Theoretical Analysis of theVibrational Dynamics of Neutron

Star Interiors

by

Jonathan M. Hartman

Dissertation

submitted in fulfilment of the requirements for the degree

Master of Science

in

Physics

in the

Faculty of Science,

University of Johannesburg

Supervisor: Dr C.A. EngelbrechtCo-Supervisor: Dr F.A.M. FrescuraCo-Supervisor: Prof. C.M. Villet

May 2009

ABSTRACT

Just as the observations of oscillations of ordinary stars can be used to determinetheir composition and structure, the oscillations of neutron stars could potentially beused to determine the nature of the dense nuclear matter from which they are made.The superfluidity of the interiors of neutron stars is normally probed by observationsof pulsar glitches. It turns out that the superfluidity affects the oscillations in aneutron star core. In particular, it results in a class of oscillation modes specificallyassociated with the superfluid core. Although these modes have not been detectedfrom observations, it is hoped by some that gravitational wave data may be usedto probe the superfluidity of neutron star cores. In this dissertation, a simpleequilibrium model is used in order to calculate the superfluid modes in the contextof newtonian gravity. The equilibrium model that is used is the same combinationof the Serot equation of state and the Harrison-Wheeler equation of state that wasused formerly by Lee and by Lindblom & Mendell. Numerical calculations of thesuperfluid modes are done for 20 different neutron star models ranging in massbetween 0.5 and 2 solar masses. The frequencies of the oscillations for the 0.5 and1.4 solar masses agree fairly well with Lees results, which strongly validates thecomputer code written for numerical calculation in this work. In all the models, theeigenfrequencies of the superfluid or s-modes are found among those of the f andp-modes. For the equation of state that is used, it is shown that the dimensionlessfrequencies of the p-modes increase with an increase in mass of the neutron starwhile those of the s-modes decrease with an increase in neutron star mass.

The plan of the dissertation is as follows. Chapter 1 gives a short introductionto stellar oscillations and mentions the oscillations of neutron stars. Chapters 2and 3 provide the general theoretical background of stellar structure and stellaroscillations respectively. Chapter 4 is a review of the equations of state of neutronstar matter derived previously in the literature. Chapter 5 provides the method ofcalculation as well as the results. Chapter 6 provides a discussion of the results.Chapter 7 briefly gives a review of a mathematical framework for fluids that couldbe used in order to calculate the oscillations in a general relativistic context andthen briefly describes the effects of rotation and magnetic fields. Appendix B liststhe source code for the programs used to do the calculations and also explains someof the extra numerical procedures used for the computation.

Acknowledgements

I wish to dedicate my dissertation to four very special people.

Judy Symons, former principal of Grantley College, where I matriculated in1999 and Prof Roux Botha, Prof Dirk van Reenen and the late Mrs Elize Albertynof RAU now known as UJ. Your faith in me has enabled me to fulfill my dreamof becoming a physicist. I will be eternally grateful to you all for those wonderfulopportunities.

To my supervisor, Chris Engelbrecht, Thank you for your patience and dedi-cation and for sharing your incredible knowledge with me, yet still allowing me todiscover things for myself.

To my co-supervisors Fabio Frescura and Prof Charles Villet, Your help andguidance was invaluable.

To Yoric Hardy, Justin Prentice and Thebe Mdupe, Thanks for your supportwhen I needed it.

To Prof Kinta Burger and staff in the Faculty of Science, especially NaomiStrydom, Karien van den Berg and Ferdi van der Walt, What would my Mom andI have done without you? Thanks a million.

To Prof Andre Strydom and the staff in the Department of Physics, Wevebeen together for so long, you guys are like family. Your encouragement has meantso much to me.

To Anlia Pretorius, Dr Paul de Wet and Brinton Spies, Im most grateful foryour guidance along my life path.

To the National Research Foundation, Im grateful for the bursaries you granted,for both my Honours and Masters degrees. They have enabled me to continue withmy studies and become more independent as a student at UJ.

To Autism South Africa for their ongoing initiative in informing the generalpublic about Autism, and in my case, Asperger Syndrome, I hope Ive helped inbreaking the mould!.

I am saddened that my grandparents, who always believed that I could achieveanything that I put my mind to, are no longer here to enjoy this achievement withme.

Finally, and most importantly, Im indebted to my parents, Gerrit and Lorraine,my sister Jaclyn and our extended family and special friends for their unconditionallove and daily encouragement. Mom, you taught me the freedom, as you alwaysput it, to fly with the angels ... and dance with the stars. Thank you to you all!.

The Cosmic religious experience is the strongest and noblest driving force behindscientific research - Albert Einstein, quoted in his obituary, 19 April 1955.

3

CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2. An overview of stellar structure . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Hydrostatic equilibrium and polytropic equations of state . . . . . . 7

3. Theoretical background of stellar oscillations . . . . . . . . . . . . . . . . 113.1 Linear theory of stellar oscillations . . . . . . . . . . . . . . . . . . . 113.2 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3 Adiabatic case and boundary conditions . . . . . . . . . . . . . . . . 173.4 Forms of oscillation equations used for calculation of models . . . . . 193.5 Numerical calculation of oscillation models . . . . . . . . . . . . . . 21

4. Dense nuclear matter inside neutron stars . . . . . . . . . . . . . . . . . . 254.1 Review of equations of state of neutron star matter . . . . . . . . . . 254.2 Oscillations in neutron stars . . . . . . . . . . . . . . . . . . . . . . . 30

5. Oscillations in the cores of superfluid neutron stars . . . . . . . . . . . . . 345.1 Equilibrium model and equation of state . . . . . . . . . . . . . . . . 345.2 Oscillations in spherically symmetric superfluid neutron stars . . . . 395.3 Form of superfluid oscillation equations used for numerical calculation 425.4 Results of the oscillation calculations . . . . . . . . . . . . . . . . . . 48

6. Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

7. Conclusions and plans for the future . . . . . . . . . . . . . . . . . . . . . 67

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Appendix 75

A. Derivation of Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . 76

B. Computer Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78B.1 Matlab/Octave programme that calculates the Lane Emden func-

tions for the polytrope . . . . . . . . . . . . . . . . . . . . . . . . . . 79B.2 Matlab/Octave programme that implements the finite-difference scheme

for the oscillations in the polytrope model . . . . . . . . . . . . . . . 81B.3 Matlab/Octave programme used for calculating the oscillations in

the polytrope model . . . . . . . . . . . . . . . . . . . . . . . . . . . 82B.4 C++ programme used for calculating equilibrium models . . . . . . 83B.5 Programme for adding neutron and proton densities to MAT file . . 101B.6 C++ programme for neutron star oscillations . . . . . . . . . . . . . 116B.7 Matlab/Octave programme used to calculate neutron star oscillations 146

1. INTRODUCTION

This chapter is a short introduction which briefly describes stellar oscillations beforementioning the oscillations of neutron stars.

In general, observations of stellar oscillations can be be used to determine thecomposition of the matter making up stars. Theoretically, stellar oscillations aremodeled in the language of fluid mechanics since stars in general are made up of gasand/or plasma. For calculations of basic stellar structure, hydrostatic equilibriumis assumed, in other words the force due to gas pressure as well as radiation pressurebalances with gravity. When there is no rotation, the star would assume a perfectspherical shape. However, when there is rotation, the centrifugal force due to therotation distorts the spherical shape by creating a bulge at the equator. In the caseof multiple star systems, when a star has a massive nearby companion, then theshape may be further distorted by tidal forces.

When the stellar oscillations are small, they may be taken into account as smallperturbations on the equilibrium configuration and this gives rise to the linear theoryof oscillations. The oscillations are modeled as waves propagating through the fluid.When the oscillations are radial, they propagate in the radial directions, in otherwords they propagate from the centre outwards to the surface and the star wouldappear to expand and contract while maintaining its shape. When the oscillationsare nonradial the waves propagate in the angular directions or along meridionaland/or azimuthal directions in the star and so the shape of the star gets distorteddue to the propagating waves. Imagine a wave that propagates in the angulardirections of the star, and say it starts at one point. It will eventually returnto that point if it moves right around the star. When thinking about nonradialoscillations in that way, it is easy to understand that only an integral number ofwaves can appear in the angular directions, otherwise destructive interference willprevail. In the case of spherical symmetry (i.e. when the star is non-rotating), thisdiscrete set of wave modes that can propagate around a sphere is described by theset of mathematical functions called the spherical harmonics (described in chapter3).

In non-rotating stars, there are in general three kinds of oscillations. They arecalled f -modes, g-modes and p-modes. If we consider a single fluid element in thestar and imagine that this fluid element moves due to a buoyant force, the gravitywill restore it to the original position and so it acts as a restoring force. Thismovement is then responsible for creating oscillations in the star, which are calledthe g-modes (the g stands for gravity waves). Pressure modes or p-modes are drivenby fluctuations of the pressure. The f -modes or fundamental modes are just surfacegravity waves. In any particular spectrum of modes for a given value of l (for themeaning of l, see the section on spherical harmonics in chapter 3), there is onlyone f -mode and its frequency lies between those of the g1-mode and the p1-mode.When the effects of rotation are taken into account, the star in the equilibriumstate is no longer spherical and Coriolis forces, due to the rotation acting on thefluid, result in the r-modes. The r-modes will not be discussed in this dissertationbecause rotation is not considered in the models described in this thesis.

Because the behavior of the waves propagating through the gas is dependenton the material that makes it up, stellar oscillations can be used to determine the

composition of matter in the star.The structure of a neutron star consists of a solid crust, inner core and outer core.

The solid crust in the neutron star is made up of a metallic lattice. The outer coreconsists of free neutrons in a superfluid state with a small admixture of electronsand protons in a superconducting state. The inner core may consist of deconfinedquarks. Although the crust is solid, oscillations can occur in the superfluid core ofneutron stars. If the oscillations in neutron star cores are detected, they might beused to deduce the composition of the dense nuclear matter that makes up the cores,just as observations of the oscillations in ordinary stars are used to determine theircomposition. It was first suggested by Epstein (1988) [24] that the superfluidityin neutron stars leads to a further class of modes. Mendell (1991) [51] developedthe adiabatic oscillation equations for the superfluid core of neutron stars and thenLindblom and Mendell (1994) [46] used these equations to calculate the effectsof superfluidity on the neutron star oscillations, both with and without rotation.Lindblom and Mendell found the oscillation modes associated with the superfluidcore (now called the superfluid modes and which they called s-modes) analyticallyin a simplified model but not in their numerical analysis. Lee (1995) [44] foundthe superfluid modes amongst the f -modes and p-modes in his numerical analysis,using the same equilibrium model as Lindblom and Mendell (1994) [46] but calledthem -modes.

Observations of the superfluid modes associated with the superfluid cores ofneutron stars have not been made thus far, so the superfluid properties of neutronstar interiors have been probed mainly by the observations of pulsar glitches [47] ,[41]. So far, all the papers dealing with observations of neutron star oscillations inthe literature, such as in X-ray bursters, refer to observations in accreting matterand confined to the surface (eg. McDermott and Taam [50]) but there is no recordof observed oscillations in the interior of neutron stars. The 424 ms period reportedby Zavlin et al (2000) [67] does not correspond to the oscillations calculated in thisdissertation since the longest calculated period is only 136 s. The interperetationby Zavlin et al that the 424 ms period is due to rotation is therefore probablycorrect. However, the periods calculated here could be different for a more realisticequation of state and may be refined when rotation and electromagnetic fields aretaken into account. Most current telescopes dont have the time resolution to seeperiods in the order of micro-seconds but MeerKAT and the SKA might be able todetect more neutron stars than the the ones currently known and interior oscillationsmight eventually be detected.

The oscillations of neutron star interiors have also been calculated in the contextof general relativity [7], [21], [9], [40]. It is hoped that gravitational wave data mayprovide a probe into neutron star superfluidity [6], [19].

This thesis describes the pulsational behaviour of a simplified two-componentmodel of a neutron star across a broad range of neutron star masses. Where theresults overlap with those published by Lindblom and Mendell, and Lee respectively,good agreement is found with their results. However, the results described here alsodemonstrate the effect of neutron star mass on the pulsational behaviour. The aimof this work was also to develop numerical tools that can be used for further studiesof neutron stars.

3

2. AN OVERVIEW OF STELLAR STRUCTURE

This chapter gives an overview of stellar structure described in the language of fluiddynamics.

In general, stars can be thought of as balls of gas which are held together bygravity, which is opposed by forces due to gas pressure and radiation pressure to keepthem from collapsing. The interior of a star can be divided into an inner core andan outer envelope. In the interior, as we increase the depth from the surface of thestar, gravity acts more strongly on the gas molecules which as a result become moreenergetic and so the temperature increases. When we get to the core, the ionisedatoms are so energetic that the nuclei fuse together forming heavier elements andreleasing radiation (a radiation pressure gradient is also one of the forces opposinggravity). In a star with a similar mass to our sun, hydrogen nuclei will fuse togetherto form helium. When the hydrogen is used up, the core of the star will initiallycollapse while the temperature rises. Since helium has a higher ignition temperaturethan hydrogen, this collapse continues until the temperature is high enough for thehelium nuclei to fuse together to form carbon. Eventually the helium will be usedup as well. When this happens, the core collapse of our low mass star continuesuntil it is stopped by electron degeneracy pressure at which point it becomes awhite dwarf. Paulis exclusion principle states that there can only be one fermionper energy state. At densities as high as those in white dwarf matter, every energystate from the ground state upward is occupied by an electron and so due to Paulisexclusion principle electrons cannot fall into lower energy states. The result of thisis that the electrons will be extremely energetic, producing a pressure strong enoughto oppose gravity. In stars that are initially much more massive than our sun, thecore collapse after the helium is used up may raise the temperature high enoughto allow the fusion of carbon into heavier elements. The heaviest element that canbe formed in these more massive stars is iron. When most of the matter in thecores these heavy stars is fused into iron then the core will continue to collapse andthis time electron degenracy pressure will not be strong enough to oppose it. Thecollapse will then force inverse beta decay between the electrons and protons in thenucleus to form neutrons. Neutron degeneracy pressure is stronger than electrondegeneracy pressure and this may be strong enough to oppose the gravitationalcollapse thus resulting in a neutron star. If the star exceeds the mass limit of aneutron star it will collapse to a black hole unless the hypothesis of quark starsproves correct. As yet no convincing evidence of the latter has been found.

2.1 Fluid Dynamics

Since stars are considered to be made up of gas, fluid dynamics would be theappropriate language to describe them on scales much larger than the mean freepath of the constituent molecules. This section is concerned with setting up thebasic equations of fluid dynamics describing the conservation of mass, momentumand energy in fluids. Hydrodynamics is concerned with the motion of fluids on amacroscopic scale. As such the fluid is described as a continuous medium, so whenspeaking of a fluid particle what is meant is not an individual atom or molecule

Fig. 2.1: Volume within a body of fluid; the mass flowing out of the closed surface is equalto the decrease in mass inside the volume

but an infinitesimal volume in the medium. In order to reconcile the fact thatreal fluids are made up of atoms or molecules, this infinitesimal volume can bethought of as being large with respect to the size of individual molecules, in factcontaining many molecules, even though it is small with respect to the total volumeof the fluid, (Landau and Lifshitz [39]). The equation of continuity which describesthe conservation of mass in fluid dynamics can be argued for as follows. Withina body of fluid, consider a closed surface containing a volume V0 of the fluid, asshown in figure 2.1. The mass of this volume is

dV where is the density of

the fluid. For mass to be conserved, the net mass of the fluid flowing out throughthe closed surface must be equal to the decrease in mass inside it. If the magnitudeof the vector d~f is taken as an area element of this surface while pointing alongthe outward normal then the mass of fluid flowing through this volume element is~v d~f . So by conservation of mass;

ddt

dV =

S

~v d~f. (2.1)

Where S is the total surface area containing the volume. By the divergence theoremS~v d~f = ~ (~v)dV , therefore

t+ ~ (~v) = 0. (2.2)

The above equation is called the equation of continuity. If we again consider someinfinitesimal volume of fluid and take p to be the fluid pressure acting on the surfacebounding the volume, then the total force acting on the volume of fluid is

Spd~f .

Remembering thatdV is the mass of the volume then by Newtons second law

of motion we haveS

pd~f =d~v

dtdV. (2.3)

By applying an extension of the divergence theorem to the pd~f term to get itas a volume integral and then equating the integrands, (2.3) can be rewritten as

d~v

dt= ~p. (2.4)

The d~vdt term describes the rate of change of the velocity of a given fluid particleas it moves from a position ~r to a position ~r in space in an infinitesimal time dt.However, given the continuum assumption, the space is filled with a material whosevelocity field also changes in time. So to write this in terms of rate of change in

5

fluid velocity at given points in space, notice that the fluid particle changes positionby an infinitesimal amount d~r = ~r ~r in the time dt, so the change in velocity ofthe fluid particle d~v when it reaches the position ~r is equal to the difference in fluidvelocity in the velocity field over the distance d~r , ie dx ~vx +dy

~vy +dz

~vz = (d~r ~)~v,

plus the change in fluid velocity at position ~r in time dt, ~vt dt, thus

d~v =~v

tdt+

(d~r ~

)~v (2.5)

or by dividing both sides by the infinitesimal value dt

d~v

dt=~v

t+(~v ~

)~v. (2.6)

We get Eulers equation by substituting (2.6) into (2.4)

~v

t+(~v ~

)~v = 1

~p. (2.7)

Eulers equation here is derived for ideal fluids, i.e. no viscosity. To take accountof viscosity, the more general Navier-Stokes equation is used:

[~v

t+(~v ~

)~v

]= ~f ~p+ ~ ~~, (2.8)

where ~~ is the viscous stress tensor. The term ~f is added to take account ofexternal forces.

In ideal fluids, the conservation of energy is expressed as

t

(12v2 +

)= ~

[~v

(12v2 + w

)], (2.9)

The left hand side is the decrease in energy per unit volume, where is the internalenergy per unit mass, and the right hand side is the energy flux through the surfacebounding the volume, where w is the specific enthalpy. When there is viscosity,there is also an energy flux resulting from internal friction given by ~v ~~. If thetemperature is not constant throughout the fluid, there will be heat transfer fromregions of higher temperature to regions of lower temperature. In this case, thereis an added term describing thermal conduction: K~T , where K is the thermalconductivity. For viscous fluids with a temperature gradient, (2.9) can be expressedmore generally as

t

(12v2 +

)= ~

[~v

(12v2 + w

) ~v ~~ K~T

]. (2.10)

Expanding the left hand side of equation (2.10) gives

t

(12v2 +

)=

12v2

t+ ~v ~v

t+

t+

t(2.11)

Then using the continuity equation (2.2) and the Navier-Stokes equation (2.8) tosubstitute for t and

~vt respectively, we have

t

(12v2 +

)= 1

2v2~ (~v)~v ~

(12v2)~v ~p+vi

ik

xk+

t ~ (~v),

(2.12)

6

where the Einstein summation convention is used in the term viikxk

i.e. sum overrepeated indices. From thermodynamics, a small change in energy, d, is given by

d = Tds+(p

2

)d, (2.13)

so

t= T

s

t+

p

2

t, (2.14)

or from (2.2),

t= T

s

t p2~ (~v). (2.15)

Note that the specific enthalpy w is given by

w = +p

(2.16)

so thatdw = d+

1dp p

2d (2.17)

or by substituting equation 2.13 for d into 2.17 we get

dw = Tds+dp

. (2.18)

we also note that~p = ~w T ~s (2.19)

and that

viikxk

= ~ (~v ~~) ikvixk

. (2.20)

Substituting the three relations (2.18), (2.19) and (2.20) plus equation (2.15) into(2.12), and adding and subtracting ~

(K~T

)on the right hand side, we get

t(12v2+) = ~

[~v

(12v2 + w

) ~v ~~ K~T

]+T

(s

t+ ~v ~s

)ik

vixk~

(K~T

).

(2.21)Substituting (2.21) into (2.10) and rearranging the equation, we arrive at a differentform for the expression for conservation of energy

T

(s

t+ ~v ~s

)= ik

vixk

+ ~ (K~T ). (2.22)

2.2 Hydrostatic equilibrium and polytropic equations of state

For a star in hydrostatic equilibrium, gravity, which pulls the gas inward toward thecenter, is counteracted by the gas pressure plus radiation pressure acting outwardfrom the center. Other forces can be neglected except under some circumstances,for example when electromagnetic phenomena need to be considered. A descriptionof the star in hydrostatic equilibrium is necessary before the pulsations of a starcan be described. We follow the treatment of Chandrasekhar [18], chapter 3, indiscussing the equilibrium conditions.

In stars, viscosity is generally small, so the viscosity terms in the fluid equations(2.8) and (2.22) can be neglected. In hydrostatic equilibrium, there is no motion in

7

the fluid, so the velocity and time derivative terms in these equations can be set tozero. The Navier-Stokes equation (2.8) then becomes

~p ~ = 0. (2.23)In the energy conservation equation, while the heat generation from internal frictiondue to viscosity is small and can be neglected, there is also a term due to generationof nuclear energy through nuclear fusion in the core. So for the energy conservation

N ~ ~F = 0, (2.24)

where ~F in this case is the flux of radiation

~F = K~T, (2.25)where the radiative conductivity K is given by

K =4ac3

T 3 (2.26)

with a being the radiation constant, c is the speed of light and is the opacity.The gravitational potential is given by the Poisson equation

2 = 4piG. (2.27)If there is no rotation, then the equilibrium configuration is spherically symmetric.If the spherical symmetry is taken into account and noting that the gravitationalacceleration is

~g = ~ = GMrr2

r, (2.28)

where Mr is the mass at a given radius, then in spherical coordinates equation(2.23) is written as

dp

dr= GMr

r2. (2.29)

A spherical distribution of matter is described by

dMrdr

= 4pir2. (2.30)

Equation (2.24) gives the rate of change of the radiative luminosity Lr = 4pir2Fwith respect to radius in spherical coordinates:

dLrdr

= 4pir2N , (2.31)

and the rate of change in temperature would be given by the equation

dT

dr= 3

4ac1T 3

Lr4pir2

. (2.32)

The equation of equilibrium is arrived at by combining (2.29) and (2.30),

1r2

d

dr

(r2

dp

dr

)= 4piG. (2.33)

In order to get the complete description, an equation of state is required in order toget a relation between the pressure p and the density . For ideal gasses, we assumethe ideal gas law pV = NkT . An adiabatic change of state is a quasi-static changein which there is no heat transferred to or from the fluid, so changes in temperature

8

are due to changes in pressure of the gas. For an adiabatic change pV = constant,where the adiabatic index is given by the ratio of the specific heats = cpcV . If we

define the specific heat when the parameter is constant as c =[dQdT

]

, then we

have c = 0 when dQ = 0 (the adiabatic case), c when dT = 0 (the isothermalcase), c = cp when the pressure remains constant and c = cV when the volumeremains constant. For a polytropic change pV = constant, where

=cp ccv c . (2.34)

An adiabatic change then is the specific case of a polytropic change with c = 0.Noting that = mV with m being the mass, we have for a polytrope:

p = Cn+1n , (2.35)

where n is the polytropic index with = n+1n and C is a constant. If we considera star consisting of an ideal gas, with gravity opposed by the gas pressure thenthe pressure would increase toward the core and decrease outward along the radius.Then, if the gas expands adiabatically it would obey the relation (2.35). In orderto write equation (2.33) in terms of dimensionless quantities, consider

= n (2.36)

where is the central density when considering a complete polytrope, ie the relation(2.33) with the same polytropic index holds throughout the star. The pressure isthen given by

p = C1+1n n+1. (2.37)

Also consider

r =[

(n+ 1)C4piG

1n1

] 12

. (2.38)

When the equilibrium equation (2.33) is rewritten in terms of the new variables, weget the Lane-Emden equation

12

d

d

(2d

d

)= n (2.39)

with index n and the boundary conditions

= 1,d

d= 0 (2.40)

at = 0. The Lane-Emden equation can only be solved analytically for threevalues of n, namely for indices 0, 1 and 5. For any other polytropic index, theLane-Emden equation has to be solved numerically. Seven-digit tables of calculatedsolutions to the Lane-Emden equation are given in a 1986 paper by G. P Horedt[31]. To solve the Lane-Emden equation, start with a series expansion near theorigin. For the series to satisfy the boundary conditions, there can only be termsof even powers of in order for dd to vanish at the origin. So then assume a seriessolution = 1 +

m=1 cm

2m, and substitute this into the Lane-Emden equation(2.39) and equate the like powers of to calculate the coefficients. The resultingseries should give accurate values of for < 1 with enough terms. To calculate thesolution for the entire star, including > 1, use the value calculated from the seriesat a close to the origin as the initial condition and then use another numericaltechnique such as the Runge-Kutta method to find the rest of the solution. Thesolution to the Lane-Emden equation of index 3 is given in figure (2.2).

9

Fig. 2.2: Solution to the Lane-Emden equation index 3. This was calculated with theMatlab code in section B.1 in appendix B

10

3. THEORETICAL BACKGROUND OF STELLAROSCILLATIONS

This chapter gives a general background of stellar oscillations.The theory of stellar oscillations was originally developed to explain the be-

haviour of variable stars, i.e. stars that appear to change their brightness periodi-cally. This is caused by the star periodically expanding and contracting due to wavesin the gas or fluid making up the star. Pulsations in a star are radial if the per-turbations only occur along the radius of the star, in other words the star expandsand contracts without altering its spherical shape; whereas non-radial oscillationsoccur along the surface as well and so parts of the star contract while other partsexpand in the angular directions. Actually radial oscillations are a special case ofthe non-radial oscillations as will be explained later on in this chapter. While os-cillations were originally used to explain variable stars, it was later discovered thatthey occur in other stars that were considered non-pulsating. Actually all stars arevariable to some degree, but the variability is not as great in some as in others.Since the pulsations are greatest in the variable stars, they were first discovered inthem. Since the behaviour of the oscillations depend on the material that make upthe star, an analysis of the oscillations can help determine the stars compositionand internal structure in a similar sense that an analysis of the waves in the Earthcaused by tremors and earthquakes can help determine the internal structure of theEarth. This subfield of astrophysics is called asteroseismology (helioseismology inthe specific case of the sun). Although they have a solid surface, neutron stars havea superfluid in the core which theoretically can also oscillate and although theseneutron star oscillations have not been observed as yet, they could give insight intothe structure of the nuclear matter that make them up. So far, the interiors ofneutron stars have usually been determined by studying a phenomenon known asglitching where the rotation of the star suddenly speeds up. Oscillations of neutronstars will be the main subject of later chapters but this chapter will be about thegeneral theoretical background of stellar oscillations as a whole. The best descrip-tion of stellar oscillations is in Ledoux and Walraven [43]. Alternative descriptionsare in Unno et al [63] and Cox [22].

3.1 Linear theory of stellar oscillations

Since, in general, stars are considered fluids in hydrostatic equilibrium, the bestway to describe their oscillations is by the use of fluid dynamics. More specifically,the oscillations are described by waves in the fluid. To do this we take the equa-tions of stellar equilibrium derived in the previous chapter and superimpose smallperturbations for each quantity. These perturbations, given a physical quantity f ,can be expressed in Eulerian or Lagrangian form. The Eulerian form describes theperturbation of the fluid at a given position (remember that the fluid is regardedas a continuum) and denoted with a prime,

f(~r, t) = f0(~r) + f (~r, t), (3.1)

while the Lagrangian form describes the perturbation of a given fluid particle, i.e.an infinitesimal volume in the fluid, and is denoted by ,

f(~r, t) = f0(~r) + f(~r, t). (3.2)

In order to relate these two forms, note that a fluid particle moves from a position~r0 to ~r in the time t due to the perturbation. Since the Eulerian perturbation isdefined as the perturbation of the fluid at a given position, it gives the change inthe quantity f in time t at a fixed position ~r, but the Lagrangian perturbation,since it describes the change in f of a fluid particle, must also take into account thedifference in f over the displacement ~ ~r ~r0 at the instant t0 = 0. The relationbetween the Lagrangian and Eulerian forms are then,

f(~r, t) = f (~r, t) + ~ ~f0(~r). (3.3)

In the previous chapter, the equation (2.5) describes the relationship between theLagrangian and Eulerian forms of the time variations of the fluid velocity, but thesame reasoning that led to this relationship will work for any physical quantity, thusfor time derivatives,

d

dt=

t+ ~v ~. (3.4)

In the linear theory of oscillations, the equations are then derived by replacing everyphysical quantity f in the equilibrium equations from chapter 2 with f + f , wherethe f represents the perturbation. The perturbations are considered small enoughthat the multiplication of two perturbations such as f g is considered negligible andignored. In the equations that follow, a subscript 0 is used to indicate a quantityin equilibrium.

Remembering that there is no motion in the equilibrium configuration in thecurrent assumptions, the perturbed form of the equation of continuity (2.2) becomes,

t+ ~ (0~v) = 0, (3.5)

the perturbed form of the Euler equation (2.7) becomes,

0~v

t+ ~p + 0~ + ~0 = 0, (3.6)

and the perturbed form of the conservation of energy 2.22 becomes

0T0

t

(s + ~ s0

)= (0N )

~ ~F . (3.7)

The perturbation in the gravitational potential given by the Poisson equation equa-tion (2.27) is given by

2 = 4piG (3.8)and that for the radiative flux (2.25) is given by

~F = K0~T K ~T0. (3.9)

Since the unperturbed state of the star is spherically symmetric under the currentassumptions, all the equilibrium state variables are functions of the radius r only.Therefore the perturbed variables can be separated as a product of their spatial andtime dependencies with the time dependency being a wave function eit where is the oscillation frequency. So the displacement is ~ = ~ (r, , ) eit and thereforethe velocity is ~v = i~ (r, , ) eit. Substituting this and = (r, , ) eit into

12

the equation of continuity (3.5) it becomes (note: the subscript 0 for equilibriumvariables from now on will be omitted for brevity),

+ ~ (~)

= 0 (3.10)

or in Lagrangian form obtained by expanding the ~ (~)

term and substitutingthe translation from the Eulerian to the Lagrangian perturbation of from (3.3):

+ ~ ~ = 0. (3.11)

When substituting ~v into the equation of motion (i.e. the Euler equation) (2.7) itbecomes

2~ + 1~p + 1

~ +

~ = 0 (3.12)

(the term ~ was added to the Euler equation to take into account the gravita-tional interaction) which can be separated into radial and tangential componentsas

2r + 1

p

r+

r+

ddr

= 0 (3.13)

and

2~ + ~(p

+

)= 0, (3.14)

where ~ = (0, , ) and ~ = 1r(

0, ,1

sin

). So therefore (3.11) can be

written as

+

1r2

r

(r2r

)+ ~ ~ = 0 (3.15)

and substituting (3.14) into this we obtain

+

1r2

r

(r2r

)+

122

(p

+

)= 0. (3.16)

Using radial and horizontal components, the Poisson equation can be written as

1r2

r

(r2

r

)+2 = 4piG. (3.17)

Using the thermodynamic relation below , where pressure and entropy are regardedas independent variables, (for a full list of thermodynamic relations relating the per-turbations of pressure, density, temperature and entropy in nonradial oscillations,section 13.4 of Unno et al [63] or sections 4.2c and 5.4c of Cox [22], the densityperturbations, and , can be eliminated from 3.13, 3.16 and 3.17 and expressedin terms of p, r and s,

=

11

p

pad T

ps (3.18)

where 1 =( ln p ln

)s

and ad =( lnT ln p

)s. If we write (3.18) in terms of the

Eulerian perturbation using the relation (3.3) (remember all equilibrium variablesare only dependent on the radius r), we get

=

11

p

pAr ad T

ps (3.19)

13

where we defineA =

d ln dr 1

1d ln pdr

(3.20)

(the Schwarzchild discriminant) which indicates the degree of convective stability.When A < 0 the star is stable against convection; when A > 0 it is unstable (seesection 75 and 78 of Ledoux and Walraven [43]). So using these to eliminate thedensity perturbations in (3.13), (3.16) and (3.17) we get

1

(

r+

g

1p

)p (2 + gA) r +

r= gad T

ps, (3.21)

1r2

r

(r2r

)+

11

d ln pdr

r +(

1p+22

)p

+

122 = ad

T

ps (3.22)

and (1r2

rr2

r+2

) 4piG

(p

1pAr

)= 4piGad

2T

ps. (3.23)

The flux perturbation can be split into radial and horizontal components as

F r = KT

rK dT

dr(3.24)

and~F = K~T . (3.25)

Since s+ ~ ~s0 = s and since the time dependency is eit, substituting equation(3.25) into (3.7) we get

iTs = (N ) 1

r2

r

(r2F r

)+2 (KT ) . (3.26)

Equations 3.21, 3.22, 3.23, 3.24, 3.26 and the thermodynamic relation

= ad p

p+s

cp, (3.27)

where cp = T(sT

)p

is the specific heat per unit mass at constant pressure, are thebasic equations of oscillation. In the adiabatic case, which we will get to later, theentropy is always constant, since the change must be reversible, and so the termscontaining the entropy perturbations are zero, so that equations (3.24) and (3.26)are not required.

3.2 Spherical Harmonics

From the forms of equations (3.21), (3.22), (3.23), (3.24) and (3.26), we see thatthe equations are again separable into radial and angular components since allthe coefficients of these differential equations are either constant or related to theequilibrium variables, which only depend on the radius r due to spherical symmetry,and derivatives with respect to the angular coordinates only occur in the form of theangular part of the Laplacian operator 2. In this case we may assume solutions ofthe form f = f (r)Y (, ) where Y (, ) is a common factor in all the perturbedvariables. Putting this into the above mentioned equations and rearranging we get

1

dp (r)dr

+g

1pp (r) (2 + gA) r (r) + d (r)

dr gad T

pS (r) = 0, (3.28)

14

r2((r)2 +

p(r)2

) ( 1r2

d

dr

(r2r (r)

)+

11

d ln pdr

r (r) +1

1pp (r)ad T

rS (r)

)

= r2

Y (, )2Y (, ) ,

(3.29)

1 (r)

d

drr2d (r)dr

4piGr2 1 (r)

(p (r)1p

Ar (r)) 4piGadr2

2T

p

1 (r)

S (r)

= r2

Y (, )2Y (, ) ,

(3.30)

F r (r)KdT (r)dr

+K (r)dT

dr= 0 (3.31)

and

r2

KT (r)

(iTS (r) + (N ) (r) 1

r2d

dr

(r2F r (r)

))= r

2

Y (, )2Y (, ) .

(3.32)Since the left hand side of the above equations are only dependent on the radiusr, while the right hand side is dependent on the angular coordinates and butindependent of the radius r, we may solve this system by separation of variables.If we let be the separation constant, we find that the angular dependency of thewave equations satisfies

(r22 +

)Y (, ) = 0 which is just the angular part of

Laplaces equation in spherical coordinates. The solution to this equation is givenby the spherical harmonics (see appendix A for a full derivation):

Y ml (, ) = (1)(m+|m|)

2

[2l + 1

2pi(l |m|)!(l + |m|)!

] 12

P|m|l (cos ) e

im. (3.33)

In this case, the spherical harmonics represent the non-radial oscillations in theangular directions of a spherical star. The spherical harmonics can be visualised aswaves along a spherical surface with a total of l nodes, i.e |m| nodes in the azimuthaland l |m| in the latitudinal directions as shown in figure (3.1). As shown, whenm = 0 the spherical harmonic functions only have a dependency on and whenl = |m| they only depend on but for all other values of l and m they depend onboth angular coordinates. Furthermore the wave equations in the radial directionsare now given as the following (all perturbed functions are functions of radius onlybut for brevity the argument (r) is suppressed, for example f is the same as f (r)),

1

dp

dr+

g

c2p +

(N2 + 2

)r +

d

dr= gad T

pS, (3.34)

1r2

d

dr

(r2r

)+

11

d ln pdr

r +(

1 L2l

2

)p

c2 l (l + 1)

2r2 = ad T

pS, (3.35)

1r2

d

dr

(r2d

dr

) l (l + 1)

r2 4piG

(p

c2+N2

gr

)= 4piGad

2T

pS, (3.36)

15

Fig. 3.1: an illustration of spherical harmonics (source - Wikipedia)

KdT

dr= F r K

dT

dr(3.37)

and

iTS = (n) 1

r2d(r2F r

)dr

l (l + 1)r2

KT (3.38)

where c is the sound velocity, Ll is the lamb frequency, and N is the Brunt-Vaisalafrequency, given respectively by

c2 =1p, (3.39)

L2l =l (l + 1) c2

r2(3.40)

andN2 = gA. (3.41)

Since the solutions in the angular directions are known, all that remains is to solvethe system of radial equations above. This requires boundary conditions which will

16

be determined in the next section. Up to now the equations were derived for thegeneral case. We now specialise to the case of adiabatic perturbation.

3.3 Adiabatic case and boundary conditions

We will now assume that the changes during oscillations are adiabatic. Then theentropy is conserved, i.e. S = 0. For adiabatic oscillations we dont need equations(3.37) and (3.38), and by putting S = 0, the thermodynamic relation (3.27) givesa relation for the density perturbation in terms of the perturbation of pressure as

=p

c2(3.42)

or in Eulerian form

=p

c2+ r

N2

g(3.43)

and then we have for equation (3.34)

1

dp

dr+

g

c2p +

(N2 2) r = d

dr, (3.44)

for (3.35) we have

1r2

d

dr

(r2r

) gc2r +

(1 L

2l

2

)p

c2=l (l + 1)2r2

(3.45)

and for 3.36 we have

1r2

d

dr

(r2d

dr

) l (l + 1)

r2 = 4piG

(p

c2+N2

gr

). (3.46)

The central boundary conditions are as follows. Towards the centre c2 varies verylittle and therefore we have for the Lamb frequency L2l 1r2 . If we take the density to be be approximately constant near the centre then noting that for a sphericalmass distribution Mr = 4pi

r0r2 (r) dr, where r a given radius inside the star,

when r 0 Mr 43pir3 or Mr 0,

A = N2

g=(

11

d ln pdr d ln

dr

) 0, (3.47)

N2 0 andg =

GMrr2 4

3piGr 0 (3.48)

as r 0. Therefore near the centre r = 0, equations (3.44), (3.45) and (3.46)reduce to

1

dp

dr 2r + d

dr 0, (3.49)

d

dr

(r2r

) l (l + 1)2

(p

+

) 0 (3.50)

andd

dr

(r2d

dr

) l (l + 1) 0. (3.51)

The differential equation (3.51) has a general solution = c1rl + c2r(l+1), wherec1 and c2 are constants, but r(l+1) as r 0 and so in order for to beregular in the centre of the star, c2 must be set to zero so that = c1rl. If we take

17

the derivative of this solution and substitute back in, then we have a boundarycondition for the perturbation of the gravitational potential at the centre r = 0,

d

dr l

r= 0. (3.52)

In order to get the second central boundary condition, we use the equations (3.50)and (3.49). If we eliminate the r term between these two equations by adding12

ddr r

2 (3.49) to (3.50), we get

d

drr2d

dr

(p

+

) l (l + 1)

(p

+

) 0 (3.53)

which has a general solution p

+ = c3rl + c4r(l+1) and this time in order for p

to be regular in the centre, we set c4 = 0 so that we get a solution of p

+ = c3rl

or p rl and r rl1. Therefore the second central boundary condition is

r l2r

(p

+

)= 0 (3.54)

at r = 0. The other two boundary conditions for the system of differential equationsare given at the surface of the star. The simplest surface boundary conditions arethe zero boundary conditions where we assume that = 0 and p = 0 at the surfacer = R (this is not generally the case if we consider the atmosphere of the star).Since by our assumption, there is no pressure acting from outside the star we havep = 0 or in Eulerian form

p + rdp

dr= 0 (3.55)

at r = R. Since the density is zero at the surface, equation (3.46) becomes

1r2

d

dr

(r2d

dr

) l (l + 1)

r2 = 0 (3.56)

at r = R, which has a general solution of = k1rl + k2r(l+1). In order to makesure that doesnt increase outward along the radius, k1 must be set to zero so that = k1r(l+1) which by taking the derivative, delivers a second surface boundarycondition

d

dr+

(l + 1)r

= 0 (3.57)

at r = R. The equations (3.44), (3.45) and (3.46) plus boundary conditions (3.52),(3.54), (3.55) and (3.57) give an eigenvalue problem i.e. boundary value problemwith discrete set of solutions, with eigenvalues as the square of the frequency. Theseequations describe adiabatic non-radial oscillations. The radial oscillations are thespecial case when l = 0 (remember l is an integer). Here there are mainly two setsof solutions. The one is the set of g-modes, where the order of the mode increasesas the frequency goes to zero, and the other set is that of the p-modes, where theorder of the p-mode increases as the frequency increases. In addition to these twosets of eigenfunctions, there also exists an extra one with intermediate characterbetween the p1-mode and the g1-mode, called the f -mode. Physically the g-modesare oscillations in the star that are driven by buoyancy. In other words, if a givenfluid particle moves due to a buoyant force, then gravity (acting on the densityperturbation) acts as the restoring force, thereby causing an oscillation. The p-modes on the other hand are standing acoustic waves, with pressure acting as therestoring force.

18

3.4 Forms of oscillation equations used for calculation of models

There are two different forms in which the oscillation equations (3.44), (3.45) and(3.46) can be written in order to make them practical for numerical computation.One is described in 79 of Ledoux and Walraven [43] and in 17.5a of Cox [22]. Itis written in terms of the variables

y =p

(3.58)

andu = r2r. (3.59)

In this format, p and r in terms of y and u are given by

p = y (3.60)

andr =

u

r2. (3.61)

Substituting 3.60 and 3.61 into 3.44, and rearranging, we obtain

dy

dr=2 +Ag

r2uAy

dr(3.62)

if we make use of the combination of the definition of A and equation (2.29) andnoting that N2 = gA. Substituting (3.60) and (3.61), as well as (3.40) for theLamb frequency Ll and (3.39) for c2, into (3.45), and then rearranging to make(du/dr), we obtain

du

dr=

g

1pu+

[l (l + 1)2

r2

1p

]y +

l (l + 1)2

. (3.63)

Finally, we can rewrite the Poisson equation in terms of y and u by substituting(3.60) and (3.61) into (3.46) to get

1r2

d

dr

(r2d

dr

)=

4piG2

1py 4piGA

r2u+

l (l + 1)

r2(3.64)

The other form for the oscillation equations, which is also used for numerical modelsand which we will be using here, appears in chapter 18 of Unno et al [63] and in17.5b of Cox [22]. In order to write the equations in the latter form, we define fourdimensionless interdependent variables as

y1 =rr, (3.65)

y2 =1gr

(p

+

), (3.66)

y3 =1gr

(3.67)

and

y4 =1g

d

dr. (3.68)

Rearranging, we then get r, p, and (d/dr) in terms of the dimensionlessvariables (3.65), (3.66), (3.67) and (3.68) as

r = ry1, (3.69)

19

p = gr (y2 y3) , (3.70)

= gry3 (3.71)

andd

dr= gy4. (3.72)

If we substitute (3.69), (3.70) and (3.71) into (3.44) and make r (dy2/dr) the subject,we get

rdy2dr

=(N2 2) r

gy1

(gr

c2+

r

g

d (g)dr

+ 1)y2 +

(gr

c2+r

d

dr

)y3 (3.73)

(the (d (gry3) /dr) term cancels). Substituting (3.69), (3.70) and (3.71) into (3.45)and making r (dy1/dr) the subject (if we take the Lamb frequency (3.40) into ac-count and then note that we have a term of

(Ll/

2)y3 on both sides, which cancel),

we get

rdy1dr

=(grc2 3)y1 +

(Ll2 1)y2 +

gr

c2y3 (3.74)

and substituting (3.69), (3.70), (3.71) and (3.72) into (3.46) and rearranging tomake r dy4dr the subject, you get

rdy4dr

= 4piGN2r2

g2y1 + 4piG

r2

c2y2 +

[l (l + 1) 4piGr

2

c2

]y3 1

gr

d (gr)dr

y4. (3.75)

We also get a relation between y3 and y4 by taking the derivative of (3.71) andsubstituting in (3.72):

rdy3dr

= 1g

d (gr)dr

y3 + y4. (3.76)

Lets define the dimensionless radius as

x =r

R, (3.77)

and dimensionless frequency as

2 =2R3

GM. (3.78)

We can also define four other dimensionless variables as

V = d ln pd ln r

, (3.79)

c1 =r3M

R3Mr, (3.80)

U =d lnMrd ln r

(3.81)

and

A = rA = r1p

dp

dr r

d

dr=rN2

g. (3.82)

Using (3.77), (3.78), (3.79), (3.80), (3.81) and (3.82), equations (3.73) - (3.75) be-come

xdy1dx

=(V

1 3)y1 +

[l (l + 1)c12

V1

]y2 +

V

1y3, (3.83)

20

xdy2dx

=(c1

2 A) y1 + (A U + 1) y2 Ay3, (3.84)xdy3dx

= (1 U) y3 + y4 (3.85)and

xdy4dx

= UAy1 +UV

1y2 +

[l (l + 1) UV

1

]y3 Uy4. (3.86)

Note that in order to get the A in the coefficients of y2 and y3 in equation (3.84),the term (gr) /c2 can be rewritten as (r/1p) (dp/dr) if we make use of equation(2.29) from the previous chapter after substituting in (GMr) /r2 for g and (1p) for c2 (in the coefficient of the y2, the (r/) (d/dr) term from (3.82) is one ofthe terms resulting from application of the product rule to the (r/g) (d (g) /dr)).When rewriting (3.52) and (3.54) in terms of the dimensionless variables y1, y2, y3and y4 we get the central boundary conditions as

ly3 y4 = 0 (3.87)for the first central boundary condition (3.52) and

c12

ly1 y2 = 0 (3.88)

for the second central boundary condition (3.54). Similarly for the surface boundaryconditions (3.55) and (3.57) we obtain

y1 y2 + y3 = 0 (3.89)for the first surface boundary condition (3.55) (substitute in dpdr = g) and

(l + 1) y3 + y4 = 0 (3.90)

for the second surface boundary condition (3.57).

3.5 Numerical calculation of oscillation models

Boundary value problems for ordinary differential equations may be solved by theshooting method. In the shooting method, the boundary value problem is turnedinto an initial value problem by choosing an initial value which satisfies the boundaryconditions at one end point but then requiring the equations to give a solutionwhich satisfies the boundary conditions at the other end. If the ordinary differentialequations are linear, then what is needed is to solve the system with differentpossible initial values which are linearly independent. The solution which satisfiesthe boundary conditions on the other end is a linear combination of these (seeAscher, Mattheij and Russell [13] and Keller [34]). Otherwise, the boundary valueproblem may also be solved by using a finite difference technique. This is done bydividing the interval of the solution into a mesh of N points, and then approximatingthe differential equations as finite difference equations at each of the N points. Wethen have for I differential equations, a set of N I algebraic equations which mustbe solved simultaneously for every point on the mesh (see Ascher, Mattheij andRussell [13]). In the case of the oscillation equations, it may be solved with shootingby integrating from each boundary to a point in between, or fitting point, by usingthe boundary conditions as the initial conditions. Since the problem is an eigenvalueproblem, i.e. a problem with a discrete set of solutions, the eigenvalues must alsobe determined (the eigenvalues are the square of the eigenfrequencies). When using

21

the shooting method, the solutions will only match at the fitting point if the correctvalues are chosen for the eigenfreqiencies. When using a finite difference scheme, itis convenient to use a so-called -method in order to write the differential equationsas a set of difference equations. If we consider the set of differential equations as

dyidx

= fi(yj ; 2

), (3.91)

for i, j = 1, 2, 3, 4 then, using the -method, the difference equations are approxi-mated as

yn+1i ynixn

= (1 i) fi(yn+1j ;

2)

+ ifi(ynj ;

2)

(3.92)

where xn = xn+1 xn, n is a given mesh point between 1 and N 1 and issome constant 0 1 and is normally taken as 0.5 for good accuracy. Theidea behind the method is to take an average of values at adjacent mesh pointsso as to evaluate the equations at a point in between the mesh points n + 1 andn. As a result the derivative would be a central difference of that point (in thecase of = 0.5 it could be thought of as the central difference at the point n + 12 )and hence gives a better approximation for the derivative and for the ys. Thenequations (3.83), (3.84), (3.85) and (3.86) may be written as the set of differenceequations

0.5 (xn+1 + xn)yn+11 yn1xn+1 xn =

0.5[(

V n+1

n+11 3)yn+11 +

[l (l + 1)cn+11

2 V

n+1

n+11

]yn+12 +

V n+1

n+11yn+13

]+0.5

[(V n

n1 3)yn1 +

[l (l + 1)cn1

2 V

n

n1

]yn2 +

V n

n1xyn3

],

(3.93)

0.5 (xn+1 + xn)yn+12 yn2xn+1 xn =

0.5[(cn+11

2 An+1) yn+11 + (An+1 Un+1 + 1) yn+12 An+1yn+13 ]+0.5

[(cn1

2 An) yn1 + (An Un + 1) yn2 Anyn3 ] ,(3.94)

0.5 (xn+1 + xn)yn+13 yn3xn+1 xn = 0.5

[(1 Un+1) yn+13 + yn+14 ]+0.5 [(1 Un) yn3 + yn4 ]

(3.95)and

0.5 (xn+1 + xn)yn+14 yn4xn+1 xn =

0.5[Un+1An+1yn+11 +

Un+1V n+1

n+11yn+12 +

[l (l + 1) U

n+1V n+1

n+11

]yn+13 Un+1yn+14

]+0.5

[UnAnyn1 +

UnV n

n1yn2 +

[l (l + 1) U

nV n

n1

]yn3 Unyn4

](3.96)

where in this case the average of xn+1 and xn was also taken on the left. The setof equations (3.93), (3.94), (3.95) and (3.96) as well as the boundary conditions

22

are needed to solve the system. The boundary conditions (3.87), (3.88), (3.89) and(3.90) are given in terms of the above scheme as

ly03 + y04 = 0 (3.97)

andc1 (0) 2

ly01 y02 = 0 (3.98)

at the centre, andyN1 yN2 + yN3 = 0 (3.99)

and(l + 1) yN3 + y

N4 = 0 (3.100)

at the surface. Since this is an eigenvalue problem, this system will only havesolutions at specific values of . In order to solve the system as well as find theeigenfrequencies, we need to impose a normalisation condition, for instance

yN1 = 1. (3.101)

We may solve the system with the normalisation condition replacing one of theboundary conditions, say (3.99). We do this by putting the coefficients of the ysinto a matrix. In this case, we get a staircase matrix. This system may be solvedby matrix inversion or by gauss elimination and since most of the elements are zero,sparse routines may be used in order to reduce memory used and to increase thespeed in computation (at the eigenfrequencies, the solution to the system gives anumerical solution to the original differential equations at every point on the mesh).The remaining boundary condition may be used to look for eigenfrequencies if wesolve the above linear system of equations while allowing to vary continuously. Inother words,in order to look for eigenfrequencies, let

D () = yN1 yN2 + yN3 , (3.102)

keep solving the system with the normalisation (3.101) by Gauss elimination whilecontinuously varying until the deviation (3.102) changes sign since D () = 0 onlywhen is an eigenfrequency. The solution at the points where D () is close to zeromay be used as an initial trial solution when applying the Henyey relaxation method(described in chapter 18 of Unno et al [63]) to improve accuracy. If the equationsdescribed above are applied to a polytrope with index = 43 , the solutions behaveas shown in figure 3.2, which shows the behaviour of the f -mode and lower orderp-modes. In this particular instance, the zeroes of D () were located by using thebisection method. The Henyey method was not used.

23

Fig. 3.2: Oscillation modes of polytrope with polytropic index 4/3. These modes werecalculated with the code in section B.3 which also requires the code in sectionB.2 of appendix B for the finite difference calculations.

24

4. DENSE NUCLEAR MATTER INSIDE NEUTRON STARS

This chapter is a review of the literature. The first section is a review of equationsof state of the nuclear matter in the interior of neutron stars and the second sectionis a literature review of neutron star oscillations.

Before we can discuss oscillations in neutron stars, like any discussion of stellaroscillation, an equilibrium model is needed. Since neutron star interiors are madeup of dense nuclear matter, this means that we need equations of state for suchmatter to replace that of gases discussed in the previous two chapters. The internalstructure of a neutron star can be divided into a crust, outer core and inner core.Generally, the crust of a neutron star consists of a metallic lattice. Deeper intothe crust, inverse beta processes between the electrons and protons in the nucleiresult in the lattice having atoms with more neutron rich nuclei. At a certain depththere is a point where there is not only a gas of free electrons but free neutrons aswell (this occurs at a density of 4 1011g cm3 and is called neutron drip, seeChapter 2 of Shapiro and Teukolsky [60]). At the depth of the outer core, there areno more atomic nuclei but rather just neutrons and some protons that interact viathe strong nuclear force forming a superfluid. The composition of the inner coreof the neutron star is not well understood but could be composed of deconfinedquarks. The simplest model for the outer core is a mixture of protons and neutronsbut a more realistic one could contain other particles such as hyperons and strangematter. If deconfined quarks which interact via the colour force compose the innercore, it is most likely to consist of strange quark matter. The following is a reviewof papers describing equations of state of the dense nuclear matter inside neutronstars.

4.1 Review of equations of state of neutron star matter

Shapiro and Teukolsky [60] have discussed the variational method for calculatingthe equation of state of dense nuclear matter. Global attributes of neutron stars,such as the radius for a given mass, depend on the internal pressure (as is alsothe case in ordinary stars). In the case of neutron stars, the equation of state andthe composition in the core depend on the nature of strong interactions in densenuclear matter. One way to calculate the equation of state of nuclear matter, takinginto account correlations between fermions, is the variational method. A variationalcalculation is usually done by constructing a trial many body wave function v (alsocalled the Jastrow trial function), which when incorporating two-body correlationsis written as a product

v (~r1, . . . , ~rN ) = Ai

value is then minimised by varying the factors fij in the wave function in order toestimate the ground state energy, since by the Rayleigh-Ritz upper bound

v|H|vv|v E0, (4.2)

where v is the trial wave function, H is the hamiltonian and E0 is the ground stateenergy.

Baym, Pethick and Sutherland (1971) [14] determined an equation of state forhigh density matter below neutron drip that takes into account shell effects andlattice energy. The BPS equation of state can describe the crust of neutron stars upto the density where neutron drip occurs. The energy density in the BPS equationof state is given by

= n (1 Yn) M (A,Z)A

+ e (ne) + n (nn) + L (4.3)

where n is the baryon number density, Yn = nn/n is the ratio of neutrons out of thenumber of baryons at a given density, A is the number of nucleons in the nucleus,M (A,Z) is the energy of a nucleus with Z protons and A Z neutrons, e is theenergy density of the electrons, n is the energy density of free neutrons and L isthe lattice energy given by

L = 1.444Z 23 e2n43e . (4.4)

The pressure is given byp = pe + pL (4.5)

where pe is the electron degeneracy pressure and

pL =13L. (4.6)

In the BPS equation of state, the composition of the matter in beta equilibrium isdetermined by the values of A and Z which minimize at a given baryon density n.Later in the paper, they use this equation of state to construct models of nonrotatingstars (both neutron stars and white dwarfs) and then they use properties from thenonrotating configuration to calculate the properties of slowly rotating stars.

Friedman and Pandharipande (1981) [25] did some variational calculations of theequation of state of nuclear and neutron matter over a wide density range. Theyuse a hamiltonian which contains both two and three nucleon interactions. Theyalso examine the effects of the three nucleon interactions.

Wiringa, Fiks and Fabrocini (1988) [64] calculated some equations of state ofdense nuclear matter from five different Hamiltonians containing potentials for two-nucleon as well as three-nucleon interactions given as

H =i

h2m2i +

i

where x = p is the proton fraction, is the total number density of nucleons, TFis the Fermi-gas kinetic energy given by

TF (, x) =35h2

2m(3pi2

) 23[x

53 + (1 x) 53

](4.9)

and V0 () and V2 () are functions obtained from the results for pure neutron matter(x = 0) and symmetric nuclear matter (equal numbers of protons and neutrons,x = 12 ) respectively. From there, they used the relation 4.8 to tabulate values ofenergy E (, x) and proton fraction x () with respect to a given number density(which they calculated by assuming beta equilibrium or n p+ e). The tableis then interpolated to get E () in order to calculate the energy density, given by:

() = [E () +mc2

]. (4.10)

They also used the interpolated function E () in order to calculate the pressure interms of number density, given by

p () = 2E ()

. (4.11)

The equation of state in the form of pressure in terms of energy density p () isobtained by combining equations 4.10 and 4.11 in order to eliminate betweenthem. Using this equation of state, they then calculated neutron star structurein the context of general relativity, in which case the stellar structure is givenby the Oppenheimer-Volkoff equation (e.g. Shapiro and Teukolsky [60]), which isderived from Einsteins field equations. Later in the paper, pion condensation isalso discussed in order to explain softening in one of the equations of state.

Akmal and Pandharipande (1997) [1] also used the variational method to domany body calculations with two and three nucleon interactions (with a hamiltonianlike that in equation 4.7) for symmetric and pure neutron matter. More specifically,they used the hamiltonians

HSNM = i

h2

4

(1mp

+1mn

)2i +

i

Serot (1979) [59] extended an earlier quantum field theory of nuclear interactionsto include interactions with pi and mesons. Serot constructed the Lagrangianfor the theory by adding terms for the pi and a doublet of scalar fields to theLagrangian of the earlier theory. A theory containing a non-abelian vector field likethis one (i.e. the field for the mesons) is nonrenormalizable but in this case it ismade renormalizable by allowing spontantaneous symmetry breaking via the Higgsmechanism in order to generate mass for the vector gauge field. In order to get anequation of state, the field operators in this theory are replaced by their expectationvalues in order to get the field equations. The field equations are linear and can besolved to get the energy density and pressure in dense nuclear matter as

=(g2v

2m2v

)2B +

(m2S2g2s

)(M M)2 +

(g2

8m2

)23

+2

(2pi)3

{ kFp0

+ kFn

0

}d3k

[k2 +M2

] 12

(4.14)

and

p =(g2v

2m2v

)2B

(m2S2g2S

)(M M)2 +

(g2

8m2

)23

+13

2(2pi)3

{ kFp0

+ kFn

0

}d3k k2

[k2 +M2

] 12 (4.15)

respectively, where M is the effective mass determined by

M = M (g2Sm2S

)2

(2pi)3

{ kFp0

+ kFn

0

}d3kM

[k2 +M2

] 12 , (4.16)M is the mass of the nucleons, mv is the mass of the vector mesons, mS is the massof the scalar mesons, m is the mass of the mesons,

B =k3Fp + k

3Fn

3pi2(4.17)

and

3 =k3Fp k3Fn

3pi2(4.18)

with kFp and kFn being the Fermi momenta of protons and neutrons respectively.Glendenning and Moszkowski (1991) [27] approached the idea of a mixture of

neutrons, protons and hyperons in the composition of neutron stars from a fieldtheoretical point of view. Their model resolves a discrepancy between previouslycalculated binding of hyperons in nuclear matter and observed neutron star masses(prior calculations with this mixture led to maximum neutron star masses whichwere too small, see [26] and [33]).

Prakash, Cooke and Lattimer (1995) [55] investigated the effects of trappedneutrinos on the composition of quark-hadron phase transitions in protoneutronstars (the collapse of the core of a massive star just before the formation of aneutron star, see Burrows and Lattimer [16]). The equation of state that theydiscussed contained a mixed phase of nucleons, hyperons and quarks. They foundthat the presence of trapped neutrinos causes a shift in the phase transition tohigher baryon densities. They also found that when negative electric charges (other

28

than the ones due to leptons) are present then the maximum masses of stars withtrapped neutrinos are higher than those without neutrinos.

Muller and Serot (1996) [53] also used a field theoretical approach to calculatethe equation of state. For the equation of state, they eventually get the followingfor the pressure and energy density,

p =1

3pi2

kFp0

dkk4

(k2 +M2) 12+

13pi2

kFn0

dkk4

(k2 +M2) 12+

12c2v

W 2 +1

2c2R2

12c2s

2 +i,j,k

aijkW 2jR2k

(4.19)

and

=1pi2

kFp0

dkk2(k2 +M2) 12 + 1

pi2

kFn0

dkk2(k2 +M2) 12 +W+ 1

2R3

12c2v

W 2 12c2

R2 +1

2c2s

i,j,k

aijkW 2jR2k,

(4.20)

where kFp and kFn are the Fermi momenta for protons and neutrons respectively,, W and R are the scaled meson fields, M M is the effective nucleon massand c2i and aijk are the coupling constants. They then extrapolate the high densitylimit in order to discuss the equation of state in relation to neutron stars. Finallythey conclude that the extrapolation of the equation of state to high densities isnot sufficient to give precise predictions of the properties of neutron stars.

Glendenning and Schaffner-Bielich (1999) [28] used a field theoretical calculationin order to get an equation of state that includes kaon condensation. They discussthe differences between a Maxwell construction and the Gibbs criteria. A Maxwellconstruction as they describe is useful when there is a single chemical potentialbecause it ensures that only one chemical potential is common to multiple phases.Since neutron star matter has two chemical potentials, a Maxwell construction cantbe used for neutron stars. For isospin asymmetric nuclear matter, Gibbs conditionscan be satisfied if the charge density in the two different phases have oppositesign at phase equilibrium. Later in the paper Glendenning and Schaffner-Bielichdiscuss the mixed phase region in neutron stars. They find that when kaons startto condense, the neutron density appears almost constant on a logarithmic scale,although it varies slowly. This is because the kaons are more favourably producedin association with protons (because of the negative charge in the K kaons) and soin order to maintain beta equilibrium, the neutron density remains almost constant.In the condensed phase, the energy favours the production of proton-K pairs overthe production of neutrons although their growth is not uninhibited.

Muther, Prakash and Ainsworth (1987) [54] extended Brueckner-Hartree-Fockcalculations for nuclear matter to dense neutron matter, taking into account inter-actions between nucleons due to the exchange of pi and mesons as well as thedependency on the asymmetry. They also discussed the role of the nuclear symme-try energy in the context of neutron star structure.

Engvik et al (1994) [23] calculated an equation of state containing just neutronsand protons using Dirac-Brueckner-Hartree-Fock analysis and not the variationalmethod as described above. For particle interactions, they use a potential describedby Machleidt [48]. They used their equation of state to calculate global propertiesof neutron stars such as mass and radius in order to compare them to the observedmasses and radii of real stars. For pure neutron matter, their equation of state gave

29

a maximum mass of MMAX 2.4M at a radius of R = 12 km but can be reducedto MMAX 2.0M at a radius of R = 10 km when including the effects of pionand kaon condensation (which increases the proton fraction up to 40%).

Prakash, Ainsworth and Lattimer (1988) [56] investigated the relationship be-tween other parameters in the equation of state, such as nuclear incompressibility,symmetric nuclear energy, and the maximum mass of neutron stars. They did thisby parameterising the function describing the energy per particle E (n) and usingthe nuclear incompressibility given as K0 as an input parameter. They found thatthe observed neutron-star masses cannot constrain K0 by a large amount for real-istic symmetry energies and therefore dont give useful constraints on the possibleequations of state. As a result they suggested a number of directions for furtherstudy, including the effect of of hyperonic interactions and whether observations ofproperties other than the mass can constrain the equation of state.

Haensel and Potekhin (2004) [29] derived analytical representations for two equa-tions of state. These equations of state are unified in the sense that they describethe crust and the core with the same model. The derivation was done by interpo-lating tabulated values of the equations of state in such a way as to respect the firstlaw of thermodynamics everywhere. They describe how the equations of state canbe parameterized for both rotating and non-rotating neutron star configurations.

Lattimer and Prakash (2001) [42] examined a wide variety of models for theequation of state of neutron star matter (originally from some of the papers re-viewed above, i.e. [1], [23], [25], [27], [28], [53], [54], [55], [56] and [64]). Theydetermined that useful constraints on the possible forms of the equations of statemay be obtained experimentally by a measurement of radius for a single neutronstar to an accuracy of 1 km with a given mass (general relativity was used forgravity in the calculation of radii). This could be done with an observation of asingle neutron star because Lattimer and Prakash were able to establish a generalcorrelation

RM ' C (n,M) [p (n)]0.230.26 , (4.21)where RM is the total radius of the star, p (n) is the pressure at a particle numberdensity (of both leptons and baryons) n and C (n,M) is a coefficient dependent onthe number density at which the pressure was calculated (using a particular equationof state) and the total mass M . This correlation was found by approximating theequations of state as polytropes (most of the ones examined turned out to have aneffective polytropic index of n = 1 in the approximations). Lattimer and Prakashalso discussed some other quantities such as moments of inertia and binding energyand found that they were mainly independent of the equations of state.

4.2 Oscillations in neutron stars

Although neutrons stars have a solid crust, the core consists of superfluid protonsand neutrons (other particles, including strange matter, may also exist at higherdensities). Oscillations may occur in these regions and have been studied in thecontext of Newtonian gravity ([5], [44], [45], [46], [57], [58], [61], [65]). They havealso been studied in the context of general relativity ([3], [4], [9], [10], [11], [21], [35],[38]).

In addition to the p-modes and the g-modes, another class of modes exist in thesuperfluid region of neutron stars due to the extra degrees of freedom introducedby two different fluids (i.e. the proton and electron fluid and the neutron fluid) aswell as the superfluidity in that region. Superfluidity also has effects on oscillationsbecause of the differences in the dynamics of a superfluid as opposed to an ordinaryfluid. The idea of a new set of modes due to the superfluidity in neutron stars was

30

first suggested by Epstein (1988) [24]. Lindblom and Mendell (1994) [46] investi-gated the effects of superfluidity on the oscillations of neutron stars using Newtoniangravity. They started off by providing some equations for the oscillations of the su-perfluid matter in neutron stars and then derived the boundary conditions for theseequations (the boundary conditions also include jump conditions at the interfacebetween the superfluid core and ordinary fluid envelope). For the equation of state,they used the Harrison-Wheeler equation of state in the low density region of thestar, 2 1012g cm3, and a variation of the Serot equation of state [59] in thehigh density region 3 1013g cm3. The main difference between the variationof the Serot equation of state used by Lindblom and Mendell and the original onegiven by Serot is that it contains both protons and neutrons, with the ratio beingdetermined by imposing beta equilibrium, whereas Serot originally assumed pureneutron matter. In the densities between these two regions, 21012 31013,they matched the two equations of state by interpolating with a polytrope

p = 8.49 10161.09. (4.22)Since it will be important in the next chapter, this version of the Serot equation ofstate will be given here. The mass densities of the proton and neutron fluids arerespectively given in terms of the Fermi momenta,kp and kn as

p =mpk

3p

3pi2h3(4.23)

and

n =mnk

3n

3pi2h3. (4.24)

The electron mass density is determined in terms of the Fermi momentum of theprotons, kp, due to the condition of charge neutrality,

e =mek

3p

3pi2h3. (4.25)

The Serot equation of state requires an effective mass, m, which in this case isdetermined in terms of kp and kn as

m = mb 534.2m3

m2b

[

(knmc

)+

(kpmc

)](4.26)

where mb 1.675 1024g is the average baryon mass and the definition of appears below. The effective mass in equation (4.26) can be solved for numericallywith a root seeking algorithm. The energy density in this case is given by

=m4c

5

h3

[

(knmc

)+

(kPmc

)]+

(k3n + k

3p

)28.950m2b h

3c+m2b (mb m)2 c5

534.2h3

+

(k3n k3p

)2128.2m2b h

3c.

(4.27)

The definition of appears below. The number of neutrons and protons are deter-mined at a given density by imposing beta equilibrium (i.e. the condition that thereaction p+ e n is in equilibrium). This requires that

mee +mpp = mnn (4.28)

where the chemical potentials are given by

n =(

n

)p

, (4.29)

31

p =(

p

)n

(4.30)

and

e = c2

1 +k2pm2ec

2(4.31)

for neutrons, protons and electrons respectively. The pressure is given by

p =m4c

5

h3

[

(knmc

)+

(kpmc

)]+m4ec

5

h3

(kPmec

)+

(k3n + k

3p

)28.950m2b h

3c

m2b (mb m)2 c5

534.2h3+

(k3n k3p

)2128.2m2b h

3c.

(4.32)

In the above equations is given by

(x) =1

4pi2[x

1 + x2 ln(x+

1 + x2

)], (4.33)

the function that determines the energy density of an ideal Fermi gas is given by

(x) =1

8pi2[x

1 + x2(1 + 2x2

) ln(x+1 + x2)] (4.34)and the function that determines the pressure of an ideal Fermi gas is given by

(x) =1

8pi2

[x

1 + x2(

2x2

3 1)

+ ln(x+

1 + x2

)]. (4.35)

The equations 4.23 to 4.35 taken together can be used to calculate the pressure atany given mass density, i.e. the total mass density given by

= n +(

1 +memp

)p. (4.36)

Lindblom and Mendell then demonstrate the existence of the new set of modes(which they called s-modes for superfluid) caused by the oscillations of the neutronand proton fluids in the core. They do this by solving the superfluid oscillationequations analytically in the special case of spatially uniform matter in the star,although they failed to find them numerically in the more realistic model.

Lee (1995) [44] also calculated the oscillations in superfluid neutron stars withNewtonian gravity and used the same equation of state as Lindblom and Mendell.Lee first provides some equations for waves propagating in superfluids. The veloc-ities of the two fluids in the neutron star core are not independent because thereis an effect called entrainment whereby the velocity of one fluid induces a currentin the other (e.g. Andreev and Bashkin (1975) [12] for the case of a mixture ofliquid helium 3 and helium 4). A full derivation of the oscillation equations will begiven in the next chapter. Lee wrote the oscillation equations in terms of dimen-sionless variables for the numerical calculations and found the superfluid modes.Lee calculated the oscillations for two different models. One of the models was aneutron star with a mass of 0.5M (M is for solar masses) and the other onehas a mass of 1.4M (same as Lindblom and Mendell). In the 0.5M model,the calculations were done assuming that the Schwarzschild discriminant and theBrunt-Vaisala frequency are both zero (i.e. A = 0 and N2 = 0). In this case nog-modes exist. In the 1.4M model, the calculation was done for two cases. Onewas for zero Schwarzchild discriminant, like in the 0.5M model, and the other one

32

was for a negative Schwarzschild discriminant which is given by taking into accountthe composition gradient of the proton and neutron fluids. Lee only calculated thecomposition gradient in the high density region 31013g cm3, where the Serotequation of state is used for the proton and neutron fluid mixture and assumed thatA = 0 in the rest of the star. In the case where A < 0, g-modes only propagate inthe region between the superfluid-normal fluid boundary and the density thresholdof 3 1013g cm3. In conclusion Lee found that the superfluidity prevents g-modesfrom propagating. Andersson and Comer (2001) [5] later confirmed the result thatg-modes dont propagate in the superfluid core of neutron stars.

33

5. OSCILLATIONS IN THE CORES OF SUPERFLUID NEUTRONSTARS

This chapter describes the neutron star models and numerical methods used in thiswork to calculate the oscillations and also shows the results of those calculations.

A neutron star consists of a solid crust and a core. In the simplest model, thecore consists of a mixture of neutrons, protons and electrons. Even when consideringthe possibility of other particles in the core such as hyperons (strange matter) anddeconfined quarks (in the inner core), these particles will still be in a superfluidstate. Superfluidity has important effects on the oscillations in the cores of neutronstars. In particular, it results in a new class of modes (called s-modes by Lindblomand Mendell [46] and -modes by Lee [44]). Up to now, the determination of thematter composition inside neutron stars has been made primarily by observations ofphenomena called glitches (when the spinning rate of a pulsar suddenly speeds up)[47], [41], and of neutron star cooling rates. Oscillations in neutron star interiors areimportant because if detected, they provide another window into the compositionof neutron star interiors. Although it is unlikely that oscillations of the interiorsmight be observed in electromagnetic radiation, gravitational wave data (if detected)might have signatures of oscillations in them (e.g. Comer [19]). This is becausethere is a class of modes (called w-modes) emitted as gravitational radiation whichare strongly related to the oscillations of the matter inside neutron stars whenthe calculations are done for rotating relativistic neutron stars ( e.g. [38], [11],[9]). This chapter shows the results of calculations done for the oscillations ofsuperfluid neutron stars. It extends the work done by Lee [44] to a more denselypacked grid of neutron star masses and oscillation modes respectively. Firstly, theequilibrium model will be described before derivations of the oscillation equationsfor a superfluid are presented.

5.1 Equilibrium model and equation of state

All the equations, except for the ones describing the Harrison-Wheeler equationof state, presented in this section are taken from Lee (1995) [44]. The Harrison-Wheeler equation of state is also given in Shapiro and Teukolsky [60]. The equi-librium model used here is the same as the one that was used by Lee (1995)[44] and Lindblom and Mendell (1994) [46]. The Serot equation of state (de-scribed by equations (4.23) - (4.36) in chapter 4) is used in the density regionof 3 1013 g cm3, while the Harrison-Wheeler equation of state is used in thedensity region of 2 1012 g cm3. In the Serot equation of state, the chemical

potentials (4.29) and (4.30) can be given explicitly as

n =4m3c

5

h3

(

(knmc

)+

(kpmc

))mn

+m2c

2

mn

1 +

(knmc

)2 c

2k3npi2h3

1 +

(knmc

)2mn

m2c

4kp

h31pi2

(kpmc

)21 +

(kpmc

)2mn

+6pi2

(k3n + k

3p

)8.950m2umnc

2m2u (mu m) c5

534.2h3mn

+6pi2

(k3n k3p

)128.2m2umnc

(5.1)

and

p =4m3c

5

h3

(

(knmc

)+

(kpmc

))mp

+mc2

mp

1 +

(kpmc

)2 c

2k3p

pi2h3

1 +

(kpmc

)2mp

m2c

4kn

h31pi2

(knmc

)21 +

(knmc

)mp

+6pi2

(k3n + k

3p

)8.950m2umpc

2m2u (mu m) c5

534.2h3mp

6pi2(k3n k3p

)128.2m2umpc

.

(5.2)

In the density region 2 1012 3 1013 an interpolation is used with apolytrope given by

p = 8.49 10161.09. (5.3)The Harrison-Wheeler equation of state describes a mixture of nuclei and free elec-trons at the lowest densities and includes free neutrons above neutron drip. TheHarrison-Wheeler equation of state (taken from 2.6 of Shapiro and Teukolsky [60])is summarised in what follows. The mass density is given by

=neM (A,Z) /Z + e + n

c2(5.4)

where ne is the electron number density given by

ne =m3ec

3

3pi2h3x3e. (5.5)

The M (A,Z) in equation 5.4 is the energy of an atomic nucleus given by the semi-empirical mass formula

M (Z,A) = mc2[b1A+ b2A

23 b3Z + b4A

(12 ZA

)2+b5Z

2

A13

], (5.6)

where A is the baryon number, Z is the number of protons and the bs are given by

b1 = 0.991749, (5.7)

b2 = 0.01911, (5.8)

b3 = 0.000840, (5.9)

b4 = 0.10175 (5.10)

35

andb5 = 0.000763. (5.11)

The electron energy density e is given by

e =m4ec

5

h3 (xe) nemec2, (5.12)

where the rest energy nemec2 is subtracted because it is included in the termM (A,Z). The neutron energy density n is given by

n =m4nc

5

h3 (xn) . (5.13)

In all the above equations, c is the speed of light. The function is given by

(x) =1pi2

{x(1 + x2

) 12(1 + 2x2

) ln [x+ (1 + x2) 12 ]} . (5.14)When beta equilibrium is assumed, then

Z =(b22b5

) 12

A12 . (5.15)

The terms xe and xn are given by

b3 + b4

(1 2Z

A

) 2b5 Z

A13

=[(

1 + x2e) 12 1

] memu

(5.16)

and

b1 +2b2A

13

3+ b4

(14 Z

2

A2

) b5Z

2

3A43

=(1 + x2n

) 12 mnmu

. (5.17)

The total pressure isp = pe + pn, (5.18)

where the electron pressure is given by

pe =m4ec

5

h3 (xe) (5.19)

and the neutron pressure is given by

pn =m4nc

5

h3 (xn) , (5.20)

where the function is given by

(x) =1

8pi2

{x(1 + x2

) 12

(23x2 1

)+ ln

[x+

(1 + x2

) 12]}

. (5.21)

The Harrison-Wheeler equation of state gives the pressure in terms of density cal-culated from equations (5.4) - (5.21) and is parameterised by A. In order to use theHarrison-Wheeler equation of state to calculate the pressure at any given density,first choose a value of A > 56 and then use it to calculate Z and xn from equations(5.15) and (5.17). If xn > 0, then neutron drip has been reached, so use the valueof xn to calculate n and pn from equations (5.13) and (5.20). If neutron drip hasnot been reached, then set n and pn to zero. Now use equation (5.16) in orderto calculate xe and from there calculate e and pe from (5.12) and (5.19). Now

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log()

lo

g(P)

Co