UNIVERSIDAD NACIONAL AUTÓNOMA DE MEXICO
PROGRAMA DE POSGRADO EN ASTROFÍSICA
INSTITUTO DE ASTRONOMÍA
ASTROFÍSICA TEÓRICA
EVOLUCIÓN TÉRMICA DE ESTRELLAS 
DE NEUTRONES EN SISTEMAS BINARIOS 
DE BAJA MASA CON ALTAS TASAS DE 
ACRECIÓN  
TESIS
QUE PARA OPTAR POR EL GRADO DE:
DOCTOR EN CIENCIAS (ASTROFÍSICA)
PRESENTA:
MARTIN JAVIER NAVA CALLEJAS
TUTOR
DR. DANY PIERRE PAGE ROLLINET
INSTITUTO DE ASTRONOMÍA - UNAM
CIUDAD UNIVERSITARIA, CDMX, DICIEMBRE 2024.
 
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To my family - mom, dad and sister - for their unconditional support and love;
to Prof. Dany, for his support and mentorship along the years;
to Macarena, for her encouragement and friendship (ten years and beyond!);
to Tais, for her support - always throughout music, lately throughout birbs, cheep;
to Anna, for her kindness and for reminding me the importance of subjectivity.
1
Acknowledgements
Given this is (probably) my last written thesis and thus the conclusion of my aca-
demic path as a learner I wish to thank to all my key mentors along this road, both
academic and personal aspects of my life. Everaldo Tapya in primary school. At
CECYT 11 - “la Willy” - Prof. Alma Benítez, for introducing me to the world of
research and for believing in me to reach this far. Prof. Ismael Sandoval (RIP), who
taught me how wonderful physics is. At University: Prof. Mario Pacheco, my men-
tor along my 8 semesters at ESFM, my alma mater ; Prof. Isaura Fuentes-Carrera,
who introduced me into the marvelous world of astrophysics; Prof. Arturo Zúñiga
Segundo, who motivated me to go beyond. To Profs. Ana Hidalgo-Gámez, Fernando
Angulo Brown, Jaime Avendaño, José Castro Quilantán and Alfredo López Ortega,
for guiding me along the different branches of physics, and to Profs. Elena Luna
Elizarrarás and Luis Cisneros Ake for making me appreciate the beauty of formal
mathematics.
My time at the IA-CU has been wonderful. Such experience wouldn’t have been
possible without the continuous support from my advisor, Prof. Dany Page, who
mentored me throughout all these years of Master and PhD, believing in what I
could do and always motivating me to keep exploring new ideas. My PhD project
wouldn’t have been the same without the collaboration of Yuri Cavecchi, who pro-
vided support and guidance as well. Thanks to Aida Kirichenko, who was always
supportive and provided me valuable advices along my path as a PhD student. I
wish to express my gratitude towards all the staff for welcoming me as a part of the
Institution. Special thanks to Bertha Vázquez and Heike Breunig for their ample
support during my time as a student, as well as to Brenda Arias. A special thank
as well to the Academic Committee of the Program for allowing me to represent the
PhD students’ for two years.
A special grazie mille to the ECT and Residenza Arcivescoville staffs and the
DTP/TALENT 2024 people I met during my stay in Trento, Italy. Forever I will
treasure the memories of my first international travel, the extraordinary three weeks
and the kind people I met, specially Nithish, Tiago, Milena and Rajesh, with whom
I spent more time hanging around the city. Special thanks to Jorge Piekarewicz
and Barbara Gazzoli for their support and attention during those days and for the
organization of the event.
2
Contents
1 Equations of Stellar Evolution 14
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2 Spacetime evolution of physical quantities . . . . . . . . . . . . . . . 15
1.3 Evolution in a static and spherically symmetric (SSS) spacetime -
fluid in radial motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4 On the numerical implementation of neutron star evolution . . . . . . 33
1.5 Thermonuclear Instabilities . . . . . . . . . . . . . . . . . . . . . . . 39
2 Microphysical aspects of stellar evolution 43
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 A brief survey on Thermodynamics . . . . . . . . . . . . . . . . . . . 44
2.3 The Neutron Star Equation of State . . . . . . . . . . . . . . . . . . . 50
2.3.1 The envelope, ρ ≤ 108 g cm−3 . . . . . . . . . . . . . . . . . . 54
2.3.2 Outer and inner crusts, ρ ∈ [108, 1014] g cm−3 . . . . . . . . . 62
2.4 Thermal conductivity and opacity . . . . . . . . . . . . . . . . . . . . 66
2.5 Particle reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.5.1 β reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.5.2 Chains and cycles . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.6 Thermal neutrinos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3 Nuclear networks for neutron stars 86
3.1 Overview of numerical methods . . . . . . . . . . . . . . . . . . . . . 87
3.2 On the networks for neutron stars. . . . . . . . . . . . . . . . . . . . 90
3.3 An overview of the rp-process . . . . . . . . . . . . . . . . . . . . . . 94
4 Neutron Star Envelopes 104
4.1 General features of the envelope . . . . . . . . . . . . . . . . . . . . . 105
4.2 Evolution of non-accreting envelopes . . . . . . . . . . . . . . . . . . 109
4.3 Evolution of accreting envelopes . . . . . . . . . . . . . . . . . . . . . 114
4.3.1 Stationary accreted envelopes . . . . . . . . . . . . . . . . . . 115
4.3.2 Intermezzo: Linear perturbations to stationary solutions and
the origin of bursts . . . . . . . . . . . . . . . . . . . . . . . . 131
3
4.3.3 Time-dependent envelopes . . . . . . . . . . . . . . . . . . . . 135
5 Conclusions 150
6 Paper 1: “The Effect of Opacity on Neutron Star Type I X-ray
Bursts Quenching” 153
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7 Paper 2 - Considered for acceptation 156
7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
8 Paper 3 (Third Author) 158
8.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
A Thermodynamics 159
A.1 Propositions and theorems . . . . . . . . . . . . . . . . . . . . . . . . 159
A.2 Maxwell relations and alternative expressions for some partial deriva-
tives of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
A.3 Heat capacity in superfluid systems . . . . . . . . . . . . . . . . . . . 163
B General Relativity 165
B.1 On the conservation laws . . . . . . . . . . . . . . . . . . . . . . . . . 165
B.2 SSS metric - Identities . . . . . . . . . . . . . . . . . . . . . . . . . . 167
B.3 Propositions and theorems . . . . . . . . . . . . . . . . . . . . . . . . 169
B.4 Radiative Zero Solution . . . . . . . . . . . . . . . . . . . . . . . . . 175
4
Abstract
The purpose of the present work is to study the envelope of neutron stars in Low-
Mass X-ray Binaries (LMXB) in the presence of high mass accretion rates, i.e. ≥
0.3ṀEddington, in order to identify the necessary conditions to treat accreted envelopes
as time-independent and thus simplify the numerical evolution of temperature and
luminosity of the whole neutron star. Considering the current understanding of
nuclear burning at such high accretion rates, for such study we required to construct
a network of nuclear reactions, sufficiently large as to simulate in detail the rapid
proton capture process (rp-process). In addition, the present study aimed to restrict
the conditions under which we could regard the leakage of thermonuclear energy due
to H and He burning, from the envelope to the core of the neutron star, as the source
of the still-unknown shallow heating.
In Chapter 1 I introduce the equations governing the structure and tempera-
ture of a neutron star undergoing mass accretion. The functionals describing the
microphysical aspects of the evolution, such as the Equation of State (EOS) or ther-
monuclear rates, are detailed in Chapter 2. In Chapter 3 we present the numerical
challenges of solving the equations of evolution for the chemical composition at the
surface of the stars, as well as further details regarding the rp-process. A detailed
study of the envelope, in both time-independent and -dependent scenarios is pre-
sented in Chapter 4, where we show the results from the simulations computed with
the stellar evolution code MESA and my own newly developed code for stationary
envelopes. In Chapter 5 I summarize the results from Chapter 4 and prospects
for future applications. In Chapter 6 I give further details on the research article
(manuscript) presenting the newly-developed code, while in Chapter 7 I present a
second research article related to modifications in opacity as a possible explanation
for the burst quenching at 0.3ṀEdd. In Chapter 8, a third-author paper is intro-
duced, in which I collaborated computing the evolution under electron captures of
oxygen and neon bubbles.
5
Preface
Neutron stars are among the most fascinating objects in the Universe. Quoting
Page, D.1, “You can ask for whatever physics you want and neutron stars have it.
General relativity? Solid state physics? You can have them all!”
The current picture of how the interior of a neutron star looks like is illustrated
in Fig. 1. The densest portion of the star, the core, is composed of neutrons, pro-
tons, electrons and muons in charge and beta equilibrium, and the baryonic sector
conforming a superfluid/superconductive state. While theoretical estimates point
towards the existence of exotic phases at the most inner portion (marked as (?)),
observational evidence is still in favor of the enlisted baryonic and leptonic compo-
sition ([AAA+17, RGH+21]), and we thus stick to this picture in the present work.
Between 1011 and 1014 g cm−3, in the region known as inner crust, neutrons can be
either free, conforming a superfluid, or bounded with protons in nucleus of exotic
shapes, going from pasta to spherical shape at low densities. At ρ ≤ 1011 we observe
a transition to “standard” matter of different aggregation state: between 109 and
1011 g cm−3 nucleus are arranged in lattices imbued in a gas of free degenerate elec-
trons, while between 108 and 109 g cm−3 nucleus are best described by the transition
from a gas to a liquid state. While the region between 108 and 1011 g cm−3 is usually
dubbed as outer crust, due to the composition and state of aggregation for some
authors the portion between 108 and 1010 g cm−3 still belongs to the envelope, the
region which in our Fig. 1 corresponds to the outermost and thin layer of the star.
While in the present work we employ ρb to denote the density boundary between
the envelope and the outer crust, to avoid confusions we always state the explicit
value of this boundary for each of the employed models.
Despite such versatility, placing “nuclear reactions” and “neutron stars” in the
same sentence seems odd at first glance: after all, it is a well stablished fact
[KKO+03] neutron stars are the last stage in the life of very massive stars, had
they not collapsed into black holes. Being the last stage means nuclear reactions no
longer provide sufficient heat to counterbalance the energy loss due to neutrino and
photon emission. Furthermore, they do not take place anymore due to the inexistent
fuel to start them.
Surprisingly, however, we can place “nuclear reactions” and “neutron stars” in
1Private communication
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the same sentence, without restricting ourselves to the subject of stellar evolution.
In the vastness of the Universe, there exists systems where nuclear reactions do
take place at the surface of neutron stars. These are the interacting binaries, self-
gravitating system where the neutron star receives mass from a main sequence or
more evolved companion star [BD23]. The transference mechanisms depends on
the mass of the young star, Mcomp. If ∼ M⊙, i.e. a lower mass than the neutron
star’s (Mneutron star ≈ 1.4M⊙), gravitational pull generates a disk around the neutron
star and eventually, via thermal-viscous instabilities (e.g. the disk-instability model
[van96, Ham20]), matter is uniformly accreted towards the surface of the compact
object. On the other hand, in the presence of a massive companions the transfer of
material towards the neutron star occurs via the stellar wind or equatorial disk of
the companion [Cha22a]. In the present work, only the former scenario is addressed.
Observationally, binary systems composed of a neutron star and a low-mass, main
sequence companion are dubbed as Low-Mass X-ray Binary (LMXB) since they were
discovered employing X-ray telescopes due to their prominent emission in said band
due to the release of gravitational energy of the in-falling material. Such was the case
of Scorpio X-I, first spotted with an X-ray luminosity of LX ∼ 104L⊙ ≈ 1037 erg s−1
[GGPR62], later identified as a LMXB [SOG+66, Shk67, Cha22b]. Following this
event, many neutron stars in LMXBs have also been identified due to the observation
of very energetic explosions in the X-ray band (Lx ∼ 1038 erg s−1 [Kdi+03, Kuu04]),
hence named as X-ray bursts.
Since then, observational capabilities in the X-ray band have significantly im-
7
proved, allowing to discover new LMXBs as well as to monitor them for either
short (∼ 1 day) or long (∼ months) periods of time [BD23]. Such explosions,
however, occur in the presence of accretion and represent an excellent opportunity
to directly observe photons coming from the neutron star surface, and not from
the output of mass accretion. This is not the only moment when the surface be-
comes visible: while for some system the mass accretion occurs continuously, there
also exist other systems where mass accretion ceases from time to time and, in
the absence of in-falling material, the surface of the neutron star becomes visible
[WMM+01, RBB+02, WDP17]. Such systems are said to exhibit transient accretion,
while the period without in-falling mass towards the surface receives the name of
quiesence. In practice, it is convenient to express all accretion rates in terms of the
Eddington accretion rate for a 10 km neutron star, ṀEdd = 1.1 × 1018 g s−1, coming
from the local rate per unit area, i.e. ṀEdd = 4πr2ṁEdd with ṁEdd = 8.8 × 104 g
cm−2 s−1. In these units, for instance, we say bursts take place for the accretion
rates between ∼ 0.01 and ṀEdd. Systems with Ṁ ≥ ṀEdd actually exist, such as
XTE J1701-462[HvF+10] and MAXI J0556-332 [HFW+14].
But, what is the origin of the X-ray bursts? As a first step, the surface of the
neutron star is enriched - via accretion - in low-Z material. For the case of main-
sequence Sun-like stars, the composition is of ∼ 70% 1H, ∼ 28% 4He and the re-
maining 2% of CNO metals, while in the case of a white dwarf or He-star companion
matter is enriched in 4He and poor in 1H ([KiA+10]). Given the extreme gravita-
tional acceleration at the neutron star surface (gs ∼ 1014 cm s−2, while at Earth
gs ∼ 9.8 cm s−2), a temperature build-up is likely to take place: from Tmax ∼ 106 K
to Tmax ∼ 108 K. Consequently, thermonuclear reactions eventually start operating
at the surface, mainly converting 1H into 4He via pp-chains and CNO cycles, al-
though synthesis of small amounts of 40Ca is still viable at moderate accretion rates
(e.g. ∼ 0.1ṀEdd). However, this burning might become unstable: as long as fuel
is added, the initial temperature build up due to accretion is further enhanced by
the deposited heat from thermonuclear burning. If the cooling mechanisms, such as
radiation and conduction transport, are not capable of distributing this heat then an
extremely fast (≤ 1s) temperature build-up takes place (from T ∼ 108.25 to 109.25 K)
and induces a thermonuclear explosion at ∼ 106 g cm−3, i.e. P ≈ 1022.5 erg cm−3.
The energy transported at the surface increases the local temperature up to 107.25
K, e.g. L ∼ 1038 erg s−1, and during the following ∼ 100 seconds the surface relaxes
and temperature decreases while the luminosity coming from the surface overcomes
that from mass accretion. In the aftermath of this explosion, the thin matter layer
of fuel at P ≈ 1022.5 erg cm−3 transformed, via thermonuclear reactions, into ashes
enriched in high-Z elements of A = 56, A = 72, and even of A = 107, albeit in little
amounts. This process of rapid captures of 1H and transformation into metals (e.g.
less than 103 seconds) is known as rp-process. Under continuous accretion these
explosions occur periodically, resulting in the accumulation of ashes at ρ ≥ 106 g
cm−3. Despite its high-Z composition, these X-ray burst ashes between 106 and 1011
8
g cm−3 are far from being inert. Instead, heavy-Z burning proceeds via pycnonu-
clear reactions, i.e. sensitive to density. As examples of these reactions we have
carbon burning (12C + 12C) or electron-captures, which enrich the densest layers
of the envelope (or the upper layers of the outer crust) with Z ≥ 20 material and
injects up to ∼ 1 MeV per accreted baryon.
Pristine 
composition
70 % 1H, 28 % 4He, 2 % Z
⊙
12C, 24Mg, 40Ca
56Fe, 64Zn, . . . , 104Cd?
56Fe → 56Co → 56Ti
20Ne → 20O
ρ ∼ 108 g cm−3
ρ ∼ 1011 g cm−3
ρ ∼ 106 g cm−3
·
M
Stable/unstable burning ashes:
Fuel:
Fusion/pycnonuclear
Figure 2: A brief look at the composition of the accreted envelope.
At this point, it would seem the picture is complete. Turns out, it is far from
such state. For instance, there are two open questions regarding the end of the
accretion phase, referred to as outburst as well, and the behavior of the bursts as a
function of the mass accretion rate. In the first case, the necessity of invoking an
additional heating source of unknown nature, between 0.1 and 10 MeV per accreted
baryon at ρ ∈ [108, 1011] g cm−3 in order to explain the cooling of the neutron star
after an outburst episode This constitutes the so-called shallow heating paradigm
[BC09, DBW11, DWB+13, DCBP15, Mei18, CFZH20, PHN+22]. A still existing
discrepancy between theory and observations is the apparent presence of a critical
mass accretion rate above which bursting behavior is suppressed: while observations
suggest 0.3ṀEdd [CiV+03, WM07, GMH+08, GK21], several theoretical models stop
bursting at around and above ṀEdd [HCW07, FGWD06a, FTG+07, GK21]. There
are, however, sources which still exhibit bursting behavior between 0.3 and 1 times
ṀEdd, such as CygnusX-2 [Kv95], GX 17+2 [KHv+02], Circinus X-1 [LWA+10] and
Terzan 5 [LAC+12]. Given the majority of simulations are in one spatial dimension
only, such discrepancies leave open the possibility that additional effects - besides
the shallow heating - such as diffusion of 4He or rotation ([PB07, IS10, CGG+20])
9
might play an important role in order to unify these apparently different bursting
regimes.
Probably the biggest open question, among the existing ones, is the absence
of a self-consistent long-term modeling of the neutron star in transient accretion.
While the existing models rely on reasonable approximations regarding the physics
of the star, some of the other open issues re-appear in these models and complicate
the picture. Thus, it is important to settle why, despite the incredibly detailed
picture given above, there still exist open questions. First of all, implementing
nuclear reactions is numerically challenging since the system of partial differential
equations (PDEs) is stiff, i.e. has many different scales involved and is thus difficult
to integrate employing explicit methods (such as the workhorse 4th order Runge-
Kutta) [PTVF92]. Assuming we find a method to integrate these equations (which
have been extensively developed in the past 50 years and belong to the category
of implicit methods), the next question to worry about is the number of species
we need to incorporate into the network to obtain as realistic values as possible of
generated energy and of chemical concentrations, which have a profound effect on
the opacity κ (thus in the energy transport) and the equation of state (EOS) of the
star. In order to have both proton- and neutron-rich isotopes, necessary to follow
the whole picture described above and illustrated in Fig. 2, we require nearly 1000
species. And there lies the next big issue: we must solve ≥ 1000 PDEs (one for each
species) at every spatial location in the star, i.e. ≥ 1000 cells. Assuming we can
neglect the evolution of the network at ρ ≥ 1011 g cm−3, i.e. above the neutron drip
point, we are still left with the task of simultaneously modeling the dense core and
the surface, each with ≥ 1000 cells. The final issue concerns the timescale: it turns
out the characteristic timescale at the surface is of ≤ hr, while the core evolves in
∼ months. Consequently, a full simulation of the whole neutron star undergoing
accretion, even in 1D, is challenging.
A quick review on the literature seems to contradict the above assertion. Be-
fore and around 1978 [WZW78, FHM81] simulations of neutron stars undergoing
accretion were indeed performed, although in retrospective their nuclear synthesis
is inaccurate. For instance, the limited size of their networks allowed large amounts
of 1H to be still present at ≥ 106 g cm−3 (deep hydrogen burning), the heaviest iso-
tope to be synthesized was 56Fe and thus the possibility of pycnonuclear reactions
was still out of the picture. Later works showed that the size of the network had
an important impact on the chemical composition of the ashes, no longer rich in
1H [vGI+94, SBCW99]. Furthermore, they put into relevance the rp-process as the
mechanism to synthesize metals far heavier than 56Fe. In addition, the enrichment
in metals of the dense layers of the surface put into question the inert character of
the crust, now possibly undergoing electron captures as a consequence of the com-
pression and thus injecting energy. These pycnonuclear reaction gained relevance
as well due to the discovery of the superbursts, which called out for the identifica-
tion of fuel capable of liberating such tremendous amount of energy in a period of
10
time of days. In the modern era of networks, computers and telescopes, a review
on the literature shows the simulation of neutron stars undergoing accretion is now
focused on punctual aspects which restrict the time interval to be simulated, and
thus alleviates the tension on the numerical resources:
• For an accurate modeling of bursts, it suffices to model the envelope, i.e.
ρ ≤ 107 g cm−3, by either multi-zone or one-zone simulations for a timespan
sufficiently large as to obtain ≈ 20 bursts, i.e. t ≤ 1 week, in order to contrast
them with observations [Mei18]. The size of the network can be as short as
300 or as large as 1500 [SBCW99, FGWD06a]. Techniques such as adaptative
networks have been developed as well [HCW07].
• The end of accretion can be followed by either a detailed and large numeri-
cal network, or by simple estimates on the secular burning of remaining fuel
[SAB+01]. Since the typical time between the end of accretion and the relax-
ation towards a “typical”, pre-accretion state, is of ∼ weeks, only the crust is
usually evolved. The core, on the other hand, remains at a fixed temperature.
• Existing long-term simulations of transient accretion and quiescent phases
assume the heating can be modeled as ∝ Ṁ [BC09, PHN+22], and thus intro-
duce this parametrization instead of solving a nuclear network. The envelope
is treated as an atmospheric-like boundary condition, that is, giving an ex-
plicit relation between the observed effective temperature Teff and the internal
temperature Tb. This treatment regards the exiting luminosity from the core
as constant in the envelope.
Objective of the present work
Then, what is the purpose of the present work? In principle, to find out if it is
possible to accelerate the simulations of long-term evolution of accreting neutron
stars by means of a Tb(Teff) scheme. This consists on replacing the whole simulation
of the envelope by an atmospheric-like boundary condition. Under the assumption
no heating and cooling sources exist in this region, we can know at all times the
temperature at the surface as a function of the temperature at the envelope-crust
boundary by simply solving dT/dP . Given the time independency of such equation,
it thus suffice to solve it once, at the beginning of the simulation, extract the desired
Ts(Tb) function (or data, and store them into a table) and implement them into the
numerical code for the evolution of the whole star. The underlying assumption of
no heating/cooling sources, however, prevents this approach to be employed in self-
consistent simulations of accreting neutron stars. However, it has not been proved
such scheme cannot be implemented for accreting neutron stars, naturally by taking
into account the existence of the heating and cooling mechanisms. Indeed, given
11
the luminosity at all points depend on these sources, we could expect to have not
only one function as boundary conditions, but two, one for Tb(Ts) and the other for
Lb(Ts), i.e. the luminosity at the boundary becomes now a function of the surface
temperature as well. As we shall see in the present work, the implementation is not
straightforward, and the main complication arises due to the nature of the time-
independent solutions to the stellar evolution equation at the neutron star envelope,
which forced us to consider the proper time evolution of the system. For that part of
the work, we employed an open-source code which is suitable for simulating neutron
star envelopes, given its similar nature as with the standard stellar matter. For
simulating the whole star, however, such publicly available code is not suited due
to the absence of the high-density sector and fully general relativistic effects. In
the process of studying time-dependent envelopes - and of course, of contrasting our
stationary state results - additional properties of the envelope were re-examinated.
For instance, given some stationary states predict Lb < 0, a quite unusual value
for the standard works on the subject, we explored for their physical meaning and,
naturally, whether they occur as part of time-dependent simulations or not.
While the present work is devised as a compilation of research papers, it is best
to provide the essential aspects of the context surrounding the results which we
present in the compilation. Therefore, Chapters 1 to 4 are devoted to cover these
essential aspects: in the first one, we summarize the equations of structure and
thermal evolution governing the neutron star in the presence of accretion and, in
particular, the surface. In Chapter 2 we analyze the microphysics of the neutron
star envelopes. This is, concepts such as the Equation of State (EOS) or the nuclear
reactions. Chapter 3 provides a brief summary on the nuclear networks employed in
the research papers, as well as their physical justification. In Chapter 4 we wrap-up
the content of the previous chapters into the study of neutron star envelopes, both
in time dependent and independent scenarios. In Chapter 5 we provide a brief word
summarizing the whole research, and the perspectives for the future.
On the numerical codes in the present work
All the contents of this work, related to the evolution of accreting neutron star
envelopes, were obtained from different numerical codes, enlisted below:
• TOV code. Developed by the author of this work, solves the relativistic struc-
ture equations for a neutron star (TOV), as well as the stationary equation for
quadrupolar deformation (see for instance [Hin08]), given as input an EOS in
tabulated form. The output from this private code was employed in [NCPB23]
for the GR structure.
• Developed code. In order to simulate one zone models and stationary neutron
star envelopes. The driver routines for adjusting the parameters, the subrou-
12
tines to implement the EOS, opacity and reaction rates were designed by the
author of this work, while subroutines related to numerical methods for solving
equations (as for instance stifbs) were taken from [PTVF92].
• MESA (Modules for Experiments in Stellar Astrophysics). Public code which
solves the time-dependent stellar evolution equations in one spatial dimension
[PBD+11, PMS+15]. Let us stress that MESA is not designed to evolve
complete neutron stars. The capabilities of this code extend to up to
ρ ≈ 1010 g cm−3, i.e. at the envelope-outer crust boundary, where relativistic
corrections of the form e±Φ are still viable.
• NSCool. Developed by Dr. Page and collaborators [Pag16]. This public code
is designed to model one-dimensional thermal evolution of isolated neutron
stars. In the presence of accretion, this code implements an approximate
treatment of the heating sources as Qheat ∝ Ṁ , with Ṁ the mass accretion
rate, and considers the envelope is described via the standard Tb−Teff relations
for non-accreting envelopes [BC09, PHN+22]. As work in progress is the im-
plementation of the results from the present thesis into NSCool, thus allowing
it to handle thermonuclear burning as self-consistently as possible.
13
Chapter 1
Equations of Stellar Evolution
“And yet [...] there are
mathematicians of extraordinary
genius who doubt the whole
universe [...] was created only
according to Euclidean geometry
[...]”
F. Dostoievsky
Abstract. In this chapter the differential equations governing the neutron
star under mass accretion - and specifically the envelope - are introduced and
discussed.
1.1 Introduction
The modeling of stellar evolution is possible through the fluid approximation, which
consists on following the spacetime evolution of a set of physical quantities A (such as
pressure, temperature and composition), characterizing the state of a fluid element,
i.e. a blob of ≳ 1023 particles, on a spatial scale much larger than the intrinsic
size of the constituent particles, which for a good approximation can be assumed as
their Compton wavelength, λc = h/mc. An advantage of this approximation is the
possibility to separately construct relations, to some extent regarded as functionals,
among the physical quantities. These are further referred to as microphysics, and
the actual values of them as locals since they are calculated and measured on an
inertial frame. As important examples of these relationships we can mention the
Equation of State (EoS), which relates the pressure, temperature and density, and
the opacity, which is a measure of how much energy photons lose when passing
through matter and depends on temperature and density.
14
To propagate A along spacetime, a set of equations compatible with general
relativity is needed. This is the task of section X, where we introduce and discuss
in detail this set of partial differential equations. A review on the indispensable
concepts and mathematical framework occupies the first subsection.
This chapter introduces the basic set of equations which are employed to describe
thermal evolution of compact objects. In this work, we stick to the metric convention
(−,+,+,+). We denote 3 and 4-vectors by V and V⃗ respectively. Similarly, we
reserve ∇µ to denote the 4-divergence.
As all the systems discussed in this work obey the conservation of baryons, for
any physical variable of interest X we adopt the following conventions: x̃(= X/Nb)
denotes X per baryon, x(= X/V ) the X-density, and x̆(= x̃/mu) the specific-X, i.e.
X per unit mass. As baryons is a surrogate name for the mixture of protons and
neutrons, it is appropriate to regard the atomic mass unit mu as the baryonic mass
since mp ≈ mn ≈ mu. Recalling the value of Avogadro’s number
NA = 6.02214076 × 1023 particles mol−1 , (1.1)
the atomic mass unit mu (or amu for simplicity) is thus defined as the 12th part of
the ratio of 12 g of 12C and 6.022×1023, which is the amount of atoms in a mole of
this ion:
mu =
1 g
6.022 × 1023 particles
=
1 g mol−1
NA
. (1.2)
1.2 Spacetime evolution of physical quantities
So far, the best theory for describing self-gravitating objects is General Relativity
(GR) [Wal84], [MTW17], which serves as a generalization of Newton’s Mechan-
ics. A geometric theory by definition, GR postulates the structure of spacetime
is completely determined by the components of a metric tensor gµν , related to the
line-element of a coordinated system by ds2 = gµνdx
µdxν , influenced by a collection
of local mass-energy fields Ψ exerting mechanical work. From a modern point of
view, the PDEs governing all these fields can be deduced from variational principles
over the associated action functional
S =
1
c
∫
d4x
√−g [Lgrav(gµν) + Lmatter(gµν ,Ψ)] . (1.3)
Indeed: from δSGR/δgµν = 0, we obtain the following set of non-linear PDEs, com-
monly referred to as Einstein’s Field Equations
Gµν =
8πG
c4
T µν . (1.4)
The left-hand side, the Einstein tensor, is purely geometric and thus involves gµν and
their partial derivatives. By construction, it satisfies ∇µG
µν = 0 and Gµν = Gνµ,
15
thus implying the existence of 10 independent PDEs in 4 dimensions. On the right
hand side we have several constants allowing to recover the Newtonian equations in
the non-relativistic limit: G, the Cavendish constant of gravitation, and c the speed
of light in vacuum. The tensor appearing in the right hand side is defined as the
energy-momentum tensor
Tµν = − 2√−g
δ(
√−gLmatter)
δgµν
, (1.5)
and by definition contains the contributions from the energy-matter fields Ψ. Due
to the the enumerated properties of the Einstein tensor, Tµν = Tνµ and
1√−g∂µ
(√−gT µν
)
= 0, (1.6)
an identity which yields four additional PDEs, for Ψ’s spacetime evolution. While
the original formulation of GR demanded T µν to solely contain fields exerting me-
chanical work [Wey22], a role heat fluxes clearly do not fulfill, more recent works
on the subject have relaxed this condition in order to self-consistently incorporate
the First and Second Laws of Thermodynamics into the grand scheme of GR, an
extension not necessarily free of controversies [GS06]. While in principle we can sep-
arate T µν as a sum T µνmechanical + T µνheat, from the classic point of view only T µνmechanical,
not the sum, satisfies Eq. 1.6, a vision we shall adopt in order to later incorporate
thermodynamical quantities into the fluid description.
The most frequently used model to describe stellar energy-matter is the perfect
fluid, an inviscid system with zero exchange of entropy, moving with unitary 4-
velocity uµ (i.e. uµu
µ = −1, or equivalently Uµ = cuµ). Its corresponding energy-
momentum tensor is
Tµν = εuµuν + PΠµν , (1.7)
where Πµν = gµν + uµuν is the orthogonal projection tensor satisfying uµΠµν = 0,
ε is the total energy density and P the thermodynamical pressure. These scalars
are interconnected via the Equation of State, a functional relating all thermody-
namical degrees of freedom with P . For instance, in the non-relativistic limit, P
is typically related to the temperature T 1 and the individual number densities ni
(i ∈ ¶1, . . . , Nspecies♢) for multi-species systems. On the other hand, for degenerate
fermionic systems (i.e. where T/TF ≪ 1, with TF the Fermi temperature) P be-
comes a functional of the total density ρ := ε/c2 only, i.e. P (ρ). Finally, for some
discussions in the latter example it is useful to consider both P and ε as explicit
functionals of the baryon number density nB, i.e. P (nB) and ε(nB). This latter
1T is also usually employed as the notation for the trace of T µν . While this quantity is not
employed in the present work, whenever potential confusions might arise we shall explicitly state
what we are denoting by T .
16
quantity plays an important role in the description of stellar matter, as in the non-
relativistic limit ρ ≈ ρB, with ρB ≈ munB. Since experimental evidence support the
notion of conservation for this quantity, in General Relativity we must also include
this fact via a covariant identity,
1√−g∂µ
[√−gnBu
µ
]
= 0 , (1.8)
an expression which can also be expressed, via the Gauss theorem, as the conserva-
tion of a “charge” on a Cauchy hypersurface Σ, the total number of baryons:
NB =
∫
Σ
d3x
√
γ nBkµu
µ, (1.9)
with kµ a unitary normal 4-vector to Σ. Another important property from Eq. 1.8 we
must emphasize is that high-density environments tend to have low fluid velocities.
Conceptually, n
(1)
B uµ(1) ∼ n
(2)
B uµ(2), hence if n
(1)
B > n
(2)
B we have ♣uµ(2)♣ > ♣uµ(1)♣ for all
µ, i.e. the fluid moves faster in low-density environments than in high-density ones.
Roughly speaking, we thus have v ∝ n−1
B , with v the norm of the 3-velocity field.
In spite of its simplicity, Eq. 1.7 has earned his place as the workhorse by excel-
lency for studying stellar structure, a situation we can easily understand by analyzing
the projections of the 4-vector expressing its conservation, Eq. 1.6. The first is along
the fluid 4-velocity, uµ∇νT
νµ = 0. Taking into consideration Eq. 1.8, we obtain
uµ∂µε =
ε+ P
nB
uµ∂µnB, (1.10)
closely resembling the First Law of Thermodynamics in the absence of terms related
to entropy exchange and variations in the individual species’ number densities, i.e.
either changes in entropy are zero or T → 0, and dni → 0.
On the other hand, the contraction Παβ∇µT
µα = 0 leads to
(ε+ P )aα = − [∂αP + uαu
µ∂µP ] , (1.11)
where we identified the 4-acceleration,
aα = uµ∇µuα. (1.12)
Eqs. 1.11 are usually referred to as momentum equations [MTW17] since they gen-
eralize the non-relativistic behavior for ρu. Notice enthalpy per unit volume, ε+P ,
serves as the relativistic generalization of the matter density, indicating pressure
plays an important role in fully-relativistic systems. Apropos of this statement, let
us examine this non-relativistic limit, where ♣u0♣ ≫ ♣ui♣ and ε ≫ P . For the spatial
components we have
ε
[
u0∂0uj + ui∂iuj + u0Γ0
0ju0 + O(u0ui)
]
= −
[
∂jP + O(u0ui)
]
ε
[
u0∂0uj + ui∂iuj + u0Γ0
0ju0
]
≈ −∂jP.
17
Identifying Γ0
0j = ∂j(Φnr/c
2), with Φnr the non-relativistic “gravitational poten-
tiall”, we obtain
ε
[
1
c2
∂vj
∂t
+
1
c2
vi∂ivj − 1
c2
∂jΦnr
]
≈ −∂jP,
ρ
[
∂t + vi∂i
]
vj ≈ ρ∂jΦnr − ∂jP,
which is the standard form of Euler’s equation. Here, we see the Christoffel symbols
give rise to the usual contribution due to the gradient of the gravitational field, while
the “kinetic” term, uν∂νuj, encloses the usual variation in time of uj but also the
contribution from the so-called ram pressure, ρvi∂ivj. Given this construction, by
looking back to Eq. 1.11, taking into consideration the explicit form of aα, Eq. 1.12,
we see the relativistic analogous of the ram pressure is given instead by (ε+P )ui∂iuj.
For the majority of applications, it is safe to neglect this term as is small in com-
parison with either temporal variations of the 4-velocity or the “gravitational field”
enclosed in the Christoffel symbols.
As EFE were envisioned to describe the structure of self-gravitating objects the
next, natural question is: how can we analyze thermal degrees of freedom within
GR? The first step is to recall in the non-relativistic (i.e.Newtonian) regime, for
each member of fluid’s collection of physical variables A we can associate a single
or a collection of 4-currents, for simplicity denoted as J µ
A , obeying
∂µJ µ
A = ΓA , (1.13)
where the right-hand side can be interpreted as a source or sink of A according to
sgn ΓA, the particular case ΓA = 0 defining the the conservation of A. Notice the
scalar character of the left-hand side demands all contributions on the other side
to be scalars as well. By inspection, Eq. 1.13 is valid within Special Relativity due
to its invariance with respect to the coordinated system, feature which allows its
incorporation into GR as long as the partial derivatives are promoted to covariant
ones,
1√−g∂µ
[√−gJ µ
A
]
= ΓA , (1.14)
as we have already seen exemplified by the conservation of baryon number density,
Eq. 1.8, expressing the zero divergence of a 4-vector, but also with Eq. 1.6, a 4-vector
resulting from the conservation of a tensor, T µν . Since Eq. 1.14 automatically sat-
isfy the requisites of being covariant and to preserve causality, within GR 4-currents
must play a critical role in describing the spacetime evolution of A. There are, how-
ever, further requisites “the correct”, fully relativistic version of Thermodynamics
must fulfill: expressions for the 4th Laws, causal propagation of heat fluxes and the
absence thereof within the non-relativistic regime (e.g. Fourier’s law).
To date, one of the most successful attempt for constructing Relativistic Thermo-
dynamics was introduced by [Isr76, IS79]. At its core, this model promotes several
18
thermodynamical quantities of interest to 4-vectors and tensors and exploits the
covariant character of Eq. 1.13 to propose a generalized version of the 1st Law of
Thermodynamics employing the classic Gibbs’ differential for a system composed of
multiple species,
dSµ =
∑
i
αidN
µ
i + βνdT
νµ. (1.15)
In this differential, Sµ and Nµ
i are the entropy and i-th species’ number density
4-vectors, αi are the generalized chemical potentials, T νµ is the complete energy
momentum tensor and βµ = T−1uµ defines the inverse temperature Killing 4-vector,
where T is the temperature measured at equilibrium. Conceptually, this relativistic
Gibbs relation is expected to be valid at both reversible and irreversible conditions
since all tensorial quantities admit an expansion of the form Tequilibrium + Tirreversible,
where Tequilibrium is typically related to the fluid’s 4-velocity uµ, and the irreversible
part Tirreversible is a sum of first, second or n-th order contributions, albeit for the
majority of applications is sufficient to retain up to first order.
As a competing formulation for relativistic thermodynamics, we must mention
the variational approach of [Car91], as well as related works on the subject concern-
ing the equations for a perfect fluid [Bro93], [KB97], [Sch70] (see also [LA11] for
an excellent review on this subject). The advantage of such scheme is the easiness
on the incorporation of additional fluid components, as well as couplings among
them as in the superfluid scenario. In spite of their apparent differences, both Is-
rael’s and Carter’s approaches are in good correspondence up to second order in
deviations from equilibrium [Pri91], although both formalism differ when significant
dissipation takes place. This similarity also extends to the heat transport, also re-
ferred to as relativistic Cattaneo equation as in the non-relativistic regime we expect
τ Ḟ + F = −K∇⃗T , expression with finite-speed propagation of heat. From the pre-
dictions of both theories, it becomes easy to see Fourier’s law can still be kept as
“correct” within relativistic and astrophysical systems as long as the flux is steady
on a thermal timescale [LA11], [LA18].
In spite of both models’ success on constructing causal thermodynamics, as well
as in their similarities under thermal equilibrium, the most prominent sources of
controversies are the apparent excess of parameters needed for both theories and
their physical interpretation, the non-mechanical terms appearing in the energy-
momentum tensor and possible instabilities due to the presence of 4-acceleration
[GS06]. While the source of these observations do not seem to be impediments for
regarding Israel & Stewart and Carter theories as correct, in particular as some
terms admit conceptual justifications [LA11], [Bro93], there exists a strong objec-
tion not only for these approaches, but for all those attempts to relativistic ther-
modynamics: the “correct” transformation law for actual temperature. This is even
more accentuated in the expressions found in both physical and astrophysical litera-
ture where, apparently, the relationship between observed and local temperatures is
T∞ = eΦTlocal, i.e. objects are colder when observed at an infinite distance, an affir-
19
mation more closer to the Einstein-Planck point of view than to Ott’s or Landsberg
[DHH09], [Bec16], [FPM17]. This “transformation law”, as we explain later in the
text, remains valid as long as one is interested in determining the temperature asso-
ciated with the peak in frequency of the Planck function. However, the postulation
of an entropy 4-current and the existence of the 4-momentum, in combination with
the First Law of Thermodynamics, lead us to the conclusion that it is safer to as-
sume the actual temperature as a scalar function. This point of view is supported by
numerical simulations of particle interactions in the relativistic regime [CCPD+07],
[FPM17] where, apparently, the definition of a frame-independent temperature is
within reach. However, we advise the reader to notice that such treatment of tem-
perature strongly depends on the set of initial assumptions, a neutral point of view
which is valid as long as controversies over the correct formulation of relativistic
thermodynamics exist [FPM17].
Let us now describe Israel & Stewart approach for relativistic thermodynamics
in terms of 4-divergences, taking uµ as the 4-velocity of the fluid element. Taking T
as the temperature measured in a local frame, βµ = T−1uµ defines a Killing vector
(e.g. ∇µβν = −∇νβµ) taken as the inverse of temperature. From EFEs, we already
know the energy-momentum obeys a conservation law, Eq. 1.6. In this regard, we
also know the baryons obey a conservation law, Eqs. 1.8 and 1.9. For systems
composed of Nspecies different species, it is necessary to attach a 4-current to each
one of them, Nµ
i := niU
µ
i +Qµ, together with a corresponding PDE of the form
1√−g∂µ
[√−g niUµ
i
]
= Gi , (1.16)
[Gi] = Baryon Volume−1 Time−1, where the non-equilibrium term has been absorbed
into the contributions from the right-hand side to emphasize Gi expresses the net
rate at which all other species contribute with creation, annihilation, scattering or
diffusion for the i-th species, which in general are functionals of temperature, rest-
mass density or the individual number densities. In general, Uµ
i ̸= Uµ, a difference
which can be accentuated in non-vacuum environments where the background den-
sity determines which species fall at faster rates than others2. However, at supersonic
velocities encountered as for instance in typical surfaces of neutron stars it is a valid
approximation to consider Uµ
i ≈ Uµ ∀i ∈ ¶1, . . . , Nspecies♢ in the left-hand side of
Eqs. 1.16. Diffusive effects can thus be treated as corrective terms and included as
a source in the right-hand side. An important consequence of this approximation is
the possibility to write Eqs. 1.16 in terms of abundances Yi := ni/nB and, employing
Eq. 1.8, obtain a scalar-like equation for each species,
Uµ∂µYi = Rlocal
i + Rdiffusive
i , (1.17)
with Rx := Gx/nB for x =local, diffusive, and where, for further convenience, the
contributions in the right hand side have been split as local if they solely depend on
2Terrestrial experiments are a quick corroboration of this assertion
20
variables such as nB, T or Yi, and diffusive if they include first or second derivatives
of any Yi. From an astrophysical point of view, Rlocal
i represents the variations in
abundances due to reaction processes, in particular, of nuclear reactions. Finally,
we must note in environments where ρ ≈ ρB holds, Eqs. 1.17 are constrained by the
mass-conservation condition, i.e.
Nspecies
∑
i
AiYi =
Nspecies
∑
i
Xi = 1, (1.18)
with Ai the number of baryons for the i-th species and Xi := AiYi denotes the mass
fraction.
The next quantity of interest is the entropy 4-current, Sµ = nBs̃U
µ +Qµ
s , where
the second term denotes all non-equilibrium contributions. While the corresponding
PDE is again of the form in Eq. 1.14, the presence of nBU
µ in Sµ allows to employ
the conservation of baryon density, Eq. 1.8, to simplify this equation into
nBU
µ∂µs̃ = Gs ≥ 0 , (1.19)
where the contribution due to ∇µQ
µ
s has been absorbed in the right-hand side term
Gs, referred to as the net entropy production rate. In order to generalize the Second
Law of Thermodynamics this term must be non-negative, hence the restriction in
Eq. 1.19. If Gs > 0 then we say our system is irreversible, and whenever the equality
holds we have a reversible one. From conceptual grounds, it is useful to explicitly
categorize all contributions to Gs as heating or cooling mechanisms, i.e. Gs = Gheat -
Gcool, in order to understand reversibility as the limiting scenario when the net rates
of heating and cooling balance each other, Gheat = Gcool. Within the astrophysical
context, as heating mechanisms we can mention the energy release from nuclear
reactions, which tends to increase the temperature of the system, or the compres-
sion of magnetic fields. For cooling mechanisms, on the other hand, we have the
emission of photons, described by radiation and conduction fluxes, and the escape
of neutrinos due to its small cross section for interactions with baryonic matter. As
a consequence of Eq. 1.15, the general relativistic version of all these processes must
be described either by variations in the species’ densities Nµ
i or the changes in the
energy-momentum tensor for the irreversible case. From these considerations, let us
write Gs = Gs,N + Gs,T and separately analyze each term.
Albeit neutrinos cannot be handled in the same foot as any of the other species
in the mixture being out-of-thermal equilibrium with the rest of the system, con-
ceptually its contribution to Gs resembles a variation in an hypothetical abundance
21
Yν . Thus
Gs,N :=
ρB
T
[
˙̆εspecies − ˙̆εν
]
,
˙̆εspecies = ˙̆εnuc + ˙̆εdiff
˙̆εx = −
Nspecies
∑
j=1
µ̆jRx
j , x ∈ ¶local, diffusive♢ ,
with µ̆j the specific chemical potential per unit baryon for the j-th species. As
previously stated, within the astrophysical context the x = local term can be taken
as the definition for the energy generation rate due to nuclear reactions, as long as
neutrino losses due to electroweak interactions are included in ˙̃εν . Since Rlocal
j is
typically < 0, the minus sign at the front allows a positive energy generation rate.
Regarding Gs,T , we have the contributions due to heat to the energy momentum
tensor. To incorporate energy diffusion due to radiative and conductive processes,
we follow the approach of Tauber, Weinberg, Eckart and Misner [Eck40], [MS69],
which is correct to first-order within the context of Relativistic Thermodynamics,
despite some differences on notation [Isr76]. Let F µ be the flux 4-vector resulting
from the addition of radiative and conductive contributions, satisfying F µFµ ≥ 0
and F µuµ = 0, i.e. heat propagates in a perpendicular direction to the motion of the
fluid element. Defining a radiation energy-momentum tensor T µνrad = F µuν + F νuµ,
it is immediate to write a frame-invariant rate of energy generation [MS69],
Gs,T = T−1uν∇µT
µν
rad
= −T−1 [∇µF
µ + F νaν ] . (1.20)
By demanding a positive-definite rate, as well as a first order expression in temper-
ature gradient ∂νT and 4-acceleration aν , Eq. 1.12, it is possible to deduce
F µ = −KΠµν [uα∇α(Tuν) − uα∇ν(Tu
α)] (1.21)
= −KΠµν [∂νT + T aν ] . (1.22)
where K is the usual thermal conductivity, expected to reproduce its non-relativistic
form 16σSBT
3/3κρB (but consequently neglecting the possible presence of anisotropies
due to magnetic fields) and Πµν is the orthogonal projection tensor already intro-
duced. To complete our discussion of radiation, we introduce the luminosity, energy
per unit time carried away by photons. For some authors [Eck40], [MTW17], as the
stellar structure is frequently regarded as purely radial the luminosity is frequently
traded by 4πr2F , with F the absolute value of FµF
µ, or by its simplified form
4πr2√grrF r. Save for r2, this term is invariant under the exchange of coordinates,
but limits the angular dependence to be constant. Instead, we adopt as definition
of luminosity the natural
L := −
∫
Σ
d3x
√
γ uν∇µT
µν
rad, (1.23)
22
which can be shown to recover the well-known relation L = 4πr2F for spherically
symmetric systems (see Proposition 7). In summary: Eq. 1.19, considering the
contributions from variations of species’ densities and radiation, can be written as
nBTU
µ∂µs̃ = ρB
[
˙̆εspecies − ˙̆εν
]
− [∇µF
µ + F νaν ] . (1.24)
On the temperature gradient. Due to the Equivalence Principle, since both
T and P are local functionals (or scalar functions), in principle we can still adopt
the non-relativistic definition for the actual temperature gradient ∇T , i.e.
∂T
∂s
:= ∇T
T
P
∂P
∂s
, (1.25)
with s a spatial coordinate (such as the radial direction r), and apply the same
formalism of standard stellar evolution to determine whether ∇T = ∇r or ∇T =
∇conv, with ∇r and ∇conv the radiative and convective gradients respectively:
∇T =



∇r, ∇ad > ∇r
∇conv, ∇ad < ∇r,
(1.26)
with the adiabatic gradient ∇ad, provided by the EoS. Notice this compels us to find
an actually local expression for ∇r, and to adopt any of the existent treatments for
convection to compute ∇conv whenever radiation exceeds the adiabatic contribution.
1.3 Evolution in a static and spherically symmet-
ric (SSS) spacetime - fluid in radial motion
Employing spherical coordinates (ct, r, θ, ϕ), the line element of this spacetime is
ds2 = −e2Φ(r)c2dt2 + e2Λ(r)dr2 + r2dθ2 + r2 sin2 θdϕ2 . (1.27)
Being part of the g00 component of the metric, and thus related to variations in time
at fixed spatial coordinates, eΦ(r) is referred to as the redshift factor, or redshift for
shortness. While eΛ does not have a specific name, in vacuum it fulfills the condi-
tion eΛ(r) = e−Φ(r), usually referred to as the Schwarzschild metric as it describes
the exterior spacetime of a black hole, and even more generally, of any spherically
symmetric object such as a neutron star or the Sun itself. For a complete list of
factors related to this metric (determinant, Christoffer symbols), see Appendix B.2.
When discussing vectors in this spacetime, it is customary to distinguish between
local, i.e. as measured by an orthonormal frame of reference, and at infinity quanti-
ties. Since the SSS metric is diagonal, we can easily construct an orthonormal frame
of reference: denoting such coordinates with hats, the relationship among basis are
eα̂ =
1
√
♣gαα♣
eα (1.28)
23
for α ∈ ¶ct, r, θ, ϕ♢ and where no summation is intended for these indices. As any
4-vector admits an expansion in terms of both basis, we have V⃗ = V µeµ = V µ̂eµ̂.
Consequently, the relationship between the components is V µ̂ =
√
♣gµµ♣V µ, where
once again no summation is intended for the repeated indices. Thus, we formally
identify local quantities as
Vlocal = V µ̂ =
√
♣gµµ♣V µ . (1.29)
As we expect these variables to represent a measurable physical quantity, a non-
orthonormal (or non-inertial) observer can only attain physical meaning if these
variables are propagated along a timelike curve. Thus, the quantities measured at a
distance, or simply measured at infinity, are related to the local ones as
V ∞ =
√
♣g00♣Vlocal . (1.30)
It is important to notice this “transformation law” is valid for vector components.
From the followed steps, we can easily see Eq. 1.28 defines the “transformation law”
for covector components.
To exemplify these concepts, let us examine three important scenarios. First of
all, for massive particles we know its line element can be also expressed as ds2 =
−c2dτ 2, where τ is defined as the proper time of the particle. Notice τ plays the
role of an orthonormal coordinate, so we can identify it as the local measured time
tlocal = τ . Now, due to the invariance of the line element, for events occurring at a
fixed location in space, i.e. dr = dθ = dϕ = 0, we have an explicit relation between
tlocal and the time measured by the t coordinate, t∞ = e−Φtlocal, hence t is the time
measured by an observer at infinity, as expected by the covector “transformation
law”, Eq. 1.28.
Now, let us study the energy for a single particle. Let p⃗ denote its 4-momentum,
p0 having units of Energy divided by c. An observer in an inertial frame of reference
seeking to measure the energy of such particle, according to the general expressions
given above, will measure Elocal = eΦp0c. On the other hand, an observer at infinity
will measure an energy E∞ = eΦElocal = e2Φp0c = −p0c, in agreement with the
“conservation of energy” arising from the time-independency of the SSS metric (see
for instance [Sch09, MTW17]). A typical application of this analysis is the case of
photon’s frequencies, in particular those coming from Wien’s displacement law, the
maximum effective temperature assuming a blackbody radiation: considering the
standard relations E = hν and νpeak = c1Teff, with c1, h and kB constants, (the
latter known as Planck and Boltzmann constants, respectively), we have
ν∞ = eΦνlocal (1.31)
T∞
eff = eΦTlocal,eff . (1.32)
24
The final scenario for consideration is the ratio of physical quantities, such as the
the luminosity, i.e. energy per unit time. Employing the identities so far deduced,
we have
L∞ =
E∞
t∞
=
eΦElocal
e−Φtlocal
= e2ΦElocal
tlocal
= e2ΦLlocal, (1.33)
i.e. the luminosity at infinity is twice redshifted with respect to the locally measured
value.
Before leaving this discussion, we must stress out that, despite the usefulness
of the local-at infinity relations so far introduced, attempts to generalize the stellar
structure and thermal evolution equations from Newton to Einstein formalism via
attachment of redshift factors has a limited range of applicability. For instance,
for fluids at rest and constraining the system to remain spherically symmetric, i.e.
most quantities depending on ct and r. As we shall see below, however, not in all
scenarios it is possible to construct such promotion.
Fluid in radial motion. Let us now deduce the aspect of the normalized 4-
velocity of a fluid element moving along the radial direction in the SSS metric3.
Due to the diagonal character of this metric, we can easily identify orthonormal
coordinates dt̂ = eΦdt, dr̂ = eΛdr. Let v be the 3-velocity measured by a local
observer at a coordinate r, e.g. v = dr̂/dt̂ = eΛ−Φ dr
dt
. Denoting the proper time by
τ , the general expression for the components of the fluid 4-velocity vector U⃗ is
(U ct, U r, U θ, Uϕ) =
dt
dτ
(
c, veΦ−Λ, 0, 0
)
. (1.34)
To obtain the front factor, we consider the normalization restriction, UµU
µ = −c2.
This leads to
dt
dτ
= γSe
−Φ, (1.35)
γS =
[
1 − v2
S
]−1/2
, (1.36)
where γS denotes the traditional Lorentz factor, vS = v/c, and we added a subindex
S for the SSS metric in order to avoid confusion with other γ’s and v’s. Thus, the
desired expression for the normalized 4-velocity uµ = Uµ/c is
(uct, ur, uθ, uϕ) = γS
(
e−Φ, e−ΛvS, 0, 0
)
. (1.37)
Eq. 1.37, upon the explicit replacement v → v(t, r), can be adopted as the definition
of the normalized fluid 4-velocity field in spherical coordinates. Notice the only
3In order to reduce the amount of mathematical details, many of the expressions employed in
this part are given without proof. For further details on their origin and intermediate steps, see
Appendix B.
25
non-zero components of the orthogonal projection tensor are:
Π00 = e−2Φγ2
Sv
2
S (1.38)
Πrr = e−2Λγ2
S (1.39)
Π0r = γ2
Se
−Φ−ΛvS. (1.40)
For further reference, it is useful to note the projection of the 4-gradient along uµ
and Πµν are
uµ∂µ =
γSe
−Φ
c
[
∂t + veΦ−Λ∂r
]
, (1.41)
Dt := e−Φ∂t + ve−Λ∂r (1.42)
Π0ν∂ν = γ2
Se
−2ΦvS
[
vS
c
∂t + eΦ−Λ∂r
]
, (1.43)
Πrν∂ν = γ2
Se
−Λ−Φ
[
vS
c
∂t + eΦ−Λ∂r
]
, (1.44)
where Dt can be identified in orthonormal coordinates t̂, r̂ as the Lagrangian deriva-
tive. From its coordinates, and the Christoffel symbols for the SSS metric, it is
straightforward to see the fluid 4-acceleration a⃗, whose components (see Prop. 5)
are given by
a0 = u0∂0u0 + ur∂ru0
= γ2
SvS
{
γSe
Φ
c
uµ∂µv − eΦ−ΛdΦ
dr
}
(1.45)
= γ2
SvS
{
γ2
S
c2
[
∂tv + eΦ−Λv∂rv
]
− eΦ−ΛdΦ
dr
}
(1.46)
ar = u0∂0ur − u0∂ru0 = −u0
ur
a0 (1.47)
= γ2
Se
Λ−Φ
{
−γSe
Φ
c
uµ∂µv + eΦ−ΛdΦ
dr
}
(1.48)
aθ = 0 (1.49)
aϕ = 0. (1.50)
Let us examine now the continuity equation. By direct computation, there are
several forms we can express it, each one useful in its on way:
∂t(nBγS) +
1
r2eΛ
∂r(e
Φr2nBγSv) = 0 (1.51)
[∂t + eΦ−Λv∂r](nBγS) +
nBγS
r2eΛ
∂r(e
Φr2v) = 0 (1.52)
[∂t + eΦ−Λv∂r]nB − γ2
SnBv
c2
[∂t + eΦ−Λv∂r]v +
nB
r2eΛ
∂r(e
Φr2v) = 0. (1.53)
26
The second form of the continuity equation, Eq. 1.52, results useful for a quick
analysis within Minkowski spacetime, where both eΦ and eΛ approach unity. Taking
the limit γS → 1 we directly recover the traditional form of the continuity equation,
i.e. ∂tnB + u · ∇⃗nB + nB∇⃗ · u = 0. A more careful approach, within a Special
Relativistic context, can be extracted from Eq. 1.53 as we have explicitly stated the
action of the partial derivatives over the velocity field v(r, t).
Assuming time-independency of both nB and v, the first of these expressions,
Eq. 1.51, implies the existence of a conserved quantity, the redshifted mass accretion
rate Ṁ∞ := −4πr2eΦmunBγSv, which encloses both special and general relativistic
effects via γS and eΦ respectively. Notice we have chosen a negative sign for the
definition as for accreting matter we expect v < 0. To consider the impact of these
terms over the surface of a typical neutron star, let us fix munB ∼ 10−3 g cm−3,
the gravitational mass and radius as m∗ ∼ 1.4M⊙ and r∗ ∼ 12 km respectively,
thus having eΦ ≈
√
1 − 2Gm∗
c2r∗
≈ 0.80 and the extreme case Ṁ∞/4πr2
∗ = 8.8 × 104
g cm−2, i.e. the Eddington rate of accreted mass per unit time and surface. These
lead to γSvS ∼ 3.67 × 10−3, i.e. vS ∼ 3.66 × 10−3. On the other hand, if
Ṁ∞/4πr2
∗e
Φ = 8.8×104 g cm−2, we obtain γSvS ∼ 2.94×10−3, i.e. vS ∼ 2.94×10−3.
Consequently, the redshift factor has a slightly more prominent impact over v than
the SR correction due to γS, at least as long as ρ ≥ 10−3 g cm−3. Due to the
smallness of vS, in this density regime it is safe to set γS ≈ 1. Furthermore, from
the continuity equation we also expect v to become progressively smaller than its
surface value as we move inwards since v ∝ ρ−1. From these considerations, we thus
adopt the
First-order approximation (FOA): Given vS ≪ 1, γS → 1 and all vS/c and v2
S
terms will be considered as small perturbations and thus neglected.
Within the FOA, the baryon conservation (continuity equation) we adopt is a re-
duced version of Eq. 1.53, namely
∂tρB − e−Λ
4πr2
∂rṀ
∞ = 0, (1.54)
with ρB = munB and
Ṁ∞ = −4πr2eΦρBv (1.55)
as the redshifted mass accretion rate.
In what follows, we employ the general expressions for the physical quantities
of interest, deduce the general expressions for the complete 4-velocity uµ, Eq. 1.37,
and then apply the FOA.
Einstein Field Equations. For the analysis of these and subsequent equations,
27
it is useful to introduce the gravitational mass function
m(r) :=
c2r
2G
(1 − e−2Λ(r)). (1.56)
For the SSS metric, the only non-vanishing components of the Einstein tensor are
G00, Grr, Gθθ and Gϕϕ. Due to the symmetries of the problem, only the first two
are necessary for the discussion, hence we shall only need T00 and Trr from the
energy-momentum tensor. Given
G00 =
2Ge2Φ
c2r2
dm
dr
Grr = −2Gme2Λ
c2r3
+
2
r
dΦ
dr
T00 = γ2
Se
2Φ
[
ε+ v2
SP
]
Trr = γ2
Se
2Λ
[
P + v2
Sε
]
,
from EFE we obtain two differential equation, one for the gravitational mass of the
star and the other for the redshift function,
dm
dr
=
4πr2
c2
γ2
S
[
ε+ v2
SP
]
(1.57)
dΦ
dr
=
Gre2Λ
c4
[
mc2
r3
+ 4πγ2
S
(
P + v2
Sε
)
]
. (1.58)
For v ̸= 0, we see the 3-velocity provides additional contributions to ε in the grav-
itational mass case, and to P for the redshift function. However, within the FOA
these can be neglected, and we thus recover the standard set of equations for the
v ≡ 0 case,
dm
dr
= 4πr2ρ (1.59)
dΦ
dr
=
geΛ
c2
G, (1.60)
where, following Thorne et al, we have introduced some factors for further conve-
nience:
g :=
GmeΛ
r2
(1.61)
G :=
[
1 +
4πPr3
mc2
]
. (1.62)
28
Euler equations. Considering the only nonzero components of the 4-acceleration
are a0 and ar, as well as the normalization condition u0u0 + urur = −1, we have
(ε+ P )a0 = ur [ur∂0P − u0∂rP ]
(ε+ P )ar = −u0 [ur∂0P − u0∂rP ] ,
in agreement with the functional relation between ar and a0, Eq. 1.47, indicating
only one equation is actually independent. Thus, it is necessary to solve only one
equation, for simplicity the first one. Plugging the expanded expression for a0, given
in Eq. 1.46, we obtain
(ε+ P )
{
γ2
Se
Λ
c2
Dtv − dΦ
dr
}
=
1
γ2
S
∂rP +
vSe
Λ
c
DtP. (1.63)
Several particular cases can be identified from Eq. 1.63:
• Taking v ≡ 0, we recover the well-known hydrostatic equilibrium relation,
∂P
∂r
= −Hc2dΦ
dr
(1.64)
H = ρ+
P
c2
. (1.65)
• For v ̸= 0 and under the FOA we have
HeΛ ¶Dtv − gG♢ = ∂rP. (1.66)
In the left-hand side we have a competition between fluid’s 3-acceleration,
Dtv, and the gravitational acceleration gG. At the surface of neutron stars,
this second quantity has an order of magnitude ∼ 1014 cm s−1. If we consider
accretion at the Eddington rate, we have seen v ∼ 10−3c ≈ 107 cm s−1 near
the surface. Thus, unless very rapid changes in velocity occurs, i.e. in time
scales of ∼ 10−7 s, this term is amply negligible with respect to the gravita-
tional acceleration. Consequently, the contribution due to ram pressure can be
ignored within the FOA. Consequently, within our first-order approximation
we can retain Eq. 1.64 as the equation describing hydrostatic equilibrium.
Thus, within the FOA we see the structure can still be described throughout the
set of ODEs which, for the strictly v ≡ 0 case, are known as Tolman-Oppenheimer-
Volkoff equations.
Thermodynamical equations. The equations for all abundances are straight-
forward considering Eq. 1.41 and the FOA,
DtYi = Rlocal
i , i ∈ ¶1, . . . , Nspecies♢ . (1.67)
29
As expected, the term in the left-hand side encloses the contributions from changes
in time at a fixed spatial location, and those due to the fluid motion in space, i.e.
advection. Besides the expected coupling with the spacetime evolution of ρB and
T , in the general case we expect the right-hand side to enclose interactions among
all Nspecies species, hence Eqs. 1.67 are collectively referred to as the network of
reactions. The largest Nspecies is, the more difficult it becomes describing in terms of
individual reactions the evolution of the whole system. Associated with the network
is the specific energy generation rate, Eq. 1.20, which besides all Rlocal
i depends
on the specific chemical potentials per unit baryon, µ̆j = µj/mu with µj the usual
(relativistic) chemical potential. For massive species, i.e. those with mj ≥ mu,
µj ≈ mjc
2. On the other hand, for particle such as electrons where me ≪ mu, the
contribution to the chemical potential coming from the internal energy might exceed
the mec
2 term, thus we must employ retain the general expression µj. From these
considerations, the specific energy generation rate due to local changes in species is
given by
˙̆εspecies = −
Nspecies
∑
j=1
µj
mu
Rlocal
j . (1.68)
In the context of purely temporal evolution, typically within the non-relativistic
regime, it is frequent to measure the impact of each reaction in the network through-
out the integrated flux for the A → B reaction, namely
FA→B =
∫
I
dt
[
∂YA
∂t
∣
∣
∣
∣
A→B
− ∂YB
∂t
∣
∣
∣
∣
B→A
]
, (1.69)
where ∂tYj♣A→B denotes the (local) contribution on the right-hand side of ∂tYj explic-
itly connecting nuclides A and B. By definition, FA→B < 0 if the A → B reaction is
favorable, and > 0 if the reverse one, B → A, is the one favorable. Despite the scalar
character of the integrand this integral is neither covariant nor can be extended via
∂t → Dt since the advection term requires integration along the spatial direction,
not the temporal one. In practice, however, we can keep Eq. 1.69 for analyzing
integrated fluxes in time, at fixed locations, and for studying the behavior along the
radial direction it suffices to employ a simple replacement dt → dx/v′, with x and
v′ a characteristic length and speed.
Now let us focus on F µ, under the assumption F θ = F ϕ = 0. Since F µuµ = 0,
F 0 = vSe
Λ−ΦF r, (1.70)
i.e. only one component is independent. In analogy with the v ≡ 0 case, we choose
to compute F r. Within the FOA (see Proposition 6), we have
F r = −Ke−2Λ−Φ∂r(e
ΦT ), (1.71)
30
from where it immediately follows (Proposition 7) the associated cooling rate and
luminosity, still within the FOA, are
Gs,T = − 1
T
{
1
e2Φ+Λr2
∂r(e
2Φ+Λr2F r)
}
(1.72)
L = 4πr2eΛF r = −4πr2Ke−Φ−Λ∂r(e
ΦT ). (1.73)
Radiative temperature gradient. Combining the expression for ∂rT , Eq. 1.73,
dΦ/dr from Eq. 1.60 and the hydrostatic equilibrium condition, Eq. 1.64, we obtain
(Proposition 9) an expression for this gradient within the FOA,
∇r = − 3κρBLPe
Λ
64πσSBT 4r2
dr
dP
+
(
1 − ρ
H
)
. (1.74)
In low-density environments, H/ρ ≪ 1 so the second contribution on the right-hand
side is only relevant as we approach the regime where P ≈ ρc2.
Let us now write down the entropy equation within the FOA. Considering
Eqs. 1.41 and 1.72,
ρBTDts̆ = ρB
[
˙̆εspecies − ˙̆εν
]
−
{
1
4πe2Φ+Λr2
∂r(e
2ΦL)
}
,
where we have introduced the luminosity, Eq. 1.73, and the definition of ρB. Some
authors prefer to recast this equation, either for theoretical or numerical discussions,
in favor of L introducing the so-called gravitational energy
˙̆εgrav = −TDts̆, (1.75)
and even labeling the temporal and radial contributions in the right-hand side with
different names, non-homologous for the ∼ ∂ts̆ term, and consequently homologous
for the ∼ ∂rs̆ part,
˙̆εgrav, n.h. = −Te−Φ∂ts̆ (1.76)
˙̆εgrav, h. = −Tve−Λ∂rs̆. (1.77)
The origin of the gravitational nickname comes from the interpretation that changes
in s̆, related to ε and P via the 1st Law of Thermodynamics Tds ≈ dε−(P/n2
B)dnB,
are a consequence of altering the internal energy via compressions and expansions.
The minus sign at the front of the definition ensures heat is released(absorbed)
during a contraction(expansion). As it is, this equation is seldom used since Dts̆,
not DtT , appears explicitly, thus demanding to account for variations of T implicitly.
As shown in Proposition 8, we can invert this situation employing thermodynamical
identities, thus obtaining
∂(e2ΦL)
∂r
= 4πe2Φ+Λr2ρB
[
˙̆εspecies − ˙̆εν + ˙̆εgrav
]
(1.78)
˙̆εgrav = −c̆P
[
DtT − T∇ad
P
DtP
]
. (1.79)
31
Here, for instance, it is possible to recover well-known expression for neutron star
cooling in the case v ≡ 0, ∂tP → 0 and the degenerate regime, c̆ = c̆P = c̆V ,
∂(e2ΦL)
∂r
= 4πe2Φ+Λr2ρB
[
˙̆εspecies − ˙̆εν − c̆e−Φ∂tT
]
. (1.80)
Another characteristic we can extract from Eq. 1.78 is that all sources/sinks on the
right-hand side give rise to different redshifted luminosities along the radial direction,
L∞
j = 4π
∫
I
dr e2Φ+Λr2ρB
˙̆εj, j ∈ ¶species, ν, grav♢ , (1.81)
where I is the appropriate integration domain. These integrals allow to interpret
Eq. 1.78 as a sum of contributions to the total luminosity.
In summary, the theoretical modeling of neutron stars in the presence of mass
accretion is governed by two sets of variables,
• Structure: m, P , Φ,
• Thermodynamical: T , L, Y
obeying, within the FOA, the following differential equations:
∂tρB − e−Λ
4πr2
∂rṀ
∞ = 0, Ṁ∞ = −4πr2eΦρBv, (1.82)
∂m
∂r
= 4πr2ρ,
dΦ
dr
=
geΛ
c2
G, (1.83)
∂P
∂r
= −geΛHG, ∂T
∂r
=
T∇T
P
∂P
∂r
, (1.84)
g =
GmeΛ
r2
, G =
[
1 +
4πPr3
mc2
]
, (1.85)
H = ρ+
P
c2
, ∇r = − 3κρBLPe
Λ
64πσSBT 4r2
∂r
∂P
+
(
1 − ρ
H
)
, (1.86)
DtYi = Rlocal
i , i ∈ ¶1, . . . , Nspecies♢ , (1.87)
∂(e2ΦL)
∂r
= 4πe2Φ+Λr2ρB
[
˙̆εspecies − ˙̆εν + ˙̆εgrav
]
(1.88)
˙̆εspecies = −
Nspecies
∑
j=1
µj
mu
Rlocal
j (1.89)
˙̆εgrav = −c̆P
[
DtT − T∇ad
P
DtP
]
(1.90)
∇T =



∇r, ∇ad > ∇r
∇conv, ∇r > ∇ad.
(1.91)
32
1.4 On the numerical implementation of neutron
star evolution
−2 0 2 4 6 8 10
log10ρ
[
g cm−3
]
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
lo
g 1
0T
[K
]
P i
de
al
=
P r
ad
P i
de
al
=
P e
,re
l,d
eg
P
id
ea
l
=
P
e,
no
n−
re
l,d
eg
Radiation
dom
ain
Id
ea
l g
as
do
m
ai
n
Electron
domain
〈m〉i = 1.320
〈m〉e = 1.177
log10P
[
erg cm−3
]
3.00
5.88
8.76
11.64
14.52
17.40
20.28
23.16
26.04
28.92
105 106 107 108 109 1010
ρ
[
g cm−3
]
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
e%
[g
s
−
g
]
Figure 1.1: Left panel: contour plots of pressure, in terms of density and tempera-
ture, for a typical stellar EOS. Right panel: Relative percentual difference between
the surface and local gravity acceleration, gs and g as a function of density for a
realistic EOS.
While Eqs. 1.82-1.91 allow to describe the whole neutron star, the numerical
implementation faces a first challenge in the selection of the appropriate number of
cells, both in space and time directions, due to the different scales at which each
functional exhibit variations. For instance, pressure typically varies from ∼ 1014 erg
cm−3 to 1025 in ∼ 10 m and with a total mass of ∼ 10−9Mstar, while between 1025
and 1036 erg cm−3 the variations in mass and radius range from 0 to ∼ Mstar and
∼ Rstar, respectively. Similarly, the characteristic timescale for heat diffusion at a
layer δr, τ = (δr)2ρc̆P/K. Near the surface, τ ≤ 1 day while above and around
1026 erg cm−3 it extends from days to weeks. Therefore, for modeling neutron stars
it is customary to employ a combined strategy of splitting the whole star into two
regions and employing a different spatial coordinate.
Core-Envelope splitting. As its name suggests, the star is divided into two
regions: the core, spanning almost 90% of the total mass and radius of the star, and
33
the envelope, the thinnest and lightest region near the surface. Formally speaking,
the numerical core comprises up to three different regions from an actual neutron
star: the physical core at ρ ≥ 1014 g cm−3; the inner crust, between 1014 and ∼ 1011
g cm−3, characterized by the presence of free neutrons, and the outer crust, between
∼ 1011 and 108 g cm−3. Whenever needed, the actual distinction between these
regions will be explicitly stated.
The boundary between the envelope and the numerical core is usually determined
by two aspects:
• The possibility to approximate m, r and Φ as almost constants, or have rel-
ative errors with respect to their surface values less than 1%. As shown in
Fig. 4.1, where we illustrate the metric function Φ and the gravitational mass
m (related to the radial function Λ) for two different core-EOS and the same
pristine crust EOS, below 1010 g cm−3 these metric functions practically be-
come constants. Similarly, in the right panel of Fig. 1.1 we can see the relative
difference between the surface gravity gs = g(Rstar,Mstar) and the local value
g(r,m), for a typical crust-envelope EOS, does not exceed 3% at around 1010
g cm−3.
• The dependence of the EOS between temperature and pressure. Indeed, if
these are independent of each other, then we can decouple the structure and
thermal equations and solve them sequentially, keeping the initial structure at
all times during the simulation. For example, considering the typical EOS for a
star (i.e. a sum of the pressure from photons, free electrons and an ideal gas of
ions), as depicted in the left panel of Fig. 1.1, we see the pressure becomes less
sensitive to changes in temperature at above and around 107 g cm−3 where
degenerate electrons provide the largest contribution to pressure. At large
density such degeneracy is unlikely to be lifted since the Fermi temperature
for neutrons and protons is far above 1010 K, out of range for typical neutron
stars.
Following these requirements, the density boundary between the envelope and
the numerical core ρb is typically set between 107 to 1010 g cm−3. In the old days of
purely isolated neutron stars (e.g. [GPE83]), however, an additional requirement for
setting the boundary was to have zero heating and cooling sources in the envelope,
thus allowing to keep L as strictly constant. However, this condition can be relaxed
as long as the characteristic thermal timescale for these sources is comparable or
inferior to the local diffusion timescale, thus allowing to approximate L as constant.
Different spatial coordinates. In one dimensional, non-relativistic stellar
evolution it is frequent to replace the radial coordinate in the PDEs by the rest-mass
MB = muNB. Physically, such replacement is suitable as intuitively it is simpler to
visualize what happens in chunks of matter than in fixed locations in space, i.e. we
are allowed to handle stellar physics in terms of intensive variable (see next Chapter
34
5 10 15
log10ρ [g cm−3]
0.55
0.60
0.65
0.70
0.75
0.80
ex
p
[Φ
(ρ
)]
C
ru
st
-C
or
e
T
ra
n
si
ti
on
MS-A1 + PC, ρc,15 = 1.9152
APR + PC, ρc,15 = 1.0037
5 10 15
log10ρ [g cm−3]
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
m
(ρ
)
[M
⊙
]
C
ru
st
-C
or
e
T
ra
n
si
ti
on
MS-A1 + PC, ρc,15 = 1.9152
APR + PC, ρc,15 = 1.0037
Figure 1.2: Metric functions in terms of local density for two stars of the indicated
EOS. In the left panel we show the redshift Φ and in the right the gravitational mass
m. The transition between the core and the crust is indicated by vertical lines, and
their corresponding central density are stated in units of 1015 g cm−3.
for a thermodynamical justification). Mathematically, this is possible due to the
inverse function theorem, i.e. we can always compute MB(t, r) from r(t,MB).
By changing the independent spatial variable, both fields and differential opera-
tors must change accordingly. Let A be any physically relevant field (temperature,
pressure, mass), x1 and x2 the variables to be exchanged, having an explicit relation
x1(t, x2) and x2(t, x1). The partial derivatives of A with respect to x1, t and x2 are
related via
∂A(t, x2)
∂x2
=
∂A(t, x1)
∂x1
∂x1(t, x2)
∂x2
(1.92)
∂A(t, x2)
∂t
=
∂A(t, x1)
∂t
+
∂x1(t, x2)
∂t
∂A(t, x1)
∂x1
. (1.93)
These transformation laws are the formal representation of what in standard fluid
mechanics is said to be at fixed x1 or x2 coordinate. For instance, if x1 = r and
x2 = MB, the rate of change in time for field A at fixed baryonic mass is
∂A(t,MB)
∂t
=
∂A(t, r)
∂t
+
∂r(t,MB)
∂t
∂A(t, r)
∂r
. (1.94)
For the SSS metric in particular, if A = MB and due to Eqs. 1.54 1.55, we arrive
to DtMB = 0, i.e. the Dt operator we have previously introduced corresponds
to the so-called total time derivative, or the rate of change in time at a fixed MB.
Mathematically, such name is ill posed since time does not have a primordial role over
any other variable in multi-dimensional calculus. Physically, on the other hand, it is
35
useful to stress the analysis is carried at fixed chunks of matter, i.e. at fixed MB as in
Eq. 1.94, and to state its relation with respect to variations at fixed location in space.
However, this is a mere particular case of the application of chain’s rule. Therefore,
Eqs. 1.93 should be taken as the formal route to exchange partial derivatives when
moving from one set of coordinates ¶t, x1, x2, x3, x4♢ to ¶t, x′
1, x2, x3, x4♢ in the case
x1 = x1(t, x
′
1).
Although the mass coordinate is the most usual system in stellar astrophysics,
it is far from being unique: for instance, in the study of atmospheres the altitude z
in a cartesian coordinate system (x, y, z) (or the radius if the system is spherically
symmetrical) is typically replaced by pressure P since variations in pressure span
several orders of magnitude in contrast to the little variations for both mass and
altitude. Another example is the modeling of accreting systems with a given analytic
expression for the net rate of mass accretion, ṀTOT(t). Here, instead of MB the
relative ratio q = MB/MTOT,B(t) is employed, with MTOT,B(t) the instantaneous
total mass of the system.
In order to employ any of these coordinates, Eqs. 1.82-1.91 must be properly
adapted. Considering the transformation laws, Eqs. 1.93, notice the major shift only
takes place in the Dt operator, which under change of spatial coordinates transforms
into a sum of a temporal evolution plus an advection term.
For completeness, let us summarize the required changes for Eqs. 1.82-1.91 under
the different coordinates so far discussed.
• Baryonic mass. Since DtMB = 0, the operator Dt already represents the rate
of change at fixed MB. For the equations involving ∂r, it suffices to consider
the first of Eqs. 1.93 with
∂MB
∂r
= 4πr2ρBe
Λ. (1.95)
• Pressure. While the change over the ∂r is immediate via chain’s rule, for the
time operator we have
Dt = e−Φ ∂
∂t
∣
∣
∣
∣
P
+ e−Φ∂P
∂t
∂
∂P
= e−Φ
{
∂
∂t
∣
∣
∣
∣
P
+
Ṁ∞gHG
4πr2ρB
∂
∂P
}
. (1.96)
• Mass ratio q = MB/MTOT,B(t). Similarly, changes in ∂r are immediate via
∂
∂r
=
∂q
∂r
∂
∂q
=
∂MB
∂r
∂q
∂MB
∂
∂q
=
1
MTOT,B
∂MB
∂r
∂
∂q
, (1.97)
36
while for Dt we have
Dt = e−Φ ∂
∂t
∣
∣
∣
∣
q
+ e−Φ∂q
∂t
∂
∂q
= e−Φ
{
∂
∂t
∣
∣
∣
∣
q
− ṀTOT
ṀTOT,B
∂
∂ ln q
}
. (1.98)
Initial and boundary conditions. Eqs. 1.82 and 1.91 are a combination of
an Initial Value and a Boundary Value Problem (IVP and BVP respectively) in
[0, rs], where rs is the total radius of the star. On the one hand, equations with
explicit ∂t terms such as the one for density, temperature and composition require
an initial profile and to fix either one or the two values at r = 0 and r = rs. On the
other hand, we have quantities such as m and P requiring only the latter boundary
conditions.
Within the core-envelope splitting method, we have the possibility of simplifying
this IVP+BVP problem. Indeed, if the EOS is temperature-independent at the
core then we can decouple pressure, mass and Φ and solve their associated PDE
only once - at the beginning of the numerical simulation - and then solve the rest
of the equations with m(t, r) = m(r), P (t, r) = P (r). Here we must note that if
one is just interested in the local and global structure of the star - as is done in
many studies contrasting theory against observations on the gravitational mass and
radius - it suffices to solve the structure equations, i.e. Eqs. 1.59, 1.60 and 1.64.
These are collectively referred to as Tolman-Oppenheimer-Volkoff (TOV) equations
of relativistic stellar structure [ST83]. As an example of this situation, in Fig. 1.3
we show the results of integrating the TOV equations with the same envelope-
EOSs and four different core-EOSs: APR [APR98] and a family of MS models
[PBG+20]. The procedure is as follows: we pick a pair of values (ρc, Pc), given by
the EOS, integrate the TOV equations until P ≤ 0, i.e. the numerical surface, and
obtain another pair of values, the gravitational mass of the star and its radius. The
procedure is repeated for all the points in the EOS table and as a result every table
generates a family of neutron stars’ mass and radius. In the left panel of Fig. 1.3,
we can see the gravitational mass Mgrav, i.e. m(r = rs), plotted against the stellar
radius, while on the right panel we see the relation between the central densities and
the corresponding gravitational mass. The four selected EOSs satisfy the different
observational constraints, such as predicting the existence of stars as massive as
2M⊙, and to have stars of 1.4M⊙ with radius between 11.5 and 13 km.
Concerning thermal evolution studies, we must solve for the instantaneous evo-
lution of the temperature, density, chemical composition and the luminosity. At the
core, it is typical to suppose matter is in equilibrium, i.e. the chemical composition
is fixed at all times, while at the envelope, specifically in the presence of mass accre-
tion, we must consider the evolution of the nuclides due to reactions among them.
Consequently, at the core we assume X(t, r) = X(0, r) for the composition, while
37
10 12 14
Radius [km]
0.0
0.5
1.0
1.5
2.0
2.5
M
g
ra
v
[M
⊙
]
0 1 2 3
ρc [10
15 g cm−3]
0.0
0.5
1.0
1.5
2.0
2.5
MS-A1
MS-B1
MS-C1
APR
Figure 1.3: Global structure properties of neutron stars, given four different core-
EOSs. In these figures, we illustrate the gravitational mass of the star as a function
of the total radius in the left panel, and against their central densities in the right
panel.
at the envelope we must provide a distribution of chemical components and specify
the composition of the accreted matter, i.e. X(t, rs). Further details concerning the
composition of the envelope and the accreted material are given in Chapters 3 and
4. Regarding the temperature it is valid to start with either T (0, r) = constant, or a
temperature distribution with a small gradient from the center to the top. Finally,
for the luminosity we only demand L(r = 0) = 0.
At the very top of the star, where the total luminosity of the object is emitted,
it is suitable to employ an atmospheric boundary condition for the envelope itself,
i.e. a relation of temperature and pressure with the effective temperature via the
optical depth τod. From Fs = σSBT
4
eff, the actual temperature at the surface is
obtained from the analytic relation Ts(Teff, τ). For simplicity, we adopt Eddington
relations [Gra08, KWW12]
Ts =
3
4
(
τod +
2
3
)1/4
Teff , (1.99)
Ps =
τodgs
κ(ρs, Ts,Xs)
, (1.100)
which constraint the temperature and the pressure at the surface. Under the as-
sumption the chemical composition Xs is known as well, we can employ the equation
of state to solve for ρs or Ps. This choice of atmospheric boundary condition is not
unique: for instance, another common choice is the radiative zero solution, which
38
constraints the temperature and pressure to the surface flux via
T 4
s =
(
3κ
16σSBgs
× 4Fs
)
Ps. (1.101)
For additional comments on this expression, see Appendix B.4.
1.5 Thermonuclear Instabilities
Around 1928, Semenov [Sem28], [ZBLM80], [Nov16] developed a criterion for char-
acterizing, in the context of chemical reactions, the stability of steady-state burning
and the potential onset of thermal explosions. The underlying physics can be syn-
thesized as follows: the temperature evolution is governed by the imbalance in the
amount of generated energy from heating mechanisms, such as chemical and nuclear
reactions, and cooling mechanisms such as escaping photons and neutrinos. If the
net change is zero, we say the system is in thermal equilibrium, as illustrated by
the crossing points A and B of the fictitious mechanisms at Fig. 1.4 [Lib10], where
we employ an adimensional temperature τ for simplicity. Due to the dependence
of these mechanisms with τ , a small perturbation in temperature induces a per-
turbation from thermal equilibrium. Suppose the system was originally at A: if
temperature is slightly increased, the rate of cooling exceeds that of heating, so our
system comes back to its equilibrium position. If temperature is slightly decreased,
the amount of heating injected overcomes the cooling mechanisms, and the system
once again comes back to equilibrium. Thus, A is a stable equilibrium. A different
scenario takes place at B: if temperature is increased by a small amount, the cooling
mechanisms are inefficient for taking away all the energy generated by the heating
mechanisms, thus system’s temperature is likely to increase almost indefinitely. On
the other hand, if temperature is slightly decreased the rate of cooling is likely to
enhance a drastic drop in temperature, taking the system back to equilibrium point
A. Therefore, B must be considered as an unstable equilibrium.
Let us now express these ideas in formal terms. Let Γ(T ) and Λ(T ) be the rates
of generation and losing of energy, respectively. The temperature of the system
evolves in time according to
cP
∂T
∂t
= Γ − Λ . (1.102)
We say T0 defines an equilibrium state of the system if Γ(T0) = Λ(T0). Due to
the explicit functional form of these mechanisms, in general we can expect more
than one equilibrium to exist. Now we pick one of them and construct a perturbed
temperature profile, T (t) = T0 + θ(t), with θ ≪ T0. Upon linearization of the ODE,
we arrive to
cP
∣
∣
∣
∣
0
∂θ
∂t
=
[
∂Γ
∂T
∣
∣
∣
∣
0
− ∂Λ
∂T
∣
∣
∣
∣
0
]
θ , (1.103)
39
τ
ǫ∗
A
B
ǫ∗heat
ǫ∗cool
Figure 1.4: Semi-realistic rates of cooling (blue line) and heating (red line) for a
chemical system as functions of adimensional temperature. The intersections of
these functions define steady-states. Based on [Lib10]
where ♣0 indicates the correspondent functional is evaluated at T = T0. As the gen-
eral solution to this ODE is θ ∝ eλt, and cP,0 is a positive quantity (see Appendix A),
a stationary state T0 is stable against small perturbations if
[
∂Γ
∂T
∣
∣
∣
∣
0
− ∂Λ
∂T
∣
∣
∣
∣
0
]
< 0, (1.104)
and we can refer to such T0 as a steady state. In case the opposite inequality holds
true, we say such equilibrium is unstable. Furthermore, the nature of Γ defines
which kind of thermal explosion we have: for instance, if heating is due to chemical
reactions, we speak of thermo-chemical explosions, while if coming from nuclear
reactions as in the case of astrophysical scenarios, such as neutron star in binary
systems, these are labeled as thermonuclear explosions.
Although Semenov’s model is quite useful to understand the physics of thermal
instabilities, the absence of spacial-dependence prevents us to apply it for systems
where dissipative fluxes are present, as is the case in stellar evolution with ∇ · F . A
first step to remedy this issue was discussed in later works by Frank-Kamenetskyy
[FK38], [ZBLM80], who studied the stability of the following PDE:
cP
∂T
∂t
= K∇2T + Γ(T ) , (1.105)
where K is regarded as a constant. Following a similar perturbative analysis, to
linear order we obtain
cP
∣
∣
∣
∣
0
∂θ
∂t
= K∇2θ +
[
∂Γ
∂T
∣
∣
∣
∣
0
]
θ, (1.106)
i.e. a Sturm-Liouville problem which can be solved by proposing a series of the form
θ(t, r) =
∞
∑
n=0
Cne
−λntψ(r) . (1.107)
40
An instability in this context is essentially the same as in the Ineq. 1.104 case.
Its identification, however, is now slightly more involved: as the eigenvalues λn for
a Sturm-Liouville problem are real and ordered with n, the identification of the
eigenmode λ = 0 must be complemented with the verification that it corresponds
to n = 0. If this second aspect is ignored, there might exists a negative eigenmode
which will render T0 as truly unstable [Kor79], [BFG83a].
In the astrophysical context, the constancy of K is objectionable by recalling
that K = (16σ/3)T 3ρ−1κ−1, i.e. it is an implicit function of position via T and ρ.
As pointed out in the context of chemical reactions [BFG83b], including this func-
tional dependence is important as variations in the critical parameters for thermal
explosions are significant. In principle, keeping the dependency of K with r for the
linear perturbation analysis leads to a second-order PDE in θ, whose eigenmodes
and eigenfunctions might be extracted by similar steps as in the K ≡ constant case.
Surprisingly, the linear perturbation analysis for the surface of neutron stars can
relatively overcomes this difficulty by imposing some locality over ∇ · F , i.e.
∇ · F ≈ KT
d2
H
≡ εcool(T ) (1.108)
with dH ≈ P/(ρgs), and performing the standard perturbation analysis to first order
leading to Eq. 1.104, in the form
∂εheat
∂T
>
∂εcool
∂T
. (1.109)
This criterion for instability practically became a standard in the field, simplifying
the process from computing eigenvalues and eigenvectors (as attempted by some
authors, e.g. ) to computing numerical or analytical partial derivatives. The ca-
pacity of this approximation to predict the critical density and temperature for
the beginning of thermonuclear explosion, provided stationary solutions have been
calculated, is relatively good for a first comparison with fully time-dependent sim-
ulations. The caveats, however, have become visible in the modern era where fast
fully-time dependent simulations are available for desktop computers.
A slightly outdated, yet complementary approach to the imbalance in perturba-
tions to the heating and cooling mechanisms is the comparison of local timescales.
The physical picture is simple: if the amount of time required to heat and cool a
bulk of matter under certain conditions of temperature and density is the same, then
the system will stay in thermal equilibrium. However, if the cooling time exceeds
the heating one then an instability will take place. Given thermonuclear insta-
bilities occur in the presence of mass accretion, for this approach three scales are
contrasted: the diffusion timescale τdif = (δr)2ρc̆P/K, associated with the radiative
and conductive fluxes; the heating timescale, principally due to nuclear reactions
and given by τnuc = Enuc/εnuc, i.e. the ratio between the energy per gram and the
specific energy generation rate, and finally the accretion timescale τacc = δM/Ṁ ,
41
measuring the amount of time required to accrete a layer of δM at an accretion rate
of Ṁ . Stability is guaranteed when τdif < τnuc and τnuc < τacc, a criterion which can
also be deduced from Ineq. 1.109 by order-of-magnitude estimations. Such enforced
locality, however, reduces the range of applicability of this criterion, although it is
useful for a back-of-the-envelope estimation.
42
Chapter 2
Microphysical aspects of stellar
evolution
“Can’t repeat the past?...Why of
course you can!”
F. Scott Fitzgerald
Abstract. In this chapter we present the characteristics which a nuclear
network of reactions must have in order to describe the envelope of the star.
2.1 Introduction
In the previous chapter we encountered several functionals, i.e. functions of func-
tions, representing physical processes such as the rate of interactions among parti-
cles or the opacity of a system. These functionals share the property of locality, i.e.
not depending on derivatives of any kind or order (partial derivatives, 4-gradients,
D’Alembertian, etc) of their functional arguments, a set of finite thermodynamical
degrees of freedom such as nB, T , and ¶Yi♢Nspecies
i=1 . From a strict point of view, their
construction is based on a statistical approximation to the underlying processes: for
instance, instead of following the instantaneous movement of each and every con-
stituent particle of the system, we compute averages over a well-posed distribution
of energies or velocities. As a consequence, this collection of functionals is referred
to as microphysics.
It is the purpose of the present chapter to provide an exposition on the enclosed
physics by each of these microphysical functionals. Given the rich vastness of tech-
niques behind their construction, as well as the extensive literature for each of the
topics, it is our scope to provide the essential features as to allow an understanding
of the following chapters.
43
In section II we discuss the fundamental thermodynamical degrees of freedom
governing all the functionals discussed in the chapter. In section III we briefly discuss
the neutron star EOS at different densities, focusing on the opacity at section IV,
followed by the neutrino emission mechanisms at section V, and closing the chapter
with the extensive subject of nuclear reactions.
2.2 A brief survey on Thermodynamics
Thermodynamics is the branch of physics concerned with the study of systems Σ(T )
with ≥ 1023 particles through a collection of variables ¶ηi♢Nd.o.f
i=1 , referred to as Degrees
of Freedom (DoF), as well as the functionals of these variables, such as the total
energy U or the EOS. The DoF can be classified according to their behavior under
the addition or remotion of thermodynamical subsystems: let Σ
(T )
3 = Σ
(T )
1 ∪ Σ
(T )
2 . If
η
(3)
j = η
(1)
j +η
(2)
j for some j ∈ ¶1, Nd.o.f.♢, we say this variable is extensive. Otherwise,
we define it as intensive. Examples of extensive variables are the baryonic mass,
number of particles, volume V , U or entropy S, while for intensive variables the
temperature T , pressure P or chemical potential µ are good examples. Related
to the concept of extensive variables is that of homogeneous function of k-degree,
which in general is any differentiable f : R
n → R, with k ∈ R and λ ∈ R
+,
satisfying f(λx) = λkf(x). In thermodynamical systems this property is crucial
for our conception of functionals of state as it suggests the existence of a certain
parameter for scaling all the extensive variables. Intuitively, this parameter can be
the total number of particles or the volume.
The mathematical foundation of Thermodynamics lies in Statistical Physics,
which is the bridge between the microscopic predictions of Quantum Mechanics or
Field Theory and the macroscopic observables, the DoF. An important question
is, to what extent are mathematically consistent the predictions of these statistical
theories? Quite frequently we talk about systems with 1023 particles or very large
in volume. Recalling the old criticisms to differential calculus’ methods, two im-
portant questions emerge here: Why 1023 particles?, and even more importantly,
how large is a large system? The first question seems to possess a rather simple
answer, as we learn in basic Chemistry that the Avogadro’s number NA, which in
numerical terms is the inverse of the atomic mass unit mu ∼ 10−24 g, represents
a useful rod to quantify the amount of particles in a system. As for the second
question, by following our most basic mathematical intuition we might be tempted
to claim that predictions are valid as long as we take limN,V→∞ for all thermody-
namical variables. Turns out this is just half the answer: while intensive properties
are indifferent to this limit, extensive ones actually depend on it as this classifica-
tion precisely follows from their proportionality with N , hence if the limit is simply
taken we might obtain infinite quantities! To precise these notions, and to make
Statistical Physics a well-posed mathematical theory, the postulate of thermody-
44
namical limit is introduced [Dor21, Kuz14, Sty04], allowing to extend our formalism
even to “trivial” problems such as the incorporation of Coulomb interactions [LL69].
In simple words, the thermodynamical limit postulates predictions from Statistical
Physics are meaningful only if the following conditions hold:
1. The particle number density
n := lim
(N,V )→(∞.∞)
N
V
(2.1)
exists and is finite.
2. For extensive variables such as X(T, V ) or X(T,N), there exist associated
functions of states referred to as X per particle and X-density (X per unit
volume), respectively, satisfying
x̃(T, n) := lim
(N,V )→(∞.∞)
X(T, V )
N
(2.2)
x(T, n) := lim
(N,V )→(∞.∞)
X(T,N)
V
. (2.3)
In the astrophysical contexts, these postulates acquire uttermost importance since
it is frequent to employ either specific quantities (i.e. X per gram of matter) x̆, X
per particle x̃ = mux̆, or X-densities (i.e. per volume) x = nx̃, taking N ≡ NB and
n ≡ nB due to the conservation of baryons and since the proton mp and neutron
mn masses, close to mu, exceeds by almost 2000 times the electron’s me. As a
consequence of this choice, for rewriting thermodynamical identities, and only within
that context, it is valid to consider the replacements V ≡ VB = NB/nB or its close
relative V ≡ Vm = MB/ρB with MB = muNB.
Thermodynamical functionals. As described in standard Statistical Mechan-
ics textbooks [LL76], the central functional in the study of an ensemble of particles
whose dynamics is governed by the Hamilton operator Ĥ and which is allowed to
exchange particles and energy with a thermal reservoir is the grand-partition func-
tion Z. Assuming the system is composed of Nspecies species, with individual particle
number operator N̂i satisfying [Ĥ, N̂i] = 0̂, this functional is given by
Z = Tr



exp

−β

Ĥ −
Nspecies
∑
i=1
µiN̂i







(2.4)
with β = (kBT )−1 and µi is the chemical potential of the i-th species. If the
eigenstates of Ĥ are known, the trace simplifies into a sum for the discrete case,
or into an integral for the continuum one. Through partial differentiation of Z, we
45
obtain the expectation value for the number of particles, energy and entropy as1
Ni =
1
β
∂lnZ
∂µi
, i ∈ ¶1, . . . , Ns♢ (2.5)
U = −∂lnZ
∂β
+ µiNi (2.6)
S = kB
∂ (T lnZ)
∂T
. (2.7)
If U is an homogeneous function of degree k = 1, as is the case in the majority of
astrophysical applications, we can obtain the pressure from Z via
P = −kBT
V
ln Z. (2.8)
Formally speaking, the quantities on the left should be denoted as ⟨X⟩ since these are
thermal expectation values for a given operator. However, we opt for the traditional
approach of writing them as X, stressing on the fact that for the systems of interest
in this work the system 2.5 - 2.7 can be written in the form
X =
∫
dE F(E)hX(E) , (2.9)
where dE denotes the appropriate measure of integration, hX is a specific function
associated with the variable X and F is the distribution function, which is inherent
to the physical system and thus common to Ni, U and S.
By construction, Eqs. 2.5 - 2.7 satisfy the First Law of Thermodynamics, in
terms of specific quantities given by
dε̆ = Tds̆+
P
ρ2
B
dρB +
Nspecies
∑
i=1
µi
mu
dYi, (2.10)
which is a statement of how variations in s̆, ρB and the abundances Yi change of the
total specific energy of the system, and the Second and Third Laws of Thermody-
namics,
ds̆ ≥ 0 (2.11)
lim
T→0
∆s̆ = 0. (2.12)
The first can be interpreted as the mathematically formal statement of the every-
day situation that heat flows from hot to cold bodies. Taking into account the
statistical foundation of Thermodynamics, however, it is equally plausible to con-
sider the Second Law as the formal statement that systems will evolve from low- to
1For systems such as a photon gas, its appropriate description is obtained via Nspecies = 1 and
µ → 0 in the expressions for N , U and S.
46
high-probability configurations. The second statement, also referred to as Nernst’s
Postulate, is a requirement emerging from quantum mechanics.
Eq. 2.10, the differential of the specific energy, also encloses the functional nature
of ε̆, namely, ε̆(s̆, ρB,Y ), and states the intensive variables T , P and µi can be
obtained via first-order partials of ε̆,
T =
(
∂ε̆
∂s̆
)
ρB,Y
, P = ρ2
B
(
∂ε̆
∂ρB
)
s̆,Y
, µi = mu
(
∂ε̆
∂Yi
)
s̆,ρB,Yk ̸=i
. (2.13)
It is important to stress that, given the functional dependence of ε̆ (as well as the
Implicit/Inverse function theorems from standard multivariable calculus), we can
expect the rest of thermodynamical variables here written to have analogous func-
tional dependence, for instance P (s̆, ρB,Y ), or even to allow inversion in terms of a
measurable quantity, such as ρB(s̆, P,Y ) or T (s̆, ρB,Y ) → s̆(T, ρB,Y ) → s̆(T, P,Y ).
Similarly to the case for ε̆, the partial derivatives of functionals such as s̆ and
P have well-defined roles in astrophysical discussions. For instance, to quantify
how much heat Σ(T ) exchanges under a variation of temperature at constant η, we
introduce specific heat at constant η as
c̆η = T
(
∂s̆
∂T
)
η,Y
. (2.14)
By virtue of the Second Law, c̆η ≥ 0, a mathematical statement of the everyday
experience that providing heating to a Σ(T ) increases its temperature, while removing
it leads to a decrement in T . The most frequently used specific heats are those at
constant volume and pressure, c̆V
2 and c̆P respectively. By virtue of the Second
and Third Laws, these satisfy c̆P - c̆V ≥ 0 and c̆P ≈ c̆V in the vicinity of T = 0,
a condition of extensive usage in degenerate systems such as the core and crust of
Neutron Stars.
Considering P (T, ρB,Y ), its adimensional partial derivatives functionals - or
sensitivities - are [Cla68, HK94, JSB+21, AJL+22]3,
χρ =
ρB
P
(
∂P
∂ρB
)
T,Y
, χT =
T
P
(
∂P
∂T
)
ρB,Y
, χYi
=
Yi
P
(
∂P
∂Yi
)
T,ρB,Yk ̸=i
. (2.15)
By definition, these functionals serve to give a qualitative measure of the EOS sen-
sitivity with each DoF present at its differential, as under a small neighborhood
around a fixed tuple (Tx, ρB,x,Yx) we have Px = P0T
χT
x ρ
χρ
B,x
∏N
s=1 Y
χYs
s with P0 some
constant. Due to the functional form of each coefficient, for improving numerical
2Properly speaking, it is at constant nB. However, we attach to the most common notation
considering the constancy of NB in the relation V ≡ VB.
3Kippenhann’s ([KWW12]) notation for these derivatives, i.e. α = 1/χρ and δ = χT /χρ,
represent an exception rather than a rule.
47
accuracy it is sometimes more convenient to express them employing the notation
(d ln Y/d lnX)Z1,...,Zs
. As shown in Appendix A, these adimensional χ can be em-
ployed to express the specific heats avoiding an explicit dependence with s̆
c̆V =
(
∂ε̆
∂T
)
ρB,Y
, (2.16)
c̆P =
(
∂ε̆
∂T
)
P,Y
+
χT
χρ
P
ρBT
(2.17)
c̆P − c̆V =
χT
χ2
ρ
P
ρBT
. (2.18)
The adiabatic gradient. A coefficient of interest mixing both P and s̆ is
the adiabatic temperature gradient, expressing how much temperature change with
respect to pressure in the absence of heat exchange,
∇ad =
P
T
(
∂T
∂P
)
s̆,Y
=
χT
χρ
P
Tρc̆P
. (2.19)
This quantity plays a central role in the study of convection, i.e. heat transport
via the movement of matter. Intuitively, this is to be expected since ∇ad physically
states how much can temperature be changed under a perturbation in the structure,
i.e. pressure, without allowing such perturbation to absorb or provide heat. It can
be shown, via stability analysis in 1D, that a sufficient condition for the onset of
convection is [KWW12, HK94]
∇ad < ∇T , (2.20)
with ∇T = (P/T ) ∂T
∂P
the actual temperature gradient. This is the celebrated
Schwarzschild’s criterion, which intuitively states that the maximum temperature
gradient a blob can reach before starting its motion is that where no heating ex-
change takes place. Since Schwarzschild’s criterion assumes composition is fixed, it
is only a limit case of Ledoux’s criterion, which states that convective effects allowing
mixing of material takes place when [HK94, BM08, GNMM14]
∇ad +
χρϕ
χT
∇µ,s < ∇T , (2.21)
with χρϕ
χT
∇µ,s the gradient of mass fraction with respect to pressure at fixed entropy.
For systems such as stellar interiors, subject to gravitational acceleration and
consequently to pressure gradients, convection is typically assumed to predominantly
operate along the direction of maximum growth of pressure, i.e along the radial
direction. This also responds to our everyday experiences at Earth (for instance,
in boiling water) where mass blobs move in the vertical direction, from the heated
base to the surface. While pressure gradients in the horizontal direction have a role
48
for the size of the blob and in current 3D models, employing Ineq. 2.20 and in
single-dimensional simulations of stars is still a reliable approach, in combination
with models computing the energy flux due to convection and an appropriate ∇T
[PBD+11, GNMM14, AJL+22, JT23].
The chemical potential. As is well-known, a typical Ĥ consists on contri-
butions from a kinetic K̂ and a potential energy V̂ operators. Regarding the first,
in momentum space it is possible to expand it as
∑Nspecies
i=1 [mic
2 + p2
i /2mi + O(p4
i )],
where the first term in the series is referred to as the “rest-mass” energy. Retaining
all O(p2n
i ) terms in the expansion, it is possible to write ε̆ = ε̆rest + ε̆int, the first
term given by
ε̆rest =
Nspecies
∑
i=1
mic
2Yi
mu
. (2.22)
The second term, the “internal energy” ε̆int, encloses the contributions to energy
given by the remaining terms in the series of p and the contributions from potential
energy. As a consequence, by explicit calculation we see the chemical potential for
the j-th species is
µj = mjc
2

1 +
mu
mjc2
(
∂ε̆int
∂Yj
)
s̆,ρ̆B,Yk ̸=j

 , (2.23)
with the leading contribution coming from particle’s rest-mass, which introduces the
notion that µj is the relativistic chemical potential while contribution due to changes
in ε̆ represent the non-relativistic chemical potential. Due to its origin as thermal
average of terms ∝ p2n, we can get further insight of its importance according to
the relativistic status of the underlying particles. For instance, for particles obeying
a Boltzmann distribution we have mu∂ε̆int/∂Yj = 3kBT/2, i.e. chemical potential
acquires a contribution from the internal energy as long as kBT ∼ mjc
2, requiring
large temperatures as long as mj ≈ mu. For these species, µj ≈ mjc
2. On the
other hand, for particles obeying the Fermi-Dirac distribution, such as e−/e+, the
contribution from the internal energy becomes prominent due to the smallness of me
at different densities and temperatures. Consequently, both rest-mass and internal
contributions must be included in the actual computation of µe.
Thermodynamical potentials. Quite often, in both theoretical and experi-
mental sides, we have systems where intensive variables, such as P or T , can be
regarded as independent instead of ρB or s̆. Furthermore, in some theoretical and
experimental cases ε̆ might not be known, a requisite in principle we must have in
order to construct, via the 1st Law of Thermodynamics, quantities such as P , s̆
and c̆P . To solve this problem, we employ Legendre Transformations to construct
adequate thermodynamical potentials regarding intensive variables as conjugated of
the extensive ones. Once constructed, we can extract the functionals of interest such
as P , s̆ or c̆P from their first and second derivatives, constituting a set of intercon-
49
nected relations referred to as the Maxwell relations. For a full list of them, we refer
the reader to Appendix A.
Given the set of variables s̆, ρB and Y , the Legendre transformations for ε̆ are:
• Helmholtz specific free energy f̆(ρB, T,Y ): defined as f̆ = ε̆ − T s̆, its differ-
ential satisfies
df̆ =
P
ρ2
B
dρB − s̆dT +
Nspecies
∑
i=1
µi
mu
dYi. (2.24)
By construction, this potential is suitable for systems where ρB and T are
accessible for measurements. In the context of standard, non-relativistic stel-
lar evolution, this potential is of frequent use as it is easy to compute from
first principles, via the standard partition function Z := Tr
{
exp[−βĤ]
}
, and
to obtain the remaining functionals such as P , s̆ and c̆P via first and sec-
ond derivatives. Due to its definition, this potential is closely related to the
maximum amount of work we can extract from Σ(T ).
• Specific enthalpy h̆(s̆, P,Y ): defined as h̆ = ε̆+ P/ρB. Correspondingly,
dh̆ = Tds̆+
1
ρB
dP +
Nspecies
∑
i=1
µi
mu
dYi. (2.25)
If the pressure of Σ(T ) is kept constant, as for instance in the case of solids,
∆h̆ =
∫
dh̆ expresses the total amount of heat injected or removed from the
system. This shows the origin of its old name, “heat function”.
• Gibbs specific free energy ğ(P, T,Y ): defined by ğ = ε̆ + P/ρB − T s̆, its
differential is
dğ =
1
ρB
dP − s̆dT +
Nspecies
∑
i=1
µi
mu
dYi, (2.26)
i.e. this potential is adequate for studying processes at constant temperature
or pressure. Additionally, for homogeneous U of degree k = 1, ğ can be shown
equal to
∑Nspecies
i=1 µiYi/mu, i.e. the Gibbs specific energy is a pondered sum of
the amount of invested energy to add/remove the Nspecies species.
2.3 The Neutron Star Equation of State
By simple order-of-magnitude estimations, at the core of a typical neutron star we
expect a large density, ∼ 1014−15 g cm−3, contrasted against stars like our own
Sun, with central densities of ∼ 102 g cm−3. From a nuclear physics point of view,
neutron stars’ core is the perfect laboratory to put into test different models of
50
nuclear matter and their interactions since the core’s density is actually comparable
to the nuclear saturation density, e.g. ρ0 = 2.5 × 1014 g cm−3, above and around
we could simply have “ordinary” particles, such as protons, neutrons, electrons and
muons, or exotic forms of matter such as de-confined quarks. This, on the other
hand, brings certain uncertainty as to which model we should opt for to describe
the inner portion of the neutron star. Additionally, since the surface of the star
should have densities around 100 g cm−3 we must have phase transitions, from the
“ordinary” stellar matter - gas like with Coulomb interactions as corrections - to
the center of the star. Fortunately for the latter scenario, the large densities and
the chemical composition4 allow to remove temperature from the thermodynamical
degrees of freedom as it suffices to consider matter at T = 0 K, i.e. in the ground
state.
So far, it is difficult to speak of a “unified” equation of state, i.e. a single
functional P (ρ, T,X) capable of describing the whole interior of the star. Some
efforts into that direction actually exist, although it is customary as well to have the
EOS in tabulated form and to implement interpolation schemes for incorporating
them into stellar evolution codes. The main difficulty, we can say, is the absence of
a unified model of matter behavior at all densities and temperatures. Additionally,
we can cite the success of dividing the star into layers, get the EOS for each of them
and then join the resulting functional/table. From a qualitative point of view, this
latter path is better than the first as we can actually comprehend which are the
underlying processes and the uncertainties in each of these models.
Despite the different names the layers of the star receive in the literature, we
must be aware there exists a simple criterion to make the distinction between one
region and another: the kind of inhabiting particles and their interactions.
Small digression: what kind of particles do we expect?
Considering we are concerned with the study of neutron stars, it is pertinent to make
a short glossary on their constituent particles in order to avoid a constant repetition
of their intrinsic properties.
By element we define any bounded array of Z protons, Z electrons and N neu-
trons named after their Z number. For instance, a system with Z = 6 is referred
to as carbon or with Z = 26 as iron, regardless of having N = 6 or N = 30. Each
element, as is well known, has its on symbol: C for carbon, Fe for iron, and so on.
Since clustered protons and neutrons provide the largest contribution to rest-mass
and they are bounded by the nuclear force, they are collectively referred to as the nu-
cleus. An isotope is an element with different number of neutrons, and are denoted
as either A(Element name) or (Element name)-A, with A = Z + N . For instance,
we have carbon-13 or 55Fe. By definition, an element is electrically neutral: when
4It is customary to say “chemical composition”, despite the particles under consideration have
little relation with everyday chemistry.
51
107Te
106Sb
Valley of stability
99Sn 100Sn 101Sn 102Sn 103Sn 104Sn 105Sn
98In 99In 100In 101In 102In 103In 104In
95Cd 96Cd 97Cd 98Cd 99Cd 100Cd 101Cd 102Cd 103Cd
nuclides
94Ag 95Ag 96Ag 97Ag 98Ag 99Ag 100Ag 101Ag 102Ag
90Pd 91Pd 92Pd 93Pd 94Pd 95Pd 96Pd 97Pd 98Pd 99Pd 100Pd 101Pd
89Rh 90Rh 91Rh 92Rh 93Rh 94Rh 95Rh 96Rh 97Rh 98Rh 99Rh 100Rh
nuclides
86Ru 87Ru 88Ru 89Ru 90Ru 91Ru 92Ru 93Ru 94Ru 95Ru 96Ru 97Ru 98Ru 99Ru
85Tc 86Tc 87Tc 88Tc 89Tc 90Tc 91Tc 92Tc 93Tc 94Tc 95Tc 96Tc 97Tc 98Tc
82Mo 83Mo 84Mo 85Mo 86Mo 87Mo 88Mo 89Mo 90Mo 91Mo 92Mo 93Mo 94Mo 95Mo 96Mo 97Mo
81Nb 82Nb 83Nb 84Nb 85Nb 86Nb 87Nb 88Nb 89Nb 90Nb 91Nb 92Nb 93Nb 94Nb 95Nb 96Nb
78Zr 79Zr 80Zr 81Zr 82Zr 83Zr 84Zr 85Zr 86Zr 87Zr 88Zr 89Zr 90Zr 91Zr 92Zr 93Zr 94Zr 95Zr
77Y 78Y 79Y 80Y 81Y 82Y 83Y 84Y 85Y 86Y 87Y 88Y 89Y 90Y 91Y 92Y 93Y 94Y
74Sr 75Sr 76Sr 77Sr 78Sr 79Sr 80Sr 81Sr 82Sr 83Sr 84Sr 85Sr 86Sr 87Sr 88Sr 89Sr 90Sr 91Sr 92Sr 93Sr
73Rb 74Rb 75Rb 76Rb 77Rb 78Rb 79Rb 80Rb 81Rb 82Rb 83Rb 84Rb 85Rb 86Rb 87Rb 88Rb 89Rb 90Rb 91Rb 92Rb
69Kr 70Kr 71Kr 72Kr 73Kr 74Kr 75Kr 76Kr 77Kr 78Kr 79Kr 80Kr 81Kr 82Kr 83Kr 84Kr 85Kr 86Kr 87Kr 88Kr 89Kr 90Kr 91Kr
68Br 69Br 70Br 71Br 72Br 73Br 74Br 75Br 76Br 77Br 78Br 79Br 80Br 81Br 82Br 83Br 84Br 85Br 86Br 87Br 88Br 89Br 90Br
65Se 66Se 67Se 68Se 69Se 70Se 71Se 72Se 73Se 74Se 75Se 76Se 77Se 78Se 79Se 80Se 81Se 82Se 83Se 84Se 85Se 86Se 87Se 88Se 89Se
64As 65As 66As 67As 68As 69As 70As 71As 72As 73As 74As 75As 76As 77As 78As 79As 80As 81As 82As 83As 84As 85As 86As 87As 88As
60Ge 61Ge 62Ge 63Ge 64Ge 65Ge 66Ge 67Ge 68Ge 69Ge 70Ge 71Ge 72Ge 73Ge 74Ge 75Ge 76Ge 77Ge 78Ge 79Ge 80Ge 81Ge 82Ge 83Ge 84Ge 85Ge 86Ge 87Ge
59Ga 60Ga 61Ga 62Ga 63Ga 64Ga 65Ga 66Ga 67Ga 68Ga 69Ga 70Ga 71Ga 72Ga 73Ga 74Ga 75Ga 76Ga 77Ga 78Ga 79Ga 80Ga 81Ga 82Ga 83Ga 84Ga 85Ga 86Ga
55Zn 56Zn 57Zn 58Zn 59Zn 60Zn 61Zn 62Zn 63Zn 64Zn 65Zn 66Zn 67Zn 68Zn 69Zn 70Zn 71Zn 72Zn 73Zn 74Zn 75Zn 76Zn 77Zn 78Zn 79Zn 80Zn 81Zn 82Zn 83Zn 84Zn
54Cu 55Cu 56Cu 57Cu 58Cu 59Cu 60Cu 61Cu 62Cu 63Cu 64Cu 65Cu 66Cu 67Cu 68Cu 69Cu 70Cu 71Cu 72Cu 73Cu 74Cu 75Cu 76Cu 77Cu 78Cu 79Cu 80Cu 81Cu 82Cu
50Ni 51Ni 52Ni 53Ni 54Ni 55Ni 56Ni 57Ni 58Ni 59Ni 60Ni 61Ni 62Ni 63Ni 64Ni 65Ni 66Ni 67Ni 68Ni 69Ni 70Ni 71Ni 72Ni 73Ni 74Ni 75Ni 76Ni 77Ni 78Ni 79Ni 80Ni
50Co 51Co 52Co 53Co 54Co 55Co 56Co 57Co 58Co 59Co 60Co 61Co 62Co 63Co 64Co 65Co 66Co 67Co 68Co 69Co 70Co 71Co 72Co 73Co 74Co 75Co 77Co
47Fe 48Fe 49Fe 50Fe 51Fe 52Fe 53Fe 54Fe 55Fe 56Fe 57Fe 58Fe 59Fe 60Fe 61Fe 62Fe 63Fe 64Fe 65Fe 66Fe 67Fe 68Fe 69Fe 70Fe 71Fe 72Fe 74Fe
46Mn 47Mn 48Mn 49Mn 50Mn 51Mn 52Mn 53Mn 54Mn 55Mn 56Mn 57Mn 58Mn 59Mn 60Mn 61Mn 62Mn 63Mn 64Mn 65Mn 66Mn 67Mn 68Mn 69Mn 70Mn
44Cr 45Cr 46Cr 47Cr 48Cr 49Cr 50Cr 51Cr 52Cr 53Cr 54Cr 55Cr 56Cr 57Cr 58Cr 59Cr 60Cr 61Cr 62Cr 63Cr 64Cr 65Cr 66Cr
43V 44V 45V 46V 47V 48V 49V 50V 51V 52V 53V 54V 55V 56V 57V 58V 59V 60V 61V 62V 63V 64V
40Ti 41Ti 42Ti 43Ti 44Ti 45Ti 46Ti 47Ti 48Ti 49Ti 50Ti 51Ti 52Ti 53Ti 54Ti 55Ti 56Ti 57Ti 58Ti 59Ti 60Ti 61Ti
39Sc 40Sc 41Sc 42Sc 43Sc 44Sc 45Sc 46Sc 47Sc 48Sc 49Sc 50Sc 51Sc 52Sc 53Sc 54Sc 55Sc 56Sc 57Sc 58Sc
35Ca 36Ca 37Ca 38Ca 39Ca 40Ca 41Ca 42Ca 43Ca 44Ca 45Ca 46Ca 47Ca 48Ca 49Ca 50Ca 51Ca 52Ca 53Ca 54Ca 55Ca 56Ca
35K 36K 37K 38K 39K 40K 41K 42K 43K 44K 45K 46K 47K 48K 49K 50K 51K 52K 53K 54K
31Ar 32Ar 33Ar 34Ar 35Ar 36Ar 37Ar 38Ar 39Ar 40Ar 41Ar 42Ar 43Ar 44Ar 45Ar 46Ar 47Ar 48Ar 49Ar 50Ar
30Cl 31Cl 32Cl 33Cl 34Cl 35Cl 36Cl 37Cl 38Cl 39Cl 40Cl 41Cl 42Cl 43Cl 44Cl 45Cl 46Cl 47Cl
27S 28S 29S 30S 31S 32S 33S 34S 35S 36S 37S 38S 39S 40S 41S 42S 43S 44S 45S 46S
26P 27P 28P 29P 30P 31P 32P 33P 34P 35P 36P 37P 38P 39P 40P 41P 42P 43P 44P 45P
22Si 23Si 24Si 25Si 26Si 27Si 28Si 29Si 30Si 31Si 32Si 33Si 34Si 35Si 36Si 37Si 38Si 39Si 40Si 41Si 42Si 43Si
22Al 23Al 24Al 25Al 26Al 27Al 28Al 29Al 30Al 31Al 32Al 33Al 34Al 35Al 36Al 37Al 38Al 39Al 40Al 41Al
20Mg 21Mg 22Mg 23Mg 24Mg 25Mg 26Mg 27Mg 28Mg 29Mg 30Mg 31Mg 32Mg 33Mg 34Mg 35Mg 36Mg 37Mg 38Mg
20Na 21Na 22Na 23Na 24Na 25Na 26Na 27Na 28Na 29Na 30Na 31Na 32Na 33Na 34Na 35Na
17Ne 18Ne 19Ne 20Ne 21Ne 22Ne 23Ne 24Ne 25Ne 26Ne 27Ne 28Ne 29Ne 30Ne 31Ne 32Ne 33Ne
17F 18F 19F 20F 21F 22F 23F 24F 25F 26F 27F 28F 29F 30F
13O 14O 15O 16O 17O 18O 19O 20O 21O 22O 23O 24O 25O 26O
12N 13N 14N 15N 16N 17N 18N 19N 20N 21N 22N 23N 24N
9C 10C 11C 12C 13C 14C 15C 16C 17C 18C 19C 20C 21C 22C
8B 9B 10B 11B 12B 13B 14B 15B 16B 17B 18B 19B 20B 21B
7Be 8Be 9Be 10Be 11Be 12Be 13Be 14Be 15Be 16Be
6Li 7Li 8Li 9Li 10Li 11Li
3He 4He 5He 6He 7He 8He 9He 10He
1H 2H 3H 4H 5H 6H 7H
N
NEUTRON RICH
P
R
O
T
O
N
 
R
I
C
H
1H
4He
N
Tz = − 1
α
Figure 2.1: Nuclide chart
one or more electrons are removed while keeping the same amount of protons and
neutrons, we speak of ions and their degree of ionization. For the densities and tem-
peratures of interest, we can assume all elements are fully ionized, i.e. the net charge
is positive and equal to Z. Finally, by nuclide we understand any bounded array of
Z protons, Z electrons and N neutrons, i.e. A = Z +N baryons, and denote them
as A(Element name). For instance, 12C, 56Fe or 80Kr. In theory, the only difference
with the concept of elements is the prominent role the number of neutrons acquire:
for nuclides it is necessary to specify it, while for elements the number of protons is
enough. A visual classification of nuclides is made by constructing nuclide charts,
such as the one illustrated in Fig. 2.1. Along the vertical axis the number of pro-
tons increase and along the horizontal axis the number of neutrons. The grey band
at the middle is named as valley of stability since at Earth these nuclides do not
undergo β+/β− decays or have estimated lifetimes exceeding thousands of years. In
addition, these are the final states of nuclides at both proton- and neutron-rich sides
undergoing β+/β− decays. Given an α particle is a bounded state of two protons
and two neutrons (i.e. the nucleus of an 4He), nuclides having 2Z protons and 2N
neutrons are classified as α nuclides. Species with (N − Z)/2 = −1 are dubbed as
Tz = −1 and we remark their presence due to their involvement in the rp-process.
While protons and neutrons conform the baryonic sector of the composition, we
still have some words regarding the leptonic sector. Besides electrons and positrons,
neutrinos and anti-neutrinos play a key role in the evolution of the neutron star.
However, once the supernovae phase has finished and the proto-neutron star is born,
52
matter is transparent for them and they practically can escape from the interior,
carrying away energy in the process. As such, they are not considered into the
construction of the equation of state due to their practically null interaction with
the rest of particles. At high densities, in order to have equilibrium the presence of
muons is required, leading to the so-called npeµ matter which conforms the most
accepted model for neutron star composition at high density. The presence of hy-
perons, while theoretically viable, is observationally dubious at the moment given
the irresolution of the hyperon puzzle, consisting on the softening of the EOS (i.e. a
decreased dP/dρ) by the inclusion of hyperons, which reduces the maximum grav-
itational mass a star can reach to Mmax < 2M⊙, in disagreement with existent
inferences of neutron stars with masses of ∼ 2.08M⊙.
From Quantum Chromodynamics (QCD) and its experimental verifications, we
know protons and neutrons, contrasted against electrons, muons or neutrinos, are
not fundamental particles but have structure on their own, i.e. quarks and gluons.
Although their de-confinement from the neutrons and protons is a possible state of
matter at ρ ≥ ρ0, from current astrophysical observations the “ordinary” matter
at the core is still preferred, so in the rest of this work we shall not consider the
possibility of quark matter.
For theoretical purposes, we separate the neutron star layers at ρ ≤ 1011 g cm−3
into different regions, ordered according to the local density:
• Atmosphere. Occupies the region ρ ≤ 10−3 g cm−3, where nuclides can
be in different stages of ionization and where radiation escapes the star. As
described in Chapter 1, typically it is approximated as a functional relation
between the variables of interest (temperature, pressure, luminosity), or it can
also be computed - based on actual observations - and stored into tabular form
[ZPS96, HH09].
• Envelope. Between 10−3 and ∼ 108 g cm−3 we have this thin layer, composed
primordially of fully-ionized nuclides, electrons and photons. The aggregation
state of the massive particles moves from a gas-like below 104 g cm−3 towards
a lattice of nuclides imbued in a gas of degenerate electrons near 108 g cm−3.
Electrostatic interactions thus become important as we move towards the high
density sector, and depending on the temperature it is feasible to have a liquid
phase near 106 g cm−3. Precisely because of this within some contexts the
envelope is divided in two parts, one belonging to the atmosphere and the other
to the “ocean” at the surface of the neutron star [MC11, GBS+07, DG09].
• Outer crust. This layer occupies the region between ∼ 108 g cm−3 and ∼ 1011
g cm−3, i.e. the neutron drip point. In the outer crust we have a lattice of
nuclides imbued in a gas of degenerate electrons.
For numerical purposes, it is suitable to split the neutron star into two regions
only, the (numerical) envelope, consisting on the actual atmosphere, envelope and a
53
portion of the outer crust below 1010 g cm−3, and the numerical core, including the
rest of the star, i.e. outer and inner crust, as well as the core.
2.3.1 The envelope, ρ ≤ 108 g cm−3
In this portion of the star - the envelope - matter is composed of photons, Ntot(≥ 1)
different species of fully ionized nuclides and electrons. i.e. the composition of a
typical Sun-like star. From 10−3 g cm−3 (at the surface) to 108 g cm−3, at envelope-
crust boundary, the aggregation state varies from a gas-like (or plasma-like) to
a jelly-like environment where electrostatic interactions among ions and electrons
become prominent, eventually allowing the conformation of ionic lattice structures
above 108 g cm−3 and at moderate temperatures (i.e. 108 K). The prominence of
electrostatic effects is quantified throughout the electrostatic-to-thermal ratio, i.e.
the coupling parameter
Γ =
Z2e2
acellkBT
, (2.27)
where acell = (3ncell/4π)1/3 is a characteristic dimension associated with the local
number density of the cell ncell and Z is a local average charge. According to
numerical simulations, the crystallization of matter takes place at Γ ∼ 200, although
the exact threshold is sensitive to composition [BP13].
The gas-like environment can be formally described via the standard non inter-
active EOS, composed of the simple addition of the individual pressures of the ions,
described as an ideal gas, the free electrons obeying the Fermi-Dirac distribution and
the photons obeying a Planck distribution. On the other hand, the jelly-like environ-
ment requires the addition of at least three terms describing Coulomb interactions
among electrons (electron-electron), between electrons and ions (electron-ion) and
among ions (ion-ion). However, the overall contribution to thermodynamical func-
tionals is still small (corrections in the order of ∼ 10%) as to allow the retaining
of the additivity property for pressure and Helmholtz free energy. Furthermore,
among the three kind of corrections, ion-ion contributions has the largest impact at
above 104 g cm−3, hence this term receives more attention than electron-electron
and electron-ion.
Mass versus number normalization. Since me ≪ mu and neutrons are
typically bounded well below 108 g cm−3, it is viable to assume all mass is contained
in protons and neutrons and to write the rest-mass density as ρB = munB, with nB
the baryonic number density, and mp ≈ mn ≈ mu as the relative difference is of
≤ 2%.
In the study of stellar plasmas, the distribution of total mass or particles among
the different Ntot species leads to two different normalization schemes. The first,
frequently encountered in the context of nuclear reactions, prioritizes the distribution
of mass by introducing the mass density ρi for each species. A single fully ionized
54
nuclide i has Zj protons and Nj neutrons, or Aj = Nj + Zj baryons. This amounts
for a total mass of Mi = Zimp + (Ai − Zi)mn ≈ Aimu. In the thermodynamical
limit,
ρi := lim
(Ni,V )→(∞,∞)
Mi
Ni
V
= Aimuni, (2.28)
related to the mass fraction Xi and the mass abundance Yi,
Xi :=
ρi
ρB
= Ai
ni
nB
, Yi :=
ni
nB
=
Xi
Ai
, (2.29)
with ni the ionic number density, obeying the mass conservation
Nspecies
∑
i=1
Xi =
Nspecies
∑
i=1
YiAi = 1. (2.30)
Considering the Xi add up to unity, we can take the collection of Nspecies fractions
as weights to compute mean values, as for instance
⟨Z⟩ =
Nspecies
∑
i=1
ZjXj, ⟨A⟩ =
Nspecies
∑
i=1
AjXj, (2.31)
⟨Z2⟩ =
Nspecies
∑
i=1
Z2
jXj, ⟨A2⟩ =
Nspecies
∑
i=1
A2
jXj. (2.32)
From the individual ionic number densities nj, it is natural to define the number
density of total (fully-ionized) nuclides, or number density of total ions, as
nion =
Nspecies
∑
i=1
ni =
nB
⟨m⟩ion
(2.33)
1
⟨m⟩ion
=
Nspecies
∑
i=1
Yi, (2.34)
where ⟨m⟩ion is referred to as the ion mean weight. For plasmas, charge neutrality
holds. This imposes an additional relation between electrons and rest-mass,
ne =
Nspecies
∑
i=1
Zini =
nB
⟨m⟩e
(2.35)
1
⟨m⟩e
=
Nspecies
∑
i=1
ZiYi := Ye, (2.36)
with ⟨m⟩e the electron mean weight. For a mixture of 70% 1H, 28% 4He and 2% 56Fe,
for instance, ⟨m⟩e ≈ 1.177, i.e. Ye ≈ 0.85, while for pure-12C or pure-4He systems
we have ⟨m⟩e = 2 and Ye = 0.5.
55
The second renormalization scheme is frequent in the study of plasma interac-
tions. Here, the ion number densities ni are renormalized by introducing the number
fraction xj := nj/nion, resulting in an analogous to the mass conservation expression
as
Nspecies
∑
j=1
xj = 1. (2.37)
What is the relation between Xj and xj? By direct substitution, it is easy to see
xj = ⟨m⟩ionYj
= Ā′Yj
=
Xj/Aj
∑Nspecies
j=1 Xj/Aj
, (2.38)
i.e. the xj can be taken as a renormalized version of Xj, and the old ⟨m⟩ion now plays
the role of an “average mass number”. However, in this renormalization scheme the
proper average for the mass number, considering the weights are now the xi, is
⟨A⟩′ =
Nspecies
∑
j=1
Ajxj. (2.39)
While charge neutrality is still given by ne =
∑
j Zjnj, the xj’s induces a new
definition for the electron mean weight, now having the physical meaning of being
the inverse of the “average charge”, i.e.
ne =
Nspecies
∑
j=1
Zjnj = nion⟨Z⟩′ (2.40)
⟨Z⟩′ =
Nspecies
∑
j=1
Zjxj =
1
⟨m⟩′
e
. (2.41)
The coupling parameter. Old studies of plasmas consisted on modeling matter as
composed of a single-species, thus receiving the name one-component-plasma, OCP.
Modern studies, in contrast, focus on mixtures of species, i.e. multi-component-
plasma MCP. Within the particle number renormalization, it is viable to define an
appropriate sum of coupling parameters [HPY07, DG09]
Γeff =
Nspecies
∑
j=1
xjΓj =


Nspecies
∑
j=1
xjZ
5/3
j

Γe (2.42)
Γe =
e2
kB
(
4π
3mu
)1/3 (ρ/⟨m⟩e)1/3
T
≈ 2.2747 × 105 K cm g−1 × (ρ/⟨m⟩e)1/3
T
, (2.43)
56
with Γe the electron coupling parameter. For a pure-12C matter, for instance, at
ρ = 106 and T = 108 K we have Γe ≈ 0.18 and Γeff ≈ 3.57, practically little effect of
electrostatic interactions, while at T = 104 K we have Γe ≈ 1805 and Γeff = 3.5×104,
i.e. a carbon crystal assuming Γcryst ≈ 200.
The EOS - Helmholtz approach. Since ρB, T and the abundances Y are
usually taken as independent variables in standard stellar astrophysics, the ade-
quate thermodynamical potential is f̆ = f̆ideal + f̆interaction, where f̆ideal includes the
ion ideal gas, free electrons and photons contributions, while f̆interaction include the
Coulomb contributions. From the partial derivatives of f̆ , we recover the required
thermodynamical functionals (e.g. [JSB+21]):
P = ρ2
B


∂f̆
∂ρB


T,Y
s̆ = −


∂f̆
∂T


ρB,Y
ε̆ = f̆ − T


∂f̆
∂T


ρB,Y
c̆V = −T


∂2f̆
∂T 2


ρB,Y
χT =
T
P
(
∂P
∂T
)
ρB,Y
χρ =
ρB
P
(
∂P
∂ρB
)
T,Y
c̆P = c̆V +
χT
χ2
ρ
P
ρBT
∇ad =
PχT
TχρρBc̆P
,
noting that for P , s̆, c̆V and ε̆ the additive property of f̆ is preserved, while for the
rest of them the interaction and ideal terms are mixed. The justification for an addi-
tive f̆ or P , despite the intrinsically intensive nature of this thermodynamical func-
tional is related to the experimental Dalton’s law [BB09], affirming such addition is
valid as long as the individual components are non-interacting or weakly-interacting.
In a substantial amount of papers, the Helmholtz free energy F and other func-
tionals, such as the heat capacity, are expressed in units of NkBT , where N is taken
as the total number of ions (see for instance [BST66, SDD80, IIT87, CP98, PC13]).
The provided functional can be easily adapted into our notation as
f̆int =
kBT
⟨m⟩ionmu
fpapers
int . (2.44)
Non-interacting sector
As anticipated, in f̆ideal we include the contributions from photons, Nspecies different
species of ions and electrons, i.e. f̆ideal = f̆photons + f̆ion + f̆e, subject to mass
conservation, Eq. 2.30, and charge neutrality, Eq. 2.36. For photons we have
f̆photons(ρB, T ) = −4σSB
3ρBc
T 4, (2.45)
independent of the ion abundances Y . Sometimes, the constant at the front, in-
volving the Stefan-Boltzmann constant σSB, is rewritten in terms of the so-called
57
−5 −3 0 2 4 6 8 10
log10ρ
[
g cm−3
]
4
5
6
7
8
9
lo
g
1
0
T
[K
] P io
n
=
P ph
ot
on
P
io
n
=
P
e,
n
r
〈m〉i = 1.296
〈m〉e = 1.176
Γ e
ff
=
20
0
log10Pideal
[
erg cm−3
]
−5 −3 0 2 4 6 8 10
log10ρ
[
g cm−3
]
4
5
6
7
8
9
η
=
−
10
η
=
−
1
η
=
10
η
=
10
3
η
=
10
5
log10|η| [Adimensional]
6.75
9.00
11.25
13.50
15.75
18.00
20.25
22.50
24.75
27.00
−2
−1
0
1
2
3
4
5
6
7
−4.32
−3.60
−2.88
−2.16
−1.44
−0.72
0.00
0.72
1.44
Figure 2.2: Left panel: contour plots for the non-interacting EOS, assuming a Solar-
like composition of 70% 1H, 28% 4He and 2% 12C. Right panel: log10♣η♣ contours,
considering the same composition as in left panel.
a-radiation coefficient, a := 4σSB/c. Via partial differentiation, we see ε̆photons =
aT 4/ρB and Pphotons = ρBε̆/3, i.e. photon pressure is independent of the local bary-
onic density and strongly dependent on temperature.
For ions’ free energy we have two terms: one from rest-mass and the other from
the kinetic terms (see the discussion around Eq. 2.22). However, the rest-mass term
solely depends on Yi, thus having no impact for other functionals such as P , c̆P and
f̆ion. Therefore, ions’ free energy is given by
f̆ion(ρB, T,Y ) =
kBT
mu
Nspecies
∑
j=1
Yj
[
ln
(
nj
nQ,j
)
− 1
]
(2.46)
nj =
ρBYj
mu
, nQ,j =
(
mjkBT
2πℏ2
)3/2
, (2.47)
with nj the j-th ion number density, and mj its corresponding mass. By partial
differentiation, we recover the standard relations [KWW12]
Pion =
ρBkBT
⟨m⟩ionmu
, ε̆ion =
3
2
Pion
ρB
. (2.48)
As discussed elsewhere [Hua87, JEL96], for free electrons there does not exist
an exact expression for functionals such as Pe(ne, T ) since (a) the Fermi-Dirac in-
tegrals take µe and T as arguments, not ne, which is actually a functional of µe
and T as well, and (b) the evaluation of these integrals must be done numerically.
As a consequence of this complication, for astrophysical calculations several options
58
have been introduced and employed, for instance (a) compute these integrals nu-
merically and store the data as tables, implementing interpolating subroutines or
fitting functionals based on them [TS00], (b) employ analytic expressions, resulting
from an exact integration of ne(µe, T ) and Pe(µe, T ) under certain limiting cases
[Nad74, Pac83] or (c) attempt to construct a fitting formulae based on the limiting
cases [EFF73, JEL96]. Among these limiting cases, the most important is the exis-
tence of a region where functionals are approximately independent of temperature,
i.e. the degenerate limit, a general feature of fermionic systems. Intuitively, it can
be understood in terms of the ratio T/TF, with TF the Fermi temperature of the
gas. At T/TF ≪ 1, thermal effects become unimportant and functionals are of the
form F (ρB, T ) ≈ F1(ρB) + c1(T/TF)s with s ∈ N, c1 constant and F1 a functional of
rest-mass. Due to the smallness of the temperature ratio, changes in temperature
are practically decoupled from changes in pressure. The dependency of TF with ρB
makes this approximation to break down as long as T/TF ≫ 1, and calls to con-
sider another parameter to quantify the degeneracy of the gas. This is done via a
redefinition of the chemical potential µe in terms of the adimensional parameter
η =
µe −mec
2
kBT
. (2.49)
We say electrons are degenerate if η ≫ 1, and otherwise if η ≪ 1, region where
thermal effects become important. Since µe can be negative, η as well, in good
agreement with the fact electrons behave as an ideal gas, i.e. Pe ≈ nekBT , as ρ ≪ 102
g cm−3 and T > 106 K. In addition to degeneracy, for electrons we can distinguish
two limits depending on whether the kinetic energy exceeds the contribution of rest-
mass or not. This is decided via the kinetic-to-rest ratio x := p/mec. If x ≪ 1,
rest-mass contribution exceeds the kinetic energy and we say electrons are non-
relativistic. On the other hand, if x ≫ 1 the energy is predominately kinetic and thus
electrons are relativistic. If electrons are highly degenerate, these relativistic limits
allow to construct exact expressions for the pressure, useful for back-of-the-envelope
calculations: in first place, ρB ∝ ⟨m⟩ex3 [Hua87], i.e. extremely relativistic electrons
are usually associated with high-density environments and viceversa. Additionally,
in the non-relativistic case, Pe ≈ 1013(ρB/⟨m⟩e)5/3 (c.g.s. units), while for the
extreme relativistic case Pe ≈ 1015(ρB/⟨m⟩e)4/3 (c.g.s. units) [Pac83]. For instance,
at ρB = 108 g cm−3 and for pure carbon-12 matter we have Pe ≈ 1025 erg cm−3.
In the present work, for the free electrons’ EOS we adopt the fitting functionals
from [JEL96], valid for a large span of densities and temperatures, i.e. ∼ [10−10, 1010]
g cm−3 and ∼ [103, 1010] K respectively due to their good agreement with respect
to numerical evaluation of the integrals. These fits, however, are still functionals
of T (via the adimensional t = kBT/mec
2) and µe via an f parameter, a strictly
positive functional of η, thus requiring the inversion of the functional ne(f, t) in order
to extract, given a fixed ρB, the correct f for the remaining functionals (pressure,
internal energy, etc).
59
In order to gain further insight on the non-interacting EOS, let us discuss its
main functionals employing a fixed, Solar-like composition of 70% 1H, 28% 4He and
2% 12C. In Fig. 2.2 we display the pressure Pideal - resulting from the addition of Pion
(Eq. 2.48), Pphotons and the fitting formulae for Pe - and the η parameter as contours
in the ρB − T plane. By inspection of both panels, we see a correlation between
electron degeneracy and the Pion − Pe,non-rel boundary. At the right hand side of
this threshold, in left panel of Fig. 2.2, electrons provide the largest contribution to
pressure. From the behavior of η in the right panel, we see electrons must have a
high degree of degeneracy to dominate the pressure over the ideal gas contribution
from ions. Moreover, since the envelope’s thermal profile resides in this region of
the ρB −T plane, we deduce electron’s pressure controls the evolution of the largest
portion of such profile. In the other extreme of the left panel, below 10−1 g cm−3
and above 106 K, photons start dominating the contribution to pressure, going as far
as 1020 erg cm−3 at extremely low densities (i.e. 10−5 g cm−3) and 109 K, although
in this region e+ − e− pair production becomes present as well.
Now let us explore the sensitivity of the EOS to temperature and density via
the χT and χρ coefficients displayed in Fig. 2.3. As expected from the behavior in
pressure, χT = 1 in a diagonal band going from 10−5 g cm−3 and ∼ 104 K to 106 g
cm−3 and 109 K. As electrons become degenerate and their contribution to pressure
exceeds that from ions, χT slowly converges towards 10−3 near the crystallization
region. On the other hand, for temperatures above this χT = 1 contour we see
photon pressure rapidly taking the leading role, thus producing a drastic decrease
in χρ, as shown in the right panel of Fig. 2.3. Regarding the degenerate electron
region, we see the contours adopting a triangular-shaped pattern.
Finally, let us comment on the heat capacity densities, Fig. 2.4. As a consequence
of electron degeneracy, we see cV ≈ cP in the electron-dominated region of the ρB−T
plane, in good agreement with the theoretical result on heat capacities. Since cP
contains non-linear terms related to partial derivatives, the behavior with respect to
cV is entirely different in the η < 0 region of the ρB − T plane, where cV exhibits a
ρ-independent tendency above 106 K as a consequence of the ideal gas behavior and
the progressive influence of the photon pressure. On the other hand, cP drastically
increases up to 1030 erg K−1 cm−3, which we can easily attribute to the smallness
of χρ.
60
−5 −3 0 2 4 6 8 10
log10ρB
[
g cm−3
]
4
5
6
7
8
9
lo
g
1
0
T
[K
] 3.
0
Id
ea
l g
as
, 1
.0
10
−
1
10
−
3
Γ e
ff
=
20
0
χT
−5 −3 0 2 4 6 8 10
log10ρB
[
g cm−3
]
4
5
6
7
8
9
Id
ea
l g
as
, 1
.0
10
−
110
−
3 4/
3
〈m〉i = 1.296
〈m〉e = 1.176
Γ e
ff
=
20
0
χρ
0.00
0.44
0.88
1.32
1.76
2.20
2.64
3.08
3.52
3.96
0.00
0.18
0.36
0.54
0.72
0.90
1.08
1.26
1.44
1.62
Figure 2.3: χT (left panel) and χρ (right panel) for the non-interacting EOS, assum-
ing a Solar-like composition of 70% 1H, 28% 4He and 2% 12C.
−5 −3 0 2 4 6 8 10
log10ρB
[
g cm−3
]
4
5
6
7
8
9
lo
g
1
0
T
[K
]
8.0
12.0 14.0 16.0
Γ e
ff
=
20
0
log10cV
[
erg K−1cm−3
]
−5 −3 0 2 4 6 8 10
log10ρB
[
g cm−3
]
4
5
6
7
8
9
8.0
12.0 14.0 16.0
〈m〉i = 1.296
〈m〉e = 1.176
Γ e
ff
=
20
0
log10cP
[
erg K−1cm−3
]
3.09
4.62
6.15
7.68
9.21
10.74
12.27
13.80
15.33
16.86
2.88
5.94
9.00
12.06
15.12
18.18
21.24
24.30
27.36
30.42
Figure 2.4: Heat capacity densities at fixed volume (left panel) and fixed pressure
(right panel) for the non-interacting EOS, assuming a Solar-like composition of 70%
1H, 28% 4He and 2% 12C.
61
2.3.2 Outer and inner crusts, ρ ∈ [108, 1014] g cm−3
Outer curst
27.5 28.0 28.5 29.0 29.5 30.0
log10P
[
erg cm−3
]
9.75
10.25
10.75
11.25
11.75
lo
g
1
0
ρ
[
g
cm
−
3
]
Neutron drip threshold
HZD
BPS, AME2020
BPS, HFB26
26 27 28 29 30
log10P
[
erg cm−3
]
18
21
24
27
30
33
36
Z
N
e
u
tr
o
n
d
ri
p
th
re
sh
o
ld
HZD
BPS, AME2020
BPS, HFB26
Figure 2.5: EOS for the outer crust near the neutron drip line, considering two
different models for nuclei masses.
Aside from present uncertainties in nuclear physics (such as binding energies),
between 108 and 1011 g cm−3 the qualitative aspects of the EOS are relatively
straightforward: ions conforming a lattice imbued in a relativistic gas of highly
degenerate electrons, according to the behavior of the η parameter in Fig. 2.2
[BPS71, HZD89, HPY07, JSB+21]. Since nuclides conform a crystalline structure,
the Wigner-Seitz approximation is employed to impose charge neutrality inside a cell
of characteristic dimension ai = [3/(4πni)]
1/3, which gives rise to a lattice pressure
PL = −fcZe2ni/a (c.g.s. units for e), where fc is a structure factor associated with
the kind of lattice chosen5. In this picture of the outer crust, however, there still
prevail several critical questions:
(i) To which extent are thermal effects negligible?
(ii) What kind of species are there and which are their mass fractions?
(iii) What kind of crystalline structure are they arranged into?
Regarding question (i): an inspection to the left panel of Fig. 2.2 shows that for
T ≤ 109 K and ρ → 1010 the pressure is practically supplied by very degenerate
electrons, as indicated by the high values of η in the right panel of Fig. 2.2. While
claiming P = Pe is inaccurate since electrostatic interactions - neglected in the
aforementioned plot - are likely to take a more prominent role in the outer crust
51.4507 for a pure spherical case, 1.4442 for bcc
62
than in the envelope due to the gas-to-liquid-to-solid phase transitions, we can expect
the outer crust EOS to actually decouple pressure from temperature, i.e. structure
becomes independent of thermal effects, an advantage usually exploited to simplify
the simulation of thermal evolution (e.g. Chapter 1). This characteristic is shared
with the inner portions of the star (inner crust and core) where neutrons and protons,
being fermions as well, have large Fermi temperatures and thus require T ≥ 1010 K
to require the presence of thermal effects in their description. Such decoupling allows
some simplification in the process of computing the EOS: regarding the pressure, we
end up with an expression of the form P (ρ), now with ρ = ε/c2 instead of ρ = ρB.
On the other hand, second order quantities such as the specific heat do retain a
dependency on temperature. Indeed, recalling at high degeneracy electron’s EOS
behaves as a series in powers of (T/TF )2, the specific heats behaves as ∝ T/TF .
However, given T/TF ≪ 1 it is customary to regard this regime as of T → 0 and
take cV,e ≈ cP,e ≡ ce, i.e. we can speak of a single heat capacity (see Appendix A.3
for further details). Such T → 0 construction is customary referred to as the ground
state of matter [BPS71].
To address questions (ii) and (iii), the knowledge on the composition at each den-
sity or pressure is required. In principle, it can actually be a mixture of species with
different abundances, as shown in the case of accreting systems [HZD89, GBS+07],
or it can be a composition enriched in only one nuclear species, as for instance the
“canonical” neutron star envelope of pure 56Fe matter.
Single nuclear species. This scenario has been kept as the standard in the
field given the simplicity of computing the EOS for single-composition layers at
ρ ≤ 1011 g cm−3. Since pressure and temperature now take a decoupled role from the
thermodynamical point of view, to construct the outer crust EOS and its derivatives
it is useful to shift from a Helmholtz to a Gibbs free energy formalism since the
independent variables are now temperature (with T → 0), pressure and composition,
for the latter assuming at each pressure there exists only one kind of nuclear species
[BPS71, HZD89, HPY07]. Given such single-species character of their ground-state
EOS, the usage of density as independent variable is discouraged given the prominent
role of discontinuities: certainly, given the continuous character of the pressure
(related to the structural Φ function, see Chapter 1), as well as the predominance of
electron’s pressure Pe = Kρ/⟨m⟩e)4/3, by computing its differential and taking the
limit dPe → 0 we obtain a density jump with each discontinuity in composition, i.e.
∆ρ/ρ = ∆⟨m⟩e/⟨m⟩e.
Concerning these single-composition approximations, it is important to notice
their ground-state search pursues to describe two limiting scenarios, which nonethe-
less cover the majority of neutron star cases for which thermal evolution can be
addressed:
(a) Composition in equilibrium [BPS71], presumed to be of isolated objects
which are cooling after their birth, i.e. matter has reached an equilibrium
63
against nuclear reactions such as β-decays or captures. For more than twenty
years, this model - shortly referred to as BPS given the initial letter of authors’
surnames - became known as the “canonical” outer crust for neutron stars.
According to BPS, for the construction of the ground-state EOS we require
a set of pairs A =
{
(A(j), Z(j))
}M
j=1
(with A(j) ≥ 56 and for which masses
are known either experimentally of theoretically), a grid of pressures P =
{
P (i)
}K
i=1
. ∀P (i) ∈ P, and to perform the following steps:
(i) n(i)
e and G(A(j), Z(j), n(i)
e )/A(j) are calculated for each member of A. The
former is obtained by solving
P (i) = Pe(n
(i)
e ) + PL(niN) , (2.50)
n(i)
e = Z(j)n
(i)
N , (2.51)
while the latter is calculated from
G(A(j), Z(j), n(i)
e ) =
1
n
(i)
N
[
ρ(i)c2 + P (i)
]
, (2.52)
ρ(i) =
1
c2
[
n
(i)
NM(Z(j), n
(i)
N ) + εe(n
(i)
e )
]
(2.53)
M(Z(j), n
(i)
N ) = WL(n
(i)
N ) +WN(A(j), Z(j)) , (2.54)
where the nuclide mass is expressed as WN , and ρ(i) is now the associated
mass-density considering relativistic terms beyond the rest-mass. For
explicit expressions of εe and Pe, see [ST83].
(ii) The pair producing the lowest G(A(j), Z(j), n(i)
e )/A(j) is chosen as the
ground state composition for P (i), and the rest of thermodynamic vari-
ables can now be deduced from such G. As ρ ̸= nbmc
2, now the baryon
number density for the j-th step is
n
(i)
b =
A(j)n(i)
e
Z(j)
. (2.55)
In Fig. 2.5 we plot the resulting BPS-EOS, as generated with the algorithm
described above6. In the left panel we show density against pressure, while in
the right panel we observe the change in charge number as a function of pres-
sure as well. For the calculations two different sets of nuclear binding energies
were employed, both combining experimental data and theoretical calcula-
tions: those from the recent AME2020 [WHK+21] and those from the old
AME2012, dubbed as HFB26 [GCP13] given the involvement of the Hartree-
Fock-Bogoliubov model in the computation of nuclear masses. In the right
6The corresponding numerical code was developed by the author of this work.
64
panel we observe some discrepancies on the Z minimizing the Gibbs free en-
ergy around 1027 erg cm−3 and at 1025 erg cm−3, where the updated data
suggests 56Ni is the ground state of matter instead of 56Fe. On the other hand,
these discrepancies have little impact over ρ(P ), which follow an almost simi-
lar track nevertheless sensitive to composition discontinuities as expected from
the ∆ρ/ρ = ∆⟨m⟩e/⟨m⟩e estimation.
(b) Composition out of equilibrium [HZD89, HPY07], for recurrently accret-
ing neutron stars where these deep layers, composed of ashes from previous
nuclear burning episodes, are compressed due to the piling up of material at
the surface and thus induce pycnonuclear reactions. While this model still
takes place within the Gibbs free energy formalism, the changes in composi-
tion are not dictated by minimization of G/A but instead by electron captures
[Sat79, HZ90a, HZ90b, HZD89]. The picture is as follows: a mass element of
fixed A is pushed inwards the star due to mass accretion. On its downward
path, it is possible that Z changes each time that a non-equilibrium beta cap-
ture (Z → Z − 1) takes place. Being an unstable state, a second electron
capture must occur at the same pressure, i.e. Z − 1 → Z − 2. This lower-Z
ion keeps traveling inwards with successive electron captures dictated by this
stability criterion, until it eventually reaches the neutron drip point, where its
chemical potential is equal to the rest-mass energy of the neutron. The nu-
merical implementation for constructing the corresponding EOS - dubbed as
HZD-EOS - goes as follows: we now consider a set with fixed A and different
Z, which we shall denote as Z. We still have a grid of pressures P , and at
each of them charge neutrality holds as well. For P (i), we impose an initial
Z(i) ∈ Z (although this proton number is not the minimum of the set), solve
for n(i)
e from Eq. 2.51 and study the sign of the beta-capture criterion (which
is valid ∀P (j) ∈ P):
Q = M(Z(i) − 1, n
(i)
N ) −M(Z(i), n
(i)
N ) − µe(n
(i)
e ) . (2.56)
If sgn(Q) > 0, we move to the next pressure in the grid with Z(i+1) = Z(i). On
the other hand, if sgn(Q) ≤ 0 Z(i) must be replaced with Z(i) − 2 and n(i)
e is
recalculated. This same procedure is repeated until the full grid of pressures
is exhausted or the neutron drip point is reached, whichever occurs first.
On Fig. 2.5 we contrast the HZ and BPS EOSs, the former constructed with the
recently described algorithm7. While their ρ(P ) exhibit minor discrepancies such
as the location of the jumps, and thus put into evidence the major role played by
the electron’s pressure, the changes in composition result in low-Z matter for the
HZD-EOS in contrast to the high-Z matter of both BPS-EOSs.
7Again, the numerical code was developed by the author of this work.
65
Multiple nuclear species at each pressure. If, however, one is following
the evolution of the chemical composition then for the outer crust it is still viable
to restore to EOSs such as Skye [JSB+21] or PC [CP98], based on the electrostatic
interactions briefly described in the previous section, given their interacting sector
take into account the transition from liquid to solids and allow multi-species sys-
tems. When employing these EOSs, we must be aware of (i) limiting their range of
validity to ρB ≤ 1011 g cm−3, given the neutron drip threshold, and (ii) the original
composition of the functionals for which crystallization effects were computed, a
limitation shared with the conductivity in the outer crust as well. Historically, this
limitation can be appreciated by the custom introduction of impurity parameters in
the description of the crust, emerging from the idea that single-composition layers
(formed during the neutron star birth) might contain small amounts of different
nuclides.
It must be stressed out that, despite the disagreement of the models on the
composition, there exists agreement on the interactions governing the outer crust,
i.e. almost free electrons and nuclides in lattices. Equally important is to stress
out that, despite such similarity on the governing interactions, quantities such as
released heat and their conductivity vary from model to model as well since, in turn,
they critically depend on the chemical composition of the outer crust, a source of
uncertainty given the complete accretion history of a given neutron star is unknown.
Inner crust
Between 1011 and 1014 g cm−3 most of the existent models predict a similar behavior
in the P − ρ plane, in spite of the discrepancies on the exact composition and in the
arrangement and shape of ions [Lat12, LRP93, NV73]. An important prediction of
these models is the gas of free neutrons, which appears since the chemical potential
of these particles is µn ≈ mnc
2. The threshold density of this process, commonly
referred as neutron drip point occurs at ∼ 1011 g cm−3, although the exact value is
model-dependent. In this region, however, temperature effects are not negligible for
the purpose of describing thermal evolution (e.g. [KWW12, JEL96]).
2.4 Thermal conductivity and opacity
In the standard non-relativistic analysis of the radiative flux, thermal conductivity
Krad is defined in terms of standard thermodynamical variables as
Krad :=
16σSBT
3
3ρBκrad(ρB, T,Y )
, (2.57)
where the functional κrad is defined as the opacity. From first principles we know the
mean free path lfree expresses the characteristic distance energy carriers can move
66
10−3 10−2 10−1 100
kBT/mec
2
10−1
100
l−
1
eγ
σ
−
1
T
h
om
so
n
n
−
1
e
10 7K
10 8K
10 9K
η ≤ −10
η = 0
η = 5
Weaver
Poutanen
Paczynski
Figure 2.6: Contours for the product of the mean free path multiplied by the cross
section and the electron number density, for different fits for the electron opacity as
a function of temperature. Based on Fig. 3 from [Pou17].
between collisions, thus we can also write opacity as κ := (ρBlfree)
−1. In the presence
of conduction, as in neutron star’s crust, the associated flux also contributes with an
associated thermal conductivity, Kcond, which can be parametrized in a similar way
as Eq. 2.57 introducing an opacity due to conduction κcond. Since fluxes are additive
quantities, in the case where both radiative and conductive fluxes are present it is
customary to define a single thermal conductivity and effective opacity,
K =
16σSBT
3
3κeffρB
(2.58)
1
κeff
=
1
κrad
+
1
κcond
, (2.59)
where the addition rule for opacities resembles that of parallel resistors.
While the distinction of radiative and conductive opacities is useful for theoretical
purpose, in practice both quantities face the same difficulties for their obtention as
opacity is not a directly observable quantity. Indeed, as lfree is completely dependent
on the exact composition of matter and all the interactions among species enclosed
in their respective cross-sections, opacity is necessarily a functional of (ρB, T,Y ),
thus demanding the usage of either purely theoretical models, or extrapolations
from experimental constraints. In astrophysical applications this problem is even
more accentuated by the apparent circularity of the situation (using for a proof
the elements which are intended to be proven). While such degeneracy can be
broken from independent analysis of stellar surfaces in objects like our own Sun,
stellar interiors are slightly more complicated as they are not directly observable.
67
Consequently, under such scenarios analytic models become indispensable tools for
stellar evolution, representing exceptional cases for the common advise of exclusively
employ first-principle, self-consisntent calculations [CKM+16], [PGC17].
As the calculation of stellar opacities is an extensive subject deserving its own
chapter due to the large amount of subtleties, practically an art by itself (e.g.
[Car76],[PGC17]), here we limit to discuss their very essential features.
• Radiative opacity. Also known as Rosseland’s, this functional takes into
account the effects of bremsstrahlung (free-free electron transition) and of
γ-electron scattering, referred to as Thomson’s and Compton’s in the non-
relativistic and relativistic regimes respectively. Formally speaking, this opac-
ity is the integral of the harmonic mean of the individual contributions, al-
though for practical purposes it is possible to compute each one of them sep-
arately, then add them up as (e.g. [PY01]) κrad = (κff + κes)A(κff, κes, T ). As
customary, free-free opacity consists on the sum of individual contributions,
one from each nuclei species and all of them factorized in terms of the Gaunt
factor gff(A,Z, ρB, T ), as reported in one of the most cited expressions, e.g.
[SBCW99], emerging from fits to old numerical calculations. Regarding γ-
electron scattering, to date one of the best fitting formulae, correctly covering
the Thomson’s and Compton’s regimes was provided by [Pou17], starting from
self-consistent calculations and leading to improvements on previous fits for
both sectors as discrepancies are considerable with old approaches, as illus-
trated in Fig. 2.6 for a selected collection of η.
Aside from these analytic fits, the results of self-consistent calculations have
been reported in tabulated form by several groups of work, such as the Opac-
ity Project [BBB+05] [MSB+07], the Lawrence Livermore National Labora-
tory OPAL [RI92], [IR96], and at Los Alamos National Laboratory [MAC+95]
[CKM+16], where the opacities correctly account for all the energy levels, ion-
ization degree and local ρB−T conditions. For astrophysics, however, the most
significant defect of these works is both abundances and isotopes are specified
beforehand, limiting their range of applicability in spite of the Principle of
Corresponding States (e.g. [Gug45] and [RI92]) as it demands the presence of
the same ions in any situation to be valid. A similar problem might arise with
the extension of the covered range in the ρB −T plane, which is not necessarily
homogeneous for all compositions.
• Conductive opacity. As previously described, this concept and its functional
form is only introduced to simplify the definition of a total opacity. However,
typical works on the subject focus on the calculation of the associated thermal
conductivity of electrons instead [SY06], [CPP+07], [DG09], i.e.
Kcond =
16σSBT
3
3ρκcond
, (2.60)
68
which can be computed from first principles via
Kcond =
aTk2
Bne
m∗
eν
(2.61)
where ν is the effective electron scattering/collision frequency, and a is a nu-
merical factor depending on the relativistic status of the electrons. In practice,
ν = νee + νei, i.e. the individual frequencies of electron-electron and electron-
ion collisions are simply added, a good approximation within the context of
neutron stars given conduction takes place at high densities, i.e. ρ ≥ 105 g
cm−3, where electrons are highly degenerate, i.e. η ≫ 1 [HL69, Zim01] (see the
right panel of Fig. 2.2). Consequently, the majority of works on this subject
provide analytic fits for the associated frequencies as we can always construct
κcond employing Eqs. 2.60 and 2.61. Thus, for κcond we consider the fit for νee
from [SY06], and the fit from [SBCW99] for νei.
2.5 Particle reactions
Let
∑N
j=1 aj → ∑M
j=1 bj be an arbitrary interaction process, where M particles, the
products, emerge as a result of the interaction among N particles, the reactants.
The associated probability per unit volume and time for this process to happen, the
reaction rate, can be written in formal terms as [GW04, dDGFJ14]
r∑N
j=1
aj→
∑M
j=1
bj
= nBR∑N
j=1
aj→
∑M
j=1
bj
=
∫ N
∏
i=1
dni P


N
∑
j=1
aj →
M
∑
j=1
bj

 , (2.62)
where the measure of integration includes the incoming particles’ number densities,
and the P functional contains all the information related to the nature of the in-
teractions among particles, and we have explicitly stated its relation with respect
to the rate R discussed in Chapter 1. Naturally, the reaction rate for the inverse
process, the inverse reaction for shortness, can be written in a similar fashion consid-
ering the replacement M → N in the measure of integration and the appropriate P .
By construction, r can be expected to be a functional of temperature T , rest-mass
density ρB, individual abundances and their charge and mass numbers (Yi, Zi, Ai)
(∀i ∈ ¶1, . . . , N +M♢). Independently of the individual abundances, by employing
dimensional analysis we can associate each reaction rate with an intrinsic lifetime
τ−1 = ρBNA⟨σv⟩T .
From a theoretical point of view, it is plausible to consider P ∝ ♣MNM ♣2, where
MNM is the matrix element of the process, as well as P ∝ σ, with σ the total cross-
section of the process due to σ ∝ ♣MNM ♣2 [GW04, dGFJ10]. As one might expect,
as N increases, r decreases, as the available surface for N particles to interact
simultaneously becomes smaller. We can get a more formal glimpse of this by
69
inspection of MNM : for two-body processes, it is equal to the sum of contributions
of the form Mif (Ei−E)−1; for three-body processes, it is the sum of products of two
such terms. This trend implies, in energy space, MNM ∝ E1−N , meaning that only
low order terms, such as one- and two-body interactions, are important. However,
since ni ∝ nb ∝ ρ in dense environments the probability of P -body interactions
(P > 2) becomes significant as the proportionality ρPB can overcome the damping
due to E1−P of the matrix element.
Although the computation of r strongly depends on the individual distribution
functions and the nature of the interactions, locally it can be expressed as
r = r0
N
∏
i=1
Y si
i ρ
ψρ
B TψT ,
where si is the total number of reactant particles from the i-th species. This form
introduces a first classification for reaction rates: if ψT ≫ ψρ, the reaction is referred
to as thermonuclear, a behavior largely seen in reactions taking place between ρB ≤
107 g cm−3 and T ≥ 104 K. On the other hand, if ψρ ≫ ψT the reaction is labeled
as pycnonuclear (from the greek πυκνoς, translated as dense or compact [Cam59]),
a behavior typically taking place above 108 g cm−3 and below 107 K. From a very
strict point of view, there exists neither a purely thermo- nor a purely pycno- nuclear
reaction, although these categories are useful to understand which thermodynamical
DoF is locally the most influential for r.
Several published works have shown the feasibility, when the involved particles
are nuclides and photons, of splitting the reaction rate into three terms: one purely
thermonuclear rthermonuclear, regarded as the main part of the rate, one purely pyc-
nonuclear, and an enhancement factor due to the electrical influence of the surround-
ing cloud of electrons, i.e. screening effects. For the majority of astrophysical appli-
cations, well below 108 g cm−3, pycnonuclear effects are almost negligible and such
contribution is simply taken as equal to 1. We must note, however, that such split-
ting might be of limited use when the number of reactants exceeds two, e.g. [FL87].
Regarding the purely thermonuclear part, an extensive compilation of them, with
excellent results for numerical applications, can be found in the [JIN22, CAF+10]
library, where the following parametrization is adopted:
rthermonuclear =
M
∑
j=1
exp[aj0 + aj1T
−1
9 + aj2T
−1/3
9 (2.63)
+ aj3T
1/3 + aj4T9 + aj5T
5/3 + aj6ln(T9)] , (2.64)
where M differs from rate to rate, depending on the amount of resonances.
Associated to each rate is the Q-value of the reaction, formally defined in terms
of the rest-mass of the reactants and products, denoted as maj
and mbj
respectively,
70
as
Q∑N
j=1
aj→
∑M
j=1
bj
=


N
∑
j=1
maj
−
M
∑
j=1
mbj

 c2, (2.65)
it serves to quantify the amount of gained (Q > 0) or lost (Q < 0) energy by the
system due to the N -body interaction. The first scenario, as it provides energy for
the system, is called exothermic, in contrast with the later which is referred to as
endothermic as it needs to be provided with energy to take place.
For nuclides, having Z protons and electrons, N neutrons (i.e. A = N + Z
baryons), their rest-mass is expressed as
mj(Aj, Zj) := Zjmp + Zjµe + (Aj − Zj)mn − B(Aj, Zj)/c
2,
with B(A,Z) the binding energy and µe the electron’s chemical potential (see Eq. 2.23
and the discussion below for the importance of keeping terms associated with mo-
mentum for electrons). Due to baryon conservation, if all reactants and products
are nuclides, the Q value is simply the difference between sums of binding energies.
For electroweak reactions, on the other hand, besides this difference the (mp−mn)c2
term emerges as a consequence of n ↔ p transmutations. An alternative to the bind-
ing energy - and reported in tables as well - is the mass excess ∆m = mj − Ajmu,
which is the difference between the actual rest-mass and the baryonic rest-mass
a particle with the same number of nucleons, all with masses mu, would have
[GCP13, WHK+21]. Due to the definition of the atomic mass unit, ∆m = 0 for
12C [Tay09].
An important classification scheme for nuclear reaction rates is set considering
the number of reactant species:
• One-body: Also referred to as particle decay. In this category we have photo-
disintegration, a + γ → ∑M
j=1 bj, and β-decays which are mediated by the
electroweak force. In a broad sense, the correspondent reaction rate adopts
the form ρBλ(ρB, T,Y ), a simplification which allows to speak in terms of
lifetimes τ = λ−1.
• Two-body interactions. Practically most of the reactions of interest belong
to this category. From the discussion regarding ♣MNM ♣2, it can be inferred P -
body (P > 2) interactions can be understood in terms of successive two-body
interactions, pointing out once more the great importance such processes have.
Considering two particles in the reactant side brings an important simplifica-
tion to the reaction rate: either from first-principles or by reasoning in terms
of fluxes and targets, it can be shown Eq. 2.62 adopts the more familiar form
r
a1+a2→
∑M
j=1
bj
=
∫
dna1
dna2
σ(♣va1
− va2
♣) ♣va1
− va2
♣. (2.66)
71
Clearly, the evaluation of this integral depends on the individual velocity dis-
tributions of a1 and a2. In this regard, the most important limit case, due
to the large range of applicability, is that when both particles obey Maxwell-
Boltzmann statistics, allowing us to work with the center-of-mass energy as
integration variable and to write the reaction rate as8 [HM06]
r
a1+a2→
∑M
j=1
bj
= Ya1
Ya2
ρ2
BN
2
A⟨σv⟩T
where
⟨σv⟩T =
(8/π)1/2
M ′1/2(kBT )3/2
∫ ∞
0
dE E σ(E) exp (−E/kBT ) ,
M ′ =
ma1
ma2
ma1
+ma2
.
Among the several possibilities of reactions between two nuclides, the most
frequently encountered in literature are those between lighter ones only, e.g.
p(p, e+νe)
2H or α(α, p) 7Li, as well as those involving one nucleon and a heavy
(m ≫ mp) nuclide, as for instance 12C(p, γ) 13N or 55Fe(n, γ) 56Fe. This can be
justified from the fact that, the heavier both nuclides’ masses are, the shorter
the probability of interaction becomes under typical conditions for temperature
(≤ 109 K) and density (≤ 108g cm−3). As we approach or exceed these limits,
reactions such as 12C +12 C → 20Ne + α adopt a more prominent role in the
evolution of the system as long as their abundances are nonzero. Nevertheless,
we can associate most of the two-body reactions among nuclides with one of
the following categories:
1. Proton captures (p,γ). As its name suggests, we have
AZ + p → A+1(Z + 1) + γ,
with
Q(p,γ) = B(Z + 1, A+ 1) − B(Z,A).
2. α captures (α,γ). Here we have
AZ + α → A+2(Z + 2) + γ,
resulting in
Q(α,γ) = B(Z + 2, A+ 2) − B(Z,A) − B(α).
8In these notes, we follow the convention of replacing mu by N−1
A as their numerical values are
reciprocal.
72
3. (α,p) transmutation. In this case,
A+3(Z + 1) + p → AZ + α,
Q(α,p) = B(A,Z) + B(α) − B(A+ 3, Z + 1).
• Three-body interactions. Properly speaking, most of the reactions in this
category emerge as subsequent two-body processes, either to reduce the size of
networks as for instance happens when one studies A(α, p)B(p, γ)C under the
assumption DtYB = 0 [WZW78, NTM85, GZ05], or due to the shortness of the
intermediate state, as in the case of the 3α reaction which is formally defined
as α + α → 8Be, 8Be + α → 12C, but for simplicity is taken as 3α → 12C
[FL87].
• N-body interactions, with N ≥ 4, usually provide smaller contributions to
the astrophysical processes, and as such can be neglected in most calculations
(for theoretical examples, see for instance [GJ21]).
In the astrophysical context, it is frequent to parametrize the cross section as
σ(E)E = S(E)P (E),
with S(E) the astrophysical S factor, available via extrapolations from terrestrial
experiments, and P (E) the penetration factor, accounting for the probability of
nuclide to overcome the Coulomb barrier. For the nuclear problem, we can generally
write it as [Sal54, HK94, Kam14]
P (E) = exp



−2
∫
L
dr
√
2m
ℏ2
[U(r) − E]



, (2.67)
where L denotes the spatial interval where integration is to be performed. In the
absence of screening effects (that is, the response of the background medium to the
presence of charged nuclides a1 and a2), this factor can be simplified into the Gamow
factor P (E) = e−2πηSomm , with the Sommerfeld parameter
ηSomm =
Za1
Za2
e2
ℏ
(
M ′
2E
)1/2
.
Enhancement due to medium: electron screening. Since nuclear reactions
are an interplay between nuclear and electromagnetic (or, more properly, electric)
forces, where the projectile is expected to have sufficient energy as to overcome
the Coulomb barrier of the target one can ask: but can the composition of the
background alter the effective Coulomb barrier? After all, in macroscopic media
we learn boundaries affect the strength and geometry of the electric field. The
73
answer in this case is yes: the background responds to the electric charge/field of
both interacting nuclides. And the barrier the projectile must overcome is in fact
influenced by this as well, such that the total potential energy of the system is given
(in c.g.s./Gaussian units) by
U(r) = UCoulomb(r) − Uscreen(r), (2.68)
UCoulomb(r) =
Z1Z2e
2
r
, (2.69)
where Z1,Z2 are the charge numbers of projectile and target, r is the distance be-
tween them, and Uscreen(r) is the screening potential, i.e. the response of the back-
ground medium’s electric field to the presence of these charged particles. From a
thermodynamical point of view, this should be expected as a functional of temper-
ature, density, composition of the background, and Z1, Z2. What do we do with r
then? In principle, this variable will be suppressed in practical applications for sev-
eral reasons, the most important being that the actual rate is constructed by thermal
averaging over the energy of the charged particles, which implies performing a set of
integrals. The second reason is that we do not necessarily have to evaluate difficult
integrals related to exp [−Uscreen(r)/kBT ]: as proven by [Sal54], under certain con-
ditions we can safely replace Uscreen(r) by Uscreen(r = 0), or Uscreen(0) for simplicity.
As a consequence, the reaction rate can be expressed as [Sal54]
R = exp[Uscreen(0)/kBT ]Runscreened,
i.e. this simplification allows to treat screening effects in the thermonuclear regime
as a multiplicative factor over the unscreened rate, i.e we must compute/attach
screening factors. Quite a nice and simple picture!
So far, however, we have said anything regarding the actual calculation of Uscreen(0),
sometimes denoted as H(0). That is, in fact, a quite broad subject, strongly de-
pendent on microphysical inputs as well as on the classical or quantum assumption
regarding the underlying distribution functions. Regardless of this, it is useful to
distinguish at least three thermonuclear regimes, determined by the ratio of electro-
static and thermal energy Γ [Sal54, AL22]:
• Strong screening. Predominant at high densities (not in excess of 109 g
cm−3), here we have Γ ≫ 1.
• Weak screening. Operating at low densities and high temperatures, it is
characterized by Γ ≪ 1.
• Intermediate screening. In this regime, which represents a transition be-
tween the first and the second of this list, we have Γ ∼ 1.
74
Probably the most famous within the astrophysical community of stellar evolution
is the fudging of weak and strong screening subroutines[PBD+11, PMS+15]9, as it
allows a smooth transition between the two above regimes and include the quantum
corrections as introduced by [Jan77, AJ78]. This formulation is not unique: we have
for instance the approach from [CDY07], which at least in MESA exhibits some issues
if employed around 1010 g cm−3 and 106 K.
2.5.1 β reactions
0 2 4 6 8 10
T/T9
−2.05
−2.00
−1.95
−1.90
−1.85
−1.80
lo
g
10
λ
[s
−
1
]
β+ dec
NuDat 3.0
JINA-CEE
0 2 4 6 8 10
T/T9
−6
−4
−2
0
2
4
6
8
10
ρYe = 1011 g cm−3
ρYe = 108 g cm−3
β+ cap
β− dec
β− cap
17F ! 17O (Oda et al, 1994)
Figure 2.7: β reactions for the 17F→ 17O process, as computed by Oda et al 1994
[OHM+94]. The different linestyles are associated with values of ρYe, as illustrated
in the right-hand panel. Left panel: rate of direct reaction, β+ decay, as a function
of temperature. The comparison with two databases (constant lines) is given. Right
panel: remaining β processes: β− capture (direct reaction), and β+ capture and β−
decay (inverse reactions, i.e. 17O→17F).
This category includes ion-lepton scattering mediated by the electroweak force.
As mlepton ≪ mproton, reaction rates involving leptons can be written as Pion,lepton =
nionλlepton, where
λlepton =
∫
dnlepton σ(vlepton)vlepton (2.70)
is the natural definition of this factor, considering Eq. 2.66.
For typical stellar interiors, ¶e−, νe♢ and their antiparticles are the only leptons of
relevance. Furthermore, as e± in these reactions were originally called β± particles,
it is frequent to label all the following as β reactions [Zub20]:
9https://cococubed.com/code_pages/codes.shtml
75
• β± decay. Consists on the emission of e± and νe/ν̄e due to the decay of a
proton(neutron) inside the nucleus,
A
ZX → A
Z−1Y + e+ + νe +Qβ+ (2.71)
A
ZX → A
Z+1Y + e− + ν̄e +Qβ− . (2.72)
• β- capture. Occurring inside neutral atoms, it consists on the capture of a
bounded e−, in the lowest orbital state, by a proton, giving rise to
A
ZX + e− → A
Z−1Y + νe +QEC . (2.73)
Similar transitions can occur for electrons in higher orbital states, although
transition probabilities are significantly reduced as they correspond to first-
and second-forbidden transitions, according to their correspondent selection
rules.
• β+ capture. In this process, a bounded neutron captures a positron and
decays into a proton and an antineutrino,
A
ZX + e+ → A
Z+1Y + ν̄e +QPC . (2.74)
Considering the rest-mass energy of a nuclei with Z protons and electrons plus
N = A− Z neutrons is
W (Z,A) = Z(mp +me)c
2 + (A− Z)mnc
2 −B(Z,A) , (2.75)
with B the binding energy, the Q value for each β-channel can be written as
Qβ− = W (Z,A) −W (Z + 1, A) −mec
2 (2.76)
= B(Z + 1, A) −B(Z,A) + 0.782 MeV −mec
2 (2.77)
Qβ+ = W (Z,A) −W (Z − 1, A) −mec
2 (2.78)
= B(Z − 1, A) −B(Z,A) − 0.782 MeV −mec
2 (2.79)
QEC = W (Z,A) −W (Z − 1, A) +mec
2 (2.80)
= B(Z − 1, A) −B(Z,A) − 0.782 MeV +mec
2 (2.81)
QPC = W (Z,A) −W (Z + 1, A) +mec
2 (2.82)
= B(Z + 1, A) −B(Z,A) + 0.782 MeV +mec
2 , (2.83)
where 0.782 MeV is the numerical value of [mp+me−mn]c2. By inspection, Qβ+/− =
QEC/PC − 2mec
2, implying the Q value of decays must be larger than 2mec
2 for
such process to dominate over electron captures. Since β+β− → 2γ is certain to
occur under typical astrophysical conditions, the energy carried away by photons
is accounted for in the Q-value simply by neglecting the −mec
2 term at the end of
76
Qβ± , indicating electron’s energy has become photon’s. As is customary, the net
amount of specific energy production can be easily estimated as Qm−1
u λ.
Although calculations for σ(vlepton) constitute a subject demanding careful ex-
planations [DLC+98], [FFN82], [JLH+10], [BRSGS+06], we can grasp the essential
features in actual expressions for λ through Fermi’s model of weak-Interaction aris-
ing in standard perturbation theory: the total probability of transition per unit
time, or essentially the rate of capture or decay of a β particle, can be expressed as
[OHM+94, Kam14]
λβ = Constant ×
∫
dx x
√
x2 − 1x2
νF (Z, x)fFD(x, T ) , (2.84)
where x = Ee/mec
2, xν = Eν/mec
2. These x-terms naturally emerge from stan-
dard calculations of three-body decay with the daughter nuclei remaining at rest;
F (Z,Ee) is the correction factor due to Coulomb interactions, and fFD is Fermi-Dirac
distribution function. The constant at the front is mainly composed of electroweak
coupling constants and electron’s mass, while the limits of integration are set by the
specific channel the particles are following. It is important to note that, in spite of
Q factors becoming independent on neutrino’s energy, for computing the net rate of
specific energy the contributions from ν/ν̄ must be subtracted. Consequently, the
instantaneous rate of specific energy loss due to β± neutrinos is given by
˙̆εν,β± = m−1
u × Constant ×
∫
dx x
√
x2 − 1x3
νF (Z, x)fFD(x, T ) . (2.85)
Although the above formalism is quite general, under certain conditions these rates
can be shown to be independent of both temperature and density, although not nec-
essarily of Z. This approximation is supported by experimental results [GGIH09]
[GNI+10], although controversies for low-Z ions and captures still prevail [RSD+20],
[LRW20], [NRB+01]. From theoretical side, it can be argued that environment can
play a role via electronic screening, altering the effective Z, and even a more promi-
nent one in the low-ρ and high-T regime, such as supernovae conditions, as electrons
become highly non-degenerate, T/TF ≥ 1 [FFN82]. As the zero-temperature approx-
imation is adequate for practical purposes, in the remaining of the text we consider
λ as T -independent. We must be aware, however, that such simplification still de-
mands the evaluation of at least the integral for ˙̆εν,β, as the decay rates for certain
ions are experimentally known. In general, there does not exist a single formula,
leading to the usage of tabulated data [GM71, OHM+94]. One of the few excep-
tions for tabulated specific neutrino energy comes from the β± decays and electron
captures occurring in the CNO cycle and pp chains respectively. Considering, from
[FCZ75]
⟨Eν⟩FCZ
β± =
W±
0
2
(
1 − 1
w2
0,±
)(
1 − 1
4w0,±
− 1
9w2
0,±
)
(2.86)
⟨Eν⟩FCZ
pp = QEC , (2.87)
77
where QEC is the Q-value of the reaction, W0,± = Qβ± ∓mec
2 and W0 = mec
2w0,±,
˙̆εFCZ
ν = Yim
−1
u τ−1⟨Eν⟩FCZ , (2.88)
where τ denotes the lifetime of the reaction and Yi is the abundance of the parent
ion.
Fig. 2.7 exemplifies the typical behavior of β reaction rates, taking the particular
case 17F → 17O as reported by [OHM+94]. Due to the difference in binding energies
and the available phase space for integrals, we see β+ decay is the most likely of the
4 reactions at Yeρ ≤ 106 g cm−3 and T ≤ 109 K, exhibiting a very weak dependence
on density and parabolic trajectory on temperature as T ≥ 4 × 109 K. As most
astrophysical scenarios of interest lie below this threshold, it is usual to consider
this rate as T - and ρ- independent. The chosen fixed value can be taken as either
the theoretically estimated one, or as the Earth-measured inverse lifetime if available,
as in the present case where the horizontal lines are taken from two database set,
NuDat3 [NuD22] and JINA-CEE [JIN22, CAF+10]. On this regard we must notice
the lack of absolute convergence between theoretical and experimental rates at Earth
conditions, a difference which becomes more accentuated at increasing parent’s Z
number. As this discrepancy might be a situation of concern, for the typical stellar
environments the Earth’s rate, due to its experimentally verifiable status, might be
regarded as the correct one.
At progressively more extreme physical conditions the β− capture takes the lead-
ing role, significantly increasing in proportion with density. As a consequence of
Pauli blocking factors, we see extreme environments are certain to reduce the likeli-
hood of the “inverse” processes, β+ capture and β− emission, justifying the approx-
imate treatment of β reactions as non-reversible rates.
2.5.2 Chains and cycles
In astrophysical scenarios, rarely just one reaction is responsible for the whole syn-
thesis of elements. Instead, chains of reactions, sometimes confined to cycles, are
the mechanisms controlling the overall evolution of the composition.
We have seen the probability of interaction among nuclides is severely restricted
by Coulomb repulsion. Considering this barrier is overcame in high-density environ-
ments, of seldom appearance in standard astrophysics, it is not surprising the most
discussed chains and cycles of reactions are those involving burning and exhaustion
of H, He, C and even O in the context of late stellar evolution. At the envelope of
accreting neutron stars, where the conditions are favorable for the nuclear burning
of these elements, it is feasible to find these chains and cycles as well. Thus, let us
review the main features of these reaction chains.
Hydrogen burning. Being the element with lowest Z number, there are many
mechanisms operating at increasing conditions of temperature and density. In pro-
gressive order, these are:
78
• pp chains. This collection of reactions, primordially operating at T ≤ 107 K
and ρB ≤ 103 g cm−3, have the net balance of converting four 1H atoms into a
single 4He, liberating ∼ 15 MeV per process [KWW12]. We speak of them as
chains since these are likely scenarios, with different probability of occurrence,
emerging from the primordial chain
1H(p, e+ν) 2H (2.89)
2H(p, γ) 3He. (2.90)
Of them, the first reaction is the slowest one not only between these two but
among all reactions composing the pp chains, having an associated lifetime
of ∼ 1010 yr. For practical purposes, this allows to approximate the specific
energy generation rate as ∼ Qall chains × r1H(p,e+ν).
The three channels (or pp chains) allowing to synthesize 4He after the oc-
currence of Eq. 2.90, labeled according to the amount of helium produced,
are:
pp - I : 3He( 3He, αp) 1H (∼ 83%) ;
pp - II : 3He(α, γ) 7Be(e−, ν) 7Li(p, α) 4He (∼ 17%) ;
pp - III : 3He(α, γ) 7Be(p, γ) 8B(e+ν) 8Be(αα) (∼ 0.02%);
pp - IV(hep) : 3He(p, e+νe)
4He.
The fourth chain, usually called hep, is regarded as the highest source of
neutrinos, although having the severe drawback of being very unlikely [BK98,
Bah02].
• CNO cycles. These cyclic chains have, once more, the net balance of trans-
forming four 1H atoms into 4He, now with C, N and O isotopes acting as
catalyzers. These cycles supersede the pp-chains as 1H burning mechanisms
at T ∈ [107, 108.25] K and ρ ≤ 106 g cm−3. Due to the interplay of temperature-
dependent reaction rates and β-decays, we can identify cold and hot versions
of the cycles, operating below and above ∼ 108.25 K respectively [WGU+10].
The chains of reactions conforming the CNO cycle are:
I : 12C(p, γ) 13N(e+ν) 13C(p, γ) 14N(p, γ) 15O(e+ν) 15N(p, α) 12C
II : 15N(p, γ) 16O(p, γ) 17F(e+ν) 17O(p, α) 14N(p, γ) 15O(e+ν) 15N
III : 17O(p, γ) 18F(e+ν) 18O(p, α) 15N(p, γ) 16O(p, γ) 17F(e+ν) 17O
IV : 18O(p, γ) 19F(p, α) 16O(p, γ) 17F(e+ν) 17O(p, γ) 18F 18O(p, α) .
In the hot version of the cycle, reactions such as 13N(p, γ) 14O, 17F(p, γ) 18Ne,
18F(p, γ) 19Ne, 18F(p, α) 15O and 15O(α, γ) 19Ne, together with β+-decays of
14O, 18Ne and 19Ne, become more likely as T exceeds 108.25 K, enhancing the
79
amount of heavy-Z ions due to the short lifetimes of these hot processes. As
T ∼ 108.40 K, the production of F and Ne isotopes is enhanced through breakout
reactions such as 14O(α, p) 17F, 15O(α, γ) 19Ne or 12C(α, γ) 16O(α, γ) 20Ne. The
left panel of Fig. 3.1 illustrate several of these temperature dependent and
independent rates at ρ = 104 g cm−3, where we can infer that between 108 and
108.6 the CNO cycles, as long as some Ne and Na are present, CNO cycles can
be broken and the reactions induce the enhancement of Mg and Al. On the
other hand, the right panel illustrates the amount of energy generated from the
CNO cycle, in the approximation introduced by [Cla68, NH03], which consists
on an almost-constant section coming from the hot CNO and dominated by
the approximately constant β+ decays of 14O and 15O (see the left panel). On
the other hand, the T sensitive region is controlled by both β+ decays and the
proton-capture reactions involving 13N - 14N.
• Rapid proton capture process. Dubbed as rp-process, it consists on the
synthesis of heavy elements (Z ≥ 24, A ≥ 50) as a consequence of many 1H
captures, practically its exhaustion, in a timespan of t ≤ 103 s, short with
respect to typical H-depletion time in standard stars, i.e. ∼ 106 yr. This
rapid burning takes place at T ≥ 108.4 and ρ ≥ 105 g cm−3 for initial mass
fractions of hydrogen of ∼ 70%. Although the name of this process suggests
(p, γ) reactions are the sole drivers, the actual rp-process is in fact a collection
of sub-processes:
– CNO-breakout. This step is crucial for switching on the whole rp-
process. As previously discussed, 14O(α, p)17F and 15O(α, γ)19Ne break
the I-IV cycles and start the enhancement of F and Ne. However, 19Ne
is more likely to decay into 19F than to produce 20Na (via p-capture) as
T <∼ 108.4 K. Once this relation is inverted, the material ceases to return
to CNO and synthesis of Z ≥ 10 becomes viable.
– Synthesis below A = 54. In this mass range, Tz = (N − Z)/2 = −1
nuclides, namely 22Mg, 26Si, 30S, 34Ar, 38Ca, 42Ti, 46Cr, 50Fe and 54Ni be-
come the main axis of hydrogen burning, which follows regular patterns.
Between each of these contiguous pairs, we observe a competition between
three chains: (i) a sawtooth-path of two p-captures, β+ decay, p-capture,
β+ decay and a final p-capture, (ii) an α-capture like path, i.e. (α, p)
followed by a p-capture, and (iii) a β-3p-β-p path, i.e. three consecutive
p-captures between two β+ decays. In the left panel of Fig. 2.8 these
three paths are illustrated with arrows over arbitrary Tz = −1 nuclides
(of A ≤ 54). Typically, chain (i) is favored among 22Mg, 26Si and 30S,
while chain (iii) is substantially prominent among the remaining Tz = −1
nuclides.
– Synthesis between A = 54 and A = 64. The main axis of H burn-
80
ing in this mass range shifts from Tz = −1 to α nuclides, suppressing
in the process the competition between chains (i)-(iii). The synthesis
in this range proceeds by p-captures and β+ decay reactions bounded
by three triangular-like structures: (i) 54Ni-54Fe-58Zn, (ii) 58Zn-58Ni-62Ge
and (iii) 60Zn-60Ni-64Ge. An example of such structure, as well as the re-
actions involved, is given in the right panel of Fig. 2.8. This latter isotope
represents the typical endpoint at the peak of thermonuclear explosions
(bursts).
– Synthesis for A ≥ 64. Once the main axis of H burning has settled
in α-nuclides, the synthesis of heavier species than A = 64 proceeds via
p-captures and β+ decays bounded in similar triangular-like structure as
in the former case. In addition, this region is characterized by cascades
of β+ decays from even-even α nuclides such as 68Se or 72Kr at the peak
of these triangular structures.
– Endpoint. The possibility of extending the rp-process beyond A = 64
up to a maximum A depends on the local temperature: at T ∼ 108.4 K,
Amax ∼ 72 while at T ∼ 108.7 K we obtain Amax ∼ 90. At the peak of
thermonuclear explosions, i.e. T ∼ 109.2 K, it is feasible to synthesize up
to 107Te as burning reaches a Sn-Sb-Te cycle endpoint at 105 ≤ A ≤ 108
due to the (p, γ) − (γ, p) equilibrium, (γ, α) and β+ decays.
In contrast to the pp-chains and the CNO cycles, an adequate simulation of
the rp-process requires the presence of many species. While a good estimation
on the specific energy generation rate is given by the combination of CNO-
breakout rates [WGS99], [CN06], it still must be checked if such approximation
is valid for all astrophysical scenarios of interest.
Helium fusion. For matter in excess of T ∼ 108 K and ρ ≥ 102 g cm−3, the
reaction 4He(α, γ) 8Be∗ becomes more likely to happen. However, such excited state
is unstable and has a large probability of decomposing into 2α particles or, if helium
abundance is significant, has a probability of advancing to an excited state of 12C,
the Hoyle state [Hoy54], via 8Be∗(α, γ) 12C∗, subsequently falling into the ground
state of 12C and liberating ∼ 7.28 MeV per α particle. As the lifetime of 8Be∗ is
∼ 10−16 s [HK94], this chain of reactions is simply written as 3α →12C and referred
to as triple α reaction, mechanism responsible for providing the nuclear energy for
evolved stars [HHIK56, KWW12]. Although the effective reaction rate is taken as
a three-body one, ⟨ααα⟩T , it can be understood as the product of consecutive two-
body reactions due to their formal origin [FL87]. From the experimental point of
view, on the other hand, it can be estimated from its inverse reaction 12C → 3α
(e.g. [FDB+05]). As we infer from the right panel of Fig. 3.1, at very dense and
hot environments with prominent presence of 4He the 3α process is the main source
81
α
α
Tz = − 1
Tz = − 1 α
α
A ≤ 54 A ≥ 60
Figure 2.8: Main flow of the rp-process, among Tz = −1 and α nuclides.
of nuclear energy, significantly exceeding the amount from the CNO cycles and pp
chains.
2.6 Thermal neutrinos
Besides the neutrinos from β-reactions, within standard stellar matter there exist
additional mechanisms - again mediated by the electroweak interaction - emitting
νν̄. In Fig. 2.9 we illustrate their corresponding Feynman diagrams. In first place we
have pair neutrinos, coming from the exchange of Z or W bosons by the e+e− and
νeν̄e pairs [Chi66]; bremsstrahlung neutrinos [Hau75], arising as a consequence of
electron scattering from a nuclide AZ; photoneutrinos, due to the absorption of one
photon by an electron and the posterior emission of a neutrino/antineutrino pair;
plasmon neutrinos, coming from the decay of plasma (quasi-particles) excitations
[ARW63, HRW94]. An additional mechanism in this category is the recombination
neutrinos, arising from the emission of a νν̄ pair upon an e− capture by a nuclide
AZ. From the microphysical point of view, their overall contribution for standard
astrophysical scenarios is to take energy away from the system due to their trans-
parency to ordinary matter. Such losses have been computed throughout the years
by different groups and eventually adjusted as functionals of temperature, density
and chemical composition throughout ⟨m⟩e, ⟨A⟩ and ⟨Z⟩. For astrophysical appli-
cations the compilation [IHNK96] is the standard reference, with the fits expressed
as Qν in units of erg cm−3 s−1 (see, however, [KPP+99] for an updated version of
the bremsstrahlung neutrinos at high densities). In the case of photo, plasma and
pair neutrino processes the fits depend on T , ρYe only, recombination additionally
depending on η and ⟨Z⟩ and bremsstrahlung requiring ⟨A⟩. It is straightforward to
82
Plasmon decay neutrinos
Pair neutrinos Bremsstrahlung neutrinos
Photo-neutrinos
Figure 2.9: Feynman diagrams for the relevant neutrino processes at the surface of
the neutron star. Left column: pair neutrinos (based on [Dic72]). Middle column,
upper diagram: bremsstrahlung neutrinos (based on [DD10]). Middle column, lower
diagram: plasmon decay (from [CCC70]). Right column, upper and lower diagrams:
photoneutrinos (based on [DRP04]).
83
7.0
7.5
8.0
8.5
9.0
9.5
10.0
lo
g
10
T
[K
]
log10 ˙̆εplasma[erg g−1s−1]
Pure 12C matter
log10 ˙̆εpair[erg g−1s−1]
7.0
7.5
8.0
8.5
9.0
9.5
10.0
lo
g
10
T
[K
]
log10 ˙̆εphoto[erg g−1s−1] log10 ˙̆εrecomb[erg g−1s−1]
−5.0 −2.5 0.0 2.5 5.0 7.5 10.0
log10 ρ [g cm−3]
7.0
7.5
8.0
8.5
9.0
9.5
10.0
lo
g
10
T
[K
]
log10 ˙̆εbremss[erg g−1s−1]
−5.0 −2.5 0.0 2.5 5.0 7.5 10.0
log10 ρ [g cm−3]
Pair
Photo
P
la
sm
on
B
re
m
ss
.
R
e
c
o
m
b
.
log10 ˙̆εν,thermal[erg g−1s−1]
−30
−25
−20
−15
−10
−5
0
5
10
15
−32
−24
−16
−8
0
8
16
24
32
40
−30
−20
−10
0
10
20
−32
−24
−16
−8
0
8
−20
−16
−12
−8
−4
0
4
8
12
16
−10
0
10
20
30
40
Figure 2.10: Contour plots of thermal neutrino processes, considering pure carbon-
12 matter, for densities and temperatures of a typical neutron star envelope. The
name of the corresponding process is indicated at the top of each panel. For display
convenience, here we took max[ ˙̆εν,process, 10−30].
84
convert these fits into a term with the same units as the output from the nuclear
generation rate, i.e. ˙̆εν = Qν/ρ, as well as to add each individual contribution, suf-
ficing to compute their individual Qν value, add them and then obtain the global ˙̆εν
dividing by ρ. In the grand scheme of stellar evolution, it is frequent to either add
both ˙̆εthermal
ν and ˙̆εβν into a single neutrino energy generation rate ˙̆εν , or reserve this
latter symbol to thermal neutrinos and to remove the neutrinos due to β-reactions
from ˙̆εnuc. In the present work, we adopt the first path and shall employ ˙̆εν to denote
all contributions from neutrinos, both thermal and due to β-reactions.
Considering the fits from [IHNK96, KPP+99], in Fig. 2.10 we display contour
plots for the different neutrino processes so far discussed, as well as for their grand
total ˙̆εthermal
ν , considering pure 12C matter. In the upper left panel we have the con-
tribution from plasmon decay, which reaches a maximum around the region where
η ≈ 0 (contrast the T − ρ panel against the η contours of Fig. 2.2). In the upper
right panel we observe the pair neutrinos, whose maximum evidently takes place
whenever the e+/e− production becomes viable as well, i.e. low densities and high
temperatures, rapidly dropping to zero outside this region. Photoneutrino contri-
bution (central left panel) follow a similar tendency as well, leading to a transition
of dominance with respect to pair neutrinos at around T ∼ 108.5 K at low densities.
Although recombination neutrinos (central right panel) reach a maximum at high
densities and temperatures - but still within the η ∼ 0 region - its contribution to
the grand total only exceed the rest of thermal processes at ρ ∼ 102.5 g cm−3 and
T ≤ 107.5 K, region where the K-shell of carbon-12 exists. In the lower left panel we
observe the energy release due to bremsstrahlung neutrinos, which exhibits a differ-
ent tendency with respect to the rest of processes by remaining above 10 erg g−1
s−1 at ρ ≥ 105 g cm−3 and the whole range of selected temperatures, thus leading
to the dominion of this process over the rest of them (lower right panel). This is a
consequence of the bremsstrahlung taking place within the liquid and solid phases
of the envelope-crust (i.e. Γ ≥ 200, Fig. 2.2).
85
Chapter 3
Nuclear networks for neutron stars
“In our time art attracts the
most noble souls. Art [...] is
possibly the last refuge of
freedom”
O. Honchar
Abstract. In this chapter we present the characteristics which a nuclear
network of reactions must have in order to describe the envelope of the star.
Introduction
At the very core of standard stellar evolution is the problem of solving a large number
of non-linear PDEs. While structure, temperature and luminosity account for 5 to
6 (depending on whether we are including relativistic effects or not, see Chapter 1),
for each one of the species in our system we must add one corresponding PDE.
For modeling the surface of neutron stars, how many species do we need? From
the nuclear physics aspects discussed in Chapter 2 concerning the rp-process, a first
look suggest we require a large number of species. According to different authors
this is indeed the case, requiring up to 100 species for a correct account on the
nuclear generated energy and a first approximation to the distribution of ashes due
to 1H and 4He burning [vGI+94, RFR+97, SBCW99, GBS+07, FST08]. Considering
we must solve every PDE of stellar evolution at each spatial point inside the star
- i.e. the grid of points - having a large system of equations requires significant
computational power unless we develop approximation methods based on the short
and long timescale evolution of the different layers inside the star, limit the spatial
resolution, or reduce the number of equations to be solved. In regard to this latter
aspect, we enter into the domain of the fine art of constructing approximative nuclear
reaction networks.
86
Another complication arising from the inclusion of the chemical evolution to the
set of structure and temperature equations is their stiffness character as a conse-
quence of the different timescales of each reaction. For instance, while 14−15O mean
lifetime against β+ decays is of the order of mili-seconds, above A = 56 we find
isotopes with lifetimes of the order of hours to days. Thus, any standard explicit
method such as the 4th order Runge-Kutta are no longer useful for modeling the
evolution of a neutron star [PTVF92], and it is imperative to study which integra-
tion methods we have in order to perform a fast and accurate solution of the PDEs
equations. These are the implicit methods which we briefly describe in this chapter
as they constitute the core part of the numerical codes employed in the following
Chapter.
It is then the purpose of the present chapter to provide (i) a concise view into
the actual networks for studying the envelope of the neutron star, and (ii) the
numerical setup employed to analyze the evolution of the chemical species, critical
part to compute an accurate nuclear energy generation rate.
We review in the first section the issues on the construction of a network, briefly
commenting on the available numerical methods to solve stiff sets of equations. In
the next section, we comment on the networks already existing in the literature and
introduce our own networks, commenting on the advantages and disadvantages of
each of them. The chapter concludes with a quick overview on the evolution of
the rp-process employing these numerical methods, leaving the implications for the
whole envelope for the next chapter.
3.1 Overview of numerical methods
Even with modern computers, the simultaneous solution of Eqs. 1.16, describing the
spacetime evolution of the Nspecies constituent species, and the PDEs for T (t, r) and
ρ(t, r) poses a numerical challenge for several reasons. In first place, the set of PDEs
DtYj = Rj is stiff : physically (or intuitively) for the solution of the set we must pay
close attention to very different timescales, ranging from 10−6 to 106 depending on
the species involved, their linking reactions and the local conditions for temperature
and density. For instance, in the left panel of Fig. 3.1 we can observe a collection of
reaction rates involved in the hydrogen burning, expressed as lifetimes, i.e. τ = R−1
since [R] = Time−1, as functions of temperature at 104 g cm−3. Even at this middle
density, notice the drastic variation in values for even a single rate, from 10−3 to
106 in the small window of 108 − 109 K. Additionally, we observe at, for instance
T = 108.5 K, the drastic difference between reactions such as 18Ne(α, p)21Na and
19Ne(p, γ)20Na.
Mathematically, a concise definition of stiffness requires to enclose several char-
acteristics [HNW93], such as the artificial oscillations appearing on attempts to solve
these PDEs via explicit methods (such as forward Euler or 4th order Runge Kutta),
87
7.0 7.5 8.0 8.5 9.0 9.5 10.0
log10T [K]
10−3
10−1
101
103
105
τ
−
1
[s
]
14O
15O
20Na
18Ne
19Ne
22Mg
ρ = 104 g cm−3
15O (α, γ) 19Ne
14O (α, p) 17F
18Ne (α, p) 21Na
19Ne (p, γ) 20Na
20Na (p, γ) 21Mg
22Mg (p, γ) 23Al
12C (α, p) 15N
6 7 8 9
log10T [K]
0
5
10
15
20
25
lo
g 1
0
˙̆ ε n
u
c
[e
rg
g−
1
s−
1
]
(X⊙, Y⊙, Z⊙)
pp
CNO
3α
ρ = 108 g cm−3
ρ = 102 g cm−3
Figure 3.1: Left panel: Selected collection of thermonuclear rates, expressed as the
inverse lifetime τ−1 at ρ = 104 g cm−3, relevant for CNO cycles and their breakout.
Right panel: energy generation rates per unit gram for pp, CNO and 3α processes,
as discussed in the main text.
and the requirement of employing either too small or too large time-steps as a con-
sequence of the different scales in the right-hand side in order to keep stability and
accuracy. While for some [HM06] stiffness can be verified via the ratio of maxi-
mum/minimum Jacobian matrix eigenvalues, a pragmatic definition of stiff PDEs
is the failure of explicit methods to provide satisfactory solutions [CH52]. It must
be noted the evolution of mass fractions is responsible for introducing the stiffness
into the system: if, for instance, we beforehand know the distribution of chemical
composition across our whole star, it is viable to solve the TOV equations for the
structure of the star restoring to explicit methods such as 4th Order Runge Kutta.
As described elsewhere [HNW93], [PTVF92], [Tim99], [THW00], the appropriate
numerical methods for solving stiff PDEs are implicit, requiring the inversion of large
and sparse matrices M whose dimension is determined by both the number of species
and the number of cells in the spatial dimension, i.e. dim(M) ∼ Ncells × Nspecies.
For astrophysical simulations, having Ncells ≥ 100 cells, the calculations become
computationally expensive as Nspecies ≫ 30. For instance, if each entry in the
Jacobian matrix is a double-precision number (i.e. 8 bytes), having 100 cells and
30 species implies a Jacobian matrix of 576 MB of space. In an optimistic note,
regularly we do not need to include ∼ 1000 species in all simulations to consider
them realistic. For instance, including Z ≥ 20 elements for simulating the evolution
of mass fractions after the Big Bang (T ∼ 109 K) is not mandatory since we are
dealing with the synthesis of the very first elements in the periodic table, i.e. He,
Li, Be, B and at most C, N and O. On the other hand, for the evolution of Main
Sequence stars p- and α- captures over Z ≥ 26 elements is quite infrequent that
88
we can neglect its presence for the major part of the simulation. Finally, unless
ρ ≥ 106 g cm−3, as for instance in simulations of supernovae or the neutron star
crust, the inclusion of isotopes in the neutron-rich sector of the nuclide chart is not
required since β+ decays of isotopes in the valley of stability cannot proceed due
to Pauli blocking, β− captures require densities near 107 g cm−3 to become viable
and neutron capture processes (n, γ) are inefficient as long as these particles are
confined in nucleii. Thus, the most crucial aspect for a self-consistent, yet simple,
computation of stellar evolution is to identify the range of temperatures and densities
of interest and to keep the indispensable amount of nuclides as to cover the processes
of interest. This, however, makes it clear the most expensive part of the computation
is the evolution of the network.
Implicit numerical methods. The conceptual idea behind this family of meth-
ods consists on solving differential equations by root-finding methods, usually involv-
ing the diagonalization of some matrix - the Jacobian one, given its multi-variable
context. Considering a general set of PDEs ∂tA = f(A), with A denoting the array
of dependent variables and f the right-hand side functions for each equation, the
idea is to approximate by finite differences the system as An+1 = An + hf(An+1)
and to solve this non-linear set to extract An+1, i.e. the new configuration at a
“time” tn+1 = tn + h. The advantage of these methods is the freedom to choose
either large or small time-steps, in contrast to the explicit methods, where f is
evaluated at An, where the time-steps must be large in order to avoid instabilities.
As concrete examples of implicit methods employed for modeling neutron stars,
we can cite:
• Henyey method. The workhorse for many stellar evolution codes [HFG64,
PBD+11, KWW12, Pag16], consisting on the construction of matrices of easy
inversion - via the Newton-Raphson method - and the convergence to the
updated values of the independent variables provided an initial guess for them.
While its implicit nature guarantees the possibility of choosing either small or
large time-steps, the initial guess for the next step must be close in order to
guarantee convergence, not only in a few iterations only, but as well given the
underlying Newton-Raphson method. An additional difficulty is to properly
write the coefficients for handling the inversion of the large resulting matrix -
which is divided into a task of inverting smaller but many sub-matrices.
• High-order, semi-implicit. These methods can be considered as general-
izations for stiff problems of methods for non-stiff ones. For instance, the
idea behind the Rosenbrock methods is to generalize the Runge-Kutta for
stiff problems, from where the subroutine stiff emerges [PTVF92]. Another
generalization is the Bader-Deuflehard semi-implicit method, which relies on
the conceptual idea of polynomial or rational extrapolation and the midpoint
method to obtain accurate solutions employing only a few steps. As it is, this
89
method seems the fastest one while keeping a relatively low amount of required
steps [Deu83], [PTVF92], [LMJ14] [FOR+00]. Consequently, we opt for con-
sidering this method, as implemented in [PTVF92] via the stifbs subroutine,
for constructing our one-zone model and our code for stationary envelopes.
3.2 On the networks for neutron stars.
Besides keeping as few nuclides as possible, further approximations can still be made
to simplify the process of solving the system of PDEs while keeping a good grasp on
the physics of interest. For neutron stars, efforts are oriented to have either a de-
tailed evolution of the chemical composition or a good estimation on the generated
energy due to nuclear reactions. In the first category we find the so-called one-zone
models (OZM), which evolve the abundances’ PDEs in a single cell, i.e. at fixed
spatial coordinate (for instance, at fixed radius or pressure). This constraints the
temperature and density to adopt either constant values or, at most, to be functions
of time. The numerical benefit of these models is that abundances’ PDEs turn into
ODEs, i.e. we go from a BVP + IVP to an IVP, allowing to reduce the dimension
of the Jacobian matrices to ∼ Nspecies instead of ∼ Ncells ×Nspecies. Physically, work-
ing under specified local conditions for temperature and density allows to identify
which reactions are operating and, consequently, which are the most important at
certain phases of the whole system evolution. The main disadvantage of OZMs is
neglecting diffusion effects, which play a role in the enhancement or diminishing of
mass fractions via matter movements as well as in the transport of the generated
energy via nuclear reactions, and thus on the overall evolution for T .
A common approximation for saving computational time while resulting in accu-
rate amounts of energy generation is to assume certain species have reached a steady
state, i.e. DtXi = 0. This translates into the trading of differential by algebraic
equations which, after being solved, define additional relations among mass frac-
tions. Although such simplification does not automatically guarantee a reduction
in the computation time, due to the delay root-finding methods might bring, for
some systems these approximations define whole new networks which are actually
fast and are suitable for simulating large periods of time. However, whenever we
restore to this approximation we must beware of physical artifacts such as low-Z
survival at high depths, underestimation of α nuclides or energy from β+ decays,
among others.
Examples of such energy-oriented approximation are:
• pp-chains and CNO cycles. A good approximation for the specific energy
generation rate of these processes is
ε̇eff ≈
[
∑
i
Qi
]
τ−1
longest , (3.1)
90
where τlongest ∝ r−1
longest is the corresponding timescale of the slowest reaction
rate from the set. For instance, the lifetime of 1H(p, e+ν)2H, denoted as τpp, is
the largest in the pp-chains, thus ε̇pp ≈ τ−1
pp ×∑iQi. Below 107.5 K, 12C(p, γ)13N
is the slowest reaction of the so-called cold cycle, allowing to write the specific
energy generation rate as ∼ QCNOr12C(p,γ) 13N, sensitive to both temperature
and density. At T ∼ 108 K, i.e. the hot CNO cycle, β+ decays from 14−15O
become the slowest reactions and thus the net production of energy becomes
independent of temperature and density. In both scenarios, it becomes viable
to trade the detailed evolution of 12−13C, 13−15N and 14−15O by an effective
rate relating 1H, 4He and 12C. Following [Cla68] [NH03],
rCNO =
(
1
τ13N
+
1
τ13N,β
)−1
×


(
τ−1
13N
τ13N + τ14N + τ14O,β + τ15O,β
)
+


τ−1
13N,β
τ14N + τ13N,β + τ15O,β



 , (3.2)
where τX and τX,β denote the lifetime of X against a thermonuclear reaction
and a β+ decay, respectively. In the right panel of Fig. 3.1 we plot these
approximations to ε̇ as functions of temperature and density. In the case of
the approximation to the pp-chains, we see the fit produces reasonable results
well below 107 K, around the temperature at which CNO cycles replace the
pp-chains as main mechanisms of 1H burning. At increasing temperatures, we
observe 4He burning via the 3α process provides the largest contribution to
nuclear energy instead of the hot CNO cycle, which provides up to 1015 erg
g−1 s−1.
For the particular case of neutron star surfaces undergoing accretion: since H
and He burning are favorable below 107 g cm−3 while heavy-Z burning requires
higher densities, at the envelope it suffices to consider species in the proton-rich
sector of the nuclide chart and the valley of stability, where rates are prominently
thermonuclear. There exist several networks of reactions which can be employed for
the description of the envelope. As stressed out above, neither is free of limitations,
so in principle it is advisable to understand the contents of each one of them and to
implement them according to the specific requirements.
• Approx21. The workhorse for MESA, this network contains 21 species, between
1H and A = 56, covering the essentials of pp-chains, CNO cycles and α-burning
up to 56Fe, as originally devised by its 19-species parent network [WZW78].
Given its reduced size, simulations are actually fast even with Ncells ≥ 1000.
As main drawbacks we can cite the artificial super-bursts that occasionally
emerge as a consequence of 1H reaching depths well below 106 g cm−3 (i.e. not
exhausting due to the absence of proton-rich nuclides), the artificial bursting
91
suppression at moderate accretion rates which, albeit observationally correct,
it is numerically artificial when contrasted against other networks, and finally
its limited range of proton-rich nuclides, essential in the understanding of the
rp-process. While it cannot be modified in MESA, it is easy to implement in
other stellar codes due to its structure.
• rp153 and rp300. These were devised by [FGWD06b, FTG+07] in order to
simulate X-ray bursts having as little isotopes as possible while guaranteeing a
good production of nuclear energy (in both networks), or allowing to have the
indispensable heavy-A species synthesized during and after the burst peak,
as for instance 107Te, according to the then stablished endpoint for the rp-
process [SAB+01]. Although rp153 gives a correct description of the energy
output, it suffers similar issues as Approx21, allowing hydrogen to reach high
depths (≤ 107 g cm−3) instead of being burnt and limiting the heaviest-Z
in the network to be in the A = 56 group. While the energy production of
rp300 - an extended version of rp153 - is similarly correct and more species
at A > 56 are included, its main drawback is the absence of A > 56 nuclides
having lifetimes against β+ decay larger than 1 day. This reduces the amount
of synthesized nuclides in the valley of stability after the bursts, particularly
at A ≥ 56, inducing artificial amounts of 107Te for instance. Both networks,
however, result in good fittings against observations for the luminosity curve
of an average bursts (see [PMS+15]). The list of species for rp153 can be seen
in Table 3.1.
• net381. As such, this network is essentially an extended version of rp300. The
nuclear energy production is accurate, and the number of species allow for an
improved distribution of ashes in terms of A and Z. For instance, the presence
of 80Kr allows the whole mass fraction of 80Y (t1/2 ≈ 30 s against β+ decay)
to go to the valley of stability. The advantage of having more species near
this zone, in contrast to the rp300 network, is the possibility of contrasting
time-dependent and -independent simulations, where the energy released by
the ashes of H-burning provide the largest contribution at ρ ≥ 106 g cm−3,
as described in the next chapter. Since this network neglects the presence
of neutron-rich nuclides below the valley of stability, its range of validity is
restricted to ρ ≤ 108 g cm−3. Similarly to rp153 and rp300, in net381 we
neglect the presence of isotopes above A = 91 near the valley of stability, thus
having artificial amounts of 107Te for instance. The most problematic aspect
of this network is the large times required for running a complete simulation in
MESA. In a time-independent scenario, however, it is feasible to run relatively
fast simulations with this network and to employ its output to understand the
mass fraction distributions and the generation of nuclear energy. Finally, let
us note the version employed in the following chapter, net380, only differs
92
from net381 by the exclusion of neutrons in the network. From this network,
we can also extract another one, net344, by removing nuclides related to the
endpoint of the rp-process but which cannot β+ decay in the absence of species
with A ≥ 90 in the valley of stability. The isotopes which can be removed from
the net381 network to conform this sub-network are enlisted in Table 3.1.
• Approx140 and Approx170. These networks were devised to (i) guarantee
a reasonable production of nuclear energy, (ii) include as much β+ decays
from heavy-Z species as possible, (iii) keep nuclides related to the primordial
paths of the rp-process between A = 20 and A = 56, and (iv) exhaust the
available hydrogen before 106 g cm−3. The complete list of species can be
found in Table 3.1. Let us now give a brief summary for our motivation to
introduce these networks. As a first step, pp chains were omitted since the
associated energy production occurs just at the very surface of neutron stars
(i.e. below 102 g cm−3), where compression is actually more energetic than
nuclear reactions. Li, Be and B nuclides are thus ignored. All C, N and O
isotopes for the hot CNO cycle are included, since this is an essential part
for the H-burning at middle (ρ ∼ 104 g cm−3) densities and temperatures
(T ≥ 107 K). Below A = 56, β+ decays take less than 1 hour to occur (a
remarkable exception is 26Al). This implies once a proton-rich A isotope has
been synthesized during the rp-process, it might decay towards the valley of
stability in just a few minutes. Due to the local maxima at α-nuclides in the
distribution of ashes due to nuclear burning (see [SBCW99]), we make the
approximation of allowing the whole chain of β+ decays towards the valley of
stability only to isotopes leading to α-nuclides for A ≤ 52. Considering that
the integrated flow of the rp-process, either in stable or explosive burning,
proceeds between Tz = −1 and α-nuclides, such neglect should not affect the
evolution of ashes well below 108 g cm−3. A secondary consequence of the
necessity of keeping α-nuclides in the network is to adequately simulate 4He
burning. Between 22Mg and 54Ni we have “box scheme” [RFR+97], where a
competition between the three chains of reactions discussed in the previous
chapter and in [FGWD06a], occurring between Tz = −1 isotopes, takes place.
To decide which chain is more relevant, we considered the results of [SBCW99,
SAB+01] as well as our own simulations (with updated reaction rates from
[CAF+10] and [JIN22]): from 22Mg to 26Si and to 30S, the sawtooth path
is the most prominent one and thus the associated isotopes are included in
the network. On the other hand, from 30S to 54Ni, the β-3p-β-p path is the
dominant one. Between 54Ni and 62Ge (Tz = −1 nuclides), the fictitious axis
of the main flow moves from Tz = −1 to α nuclides and those at two β+
decays of separation from them, as for instance 60Ni and 64Zn. At A = 64,
specifically at 64Ge synthesized during the peak of the bursts, the main flow
now follows a triangular-like structure: a cascade of proton captures and β+
93
decays connect α-nuclides, while heavy isotopes such as 64Zn, 68Ge and 72Se
are synthesized and do not further decay due to their long lifetimes. To reduce
as much species in the network as possible, we simulate an endpoint to the rp
process following two basic criteria: (i) H is fully exhausted at around 106 g
cm−3, and (ii) the heaviest nuclide must have a large (above 6 days) lifetime
against β+ decay. We find 80Kr, and specifically the A = 80 family, as a
suitable artificial endpoint for the process. The A = 84 family is a suitable
endpoint as well, although it increases the amount of nuclides.
In the particular case of the present work, only the latter networks will be amply
discussed. net380 allows a better distribution of species and energy production
in stationary state, net344 allows to understand the essential features of the rp-
process and Approx140 (or Approx170 as well) allow fast numerical simulations while
fulfilling the requirements for an approximately good modeling of the rp-process.
3.3 An overview of the rp-process
0.0 0.2 0.4 0.6 0.8 1.0
Time [day]
10−14
10−12
10−10
10−8
10−6
10−4
10−2
100
M
as
s
F
ra
ct
io
n
12C
1H
stifbs
eps = 10−1, N = 30
eps = 10−3, N = 29
eps = 10−4, N = 37
eps = 10−6, N = 34
0.0 0.2 0.4 0.6 0.8 1.0
Time [day]
10−14
10−12
10−10
10−8
10−6
10−4
10−2
100
Z⊙, ρ4 = 7.08, T8 = 3.86
12C
1H
stiff
eps = 10−1, N = 48
eps = 10−3, N = 153
eps = 10−4, N = 549
eps = 10−6, N = 5176
Figure 3.2: Comparison of stiff integrators for a OZM with different accuracies eps.
The number of steps until tmax is shown as N . Left panel: stifbs. Right panel:
stiff.
Numerical tests. A comparison of results from two different stiff integrators
is shown, for the net380 network at fixed conditions of temperature and density, in
Fig. 3.2. For simplicity we adopted the solar composition at t0 = 0 s. Given both
integration methods require the user-provided parameter eps, we have selected a
94
collection of values ranging from 10−1 to 10−6. In the left panel we illustrate the
results from the stifbs subroutine, and in the right panel those from stiff. In
both panels, we observe a correlation between the size of eps and the total number
of steps taken by the subroutine, although for stifbs we see the occurrence of a
local maximum of N at eps= 10−4. While stiff shows a better consistency in
the mass fractions among the four models, in clear contrast with the discrepancies
among different eps for stifbs, the number of steps to be taken with stiff is
always larger than the other subroutine. Furthermore, the accuracy for keeping a
low N seems adequate for a pure mass-fraction evolution, although it is dubious
whether it would work when incorporating the rest of equations, each with their
proper scale and thus with different needs for eps. Notice that stifbs, on the other
hand, yields consistent results as long as one restricts the eps parameter around
10−4, while N always remain smaller than the values obtained from stiff. In
the long run, this suggests stifbs provides a faster and accurate performance than
stiff. Consequently, we opt for employing this subroutine for our time-independent
numerical code.
Evolution of the network. In order to exemplify the development of the
rp-process, let us now describe in detail one simulation at fixed density and temper-
ature, ρ = 105.5 g cm−3 and T = 108.7 K. For simplicity, we employed the net344,
composed of the nuclides depicted in Fig. 3.4, running for t ∈ [0, 105] s. As initial
conditions we chose the simplest scenario: X1H = 0.70 and X4He = 0.30, composition
which allows to observe how the rest of nuclides are synthesized as a consequence of
hydrogen and helium burning. In Fig. 3.3, we illustrate the evolution of the mass
fraction of a selected subset of nuclides, participating at different stages of the rp-
process. In Figs. 3.4 - 3.7, the integrated reaction flow at different intervals of time
are illustrated, in order to identify the main reactions controlling the evolution of
the species.
In the upper left panel of Fig. 3.3 we appreciate the evolution of the chemical
composition at t ≤ 1 s. Here we see that, as a consequence of the temperature, 12C
has been synthesized and exhibits an almost constant mass fraction. By inspection
of Fig. 3.4, we infer this is due to an equilibrium between the rate of creation, 3α
process, and that of destruction via 12C(p, γ)13N(p, γ)14O. Such chain also explains
the rapid formation of 14O. The integrated flow and the mass fractions also shows
the operation of the 14O(α, p)17F and 17F(p, γ)18Ne reactions as we observe synthesis
of 17F and 18Ne. On the other hand, time is still too short as to further proceed into
A ≥ 18 and the carbon-12 burning results in enhancement of 15O, a sign that 18Ne
β+ decay and the subsequent 18F(p, α)15O is favored over the 18F(p, γ)19Ne reaction.
This building-up of 15O, as we see in the upper right panel, is fundamental for the
production of 19Ne via the breakout reaction 15O(α, γ)19Ne.
At t ≤ 10 s, the mass fractions of 14−15O, 18Ne, 17F and 19Ne reach a maxi-
mum, followed by a rapid decreasing: the 15O(α, γ)19Ne(p, γ)20Na chain has had
sufficient time as to become efficient and the synthesis of A > 20 nuclides takes
95
place, as indicating by both the upper right panel of Fig. 3.3 and the integrated
flow at 10 s, Fig. 3.5. The enhancement of proton-rich nuclides occurs among
contiguous nuclides, such as 24−25Si, 33−34Ar or the endpoint at this time, 37−38Ca.
These species, however, are short-lived: in the mass fraction diagram, we can observe
their rapid decrease just some seconds after their synthesis. Let us also note, from
the integrated reaction flow, that low-Z burning plays at this stage a more promi-
nent role than the synthesis of A > 20 material as for the production of heavy-Z
we require 15O, which the chain of reactions 3α → 12C(p, γ)13N(p, γ)14O(α, p)17F
(p, γ)18Ne (β+νe)
18F (p, α)15O keeps supplying at the cost of temporarily decrease
the mass fraction of carbon-12. In the integrated flow, we can also identify some of
the discussed paths for the rp-process, such as the sawtooth one between 26Si and
30S.
At t ≤ 102 s (lower left panel of Fig. 3.3 and Fig. 3.6), we observe a fast synthesis
of material, up to 60Zn, 63Ga and 64Ge, as well as some enhancement of 68Se. Below
A = 56, we observe the presence of the sawtooth path between 30S and 34Ar, as well
as the β−3p−β−p path among 34Ar-38Ca, 38Ca-42Ti, 42Ti-46Cr and 46Cr-50Fe. In the
intermediate region between Fe and Ga, we observe (p, γ) and β+ decays dominating
the integrated reaction flow across the whole nuclide chart. Most of the proton-rich
nuclides, such as 64Ge, are even shorter-lived in contrast to those between A = 20
and A = 56, and quickly decay as the system approaches the hydrogen exhaustion
point, whose occurrence can be anticipated from the integrated flow, as we see
15N(p, α)12C has started to close the CNO cycle.
Around 103 s, hydrogen has been exhausted. The rapid decrease of hydrogen, in
contrast to standard astrophysical scenarios, does not lead to an increasing in the
amount of helium-4. Instead, 12C is synthesized, as well as other heavy elements
such as 68Ge. At this point, those species at A > 56 initiate its β+ decay towards
the valley of stability, as shown in both distribution of mass fractions for A = 70
in the lower right panel of Fig. 3.3 and in the integrated nuclear flow of Fig. 3.7.
Given the exhaustion of hydrogen below 103 s, this chart can be useful to identify
which reactions contribute with the largest portion of the rp-process. Clearly, the
CNO breakout reactions play a critical role, as well as the further proton captures
and the β+ decays.
In Fig. 3.8, we plot the distribution of mass fractions per element (i.e. adding up
the mass fractions of the individual isotopes of each element) at the beginning and
at the end of the simulation. In the top panels, corresponding to our example with
ρ = 105.5 g cm−3 and T = 108.7 K, we observe that in 105 seconds the system went
from almost pure hydrogen to matter rich in Ge, Fe but also in C and Mg. In the
lower panels, we appreciate how the synthesis changes as we reach a slightly higher
temperature, typically found at the peak of the unstable thermonuclear burning.
At these conditions, matter becomes deficient in C or Mg but is enriched in Ge,
Zr and the elements between them. In both cases, no metals were present in the
initial composition, thus it is via hydrogen and helium burning, as well as from
96
the adequate conditions of temperature, that we can synthesize, via the rp-process,
heavy elements such as Ge or Zr.
−10
−8
−6
−4
−2
0
lo
g
10
X
log10ρ = 5.5, log10T = 8.7
1H
4He
12C
68Ge
14O
15O
17F
18Ne
19Ne
(XH,0, Xα,0) = (0.70, 0.30)
1H
4He
12C
68Ge
24Si + 25Si
29S + 30S
33Ar + 34Ar
37Ca + 38Ca
−5 −3 −1 0 1 3 5
log10Time [s]
−10
−8
−6
−4
−2
0
lo
g
10
X
1H
4He
12C
68Ge
37Ca + 38Ca
58Cu + 59Cu
60Zn
63Ga
64Ge
−5 −3 −1 0 1 3 5
log10Time [s]
1H
4He
12C
68Ge
70Br
70Se
70As
70Ge
Figure 3.3: Mass fractions as a function of time for the rp-process example. This
simulation employed the net344 network, and evolved the system at fixed tempera-
ture and density. The initial conditions for the mass fractions are given in the title
of the plot.
97
0 1
2
5
21
25
35
39
45
6
22
26
36
40
46
7
23
27
37
41
47
8
24
28
38
42
48
3
9
11
13
15
17
19
29
31
33
43
49
4
10
12
14
16
18
20
30
32
34
44
50
H(1)
F(9)
Cl(17)
He(2)
Ne(10)
Ar(18)
Li(3)
C(6)
Na(11)
Si(14)
Be(4)
N(7)
Mg(12)
P(15)
B(5)
O(8)
Al(13)
S(16)
K(19)
Ca(20)
Sc(21)
Ti(22)
V(23)
Cr(24)
Mn(25)
Fe(26)
Co(27)
Ni(28)
Cu(29)
Zn(30)
Ga(31)
Ge(32)
As(33)
Se(34)
Br(35)
Kr(36)
Rb(37)
Sr(38)
Y(39)
Zr(40)
Nb(41)
Mo(42)
Tc(43)
Ru(44)
Rh(45)
log10 ρ = 5.5
log10 T = 8.7
~Z = (XH,0, Xα,0) = (0.70, 0.30)
∆t = 1 s
Figure 3.4: Time-integrated network flow at fixed ρ and T , corresponding to the
rp-process example with net344, from t0 to 1 s. Red lines represent the reactions
satisfying ♣Fij♣ ≥ 10−1♣Fmax♣, blue lines standing for 10−3♣Fmax♣ ≤ ♣Fij♣ ≤ 10−1♣Fmax♣.
98
0 1
2
5
21
25
35
39
45
6
22
26
36
40
46
7
23
27
37
41
47
8
24
28
38
42
48
3
9
11
13
15
17
19
29
31
33
43
49
4
10
12
14
16
18
20
30
32
34
44
50
H(1)
F(9)
Cl(17)
He(2)
Ne(10)
Ar(18)
Li(3)
C(6)
Na(11)
Si(14)
Be(4)
N(7)
Mg(12)
P(15)
B(5)
O(8)
Al(13)
S(16)
K(19)
Ca(20)
Sc(21)
Ti(22)
V(23)
Cr(24)
Mn(25)
Fe(26)
Co(27)
Ni(28)
Cu(29)
Zn(30)
Ga(31)
Ge(32)
As(33)
Se(34)
Br(35)
Kr(36)
Rb(37)
Sr(38)
Y(39)
Zr(40)
Nb(41)
Mo(42)
Tc(43)
Ru(44)
Rh(45)
log10 ρ = 5.5
log10 T = 8.7
~Z = (XH,0, Xα,0) = (0.70, 0.30)
∆t = 10 s
Figure 3.5: Same caption as in Fig. 3.4, now for ∆t = 10 s.
99
0 1
2
5
21
25
35
39
45
6
22
26
36
40
46
7
23
27
37
41
47
8
24
28
38
42
48
3
9
11
13
15
17
19
29
31
33
43
49
4
10
12
14
16
18
20
30
32
34
44
50
H(1)
F(9)
Cl(17)
He(2)
Ne(10)
Ar(18)
Li(3)
C(6)
Na(11)
Si(14)
Be(4)
N(7)
Mg(12)
P(15)
B(5)
O(8)
Al(13)
S(16)
K(19)
Ca(20)
Sc(21)
Ti(22)
V(23)
Cr(24)
Mn(25)
Fe(26)
Co(27)
Ni(28)
Cu(29)
Zn(30)
Ga(31)
Ge(32)
As(33)
Se(34)
Br(35)
Kr(36)
Rb(37)
Sr(38)
Y(39)
Zr(40)
Nb(41)
Mo(42)
Tc(43)
Ru(44)
Rh(45)
log10 ρ = 5.5
log10 T = 8.7
~Z = (XH,0, Xα,0) = (0.70, 0.30)
∆t = 102 s
Figure 3.6: Same caption as in Fig. 3.4, now for ∆t = 102 s.
100
0 1
2
5
21
25
35
39
45
6
22
26
36
40
46
7
23
27
37
41
47
8
24
28
38
42
48
3
9
11
13
15
17
19
29
31
33
43
49
4
10
12
14
16
18
20
30
32
34
44
50
H(1)
F(9)
Cl(17)
He(2)
Ne(10)
Ar(18)
Li(3)
C(6)
Na(11)
Si(14)
Be(4)
N(7)
Mg(12)
P(15)
B(5)
O(8)
Al(13)
S(16)
K(19)
Ca(20)
Sc(21)
Ti(22)
V(23)
Cr(24)
Mn(25)
Fe(26)
Co(27)
Ni(28)
Cu(29)
Zn(30)
Ga(31)
Ge(32)
As(33)
Se(34)
Br(35)
Kr(36)
Rb(37)
Sr(38)
Y(39)
Zr(40)
Nb(41)
Mo(42)
Tc(43)
Ru(44)
Rh(45)
log10 ρ = 5.5
log10 T = 8.7
~Z = (XH,0, Xα,0) = (0.70, 0.30)
∆t = 103 s
Figure 3.7: Same caption as in Fig. 3.4, now for ∆t = 103 s.
101
He CO Mg Ca Fe Ge Zr
10−4
10−3
10−2
10−1
100
lo
g 1
0
X
t = 10−20 s
log10ρ = 5.5, log10T = 8.7
He CO Mg Ca Fe Ge Zr
10−4
10−3
10−2
10−1
100
t = 105 s
(XH,0, Xα,0) = (0.70, 0.30)
He CO Mg Ca Fe Ge Zr
10−4
10−3
10−2
10−1
100
lo
g 1
0
X
t = 10−20 s
log10ρ = 5.5, log10T = 9.0
He CO Mg Ca Fe Ge Zr
10−4
10−3
10−2
10−1
100
t = 105 s
(XH,0, Xα,0) = (0.70, 0.30)
Figure 3.8: Distribution of mass fractions per element at the beginning (left panel)
and at the end (right panel) of the simulation of the rp-process with the net344
network, for the temperature and density conditions indicated in the title of the
plot. Upper panels: example of the rp-process evolution, as described in the main
text and in Figs. 3.3 and 3.4 - 3.7. Lower panels: evolution of the rp-process at a
slightly larger temperature than for the upper panels.
102
Table 3.1: List of nuclides in the networks Approx140 and rp153.
Approx140 rp153
Z A Z A Z A Z A
N 1 V 44-46 N 1 Ca 36-44
H 1 Cr 45-48 H 1-3 Sc 39-45
He 4 Mn 48-50 He 3-4 Ti 40-47
C 12 Fe 49-52,54 Li 7 V 43-49
N 13-15 Co 52-56 Be 7-8 Cr 44-52
O 14-18 Ni 53-58,60 B 11 Mn 47-53
F 17-19 Cu 56-60 C 9,11,12 Fe 48-56
Ne 18-20 Zn 58-62,64 N 12-15 Co 51-56
Na 20-21 Ga 61-65 O 13-18 Ni 52-56
Mg 21,22,24 Ge 62-66,68 F 17-19
Al 23-25 As 66-69 Ne 18-21
Si 24-26,28 Se 68-70,72 Na 20-23
P 27-30 Br 70-73 Mg 21-25
S 28-32 Kr 72-74,76,80 Al 22-27
Cl 32-34 Rb 74-77,80 Si 24-30
Ar 33-36 Sr 76-78,80 P 26-31
K 36-38 Y 78-80 S 27-34
Ca 37-40 Zr 80 Cl 30-35
Sc 40-42 Ar 31-38
Ti 41-44 K 35-39
Nuclides for the rp-process endpoint
(Excluded from net344)
Z A Z A Z A Z A
Tc 85 Pd 90-94 In 98-104 Te 107
Ru 86 Ag 94-98 Sn 99-105
Rh 89,91-93 Cd 95-99 Sb 106
103
Chapter 4
Neutron Star Envelopes
“La libertad, Sancho, es uno de
los más preciosos dones que a
los hombres dieron los
cielos[...]y, por el contrario, el
cautiverio es el mayor mal que
puede venir a los hombres.”
M. de Cervantes Saavedra
Abstract. This chapter discusses the evolution of neutron star envelopes in
the presence and absence of accretion. After introducing the conditions under
which the envelopes can be treated as time independent - i.e. the stationary
approach - we analyze their characteristics and potential implications for the
evolution of the rest of the neutron star.
Introduction
Modeling thermal evolution of isolated neutron stars requires the simultaneous solu-
tion of the PDEs of structure and energy transport equations introduced in Chapter
1. The large variety of scales involved, both in the microphysical sector and on the
spacetime coordinates, make this a challenging task as early works on the subject
pointed out [NT81]. As stated in Chapter 1, for thermal evolution it is suitable to
split the star into two regions, the envelope and the numerical core. In the absence
of mass accretion, such splitting actually accelerates the numerical simulations: as
first pointed out by [GPE83], since the envelope practically moves between steady-
states of constant luminosity it is feasible to replace the evolution of the envelope
by analytic or tabulated relations between the effective temperature at the surface
and the temperature at the envelope-core boundary, i.e. Teff − Tb relations.
104
In the presence of mass accretion, on the other hand, we must consider the
evolution of the mass fractions and, due to the onset of thermonuclear instabilities
below 0.3ṀEdd (which produces drastic changes in the effective temperature within
the interval of minutes), so far no attempt to produce an analogous Tb−Teff relation
has been attempted. Another reason we could cite for the absence of such scheme is
the usual assumption that once the envelope has ceased bursting the base luminosity
becomes unique and positive, typically adopting values between 150 keV to 1 MeV
per baryon. Observationally, there are systems for which we could pose into question
that canonical picture: for instance, MAXI J0556-332 is presumed to have undergone
accretion episodes while having a cold numerical core (e.g. Tb ∼ 106 K), and we could
thus expect some heating to flow from the envelope towards the crust and core of
the star. Naturally, the presence of heating sources in the right hand side of dL/dP
demands to consider Lb(̸= Ls) as well. Consequently, the extension of the Tb(Teff)
scheme in the presence of accretion requires it to become a Teff −Tb−Lb relation, and
it is thus imperative to determine under which conditions we can actually construct
and implement them into neutron star evolution codes.
In the present chapter we review neutron star envelopes, both in the presence and
in absence of mass accretion and within time dependent and independent scenarios,
in order to determine the possibility to construct such a Teff − Tb − Lb relation
at nonzero Ṁ . In Section 4.1 we review the general properties of the envelope.
On Section 2 the case of envelopes without mass accretion is reviewed, and the
conditions for the viability of the Tb(Teff) relations are stablished. In the next section,
we analyze the envelope in the presence of mass accretion, first in time-dependent
scenarios and then we move into the time-independent sector.
4.1 General features of the envelope
By envelope we shall denote the section of the star between ρb = 108 g cm−3, i.e.
the boundary with the numerical core, and the density at the surface ρs, set by the
atmospheric boundary condition and with typical values of 10−1 to 10−3 g cm−3.
Their main features can be enlisted employing the PDEs governing the structure
and temperature derived in Chapter 1, as well as the thermodynamical functionals
described in Chapter 2.
• Mass and radius constancy. For a typical neutron star, we have gs ∼
GM/R2 ≈ 1014 cm s−2, with M and R the gravitational mass and radius of
the star, respectively. Assuming hydrostatic equilibrium (Eq. 1.64) with ∆P ≈
⟨ρ⟩gsdEnvelope, at ∆P = Pb − Ps ≈ 1024 erg cm−3 we have dEnvelope ∼ 10 m, i.e.
≈ 10−3R. Similarly, from the mass equation (Eq. 1.59), the enclosed mass in
the envelope is Menvelope ≈ 4πR2dEnvelope ≈ 10−9M . Therefore, the thickness
of the envelope and the enclosed mass are actually small with respect to the
overall values of the star, and we can set m ≈ M and r ≈ R in the PDEs of
105
5 10 15
log10ρ [g cm−3]
0.55
0.60
0.65
0.70
0.75
0.80
ex
p
[Φ
(ρ
)]
C
ru
st
-C
or
e
T
ra
n
si
ti
on
MS-A1 + PC, ρc,15 = 1.9152
APR + PC, ρc,15 = 1.0037
5 10 15
log10ρ [g cm−3]
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
m
(ρ
)
[M
⊙
]
C
ru
st
-C
or
e
T
ra
n
si
ti
on
MS-A1 + PC, ρc,15 = 1.9152
APR + PC, ρc,15 = 1.0037
Figure 4.1: Metric functions in terms of local density for two representative stars of
the indicated core EOS. The transition between the core and the crust is indicated
by vertical lines.
stellar evolution at the envelope. In the presence of mass accretion, the validity
of these approximations depends on the time interval of interest. For instance,
at a typical accretion rate of Ṁ ≈ 0.1ṀEdd ≈ 1.1 × 1017 g s−1, the aggregated
baryonic mass during one year is 3.5×1024 g ≈ 1.25×10−9M (at M = 1.4M⊙),
i.e. the mass of the envelope is practically duplicated, yet remains small with
respect to M . The constancy of R might be put into question considering
the observational evidence for the photospheric radius expansion during the
thermonuclear explosions. Typically, this changes the observed radius from
∼ 12 km to ∼ 19 km during a few seconds around the burst peak. From
Chapter 1 we know low-density environments favor fluid motion, hence the
outer layers of the envelope are displaced during this expansion and thus r
cannot be approximated by R anymore, although for the innermost layers it
is still viable to consider r ≈ R. Furthermore, since the expansion takes place
during a few seconds only, for long timespans (hours to years) r ≈ R remains
a good approximation for modeling envelopes. An interesting discrepancy
between numerical simulations and observations is the absence of a prominent
increase in radius despite the good agreement in the rest of characteristics of
the bursts (peak luminosity, recurrence time, cooling phase after peak). For
instance, at MESA we might go from r ∼ 11.53 km in the inter-bursting lapse
to r ∼ 11.60 km at burst peak.
• Constancy of the metric functions Λ(r), Φ(r). Due to the explicit relation
between m(r) and Λ(r) (Eq. 1.56), the constancy of Λ is a corollary from the
previous point, i.e. eΛ(r,m) ≈ eΛ(R,M). The same constancy in the envelope
106
holds for Φ(r,m) due to eΦ(R,M) =
√
1 − 2GM
c2R
. Moreover, since Λ(R,M) =
−Φ(R,M), the combination of both results leads to the simplification that
at the envelope it is viable to replace all eΛ by e−Φ or viceversa, i.e. general
relativistic effects can be simply accounted by the introduction of additional
redshift factors. The constancy of the metric functions near the surface of
the star is nicely illustrated in Fig. 4.1, where for two different core-EOS,
albeit with same crust-EOS, both eΦ and eΛ are essentially constant below
1010 g cm−3. Such result should not differ had we used another crust-EOS
since between 104 and 1010 g cm−3, despite variations in the actual chemical
composition, pressure and density are related via electron’s degenerate EOS.
• Alternative spatial coordinates. Since variations in mass and radius
are actually small throughout the envelope, as discussed above, neither of
them is suitable as spatial coordinate. Reasonable alternatives are the rel-
ative mass ln q1 = ln(MB/MB, TOT) or its modified version ln q2 = [(MB −
MB, TOT)/MB, TOT], which have been employed in time-dependent codes for
simulating stellar evolution in the presence of mass accretion with good re-
sults. Another option for spatial coordinate is the pressure due to its promi-
nent variations, from Pb ∼ 1025 erg cm−3 to Ps ∼ 1014 erg cm−3, as well as
its continuous character. A variation of this choice, considering the constancy
of M and R, is the column density coordinate y := P/gs, which allows to
absorb geometrical factors in the PDEs. While variations in ρB are of 9 orders
of magnitude as well, discontinuities due to changes in composition make it
unsuitable as spatial coordinate.
The above considerations allow for several simplifications in the discussion of stellar
evolution equations at the envelope. These are useful for numerical implementations
as well, albeit taking into consideration the limitations of these approximations.
Thus, in order to proceed with our discussion let us first introduce the following
notation:
(Local mass accretion per unit surface) ṁ =
Ṁ∞e−Φ
4πR2
(4.1)
(Local surface gravity) gs =
GMe−Φ
R2
(4.2)
(Proper time operator) ∂τ = e−Φ∂t (4.3)
(Radial to pressure coordinate) eΦ∂r = −ρBgs∂P (4.4)
(Eulerian operator for pressure) Dt = ∂τ + ṁgs∂P (4.5)
(Local flux) F = −KeΦ∂rT = KρBgs∂PT (4.6)
(Local luminosity) L = 4πR2F. (4.7)
As a result of all these considerations, the evolution of the envelope is fundamen-
tally governed by the temperature and the mass fractions for the Nspecies, according
107
to
cP
∂T
∂τ
= ρB
(
˙̆εnuc − ˙̆εν
)
+
cPT
P
ṁgs(∇ad − ∇T ) + ρBgs
∂F
∂P
(4.8)
∂Xi
∂τ
= −ṁgs
∂Xi
∂P
+ AiRi, i ∈ ¶1, . . . , Nspecies♢ . (4.9)
By inspection of Eq. 4.8, we see temperature is governed by the interplay among
three fundamental timescales:
• Heat diffusion. This scale is associated with the amount of time required
for radiative and conductive fluxes to transport thermal energy. Consider-
ing Eq. 4.8, as well as the transformation law between P and r (hydrostatic
equilibrium), this time scale can be expressed as
τdiff :=
d2cP
K
. (4.10)
Here, cP is the heat capacity at constant pressure per unit volume, K is the
thermal conductivity, and d is the thickness of the layer of material under
consideration. At ρb, for instance, considering Kb ≈ 1016 erg cm−1 s−1 K−1,
cP ≈ 1014 erg cm−3 K−1 and d = dEnvelope, we get τdiff ≈ 104 s, i.e. a few
hours only. As illustrated in Fig. 4.3 for different envelope models, we see this
back-of-the-envelope estimation is independent of the chemical composition
chosen, as well as of gs. Notice as we move from 108 g cm−3 to 1010 g cm−3,
the diffusion timescale moves from days to years.
• Mass accretion. Typically defined as τacc = δM/Ṁ , this scale quantifies the
required amount of time to accrete a layer of mass δM at an accretion rate
Ṁ . For instance, to completely replace the envelope, i.e. δM = 10−9M , at
the Eddington rate ṀEdd ∼ 1018 g s−1 we must wait approximately 20 days.
From Eq. 4.8, we see hydrostatic equilibrium allows to consider an alternative
definition for this timescale,
τacc :=
P
ṁgs
, (4.11)
which serves as a local indicator of the required amount of time for accreted
matter to reach a certain pressure in the envelope. For instance, at Pb ≈ 1025
erg cm−3, gs ∼ 1014 cm s−2 and ṁ ≈ 105 g cm−2 s−1, we obtain τacc ≈ 11 days.
• Nuclear reactions. Assuming the total amount of energy generated by nu-
clear reactions is Enuc (with units of erg g−1), it is customary to associate
τnuc := Enuc
ε̇nuc
as the characteristic timescale for nuclear reactions at fixed den-
sity and temperature (via ε̇nuc). For instance, H burning via CNO cycles yields
τnuc ∼ 103 s, considering ˙̆εCNO ≈ 1015 erg g−1 s−1 and Enuc ≈ 1018 erg g−1.
108
Typically, this occurs at around ρ ∼ 106 g cm−3. On the other hand, Eq. 4.8
suggests to consider
τnuc := cPT/ρB
˙̆εnuc (4.12)
as an alternative definition for the nuclear timescale. As we shall illustrate
below, this definition emerges in the context of thermonuclear instabilities,
and we shall thus adopt it.
4.2 Evolution of non-accreting envelopes
The main characteristic of these envelopes, as their name suggest, is the absence of
mass increase. This state can be steady, as in the case of isolated neutron stars, or
transient, as for instance during quiescent phases of binary systems. Without com-
pression, the chemical composition is expected to undergo a sedimentation process,
with high-Z elements displacing towards high-density layers and light elements such
as H and He remaining at low densities. For neutron stars in binary systems, this
sedimentation displaces the ashes from nuclear burning toward high-density layers
inside the envelope. In the case of isolated stars, such sedimentation might be ex-
pected during the first years after birth while at latter epochs it is reasonable to
consider these layers of well-defined ⟨Z⟩ have already been settled and will remain
undisturbed.
In the absence of accretion, secular nuclear burning is viable as long as fuel is
present and adequate conditions of temperature and density for starting ignition
still exist. Keeping those conditions, however, becomes challenging and unlikely as
t > 0. For instance, at T ∼ 108.25 K and ρ ≥ 104 g cm−3, 1H burning via hot
CNO cycle leads to complete exhaustion of fuel in ∼ 104 s ≈ 0.11 day. Without
1H replenishment, the ashes remain inert to nuclear burning and thus heating is no
longer injected at this density. Thus, any remaining fuel for thermonuclear reactions
should be eventually exhausted and ε̇nuc → 0 in less than 1 day. On the other hand,
electroweak reactions operating at ρ ≤ 106 g cm−3 are still likely to operate in the
absence of accretion, although in the long term (i.e. timescale of years), such heating
injection completely ceases.
What about neutrinos? Inferred effective temperatures from isolated neutron
stars, as well as during quiescent phases at binary systems, yield Teff ∼ 5 × 106 K,
i.e. L ∼ 1035 erg s−1. On the other hand, thermal neutrinos at the surface contribute
with Lν ≈ ε̇νMenvelope ≤ 1033 erg s−1, i.e. their contribution to the total luminosity
is significantly small and their presence in the envelope can thus be neglected.
We thus see the evolution of non-accreting envelopes is governed by heat diffu-
sion. From Eq. 4.8, the temperature profile evolves according to a diffusion process
with non-constant K,
cP
∂T
∂τ
≈ ρBg
2
s
∂
∂P
(
KρB
∂T
∂P
)
. (4.13)
109
Given the smallness of τdiff, i.e. < 1 day, it turns out any stationary solution to
Eq. 4.13, upon setting the appropriate boundary conditions, is a steady-state so-
lution. The proof relies on linear perturbation theory, where it is shown thermal
perturbations decay as e−t/τdiff . In other words, non-accreting envelopes are
stable against thermal perturbations. Furthermore, this stability is the mile-
stone for simplifying the analysis of non-accreting envelopes, as summarized in the
following statement:
Steady-state approximation (SSA). In the absence of accretion and nuclear
burning, thermal evolution of the envelope proceeds among stationary states of
constant luminosity.
0.0 2.5 5.0 7.5 10.0
log10 ρ [g cm−3]
6.0
6.5
7.0
7.5
8.0
lo
g 1
0
T
[K
]
ρb = 1010 g cm−3
gs,14 = 1.73
Teff = 106 K
56Fe
12C
1H
4He
0.0 2.5 5.0 7.5 10.0
log10 ρ [g cm−3]
6.0
6.5
7.0
7.5
8.0
gs,14 = 1.26
gs,14 = 3.74
56Fe
1H
ρb = 1010 g cm−3
Teff = 106 K
Figure 4.2: Temperature as a function of density for a selected collection of envelopes
in steady state. Left panel: envelopes at fixed gs,14 = gs/1014 cm s−2 composed of a
single species. Right panel: envelopes of the indicated compositions with different
values of gs,14.
Such result, first noticed by [GPE83], introduced a new paradigm in the modeling
of neutron star cooling. Indeed: combining the constancy of L = 4πR2T 4
eff across the
envelope with the approximations over mass and radius, temperature profile obeys
dT
dP
=
3κ
16T 3
T 4
eff
gs
. (4.14)
Since structure remains unchanged with time, for a given Teff we obtain a unique
T (P ) regardless of the time at which Teff occurs. Therefore, to simplify the simu-
lations at all times [GPE83] proposed to replace the envelope’s evolution, via the
110
0 2 4 6 8 10
log10 ρ [g cm
−3]
0
1
2
3
4
5
6
7
lo
g 1
0
τ
[s
]
1 day
ρb = 1010 g cm−3
gs,14 = 1.73
Teff = 106 K
1H
4He
12C
56Fe
0 2 4 6 8 10
log10 ρ [g cm
−3]
0
1
2
3
4
5
6
7
ρb = 1010 g cm−3
Teff = 106 K
gs,14 = 1.26
gs,14 = 3.74
56Fe
1H
1 day
Figure 4.3: Heat diffusion timescale for the envelope models in Fig. 4.2.
explicit solution of the corresponding PDEs of structure and temperature, by Tb−Teff
relations, in a sense mimicking the procedure of employing pre-calculated tables or
analytic fits as atmosphere boundary conditions.
The recipe to construct Tb − Teff relations goes as follows:
1. Fix a set of values for Teff, i.e. Teff. Typically, in the absence of accretion it is
usual to consider Teff = [104, 107] K.
2. For each Teff ∈ Teff, solve Eq. 4.14 in [Ps, Pb] and obtain a unique Tb.
3. Use the resulting data, in either tabular or fitting form, as surface boundary
condition for the numerical core at all times.
Within the Tb − Teff scheme, what is the composition of the envelope? At the
time the GPE approximation was proposed [GPE83], neutron star envelopes were
thought to be constituted of pure 56Fe, given its stability under β+ decays and its
role as the ground state of matter, at T → 0, below 1010 g cm−3 (e.g. [BPS71]).
A modern picture of the composition, taking into consideration the composition
after accretion episodes and the sedimentation process, consists on separating the
envelope into layers of well-defined composition, i.e. an onion-like structure (e.g.
Potekhin and Yakovlev, PY [PCY97]). Quite often the amount of layers employed
range from 1 to 3, so in general we can write the mass and proton numbers, as well
111
as their correspondent mass fractions as
A(P ) =
4
∑
i=1
Aiθ(P − Pi)θ(Pi+1 − P ) (4.15)
Z(P ) =
4
∑
i=1
Ziθ(P − Pi)θ(Pi+1 − P ) (4.16)
X(P ) =
4
∑
i=1
Xiθ(P − Pi)θ(Pi+1 − P ) , (4.17)
where i = 4 corresponds to Pb, θ(x) is the Heaviside function and each Ui is a
constant vector in R
Nion . Albeit modern and reliable for modeling non-accreting
envelopes, it is still an approximation which leaves several parameters unconstrained,
as for instance
• The transition depths Pi (i ∈ ¶1, . . .♢) at which jumps from low- to high-Z
matter take place. A good estimation for the 1H/ashes boundary, for example,
comes from thermonuclear burning simulations and yields Pthreshold ≈ 1022.5
erg cm−3. In contrast, a 4He/ashes boundary is considerably more challenging
since the amount of 4He depends on the history of mass accretion (since at
stationary conditions helium-4 survives but during thermonuclear explosions
it is completely destroyed) and the maximum internal temperature reached
(due to the sensitivity of the 3α reaction with temperature).
• The distribution of metals in a layer. Although we can artificially set X ≈ 1
for a given species, actual simulations of explosive nuclear burning indicate
the ashes are actually a mixture of metals, neither of which possess X ≈ 1.
Since different mixtures of metals yield different values for the opacity and
the specific heat, the composition of the layers we impose by hand should be
based on the results from time-dependent simulations of accreting episodes.
On Fig. 4.2 we illustrate a collection of temperature profiles for envelopes with
same Teff but different composition and surface gravity. For these models, we adopted
the predicted values for masses and radii considering the APR-core EOS and the
TOV equations of structure. As predicted by Eq. 4.14 and exemplified on the right
panel of Fig. 4.2, at fixed composition we see large values of gs translate into small
temperature gradients and, consequently, into low values for Tb. Here, gs,14 = 3.74
corresponds to a 2M⊙ APR-EOS star and gs,14 = 1.26 to a M⊙ star with the same
core-EOS. In this same panel, we see a low-Z envelope is considerably colder than
one with high-Z, a direct consequence of having dT/dP ∝ κ since κ ∝ Z2. This
behavior is also exemplified, at a gs corresponding to an 1.4M⊙ APR-EOS star, in
the left panel of Fig. 4.2. Here we see an approximate progression of slopes in the
temperature profiles as we move from 4He (Z = 2) to 12C (Z = 6) and finally to 56Fe
112
(Z = 26). Artificial behavior of the pure-1H envelope might be anticipated since
we are placing hydrogen at unlikely depths, according to the nuclear burning and
sedimentation models.
On Fig. 4.3 we illustrate the behavior of τdiff as a function of density for the
same steady-state envelopes in Fig. 4.2. Since dEnvelope ∝ P/ρBgs, the curves on
the right panel of this figure exhibits a similar tendency as the temperature profiles
of Fig. 4.2, i.e. high-gravity environments lead to both small τdiff and small T (ρ).
This also implies high-gravity environments need less time to dissipate a thermal
perturbation than low-gravity ones. A similar conclusion arises by changing chemical
composition, as shown in the envelope models in the left panel of Fig. 4.3: a high-Z
(metallic) envelope requires more time to dissipate a thermal perturbation than a
low-Z one, 1H representing an exception to this rule at high densities due to the
unphysical situation of pushing this element far beyond its exhaustion point by
thermonuclear burning. By inspection of both panels, we are led to the universal
conclusion that treating thermal evolution of neutron star envelopes as the succession
of steady-states is an excellent approximation as long as heating or cooling sources
are absent, or as long as such processes occur in a timescale much shorter than the
local τ [BC09].
Following the steps in the recipe 4.2, we constructed several Tb - Teff relations
with different chemical composition, which can be seen in Fig, 4.4. For simplicity, we
adopted the surface gravity corresponding to a 2M⊙ APR-core EOS, and adopted
ρb = 1010 g cm−3. The model with pure 56Fe corresponds to the GPE model, while
the mixtures follow the layered description of PY (e.g. Eqs. 4.15-4.17). The impact
of chemical composition on the Tb−Teff relations is quite noticeable: for a fixed value
of Tb, envelopes of pure 12C have a larger effective temperature than those with pure
56Fe. Similarly, replacing carbon-12 with helium-4 leads to envelopes with even
higher effective temperatures than those with pure iron-56. Physically, this means
that if we have two neutron stars with same core temperature, the star with low
metallicity at the surface would look brighter than the star with high metallicity at
the envelope. On the other hand, at a fixed Teff we see high-Z envelopes have higher
Tb than those with low-Z composition, i.e. the numerical core remains hotter as a
consequence of an increased opacity at the surface (recall κ ∝ Z2). Conversely, we
do not require a hot core to have a bright object, only to have a low metalicity at the
envelope. Having discussed the single-species Tb − Teff relations, let us inspect the
impact of having two layers, i.e. the bounded, dashed or dotted curves in Fig. 4.4.
For these curves, the transition between the low and high Z species takes place at
ρx = 10x g cm−3. The first noticeable change is that the lower ρx is the closest the
two-layered envelopes’ Tb − Teff relation remains to the curve corresponding to the
highest-Z envelopes. For instance, in the left panel of Fig. 4.4, at ρ4 the Tb − Teff
relation for the two-layered envelope approaches the curve corresponding to the pure
carbon-12 at around Teff ∼ 105.9 K. On the other hand, at ρ6 and ρ8 the proximity of
the two-layered envelope’s relation with the pure helium-4 curve is more prominent
113
than with the pure carbon-12 curve. By comparison with the right panel, we see the
higher the jump of Z in the two-layered envelope is, the closest its Tb − Teff relation
is with respect to the high-Z relation. Physically, this implies two neutron stars
of identical core temperature Tb ∼ 108 will look brighter if the surface is composed
of pure carbon-12 than an envelope with a mixture of carbon-12 and iron-56 and,
similarly, such stars would be brighter than one with a pure iron-56 envelope.
5.6 5.8 6.0 6.2 6.4
log10 Teff [K]
7.0
7.5
8.0
8.5
lo
g 1
0
T
b
[K
]
gs = 3.23×1014 cm s−2
ρb = 1010 g cm−3
12C
4He to 12C at ρ4
4He to 12C at ρ8
4He to 12C at ρ6
4He
5.6 5.8 6.0 6.2 6.4
log10 Teff [K]
7.0
7.5
8.0
8.5
9.0
56Fe
56Fe to 12C at ρ4
56Fe to 12C at ρ6
56Fe to 12C at ρ8
12C
Figure 4.4: Tb − Teff relations for representative composition configurations.
4.3 Evolution of accreting envelopes
In the presence of low-Z rich matter accretion, conditions become viable for the
onset of thermonuclear burning via the processes discussed in the previous chap-
ter. In principle, the evolution is governed by all terms in the right-hand side of
Eq. 4.8 and Eq. 4.9. The physics of the system is simple: matter is accreted at a
rate ṁ, temperature increases at the surface, allowing for thermonuclear burning
to transform the low-Z fuel into high-Z ashes, which accumulate upon the pris-
tine surface and, under constant accretion for t ∼ months, both ashes and pristine
material are compressed and ignite via pycnonuclear reactions. Depending on ṁ,
from observations we know thermonuclear burning takes place with or without pe-
riodic explosions (short-termed increase of temperature and luminosity). Despite
the qualitative simplicity of this picture, for its quantitative side we must solve the
stellar evolution equations in the envelope together with the evolution of the nu-
merical core. To model transient accretion episodes, such task is complicated due
to the large difference in the cooling timescale: while at the surface we have τdiff ∼
seconds, at the numerical core we have τdiff ∼ yr, i.e. we must either take small
time-steps to resolve the bursts at Ṁ ≤ 0.3ṀEdd or we must find a way to simulate
114
the temperature and luminosity of the envelope while correctly accounting for their
evolution at the core-envelope boundary, which is no longer Teff = Ts due to the
heating sources.
A priori, we should restore to fully time-dependent simulations to answer the
former questions. However, to fully grasp the physics involved in their output it is
suitable to divide the discussion of accreting envelopes into steps. First, we look
at the equilibrium points of the system, i.e. solutions of stellar equations at the
envelope setting all ∂t terms to zero. Following the standard theory of ODEs and
PDEs, the equilibrium points are useful to understand the dynamics of the system.
In the next step, we look at the evolution of perturbations to equilibrium curves. As
we shall see, they enclose the core aspects of the observed bursts. Finally, we discuss
the numerical output from fully-time dependent simulations, and the implications for
the long term evolution of the whole envelope and the characteristics an acceptable
Tb − Teff − Lb relation in the presence of accretion must have.
4.3.1 Stationary accreted envelopes
By setting all ∂t terms to zero, the envelope is described by a system of first-order
ODE in the spatial variable - the pressure, for numerical convenience - and sim-
plifies into a BVP. Moreover, the redshifted accretion rate Ṁ∞ becomes constant
throughout the whole envelope. The resulting equations are:
dr
dP
= − 1
geΛHG (4.18)
dm
dP
= 4πr2ρ
dr
dP
(4.19)
dΦ
dP
= − 1
Hc2
(4.20)
dT
dP
=
T
P
∇T (4.21)
∇T = − 3κρBLPe
Λ
64πσSBT 4r2
dr
dP
+
(
1 − ρ
H
)
(4.22)
dXi
dP
=
4πr2ρB
Ṁ∞e−ΦgHGAiRi, i ∈ ¶1, . . . , Nspecies♢ (4.23)
dL
dP
=
2L
c2H + 4πr2ρBe
Λ dr
dP
(
˙̆εnuc + ˙̆εgrav,h − ˙̆εν
)
(4.24)
˙̆εgrav,h = −Ṁ∞e−Φ−Λc̆PT
4πr2ρBP
dP
dr
(∇ad − ∇T ) (4.25)
where, following [FHM81, PMS+15], we have denoted as homogeneous (subindex h)
the contribution of heating due to mass compression.
115
Consideringthat composition of accreted material for LMXB is known (i.e. Z ≈
Z⊙), we opt for specifying the boundary conditions (now turned into initial condi-
tions in P ) for Eqs. 4.18 - 4.25 at the surface, which we set at the optical depth
τs = 2/3. This allows us to have freedom on choosing an arbitrary - yet physically
reasonable - value for Teff. In what follows, we restrict this parameter to the range
[106, 107] K. Regarding the rest of surface values: we mostly employ the network
with Nspecies = 380 (see Chapter 3), and unless otherwise stated, we take Z = Z⊙,
X1H = 0.70 and X4He = 0.28 as the surface composition. The gravitational mass
ms = M and radius rs = R are specified according to the standard APR-Core EOS
M −R relation (see Chapter 1, specifically Fig. 1.3), which defines Φs as well (recall
eΦs(M,R) =
√
1 − 2GM
c2R
). Throughout the Eddington relation (Chapter 1), provided
τs, Teff, gs and Xs we obtain Ts, as well as Ps and ρs by a combination of bisection
and Newton-Raphson methods. From R and Teff, the surface luminosity is obtained
via Ls = 4πR2σSBT
4
eff.
Given these initial conditions, Eqs. 4.18 - 4.25 are solved in Psb := [Ps, Pb], with
Pb corresponding to the pressure at which ρb = 107 g cm−3. The constant Ṁ∞ is
expressed in units of ṀEdd = 1.1 × 1018 g s−1.
Let us emphasize that, contrary to most - if not, all - approaches to this subject,
we are neither fixing nor shooting for a single, specific value of Lb, unless otherwise
stated. Given the nature of Eqs. 4.18 - 4.25 as IVP in P , doing so would imply
aiming for a single, very concrete Teff yielding the desired Lb. Instead, our main goal
is to analyzing the temperature, luminosity and composition profiles when we take a
collection of effective temperatures Teff, i.e. their corresponding Lb’s are determined
by the integration of Eqs. 4.18 - 4.25.
Nuclear output at different accretion rates
Let us first explore the output from full nuclear burning. Within the classical
paradigm, this requires to consider a single model with Lb > 0: we opt to fol-
low this approach, albeit we only demand an Lb sufficiently high as to produce
an envelope profile of almost flat slope near ρb. By performing a manual shooting
method in order to find such acceptable Teff to produce the desired positive base
luminosity, the stationary equations were integrated for gs = 1.735, corresponding
to a 1.4 M⊙ APR-Core EOS for four accretion rates: 0.01, 0.10, 1.0 and 5.0 ṀEdd.
In Figs. 4.5, 4.6 and 4.7 we show three panels - functions of the local density - per
accretion rate: in the upper one, we plot the temperature profile, indicating at the
headers their corresponding accretion timescale (e.g. Eqs. 4.11) and column depth
y. In the middle panel a collection of mass fractions for relevant species are shown,
and in the lower panel the specific energy generation rate due to the whole and
particular nuclear processes are plotted. Let us emphasize our choice of Lb (or its
corresponding Teff) is only a requirement for an adequate comparison with existent
models in the literature: we could also have started with Lb < 0, albeit such models
116
are reserved for the next subsection.
At 0.01ṀEdd, left panels of Fig. 4.5, we observe hydrogen being transformed into
4He via the cold and hot CNO cycles as we move from log10 ρ = 3 to ∼ 5.6 (in g cm−3
at panel (c)). The shape of the specific energy generation rate is very distinguishable:
at first, we observe a rapid increase in the energy production (between 3.0 and 4.0,
in log10 ρ scale), sign of the presence of 12C(p, γ)13N, followed by a constant region
of ∼ 1014 erg g−1 s−1, consequence of the β+ decays of 14−15O. Despite the tiny
fractions of accreted A ≥ 40 isotopes, the temperature is still not sufficiently high
(T < 108.25 K) as to allow proton captures over these nuclides, although we observe
some captures between A ≥ 20 and A < 40 in panel (c), which are closely followed by
the β+ decays due to the short lives of these proton-rich nuclides, as also evidenced
by the rapid decrease in the Tz = −1 mass fractions in panel b. At log10 ρ = 6.25 g
cm−3 helium-4 is transformed into carbon-12 via the 3α process (secondary peak in
panel (c)), while α-captures occur in a minor scale, thus leading to high amounts of
carbon-12 near ρb.
At a ten times large accretion rate, 0.1ṀEdd, hydrogen is still transformed into
4He via CNO cycles near log10 ρ ∼ 5.5 g cm−3, although the resulting layer occupies
a smaller region in the density space than its 0.01ṀEdd analogue. As shown in panels
(a)/(a’) and (c)/(c’) of Fig. 4.5, this is due to the large temperature at the envelope
- proportional with the accretion rate - which enhances the helium-4 burning via
the 3α reaction. While its specific energy generation rate does not exceed that from
the hot CNO cycle, its influence extends from log10 ρ ∼ 4.5 to 6.5 g cm−3, turning
carbon-12 as one of the most prominent nuclides in the composition of the ashes. At
0.1ṀEdd we start observing the presence of proton captures over A ≥ 40 nuclides,
closely followed by subsequent β+ decays. In contradistinction with the 0.01ṀEdd
case, we now observe that nuclear energy production does not abruptly decay after
the hydrogen exhaustion, a direct consequence of different processes, as for instance
3α (burning helium-4) or β+ decays.
At slightly higher accretion rates, 0.25 and 0.35 ṀEdd in Fig. 4.6, we do not see a
significant production of helium-4 right at the hydrogen exhaustion point. Instead,
we see helium-4 already being burned via 3α (panels (c) and (c’)) due to T ∼ 108.5
K (panels (a) and (a’)). Moreover, we now identify an helium-4 exhaustion point
near log10 ρ ∼ 6.75 g cm−3. By inspection of the accreting timescale at the top of
Fig. 4.6, hydrogen is completely burnt in nearly a few hours while helium should be
completely burnt in approximately 1 day. Similarly, the associated column depth
differs in only an order of magnitude, i.e. y ∼ 108 at hydrogen exhaustion while
y ∼ 109 at helium exhaustion. Regarding the rp-process, in the specific energy
generation rate we observe a richer structure as a consequence of T ≥ 108.25 K. In
first place, although the CNO cycle is still dominating the nuclear energy between
log10 ρ = 3 and 5 g cm−3, proton captures over 20 < A < 40 species enrich the
A = 40 sector. At these densities, however, temperature is still not high enough as
to allow proton captures over A > 40 nuclides. We can also argue the occurrence of
117
the CNO cycles block the possibility to synthesize further species. However, once
log10 ρ = 5 and T ∼ 108.4 K is reached, CNO cycles are no longer the primary
source of energy and are overcome by proton captures over A ≥ 20. This, we infer,
is a consequence of the 19Ne(p, γ)20Na(p, γ)21Mg chain, occurring right after the
15O(α, γ)19Ne CNO-breakout reaction. Once started, we see hydrogen progressively
starting to decrease while proton captures rapidly occur in the A ≥ 40 sector, and in
a minor scale over A ≥ 60 species. The Tz = −1 nuclides, auxiliary in these chains
of proton captures and subsequent β+ decays, eventually reach a mass fraction
comparable to that of the A ≥ 40 nuclides and enrich them up to the hydrogen
exhaustion point, at log10 ρ ∼ 5.75 g cm−3. It is noticeable the maximum of nuclear
energy due to the rp-process is an order of magnitude larger than the production
due to CNO, i.e. 1016 erg g−1 s−1. At the hydrogen exhaustion point we observe a
drastic decrease in the proton capture reactions and the predominance of β+ decays
to the contribution of nuclear energy. Such tendency is accentuated as we move
from 0.25 to 0.35 ṀEdd, where the synthesis of A ≥ 60 nuclides is followed by the
subsequent decays towards the valley of stability, as suggested by panels (c) and (c’),
where in the latter we appreciate the β+ decays over A ≥ 20 nuclides practically
provide the majority of nuclear energy.
Finally, let us analyze the mass fractions and nuclear energy output at 1 and 5
ṀEdd, depicted in the eight panels of Fig. 4.7. At the Eddington rate, practically no
helium-4 is synthesized as a consequence of hydrogen burning, which leads to the
small contribution to the nuclear energy from the 3α process in comparison with
the β+ decays from A ≥ 20 nuclides. Similarly, the hot CNO cycle is no longer
the main contributor to the energy output between log10 ρ = 3.0 and 5.0. Instead,
proton captures over A ≥ 20 species and their β+ decays are the main contributors
to the energy output. Notice, however, that only 12C is exhausted as a consequence
of these captures, with little enhancement of 40Ca but increased amounts of 15O and
17F, an evidence in favor that CNO cycles are less energetic but they still control,
between log10 ρ = 4.0 and 5.0, the production of nuclear species. It is only when
temperature reaches and exceeds 108.5 K and the density is ∼ 105 g cm−3 that CNO
cycles are finally broken and a cascade of proton captures - almost immediately
followed by β+ decays - lead to hydrogen exhaustion near 105.75 g cm−3. Counter-
intuitively, such process leads to the enhanced production of 12C (panel (b)). The
explanation, however, relies on the interplay between hydrogen and helium burning:
the rupture of the CNO cycles implies that whatever amounts of produced 12C - via
3α reaction - is immediately converted into 14O and reaches 19Ne, 20Na and beyond
via proton captures. However, when the system starts running out of hydrogen, yet
possesses sufficient amounts of 4He, 12C(p, γ)13N is no longer efficient and carbon-12
starts accumulating. As T ≥ 108.6 K, the remaining helium-4 is then transformed
into carbon-12 or 16O via α captures, inducing an helium-exhaustion point around
106.5 g cm−3. As a consequence of the rp-process, nuclides in the range A ≥ 60
are produced and practically dominate the composition of the ashes. The long-
118
lived isotopes under β+ decays, above 105.75 g cm−3, start decaying and produce the
largest contribution to the specific energy generation rate.
The scenario at 5ṀEdd shares many aspects with that at ṀEdd, such as hydrogen
and helium-4 exhaustion points, the generation of considerable amounts of 12C at
the high-density sector and the largest contribution to the nuclear energy - in the
aforementioned sector - coming from the β+ decays over A ≥ 20 nuclides. As
differences we can cite the maximum temperature at the envelope, ∼ 108.8 K at
5ṀEdd against the 108.75 K at ṀEdd, the enhanced production of A ≥ 70 species
instead of A ≥ 60, and the synthesis of 40Ca, together with the bypass of the
bottleneck near 43Ti, at lower densities in comparison to the ṀEdd case.
At fixed gs: different accretion rates.
Having acquired a first insight to the nuclear processes in the envelope, let us now
explore the impact of changing the effective temperature at the surface, an appropri-
ate scenario for our final goal. Taking Teff = [106, 107] K, the stationary equations
were solved for gs = 1.735, corresponding to a 1.4 M⊙ APR-Core EOS, for four
accretion rates: 0.03, 0.10, 0.30 and 0.60 ṀEdd. In Figs. 4.9 and 4.10 we display
four panels for each accretion rate. In the upper left panel, we show the resulting
temperature profiles for each effective temperature in Teff. In the lower left panel, we
display the specific energy generation rate due to nuclear reactions for four selected
models, as well as the generation rate from accretion and neutrinos for a model with
Lb > 0. In the upper right panel we have the luminosity at the base of the envelope
(expressed in units of MeV per baryon), for each of the profiles in the upper left
panel, against the surface temperature. Finally, in the lower right panel we display
the phase space, i.e. T against dT/dP , for each temperature profile.
The first and most notorious characteristic of the T vs ρ panel, at any of the
selected accretion rate, is not all numerical solutions qualify as “complete” or phys-
ical envelopes: there exists a subset T − ⊂ Teff whose corresponding temperature
profiles reach a maximum value at ρ < ρb and then start decreasing inwards. To
proceed with the numerical integration, all these models require T → 0. Although
we could opt for setting a minimum temperature to proceed the integration, for
instance Tmin = 104 K, there is not a physical justification for keeping a constant T
until Tb. To prevent ambiguities, these models were simply cut at ρcut < ρb such that
d log10 T/d log10 ρ = −101. Mathematically, on the other hand, the T → 0 behavior
is consistent with the input: notice dL/dP (Eq. 4.24) is controlled by two heating
sources (nuclear burning and mass accretion) and only one cooling mechanism, neu-
trino emission. Since ˙̆εν ≪ ˙̆εnuc, whenever L becomes negative the stationary profile
does not have additional mechanisms to return to L > 0 at higher densities. Due
to ∇T ∝ L, we see dT/dP is and remains negative. The presence of convection in
1In what follows, we refer to those quantities at ρcut by the subindex cut, i.e. Tcut, Lcut, etc.
119
7.50
7.75
8.00
8.25
8.50
8.75
9.00
lo
g
1
0
T
[K
]
log10y [g cm−2]
5.0 6.0 7.0 8.0 9.0
Time [y/ṁ]
1 min 10 min 1 h 10 h 1 day
Teff = 5.200× 106 K
−10
−8
−6
−4
−2
0
lo
g
10
X
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
log10ρ [g cm−3]
8
9
10
11
12
13
14
15
16
17
lo
g 1
0
ε̇ n
u
c
[e
rg
s−
1
g−
1
]
Hot CNO
(α, p)
(α, γ)
ε̇3α
A ≥ 20
A ≥ 40
A ≥ 60
A ≥ 20
A ≥ 40
A ≥ 60
(p, γ)
(e+νe)}
}
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
log10ρ [g cm−3]
8
9
10
11
12
13
14
15
16
17
lo
g 1
0
ε̇ n
u
c
[e
rg
s−
1
g−
1
]
Hot CNO
(α, p)
(α, γ)
ε̇3α
A ≥ 20
A ≥ 40
A ≥ 60
A ≥ 20
A ≥ 40
A ≥ 60
(p, γ)
(e+νe)}
}
b
c
b'
c'
0.01
·
MEdd 0.10
·
MEdd
CNO
Tz=-1
H 4He
A=20-39
A=40-59
H
4He
CNO
A=20-39
Tz
=-1
A=40-59
A ≥60
7.50
7.75
8.00
8.25
8.50
8.75
9.00
lo
g
10
T
[K
]
log10y [g cm−2]
5.0 6.0 7.0 8.0 9.0
Time [y/ṁ]
1 min 10 min 1 h 10 h 1 day 10 day
Teff = 3.000× 106 K
−10
−8
−6
−4
−2
0
lo
g
10
X
a'a
A ≥60
Figure 4.5: Envelope models at Ṁ∞ = 0.01ṀEdd (left panels) and 0.10ṀEdd (right
panels) as a function of ρ. The upper scales indicate the corresponding column
depth y and time spent by the accreted matter since it started its journey from
the neutron star surface. Panels (a) and (a’): temperature; panels (b) and (b’):
mass fraction of selected nuclei; panels (c) and (c’): specific energy generation of
dominant processes, as indicated, and with the upper thick line showing the total
energy generation. The corresponding values of Qb for 0.01 and 0.10 ṀEdd are 1.09
and 0.15 MeV per baryon, respectively.
120
7.50
7.75
8.00
8.25
8.50
8.75
9.00
lo
g
1
0
T
[K
]
log10y [g cm−2]
5.0 6.0 7.0 8.0 9.0
Time [y/ṁ]
1 min 10 min 1 h 10 h 1 day
Teff = 6.500× 106 K
−10
−8
−6
−4
−2
0
lo
g
10
X
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
log10ρ [g cm−3]
8
9
10
11
12
13
14
15
16
17
lo
g 1
0
ε̇ n
u
c
[e
rg
s−
1
g−
1
]
Hot CNO
(α, p)
(α, γ)
ε̇3α
A ≥ 20
A ≥ 40
A ≥ 60
A ≥ 20
A ≥ 40
A ≥ 60
(p, γ)
(e+νe)}
}
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
log10ρ [g cm−3]
8
9
10
11
12
13
14
15
16
17
lo
g 1
0
ε̇ n
u
c
[e
rg
s−
1
g−
1
]
Hot CNO
(α, p)
(α, γ)
ε̇3α
A ≥ 20
A ≥ 40
A ≥ 60
A ≥ 20
A ≥ 40
A ≥ 60
(p, γ)
(e+νe)}
}
b
c
b'
c'
0.25
·
MEdd 0.35
·
MEdd
a'
CNO
Tz=-1
H
4He
A=20-39
A
=
40
-5
9
H 4He
CNO
A=20-39
T
z=
-1
A
=
40
-5
9
A ≥60
A ≥60
a
7.50
7.75
8.00
8.25
8.50
8.75
9.00
lo
g
10
T
[K
]
log10y [g cm−2]
5.0 6.0 7.0 8.0 9.0
Time [y/ṁ]
1 min 10 min 1 h 10 h 1 day
Teff = 7.050× 106 K
−10
−8
−6
−4
−2
0
lo
g
10
X
Figure 4.6: Same as Fig. 4.5, for Ṁ∞ = 0.25ṀEdd (left panels) and 0.35ṀEdd (right
panels). The corresponding values of Qb for 0.25 and 0.35 ṀEdd are 0.204 and 0.172
MeV per baryon, respectively.
121
8.25
8.50
8.75
9.00
lo
g
10
T
[K
]
log10y [g cm−2]
6.0 7.0 8.0 9.0
Time [y/ṁ]
1 s 10 s 1 min 10 min 1 h 10 h
Teff = 9.100× 106 K
−4
−2
0
lo
g
10
X
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
−10
−5
0
lo
g
10
X
8.25
8.50
8.75
9.00
lo
g
1
0
T
[K
]
Teff = 1.3628× 107 K
log10y [g cm−2]
6.0 7.0 8.0 9.0
Time [y/ṁ]
1 s 10 s 1 min 10 min 1 h
−6
−4
−2
0
lo
g
1
0
X
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
−10
−5
0
lo
g
10
X
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
log10ρ [g cm−3]
8
9
10
11
12
13
14
15
16
17
lo
g 1
0
ε̇ n
u
c
[e
rg
s−
1
g−
1
]
Hot CNO
(α, p)
(α, γ)
ε̇3α
A ≥ 20
A ≥ 40
A ≥ 60
A ≥ 20
A ≥ 40
A ≥ 60
(p, γ) (e+νe){ {
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
log10ρ [g cm−3]
8
9
10
11
12
13
14
15
16
17
lo
g 1
0
ε̇ n
u
c
[e
rg
s−
1
g−
1
]
Hot CNO
(α, p)
(α, γ)
ε̇3α
A ≥ 20
A ≥ 40
A ≥ 60
A ≥ 20
A ≥ 40
A ≥ 60
(p, γ) (e+νe){ {
d
b
c
d'
b'
c'
1
·
MEdd 5
·
MEdd
a'a
α β γ δ ε ζ α
CNO
Tz=-1
H
4He
A=20-39
A ≥60
A=40-59
H
4He
CNO
A=20-39
A=40-69 A ≥70
Tz
=-1
β δγ ε ζ
12C
16O 15O17F
12C
16O 15O17F
40Ca 40Ca
72Kr
64Ge
43Ti
43Ti
56Ni
56Ni
Figure 4.7: Envelope models at Ṁ∞ = ṀEdd (left panels) and 5ṀEdd (right panels)
as a function of ρ. The upper scales indicate the corresponding column depth y
and time spent by the accreted matter since it started its journey from the neutron
star surface. Panels (a) and (a’): temperature; panels (b) and (b’): mass fractions
of selected light nuclei; panels (c) and (c’): mass fractions of selected heavy nuclei;
panels (d) and (d’): specific energy generation of dominant processes, as indicated,
and with the upper thick line showing the total energy generation. Greek labels
at the top of panels (d) and (d’) indicate specific positions of events discussed in
the text. Qb at the base are 32.59 and 25.54 keV per baryon for ṀEdd and 5ṀEdd,
respectively.
122
0 1
2
5
21
25
35
39
45
6
22
26
36
40
46
7
23
27
37
41
47
8
24
28
38
42
48
3
9
11
13
15
17
19
29
31
33
43
49
4
10
12
14
16
18
20
30
32
34
44
50
H(1)
F(9)
Cl(17)
He(2)
Ne(10)
Ar(18)
Li(3)
C(6)
Na(11)
Si(14)
Be(4)
N(7)
Mg(12)
P(15)
B(5)
O(8)
Al(13)
S(16)
K(19)
Ca(20)
Sc(21)
Ti(22)
V(23)
Cr(24)
Mn(25)
Fe(26)
Co(27)
Ni(28)
Cu(29)
Zn(30)
Ga(31)
Ge(32)
As(33)
Se(34)
Br(35)
Kr(36)
Rb(37)
Sr(38)
Y(39)
Zr(40)
Nb(41)
Mo(42)
Tc(43)
Ru(44)
Rh(45)
Pd(46)
Ag(47)
Cd(48)
In(49)
Sn(50)
Sb(51)
Te(52)
1ṀEdd
0 1
2
5
21
25
35
39
45
6
22
26
36
40
46
7
23
27
37
41
47
8
24
28
38
42
48
3
9
11
13
15
17
19
29
31
33
43
49
4
10
12
14
16
18
20
30
32
34
44
50
H(1)
F(9)
Cl(17)
He(2)
Ne(10)
Ar(18)
Li(3)
C(6)
Na(11)
Si(14)
Be(4)
N(7)
Mg(12)
P(15)
B(5)
O(8)
Al(13)
S(16)
K(19)
Ca(20)
Sc(21)
Ti(22)
V(23)
Cr(24)
Mn(25)
Fe(26)
Co(27)
Ni(28)
Cu(29)
Zn(30)
Ga(31)
Ge(32)
As(33)
Se(34)
Br(35)
Kr(36)
Rb(37)
Sr(38)
Y(39)
Zr(40)
Nb(41)
Mo(42)
Tc(43)
Ru(44)
Rh(45)
Pd(46)
Ag(47)
Cd(48)
In(49)
Sn(50)
Sb(51)
Te(52)
5ṀEdd
Neutrons
P
ro
to
n
s
Figure 4.8: Space-integrated reaction flows for Ṁ∞ = ṀEdd and ṀEdd, in units of
the corresponding value of the 3α reaction. Solid lines: ≥ 0.3; dashed lines: between
0.3 and 0.01; dotted lines correspond to values between 0.01 and 0.005. Nuclides
colored in lightblue illustrate the “sawtooth” path, while those in green correspond
to the β − 3p − β path (see main text for further details). Some Tz = −1 nuclides
are colored in blue.
123
0 2 4 6
6
7
8
9
lo
g
1
0
T
[K
]
6.00 6.25 6.50 6.75 7.00
log10Ts [K]
−4
−2
0
2
4
Q
L
,b
[M
eV
b
ar
yo
n
−
1
] Ṁ∞ = 0.03ṀEdd
gs,14 = 1.735, Zs = Z⊙
net380
0 2 4 6
log10ρ [g cm−3]
8
10
12
14
16
lo
g
1
0
˙̆ ε n
u
c
[e
rg
g
−
1
s−
1
] ˙̆ε
grav
˙̆εν
6 7 8 9
log10T [K]
−0.04
−0.02
0.00
0.02
0.04
(d
T
/d
P
)/
10
−
1
3
0 2 4 6
6
7
8
9
lo
g
10
T
[K
]
6.00 6.25 6.50 6.75 7.00
log10Ts [K]
−4
−2
0
2
4
Q
L
,b
[M
eV
b
ar
yo
n
−
1
] Ṁ∞ = 0.10ṀEdd
gs,14 = 1.735, Zs = Z⊙
net380
0 2 4 6
log10ρ [g cm−3]
8
10
12
14
16
lo
g
10
˙̆ ε n
u
c
[e
rg
g
−
1
s−
1
] ˙̆ε
grav
˙̆εν
6 7 8 9
log10T [K]
−0.04
−0.02
0.00
0.02
0.04
(d
T
/d
P
)/
10
−
13
Figure 4.9: Four panels, at fixed accretion rate, illustrating the envelope profiles
in stationary state for different values of Teff. The upper figure corresponds to
Ṁ = 0.03ṀEdd while the lower one to Ṁ = 0.10ṀEdd. Upper left panel: temperature
as a function of density. Upper right panel: base or cutoff luminosity as a function
of the surface temperature. Lower left panel: specific energy generation rate as a
function of density, for four selected curves with the same colors as in the upper left
panel. The heating/cooling contributions due to neutrinos and mass accretion are
illustrated only for the red-curved envelope. Lower right panel: phase space of the
temperature profile.
124
0 2 4 6
6
7
8
9
lo
g
1
0
T
[K
]
6.00 6.25 6.50 6.75 7.00
log10Ts [K]
−4
−2
0
2
4
Q
L
,b
[M
eV
b
ar
yo
n
−
1
] Ṁ∞ = 0.30ṀEdd
gs,14 = 1.735, Zs = Z⊙
net380
0 2 4 6
log10ρ [g cm−3]
8
10
12
14
16
lo
g
10
˙̆ ε n
u
c
[e
rg
g
−
1
s−
1
] ˙̆ε
grav
˙̆εν
6 7 8 9
log10T [K]
−0.04
−0.02
0.00
0.02
0.04
(d
T
/d
P
)/
10
−
1
3
0 2 4 6
6
7
8
9
lo
g
10
T
[K
]
6.00 6.25 6.50 6.75 7.00
log10Ts [K]
−4
−2
0
2
4
Q
L
,b
[M
eV
b
ar
yo
n
−
1
] Ṁ∞ = 0.60ṀEdd
gs,14 = 1.735, Zs = Z⊙
net380
0 2 4 6
log10ρ [g cm−3]
8
10
12
14
16
lo
g
10
˙̆ ε n
u
c
[e
rg
g
−
1
s−
1
] ˙̆ε
grav
˙̆εν
6 7 8 9
log10T [K]
−0.04
−0.02
0.00
0.02
0.04
(d
T
/d
P
)/
10
−
13
Figure 4.10: Same caption as in Fig. 4.9, now for 0.30ṀEdd and 0.60ṀEdd.
125
the model is irrelevant at the T → 0 region since, given L < 0, ∇rad < ∇ad and the
system remains stable against convection.
While having L < 0 at some subset P− ⊂ Psb in the envelope is a physical
scenario, T → 0 should be taken with care. It might well be the case that, at
most, the crust and nucleus of the star are cold (∼ 106 K), and these cut solutions
correspond to transient states where matter is accreted and deposits energy at P−.
Notice, however, the solutions to Eqs. 4.18 - 4.25 do not presuppose or require
any information regarding Lb/Lcut or Tb/Tcut since they are obtained by integration
(given its IVP status). In addition, Lb < 0 but “complete” envelopes actually exist
at the four accretion rates (see the curves below the red profiles at the upper right
panel of Figs. 4.9 and those lying between the red and the immediately next black
curve of Figs. 4.10). Thus, we cannot discard solutions only by having Lb < 0
or Lcut < 0. Additionally, the envelopes with Teff ∈ T − should not be regarded
as completely unphysical: it is important to notice the choice of ρb = 107 g cm−3
as envelope-crust boundary is an artificial construction, where we could also have
chosen 105 or 106 and these solutions would have remained Lb < 0 but “complete”,
or 108 g cm−3 and more Lb < 0 solutions around the red curves would have been
discarded due to their T → 0 behavior.
The second characteristic of relevance in the T vs ρ panel is the existence, below
0.30ṀEdd, of two branches of fully-accreted envelopes, one below and the other above
the region of the Teff ∈ T − models. All envelopes in the branch with Teff < 106.5 K
have QLb
< 0, as shown in the upper right panel. In this same figure we observe QLb
follows a concave tendency, reaching a global minimum and then moving to positive
values at increasing Ts(Teff), regardless of the existence of two branches of fully-
accreted envelopes. The location of the minimum in the QLb
vs Ts plane is sensitive
to Ṁ∞, moving from 106.5 K at 0.03 to 106.75 K at 0.6 ṀEdd. This minimum is
closer to the second branch of fully-accreted solutions than to the first, low-T one,
and the behavior of QLb
in this second, high-T branch is almost exponential.
Similarly to the existence of T − ⊂ Teff, it could be argued the emergence of
two branches of solutions is an artificial construction, consequence of choosing ρb.
For instance, at the very 0.3ṀEdd (Fig. 4.10) we see by choosing ρb = 108 g cm−3
the low branch is absorbed into the Teff ∈ T − subset. In clear contrast, we can
suspect the upper branch is not an artifact of choosing ρb since the slope of the
envelope is related to the nuclear burning: at high densities and temperatures (e.g.
lower left panel of Figs. 4.9 and 4.10), ˙̆εnuc decreases from ∼ 1015 erg g−1 s−1 at its
maximum to 1011 erg g−1 s−1 as a consequence of 1H and 4He - the main source of
fuel - exhaustion. Here we must also consider that below 108 g cm−3 electroweak
reactions do not provide sufficient energy as to significantly alter dL/dP , and higher
temperatures (e.g. above ∼ 108.75 K) are required to start the fusion of 12C and
16O. Consequently, models with Lb > 0 should exist regardless of our choice of ρb, a
scenario which is in agreement with the non-accreting case.
What can the phase space tell us regarding the numerical solutions? From the
126
behavior of the envelopes at the four selected accretion rates, we deduce the system
has a separatrix since any temperature profile either exhibits T → 0 (going to the
left direction in the phase space) or T ≫ Ts (i.e. moving to the right direction of the
phase space). The location of the Ts for this almost-vertical separatrix depends on
the Ṁ∞: while at 0.03ṀEdd we have Ts,separatrix ∼ 108.25 K, at 0.6ṀEdd we observe
Ts,separatrix ∼ 108.75 K. However, due to the density of curves at each side of the
separatrix we can deduce solutions starting with low-Teff will reach a maximum and
then T → 0, while those starting at high-Teff will remain with finite temperature
throughout the whole profile. Without further information, we cannot regard ei-
ther of these sides as stable or unstable although, intuitively, we could expect any
solution close to the separatrix might become unstable under the presence of small
perturbations. Furthermore, the phase space corroborates our initial assumption of
the “lower” branch of solutions as a mere artifact of imposing ρb since all solutions
in this branch, regardless of the accretion rate, move towards the left direction.
Finally, let us discuss the specific generation rate, e.g. lower left panels of
Figs. 4.9 and 4.10. Here, the red curves correspond to the model of the same color
in the T vs ρ diagram. Below 103 g cm−3, the main contribution to the luminosity
comes from the compression of accreted material, although the rather small mass
of these outer layers produces little impact over L. In the specific generation rate
due to nuclear reactions we observe a first maximum, corresponding to the pp chain
processes, in particular to the 2H(p, γ)3He reaction. Its peak contribution (∼ 1014
g cm−3), however, is small in comparison with the heating from compression. A
different situation takes place between 104 and 106 g cm−3, where the CNO cycles
(both cold and hot versions) overtake the contribution from matter compression and,
in contrast to the outermost layers, now the decreasing in luminosity can be large
due to the moderate densities. This is corroborated by the blue, green and purple
temperature profiles which, after reaching a global maximum, exhibit the T → 0
behavior discussed above. This feature impacts the specific energy generation rate
due to nuclear reactions since, as temperature decreases, thermonuclear reactions
produce progressively less energy. In clear contrast, for the red curves we see the hy-
drogen burning has been completed near 106 g cm−3 and the remaining contribution
to the energy comes from the electroweak sector, which is temperature-independent
but is limited in extension by the composition of the ashes.
Same accretion rate, different gs
While we have observed the low-temperature branch of “fully accreted” envelopes is
an artifact of choosing ρb, it is nevertheless important to observe how different are
envelopes when we change gs. In Fig. 4.11 we fix the composition at the surface and
the accretion rate at ṀEdd and change the values of stellar radius from the 1.4M⊙,
APR-core EOS star, as to those corresponding to 1.1M⊙ and 2.0M⊙ for the same
core EOS, thus having a smaller and larger surface gravity than the value for the
127
0 2 4 6
6
7
8
9
lo
g
1
0
T
[K
]
6.00 6.25 6.50 6.75 7.00
log10Ts [K]
−4
−2
0
2
4
Q
L
,b
[M
eV
b
ar
yo
n
−
1
] Ṁ∞ = 0.30ṀEdd
gs,14 = 1.262, Zs = Z⊙
net380
0 2 4 6
log10ρ [g cm−3]
8
10
12
14
16
lo
g
10
˙̆ ε n
u
c
[e
rg
g
−
1
s−
1
] ˙̆ε
grav
˙̆εν
6 7 8 9
log10T [K]
−0.04
−0.02
0.00
0.02
0.04
(d
T
/d
P
)/
10
−
1
3
0 2 4 6
6
7
8
9
lo
g
10
T
[K
]
6.00 6.25 6.50 6.75 7.00
log10Ts [K]
−4
−2
0
2
4
Q
L
,b
[M
eV
b
ar
yo
n
−
1
] Ṁ∞ = 0.30ṀEdd
gs,14 = 3.238, Zs = Z⊙
net380
0 2 4 6
log10ρ [g cm−3]
8
10
12
14
16
lo
g
10
˙̆ ε n
u
c
[e
rg
g
−
1
s−
1
] ˙̆ε
grav
˙̆εν
6 7 8 9
log10T [K]
−0.04
−0.02
0.00
0.02
0.04
(d
T
/d
P
)/
10
−
13
Figure 4.11: Same caption as in Fig. 4.9, now for a fixed accretion rate, 0.30ṀEdd,
and two different values of surface gravity gs.
128
0 20 40 60 80 100
Mass Number
−10
−8
−6
−4
−2
0
lo
g 1
0
[A
b
u
n
d
an
ce
]
gs,14 = 1.74
ρb,7 = 1
Tb,8 = 5.45
1.0ṀEdd
XH = 0.01, 6.35×106 K
XH = 0.10, 7.15×106 K
XH = 0.20, 7.60×106 K
XH = 0.30, 8.00×106 K
XH = 0.40, 8.30×106 K
XH = 0.50, 8.60×106 K
XH = 0.60, 8.87×106 K
XH = 0.70, 9.10×106 K
0 20 40 60 80 100
Mass Number
−10
−8
−6
−4
−2
0
lo
g 1
0
[A
b
u
n
d
an
ce
]
gs,14 = 1.74
ρb,7 = 1
Tb,8 = 4.87
0.3ṀEdd
XH = 0.01, 5.000×106 K
XH = 0.10, 5.390×106 K
XH = 0.20, 5.750×106 K
XH = 0.30, 6.040×106 K
XH = 0.40, 6.285×106 K
XH = 0.50, 6.470×106 K
XH = 0.60, 6.635×106 K
XH = 0.70, 6.800×106 K
Figure 4.12: Distribution of abundances for different accreted fractions of hydrogen,
denoted as XH, and helium at the surface with XHe = 0.98−XH. Left panel: Ṁ∞ =
ṀEdd, right panel: Ṁ∞ = 0.30ṀEdd. Here, ρb,7 = ρb/107 g cm−3 and Tb,8 = Tb/108
K. At ṀEdd, Qb ranges from 0.58 (at XH = 0.01) to 0.03 (at XH = 0.70) MeV per
baryon. At 0.3ṀEdd, Qb ranges from 1.0 (at XH = 0.01) to 0.15 (at XH = 0.70)
MeV per baryon.
1.4M⊙ star. In the phase-space diagram (lower right panel of Fig. 4.11) we still infer
the presence of a separatrix curve, although its location in Ts is directly affected by
gs, similarly to the location and the value of the minimum in the QL,b
vs Ts diagram.
Within the context of stable burning, we could argue a higher internal temperature
is required in the presence of a strong gravitational field in order to stabilize the
burning. Regarding the nuclear energetic output, we observe high surface gravity
environments allow to reach the rp-process even for those incomplete envelopes of
T → 0 behavior, while their counterparts at low surface gravity are starting to reach
the CNO-breakout point.
In the absence of accretion, at fixed chemical composition and at fixed effective
temperature (Fig. 4.2), we observed an inverse correlation between Tb and gs. For
the Lb > 0 envelopes in Fig. 4.11 we still observe this correlation: take the Ts = 107
K for instance. At gs,14 = 1.262, we see Tb ∼ 109 K, while at gs,14 = 3.238 we have
Tb < 109 K, from where we can infer a similar correlation for all “fully accreted”
envelopes below the red curves.
Does the rp-process depends on the metals at the surface?
Short answer: no. Figs. 4.12 illustrate this point very well: here we see the distri-
bution of ashes at ρb = 107 g cm−3 for two collections of models, each panel with
the indicated mass accretion rate and gs. Common to all curves is the amount and
composition of metals at the surface, Z⊙ = 0.02, while specific to each model is the
effective temperature and the mass fractions of hydrogen XH and helium, chosen
129
as X4He = XH+4He,⊙ − XH with XH+4He,⊙ = 0.98. In both panels we observe that
H-deficient matter is less likely to change the initial distribution of metals. On the
other hand, even with as small fractions of XH as 10% we start noticing the change
in composition below A = 56. As the hydrogen fraction exceeds 30% the disap-
pearance of 40Ca in favor of A ≥ 56 metals becomes progressively significant until
reaching the distribution discussed in the first subsubsection. From these curves, we
can deduce that as long as hydrogen fraction is ≈ 20% some metals can actually be
synthesized, although for having a basic form of rp-process we require between 30%
to 40% percent of hydrogen in the accreted matter.
First wrap-up: are Tb − Lb − Teff viable?
In the absence of time-dependent terms in the differential equations, we were led
to Eqs. 4.18 - 4.25. Contrary to the intuitive expectation of having all models in
a given Teff set with Lb > 0, we found stationary solutions with Lb < 0, as well as
other curves which behaved as T → 0 at some ρcut < ρb, which we could not regard
as fully-accreted envelope given their physical behavior, despite its mathematical
consistency with the input functionals.
In principle, at any Ṁ∞ ≥ 0.3ṀEdd (in order to be consistent with observations),
we might construct Tb − Teff − Lb relations. It suffices to consider only the fully-
accreted envelopes, regardless of sgnLb, and store in tables (or attempt an analytic
fit) their values of Teff and the pair Tb, Lb. However, without further information
regarding the time-evolution behavior of an actual envelope, the existence of an
artificial category of T → 0 curves as “incomplete” envelopes prevents us from im-
plementing such straightforward scheme. In particular, we could simply argue there
exists certain values of Teff which the system cannot adopt, although without the
temporal terms and within the given physics there does not exist a strong physical
reason as to why it should be so. On the other hand, there is a physical reason to
question why we cannot rule out these solutions without restoring to time-dependent
simulations: the existence of the “secondary branch”, despite being artificial, points
out that the equations allow envelopes of low effective temperature to inject energy
to the interior (i.e. Lb < 0), which in time-dependent scenarios should lead to the
heating of the core. This point is important given that, a priori, we do not know
the temperature of the core and, if it starts - before accretion - well below 107 K,
implementing a Tb −Teff −Lb relation only with fully-accreted envelopes lead to the
existence of a gap in accessible temperatures Tb. Furthermore, directly ruling out
the “secondary” branch suggests all cores should start with at least Tb ∼ 107 K and
with a high effective temperature, close to 107 K as well.
A secondary characteristic which requires clarification is the steep slope of Lb(Teff).
Leaving aside the potential numerical difficulties of having a large difference in Lb
within a small range of Teff, the behavior in the phase-space of the stationary solu-
tions poses a question: are the Lb ≈ 0 curves approaching an unstable equilibrium
130
point? In that case, what is the role of the “discarded” solutions with T → 0? What
is the response of the rest of the start to such unstable equilibrium?
As a first patch, we could restore to instability arguments in order to impose
a different cutoff to the T → 0 models. Indeed: by comparison with existent
diagrams on the subject, we see these curves correspond to the “envelope just before
the explosion” models described elsewhere. In these works, the main idea was to
find out the location at which the classical instability criterion, e.g. ∂ϵheat/dT >
∂ϵcool/dT , predicts the stationary system is unstable against small perturbations in
temperature. Coincidentally, we see the location of such instability point agrees well
with the location of the maximum in temperature. However, this criterion is not
exact, and the uncertainty in the location of the explosion density suggests we have
to either cut at L > 0 or L < 0. Given ρcut < ρb, for the formal Tb − Teff − Lb
scheme we require to join this piece of solution to another portion, not necessarily
the numerical core. In the first case, Lcut > 0, joining the cut accreted envelope with
a “compressed”-like solution induces a higher Tb - as in those non-accreted models
of the past section. On the other hand, having Lcut < 0 is likely to lead towards the
same T → 0 behavior given the absence of heating or cooling sources at ρ ∈ [ρcut, ρb].
Given the behavior of T (P ) and L(P ), without time-dependence one could also
suspect the stationary solutions represent the different states the envelope takes
during constant accretion, i.e. starting at low-Teff and then moving to high-Teff
while injecting energy at ∼ 106 g cm−3. However, such picture would indicate the
crust and core are colder than the surface at all times, and only latter they become
hotter. There is, however, not sufficient evidence as to claim this is a universal
behavior, as it could be the case to start with a hot core but cold exterior (as
suggested by the non-accreting envelopes in the past section).
In summary: while the stationary states allowed us to have a deeper grasp on
the underlying physics and the overall behavior of the envelope, we still require to
restore to the fully time-dependent simulations in order to understand how we can
- if possible - implement adequate Tb − Teff − Lb relations since we require further
knowledge on the nature of those curves of effective temperature in T −.
4.3.2 Intermezzo: Linear perturbations to stationary solu-
tions and the origin of bursts
Before embarking in a fully time-dependent trip, it is important to understand the
underlying behavior of these simulations. In order to do so, let us now discuss per-
turbations to the stationary solutions. From the general properties of the envelope
(Subsection 1), the discussion of time-dependent perturbations can be carried em-
ploying Eqs. 4.8 and 4.9 only - we see structure plays a minor role in contrast to
temperature and the evolution of chemical composition.
Given the instability analysis to the heat diffusion equation of Chapter 1, it is
131
2 3 4 5 6
log10ρ [g cm−3]
−10.0
−7.5
−5.0
−2.5
0.0
2.5
5.0
7.5
10.0
lo
g
10
τ
[s
]
Ṁ∞ = 0.30ṀEdd
gs,14 = 1.735, Zs = Z⊙
net380
τdiff τacc τnuc
Figure 4.13: Different timescales for the same models in Fig. 4.10.
useful to classify the sources at the right hand side of Eq. 4.8 as heating or cooling
mechanisms, depending on their overall sign. From the stationary solutions, we
know ˙̆εnuc − ˙̆εν remains positive in practically all scenarios of interest, hence it is
a heating mechanism. For simplicity, let us enclose them into a single term, ˙̆ε :=
ρB( ˙̆εnuc − ˙̆εν). Since ∇ad − ∇T > 0 (c.f. Chapter 2), the source term related to
accretion in Eq. 4.8 acts as a heating mechanism as well. Finally, we have the flux
gradient term. While its tendency in stationary state is that of a cooling mechanism,
whenever dF/dP > 0 this term must be taken as a heating mechanism instead.
Now, let us look at the timescales. Given heat diffusion is the only cooling
mechanism, a separate comparison between τdiff with τnuc and of τacc with τdiff allows
to qualitatively estimate the interplay between heating and cooling mechanisms,
while comparing τnuc with τacc allows to determine which heating mechanism is
more important than the other at different conditions throughout the envelope.
Let us first compare τdiff and τacc, Eqs. 4.10 and 4.11. At ρb = 108 g cm−3 and
ṀEdd, we have τacc ∼ 12 days, while the local diffusion timescale is approximately a
12th part of this value. Consequently, if mass accretion heats the envelope, radiation
and conduction have ample time to transport the energy towards the coldest region
of the star, either the surface or the deep layers. Notice in this assertion we have
not made further assumptions regarding the temperature of the core just before
the start of accretion: while the majority of simulations presuppose Tb ∼ 108 K,
it might be the case that Tb ∼ 106 K. Under the latter scenario, some fraction of
nuclear heating (generated as a consequence of mass accretion) might leak towards
the interior of the star rather than to the exterior. However, the propagation of heat
is still controlled by τdiff, which goes from days to weeks as we move from 107 to
132
1010 g cm−3, while at the surface this value is of ∼ hr to seconds (see also Fig. 4.3).
A different scenario takes place between τnuc and τacc, Eqs. 4.12 and 4.11. At the
outermost layers of the envelope (≤ 103 g cm−3), nuclear energy is predominantly
generated by pp-chains, thus ˙̆εnuc ≈ 1011 g cm−3. Since the EOS is almost of an
ideal gas, i.e. cP ≈ 5ρkB/2⟨m⟩imu, by definition of the nuclear timescale at T ∼ 108
K we have τnuc ≈ 105 s. On the other hand, near the conditions for 1H exhaustion
(via CNO or rp-process) we know τnuc ≈ 103 s, thus nuclear timescale becomes
short at increasing depths. However, τacc exhibits the opposite behavior, going from
small values at the surface to high values (e.g. ∼ hrs to days) at ρB ≥ 106 g cm−3.
Therefore, at the outermost layers of the envelope it takes more time to burn than
to accrete fuel, while at deep layers the nuclear burning of matter is faster than the
accretion.
Finally, we have the competition between nuclear and diffusion timescales. From
the above discussion, as well as the considerations on stability of Chapter 1, as long
as τdiff < τnuc the stability against thermonuclear perturbations is guaranteed in the
envelope. In the opposite limit, such thermonuclear perturbations lead to a drastic
increase in the local temperature, which propagates towards the surface via radiation
or convection, depending on the sign of ∇T − ∇ad. Such increase is short-lived with
respect to the duration of accretion episodes, and as such is the origin of the observed
X-ray bursts. The ashes of these bursts accumulate at the envelope-crust boundary,
compress the pristine outer crust and eventually replace it.
We can also take a look at these timescales from the actual output of our time-
independent simulations. In Fig. 4.13, we observe the three timescales for the four
selected envelope models from Fig. 4.10. In first place, we observe little discrepancy
among the four displayed τacc, as expected since this term is practically controlled
by the accretion rate and the surface gravity - equal in the four scenarios - with
small influence of pressure only, as evidenced by the cut-off in the purple curve. A
similar pattern is followed by τdiff, timescale which is primarily controlled by the
thickness of the envelope d - which is rather small for the selected models - and
secondary by the chemical composition throughout the specific heat and thermal
conductivity. Given the actual temperature is higher for the red curve than for the
purple one, we expect the observed inverse correlation, e.g. the timescale is larger
for the coldest model, the purple one. Notice τdiff < τacc well below 105 g cm−3,
indicating the envelope has ample time to relax into equilibrium. Furthermore, in
this density region we observe diffusion timescale is actually the shortest among the
three, allowing us to conclude the envelope there has ample time to relax into an
equilibrium state.
An interesting scenario takes place for the green and red curves, and consequently
for all intermediate envelope models between them. As a consequence of the nuclear
burning, we see an inversion of the relation between τacc and τnuc near 105 g cm−3.
Having a short nuclear timescale implies more time is required into accreting than
into burning it. Following this inversion trend, between 105 and 106 g cm−3 we
133
observe nuclear and diffusion timescales are equal. In this region an instability is
likely to take place: if matter requires more time to diffuse than to produce heating,
a temperature build up might follows and, consequently, a thermonuclear explosion.
The former qualitative discussion can be put into formal terms employing the
one zone model, which focuses on analyzing the stability of temperature and com-
position under small perturbations. In this approximation, spatial derivatives are
approximated by local values (i.e. ∂PA ≈ A/P ), and taking the flux contribution
to Eq. 4.8 as a local term, i.e.
ε̇cool := −ρBgs
∂F
∂P
≈ 16σSBg
2
sρBT
4
3P 2κ
. (4.26)
Given Nspecies ≫ 1, at the perturbative level it suffices to consider only one species
as responsible for the perturbations to the system. Its mass fraction will be denoted
as Xc. Under these considerations, in the one zone model approximation Eqs. 4.8
and 4.9 reduce to
cP
∂T
∂τ
= ε̇c − ε̇cool (4.27)
∂Xc
∂τ
= −ṁgs
Xc
P
+ AcRc. (4.28)
In stationary state, e.g. ∂τT = 0 and ∂τXc = 0, we obtain two expressions indicat-
ing the equilibrium between heating and cooling mechanisms for the temperature
equation, i.e. ε̇c = ε̇cool, and the balance between mass accretion and the rate of
burning, ṁgsXc/P = AcRc. For the linear perturbation analysis, let the local rate,
energy generation rate and opacity be approximated as2
Rc = R0X
α1
c Tα2 (4.29)
ε̇c = R0X
α1
c Tα2 (4.30)
κ = κ0X
β1
c T
β2 (4.31)
By introducing T (t, P ) = T0(P ) + δT (t, P ) and Xc(t, P ) = X0(P ) + δXc(t, P ) into
Eqs. 4.27 and 4.28, keeping the linear terms, employing the stationary relations
among mechanisms and defining θ := δT/T and χc := δXc/Xc, we arrive to
∂θ
∂τ
=
1
τnuc
¶[α2 + β2 − 4] θ + (α1 + β1)χc♢ (4.32)
∂χc
∂τ
=
1
τacc
¶α2θ + (α1 − 1)χc♢ , (4.33)
i.e. a coupled system of linear equations. While these can be combined into a
single ODE for either θ or χc (and thus admitting solutions of the form ∝ exp(λ′τ)),
2Notice perturbations in density can easily be absorbed as perturbations in κ since, at constant
pressure, ρB(T, Xc).
134
its linearity make suitable to analyze the behavior of the system by looking at its
eigenvalues. By explicit calculations, it can be shown these are3
λ± =
1
2
[
α2 + β2 − 4
τnuc
+
α1 − 1
τacc
]
±



1
4
[
α2 + β2 − 4
τnuc
− α1 − 1
τacc
]2
+
α2(α1 + β1)
τnucτacc



1/2
.
(4.34)
We thus see the system admits two kinds of solutions, oscillatory and exponentials,
depending on the sign of the term inside the square root. The first case corresponds
to the observed thermonuclear explosions due to the recurrence time. The second
scenario corresponds to systems where nuclear burning does not induce explosions,
i.e. as in sources at Ṁ ≥ 0.3ṀEdd.
4.3.3 Time-dependent envelopes
In order to model the time-dependent evolution of an envelope, we employed the
publicly available code MESA. Given our interest in varying parameters such as the
base luminosity Lb and the boundary density ρb, we provided as initial conditions
to this program the output from our time-independent envelope code in the absence
or presence of accretion. In the former case, an artificially layered composition was
set, prioritizing the presence of hydrogen at ρ ≤ 104 g cm−3 and of metals at higher
densities. Considering the surface boundary condition is written in terms of Teff,
and for non-accreting envelopes Ls = Lb, in addition to the MeV per baryon unit
for the luminosity we employ as well the effective temperature. For instance, as
a “cold” model we refer to the initial envelope from our stationary code having
Teff = 105 K, i.e. ≈ 2.59 × 10−5L⊙. To be self-consistent with the stationary codes
we employed in the past sections, we incorporated into MESA additional subroutines
for the radiative opacity and called them during the execution of the simulations
instead of the tabulated opacities which come by default. These fits correspond to
the updated version of the electron scattering opacity from [Pou17], and the electron-
ion interaction from [SBCW99]. Regarding the conductive opacity, we employed
the tables provided by MESA considering they share the same formalism as that we
employ in our stationary code.
Simple experiments
As a first step towards the understanding of the different branches and curves we
found in the stationary approximation, we implemented accreted envelopes with
3Our expression for λ± differs from Heger 2007 for two reasons: (i) we consider the evolution
of the mass fraction instead of the column density y = P/gs and (ii) dependency of κ is explicitly
addressed. In order to recover their expressions, it suffices to consider α1 = β2 = 0 and β1 = −2.
This apparent dependency of κ with y is only a consequence of choosing y instead of X at ε̇cool
(e.g. Eq. 4.26): notice ∂y ε̇cool ∝ −2κ−1y−3 = β1κ−1y−3 despite κ ≡ constant, as done by Heger
2007.
135
0
2
4
lo
g 1
0
L
/L
⊙
0 1 2 3 4 5 6
t [hr]
7
8
lo
g 1
0
T
b
[K
] 〈gs,14〉 = 2.167, net381
0.30ṀEdd, Zs = Z⊙
Lb =-1.926 L⊙, ρ6.99
Lb =-56.25 L⊙, ρ5.93
Lb =-13.07 L⊙, ρ6.93
2
4
lo
g 1
0
L
/L
⊙
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
t [hr]
8.0
8.5
lo
g 1
0
T
b
[K
]
〈gs,14〉 = 2.167, Approx140
ṀEdd, Zs = Z⊙
Lb =-11.17 L⊙, ρ7.00
Lb =-2.413 L⊙, ρ8.00
Lb =-11.17 L⊙, ρ8.00
Lb =-48.93 L⊙, ρ7.00
Figure 4.14: Upper panels: Surface luminosity and bottom temperature for 0.3ṀEdd
models with different Lb. Their respective ρb are indicated in the labels. Lower
panels: same as upper ones, but for ṀEdd.
different Teff as initial conditions for MESA. Considering the observed two-branched
behavior of the stationary envelopes at 0.3ṀEdd, we selected three envelopes from
this family of solutions: one from the “artificial”, low-Teff branch, the second from
the cut-off models with T → 0 at ρcut < 107 g cm−3, and the third one from those
fully-accreted envelopes (i.e. high-Teff branch) with Lb < 0.
The temperatures at the bottom of the envelope as functions of time, as well as
the surface luminosity, are displayed in Fig. 4.14. The models were labeled according
to their base luminosity, expressed in units of L⊙, and are ordered from bottom to
top according to their Teff at t = 0 s. The most remarkable feature of this plot is
the rapid decrease in temperature from the T → 0 model - purple curves. Once
reached Tb ∼ 103 K, MESA did not proceed with the integration due to the lack of
convergence towards a satisfactory model. Although its base density is < 107 g
cm−3, and we could actually expect this behavior since we are providing a negative
value at the base - fixed by MESA construction - its lack of further advance in contrast
to the other models (with negative luminosity as well) suggests the existence of a
minimum luminosity we can inject at a certain ρ.
136
In contradistinction with the T → 0 model, we observe the remaining two proceed
normally for more than one hour. In the case of the one belonging to the “low
temperature branch”, we see a slow build-up of temperature, both at surface and
at the bottom, until eventually a sudden increase takes place after ∼ 5 hours, i.e.
a thermonuclear explosion as a consequence of having hydrogen up to 107 g cm−3.
Finally, we observe the model from the “high branch” of solutions cools down for
one hour before starting a thermonuclear instability. Given ρb = 107 g cm−3 for
this latter model, we deduce models with Lb < 0 are indeed viable scenarios for
representing the envelope of a neutron star. However, one could pose into question
the viability of having a fixed (e.g. time-independent) amount of energy flowing from
the surface to the interior since this scenario does not take into consideration the
reaction from the numerical core, which must be heated as a consequence. Another
characteristic to take into consideration is that injecting energy from the envelope
to the core does not lead to a bursting suppression behavior - bursts still occur
even with Lb < 0. And then another question emerges: how much energy do bursts
actually deposit towards the interior of the star? How is this compatible with the
classic picture of Lb > 0?
In the lower panels of Fig. 4.14, we show the results of performing a similar
experiment, now with stationary models corresponding to the ṀEdd mass accretion
rate. Here we also allowed for different location of the negative luminosity, either
ρ = 107 g cm−3 or 108 g cm−3. The cost to pay for increasing the boundary density
is to have a slow numerical simulation in proportion to the amount of species in the
network. Consequently, we opted for changing the net381 network by Approx140.
Once again, the most notorious of the models is that where internal temperature
drastically converges to zero in less than one hour, a consequence of keeping a
negative luminosity at all times and, similarly as in the 0.3ṀEdd case, we can infer
the existence of a minimum amount of heat that can be injected at certain densities.
By keeping the same luminosity at different ρb, we observe a displacement in the
rising phase of the first burst of the simulation, starting first for the ρb = 108 g cm−3
model. The variations in the recurrence time, however, are almost negligible. The
second characteristic of importance is the small variation in Tb due to our selection of
ρb. Having a short boundary density but keeping the luminosity as constant implies
the inner temperature at such ρb varies drastically, as shown in the lowest panel
of Fig. 4.14. On the other hand, the variations in Tb at the ρb = 108 g cm−3 are
practically non-existing in the selected interval of time, suggesting the injection of
energy from the envelope to the outer crust cannot proceed between 107 and 108 g
cm−3. However, this is also a consequence of the different timescales (see Fig. 4.10),
where the dissipation of a perturbation to the temperature profile requires scales in
the order of days to weeks, hence the possibility of keeping Tb as practically constant
for the ρb = 108 g cm−3 model.
137
Time-independent against time-dependent envelopes
8
9
lo
g
10
T
[K
]
4.0 4.5 5.0 5.5 6.0 6.5 7.0
log10ρ [g cm−3]
−200
0
200
400
600
Q
L
[M
eV
b
ar
yo
n
−
1
]
Approx140, log10 ρb = 7.11
0.3ṀEdd, gs,14 = 1.73
70% 1H, 28% 4He, 2% 12C
8.0
8.5
lo
g
10
T
[K
]
Approx140, log10 ρb = 7.11
0.3ṀEdd, gs,14 = 1.73
70% 1H, 28% 4He, 2% 12C
4.0 4.5 5.0 5.5 6.0 6.5 7.0
log10ρ [g cm−3]
−5
0
5
Q
L
[M
eV
b
ar
yo
n
−
1
]
Figure 4.15: Temperature and luminosity (in MeV per baryon) against density for
several snapshots of a time-dependent simulation with MESA (colored curves), con-
trasted against the output from our time-independent envelope code (red, dashed
curves). Upper panel: comparison during the rising phase of an arbitrary burst.
Lower panel: comparison during the inter-bursting phase.
The next logical step, given the time evolution of these “fully accreted envelopes”
with negative base luminosity, is to ponder whether such states can actually be
reached if we start accreting material over a pristine envelope. A related question,
given the unstable nature of the mass accretion at 0.3ṀEdd, the origin of the explo-
sion at around 106 g cm−3 and the similarity of the timescales - nuclear, accretion
and heat diffusion - at said density, is: to what extent our accreting envelopes match
with the time-evolution output from MESA? In order to answer these questions, we
ran a simulation, employing the Approx140 network, of constant mass accretion at
0.3ṀEdd of a simple mixture of 70% 1H, 28% 4He and 2% 12C. The composition of
the pristine envelope was chosen as pure-hydrogen below 104 g cm−3 and of pure
80Kr above this density. For simplicity, we chose ρb = 107 g cm−3 and Lb correspond-
ing to an effective temperature of 105 K for a 11.57 km neutron star. As is usual, we
regarded the output from the first burst as unphysical and focused on the following
ones. To compare the MESA models with our own, we took their corresponding Teff
and performed the usual integration until either ρ = ρb or ρ = ρcutoff, depending on
138
whether the curves exhibited a T → 0 behavior or not.
In Fig. 4.15, we can see a comparison between the time-dependent and indepen-
dent temperature and luminosity profiles at two different stages during this accretion
episode. In the upper panels, we contrast the profiles during the rising phase of a
thermonuclear instability, while in the lower panels the envelopes are contrasted dur-
ing the stable mass accretion in-between bursts. The most prominent characteristic,
common to both episodes, is the similarity in the temperature profile at ρ ≤ 105 g
cm−3. At large densities, however, the discrepancies are notorious in both cases as
well: at the explosion episode, the stationary temperature profiles with high Ts have
larger inner temperature than their time-dependent counterparts, while those ex-
hibiting T → 0 behavior underestimate the temperature - when contrasted against
the time-dependent models - near 106 g cm−3. On the other hand, all stationary
temperature profiles in-between bursts have the T → 0 behavior at 106 g cm−3 while
their time-dependent counterparts follow the same trend dictated by the Lb > 0.
Nevertheless, in principle we could argue in favor of setting ρb = 105 g cm−3 in
order to employ these time-independent temperature profiles to construct a Tb(Teff)
relation. However, not all stationary luminosity profiles match their time-dependent
counterparts: for instance, during the explosion both profiles diverge from one an-
other at ρ ≥ 104 g cm−3, while during the in-between bursting period the highest Ts
stationary profile predicts a larger injection of energy near 105.5 g cm−3 than what
is actually obtained by the time-dependent simulation.
An important result to be noted from Fig. 4.15 is the amount of energy injected
towards the interior during the burst episode, ≈ 104L⊙. Since MESA did not find
further issues on running and completing this simulation, we might argue such results
are self-consistent and do not fail to converge due to the presence of the Lb > 0
source at ρb, which prevents the temperature profile to drastically drop towards
zero. However, we can also raise the question: does this amount of heat depends on
our choice of ρb?
Energy flowing to the interior
In the previous subsection we have seen energy can flow from the envelope towards
the numerical core of the star. Is there a maximum density such injection of energy
can reach? Is the size of the network important for the correct account of the energy?
In Figs. 4.16 4.17 we see the output of four simulations which have a common
value for ρb, 109.58 g cm−34. Such value of boundary density is suitable for both
observing the distribution of heat from the bursts and to relax the condition on the
amount of heating from the base, as we have set it to 2.59 × 10−5L⊙ in the four
models. The accretion rates we have selected are 0.3 and 1 times ṀEdd, and employed
4The original envelope, generated with our stationary code, had a boundary of 3 × 109 g cm−3.
However, during the adjustment of density and mass performed by MESA this boundary density
was modified.
139
two different reaction networks, Approx21 with a base of 80Kr and Approx140. Solar
composition was accreted at the surface, and in the four models we ran the simulation
for 8 hours (MESA time).
In the four envelopes, we see the bursts deposit a considerable amount of energy
at 107 g cm−3. In the case of recurring explosions, after the first one we observe an
almost constant amount: between 1 and 4 MeV per baryon in the Approx140 network
and between 0.2 and 0.4 in the Approx21 case. By comparing the evolution of the
models from each network at same accretion rate, we appreciate the importance
of having a sufficient amount of nuclides as to not underestimate the amount of
heating flowing towards the interior. In addition to such punctual deposition of
energy at 107 g cm−3, we observe that as t ≫ 0 energy progressively flows towards
the interior, as evidenced by the QL curves (lower panels of each pair of figures) at
108 and 109 g cm−3, and the increasing in temperature (upper panels of each pair
of figures). As a consequence of having more explosion episodes - and thus more
heating injection - we observe that regardless of the network choice the temperature
at 108 g cm−3 increased at least one order of magnitude, while temperature at 109 g
cm−3 remained almost constant. While this is partially a consequence of choosing a
same base - composed of 80Kr - the network also plays a role since the exact values
of this temperature gradient are clearly network dependent. An additional network-
dependent effect that we observe is the bursting suppression for the Approx21 net
at ṀEdd, while for the Approx140 net the bursts continue to occur at a regular
rate. Despite such suppression, the luminosity remains negative between 107 and
109 g cm−3: eventually, such heating will flow towards the interior and will increase
the temperature. Given the Lb > 0 condition cannot be relaxed in MESA, and the
evolution of the envelope with such restriction becomes less valid as the time of the
simulation approaches and exceeds ∼ 1 week (typical heat diffusion scale for the
outer crust, as inferred from Fig. 4.13), we cannot rule out the possibility that this
stored heat at ≈ 108.5 g cm−3 will eventually flow until ρb in a simulation of the full
neutron star.
5
6
7
8
lo
g
10
T
[K
]
log10Teff
0.3ṀEdd, 〈gs,14〉 = 2.176
80Kr base, log10ρb =9.582
log10ρ = 7.0
log10ρ = 7.5
log10ρ = 8.0
log10ρ = 9.0
0 1 2 3 4 5 6 7 8
Time [hr]
−0.4
−0.2
0.0
Q
L
[M
eV
b
ar
yo
n
−
1
]
Zs = Z⊙
Approx21 + 80Kr
5
6
7
8
lo
g
10
T
[K
]
log10Teff
log10ρ = 7.0
log10ρ = 7.5
log10ρ = 8.0
log10ρ = 9.0
0 1 2 3 4 5 6 7 8
Time [hr]
−0.4
−0.2
0.0
Q
L
[M
eV
b
ar
yo
n
−
1
]
ṀEdd, 〈gs,14〉 = 2.176
Zs = Z⊙
80Kr base, log10ρb =9.583
Approx21 + 80Kr
Figure 4.16: Temperature and luminosity, at specific values of log10ρ, as a function
of time for accretion over a pure 80Kr envelope employing the Approx21 network,
for two selected accretion rates, 0.3 and 1 times ṀEdd.
140
5
6
7
8
9
lo
g
10
T
[K
]
log10Teff
log10ρ = 7.0
log10ρ = 7.5
log10ρ = 8.0
log10ρ = 9.0
0 1 2 3 4 5 6 7 8
Time [hr]
−4
−2
0
Q
L
[M
eV
b
ar
yo
n
−
1
]
0.3ṀEdd, 〈gs,14〉 = 2.176
Zs = Z⊙
80Kr base, log10ρb =9.582
Approx140
5
6
7
8
9
lo
g
10
T
[K
]
log10Teff
log10ρ = 7.0
log10ρ = 7.5
log10ρ = 8.0
log10ρ = 9.0
0 1 2 3 4 5 6 7 8
Time [hr]
−2
−1
0
Q
L
[M
eV
b
ar
yo
n
−
1
]
ṀEdd, 〈gs,14〉 = 2.176
Zs = Z⊙80Kr base, log10ρb =9.583
Approx140
Figure 4.17: Same caption as in Fig. 4.16, now for the Approx140 network.
Increased opacity: a mechanism to stop bursts?
Observationally, no bursts have been registered at Ṁ ≥ 0.3ṀEdd [BHM+07]. Al-
though the analysis of the previous sections suggest this is feasible since temperature
obeys a diffusion equation, fully time-dependent simulations have shown bursting
suppression occur, theoretically, at Ṁ ≥ ṀEdd [HCW07, FTG+07]. In addition,
the envelope requires very high values for Lb to suppress the oscillations. Several
mechanisms have been invoked to explain either this large value of Lb or potential
mechanisms for stabilization of bursts at exactly 0.3ṀEdd, such as
• 2D/3D effects. So far, the majority of bursts simulation have considered a
single spatial direction. There are hints that spreading of material might have
an impact for their stabilization, taking into consideration gradients in the
surface gravity [CWG17]. Another mechanisms might be chemical diffusion as
a consequence of rotation: models in this direction have reported a reduction
in the required value for Lb [KLi09].
• Different reaction rates. The 15O(α, γ)19Ne reaction has received special
attention since it was found to be one, if not the most important, of the CNO-
breaker reactions [FGWD06a, TFG+07]. For this reaction to take place, tem-
peratures between ∼ 108.4 K (around 107 g cm−3) and ∼ 108.6 K (for 104
g cm−3) are required, becoming competitive with respect to the density and
temperature independent 15O(β+νe)
15N. In the left panel of Fig. 4.18 we illus-
trate several versions of this thermonuclear rate as a function of temperature,
at two typical densities at the surface of neutron stars, and we can see their
rather good agreement below ∼ 109.25 K. Regardless of the importance of this
rate, however, the CNO-breaking must be complemented with the subsequent
proton capture over 19Ne in order to prevent the return to the CNO cycle via
19Ne(β+νe)
19F(p, α)16O. As illustrated in the right panel of Fig. 4.18, in con-
trast to the α-capture, this p-capture breaking path is favorable even below
108.25 K, enhanced by density around 107 g cm−3, which favors the subse-
quent proton captures over progressively heavier nuclides. This in principle
141
explains why, even if CNO cycles are not broken, we can still synthesize mod-
erate amounts of 40Ca at low accretion rates (e.g. ∼ 0.1ṀEdd). Going back to
our main reaction of interest, if the CNO breaking occurs, we can go beyond
40Ca and 56Fe synthesis, reaching until ∼ 64Zn in the rp-process. Current
experimental knowledge of this rate is limited due to the absence of strong
15O beams permitting a direct measurement. The explanation of this is due to
t1/2 =122.22 s, which makes “impossible” to sustain these beams long enough
as to allow significant α-captures to be detected [HdS21], although some evi-
dence also exists in favor of a direct measurement [TWK+17]. Instead, some
ongoing and recent collaborations have focused on 15O(7Li,t)19Ne reaction,
mimicking an α-capture, from where it should be possible to pose further con-
straints on the α-capture reaction [ACL+21]5. Nevertheless, available rates
are usually constrained from indirect mechanisms such as the 19Ne energy lev-
els [HBB+20], 19Ne(γ, α) decay [TGB+09] or the outcome from 20Mg(βp)19Ne
path [GPLW+19] provide information to construct theoretical version of this
rate, such as those in Fig. 4.18. Over the years, this absence of direct data
has opened the possibility for theoretical speculation, probably the most no-
torious in the astrophysical context is that the actual rate is few times smaller
than estimated, as this would allow to reconcilie the absence of X-ray bursts
above 0.3ṀEdd with theoretical predictions, pointing that bursting behavior
cease at ∼ 3ṀEdd. This argument was primarily pointed out by Fisker et al
[FGWD06a, FTG+07], whose PhD dissertation focused on the reaction flow
during an X-ray bursts and represents a modern understanding on the rp
process. Later theoretical models [CN06], recovering linear stability analysis
[CN05] (which in their own words could be of “obscure” meaning), showed
this argument could be used to construct an approximate network of three-to-
four nuclides for simulating the rp process, and effectively found that bursting
behavior should stop at ∼ 0.3ṀEdd. We must note, however, this approach
overlooks the importance of the 18Ne (α, p) 21Na rate which is important for
X-ray bursts as well. From stationary simulations we see this rate replaces the
15O (α, γ) as the leading CNO-breakout for accretion rates comparable with or
greater than 5ṀEdd, a behavior resembling the evolution during a burst (see
[SAB+01]), thus a simplification requires the presence of the 18Ne(α, p) rate.
In spite of how good such argument could be, particularly in view of the
observational evidence from ROSSI X-Ray Timing Explorer [GMH+08], this
argument is far from being perfect. For instance, in [DCJM11], a Monte Carlo
simulation simulation of the astrophysical rate was made, and this used as
input for simulating X-ray bursts with a different code, and the results were
5See for instance https://indico.cern.ch/event/1013634/contributions/4313739/
or https://agenda.infn.it/event/27358/contributions/145695/contribution.pdf for
examples on recent diffusion of these ongoing projects.
142
quite different with respect to [FTG+07]. Their networks were indeed different,
324 in contrast to Fisker’s 298, which could also introduces differences with
respects to β+ decays due to the inclusion of isotopes in or nearby the valley
of stability. But notoriously, the conclusion is overall different: no suppressing
was found even for low values of the reaction rate. Finally, a critic point of
view came some years later, from the same group of research who proposed
this argument: in [TGB+09], taking as basis experimental measurements on
excited states of 19Ne, favorable to decay into 15O + α, these authors pro-
posed a revisioned rate and found their results were quite similar as those by
Langanke et al. Moreover, by recalculating some models with Fisker’s code
they found the unstable-stable transition in accreting systems, assuming solar
composition, took place at ∼ 2 × 1018 g s−1 ≈ 1.8 ṀEdd. Considering the
small deviations among versions of this rate (e.g. left panel of Fig. 4.18), we
must infer this argument cannot be the “correct” answer to this puzzle, unless
future experiments show otherwise. Nevertheless, as pointed out by modern
simulations on X-ray bursts (e.g.[MMM19, dCC+21]), uncertainty in certain
thermonuclear rates can affect the overall evolution of the accreting process.
8.0 8.5 9.0 9.5 10.0
log10T [K]
10−4
10−1
102
105
108
ρ
N
A
〈σ
v
〉 T
[s
−
1
]
15O(β+νe)
15N
ρ = 104 g cm−3ρ = 107 g cm−3
15O(α, γ) 19Ne
dc11
fs07
Ha96
il10
7 8 9 10
log10T [K]
10−4
10−1
102
105
108
ρ
N
A
〈σ
v
〉 T
[s
−
1
]
19Ne(β+νe)
19F
ρ = 104 g cm−3ρ = 107 g cm−3
19Ne(α, γ) 20Na
cf88
rath
ths8
il10
Figure 4.18: Different versions of reaction rates, in units of s−1, as functions of
temperature. These fits were taken from JINA-CEE Reaction Library (REACLIB)
[CAF+10], preserving the assigned labels to each rate: ths8 are from T. Rauscher
and part of REACLIB v1.0, cf88 from the classic [CF88], rath from [RT00], fs07
is based on [FTG+07], dc11 comes from [DCJM11], il10 is based on [ILC+10]
and Ha96 is from [HGA+96]. The thickest lines correspond with the recommended
version by JINA-CEE library. The horizontal lines represent the β+ decay of the
parent nuclides (rates also taken from JINA-CEE). Left panel: 15O(α, γ)19Ne. Right
panel: 19Ne(p, γ)20Na.
We observed in the past subsection that burst suppression can be in theoretical
simulations merely an artifact of the size of the network chosen. Thus, any model
143
that we propose for bursting suppression must be free of this degeneracy - i.e.,
the model must exhibit bursting suppression employing a large network such as
Approx140 or net380. Under these considerations, in this subsection we introduce
another possibility to those enlisted above, found by exploring the available options
to change parameters in MESA: an increased opacity, which allows to suppress the
bursting behavior near 0.3ṀEdd.
As inner boundary conditions for MESA at ρb one needs to fixed and inner lumi-
nosity Lb, coming from the stellar interior and we chose two extreme values, a very
low one of Lb = 2.59 × 10−5L⊙ and high one of Lb = 2.49 × 10−1L⊙.
The amount of cells of the spatial grid, as well as the time-step, are factors which
might have some impact on the simulation results. In MESA, mesh_delta_coeff
parameter controls the mesh reĄnement during a simulation: above 1.0, the number
of grid cells tends to be smaller, while below 1.0 the number might reach up to
3000 cells. Unless explicitly stated, we adopt 5.0 for this coefficient. Besides the
size of the mesh, MESA allows to have some control over the chosen time step. With
time_delta_coeff (hereafter tdc), the user can ask for overall large steps in time,
while min_timestep_factor (hereafter mtf) controls the minimum ratio between
the new and previous time step. By default, their respective values are 1.0 and
0.8. By trial and error, for some simulations we have found suitable to replace
these defaults by a customized conĄguration of 5.0 and 1.2, which we employ unless
another combination is explicitly stated.
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
t [hr]
0
2000
4000
6000
8000
L
/L
⊙
Ṁ = 5.26× 10−9 M⊙ yr−1
Ṁ = 6.26× 10−9 M⊙ yr−1
Ṁ = 7.26× 10−9 M⊙ yr−1
Ṁ = 9.26× 10−9 M⊙ yr−1
Ṁ = 1.56× 10−8 M⊙ yr−1
Figure 4.19: Luminosity (in units of L⊙) as a function of time for different values
of mass accretion rate. The luminosity at the base is equal to 2.59 × 10−5L⊙, the
opacity is globally increased by a factor of 10, log10ρb = 9.57, gs,14 = 2.14, approx140
network, mesh= 5.0 and 80Kr at the base.
One concern could be that the bursting suppression is also a consequence of our
other choices for the parameters, not only of the change in opacity. For example,
setting mtf = 1.2 leads to progressively longer time steps, which may inĆuence the
144
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
t [hr]
0
2000
4000
6000
8000
L
/L
⊙
Ṁ = 5.26× 10−9 M⊙ yr−1
Ṁ = 6.26× 10−9 M⊙ yr−1
Ṁ = 7.26× 10−9 M⊙ yr−1
Ṁ = 9.26× 10−9 M⊙ yr−1
Ṁ = 1.56× 10−8 M⊙ yr−1
Figure 4.20: Same as Fig. 4.19 but for Lb = 2.4910−1L⊙.
evolution of the column by skipping important variations in the burning. To test
whether this is the case, we ran a few simulations, keeping the accretion rate Ąxed
at 7.26 × 10−9 M⊙ yr−1 (∼ 0.4ṀEdd) as well as an increased opacity of 10κ, but
chaniging other parameters. In particular, we focused on the following:
• Limits to the time step in the mesh. We modiĄed both time_delta_coeff
and min_timestep_factor to favour longer or shorter time steps.
• Number of cells in the mesh. This is partially controlled by the user via
the mesh_delta_coeff parameter. In our simulations, a value around and
above 5 restricts the amount of cells below 500, while with a value around 1
the cells can be as many as 2000 to 3000.
• Number of species in the network. To rule out the possibility of the
electroweak reactions from the omitted nuclides in the approx140 network
altering the bursting behavior, we employed the same network of 380 species,
adding neutrons for a fair comparison with the approx140, and thus resulting
in 381 species. We refer to this larger network as net381.
• Composition of the base. We chose two compositions: either the rp-ashes
mixture, or a single-species with a heavy nucleus, i.e. 80Kr, common to both
networks we considered.
• Composition of accreted material. We explored two compositions: the
Solar-like provided by MESA and a simpler mixture of 70% 1H, 28% 4He and 2%
12C, key species in the actual synthesis of heavier elements via the rp-process.
12C is necessary for a fair comparison between net381 and approx140 due to
the absence of Li, Be and B isotopes directly connecting 4He with C, N and
O in the latter network.
145
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
t [hr]
6.0
6.2
6.4
6.6
6.8
7.0
7.2
lo
g 1
0
T
[K
]
log10ρb[g cm−3] = 9.571
〈gs,14〉 = 2.162
Ṁ = 7.26× 10−9 M⊙ yr−1
approx140, mesh = 5.0
80Kr base
70% 1H, 28% 4He, 2% 12C
10κ
Tb
Teff
mtf = 1.2, tdc = 5.0
Default mtf, tdc = 1.1
Default mtf, tdc = 2.0
Default mtf, tdc = 5.0
Figure 4.21: Effective temperature as a function of time, at Ąxed opacity factor and
accretion rate, for different setups of timestep.
The resulting light curves can be found in Figs. 4.21 and 4.22.
To test the impact of the time-step controls over the stabilization, we ran simu-
lations altering both mtf and tdc, employing the approx140 network, 80Kr as the
composition of the base and the simpler accretion mixture. In Fig. 4.21 we also plot
the temperature at the base of our models. The models discussed in the main text
have mtf = 1.2 and tdc = 5.0 (this is the red curve in the Figures), while we test
combinations with MESAŠs default value of mtf = 0.8 and tdc = 1.1, 2, 5. The overall
similarity of all results with respect to our Ąducial strongly suggest that our previous
results are solely due the the change in opacity and not an artifact of the numerical
integration. Some discrepancies can still be seen, though they donŠt change the
conclusions. For instance, employing the default value for mtf in combination with
a large tdc induces an additional burst and delays the stabilization for 0.5 hr. The
equilibrium temperature, however, is similar to the rest of the models. The second
difference is the time span of the decay phase after the burst peak, which is slightly
shorter in the mtf = 1.2 case than in the rest of the simulations.
The other important numerical parameter is resolution. The difference in size of
the mesh does not produce appreciable deviations, neither in the Ąrst ŞnumericalŤ
burst nor in the equilibrium temperature, suggesting that our Ąducial parameters
for the resolution are high enough (Fig. 4.22).
We now turn to more physical parameters (Fig. 4.22). Regarding the size of
the network, although the decay after the peak of the Ąrst burst using net381 is
faster than for the approx140 model, the overall behavior is the same: only one
burst occurs, followed by a damping process Ąnally converging to a stable state.
The oscillations from the net381 network around the equilibrium value are slightly
more visible than those generated by the approx140 network, but this difference is
146
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
t [hr]
6.00
6.25
6.50
6.75
7.00
lo
g 1
0
T
eff
[K
]
log10ρb[g cm−3] = 9.571
〈gs,14〉 = 2.162
Ṁ = 7.26× 10−9 M⊙ yr−1
10κ
approx140 , mesh = 5.0 , rpashes
approx140 , mesh = 5.0 , 80Kr
approx140 , mesh = 1.0 , 80Kr
approx140 , mesh = 5.0 , 80Kr, HHeC
net381 , mesh = 5.0 , 80Kr
Figure 4.22: Effective temperature as a function of time, at Ąxed opacity factor and
accretion rate, for different combinations of parameters.
minor.
With respect to the base composition we observe little difference in the stabi-
lization properties when using either of the ashes or pure 80Kr mixtures, although
some differences in the rise and decay phases of the Ąrst burst are visible. The over-
all simulation reaches stabilization nonetheless and there is little difference in the
equilibrium temperature with respect to the rest of the models. When considering
different accreted composition, we notice that the absence of Z > 6 species in the
fuel material delays the stabilization process until a second, less-energetic burst have
occurred. However, no major changes are noticeable when the burning turns stable.
In order to check if there is a dominant term which is responsible for the stabi-
lization of the burning, we applied the factor of 10 increase only to κrad as a whole,
and then to κff and κes separately. We show also these results in Figs. 4.23 and ??.
Modifying only κrad simply delays the Ąrst burst, but does not avoid stabilization.
However, noticeably, a global increase only to κff seems to be unable to prevent the
instability and we observe multiple bursts. On the other hand, changing the electron
scattering opacity by a factor of 10 produces again the observed suppression as in
the global 10κrad model. To rule out this as a mere consequence of employing an
old Ąt for opacity, we tested both Paczynsy and Poutanen Ąts [Pac83, Pou17] and
found out that despite small differences in the rising time and cooling phase after
the Ąrst burst, the stabilization still took place.
In order to interpret these result, it is useful to look at the depths where each term
is dominant. A typical proĄle is shown in Fig. 4.24. Incrementing only κrad has the
same effect as multiply the whole κ function because the conduction term dominates
only at the highest depths, after ignition and is thus not too relevant for the bursting
behaviour. The distinction between κff and κes is more interesting: changing only
147
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
t [hr]
6.00
6.25
6.50
6.75
7.00
7.25
lo
g 1
0
T
eff
[K
]
Ṁ = 7.26× 10−9 M⊙ yr−1
mesh = 2.0 , 10κMESA
mesh = 2.0 , 10κrad
mesh = 2.0 , 10κff
mesh = 2.0 , 10κes (Paczynsky)
mesh = 2.0 , 10κes (Poutanen)
Figure 4.23: Effective temperature as a function of time, for different opacities.
Here, log10 ρb = 9.57, gs,14 = 2.14, 80Kr as composition of the base and Approx140.
As base luminosity we employ 2.59 × 10−5L⊙.
κff re-introduced the bursts indicating that for the opacity to affect the burst, what
is really important is an increase at lower densities, above the ignition depth. We
do not expect the calculations of κes to be off by a factor of 10, but our results may
indicate that if a different mechanism is at work, which increases the opacity in the
most superĄcial layers, this could contribute to reconcile the observations with the
numerical predictions.
However, is ten the minimum increase an opacity should have to suppress bursts?
In Fig. 4.25 we show the luminosity curves for two families of models, with different
global opacity factors, 2 and 4. Here, the base luminosity is expressed in terms of Teff,
via Lb = 4πσSBR
2T 4
eff,b, taking R = 11.57 km - i.e. the radius of an 1.4M⊙ APR-core
EOS. The accretion rates correspond to 0.2, 0.3 and 0.4 ṀEdd - all of them around
the critical 0.3ṀEdd at which bursts observationally disappear. For simplicity, we
adopted ρb = 107 g cm−3. What we observe is that doubling the opacity conĄnes the
stabilization of bursts to the interval between 6.75 and 7.0 in log10Teff,b - i.e. we still
require large amounts of heat coming from the base to produce bursting suppression.
A four-times large opacity, however, reduces the amount of base heating required
for burst suppression by 0.25 in log10Teff,b. In terms of Qb, this implies at 4κ we
require between 0.3 and 3 MeV per baryon at 0.3ṀEdd to suppress bursts, while at
2κ we require between 3 and 30 MeV per baryon.
148
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
log10ρ [g cm−3]
0.00
0.05
0.10
0.15
0.20
0.25
0.30
κ
[c
m
2
g−
1
] κtot
κrad
κcond
κff
κes
Figure 4.24: Opacity proĄle for a typical stationary accreted envelope at ṀEdd.
10−1
103
Ṁ =3.5071×10−9 M⊙ yr−1
80Kr, approx140, opacity factor = 2.0
10−1
103
L
/L
⊙
Ṁ =5.2606×10−9 M⊙ yr−1
log10Teff,b = 5.00
log10Teff,b = 6.00
log10Teff,b = 6.50
log10Teff,b = 6.75
log10Teff,b = 7.00
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
t [hr]
10−1
103
Ṁ =7.0142×10−9 M⊙ yr−1
10−1
103
Ṁ =3.5071×10−9 M⊙ yr−1
80Kr, approx140, opacity factor = 4.0
10−1
103
L
/L
⊙
Ṁ =5.2606×10−9 M⊙ yr−1
log10Teff,b = 5.00
log10Teff,b = 6.00
log10Teff,b = 6.50
log10Teff,b = 6.75
log10Teff,b = 7.00
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
t [hr]
10−1
103
Ṁ =7.0142×10−9 M⊙ yr−1
Figure 4.25: Lightcurves for models with different base luminosity - parametrized
in terms of Teff. The upper panels, at accretion rates 0.2, 0.3 and 0.4 ṀEdd, corre-
spond to models with a doubled-opacity. The lower panels, at same accretion rates,
correspond to models with a factor of 4 for the global opacity.
149
Chapter 5
Conclusions
“Books are not made to be
believed, but to be subjected to
inquiry. When we consider a
book, we mustn’t ask ourselves
what it says but what it means.”
U. Eco
Abstract. This chapter outlines the main findings of the present thesis, as
well as the work in progress for future research.
Are Tb − Lb − Teff relations viable?
In Chapter 4 we observed that, upon contrast between time-independent and depen-
dent simulations, their region of agreement extended from the surface of the star up
to ∼ 105 g cm−3 for the temperature and luminosity. Between 105 and 108 g cm−3,
however, the timescales of the different processes are actually similar and it is thus
necessary to restore to fully time-dependent simulations in order to obtain a correct
distribution of energy, and speciĄcally on the amount Ćowing from the surface to
the numerical core. Consequently, for the implementation of accreted envelopes at
NSCool we can proceed as follows:
• Compute, with our stationary code, the Teff − Lb − Tb relations considering
ρb ∼ 105 g cm−3.
• Implement these relations as boundary conditions for NSCool.
• Incorporate the equations for the evolution of the mass fractions into NSCool,
in order to account for the speciĄc energy generation rate due to nuclear re-
actions as well.
150
These tasks are currently a work in progress.
On the energy flowing from the surface to the core
From the numerical experiments concerning the change in ρb, we observed that keep-
ing a constant Lb had an impact on the distribution of energy Ćowing to the interior
of the star. While the usage of small networks such as Approx21 provide a decent
Ąrst approximation to this modeling, it is advisable to employ other approximations
such as Approx140 in order to check whether bursting behavior ceases or not and,
more importantly, to account for the actual energy deposited between ∼ 107 and
109 g cm−3, which can be as large as ∼ 4 MeV per baryon, in contrast to the ∼ 1
MeV per baryon reported by Approx21. Another reason to look for an improvement
in the simulation is to extend the simulations beyond ∼ 1 week, i.e. the range of
validity for the assumption of Lb constant, which could have an impact on the evolu-
tion of the numerical coreŠs temperature given the observed increase in temperature
between ∼ 107 and 109 g cm−3, even in the simplest case of the Approx21 network.
Given the observed amount of injected energy and its location, ∼ 1 MeV per
baryon between 107 and 109 g cm−3, we might argue this could be an additional
heating source, similar as to the one required to explain the shallow heating. We
must observe, however, that our reported energy emerges during accretion episodes.
It might be the case, however, that such injection of energy actually takes place as
the mass accretion rate decreases: if, for instance, the inner and outer crust have had
ample time to cool down, they will be at lower temperature than the envelope. Thus,
with the occurrence of thermonuclear explosions - elevating the temperature at 107
g cm−3 up to 109 K - some energy might Ćow towards the interior and increase its
temperature in a range of ∼ 1 day, as observed in our theoretical models for constant
accretion. An important take away message from these numerical simulations is that
the assumption of Lb > 0, while standard in practice, is not necessarily the only one
which we should consider, as states with Lb < 0 are physically viable as well and
might account for the distribution of energy around 107 g cm−3, which is also the
typical boundary density employed in numerous studies of accreted neutron star
envelopes.
Below 0.3ṀEdd, a self-consistent modeling of the envelope requires the inclusion
of time-dependent terms in the PDEs in order to resolve the bursting behavior and
a realistic evolution of the ashes. Above this threshold for mass accretion, however,
we can expect a steady-state to be reached in order to explain the absence of bursts,
considering the observational basis. However, given the future incorporation of the
network into NSCool, it is still necessary to check whether the full evolution of the
star actually reaches the steady states predicted by our stationary code. However, we
have seen that artiĄcially increasing the opacity, albeit physically questionable due
to the absence of a straightforward mechanism to explain such increase, actually
151
allows to suppress the bursting behavior at accretion rates near the critical one,
0.3ṀEdd.
152
Chapter 6
Paper 1: “The Effect of Opacity
on Neutron Star Type I X-ray
Bursts Quenching”
6.1 Overview
The main motivation behind this paper was to test whether changes in opacity could
suppress the bursting behavior observed in MESA or not. Part of the contents in this
manuscript was introduced at the end of Chapter 4.
Link: https://www.astroscu.unam.mx/RMxAA/accepted.html
As mentioned in the Preface and in Chapter 4, there exists some tension between
theory and observations regarding the Ṁcrit above which bursting suppression - also
referred to as burst quenching - takes place. To resolve this issue, additional heating
sources are typically invoked at the base of the envelope. Given my present work
focused on examining the microphysics of the envelope, as well as in the compari-
son of time dependent-independent simulations, eventually the idea of changing the
opacity in MESA emerged. From one side, most of - if not all - radiative opacities
provided by MESA in tabulated form assume the composition is of primary hydrogen
or helium with small traces of carbon and oxygen. Similarly, the analytic Ąt for
free-free opacity we employ was constructed by considering computations of hydro-
gen, helium, carbon and oxygen as well. Thus, some uncertainty regarding high-Z
nuclides exists on these radiative opacities, which is critical for the accreted envelope
given the composition of matter after bursts is enriched in these high-Z nuclides.
The simulations in this manuscript - all employing MESA - thus had the goal of
testing the impact of such opacity uncertainties over the burst quenching. In partic-
ular, to check whether an altered opacity could reduce the gap between Ṁcrit from
theory and observations. Our results indicate that increasing the opacity by a factor
∼ 10 actually reduces the gap: bursts are no longer observed as Ṁ ≈ 0.40ṀEdd.
153
From our models we infer this behavior is independent of the base luminosity Lb,
a situation which also alleviates the tension with respect to the large amounts of
base heating required for the burst quenching. By running additional models, we
also corroborated this suppression was not a numerical effect but physical, as long
as one accepts a dramatic increase in the electron scattering opacity, not in the
free-free one. Such result was surprising taken into account the uncertainties above
described.
Despite the robustness of our theoretical results, at present we could not came up
with a physical mechanism responsible of increasing the opacity up to a factor of 10.
On one hand, we can relax such stringent factor by invoking a moderate value for Lb
with respect to the typical ones. On the other, we are still left without explanation
for the ∼ 2 factor. After some research, we noticed our primary candidate - magnetic
Ąeld - cannot be responsible as its presence decreases the opacity.
The manuscript was submitted to RMxAA on September 17th 2024. On October
16th 2024 a response was received by the Reviewer, stating the manuscript was
accepted with minor corrections. On November 4th 2024 the paper was officially
accepted, to be published on April 2025.
154
Chapter 7
Paper 2 - Considered for
acceptation
7.1 Overview
This paper had the purpose of introducing the numerical code I developed for solving
the time-independent envelope equations (see Chapter 4).
Link: https://arxiv.org/abs/2403.13994
The manuscript was submitted to RASTI (Royal Astronomical Society - Tech-
niques and Instruments) on May 28th 2024. On October 21st 2024 a response of
ŞConsidered to be accepted under minor correctionsŤ was sent by the Reviewer
throughout the Editors. On October 22nd 2024 a revisited version was submit-
ted. By November 4th 2024, the status on RASTIŠs ScholarOne Manuscript site is
still ŞAwaiting Reviewer ScoresŤ. The requested changes have been kept in red for
illustrative purposes.
• Introduction. Here we provide a brief review on the state-of-art regarding
the modeling of thermal evolution for accreting neutron stars, and present our
plans for the present paper and its relevance for future works, where we are
employing the output from this time-independent code for computing the tem-
perature and luminosity of the envelope in combination with a time-dependent
code for the core, NSCool.
• Stationary accreting neutron star envelopes. In the Ąrst part of this
section we delimit the scope of our code. Since it is time-independent, it
is important to state at which stages of the evolution of the whole star our
approximations hold. In the second part, we introduce the ODEs governing
the structure, temperature and chemical composition of the envelope. For the
reasons exposed in Chapters 1 and 4, we restore to pressure as our independent
variable instead of radius or mass. Further details on the derivation of these
equations can be found in Chapter 1 or in the Appendix of this manuscript.
155
• Methodology. Here we enumerate the boundary conditions for the differ-
ential equations and the network of reactions to be employed, net380. As
mentioned in Chapter 3, this network is suitable for stationary states stud-
ies since we can correctly account for both the distribution of mass fractions
and the nuclear energy output. Despite the apparent computational cost (i.e.
384 ordinary differential equations), the absence of the temporal coordinate
transform this IVP+BVP into a IVP, where we require to specify the condi-
tions at the surface and integrate 384 equations, not 384 x Ncells as is done in
time-dependent codes such as MESA.
• Results. (i) Our code correctly reproduces the output from the basic refer-
ence in the Ąeld, i.e. [SBCW99], although employing a smaller network with
updated reactions and binding energies. In our corroboration, we expanded
on the contributions to the total nuclear energy from different processes, such
as proton captures or the β+ decay. (ii) We showed that variations in the
amount of accreted hydrogen, below 50%, could still lead to the occurrence of
the rp-process. Furthermore, in (iii) we illustrated that only the presence of
hydrogen and helium was sufficient to synthesize the rest of metals, as long as
Ṁ ≥ 0.1ṀEdd. While typical approximations at such low accretion rates focus
on the CNO and 3α process, here we illustrate some rp-process might still
occur. Finally, in (iv) we analyzed some envelopes at 1 and 5 ṀEdd accreting
helium, and found out that moderate amounts of carbon-12 and heavier α
nuclides were produced at these high accretion rates.
• Conclusions. We closed the paper making some punctual observations on the
results found. Having a stationary code capable of calculating the temperature,
chemical composition and luminosity of envelopes undergoing high accretion
rates will allow us to incorporate them into time-dependent codes such as
NSCool. Finally, the amounts of A = 24 and A = 28 predicted by helium-
4 burning at high accretion rates provide a theoretical explanation on the
possible origin of such α nuclides at high depths. However, it still requires the
observational corroboration of having helium, not hydrogen, accretion over the
surface of the neutron star.
156
Chapter 8
Paper 3 (Third Author)
8.1 Overview
In this paper I collaborated as third author.
Link: https://iopscience.iop.org/article/10.3847/1538-4357/ac72a8
My main contribution to this paper was the computation of the electron captures
over the α nuclides. The methodology employed comes from [HZ90b], and is brieĆy
addressed in Chapter 2 of the present work as well. In summary: under constant
accretion, the original composition of the neutron star crust, enriched in metals such
as 12C, 16O or 32S, undergoes compression as new material is accreted and burnt.
Such compression increases the probabilities for pycnonuclear reactions, such as
electron captures, to take place over this compressed matter. Besides liberating
heat in the process, it changes the chemical composition by generating neutron-rich
species such as 24O. According to the Monte Carlo Markov Chain simulations in
the paper, such species should be the fuel for the theoretical explosion that Ąts the
observational data. Hence, to corroborate or refute such assertion we restored to
the model of Haensel and Zdunik, and found out that species such as 24O or 28Ne
could actually exist at the densities where the explosion is likely to take place. Their
parent nuclides are 24Mg and 28Si, thus another question is whether we can actually
synthesize signiĄcant mass fractions of said nuclides in order to push them up to
1011 g cm−3. According to the models from Chapter 4, the synthesis of 24Mg and
28Si does take place at hydrogen and helium burning conditions, albeit the resulting
mass fractions do not exceed 10−2.
157
Appendix A
Thermodynamics
This appendix is devoted to prove several of the thermodynamical identities and
claims employed along the Chapters. As such, it is divided in three sections: in
the Ąrst, several propositions and theorems which are more general are stated and
proved. The second is devoted to identities which are invoked in the main discussion.
A.1 Propositions and theorems
The common hypothesis to all the following propositions and theorems is a thermo-
dynamical system Σ(T ) with a constant number of baryons NB
Proposition 1. Considering fixed abundances,
c̆V =
(
∂ε̆
∂T
)
ρB,Y
, (A.1)
c̆P =


∂h̆
∂T


P,Y
(A.2)
=
(
∂ε̆
∂T
)
P,Y
+
χT
χρ
P
ρBT
. (A.3)
Proof. The Ąrst identity directly follows from the First Law, working at dρB = 0
and dYi = 0 for all species, i.e. dε̆ = Tds̆. By taking its ratio with dT , and the limit
dT → 0, we see
c̆V =
(
∂ε̆
∂T
)
ρB,Y
.
Now let us consider dh̆ at dP = 0 and dYi = 0 for all species, i.e. dh̆ = Tds̆. Taking
its ratio with dT and the limit dT → 0,
c̆P =


∂h̆
∂T


P,Y
.
158
Replacing h̆ = ε̆+ P/ρB and the deĄnition of χT and χρ, we obtain
c̆P =
(
∂ε̆
∂T
)
P,Y
+
χT
χρ
P
ρBT
.
■
Proposition 2. The relation between our f̆ and those in the papers is given by
f̆int =
kBT
⟨m⟩imu
fpapers
int . (A.4)
Proof. Considering the total number of ions divided by volume is given by N/V =
ρB/⟨m⟩imu, we have
fpapers
int =
Fint
NkBT
=
1
kBT
V
N
Fint
V
=
⟨m⟩imu
ρBkBT
fint =
⟨m⟩imu
kBT
f̆int.
■
The second aspect of importance with respect to fpapers
int is its implicit depen-
dence with respect to both ρB and T , in concrete throughout the ŞΓ factorsŤ, the
ratio between electrostatic and thermal energies, as well as the Şelectron degener-
acyŤ τ = kBT/EF (following IchimaruŠs notation. Eq. (3.83) of his expresses the
exchange energy due to electrons) with EF electronŠs Fermi energy. For simplicity,
let us denote each of these parameters as Γi with i ∈ ¶1, . . . , Nθ♢. Such functional
dependence is very useful for numerical implementations, as we can see from the
following propositions. For simplicity of notation, fpapers
int = f .
Proposition 3.
Proof. Considering f̆ = f̆ideal + f̆interaction, we see
P = ρ2





∂f̆ideal
∂ρ


T,Y
+


∂f̆interaction
∂ρ


T,Y



= Pideal + ρ2


∂f̆interaction
∂ρ


T
.
From chainŠs rule,
P = Pideal +
ρ2kBT
⟨m⟩imu
Nθ
∑
j=1
(
∂f
∂Γj
)(
∂Γj
∂ρ
)
T
.
159
Regarding the partial derivatives: as each ∂f/∂Γj is in itself a functional of the Nθ
factors,
(
∂P
∂ρ
)
T
=
(
∂Pideal
∂ρ
)
T
+
2ρkBT
⟨m⟩imu
Nθ
∑
j=1
(
∂f
∂Γj
)(
∂Γj
∂ρ
)
T
+
ρ2kBT
⟨m⟩imu



Nθ
∑
j=1
(
∂f
∂Γj
)(
∂2Γj
∂ρ2
)
T
+
Nθ
∑
j=1
(
∂Γj
∂ρ
)
T
Nθ
∑
k=1
∂2f
∂Γk∂Γj
(
∂Γk
∂ρ
)
T



(
∂P
∂T
)
ρ
=
(
∂Pideal
∂T
)
ρ
+
ρ2kB
⟨m⟩imu
Nθ
∑
j=1
(
∂f
∂Γj
)(
∂Γj
∂ρ
)
T
+
ρ2kBT
⟨m⟩imu



Nθ
∑
j=1
∂f
∂Γj
∂2Γj
∂T∂ρ
+
Nθ
∑
j=1
(
∂Γj
∂ρ
)
T
Nθ
∑
k=1
∂2f
∂Γk∂Γj
(
∂Γk
∂T
)
ρ



.
Finally, for constructing the speciĄc heat at constant volume we have


∂f̆int
∂T


ρ
=
kBT
⟨m⟩imu
Nθ
∑
j=1
∂f
∂Γj
(
∂Γj
∂T
)
ρ
+
kBf
⟨m⟩imu


∂2f̆int
∂T 2


ρ
=
2kB
⟨m⟩imu
Nθ
∑
j=1
∂f
∂Γj
(
∂Γj
∂T
)
ρ
+
kBT
⟨m⟩imu



Nθ
∑
j=1
∂f
∂Γj
(
∂2Γj
∂T 2
)
ρ
+
Nθ
∑
j=1
(
∂Γj
∂T
)
ρ
Nθ
∑
k=1
∂2f
∂Γk∂Γj
(
∂Γk
∂T
)
ρ



.
We can notice the contributions to the partial derivatives from these interacting
terms, for clarity in the discussion, can be written as
(
∂Pint
∂ρ
)
T,Y
= ξiρT [2T1 + ρT2]
(
∂Pint
∂T
)
ρ,Y
= ξiρ [ρT1 + ρTT3]
c̆
(int)
V = −ξiT [2T5 + TT4] ,
160
where ξi = kB⟨m⟩−1
i m−1
u and
T1 =
3
∑
l=1


Nl
∑
j=1
∂fl
∂Γj
(
∂Γj
∂ρ
)
T,Y


T2 =
3
∑
l=1



Nl
∑
j=1
∂fl
∂Γj
(
∂2Γj
∂ρ2
)
T,Y
+
Nl
∑
j=1
(
∂Γj
∂ρ
)
T,Y
Nl
∑
k=1
∂2fl
∂Γk∂Γj
(
∂Γk
∂ρ
)
T,Y



T3 =
3
∑
l=1



Nl
∑
j=1
∂fl
∂Γj
∂2Γj
∂T∂ρ
+
Nl
∑
j=1
(
∂Γj
∂ρ
)
T,Y
Nl
∑
k=1
∂2fl
∂Γk∂Γj
(
∂Γk
∂T
)
ρ,Y



T4 =
3
∑
l=1



Nl
∑
j=1
∂fl
∂Γj
(
∂2Γj
∂T 2
)
ρ,Y
+
Nl
∑
j=1
(
∂Γj
∂T
)
ρ,Y
Nl
∑
k=1
∂2fl
∂Γk∂Γj
(
∂Γk
∂T
)
ρ,Y



T5 =
3
∑
l=1


Nl
∑
j=1
∂fl
∂Γj
(
∂Γj
∂T
)
ρ,Y


On the other hand, by direct differentiation we obtain
s̆ = s̆ideal − kBT
⟨m⟩imu
[
f
T
+ T5
]
■
A.2 Maxwell relations and alternative expressions
for some partial derivatives of interest
From the deĄnitions of dε̆, df̆ , dh̆, dğ and dω̆ respectively, we can show the following
relations between partial derivatives hold:
(
∂T
∂ρ
)
s̆,Y
=
1
ρ2
(
∂P
∂s̆
)
ρ,Y
(A.5)
(
∂s̆
∂ρ
)
T,Y
= − 1
ρ2
(
∂P
∂T
)
ρ,Y
(A.6)
(
∂T
∂P
)
s̆,Y
= − 1
ρ2
(
∂ρ
∂s̆
)
P,Y
(A.7)
(
∂s̆
∂P
)
T,Y
=
1
ρ2
(
∂ρ
∂T
)
P,Y
(A.8)
(
∂s̆
∂ρ
)
T,µ
= − 1
ρ2
(
∂P
∂T
)
ρ,µ
. (A.9)
161
A.3 Heat capacity in superfluid systems
In this appendix we reĄne the argument of [LL76] for the heat capacities at constant
pressure and volume, namely, that their difference converges to zero as T → 0 more
rapidly than them separately. For this purpose, it is necessary to deĄne what we
must understand as rapid convergence, and then move to the physical aspects of the
problem.
Definition 1. Let f, g : (0, δ) → R
+ ∪ ¶0♢ be two functions, with δ ∈ R
+. We say
that f converges more rapidly than g to zero if the following conditions are met:
• Their right limits exist:
lim
x→0+
f(x) = lim
x→0+
g(x) = 0 ; (A.10)
• ∀x ∈ (0, δ),
0 < f(x) < g(x) . (A.11)
We are now ready to discuss the problem of heat capacity. As basic facts we
consider that, according to [LL76]:
• NernstŠs postulate (or the Third Law of Thermodynamics) holds: the entropy
of the system goes to zero as T → 0.
• The heat capacity at constant X, being this variable a thermodynamical one,
can always be written as
CX = T
(
∂S
∂T
)
X
(A.12)
• The difference between heat capacities can be expressed as
CP − CV = −T
(
∂P
∂V
)
T
(
∂S
∂P
)2
T
(A.13)
From the Ąrst point, we replace the argument in [LL76] of considering S ∝ T b,
b ∈ N, by a more general expansion around T = 0:
S(P, V, T ) =
∞
∑
n=1
sn(P, V )τn (A.14)
with
sn(P, V ) =
(
∂S
∂T
)
T=0,P,V
, (A.15)
162
and τ = T/TF , with TF a positive scale parameter (which can be identiĄed with
the Fermi temperature for fermion systems), [TF ] = K, that serves to work with a
dimensional variables.
By replacing the series expansion into the heat capacities, we can see that
CP,V =
∞
∑
n=1
sn(P, V )τn (A.16)
CP − CV =
∞
∑
n=1
σnτ
n+1 (A.17)
σn = −
n
∑
m=1
TF
(
∂P
∂V
)
T
(
∂sm
∂P
)
V
(
∂sn−m
∂P
)
V
. (A.18)
Now, if τ < 1, τn+1 < τn holds (Proof: Being τn positive, multiplying the inequality
τ < 1 by this value does not alter the order of the terms. Thus, τn+1 < τn. ■).
All that is left is to prove that
(
∂P
∂V
)
T
is negative. This can be seen from physical
argumentation: if we increase the pressure (δP > 0), the occupied volume tends
to decrease (δV < 0). Similarly, if we expand the region where the system exists
(δV > 0), the pressure exerted diminishes (δP < 0). Thus, σn is positive ∀n ∈ N
and the following inequalities hold:
CP > CP − CV , (A.19)
CV > CP − CV . (A.20)
By deĄnition, this means that the difference between heat capacities tend to zero
more rapidly than the individual capacities. In the particular case of fermion sys-
tems, this result implies that the heat capacities are almost indistinguishable from
one another, and it is safe to assume that CP ≈ CV .
Before claiming that the proof is complete, it is necessary to examine the follow-
ing subtlety: if we replace the expansions for CP,V into the difference CP − CV and
compare coefficients with the expansion involving the σn-coefficients, we obtain
sn(Pfixed, V ) − sn(P, Vfixed) = −T
n
∑
m=1
(
∂P
∂V
)
T
(
∂sm
∂P
)
V
(
∂sn−m
∂P
)
V
. (A.21)
Apparently, the right hand side is not independent of temperature. We can be sure
that it actually is: the EoS opens the possibility to write T as a function of P and
V , i.e. T = f1(P, V ). In addition, the evaluation of the partial derivative of P
with respect to V at a Ąxed temperature is mathematically equivalent to introduce
a function f2(P, V, Tfixed) = f2(P, V ). Therefore, the right hand side is independent
of temperature, as the left hand side demands, and our proof is thus consistent and
complete.
163
Appendix B
General Relativity
This appendix introduces very speciĄc aspects of General Relativity which are in-
dispensable for deducing and understanding the Ćuid-element expressions discussed
in the main text. As such, they cannot be considered a review on the subject and
we urge readers not familiar with the terminology to consult the excellent books on
the subject.
B.1 On the conservation laws
When postulating the existence of physical conserved quantities, it is of universal
agreement that we must start by associating to each of the variables of interest a
4-current Jµ and understand as conservation the identity
1√−g∂α
(√−gJα
)
= 0 , (B.1)
i.e. the 4-current must have zero divergence. In the astrophysical context, for
example, it can be relevant to seek the inverse approach, that is: can we deĄne
a charge and claim that it is conserved? For instance, we can be interested on
extending the baryon number conservation, and verify if our intuition that the total
number of them can be written as
Nb =
∫
Σ
d3x
√
γ nb , (B.2)
i.e., as an integration of the baryon number density n over the proper volume element
at a fixed time, i.e. on a Cauchy hyper-surface.
As a starting point, let us review the conservation of the total electric charge.
We can postulate (or derive, via NoetherŠs Theorem) the existence of a conserved
4-current further referred as the current density 4-vector,
∂µJ
µ = 0 . (B.3)
164
Now let us consider a Cauchy hyper surface Σ, with boundary ∂Σ where the spatial
part of the 4-current identically vanishes. By virtue of GaussŠ Theorem,
∫
Σ
d3x ∂0J
0 = −
∫
Σ
d3x ∂iJ
i
= −
∫
∂Σ
d2x kiJ
i
= 0 , (B.4)
where ki is the 1-form associated with the unitary normal vector to ∂Σ. Thus, we
can see that
∂Q
∂x0
= 0 , (B.5)
where we have introduced the conserved electric charge as
Q =
∫
Σ
d3x J0 . (B.6)
Now let us take an analogous route for the curvilinear coordinated case. Typically,
we can expect conservation with respect to the time coordinate, so the Cauchy
surface must be chosen such that ct is constant. In addition, the invariance of the
integrals (with the appropriate measures) with respect to the coordinated system
allows us to express the quantities involved in an orthogonal frame. This implies
that the determinant of the metric tensor gαβ is related to that for the spatial-one
γαβ by √−g =
√
♣ − g00♣
√
γ . (B.7)
Note that
√
g00 is the 0 component of the 4-dual associated with the normal unitary
vector to Σ, i.e.
k0 =
√
♣ − g00♣ . (B.8)
Thus, our discussion can start from
A =
∫
Σ
d3x
√
γ
[
k0√−g∂0
(√−gJ0
)
]
. (B.9)
The reason to choose this form is twofold: Ąrst, the factor outside ∂0 yields 1/
√
γ
while the one inside simpliĄes to
√
γk0J
0, and we can now introduce the spatial part
of the conserved 4-current. With these considerations, it is more evident that we
now have a true 3-divergence which can be cast into a 2-surface integral by virtue
of GaussŠ Theorem. Denoting the induced 2-metric and orthogonal vector with a
165
super-index (2), we have
A =
∫
Σ
d3x
√
γ
[
1√
γ
∂0
(√
γk0J
0
)
]
=
∫
Σ
d3x
√
γ
[
1√
γ
∂i
(√
γk0J
i
)
]
=
∫
∂Σ
d2x
√
γ(2)k0k
(2)
i J i . (B.10)
Under the assumption that J i identically vanishes at ∂Σ, it follows that A = 0. From
its deĄnition, it is evident that the conserved quantity with respect to x0 variations
is
Q =
∫
Σ
d3x
√
γk0J
0 . (B.11)
Now, this expression was derived in an orthogonal frame. In order to be valid in
any coordinated system, we can extend the product of 0 components to the scalar
product of the unitary normal vector to Σ with the 4-current:
Q =
∫
Σ
d3x
√
γkµJ
µ . (B.12)
For a more formal discussion - leading to this same expression - see [CCAW04].
B.2 SSS metric - Identities
In matrix notation, the SchwarzschildŠs metric is deĄned as
[gαβ] :=





− exp(2Φ(r)) 0 0 0
0 exp(2Λ(r)) 0 0
0 0 r2 0
0 0 0 r2 sin2 θ





Some of its properties are:
• Inverse
[gαβ] :=





− exp(−2Φ(r)) 0 0 0
0 exp(−2Λ(r)) 0 0
0 0 r−2 0
0 0 0 r−2 sin−2 θ





.
• Christoffel Symbols. The entries in the following matrices respect the order
of the coordinated system, i.e. (ct, r, θ, ϕ), going from left to right and from
166
top to bottom. For instance, the 1, 0 entry of [Γctαβ] is equivalent to Γctr ct.
[Γtαβ] =





0 ∂rΦ 0 0
∂rΦ 0 0 0
0 0 0 0
0 0 0 0





[Γrαβ] =





exp(2(Φ − Λ))∂rΦ 0 0 0
0 ∂rΛ 0 0
0 0 −r exp(−2Λ) 0
0 0 0 −r sin2 θ exp(−2Λ)





[Γθαβ] =





0 0 0 0
0 0 1/r 0
0 1/r 0 0
0 0 0 − sin θ cos θ





, [Γϕαβ] =





0 0 0 0
0 0 0 1/r
0 0 0 cot θ
0 1/r cot θ 0





.
• Metric determinant. Being a diagonal metric, it is given by the product
of all entries, g = −e2Φ+2Λr4 sin2 θ. Consequently,
√−g = eΦ+Λr2 sin θ. The
spatial determinant obeys
√
γ = eΛr2 sin θ. Clearly,
√−g = eΦ√
γ.
• 4-divergence. In terms of both coordinated and non-coordinated (denoted
with hats) systems, for an arbitrary 4-vector Aα we have
∇αA
α =
1√−g∂α
(√−gAα
)
= ∂0A
0 +
1
eΦ+Λr2
∂r
(
eΦ+Λr2Ar
)
+
1
sin θ
∂θ
(
sin θAθ
)
+ ∂ϕA
ϕ (B.13)
= e−Φ∂0
(
A0̂
)
+
e−Φ−Λ
r2
∂r
(
eΦr2Ar̂
)
+
1
r sin θ
[
∂θ
(
sin θAθ̂
)
+ ∂ϕA
ϕ̂
]
.
(B.14)
167
B.3 Propositions and theorems
Proposition 4. The SSS metric and the normalized 4-velocity of the fluid satisfy
the following properties:
(a)
[
g00Γ0
r0 + grrΓr00
]
= 0
(b) u0∂rΦ = e−Φ∂r(eΦu0) − ∂ru
0
(c) ur∂rΛ = e−Λ∂r(eΛur) − ∂ru
r
(d) u0e
−Φ = −eΦu0
(e) ure
−Λ = eΛur
(f)
[
u0u0Γ0
0r + ururΓrrr
]
= −u0∂ru
0 − ur∂ru
r
(g) u0∂0u0 = −ur∂0ur.
Proof. For (a): employing the deĄnitions of the metric and uµ, by direct computa-
tion we have
[
g00Γ0
r0 + grrΓr00
]
= −e2Φ∂rΦ + e2Λe2Φ−2Λ∂rΦ
= 0.
For (b) and (c), we start from ∂ru
0 and ∂ru
r and rearrange terms in favor of the
partial derivatives of Φ and Λ:
∂ru
0 = e−Φ∂r(eΦu0) − u0∂rΦ
u0∂rΦ = e−Φ∂r(eΦu0) − ∂ru
0
∂ru
r = e−Λ∂r(eΛur) − ur∂rΛ
ur∂rΛ = e−Λ∂r(eΛur) − ∂ru
r.
It is easy to see (d) and (e) are simple consequences of the deĄnition of the 4-
velocity Ąeld and the raising and lowering of components in the SSS metric, which
is a diagonal one. The properties so far discussed can be now used to prove (f):
[
u0u0Γ0
0r + ururΓrrr
]
= u0∂rΦu0 + ur∂rΛur
= −u0∂ru
0 − ur∂ru
r + u0e
−Φ∂r(eΦu0) + ure
−Λ∂r(eΛur)
= −u0∂ru
0 − ur∂ru
r − 1
2
∂r
[
eΦu0
]2
+
1
2
∂r
[
eΛur
]2
.
Notice
[
(eΦu0)2 − (eΛur)2
]
= γ2
S
[
1 − v2
c2
]
= 1,
168
hence
[
u0u0Γ0
0r + ururΓrrr
]
= −u0∂ru
0 − ur∂ru
r.
Finally, (g) follows from the normalization of the 4-velocity,
g00∂0(u0)2 + grr∂0(ur)2 = 0,
u0∂0u0 = −ur∂0ur.
■
Proposition 5. The components of the correspondent 4-acceleration to the 4-velocity
field are
a0 = u0∂0u0 + ur∂ru0
=
γ2
Sv
c
{
γ2
S
c2
[
∂tv + eΦ−Λv∂rv
]
− eΦ−ΛdΦ
dr
}
ar = u0∂0ur − u0∂ru0 = −u0
ur
a0
aθ = 0
aϕ = 0.
Proof. Let us Ąrst deduce the general expressions for the 4 components of aµ. Em-
ploying the results of Proposition 4, it is easy to see
a0 = u0∂0u0 + ur∂ru0 −
[
u0Γσ00 + urΓσr0
]
uσ = u0∂0u0 + ur∂ru0 − uru0
[
g00Γ0
r0 + grrΓr00
]
= u0∂0u0 + ur∂ru0,
ar = u0∂0ur + ur∂rur −
[
u0Γσ0r + urΓσrr
]
uσ = u0∂0ur + ur∂rur −
[
u0u0Γ0
0r + ururΓrrr
]
= u0∂0ur + ur∂rur + u0∂ru
0 + ur∂ru
r = u0∂0ur + u0∂ru
0 + ∂r(urur)
= u0∂0ur + u0∂ru
0 − ∂r(u0u0) = u0∂0ur − u0∂ru0 = −u0
ur
[−ur∂0ur + ur∂ru0]
= −u0
ur
[
u0∂0u0 + ur∂ru0
]
= −u0
ur
a0.
For aθ and αϕ: due to uθ = uϕ = 0, the 8 partials ∂µuθ and ∂µuϕ are identically zero.
On the other hand, the terms associated to the Christoffel symbols vanish as well
since Γµθν and Γµϕν are identically zero for all µ, ν combinations. Thus, aθ = aϕ = 0.
Finally, to show the explicit version of a0 it suffices to replace the components of
the normalized 4-velocity and to consider
∂µγS = −γ3
S
v
c2
∂µv
169
for all 4 coordinates µ:
a0 = −
[
u0∂0 + ur∂r
]
(γSeΦ)
= −
{
γS
[
u0∂0 + ur∂r
]
(eΦ) + eΦ
[
u0∂0 + ur∂r
]
(γS)
}
= −
{
γSu
reΦdΦ
dr
− eΦγ3
S
v
c2
[
u0∂0 + ur∂r
]
v
}
=
γ2
Sv
c
{
γ2
S
c2
[
∂t + veΦ−Λ∂r
]
v − eΦ−ΛdΦ
dr
}
.
■
Proposition 6. The F r component, in the first-order approximation, is given by
F r ≈ −Ke−2Λ−Φ∂r(eΦT ).
Proof. By straightforward calculation, we have
F r = −K ¶Πrν∂νT + ΠrνTaν♢
= −K
{
γ2
Se
−Λ−Φ
[
vS
c
∂t + eΦ−Λ∂r
]
T + Tgrrar
}
= −Kγ2
Se
−Λ−Φ
{
[
vS
c
∂t + eΦ−Λ∂r
]
T + T
[
−γSe
Φ
c
uµ∂µv + eΦ−ΛdΦ
dr
]}
.
Neglecting the vs/c term (Ąrst-order approximation), setting γS ≈ 1 and rearranging
terms, we obtain
F r ≈ −Kγ2
Se
−Λ−Φ
{
e−Λ∂r(eΦT ) − Tγ2
S/c
2
[
∂t + veΦ−Λ∂r
]
v
}
≈ −Ke−Λ−Φ
{
e−Λ∂r(eΦT ) − T
c2
[
∂t + veΦ−Λ∂r
]
v
}
.
Notice the last term in the right-hand side is also of order vS/c, hence we can neglect
it in our Ąrst-order approximation. Consequently,
F r ≈ −Ke−2Λ−Φ∂r(eΦT ).
■
Proposition 7. Within the FOA, the contribution of the 4-flux to the entropy equa-
tion and the corresponding luminosity are given by
Gs,T = − 1
T
{ 1
e2Φ+Λr2
∂r(e2Φ+Λr2F r)
}
L = −4πr2Ke−Φ−Λ∂r(eΦT ).
170
Proof. Let us separately compute each term in Gs,T . It is easy to see
∇µF
µ =
1√−g
[√−geΛ−Φ∂0(vSF r) + ∂r(
√−gF r)
]
.
Regarding the projection of F µ along aµ, we have:
F νaν = vSe
Λ−ΦF ra0 − F ru
0a0
ur
= F ra0e
Λ−Φ
[
vS − 1
vS
]
= −F r e
Λ−Φ
γ2
SvS
a0
= −F reΛ−Φ
{
γSe
Φ
c
uµ∂µv − eΦ−ΛdΦ
dr
}
≈ F r dΦ
dr
.
Notice we have employed the Ąrst-order approximation for the last line. Since
1
e2Φ+Λr2
∂r(e2Φ+Λr2F r) =
1
eΦ+Λr2
∂r(eΦ+Λr2F r) + F r dΦ
dr
,
we see
∇νF
ν + F νaν =
eΛ−Φ
c
∂t(vSF r) +
1
e2Φ+Λr2
∂r(e2Φ+Λr2F r).
Finally, the presence of vS/c in the Ąrst term allows us to write, within this Ąrst-order
approximation,
Gs,T = − 1
T
{ 1
e2Φ+Λr2
∂r(e2Φ+Λr2F r)
}
.
Notice our expression for Gs,T , to Ąrst order in vS, is equivalent as the one we
obtain for a Ćuid at rest, uct = e−Φ, where the only non-vanishing component of the
associated energy-momentum tensor is T r0rad, i.e.
uν∇µT
µν
rad =
uν√−g∂µ(
√−gT µν)
=
u0√−g∂r(
√−gT r0)
=
u0√−g∂r(
√−gF ru0).
171
For computing the luminosity, we exploit this equivalence:
L = −
∫
Σ
d3x
√
γ
u0√−g∂r(
√−gF ru0)
=
∫
Σ
d3x
eΛr2 sin θeΦ
eΦ+Λr2 sin θ
∂r(eΦ+Λr2 sin θF re−Φ)
=
∫
Σ
dϕdθdr sin θ∂r(eΛr2F r)
= 4πr2eΛF r
= −4πr2Ke−Φ−Λ∂r(eΦT ).
where the angular integral was immediate due to the absence of dependence on θ
and ϕ of F r, and in the last line we employed the result from Proposition 6. ■
Proposition 8. Considering s̆(P, T, ¶Yi♢Nspecies
i=1 ), and under the assumption (∂s̆/∂Yi)P,T,Yk ̸=i
are negligible12,
TDts̆ = c̆P
[
DtT − T∇ad
P
DtP
]
.
Proof. Since s̆ is a functional of several variables, by direct calculation we have
TDts̆ = T
(
∂s̆
∂P
)
T,Y
DtP + T
(
∂s̆
∂T
)
P,Y
DtT + T
Nspecies
∑
i=1
(
∂s̆
∂Yi
)
T,P,Yk ̸=i
DtYi.
Since (∂s̆/∂Yi)P,T,Yk ̸=Yi
are assumed negligible, it suffices to employ thermodynamical
identities to simply the remaining coefficients:
TDts̆ = T
(
∂s̆
∂P
)
T,Y
DtP + T
(
∂s̆
∂T
)
P,Y
DtT
= c̆P
[
DtT − T∇ad
P
DtP
]
.
■
Proposition 9. Within the FOA, the radiative temperature gradient is
∇r = − 3κρBLPeΛ
64πσSBT 4r2
dr
dP
+
(
1 − ρ
H
)
.
1Or already part of ˙̆εspecies.
2Depending on the context, it is sometimes necessary to consider s̆(ρB , T, ¶Yi♢Nspecies
i=1 ). The
steps are analogous, employing the appropriate thermodynamical coefficient for DtρB .
172
Proof.
∂rT = − L
4πr2e−ΛK
− T
dΦ
dr
dT
dP
= − L
4πr2e−ΛK
dr
dP
+
T
Hc2
∇r =
P
T
dT
dP
= − LP
4πr2e−ΛKT
dr
dP
+
P
Hc2
= − LP
4πr2e−ΛKT
dr
dP
+
(
1 − ρ
H
)
= − 3κρBLPeΛ
64πσSBT 4r2
dr
dP
+
(
1 − ρ
H
)
.
■
Proposition 10. For a fluid at rest, and within the SSS metric, the cooling rate
can be written as
Γcool = − 1
T
{ 1
e2ΦeΛr2
∂r
(
e2ΦeΛr2F r
)
+
1
sin θ
∂θ
(
sin θF θ
)
+ ∂ϕF
ϕ
}
.
Proof. From repetitive application of the product rule, we have
F r∂rΦ = F re−ΦeΦ∂rΦ
= e−Φ∂r(eΦ)F r
= e−Φe−Λr−2∂r(eΦ)F reΛr2
= e−Φe−Λr−2
[
∂r
(
F reΦeΛr2
)
− eΦ∂r
(
F reΛr2
)]
= e−Φe−Λr−2
[
∂r
(
eΦA
)
− eΦ∂rA
]
, (B.15)
where we have deĄned A = r2eΛF r for simplicity. Back to the cooling rate, we have
Γcool =
1
T
{ 1
eΦeΛr2
[
−2∂r(eΦA) + eΦ∂rA
]
− ∆
}
, (B.16)
∆ =
1
sin θ
∂θ
(
sin θF θ
)
+ ∂ϕF
ϕ . (B.17)
From
∂r(e2ΦA) = e2Φ∂rA+ 2eΦ
(
∂re
Φ
)
A (B.18)
= eΦ∂r
(
eΦA
)
+
(
∂re
Φ
)
eΦA , (B.19)
we can see that
Γcool = − 1
T
{ 1
e2ΦeΛr2
∂r
(
e2ΦeΛr2F r
)
+
1
sin θ
∂θ
(
sin θF θ
)
+ ∂ϕF
ϕ
}
. (B.20)
This completes the proof. ■
173
B.4 Radiative Zero Solution
Let us go back to the differential equation for the temperature,
dT
dP
=
3κ
16T 3
F
σgs
. (B.21)
Under the assumption that the right hand side becomes temperature but not pres-
sure or density dependent, we can immediately integrate this equation. If we addi-
tionally assume that the lower limits of integrations are negligible in comparison to
the upper ones, we arrive to
T 4 =
(
3κ
16σgs
× 4F
)
P . (B.22)
This is known as the radiative zero solution, and it expresses the relation between
pressure and temperature in the case radiation transport dominates, but hydrostatic
equilibrium still holds.
174
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