1

UNIVERSITY of CALIFORNIA

SANTA CRUZ

Neutron Star Structure, Equations of State, and 1D Modeling

A thesis submitted in partial satisfaction of the requirements for the degree of BACHELOR OF SCIENCE

in

ASTROPHYSICS

by

Andrew Stine

June 2015

The thesis of Andrew Joseph Stine is approved by:

Professor Jonathan Fortney Professor David P. Belanger Advisor Senior Theses Coordinator

Professor David P. Belanger Chair, Department of Physics

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Acknowledgments

I would like to thank Daniel Thorngren for sacrificing his time to teach me the computational

methods needed to create this simulation. Without his help I would most likely still be debugging. I

would also like to thank Professor Jonathon Fortney for consistently sparking my interest in

astrophysics. His friendly demeanor and passion has brought new life to the subject for me, and for

that I’m grateful.

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I’d like to dedicate this thesis to my parents, who have shown me nothing but love and

opportunities my whole life. Thank you!

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Copyright © by

Andrew Joseph Stine

2015

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Abstract

The purpose of this thesis is to explicate the main theory of neutron star composition and evolution, with a

focus on potential equations of state (EOSs). While much is known about electron-degenerate matter, the

density required to create neutron-degenerate matter is too large to be reproduced in the lab. As such,

models of the neutron star EOS must be grounded in observation. However, the compactness of neutron

stars has made accurate observations difficult, leaving uncertainty in the maximum and minimum mass

values of neutron stars to this day. A suite of EOSs have been published to model the stellar interior

under any of the following unproven theoretical paradigms: presence of exotic matter, differing nuclear

interactions, varying nuclear symmetry energies, and various phase transitions. In this thesis I present a

1-D neutron star model that solves the appropriate stellar structure equations in conjunction with two

unified EOSs, BPS and SLy. The results of this simulation exhibit predicted degrees of stiffness for both

EOSs from the outer crust to the core, in accordance with the published results of Haensel and Potekhin

(2004). I predict the maximum mass of a neutron star’s core and inner crustal region to be 2.05M☉ for the

SLy EOS and 1.8M☉ for the FPS EOS, in perfect agreement with Haensel (2003).

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Contents 1. Neutron Star Theory

1.1 Birth, Evolution……………………………………………………………………………….........................7

1.2 Schematic Composition…………………...………………………………………………..............................8

2. Equations of State

2.1 History, Theory……………………………………………………………………………………................11

2.2 FPS and SLy EOSs…………………………………………………………………………………..............14

3. Neutron Star Model Theory and Numerical Analysis

3.1 Stellar Structure Equations………………………………………….……………………………………….17

3.2 Finite Difference Method, Solution….………………………………………………………………………18

3.4 Sources of Error……………..……………………………………………………………………………….19

4. Results and Concluding Remarks……………………………………………………………………………………20

Appendix I: Complete Python code for 1-D neutron star model 23

Appendix II: Tabulated EOS values for FPS and SLy models 24

Bibliography 27

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1 Neutron Star Theory

1.1 Birth, Evolution

Fig. 1.1 Six main stages of neutron star evolution from birth to old age (Prakash, 2010)

This paper surveys the overall evolution of neutron stars, and the analytical tools that enable a

mass-radius estimation of these stars under some simplifying approximations. I begin with an overview

of neutron star creation and evolution that mirrors Fig. 1.1 (Prakash, 2010). The progenitor of every

neutron star is a massive star that has gone supernova. In Type II supernovae, stars of masses 8-50 times

the mass of the sun (M☉) violently collapse when their iron-dominant core surpasses the Chandrasekhar

limit of 1.4 M☉. Beyond this mass, the core is unable to support the immense pressure via electron

degeneracy pressure. With rapid collapse of the core comes an associated ejection of the outer mantle, as

dictated by Virial’s theorem. These extreme pressures also produce photons of high enough energy to

drive the photodisintegration of iron. This robs the iron of the fusion energy in an instant that took the star

a lifetime to create. The products of iron photodisintegration are helium and free neutrons, mediated by

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the reaction 56Fe→134He+4n−124MeV (Prialnik, 2010). With such rapidly increasing densities, it is

possible that the core could surpass the Tolman-Oppenheimer-Volkoff limit, the maximum pressure a

neutron degenerate gas is able to withstand before collapsing into a black hole. Assuming it does not, the

resultant 4He is further deconstructed into a sea of elementary particles as high densities and

temperatures drive the photodisintegration of helium, 4He+γ→ 2p+ 2n and then p+ e− → n+νe−

.

Because all these processes are highly endothermic they drive an increase in density. In this final process

the majority of the supernova energy is released in the form of high-energy electron neutrinos of 200-300

MeV (Prakash, 2010). The weakly interacting particles are released in high enough fluxes (~1057υe− over

~15 seconds) to further jettison the outer envelopes of the star and spur heavy-element nucleosynthesis.

Some equations of state for the neutron star core allow for the creation of strange matter in the form of a

Bose-Einstein condensate, quark matter, or hyperons. As these particles are not as repulsive as neutrons at

such large pressures, it is possible that the emergence of strange matter in this stage could push the

neutron star over the Tolman-Oppenheimer-Volkoff limit

In the wake of such an explosion our neutron star, of approximate mass range .1 – 3 M☉ and

radius range of tens of kilometers, is done producing energy and will begin to lose energy. For the first

minute, most neutron stars will be hot enough to remain opaque to its neutrino emissions. When the mean

free path of an emitted neutrino is comparable to the mass of the star, the core will begin to cool by

neutrino emission, while the crust will maintain at T≈ 3*106K for about 100 years. After this time the

young neutron star is isothermal. In the proceeding 100yr---3Myr, cooling is dominated by neutrino

emission via the modified Urca process, n→ p+ l +υ or its inverse, p+ l→ n+υ for densities ≥

1015( gcm3 )

, and the weaker direct Urca process, n+ N→ p+ N + l +υ or its inverse, p+N+l→n+N+υ ,

where N refers to nucleons (either protons or neutrons), and l to leptons (any of the six flavors of

neutrinos and muons)(Nomoto, 1986). This final phase of cooling is accompanied by less intense infrared

photon emission, which is gravitationally redshifted into the soft X-ray range. After approximately a

million years, the photon emissions will become the dominant mode of cooling.

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1.2 Schematic Composition

For the vast majority of a neutron star’s life, the interior of the star is fully catalyzed and “cold”.

It is for this epoch that I give an overview of the main phases of nuclear matter in the interior. Indeed, the

equations of state described in section two extend this coldness assumption, approximating T=0°K. The

large-scale structures of a neutron star are, moving from the surface to the core: the atmosphere, outer

crust, inner crust, outer core, and inner core. These regions are displayed below.

Fig 1.2 Schematic of neutron star structure and composition (Haensel, 2007)

Atmosphere: The thinnest region of the star, the atmosphere is the only region that may behave

as a completely ideal gas. It is comprised of a plasma, although it has been theorized that very cold, or

ultramagnetized neutron stars may have a liquid or solid surface.

Outer Crust: Extending approximately 100 meters below the atmosphere, this shell is comprised

of heavy nuclei, neutrons and electrons. This outer shell possesses a typical range of density found in

white dwarfs. It exhibits a thin shell of ideal electron gas atop an increasingly degenerate gas mix. Below

the thin top layer, a solidification of the crust ensues. Under these increasing densities, the EOS of this

shell shifts from being dominated by the ideal gas pressure to the electron-degenerate pressure (Haensel.

2007). Let it be noted that the internal forces are not yet high enough to engage the repulsive neutron

degenerate pressure, however they do encourage increased neutronization via electron captures. This

neutronization becomes increasingly significant at the base of the outer crust where the neutron drip

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density, ρND , is reached (approximately 5*1011( gcm3 ) as averaged between the FPS and SLy EOSs).

The neutron drip occurs when the lattice of heavy nuclei begin to leak free neutrons. This serves to make

the material more compressible (and corresponds to a “softer” EOS), as it removes electrons that supplied

the supportive pressure (Kippenhahn, 2012).

Inner Crust: About one kilometer thick, the densities here range from ρND ≤ ρ ≤ .5ρ0 where

ρ0 = 2.85*1014 ± 5*1012 ( g

cm3 ) is the nuclear saturation density (Haensel, 1981). At the top of this

shell the relativistically degenerate electron pressure dominates the EOS. At the bottom the neutron

degeneracy pressure becomes dominant. Matter in this shell is composed of free neutrons, free electrons,

and neutron-rich atomic nuclei.

Outer Core: At this depth all heavy nuclei are neutronized. Densities range from

.5ρ0 ≤ ρ ≤ 2ρ0 (Haensel. 2007). Correspondingly, this shell is almost completely comprised of

neutrons, with a several percent admixture of protons, electrons, and possibly muons. The electrons and

muons form a nearly ideal Fermi gas. The neutrons and protons form a strongly interacting Fermi

fluid/superfluid, mediated by nuclear forces. The EOS of the outer core is formulated in theory by

applying the conditions of charge neutrality and beta equilibrium (the process of muon-neutron decay and

creation), supplemented by a microscopic model of many-body nucleon interaction.

Inner Core: Relevant for high-mass neutron stars, densities range ρ ≥ 2ρ0 . At such high

densities, the creation of strange matter and the emergence of strange nuclear interactions becomes a

distinct possibility. Accordingly, compositional descriptions and EOSs of the inner core are model

dependent and highly theoretical. The main classes of models revolve around: the creation of hyperons,

the boson condensation of pions, the Bose-Einstein condensation of kaons, and the phase transition to

quark matter. Note that evidence of this quark matter is scarce, and is non-existant for kaon condensates.

This is reflected in the reliability of corresponding EOSs and their fit to observational data.

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2

Equations of State

2.1 History, Theory

At its most basic level, an EOS relates a pressure to the energy density of the matter in question.

These relations summarize the fundamental interactions at play in the matter that may arise from

compositional properties like those discussed in Section 1.2 (of particular interest to this paper are the

nucleon-nucleon interactions and the nuclear drip density). The resulting variety of EOSs is valuable for

two reasons. Firstly, it provides a contrast between different theoretical paradigms. Secondly, it spurs the

discovery of more general correlations of the stellar structure that may hold regardless the theoretical

paradigm employed in a given EOS. For example, the AP4 EOS of Akmal and Pandharipande

incorporates AP1-3. While AP1-3 are founded in differing theoretical paradigms, they are included to

reinforce correlations between stellar structure and microscopic physics (Prakash, 2010)

An important quality of any EOS is its relative “stiffness”, which qualitatively describes how

compressible the bulk matter is. A stiffer EOS will support more mass at the same density. A softer EOS of

a neutron star will precipitate the gravitational collapse to form a black hole at a lower mass. While it is

generally assumed that a stiff EOS implies both a large maximum mass and a large radius, many counter

examples exist. For example, the GM3, MS1, and PS EOSs have relatively small maximum masses but large

radii compared to most other EOSs with larger maximum masses. Also, not all EOSs with extreme softening

have small radii for M > 1 M (the GS2, PS EOS)(Lattimer, 2000).

The progenitor of a neutron star EOS came from Landau in 1932. Landau arrived at the

postulation of the neutron star description of state by extending the treatment of the white dwarf stars and

the electron degeneracy pressure that won Chandrasekhar the physics Nobel Prize in 1983 to nuclear matter.

Interestingly, Landau’s development of this analysis occurred independently and concurrently of

Chandrasekhar’s. He postulated the existence of one gigantic nucleus, expecting that in such stars “the

density of matter becomes so great that atomic nuclei come in close contact forming one gigantic nucleus.”

This description is impressively apt considering it was published before the discovery of the neutron (by

Chadwick in 1932), and 35 years before the first observation.

This description was quickly progressed by Baade and Zwicky in 1934, which proposed the birth

of neutron stars from supernovae, similar to the description in Section 1.1. They also postulated several

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theoretical descriptions of state to explain the densities in the core, which they correctly predicted would

exceed the typical nuclear density. These descriptions were wildly inaccurate, assuming no photon pressure

and predicting a type of neutron “rain”.

The next notable development in the EOS came from Tolman Oppenheimer and Volkoff (or TOV)

in 1939. They derived a generally relativistic formulation of hydrostatic equilibrium for a spherically

symmetric star. This is especially applicable to neutron stars, whose great gravitational forces warp

spacetime in ways that disagree with the Newtonian formulation. The shortfall of this EOS was that it

neglected local neutron-neutron and many-body interactions, making their EOS very soft. Accordingly,

TOV underestimated the maximum mass of a neutron star to be ~0.7 M☉. It is worth noting that

Chandrasekhar and von Neumann obtained identical results in 1934, but failed to publish their results.

Between 1939 and the first published observation in 1968, the advancement of particle physics

spurred EOSs to describe the high density shells of neutron stars are too dense to determine empirically. In

the 1950’s, Wheeler built from TOV’s model by including nuclear interactions, effectively stiffening the

EOS. This reformulation raised the maximum neutron star mass to 2M☉. Zeldovich, 1961, published an

even stiffer EOS by modeling baryon interactions in the highly relativistic limit (such that the speed of

sound approached lightspeed). Bardeen, Cooper and Schrieffer were the first to propose the superfluidity

baryonic matter in the crust and core (BCS theory). Throughout the 1960’s, postulates of strange matter

(hyperons, mesons, pions and muons) in neutron stars were put forward to develop unique EOSs.

In light of these numerous, distinct EOSs, the value of a unified EOS in forming a stellar model

becomes apparent. A unified EOS is one that describes the different schematic mass shells without major

discrepancies in pressure or density on either sides of the interfaces. In forming a unified EOS, more

localized EOSs are conjoined using simplifying approximations, many-body calculations, or interpolation

at boundaries (Haensel, 2004). It is possible for some unified EOSs to be described in a completely

analytical method, due to their mathematical cleanliness.

In August 1967 the first observation of a neutron star was inferred via the periodic radio emissions.

Quickly following the announcement of the observation in 1968, it was determined that the short rotational

period of 0.33 ms could not be produced by a rotating white dwarf and thus must be a rotating neutron star,

or a pulsar.

Since then, new observational techniques have allowed theoretical and observational

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astrophysicists to rule out many EOSs, and broaden the possible mass range of neutron stars. However

because of their small radius and faint emissions, a degree of uncertainty to these measurements persists.

This ambiguity is manifest in the number of EOSs that conform to observations.

Fig. 2.1 Above, a survey of all neutron star mass measurements and uncertainties, as of November 10th,

2010. The four differently colored regions refer to the methods of observation (Lattimer, 2011)

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Fig 2.2 The mass-radius plot shows an assortment of EOSs in lines, and three constraining pulsar mass

observations in horizontal bands. The blue lines refer to EOSs dominated by nucleon interactions, the pink to EOSs including nucleon interactions and the emergence of exotic matter, and the green to EOSs

concerning strange quark matter. The J1614-2230 pulsar measurement of 1.97± .04M☉ rules out EOSs that do not intersect the horizontal band. The grey regions show parameter space that is ruled out by other

theoretical or observational constraints.

2.2 FPS and SLy EOSs

In this section I will summarize the main features and considerations associated with the two

EOSs I chose to apply in my 1-D neutron star model. Both the BPS and SLy unified EOSs are founded in

the theory of neutron-neutron interactions at high densities, but are supplemented by the HP94 and BPS

EOS at lower densities. The FPS EOS was presented by Pandharipande and Revenhall in 1989 and the Sly

EOS by Douchin and Haensel in 2001. The FPS or SLy theory becomes dominant at ρ > 5*1010 ( gcm3 ) ,

in the core. The HP94 EOS becomes dominant at densities of 108( gcm3 ) ≤ ρ < 5*10

10 ( gcm3 ) ,

describing the majority of the crust. At still lower densities, ρ ≤108( gcm3 ) , the EOS is described by the

BPS model. Below densities of 108( gcm3 ) ,the EOS is no longer temperature independent, and the BPS

EOS begins to underestimate pressures. For this reason, densities below this value are neglected in their

unified forms.Both the HP94 and BPS models are experimentally grounded, deriving their EOS from

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interpreting neutron-rich scattering data and a semi-empirical nuclear mass formula (Haensel, 2008). This

enables a check on the theoretical FPS and SLy models in the smoothness of the EOS around the neutron

drip density.

Note that the FPS and SLy EOSs discard BPS EOS data below ρ <105( gcm3 ) because they

have a strong temperature dependence. This is expected as these densities correspond to the only ideally

behaving region of a neutron star, the outer envelopes. These models, like most others, assume that the

neutron star matter is cold and catalyzed. This enables the assumption that T=0. This is important to most

EOSs because it allows the pressure to be only density dependent (much like polytropic EOSs). Rejection

of these data has little effect on the stellar model because the outer envelopes contribute a negligible

amount of mass to the star.

The main difference between the FPS and SLy models is in their treatment of the crust-core

interface. We know from Section 1.2 that the heavy nuclei within the crust are neutronized as we move

deeper into the core. The SLy model treats this interface as a weak first-order phase transition with a

relative density jump of ~1%. Additionally the SLy model uses a neutron drip threshold value of

ρND (SLy) ≅ 4*1011(g*cm−3) (in agreement with the semi-empirical HP94 EOS), while the FPS model

uses ρND (FPS) ≅ 6*1011(g*cm−3) (Haensel, 2007).

The FPS model employs a more gradual transition theory, drawing from the transition of nuclei

dimensionality under increasing pressures. For densities less that the saturation density we expect to find

the typical spherical nuclei. However, if the fraction of volume occupied by nuclear matter exceeds 50%

the nuclei will invert, forming bubbles of neutron gas. This is referred to as the bubbular phase of spherical

nuclei. As densities surpass the saturation density, it becomes energetically favorable for nuclei to change

their dimensionality. The dimensions, d = 3, 2, 1, correspond to a basic set of spherical, cylindrical, and

planar geometries respectively. Each set contains two geometries, one being the “inside out”, or bubbular

form of the other. Let us index the possible phase geometries by dimensionality and inversion: sphere (3A),

bubble (3B), cylinder (2A), cylindrical shell (2B), planar (1A) and inverted planar (1B). The transitions

between these phases of matter occur in this order as density increases: 3Aè2Aè1Aè1Bè2Bè3B

(Ravenhall, 1983). These various shapes of “nuclear pasta” serve to smoothen the transition to the uniform

plasma phase of matter in the inner core. Note that analytic derivations of the BPS and SLy core EOSs

from basic principles are outside of the scope of this paper, and involve the presence of hyperons,

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superfluidity, and stellar pulsations. My main motivations for choosing these EOSs was their unified

quality and the availability of tabulated EOS values to be used in my model.

Fig 2.2 Unit cells for three nuclear shape geometries of size rc . Hatched regions show nuclear matter

while white regions show neutron gas. In the bubbular phases, the hatched and blank regions are

exchanged.

Fig. 2.3 Above, a comparison of the SLy and FPS EOSs around the crust-core boundary. Thick solid lines

refer to the inner crust of phase 3A. The thick dashed line sets the “nuclear pasta” phase transition range.

Thin solid lines refer to uniform neutron plasma.

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3

Neutron Star Model Theory and Numerical Analysis

3.1 Stellar Structure Equations

In order to create a model that determines the pressure, density, and enclosed mass as function of

radius for a neutron star, the stellar structure equations must be solved. They are as follows:

ρ =mV=34mπr3

(1)

Equation one is the simple density relation, applied to a spherical geometry. Putting (1) in terms

of radius we have equation (2)

r = 34π

mρ

3 (2)

(3)

Equation three is the Tolman-Oppenheimer-Volkoff equation (TOV) mentioned in 2.1. It is

derived from solving the Einstein equations under a time-invariant, spherically symmetric metric.

Derivation of (3) is beyond the scope of this paper. However it is worth noting that if removing the terms

in order of 1c2

(3) becomes the typical hydrostatic equilibrium equation.

EOS = ρ(P) (4)

Equation four refers to the specific EOS to be supplemented into the (3). For my model, I

consider the EOSs, FPS and SLy (and by extension HP94 and BPS). The form of these EOSs is discussed

in Section 2.2. The values used in my numerical calculations are tabled in Appendix II.

Because there are no exact analytical solutions to this set of equations with my semi-empirical,

semi-theoretical unified EOSs, I solve these numerically using the finite difference method.

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3.2 Finite Difference Method, Application

This method is used to numerically solve differential equations by discretizing the ranges of the

equation in question and approximating a derivative by Taylor expansion. To apply this method, the

derivatives of the (3) must be able to be approximated by a Taylor series expansion:

(5)

Where Rn(x) is the remainder term, h a step value applied over the range of (3) to discretize the

results of (3), and x0 the initial guess input for (3). In my model, x0 takes the form of a constant density

guess for (3), equated by (2) in conjunction with an evenly spaced mass array to give radii as a discretized

function of constant density and discrete mass values. This guess x0 = a, we have (to first order):

(6)

and solving for f’(a)…

(7)

Finally, in assuming a negligible remainder we arrive at the finite difference approximation of a derivative.

(8)

Convergence of stellar structure relations using the finite difference method is made evident

stepping through the procedure of my model for a single iteration. Beginning with the (clearly incorrect)

guess of a constant density stellar interior, and an array of mass shells beginning at the core (m=0) and

ending at the effective surface (m=M), I substitute into (2) to create an array of radii corresponding to the

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array of masses at constant density. At this point I substitute (2) and my density guess into (3). After

integrating, we are returned pressure as a function of radius at our discrete values. Finally, to converge

towards the true density profile, I substitute our pressure values into an interpolated EOS (either FPS or

SLy) to be returned density values. These density values will not be constant over radius as before and

will converge towards the exact solution with repeated iteration.

3.4 Adiabatic Index

Defined as the ratio of specific heat at constant pressure to specific heat at constant volume, the

adiabatic index is an important unitless value for determining the stiffness of an EOS. The formulation of

the adiabatic index in terms of variables used in my model is dependent upon the presence of hyperons,

stellar pulsations and baryon superfluidity (Haensel, 2002). Analysis of these variables resulted in the

following equation for the adiabatic index,

(9)

where n refers to the neutron number density, and c refers to lightspeed. I analyze the relative

stiffness of the FPS and SLy models by plotting their adiabatic index as a function of density in the core.

3.5 Sources of Error

As with any converging numerical solution to differential equations, there will be

sources of irreducible error. In this model, the two sources of error are round-offs and the finite step size.

Both these sources of error are propagated in interpolation, while only the round-off error is involved in

numerical manipulation. Interpolation is an important tool of numerical analysis that allows a discretized

function to return a value for an argument that is not one its discrete values. In my model I made use of

linear interpolation in producing ρ (P) values that lay in between the tabulated values. For an intermediate

( ρ ,P ) value in between (ρa,Pa ) and (ρb,Pb )

ρ − ρaρb − ρa

=P −PaPb −Pa

(10)

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This linear interpolant pair is subject to an irreducible uncertainty proportional to the square of

the step size. If we define our step size as h = ρb − ρa then our error, χ ∝ (h)2 , or h∝ χ . Note that

polynomial interpolation carries a higher degree of error from step size than linear interpolation, and is

thus unwanted in accurate simulations.

The other source of error comes from the computer’s finite memory and the limiting number of

bytes allowed for any given number. Because the numbers and equations dealt with in this model are not

comprised of integers and linear operations, any operations carried out on them will produce an

approximate value out to a finite number of decimal points. The computer will round the final decimal

place up or down. Carrying out further operations with this result will compound the error further (for

example consider 4.95*5.00 = 24.75 versus 5.0*5.0=25).

These sources of error are of particular concern for neutron star models, as very few analytical

unified EOSs exist. So in most cases, physicists must interpolate from tabulated EOS values. While these

methods are used to produce reproducible models, the degrees of error may be unique to the computer it

was run on. This is because different machines may carry out different interpolation methods, and allot

different amounts of memory to a number. As of yet there are no agreed-upon conventions to systematize

these sources of error, so differing results of identical methods and data are to be expected.

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Results and Concluding Remarks

The results of my 1D simulation are meant to affirm the theory presented in Section 2.2, as well

as agree with the results of Haensel. Here I present four pieces of comparative information to aid analysis:

a graph of the EOS pressures around the neutron drip density, a graph of the adiabatic index as a function

of density, mass-radius relationships of the EOSs, and the maximum masses of the EOSs.

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Fig 4.1 FPS and SLy EOSs between the neutron drip density (approximately 11.5 g/cm^3 on this scale)

and the crust-core interface (approximately 14.2 g/cm^3 on this scale)

Fig. 4.2 Adiabatic index profile of the FPS and SLy EOSs from the neutron drip density (approximately

11.5 g/cm^3) and the crust-core interface (approximately 14.2 g/cm^3)

From these two plots we may compare the stiffness of these two EOSs in the density ranges for

which they differ. The adiabatic index plot exhibits a steeper, and deeper softening of the SLy EOS at the

lower neutron drip density of ρND (SLy) ≅ 4*1011(g*cm−3) .This is in agreement with the first-order

phase transition utilized. Note also that the slight leveling of the SLy line in Fig. 4.1 at this value is

consistent with the ~1% density decrease at this interface. The FPS EOS, on the other hand, displays more

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gradual softening in line with the bubbular nuclear phase transitions discussed earlier. Both EOSs display

extreme stiffening around the crust-core interface, where strong nuclear, many-body interactions dominate.

The nearly vertical softening/stiffening of the BPS EOS around the nuclear saturation density is an artifact

of the finite number of bins in the simulation.

Fig 4.3 Mass-Radius relations of the FPS and SLy EOSs for a 1.8M☉ neutron star

The steeper FPS mass-radius curve reflects its relatively softer EOS, for with a less supportive

pressure, more mass will be squeezed within an equivalent radius. The flattened regions of both curves

near their surfaces are evidence of corresponding low surface densities. This low-density regions grows

with decreasing neutron star mass.

The maximum mass calculations of 1.8M☉ for the FPS model and 2.05M☉ for the SLy model

were derived by keeping tabs on the rest-mass energy of the stars while running my iterative numerical

solution algorithm. The conditions for neutron star core collapse are met when the neutrons become so

relativistic that their energy density grows larger than their rest mass density. I checked this condition by

comparing the non-relativistic results of hydrostatic equilibrium with those of the TOV equation. When

the difference between the results grew larger than a factor of 10, I deemed the star unstable. This factor

was a matter of preference, chosen because values much larger caused the stellar radius to blow up to

unphysical values, and the interpolator to be given pressure values exceeding those given by the tabulated

EOSs.

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The limiting approximations used in this model parallel those made by Haensel and Potekhin

(2004). Our shared approximations were that the star is non-rotating, and has zero temperature. These

approximations greatly simplify the EOS, but lead to further approximations. A rejection of temperature

dependence entails the rejection of stellar regions that have a temperature dependent EOS. Haensel and

Potekhin chose to dismiss densities less than 105( gcm3 ) in their unified EOS, as it is misrepresented by

the BPS EOS (which does contain values below this threshold). I applied this boundary condition in the

form of a constant surface pressure of 5*1022 (Ba) , which was the associated pressure of the SLy and

FPS density threshold. This approximation allowed me to propagate the pressures from the outer

envelopes down into the core without having to describe their EOS. This approximation has a negligible

effect on the determination of total mass and radius, as the outer envelopes only constitute a few hundred

meters of the star.

Appendix I: Complete Neutron Star Model Code

The following code was written for Python 2.7.6, with the use of the additional Scipy, Numpy,

and Matplotlib library packages. The FPS and SLy EOSs were supplied by Haensel and Potekhin, 2007

from their website http://www.ioffe.ru/astro/NSG/NSEOS/. The .txt files retrieved were edited to be

readable by the program. The number of lines before the tabulated EOS values began were made even,

and a row of zeros was put at the start to avoid interpolation errors. Note that values of mass and density

were interpolated because the integrator sp.integrate.odeint() required continuous values in order to

integrate.

import numpy as np import scipy as sp from scipy.interpolate import interp1d import matplotlib.pyplot as plt def interpmaker(filepath):

data = np.loadtxt(filepath, skiprows=7) return interp1d(data[:,3], data[:,2])

FPS = getDensityFPS = interpmaker("/Users/AndyStine/Desktop/fps.txt") SLy = getDensityFPS = interpmaker("/Users/AndyStine/Desktop/sly.txt") mSun = 1.989e33 # g km = 100000 # cm G = 6.67e-8 # cm^3*g^-1*s^-2 c = 2.998e10 # cm*s^-1 def TOV(m,r,rho,p): # function of mass, radius, density, pressure

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adjustedGravity = - G / r**2 * (m + 4 * np.pi * p * r**3 / c**2) adjustedDensity = rho + p / c**2 correction = (1 - 2 * G * m / (r * c**2))**-1.

result = adjustedGravity * adjustedDensity * correction if result/HSE(m,r,rho) > 10: print "TOV Exploding!" #Alerts when TOV becomes unphysical, M>Mmax return result

def HSE(m,r,rho): #Hydrostatic Equilibrium EQ

return -(G*m*rho)/(r**2) # let m=x*mSun, loops determine number of iterations, choose either FPS or SLy for EOS def model(m, loops, EOS,s=100000, p0=1e29,radGuess=1e6, useTOV=True):

denGuess = (3*m)/(4 * np.pi * ((radGuess)**3)) mass = np.linspace(0, m, s) dMass = np.diff(mass) radius = np.zeros(np.shape(mass)) radius[1:] = 3*dMass/(4*np.pi*denGuess) radius = np.cumsum(radius)**(1./3) density = np.ones(np.shape(dMass)) density = density * denGuess

for i in xrange(loops):

# TOV lambda function if useTOV:

dpdr = lambda p, r: TOV(np.interp(r, radius, mass), r, np.interp(r, radius[:-1], density),p) else:

dpdr = lambda p, r: HSE(np.interp(r, radius, mass), r, np.interp(r, radius[1:], density))

# Pressure integrates the TOV EQ from p0 to core pressure = sp.integrate.odeint(dpdr,p0,radius[:0:-1])[::-1].flatten() density = EOS(pressure) radius[1:] = .75 * dMass/(np.pi * density) radius = np.cumsum(radius)**(1./3) outer = radius[-5]

return outer/km

Appendix II: Tabulated Values for the SLy and FPS EOSs

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Table 1: Discreet values of the SLy EOS wrapped around in three columns, each with four columns denoting (from left to right): an

indexing number, the number density in fm−3 , density in ( gcm3 ) ,

and pressure in Ba. Densities range from

45.1≤ ρ ≤ 6.749*1015( gcm3 ) . Indices 44-65 were provided by a fit

between EOSs Sly4n (2003) and Sly4 (2000). Entries with densities

less than 5*1010 (gcm3 ) were supplemented by the BPS and HP94

EOSs (Haensel and Potekhin, 2007).

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Table 2: Discreet values of the FPS EOS in two columns, each with four columns denoting (from left to right): an indexing number, the number density in

fm−3 , density in ( gcm3 ) , and pressure in Ba.

Densities range from 45.1≤ ρ ≤1.05*1017( gcm3 ) .

Values with densities less than Γ ( gcm3 ) were

supplemented by the BPS EOSs (Haensel and Potekhin, 2007).

27

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