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ACCELERATION DUE TO GRAVITY ON A RAPIDLY ROTATING
NEUTRON STAR
by
Mohammad F. AlGendy
A thesis submitted to the Faculty of Graduate Studies and Research
in partial fulfillment of the requirements for the degree of
Master of Science
Department of Physics
c
Mohammad
F. AlGendy
Spring 2012
Edmonton, Alberta
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To My Father, Fawzy AlGendy. The Most Honorable Man I ever Known
Abstract
In this thesis I am going to calculate the acceleration due to gravity on axisymmetric neutron
stars using the rapidly rotating neutron star code (RNS). I modified the original RNS code
so that it can compute the acceleration at different latitudes of the star. I am calculating
different stellar models, and I was able to get physical quantities like the maximum mass
for each spin frequency.
The code that I am using solves for the geometry of an axisymmetric rotating neutron star
described by Komatsu, Eriguchi and Hachisu (1989). The metric of the star has four potentials which are used in the calculation of acceleration due to gravity. The code numerically
solves the four field equations and the integrated equation of hydrostatic equilibrium. I find
the acceleration due to gravity for different equation of states (EOS). The EOS range from
the soft EOS BBB2, stiff (using equation of state L) and intermediate stiffness (using equation of state APR). Lastly I will plot the equatorial and polar acceleration due to gravity
for the different equations of state. I will attempt empirical fit for those curves based on
the stars’ physical parameters such as mass, radius and spin. I show also the acceleration
gradient at stellar latitudes at different spin frequencies for the three different equations of
states.
Acknowledgements
First and foremost am thankful to my Lord Almighty for everything I have.
I am truly full of gratitude to my supervisor Dr Sharon Morsink, without her kind personality
and support I wouldn’t be able to finish my master degree, she was extremely supportive
throughout my studies and especially kind at in my period of sickness. She helped me a lot
understanding the physical ideas, and with programming as well. Dr Morsink helped me a
lot me financially as well throughout my master, she hepled me even with the english of my
thesis.
I am thankful to my parents for continuous support,love and care.
Contents
1 Introduction
1.1
2
Where neutron stars come from? . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Properties of a neutron star
. . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3
The Acceleration due to gravity in Newtonian physics . . . . . . . . . . . . .
5
1.4
The RNS code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2 Spherically symmetric neutron stars in relativity
2.1
8
Equilibrium structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.1.1
Einstein field equations . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2
The equations of stellar structure . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.3
Equations of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4
Acceleration due to gravity for a spherical star in general relativity . . . . . . 12
2.5
Coordinate notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Rotating neutron stars in general relativity
3.1
3.2
15
Properties of relativistic rotating stars of perfect fluid . . . . . . . . . . . . . 15
3.1.1
Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.2
Mass-Radius curves for rotating star . . . . . . . . . . . . . . . . . . . 17
Acceleration due to gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.1
Hydrostatic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4 Results
4.1
The dependence of the acceleration on latitude . . . . . . . . . . . . . . . . . 27
4.1.1
4.2
4.3
27
Accuracy used in computing our models . . . . . . . . . . . . . . . . . 29
Detailed results for EOS APR . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2.1
Fitting for equations used . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2.2
Low energy density values for EOS APR . . . . . . . . . . . . . . . . . 31
4.2.3
Intermediate energy density values for EOS APR . . . . . . . . . . . . 32
4.2.4
High energy density values for EOS APR . . . . . . . . . . . . . . . . 32
4.2.5
Results of the acceleration gradient for EOS L . . . . . . . . . . . . . 34
4.2.6
Results of the acceleration gradient for EOS BBB2 . . . . . . . . . . . 35
Dependence of the star acceleration on the star’s properties . . . . . . . . . . 37
4.3.1
Results for EOS APR . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3.2
Results for EOS L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3.3
Results for EOS BBB2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3.4
Results for EOSs combined . . . . . . . . . . . . . . . . . . . . . . . . 44
5 Conclusion
Appendices
46
List of Tables
2.1
Maximum mass for non-rotating stars of 3 different equation of states . . . . 12
4.1
EOS APR, RNS code output for different grid sizes for a value of central
energy density value of εc = 0.8 × 1015 g/cm3 and rratio = 0.8 . . . . . . . . . 29
4.2
EOS APR, RNS code output for low ε= 0.7 × 1015 g/cm3 . . . . . . . . . . . 31
4.3
EOS APR, RNS code output for intermediate energy density ε = 1.0 ×
1015 g/cm3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.4
EOS APR, RNS code output for high energy density value of ε = 2.5 ×
1015 g/cm3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.5
Best fitting parameters for EOS APR . . . . . . . . . . . . . . . . . . . . . . 35
4.6
RNS code output for EOS L. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.7
Best fitting parameters for EOS L . . . . . . . . . . . . . . . . . . . . . . . . 37
4.8
EOS BBB2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.9
Best fitting parameters for EOS APR . . . . . . . . . . . . . . . . . . . . . . 41
4.10 EOS L, RNS code fitting parameters output for different energy density values. 43
4.11 EOS BBB2, RNS code fitting parameters output for different energy density
values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.12 Universal best fitting parameters for the 3 EOSs combined . . . . . . . . . . . 44
List of Figures
1.1
Newtonian non rotating spherical star representation . . . . . . . . . . . . . .
5
2.1
Logarithmic plot of pressure versus density values originally found at the 3
EOSs used in this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2
Mass-Radius curves for zero spin star with different EOSs. . . . . . . . . . . . 11
3.1
The mass radius curves for different spinning frequencies for EOS BBB2. . . . 17
3.2
The mass radius curves for different spinning frequencies for EOS APR . . . 18
3.3
The mass radius curves for different spinning frequencies for EOS L . . . . . 18
4.1
The scaled acceleration versus µ for different grid accuracies . . . . . . . . . . 30
4.2
Dependence of acceleration on the colatitudes. The scaled acceleration is
displayed on the vertical axis and on the horizontal axis are the values of
µ = cos θ. The thick dotted curves are the curves produced by the RNS code
and thin dotted line is the one produced by the quartic fit. The EOS used is
APR, and the stellar models are shown in table . The central energy density
for all models 0.7 × 1015 g/cm3 . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3
The scaled effective acceleration versus µ for EOS APR for medium energy
density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.4
The scaled effective acceleration versus µ for EOS APR for high energy
densityε = 2.5 × 1015 g/cm3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.5
The scaled effective acceleration versus µ for EOS L for low energy density . 36
4.6
The scaled effective acceleration versus µ for EOS L for medium energy density 36
4.7
The scaled effective acceleration versus µ for EOS L for high energy density . 37
4.8
This scaled acceleration versus µ for low value of energy density for EOS BBB2 38
4.9
This scaled acceleration versus µ for medium value of energy density for EOS
BBB2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.10 This scaled acceleration versus µ for high value of energy density for EOS
BBB2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.11 A plot of the scaled equatorial acceleration versus ζ and for EOS APR.
We start by plotting a set of energies densities (Min energy density = 0.8 ×
1015 g/cm3 to Max energy density of 2.5×1015 g/cm3 in sets of 13 increments).
The equation of the fitting surface is given byc0 + c1 ζ + c2 ζ 2 + c3 2 + c4 ζ 3 . 40
4.12 Plot of the scaled polar acceleration versus ζ and for EOS APR . . . . . . . 41
4.13 A plot of the scaled equatorial acceleration versus ζ and for EOS L. . . . . 42
4.14 A plot of the scaled polar acceleration versus ζ and for EOS L . . . . . . . . 42
4.15 A plot of the scaled Equatorial acceleration versus ζ and for EOS BBB2. . . 43
4.16 A plot of the scaled polar acceleration versus ζ and for EOS BBB2. . . . . . 44
4.17 A plot of the equatorial acceleration scaled versus ζ versus for all EOSs.. . . 45
4.18 A plot of the scaled polar acceleration versus ζ and for all EOSs. . . . . . . 45
List of Symbols
: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Central energy density
p: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure
h: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plank constant
mn : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neutron mass
a: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acceleration due to gravity
ar : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radial components of acceleration due to gravity
aθ : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Angular components of acceleration due to gravity
v : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . velocity
R: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Star radius at equator
t: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time coordinate
J : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Angular momentum
I: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rotational inertia
Ω : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Angular speed
M : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass
gαβ : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spacetime metric
Rαβ : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ricci curvature tensor
M : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solar mass
δ : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kronecker delta
γ : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Metric potential function
ρ: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Metric potential function
α : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Metric potential function
ω : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Metric potential function
N : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Normalization factor
h : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enthalpy
SDIV : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The code divisions in the s direction
1
M DIV : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The code division in the angular direction
M0 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Star rest mass
µ : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cosin θ
e15 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Central energy density times 1015
b0 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fitting parameters for acceleration gradient curve
b2 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fitting parameters for acceleration gradient curve
b4 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fitting parameters for acceleration gradient curve
rratio : . . . . . . . . . . . . . . . . . . Ratio of the polar stellar radius over the equatorial stellar radius
c0 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fitting parameters for acceleration versus ζ and c1 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fitting parameters for acceleration versus ζ and c2 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fitting parameters for acceleration versus ζ and c3 : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fitting parameters for acceleration versus ζ and CHAPTER 1
Introduction
In this chapter I am going to introduce a basic overview about neutron stars and their
physical properties and the geometrical representation that we are going to use in this
thesis. The ultimate goal is to calculate the acceleration due to gravity at the surface of
rotating, relativistic neutron stars. I am making using of a code that already exists. The
code is called Rapidly Rotating Neutron Star Code (RNS) (Stergioulas & Friedman 1995).
I have made changes so that the code will calculate the desired physical quantities. In this
thesis, I will use these results to show how the acceleration due to gravity depends on the
star’s mass, radius and spin rate.
The content of this thesis is as follows. In Chapter 1, I will provide a brief introduction to
the properties of neutron stars, and introduce the acceleration due to gravity in Newtonian
physics. In Chapter 2, background material will be presented on non-rotating neutron stars
in general relativity. Chapter 3 will introduce the description of rotating stars in general
relativity. In this chapter I will derive a formula for the acceleration due to gravity on
the surface of a rotating, relativistic star, and outline the method used to calculate it. In
Chapter 4, I will present my numerical results showing how the acceleration due to gravity
depends on latitude. In this chapter I present results for a number of models computed
with different physical properties, and then introduce an empirical formula that describes
the results for astronomically observed spin rates. Finally, in Chapter 5, a brief conclusion
will be presented.
1.1
Where neutron stars come from?
Evolution of a star is dependent on its mass. For a massive (M > 8M ) star going into
its end, the star is compressed during supernova and collapses to a neutron star. The star
2
CHAPTER 1. INTRODUCTION
3
will lose most of its moment of inertia during the collapse but retains most of the angular
momentum, so the newly formed star will have a very high rotation speed.
As the star rotates it emits magnetic dipole radiation and loses some of its rotation speed.
The binding energy of iron
56
Fe has the highest binding energy per nucleon. When all
the mass in the stellar core is converted to Iron and other stable elements like nickel, the
nuclear reactions stop and the core will start to cool down and the thermal pressure will
no longer be able to support it. The potential energy is released during the collapse. As
the iron nuclei dissolve, protons are released from the nuclei, after which electrons combine
with these protons forming neutrons and neutrinos (Kutnere, 1998) through the reaction
e− + p → n + ν.
(1.1)
The explosion will result in a supernova type II event where most of the original star mass will
be blown away and the dense solid core will remain. The collapse will continue even to high
density than that associated with a white dwarf. The neutrons will produce a degeneracy
pressure that will stop the collapse. During the supernova explosion heavier elements are
formed. Neutron rich nuclei are formed and neutrons are released in the process of neutron
drip. As long as the star is below the Oppenheimer Volkoff limit for the neutron star then
the star will be stable. Otherwise further collapse will occur to a black hole (Kutnere, 1998).
1.2
Properties of a neutron star
Normal gas pressure cant support the core left behind, supernova explosion. If the mass
is more than ≈1.44M electron degeneracy pressure can’t support it. The core can’t be
a white dwarf and as the density increases electrons and protons are forced together to
make neutrons, which in turn form a neutron star. There have been different equations
of state (EOS) proposed to fit to the observed physics quantities from neutron star. The
matter at the surface of neutron star is composed of crushed atomic nuclei surrounded by
electrons which fill gaps between them. The Pauli Exclusion Principle controls the neutron
in the neutron stars. As a first approximation, non-relativistic neutron degeneracy pressure
supports the neutron star and is given by
p'2
h 2 5/3
,
2π m8/3
n
where mn is the neutron mass and is the neutron star density.
(1.2)
CHAPTER 1. INTRODUCTION
4
According to the Pauli exclusion principle, two or more neutrons are denied from occupying
the same physical space time, so the effective pressure generated by neutrons on compact
objects like neutron star is referred as the neutron degeneracy pressure.
The gravitational acceleration g for a Newtonian gravitational field is given by
g=
GM
.
R2
(1.3)
The gravitational field on the surface of a neutron star is so strong that the acceleration
due to gravity is expected to be about 1011 times the acceleration due to gravity on the
Earth’s surface (Hansen & Kawaler & Trimble 1994). That acceleration would make any
astronaut thinking of landing on a neutron star a suicidal mission because an astronauts
feet will experience much greater force of gravity than his head and will tear him apart. As
well, the huge gradient of gravitational force will lead to a huge decrease in the atmospheric
pressure at the surface as expected from the equation of hydrostatic equilibrium, since
dP
dR
is proportional to g. This leads to an exceptionally thin atmosphere of a few centimeters.
After the supernova explosion and formation of the neutron star, if the angular momentum
of the core is conserved, then the core must rotate faster as it becomes smaller. From
Newtonian physics we know that the angular momentum is given by
J = IΩ,
(1.4)
where I is the rotational inertia and Ω is the angular speed. For a uniform sphere we have
J=
2
M R2 Ω.
5
(1.5)
A neutron star has a strong magnetic field. If a neutron star started with a magnetic field of
about the same magnitude as our sun, it will end up with 2 × 109 times the solar magnetic
field. From observations the magnetic fields span from 108−15 G.
Due to the violent formation process in a supernova, neutron stars can have high temperatures of more than a million Kelvin. As a result, they mainly emit X-rays. In visible
CHAPTER 1. INTRODUCTION
5
light, neutron stars probably radiate approximately the same energy in all parts of visible
spectrum, and therefore appear white.
Rotation can support stars with higher mass than the maximum static limit (Shapiro &
Teukolsky 1983) Such high mass stars can be created when a neutron star accretes gas from a
normal binary companion. This scenario can also lead to recycled (rapidly rotating) pulsars.
Alternatively, high mass stars can be produced in the merger of binary neutron stars. Pulsars
are believed to be uniformly rotating. Eventually, viscosity will drive any equilibrium star
to uniform rotation. Uniformly rotating configurations with sufficient angular momentum
will be driven to the mass shedding limit.
1.3
The Acceleration due to gravity in Newtonian physics
Understanding the acceleration due to gravity is essential for my thesis and my research.
When neutron star rotates it exerts an effective outwards centrifugal force that reduces the
effective force due to gravity acting on a particle at the star’s surface. For simplicity, we will
define the acceleration due to gravity to mean the net acceleration. Before computing this
net acceleration felt by a particle on a rotating relativistic star, we first consider the simpler
case in Newtonian gravity. Although a realistic rotating star will have an oblate shape, it
is useful to first derive the acceleration on the surface of an artificial spherically symmetric
star.
Ω
ac
θ
r
θ
R
θ ar
Equator
a)r = R sin θ
b) Components of centripetal acceleration
Figure 1.1: Newtonian non rotating spherical star representation
Consider a point particle at an angle θ from the spin axis as shown on Figure 1.1. From
this figure, we can find the component of the net acceleration in the radial direction, given
by
a=
GM
− ar ,
R2
(1.6)
CHAPTER 1. INTRODUCTION
6
where ar is the radial component of the centripetal acceleration. But we know that
ar = ac sin θ
. The net acceleration is
(1.7)
GM
− ac sin θ.
R2
(1.8)
2πR sin θ
2πR sin θΩ
=
= R sin θΩ,
P
2π
(1.9)
a=
The velocity at θ is
V =
where P is the spin period. But the centripetal acceleration is given by
ac =
R2 sin θ2 Ω2
V2
=
= R sin θΩ2 ,
r
r
(1.10)
since r = R sin θ. Therefore, the net acceleration is
GM
− RΩ2 sin2 θ,
R2
(1.11)
GM
R3 Ω2 sin2 θ
(1 −
).
2
R
GM
(1.12)
a=
which can also be written as
a=
In this thesis I will deriv a similar expression for a relativistic rotating star that also includes
the star’s oblate shape.
1.4
The RNS code
The RNS code, is a code based on a numerical method described by (Komatsu, Eriguchi
and Hachisu), which describes the geometry of axisymmetric rotating Neutron star The
neutron star spin about certain axis and the speed by which this star is spinning called
the spin frequency, I am using a different spin frequency. The geometrical representation
of the neutron star is given by a metric which is provided to us by other theory of general
relativity, the metric has basic quantities that it depends on which is called the metric
potentials, by changing the value of the metric potential I can get other physical quantities.
the physical properties describing neutron stars are described by Equation Of State EOS,
there are so many EOSs which describe different model of neutron stars, since we have not
yet determined what is the correct EOS and it is still under physical observation, I use 3
EOS which still valid by our theoretical and observational values. The equation of state is
called either soft or stiff according to whether or not it is compressible with spinning, the
CHAPTER 1. INTRODUCTION
7
softer the EOS the less the compressibility and then the higher the radius will be for the
certain spin frequency, and I am using EOS BBB2(Baldo et al. 1997) as an example of that
sort of EOS. The stiffer the EOS the less the compressibility the EOS will be and smaller the
expansion in the radius will be when the star spins fast, I am using EOS L(Pandharipande
et al. 1976) as an example of this sort of EOSs. I have used EOS APR (Akmal et al. 1998)as
an example of intermediate stiffness EOS. The ultimate goal of this thesis is to find the
acceleration due to gravity for these different EOSs and find a curve fitting the theoretical
data of the output given by the RNS code.
CHAPTER 2
Spherically symmetric neutron stars in relativity
2.1
Equilibrium structure
We need to construct a well-defined coordinate system to find the good analytic description
of the gravitational field of a non-rotating star. Then we need to construct the metric tensor, gµν which determines the geometry of space-time. The non-rotating equilibrium stellar
configurations are spherically symmetric.
We shall use spherical coordinates (r, θ, φ) where 4πr2 is the surface area of the sphere about
the center of star, and (θ, φ) are the angular coordinates on the sphere. The time coordinate
t, is chosen such that, the geometry of space-time is independent of t and invariant under
time reversal. Very far from the star at (r → ∞) the coordinate time t, is identical to the
proper time measured by the clock of an observer at rest with respect to the star.
In order to find the gravitational field of a star, we will define two regions, the interior region
which is composed of a perfect fluid, and the exterior region which is vacuum. Applying the
boundary condition will enable us match the interior and exterior region at the surface of
the star, this is defined at coordinate where r = R. If the star is static, the interior metric
can be written as (Misner, Thorne, & Wheeler 1973)
ds2 =
X
gµν dxµ dxν = −e2Φ dt2 + (1 − 2m(r)/r)−1 dr2 + r2 dθ2 + r2 sin2 θdφ2 ,
(2.1)
µ,ν
where the geometry of space-time depends upon two gravitational potentials Φ(r) and m(r).
8
CHAPTER 2. SPHERICALLY SYMMETRIC NEUTRON STARS IN RELATIVITY
9
Birkhoff’s theorem makes us know that for any spherically symmetric asymptotically flat
vacuum gravitational field is always static and always Schwarzschild solution. (Misner,
Thorne, & Wheeler 1973) The exterior of the neutron star is therefore a Schwarzschild
geometry. The metric for a spherically symmetric neutron star of a mass M and a radius
R can be written for the exterior of the star in the following form:
ds2 = −(1 −
2M 2
dr2
+ r2 dΩ2 .
)dt +
r
1 − 2M
r
(2.2)
Where dΩ2 = dθ2 + sin2 θdφ2 Which must be matched to the interior metric equation 2.1,
at the surface of the neutron star.
2.1.1
Einstein field equations
Using geometrized units where G = c = 1, Einstein field equation can be written as
1
Gαβ = Rαβ − gαβ R = 8πTαβ ,
2
(2.3)
where Rαβ and R are Ricci curvature tensor and scalar curvature derived from the metric
gαβ . The expression on the left represents the curvature of space-time as determined by the
metric and the expression on the right represents the matter-energy content of space-time.
The Einstein Field Equations can then be interpreted as a set of equations dictating how
the curvature of space-time is related to the matter-energy content of the universe. The
energy-momentum tensor for a perfect fluid is given by
Tαβ = ( + p)uα uβ + pgαβ ,
(2.4)
where uα is a unit timelike vector field representing the 4-velocity of the fluid, is the energy
density, p is the pressure and gαβ is the spacetime metric. Since the space outside the stellar
distribution is empty, the energy-momentum tensor Tαβ vanishes so we get the Einstein field
equation for a Schwarzschild spherically symmetric metric is given by
Rαβ = 0.
2.2
(2.5)
The equations of stellar structure
The equations of stellar structure are the equations that describe the physical situation of
the star. It consists of a set of differential equations involving mass, pressure, temperature
and density. The simplest approximation is when spherical symmetry is considered. The
CHAPTER 2. SPHERICALLY SYMMETRIC NEUTRON STARS IN RELATIVITY 10
Einstein field equations lead to the following set of equations which are known as the equations of stellar structure (Thorne 1967).
The mass equation is
dm
= 4πr2 , m(0) = 0,
dr
(2.6)
where is the density of mass energy. At the surface of the star, we have the matching condition m(R) = M . The Tolman−Oppenheimer−Volkoff equation of hydrostatic equilibrium
is
dp
−( + p)(m + 4πr3 p)
=
.
dr
r(r − 2m)
(2.7)
The source equation for the gravitational potential, Φ, is
m + 4πr3 p
dΦ
=
,
dr
r(r − 2m)
(2.8)
which always satisfies Φ(0). (Thorne 1967).
2.3
Equations of state
The equation of state (EOS) is a relation between the fluid’s pressure, density and temperature T. Since we are interested in degenerate stars, we will work in the T = 0 approximation,
so the pressure is just a function of density.
The EOS of dense matter is very important for neutron star structure calculations, in normal situation the neutron star is strongly degenerate, which means the matter pressure is
temperature independent except for newly born stars. The correct EOS is not known. That
is why we try different EOSs and compare them to the observational data. At high density
the equation of state is not affected by the magnetic field or by temperature. To find a
neutron star structure and its maximum allowable mass we use the tabulated EOS.
The sets of outputs from the tabulated EOS give us an idea of what the mass of a neutron
star will be if it has the physical properties specified by the EOS tabulated. This gives us
an approximation but not exact answer since the EOS is still not well known. EOSs are
usually tabulated and we have to use interpolation between constraints set by observations.
I use three different equation of states: APR, BBB2, and L.
1-EOS BBB2, (Baldo et al. 1997) is one of the softest EOS allowed by observations.
CHAPTER 2. SPHERICALLY SYMMETRIC NEUTRON STARS IN RELATIVITY 11
2-EOS APR, (Akmal et al. 1998) is of intermediate stiffness.
3-EOS L, (Pandharipande et al. 1976) is a very stiff equation of state.
Logarithmic plot of the pressure vs. the density tabulated in the 3 EOSs
40.0
EOS APR
EOS BBB2
EOS L
35.0
logP
30.0
25.0
20.0
15.0
10.0
5.0
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
log
Figure 2.1: Logarithmic plot of pressure versus density values originally found at
the 3 EOSs used in this thesis
Mass - Radius Curve For Zero Spin Stars
2.8
2.6
2.4
2.2
M(M_Sun)
2
1.8
1.6
1.4
1.2
1
0.8
eosBBB2
eosAPR
eosL
9
10
11
12
13
14
15
16
R(Km)
Figure 2.2: Mass-Radius curves for zero spin star with different EOSs.
CHAPTER 2. SPHERICALLY SYMMETRIC NEUTRON STARS IN RELATIVITY 12
Figure 2.2 shows the mass-radius curves for the three EOS used in this thesis. Each point on
one of the curves corresponds to a solution to the equations of stellar structure (2.6)-(2.8).
Just as white dwarf stars have a maximum mass,(Schutz 2009) know as the Chandrasekhar
limit, neutron star also have a maximum mass. The maximum mass depends on the EOS.
Since it is not known what the correct EOS is, the maximum mass for neutron stars is
not known. However, causality limits the mass to be M ≤ 3.0Msun (Chitre D. M.). We
can see that the maximum mass can be obtained from the stiffest EOS L. An intermediate
maximum mass is obtained by the intermediate stiffness EOS APR. The stable region is on
the right hand side of the maximum for each of the curves presented in Figure 2.2. Radius
increase when M decreases as a result from matter degeneracy that can be deduced from
mass volume relation.
1015 e15
gm/cm3
3.1
2.6
1.5
EOS
BBB2
APR
L
MM ax
M
1.9
2.2
2.6
R
km
9.5
10.1
13.7
Table 2.1: Maximum mass for non-rotating stars of 3 different equation of states
According to observation we adjust our sets of neutron star EOS used, for example the EOS
A (Arnett & Bowers 1977) is no longer valid because observation indicates that there are
higher mass neutron stars that can’t be formed by EOS A. EOS BBB2 is the softest EOS
that allows a star with M = 1.93M and spin frequency to 317 Hz (Demorest et al. 2010)
2.4
Acceleration due to gravity for a spherical star in
general relativity
The proper distance is r0 defined by
dr0 =
dr
.
1/2
(1 − 2 M
r )
(2.9)
Proper distance is the invariant spacelike path of simultaneous events, in which distance
measured in an inertial frame of reference. The buoyant force density, which is responsible
for lift fluid through a proper distance r0 , as measured by local observer at the surface of
the star is (Thorne 1967)
Fbouy =
dp
.
dr0
(2.10)
CHAPTER 2. SPHERICALLY SYMMETRIC NEUTRON STARS IN RELATIVITY 13
On the other hand in general relativity and in the spherical star approximation the equation
of hydrostatic equilibrium for outside the star is given by (2.7), which gives the pressure
gradient inside the neutron star
dp
( + p)(m + 4mπr3 p)
=−
.
dr
r(r − 2m)
(2.11)
Then the force density can be written as
Fbouy = (1 −
2M 1/2 dp
)
.
r
dr
(2.12)
In Newtonian physics
Fbouy
= g,
(2.13)
so using a similar definition of Fbouy in relativity
Fbouy
= g.
+p
(2.14)
At the surface of the star
g=G
(M + 4πR3 p(R))
GM
1
= 2
,
M 1/2
1/2
R
R2 (1 − 2 M
)
(1
−
2
R
R)
(2.15)
since we are using the ideal star approximation at p(R) = 0.
2.5
Coordinate notes
It is useful to introduce a different radial coordinate r̄, known as the isotropic radial coordinate. The isotropic radial coordinate, is related to the Schwarzschild radial coordinate r,
by
r = r̄(1 +
M 2
) .
2r̄
(2.16)
As r̄ → ∞, the two radial coordinates coincide.
The differential of the isotropic coordinate is
dr = dr̄((1 +
M 2
M
M
) + 2r̄(1 +
)(− 2 )),
2r̄
2r̄
2r̄
(2.17)
M
M
)(1 −
).
2r̄
2r̄
(2.18)
so
dr = dr̄((1 +
CHAPTER 2. SPHERICALLY SYMMETRIC NEUTRON STARS IN RELATIVITY 14
At r = 2M at the horizon, we have
2M = r̄(1 +
M 2
M
M2
) = r̄(1 +
+ 2 ).
2r̄
r̄
4r̄
It follows that
2M r̄ = r̄2 + M r̄ +
and
0 = r̄2 − M r̄ +
So the horizon is at r̄ =
M2
4r̄
M 2
M2
= (r̄ −
) .
4
2
(2.19)
(2.20)
(2.21)
M
2 .
Since,
1−
therefore
2M
2M
=1−
,
2
r
r̄(1 + M
2r̄ )
1
M 4 2
dr2 = (1 +
) dr̄ .
2M
2r̄
(1 − r )
(2.22)
(2.23)
The Schwarzschild metric is given by
ds2 = −(1 −
2M 2
1
)dt +
dr2 + r2 dΩ2 ,
r
(1 − 2M
)
r
(2.24)
so using the isotropic radial coordinate, the metric is
ds2 =
2
−(1 − M
M 4 2
2r̄ )
) [dr̄ + r̄2 dΩ2 ]
dt2 + (1 +
M 2
2r̄
(1 + 2r̄ )
(2.25)
The last term in the last equation [dr̄2 + r̄2 dΩ2 ] is the metric of three dimensional Euclidean
space.
CHAPTER 3
Rotating neutron stars in general relativity
3.1
Properties of relativistic rotating stars of perfect
fluid
In this chapter to find the mathematical formulation of my thesis, we need to calculate the
acceleration components. The assumption of stationary means that there is time translation
symmetry generated by the Killing vector
tα = δtα ,
(3.1)
where δβα is the Kronecker delta. Axial symmetry is generated by the second Killing vector,
φα = δφα .
(3.2)
As a result of the two killing vectors, the metric component will have the property that,
∂
gαβ
∂t
∂
gαβ
∂φ
=
0
(3.3)
=
0.
(3.4)
The metric for an axi-symmetric and stationary space-time is given by (Komatsu et al. 1989)
ds2 = −eγ+ρ dt2 + eγ−ρ r2 sin2 θ(dφ − ωdt)2 + e2α (dr2 + r2 d2 θ),
(3.5)
where the metric potentials γ, ρ, α, ω are independent of t, φ. In the limit of zero rotation
the metric (3.5) reduces to the isotropic Schwarzschild metric (2.17).
15
CHAPTER 3. ROTATING NEUTRON STARS IN GENERAL RELATIVITY
16
The stress tensor for a perfect fluid is
T αβ = ( + p)uα uβ + pg αβ ,
(3.6)
where p is the pressure and is the density. The fluid four velocity is defined by
uα = N (tα + Ω∗ φα ),
(3.7)
where N is the normalization factor and Ω∗ is the constant angular velocity as measured by
an observer at infinity.
3.1.1
Numerical method
We use a code that utilizes an integration method based on Green function theory. A
Green’s function is a type of function used to solve inhomogeneous differential equations
subject to specific initial conditions or boundary conditions. First we have to transform
the differential form of the basic equations into integral form, which enables us to handle
boundary conditions in a much easier way.
The equations for the potential are the non-linear Einstein field equation and can be represented by differential equations, for example the equation for metric potential ρ is of the
form (Komatsu et al. 1989)
∇2 ρ = Sρ (r, θ, , p, α, ρ, γ, ω),
(3.8)
where Sρ (r, θ) is the source term. There are similar equations for the other potentials. The
source terms depend on the energy density , the pressure p, and the angular momentum of
the fluid j which vanishes outside the stellar surface, the gravitational potentials and their
derivatives which do not vanish outside stellar surface up to the quadratic order. For a full
discussion please see (Cook et al. 1992),(Komatsu et al. 1989).
We prepare at the beginning an initial guess for the metric potential by using the simplest
case possible. For a star with no rotation and spherical symmetry and get a first guess of
ρ, γ, α, ω and the energy density and the angular velocity Ω. Then, substituting them into
the integrand, we obtain new value ρ, γ, α, ω using the newly obtained values of ρ, γ, α, ω we
calculate new energy density and new angular velocity Ω. This is one cycle of iteration.
This newly obtained value of α, β, γ, α and Ω, is an improved set of new guess in the next
iteration cycle. As the iterations continue, finally the differences between each value of
calculated values become very small. When the difference is small enough this is when we
stop. In this method the quantities held fixed at each iteration are the ratio of
rp
re
of values
of the coordinate r at the pole and equator as well as the central energy density .
CHAPTER 3. ROTATING NEUTRON STARS IN GENERAL RELATIVITY
17
If one does not follow the above procedure to assure the smoothness of the potentials,
errors introduced in the physical quantities can range from ∼ 0.5 in the mass and radius to
∼ 2 in other quantities such as the angular velocity, angular momentum, and redshifts for
maximum mass models since we have four gravitational potential ρ, γ, α, ω.
3.1.2
Mass-Radius curves for rotating star
The mass-radius curves for a rotating star give us an explanation of the physical properties
of the star with different spin frequencies. For example from the curves shown in Figures
3.1 to 3.3 we can see that as the frequencies increases the stable configuration of the star
shifts to higher mass and greater radius. The Kepler limit is defined to be the highest spin
frequency that a star can have before the centrifugal force tears it apart. The bulge starts to
be bigger with higher frequencies close to the Kepler frequencies which is a normal output
of the centrifugal force. Each point on the figure represents a solution of the Einstein Field
equation.
The mass radius curves for EOS BBB2 shown in Figure 3.1.
Mass - Radius Curve For Spining Stars EOS BBB2
2.4
2.2
2
M(MSun)
1.8
1.6
1.4
1.2
0 Hz
300 Hz
600 Hz
Kepler limit
1
0.8
9
10
11
12
13
14
15
16
R(Km)
Figure 3.1: The mass radius curves for different spinning frequencies for EOS BBB2.
The mass radius curves for EOS APR shown in Figure 3.2.
The mass radius curves for EOS L shown in Figure 3.3. It is noteworthy that, most observed
stars have spin frequencies less than 600Hz, and the highest observed spin frequency up to
date is 716 Hz.(Hessels et al. 2006)
CHAPTER 3. ROTATING NEUTRON STARS IN GENERAL RELATIVITY
18
Mass - Radius Curve For Spining Stars EOS APR
2.8
2.6
2.4
2.2
M(MSun)
2
1.8
1.6
1.4
1.2
0 Hz
300 Hz
600 Hz
Kepler limit
1
0.8
10
11
12
13
R(Km)
14
15
16
Figure 3.2: The mass radius curves for different spinning frequencies for EOS APR
Mass - Radius Curve For Spining Stars EOS L
3.5
3
M(MSun)
2.5
2
1.5
1
0 Hz
300 Hz
600 Hz
Kepler limit
12
13
14
15
16
17
R(Km)
18
19
20
21
22
Figure 3.3: The mass radius curves for different spinning frequencies for EOS L
3.2
Acceleration due to gravity
The use of the two killing vectors implies that metric components are independent of t and φ.
The freedom in the normalization of uα allows us to write down the standard normalization,
CHAPTER 3. ROTATING NEUTRON STARS IN GENERAL RELATIVITY
− 1 = uα uα .
19
(3.9)
Substituting (3.7) into equation (3.9) we have
− 1 = N 2 (tα tα + Ω∗ tα φα + Ω∗ tα φα + Ω2∗ φα φα ).
(3.10)
tα tα = tα tβ gαβ = gtt ,
(3.11)
tα φα = tα φα = gtφ ,
(3.12)
φα φα = gφφ .
(3.13)
− 1 = N 2 (gtt + 2Ω∗ gtφ + Ω2∗ gφφ ).
(3.14)
Since
and similarly
and
Equation 3.10 can be simplified to
The relevant metric components are
gtt = e(ρ+γ) (−1 + e−2ρ ω 2 r2 sin2 θ),
(3.15)
gtφ = −e(γ+ρ) (e−2ρ r2 sin2 θ)ω,
(3.16)
gφφ = eγ+ρ e−2ρ r2 sin2 θ,
(3.17)
and gφφ
which come from equation (3.5). After substituting equations (3.15) and (3.17)-(3.14) we
can calculate the normalization function N (r, θ).
The normalization factor, in terms of the metric function is given by
N −2 = eγ+ρ (1 − e−2ρ ω 2 r2 sin2 θ − Ω2∗ e−2ρ r2 sin2 θ + 2ωΩ∗ e−2ρ r2 sin2 θ).
(3.18)
We can define a speed by the following form
v = (Ω∗ − ω)e−ρ r sin θ
(3.19)
The circumference of the star at some latitude θ is given by C = 2πr sin θe−ρ . Squaring the
speed we get
v 2 = (Ω2∗ + ω 2 − 2ωΩ∗ )e−2ρ r2 sin2 θ
(3.20)
CHAPTER 3. ROTATING NEUTRON STARS IN GENERAL RELATIVITY
20
So we can rewrite the normalization factor with this new formula
N −2 = (1 − v 2 )eγ+ρ ,
(3.21)
for any Killing vector ζa , has to satisfy the Killing equation
∇(α ζβ) = 0,
(3.22)
where the symmetrization is denoted by parentheses and anti-symmetrization is denoted by
square brackets, defined by
∇(α ζβ) =
1
(∇α ζβ + ∇β ζα );
2
∇[α ζβ] =
1
(∇α ζβ − ∇β ζα ).
2
(3.23)
We can introduce a new killing vector ζ which is a linear combination of tα and φα
ζ α = t α + Ω ∗ φα .
(3.24)
The four velocity of can be defined as
uα = N ζ α .
(3.25)
− N −2 = ζ α ζα ,
(3.26)
Noting that
we have
uα =
ζα
1
(−ζγ ζ γ ) 2
.
(3.27)
In order to calculate the acceleration, we first make use of the definition
aβ = uα ∇α uβ .
(3.28)
Substituting equation 3.27 into 3.28 we find that
aα = uβ ∇β uα = ζ β
3
∇β ζ α
ζβ
+ ζα ζ γ ∇β ζγ (−ζξ ζ ξ )− 2
1 .
γ
(−ζγ ζ )
(−ζµ ζ µ ) 2
(3.29)
This can be simplified to
aα = uβ ∇β uα = uβ
∇β ζα
(−ζγ
1
ζγ) 2
+ ζα ζ β ζ γ ∇β ζγ (−ζξ ζ ξ )−2 .
(3.30)
CHAPTER 3. ROTATING NEUTRON STARS IN GENERAL RELATIVITY
21
Since ζ α is a Killing vector then
ζ γ ζ β ∇β ζγ = ζ(γ;β) ζ γ ζ β = 0,
(3.31)
since ζ γ ζ β is symmetric under interchange of indices. This causes the second term in
equation (3.30) to vanish, so that the acceleration is,
aα = −
ζ β ∇β ζ α
.
ζ γ ζγ
(3.32)
Since
∇β ζα + ∇α ζβ = 0,
(3.33)
due to Killings equation it follows that
∇β ζα = −∇α ζβ .
(3.34)
Then, substituting (3.24) into (3.32), the acceleration is
aα =
∇α ζ β ζ β
.
ζ γ ζγ
(3.35)
A useful property of any vector ζ α is
∇α (ζβ ζ β ) = ζ β ∇α ζβ + ζβ ∇α ζ β = 2ζ β ∇α ζβ .
This allows us to write
aα =
1
1 ∇α (ζβ ζ β )
= (ln(ζβ ζ β )),α .
γ
2 ζγ ζ
2
(3.36)
(3.37)
The acceleration vector aα , using equation (3.21) is,
aα =
1
1
∂(γ + ρ)
∂v
∂α ln[N −2 ] =
[
(1 − v 2 ) − 2v α ]
2
2(1 − v 2 ) ∂x α
∂x
(3.38)
This can be decomposed in two components the angular and radial components. Since the
four metric potentials γ,ρ,ω,α depend only r and θ, from equation (3.19)
v,r = e −ρ r sin θ(−ρ,r (Ω∗ − ω) +
1
(Ω∗ − ω) − ω,r ).
r
(3.39)
The angular derivatives of the velocity contributes to the angular component of the acceleration due to gravity. The angular derivative of velocity in the angular component is given
CHAPTER 3. ROTATING NEUTRON STARS IN GENERAL RELATIVITY
22
by the following equation
v,θ = e −ρ r sin θ(−ρ,θ (Ω∗ − ω) +
cos θ
(Ω∗ − ω) − ω,θ ).
sin θ
(3.40)
Then the components of the acceleration are given by the radial and angular components.
The radial component is given by
ar =
1 ∂(γ + ρ)
v
1
−
(e−ρ r sin θ(−ρ,r (Ω − ω) + (Ω − ω) − ω,r )).
2
∂r
(1 − v 2 )
r
(3.41)
The angular component is given by
aθ =
v
cos θ
1 ∂(γ + ρ)
−
(e−ρ r sin θ(−ρ,θ (Ω − ω) +
(Ω − ω) − ω,θ ),
2
∂θ
(1 − v 2 )
sin θ
(3.42)
which vanishes at the poles. The magnitude of the acceleration is given by
a2 = gαβ aα aβ .
(3.43)
The direction of acceleration is normal to the surface. Finding the surface of the star from
the RNS code is a task that requires me to make the code calculate the value of the zero
enthalpy, which correspond to the stellar surface, I used the four points interpolation routine
to get the best approximation of where is the stellar surface.
I modified the code so that it not only calculates the equilibrium model of sequences, a
routine which finds the value of zero enthalpy which is a new defined quantity. Normally
in Newtonian physics, we define surface of star where the pressure vanishes at p = 0. In
relativity the quantity
dp
,
p+
(3.44)
appears in the equation of hydrostatic equilibrium. In relativistic computations a relativistic
quantity called ”enthalpy”, h, is defined by
dh =
dp
+p
(3.45)
This enthalpy is not exactly the same as the quantity normally defined in Newtonian physics.
I used the four point interpolation technique to find the zero enthalpy at various θ on the
surface of neutron star. This step is to find the values on the surface which we will be used
later to find other values like the acceleration. By using the same technique we should be
able to find the potential functions at the surface of the neutron star, which we needed to
and calculate the values of the acceleration due to gravity.
CHAPTER 3. ROTATING NEUTRON STARS IN GENERAL RELATIVITY
23
The original RNS code utilizes 2 new coordinates, the variables s and µ defined by
s=
r
,
r + re
(3.46)
and
µ = cos θ.
(3.47)
The model center is defined at s = 0. The surface of the star at the equator is s = 0.5 and
infinity at s = 1. The equatorial plane is at θ =
π
2,
and the spin axis is at θ = 0. The
code uses SDIV divisions in the s-direction and MDIV divisions in the µ direction. The grid
points is defined by MDIV × SDIV, the default is 65 × 129. I used higher accuracy for my
calculation (higher grid sizes means better accuracy) as we will see in chapter four.
I made a modification so that we can apply different spin frequencies giving us a series of
equilibrium values, This can be easily done by giving the code instructions on the lowest and
the highest spin frequency that we need the code to use. The original code calculates the
potentials derivatives with respect to code coordinates s and µ, but I want to calculate my
differentials with respect to r for the radial coordinates (3.46) and with respect to θ instead
of µ in the angular coordinates (3.47). I made the conversion between the code coordinate
and the physical coordinate so we can get the desired values.
Then I have to find the derivatives of the metric four functions α, ω, γ, ρ with respect to
both radial and angular co-ordinates. a conversion has to be done in order to compute the
correct derivatives with respect to the radial components.
Then I have to find the velocity values that corresponds to Zero Angular momentum Observer (ZAMO) (3.19) and find the normalization factor. N defined in equation (3.21) which
will be used to find the values of the acceleration. I then used the routine that I have used
earlier to interpolate the surface of star to find the values of the four metric potentials
and their derivatives on the surface. Then after calculating the potential values and their
derivatives on the surface of the star we can implement these results into the code to find
the acceleration on the surface of the star. The code outputs different files according to the
stellar rotation and outputs two separate output files for values of the acceleration on the
equator and on the pole. I use those values to plot graphs of the acceleration variation over
the latitude.
CHAPTER 3. ROTATING NEUTRON STARS IN GENERAL RELATIVITY
3.2.1
24
Hydrostatic Equilibrium
To motivate the acceleration due to gravity we start by using conservation law. For a perfect
fluid the stress tensor is given by
T αβ = ( + p)uα uβ + pg αβ ,
(3.48)
where p is the pressure, is density and the conservation law is
0 = ∇β T αβ .
(3.49)
q αβ = g αβ + uα uβ
(3.50)
We define the projection tensor
where q αβ used to project tensor to the subspace perpendicular to uα . The projection tensor
has a property that it is orthogonal to uα since we have
uα q αβ = 0.
(3.51)
Using q αβ ,we can rewrite the stress tensor in the following form
T αβ = uα uβ + pq αβ .
(3.52)
Then
∇α T αβ = ∇α (uα uβ +pq αβ ) = ∇α uα uβ +uα ∇α uβ +uα uβ ∇α +q αβ ∇α p+p∇α q αβ . (3.53)
Since uα is proportional to a Killing vector, that
uα ∇α = uα ∇α p = 0.
(3.54)
This allows the simplification of (3.53)
∇α T αβ = uβ ∇α uα + uα ∇α uβ + g αβ ∇α p + p∇α (g αβ + uα uβ ).
(3.55)
Using the basic property of covariant differentiation
∇α g βγ = 0,
(3.56)
CHAPTER 3. ROTATING NEUTRON STARS IN GENERAL RELATIVITY
25
equation (3.55) can be simplified to
∇α T αβ = uβ ∇α uα + uα ∇α uβ + g αβ ∇α p + puβ ∇α uα + uα p∇α uβ .
(3.57)
As a result
0 = ∇α T αβ = (p + )(uβ ∇α uα + aβ ) + q αβ ∇α p.
(3.58)
There are two projection of the conservation law. The timelike projection is
uβ ∇α T αβ = 0.
(3.59)
0 = uβ ((p + )(uβ ∇α uα + uα ∇α uβ ) + q αβ ∇α P ).
(3.60)
0 = (p + )(uβ uβ ∇α uα + uα uβ ∇α uβ ).
(3.61)
∇α (uβ uβ ) = uβ ∇α uβ + uβ ∇α uβ = 2uβ ∇α uβ
(3.62)
Expanding (3.59) using (3.58)
A simple identity is
Since uα uα = −1, ∇α (uβ )(uβ ) = 0 so uβ ∇α uβ = 0. Hence (3.62) implies that ∇α uα Since
∇α uα = 0, and ∇α (uβ uβ ) = 0 it follows that from equation (3.58) reduce to
0 = ∇α T αβ = (P + )aβ + q αβ ∇α P.
(3.63)
From equation (3.63), the spacelike projection of the conservation law is
0 = qβγ ∇α T αβ = (p + )qγβ aβ + qγβ q αβ ∇α P.
(3.64)
qγβ q αβ = qγα .
(3.65)
aα uα = 0,
(3.66)
q αβ aβ = aα .
(3.67)
Simplify using (3.50)
Since
Therefore the equation of the hydrostatic equilibrium results:
q αβ ∇β p = ( + p)aα .
(3.68)
CHAPTER 3. ROTATING NEUTRON STARS IN GENERAL RELATIVITY
26
Using equation (3.50)
1
∇β p = −∇β ln N,
(p + )
(3.69)
which is the equation of hydrostatic equilibrium. We started with the perfect fluid stress
tensor to the equation of hydrostatic equilibrium, which has a term for the acceleration due
to gravity aα
The importance of this equation is that it is an equivalent formulae of the Newtonian form
of the hydrostatic equilibrium and that the form of the acceleration due to gravity can be
deducted by using equation(2.15).
CHAPTER 4
Results
4.1
The dependence of the acceleration on latitude
In this chapter I will present my results. The effective acceleration due to gravity is the net
acceleration that an object will feel on the surface of neutron star. It is the resultant from
the outward centrifugal force and the inward acceleration due to gravity. I will present a
series of data and results for different EOSs.
The effective acceleration due to gravity at the surface of a Newtonian spherical neutron
star that is rotating depends on the colatitudes θ and is given by
gef f (θ) =
GM
(1 − sin2 θ),
R2
(4.1)
where is defined by
=
Ω2 R 3
.
GM
(4.2)
Equation (4.1) suggests a quadratic dependence on cos θ.
We expect the equatorial acceleration to be given by the following equation at θ = 90◦
geq =
GM
(1 − )
R2
(4.3)
and the polar acceleration is given by the following equation at θ = 0
gpole =
.
27
GM
R2
(4.4)
CHAPTER 4. RESULTS
28
For a more realistic Newtonian rotating star, we expect a more complicated expression since
the star’s shape is distorted by with rotation into an oblate spheroid. The shape is given by
an equation of the form (Morsink et al. 2007)
R(θ) = Req (1 − a0 cos2 θ) + O(2 )
(4.5)
where a0 is a constant that could be determined by numerical modeling. From this we can
find the effective acceleration at any angle by the following equation
g=
GM
Ω2 R3 (θ) sin2 θ
(1
−
),
R2 (θ)
GM
(4.6)
which will yield an equation quadratic in cos θ. which can be expanded for small g=
GM
(1 − (1 − cos2 θ)(1 − 3a0 cos2 θ)).
R2 (1 − 2a0 cos2 θ)
(4.7)
For small we expect an expression of the the form
g=
GM X
b2n cos2n θ)
(
R2 n
(4.8)
Now consider a relativistic nonrotating neutron star with an acceleration given by equation
(3.30),
g=
GM
1
1 .
2
R (1 − 2 M
2
R)
(4.9)
We expect adding rotation to a relativistic star will add terms proportional to cos2n θ just
as in equation (4.8). For this reason, we expect that
g=
GM
1
q
R2
1−
(
2M
R
X
b2n cos2n θ) =
n
GM
1
q
R2
1−
X
2M
R
b2n µ2n .
(4.10)
n
will be a good approximation for small . In the thesis we use µ = cos θ in our calculations.
Now introduce the scaling factor
R2
GM
√
and define a dimensionless normalized acceler2m
1−
ation, ḡ, given by
ḡ =
R
g
q
GM
1−
R2
.
(4.11)
2m
R
In the limit of zero rotation we have
lim ḡ → 1
Ω→0
(4.12)
CHAPTER 4. RESULTS
29
everywhere on the star’s surface.
Note on the symbols used
We use µ = cos θ .
ε is used to indicate the star’s central energy density.
M0 is the rest mass, means with no rotation or gravitational effects.
M is the gravitational mass. ΩK is Kepler angular velocity.
4.1.1
Accuracy used in computing our models
In order to test the dependence of the code on the grid size, I calculated the effective
acceleration using the formulae in chapter 3 for one stellar model using EOS APR, with a
central energy density ε = 0.8 × 1015 g/cm3 and rratio = 0.8 using a variety of different grid
sizes. The results are shown in table 4.1 and figure 4.1. When moving from a grid size of
MDIV×SDIV=101×201 to one almost 50% larger with a grid size of 151 ×301, the relative
change in the value of aeq is one part in 105 , as we can see in Table 4.1.
Due to small changes in the acceleration when larger grid size are used, I decided that 101 ×
201 was the smallest grid size that could be used without causing numerical inaccuracies.
My data output is based on the public domain code RNS with some modification so that it
can compute the effective acceleration due to gravity. The difference that I’ve noticed with
different grid-sizes, are the values of M and R that you get when you change the grid size.
This is where I’ve noticed that it is best to use 101×201 instead of the 65×129 this is shown
for the table 4.1.
Grid Size
MDIV×SDIV
41×71
51×101
65×129
101×201
151×301
201×401
M
M
1.607
1.606
1.605
1.605
1.604
1.604
Req
km
12.627
12.626
12.627
12.629
12.630
12.630
Spin
Hz
935.8
935.7
937.7
938.0
938.1
938.1
geq
Scaled
7.392943e-01
7.415181e-01
7.410094e-01
7.407124e-01
7.405804e-01
7.404662e-01
gP ole
Scaled
1.321275e+00
1.319250e+00
1.319494e+00
1.319434e+00
1.319897e+00
1.320140e+00
Table 4.1: EOS APR, RNS code output for different grid sizes for a value of central
energy density value of εc = 0.8 × 1015 g/cm3 and rratio = 0.8
CHAPTER 4. RESULTS
30
The figure for scaled acceleration versus µ for various accuracy with different grid sizes with
accuracy shows no obvious deflection in accuracy shown in the figure 4.1
EOS APR, c=0.8x1015g/cm3
1.4
Scaled Acceleration
1.3
Grid size=101x201
Grid size=65x129
Grid size=151x301
Grid size=201x401
Grid size=51x101
Grid size=41x71
1.2
1.1
1.0
0.9
0.8
0.7
0.0
0.2
0.4
0.6
0.8
1.0
µ
Figure 4.1: The scaled acceleration versus µ for different grid accuracies
4.2
Detailed results for EOS APR
In this section I display detailed results of the RNS code output of neutron star models
with different central energy densities. The results will be displayed in sequences of tables
and graphs showing the dependence of acceleration on latitude. I use three different energy
densities, low, central, and high. The code is capable of producing outputs of data up to
just below Kepler spin frequency.
4.2.1
Fitting for equations used
The fitting equation (4.11) is expected to have the form to be quadric equation in µ, similar
equation (4.1) in the limits of slow rotation. For more rapid rotation we expect that the
terms higher order in µ will contribute in equation (4.11). So we begin by trying out the
quartic fit of the equation
ḡ(µ) = b0 + b2 µ2 + b4 µ4 ,
(4.13)
CHAPTER 4. RESULTS
31
where b0 , b2 , b4 are the fitting parameters that are determined by the numerical computer
program. A measure of the rotation can be given by rratio , which is the ratio of polar radius
over the equatorial radius. A value of rotation close to 1 means that the star is spinning
very slow. In the verge of very high rotation at rratio = 0.65 the star approaches the Kepler
frequency.
4.2.2
Low energy density values for EOS APR
Let’s begin with a low central energy density equal to 0.7×1015 g/cm3 . Table 4.2 shows the
values of mass, radius and spin for a sequence of stars computed with this equation of state
and central energy density. The data in table 4.2 was generated by the RNS code.
rratio
e15
10 g/cm3
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
0.7
15
1.000
0.950
0.900
0.850
0.800
0.750
0.700
0.650
0.600
M
M
0.839
0.861
0.884
0.910
0.938
0.967
0.996
1.021
1.037
M0
M
0.886
0.910
0.936
0.963
0.993
1.024
1.055
1.083
1.100
R
km
11.371
11.677
12.019
12.405
12.847
13.360
13.969
14.715
15.670
spin
Hz
0.0
367.8
517.2
628.2
717.6
790.9
849.7
893.1
917.0
ΩK
Hz
1384.2
1346.2
1311.6
1273.8
1231.3
1182.3
1124.1
1053.0
964.0
Table 4.2: EOS APR, RNS code output for low ε= 0.7 × 1015 g/cm3
In Figure 4.2, plots of the scaled acceleration versus µ = cos θ for 3 values of rratio are
shown. Due to rotation the effective acceleration is not constant and varies according to the
latitude. The curve labeled rratio = 0.95 corresponds to the slow rotation of 367.8 Hz, a
mass of 0.861M , and an equatorial radius of 11.7 km, as can be seen from Table 4.2. The
curve labeled rratio = 0.65 corresponds to very high rotation with spin frequency=893.1 Hz,
at which spin the star has a huge bulge. As expected at the higher the spin frequency the
quartic fit is not as good fitting as at lower frequencies. At the equator the acceleration is
lower than the acceleration at the pole as expected and the difference between the two values
of acceleration because the effective acceleration (acceleration due to gravity-centrifugal
acceleration). The fitting parameters’ values are given in table 4.5.
CHAPTER 4. RESULTS
32
EOS APR, c=0.7x1015g/cm3
1.8
1.6
rratio=0.95
rratio=0.75
rratio=0.65
g(µ,0.95)
g(µ,075)
g(µ,0.65)
Scaled Acceleration
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
µ
Figure 4.2: Dependence of acceleration on the colatitudes. The scaled acceleration
is displayed on the vertical axis and on the horizontal axis are the values of µ = cos θ.
The thick dotted curves are the curves produced by the RNS code and thin dotted
line is the one produced by the quartic fit. The EOS used is APR, and the stellar
models are shown in table . The central energy density for all models 0.7×1015 g/cm3
4.2.3
Intermediate energy density values for EOS APR
For the next set of results we picked an intermediate energy density of 1 × 1015 g/cm3 . Now
we present the results for the acceleration gradient over the latitude. The stars values for
the intermediate central energy density, given by the RNS code, is shown in table 4.3.
The graphs of acceleration versus the latitude for the intermediate value of central energy
density for 3 values of spin are shown in figure 4.3. The values for the best-fit parameters
of the quartic fit are shown in table 4.5. It can be seen that the results for intermediate
energy density show similar trends as the low energy density results.
4.2.4
High energy density values for EOS APR
The results for EOS APR and a high value of energy density of 2.5 × 1015 g/cm3 are shown
in table 4.4.
The acceleration gradient for this high energy density with the latitude is shown in figure
4.4.
CHAPTER 4. RESULTS
rratio
33
e15
1015 g/cm3
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.000
0.950
0.900
0.850
0.800
0.750
0.700
0.650
0.600
M
M
1.441
1.476
1.515
1.557
1.605
1.656
1.710
1.764
1.809
M0
M
1.603
1.643
1.686
1.735
1.788
1.847
1.908
1.969
2.020
R
km
11.378
11.641
11.933
12.259
12.629
13.052
13.545
14.135
14.867
spin
Hz
0.0
475.8
671.4
818.2
938.0
1037.9
1121.1
1187.4
1233.4
ΩK
Hz
1812.4
1758.0
1718.9
1678.6
1634.6
1585.1
1527.0
1456.1
1365.6
Table 4.3: EOS APR, RNS code output for intermediate energy density ε = 1.0 ×
1015 g/cm3 .
EOS APR, c=1.2x1015g/cm3
1.8
Scaled Acceleration
1.6
rratio=0.95
rratio=0.75
rratio=0.65
g(µ,0.95)
g(µ,075)
g(µ,0.65)
1.4
1.2
1.0
0.8
0.6
0.4
0.0
0.2
0.4
0.6
0.8
1.0
µ
Figure 4.3: The scaled effective acceleration versus µ for EOS APR for medium
energy density
The fitting parameters for the EOS APR with the different energies densities are shown in
table 4.5.
CHAPTER 4. RESULTS
rratio
34
e15
1015 g/cm3
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
1.000
0.950
0.900
0.850
0.800
0.750
0.700
0.650
0.600
M
M
2.226
2.270
2.315
2.366
2.424
2.480
2.542
2.598
2.646
M0
M
2.711
2.756
2.802
2.855
2.916
2.975
3.041
3.102
3.158
R
km
10.152
10.320
10.505
10.716
10.960
11.233
11.557
11.940
12.416
spin
Hz
0.0
779.4
1102.8
1343.0
1536.9
1694.9
1821.1
1914.9
1971.3
ΩK
Hz
2673.8
2550.6
2495.3
2445.7
2394.9
2339.5
2273.6
2192.2
2083.6
Table 4.4: EOS APR, RNS code output for high energy density value of ε = 2.5 ×
1015 g/cm3 .
EOS APR, c=2.5x1015g/cm3
1.6
Scaled Acceleration
1.4
rratio=0.95
rratio=0.75
rratio=0.65
g(µ,0.95)
g(µ,075)
g(µ,0.65)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
µ
Figure 4.4: The scaled effective acceleration versus µ for EOS APR for high energy
densityε = 2.5 × 1015 g/cm3
4.2.5
Results of the acceleration gradient for EOS L
We now turn to the stiffer EOS, EOS L. Table 4.6 shows the values of mass, radius and spin
for a few values of central energy density and rratio .
The plots of acceleration versus the latitude are shown in figures 4.5 (low density), 4.6
(medium density), and 4.7. The figures show trends similar to those seen for EOS APR.
CHAPTER 4. RESULTS
εc
0.7
1.2
2.5
0.7
1.2
2.5
0.7
1.2
2.5
35
r-ratio
0.95
0.95
0.95
0.75
0.75
0.75
0.65
0.65
0.65
b0
0.944738
0.954168
0.950383
0.62903
0.689017
0.659544
0.401597
0.490804
0.452541
b2
0.138964
0.118277
0.119064
1.3235
1.087
1.14817
2.61915
2.10982
2.22351
b4
-0.013023
-0.0125854
-0.0127808
-0.514094
-0.422196
-0.482097
-1.3107
-1.05851
-1.18722
Table 4.5: Best fitting parameters for EOS APR
rratio
1
0.95
0.75
0.65
1
0.95
0.75
0.65
1
0.95
0.75
0.65
e15
1015 g/cm3
0.4
0.4
0.4
0.4
1.0
1.0
1.0
1.0
1.6
1.6
1.6
1.6
M
M
1.227
1.262
1.435
1.539
2.638
2.694
2.983
3.161
2.709
2.763
3.03
3.186
M0
M
1.307
1.344
1.533
1.647
3.119
3.182
3.511
3.715
3.227
3.285
3.578
3.753
R
km
14.696
15.079
17.144
18.737
14.428
14.709
16.209
17.352
13.556
13.801
15.135
16.173
spin
Hz
0
294.3
639.3
731
0
469.2
1026.2
1171.8
0
540.3
1174
1330.2
ΩK
Hz
1139.4
1110
993.9
905
717.6
1652.3
1506.7
1405
1911.5
1831.6
1667.5
1552.5
Table 4.6: RNS code output for EOS L.
The values of the best-fit parameters for EOS L are shown in table 4.7. From the table
we can see that the fitting parameters values trends are similar to value presented to EOS
APR, which suggest a universal fitting equation that is good for all.
4.2.6
Results of the acceleration gradient for EOS BBB2
Finally, we present similar results for the softest EOS, EOS BBB2. The plots of the acceleration versus the latitude are shown for the low energy density in the figure 4.8.
The intermediate energy density plots are shown in figure 4.9.
The high energy density plots are shown in figure 4.10.
CHAPTER 4. RESULTS
36
EOS L, c=0.4x1015g/cm3
1.8
Scaled Acceleration
1.6
rratio=0.95
rratio=0.75
rratio=0.65
g(µ,0.95)
g(µ,075)
g(µ,0.65)
1.4
1.2
1.0
0.8
0.6
0.4
0.0
0.2
0.4
0.6
0.8
1.0
µ
Figure 4.5: The scaled effective acceleration versus µ for EOS L for low energy
density
EOS L, c=1.0x1015g/cm3
1.6
Scaled Acceleration
1.4
rratio=0.95
rratio=0.75
rratio=0.65
g(µ,0.95)
g(µ,075)
g(µ,0.65)
1.2
1.0
0.8
0.6
0.4
0.0
0.2
0.4
0.6
0.8
1.0
µ
Figure 4.6: The scaled effective acceleration versus µ for EOS L for medium energy
density
The best-fit parameters for EOS BBB2 are shown in table 4.8.
CHAPTER 4. RESULTS
37
EOS L, c=1.6x1015g/cm3
1.6
rratio=0.95
rratio=0.75
rratio=0.65
g(µ,0.95)
g(µ,075)
g(µ,0.65)
Scaled Acceleration
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
µ
Figure 4.7: The scaled effective acceleration versus µ for EOS L for high energy
density
c
0.4
1.0
1.6
0.4
1.0
1.6
0.4
1.0
1.6
r-ratio
0.95
0.95
0.95
0.75
0.75
0.75
0.65
0.65
0.65
b0
0.951572
0.955002
0.951735
0.675733
0.688636
0.668544
0.471293
0.491443
0.463983
b2
0.124734
0.113661
0.119085
1.15847
1.07031
1.13624
2.26269
2.07084
2.202
b4
-0.0113135
-0.0102174
-0.0113894
-0.442206
-0.421538
-0.459647
-1.11718
-1.05263
-1.14194
Table 4.7: Best fitting parameters for EOS L
.
4.3
Dependence of the star acceleration on the star’s
properties
We noticed many similar trends in tables 4.5, 4.7 and 4.8. We now explore the dependence
of the value of ḡ on the star’s physical parameters. There are two dimensionless parameters
CHAPTER 4. RESULTS
38
EOS BBB2, c=0.7x1015g/cm3
1.8
1.6
rratio=0.95
rratio=0.75
rratio=0.65
g(µ,0.95)
g(µ,075)
g(µ,0.65)
Scaled Acceleration
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
µ
Figure 4.8: This scaled acceleration versus µ for low value of energy density for
EOS BBB2
EOS BBB2, c=2.0x1015g/cm3
1.8
1.6
rratio=0.95
rratio=0.75
rratio=0.65
g(µ,0.95)
g(µ,075)
g(µ,0.65)
Scaled Acceleration
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
µ
Figure 4.9: This scaled acceleration versus µ for medium value of energy density
for EOS BBB2
ζ and , given by
ζ=
GM
,
Req c2
(4.14)
CHAPTER 4. RESULTS
39
EOS BBB2, c=3.0x1015g/cm3
1.8
1.6
rratio=0.95
rratio=0.75
rratio=0.65
g(µ,0.95)
g(µ,075)
g(µ,0.65)
Scaled Acceleration
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
µ
Figure 4.10: This scaled acceleration versus µ for high value of energy density for
EOS BBB2
εc
0.7
2.0
3.0
0.7
2.0
3.0
0.7
2.0
3.0
r-ratio
0.95
0.95
0.95
0.75
0.75
0.75
0.65
0.65
0.65
b0
0.944738
0.95212
0.949406
0.62903
0.669138
0.65415
0.401597
0.463014
0.443784
b2
0.138964
0.120127
0.123807
1.3235
1.14902
1.19599
2.61915
2.23669
2.3244
b4
-0.013023
-0.0106903
-0.0120492
-0.514094
-0.456549
-0.489898
-1.3107
-1.14334
-1.21557
Table 4.8: EOS BBB2
where Req is the radius of the rotating star measured at the equator, and
=
3
2
Ω2 Req
Ω2 Req
=
,
GM
c2 ζ
(4.15)
where Ω = 2π/P , and P is the spin period.
Our hypothesis is that the effective acceleration due to gravity depends on and ζ. In
this section we will test a simple polynomial dependence of the acceleration on these two
parameters. We will use a simple empirical equation for the effective acceleration due to
CHAPTER 4. RESULTS
40
gravity of the form
ḡ(µ) = c0 + c1 ζ + c2 ζ 2 + c3 2 + c4 ζ 3 ,
(4.16)
where the cn are unknown coefficients that may depend on the EOS. For zero rotation = 0
and ḡ = c0 , so we expect c0 = 1, although we will allow this parameter to be freely varied.
For each EOS, we test the empirical formula at two points on the star, the spin axis and
the equator. The values of the acceleration at the two points for all of the models computed
in the previous sections were plotted versus and ζ and a best fit for the parameters in
equation (4.16) were found.
4.3.1
Results for EOS APR
The equatorial acceleration versus ζ and is shown in figure 4.11. The spin rates range
from zero spin up to the Kepler limit.
Equatorial Acceleration, EOS APR
1.1
1.0
Accel 0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.34
0.32
0.30
0.28
0.26
0.24
0.22
0.20
0.18
0.16
0.14
0.12 0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Figure 4.11: A plot of the scaled equatorial acceleration versus ζ and for EOS
APR. We start by plotting a set of energies densities (Min energy density = 0.8 ×
1015 g/cm3 to Max energy density of 2.5 × 1015 g/cm3 in sets of 13 increments). The
equation of the fitting surface is given byc0 + c1 ζ + c2 ζ 2 + c3 2 + c4 ζ 3 .
The fitting surface is a very good fit according to the fitting parameters value indicated in
the table below. We find that the 4th term is required to describe the shape of the largest,
most rapidly rotating stars. However, higher order terms are not required.
The polar acceleration for EOS APR for the same stars is shown in figure 4.12.
CHAPTER 4. RESULTS
41
Polar Acceleration, EOS APR
2.0
1.9
Accel 1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1.0
0.34
0.32
0.30
0.28
0.26
0.24
0.22
0.20
0.18
0.16
0.14
0.12 0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Figure 4.12: Plot of the scaled polar acceleration versus ζ and for EOS APR
The fitting parameters for the graphs for the equatorial and polar accelerations are shown
in table 4.9.
Plane
Equatorial
Polar
c0
0.994+/-0.003
1.005+/-0.002
c1
-7.063+/-0.347
12.289+/-0.260
c2
31.452+/-2.359
-58.516+/-1.765
c3
-0.539+/- 0.020
0.300+/-0.015
c4
-44.035+/-4.559
80.615+/-3.411
Table 4.9: Best fitting parameters for EOS APR
.
4.3.2
Results for EOS L
The graphs of equatorial acceleration for EOS L for different energy densities, are shown in
figure 4.13.
The graph for the polar acceleration is shown in figure 4.14.
The values for the fitting parameters used to generate the fitting surfaces for EOS L are
shown in table 4.10.
CHAPTER 4. RESULTS
42
Equatorial Acceleration, EOS L
1.1
1.0
Accel 0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.05
0.10
0.15
0.20
0.25
0.30 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Figure 4.13: A plot of the scaled equatorial acceleration versus ζ and for EOS L.
Polar Acceleration, EOS L
Accel
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1.0
0.05
0.10
0.15
0.20
0.25
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.30
Figure 4.14: A plot of the scaled polar acceleration versus ζ and for EOS L
4.3.3
Results for EOS BBB2
The fitting parameters table that used to generate the fitting surfaces for the EOS BBB2 is
given by table 4.11.
CHAPTER 4. RESULTS
Plane
Equatorial
Polar
c0
0.987+/-0.004
1.025+/-0.007
43
c1
-11.238+/- 0.773
13.784+/-0.868
c2
71.077+/-5.568
-85.162+/-6.215
c3
-0.567+/- 0.048
0.468+/-0.040
c4
-130.172+/-11.59
151.717+/-13.190
Table 4.10: EOS L, RNS code fitting parameters output for different energy density
values.
Plane
Equatorial
Polar
a
0.992+/0.002
1.005+/-0.001
b
-7.543+/-0.3444
-1.489+/-0.151
c
37.066+/-2.346
7.295+/-1.029
d
-0.577+/-0.018
0.564+/-0.008
e
-57.579+/-4.641
-13.223+/-2.036
Table 4.11: EOS BBB2, RNS code fitting parameters output for different energy
density values.
.
The equatorial acceleration graph versus and ζ is shown in figure 4.15. The polar acceler-
Equatorial Acceleration, EOS BBB2
Accel
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.05
0.10
0.15
0.20
0.25
0.30 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Figure 4.15: A plot of the scaled Equatorial acceleration versus ζ and for EOS
BBB2.
ation graph versus and ζ is shown in figure 4.16.
CHAPTER 4. RESULTS
44
Polar Acceleration, EOS BBB2
Accel
2.2
2.0
1.8
1.6
1.4
1.2
1.0
0.05
0.10
0.15
0.20
0.25
0.30 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Figure 4.16: A plot of the scaled polar acceleration versus ζ and for EOS BBB2.
4.3.4
Results for EOSs combined
We now wish to test if the depends of ḡ onζ and is independent of the EOS. To see if
our fitting equation works on all EOSs, I have tested the fitting equation by plotting the
acceleration for all three EOS and finding the best-fit surface for the polar and equatorial
accelerations.
The equatorial acceleration for all 3 EOS is shown in figure 4.17 and the polar acceleration
is shown in figure 4.18.
As we can see from the graph that the fitting graph is almost perfect for all equation of
state used and I expect it to be the same way with a wide variety of EOSs with different
stiffness.
The universal fitting parameters for the graphs for the equatorial and polar acceleration are
shown in table 4.12.
Plane
Equatorial
Polar
c0
0.994+/-0.003
1.005+/-0.002
c1
-7.037+/-0.353
12.255+/-0.263
c2
31.134+/-2.418
-58.154+/-1.806
c3
0.539+/-0.020
0.300+/-0.015
Table 4.12: Universal best fitting parameters for the 3 EOSs combined
.
c4
-43.302+/-4.698
79.803+/-3.511
CHAPTER 4. RESULTS
45
Equatorial Acceleration, EOSs(APR, L, and BBB2)
Accel
1.1
1.0
1.0
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.16
0.18
0.20
0.22
0.24
0.26
0.28
0.0
0.30
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.4
0.4
0.5
Figure 4.17: A plot of the equatorial acceleration scaled versus ζ versus for all
EOSs..
Polar Acceleration, EOSs(APR, L, and BBB2)
Accel
1.5
1.5
1.4
1.4
1.3
1.3
1.2
1.2
1.1
1.1
1.0
0.16
0.18
0.20
0.22
0.24
0.26
0.28
0.5
0.4 0.4
0.3 0.3
0.2 0.2
0.2
0.1
0.30 0.0 0.1
Figure 4.18: A plot of the scaled polar acceleration versus ζ and for all EOSs.
We found that this fit is a good approximation to our graphs and the accuracy will even
become better if we added some extra fitting parameters and the margin of error become
very low. The output values are for < 0.5 for higher values of the surface starts to differs.
CHAPTER 5
Conclusion
In this thesis I investigated the effective acceleration due to gravity on the surface of a
rapidly rotating relativistic neutron star. I first reviewed the definition of acceleration on
the surface of a star with a Newtonian gravitational field in the introductory chapter. In
the second chapter I reviewed the definition of acceleration due to gravity on the surface of
a relativistic star which is not rotating.
In the third chapter, I used the equation of hydrostatic equilibrium and relativistic physics to
extend the definition of acceleration to a relativistic spinning star. The metric and structure
of a rotating relativistic star are computed by an existing computer code, RNS (Stergioulas
& Friedman 1995). The modifications I made for the RNS code enabled me to make use
of the output which yields stable stellar models. I modified the code to control the spin
frequency and get the star’s equilibrium values for a certain spin frequency up to the Kepler
frequency. Those modifications enabled me to do further modifications to get the values
of the metric potentials (and their derivatives) at the surface of a rotating neutron star by
doing four-point interpolation. Assuming the enthalpy value at the surface goes to zero, I
was able to implement the four point interpolation to calculate the stellar surface at different
latitudes and different spin values. was used to calculate the acceleration components and
the effective acceleration at the different latitudes on the stellar surface.
In the fourth chapter, I presented the results of my numerical calculations of the effective
acceleration. I presented my data for a wide spectrum of equations of states (EOS) represented by a soft, intermediate, and stiff EOS. I was able to get the maximum possible values
of the acceleration by choosing models with the highest energy density possible and highest
spin. I showed how the effective acceleration depends on latitude on the star’s surface and
how it depends on the properties of the star, such as the mass, radius and spin frequency. I
was able to find an empirical equation that yields a good fit to my numerical results at low
46
CHAPTER 5. CONCLUSION
to medium high spinning frequencies.
47
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