1. Reaction Rate Studies in Novae with an
Emphasis on the Astrophysically Relevant
𝑭𝟏𝟖
𝒑, 𝜶 𝑶𝟏𝟓
Reaction
by
Matthew Thickitt
A project submitted to
partially fulfil the
requirements for the degree of
Master of Physics
Department of Physics
University of York
Heslington
York
YO10 5DD
May 2013

2. 2
Abstract
Computational studies into the nuclear behaviour of novae are at the forefront of nuclear
astrophysics and of particular interest is the 𝐹!"
𝑝, 𝛼 𝑂!"
reaction. The radioisotope 𝐹!"
is a
strong source of observable gamma-rays in novae, yet the reaction rate of its destruction is
poorly understood. This work aimed to study the influence of reaction rates on the nuclear
behaviour of novae. Initially, the Webnucleo post-processing codes were used to simulate the
reaction networks for the HCNO-II/ III cycles in two theoretical 1 MSun CO and 1.25 MSun ONe
novae. The product abundances for each reaction in the cycle were explored at three different
rates (x0.01, unchanged, x100) to investigate their rate sensitivity. It was found that the
abundance of daughters from the proton-capture reactions varied linearly with reaction rate,
whereas the daughters of the 𝛽!
-decay reactions were not reaction rate sensitive at all. An
analysis into the temperature dependence and reaction behaviour is also presented. The results
from these simulations were as expected from theory and so support the code’s validity. A
study into the 𝐹!"
𝑝, 𝛼 𝑂!"
reaction by Laird et al (2013) was performed by calculating the
normalised reaction rate as a function of nova temperature (0.1-0.4GK) for the narrow
resonances in the 𝐹!"
𝑝, 𝛼 𝑂!"
reaction. The results presented support the analysis from Laird
et al that the 48keV and 331keV resonances dominate over nova temperatures. The spin-parity
assignment of the 331keV resonance was also explored with a sensitivity study into the rate
dominance when it is assigned either 𝐽!
= !
!
!
or 𝐽!
= !
!
!
. The study shows that changing the
assignment has an insignificant effect on the rate dominance.

3. 3
Acknowledgements
The biggest acknowledgment must go to my supervisor, Dr Alison Laird, for her continued
support with the project and my other endeavours in life, as well as her tolerance for my (more
than occasional) ignorance! Additional thanks must go to Ben Shaw for his help with the
computational aspect of the project. I greatly appreciated his continued support at all times,
especially during the beginning of the project when I felt like a fish out of water. Thanks must
also be extended to Trystan Coulson Thomas. Last but certainly not least, I would like to thank
my ever loving family for their continued support throughout the last four years at university:
Mum, Dad, Jess, Bud, Pat, Callum, Hayden and Kyra, I couldn’t have done it without you!

4. 4
Contents
Abstract ............................................................................................................. 2
Acknowledgements........................................................................................... 3
1. Introduction .................................................................................................. 6
1.1 Introduction ...............................................................................................................6
1.2 Motivations and Objectives.......................................................................................6
1.2.1 Objectives of the Project .........................................................................6
1.2.2 Motivation ...............................................................................................7
1.1.2.1 General Motivations .....................................................7
1.1.2.2 Importance of the F!"
p, α O!"
Reaction.....................7
1.3 Previous and Current Studies ....................................................................................8
2. Theoretical Considerations.......................................................................... 9
2.1 Scientific Background ...............................................................................................9
2.1.1 An Introduction to Novae........................................................................9
2.1.2 The Hot-CNO (HCNO) Cycle in Novae ...............................................11
2.1.3 The Physics of Reactions.......................................................................12
2.1.3.1 Resonant Reactions ....................................................14
2.2 Methodology............................................................................................................15
2.2.1 The ‘libnucnet’ Module.........................................................................15
2.2.2. Running Simulations to find the Abundance of the HCNO-II/III
Products for the 1 MSun CO and 1.25 MSun ONe Novae Profiles ..........16
2.2.3 Calculating Reaction Rates for the F!"
p, α O!"
Reaction from
Various Parameters................................................................................16
2.3 Error Considerations................................................................................................17
3. Results and Discussion ............................................................................... 17
3.1 Investigation into the HCNO-II and II Cycle Abundance Evolution......................17
3.1.1 Abundance of HCNO-II Cycle Products in a 1 MSun CO Novae and
Analysis .................................................................................................18

5. 5
3.1.2 Abundance of HCNO-III Cycle Products in a 1 MSun CO Novae and
Analysis ..........................................................................................................20
3.1.3 Abundance of HCNO-II Cycle Products in a 1.25 MSun ONe Novae and
Analysis ..........................................................................................................22
3.1.4 Abundance of HCNO-III Cycle Products in a 1.25 MSun ONe Novae and
Analysis ..........................................................................................................24
3.2 Investigation into the F!"
p, α O!"
Reaction..........................................................25
3.2.1 Investigating the Narrow Resonances of the F!"
p, α O !"
Reaction....25
3.2.2 Investigating the 331 keV Resonance J!
assignment...........................26
3.2.3 Other Studies .........................................................................................27
4. Conclusions.................................................................................................. 28
Bibliography.................................................................................................... 29
Appendix.......................................................................................................... 31
Appendix A: Finding the Gamow Window Peak and Width........................................31
Appendix B: Deriving the Reaction Rate for a Narrow Resonance..............................31
Appendix C: Running a Simulation ..............................................................................33
Appendix D: Changing Reaction Rate files ..................................................................35
Appendix E: Breit-Wigner/ Resonance Strength Solver...............................................36

6. 6
Chapter 1
Introduction
1.1 Introduction
Nuclear astrophysics is one of the most all-encompassing fields within modern science today.
The analysis of how the smallest of processes can drive some of the most volatile physical
environments imaginable has cemented its status as one of the most revered branches of
physics. From the early works of Hoyle in the 1950s to the contributions to neutrino physics by
Ray Davis in the late 1960s [1], such feats represented huge leaps in not only their respective
branches, but to physics as a whole. The continual strive to explain the unreachable has paved
the way for countless important innovations. The discoveries of the intricate fusion processes
that sustain solar furnaces, solving the solar neutrino problem and explanations for the most
cataclysmic of events in the universe are only a few of countless many examples of the
overwhelming contribution of this field to human knowledge. Yet the implications of nuclear
astrophysics are not only constrained to this field. For example, the use of radioactive
fluorine-18 beams (rife in reaction rate studies) can also be applied to research into the
radionuclides used in Positron Emission Tomography (PET) scanners, which are used
extensively in the medical world [2].
The study of the nuclear behaviour governing the initiation and evolution of cataclysmic end-
point events has been at the forefront of both experimental and theoretical work for many years.
Trying to understand the nuclear intricacies of extremely exotic events, such as novae,
supernovae and X-Ray Bursts (XRBs), still remains a colossal challenge. The introduction of
computational studies coinciding with the ever-increasing processing power of today’s research
computers has made unravelling this picture much more tangible. Yet, uncovering some secrets
of the universe only exposes us to more challenging ones, and this field is no exception to that.
1.2 Motivations and Objectives
1.2.1 Objectives of the Project
This project looks into utilising post-processing codes to study how the reaction rate of
explosive nucleosynthesis processes can govern the behaviour of novae, a cataclysmic
explosion stemming from the evolution of a binary system of a degenerate white dwarf
receiving hydrogen rich material from a companion star. The coding package ‘libnucnet’,
which is a part of the Webnucleo code library [3], will be used to study several scenarios and
reactions within different types of novae. This projects aims to explore how changing the rates
affects the evolution and abundance of the relevant HCNO (Hot Carbon-Nitrogen-Oxygen)
cycles that operate within novae. The production and destruction of the astrophysically relevant
radioisotopes in this cycle will be simulated and analysed thoroughly to gain an insight into
how influential the reaction rates really are. The project will also investigate how parameters
determining the reaction rate can be studied by exploring real data for the relevant
𝐹!"
𝑝, 𝛼 𝑂!"
reaction. By carrying out these investigations, it is hoped that a critical insight

7. 7
into the nuclear behaviour of novae will be obtained by building upon current experimental
research; so giving a clearer indication of how important the reaction rate is to nuclear
evolution.
The project is broad in the sense that there was not one significant aim, yet it was a
comprehensive study of several interesting properties of novae. The main objectives included
evaluating the abundance behaviour over a nova timescale for the HCNO-II/III cycles and the
nuclei created in them in CO and ONe novae, as well as investigating the temperature
dominances and dependencies of the narrow resonances in the 𝐹!"
𝑝, 𝛼 𝑂!"
reaction. The spin-
parity dependence of the 331keV resonance within this reaction and its contribution to the
reaction rate was also investigated. The effects of varying the reduced proton width of the
48keV resonance was explored briefly, as this was identified as a key source of the reaction
rate uncertainty in Laird et al [9]. Additionally, it hopes to act as a platform to show that the
results gathered from the code are accurate and reliable, as well as supporting the results
gathered in current and relevant publications.
1.2.2 Motivation
1.2.2.1 General Motivations
One of most profound problems with studying the nuclear behaviour of novae is that direct
experiments cannot be performed; instead physicists must rely on models to explain the
behaviour that is seen. However, these models are theoretical and understandably not always
precise. One motivation within the project is to use the best models available to study how
changing the reaction rate affects the evolution and abundance of certain nuclear networks, and
to see if the library of codes being used produce the results that one would expect theoretically.
1.2.2.2 Importance of the 𝐹!"
𝑝, 𝛼 𝑂!"
Reaction
The main motivation is the analysis of the astrophysically relevant 𝐹!"
𝑝, 𝛼 𝑂!"
reaction,
which is one of the most widely studied reactions within nova studies. Through observation and
experimentation, one finds that the current models of novae largely underestimate the amount
of material ejected. Many studies have been conducted to attempt to find the reaction rate to
within 30% precision [4,5,6], yet it still remains one of the biggest problems in the field. A
possible mode of studying this is through gamma-ray detection by space-based satellites (e.g.
INTEGRAL/ GLAST [7]). In the early stages of the nova expansion (t < 110 minutes [8]), the
expanding envelope is optically thick and opaque to gamma-rays. After this time, the envelope
becomes optically thin enough to be transparent to gamma-rays, so mediating their escape [9].
Of particular importance are gamma-rays that have an energy of 511keV. These are created by
electron-position annihilations that feed from positrons created by the 𝛽!
-decay of 𝐹!"
.
Comptonization of the newly created gammas results in their characteristic energies. The half-
life of 𝐹!"
(109 minutes [8]) means that the positrons are emitted when the expanding nova
envelope becomes transparent to gamma-rays, meaning that 𝐹!"
is one of strongest observable
sources of gamma-rays in novae. Importantly, the number of gamma-rays detected depends on
the abundance of 𝐹!"
left over from the novae explosion [4]. The detectability distance to
novae can also be determined from this information, and gives experimentalists an indication to
within what distance such observations can be made and thus how many novae can be observed

8. 8
using such methods [7]. Hence, studying this reaction has important implications to the nuclear
model and observation of novae, and this project aims to do this through evaluating its
production and destruction via the HCNO-II/ III cycles. However, this project does not aim to
specifically address this problem, but to act as a supplementary study to which it could be
applied.
1.3 Previous and Current Studies
There are no dedicated publications to computationally varying the rates of the HCNO
reactions solely. Thus, only the work investigating the reaction rate of the 𝐹!"
𝑝, 𝛼 𝑂!"
is
covered. The first works in the field were performed by Wiescher & Kettner in 1982 [11] and
offered a primitive stepping-stone for further works despite the lack of experimental data. The
introduction of radioactive 𝐹!"
beams enabled physicists to conduct fully configurational
experiments and in 1995, Cozsach et al [12] made headway in determining the cross section
and reaction yields for the reaction within an astrophysically relevant regime. This led to the
works of Rehm et al [13,14,15] in the mid-90s who utilised this beam to study the reaction
using a gas-filled magnetic technique, finding the first evidence for the 𝐽!
= !
!
!
660keV
resonance [13,14].
Experimentalists then began to hone in on reducing the reaction rate uncertainties. In 2000,
Coc et al [8] used a hydrodynamic code ‘SHIVA’ to study the 𝐹!"
𝑝, 𝛼 𝑂!"
and
𝐹!"
𝑝, 𝛾 𝑁𝑒!"
reactions to obtain low, nominal and high calculations of the rates, reducing
uncertainty from Wiescher & Kettner [11] by a factor 60. Bardayan et al [16] (2002) extended
on this work by measuring the strength of the important 𝐽!
= !
!
!
330keV resonance in the
𝐹!"
𝑝, 𝛼 𝑂!"
reaction, finding rates to be lower by a factor 2 from previous studies. A recent
study followed in 2010 by Iliadis et al [17,18,19], who used a Monte Carlo Code ‘RatesMC’
to calculate the rates for the 8, 38 and 665keV resonances using strict statistical criterion based
on probability density functions. Former hydrodynamical techniques were ignored, as they
offered “no rigorous statistical meaning” according to [17], yet the uncertainties remained at a
factor 2 despite the improved statistical treatment. Adekola et al [20] used the proton-transfer
reaction 𝐹!"
𝑑, 𝑛 to study several important resonances in the 𝐹!"
𝑝, 𝛼 𝑂!"
reaction. Using
this experimental technique, they found that the rate produces 𝐹!"
at ~15% more than the
previous works by Chae et al [21].
In 2011, Beer et al [22] explored uncertainties stemming from the 8 and 38keV resonances and
their interaction with the well known 𝐽!
= !
!
!
665keV resonance. This was done experimentally
at TRIUMF at various energies (250-673keV). They found constructive interference between
𝐽!
= !
!
!
states in the resonances at 38 and 665keV, hinting that there may also be strong
influences from 𝐽!
= !
!
!
states. This implied stronger rates of destruction of 𝐹!"
in novae,
equating to smaller abundances. In 2013, Laird et al [9] used the experimental approach of
measuring states in the relevant energy regime using the 𝐹!"
𝐻𝑒!
, 𝑡 𝑁𝑒!"
reaction to yield
information about the mirror nucleus 𝑁𝑒!"
that can then be applied to the 𝐹!"
𝑝, 𝛼 𝑂!"
reaction. They concluded that the main uncertainty stems not from the poor constraint of
the 𝐽!
= !
!
!
331 keV resonance, but from the unknown proton-width for the 48keV resonance
(assigned to be 0.014). This yields a factor of ~2 uncertainty in the abundance of the 𝐹!"
from
the uncertainty in the reaction rate. The data from this paper is used throughout the project and
forms the basis of the in-depth analysis of the 𝐹!"
𝑝, 𝛼 𝑂!"
reaction.

9. 9
Chapter 2
Theoretical Considerations
2.1 Scientific Background
2.1.1 An Introduction to Novae
Of particular interest in this project are binary systems of stars, whereby two stars bound by
mutual gravitation orbit a common centre-of-mass whose position is governed by their relative
masses and positions. Physically, the evolution of a binary system is complex and one must
evaluate many different properties to gain a full picture (e.g. orbital mechanics, mutual
gravitational effects, perturbations etc.). Within this project, only the nuclear evolution of the
system, and particularly the white dwarf progenitor, will be treated. Despite this, it is important
to gain a brief picture as to how novae form from these binary systems.
The initial binary system that leads to a nova contains a main-sequence star of mass M1 and a
white dwarf progenitor of mass M2 (Figure 2.1). The white dwarf is formed when a dying main
sequence star ends it predominant hydrogen fusion and expands to a red giant and cools,
whereby helium core burning initiates. By the same process, when this fuel terminates, other
burning processes begin, such as carbon and oxygen burning (as initiated by the fusion of three
𝐻𝑒 !
nuclei to form carbon; the triple-alpha process [23]). The outer layer of this evolving star
is then shed to form a planetary nebula, leaving the white dwarf remnant (see figure 2.2).
Depending on the size of the original main sequence host, either the core comprises
predominantly of carbon and oxygen (leading to CO novae) or, for higher mass host stars,
oxygen, neon and magnesium (as at higher temperatures, carbon can fuse which leads to neon
and magnesium synthesis), forming ONe novae. This makeup is influential on the conditions
under which the nova operates.
Figure 2.1. In this binary system, the dumbbell shapes containing the low mass main-sequence companion of
mass M1 and the white dwarf of mass M2 about the Lagrange point L1 represent the Roche-lobes. During
Roche-lobe overflow, material from the donor star is captured by the gravitational field of the acceptor star
and accretes through the Lagrangian point L1. Figure adapted from [24] with analysis from [25].
Hydrogen-rich material is accreted from the companion star onto the surface of the white dwarf
by Roche lobe overflow. It occurs when the companion star fills its Roche lobe (region in which
any orbiting material is bound gravitationally to that star), so that the gravitational field of the
white dwarf captures the matter from the companion star. Material then passes through the
Roche
Lobes

10. 10
Lagrangian point L1, which is the point where both bodies are in gravitational equilibrium. This
allows a steady accretion of material from the companion into the white dwarf’s Roche lobe,
which creates an accretion disk due to the inflowing mass having a high angular momentum-to-
mass ratio. Dissipative processes cause kinetic energy losses of the material through radiation;
so meaning some material will spiral inward [26] to settle on the white dwarf surface.
As the white dwarf is such a dense object (~10!
𝑘𝑔/𝑐𝑚!
[27]), it means it has a significant
surface gravity that causes its constituent electrons to be highly compressed. From the Pauli
Exclusion Principle, only two electrons (fermions) can occupy the same energy level, both with
opposite half-integer spins. Under the high compression conditions of a white dwarf, the energy
levels are soon occupied. So gravity is compressing the electrons, but there are no available
energy levels to be occupied and hence the white dwarf becomes degenerate.
The strong surface gravity compresses the material settling on the white dwarf surface, heating
it to high temperatures as further material is funnelled in. Due to degeneracy, the conditions of
the white dwarf are such that the pressure and temperature are now independent, contrary to the
laws governing ideal, non-degenerate materials. As the temperature increases, the Carbon-
Nitrogen-Oxygen (CNO) cycle begins on the white dwarf surface, leading to the synthesis of
heavy elements and releasing energy. But thanks to the degenerate properties of the white
dwarf, there is no expansion to provide cooling when the energy/ temperature increases. Thus,
the temperature increases uncontrollably and initiates a thermonuclear runaway on the surface
via the Hot-CNO cycle.
A vast amount of energy is created from these processes, which is then expelled from the white
dwarf as the cataclysmic nova event. A huge quantity of material is ejected, and is known as
the ejecta envelope. It is in this fast moving (~1000 kms-1
[28]), expanding envelope that many
interesting nuclear processes occur thanks to its high temperature and dynamic environment
over a duration of ~1000-2000 seconds [4]. In this project, only classical novae will be
considered thanks to their regularity within the galaxy (~35 per year [29]) making them
attractive observational phenomena. Typically, classical novae only have one observed eruption
while recurrent novae can have several. Figure 2.2 shows the nova remnant Nova V1974 Cygni
1992.
Figure 2.2. An image of the nova V1974 Cygni 1992 in the ultraviolet regime, showing the ring-like gaseous
ejecta surrounding the white dwarf. Credit: F. Paresce, R. Jedrzejewski (STScl) NASA/ ESA. Adapted from
[25].

11. 11
2.1.2 The Hot-CNO (HCNO) Cycle in Novae
In novae, the predominant energy production process is the Hot Carbon-Nitrogen-Oxygen
(HCNO) cycle. This is an evolution of hydrogen burning in environments with an adequate
metallicity that there are sufficient carbon, oxygen and nitrogen catalyst nuclei to sustain the
cycle. It is dominant within explosive events such as novae due to the higher temperatures
(100 ≤ 𝑇! 𝐾 ≤ 400) and pressures present.
The HCNO cycle is a derivative of the usual ‘cold’ CNO cycle (dominant from
15 ≤ 𝑇! (𝐾) ≤ 100) [29], which is an alternative hydrogen-burning mode to the proton-
proton chain (dominant at lower temperatures). Although both cycles effectively convert 4
protons into one 𝐻𝑒 !
nucleus (along with other products), the CNO cycle acts as a catalyst to
this process. At the temperatures under which the CNO cycle operates, the beta-decays
(temperature independent) tend to occur faster than the proton-capture (p-capture) reactions.
The CNO cycle is shown below:
𝐶!"
(𝑝, 𝛾) 𝑁!"
(𝛽!
, 𝜐!) 𝐶!"
(𝑝, 𝛾) 𝑁 𝑝, 𝛾 𝑂!"!"
(𝛽!
, 𝜐!) 𝑁!"
(𝑝, 𝛼) 𝐶!"
The HCNO cycle differs as, due to the higher temperatures involved, the p-capture reactions
occur quicker than the competing beta-decays. The higher temperature means the protons
possess higher energies, meaning they can penetrate the Coulomb barrier inhibiting their fusion
to nuclei more readily. However, the fundamental goal of converting four protons to
a 𝐻𝑒 !
nucleus remains, as well as the fact that the energy generation rate depends on the
abundances of the catalyst C, N and O nuclei. The HCNO-I cycle is represented below and in
Figure 2.3:
𝐶(𝑝, 𝛾) 𝑁!"!"
(𝑝, 𝛾) 𝑂(𝛽!
, 𝜐!)!"
𝑁(𝑝, 𝛾) 𝑂(𝛽!
, 𝜐!) 𝑁(𝑝, 𝛼)!"!"!"
𝐶!"
Figure 2.3. A schematic of the CNO region of the Segrè Chart showing the path of the main HCNO-I cycle.
One can see it differs from the main cold CNO cycle via the proton capture onto 𝑁!"
instead of the beta-plus
decay seen in the cold CNO version.
This differs from the CNO cycle through a p-capture onto 𝑁!"
instead of it 𝛽!
-decaying, thus
demonstrating that at higher temperatures, p-captures occur before beta-decays can intervene.
For this project, the abundance behaviour of newly created HCNO-II/III products will be
scrutinised at different reaction rates. These cycles arise from competing (𝑝, 𝛾) and (𝑝, 𝛼)
reactions, and although the probability of such breakouts occurring are low compared to the
HCNO-I cycle, they are still significant in novae [29].

12. 12
The HCNO-II/ III cycles are represented below and in Figure 2.4.
HCNO-II - 𝑁(𝑝, 𝛾) 𝑂(𝑝, 𝛾) 𝐹(𝑝, 𝛾) 𝑁𝑒(𝛽!
, 𝜐!) 𝐹(𝑝, 𝛼)!"!"!"!"!"
𝑂(𝛽!
, 𝜐!)!"
𝑁!"
HCNO-III - 𝐹(𝑝, 𝛾) 𝑁𝑒(𝛽!
, 𝜐!) 𝐹(𝑝, 𝛼) 𝑂(𝑝, 𝛾) 𝐹(𝑝, 𝛾) 𝑁𝑒(𝛽!
, 𝜐!)!"!"!"!"!"!"
𝐹!"
Figure 2.4. The HCNO-II and HCNO-III cycles adapted from [4]. The HCNO-II/III cycles create 𝐹!"
via the
β+
-decay of 𝑁𝑒!"
. In the HCNO-II cycle, 𝐹!"
is destroyed when it captures a proton and emits an alpha,
leaving 𝑂!"
. The HCNO-III cycle destroys 𝐹!"
through a proton-capture to produce 𝑁𝑒!"
(analysis adapted
from [25]).
2.1.3 The Physics of Reactions
The two main categories of reaction that dominate nuclear astrophysics are resonant and non-
resonant reactions. This project focuses on the resonant type, especially when considering
the 𝐹!"
𝑝, 𝛼 𝑂!"
reaction. It is commonplace to write a compound-nucleus reaction of the form
𝑋 + 𝑎 → 𝑌 + 𝑏 as 𝑋 𝑎, 𝑏 𝑌. Where X is the target nucleus, a the initial interacting projectile, Y
the daughter nucleus and b the reaction product particle. For fusion to occur between X and a, a
substantial potential barrier (the Coulomb Barrier) must be overcome. This represents the
superposition of an attractive potential well and a repulsive Coulomb potential, which stems
from the repulsive force experienced by charged particles as they move closer together. The
height of this barrier is given by Coulomb’s law and classically, the temperature (and thus
energy) needed to overcome this potential barrier is substantial and non-viable. However,
penetration can occur via quantum mechanical tunnelling [30], i.e. the particle wavefunction
has a small probability of passing through a spatial domain not accessible classically. The
probability 𝑃 of tunnelling through the barrier is (1):
𝑃 ∝ 𝑒𝑥𝑝
−2𝜋𝑍! 𝑍! 𝑒!
ℏ𝑣
Where 𝑍! 𝑍! are the proton numbers of the target and projectile respectively, 𝑒 the electron
charge, ℏ the reduced Planck constant and 𝑣 the relative velocity between the projectile/ target.
(1)
HCNO-IIIHCNO-II

13. 13
Additionally, the particles will have a velocity distribution that follows a Maxwell-Boltzmann
distribution (2):
𝑃 ∝ 𝑒𝑥𝑝 −
!
!"
Which gives the distribution of particle velocities as a function of the energy 𝐸, the Boltzmann
constant 𝑘 and the temperature 𝑇. Superimposing these two factors will give an important
concept within nuclear astrophysics; the Gamow Window. This gives an indication of the
optimum energy region where reactions are most likely to occur, such that at higher energies
there are insufficient particles present and lower energies result in a smaller barrier penetration
probability (Figure 2.5 [25]).
Figure 2.5. A plot of the Gamow peak, seen as a superposition of the Maxwell-Boltzmann particle velocity
distribution and the Coulomb Barrier tunnelling probability. This region represents the range of energies that
gives the most probable chance of a reaction occurring. E0 represents the peak of the distribution (the Gamow
Energy), with ΔE0 the energy/ distribution width. Information on calculating the Gamow Window is
expressed in Appendix A. Figure taken from [4].
Knowledge of the Gamow window is important when seeing if certain reactions are viable
within a given energy range and its position indicates which reactions will dominate. This
implies where to focus an experimental study without wasting time and resources investigating
a reaction that is energetically improbable.
It should be noted that the Astrophysical S-factor 𝑆(𝐸) is a constant of proportionality relating
the cross section 𝜎 𝐸 (the reaction probability as dictated by the effective area of the
reactants) to the penetration probability (1) and the energy 𝐸 of the reaction, giving (3):
𝜎 𝐸 = 𝑆(𝐸)
1
𝐸
𝑒𝑥𝑝
−2𝜋𝑍! 𝑍! 𝑒!
ℏ𝑣
This S-factor is useful as it can be used to study large variations in the energy of a reaction, as
well as being extrapolated to estimate quantities not attainable from experiment (i.e. low energy
reactions). More importantly for this project, non-resonant reactions can be defined as the S-
factor varying smoothly with energy, while resonant reactions vary the S-factor sharply at
resonant energies.
This brief overview has omitted important concepts such as the cross section treatment,
kinematics etc. Analysis of these can be found in Iliadis’ Nuclear Physics of Stars [29] and
Krane’s Introductory Nuclear Physics [23].
(2)
(3)

14. 14
2.1.3.1 Resonant Reactions
Within a compound nucleus, there exist discrete nuclear states that are spaced closely to each
other, so forming a continuum [23]. These states of width Γ are unstable against decay and lie
within the ‘resonance region’. It is in this region that such discrete levels have a high
probability of formation due to their inherently large cross sections and small widths (from low
incident energies).
If assuming a captured particle (e.g. proton) observing a square well nuclear potential, at
certain energies, the interior wavefunction (inside the well) matches the amplitude and phase of
the exterior wavefunction of the incident particle, so maximising the cross section and
increasing the reaction probability greatly. When the interior and exterior wavefunctions match,
it represents the correct energy of the incoming particle needed to excite the corresponding
excited state of the reacting nucleus. This is known as a resonant reaction.
Of interest to this project are narrow resonances, which are defined by [29] as having partial
widths that are approximately constant over the total resonance width. Partial widths represent
the widths of the individual particle or nuclear states, whereas the total width is the sum of
these. Hence, broad resonances do not fulfil this criterion. The cross section for an isolated
resonance is described by the one-level Breit-Wigner formula for a target {1} and projectile
{0} (4):
𝜎 𝐸 =
!ℏ!
!!"
(!!!!)
(!!!!!)(!!!!!)
!!!!
(!!!!)!!(! !)
!
Where 𝜎 𝐸 is the reaction cross-section in barns, 𝜇 the reduced mass of the target and
projectile, E the energy of the system, J and Er the spin and energy of the resonance, j0,1 are the
spins of the target/ projectile, 𝛤 the total resonance width and 𝛤! 𝛤! are the partial resonance
widths of the reaction entrance and exit channels.
This Breit-Wigner expression (4) is of Lorentzian form and dictates the resonance’s Lorentzian
profile (Figure 2.6).
Figure 2.6. The general Lorentzian distribution that represents the shape of a narrow resonance. At certain
resonance energies, the cross section increases greatly, as expected. The indicated Full Width at Half
Maximum (FWHM) represents the partial widths (of the reacting resonance particles), and the total resonance
width is the sum of these. Plotted in Maple [31].
(4)
Energy (MeV)
CrossSection
FWHM

15. 15
This gives an important result that is used in this project to calculate the reaction rate of a
narrow resonance (5) (derivation in Appendix B):
𝑁! 𝜎𝑣 =
1.5399 × 10!!
𝜇𝑇!
!/!
(𝜔𝛾)! 𝑒!!!.!"# !! !!
!
Where 𝑁! 𝜎𝑣 is the reaction rate in 𝑐𝑚!
𝑚𝑜𝑙!!
𝑠!!
, T9 the temperature in GK, i labels the
different resonances, (𝜔𝛾) is the resonance strength (where the spin factor 𝜔 =
(!!!!)
(!!!!!)(!!!!!)
and 𝛾 =
!!!!
!
) and Ei is the energy of the given resonance (both in units of MeV). Resonances
are labeled with their respective energies and spin-parities 𝐽!
[25].
2.2 Methodology
2.2.1 The ‘libnucnet’ Module
Within the comprehensive Webnucleo [3] package (created by the Nuclear Astrophysics Group
at Clemson University, South Carolina, USA) is embedded a library of codes that evaluates,
manages and stores nuclear reaction networks under a number of astrophysical environments
and scenarios. This is named the ‘libnucnet’ module, and was run within a Secure Shell (SSH)
connection to the University of York’s ‘npp2’ server. This allowed access to the codes from
any desktop with a Linux based platform, as well as providing a powerful platform to perform
the calculations.
The‘libnucnet’code simulates and manages various nuclear reaction networks based on several
different initial parameters. The library of codes are known as ‘Post-Processing’ codes that use
known temperature/ density profiles as a function of time to run the simulation. The profiles are
based on the results of full hydrodynamic simulations or real data taken from literature.
Hydrodynamic simulations use both one-dimensional and multi-dimensional approximations to
calculate several different parameters (conduction, rotation etc.) within the star and use these in
the simulation, and consequently are computationally intensive. Conversely, post-processing
codes assume that the temperature and density of the scenario being simulated are constant
throughout, and other physical effects (such as conduction, convection etc) are ignored or
assumed contained within the profiles. Despite these post-processing codes being
comprehensive, uncertainties in some reactions arise from the input nucleosynthesis data
originating from estimates based on statistical models and not experimental data [32].
Moreover, they do not provide the accuracy or complexity of a hydrodynamic simulation. Yet,
they reproduce results to an acceptable level of precision without the need for a great deal of
computational power.
The Webnucleo library of codes have been utilised in many studies [33] and this indicates it is a
worthy resource to employ. When comparing them to hydrodynamic codes used by some teams
(e.g. the NOVA code [34]), they are clearly not as comprehensive or accurate. However, given
the computational power available and the necessary precision of the project, the Webnucleo
codes are sufficient.
(5)

16. 16
2.2.2. Running Simulations to Find the Abundance of the
HCNO-II/III Products for the 1 MSun CO and 1.25 MSun
ONe Novae Profiles
The process of setting up and running the simulation requires the user to be connected via
Secure Shell to a server with access to a sufficiently powerful computational setup. The user
must choose an initial companion star profile (abundance file), assign a nova trajectory (the
pre-assigned temperature and density profile) and choose how many timesteps are needed. For
each simulation, a sun-like companion star was used with both the 1 MSun CO and 1.25 MSun
ONe trajectories for 10 timesteps (this number offered sufficient precision in the shortest
simulation time). The simulation was restricted to drastically cut simulation run-time (hours to
minutes) without significant loss of precision [35]. Graphical notes on configuring the
simulation are in Appendix C.
The next stage involved applying the user modified reaction rates to the simulation. The pre-
loaded reaction rate text files were accessed within the relevant server branch, manipulated
accordingly (e.g. multiplied by 100) in a spreadsheet and then exported to a new file with the
modified reaction rates as a function of the same temperatures as the original rates. The new
rate files were then assigned to the simulation. After the simulation was run, the outputs were
stored in a private branch on the server where various manipulations were applied. This
important process can be viewed in Appendix D
One of the most powerful aspects of the Webnucleo codes was the ability to view a wealth of
information garnered from the simulation. Relevantly, one has the option to print all the
reactions from the nova simulation, the rates of a certain reaction at different temperatures and
the rates for all reactions involving a certain reactant (e.g. 𝐹!"
) at differing temperatures. Final
abundances and reaction rate data were printed in the terminal and exported to a spreadsheet
where further numerical analysis was applied to the data [35]. The analysis consisted of
converting the raw rates into a base-10 logarithm to allow for a greater study into variations
between daughter abundances (this would be difficult if the small numbers presented in the raw
data were used alone). Plotting of the data was performed with the graphical package, proFit
[36] and numerical manipulation was performed in Microsoft Excel. This whole process was
repeated for each reaction in the HCNO-II/III cycles for the 1 MSun CO and 1.25 MSun ONe
novae (section 3.1).
2.2.3 Calculating Reaction Rates for the 𝐹!"
𝑝, 𝛼 𝑂!"
Reaction
from Various Parameters
No simulations were necessary for this part of the study. Instead, small programmes in Excel
were created to calculate narrow resonance rates (as a function of temperature) quickly through
many smaller calculations. The relevant data from literature only needed to be input into the
spreadsheet once for the programmes to quickly calculate the rates (Appendix E). From Laird
et al [9], the resonance parameters (resonance strength, proton/ alpha widths, spins) for the
𝐹!"
𝑝, 𝛼 𝑂 !"
reaction were used to calculate the reaction rate as a function of novae
temperature using the Breit-Wigner and Resonance Strength solvers. Initially, the resonance
strengths were calculated from these partial widths and spins using the Resonance Strength
Solver. The rates were then calculated using the Breit-Wigner Solver over temperatures of
0.01GK – 0.1GK in order to fully encompass the temperatures that novae operate. Doing this

17. 17
allowed the respective dominances of the resonances to be explored, and at the time (early
2013) offered the most up-to-date investigation into this reaction. Calculating the normalised
rates consisted of dividing the individual resonance rates by the sum of all resonance rates for
each temperature specified. These were then plotted again in proFit.
2.3 Error Considerations
In this study, the treatment of errors is different to an experimental project. As this is a
simulation package using pre-defined codes with no error output, it was impossible to form any
quantifiable errors for the abundance data produced. Hence, no quantitative approach to errors
is presented in this study as advised by Dr Alison Laird [37]. It is important to consider errors
qualitatively though. When calculating the abundance of the HCNO products, it must be noted
that the pre-loaded reaction rate files (and such the modified ones) have not changed since the
initial release in 2007 [3]. Thus, the abundance data calculated did not stem from the latest
reaction rates found by the most recent studies, although this does not represent an error per
say. Importantly, as this is a simulation, errors in the calculated abundance would be present
from a lack of complexity within the code, especially as the codes used are post-processing
ones. For the most accurate data, a full hydrodynamic approach would be needed, although this
is not viable given the available resources. Additionally, as each simulation is independent,
performing statistical analysis on a number of runs would not provide any meaningful error
analysis (such as variance, standard deviation treatments). But it must be considered that these
properties did play a part, although they cannot be considered quantitatively.
When calculating the reaction rates for the narrow resonances in the 𝐹!"
𝑝, 𝛼 𝑂!"
reaction,
there are tolerances within the resonance strength and partial width parameters determined
within the literature (see Table I [9]). Through a sensitivity study within the project, it was
found that the upper and lower limits on the errors have a negligible effect on the rate over the
range covered, and so they were ignored. The error within the narrow resonance formula
(equation 5) is assumed to be negligible as it is a well-known result. Consequently, there will
be no error analysis of the results based on this, again as advised by Dr Laird.
Chapter 3
Results and Discussion
3.1 Investigation into the HCNO-II and II Cycle
Abundance Evolution
As the nucleosynthesis and destruction of the relevant 𝐹!"
nuclei is constrained to the HCNO-
II and III cycles in novae, only these have been simulated and evaluated. All half-life values
have been taken from the NNDC database [38].

18. 18
3.1.1 Abundance of HCNO-II Cycle Products in a 1 MSun CO
Novae and Analysis
Figure 3.1. Plots of Log Yield of the HCNO-II products (seen in legend) against time (seconds) over the
1750 seconds that the simulation was run for a 1 MSun CO nova. It shows the abundance evolution for
(a) x0.01 original reaction rate; (b) unchanged reaction rate; (c) x100 original reaction rate. This nova
corresponds to a temperature range of 0.08-0.17GK, with the temperature peaking between 285-510 seconds.
proFitTRIALversion
0 500 1000 1500
–15
–10
–5
0
Time (s)
LogYield
Plot showing Log Yields of HCNO2 Products
for a 1Msun CO nova at x0.01 reaction rate!
16O
17F
18Ne
18F
15O
15N
proFitTRIALversion
0 500 1000 1500
–15
–10
–5
0
Time (s)
LogYield
Plot showing Log Yields of HCNO2 Products
for a 1Msun CO nova at x1 reaction rate!
16O
17F
18Ne
18F
15O
15N
proFitTRIALversion
0 500 1000 1500
–15
–10
–5
0
Time (s)
LogYield
Plot showing Log Yields of HCNO2 Products
for a 1Msun CO nova at x100 reaction rate!
16O
17F
18Ne
18F
15O
15N
(a)
(b)
(c)

19. 19
In Figure 3.1, it is clear that the majority of the abundance changes happen very quickly at a
time corresponding to ~ 300 seconds (s), where the temperature of the nova reached its peak at
0.17GK. As the reaction rate is exponentially dependant on the temperature (equation 5), any
increase in temperature will hugely increase the reaction rate, so corresponding to a spike in
production/ destruction events.
From Figure 3.1, the abundance of the doubly-magic nucleus 𝑂!"
(created by the 𝑁(𝑝, 𝛾) 𝑂!"!"
reaction) remains almost constant and has the highest yield throughout the cycle regardless of
reaction rate. 𝑂!"
is an abundant isotope thanks to its production directly from triple-alpha
processes in hydrogen rich stars as well as the high oxygen abundance in the progenitor CO
white dwarf. This high abundance is very much dependant on that instead of the 𝑁(𝑝, 𝛾) 𝑂!"!"
reaction, which would contribute relatively little.
The yield of 𝐹!"
from the 𝑂(𝑝, 𝛾) 𝐹!"!"
reaction shows significant increases in abundance at
~300s for the x0.01 and unchanged rates. This corresponds to the decrease in 𝑂!"
at this time,
and although it only appears to be a small decrease, thanks to the log scale, this small change in
𝑂!"
would represent the large increase in 𝐹!"
. The decrease in 𝐹!"
from ~1000s onwards could
correspond to other destruction reactions, such as 𝐹(𝛽!
, 𝜐!) 𝑂!"!"
(𝑡!/! = 64s). In Figure 3.1.c
(x100), there is large increase in the 𝐹!"
abundance over the beginning of the event. With no
decrease in 𝑂!"
abundance, this increase must be due to another production reaction, such as
𝑁𝑒(𝛽!
, 𝜐!) 𝐹!"!"
(𝑡!/! = 17s). More likely is that this is due to a simulation anomaly, as it
would be expected to follow the same shape as the x0.01/ unchanged reaction rate scenarios, as
it is a p-capture reaction (whose abundance would vary linearly with changing rate).
The abundance of 𝑁𝑒!"
is sensitive to reaction rate (peak increasing by ~4 orders between
x0.01 and x100 rate) and varies hugely over the nova timescale. The peak at ~300s corresponds
to the quick onset of the 𝑁(𝑝, 𝛾) 𝑂(𝑝, 𝛾) 𝐹(𝑝, 𝛾) 𝑁𝑒!"!"!"!"
reactions, and the higher
temperature at this point gives rise to the quicker p-captures. According to [8], p-captures
(temperature dependant) dominate corresponding 𝛽!
-decays (temperature independent) when
T > 0.2GK, such 𝑁𝑒!"
abundances rise at this temperature (at t~300s) as the p-captures onto
𝐹!"
become much quicker than the alternate path, 𝐹(𝛽!
, 𝜐!) 𝑂!"!"
, which would stunt 𝑁𝑒!"
nucleosynthesis. No noticeable decrease in 𝐹!"
abundance is seen at any reaction rate as the
abundance of 𝐹!"
is much greater relative to 𝑁𝑒!"
, as demonstrated by the log scale, explaining
why at later times/ lower temperatures the 𝑁𝑒!"
abundance steadies and decreases. 𝛽!
-decays
(and electron captures) once again dominate over p-captures, leading to 𝐹!"
production via
𝑁𝑒(𝑒!
, 𝜐!) 𝐹!"!"
(𝑡!/! = 1.6s). This is the dominant 𝐹!"
production reaction at the outer-
envelope shells of the novae and explains why its abundance remains steady over the later
portions of the nova for all reaction rate scenarios. This reaction is temperature independent
and supports why it is steady with varying rate.
This leads to the analysis of the important 𝐹!"
production and destruction, which is only very
slightly sensitive to the reaction rate. At ~300s, all reaction rate scenarios in Figure 3.1 show a
quick decrease before a sudden increase in yield. At the peak temperature, the destruction
reactions 𝐹(𝑝, 𝛾) 𝑁𝑒!"!"
, 𝐹!"
𝑝, 𝛼 𝑂!"
and 𝐹(𝛽!
, 𝜐!) 𝑂 !"!"
would have dominated over the
𝑁𝑒(𝑒!
, 𝜐!) 𝐹!"!"
reaction, so leading to the destruction of the isotope (although the long half
life of 𝐹!"
(109 minutes) may cause 𝐹!"
to build up as it awaits decay, perhaps indicating why
the abundance remains consistently steady). But this shape indicates that the temperature

20. 20
dropped enough over this period that the dominant reaction would have switched to the
𝑁𝑒(𝑒!
, 𝜐!) 𝐹!"!"
reaction, explaining the sudden increase in its yield.
The destruction reactions of 𝐹(𝑝, 𝛾) 𝑁𝑒!"!"
, 𝐹(𝛽!
, 𝜐!) 𝑂 !"!"
and 𝐹!"
𝑝, 𝛼 𝑂!"
produces 𝑂!"
.
This is created abundantly and does not change significantly with the reaction rate as explained
before. However, its high abundance cannot be described by 𝐹!"
destruction alone, and may be
due to the 𝑁(𝑝, 𝛾) 𝑂 !"!"
reaction in the HCNO-I cycle, although independent simulations
would need to confirm this.
The production of 𝑁!!
via the 𝑂(𝛽!
, 𝜐!) 𝑁 !"!"
reaction (𝑡!/! = 122s) is interesting. At all
rates, it shows an initial spike and decrease in abundance over the first few seconds
(T ~ 0.07GK) and there may destruction from the CNO/ HCNO-I reaction 𝑁!"
𝑝, 𝛼 𝐶!"
. The
sharp decrease in abundance in the x0.01 and unchanged rate graphs (figs 3.1.a and 3.1.b) at
t ~300s may be down to the acceleration of the 𝑁(𝑝, 𝛾) 𝑂!"!"
reaction, with the marked
increase in abundance after this due to the 𝑂(𝛽!
, 𝜐!) 𝑁 !"!"
reaction dominating over the
p-capture reaction due to the decrease in temperature.
3.1.2 Abundance of HCNO-III Cycle Products in a 1 MSun CO
Novae and Analysis
The HCNO-III (Figure 3.2, page 21) cycle breaks out from the HCNO-II due to the
𝐹(𝑝, 𝛾) 𝑁𝑒!"!"
reaction replacing the 𝐹!"
𝑝, 𝛼 𝑂!"
reaction. The abundance of 𝑁𝑒!"
from this
reaction is dependant on reaction rate, and increases linearly with rate (changing ~5 orders of
magnitude between x0.01 and x100). From the previous section, the increase in production at
~300s corresponds to the 𝑁𝑒!"
production reaction 𝐹(𝑝, 𝛾) 𝑁𝑒!"!"
occurring faster than the
𝐹(𝛽!
, 𝜐!) 𝑂 !"!"
reaction, so the 𝐹!"
is more likely to p-capture and produce the 𝑁𝑒!"
at these
temperatures than decay, so not producing 𝑁𝑒!"
at all. As time progresses/ temperature cools,
its abundance decreases as the 𝑁𝑒(𝛽!
, 𝜐!) 𝐹 !"!"
reaction becomes dominant and quickly
(𝑡!/! = 17s) destroys 𝑁𝑒!"
. The abundance of 𝐹 !"
is sensitive to the rate, and shows a sudden
decrease at the peak temperature time (~6 orders of magnitude). This may be due to its
destruction by the 𝐹!"
𝑝, 𝛼 𝑂!"
reaction in the next step, or the 𝐹(𝑝, 𝛾) 𝑁𝑒!"!"
reaction due to
the p-capture dominance at this temperature. It is known that 𝑁𝑒!"
is abundant in novae ejecta,
so the destruction of 𝐹!"
may be down to both of these factors, with the 𝑝, 𝛼 reaction
dominating.
Much like the 𝐹!"
𝑝, 𝛼 reaction in the HCNO-II cycle, the yield of 𝑂!"
varies slowly over
time and is sensitive to reaction rate from the 𝐹!"
𝑝, 𝛼 𝑂!"
reaction in the HCNO-III cycle,
although this sensitivity is not visible thanks to its high relative abundance. Noting this, the
abundance behaviour of the other products from this point is the same as in the HCNO-II cycle
described in section 3.1.1.

21. 21
Figure 3.2. Plots of Log Yield of the HCNO-III products (seen in legend) against time (seconds) over the
1750 seconds that the simulation was run for a 1 MSun CO nova. It shows the abundance evolution for
(a) x0.01 original reaction rate; (b) unchanged reaction rate; (c) x100 original reaction rate. This nova
corresponds to a temperature range of 0.08-0.17GK, with the temperature peaking between 285-510 seconds.
proFitTRIALversion
0 500 1000 1500
–15
–10
–5
0
Time (s)
LogYield
Plot showing Log Yields of HCNO3 Products
for a 1Msun CO nova at x100 reaction rate!
19Ne
19F
16O
17F
18Ne
18F
proFitTRIALversion
0 500 1000 1500
–15
–10
–5
0
Time (s)
LogYield
Plot showing Log Yields of HCNO3 Products
for a 1Msun CO nova at x0.01 reaction rate!
19Ne
19F
16O
17F
18Ne
18F
proFitTRIALversion
0 500 1000 1500
–15
–10
–5
0
Time (s)
LogYield
Plot showing Log Yields of HCNO3 Products
for a 1Msun CO nova at x1 reaction rate!
!
19Ne
19F
16O
17F
18Ne
18F
(a)
(b)
(c)

22. 22
3.1.3 Abundance of HCNO-II Cycle Products in a 1.25 MSun
ONe Novae and Analysis
Figure 3.3. Plots of Log Yield of the HCNO-II products (seen in legend) against time (seconds) over the
1750 seconds that the simulation was run for a 1.25 MSun ONe nova. It shows the abundance evolution for
(a) x0.01 original reaction rate; (b) unchanged reaction rate; (c) x100 original reaction rate. This nova
corresponds to a temperature range of 0.05-0.25GK, with the temperature peaking between 380-850 seconds.
proFitTRIALversion
0 200 400 600 800 1000
–25
–20
–15
–10
–5
0
Time (s)
LogYield
Untitled Data 2
16O
17F
18Ne
18F
15O
15N
proFitTRIALversion
0 200 400 600 800 1000
–25
–20
–15
–10
–5
0
Time (s)
LogYield
Untitled Data 2
16O
17F
18Ne
18F
15O
15N
proFitTRIALversion
0 200 400 600 800 1000
–25
–20
–15
–10
–5
0
Time (s)
LogYield
Untitled Data 2
16O
17F
18Ne
18F
15O
15N
(a)
(b)
(c)

23. 23
Thanks to the typically higher temperatures, higher mass and different constituent make-up of
the ONe novae compared to the CO one, the abundance evolution is unsurprisingly different
from those investigated in the CO nova scenario. The peak temperature of this scenario occurs
at t ~400s and is 0.25GK.
The actual destruction and production processes/ reactions for both scenarios are the same as
outlined in section 3.1.1, so the processes will not be repeated here. The variation in evolution
will be discussed however.
The 𝑂!"
abundance varies even slower over the timescale compared to the CO nova. For the
CO scenario, the abundance varies over a log range of -2.0 to -2.2, whereas in the ONe scenario
in Figure 3.3, all rates vary from -2.0182 to -2.0186. The 𝑂!"
abundance is high in both novae
thanks to both nova progenitors containing vast amounts of oxygen from the burning.
The 𝐹!"
yield varies linearly with reaction rate and follows the same shape as in the HCNO-II
cycle in the CO nova (Figure 3.1), except that the peak increase in abundance corresponds to
t~400s at the higher peak temperature of 0.25GK.
The 𝑁𝑒!"
abundance increases in yield more sharply at the peak temperature than in the cooler
CO scenario yet the reaction rate still affects abundances linearly. This sharper increase is due
to the higher temperature of this nova, increasing the rates at that point more significantly in the
ONe nova (recall exponential temperature dependence).
The 𝐹!"
abundance does not reproduce the kink that is seen in the CO nova, but does still
remain steady and independent on reaction rate as before. Instead there is a steady increase in
abundance, which would be dictated by the temperature independent 𝑁𝑒(𝑒!
, 𝜐!) 𝐹!"!"
reaction.
Again, the 𝑂!"
is not sensitive to the rate, but is less abundant in the ONe nova than the CO
nova at the peak temperature (by ~2 orders of magnitude), possibly due to the higher peak
temperatures in the ONe nova destroying 𝑂!"
via p-captures more substantially than the
temperature independent 𝛽!
-decay.
The different shape in the 𝑁!"
abundance compared to the CO nova is puzzling. In this
scenario it exhibits a sharp increase (which is independent of rate) and then immediate decrease
(which is rate dependant) at the peak temperature. The sharp increase in 𝑁!"
may arise from
the 𝑂!"
𝑝, 𝛼 𝑁!"
reaction being significant at lower temperatures and so occurring at
temperatures <0.25GK, corresponding to the sharp increase before the peak. 𝑁!"
is stable, so
the sharp decrease must be from a sudden increase in p-captures at this high temperature. An
increase in abundance from then on corresponds to 𝑂(𝛽!
, 𝜐!) 𝑁 !"!"
reactions, as the 𝑂!"
(𝑡!/! = 122s) abundance corresponds to this (as does the very slight decrease in abundance at
this time).

24. 24
3.1.4 Abundance of HCNO-III Cycle Products in a 1.25 MSun
ONe Novae and Analysis
Figure 3.4. Plots of Log Yield of the HCNO-III products (seen in legend) against time (seconds) over the
1750 seconds that the simulation was run for a 1.25 MSun ONe nova. It shows the abundance evolution for
(a) x0.01 original reaction rate; (b) unchanged reaction rate; (c) x100 original reaction rate. This nova
corresponds to a temperature range of 0.05-0.25GK, with the temperature peaking between 380-850
seconds.
The shape of the 𝑁𝑒!"
abundance follows that of the CO nova, yet the increase at the peak
temperature is sharper due to the temperature dependence of the rates (the ONe nova has a
higher temperature). The shape of the 𝐹!"
abundance is different in both novae. It is not
sensitive to the rate (as expected) and varies more than the CO nova (from -14 to -7 bottom-to-
peak abundance). At these temperatures, the p-captures are quick and dominant, so the spike in
abundance probably comes from the 𝑂(𝑝, 𝛾) 𝐹!"!"
reaction, supplemented by the quick decay
via 𝑁𝑒(𝛽!
, 𝜐!) 𝐹 !"!"
(𝑡!/! = 17s) with the decrease in 𝐹 !"
abundance via the 𝐹!"
𝑝, 𝛼 𝑂!"
reaction.
proFitTRIALversion
0 200 400 600 800 1000
–30
–20
–10
0
Time (s)
LogYield
Plot showing Log Yields of HCNO3 Products
for a 1.25Msun ONe nova at x100 reaction rate!
!
19Ne
19F
16O
17F
18Ne
18F
proFitTRIALversion
0 200 400 600 800 1000
–30
–20
–10
0
Time (s)
LogYield
Plot showing Log Yields of HCNO3 Products for a
1.25Msun ONe nova at x0.01 reaction rate!
19Ne
19F
16O
17F
18Ne
18F
proFitTRIALversion
0 200 400 600 800 1000
–30
–20
–10
0
Time (s)
LogYield
Plot showing Log Yields of HCNO3 Product for
a 1.25Msun ONe nova at x1 reaction rate!
19Ne
19F
16O
17F
18Ne
18F
(a)
(b)
(c)

25. 25
From all of the scenarios, one can see that varying the reaction rate by x0.01 to x100 does
affect the abundance linearly for products created by p-captures onto the parent nucleus, bar
one suspected simulation anomaly for the yield of 𝐹 !"
in the CO nova. Changing the reaction
rate does not affect abundances of products created by 𝛽!
-decays. Both of these results are
expected, as p-capture rates are temperature dependant (higher temperature equates to protons
having higher energies and so a higher probability of penetrating the Coulomb barrier to react
and initiate fusion), whereas 𝛽!
-decays are temperature independent and will have the same
reaction rate (or decay time) regardless of their environment. This is important as the reaction
rate is so sensitive to the temperature, and this has been shown in figures 3.1-3.4. As the ONe
nova operates at higher temperatures, the rate of change of abundances is greater at the peak
temperatures compared to the CO nova, which operate at lower peak temperatures. Despite this,
the abundance evolution is largely similar for both scenarios, which is anticipated seeing as
both systems are not vastly different (similar masses, temperatures, oxygen abundance etc).
3.2 Investigation into the 𝐹!"
𝑝, 𝛼 𝑂!"
Reaction
3.2.1 Investigating the Narrow Resonances of the 𝐹!"
𝑝, 𝛼 𝑂!"
Reaction
From Laird et al [9], the resonance parameters (resonance strength, proton/ alpha widths) for
the 𝐹!"
𝑝, 𝛼 𝑂 !"
reaction were used to calculate the reaction rate as a function of novae
temperature (Appendix D and Section 2.2.3). Where data was not available from Laird et al [9],
Nesaraja et al [6] was used. Figure 3.5 maps the normalised rate against temperature:
Figure 3.5. Plot of normalized reaction rate (reaction rate of resonance/ sum of reaction rate of all
resonances) against temperature (GK) for the 5 keV (𝐽!
= !
!
!
, !
!
!
), 29 keV (𝐽!
= !!
!
!
), 48 keV (𝐽!
= !
!
!
),
289 keV (𝐽!
= !
!
!
), 331 keV (𝐽!
= !
!
!
, !
!
!
) and 451 keV (𝐽!
= !
!
!
) resonances taken from Laird et al [9].
0.01 0.1 1
0.0
0.2
0.4
0.6
0.8
1.0
Temperature (GK)
NA⟨σv⟩/NA⟨σv⟩total
Plot of normalised reaction rates for the 5 keV, 29 keV, 48 keV,
289 keV, 331 keV and 451 keV resonant contributions
in the NA"
5 keV
29 keV
48 keV
289 keV
331 keV
451 keV

26. 26
It is evident that the 5keV resonance has a negligible effect on the overall reaction rate over all
temperatures. Numerically, the reaction rate was ~ 10!!"
𝑐𝑚!
𝑚𝑜𝑙!!
𝑠!!
, which is insignificant.
The reaction rate (equation 5) is sensitive to the resonance strength that is proportional to
!!!!
!!"!
.
According to [9], the proton width Γ! for this is approximately 10!!"
𝑘𝑒𝑉, which is very
narrow and so is solely responsible for the small reaction rate seen.
At temperatures < 0.015GK, the 29keV resonance dominates, but this is soon made negligible
at temperatures > 0.011GK by the 48keV resonance. Over the temperature range that this
dominates (0.011GK-0.1GK), the Gamow Window is 44-268 keV. As this falls within the
Gamow window, the probability of the reaction occurring is at its optimum (see section 2.1.3).
The proton and alpha widths are also much more significant (10!!"
𝑘𝑒𝑉, 5.5 𝑘𝑒𝑉), so
supporting the higher reaction rate.
The 289 keV resonance has a noticeable effect at ~0.105GK, but this is only small compared to
the 331 keV resonance (although significant comparatively). The 331keV resonance soon
dominates at all temperatures above 0.1GK. At this temperature, the Gamow peak is at 276keV
(width of 115keV), which would suggest that the 289keV resonance would be more dominant.
This is not the case however, and this may be down to the centrifugal barrier. This is a potential
barrier that affects particles with non-zero orbital angular momentum and so the system
acquires an effective barrier that is the sum of the Coulomb barrier and the centrifugal barrier
(proportional to 𝑙(𝑙 + 1) [23]). A larger 𝐽!
value signifies a larger 𝑙 value (as 𝐽 = 𝑙 ± !
!
) and
such a larger barrier to overcome to initiate fusion. As the 289keV resonance has 𝐽!
= !
!
!
and
the resonance at 331keV has 𝐽!
= !
!
!
, it is expected that the 331keV resonance will dominate
here, as not only does it have a smaller centrifugal barrier to overcome, but it possesses a
higher energy to more readily overcome this barrier. When the 331keV is fully dominant
(>0.2GK), the Gamow peak is at 425keV (width of 197keV), and so the resonance falls
comfortably in this window. This dominance is aided by the comparatively large proton
(10!!
𝑘𝑒𝑉) and alpha (5.2 𝑘𝑒𝑉) widths, which provides a larger resonance strength. The
451keV resonance is negligible throughout due to its comparatively large spin-parity/
centrifugal barrier.
In this study, the inclusion of direct capture reactions and broad resonances has not been
included. This would undoubtedly affect the results gathered, with broader resonances being
more dominant overall thanks to their larger proton and alpha widths. The study into broader
resonances was considered, but due to computational complications could not be completed. It
would be a consideration if extending the study however.
In summary, this study supports the analysis in Laird et al [9] that the 48keV and 331keV
resonances dominate those experimentally studied, with the 331keV being especially relevant
over nova temperatures.

27. 27
3.2.2 Investigating the 331 keV Resonance 𝐽!
Assignment
Several studies have been conducted into experimentally determining the spin-parity
assignment of the 331keV resonance, with it having an assignment of both 𝐽!
= !
!
!
and
𝐽!
= !
!
!
in literature (Garrett et al [39]). As the rates are so sensitive to the spin-parity
assignments (especially the spin factor within the resonance strength), both assignments will
have an affect on the reaction rate and the dominance of the 331keV resonance (figure 3.6):
Figure 3.6. Plot showing the resonance dominances when the 331keV resonance is assigned with a spin-
parity of 𝐽!
= !
!
!
(dashed line) and 𝐽!
= !
!
!
(solid line).
If the 𝐽!
= !
!
!
assignment is used, the 289keV resonance becomes more substantial as the
331 keV resonance drops slightly, although not significantly. With the 𝐽!
= !
!
!
assignment, the
centrifugal barrier for the 331keV resonance will be lower, so one would expect to see an even
higher dominance of the rate. It must be considered though, at lower 𝑙 values, the centrifugal
barrier is not as considerable. Additionally, as the reaction rate is proportional to the spin factor
𝜔 =
(!!!"!!!)
(!!!!!)(!!!!!)
, it is such that a lower 𝐽 value of the 331keV resonance gives a lower
reaction rate thanks to this proportionality. These properties combined explain why the reaction
rate is lower when the resonance is assigned 𝐽!
= !
!
!
rather than 𝐽!
= !
!
!
. Despite this,
changing the assignments does not have a significant impact on the reaction rate, and so is not a
concern to the validity of the study. Note, studies such as D.W. Visser et al [40] support the
𝐽!
= !
!
!
assignment experimentally.
3.2.3 Other Studies
Laird et al [9] claimed the main uncertainty in the rate of the 𝐹!"
𝑝, 𝛼 𝑂 !"
reaction stemmed
from the unknown proton-width of the 48keV resonance. The proton width Γ! is dependant on
the spectroscopic factor and reduced proton-width of the resonance [29], and it is within the
reduced proton-width that the rate uncertainty stems. This study aimed to use various reduced
proton-widths to see how this affected the reaction rate dominances. However, this was not
completed due to issues in determining spectroscopic factors. Supplementary to this was
neglecting the 48keV resonance and seeing how this changed the abundance of the 𝑂!"
product
proFitTRIALversion
0.01 0.1 1
0.0
0.2
0.4
0.6
0.8
1.0
Temperature (GK)
NA⟨σv⟩/NA⟨σv⟩total
Plot showing the normalised reaction rates from several contributing
narrow resonances with the 331 keV resonance having both
1/2- and 3/2- spin-parity assignments
5 keV
29 keV
48 keV
289 keV
331 keV
𝐽!
=
!
!
!
𝐽!
=
!
!
!

28. 28
over a nova timescale. For both the CO and ONe novae, neglecting it had a very small impact
on the abundance compared to including it.
Chapter 4
Conclusions
The main objective of this study was to use external nuclear reaction network codes from the
Webnucleo package to investigate how influential the reaction rate was in affecting and
evolving the abundances of nuclei formed in the important HCNO-II/ HCNO-III cycles over a
nova lifetime (~1000 seconds). For each of the reactions in the two cycles, the abundance of
each product was determined by simulation for both a 1 MSun CO and 1.25 MSun ONe nova. This
was done initially with an unchanged reaction rate (i.e. using the rate pre-loaded into the
codes), and then run two more times with the rate changed to x0.01 and x100 of the original
rate. The study found that, in line with theory, the abundance of the daughters from the
p-capture reactions in the cycles varied linearly with rate (bar one suspected simulation
anomaly), i.e. an increase/decrease in rate corresponded to a linear increase/ decrease in
abundance. Indeed, increasing the reaction rate would inherently increase the rate in which
protons capture onto parent nuclei to form the daughter, and vice versa. Additionally, this study
has shown that the abundance evolution for the 𝛽!
-decay reactions are not rate dependant,
which is expected as the time in which these reactions occur depend on the half-life of the
decaying parent radioisotope. An in-depth analysis into the reactions dictating the abundance
evolution was also addressed. This study has acted as a mode of confirming this expected
behaviour as opposed to finding completely unexpected results from the simulations, thus
supporting the validity of the code.
The second half of the study used known narrow resonance expressions to calculate the
reaction rate of the astrophysically relevant 𝐹!"
𝑝, 𝛼 𝑂 !"
reaction over a range of nova
temperatures based on the recent experimental parameters determined by Laird et al [9] (2013).
By calculating and plotting the normalised rates for the main narrow resonances in this
reaction, it has provided a graphical means to clearly identify what resonances dominate at
certain temperature stages of the novae. This has also supported the analysis in [9] that the
48keV and 331keV resonances dominate those experimentally studied, with the 331keV
resonance being especially relevant over nova temperatures. The additional study of finding the
reaction rate dependence to the spin-parity assignment of the 331keV concluded that, although
some variation in dominance was present, it was not hugely significant over nova temperatures.
The significance of this study is such that it supports the grounded theoretical and experimental
work that has been carried out recently. Additionally, it shows that the codes used within this
study are accurate and replicate the results expected from theory, as well as supporting the
studies of Laird et al.
Further investigations could involve investigating the rate dependence of other burning cycles
within novae, such as the CNO cycle, as well as updating the code to use the latest reaction rate
data to increase the accuracy of the study. When investigating the 𝐹!"
𝑝, 𝛼 𝑂 !"
reaction,
studying the effect of broad resonances and their temperature dominance would enable a more
complete picture to be analysed. Moreover, extension of the study into the proton-width of the
48 keV resonance would be very relevant, as it would enable a study into the rate uncertainty
presented by Laird et al [9]. This would require being able to calculate/ acquire the
spectroscopic factors, which was attempted but proved problematic.

29. 29
Bibliography
[1] R. Freedman and W.J. Kaufmann., Universe (8th
Edition), W.H. Freeman and Company
(2008).
[2] A. D. Roberts., 18F Production for Pet and an Investigation of the 18F(p,α)15O Reaction
with a Radioactive Beam, University of Wisconsin—Madison (1995).
[3] Webnucleo website, Clemson University. http://www.webnucleo.org
[4] C. Rolfs., and W. Rodney., Cauldrons in the Cosmos: Nuclear Astrophysics, University of
Chicago Press p. 156 (2006).
[5] C. Beer et al., Phys.Rev.C 83, 042801 (2011).
[6] C. Nesaraja et al., Phys. Rev. C 75, 055809 (2007).
[7] Hernanz, M et al., Prospects for detectability of classical novae with INTEGRAL, from ESA
SP-459:Exploring the Gamma-Ray Universe, 133, 65-68 (2001).
[8] A. Coc et al., Astron. Astrophys 357, 561 (2000).
[9] A. M. Laird et al., Phys. Rev. Let 110, 032502 (2013).
[10] Neng-Chuan et al., Chin.Phys.Lett 20, No.9, 1470 (2003).
[11] M. Wiescher., and K. Kettner., Astrophys. J 263, 891 (1982).
[12] R. Cozsach et al., Phys.Lett.B 353, 184 (1995).
[13] K. Rehm et al., Phys.Rev.C 52, R460 (1995).
[14] K. Rehm et al., Phys.Rev.C 53, R1950 (1996).
[15] K. Rehm et al., Phys.Rev.C 55, R566 (1997).
[16] D. Bardayan et al., Phys.Rev.Lett 89, 262501 (2002).
[17] C. Iliadis et al., Nucl. Phys. A 841, 2 (2010a).
[18] C. Iliadis et al., Nucl. Phys. A 841, 80-102 (2010b).
[19] C. Iliadis et al., Nucl. Phys. A 841, 17 (2010c).
[20] A. Adekola et al., Phys. Rev. C 83, 052801(R) (2011).
[21]
K. Y. Chae et al., Phys. Rev. C. 74, 012801(R) (2006).
[22] C. Beer et al., Phys.Rev.C 83, 042801 (2011).
[23] K. S. Krane., Introductory Nuclear Physics, John Wiley Sons (1988).

30. 30
[24] S. Biswas., T. Chattopadhyay., and B. Basu., Introduction to Astrophysics, Prentice-Hall
of India Pvt. Ltd, p. 42 (2011).
[25] M. Thickitt., Reaction Rate Studies in Novae, Unpublished Work (Professional Skills
Essay) (2012).
[26] S. Bishop., Direct Measurement of the 21
Na(p,𝛾)22
Mg Resonant Reaction Rate in Nova
Nucleosynthesis. PhD Thesis (2003).
[27] M. Camenzind., Compact Objects in Astrophysics: White Dwarfs, Neutron Stars and Black
Holes, Springer (2007).
[28] M. Bode et al., Astrophys. J 600, L63 (2004).
[29] C. Iliadis., Nuclear Physics of Stars, Wiley VCH (2007).
[30] H. J. W. Muller-Kirsten., Introduction to Quantum Mechanics: Schrodinger Equation and
Path Integral, World Scientific Publishing Co Pte Ltd (2006).
[31] Maple 15, Maplesoft, Waterloo, ON, Canada. www.maplesoft.com
[32] A. Parikh et al., Astrophys. J. Supp. Series 178, 110 (2008).
[33] List of Publications from the Webnucleo page
http://www.webnucleo.org/home/publications/
[34] S. Starrfield., Hydrodynamic Studies of the Evolution of Recurrent Novae to Supernova Ia
Explosions, Proceedings of the International Astronomical Union 7, 166-171 (2011).
[35] B. Shaw., An Introduction to Webnucleo, University of York (2012).
[36] pro Fit graphical package from QuantumSoft, https://www.quansoft.com
[37] A. Laird, Private communication.
[38] National Nuclear Data Center Chart of Nuclides, Brookhaven National Laboratory.
http://www.nndc.bnl.gov/chart/
[39] J. D. Garrett et al., Phys. Rev. C 2, 1243 (1970).
[40] D. W. Visser et al., Phys. Rev. C 69, 048801 (2004).
Word count (not including title page, acknowledgements, abstract, contents, titles,
figures, bibliography and appendices, as advised by supervisor): 7999

31. 31
Appendix
Appendix A: Finding the Gamow Window Peak and
Width
An important aspect of any reaction rate study is the calculation of the Gamow Peak, 𝐸! and
width, ΔE! (as noted on Figure 2.5). These are dependant on not only the Maxwell-Boltzmann
particle distribution and the tunnelling probability, but also several different parameters within
the physical reaction itself.
One can find the peak of the Gamow window for a reaction in the form 𝑋 𝑎, 𝑏 𝑌 using the
following expression:
𝐸! = 1.22(𝑍!
!
𝑍!
!
𝜇𝑇!
!
)! !
𝑘𝑒𝑉
Where 𝑍! is the proton number of the target, 𝑍! is the proton number of the projectile, 𝜇 is the
reduced mass
!!!!!
!!!!
in atomic mass units and 𝑇! is the temperature in Mega-Kelvin.
The width of the Gamow window is found using the following expression:
∆𝐸! = 0.749(𝑍!
!
𝑍!
!
𝜇𝑇!
!
)! !
𝑘𝑒𝑉
Where all expressions have the same meaning as before. These important expressions have
been taken from [29].
Appendix B: Deriving the Reaction Rate for a Narrow
Resonance
Note all notation is the same as defined in section 2.1.3.1. From the one-level Breit-Wigner
formula, the cross section is given by:
𝜎! 𝐸 =
!ℏ!
!!
(!!!!)
(!!!!!)(!!!!!)
!!!!
(!!!!)!!(! !)
!
Where 𝑁! 𝜎𝑣 is the reaction rate in 𝑐𝑚!
𝑚𝑜𝑙!!
𝑠!!
, T9 is the temperature of the environment
in GK, i labels different resonances, (𝜔𝛾) is the resonance strength (where the spin factor
𝜔 =
(!!!!)
(!!!!!)(!!!!!)
and 𝛾 =
!!!!
!
) and Ei is the energy of the given resonance (both in units of
MeV).
From Maxwell-Boltzmann arguments, the reaction rate at a given temperature T is given by
B.2:
𝑁! 𝜎𝑣 =
!.!#$ × !!
!!
!/! 𝜇 𝐸𝜎 𝐸 𝑒𝑥𝑝
!!!.!#!!
!!
𝑑𝐸
!
!
(B.1)
(B.2)
(A.1)
(A.2)

32. 32
Combining equations (B.1) and (B.2) yields:
𝑁! 𝜎𝑣 =
!
!
!!
(!)!/!
𝐸𝜎! 𝐸 𝑒𝑥𝑝
!!
!
𝑑𝐸
!
!
𝑁! 𝜎𝑣 = 𝑁!
(!!!!)
(!!!!!)(!!!!!)
!!ℏ!
!# !/!
!!!!
(!!!!)!!!!/!
𝑒𝑥𝑝
!!
!
𝑑𝐸
!
!
If the resonance is sufficiently narrow, the Maxwell-Boltzmann factor 𝑒𝑥𝑝
!!
!
and the partial
widths are constant over the total width of the resonance, and so can be replaced by their
implicit value at the resonance energy Er. Such, equation (A.4) can be solved analytically:
𝑁! 𝜎𝑣 =
(!!!!)
(!!!!!)(!!!!!)
!! !!ℏ!
!# !/!
!!!!
!
𝑒𝑥𝑝
!!!
!
!/!
(!!!!)!!!!/!
𝑑𝐸
!
!
𝑁! 𝜎𝑣 =
(2𝐽 + 1)
(2𝑗! + 1)(2𝑗! + 1)
𝑁! 2𝜋ℏ!
𝜇𝑘𝑇 !/!
2Γ!Γ!
Γ
𝑒𝑥𝑝
−𝐸!
𝑘𝑇
𝜋
𝑁! 𝜎𝑣 = 𝑁!
!!
!#
!/!
ℏ!
(𝜔𝛾)𝑒𝑥𝑝
!!!
!
Where 𝜔 =
(!!!!)(!!!!)
(!!!!!)(!!!!!)
and 𝛾 =
!!!!
!
as before.
Substitute in NA and k into (B.7) to get:
𝑁! 𝜎𝑣 = 1.8487 𝑥 10!
𝑚𝑜𝑙!!
𝑚!!
𝑘𝑔!!/!
𝑠!
𝐾!/!
𝜇𝑇 !!/!
ℏ!
(𝜔𝛾)𝑒!!!/!
Substitute in ℏ and cancel units in (B.8):
𝑁! 𝜎𝑣 = 2.05 𝑥 10!!
𝑚𝑜𝑙!!
m 𝑘𝑔!/!
𝑠 𝐾!/!
𝜇𝑇 !!/!
(𝜔𝛾)𝑒!!!/!
Convert (𝜔𝛾) from Joules to MeV (1 MeV = 1.6022 x 10-13
J) where 1 J = 1 𝑘𝑔𝑚!
𝑠!!
to get:
𝑁! 𝜎𝑣 = 3.2845 𝑥 10!!!
𝑚𝑜𝑙!!
𝑚!
𝑠!!
𝑘𝑔!/!
𝐾!/!
𝜇𝑇 !!/!
(𝜔𝛾)𝑒!!!/!
Convert the reduced mass from kg to atomic mass units (amu) and the temperature to Giga-
Kelvin in (B.10) to get:
𝑁! 𝜎𝑣 = 1.5399 𝑥 10!
𝑚𝑜𝑙!!
𝑚!
𝑠!!
𝜇𝑇!
!!/!
(𝜔𝛾)𝑒!!!/!
(B.3)
(B.4)
(B.5)
(B.6)
(B.7)
(B.8)
(B.9)
(B.10)
(B.11)

33. 33
Convert from m3
to cm3
by multiplying (B.11) by 1003
:
𝑁! 𝜎𝑣 = 1.5399 𝑥 10!!
𝑚𝑜𝑙!!
𝑐𝑚!
𝑠!!
𝜇𝑇!
!!/!
(𝜔𝛾)𝑒!!!/!
Using the Boltzmann constant of k = 8.6173 x 10-11
MeV K-1
, substitute this into the
exponential, as well as convert temperature to Giga-Kelvin:
𝑒!!!.!#!!/!!
Substituting the exponential in (B.13) into (B.12) gives the final result (B.14):
𝑁! 𝜎𝑣 = 1.5399 𝑥 10!!
𝜇𝑇!
!!/!
𝜔𝛾 𝑒!!!.!#!!/!! c𝑚!
𝑚𝑜𝑙!!
𝑠!!
Appendix C: Running a Simulation
This is just a small graphical representation of how the simulations were run throughout the
project for clarity.
Initially, one would log onto their specific ‘user’ branch within the SSH server they were
connecting to. In this project, this is the ‘npp2’ server in the Nuclear Physics Group server
‘npg’. In this server, one must run all simulations within their user branch, mainly so the output
files can be easily found once the simulation has completed. Below shows the form of the input
needed to run a simulation.
1)
This is the user branch under which the simulations are run, and it is within this directory path
that the output files will be placed.
2)
This is the form of the run script used to initiate the simulation
3)
This is the file that governs the initial conditions of the companion star accreting onto the white
dwarf. There are several different options for this so to diversify the scenarios possible. Here,
the companion is a main-sequence, sun like star that possesses the initial mass fractions of the
sun.
(B.14)
(B.12)
(B.13)

34. 34
4)
This is the output file of the data in a ‘.xml’ format. It can be renamed anything, but for ease,
it was named as ‘out.xml’ for the duration of the project.
5)
This portion of the input gives the trajectory of the simulation, i.e. the density and temperature
profile to be fed into the post-processing code. There were several different options that could
be chosen, and the project focused on the 1 and 1.25 solar mass CO and ONe white dwarf
trajectories respectively. The number at the end is the number of timesteps to be performed.
Upon entering this line of code, the following prompts appear:
The restrict prompt allows the user to place a restriction on the simulation with negligible loss
of precision, and is encouraged to be used as it dramatically shortens simulation time (from
over an hour to ~ 3 minutes).
The next two allow for the mass excess to be changed, or the reaction rate to be changed in the
simulation. The reaction rate change was used extensively, and is outlined in Appendix C.
After this, the simulation runs.
This shows the prompt to the user once the simulation has been run, exclaiming where the
output files have been deposited within the user branch.
By changing directories within the user branch to the OUTPUT directory and using the Linux
command ‘ls’ (to list the files) the output files can be seen. The other directories seen contain
all of the other simulations run throughout the project and were named in an organised fashion
to help with identification. Using the ‘mv’ Linux command, these output files could be moved
to any directory or sub-directory.
Certain outputs can then be applied to these output files, and a full list with excellent
explanations can be found in Ben Shaw’s comprehensive guide [35].

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Appendix D: Changing Reaction Rate files
As a hugely important part of the project, it is important to cover it within the appendices. This
example will only cover changing the reaction rates for the 𝐹!
𝑝, 𝛼 𝑂!
portion of the project,
although all of the reactions within the HCNO-II and III cycles were varied.
Within the user branch, changing the directories to the ‘reaction_rate/Laird’ directory,
which contains the different reaction rate files based on Dr Laird’s paper [9] yields:
This lists clearly the different reaction rate files, with each named accordingly to see by what
factor the original rates are changed (x0.01,x0.1 etc). For this example, a new file, the x1000
reaction rate file will be created. For this, the ‘cat’ Linux command must be used to create a
file.
After the ‘cat’ command, the name of the file must be entered, here ‘f18pa_x1000’ to
represent the correct reaction and then the amount the original rate is changed (as in line with
the convention used throughout). The next line denotes how many lines are to be in the file,
here 23. The two columns then represent the temperature in Giga-Kelvin on the left and the
reaction rate in c𝑚!
𝑚𝑜𝑙!!
𝑠!!
on the right. These were just exported straight from an Excel
spreadsheet. The user must then press ‘Ctrl+D’ to create the file. When running the simulation,
when prompted to change the reaction rate file, if ‘y’ is chosen, the following will appear:
f18 + h1 - o15 + he4

36. 36
By entering the path to the new file, and entering the reaction in the form of:
‘f18 + h1 - o15 + he4’
The reaction rate for that specific reaction if then changed within the simulation.
Appendix E: Breit-Wigner/ Resonance Strength Solver
To calculate the narrow reaction rates quickly, accurately and easily, a small programme within
Excel was created to do so. When investigating the 𝐹!
𝑝, 𝛼 𝑂!
reaction from Dr Laird’s
paper, the reaction rates needed to be calculated from several different parameters. To avoid
laborious manual calculation each time, the Breit-Wigner solver only required the parameters
to be input and the rates were instantly calculated. Figure E.1. shows the interface of the
programme:
Figure E.1. Screenshot of the Breit-Wigner solver, showing the input of the target mass (A) and projectile
mass (B) in atomic mass units, the temperature in GK, the resonance strength in MeV, resonance energy in
MeV to give the reaction rate in the appropriate units.
Using a number of hidden calculations (for example calculation of the reduced mass), the
reaction rate could be easily calculated. The following input formula was used to calculate it
(for a set of general cells):
Which is just a form of equation B.8. It was also important to calculate the resonance strength
given partial width parameters, especially relevant when considering Dr Laird’s paper. The
interface for this is shown in figure E.2.
Figure D.2. Screenshot of the Resonance Strength solver, showing the resonance energy (keV), the total
angular momentum quantum number J of the resonance (𝐽!), proton (𝐽!) and alpha (𝐽!), the partial widths of
the proton (Γ!) and alpha (Γ!) and the sum of these, (Γ!!#$) (all in keV) to calculate the resonance strength.
The formula used to calculate the resonance strength is simply:
𝜔𝛾 =
(!!!!!)
(!!!!!)(!!!!!)
!!!!
!!!#$
(E.1)

37. 37
In terms of the spreadsheet, this is given by the formula (for a set of general cells):
This is just equation (E.1) in terms of the spreadsheet layout.

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