rotating neutron stars will be calculated
in the Hartle–Thorne approximation [2, 3, 31]. The detailed theory presented in
this chapter can be found in standard textbooks (e.g. [8, 17, 22]) and we will
refer to original sources only in special cases. In this chapter and in the following
the geometrical units G = c = 1 are used.
3.1. Static, spherically symmetric neutron stars
3.1.1. Equations of Structure
Neutron stars are relativistic objects. Thus all calculations have to be done in
the framework of general relativity. The interior spacetime of static spherically
symmetric objects could be written in the Schwarzschild coordinates t, r, θ , φ as
ds2 = −e2ν dt 2 + e2λ dr2 + r2 (dθ 2 + sin2 θ dφ 2 ),
(3.1)
where the metric functions ν (r) and λ (r) are functions of radial coordinate r only.
Line element could be written in the standard way as ds2 = gµν xµ xν , where xµ
are the coordinates and gµν is the metric tensor. The geometry is connected to
matter via Einstein field equations.
1
Rµν − gµν R = Tµν ,
2
(3.2)
19
20
Chapter 3. Neutron Star Models
µ
where Rµν is the Ricci tensor, R = R µ the Ricci scalar (scalar curvature) and
Tµν energy-momentum tensor.
The neutron star matter could be considered as perfect fluid, so we can write
the energy-momentum tensor as
T µν = (E + P)U µ U ν + Pgµν ,
(3.3)
where E is the energy density, P pressure of the matter and U µ the four-velocity of
µν
the fluid. The energy-momentum tensor satisfies the conservation law T ;ν = 0
that together with the Einstein equations (3.2) leads to the equation of hydrostatic
equilibrium, the so-called Tolman-Oppenheimer-Volkoff (TOV) equation [15,
16]
dP
m(r) + 4π r3 P
= −(E + P)
,
dr
r(r − 2m(r))
(3.4)
where m(r)
Zr
4π E r02 dr0
m(r) =
(3.5)
0
is the mass inside a sphere of radius r. Equations for metric functions than could
be expressed as
dν
dr
e2λ
1 dP
= −
,
E + P dr
µ
¶
2m(r) −1
=
1−
.
r
(3.6)
(3.7)
The TOV equation (3.4) is solved for given central parameters, using the relation
between energy density E and pressure P that is given by the considered equation
of state and follows from assumed description of nuclear matter. Integration starts
from center for given central pressure Pc and corresponding energy density Ec
and ends at the surface of the neutron star r = R, where pressure vanishes. The
total mass is than given as M = m(R), where R is the radius of the neutron star.
External spacetime of static spherically symmetric body is the well-known
Schwarzschild metric
µ
¶
µ
¶
2M
2M −1 2
2
ds = − 1 −
dt + 1 −
dr + r2 (dθ 2 + sin2 θ dφ 2 ).
r
r
2
(3.8)
3.1. Static, spherically symmetric neutron stars
21
Interior metric function λ (r) is calculated simply from eq. (3.7) and is automatically matched to its external form, since at the surface of the neutron star
M = m(R). On the other hand equation for metric function ν must be solved
individually and finally it has to be re-scaled to match its external form.
3.1.2. Gross properties of neutron stars
We have shown how neutron star mass M and radius R are calculated, however
there exist other quantities that could be also relevant from astrophysical point of
view and it is useful to mention them. The total number of baryons A contained
in the neutron star, that is the integrated particle number density over proper
volume of neutron star
ZR
A=
√
4π nB (r) grr r2 dr,
(3.9)
0
where nB (r) is the baryon number density at radius r.
The baryon mass MB gives the total rest mass of particles in the neutron
star and is given by MB = uA, where u = 939.5 MeV/c2 = 1.66 × 10−24 g is the
atomic mass unit∗ .
The proper mass MP corresponds to the mass-energy of neutron star and is
given as
ZR
MP =
√
4π E (r) grr r2 dr,
(3.10)
0
where the E (r) denotes energy density at radius r.
The binding energy of the neutron star Eb is given as a difference between its
baryon mass MB and its mass M
Eb = (M − MB )c2 .
∗ Instead
(3.11)
of the atomic mass unit the neutron mass could be used, however we choose the
atomic mass unit here
22
Chapter 3. Neutron Star Models
Binding energy Eb could be divided into parts. The nuclear (internal) binding
energy EN and the gravitational binding energy EG . The relations giving these
binding energies read
ZR
EN =
√
4π (E − nB (r)u) grr r2 dr = (MP − MB )c2 ,
(3.12)
0
EG = (M − MP )c2 ,
(3.13)
Eb = EG + EN .
(3.14)
The binding energy Eb is negative and corresponds to energy released during
neutron star formation. The binding energy EG is also negative, however, the
internal binding energy EN is positive and is given mostly by nuclear energy of
nucleon–nucleon interaction.
3.1.3. Surface properties
From astrophysical point of view the parameter that plays an important role is the
surface redshift zsurf . It gives the relative change of frequency of light traveling
from surface of the neutron star to the distant observer∗∗ and is related to its mass
and radius
µ
¶
2M 1/2
ν∞ − ν R
zsurf = 1 −
−1 =
.
R
νR
(3.15)
We can see that the redshift measurements could serve rather as an estimate of
the mass-radius relation for neutron stars.
Apparent radius R∞ is the radius of neutron star as it will be seen by distant
observer. It is related to the neutron star radius and mass by the relation
R∞ =
R
(1 − 2M/R)1/2
.
(3.16)
The apparent radius can be estimated from observations of thermal radiation of
neutron stars. It depends on distance measurements and on the model of neutron
∗∗ Note
that the frequency will be shifted if the light is emitted at any place in the gravitational
field, but we are concentrated on the light emitted at the surface of the object.
3.1. Static, spherically symmetric neutron stars
23
star atmosphere. Estimate of the apparent radius R∞ leads to the mass-radius
relation
µ
¶
M
R
R2
=
1− 2 .
M¯ 2.95 km
R∞
(3.17)
3.1.4. Numerical solution
In our calculations, the equation of state enters as a table from external file.
The logarithmic interpolation between table rows have been chosen to calculate
corresponding energy density for non-tabulated pressure, i.e. we assume the
equation of state could be approximated as a polytropic on the small range
between lines in table.
TOV equation (3.4) is solved numerically for realistic equations of state
introduced in Chapter,2. We used 4th order Runge-Kutta method with constant
stepsize in radial coordinate that was set to 120cm.
The appropriate boundary conditions at the center (r → 0) read
P = Pc ,
dP
= 0,
dr
E = Ec ,
4
m(r) = π r3 Ec
3
(3.18)
for functions explicitly present in TOV eq. while boundary conditions for metric
functions at center reads
dν
= 0,
dr
e2λ = 1.
(3.19)
At the surface (r = R) the boundary conditions are
P = 0, E = ES ,
¶
µ
2M
2ν
e
=
1−
,
r
µ
¶
2M −1
2λ
=
1−
e
,
r
(3.20)
where ES is the energy density at the surface which usually corresponds to mass
energy of iron.
24
Chapter 3. Neutron Star Models
M[MSUN]
2.5
SkT5
SLy4
SV
H
L
2
M
APR
APR2
APR3
1.5 APR4
1
0.5
6
8
10
12
14
R[km]
Figure 3.1. Mass-Radius relation for selected set of equations of state.
3.1.5. Maximum mass and stability of neutron stars
The key difference between theory of stars and physics of neutron stars (or even
white dwarfs) is the existence of maximal possible mass of neutron star model.
The maximum mass serves as a first test of equation of state. The extremely
accurate measurements of neutron star masses coming from observation of double
pulsar systems necessitates the allowance of Mmax > 1.4M¯ .
The maximum mass of a neutron star corresponds to the change in its stability.
With increasing central energy density Ec , the mass M of the neutron star model
must also increase. This is called the static stability criterion
dM
>0
dEc
(3.21)
and is related to the behavior the of n = 0 mode of the neutron star radial
oscillation. This criterion is necessary but not sufficient. The stability criterion
with respect to all oscillation modes could be derived from mass-radius relation
and deals with possibilities of spiral behavior of this relation. However, since
25
3.1. Static, spherically symmetric neutron stars
2.5
SkT5
SLy4
SV
H
L
M
APR
APR2
APR3
APR4
M0[MSUN]
2
1.5
1
0.5
0
1
2
3
4
-3
ec[10 g.cm ]
5
15
Figure 3.2.
Mass of the neutron star as it depends on central energy density for our
set of equations of state.
we do not have to deal with such situation here, we only refer on section 6.5.2 of
[17].
Table 3.1. Neutron star properties for selected equations of state. First column contains
maximal masses of neutron star model using particular equation of state. Energy density
is in units of 1015 g.cm−3 and pressure in 1035 dyne.cm−2 .
EoS
SkT5
SLy4
SkT5
H
L
M
APR
APR2
APR3
APR4
Mmx [M¯ ] EcMmax
1.82
3.10
2.04
2.88
2.38
1.83
2.20
1.34
1.92
2.26
1.62
2.56
2.21
2.70
2.39
2.30
1.81
3.68
1.66
4.22
M=1.4M¯
Ec
1.07
1.00
0.61
0.74
0.78
1.06
0.95
0.78
1.46
1.91
M=1.4M¯
Pc
1.38
1.38
0.65
0.92
0.90
1.30
1.33
1.00
2.47
3.85
26
Chapter 3. Neutron Star Models
Calculated masses and radii are shown on Fig. 3.1 where mass-radius relation is depicted and dependency of the neutron star mass on the central energy
density is presented on Fig. 3.2. Table 3.1 summarizes maximum masses† and
corresponding central energy density together with central properties of neutron
stars with M = 1.4M¯ .
3.2. Rotating Neutron Stars
Real neutron stars are rotating at rotational frequencies that are ranging from
≈ 0.1mHz up to ≈ kHz [10]. The fastest observed rotational frequency is 716 Hz
of pulsar PSR J1748-2446ad [32]. Hartle and Thorne [2, 33] developed a method
for calculation of rotating neutron star models in the slow rotation approximation.
This approximation is valid for angular velocities Ω2 ¿ GM/R3 . It has been
shown that this approximation could be used for all current astrophysical objects,
even for millisecond pulsars [34].
3.2.1. Hartle - Thorne approximation
Hartle - Thorne metric is the perturbation of spherically symmetric Schwarzschild
metric (3.1) given in the form
ds2 = −e2ν0 [1 + 2h0 (r) + 2h2 (r)P2 (cos θ )]dt 2
(
)
2λ0
e
[2m0 (r) + 2m2 (r)P2 (cos θ )] dr2
+ e2λ0 1 +
r
©
ª
+ r2 [1 + 2k2 (r)P2 (cos θ )] dθ 2 + [dφ − ω (r)dt]2 sin2 θ . (3.22)
One can see that the perturbation from Schwarzschild metric takes place in
all metric functions gtt , grr , gθ θ and gφ φ †† and the additional term gt φ is the
term common for axially symmetric spacetimes. The perturbation functions
h0 (r), h2 (r), m0 (r), m2 (r), k2 (r) are quantities of order Ω2 and are functions of
radial coordinate r only. The function ω (r) is of the order Ω and describes the
so-called dragging of the inertial frame. The deviation from spherical symmetry
in diagonal part of metric of the gµν is given by the Legendre polynomial of 2nd
order P2 (cos θ ) = (3 cos2 θ − 1)/2.
† Maximal neutron star mass for equation of state H presented in this work does not correspond
to maxima in mass–radius relation but to highest central density (see [1]).
†† The
subscript 0 in the metric functions ν and λ refers to the unperturbed Schwarzschild
geometry. In the following we will use this notation also for mass.
27
3.2. Rotating Neutron Stars
All perturbation functions are calculated with appropriate boundary conditions at the center and at the surface of configuration. The perturbation functions
are label by the lower index that corresponds to the order of perturbation. We
then have the l = 0 perturbation functions that describe the monopole (spherical)
deformations
Equation of hydrodynamical equilibrium could be written in the form [3]
·
P,α = (E + P) log
e−ν
(1 −V 2 )1/2
¸
,
(3.23)
,α
where α = r, θ and
V = eψ −ν (Ω − ω ),
(3.24)
eν = eν0 [1 + h0 (r) + h2 (r)P2 (cos θ )],
(3.25)
e
ψ
= r sin θ [1 + k2 (r)P2 (cos θ )].
(3.26)
Note that for non-rotating configurations Ω = ω (r) = 0 equation (3.23) transforms to equations (3.6). Equation of hydrodynamic equilibrium 3.23 to give
1
ν + log(1 −V 2 ) + p = const.,
2
(3.27)
where p is related to the pressure P by
dP = (E + P)dp
(3.28)
and could be expanded in the slow rotation approximation in the form
p(r) = p0 (r) + δ p0 (r) + δ p2 (r) cos θ .
(3.29)
In the following, we will leave out δ in the previous equation and we will write
p0 and p2 instead of δ p0 and δ p2 ‡ . Equation (3.27) gives the relation of p0 (r)
and p2 (r) to h0 (r) and h2 (r)
dh0
dp0 1 d ¡ 2 −2ν0 2 ¢
ω̃ ,
= −
+
r e
dr
dr
3 dr
1
h2 (r) = −p2 (r) − r2 e−2ν0 ω̃ 2 ,
3
‡ We can do this notation change with no worries since the unperturbed term
equation (3.29) does not appear in the following text
(3.30)
(3.31)
p0 in the original
28
Chapter 3. Neutron Star Models
where ω̃ (r) = Ω − ω (r). Field equations lead to equations for perturbation
functions h0 , h2 , m0 , m2 and k2 and for ω̃ (r). The equation corresponding to the
t φ component of the Einstein field equations (3.2) is the equation for ω̃ and reads
µ
¶
1 d
dω̃
dj
4
r j(r)
+ 4 ω̃ = 0,
3
r dr
dr
dr
(3.32)
where j = e−(λ0 +ν0 ) and the boundary conditions at the center reads ω̃ =
ω̃c , dω̃ /dr = 0. Outside the star, the solution has the form
ω̃ (r) = Ω −
2J
,
r3
(3.33)
where the constant J is the angular momentum of the rotating neutron star [33].
This equation leads to the equation that is used to calculate the angular momentum
J
R4
J=
6
µ
dω̃
dr
¶
.
(3.34)
r=R
Moment of inertia I is given by the standard relation I = J/Ω. Because we
usually want to put the rotational frequency Ω as an input parameter, we have to
re-scale the ω̃ using the relation
ω̃new (r) = ω̃old (r)
ΩWE WANT
ΩWE GET
(3.35)
to obtain proper ω̃ (r).
Other field equations lead to the following equations for l = 0 perturbation
functions
µ ¶
dE
1 4 2 dω̃ 2 1 3 2 d j2
dm0
2
= 4π r (E + P)
p0 + r j
,
− r ω̃
dr
dP
12
dr
3
dr
µ ¶2
dp0
m0 (1 + 8π r2 P) 4π (E + P)r2
1 r4 j2
dω̃
= −
−
p
+
0
dr
(r − 2m)2
r − 2m
12 r − 2m dr
µ 3 2 2¶
1 d r j ω̃
(3.36)
+
,
3 dr r − 2m
3.2. Rotating Neutron Stars
29
and for l = 2 perturbation functions, where we use v2 = h2 + k2 instead of k2 , we
find
#
µ
µ ¶
¶"
dν0
1 dν0
dv2
1 4 2 dω̃ 2 1 3 2 d j2
= −2
h2 +
+
r j
− r ω̃
, (3.37)
dr
dr
r
dr
6
dr
3
dr
dh2
2v2
= −
dr
r(r − 2m(r))dν0 /dr
·
¸¾
½
dν0
r
4m(r)
h2
+
−2
+
8π (E + P) −
dr
2(r − 2m(r))dν0 /dr
r
·
¸
µ ¶2
1 dν0
1
3 2 dω̃
+
r
−
r j
6
dr
2(r − 2m(r))dν0 /dr
dr
·
¸
1 dν0
d j2
1
−
r
(3.38)
+
r2 ω̃ 2
.
3
dr
2(r − 2m(r))dν0 /dr
dr
The perturbation function m2 is given by
µ ¶
m2
1 4 2 dω̃ 2 1 3 2 d j2
= −h2 + r j
− r ω̃
.
r − 2m(r)
6
dr
3
dr
(3.39)
In the external vacuum, where no matter is present, i.e., E = P = 0, m(r) =
m(R) = M0 , and j = 1, the equations lead to
J2
,
r3
δM
J2
= −
+ 3
r − 2M0 r (r − 2M0 )
µ
¶
µ
¶
1
1
r
2
2
= J
+
+ KQ2
−1 ,
M0 r3 r4
M0
µ
¶
2M0
r
J2
1
+ Q2
−1 ,
= 4 +K
r
M0
[r(r − 2M0 )]1/2
m0 = δ M −
(3.40)
h0
(3.41)
h2
v2
(3.42)
(3.43)
where δ M is the total mass addition coming from rotation, K is a constant and
Qab are associated Legendre functions of the second kind (see equations (137)
and (141) of the Hartle original paper [33] for explicit formulas). The constant
K enters the equation for quadrupole moment of neutron star Q
Q=
J2 8
+ KM 3 .
M 5
(3.44)
30
Chapter 3. Neutron Star Models
The total mass of the neutron star is given by relation
M = M0 + δ M = M0 + m0 (R) + J 2 /R3 .
(3.45)
How to perform practical calculation of the neutron star models with appropriate expansion of key functions around the center of the star has been worked
out by Miller (see Section 6 in [31] and we use the same procedure here.
We have not mentioned yet, how the shape of star is affected, and how is
changed the total number of particles contained in the star. Since the parameter
that is kept to be constant during the perturbation is central energy density
(pressure), all other neutron star properties have to be calculated.
The equation for the isobaric surfaces in spherically symmetric neutron stars
is naturally the surface of constant radial coordinate. If the star is rotating the
perturbed equation of constant isobaric surfaces takes form
r(P = const.) = r0 (P) + ξ0 (r0 ) + ξ2 (r0 )P2 (cos θ ),
(3.46)
where r0 is the spherical coordinate and functions ξ0 and ξ2 are related to the
perturbation functions p0 and p2 through the relations
r[r − 2m(r)]
p0 ,
4π r3 P + m(r)
r[r − 2m(r)]
ξ2 (r) =
p2 .
4π r3 P + m(r)
ξ0 (r) =
(3.47)
(3.48)
Equatorial and polar radii Req , Rp of the rotating neutron star are then given by
1
Req = R + ξ0 (R) − ξ2 (R),
2
Rp = R + ξ0 (R) + ξ2 (R).
(3.49)
(3.50)
Change of the baryon number δ A in the neutron star due to rotation could be
calculated using the binding energy Eb = uA − M0 . Let us introduce the density
of internal energy as ε = E − unB . The change in binding energy δ Eb is given
by
J2
δ Eb = − 3 +
R
ZR
4π r2 B(r)dr,
0
(3.51)
31
3.3. Results
where
(
#
)
¶
µ
¶
2m(r) −1/2
dε
2m(r) −1/2
B(r) = (E + P)p0
1−
−1 −
1−
r
dP
r
¶
·
µ
¸
2m(r) −3/2 m0 1 2 2 2
+ (E − ε ) 1 −
+ j r ω̃
r
r
3
"
#
µ ¶2
1 2 4 dω̃
1 d j2 3 2
1
j r
+
r ω̃ .
−
(3.52)
4π r2 12
dr
3 dr
dE
dP
"µ
3.3. Results
We have calculated models of rotating neutron star using the Hartle - Thorne
approximation for all equations of state described in Chapter 2. We have calculated neutron star models for rotational frequency f = 716Hz and different
central parameters. Rotational frequency corresponds to the spin of the pulsar J1748-2446ad, being the highest known pulsar spin at present [32]. This
frequency, being as high as it is, still satisfies the condition of slow rotation approximation (Ω ¿ GM/R3 ). Assuming standard neutron star mass M = 1.4M¯
and radius R = 10 km, we find for the fastest pulsar R3 Ω2 /GM ' 0.11.
3.3.1. Moment of Inertia
By solution of equation 3.32 we calculate angular momentum J and moment of
inertia I = J/Ω. Since J is of the order Ω, the moment of inertia keeps constant
(frequency independent) and is function of central parameters only. In last years,
a particular interest was focused on the possibility that I/M0 R2 could be expressed
as a function of compactness x = 2M0 /R = rG /R = 1/rg , where rG = 2M0 is the
gravitational radius of given object and rg = R/rG is the radius of the object in
terms of its gravitational radius and equals unity for the Schwarzschild black
hole. Different authors [35, 36] used different formula that hold well for all
equations of state
x
, x ≤ 0.3,
0.295 + 2x
2
=
(1 + 1.69x), x > 0.3.
9
I/MR2 =
I/MR2
(3.53)
We present both functions (both on whole interval) together with moment of
inertia dependency on rg on Fig. 3.3
32
Chapter 3. Neutron Star Models
0.6
SkT5
SLy4
SV
H
L
M
APR
APR2
APR3
APR4
I/MR2
0.5
0.4
0.3
0.2
1
2
3
rg
4
5
Figure 3.3. Moment of inertia factor I/MR2 of neutron star versus rg = R/2M0 for all
equations of state. Analytic functions that approximate exact values are shown as black
lines (see text for details).
3.3.2. Gravitational radius, Compactness
Since we have introduced compactness (for shortness we will use now and on this
name for rg = R/2M0 instead of x = 1/rg ), and we have shown how compactness
could be used as variable to eventually omit the dependency on equation of state,
it could be useful to relate it to the mass of the neutron star. This relation is
shown on Fig. 3.4, where the horizontal line corresponds to mass M = 1.4M¯
and vertical line indicate radius of neutron star to be (R = 3M) i.e. the radius of
neutron star is at position of circular photon orbit limiting so called extremely
compact stars. It can be seen that none of the selected equations of state enables
the existence of neutron stars compact enough to have surface below the circular
photon orbit - such a neutron star could in principle show special cooling scenario
since trapped null geodesics exist in its interior (see [6] for more details).
We can see that compactness of 1.4M¯ neutron star is ≈ 2.2 to ≈ 3.4 for
equations of state selected in this work.
33
3.3. Results
SkT5
SLy4
SV
H
L
M
APR
APR2
APR3
APR4
2.5
M[MSUN]
2
1.5
1
0.5
1
Figure 3.4.
2
3
rg
4
5
Mass and compactness of static non-rotating neutron star, see text for
discussion.
3.3.3. Quadrupole moment
We have seen that taking compactness as a variable, moment of inertia (resp.
the factor in front of M0 R2 ) could be approximated with analytical function that
holds for all equations of state. Let us now try to find a parameter related to
quadrupole moment and check, if it can be approximated for all equations of
state by the same function. The natural choice is q̃ = QM0 /J 2 . This parameter
is frequency independent (since Q is of order Ω2 and J of order Ω and both are
zero for non-rotating stars), and taking it equal unity, the external spacetime is
of Kerr type‡‡ making identification a = J/M.
Because of the connection to Kerr metric we made presumption that as rg → 1,
i.e., as the non-rotating neutron star is closer to the Schwarzschild black hole,
the factor q̃ reaches unity, i.e., the rotating neutron star is closer to the Kerr black
hole state. Figure 3.5 shows that this assumption holds well assuming quadratic
‡‡ It
is exactly same as Kerr metric expanded up to second order in terms of a.
34
Chapter 3. Neutron Star Models
15
13
QM0/J2
11
9
7
SkT5
SLy4
SV
H
L
M
APR
APR2
APR3
APR4
5
3
1
1
2
3
rg
4
5
Figure 3.5. Dimensionless parameter Q/M0 J 2 versus compactness rg = R/2M0 .
Analytical functions approximating calculated values are plotted using black lines, and
both of them are presented.
behavior for low rg . For higher values of rg the dependency is linear. Relations
read in their general form
q̃ = a1 rg + a0 , rg > r0 ,
q̃ = b(rg − 1)2 + 1, rg ≤ r0 ,
(3.54)
where a1 and a0 are fitted constants and b is calculated assuming that the function
is continuous and smooth at the point where functions are matched. Under these
circumstances the relation for the constant b reads
b = a12 /[4.(−a1 − a0 + 1)],
(3.55)
and the matching point r0 is given by the relation
r0 =
2(1 − a0 )
− 1.
a1
(3.56)
We have found that for a1 = 3.64 and a0 = −5.3 approximate relation fits the
exact one very well (see Fig. 3.5).
35
3.3. Results
0.6
SkT5
SLy4
SV
H
0.5
L
M
APR
APR2
0.4 APR3
APR4
SkT5
SLy4
SV
H
L
M
APR
APR2
APR3
APR4
0.5
j
j
0.6
0.3
0.4
0.3
0.2
1
2
3
rg
4
5
0.2
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
M
Figure 3.6. Specific angular momentum j as a function of compactness rg (Left) and
total mass M in terms of solar masses (Right) of neutron star rotating with frequency
716 Hz.
For 1.4M¯ neutron stars the parameter q̃ is approximately in the range q̃ =
3 − 7.
3.3.4. Angular momentum
Instead of angular momentum of neutron star J we introduce dimensionless
angular momentum j = J/M02 . Figure 3.6 shows how j is related to rg (left) and
total mass of neutron star M for object spinning with frequency f = 716Hz. If
one is interested in value of j for another frequency of rotation f0 it can be found
through simple relation
j( f0 ) = j( f )
f0
,
f
(3.57)
since j is linear in f (Ω). We can see that for 1.4M¯ neutron star and spin 716Hz,
it is in the range j = 0.25 up to j = 0.5 for equations of state considered in this
work.
3.3.5. Mass change
Change in mass δ M due to rotation of neutron star is, again, calculated for star
rotating with frequency f = 716Hz to estimate recent upper limit for astrophysical
situations. Results are presented at Fig. 3.7. The upper limit on the relative mass
change due to rotation for pulsar spinning with highest frequency is ≈ 15% for
masses close around 1M¯ and is less than 5% if the mass is close to its maximal
value for given equation of state (left ends of curves).
36
Chapter 3. Neutron Star Models
δ M/M0
0.2
SkT5
SLy4
SV
H
0.15
L
M
APR
APR2
0.1 APR3
APR4
0.05
0
1
2
3
rg
4
5
Figure 3.7. Relative ass change δ M/M0 originated from rotation related to compactness rg = R/2M0 for star rotating with frequency f = 716Hz.
Chapter 4
Strange Stars
In this chapter we will deal with much more hypothetical objects that are called
strange stars. The hypothesis starts with assumption that under certain circumstances baryons are melted to form a matter consisting of u, d, and possibly
s quarks being energetically preferred state of matter. Important step forward
is hypothesis that this matter is preferred also at zero pressure [37]. Because
our experience with matter at zero pressure is different, the hypothesis provides
explanation that this is because the time needed to transform to strange matter is
longer than the age of universe. However, in the center of neutron star, if pressure reaches certain value, the timescale could be shortened and the transition of
nuclear matter to strange one could happen∗ . This is how strange star could be
born. We will deal with the so-called bare strange stars that are objects composed
of the strange matter also on the surface.
4.1. MIT Bag Model
The description of strange matter could be done using the very simple MIT Bag
model [38] and has been used to model both static [39] and rotating [40] strange
stars. The pressure P and baryon number density nB are related to energy density
E
1
(E − 4B) ,
3
"
#3/4
4(1 − 2αc /π )1/3
=
(E − B)
,
9π 2/3 h̄
P =
nB
where B is the bag constant that is connected to energy density of matter at zero
pressure and αc is the strong interaction coupling constant. We use MIT Bag
∗ An
alternative scenario deals with strangelet traveling in universe and hitting the neutron
star to start the process of transformation to strange star.
37
38
Chapter 4. Strange Stars
2.5
SkT5
SLy4
SV
H
L
M
APR
APR2
APR3
APR4
Strange
M [MSUN]
2
1.5
1
0.5
8
12
16
R [km]
Figure 4.1. Mass–radius relation for non-rotating neutron stars compared with models
of strange stars.
Model with B = 1014 g.cm−3 and αC = 0.15 in our calculations. More detailed
and realistic models of strange matter were developed by Farhi and Jaffe [41]
and later on by Alford, Rajagopal and Wilczek who introduced Color Flavour
Locked (CFL) model [42].
4.2. Strange star models
Strange star models are calculated by the same procedure as described in Chapter 3 with the only difference that in calculation in this chapter, equation of state
does not enter as external file but the analytical relations (4.1) are included directly into the code. We will show three different properties of relativistic objects
where the behavior of strange star differs significantly from neutron stars.
Mass–radius relation for neutron stars is compared with those for strange stars
on Fig. 4.1. We can see the different behavior for low mass objects. Radius of
strange stars is decreasing with decreasing mass and close to origin the behavior
is M ∼ R3 that originates in the fact that low mass strange stars are objects with
almost constant energy density profiles.
39
4.2. Strange star models
0.6
0.5
I/M0R2
Strange stars
0.4
0.3
Neutron stars
0.2
0.1
1
2
3
4
rg
5
6
7
Figure 4.2. Moment of inertia factor I/M0 R2 related to compactness for neutron and
strange stars.
20
15
QM0/J2
Strange stars
10
Neutron stars
5
1
2
3
4
rg
5
6
7
Figure 4.3. Quadrupole moment parameter QM0 /J 2 as it depends on compactness of
neutron and strange stars.
40
Chapter 4. Strange Stars
This fact could be seen also in Fig. 4.2, where the dependency of moment of
inertia factor I/MR2 on compactness rg is illustrated. For larger values of rg that
are corresponding to lower masses, the moment of inertia factor of strange stars
shows tendency I/MR2 → 0.4 corresponding to well-known value for spheres
in newtonian theory. This behavior also originates in the almost constant energy
density profile of low mass strange star. These facts were well known and
described in literature (see e.g. [17] for review).
For rotating strange stars quadrupole moment factor q̃ = QM0 /J 2 does not
hold the analytical relation (3.54) found in Chapter 3 and shows different behavior
than for neutron stars. Relations for neutron and strange stars, for range of
compactness rg are shown on Fig. 4.3 and represent an interesting new result.
4.2.1. Neutron to strange star transition
Transition of neutron star to strange star has been studied from different points
of view (see, e.g., [17] for review of the recent status). We will focus here on
relatively simple model assuming that during the transition from rotating neutron
star to the rotating strange star no matter is blown away, i.e., particle number A
is conserved. We also assume that angular momentum J = IΩ carried away is
small enough that we can take angular momentum as a conserved quantity as
well. We can write conservation of angular momentum in the form
fN IN = fS IS
(4.1)
and therefor the rotational frequency of the strange star is
fS = fN
IN
,
IS
(4.2)
where N denotes quantities of neutron star and S of strange star and f = Ω/2π .
We can see that if the moment of inertia of neutron star IN is smaller than moment
of inertia of strange star IS newly formed strange star will rotate with frequency
smaller than was the frequency of neutron star and vise versa.
Figure 4.4 shows moment of inertia of neutron (strange) stars as it depends
on total number of particles of the star rotating at frequency 716 Hz. We can
see that for equations of state SV, H, APR2 moment of inertia of neutron star
is always larger than moment of inertia of strange star which leads to larger
rotational frequency of strange star after the transition. The situation is entirely
opposite for equations of state APR3 and APR4. For others equations of state
both scenarios are possible depending on the mass of the neutron star. For lower
41
4.2. Strange star models
I [1045 g.cm2]
3.5
SkT5
SLy4
3
SV
H
L
2.5
M
APR
APR2
2 APR3
APR4
MIT
1.5
1
0.5
1
2
3
A [1057]
Figure 4.4.
Moment of inertia I of neutron and strange stars versus total number of
baryons.
neutron star masses the newly formed strange star will rotate slower and for
higher values of neutron star mass, strange star formed after the transition will
rotate faster. For these equations of state (SkT5, Sly4, L, M, APR) there also
exist specific situation when the rotational frequency remains unchanged. Mass
of rotating neutron star that collapse onto strange without change of rotational
frequency depends on equations of state.
Figure 4.5 shows relation between mass of the neutron (strange) star and the
total number of baryon of the star for stars rotating with frequency 716 Hz.
Difference between mass of the neutron star and strange star for given baryon
number and equation of state corresponds to energy that is released during the
transition if the rotational frequency remains unchanged.
42
Chapter 4. Strange Stars
2.5
M [MSUN]
2
SkT5
SLy4
SV
H
L
M
APR
APR2
APR3
APR4
MIT
1.5
1
0.5
1
2
3
57
A [10 ]
Figure 4.5. Mass M of neutron and strange stars versus total number of baryons.
Chapter 5
Theory Versus Observations
In previous chapters we have seen how properties of neutron star are affected
by equation of state of neutron star matter. In the first part of this chapter we
will se how this properties and their mutual relations could be estimated from
astrophysical observations, and how could they give us an information about the
validity of equation of state. Second part is focused on the influence of neutron
star properties on the orbital frequency of particle at the innermost stable circular
orbit (ISCO) around rotating neutron (strange) star.
5.1. Astrophysical observations testing equations of state
Astrophysical observations has been used to put constraints on equations of state
by different authors, see, e.g. [9] for overview of current status. We will use
here the maximal mass test that necessitates maximal mass enabled by neutron
star equation of state to be higher than observed mass of particular source.
Observations of thermal radiation of the source RX J1856-3754 and its analysis
lead to estimate of apparent radius and connected mass–radius relation. Last test
that we use to constrain neutron star equations of state was introduced by Podsiadlowski et al. [43] and is based on the analysis of possible formation of double
pulsar system J0737–3039, that could be used to put limits on mass–baryon mass
relation.
5.1.1. Maximum mass
One of particular interests of astrophysical observations connected to neutron
stars is the search of the mass of neutron star high enough that could be used to
eliminate some models of the equations of state. First interpretations of pulsar
PSR J0751+1807 observations were giving promising value M = (2.1 ± 0.2)M¯ ,
however this value was lowered to 1.26 ± 0.14 [44]. Another value, namely
1.9 − 2.1M¯ , was estimated by Barret, Olive and Miller [45] who analyzed
43
44
Chapter 5. Theory Versus Observations
M [MSUN]
1.5
1
f=0Hz
f=465Hz
0.5
10
12
14
16
18
R [km]
Figure 5.1. Mass-radius of neutron star rotating with frequency 465Hz (dashed line)
compared with static configuration (full line). Horizontal line indicates measured mass
of PSR J1903+0327. For rotating configurations equatorial radius is plotted on x-axis.
quasiperiodic oscillations observed in power spectra of the atoll source 4U
1636-536. They were using model of Miller, Lamb and Psaltis [46] that identifies
the highest observed frequency with the frequency of test particle orbiting at the
innermost stable circular orbit (ISCO). Unfortunately in their model the formula
used for ISCO frequency omit influence of quadrupole moment, and also the
estimate should be used rather as an upper limit on the mass than as its estimate
(see [46] for underlying theory). Therefore we decide to use observations of
PSR J1903+0327 that suggest mass of neutron star > 1.66M¯ on 2σ confidence
level [47, 48]. This result is not in an agreement with equation of state M (see
Tab. 3.1), however since this source is rotating with high frequency f = 465 Hz
[47] it is necessary to check whether or not this will change the mass enough
to get over the measured value. As we can see on Fig. 5.1 mass change due to
rotation with such frequency is too small to get over measured value.
45
5.1. Astrophysical observations testing equations of state
M [MSUN]
1.27
SkT5
SLy4
SV
H
1.26
L
M
APR
APR2
1.25 APR3
APR4
1.24
1.23
1.32
1.34
1.36
MB [MSUN]
1.38
1.4
Figure 5.2. Mass–baryon mass relation of static non-rotating neutron stars. Rectangle
indicates limits given by double pulsar J0737–3039. See text for details.
5.1.2. Double pulsar J0737–3039
Podsiadlowski et al. [43] developed a very accurate test of equation of state
based on assumption that pulsar B of double pulsar system J0737–3039 was born
via electron capture supernova. This scenario is supported by low pulsar mass
M = 1.2489 ± 0.0007 M¯ [49]. Authors also argue that the mass loss during
the transformation should be very low (≈ 10−3 M¯ ). If this pulsar is born under
the presented scenario, its baryonic mass MB should be in the range 1.366 to
1.375 M¯ . Under these assumptions limits on mass–baryon mass could be done.
We illustrate the result on Fig. 5.2, where we also extend the box for the case of
0.03MB mass loss. We can see that equations of state H, APR and APR2 meet
requirement even if no mass loss is assumed. Equations of state L and M are
able to fit extended box only and rest of tested equations of state do not meet
requirements of this test.
46
Chapter 5. Theory Versus Observations
Figure 5.3. Mass–radius relation for static non-rotating neutron stars. Lines corresponding to constant apparent radius R∞ are depicted by black lines. See text for
details.
5.1.3. Isolated neutron star RX J1856.5–3754
Thermal radiation observed for the neutron star source RX J1856.35–3754 could
be used as an estimate of apparent radius R∞ and therefore we could put limits
on equations of state from the mass–radius relation of the neutron star. Trümper
et al. [50] used two different models to explain the spectral feature for this
specific source and found its apparent radius that represents the radius of the
neutron star as seen by a distant observer. Estimated radius is proportional to the
distance from Earth to the source and has been discussed in the literature [51–53].
The distances obtained for RX J1856.5-3754 range from D = 61+9
−8 pc [51] to
+18
D = 161−14 pc [52]. The derived apparent radius R∞ is given by the model of
the atmosphere and distance measurements D. The original model by Pons [54]
resulted in R∞ /D = 0.13 km.pc−1 . Trümper et al. [50] presented new models
of atmosphere leading to the estimates of R∞ = 16.5 km for the two component
model of spectra and R∞ = 16.8 km assuming continuous temperature distribution
model, the latter value was used in the set of neutron star equation of state tests
5.2. ISCO frequency in the field of rotating neutron stars
47
by Klähn et al [55]. If the distance derived by van Kerkwijk and Kaplan [52] and
the original model of Pons [54] are used together, they lead to unexpectedly high
estimate of apparent radius R∞ = 20.9 km. Recently Steiner, Lattimer and Brown
[53] presented results based on new analysis of data giving the distance 119±5 pc
that together with the original model for atmosphere implies R∞ = 15.47 km. We
decided to use three different values namely R∞ = 15.5, 16.8, 20.9 km to put
limits on neutron star equation of state based on relativistic mean field equations
of state in [1] and we extend this result here to all selected neutron star equations
of state. Results are presented on Fig. 5.3. None of tested equations of state is able
to meet mass and radius requirements given by apparent radius R∞ = 20.9 km.
Apparent radius R∞ = 16.8 km could be modeled using equations of state SV,
APR, APR 2 and H. Equations of state SkT5, APR3 and APR4 do not meet the
requirements by the lowest value of apparent radius R∞ = 15.5 km.
5.2. ISCO frequency in the field of rotating neutron stars
Innermost stable circular orbit (ISCO) is the nearest orbit where test particle could
move on circular geodesic around central object. Closer to the compact object,
the gravitational attraction is too strong to be balanced by the centrifugal force.
Frequency of the particle at this orbit (ISCO frequency) could be connected to
astrophysical phenomena namely to those related to accretion processes and high
frequency quasi-periodic oscillations [46, 56]. Approximate formula for ISCO
frequency was found by Kluz̀niak and Wagoner [57]
fISCO (M, j) =
M
(1 + 0.749 j) × 2198 Hz.
M¯
(5.1)
We can see that this formula is of first order in specific angular momentum j =
J/M 2 . External Hartle–Thorne spacetime [2] deals with terms up to second order
in j and includes also influence of the specific quadrupole moment q = Q/M 3 .
Formula for angular frequency Ω = 2π f at given radius r reads [58]
#
"
µ ¶3/2
M 1/2
M
Ω(r) = 3/2 1 − j
+ j2 F1 (r) + qF2 (r) ,
r
r
(5.2)
48
Chapter 5. Theory Versus Observations
where
h
48M 7 − 80M 6 r + 4M 5 r2 − 18M 4 r3 + 40M 3 r4 + 10M 2 r5
i¡
¢−1
6
7
(5.3)
+ 15Mr − 15r 16M 2 (r − 2M)r4
+ A(r),
F1 (r) =
5(6M 4 − 8M 3 r − 2M 2 r2 − 3Mr3 + 3r4 )
− A(r),
16M 2 r(r − 2M)
µ
¶
15(r3 − 2M 3 )
r
ln
A(r) =
.
32M 3
r − 2M
F2 (r) =
(5.4)
(5.5)
Substituting radius of ISCO
"
rISCO
µ
µ ¶¶
µ ¶3/2
3
2
2 251647
= 6M 1 − j
+j
− 240 ln
3
2592
2
µ
µ ¶¶¸
−9325
3
+ q
+ 240 ln
96
2
(5.6)
to eq. 5.2 gives ISCO frequency fISCO = Ω(rISCO )/2π .
We calculated ISCO frequency for different values of q/ j2 and illustrated the
results on Fig. 5.4, where we have also plotted linear approximation (5.1) (black
dashed line) and ISCO frequency of particle orbiting around rotating black hole
described by Kerr spacetime [59] (red line). Note, that expanding formula for
ISCO frequency in Kerr spacetime up to second order in j = a/M will meet the
Hartle-Thorne ISCO frequency for j = q2 . The difference between red and green
line on Fig. 5.4 therefore origins in higher order terms in Kerr spacetime.
49
5.2. ISCO frequency in the field of rotating neutron stars
4000
Kerr
q=1j2
q=2j2
q=3j2
q=4j2
q=5j22
q=10j
Kluzniak
3800
M/Msun fISCO[Hz]
3600
3400
3200
3000
2800
2600
2400
2200
2000
1800
0
0.2
0.4
0.6
0.8
1
j
Figure 5.4. Frequency of particle orbiting at the innermost stable circular geodesics for
selected values of q/ j2 is plotted against specific angular momentum of central object.
Linear approximation is illustrated by black dashed line and red line represents the
frequency of particle orbiting Kerr black hole on the innermost stable circular geodesic.
Chapter 6
Summary and Discussion
In Chapter 3 we have shown properties of both static and rotating neutron stars
for selected set of neutron star equations of state presented in Chapter 2. We
compared these results with results obtained by modelling strange stars using the
MIT Bag model equations of state in Chapter 4. These result were in some cases
compared to various kind of astrophysical observations as presented in Chapter 5.
The numerical code was developed and used for rotating neutron (strange) stars
and some interesting effects were found that could be of significant astrophysical
relevance.
6.1. New results
We will briefly summarize results that were obtained during work on this thesis
and that remained previously unpublished.
6.1.1. Test of equation of state
We have used parameterizations obtained by Kotulič Bunta and Gmuca [26] for
description of relativistic mean field equations of state of neutron stars and we
compared calculated neutron star properties with set of astrophysical observations
[1]. We have shown that parameterization H seems to be in agreement with all
the observations we selected. On the other hand, parameterizations L and M
posse some difficulties to explain current interpretation of data.
6.1.2. Analytical representation of external HT metric
In the field of rotating neutron stars we have found that quadrupole moments
of rotating neutron stars could be interpolated by an analytical function. We
presented this result in Section 3.3.3 and on Fig. 3.5. This result could be very
useful for checking the validity of models of quasiperiodic oscillations (see, e.g.,
51
52
Chapter 6. Summary and Discussion
[60–62] and references therein) as shown in papers [4, 5]. Of high relevance is
the fact that the dimensionless parameter characterizing the quadrupole moment
q̃ = QM0 /J 2 of the Hartle - Thorne geometry representing external gravitational
field of rotating neutron stars is very close to the value corresponding to the Kerr
spacetime for near-maximal masses admitted by any equation of state, while
they take large values (up to q̃ ∼ 10) for low mass neutron star models M ∼ M¯ .
Further it is demonstrated that dependence of the q̃ factor on other characteristics
are different for neutron stars as compared with strange stars.
6.2. Future research
For future we plan to focus our research on several areas that seems to be
promising at present.
Impact of the presence of hyperons in neutron star interior on properties of
neutron star models and comparison of these models with observations could lead
to estimation of threshold density, where hyperons start to appear. For example
double pulsar J0737–3039 test (see 5.1.2 or [43]) could be used since it is very
strong and accurate test of equations of state.
Possible measurements of q̃ in future could serve as an efficient estimation
method of R/M for neutron stars, without worries about what specific equation
of state should be chosen. Another area, where knowledge of relation between
compactness and quadrupole momentum factor q̃ (3.54) would be usefull is
improved implications of ISCO frequency estimates on the models of equations
of state.
We would like also to focus our attention on transition of neutron stars to
strange stars and impact of this transition on rotational frequency. We plan to
develop a model assuming no mass loss and conserved angular momentum and
find what happen with frequency of neutron star. It was shown that both scenarios
i.e. spindown and spinup are possible depending on equation of state of neutron
star matter see Fig. 4.4 for details and we plan to improve our model.
Appendix A
Nuclear matter at β -equilibrium
Assume nuclear matter consisting of neutrons, protons and electrons in β -equilibrium
i.e. in equilibrium with respect to reactions
n ↔ p + e− .
(A.1)
The condition of β -equilibrium could be written using chemical potentials as
µn = µp + µe ,
(A.2)
where for all particles
µi =
∂E
.
∂ ni
(A.3)
The chemical potential of electrons reads
µe = µe0 = EF(e) =
q
m2e c4 + p2F(e) c2 .
(A.4)
Because electrons are extremely relativistic, the equation (A.4) could be written
in the form
µe = pF(e) c = ch̄(3π 2 ne )1/3 = ch̄(3π 2 np )1/3 = ch̄(3π 2 xp nB )1/3 ,
(A.5)
where we have used the condition of charge neutrality np = ne .
The chemical potential of protons and neutrons can be expressed as
µb =
∂
(nB EB ) ,
∂ nb
(A.6)
53
54
Appendix A. Nuclear matter at β -equilibrium
where b stands for n or p. We will now calculate the chemical potential of protons
µp =
=
=
+
=
¤
∂
∂ £
∂ EB
(nB EB ) =
(np + nn )EB = EB + nB
∂ np
∂ np
∂ np
#
"
µ
¶
np 2
∂
EB + nB
E0 (nB ) + 1 − 2
S0 (nB )
∂ np
nB
"
#
µ
¶
np 2 ∂ S0 (nB ) ∂ nB
∂ E0 (nB ) ∂ nB
EB + nB
+ 1−2
∂ nB ∂ np
nB
∂ nB ∂ np
µ
¶µ
¶
np
−2nn
S0 (nB )
2nB 1 − 2
nB
n2B
"
#
µ
¶
µ
¶
np 2 ∂ S0 (nB )
np nn
∂ E0 (nB )
EB + nB
−4 1−2
S0 (nB ),
+ 1−2
∂ nB
nB
∂ nB
nB nB
(A.7)
and of neutrons
µn =
=
=
+
=
¤
∂ EB
∂
∂ £
(nB EB ) =
(np + nn )EB = EB + nB
∂ nn
∂ nn
∂ nn
"
#
µ
¶2
∂
nB − nn
EB + nB
E0 (nB ) + 1 − 2
S0 (nB )
∂ nn
nB
"
#
µ
¶
np 2 ∂ S0 (nB ) ∂ nB
∂ E0 (nB ) ∂ nB
EB + nB
+ 1−2
∂ nB ∂ np
nB
∂ nB ∂ np
µ
¶µ
¶
np
2np
2nB 1 − 2
S0 (nB )
nB
n2B
#
"
µ
¶2
µ
¶
np np
n
∂ S0 (nB )
∂ E0 (nB )
p
EB + nB
+ 1−2
+4 1−2
S0 (nB ).
∂ nB
nB
∂ nB
nB nB
(A.8)
The difference between neutron and proton chemical potentials is then given by
¶
np
µn − µp = 4S0 (nB ) 1 − 2
,
nB
µ
(A.9)
and is equal to electron chemical potential
¢
¡
4S0 (nB ) 1 − 2xp = ch̄(3π 2 xp nB )1/3 .
(A.10)
55
This is the equation 2.10.
Lets make now step forward and include also muons. The β equilibrium of
n-p-e-µ matter is given by equations
µn = µp + µe ,
(A.11)
µe = µµ ,
(A.12)
that indicates the equilibrium with respect to reactions
n ↔ p + e− ↔ p + µ − .
(A.13)
The charge neutrality is now given by
np = ne + nµ .
(A.14)
The equality of electron and muon chemical potentials implies
µe = µµ
q
pF(e) c =
m2µ c4 + p2F(µ ) c2
(A.15)
(A.16)
p2F(e) = m2µ c2 + p2F(µ )
(A.17)
h̄2 (3ne π 2 )2/3 = m2µ c2 + h̄2 (3nµ π 2 )2/3
(A.18)
(3ne π 2 )2/3 − (3nµ π 2 )2/3 =
2/3
2/3
ne − n µ
=
m2µ c2
h̄2
m2µ c2
h̄2 (3π 2 )2/3
(A.19)
(A.20)
The number density of muons is then
"
nµ =
2/3
ne − 2
#3/2
m2µ c2
h̄ (3π 2 )2/3
,
(A.21)
and from charge neutrality the number density of protons is
"
2/3
np = ne + nµ = ne + ne −
m2µ c2
h̄2 (3π 2 )2/3
#3/2
.
(A.22)
56
Appendix A. Nuclear matter at β -equilibrium
The condition of β -equilibrium µn − µp = µe then reads
µ
np
4S0 (nB ) 1 − 2
nB
¶
= ch̄(3π 2 ne )1/3 .
(A.23)
For given baryon number density one is then solving eq. (A.23) enabling us
to calculate the number density of electrons, and thus also the density of other
particles, due to simple relations (A.21) and (A.22) and nn = nB − np .
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Part II
Individual Papers
IOP PUBLISHING
CLASSICAL AND QUANTUM GRAVITY
Class. Quantum Grav. 26 (2009) 035003 (17pp)
doi:10.1088/0264-9381/26/3/035003
Neutrino trapping in extremely compact objects: I.
Efficiency of trapping in the internal Schwarzschild
spacetimes
Zdenčk Stuchlı́k, Gabriel Török, Stanislav Hledı́k and Martin Urbanec
Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava,
Bezručovo nám. 13, CZ-746 01 Opava, Czech Republic
E-mail: [email protected], [email protected], [email protected] and
[email protected]
Received 12 June 2008, in final form 24 June 2008
Published 13 January 2009
Online at stacks.iop.org/CQG/26/035003
Abstract
Extremely compact objects (9GM/4c2 < R < 3GM/c2 ) contain trapped null
geodesics. When such objects enter the evolution period corresponding to the
geodetical motion of neutrinos, a certain part of neutrinos produced in their
interior will be trapped influencing thus neutrino luminosity of the objects
and consequently their thermal evolution. The existence of trapped neutrinos
indicates possibility of ‘two-temperature’ cooling regime of the extremely
compact objects. We present upper estimates on the efficiency of the neutrino
trapping effects obtained in the framework of the simplest model of the internal
Schwarzschild spacetime with uniform distribution of energy density, assuming
uniform distribution of neutrino emissivity. We introduce a ‘global’ luminosity
trapping coefficient representing influence of the trapping effect on the total
neutrino luminosity of the extremely compact objects and cooling trapping
coefficients of both ‘local’ and ‘global’ kinds characterizing influence of the
trapping on the cooling process. It is shown that the trapping of neutrinos can
be relevant to moderately or even slightly extremely compact objects.
PACS numbers: 95.30.Sf, 97.60.Jd
1. Introduction
It is well known that in the internal Schwarzschild spacetimes of uniform energy density
[1] with radius R < 3GM/c2 , bound null geodesics must exist being concentrated around
the stable circular null geodesic [2, 3]. It follows immediately from the behaviour of
the effective potential of null geodesics in the exterior, vacuum Schwarzschild spacetimes,
0264-9381/09/035003+17$30.00 © 2009 IOP Publishing Ltd Printed in the UK
1
Class. Quantum Grav. 26 (2009) 035003
Z Stuchlı́k et al
determining the unstable null circular geodesics at the radius rph = 3GM/c2 (see, e.g., [4]),
that any spherically symmetric, static non-singular interior spacetime with radius R < rph
admits existence of bound null geodesics. We call objects (stars) admitting existence of
bound null geodesics—extremely compact objects (stars). Note that, in principle, the bound
null geodesics could exist also in objects having R > 3GM/c2 , e.g., in some composite
polytropic spheres [5]. The realistic equations of state admitting the existence of the extremely
compact objects were found and investigated, e.g., in [5–7], for both neutron stars and quark
stars.
The existence of bound null geodesics in extremely compact objects has interesting
astrophysical consequences. For example, trapped modes of gravitational waves could
influence some instabilities in these objects as shown by Abramowicz and his collaborators
[8, 9] in an approach based on the optical reference geometry which brings a new insight into
the properties of extremely compact objects.
We shall consider another interesting problem related to the existence of bound null
geodesics—namely, the problem of neutrinos trapped by the strong gravitational field of
extremely compact objects. The trapped neutrinos can be important at least for two reasons.
First, they will suppress the neutrino flow from extremely compact stars as measured by distant
observers. Second, trapped neutrinos, being restricted to a layer extending from some radius,
depending on details of the structure of the extremely compact stars, up to their surface, can
influence cooling of the extremely compact stars. The cooling process could even be realized
in a ‘two-temperature’ regime, when the temperature profile in the interior of the star with
no trapped neutrinos differs from the profile established in the external layer with trapped
neutrinos. For the neutrino-dominated period of the cooling process, one can speculate that
some part of the external layer near the radius of the stable null circular geodesic, where
the trapping of neutrinos reaches highest efficiency, will reach a higher temperature than is
the temperature in the interior of the star. This effect can lead to an inflow of heat from the
‘overheated’ external layer to the interior of the star through other ‘agents’ than the neutrino
flow. Such a heat flow could influence the structure of extremely compact stars, maybe,
some special ‘self-organized’ structures could develop due to the assumed heat flow. Then
properties of the extremely compact stars could be modified in comparison with the standard
picture given in [10–12].
Of course, all of these ideas deserve very sophisticated analytical estimates and detailed
numerical simulations. Here, we restrict our attention to the first step in considering the role
of trapped neutrinos in extremely compact stars. We shall determine efficiency of the trapping
effect by considering the number of trapped neutrinos in comparison to all neutrinos produced
in the extremely compact objects. The influence on the neutrino luminosity of the star is given
by a luminosity trapping coefficient relating the total number of trapped neutrinos and the total
number of radiated neutrinos (per unit time of distant observers). The influence on the cooling
process is given by two ‘cooling’ trapping cefficients: a ‘local’ one given by ratio of trapped
and radiated neutrinos at any radius where the trapping occurs, and the ‘global’ one giving ratio
of trapped and radiated neutrinos (per unit time of distant observers) integrated over the whole
region where the trapping occurs. For simplicity, we shall consider the internal Schwarzschild
spacetime with uniform distribution of energy density (but a non-trivial pressure profile) and
isotropic and uniform distribution of local neutrino luminosity, when all the calculations can
be realized in terms of elementary functions only. We would like to recall that the internal
Schwarzschild spacetime can well represent the spacetime properties of realistic extremely
compact stars [10] which are crucial in estimating the role of neutrino trapping. Such a simple
spacetime could serve quite well as a test bed for realistic models of extremely compact
objects.
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Class. Quantum Grav. 26 (2009) 035003
Z Stuchlı́k et al
Clearly, it is worth of studying the trapping effect, if the extremely compact stars can be
constructed for some of the realistic equations of state. In fact, there are a wide variety of
equations of state (EOS) used to describe interior of neutron stars [7, 13, 14]. The standard
EOS of the Skyrmion type predict usually stable configurations with compactness R/M > 4,
i.e., substantially above the critical value of R/M = 3. Indeed, some observational data
indicate non-extreme neutron stars in the case of some pulsars [15]. Similar results are
implied by some versions of the mean field theories giving EOS [16]. However, some stiff
equations of state predict stable configurations just near the critical value [7].
The maximum compactness of neutron stars was estimated by several authors as
summarized in [14, 17]. In the case of supernuclear densities ρ 3 × 1014 g cm−3 the
lowest restriction R 2.83M was obtained by Lindblom [18]. The causal limits of Koranda
et al [19], obtained for the so-called minimum period equations of state give R > 2.87M.
These limits are in accord with empirical limits obtained by Glendenning [10, 20]. Similarly,
the quark stars (or hybrid stars with quark cores) could be relevant candidates of slightly
extreme compact stars [10, 12]. Especially the so-called strange stars [13, 21] could be
candidates for the extremely compact stars. The strange stars are based on the hypothesis that
strange quark matter, i.e. matter containing substantial part of strange quarks, might be the
absolute ground state of baryonic matter. The strange stars, kept together by strong forces only,
not necessarily by gravity, could then be considered as giant nuclei (having baryon number
∼1057 ) with the confined quarks freely moving through the false vacuum at the star interior.
The most compact stars predicted by realistic EOS are probably the so-called Q-stars
introduced by Bahcall, Lynn and Selipsky [22, 23] in close connection to the ideas behind
possible existence of the strange matter and strange stars. The Q-stars are based on ideas
allowed by some effective field theories of the strong force that imply the possibility
of confining nucleons (protons and neutrons) at densities below that of nuclear matter
ρn ∼ 3 × 1014 g cm−3 to form bulk nuclear matter, which is kept together by strong forces,
similarly to strange stars. Because laboratory experiments on ordinary nuclei do not constrain
the properties of such bulk baryonic matter, there are a wide range of theoretical possibilities
for properties of compact objects. In fact, this model could even be applied to quark matter,
but on different scales, and the quark–baryon matter can have the same equation of state as
strange nuggets and strange stars [22, 23]. In its simplest form, the EOS of Q-stars reads
ρ − 3p − 4U0 + αv (ρ − p − 2U0 )3/2 = 0 with ρ(p) being density (pressure), U0 the energy
density of confining scalar field and αv the strength of the repulsive interaction between
nucleons. Such EOS represents chiral Q-matter with particles of zero mass confined in the
false vacuum, but the results for non-chiral matter are only slightly different [22, 23]. The
compactness of the Q-stars can approach R/M ∼ 2.8 and could be even slightly smaller for
slowly rotating Q-stars [24]. There is another relevant point of the Q-star models, namely that
the lowest-mass models (still able to give extremely compact stars) have an almost uniform
distribution of the density, and their spacetime should be very close to our test bed model.
Our paper is organized as follows. In section 2, we summarize properties of the internal
Schwarzschild spacetime. In section 3, null geodesics of the spacetime are described in terms
of properly given effective potential. In section 4, the trapping of neutrinos is determined.
In section 5, the efficiency coefficients of the trapping are defined for both the total neutrino
luminosity and neutrino cooling process, and determined in terms of elementary functions for
the internal Schwarzschild spacetime. In section 6, concluding remarks are presented. We
shall use the geometric units, if not stated otherwise. For simplicity, we assume zero rest
energy of neutrinos and the period of evolution of the compact stars, when the temperature
is low enough that the motion of neutrinos is determined by the null geodesics of the
spacetime.
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Class. Quantum Grav. 26 (2009) 035003
2. Internal Schwarzschild spacetime
In the standard Schwarzschild coordinates and the geometric units with c = G = 1, the line
element for the internal Schwarzschild spacetime of uniform energy density ρ reads
ds 2 = −e2(r) dt 2 + e2(r) dr 2 + r 2 (dθ 2 + sin2 θ dφ 2 ).
(1)
The temporal and radial components of the metric tensor are given by the formulae
(−gtt )1/2 = e = 32 Y1 − 12 Y (r),
(grr )1/2 = e = 1/Y (r),
(2)
where
1/2
r2
Y (r) = 1 − 2
,
a
1/2
R2
Y1 ≡ Y (R) = 1 − 2
,
a
1
8
2M
= πρ = 3 .
2
a
3
R
(3)
(4)
R is the radius of the internal spacetime, M is the mass parameter of the internal spacetime,
which coincides with the mass parameter of the external, vacuum Schwarzschild spacetime.
The parameter a represents the curvature of the internal Schwarzschild spacetime; it is the
radius of the embedding diagram of its equatorial plane t = const section into the 3D Euclidean
space [2]. The pressure profile of the internal Schwarzschild spacetime is given by the formula
p(r) = ρ
(1 − 2Mr 2 /R 3 )1/2 − (1 − 2M/R)1/2
.
3(1 − 2M/R)1/2 − (1 − 2Mr 2 /R 3 )1/2
(5)
It can be shown that the internal Schwarzschild spacetimes are allowed for R > 9M/4
only, since for R = 9M/4, the central pressure p(0) diverges, see, e.g. [2, 25] for details. The
extremely compact internal Schwarzschild spacetimes are those with R ∈ (9M/4, 3M).
In terms of the tetrad formalism the metric (1) reads
ds 2 = −[ω(t) ]2 + [ω(r) ]2 + [ω(θ) ]2 + [ω(φ) ]2 ,
(6)
where
ω(t) = e dt,
ω(r) = e dr,
ω(θ) = r dθ,
−1
μ
is then given by
The tetrad of 4-vectors e(α) = ωμ(α)
e(t) =
1 ∂
,
e ∂t
e(r) =
1 ∂
,
e ∂r
e(θ) =
1 ∂
,
r ∂θ
ω(φ) = r sin θ dφ.
e(φ) =
∂
1
.
r sin θ ∂φ
(7)
(8)
Tetrad components of 4-momentum of a test particle or a photon are determined by the
μ
projections p(α) = pμ e(α) , p(α) = pμ ωμ(α) which give quantities measured by the local
observers.
3. Null geodesics and effective potential
We consider the period of evolution and cooling of extremely compact stars when their
temperature falls down enough that the motion of neutrinos can be considered free, i.e.,
geodetical. We can assume this period starts at the moment when mean free path of neutrinos
becomes comparable to the radius R, i.e., in hours after the gravitational collapse creating the
compact object [10, 12, 26]. In fact, there are arguments that this condition starts to be fulfilled
about 50 s after collapse to a proto-neutron star [14, 17]. Weak interaction of ultrarelativistic
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Class. Quantum Grav. 26 (2009) 035003
(massless) neutrinos thus implies their motion along null geodesics obeying the equations (λ
is an affine parameter):
Dpμ
(9)
= 0,
pμ pμ = 0.
dλ
Due to the existence of two Killing vector fields: the temporal ∂/∂t one, and the azimuthal
∂/∂φ one, two conserved components of the 4-momentum must exist:
E = −pt
(energy),
L = pφ
(axial angular momentum).
(10)
Moreover, the motion plane is central. For a single-particle motion, one can set θ = π/2 =
const, choosing the equatorial plane.
The motion along null-geodesics is independent of energy (frequency) and can
conveniently be described in terms of the impact parameter
L
= .
(11)
E
Then (9) yields the relevant equation governing the radial motion in the form
2
(pr )2 = e−2(+) E 2 1 − e2 2 .
(12)
r
Clearly, the energy E is irrelevant and can be used for rescaling of the affine parameter λ. The
radial motion is restricted by an effective potential defined by the relations
⎧
4a 2 [1 − Y 2 (r)]
⎪
int
⎪
⎨Veff
=
for r R
[3Y1 − Y (r)]2
(13)
2 Veff =
3
⎪
r
⎪
ext
⎩Veff
for r > R.
=
r − 2M
int
Veff
is the effective potential of the null-geodetical motion in the internal Schwarzschild
ext
is the effective potential of the null-geodetical motion in the external,
spacetime and Veff
vacuum Schwarzschild spacetime [4]. The effective potential is thus related to the impact
parameter.
Circular null geodesics are given by the local extrema of the effective potential
(∂Veff /∂r = 0), which in the internal spacetime yields for their radius and impact parameter
the relations
1
4a 2
,
2c(i) =
.
(14)
Y (rc ) =
3Y1
9Y12 − 1
The radius rc(i) is explicitly given by
R
8a 2
− R2
−1
R 3 49 M
2
2 9
R
.
=
rc(i) = a
(a 2 − R 2 )
2M 12 M
−1
(15)
Typical behaviour of the effective potential of the null-geodetical motion Veff is illustrated in
figures 1 and 2. Here, and henceforth, we put M = 1 for simplicity, i.e., we express the radii
in units of the mass parameter M.
4. Trapping of neutrinos
In the case of extremely compact static objects described by the internal Schwarzschild
spacetime (R < 3), stable bound null geodesics exist (figure 3), i.e., some part of produced
neutrinos is prevented from escaping these static objects. For the unit mass M = 1, the relation
(14) implies the impact parameter which corresponds to the local maximum of the effective
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Class. Quantum Grav. 26 (2009) 035003
40
30
30
Veff
Veff
40
20
interior
20
exterior
10
10
R = 2.5 2.8
R
0
1
2
r
3
4
5
0
1
2
r
3
3
4
4
5
Figure 1. Effective potential (M = 1) for R = 2.5 (left) and for several values of R (right). The
inner bound geodesics exist for R < 3 only. The effective potential of the internal and external
Schwarzschild spacetimes are smoothly matched at R.
int (M = 1) for 9/4 < R < 2.3 and 0 r 2.5. The
Figure 2. Segment of 3D plot of Veff
maximum of the effective potential diverges when R → 9M/4 and shifts to r = 0.
int
potential Veff
at rc(i) , where the stable circular null geodesics of the internal Schwarzschild
spacetime are located, to be given by
2c(i) =
R4
.
4R − 9
(16)
ext
at rc(e) = 3 corresponds to the unstable circular null geodesics of
The local minimum of Veff
the external vacuum Schwarzschild spacetime, with 2c(e) = 27 (see figure 3).
4.1. Regions of trapping
Bound neutrinos (depicted by the shaded area in figure 3) may generally appear outside the
extremely compact objects, but they are trapped by the strong gravitational field of these objects
and they enter them again. Therefore, we divide the trapped neutrinos into two families:
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Class. Quantum Grav. 26 (2009) 035003
r
0
1
2
3
interior
exterior
4
5
50
40
Veff
2
c(i)
2
int ( R)
30
2
c(e)
20
10
R
rb(e) rb(i)
rc(i)
rc(e) = 3
0
Figure 3. Detailed behaviour of Veff (M = 1) for R = 2.5.
• internal bound neutrinos (upper (shaded) part of the shadow area with impact parameter
between 2int (R) and 2c(i) ): their motion is restricted inside the object.
• external bound neutrinos (lower part of the shadow area with impact parameter between
2c(e) and 2int (R)): may leave the object, but they re-enter the object.
Pericentra for both the marginally bound (rb(e) ) and ‘internal’ marginally bound neutrinos
(rb(i) ) can be expressed in the form
1/2
81 1 − R2
± 2R(R 4 − 108R + 243)1/2
,
(17)
Y± (rb(e) ) =
2R 3 + 27
2 1/2 9 − 2R
Y (rb(i) ) = 1 −
.
(18)
R
2R − 3
The radii rb(e) and rb(i) are then explicitly given by the formulae
r 2
27{10R 4 − 18R 3 − 108 + 243 − [6R 3/2 (R − 2)1/2 (R 4 − 108R + 243)1/2 ]}
b(e)
=
,
R
(2R 3 + 27)2
(19)
rb(i)
(32R − 144R + 162)
=
,
(20)
R
21/2 (2R − 3)
see figure 4 for the graphical representation. For completeness, we show also loci rc(i) of the
stable circular null geodesic.
2
1/2
4.2. Mean free path of neutrinos
The approximation of free, geodetical motion of neutrinos in the internal spacetime could be
used when the mean free path of neutrinos λ > R. As discussed by Shapiro and Teukolsky
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Class. Quantum Grav. 26 (2009) 035003
3
R/M
2.6
r/M
2.2
rc(i)
1.8
r b(i)
1.4
r b(e)
1
0.6
0.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
R/M
Figure 4. The dependence of the radii determining the trapped neutrinos rb(e) , rb(i) and rb(i) on the
radius R. The relations for the variable Y (r) are converted into relations for r.
[26] neutrinos have inelastic scatter on electrons (muons) and elastic scatter on neutrons. The
scatter cross section on electrons (neutrons) σe (σn ) determines the mean free path by formula
λ = (σi ni )−1 where ni (i = (e, n)) denotes the number density of electrons (neutrons). It was
shown [26] that
λe ∼ 9 × 10
7
ρnucl
ρ
4/3 100 keV
Eν
3
km,
(21)
while
ρnucl
λn ∼ 300
ρ
100 keV
Eν
2
km.
(22)
There is λe 10 km for Eν 20 MeV and λn 10 km for Eν 500 keV. Clearly, at
temperatures T 109 K (Eν ∼ 100 keV) the neutrino motion could be considered geodetical
through the whole internal spacetime.
Bound neutrinos with mean free path R (as shown above, this condition can be fulfilled
in a few hours old neutron star, see [14, 26]) will slow down the cooling. Of course, they
will be re-scattered due to finiteness of the mean free path. An eventual scattering of trapped
neutrinos will cause change of their impact parameter, therefore, some of them will escape the
extremely compact star, suppressing thus the slow down of the cooling process in the region of
neutrino trapping. However, the ‘external’ bound neutrinos have certain portion of their orbit
outside the compact star where no interaction with matter is possible; this fact, on the other
hand, ‘suppress the suppression’ of the cooling timescale retardation. Clearly, the scattering
effect of the trapped neutrinos is a complex process deserving sophisticated numerical code
based on the Monte Carlo method (we expect modelling of this effect in future). Only neutrinos
produced above or at rb(e) are subject to this effect; those produced below rb(e) freely escape
to infinity.
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Class. Quantum Grav. 26 (2009) 035003
4.3. Directional angles
Considering (without loss of generality, as stated just above equation (11)) an equatorial
motion, we can define the directional angle relative to the outward pointed radial direction
measured in the emitor system (i.e., the local system of static observers in the internal
Schwarzschild spacetime) by the standard relations
sin ψ =
p (φ)
,
p(t)
cos ψ =
p(r)
,
p(t)
(23)
where
p(α) = pμ ωμ(α) ,
μ
p(α) = pμ e(α)
(24)
are the neutrino momentum components as measured by the static observers. Besides
conserving components (10), and pθ = 0, equation (12) implies
2
pr = ±Ee− 1 − e2 2 .
(25)
r
For the directional angles we thus obtain relations
sin ψ = α(r, R) ,
cos ψ = ±(1 − sin2 ψ)1/2 ,
(26)
r
where
2 1/2 1
2 r 2 1/2
3
1−
1−
−
.
(27)
α(r, R) =
2
R
2
R R
The interval of relevant radii is given by r ∈ (rb(e) , R). The directional angle limit for the
bound neutrinos is determined by the impact parameter 2c(e) = 27. We arrive at the relations
3(3)1/2
,
sin ψe (r, R) = α(r, R)
r
(28)
1/2
27α 2 (r, R)
cos ψe (r, R) = ± 1 −
.
r2
The directional angle limit for the ‘internal’ bound neutrinos is determined by equation (16)
and yields the relations
R 3/2
,
r(R − 2)1/2
1/2
α 2 (r, R)R 3
.
cos ψi (r, R) = ± 1 −
(R − 2)r 2
Apparently, the condition ψi > ψe holds at any given radius r < R.
sin ψi (r, R) = α(r, R)
(29)
4.4. Local escaped-to-produced neutrinos ratio
We assume that neutrinos are locally produced by isotropically emitting sources. Then the
escaped-to-produced-neutrinos ratio depends on a geometrical argument only. It is determined
by the solid angle 2 corresponding to escaping neutrinos (also inward emitted neutrinos must
be involved because even these neutrinos can be radiated away), see figure 5.
Let Np , Ne and Nb denote, respectively, the number of produced, escaped and trapped
neutrinos per unit time of an external static observer at infinity. In order to determine the
global correction factors
Ne (R)
Nb (R)
E (R) ≡
B(R) ≡
(30)
,
= 1 − E (R),
Np (R)
Np (R)
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Class. Quantum Grav. 26 (2009) 035003
Figure 5. Schematic illustration of the bound-escape ratio at a radius r ∈ (rb(e) , R) of an internal
Schwarzschild spacetime. Direction of the neutrino motion with respect to the static observers is
related to e(r) giving the outward oriented radial direction.
it is necessary to introduce the local correction factor for escaping neutrinos at a given radius
r ∈ (rb(e) , R). Because of the assumption of isotropic emission of neutrinos in the frame of
the static observers, the solid angle1 e (e ) determines fully the ratio of escaped–produced
neutrinos. The escaping solid angle is given by
e 2π
e (e ) =
sin ddφ = 2π(1 − cos e )
(31)
0
0
and the escaping correction factor
(r, R) =
2(ψe (r, R))
dNe (r)
=
= 1 − cos ψe (r, R),
dNp (r)
4π
(32)
while the complementary factor for trapped neutrinos
β(r, R) = 1 − (r, R) =
dNb (r)
= cos ψe (r, R).
dNp (r)
(33)
Note that we consider production and escaping rates at a given radius r, but the radius R of
the compact object enters the relation as it determines the escaping directional angle. The
coefficient β(r, R) determines local efficiency of the neutrino trapping, i.e., the ratio of the
trapped and produced neutrinos at any given radius r ∈ (rb(e) , R). Its profile is shown for
some representative values of R in figure 6. The local maxima of the function β(r, R) (with
In the case of non-isotropic emission of neutrinos, we should take e (e ) =
p() being the directional function of the emission (scattering) process.
1
10
e 2π
0
0
p() sin d dφ with
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Class. Quantum Grav. 26 (2009) 035003
1
β = b(c)
0.8
β max
R
M
0.6
2.3
0.4
2.5
0.2
0
2.8
2.9
2.95
0.2
0.4
0.6
0.8
1
r/R
Figure 6. The local trapping coefficient β(r, R) ≡ b for several values of R. This coefficient
represents the local effect of neutrino trapping on the cooling process.
R being fixed) are given by the condition ∂β/∂r = 0 which is satisfied at radius r = rc(i)
with rc(i) being determined by equation (15). This implies coincidence with the radius of the
stable circular null geodesic, as anticipated intuitively. In figure 6, the maxima are depicted
explicitly.
4.5. Neutrino production rates
Generally, the neutrino production is a very complex process depending on the detailed
structure of an extremely compact object. We can express the locally defined neutrino
production rate in the form
dN(r{A})
,
(34)
I (r{A}) =
dτ (r)
where dN is the number of interactions at radius r, τ is the proper time of the static observer at
the given r, {A} is the full set of quantities relevant for the production rate. We can write that
dN (r) = n(r)(r) dV (r),
(35)
where n(r), (r) and dV (r) are the number density of particles entering the neutrino
production processes, the neutrino production rate and the proper volume element at the
radius r, respectively. Both n(r) and (r) are given by detailed structure of the extremely
compact objects, dV (r) is given by the spacetime geometry.
Here, considering the uniform energy density internal Schwarzschild stars [7, 12] (for
requirements of more realistic model see, e.g. [7, 12]), we shall assume the local production
rate to be proportional to the energy density, i.e., we assume uniform production rate as
measured by the local static observers; of course, from the point of view of static observers at
infinity, the production rate will not be distributed uniformly. (According to [10], such a toy
model could be a reasonably good starting point for more realistic calculations.)
Therefore, in internal Schwarzschild spacetime we can write the local neutrino production
rate in the form
I (r) ∝ ρ = const
(36)
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Class. Quantum Grav. 26 (2009) 035003
or
dN
dN (r)
,
∝ ρ(r) ∝ const.
(37)
dτ
dτ
The local neutrino production rate related to the distant static observers is then given by
the relation including the time-delay factor
I=
dN
= I e(r) .
(38)
dt
Now, the number of neutrinos produced at a given radius in a proper volume dV per unit time
of a distant static observer is given by the relation
I=
dNp (r) = I (r) dV (r) = 4π I e(r)+(r) r 2 dr.
(39)
Integrating through the whole compact object (from 0 to R) and using (2), we arrive at the
global neutrino production rate in the form
R
3R(R − 2)1/2
−
1
r 2 dr.
(40)
Np (R) = 4π I
(R 3 − 2r 2 )1/2
0
In an analogical way, we can give the expressions for the global rates of escaping and trapping
of the produced neutrinos,
3R(R − 2)1/2
−
1
r 2 dr + Np (rb(e) ),
(R 3 − 2r 2 )1/2
rb(e)
R
3R(R − 2)1/2
Nb (R) = 4π I
cos ψe (r, R)
− 1 r 2 dr,
(R 3 − 2r 2 )1/2
rb(e)
Ne (R) = 4π I
R
(1 − cos ψe (r, R))
(41)
(42)
where rb(e) is the radius given by equation (17) and cos e (r, R) is determined by
equations (28).
5. Efficiency of neutrino trapping
In order to characterize the trapping of neutrinos in extremely compact stars, we introduce
some coefficients giving the efficiency of the trapping effect in connection to the total neutrino
luminosity and the cooling process in the period of the evolution of the star corresponding to
the geodetical motion of neutrinos.
5.1. Trapping coefficient of total neutrino luminosity
The influence of the trapping effect on the total neutrino luminosity of extremely compact stars
can appropriately be given by the coefficient BL relating the number of neutrinos produced
inside the whole compact star during unit time of distant observers and the number of those
produced neutrinos that will be captured by the extremely strong gravitational field of the star.
The luminosity trapping coefficient is therefore given by the relation
R
27α 2 (r,R) 1/2 2
r dr
rb(e) γ (r, R) 1 −
r2
BL (R) =
,
(43)
R
2
0 γ (r, R)r dr
and the complementary luminosity ‘escaping’ coefficient is determined by the simple formula
EL (R) = 1 − BL (R),
12
(44)
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Class. Quantum Grav. 26 (2009) 035003
where
3R(R − 2)1/2
− 1,
(45)
(R 3 − 2r 2 )1/2
and α(r, R) is given by (27).
We can, moreover, define other global characteristic coefficients. For the ‘internal’
neutrinos with motion restricted to the interior of the star, we introduce a coefficient
R
1/2 2
R3
2
r dr
rb(i) γ (r, R) 1 − α (r, R) (R−2)r 2
QL (R) =
(46)
R
2
0 γ (r, R)r dr
γ (R, r) =
and for the ‘external’ neutrinos, we can use a complementary coefficient
Next
XL =
= BL − QL .
Np
(47)
In the special case of the internal Schwarzschild spacetime, the integrals can be given in
terms of the elementary functions only, and we arrive at the final formula for the total neutrino
luminosity trapping coefficient in the form
Nt
BL (R) =
,
(48)
Np
where
Nt
1
3Y 2 (1 − 2k) 1/2
[Z 3/2 (Ye ) − Z 3/2 (Y1 )] − 1
=
Z (Y1 )
3
a
6(1 + k)
4(1 + k)2
3Y1 1 + k − 9kY12
3Y1 [Ye (1 + k) − 3kY1 ] 1/2
+
Z (Ye ) +
4(1 + k)2
4(1 + k)5/2
Y1 (2k − 1)
3kY1 − (1 + k)Ye
× arcsin 1/2 − arcsin 1/2 ,
1 + k − 9kY12
1 + k − 9kY12
1/2 Np
R
R2
3
R
R3
Y
−
R
−
=
,
arcsin
−
1
a3
4
a
a
a2
6a 3
(49)
(50)
Y1 and a are given by equations (3) and (4), while
Z(Ye ) = 1 − Ye2 − k(Ye − 3Y1 )2 = 0,
(51)
where Ye = Y (rb(e) ) defined by (17), and
27
.
(52)
4a 2
The total luminosity coefficient for the ‘internal’ neutrinos can be expressed in the form
Ni
QL (R) =
(53)
Np
Z(Y1 ) = 1 − Y12 (1 + 4k),
k=
with
Ni
1
3Y12 (1 − 2k̃) 1/2
3/2
3/2
Z̃ (Y1 )
=
(Y
)
−
Z̃
(Y
)]
−
[
Z̃
i
1
a3
6(1 + k̃)
4(1 + k̃)2
3Y1 1 + k̃ − 9k̃Y12
3Y1 [Yi (1 + k̃) − 3k̃Y1 ] 1/2
Z̃ (Yi ) +
+
4(1 + k̃)2
4(1 + k̃)5/2
Y1 (2k̃ − 1)
3k̃Y1 − (1 + k̃)Yi
× arcsin 1/2 − arcsin 1/2 ,
1 + k̃ − 9k̃Y12
1 + k̃ − 9k̃Y12
(54)
13
Z Stuchlı́k et al
Class. Quantum Grav. 26 (2009) 035003
Figure 7. Dependence of EL , BL , QL and XL on total radius R.
where
k̃ =
1
2(R − 2)
(55)
and
Z̃(Y1 ) = 1 − Y12 (1 + 4k̃),
Z̃(Yi ) = 1 − Yi2 − k̃(Yi − 3Y1 )2
(56)
with Yi = Y (rb(i) ) being given by equation (18). The results are illustrated for all the
coefficients BL (R), EL (R), QL (R) and XL (R) in figure 7.
5.2. Trapping coefficient of neutrino cooling process
The efficiency of the influence of neutrino trapping on the cooling process is most effectively
described by the local coefficient of trapping bc relating the captured and produced neutrinos
at a given radius of the star, i.e., we can define
bc ≡ β(r; R).
(57)
The local cooling coefficient is therefore given in figure 6 for some typical values of R. As
intuitively expected, the maximum of b (r; R) for a given R is located at the radius of the
stable null circular geodesic.
Further, the cooling process could be appropriately described in a complementary manner
by a global coefficient for trapping, restricted to the ‘active’ zone, where the trapping of
neutrinos occurs. The cooling global trapping coefficient is thus defined by the relation
R
27α 2 (r,R) 1/2
dr
rb(e) γ (r, R) 1 −
r2
Bc (R) ≡
.
(58)
R
2
rb(e) γ (r, R)r dr
The integrals are given in terms of the elementary functions in the form
Bc (R) =
Nt
,
Np(red)
(59)
where Nt is given by (49), while Np(red) reads
Np(red) = Np (R) − Np (rb(e) ),
14
(60)
Class. Quantum Grav. 26 (2009) 035003
Z Stuchlı́k et al
Figure 8. Behaviour of the coefficient Bc . It is explicitly shown that Bc ∼ 10% for R = 2.87M.
where Np (R) is given by equation (50) and
⎡
1/2 ⎤
2
r r
rb(e)
r3
Np (rb(e) )
3 ⎣
b(e)
b(e)
⎦ − b(e)
Y
−
=
−
arcsin
r
1
b(e)
a3
4
a
a
a2
6a 3
(61)
with rb(e) being given by equation (19).
In an analogical way, we can define the global cooling trapping coefficient for the ‘internal’
neutrinos by the relation
Ni
Qc (R) =
.
(62)
Np(red)
The behaviour of the global ‘cooling’ coefficient Bc is shown in figure 8.
6. Conclusions
Efficiency of the neutrino trapping in the extremely compact objects described by the internal
Schwarzschild spacetime grows with radius R descending from R = 3M down to the limiting
critical value of R = 9M/4. The local efficiency factor β(r, R) = bc gives insight into the
influence of the trapping effect on the neutrino cooling and has its maximum at the radius of
the stable null circular geodesic. Note that βmax (R = 2.9M) ∼ 0.1, and it grows strongly
with descending R, as βmax (R = 2.5M) ∼ 0.5 and βmax (R → 9M/4) → 1. Therefore, the
trapping can be locally important for even slightly extremely compact objects with R ∼ 2.9M.
The global efficiency factor of the trapping BL (R), determining efficiency of the trapping
effect on the total neutrino luminosity, grows almost linearly with R descending from the
limiting value of R = 3M. We can see that the value of the total luminosity trapping factor
BL ∼ 0.1 is reached for R ∼ 2.8M, and BL > 0.2 for R < 2.7M. We can conclude that
globally the trapping of neutrinos becomes relevant for moderately extremely compact objects.
Moreover, considering only the active zone of the trapping as applied in the definition of the
coefficient Bc , we obtain even higher values of the trapping factor related to the neutrino
cooling process. For example, we deduce from figure 8 that Bc > 0.1 for R < 2.87M.
It is important that the trapping of neutrinos is shown to be relevant even for the internal
Schwarzschild spacetimes with radius only moderately smaller than Rcrit = rph = 3GM/c2 .
15
Class. Quantum Grav. 26 (2009) 035003
Z Stuchlı́k et al
Therefore, it is worth continuing detailed studies of trapped neutrinos in realistic models of
extremely compact neutron stars or quark stars, when we usually expect radii R moderately
smaller than rph . The surface redshift for the extremely compact stars with R = 3M is
zmin = 0.732; the realistic models give maximum value of z ∼ 0.8 [12]. It is quite relevant
that for realistic models of Q-stars we are able to obtain R = 2.8M [24], i.e., compact
stars where the neutrino trapping could have efficiency leading to effects that are quite well
observable. (Of course, some models admit existence of objects with radii R close to the
critical value of 9GM/4c2 , see, e.g., [5].) Recently, we have extended the estimates of the
trapping process to the cases of the polytropic and adiabatic spherical objects and realistic
models of extremely compact Q-stars and quark stars. On the other hand, the neutron star
models based on the Skyrmion EOS closely related to the nuclei modeling, do not predict
the existence of the extremely compact stars, giving in the most efficient cases values of
R = 3.4M [16].
Because the effect of trapping of neutrinos is a cumulative one, we can expect its relevance
in realistic models of extremely compact objects. It is under study now, how the trapping will
influence the cooling process in some simple models of quark stars with a relatively simple
‘bag’ equation of state, and, especially, in models describing Q-stars predicting the moderately
extremely compact stars, and how the cooling, i.e., time evolution of temperature profile of
such a quark star or Q-star will be modified by cumulation of neutrinos in the zone of trapping.
Acknowledgments
The present work was supported by the Czech grants MSM 4781305903 and LC06014. Two
of authors (SH and ZS) would like to express their gratitude to the Theory Division of CERN
for perfect hospitality.
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17
The Astrophysical Journal, 714:748–757, 2010 May 1
C 2010.
doi:10.1088/0004-637X/714/1/748
The American Astronomical Society. All rights reserved. Printed in the U.S.A.
ON MASS CONSTRAINTS IMPLIED BY THE RELATIVISTIC PRECESSION MODEL OF TWIN-PEAK
QUASI-PERIODIC OSCILLATIONS IN CIRCINUS X-1
Gabriel Török, Pavel Bakala, Eva Šrámková, Zdeněk Stuchlı́k, and Martin Urbanec
Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava Bezručovo nám. 13, CZ-746 01 Opava, Czech Republic; [email protected],
[email protected], [email protected], [email protected], [email protected]
Received 2009 October 8; accepted 2010 March 15; published 2010 April 14
ABSTRACT
Boutloukos et al. discovered twin-peak quasi-periodic oscillations (QPOs) in 11 observations of the peculiar
Z-source Circinus X-1. Among several other conjunctions the authors briefly discussed the related estimate of
the compact object mass following from the geodesic relativistic precession model for kHz QPOs. Neglecting the
neutron star rotation they reported the inferred mass M0 = 2.2 ± 0.3 M . We present a more detailed analysis
of the estimate which involves the frame-dragging effects associated with rotating spacetimes. For a free mass
we find acceptable fits of the model to data for (any) small dimensionless compact object angular momentum
j = cJ /GM 2 . Moreover, quality of the fit tends to increase very gently with rising j. Good fits are reached when
M ∼ M0 [1 + 0.55(j + j 2 )]. It is therefore impossible to estimate the mass without independent knowledge of the
angular momentum and vice versa. Considering j up to 0.3 the range of the feasible values of mass extends up to 3 M .
We suggest that similar increase of estimated mass due to rotational effects can be relevant for several other sources.
Key words: stars: neutron – X-rays: binaries
Online-only material: color figure
2005; Zhang et al. 2007a, 2007b), oscillations that arise due
to comptonization of the disk–corona (Lee & Miller 1998) or
oscillations excited in toroidal disk (Rezzolla et al. 2003; Rezzolla 2004; Šrámková 2005; Schnittman & Rezzolla 2006; Blaes
et al. 2007; Šrámková et al. 2007; Straub & Šrámková 2009) are
considered as well. At last but not least, already the kinematics
of the orbital motion itself provides space for consideration of
“hot-spot-like” models identifying the observed variability with
orbital frequencies. For instance, recent works of Čadež et al.
(2008) and Kostić et al. (2009) deal with tidal disruption of
large accreted inhomogenities. Among the same class of (kinematic) models belongs also the often quoted “relativistic precession” (RP) kHz QPO model that is the focus of our attention
here.
The RP model has been proposed in a series of papers by
Stella & Vietri (1998, 1999, 2002). It explains the kHz QPOs as a
direct manifestation of modes of relativistic epicyclic motion of
blobs arising at various radii r in the inner parts of the accretion
disk. The model identifies the lower and upper kHz QPOs with
the periastron precession νp and Keplerian νK frequency,
1. INTRODUCTION
Quasi-periodic oscillations (QPOs) appear in variabilities
of several low-mass X-ray binaries (LMXBs) including those
which contain a neutron star (NS). A certain type of these
oscillations, the so-called kHz (or high-frequency) QPOs, often come in pairs with frequencies νL and νU typically in
the range ∼50–1300 Hz. This is of the same order as the range
of frequencies characteristic for orbital motion close to a compact object. Accordingly, most kHz QPO models involve orbital
motion in the inner regions of an accretion disk (see van der
Klis 2006; Lamb & Boutloukos 2007, for a recent review).
There is a large variety of QPO models related to NS sources
(in some but not all cases they are applied to black hole (BH)
sources too). Concrete models involve miscellaneous mechanisms of producing the observed rapid variability. One of the first
possibilities proposed represents the “beat frequency” model assuming interactions between the accretion disk and spinning
stellar surface (Alpar & Shaham 1985; Lamb et al. 1985).
Many other models primarily assume accretion disk oscillations. For instance, non-linear resonance scenarios suggested
by Abramowicz, Kluźniak and Collaborators (Abramowicz &
Kluźniak 2001; Abramowicz et al. 2003b, 2003c; Horák 2008;
Horák et al. 2009) are often debated. A set of the later models join the beat frequency idea, magnetic field influence, and
presence of the sonic point (Miller at al. 1998b; Psaltis et al.
1999; Lamb & Miller 2001). Some of the numerous versions of
non-linear oscillation models and the late beat frequency models rather fade into the same concept that commonly assumes
the NS spin to be important for excitation of the resonant effects
(Kluźniak et al. 2004; Pétri 2005a, 2005b, 2005c; Miller 2006;
Kluźniak 2008; Stuchlı́k et al. 2008; Mukhopadhyay 2009).
Resonance, influence of the spin, and magnetic field also play
a role in the ideas discussed by Titarchuk & Kent (2002) and
Titarchuk (2002). Other resonances are accommodated in models assuming deformed disks (Kato 2007, 2008, 2009a, 2009b;
Meheut & Tagger 2009). Further effects induced in the accreted
plasma by the NS magnetic field (Alphén wave model, Zhang
νL (r) = νp (r) = νK (r) − νr (r),
νU (r) = νK (r),
(1)
where νr is the radial epicyclic frequency of the Keplerian motion. (Note that, on a formal side, for Schwarzschild spacetime
where νK equals a vertical epicyclic frequency this identification
merges with a model assuming m = −1 radial and m = −2
vertical disk-oscillation modes).
In the past years, the RP model has been considered among
the candidates for explaining the twin-peak QPOs in several
LMXBs and related constraints on the sources have been
discussed (see, e.g., Karas 1999; Zhang et al. 2006; Belloni et al.
2007a; Lamb & Boutloukos 2007; Barret & Boutelier 2008a;
Yan et al. 2009). While some of the early works discuss these
constraints in terms of both NS mass and spin and include also
the NS oblateness (Morsink & Stella 1999; Stella et al. 1999),
most of the published implications for individual sources focus
on the NS mass and neglect its rotation.
748
No. 1, 2010
ON MASS OF CIRCINUS X-1
Two simultaneous kHz QPOs with centroid frequencies
of up to 225 (500) Hz have also recently been found by
Boutloukos et al. (2006a, 2006b) in 11 different epochs of
RXTE/Proportional Counter Array observations of the peculiar
Z-source Circinus X-1. Considering the RP model they reported
the implied NS mass to be M ∼ 2.2 M . The estimate was
obtained assuming the non-rotating Schwarzschild spacetime
and was based on fitting the observed correlation between the
upper QPO frequency and the frequency difference Δν = νU −νL .
In this paper, we improve the analysis of mass estimate carried
out by Boutloukos et al. In particular, we consider rotating
spacetimes that comprehend the effects of frame-dragging and
fit directly the correlation between the twin QPO frequencies.
We show that good fits can be reached for the mass–angular
momentum relation rather than for the preferred combination of
mass and spin.
2. DETERMINATION OF MASS
Spacetimes around rotating NSs can be approximated with
a high precision by the three-parametric Hartle–Thorne (HT)
solution of Einstein field equations (Hartle & Thorne 1968;
see Berti et al. 2005). The solution considers mass M, angular
momentum J, and quadrupole moment Q (supposed to reflect
the rotationally induced oblateness of the star). It is known that
in most situations modeled with the present NS equations of
state (EoS) the NS external geometry is very different from
the Kerr geometry (representing the “limit” of HT geometry
for q̃ ≡ QM/J 2 → 1). However, the situation changes
when the NS mass approaches maximum for a given EoS.
For high masses the quadrupole moment does not induce large
differences from the Kerr geometry since q̃ takes values close
to unity (Appendix A.1).
The previous application of the RP model mostly implied
rather large masses (e.g., Belloni et al. 2007a). These large
masses are only marginally allowed by standard EoS. Also
the mass inferred by Boutloukos et al. (2006a, 2006b) takes
values above 2 M . Motivated by this we use the limit of twoparametric Kerr geometry to estimate the influence of the spin
of the central star in Circinus X-1 (see Appendix A.1 where we
pay a more detailed attention to rationalization and discussion of
this choice allowing usage of simple and elegant Kerr formulae).
2.1. Frequency Relations
Assuming a compact object of mass MCGS = GM/c2 and
dimensionless angular momentum j = cJ /GM 2 described by
the Kerr geometry, the explicit formulae for angular velocities
related to Keplerian and radial frequencies are given by the
following relations (see Aliev & Galtsov 1981; Kato et al. 1998,
or Török & Stuchlı́k 2005):
6
8j
3j 2
ΩK = F (x 3/2 + j )−1 , ωr2 = Ω2K 1 − + 3/2 − 2 , (2)
x x
x
where F ≡ c3 /(2π GM) is the “relativistic factor” and x ≡
r/MCGS . Considering Equations (1) and (2), we can write for νL
and νU , both expressed in Hertz (see also Appendix A.1.2 where
we discuss a linear expansion of this formula),
2/3
8j νU
νU
νL = νU 1 − 1 +
−6
F − j νU
F − j νU
4/3 1/2 νU
.
(3)
− 3j 2
F − j νU
749
In the Schwarzschild geometry, where j = 0, Equation (3)
simplifies to
νL = νU
ν 2/3 1/2
U
1− 1−6
F
(4)
leading to the relation
Δν = νU 1 − 6 (2π GMνU )2/3/c2
(5)
that was used by Boutloukos et al. for the mass determination.
2.2. “Ambiguity” in M
There is a unique curve given by Equation (3) for each
different combination of M and j (see Appendix A.2 for the
proof). The frequencies νL and νU scale as 1/M and, as illustrated
in the left panel of Figure 1, they increase with growing j.
Naturally, one may ask an interesting question whether for
different values of M and j there exist some curves that are
similar to each other. We investigate and quantify this task in
Appendix A.2.
There we infer1 that for j up to ∼0.3 one gets a set of nearly
identical integral curves where M, j, and M0 roughly relate as
follows:
(6)
M = [1 + k(j + j 2 )]M0
with
k = 0.7.
This result is illustrated in the right panel of Figure 1. Clearly,
when using relation (6), any curve plotted for a rotating star
of a certain mass can be well approximated by those plotted
for a non-rotating star with a smaller mass, and vice versa.
Furthermore, we find that (see Appendix A.2) when the top parts
of the curves (corresponding to νU /νL ∼ 1–1.5) are considered
only, the best similarity is reached for
k = 0.75.
These parts of the curves are potentially relevant to most of
the atoll and high-frequency Z-sources data. On the other hand,
for the (bottom) parts of the curves that are potentially relevant
to low-frequency Z-sources including Circinus X-1, the best
similarity is achieved for
k = 0.65 (0.55, 0.5)
when
νU /νL ∼ 2 (3, 4).
Taking into account the above consideration we can expect that
the single-parameter best fit to the data by relation (4) roughly
determines a set of mass–angular-momentum combinations (6)
with similar χ 2 . The result of Boutloukos et al. then implies that
good fits to their data, displaying νU /νL ∼ 3, should be reached
for M ∼ 2.2 M [1 + 0.55(j + j 2 )]. In what follows we fit the
data and check this expectation.
1
We first consider a special set of apparently similar curves sharing the
terminal points. The set is (numerically) given by the particular choice of M,
for any j implying the same orbital frequency at the marginally stable circular
orbit. The curves then only slightly differ in their concavity that increases with
growing j.
750
TÖRÖK ET AL.
Vol. 714
Figure 1. Left: relation between the upper and lower QPO frequency following from the RP model for the mass M = 2.5 M . The consecutive curves differ in
j ∈ (0, 0.3) by 0.05. Right: relations predicted by the RP model vs. data of several NS sources. The curves are plotted for various combinations of M and j given by
Equation (6) with k = 0.7. The datapoints belong to Circinus X-1 (red/yellow color), 4U 1636-53 (purple color) and most of other Z- and atoll-sources (black color)
exhibiting large population of twin-peak QPOs.
Figure 2. Left: χ 2 dependence on the parameters M and j assuming Kerr solution of Einstein field equations. The continuous white curve indicates the mass–angular
momentum relation (7). The continuous thin green curve denotes j giving the best χ 2 for a fixed M. The dashed and thick green curve indicates the same dependence but
calculated using formulae (A2) and (A6) linear in j, respectively. The reasons restricting the calculation of the thick curve up to j = 0.4 are discussed in Section A.1.2.
Right: related profile of the best χ 2 for a fixed M. The arrow indicates increasing j.
2.3. Data Matching
In the right panel of Figure 1, we show the twin-peak frequencies measured in the several atoll and Z-sources2 together
with the observations of Circinus X-1. For the Circinus X-1
data we search for the best fit of the one-parametric relation (4).
Already from Figure 1, where these data are emphasized by the
red/yellow points, one may estimate that the best fit should arise
for M0 ∈ 2–2.5 M . Using the standard least squares method
.
.
(Press et al. 2007) we find the lowest χ 2 = 15 = 2 dof for
.
the mass M0 = 2.2 M which is consistent with the value reported by Boutloukos et al. The symmetrized error corresponding to the unit variation of χ 2 is ±0.3 M . The asymmetric
evaluation of M0 reads 2.2[+0.3; −0.1] M . The white curve in
Figure 2 indicates the mass–angular momentum relation implied
by Equation (6),
M = 2.2 M [1 + k(j + j 2 )],
k = 0.55.
(7)
For the exact fits in Kerr spacetime we calculate the relevant
frequency relations for the range of M ∈ 1–4 M and j ∈ 0–0.5.
These relations are compared to the data in order to calculate
2
After Barret et al. (2005a, 2005b), Boirin et al. (2000), Belloni et al.
(2007a), di Salvo et al. (2003), Homan et al. (2002), Jonker et al. (2002a,
2002b), Méndez & van der Klis (2000), Méndez et al. (2001), van Straaten
et al. (2000, 2002), Zhang et al. (1998).
the map of χ 2 . We use the step equivalent to a thousand points
in both parameters and obtain a two-dimensional map of 106
points. This color-coded map is included in the left panel of
Figure 2. One can see in the map that the acceptable χ 2 is
rather broadly distributed. The thin solid green curve indicates
j corresponding to the best χ 2 for a fixed M. It agrees well with
the expected relation (7) denoted by the white curve. The right
panel of Figure 2 then shows in detail the dependence of the
best χ 2 for the fixed M. It is clearly visible that the quality of
the fit tends to very gently, monotonically increase with rising j
and it is roughly χ 2 ∼ 15 for any considered j.
3. DISCUSSION AND CONCLUSIONS
The quality of the fit tends to very gently, monotonically
increase with rising j and it is roughly
χ 2 ∼ 2 dof ⇔ M ∼ 2.2[+0.3, −0.1] M × [1 + 0.55(j + j 2 )].
(8)
Therefore, one cannot estimate the mass without independent
knowledge of the spin or vice versa, and the above relation
provides the only related information implied by the geodesic
RP model.
To obtain relation (8), the exact Kerr solution of Einstein field
equations was considered. The choice of this two-parametric
spacetime description and related formulae (2) is justified by
No. 1, 2010
ON MASS OF CIRCINUS X-1
a large value of the expected mass M0 (see Appendix A.1 for
details). In Appendix A.1.2 we discuss the utilization of the
linearized frame-dragging description. Figure 2 includes the
mass–spin dependence giving best χ 2 resulting when the fitting
of datapoints is based on the associated formulae (A2) and (A6),
respectively. Considering that νL (νU ) formula (3) merge up to the
first order in j with the νL (νU ) relation (A6) linear in j one can
expect that the associated M(j ) relations obtained from fitting
of data should roughly coincide up to j ∼ 0.1–0.2. From the
figure we can find that there is not a big difference between the
resulting M(j ) relations even up to much higher j. The extended
coincidence can be clearly explained in terms of the kHz QPO
frequency ratio R ≡ νU /νL .3
Observations of Circinus X-1 result to R ∼ 2.5–4.5 while
usually it is R ∼ 1.2–3 (and most often R ∼ 1.5; Abramowicz
et al. 2003b; Török et al. 2008a; Yan et al. 2009). Assuming the
RP model along with any j ∈ (0, 1), the ratio R = 2 corresponds
with good accuracy to radii where the radial epicyclic frequency
reaches its maximum (Török et al. 2008c). Only values lower
than R ∼ 2 are then associated with the proximity of the
innermost stable circular orbit (ISCO) where the effects of
frame dragging come to be highly non-linear in both j and r.
Accordingly, for a given j, in the case when R ∼ 3, the individual
formulae restricted up to certain orders in j are already close to
their common linear expansion in j and differ much less than
for R ∼ 1.5 (see Appendix A.1).
The rarely large R and associated high radial distance (both
already remarked by Boutloukos et al. 2006a, 2006b, although in
a different context) in addition to large M0 warrant the relevance
of relation (8) for rather high values of the angular momentum.
Consequently, we can firmly conclude that the upper constrained
limit of the mass changes from the value 2.5 M to 3 M for
j = 0.3 and even to 3.5 M for j = 0.5. The value of M0
that is above 2 M and the increase of M with growing j for
corotating orbits elaborated here are challenging for the adopted
physical model. Further detailed investigation involving realistic
calculations of the NS structure can therefore be effective in
relation to EoS selection or even falsifying the RP model.
Finally, we note that the discussed trend of increase of
estimated mass arising due to rotational effects should be
relevant also for several other sources. Of course, many systems
display mostly low values of R. These low values of R are
in context of the RP model suggestive of proximity of ISCO.
Török (2009) and Zhang et al. (2009) pointed that under
the consideration of the RP model and j = 0, most of the
high-frequency sources data are associated with radii close to
r = 6.75M. Possible signature of ISCO in high frequency
sources data has been also reported in a series of works by
Barret et al. (2005a, 2005b, 2006) based on a sharp drop in
the frequency behavior of the kHz QPO quality factors (for
instance the atoll source 4U 1636-53 denoted by “blueberry”
points in Figure 1 clearly exhibits both low R and a drop of
QPO coherence, see Boutelier et al. 2010). Considering the
proximity of ISCO, high-order non-linearities in both j and r are
important and even small differences between the actual NS and
Kerr metric could have certain relevance. For this reason some
caution is needed when applying our results to high frequency
sources.
3
Orbital frequencies scale with 1/M. For any model considering νL and νU
given by their certain combination, the ratio R represents the measure of radial
position of the QPO excitation (provided that the NS spin and EoS are fixed).
751
This work has been supported by the Czech grants MSM
4781305903, LC 06014, and GAČR 202/09/0772. The authors
thank the anonymous referee for his objections and comments
which helped to greatly improve the paper. We also appreciate
useful discussions with Milan Šenkýř.
APPENDIX
APPROXIMATIONS, FORMULAE, AND EXPECTATIONS
A.1. Matching Influence of Neutron Star Spin
Rotation and the related frame-dragging effects strongly
influence the processes in the vicinity of compact objects and
there is a need of their reflection in the appropriate spacetime
description. External metric coefficients related to up-to-date
sophisticated models of rotating NS are taken out of the model
in two distinct ways. In the first way, the coefficients are obtained
“directly” from differential equations solved inside the numeric
NS model, while in the second (more usual) way, they are
inferred from the main parameters of the numeric model (mass,
angular momentum, etc.) through an approximative analytic
prescription. Several commonly used numerical codes related
to rotating NS have been developed and discussed (see, RNS,
Stergioulas & Morsink 1997; LORENE: Gourgoulhon et al.
2000; and also Nozawa et al. 1998; Stergioulas & Friedman
1995; Cook et al. 1994; Komatsu et al. 1989).
A.1.1. Analytical Approximations and High-mass Neutron Stars
In the context of a simplified analysis of NS frame-dragging
consequences, an approximation through two solutions of
Einstein field equations is usually recalled: Lense–Thirring metric also named linear-Hartle metric (Thirring & Lense 1918;
Hartle & Sharp 1967; Hartle 1967) and Kerr-black-hole metric
together with related formulae (Kerr 1963; Boyer & Lindquist
1967; Carter 1971; Bardeen et al. 1972). It is expected that
the Lense–Thirring metric fits well the most important changes
(compared to the static case) in the external spacetime structure
of a slowly rotating NS. This expectation is usually assumed
for j < 0.1–0.2.4 Due to asymptotical flatness constraints the
formulae related to Lense–Thirring, Kerr and some other solutions considered for rotating NS merge when truncated to the
first order in j. Accordingly, for astrophysical purposes there
is a widespread usage of the approximate terms derived with
the accuracy of the first order in j. While these approximations
are two-parametric, the more realistic approximations—for instance, those given by the HT metric (Hartle & Thorne 1968) and
related terms (Abramowicz et al. 2003a), relations of Shibata
& Sasaki (1998) or the solution of Pachón et al. (2006)—deal
with more parameters and provide less straightforward formulae. Perhaps also because of that they are not often considered
in discussions of concrete astrophysical compact objects.
Astrophysical applicability of the above analytical approaches has been extensively tested in the past 10 years. Criteria
based on the comparison of miscellaneous useful quantities have
The interval 0 < j < 2 × 10−1 is often assumed as one of the several
possible definitions of “slow rotation”. However, in relation to implications of
the frame-dragging effects, the effective size of this interval depends on the
radial coordinate. For x close or below xms the interval in j rather reduces to
low values. On the other hand for x above the radius of the maximum of νr the
interval can be extended to j higher than j = 0.2. The term slow rotation is
also frequently considered in another context. For instance, when using the Ht
metric in NS models the slow rotation is usually associated with the
applicability of the metric and consequently to spins up to ∼800 Hz for most
EOS and NS masses. For these reasons we do not use the term elsewhere in the
paper.
4
752
TÖRÖK ET AL.
Vol. 714
Figure 3. Left: parameter q̃ for several EoS. Shaded areas denote q̃ = 6 and q̃ = 3. Right: ISCO frequencies for the same EoS as used in the left panel. The curves are
calculated for mass 1.4 M and a relevant maximal allowed mass. The curves following from the exact Kerr solution and linear relation (A4) are displayed as well.
The quadratic relation denoted by the black-dashed curve is discussed later in Section A.2.1.
Figure 4. Frequencies of the perturbed circular geodesic motion. Relations for the Kerr metric given by Equation (A2) are denoted by blue and dashed-blue curves.
Relations (A2) are indicated by red curves, while relation (A5) is plotted using the green color. Dotted relations denote the Kerr- and linearized-vertical frequencies
that are not discussed here (see Morsink & Stella 1999; Stella et al. 1999). Inset emphasizes a difference between the radii fulfilling the ISCO condition νr = 0 for the
relations ((2), explicitly given by Equation (A1)), Equation (A2), and the ISCO-radius given by (A3).
Figure 5. Left: the RP model frequency relations given by Equation (3), blue curves; formulae (A2), red curves; relations (A6), green curves. Relation (A7) roughly
determining the applicability of Equation (A6) is denoted by the dashed black/yellow curve. Right: related differences Δν between the lower QPO frequency implied by
the Kerr formulae (3) and those following from Equation (A2) and (A6), respectively indicated by continuous respectively dashed curves. Different colors correspond
to different frequency ratio R. Shaded areas indicate Δν < 5% and Δν < 2%.
been considered for these tests (e.g., Miller et al. 1998a; Berti
et al. 2005). It has been found that spacetimes induced by most
up-to-date NS EoS without inclusion of magnetic field effects
are well approximated with the HT solution of the Einstein field
equations (see Berti et al. 2005, for details). The solution re-
flects three parameters: NS mass M, angular momentum J, and
quadrupole moment Q. Note that Kerr geometry represents the
“limit” of the HT geometry for q̃ ≡ Q/J 2 → 1. The parameter
q̃ then can be used to characterize the diversity between the NS
and Kerr metric.
No. 1, 2010
ON MASS OF CIRCINUS X-1
753
The left panel of Figure 3 displays a dependence of q̃ on
the NS mass. This illustrative figure was calculated following
Hartle (1967), Hartle & Thorne (1968), Chandrasekhar & Miller
(1974), and Miller (1977). The considered EoS are denoted as
follows (see Lattimer & Prakash (2001, 2007) for details):
[EoS1] SLy 4, Rikovska Stone et al. (2003).
[EoS2] APR, Akmal et al. (1998).
[EoS3] AU (WFF1), Wiringa et al. (1988); Stergioulas &
Friedman (1995).
[EoS4] UU (WFF2), Wiringa et al. (1988); Stergioulas &
Friedman (1995).
[EoS5] WS (WFF3), Wiringa et al. (1988); Stergioulas &
Friedman (1995).
Inspecting the left panel of Figure 3 we can see that for
EoS configurations resulting in low or medium mass of the
central star (M up to 0.8Mmax , i.e., roughly up to 1.4M–1.8 M )
depending on EOS, the implied HT geometry is rather different
from the Kerr geometry. More specifically, for a fixed central
density, q̃ strongly depends on the given EoS and substantially
differs from unity. On the contrary, for high mass configurations
q̃ approaches unity implying that the actual NS geometry is
close to Kerr geometry. One can expect that in such cases
formulae related to the Kerr geometry should provide better
approximation than for low values of M. Next, focusing on
high-mass NS, we briefly elaborate some points connected to
the applicability of the Kerr formulae and related linearized
terms.
Note that the root of the expression for ωr2 from Equation (A2)
is of higher order in j so that the exact radius where ωr vanishes
agrees with the solution (A3) only in the first order of j.
The related ISCO frequency can be evaluated as (Kluźniak &
Wagoner 1985; Kluźniak et al. 1990)
A.1.2. Kerr and Linearized Kerr Formulae: Comparison, Utilization,
and Restrictions
When the expression for the radial epicyclic frequency given
by Equation (2) or (A2) is fully linearized in j, it leads to
The radial epicyclic frequency goes to zero on a particular,
so-called marginally stable circular orbit xms (e.g., Bardeen et al.
1972). In Kerr spacetimes it is given by the relation (Bardeen
et al. 1972)
xms = 3 + Z 2 − (3 − Z 1 )(3 + Z 1 + 2Z 2 ),
(A1)
where
Z 1 = 1 + (1 − j 2 )1/3 [(1 + j )1/3 + (1 − j )1/3 ],
Z 2 = 3j 2 + Z 21 .
Below xms there is no circular geodesic motion stable with
respect to radial perturbations. The orbit is often named ISCO
and determines the inner edge of a thin accretion disk. The
corresponding ISCO orbital frequency νK (xms ) represents the
highest possible orbital frequency of the thin disk and the related
“spiraling” inhomogenities (Kluźniak et al. 1990). Dependence
of ISCO frequency on j following from Equation (A1) is shown
in the right panel of Figure 3.
Assuming the description of geodesic motion accurate in the
first order of j, using Taylor expansion around j = 0, one may
rewrite the explicit terms in Equation (2) as
1
6
8j
j
2
2
ΩK = F
− 3 , ωr = ΩK 1 − + 3/2 .
x 3/2
x
x x
(A2)
Consequently, linearized formula for the ISCO radius can be
expressed as
2
xms = 6 − 4
j.
(A3)
3
νK (xms ) = (M /M) × (1 + 0.749j ) × νK (xms , M = M , j = 0)
.
= (M /M) × (1 + 0.749j ) × 2197 Hz.
(A4)
This frequently considered relation is included in the right panel
of Figure 3.
In the right panel of Figure 3 we integrate the ISCO frequencies plotted for several EoS (the same as in the left panel).
We choose two groups of models—one calculated for the set
of five different EoS and “canonic” mass 1.4 M , the other
one for the same set of EoS but considering a maximal mass
allowed by each individual EoS. This choice allows for the illustration of medium and high mass behavior of ISCO relations
and comparison of their simple approximations. Clearly, for
medium mass configurations Equation (A4) provides better approximation than using the Kerr-spacetime formulae (see also
Miller et al. 1998a). On the other hand, when the high mass
configurations are considered, the Kerr solution provides better approximation than Equation (A4). Moreover, its accuracy is
higher than the accuracy of both approximations for middle mass
configurations.
A.1.3. Geodesic Frequencies and RP Model
ωr =
(x − 6)x 3/2 + 3j (x + 2)
.
(x − 6)x 7
(A5)
Relation (A5) provides a good approximation except for the
vicinity of xms (j ) as it diverges at x = 6. Note that this
divergence arises only for corotating but not counterrotating
orbits (which we however do not discuss in this paper). For any
positive j < 0.5 the fully linearized frequency (Equation (A5))
does not differ from ωr given by Equation (A2) for more than
about 5% when x 6 + 4j . The left panel of Figure 4 compares
the frequencies of geodesic motion associated directly with Kerr
metric to formulae (A2) and (A6), respectively.
Assuming linearized Keplerian frequency given by
Equation (A2) and the radial epicyclic frequency
(Equation (A5)), we can write for the RP model the relation
between νL and νU as
ν 2/3
√
2j νU (α − 2)
U
νL = νU 1 − 1 − 6α + √
,
, α=
F
F 1 − 6α
(A6)
which equals the first-order expansion of Kerr spacetime
Equation (3) and also to the first-order expansion of the same
relation if it would be derived for Lense–Thirring or HT metric. Similarly to relation (A5), relation (A6) loses its physical
meaning for frequencies close to νK (ISCO) since it reaches a
maximum at frequencies that can be expressed with a small
inaccuracy as
M
νU
(Hz).
(A7)
νL =
12 − νU /200 M
The left panel of Figure 5 compares the frequency relations (A6)
to relations (3) and those following from formulae (A2). It is
754
TÖRÖK ET AL.
useful to discuss their differences in terms of the frequency
ratio R = νU /νL . For a fixed j the frequencies νL and νU scale
with 1/M. The ratio R then represents a “measure” of the radial
position of the QPO excitation. It always reaches R = 1 at
ISCO where the non-linear j terms are important and R = 0
at infinity where the spacetime is flat. Note that R = 2 almost
exactly corresponds to the maximum of νr for any j (Török et al.
2008c).
The right panel of Figure 5 quantifies differences between the
QPO frequency implied by the Kerr formulae (3), relations (A2),
and relation (A6). We can see that differences between the
Kerr relations (3) and those implied by formulae (A2) become
small when R 2 (Δν 5% for j 0.5). For R ∼ 3
and higher, relations (3) and those implied by Equation (A2)
are almost equivalent nearly merging to their common linear
expansion (A5). Note that taking into account relation (A7) the
linear expansion (Equation (A6)) provides reasonable physical
approximation for spins and frequency ratios roughly related as
j 0.3(R − 1).
(A8)
A.1.4. Applications
Several values of NS mass previously reported to be required
by the RP model, including the estimate of Boutloukos et al.,
belong to the upper part of the interval allowed by standard
EoS. We can therefore expect low q̃ and take advantage of the
exact Kerr solution for most of the practical calculations needed
through the paper. Unlike formulae truncated to certain order, all
the formulae derived from the exact Kerr solution are from the
mathematical point of view fully self-consistent for any j. This
allows us to present the content of Appendix A.2 in a compact
and demonstrable form.
In Section 3, we finally compare the results of QPO frequency
relation fits for Circinus X-1 using the Kerr solution and those
done assuming Equations (A2) and (A6), respectively. From the
previous discussion it can be expected that for Circinus X-1,
due to its exceptionally high R, the fits obtained with the Kerr
formulae (3) and “linear” formulae (A2) should nearly merge
with the fits obtained assuming the common linear expansion
(Equation (A6)). Note also that, on a technical side, the linear
expansion can be used up to j ∼ 0.3–0.4 since the highest R in
the Circinus X-1 data is R ∼ 2–2.5 (Equation (A8)).
A.2. Uniqueness of Predicted Curves and “Ambiguity” in M
The radial epicyclic frequency vanishes at xms . In the RP
model it is then νUmax = νLmax = νK (xms ). Obviously, if there are
two different combinations of M and j which, based on the RP
model, imply the same curve νU (νL ), such combinations must
also imply the same ISCO frequency.
In the left panel of Figure 6 we show a set of curves
constructed as follows. We choose M0∗ = 2.5 M and j ∈
(0, 0.5) and for each different j we numerically find M such
that the corresponding ISCO frequency is equal to those for M0∗
and j = 0. Then we plot the νU (νL ) curve for each combination
of M and j. We can see that except for the terminal points the
curves split. The frequencies in the figure can be rescaled for any
“Schwarzschild” mass M0 as M0∗ /M0 . Thus, the scatter between
the curves provides the proof that one cannot obtain the same
curve for two different combinations of M and j.
On the other hand, the discussed scatter is apparently small
and the curves differ only slightly in the concavity that grows
with increasing j. This has an important consequence. The
curves are very similar with respect to the typical inaccuracy
Vol. 714
of the measured NS twin-peak data and there arises a possible
mass–angular momentum ambiguity in the process of fitting the
datapoints. Next, we derive a simple relation approximating this
ambiguity.
A.2.1. Formulae for ISCO Frequency
The ambiguity recognized in the previous section is implicitly
given by the dependence of the ISCO frequency on the NS
angular momentum which for the Kerr metric follows from
relations (2) and (A1). In principle we can try to describe the
ambiguity starting with these exact relations. The other option
is to assume an approximative formula for the ISCO frequency.
One can expect that this formula should be at least of the second
order in j if consideration of spin up to j = 0.5 is required. We
check an arbitrarily simple form
νK (xms ) = (M /M) × [1 + k(j + j 2 )] × 2197 Hz. (A9)
The right panel of Figure 6 indicates the square of difference
between the exact ISCO frequency in Kerr spacetimes following
from Equation (A1) and the value following from Equation (A9).
Inspecting the figure we can find that the particular choice of
k = 0.75 provides a very good approximation.
Figure 7 then directly compares the exact relation and relation (A9) with k = 0.75. For comparison, the first-order
Taylor expansion formula (A4) is indicated. Clearly, using
Equation (A9) one may well approximate the Kerr-ISCO frequency up to j ∼ 0.4 and describe the discussed ambiguity in
terms of Schwarzschild mass M0 as
M ∼ [1 + k(j + j 2 )]M0 ,
(A10)
where k = 0.75. In further discussion we therefore assume this
formula.
A.2.2. Comparison Between Curves
The curves given by Equation (A10) with k = 0.75 are
illustrated in the left panel of Figure 8. Here we quantify their
(apparent) conformity and investigate its dependence on k. It
is natural to consider the integrated area S between the curve
for M0 , j = 0 and the others as the relevant measure. The
right panel of Figure 8 shows this area as the function of k in
Equation (A10) for several values of j. The same panel also
indicates the values related to the set of curves for mass found
numerically from the exact Equations (A1), i.e., curves in the left
panel of Figure 6. We can see that values of S for k = 0.75 are
comparable to those related to Figure 6. Moreover, for a slightly
different choice of k = 0.7, all the values are smaller. The
ambiguity in mass with relation (A10) is therefore best described
for k ∼ 0.7 when the data uniformly cover the whole predicted
curves.
The available data are restricted to certain frequency ranges
and often exhibit clustering around some frequency ratios νU /νL
(see Abramowicz et al. 2003b; Belloni et al. 2007b; Török et al.
2008c, 2008a, 2008b; Barret & Boutelier 2008b; Török 2009;
Boutelier et al. 2010; Bhattacharyya 2009). It is then useful
to separately examine the mass ambiguity for related segments
of the curves. Such investigation is straightforward for small
segments. Let us focus on a single point [νL , νU ] representing
a certain frequency ratio for a non-rotating star (j = 0) of
mass M0 . Assuming relation (A10) one may easily calculate
the value of k which rescales the mass to M = M0 for a fixed
No. 1, 2010
ON MASS OF CIRCINUS X-1
755
Figure 6. Left: set of curves plotted for various combinations of M and j giving identical ISCO frequency. Right: the square of difference between the exact ISCO
frequency and the frequency given by Equation (A9).
Figure 7. Left: ISCO frequency calculated from Equation (A9) vs. exact relation implied by the Kerr solution (dashed vs. thick curve). The linear relation (A4) is
shown as well for comparison (dotted curve). Right: the related relative difference from ISCO frequency in Kerr spacetime.
Figure 8. Left: the set of curves plotted for combinations of M and j given by Equation (A10) with k = 0.75. Right: the integrated area S related to Equation (A10).
Different values of j are color-coded. The same color code is relevant for horizontal lines. These lines denote the values of S arising for the set of curves numerically
found from Equation (A1) and plotted in the left panel of Figure 6. The two red vertical lines denote the case of k = 0.75 (curves νU (νL ) shown in the left panel of this
figure) respectively k = 0.7 (see the text for explanation).
Table 1
The Coefficient k Representing Mass–Angular Momentum Ambiguity (A10)
Segment
νL /νU
νL /νU
νL /νU
νL /νU
νL /νU
νL /νU
∼ 1.5
∼2
∼3
∼4
∼5
∼6
Whole curve
k in M ∼ [1 + k(j + j 2 )]M0
l (%)
Distance from ISCO × M /M (km)
0.75
0.65
0.55
0.50
0.45
0.40
25
50
70
80
83
85
1
3
7
12
16
20
0.7
756
TÖRÖK ET AL.
Figure 9. Values of k approximating the M – j ambiguity for the individual
segments. The upper axes indicate the length of the curve νU (νL ) integrated from
the ISCO point to the relevant frequency ratio.
(A color version of this figure is available in the online journal.)
non-zero j in order to get exactly the same point [νL , νU ]. We
applied this calculation for νU /νL ∈ (1, 10) and j ∈ (0, 0.5).
The output is shown in Figure 9. From the figure, it is possible
to find k that should best describe the ambiguity for a given
frequency ratio (and thus for a small segment of data close
to the ratio). It also indicates the length of the curve νU (νL )
integrated from the terminal (i.e., ISCO) point to the relevant
frequency ratio (assuming j = 0). This length is given in terms
of the percentage share l on the total length L of the curve νU (νL ),
whereas the absolute numbers scale with 1/M.
Apparently, the segments with νU /νL ∈ (1, 2) cover about
50% of the total length L while k only slightly differs from the
value of 0.7. We recall that this top part corresponds to most of
the atoll and Z-sources data. For the segments of curves related
to the sources exhibiting high frequency ratios such as Circinus
X-1, there is an increasing deviation from the 0.7 value and the
coefficient reaches k ∼ 0.6–0.5. More detailed information is
listed up to νU /νL = 5 in Table 1 providing the summary of this
section.
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L171
ACTA ASTRONOMICA
Vol. 0 (0) pp. 0–0
Observational Tests of Neutron Star Relativistic Mean Field Equations
of State
M. U r b a n e c 1 , E. B ě t á k 1,2 , and Z. S t u c h l í k 1
1 Institute
2
of Physics, Silesian University, 74601 Opava, Czech Republic
Institute of Physics, Slovak Academy of Sciences, 84511 Bratislava, Slovakia
ABSTRACT
Set of neutron star observational results is used to test some selected equations of state of dense
nuclear matter. The first observational result comes from the mass–baryon number relation for pulsar
B of the double pulsar system J 0737–3039. The second one is based on the mass–radius relation
coming from observation of the thermal radiation of the neutron star RX J 1856.35–3754. The third
one follows the population analysis of isolated neutron star thermal radiation sources. The last one
is the test of maximum mass. The equation of state of asymmetric nuclear matter is given by the
parameterized form of the relativistic Brueckner-Hartree-Fock mean field, and we test selected parameterizations that represent fits of full relativistic mean field calculation. We show that only one
of them is capable to pass the observational tests. This equation of state represents the first equation
of state that is able to explain all the mentioned observational tests, especially the very accurate test
given by the double pulsar even if no mass loss is assumed.
Key words: Stars: neutron, Equations of state, Dense matter
1. Introduction
Neutron stars are compact objects that play important role in different areas
of modern physics. Here we concentrate our attention on the possibility that phenomena related to neutron stars can be used as tests of equation of state (EoS) of
asymmetric nuclear matter. The tests used in this paper represent a subset of tests
used previously by Klähn et al. (2006). We focus our attention on tests that come
from astronomical observations. However we have not applied the very promising
test coming from the observations of quasiperiodic oscillations (QPOs), since the
theory and data interpretation is still in progress (see e.g. Török et al. (2008a,b,c,
2010) van der Klis (2004)). The QPO test applied on the 4U 1636–536 object in
Klähn et al. (2006) represents the maximum mass test in the present paper, and
another test following the QPO phenomena observed in the 4U 0614+09 object do
not provide a strong test and all the EoS tested in this paper pass it.
A wide spectrum of different equations of state of nuclear matter and their
applications to astrophysical problems has been reported in literature (see, e.g.,
Vol. 0
1
Haensel, Zdunik and Douchin 2002, Rikovska Stone et al. 2003, Weber, Negreiros
and Rosenfeld 2007, Lattimer and Prakash 2007, Burgio 2008). Some of the EoS
collections (even though not all of them are up-to-date already) give an amazingly
rich general overview of the state-of-the-art, whereas the others emphasize some
specific aims. All these EoS yield (nearly) the same properties close to the standard
nuclear density (ρN ≈ 0.16 nucleon/fm 3 ≈ 2.7 × 1014 g/cm 3 ), but when one is
far off this value, s/he has to rely more on underlying principles than on possible
experimental verification of predicted physical observables.
Here we concentrate our attention on relativistic asymmetric nuclear matter
where the EoS stem from an assumed form of the interaction Lagrangian. The
calculations use the relativistic mean-field theory with allowance for an isospin degree of freedom (Kubis and Kutchera 1997, Müler and Serot 1996). We employed
the Dirac-Brueckner-Hartree-Fock mean-field approach in its parameterized form
suggested in Gmuca (1991) which reproduces the nuclear matter results of Huber,
Weber and Weigel (1995). That has been used to calculate high-density behavior
of asymmetric nuclear matter with varying neutron-to-proton ratio (Gmuca 1992).
The proton fraction has been determined from the condition of β-equilibrium and
charge neutrality, and it is density-dependent. We have extended our calculations
for densities up to 4 × ρN and if there was an astrophysical motivation even higher.
The EoS is used to model the static, spherically symmetric neutron star in the
framework of general relativity. The equation of hydrostatic equilibrium is solved
for different central parameters (pressure, energy density, baryon number density).
The radius of the neutron star model is then given by the condition of vanishing
pressure. The resulting properties of the neutron star model are then compared with
observational data. From the test ensemble presented by Klähn et al. (2006) we
choose four astrophysical observations to test our selected parameterizations that
have been found to be a good description of nuclear matter at subnuclear densities
for pure neutron matter and up to 2 × ρN for symmetric nuclear matter (Kotulič
Bunta and Gmuca 2003).
The maximum mass test is the standard way to test the EoS of asymmetric nuclear matter (see e.g. Haensel, Potekhin and Yakovlev 2007, Lattimer and
Prakash 2007, Klähn et al. 2006). The usual value to constrain the maximal mass
of neutron star comes from observations of double pulsar PSR 0751+1807 giving
M = (2.1 ± 0.2) M¯ , with M¯ being the solar mass. This value was, however,
lowered to M = (1.26 ± 0.14) M¯ (Nice, Stairs and Kasian 2008) and could not be
used as maximum mass test anymore. Another value that could serve as maximum
mass test comes from the observation of QPOs. The mass is constrained on the
basis that the observed frequency corresponds to the frequency at innermost stable
circular orbit (Barret, Olive and Miller 2005, Belloni, Mendez and Homman 2007,
van der Klis 2004).
Popov et al. (2006) used the population synthesis of the isolated neutron star
sources of thermal radiation and concluded that the neutron stars with mass M <
2
A. A.
1.5 M¯ could not cool via direct URCA reactions. This conclusion follow from the
fact that all observed sources of thermal radiation have masses bellow the quoted
value. This could be explained by the fact that more massive objects cool via
the direct URCA reactions which represents the fast cooling scenario and thus the
thermal radiation could not be detected. These arguments were used by Klähn et
al. (2006) to build a strong and a weak test on EoS.
A very accurate test of EoS was developed by Podsiadlowski et al. (2005). They
based it on the model of double pulsar system J 0737–3039 formation. The model
predicts the pulsar B of this system to be born via the electron capture supernova
what suggests extremely low mass loss and thus the number of particles conserved
during the progenitor collapse to neutron star. This put limits on the mass–baryon
number relation. Instead of the baryon number that represents the total number of
baryons contained in the neutron star, the baryon mass could be used equally.
The thermal radiation coming from the neutron star source RX J1856.35–3754
could be used to put limits on the mass–radius relation of the neutron star model.
Trümper et al. (2004) used two different models to explain the spectral feature for
this specific source and found its apparent radius that represents the radius of the
neutron star as seen by a distant observer. The analyses of data to obtain the isolated
neutron star radius strongly depend on the radiation spectrum emitted by the object
and the estimated radius is proportional to the distance from Earth to the source.
The distances obtained for RX J1856.5-3754 range from D = 61+9
−8 pc (Walter and
Matthews 1997) to D = 161+18
pc
(van
Kerkwijk
and
Kaplan
2007).
The derived
−14
apparent radius R∞ is given by the model of the atmosphere. The original model
by Pons et al. (2002) resulted in R∞ /D = 0.13 km.pc −1 . Trümper et al. (2004)
presented new models of atmosphere leading to the estimates of R∞ = 16.5 km
for the two component model of spectra and R∞ = 16.8 km assuming continuous
temperature distribution model. If the distance derived by van Kerkwijk and Kaplan
(2007) and the original model of Pons et al. (2002) are used together, they lead to
unexpectedly high estimate R∞ = 20.9 km. Recently Steiner, Lattimer and Brown
(2010) presented results based on new analysis of data giving the distance 119 ±
5 pc and the original model for atmosphere (Pons et al. 2002) then implies R∞ =
15.47 km. We decided to use the three values R∞ = 15.5, 16.8, 20.9 km to put
limits on neutron star equation of state.
Another promising way to constrain the equation of state are the moment of
inertia measurements (see e.g. Lattimer and Prakash 2007 and references therein).
Two ways have been proposed quite recently. One for the Crab pulsar (Bejger
and Haensel 2002,2003) following observations of the pulsar-nebula system, and
the other for the pulsar A of the double pulsar system J0737–3039 (Bejger, Bulik
and Haensel 2005) based on the measurements of the second order post Newtonian
parameters of the binary system. Even thought both ways could provide strong
limits on the equations of state in principle, they need more accurate observational
inputs. We need better estimates of the mass of Crab nebula in the first case and
Vol. 0
3
very accurate measurements of orbital parameters are necessary to calculate the
moment of inertia in the second case. For these reasons we do not include these
tests to our calculations. The measurements of moment of inertia of the neutron
star together with its mass put limits on the radius of the neutron star that is crucial
for the cooling scenarios (see e.g. Lattimer and Prakash 2007, Stuchlík et al. 2009).
The paper is organized as follows. In section 2 we present our EoS and details
of the neutron star matter description. Section 3 briefly summarizes the model of
static spherically symmetric neutron star. We present our results and compare them
to observations in section 4. The paper is closed by conclusions in section 5.
2. Equation of state of neutron star matter
2.1.
Asymmetric nuclear matter in relativistic mean-field approach
We follow the Dirac-Brueckner-Hartree-Fock (DBHF) mean field (see Weber
1999, Walecka 2004, de Jong and Lenske 1998, Krastev and Sammarruca 2006
for underlying theories), which easily allows to consider different neutron-proton
composition of the neutron star matter, and also the inclusion of non-nucleonic
degrees of freedom.
The full mean-field DBHF calculations of nuclear matter (Huber, Weber and
Weigel 1995, Lee et al. 1998, Li, Machleidt, and Brockmann 1992) have been parameterized by Kotulič Bunta and Gmuce (2003), and we employ their parameterization with one-boson-exchange (OBE) potential A of Brockmann and Machleidt
Li, Machleidt, and Brockmann (1992). We refer to the paper of Kotulič Bunta
and Gmuca (2003) for the explicite set of values of the corresponding parameters.
The model Lagrangian density includes the nucleon field ψ, isoscalar scalar meson field σ, isoscalar vector meson field ω, isovector vector meson field ρ ,and
isovector scalar meson field δ, including also the vector cross-interaction. The
Lagrangian density in the form used by Kotulič Bunta and Gmuca (2003) reads
L (ψ, σ, ω, ρ, δ) = ψ̄[γµ (i∂µ − gω ωµ ) − (mN − gσ σ)]ψ
1
1
1
+ (∂µ σ∂µ σ − mσ 2 σ2 ) − ωµν ωµν + mω 2 ωµ ωµ
2
4
2
1
1
1
3
4
− bσ mN (gσ σ) − cσ (gσ σ) + cω (gω 2 ωµ ωµ )2
3
4
4
1
1
1
+ (∂µ δ∂µ δ − mδ 2 δ2 ) + mρ 2 ρµ ρµ − ρµν ρµν
2
2
4
1
+ ΛV (gρ 2 ρµ ρµ )(gω 2 ωµ ωµ ) − gρ ρµ ψ̄γµ τψ + gδ δψ̄τψ,
2
(1)
where the antisymmetric tensors are
ωµν ≡ ∂ν ωµ − ∂µ ων ,
ρµν ≡ ∂ν ρµ − ∂µ ρν ;
(2)
4
A. A.
the strength of the interactions of isoscalar and isovector mesons with nucleons is
given by (dimensionless) coupling constants g’s and the self-coupling constants
(also dimensionless) are bσ (cubic), cσ (quartic scalar) and cω (quartic vector).
The second and the fourth lines represent non-interacting Hamiltonian for all mesons, ΛV is the cross-coupling constant of the interaction between ω and ρ mesons.
Furthermore, mN is the nucleon mass, ∂µ ≡ ∂x∂µ and γ’s are the Dirac matrices
(Kotulič Bunta and Gmuca 2003, Serot and Walecka 1986, Weber 1999).
We choose here three following parameterizations, which were shown to yield
the best fits to the well-known properties of nuclear matter
H HA in Kotulič Bunta and Gmuca (2003) represents the best RMF fit to results
obtained by Huber, Weber and Weigel 1995.
L LA in Kotulič Bunta and Gmuca (2003) represents the best RMF fit to results
obtained by Lee et al. 1998, but does not include the δ mesons to nucleons
coupling.
M MA in Kotulič Bunta and Gmuca (2003) represents the best RMF fit to results
obtained by Li, Machleidt, and Brockmann 1992, but does not include the δ
mesons to nucleons coupling.
The EoS of Kotulič Bunta and Gmuca which have been found to be a good
description of asymmetric nuclear matter, are easily expressed up to about 4 × ρN
(parameterization H ) or even higher (parameterizations L and M ).
2.2.
β-equilibrium
The total energy density of n-p-e-µ matter is given by
E = EB (nB , xp ) + Ee (ne ) + Eµ (nµ ),
(3)
where EB (nB , xp ) is the binding energy density of asymmetric nuclear matter, ni is
the number density of different particles (i = n, p, e, µ), nB = np + nn is the baryon
number density and xp = np /nB is the proton fraction. The leptonic contributions
El (nl ) (l = e, µ) to the total energy density are given by
2
El (nl ) = 3
h
pF(l)
Z ¡
¢1/2
m2l c4 + p2 c2
4πp2 dp,
(4)
0
where pF(l) is the Fermi momentum of l-th kind of particle.
The matter in neutron stars is in β-equilibrium, i.e. in equilibrium with respect
to n ↔ p + e− ↔ p + µ. The (anti)neutrinos contribution could be neglected, because the matter is assumed to be cold enough that they can freely escape. The
equilibrium is given by equality of chemical potentials µn = µp + µe = µp + µµ ,
where the chemical potential of each kind of particle is given by µi = ∂E /∂ni .
Vol. 0
5
The chemical potentials of electrons and muons are simply µl =
while the chemical potentials of nucleons are
µ(p,n) =
∂
(EB ) .
∂n(p, n)
q
m2l c4 + p2F(e) c2 ,
(5)
The binding energy density of asymmetric nuclear matter could be expressed
in terms of proton fraction xp (Danielewicz and Lee 2009)
EB (nB , xp ) = ESNM (nB ) + (1 − 2xp )2 S(nB ),
(6)
where ESNM is the energy density of symmetric nuclear matter (xp = 0.5) and
S(nB ) is the symmetry energy density, that corresponds to the difference of binding
energy density between pure nuclear matter and symmetric nuclear matter
The symmetry energy S(nB ) is the factor corresponding to the second order
term in expansion of binding energy density in terms of asymmetry parameter δ =
(nn − np )/(nn + np ) = 1 − 2xp and reads
¯
1 ∂2 EB (nB , δ) ¯¯
S(nB ) =
(7)
¯ .
2
∂δ2
δ=0
From equation (6) one can see that symmetry energy is the difference of binding
energy per particle between pure nuclear matter and symmetric nuclear matter.
S(nB ) = EB (nB , xp = 0) − EB (nB , xp = 0.5).
(8)
The condition of β-equilibrium then reads
µe = µµ = µn − µp = 4
¢
S(nB ) ¡
1 − 2xp .
nB
(9)
and it is solved together with condition of charge neutrality (np = ne + nµ ) to obtain
the proton fraction of neutron star matter. The binding energy per baryon in dependence on the baryon number density is illustrated in Figure 1. The proton fraction
of matter at the beta-equilibrium is given, for the chosen three EOS parameterizations, as a function of the baryon number density depicted in Figure 2.
2.3.
EoS for low densities
The nuclear EoS have been the dominant input for the calculations in the highdensity region, namely ρ ≥ 1014 g/cm3 . For lower densities, the EoS used are the
following:
• Feynman-Metropolis-Teller EoS for 7.9 g/cm3 ≤ ρ ≤ 104 g/cm3 where matter consists of e− and 56
26 Fe, Feynman, Metropolis and Teller (1949);
• Baym-Pethick-Sutherland EoS for 104 g/cm3 ≤ ρ ≤ 4.3 × 1011 g/cm3 with
Coulomb lattice energy corrections Baym, Pethick, and Sutherland (1971);
6
A. A.
300
H
L
250 M
H
L
M
H
L
M
[MeV]
200
150
100
50
0
-50
0
0.3
0.6
nB [fm-3]
0.9
0.3
0.6
nB [fm-3]
0.9
0.3
0.6
nB [fm-3]
0.9
Figure 1: Binding energy per particle of different types of nuclear matter for used
parameterizations. Left Matter at β–equilibrium, Middle symmetric nuclear matter,
and Right pure neutron matter
• Baym-Bethe-Pethick EoS for 4.3 g/cm3 × 1011 ≤ ρ ≤ 1014 g/cm3 : here,
e − , neutrons and equilibrated nuclei calculated using the compressible liquid
drop model Baym, Bethe, and Pethick (1971).
3. Neutron star models
We consider static spherically symmetric models of neutron stars. The interior spacetime is described by the internal Schwarzschild metric (see, e.g., Misner,
Thorne and Wheeler 1973, Haensel, Potekhin and Yakovlev 2007) that can be written in geometrical units (c = G = 1) as
ds2 = −e2ν dt 2 + e2λ dr2 + r2 (dθ2 + sin2 θdφ2 ),
(10)
where the radial component of metric can be expressed as a function of energy
density ρ
Z r
r
e2λ =
, m(r) = 4π ρr12 dr1 .
(11)
r − 2m(r)
0
The matter is assumed to be perfect fluid described by the energy momentum
tensor
T µν = (P + ρ)uµ uν + Pgµν ,
(12)
Vol. 0
7
0.18
0.16
0.14
xp
0.12
0.1
0.08
H
L
M
0.06
0.04
H xDU
L xDU
M xDU
0.02
0
0
0.2
0.4
0.6
nB [fm-3]
0.8
1
1.2
Figure 2: Proton fraction of matter being at β - equilibrium for used parameterizations. Also lines of direct URCA threshold (marked with xDU ) for all parameterizations are depicted.
(Misner, Thorne, and Wheeler 1973). where P is the pressure, uµ is the 4-velocity
of matter and gµν is the metric tensor. The energy momentum tensor satisfies the
µν
conservation law T ;ν = 0.
The hydrostatic equilibrium is in general relativity given by the Tolman-Oppenheimer-Volkoff equation (TOV) (Oppenheimer and Volkoff 1939, Tolman 1939),
which reads
dP
m(r) + 4πr3 P
= −(ρ + P)
.
(13)
dr
r(r − 2m(r))
Integration of TOV starting from given central energy density ρc uses the EoS and
finally yields the radius R, given by the boundary condition P(R) = 0, and the
gravitational mass M = m(R) of the neutron star.
Another useful quantity to calculate is the so-called baryonic mass MB that
represents the total number of baryons contained in the neutron star multiplied by
the atomic mass unit u. The baryonic mass is then expressed as
ZR
MB = 4πu
0
·
¸
2m(r) −1/2 2
nB (r) 1 −
r dr,
r
where nB (r) is the baryon number density at the radius r .
(14)
8
A. A.
2.5
M [MSUN]
2
H
L
M
1.5
1
0.5
0
0
0.2
0.4
0.6
0.8
-3
nB [fm ]
1
1.2
1.4
Figure 3: Mass given as a function of central baryon number density for different
parameterizations. The stars correspond to the minimum mass of a neutron star that
could cool via direct URCA reactions.
4. Results versus observations
Several dozens of neutron stars and/or similar objects have their masses reported; a great majority of them is in very close vicinity of 1.4 M¯ , and only
very few are significantly above (see, e.g., the compilations in Bethe, Brown and
Lee (2007), Lattimer and Prakash 2007) and observations and analyses (see, e.g,
Rikovska Stone et al. 2003, Weber, Negreiros and Rosenfeld 2007, Podsiadlowski
et al. 2005, Trümper et al. 2004, Pons et al. 2002, Kramer and Wex 2009, Krastev
and Sammarruca 2006, Lattimer and Prakash 2007, Blaschke, Klähn and Sandin
2008, Dexheimer, Vasconcellos and Bodmann 2008, Klähn et al. 2006, Nice, Stairs
and Kasian 2008, Rikovska Stone et al. 2007). However, recent results of the data
fitting of kHz quasiperiodic oscillations observed in the low-mass X-ray systems
containing neutron stars indicate relatively high masses of M > 2 M¯ (Belloni,
Mendez and Homan 2007, Török et al. 2008a,b,c, Barret, Olive and Miller 2005,
Boutelier et al. 2010, Boutloukos et al. 2006) which could provide very strong
constraint on the EoS. On the other hand, modification of the characteristic orbital
frequencies by a magnetic repulsion caused by the interaction of slightly charged
matter in accretion disc in vicinity of a neutron star with dipole magnetic field could
Vol. 0
9
shift the mass estimates to lower values close to canonical 1.4 M¯ (Bakala et al.
2008). Our calculations with parameterization H allow for the existence of neutron
stars even for so heavy masses.
4.1.
Direct URCA constraints
The proton fraction xp of matter in β equilibrium is presented in Figure 2 together with the direct URCA threshold. The direct URCA reactions n → p + e− +
ν̄e could operate only if the proton fraction exceeds the threshold given by the
condition
1
xDU =
(15)
³
´ ,
1/3 3
1 + 1 + xe
where xe = ne /(ne + nµ ). One can see that only parameterizations L and M enable
rapid cooling. The threshold densities are nDU = 0.457 fm−3 in the case of parameterization L and nDU = 0.571 fm−3 in the case of parameterization M . These
values correspond (see Figure 3) to neutron star masses M = 1.47 M¯ (parameterization L) and M = 1.39 M¯ (parameterization M ). Parameterizations L and M
thus do not fulfill the direct URCA constraints, however Klähn et al. 2006 used also
the value 1.35 M¯ as a weaker test that is passed also by parameterizations L and
M.
4.2. Maximum mass
The maximum mass limit is probably the most often used test of the equation of
H = 2.18 M ,
state. The maximum masses given by EoS used in this paper are Mmax
¯
L = 1.92 M and M M = 1.62 M . The maximum mass obtained for objects
Mmax
¯
¯
max
containing matter described by parameterizations L and M follows the requirements of stability with respect to radial oscillations (∂M/∂nc > 0). In the case H
we used the values corresponding to the central density nB (r = 0) = 0.66 fm−3 ,
because for the densities above this value the model used for the EoS is not without questions and also because only with central densities up to about 4× normal
nuclear density we were able to explain masses of neutron stars that meet the observational requirements. With some extrapolations, higher masses could be in principle modelled, but we decided to use parameterization H up to the quoted density
only since there is no current astrophysical observation of such a high mass. The
observation of high mass is however crucial and very promising issue of astrophysical observations. It should be noted that Klähn et al. 2006 used the value that
could not be used anymore. Also the result for the source 4U 1636–536 that gives
M = (1.9 − 2.1) M¯ as proposed by Barret, Olive and Miller (2005) should be used
rather as an upper limit of the neutron star mass than as its estimate, see, e.g. Miller,
Lamb and Psaltis (1998) for underlying theories. The neutron star mass is inferred
due to the highest observed frequency of QPOs observed in the system, under the
assumption of identifying the highest frequency with the Keplerian frequency of
the innermost stable circular orbit (ISCO). Clearly this gives an upper limit on the
10
A. A.
1.27
H
L
M
M [MSUN]
1.26
1.25
1.24
1.23
1.32
1.34
1.36
MB [MSUN]
1.38
1.4
Figure 4: Relation of calculated gravitational mass M and the baryonic one MB
for different parameterizations. The limitations imposed by the analysis of the
J0737-3039 double pulsar are drawn as a small rectangle. The box is also extended
to the left by 0.003M¯ indicating the possible mass loss.
mass, and the real neutron star mass has to be expected smaller because the QPOs
have to be excited above ISCO. Up to date, one of the two pulsars Ter 5 I and J has
a reported mass larger than 1.68 M¯ to 95% confidence level (see, e.g. Lattimer
and Prakash (2007) and references therein)1 . Champion et al. (2008) predicted
mass of PSR J1903+0327 to be M = 1.74 ± 0.04M¯ . Freire (2009) estimated the
mass for the same source to be M = 1.67 ± 0.01 M¯ . These values, even if they
are different, give approximately the same limit on mass when they are combined
together, namely & 1.66 M¯ at 2 σ level. These predictions are not in favour the
M = 1.62 M .
parameterization M with Mmax
¯
4.3. Double pulsar J0737–3039
Podsiadlowski et al. (2005) investigated possible formation scenarios of double
pulsar J0737–3039. They have shown that one can test EoS assuming the pulsar B
is formed by an electron-capture supernova. Such scenario enables formation of the
1 The individual pulsar masses unfortunately are not assumption-independent. In our discussion,
we adhere to the value 1.68 M¯ reported by Lattimer and Prakash, but bearing in mind the possible
uncertainty in its derivation.
Vol. 0
11
Figure 5: Mass–radius relation for different parameterizations. The lines corresponding to RX J 1856.5–3754 gives a lower mass limit, that should the given EoS
get over.
pulsar B that has low but very accurately measured mass M = 1.2489 ± 0.0007 M¯
(Kramer and Wex 2009). If this pulsar is born under the presented scenario, its
baryonic mass MB should be in the range 1.366 to 1.375 M¯ . The authors also
argue the matter loss being low (the matter loss they give is few times 10−3 M¯ ).
The relation between the gravitational and the baryonic masses together with the
limitations derived from the double pulsar observations are presented in Figure 4.
One can see that the only parameterization that meets requirements assuming no
mass loss is the parameterization H . The parameterization M is able to explain the
results if one includes mass loss predicted by Podsiadlowski et al. (2005). Unfortunately this parameterization was ruled out by the maximum mass test.
4.4.
Isolated neutron star RX J1856.5–3754
Several authors (see, e.g., Trümper et al. 2004, Pons et al. 2002, van Kerkwijk
and Kaplan 2007, Steiner et al. 2010) discussed observations of the isolated neutron
star RX J1856.5–3754 and they found constraints on the mass-radius relation of this
particular neutron star. They found the limits of the apparent radius being given byt
12
A. A.
the mass-radius relation
µ
¶
M
R
R2
=
1− 2 ,
M¯ 2.95 km
R∞
(16)
that could serve as a test of equation of state. We have used three different values for R∞ namely R∞ = 15.5, 16.8, 20.9 km. None of tested parameterizations
is able to explain the apparent radius R∞ = 20.9 km. Parameterization H is the
only one capable of explaining the apparent radius R∞ = 16.8 km estimated by
Trümper et al. (2004). The lowest predicted apparent radius could be modeled by
all parameterizations considered in this paper. The mass–radius relations for all
parameterizations together with observational limits are illustrated in Figure 5.
5.
Conclusions
We have employed the parameterized form of the relativistic mean-field EoS
for asymmetric nuclear matter with vector cross interaction. The proton fraction
was varied in accord with the need of the β-equilibrium and charge neutrality. Assuming spherically symmetric geometry and using TOV equation, we constructed
models of neutron stars for different central parameters. We have used set of observational data to test EoS of nuclear matter represented by three different parameterizations of relativistic Brueckner-Hartree-Fock equation.
We have shown that only the parameterization H is able to pass almost all the
tests considered in this paper. The only exception is the apparent radius R∞ =
20.9 km estimation for the isolated neutron star RX J1856.5-3754; however this
estimate is based on distance measurements being still widely discussed. This parameterization also represents the only EoS based on the relativistic BruecknerHartree-Fock theory that could explain the formation of pulsar B in the double
pulsar system J 0737–3039 without mass loss.
Our present calculations have been done considering only neutrons and protons
in β-equilibrium with electrons and muons. We aim to continue in tests of given
EoS in future. One of our plans is to include hyperons. Another is to perform
more detailed tests based on the promising fitting of observational data of quasiperiodic oscillations in low-mass X-ray systems measurements. This necessitates
to investigate the rotational effects on neutron star models based on the HartleThorne metric reflecting mass, spin and the quadrupole moment of the neutron star
(Hartle 1967, Hartle and Thorne 1968). Our preliminary results indicate that these
improvements could bring a new information on the validity of EoS (Stuchlík et al.
2007). The important role of the neutron star spin is demonstrated in the case of
Circinus X–1 (Török et al. 2010).
Acknowledgements. The authors are grateful to J. Kotulič Bunta and Š. Gmuca
for the availability of their computer codes and to F. Weber for sending some
Vol. 0
13
of his collected data. The work has been supported by the Czech grants MSM
4781305903 (EB and ZS) and LC 06014 (MU) and by the VEGA grant 2/0029/10
(EB). One of the authors (ZS) would like express his gratitude to Czech Committee for Collaboration with CERN for support and the CERN Theory Division for
perfect hospitality. Authors are also grateful to anonymous referee for his/her comments and suggestions.
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Astronomy & Astrophysics manuscript no. epicyclic˙revI2˙new˙last
July 2, 2010
c ESO 2010
Disc-oscillation resonance and neutron star QPOs:
3:2 epicyclic orbital model
Martin Urbanec∇ , Gabriel Török∇ , Eva Šrámková∇ , Petr Čech∇ , Zdeněk Stuchlı́k∇ , Pavel Bakala∇
∇ Institute of Physics, Faculty of Philosophy and Science, Silesian University in Opava, Bezručovo nám. 13, CZ-74601 Opava, Czech
Republic
Received / Accepted
ABSTRACT
High-frequency quasi-periodic oscillations (HF QPOs) which appear in the X-ray fluxes of low-mass X-ray binaries remain an unexplained phenomenon. Among other ideas it was suggested that a non-linear resonance between two oscillation modes in an accretion
disc orbiting a black hole or a neutron star plays role in exciting the observed modulation. Several possible resonances have been discussed. A particular model assumes disc-oscillation modes with the resonant eigenfrequencies equal to radial and vertical epicyclic
frequency of geodesic orbital motion. This model has been discussed for black hole microquasar sources as well as for a group of
neutron star sources. Assuming several neutron (strange) star equations of state and Hartle-Thorne geometry of rotating stars, we
briefly compare the frequencies expected from the model to those observed. Our comparison implies the neutron star radius RNS to
be larger than is the related radius of the marginally stable circular orbit rms for nuclear matter equations of state and spin frequencies
up to 800Hz. For the same range of spin and a strange star (MIT) equation of state, it is RNS ∼ rms . The ”Paczyński modulation”
mechanism considered within the model requires RNS < rms . However, we have found this condition to be fulfilled only for the strange
matter equation of state, masses below 1M , and spin frequencies above 800Hz. This result most likely falsifies the postulation of the
neutron star 3:2 resonant eigenfrequencies being equal to the frequencies of geodesic radial and vertical epicyclic modes. We suggest
that the 3:2 epicyclic modes could stay among the possible choices only if a fairly non-geodesic accretion flow is assumed, or if a
different modulation mechanism operates.
Key words. X-rays:binaries — Stars:neutron
1. Introduction: HF QPOs and desire for
strong-gravity
Galactic low mass X-ray binaries (LMXBs) display quasiperiodic oscillations (QPOs) in their observed X-ray fluxes (i.e.,
peaks in the X-ray power density spectra). Characteristic frequencies of these QPOs range from ∼ 10−2 Hz to ∼ 103 Hz. Of a
particular interest are the so-called high-frequency (HF) QPOs
having their frequencies typically in the range 50 – 1300 Hz
which is roughly of the same order as the range of frequencies
characteristic for orbital motion close to a low mass compact
object. We briefly remind that there is a crucial difference between HF QPOs observed in black hole (BH) and neutron star
(NS) systems. In BH systems, the HF QPO peaks are commonly
detected at constant (or nearly constant) frequencies which are
characteristic for a given source. When two or more QPO frequencies are detected, they usually come in small-number ratios, typically in a 3 : 2 ratio (Abramowicz & Kluźniak, 2001;
Kluźniak & Abramowicz, 2001; McClintock & Remillard, 2004;
Török et al., 2005). For NS sources, on the other hand, HF (or
kHz) QPOs often appear as twin QPOs. This features, on which
we focus here, consist from two simultaneously observed peaks
having distinct actual frequencies that substantially change over
time. The two peaks forming twin QPO are then standardly refered to as the lower and upper QPO in accord to the inequality
of their frequencies.
The amplitudes of twin QPOs in NS sources are typically
much stronger and their coherence times are often much higher
than those in BH sources (e.g. McClintock & Remillard, 2004;
Barret et al., 2005a,b; Barret et al., 2006; Méndez, 2006). It is
however interesting that most of the twin QPOs having high statistical significance have been detected around lower QPO frequencies 600 – 700Hz vs. upper QPO frequencies 900 – 1200Hz.
Because of that the twin QPO frequency ratio clusters mostly
around ≈ 3:2 value posing thus some analogy to BH case (see
Abramowicz et al., 2003a; Belloni et al., 2007; Török et al.,
2008a; Török et al., 2008b,c; Boutelier et al., 2009, for details
and a related discussion). It has been also only recently noticed
that in several NS sources the difference in the amplitudes of the
two peaks changes sign as their frequency ratio passes through
the (same) 3:2 value (Török, 2009). A detailed review on the
other similarities and differencies of the HF QPOs features can
be found in van der Klis (2006).
1.1. HF QPO interpretation
There is a strong evidence supporting the origin of the twin
QPOs inside less than 100 gravitational radii, rg = GMc−2 ,
around the accreting compact objects (e.g., van der Klis, 2006).
At present there is no commonly accepted QPO theory. It is
even not clear whether such a theory could involve the same
phenomena for both BH and NS sources. Several models have
been proposed to explain the HF QPOs, most of which involve
orbital motion in the inner regions of an accretion disc. When
describing the orbital motion, Newtonian approach necessarily
fails close to the compact object. Two most striking differences
arise from the relevant general relativistic description: Einstein’s
strong gravity cancels the equality between the Keplerian and
2
M. Urbanec et al.: On 3:2 epicyclic resonance in NS kHz QPOs
epicyclic frequencies, and (due to the existence of the marginally
stable circular orbit rms ) it gives a limit on the maximal allowed
orbital frequency. Several further effects such as the relativistic precessions of orbits then pop up in the inner accretion region. Finding a proper QPO model thus could help to test the
strong field regime predictions of general relativity and, in the
case of NS sources, also the models of high-dense matter (see
van der Klis, 2006; Lamb & Boutloukos, 2007, for a recent review).
1.2. Non-linear resonances between “geodesic and
non-geodesic” disc-oscillations
Numerous particular ideas explaining the observed lower and
upper HF modulation of the X-ray flux has been suggested while
hypothetical resonances between the two QPO oscillatory modes
are often assumed. Specific ideas of non-linear resonances between disc-oscillation modes has been introduced and extensively investigated by Abramowicz, Kluźniak and collaborators (Kluźniak & Abramowicz, 2001; Abramowicz & Kluźniak,
2001; Abramowicz et al., 2003a,b; Rebusco, 2004; Török et al.,
2005; Horák, 2008; Stuchlı́k et al., 2008; Horák et al., 2009, and
others; see also Aliev & Galtsov, 1981 and Aliev, 2007). These
ideas reached a high popularity and entered numerous individual
disc-oscillation models.
The subject of disc-oscillations and their propagation has
been extensively analytically studied for thin disc (i.e., nearly
geodesic, radiatively efficient) configurations (Okazaki et al.,
1987; Kato et al., 1998; Wagoner, 1999; Wagoner et al., 2001;
Silbergleit et al., 2001; Ortega-Rodrı́guez et al., 2002; Wagoner,
2008). Obtained results were compared to “thick” (radiatively
inefficient, slim-disc or toroidal) configurations whereas analytical (Blaes, 1985; Šrámková, 2005; Abramowicz et al., 2006;
Blaes et al., 2006, 2007; Straub & Šrámková, 2009) and numerical (Rezzolla et al., 2003a,b; Rezzolla, 2004; Montero et
al., 2004; Zanotti et al., 2005; Šrámková et al., 2007) approach
has been performed. Several consequences for disc-oscillation
QPO models have been sketched, some of them having direct
relevance for non-linear resonance hypotheses. Especially, it has
been found that, due to pressure effects, values of the frequencies
at radii fixed by a certain frequency ratio condition can differ between the geodesic and fairly non-geodesic flow of factors like
15% (Blaes et al., 2007).
1.3. Aims and scope of this paper
Gondek-Rosińska & Kluźniak (2002) suggested that the resonance theory of kHz QPOs can provide a discrimination between
quark (strange matter) stars and neutron stars. In spirit of this
suggestion, we examine a particular, often quoted “3:2 epicyclic
resonance model” (or rather class of such models). The paper is
arranged as follows.
In Section 2 we briefly recall some important points of nonlinear resonance models specific to neutron stars and briefly recall 3:2 epicyclic resonance model. In Section 3 we compare the
model to the HF QPO observations of a group of NS sources displaying the 3:2 ratio. Restrictions to the mass and radius following from equations of state for non-rotating NS are included. In
Section 4 we explore the corrections arising due to the effects of
NS rotation and again include the consideration of the equations
of state. In Section 5 we assign some consequences and discuss
possible falsification of the model, whereas the nearly geodesic
and fairly non-geodesic cases are considered separately.
Throughout the paper we use the standard notation where
νL , νU stand for observed lower and upper QPO frequencies,
while νK , νr , νθ stand for Keplerian, radial epicyclic and vertical
epicyclic frequencies that are given by considered spacetime and
its parameters.
2. Resonances in disc around neutron stars
Miscellaneous variations of the non-linear, disc-oscillation resonances have been discussed in the past (see, e.g., Abramowicz
& Kluźniak, 2001, 2004; Török, 2005a; Horák & Karas, 2006).
While the basic ideas are common to both, black hole and neutron star, models, several differences between the two classes of
sources have been considered as well. In particular, it has been
suggested that, in a turbulent NS accretion flow, the resonant
eigenfrequencies are not fixed (e.g., when oscillations of a tori
changing its position are assumed; Zanotti et al., 2003; RubioHerrera & Lee, 2005; Abramowicz et al., 2006; Török et al.,
2007; Kluźniak et al., 2007), or that the resonant corrections to
eigenfrequencies reach high values (Abramowicz et al., 2003b,
2005a,b). Both possibilities are taken into account in Section 3.
2.1. Modulation
One more important difference is considered for the QPO
modulation mechanism (Bursa et al., 2004; Horák, 2005a;
Abramowicz et al., 2007; Bursa, 2008). In the black hole case
the weak modulation is assumed to be primarily connected to
radiation of the oscillating disc and the related relativistic lensing, light-bending and Doppler effects. In the neutron star case
the expected modulation is connected to the flux emitted from a
hot spot on the NS surface causing a strong QPO amplitude.
We shortly recall a “Paczyński-modulation” mechanism
(Paczyński, 1987) which has been investigated by Horák (2005a)
and Abramowicz et al. (2007). Schematic Figure 1 displays the
considered situation. The expected mass-flow is described by the
Bernoulli equation while surfaces of constant enthalpy, pressure
and density coincide with surfaces of constant effective potential U(r, z) = const. (Abramowicz, 1971). The disc equilibrium
can exist if the disk surface corresponds to one of the equipotentials inside the so-called Roche lobe (region indicated by
the yellow colour). No equilibrium is possible in the region of
r < rin . Dynamical mass loss corresponding to a certain accretion rate arises when the fluid distribution overflows the surface
of the disk for U0 = U(rin ). When the accretion disc oscillates
it slightly changes its position with respect to the equipotencial surfaces. At a particular location corresponding to crossing
of the equipotentials, the so-called cusp, even a small displacement of the disc causes a large change of the accretion rate. The
change of the accretion rate is then nearly instantly reflected by
the hot-spot temperature leading to an enhanced X-ray emission
(Paczyński, 1987; Horák, 2005a; Abramowicz et al., 2007).
The existence of the surface U0 above the neutron star is
crucial for the model. Therefore, as a necessary condition for its
applicability, it is required that
RNS /rms < 1,
(1)
where RNS denotes the neutron (strange) star radius (“accretion
gap paradigm”, Kluźniak & Wagoner, 1985; Kluźniak et al.,
1990). Note that this is the necessary, but not sufficient condition, since the inner radius rin is located between the marginally
stable and the marginally bound circular orbit (Kozlowski et al.,
1978).
M. Urbanec et al.: On 3:2 epicyclic resonance in NS kHz QPOs
3
one can relate the black hole spin or mass to the observed frequencies. Such a procedure has been done by Abramowicz &
Kluźniak (2001) and later by Török et al. (2005) and Török
(2005b) for various resonances and set of sources. In principle,
similar calculations can be done also for resonance models of
NS QPOs. In the case of neutron stars the observed frequencies,
however, change over time and, moreover, monotonic positive
frequency correlations are similar, but specific across the individual sources. Within the framework of the resonance models
two distinct simplifications can be considered for the observed
frequency correlations when inferring the neutron star mass.
a) The observed frequencies are roughly equal to the resonant
eigenfrequencies and the observed frequency correlation follows from the changes of eigenfrequencies,
νL = νL0 (r3:2 + ∆r),
νU = νU0 (r3:2 + ∆r),
(3)
implying for the 3:2 epicyclic model
νL = νr (r3:2 + ∆r),
νU = νθ (r3:2 + ∆r).
(4)
b) The eigenfrequencies are constant and the correlation arises
due to the resonant corrections
νL = νL0 + ∆νL ,
νU = νU0 + ∆νU ,
(5)
implying for the 3:2 epicyclic model
νL = νr (r3:2 ) + ∆νL ,
Fig. 1. Mass-flow leaving the disk and crossing the relativistic accretion
gap (after Abramowicz et al., 2007). Top: Keplerian angular momentum
vs. the angular momentum in the flow. Bottom: The equipotential surfaces and the distribution of fluid in a meridional cross-section of the
disc-configuration. The yellow area denotes the fluid in the disc, while
the orange area corresponds to the overflow modulated by the oscillations. Enhanced luminosity arises as the flow enters the boundary layer
(light-blue colour).
2.2. Epicyclic resonance
A particular example of the non-linear resonance between discoscillation modes is represented by the concept of the “3:2
epicyclic internal resonance”. This hypothesis is widely discussed (e.g., Abramowicz et al., 2002; Kluźniak & Abramowicz,
2002, 2005; Horák, 2004, 2005b; van der Klis, 2005; Török &
Stuchlı́k, 2005; Vio et al., 2006; Rebusco, 2008; Reynolds &
Miller, 2009, among the other references in this paper). It assumes that the resonant modes have eigenfrequencies equal to
radial and vertical epicyclic frequency of geodesic orbital motion,
νL0 = νr (r3:2 ),
νU0 = νθ (r3:2 ),
(2)
associated with the orbital radius r3:2 where νθ /νr = 3/2. We
stress that models considers oscillations of fluid configurations
rather than test particle motion (see, e.g., Kluźniak, 2008, for
some details and related references). In next we assume the 3:2
epicyclic model and elaborate whether the Paczyński modulation
mechanism can be at work.
3. NS mass and radius implied by 3:2 epicyclic
resonant model
In the case of resonance models for BH QPOs the observed
constant frequencies are expected to coincide with the resonant eigenfrequencies. Then, assuming a particular resonance,
νU = νθ (r3:2 ) + ∆νU .
(6)
Note that for a) the resonance plays rather secondary role in producing QPOs while for b) it represents their generic mechanism.
3.1. Mass
In this Section we neglect the effects of neutron star spin and assume the 3:2 epicyclic resonance model in the Schwarzschild
spacetime.1 Introducing a relativistic factor F ≡ c3 /(2πGM),
equation (4) reads
νU = νθ = νK ≡ x−3/2 F , x ≡ r/M,
r
6
νL = νr ≡ νK 1 − ,
(7)
x
implying
r
ν 2/3
U
νL = νU 1 − 6
.
(8)
F
It has been previously discussed in terms of correlation between the QPO frequency (νL or νU ) and frequency difference
∆ν = νU − νL that the correlation (8) clearly disagrees with the
observations of NS sources (e.g., Belloni et al., 2005). The 3:2
epicyclic resonance model fully based on (3) is therefore excluded. Consequently, in the rest we focus on the option (5).
Relation of the option (5) to the observation of several NS
sources has been elaborated in works of Abramowicz et al.
(2005a,b). They assume that the corrections ∆ν in equation (6)
vanish when the observed frequency ratio νU /νL reaches the 3/2
value. Consequently, they suggest that the resonant eigenfrequencies [νL0 , νU0 ] in a group of twelve NS sources roughly equal
to [600Hz, 900Hz]. For the 3:2 epicyclic model it is then
νr3:2 = 600Hz,
νθ3:2 = 900Hz
(9)
1
We consider here this standard spacetime description for nonrotating neutron stars although some alternatives have been recently
discussed in a similar context (see Kotrlová et al., 2008; Stuchlı́k &
Kotrlová, 2009).
4
M. Urbanec et al.: On 3:2 epicyclic resonance in NS kHz QPOs
which, from terms given in equation (7), implies that the relevant
mass must be around M = 1M (first noticed by Bursa 2004
unpublished).
3.2. Radius
Modeling of NS equations of state (EoS) have been extensively
developed in the second half of 20th century resulting to numerous published methods and codes (see, Lattimer & Prakash,
2001 and Lattimer & Prakash, 2007 for a review). Here we calculate NS radii following the approach of Hartle (1967), Hartle
& Thorne (1968), Chandrasekhar & Miller (1974) and Miller
(1977). In the Figure 2 we plot the mass-radius relations for several EOS.
Skyrme denote nine different EOS (namely SkT5, SkO’,
SkO, SLy4, Gs, SkI2, SkI5, SGI, SV) given by the different parameterizations of the effective Skyrme potentials, see Řı́kovská
Stone et al. (2003) and references therein. DBHF stands for
four different parameterizations, that have been chosen to describe matter in the framework of Dirac-Brueckner-HartreeFock theory. Specifically we choose the parameterizations labeled HA, HB, LA, MA in Kotulič Bunta & Gmuca (2003) used
by Urbanec et al. (2010) to describe properties of static neutron
stars. The EOS labeled APR was used very often in past years.
We have chosen the model labeled A18 + δv + UIX∗ in the original paper (Akmal et al., 1998). The rest of pure neutron star
equations of state are FPS (Pandharipande & Ravenhall, 1989)
and BBB2 (Baldo et al., 1997). The model labeled GLENDNH3
includes also hyperons (Glendenning, 1985).
MIT model denotes strange stars calculated using the so
called MIT Bag model (Chodos et al., 1974), where we have
used standard values B = 1014 g.cm−3 for Bag constant and
αc = 0.15 for strong interaction coupling constant.
From the Figure 2 we can see that for M ∼ 1M the NS radii
are in all the cases above rms . Thus assuming the Schwarzschild
metric the condition (1) is not fulfilled for the X-ray modulation
given by the 3:2 epicyclic model.
4. Effects related to NS spin
So far we have restricted attention to implications of (9) for nonrotating NS. The spin of the astrophysical compact objects and
related oblateness however introduce some modifications of the
Schwarzschild spacetime geometry. It has been found that, without the inclusion of magnetic field effects, the rotating spacetimes induced by most of the up-to-date neutron star equations
of state (EoS) are well approximated with the solution of Hartle
& Thorne (1968), see Berti et al. (2005) for details. We use this
solution (in next HT) to discuss the spin corrections to the above
results.
The HT solution reflects three parameters, the neutron star
mass M, angular momentum J and quadrupole moment Q. Note
that, up to the second order in J, Kerr geometry represents the
“limit” of the HT geometry for q̃ ≡ QM/J 2 → 1. The formulae for Keplerian and epicyclic frequencies in the HT spacetime
were derived by Abramowicz et al. (2003c). We applied these
formulae to solve equation (9). Figure 3 displays the resulting
surface colour-scaled in terms of M/M , j = cJ/GM 2 and q̃.
We can see that for low values of q̃ and any j the implied M increases with growing j, while exactly opposite dependence M( j)
occurs for high values of q̃ and j ≥ 0.2.
Fig. 2. Mass-radius relations for several EoS assuming a non-rotating
star. The shadow area indicates the region with NS radii higher than
the radius of the marginally stable circular orbit (no accretion gap). The
mass M = 1 ± 0.1M is denoted by the dashed and dotted horizontal
lines.
4.1. EoS and radii
For a given EoS the parameter q̃ decreases with increasing
M/Mmax . In more detail, it is usually q̃ ∼ 10 for 1M while
q̃ ∈ (1.5, 3) for the maximal allowed mass (e.g. Török et al.,
2010, Figure 3 in their paper). Since the non-rotating mass following from the model is about 1M , one can expect that realistic NS configurations will be related to M – j solutions associated with high q̃. These are denoted in Figure 3 by colours of the
yellow-red spectrum.
We check this expectation using the same set of EoS like
in Section 3.2. We calculated the configurations for each EoS
covering the range of the central density ρc implying M ∈ (≈
0.5M , Mmax ) and the spin frequency νs ∈ (0, νmax ) with the
step giving thousand bins in each of both independent quantities. The mass Mmax is the maximal mass allowed for a given
EoS. The frequency νmax is the maximal frequency of given
neutron star and equals to the Keplerian frequency at the surface of the neutron star at the equator, corresponding to the socalled mass-shedding limit. In this way we obtain a group of
15 × 10002 ≈ 107 configurations. From these we keep only
those fulfilling the condition (9) for the epicyclic frequencies
(ν = ν (M, j, q), Abramowicz et al., 2003c) extended to
2/3νθ3:2 = νr3:2 ∈ (580Hz, 680Hz).
(10)
Note that this range of considered eigenfrequencies is roughly
based on the range of the observed 3:2 frequencies (Abramowicz
et al., 2005a,b). The combinations of mass and angular momentum selected in this way are displayed in the Figure 4a.
Inspecting the Figure one can see that the mass decreases with
increasing j above j ∼ 0.3.
The Figure 4b indicates the ratio RNS /rms for the selected configurations (shown in the Figure 4a). Apparently, the
modulation-condition (1) is fulfilled only for MIT-EoS and high
spins above 800Hz.
5. Discussion and conclusions
The neutron star masses restricted for the 3:2 epicyclic resonance model by the considered EoS (Figure 4a) are very low
M. Urbanec et al.: On 3:2 epicyclic resonance in NS kHz QPOs
5
a)
Fig. 3. Solution of the 3:2 frequency equation (9) projected onto M − j
plane and colour-scaled in terms of q̃.
b)
in comparison with the ”canonical” value 1.4M . Moreover, for
the non-rotating case the implied NS configurations are not compact enough to fulfill the modulation condition (1). We find that
this condition is satisfied only for high spin values, above 800Hz,
and strange matter EoS (MIT) - see the shadded region in the
Figure 4b.
Searching through the region, we find the highest mass satisfying (1) to be M = 0.97M . The related NS spin is 960Hz.
This mass and spin correspond to νL3:2 = 580Hz. For higher frequencies νL3:2 the required mass is even lower. For νL3:2 = 630Hz
it is M = 0.85M whereas the related NS spin is 900Hz.
For compact objects in the NS kHz QPO sources there
are at present no clear QPO independent mass estimates. On
the contrary, there is a convincing evidence on the spin of
several sources from the X-ray burst measurements (see, e.g,
Strohmayer & Bildsten, 2006). In the group of sources discussed
by Abramowicz et al. (2005a,b) that are of consideration in this
paper there are several that have spins between ∼250–650Hz.
The NS parameters implied by the 3:2 epicyclic model therefore
include not only very low masses, but also spins excluded by the
QPO independent methods.
The obtained results that falsify the epicyclic hypothesis are
doubtless as far as
i) The Paczyński modulation mechanism is involved, implying
that the inequality RNS < rms is requiered.
ii) The eigenfrequencies considered within the model equal to
(nearly) geodesic frequencies.
The following remarks should be considered.
Ad i. The amplitudes of NS twin peak QPOs are often very
high in comparison to BH amplitudes. The lensing effects are
then insufficient for the observed modulation (see e.g. Bursa et
al., 2004; Schnittman & Rezzolla, 2006). Neverthless, one can
still imagine other mechanisms alternative to Paczyński modulation (that do not neccessarily require the RNS < rms condition).
One example2 may be an equilibrium torus which does not have
a cusp and oscillates with a high oscillation amplitude. In such
case accretion onto the neutron star can yet occur due to random
overflow of the critical equipotential. The largest configurations
fullfilling the 3:2 frequency condition for a non-rotating NS have
2
The authors thank L. Rezzolla for suggesting this possibility during
a discussion at the Relativistic Whirlwind conference in Trieste.
Fig. 4. a) NS configurations fulfilling the 3:2 frequency condition (10).
b) Related relationships between NS spin and radii evaluated in terms
of ISCO radii rms (M, j, q). Only subset of configurations obeying the
condition (10) are depicted in the figures for clearness. Lines tending to
appear on both figures correspond to configurations with same central
parameters and different rotational spin.
the radius 1.8×6M = 10.8M which equals to the radius of the 3:2
resonant orbit for j = 0. Since all the considered EoS give NS
radius below the resonant radius, the model would, in principle,
work for such a hypothetic mechanism. In addition, interference
between a terminating disc and spinning NS surface near the 3:2
resonant orbit could represent a powerfull excitation mechanism
(see also Lamb & Coleman, 2003). Nevertheless, some difficulties clearly arise. Mainly, the (unexplored) mechanism should
be well-tuned with the flow coming from the binary companion.
The twin QPO peaks sometimes appear in the NS PDS for a few
tens of minutes representing roughly 106 oscillations. Note that
there is an intrinsical instrumental fragmentation of observations
which occurs on the same timescales and it is then assumed that
QPOs often survive even longer (van der Klis, 2006). It would
thus be neccesary to have an accretion flow with neither low,
nor high accretion rate that will support the considered toruslike configuration for a very long time.
Ad ii. As quoted in Section 1, for the 3:2 epicyclic resonance
the values of the resonant eigenfrequencies for a non-geodesic
flow are higher than those calculated for a nearly geodesic motion (Blaes et al., 2007; Straub & Šrámková, 2009). It has been
6
M. Urbanec et al.: On 3:2 epicyclic resonance in NS kHz QPOs
shown that in a certain case the difference can reach about 15%.
This would change the non-rotating mass to a higher value,
∼ 1.2M . From Figure 2 we can find that this value rather barely
fits the modulation condition for the MIT EoS. It thus cannot be
fully excluded that the model could be compatible with observations if the flow was fairly non-geodesic. However, needless
to say that a serious treatement of such possibility will require
investigation of the related pressure effects on the disc structure in the Hartle-Thorne geometry, as the above-noted studies
only consider Swarzchild (in a pseudo-Newtonian approximation) and Kerr geometry. Moreover, they assume a constant specific angular momentum distribution within the disc while there
is evidence from numerical simulations of evolution of accreting
tori that real accretion flows tend to have rather near-Keplerian
distribution (e.g. Hawley, 2000; De Villiers & Hawley, 2003).
In such case one would expect that the pressure corrections will
be rather smaller than those calculated for the marginal case of
a constant angular momentum torus. However, there is no clear
guarantee of this expectation and further investigation will be
neccessary to resolve this issue.
We can conclude with a final statement that the resonance
model for NS kHz QPOs should involve a combination of discoscillation modes different from the geodesic radial and vertical epicyclic modes, or a modulation mechanism different from
the Paczyński modulation. The results also suggest that the two
modes together with the considered modulation could be at work
as long as a fairly non-geodesic accretion flow is assumed for
strange- or some of nuclear- matter EoS.
Acknowledgments
This work has arisen from several debates iniciated and richly
contributed by Marek Abramowicz and Wlodek Kluźniak during the past few years. We also appreciate useful suggestions
and comments of an anonymous referee which helped to improve it. The paper has been supported by the Czech grants
MSM 4781305903, LC 06014, and GAČR 202/09/0772. The
authors further acknowledge the internal student grant of the
Silesian University in Opava, SGS/1/2010.
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