THE EFFECTS OF PROTO-NEUTRON STAR CONVECTION ON THE DYNAMICS AND
NUCLEOSYNTHESIS OF THE NEUTRINO-DRIVEN WIND

By

Brian Powers Nevins

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Physics—Doctor of Philosophy

2024

ABSTRACT

The neutrino-driven wind from proto-neutron stars has long been considered a possible site for

interesting nucleosynthesis of heavy elements, via either an r-process, in which heavy elements are

made via rapid neutron captures onto seed nuclei, or a 𝜈p-process, in which heavy elements are

forged by neutrino-assisted proton captures onto seed nuclei. Many recent studies have indicated

that a fiducial wind heated only by neutrino interactions does not attain a high enough entropy

or a sufficiently fast expansion to allow these processes to build the heaviest elements found in

nature, despite promising results in the early literature. However, a secondary heating source in

the wind could strongly enhance heavy nucleosynthesis by disrupting seed nucleus formation, and

allowing extended nucleon captures to build heavier elements. A promising physical source of such

secondary heating is gravito-acoustic waves, launched by vigorous convection in the proto-neutron

star, which form shocks and deposit energy into the wind.

In this thesis, we derive a numerical model for examining the effects of such propagating

waves on both the dynamics and the nucleosynthetic processes active in the wind. We explore,

via parameter studies, the effects of these wave on both neutron- and proton-rich winds, and how

potential r- and 𝜈p-processes are affected. Finally, we compute the integrated nucleosynthesis from

a realistic proto-neutron star evolution to examine the effects of propagating waves on the different

stages of nucleosynthesis a time-dependent wind undergoes, and to predict the full nucleosynthetic

yield of realistic winds. We find that gravito-acoustic waves propagating through the wind have

a substantial impact on both r- and 𝜈p-processing, via heating due to shock formation as well

as the momentum flux carried by the waves. The combination of these effects also reduces the

asymptotic electron fraction of the wind by up to 20%. We even find that, for winds in which

the wave luminosity is ≳ 1% of the wave luminosity, a third-peak, albeit suppressed, r-process

attains despite proton-rich conditions in the wind. Though a full r-process is not realized in the

realistic, time-integrated wind from a simulated proto-neutron star, this is likely due to the low-mass

(∼ 1.35 𝑀⊙) neutron star formed. If a heavier (e.g., 1.6 𝑀⊙) proto-neutron star forms with a similar

neutrino spectrum and convective properties, our results suggest that a full r-process can proceed.

Copyright by
BRIAN POWERS NEVINS
2024

To my beloved wife, Sarah, and to the saints in Christ at University Reformed Church.

iv

ACKNOWLEDGEMENTS

This work was completed at Michigan State University during the years 2019-2024, with support

from a University Distinguished Fellowship and from the College of Natural Sciences. This work

was made possible in large part by the computational resources and services provided by the

Institute for Cyber-Enabled Research at MSU.

I am deeply grateful to my lovely wife, Sarah, and to my parents for their love and support as I

have pursued this goal. Also to my many dear friends and mentors at University Reformed Church,

who have refreshed my soul time and again over the last 5 years.

I thank my advisors, Ed Brown and Luke Roberts, for their patience, help and encouragement in

completing this work through a complicated advising situation; Luke, for continuing to work with

me after moving across the country; and Ed, for adopting me into his research group and taking

the time to learn about and help me with my project, which was outside his typical research area.

Thanks also to Sean Couch for his hard work getting STIR up and running, and helping put chapter

5 together; to the rest of my committee members, Filomena Nunes, Joey Rodriguez, and Betty

Tsang, for their help and support; also to Stan Woosley and Brian Metzger, who provided helpful

comments and discussion regarding chapter 3. Further thanks to Kim Crosslan for her invaluable

administrative assistance in navigating the PhD program.

I am also grateful to my friends and

colleagues in the department: Jordan Purcell, Sheng Lee, Erin White, and many others. Thank you

to Dr. Amy Kolan, Dr. Prabal Adhikari, and the rest of the physics department at St. Olaf College,

who prepared me so well for my time at MSU.

My greatest thanks and praise are to the Lord my God, who has kept me through this journey.

He has written His faithfulness into the fixed order of the stars, and His kindness into their beauty.

v

Jeremiah 31:33-36 (English Standard Version)

"For this is the covenant that I will make with the house of Israel after those days, declares the

Lord: I will put my law within them, and I will write it on their hearts. And I will be their God, and

they shall be my people. And no longer shall each one teach his neighbor and each his brother,

saying, ‘Know the Lord,’ for they shall all know me, from the least of them to the greatest, declares

the Lord. For I will forgive their iniquity, and I will remember their sin no more.”

Thus says the Lord,

who gives the sun for light by day

and the fixed order of the moon and the stars for light by night,

who stirs up the sea so that its waves roar—

the Lord of hosts is his name:

“If this fixed order departs

from before me, declares the Lord,

then shall the offspring of Israel cease

from being a nation before me forever.”

Psalm 19:1 (English Standard Version)

The heavens declare the glory of God,

and the sky above proclaims his handiwork.

Psalm 148:3 (English Standard Version)

Praise him, sun and moon,

praise him, all you shining stars!

vi

TABLE OF CONTENTS

CHAPTER 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1 Supernovae . .
1.2 The Neutrino-Driven Wind . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.

. .

.

1
1
3

CHAPTER 2

9
MATHEMATICAL FORMALISM AND MODELING METHOD . . . .
9
2.1 Fundamental Equations of the Neutrino-Driven Wind . . . . . . . . . . . . . .
2.2 Computational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

CHAPTER 3

.

.

.
3.1 Background .
3.2 Wind Dynamics . .
3.3 Nucleosynthesis .
.
3.4 Parameter Study . .
. .
3.5 Conclusions . .

PROTO-NEUTRON STAR CONVECTION AND THE R-PROCESS . . 28
. 28
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.

CHAPTER 4

PROTO-NEUTRON STAR CONVECTION
AND THE 𝜈P-PROCESS . . . . . . . . . . . . . . . . . . . . . . . . . 47
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
.

4.1 Background .
.
4.2 Wind Dynamics .
4.3 Nucleosynthesis .
4.4 Parameter Study .
. .
4.5 Conclusions .

.
.
.
.
.

.
.
.
.
.

.

CHAPTER 5

INTEGRATED NUCLEOSYNTHESIS FROM CONVECTIVE
PROTO-NEUTRON STARS . . . . . . . . . . . . . . . . . . . . . . . . 66
5.1
.
Introduction .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2 Overview of STIR .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Integrating STIR .
5.3
5.4 Results from STIR: Model z9.6 . . . . . . . . . . . . . . . . . . . . . . . . . . 74
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.5 Conclusions .

.
.
.

.

.

.

.

.

.

CHAPTER 6

SUMMARY AND FUTURE DIRECTIONS . . . . . . . . . . . . . . . 92
6.1 Summary . .
. 92
6.2 Future Research Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. .

. .

BIBLIOGRAPHY .

.

.

.

.

. .

. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

vii

CHAPTER 1

INTRODUCTION

1.1 Supernovae

After formation, stars maintain their interior temperature against radiative losses by fusing

hydrogen into helium. As a massive (≳ 8 𝑀⊙) star reaches the end of its main-sequence phase,

the supply of hydrogen in its core is depleted, and progressively heavier isotopes begin to fuse in

its core. Carbon, neon, oxygen, silicon, and finally iron are formed. Burning stalls when the core

reaches nuclear statistical equilibrium (NSE), which favors nuclei with high binding energies near

the iron peak. Fusing these elements no longer produces energy, and an inert core of iron-peak

elements, held up by electron degeneracy pressure, is formed. As fusion continues in the layers

surrounding the core, more iron is added, until the core approaches the Chandrasekhar mass. As

the core grows, the rising density causes the electrons to become increasingly relativistic, which

lowers the adiabatic index 𝛾 from the 5/3 of an ideal gas towards 4/3. At the Chandrasekhar mass,

when 𝛾 = 4/3, the core becomes mechanically unstable and begins to collapse. The density-

sensitive electron-capture reaction rate then increases, and electrons and protons rapidly combine

into neutrons. These electron captures further decrease 𝛾 and rob the core of support, and the

material begins to free fall towards the center of the star as it becomes more and more neutron-rich.

The free fall is halted when neutron degeneracy pressure becomes significant; the abrupt stiffening

of the core launches a shockwave toward the stellar surface (Weaver et al., 1978; Woosley & Janka,

2005). The shock stalls eventually, but is re-invigorated by the neutrinos released by the ongoing

electron captures in the core (Burrows & Vartanyan, 2021). Upon reaching the surface, the shock

violently disrupts the star and creates the luminous event observed as a supernova.

This picture describes the process by which a neutron star is formed in a core-collapse supernova

(CCSN), and is born out in modern simulations (e.g., Burrows et al., 2019). The collapsing iron

core rapidly neutronizes as the extreme density drives electron capture reactions, and neutron

degeneracy pressure halts its collapse at a radius of order ∼ 50 km, from an original radius of

order ∼ 2000 km (Couch, 2017). About 1053 erg of gravitational potential energy are released in

1

Figure 1.1 Schematic showing the dominant reactions and rough temperature ranges of the various
nucleosynthetic stages the NDW undergoes during its outflow. The wind initially is extremely hot,
but cools rapidly once it leaves the PNS.

the process, resulting in an extremely hot and dense proto-neutron star (PNS) that is opaque to

neutrinos. The PNS then radiates this energy as a massive flux of neutrinos and antineutrinos over

the course of ∼ 20 s (Qian & Woosley, 1996; Woosley & Janka, 2005), while shrinking towards the

fiducial 10 km cold neutron star radius. As the PNS cools, the neutrinosphere (the high temperature

region opaque to neutrinos) shrinks until the star is fully transparent to neutrinos and becomes a

mature (though hot) neutron star (Camelio, 2018).

2

1.2 The Neutrino-Driven Wind

As the PNS cools, the immense neutrino flux interacts with its outermost layers and is sufficient

to unbind some (∼ 10−3 𝑀⊙, Wanajo, 2013) material from the surface in what has been called

the neutrino-driven wind (NDW, Duncan et al., 1986). Initially, the material in the NDW is hot

(𝑇 ∼ 3 MeV), dense (𝜌PNS ∼ 1012 g cm−3), and neutron-rich (electron fraction 𝑌𝑒 ≲ 0.3) (Qian &

Woosley, 1996; Couch, 2017). These conditions are such that the wind material begins in NSE

and proceeds through several phases of nucleosynthesis as it cools (see Figure 1.1). While in

NSE, the material primarily consists of free protons and neutrons. As the temperature drops below

𝑇 ∼ 1 MeV, the wind is expected to undergo 𝛼-recombination, in which all free nucleons possible

are bound into 𝛼 particles.

As the temperature approaches 𝑇 = 0.5 MeV, these 𝛼 particles combine into successively

heavier elements up to the iron peak. The starting point for this process is 3𝛼 → 12C or, in neutron-

rich conditions, 3𝛼 + n → 12C + n. Neutron-rich conditions are not guaranteed or necessarily even

expected at this point; the electron fraction in the wind is altered significantly by charged-current

neutrino and antineutrino reactions during these initial moments while the material is close to the

PNS surface. The neutrino spectrum of the PNS is the most important factor in determining the

asymptotic electron fraction of the wind. Assuming other factors are negligible, the equilibrium

electron fraction in the wind is reached when 𝑇 ≈ 1 MeV and is (Qian & Woosley, 1996)
𝐿 ¯𝜈𝑒
𝐿𝜈𝑒

(cid:0)𝜖 ¯𝜈𝑒 − 2Δ + 1.2Δ2/𝜖 ¯𝜈𝑒
(cid:0)𝜖𝜈𝑒 + 2Δ + 1.2Δ2/𝜖𝜈𝑒

𝑌𝑒,eq ≈

1 +

(cid:34)

(cid:35)

(cid:1)

(cid:1)

(1.1)

where 𝐿𝜈𝑒 and 𝐿 ¯𝜈𝑒 denote the electron neutrino and antineutrino luminosities respectively, the

𝜖𝜈 denote the electron neutrino and antineutrino energies in MeV, and Δ represents the positive

mass difference between protons and neutrons, approximately equal to 1.293 MeV. Most modern

simulations predict a very slightly neutron-rich or proton-rich NDW, with 0.45 ≲ 𝑌𝑒,eq ≲ 0.65 (e.g.,

Fischer et al., 2010; Roberts et al., 2012a; Hüdepohl, 2013; Martínez-Pinedo et al., 2014; Xiong

et al., 2020; Pascal et al., 2022). As the range suggests, there is significant uncertainty in 𝑌𝑒,eq.

The efficiency of this 𝛼-process, and specifically the 3- and 4- body reactions that form the

initial 12C nuclei, determine the character of the nucleosynthesis going forward. Once the wind

3

cools below 𝑇 ≈ 5 GK, 𝛼 capture reactions freeze out, and nucleosynthesis is dominated by nucleon

captures onto the heavy (𝐴 ≥ 12) nuclei formed by the 𝛼-process. The number and type of free

nucleons in the wind is determined by the electron fraction, and the number of capture targets is set

by the efficiency of the 𝛼-process. If a large number of heavy nuclei were formed, the free nucleons

are spread thinly and nucleosynthesis remains concentrated near the iron peak. If the 𝛼-process

were inefficient and only a few heavy nuclei were formed, each one could undergo a large number

of nucleon captures to build much heavier nuclei as the wind continues outward.

Two main parameters, the entropy (𝑠) and the dynamical timescale (𝜏𝑑), determine the efficiency

of the 𝛼-process (Hoffman et al., 1997). The 12C nuclei are formed via 3- and 4-body reactions

which are highly dependent on both density and temperature. These seed nuclei are formed when

𝑇 ≈ 0.5 MeV, and at a fixed temperature, a higher entropy corresponds to a lower density. High

entropies therefore inhibit seed formation by suppressing the 12C-forming reaction rates. The

dynamical timescale of the wind (defined here as 𝜏𝑑 = 𝑇/ (cid:164)𝑇, similar to the 𝑟/ (cid:164)𝑟 of Qian & Woosley,

1996) describes how quickly the wind is expanding outwards, and is a helpful indicator of how

quickly the wind is cooling. Temperatures in excess of 2.5 GK are required to facilitate the 𝛼-

capture reactions that form the heavy seed nuclei, and a fast expansion (i.e., a short dynamical

timescale) means the wind material will move through this temperature range much faster. The

reaction rate is not necessarily affected, but the amount of time over which the reactions can

take place is diminished.

In the case of a neutron-rich wind (𝑌𝑒 < 0.5), the neutron-catalyzed

4He(𝛼𝑛, 𝛾)9Be(𝛼, 𝑛)12C becomes the dominant seed-forming channel, and so the electron fraction

enters as a third parameter determining the efficiency of the 𝛼-process. A higher abundance of free

neutrons allows for more efficient seed formation.

Once the wind cools below 2.5 GK, 𝛼 capture reactions freeze out, and subsequent nucleosyn-

thesis depends primarily on the electron fraction of the wind. At this point, the wind material

comprises free nucleons (either protons or neutrons), 𝛼 particles, and whatever heavy seed nuclei

were formed in the 𝛼-process. At these temperatures, the 𝛼 particles are essentially inert, and

nucleosynthesis is driven by nucleon captures onto the seed nuclei. This can proceed via two

4

general branches depending on the electron fraction. If the wind is neutron-rich, neutron captures

and subsequent 𝛽-decays proceed freely, building heavier elements from the initial seed nuclei in

an r-process (Burbidge et al., 1957). If the wind is proton-rich, the 𝜈p-process operates, building

heavier nuclides via proton captures facilitated by a small abundance of free neutrons created in-situ

by charged-current neutrino reactions (Fröhlich et al., 2006). How far these processes can proceed

(i.e., the heaviest elements that can be synthesized) depends mainly on the ratio of free nucleons

to seed nuclei, which is set by the electron fraction and the efficiency of the 𝛼-process. Naturally,

then, most research on the NDW in recent decades has focused on these two factors. Much work has

been done to explore the neutrino spectrum of a cooling PNS, and the electron fraction it predicts in

the wind (e.g. Hüdepohl et al., 2010; Roberts, 2012; Roberts et al., 2012a; Martínez-Pinedo et al.,

2012; Hüdepohl, 2013; Martínez-Pinedo et al., 2014; Xiong et al., 2019, 2020). This work joins

the body of literature studying the variety of phenomena that affect the efficiency of the 𝛼-process,

and the subsequent nucleosynthesis.

1.2.1 Physical Phenomena Affecting NDW Nucleosynthesis

The efficiency of the 𝛼-process depends mainly on the entropy and the dynamical timescale

of the wind, as described above. The asymptotic entropy of the wind is individually affected by a

number of factors. It scales positively with PNS mass and inversely with the neutrino luminosity 𝐿𝜈

(Thompson et al., 2001). The inclusion of general relativistic effects therefore has a substantially

positive effect on entropy (Cardall & Fuller, 1997; Otsuki et al., 2000; Wanajo, 2013). Strong

(magnetar-strength) magnetic fields have been a subject of much study; they tend to increase

entropy by trapping material closer to the neutron star, thereby increasing the neutrino heating the

material receives (Thompson, 2003). The wind termination shock, when the fast-moving wind

material impacts slower-moving material behind the primary supernova shock, can also affect

nucleosynthesis based on the radius and temperature at which it occurs. A termination shock close

to the PNS can dramatically increase the amount of neutrino heating the material receives, raising

the entropy, but will also greatly increase the dynamical timescale (Arcones et al., 2007; Arcones

& Thielemann, 2012). The most-studied factor affecting the dynamical timescale of the wind is

5

PNS rotation, which provides an additional centrifugal acceleration to the wind material (Metzger

et al., 2007).

The focus of this work is a phenomenon that can affect both the entropy and the dynamical

timescale of the wind simultaneously:

the generation of waves propagating through the wind.

Suzuki & Nagataki (2005) considered Alfvén waves generated by strong magnetic fields; our focus

is the more ubiquitous phenomenon of gravito-acoustic waves briefly considered by Metzger et al.

(2007). These waves are generated by convection in the PNS, which has been shown to be common

across a broad range of progenitors (e.g., Nagakura et al., 2020; Nagakura et al., 2021), unlike the

magnetar-strength magnetic fields required for significant Alfvén wave generation. PNS convection

is predicted to be most vigorous 1–2 seconds post-bounce, and gradually tapering off thereafter.

The convective region in the outer layers of the PNS launches density waves into the adjacent

radiative region, the outflowing wind, and these waves continue to propagate outward in the wind.

Gravity, acoustic, and non-propagating wave modes will all be excited, but the energy flux will be

dominated by the gravity mode branch (Goldreich & Kumar, 1990). The luminosity carried by the

gravity waves is approximately

𝐿𝑔 ≈ 𝑀con𝐿con ≈ 𝑀con𝐿𝜈,

(1.2)

where 𝑀con is the convective Mach number in the PNS convective zone, 𝐿con is the convective

luminosity, and 𝐿𝜈 is the total neutrino luminosity of the PNS. Convection in the PNS is expected

to be efficient, so 𝐿con ≈ 𝐿𝜈. Simulations by Dessart et al. (2006) and Gossan et al. (2020) indicate

that 10−2 ≲ 𝑀con ≲ 10−1. Though the waves begin primarily in the gravity mode branch, they

will encounter an evanescent region close to the surface of the PNS. They will tunnel through this

region with some efficiency T , emerging in the acoustic wave branch and continuing out into the

wind (see figure 1.2). The total wave luminosity actually reaching the wind is then given by

𝐿𝑤 ≈ 𝑀conT 𝐿𝜈.

(1.3)

Simulations by Gossan et al. (2020) indicate that 0.01 ≲ T ≲ 0.2 in the pre-supernova explosion

environment, and our models find similar values in the NDW phase (see chapter 5). The wave

6

Figure 1.2 An approximate schematic of gravito-acoustic wave emission and propagation inside
and near the PNS. Gravity waves (dashed) are generated by the convective region, attenuate in the
evanescent region, and re-emerge as acoustic waves (solid) near the surface of the PNS. They then
propagate outward through the wind until they form shocks and dissipate. The region of possible
shock formation and heat deposition overlaps with the 𝛼-forming region. If the waves shock before
or during 𝛼 recombination, the additional heating will inhibit seed formation, making a strong
r-process more likely.

luminosity can therefore reasonably fall in the range of 10−5 to 10−2 𝐿𝜈, enough to significantly

alter the dynamics of the wind. Wave angular frequencies of 102 to 104 s−1 are expected, and tend

to increase with time. If information about the convective velocity of the PNS is known, 𝐿𝑤 and 𝜔

can be calculated explicitly (see chapter 5).

7

As the waves propagate through the wind, they impose two main effects. First, these waves carry

momentum, and this momentum flux accelerates the wind and reduces the dynamical timescale.

Second, due to the density dependence of the sound speed (𝑐𝑠), the peaks of these waves travel

slightly faster than the troughs. Eventually, the waves become non-linear and form shocks, which

efficiently deposit their energy into the wind as heat (Mihalas & Mihalas, 1984). This acts to raise

the entropy of the wind. Depending on how long it takes these shocks to form, this heat could

be deposited while the 𝛼-process is ongoing or after it has already ended. In the latter case, this

additional heating will have little if any effect on the nucleosynthesis in the wind. In the former,

however, it could substantially affect the efficiency of the 𝛼-process. A third effect emerges as a

consequence of these two primary effects: the additional energy and momentum deposited by the

waves reduces the amount of neutrino heating experienced by the material as it is ejected, which

in turn affects 𝑌𝑒. The material is initially extremely neutron-rich, and the neutrino heating it

experiences drives it toward 𝑌𝑒 = 0.5. This reduction in neutrino heating causes the material to

remain more neutron-rich than the neutrino spectrum of the PNS would predict. This also reduces

the asymptotic entropy of the wind prior to shock heating, as neutrino heating is the primary source

of entropy in the wind.

The objective of this work is to determine how the presence of gravito-acoustic waves affects

nucleosynthesis in the NDW, in both the neutron-rich and proton-rich branches. The mathematical

formalism and modeling methods used to this end are described in chapter 2. In chapters 3 and 4,

a brief history of the r-process and 𝜈p-process in the NDW is provided, and the impact of gravito-

acoustic waves on these processes is described. Finally, in chapter 5 the total, time-integrated

nucleosynthetic yield of a NDW derived from a self-consistent supernova simulation is explored.

8

CHAPTER 2

MATHEMATICAL FORMALISM AND MODELING METHOD

While a supernova is an inherently three-dimensional event, with turbulence and stellar anisotropies

playing vital roles, the computational power and complexity required to simulate these events in

3D can be prohibitive. There have been many fruitful efforts to model supernovae generally (e.g.,

O’Connor, 2015; Curtis et al., 2018; Couch et al., 2020), and the NDW specifically (e.g., Qian &

Woosley, 1996; Thompson et al., 2001; Wanajo et al., 2001), in spherical symmetry. The reduced

complexity of a spherically symmetric model allows for large parameter studies such as the ones

described in chapters 3 and 4, and is the most commonly used tool for examining the NDW.

A further approximation of time-independence is also often used when modeling the NDW.

After the supernova shock has been launched and during the cooling phase of the PNS (while the

wind is being launched), both the physical properties and the neutrino spectrum of the PNS vary

slowly compared to the dynamical timescale of the wind, which makes such a steady-state treatment

reasonable. This steady-state assumption reduces the equations describing the wind to depend only

on the properties of the PNS, the local conditions and equation of state of the wind, and the radial

distance of a fluid element in the wind from the PNS.

2.1 Fundamental Equations of the Neutrino-Driven Wind

The NDW is described by the equations of inviscid hydrodynamics. We denote the velocity,

density, pressure, and specific internal energy of the wind with (cid:174)𝑣, 𝜌, 𝑃, and 𝜖 respectively. The

equations for conservation of mass, momentum, and energy, including the energy and momentum

of the propagating waves (Jacques, 1977), are respectively

𝜕 𝜌
𝜕𝑡
𝜕
𝜕𝑡
𝜕
𝜕𝑡

+ ∇ · (𝜌(cid:174)𝑣) = 0

(𝜌(cid:174)𝑣) + ∇ · (T + 𝚷) = 𝜌 (cid:174)𝑔
(cid:21)
(cid:20)

(cid:19)

(cid:18)

𝜌

𝜖 +

𝑣2 + Φ

+ E

+ ∇ ·

1
2

(cid:18)

(cid:20)

𝜌(cid:174)𝑣

ℎ +

(cid:19)

𝑣2 + Φ

1
2

+ (cid:174)𝑣𝑔E + 𝚷 · (cid:174)𝑣

(cid:21)

= 𝜌 (cid:164)𝑞𝜈

with the ideal fluid stress tensor T given by

𝑇𝑖 𝑗 = 𝑃𝛿𝑖 𝑗 + 𝜌𝑣𝑖𝑣 𝑗 .

9

(2.1)

(2.2)

(2.3)

(2.4)

Here ℎ = 𝜖 + 𝑃/𝜌 is the specific enthalpy, Φ is the Newtonian gravitational potential, and (cid:174)𝑔 is the

Newtonian gravitational acceleration. The energy density and stress tensor of the waves are E and

𝚷, respectively. The volumetric heating rate per unit mass due to neutrino interactions is denoted

by (cid:164)𝑞𝜈; as is customary in studies of the NDW, we neglect the momentum flux of the neutrinos as it

is expected to be negligible (e.g., Qian & Woosley, 1996). By expanding equations (2.2) and (2.3),

using equation (2.1) to eliminate terms, subtracting the dot product of (cid:174)𝑣 with equation (2.2) from

equation (2.3), and taking 𝜕Φ/𝜕𝑡 = 0, we reduce equations (2.1)–(2.3) to Lagrangian form:

𝐷 𝜌
𝐷𝑡
𝐷 (cid:174)𝑣
𝐷𝑡
𝐷𝜖
𝐷𝑡

= −𝜌∇ · (cid:174)𝑣

= −

1
𝜌

= −𝑃

1
𝜌
(cid:19)

∇𝑃 −

𝐷
𝐷𝑡

(cid:18) 1
𝜌

∇ · 𝚷 − (cid:174)𝑔

(cid:20) 𝜕E
𝜕𝑡

1
𝜌

−

+ ∇ · (cid:0)(cid:174)𝑣𝑔E + 𝚷 · (cid:174)𝑣(cid:1)

(cid:21)

+ (cid:174)𝑣 · ∇ · 𝚷 + (cid:164)𝑞𝜈.

(2.5)

(2.6)

(2.7)

We use the notation

𝐷
𝐷𝑡

≡

𝜕
𝜕𝑡

+ (cid:174)𝑣 · ∇

to denote the material (or convective) derivative, which describes the Lagrangian change in a

quantity moving with velocity (cid:174)𝑣. No source term for mass exists in the outflowing wind, so

the evolution of density is determined strictly by the divergence of the velocity. The velocity is

determined by the internal thermodynamic state of the wind, but is also affected by gravity and

stresses exerted by the waves. The specific internal energy of the wind can be converted into kinetic

energy as the pressure does work on the wind material, and can also be affected by changes in the

energy of the waves due to, e.g., shock formation, and by neutrino interactions that heat or cool the

wind. Since several of these equations depend on the wave treatment, we will consider that first,

before addressing each of these equations in turn.

2.1.1 Gravito-Acoustic Wave Propagation

In this work, we follow Jacques (1977) in treating the waves in the eikonal approximation. In

this approximation, while the energy density of the waves is not conserved due to adiabatic losses,

the number density of wave quanta (often referred to as the wave action) N = E/𝜔0 is conserved

10

in the absence of dissipation, i.e.,

𝜕N
𝜕𝑡

+ ∇ · ((cid:174)𝑣𝑔N ) = 0.

(2.8)

Here 𝜔0 = 𝜔𝑐𝑠/𝑣𝑔 denotes the wave frequency in the fluid frame, with 𝑐𝑠 and 𝜔 representing the

sound speed and the wave frequency in the rest frame of the PNS, and (cid:174)𝑣𝑔 = (cid:174)𝑣 + 𝑐𝑠 (cid:174)𝑒𝑘 being the

group velocity. When dissipation is present, we parameterize the energy loss, and thus the change

in the wave action, with a characteristic dissipation length 𝑙𝑑:

𝜕N
𝜕𝑡

+ ∇ · ((cid:174)𝑣𝑔N ) = −

𝑣𝑔N
𝑙𝑑

.

In this treatment, the wave frequency 𝜔 is constant along a ray,

𝜕𝜔
𝜕𝑡

+ (cid:174)𝑣𝑔 · ∇𝜔 = 0;

(2.9)

(2.10)

thus in steady-state and spherical symmetry, the wave frequency in the PNS frame is constant. With

these approximations, we can now examine the hydrodynamic equations.

2.1.2 Mass Conservation

The mass conservation equation can be integrated to provide an expression for the mass-loss

rate from the PNS due to the wind. Rewriting equation (2.1) in the steady-state approximation, we

have

∇ · (𝜌(cid:174)𝑣) = 0.

Integrating over a sphere of radius 𝑟, assuming spherical symmetry, and applying Gauss’s law, this

becomes

∫

𝑉

∇ · (𝜌(cid:174)𝑣)𝑑3𝑟 =

∮

𝐴

𝜌𝑣𝑑2𝑟 = 4𝜋𝑟 2𝑣𝜌 = 𝐶.

(2.11)

The integration constant 𝐶 is immediately recognizable as the mass-loss rate of the PNS due to the

wind:

(cid:164)𝑀 = 4𝜋𝑟 2𝑣𝜌.

(2.12)

The steady-state approximation thus essentially states that the mass-loss rate is constant in the wind.

11

2.1.3 Momentum Conservation

The momentum conservation equation can be expanded in the following way. Applying the

one-dimensional and steady-state approximations transforms the material derivative to

𝐷
𝐷𝑡

= 𝑣

𝑑
𝑑𝑟

.

With this transformation, Equation (2.6) becomes

𝑣

𝑑𝑣
𝑑𝑟

= −

1
𝜌

𝑑𝑃
𝑑𝑟

−

1
𝜌

(∇ · 𝚷 + 𝜌 (cid:174)𝑔) · (cid:174)𝑒𝑟 .

The stress tensor for radially propagating acoustic waves is (Jacques, 1977)

𝚷 =

Π𝑟𝑟

0

0 Π𝜃𝜃

0

0

0

0 Π𝜙𝜙

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

= E

𝑎1

0

0

(cid:169)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:173)
(cid:171)

0

𝑎2

0

,

0

0

𝑎3

(cid:170)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:174)
(cid:172)

where E is again the energy density of the waves, and

(𝛾 + 1),

𝑎1 =

1
2
𝑎2 = 𝑎3 =

(𝛾 − 1).

1
2

(2.13)

(2.14)

(2.15)

(2.16)

Here 𝛾 ≡ (𝜕 ln 𝑃/𝜕 ln 𝜌)𝑠 is the adiabatic index. The divergence of 𝚷 in spherical symmetry, i.e.,

the (negative) effective force exerted by the waves on a fluid element, is (assuming constant 𝛾)

∇ · 𝚷 =

=

=

(cid:20) 1
𝜕
𝜕𝑟
𝑟 2
(cid:20) 𝜕Π𝑟𝑟
𝜕𝑟
𝜕E
𝜕𝑟

𝑎1

(cid:20)

(cid:16)

𝑟 2Π𝑟𝑟

(cid:17)

−

(cid:21)

Π𝜃𝜃 + Π𝜙𝜙
𝑟

(cid:174)𝑒𝑟 +

(cid:20)

1
𝑟 sin 𝜃

𝜕
𝜕𝜃

(Π𝜃𝜃 sin 𝜃) −

(cid:21)

Π𝜙𝜙 cot 𝜃
𝑟

(cid:174)𝑒𝜃

+

2Π𝑟𝑟 − Π𝜃𝜃 − Π𝜙𝜙
𝑟

(cid:21)

(cid:174)𝑒𝑟

(cid:21)

2E
𝑟

+

(cid:174)𝑒𝑟 .

The equality of the transverse components Π𝜃𝜃 and Π𝜙𝜙 ensures that ∇ · 𝚷 is purely radial. With

(cid:174)𝑔 = (𝐺 𝑀/𝑟 2) (cid:174)𝑒𝑟, the spherically symmetric, steady-state momentum conservation equation (2.6)

thus becomes

𝑣

𝑑𝑣
𝑑𝑟

= −

1
𝜌

𝑑𝑃
𝑑𝑟

−

𝐺 𝑀
𝑟 2

−

𝑎1
𝜌

𝑑E
𝑑𝑟

−

2E
𝜌𝑟

.

(2.17)

12

Most prior work on the NDW only includes the pressure and gravitational terms; the others arise

specifically from wave contributions.

To describe the evolution of the energy density of the waves, recalling that E = 𝜔0N , we make

use of equation (2.9) in the steady-state approximation:

∇ · ((cid:174)𝑣𝑔N ) = −

𝑣𝑔N
𝑙𝑑

,

(2.18)

which is again analogous to the conservation of quantized wave packets. For non-dissipative waves,

𝑙𝑑 → ∞ and N is conserved. The process by which the waves become dissipative is described in

section 2.1.4.2 below. Expanding and solving for 𝑑N /𝑑𝑟 in spherical symmetry, equation (2.18)

becomes

𝑑N
𝑑𝑟

= −N

(cid:18) 1
𝑙𝑑

+

2
𝑟

+

1
𝑣𝑔

(cid:19)

.

𝑑𝑣𝑔
𝑑𝑟

(2.19)

Note that, for the sake of simplifying computation, equation (2.19) can be written in terms of

L = 4𝜋𝑟 2𝑣𝑔𝜔N as

𝑑 ln L
𝑑𝑟

= −

.

1
𝑙𝑑

(2.20)

L can be thought of as an effective wave luminosity remaining in the wind at a given radius, and

thus accounts for any dissipation that has taken place. In the absence of dissipation (𝑙𝑑 → ∞), L

is a conserved quantity. For the wave energy density, we now have

𝑑E
𝑑𝑟

= 𝜔0

𝑑N
𝑑𝑟

+ N

𝑑𝜔0
𝑑𝑟

= 𝜔N

𝑐𝑠
𝑣𝑔

(cid:18) 1
𝑐𝑠

𝑑𝑐𝑠
𝑑𝑟

−

1
𝑙𝑑

−

2
𝑟

−

(cid:19)

.

𝑑𝑣𝑔
𝑑𝑟

2
𝑣𝑔

(2.21)

We can then rewrite equation (2.17) as

𝑣

𝑑𝑣
𝑑𝑟

= −

= −

1
𝜌
𝑃
𝜌

−

𝑑𝑃
𝑑𝑟
𝑑 ln 𝑃
𝑑𝑟

𝐺 𝑀
𝑟 2

−

−

𝐺 𝑀
𝑟 2

(cid:20) 1
𝑎1𝑐𝑠𝜔N
𝜌𝑣𝑔
𝑐𝑠
𝑎1𝑐𝑠𝜔N
𝜌𝑣𝑔

−

−

𝑑𝑐𝑠
𝑑𝑟
(cid:20) 𝑑 ln 𝑐𝑠
𝑑𝑟

1
𝑙𝑑

−

2
𝑟

(cid:18)

1 −

(cid:19)

1
𝑎1

−

(cid:18)

−

1
𝑙𝑑

−

2
𝑟

1 −

1
𝑎1

2
𝑣𝑔
(cid:19)

−

(cid:21)

𝑑𝑣𝑔
𝑑𝑟

(cid:18)

𝑐𝑠

𝑑 ln 𝑐𝑠
𝑑𝑟

(cid:19)(cid:21)

.

𝑑𝑣
𝑑𝑟

+

2
𝑣𝑔

(2.22)

13

It will be helpful to cast the derivatives of pressure and sound speed in terms of density, entropy,

and electron fraction:

𝑑 ln 𝑃
𝑑𝑟

𝑑 ln 𝑐𝑠
𝑑𝑟

=

=

=

=

(cid:19)

(cid:18) 𝜕 ln 𝑃
𝜕 ln 𝑠

1
𝑠
Γ𝑃𝑠
𝑑𝑠
𝜌𝑌𝑒
𝑑𝑟
𝑠
(cid:18) 𝜕 ln 𝑐𝑠
𝜕 ln 𝑠

1
𝑠

−

𝜌,𝑌𝑒
𝜌𝑐2
𝑠
𝑃

(cid:19)

𝜌,𝑌𝑒

Γ

𝑐𝑠 𝑠
𝜌𝑌𝑒
𝑠

𝑑𝑠
𝑑𝑟

− Γ

𝑐𝑠 𝜌
𝑠𝑌𝑒

𝑑𝑠
𝑑𝑟

+

1
𝜌

(cid:19)

(cid:18) 𝜕 ln 𝑃
𝜕 ln 𝜌
(cid:19)

𝑑𝑣
𝑑𝑟
(cid:18) 𝜕 ln 𝑐𝑠
𝜕 ln 𝜌

+

𝑠,𝑌𝑒
𝑃𝑌𝑒
Γ
𝑠𝜌
𝑌𝑒
(cid:19)

+

+

1
𝑣

1
𝜌

𝑑𝑌𝑒
𝑑𝑟
𝑑𝜌
𝑑𝑟

(cid:19)

1
𝑣

𝑑𝑣
𝑑𝑟

+

+

𝑠,𝑌𝑒
𝑐𝑠𝑌𝑒
Γ
𝑠𝑌𝑒
𝑌𝑒

𝑑𝑌𝑒
𝑑𝑟

,

(cid:18) 2
𝑟
𝑑𝑠
𝑑𝑟

(cid:18) 2
𝑟

𝑑𝜌
𝑑𝑟

+

1
𝑌𝑒

(cid:19)

(cid:18) 𝜕 ln 𝑃
𝜕 ln 𝑌𝑒

𝑠,𝜌

𝑑𝑌𝑒
𝑑𝑟

+

1
𝑌𝑒

(cid:19)

(cid:18) 𝜕 ln 𝑐𝑠
𝜕 ln 𝑌𝑒

𝑠,𝜌

𝑑𝑌𝑒
𝑑𝑟

(2.23)

(2.24)

with Γ𝑤𝑥

𝑦𝑧 = (𝜕 ln 𝑤/𝜕 ln 𝑥) 𝑦,𝑧. Substituting these into equation (2.22) and solving for 𝑑𝑣/𝑑𝑟 yields

the critical form of the momentum conservation equation,

(cid:20)

𝑣2 − 𝑐2

𝑠 +

𝑎1𝑣𝑐𝑠𝜔N
𝜌𝑣𝑔

(cid:16) 𝑐𝑠 − 𝑣
𝑣
𝑎1𝑣𝑐𝑠𝜔N
𝜌𝑣𝑔

Γ

− 2

𝑐𝑠 𝜌
𝑠𝑌𝑒
(cid:20) 𝑐𝑠 − 𝑣
𝑣𝑔

+

(cid:17)(cid:21) 𝑑𝑣
𝑑𝑟

= −

𝑣
𝜌

(cid:18) 𝑃
𝑠

Γ𝑃𝑠
𝜌𝑌𝑒

(cid:18) 1
𝑠

Γ

𝑐𝑠 𝑠
𝜌𝑌𝑒

𝑑𝑠
𝑑𝑟

+

1
𝑌𝑒

𝑐𝑠𝑌𝑒
𝑠𝜌

Γ

+

𝑑𝑠
𝑑𝑟
𝑑𝑌𝑒
𝑑𝑟

𝑃
𝑌𝑒

−

Γ𝑃𝑌𝑒
𝑠𝜌

(cid:19)

+

𝑑𝑌𝑒
𝑑𝑟
(cid:19)

2
𝑟

Γ

𝑐𝑠 𝜌
𝑠𝑌𝑒

+

1
𝑙𝑑

𝑣
𝑟

+

(cid:18)

2𝑐2

𝑠 −

(cid:18)

2
𝑟

1 −

(cid:19)

.

𝐺 𝑀
𝑟
(cid:19)(cid:21)

1
𝑎1

(2.25)

This form clearly shows the singularity, or critical point, in the momentum equation. In the absence

of wave effects (i.e., N = 0), the singularity occurs when 𝑣 = 𝑐𝑠, as shown by Thompson et al.

(2001). To obtain a physical wind solution, the right-hand side of the critical form equation must go

through zero exactly when 𝑣 = 𝑐𝑠. The inclusion of wave effects modifies the location of the critical

point but does not remove it. To simplify notation, we introduce the dimensionless 𝑓 functions:

𝑓1 = 1 −

,

𝑣2
𝑐2
𝑠
𝑎1𝜔N
𝜌𝑐𝑠𝑣𝑔
(cid:18) 𝑃
𝑟
𝑠
𝜌𝑐2
𝑠
𝑎1𝑟𝜔N
𝜌𝑐𝑠𝑣𝑔

(cid:16) 𝑐𝑠 − 𝑣
𝑣
𝑑𝑠
Γ𝑃𝑠
𝜌𝑌𝑒
𝑑𝑟
(cid:20) 𝑐𝑠 − 𝑣
𝑣𝑔

+

𝛿 𝑓1 = −

𝑓2 =

𝛿 𝑓2 = −

Γ

𝑐𝑠 𝜌
𝑠𝑌𝑒

(cid:17)

,

− 2

𝑃
𝑌𝑒
(cid:18) 1
𝑠

Γ𝑃𝑌𝑒
𝑠𝜌

Γ

𝑐𝑠 𝑠
𝜌𝑌𝑒

𝑑𝑌𝑒
𝑑𝑟
𝑑𝑠
𝑑𝑟

(cid:19)

− 2 +

+

1
𝑌𝑒

𝑐𝑠𝑌𝑒
𝑠𝜌

Γ

,

𝐺 𝑀
𝑟𝑐2
𝑠
𝑑𝑌𝑒
𝑑𝑟

−

2
𝑟

Γ

𝑐𝑠 𝜌
𝑠𝑌𝑒

(cid:19)

+

1
𝑙𝑑

+

2
𝑟

(cid:18)

1 −

(cid:19)(cid:21)

,

1
𝑎1

so that equation (2.25) becomes

= 𝑓2 + 𝛿 𝑓2,

( 𝑓1 + 𝛿 𝑓1)

𝑑 ln 𝑣
𝑑 ln 𝑟

14

(2.26)

(2.27)

where the wave contributions are entirely contained in the 𝛿-terms.

2.1.4 Energy Conservation

The energy conservation equation can also be expanded. In steady-state and spherical symmetry,

equation (2.7) becomes

𝑣

𝑑𝜖
𝑑𝑟

= −𝑣𝑃

𝑑
𝑑𝑟

(cid:19)

(cid:18) 1
𝜌

−

1
𝜌

(cid:2)∇ · (cid:0)(cid:174)𝑣𝑔E + 𝚷 · (cid:174)𝑣(cid:1) − (cid:174)𝑣 · ∇ · 𝚷(cid:3) + (cid:164)𝑞𝜈.

(2.28)

The total derivative of the specific internal energy is given by the first law of thermodynamics,

𝑑𝜖 = 𝑇 𝑑𝑠 − 𝑃𝑑 (1/𝜌), so equation (2.28) becomes

𝑣

𝑑𝑠
𝑑𝑟

= −

1
𝜌𝑇

(cid:2)∇ · (cid:0)(cid:174)𝑣𝑔E + 𝚷 · (cid:174)𝑣(cid:1) − (cid:174)𝑣 · ∇ · 𝚷(cid:3) +

(cid:164)𝑞𝜈
𝑇

.

Evaluating the wave stress terms, this becomes

𝑣

𝑑𝑠
𝑑𝑟

= −

(cid:20)

1
𝜌𝑇

∇ · (cid:0)(cid:174)𝑣𝑔E(cid:1) −

(𝛾 + 1)E

1
2

𝑑𝑣
𝑑𝑟

− (𝛾 − 1)

(cid:21)

E𝑣
𝑟

+

(cid:164)𝑞𝜈
𝑇

.

(2.29)

(2.30)

This equation shows the two main energy sources in the wind. One of these, neutrino heating,

is necessarily included in every similar model of the NDW. Neutrinos interacting with the wind

material, primarily via the charged-current 𝜈𝑒 + 𝑛 → 𝑝 + 𝑒− and ¯𝜈𝑒 + 𝑛 → 𝑝 + 𝑒− reactions, deposit

some of their energy as heat. This is the dominant energy source in the wind. Wave effects will also

contribute to heating the wind, via shock heating and also via the wave stress. Applying equation

(2.18), this becomes

𝑣

𝑑𝑠
𝑑𝑟

𝑣𝑔E
𝜌𝑇

=

+

(cid:20) (cid:18) 1
𝑑 ln 𝜔0
𝑙𝑑
𝑑𝑟
(cid:18) 𝑑 ln 𝑣𝑔
𝑑𝑟

−

+

(cid:19)

𝑑 ln 𝑐𝑠
𝑑𝑟

+

𝛾 + 1
2𝑣𝑔

𝑑𝑣
𝑑𝑟

+

(𝛾 − 1)𝑣
𝑟𝑣𝑔

(cid:19) (cid:21)

+

(cid:164)𝑞𝜈
𝑇

.

(2.31)

In the isentropic limit, i.e., 𝑙𝑑 → ∞ and (cid:164)𝑞 → 0, it can be shown that the logarithmic terms exactly

cancel with the wave stress terms, so that 𝑑𝑠/𝑑𝑟 = 0 self-consistently. This also ensures that, in

the absence of dissipation prior to shock formation, there is negligible change in entropy due to

the waves. Because the waves shock well before the transonic point in our simulations, 𝑣 ≪ 𝑐𝑠,

𝑣𝑔 ≈ 𝑐𝑠, and 𝜔0 ≈ 𝜔 at the relevant radii, so the derivatives of 𝑣𝑔 and 𝑐𝑠 cancel, the derivative of 𝜔0

15

vanishes, and the wave stress terms will be negligible. For computational simplicity, therefore, we

neglect all but the dissipation length term, which will be dominant when shock formation occurs.

Doing so will only lower the entropy of the wind, so our results in this work should be considered

a lower bound on the wind entropy and ensuing nucleosynthesis. Inserting the definition of N , this

leaves us with

as the final form of the entropy equation. The remaining heating terms, from neutrinos and from

𝑣

𝑑𝑠
𝑑𝑟

=

𝑐𝑠
𝜌𝑇 𝑙𝑑

𝜔N +

(cid:164)𝑞𝜈
𝑇

(2.32)

wave shocks, will be examined further in turn.

2.1.4.1 Neutrino heating and cooling

The primary energy source in the wind is heating by neutrinos. The derivation of the reaction

and heating/cooling rates is quite involved, and rather than attempting to repeat it, we refer the

reader to appendix C of Lippuner & Roberts (2017). Neutrino and antineutrino capture reactions

will heat the wind, as the energy of the (anti) neutrino is added to the wind material. (Anti) neutrino

emission reactions, on the other hand, will cool the wind. The net neutrino heating is given by

(cid:164)𝑞𝜈 = ( (cid:164)𝑞 ¯𝜈𝑒 − (cid:164)𝑞𝑒− )𝑌𝑝 + ( (cid:164)𝑞𝜈𝑒 − (cid:164)𝑞𝑒+)𝑌𝑛.

(2.33)

The (cid:164)𝑞𝑖 represent the heating/cooling rates1 due to the following charged-current neutrino reactions:

𝜈𝑒 + 𝑛 ←→ 𝑒− + 𝑝

¯𝜈𝑒 + 𝑝 ←→ 𝑒+ + 𝑛.

(2.34)

Because the neutrino flux experienced by the wind falls off as 𝑟 2, the heating rates (and thus

the entropy deposition) are very high while the wind is close to the PNS surface. Cooling via

neutrino emission is fairly weak throughout the wind, so if only neutrino heating contributions are

considered, the entropy of the wind quickly reaches an asymptotic value. Because the electron

fraction of the wind also asymptotes early (as discussed in section 2.1.5), we approximate 𝑌𝑝 = 𝑌𝑒

and 𝑌𝑛 = 1 − 𝑌𝑒, which yields

(cid:164)𝑞𝜈 = ( (cid:164)𝑞𝜈𝑒 − (cid:164)𝑞𝑒+) + ( (cid:164)𝑞 ¯𝜈𝑒 + (cid:164)𝑞𝑒+ − (cid:164)𝑞𝑒− − (cid:164)𝑞𝜈𝑒)𝑌𝑒

(2.35)

1We have made the notation change (cid:164)𝑞𝑖 = | (cid:164)𝜖𝑖 | from equation C19 of Lippuner & Roberts (2017).

16

as the total energy source term due to neutrino interactions.

2.1.4.2 Gravito-acoustic wave heating

Gravito-acoustic waves provide a secondary heat source in the wind as they steepen into shocks

and deposit their energy. These waves propagate through a fluid element at the speed of sound.

Because the waves are density perturbations and the sound speed is density-dependent, the peaks

of the waves travel slightly faster than the troughs. This causes the waves to steepen into a sawtooth

shape and form shocks, which efficiently thermalize the wave energy. Shocks will form at a radius

𝑅𝑠 given by the time needed for the peak to catch up with a preceding trough (Mihalas & Mihalas,

1984):

(𝛾 + 1)𝑐−1
𝑠

1
4

∫ 𝑅𝑠

0

𝑢0(𝑟′)𝑑𝑟′ =

𝜋𝑐𝑠
2𝜔

,

(2.36)

which, with 𝑢0 = √︁2E/𝜌 representing the velocity perturbation of the waves (Jacques, 1977),

becomes

∫ 𝑅𝑠

√︄

0

2𝑐𝑠𝜔N
𝑣𝑔 𝜌

𝑑𝑟′ =

2𝜋𝑐2
𝑠
𝜔(𝛾 + 1)

.

(2.37)

We can then use the shock jump conditions in combination with a model for the shock profile

to obtain an estimate for 𝑙𝑑. A planar shock propagating in an ideal gas will be governed by the

following shock jump conditions (Zel’dovich & Raizer, 1967):

𝑀 2

0 ≡

(cid:19) 2

(cid:18) 𝑣0
𝑐𝑠,0

𝜌1
𝜌0

=

=

𝑠1 − 𝑠0 = 𝑐𝑣 ln

(𝛾 + 1)𝑃0 + (𝛾 − 1)𝑃0
2𝛾𝑃0
(𝛾 + 1)𝑃0 + (𝛾 − 1)𝑃0
(𝛾 − 1)𝑃0 + (𝛾 + 1)𝑃0
(cid:18) 𝜌0
𝜌1

(cid:20) 𝑃1
𝑃0

(cid:19) 𝛾(cid:21)

,

(2.38)

(2.39)

(2.40)

representing the upstream Mach number and the compression and entropy changes across the shock.

Upstream and downstream quantities are indicated by a "0" and "1" subscript, respectively. In a

weak shock, the fractional pressure perturbation is small, i.e., 𝑝 ≡ (𝑃1 − 𝑃0)/𝑃0 ≪ 1. Expanding

equations (2.38)–(2.40) to lowest order in 𝑝 and expressing them in terms of the reduced Mach

17

number 𝑚 ≡ 𝑀 2

0 − 1, we have

𝑚 =

− 1 =

𝜌1
𝜌0
𝑠1 − 𝑠0 =

𝑝 ≪ 1

𝛾 + 1
2𝛾
2𝑚
𝑝
𝛾
𝛾 + 1
2𝑐𝑣𝛾(𝛾 − 1)
3(𝛾 + 1)2

=

(2.41)

(2.42)

(2.43)

𝑚3.

Multiplying the entropy jump by the temperature 𝑇, and using the relation (𝛾 − 1)𝑐𝑣𝑇 = 𝑃/𝜌, we

obtain an expression for the heat deposition per unit mass due to a single shock:

Δ𝑞 = 𝑇 (𝑠1 − 𝑠0) =

2𝛾𝑃0
3𝜌(𝛾 + 1)2

𝑚3.

(2.44)

The change in heat per unit area 𝑄 as a fluid element traverses the shocks is given by Δ𝑞 multiplied

by the mass flux 𝜌𝑣0:

𝐷𝑄
𝐷𝑡

= 𝑣0

𝑑𝑄
𝑑𝑟

=

2𝛾𝑃0𝑣0
3(𝛾 + 1)2

𝑚3,

(2.45)

where we have applied the steady-state form of equation (2.13).

We now seek an expression for the mean energy per area 𝑄 carried by the waves. The phase

velocity 𝑣 𝑝 as a function of position (𝑧) in the wave is given by

𝑣 𝑝 (𝑧) = 𝑐𝑠 +

𝛾 + 1
2

𝑢(𝑧)

(2.46)

where 𝑢(𝑧) is the velocity perturbation of the wave. For an initially sinusoidal wave packet,

𝑢(𝑧) = 𝑢0 sin(𝑘 𝑧)/2. The variation in 𝑢(𝑧) causes the waves to steepen into a sawtooth profile,

with velocity perturbation

𝑢(𝑧) =

1
2

𝑢0

(cid:18) 2𝑧
𝜆

(cid:19)

,

− 1

0 < 𝑧 ≤ 𝜆.

The shock, located at the leading edge of the wave packet, moves at a speed

𝑣0 = 𝑣 𝑝 (𝜆) = 𝑐𝑠 +

𝛾 + 1
4

𝑢0.

We can then express the reduced Mach number as

𝑚 =

(cid:19) 2

(cid:18) 𝑣0
𝑐𝑠

− 1 ≈

𝛾 + 1
2

𝑢0
𝑐𝑠

.

18

(2.47)

(2.48)

(2.49)

A propagating wave will do work at a rate (𝑃1 − 𝑃0)𝑢 ≈ 𝛾𝑃0𝑢2/𝑐𝑠. Integrated over one period

(using 𝑧 = 𝑐𝑠𝑡), the energy flux transmitted by the wave is

𝑄 =

𝛾𝑃0
𝑐𝑠

∫ 𝜆/𝑐𝑠

0

𝑢2𝑑𝑡 =

𝛾𝑃0𝑢2
0
12𝑐2
𝑠

𝜆 =

𝛾𝑃0
3(𝛾 + 1)2

𝜆𝑚2,

(2.50)

where we use equation (2.49). Note that 𝑄 = E/𝜆, so this also provides an expression for the energy

density of the waves. Using equations (2.45) and (2.50), we can write down the characteristic

dissipation length of the waves:

𝑙𝑑 =

(cid:19) −1

(cid:18) 1
𝑄

𝑑𝑄
𝑑𝑟

=

𝜆
2𝑚

𝜋𝑐𝑠
𝑚𝜔

.

=

We can thus write the entropy evolution equation (2.29) as

𝑣

𝑑𝑠
𝑑𝑟

=

𝑆𝐸
𝑇

,

where

(cid:20)

𝑆𝐸 =

( (cid:164)𝑞𝜈𝑒 − (cid:164)𝑞𝑒+) + ( (cid:164)𝑞 ¯𝜈𝑒 + (cid:164)𝑞𝑒+ − (cid:164)𝑞𝑒− − (cid:164)𝑞𝜈𝑒)𝑌𝑒 +

(cid:21)

𝜔N

𝑐𝑠
𝜌𝑙𝑑

(2.51)

(2.52)

(2.53)

represents the full energy source term including both wave and neutrino contributions, with 𝑙𝑑

given by equation (2.51) and the (cid:164)𝑞𝑖 calculated according to Lippuner & Roberts (2017).

2.1.4.3 Scaling relationships

Using the equations above, we can derive a number of scaling relationships and order-of-

magnitude estimates to describe what changes we should observe in the wind as the wave properties

are changed. Inserting equation (2.50) into equation (2.51) with the definition of N yields

𝑙𝑑 =

𝜋𝑐𝑠
𝜔(𝛾 + 1)

(cid:18) 𝛾𝑣𝑔𝑃
3𝑐𝑠𝜔N

(cid:19) 1/2

.

(2.54)

In the limit 𝑣 ≪ 𝑐𝑠, and assuming a radiation-dominated 𝛾 = 4/3, we can use this to obtain an

order-of-magnitude estimate of the dissipation length:

𝑙𝑑 ≈ (1.64 × 106 cm)

(cid:19) 1/2

𝑐𝑠,9
𝜔3

(cid:18) 𝑈
10E

(2.55)

where 𝜔3 and 𝑐𝑠,9 denote the wave frequency in units of 103 s−1 and the sound speed in units of

109 cm s−1, and 𝑈 = 𝜌𝜖 ≈ 𝑃/(𝛾 − 1) is the internal energy density of the wind. The factor of 10

19

accounts for the expected difference in scale between the wave and internal energy densities. In

terms of the wave luminosity emerging from the evanescent region prior to dissipation 𝐿𝑤 (see

Equation 1.3), the dissipation length will scale as

𝑙𝑑 ∝ 𝐿−1/2

𝑤 𝜔−1.

(2.56)

Approximating density and pressure with power law expressions derived from our numerical results,

we estimate the radius of shock formation will scale as

𝑅𝑠 ∝ 𝐿−2/11

𝑤

𝜔−4/11.

(2.57)

Therefore, as the wave luminosity is increased, we expect the radius of shock formation to move

inward, making it more likely that the energy deposited by the shocks will affect seed nucleus

formation. The dissipation length will respond more strongly, causing the waves to deposit their

energy faster and increasing the effect on seed formation. Increasing the wave frequency will have

a similar and stronger effect: the shocks will form earlier and deposit their energy more quickly at

higher frequencies.

2.1.5 Lepton Number Conservation

Though it does not appear in the standard Euler equations, tracking the evolution of the electron

fraction within a fluid element will be necessary to obtain accurate nucleosynthesis results. We can

write down the material derivative

𝐷𝑌𝑒
𝐷𝑡

= (𝜆𝜈𝑒 + 𝜆𝑒+)𝑌𝑛 − (𝜆 ¯𝜈𝑒 + 𝜆𝑒− )𝑌𝑝

(2.58)

where the 𝜆𝑖 represent the rates of the neutrino reactions in equation (2.34). Again, these rates

rapidly fall off with the neutrino flux experienced by the fluid element as it moves outward in the

wind, causing the electron fraction to asymptote to a certain value very early on. In a wind purely

affected by neutrinos (no waves, magnetic fields, etc.), 𝑌𝑒 approaches the equilibrium value of

equation (1.1). Taking the approximation 𝑌𝑛 = 1 − 𝑌𝑝 and 𝑌𝑝 = 𝑌𝑒 and applying equation (2.13),

we find

𝑣

𝑑𝑌𝑒
𝑑𝑟

= 𝜆𝜈𝑒 + 𝜆𝑒+ − (𝜆 ¯𝜈𝑒 + 𝜆𝑒− + 𝜆𝜈𝑒 + 𝜆𝑒+)𝑌𝑒 = 𝑆𝑌𝑒

(2.59)

20

for the final form of the lepton number conservation equation.

2.1.6 Corrections from General Relativity

The extreme density of the PNS makes relativistic corrections important for accurately modeling

the wind (Cardall & Fuller, 1997; Otsuki et al., 2000). Because the total mass of the wind is far less

than the PNS mass, we treat the system in the Schwarzschild limit, in which the mass of the system

is treated as a point source at the center of the PNS. The derivation is quite involved; following a

similar process to Thompson et al. (2001), we find adjusted forms of the mass, momentum, energy,

and lepton number conservation equations (absent wave effects):

(cid:164)𝑀 = 4𝜋𝑟 2𝑒Λ𝑊𝑣𝜌
Γ𝑃𝑠
𝜌𝑌𝑒
𝑊 2ℎ

(cid:32) 𝑃
𝑠

= −

𝑑𝑣
𝑑𝑟

𝑣
𝜌

(𝑣2 − 𝑐2
𝑠)

𝑑𝑠
𝑑𝑟

+

𝑃𝑌𝑒
Γ
𝜌𝑠
𝑊 2ℎ

𝑑𝑌𝑒
𝑑𝑟

𝑃
𝑌𝑒

(cid:33)

𝑣
𝑟

(cid:18) 2𝑐2
𝑠
𝑊 2

+

𝐺 𝑀
𝑟 2

𝑐2 − 𝑐2
𝑠
𝑐2𝑒2Λ𝑊 2

(cid:19)

−

𝑣

𝑑𝑠
𝑑𝑟
𝑑𝑌𝑒
𝑑𝑟

𝑣

=

=

𝑆𝐸
𝑒Λ𝑊𝑇
𝑆𝑌𝑒
𝑒Λ𝑊

.

For the mass, energy, and lepton number conservation equations, the corrections take the form

of a simple prefactor in terms of the Lorentz factor 𝑊 =

(cid:16)

1 − 𝑣2
𝑐2

(cid:17) − 1

2 and the metric function

√︃

𝑒Λ =

1 − 2𝐺 𝑀

𝑟𝑐2 . The critical form of the momentum equation is somewhat more involved, using
the dimensionless form of the specific enthalpy ℎ = 𝜌(𝑐2 + 𝑒) + 𝑃/𝜌𝑐2, with 𝑒 representing the

energy density of the wind. Ignoring the wave contribution terms, we can recognize the form of

equation (2.25), and so we can adjust our definition of 𝑓2 from equation (2.26) to incorporate the

relativistic corrections:

𝑓2 =

𝑟
𝜌𝑐2
𝑠

(cid:32) 𝑃
𝑠

Γ𝑃𝑠
𝜌𝑌𝑒
𝑊 2ℎ

𝑑𝑠
𝑑𝑟

(cid:33)

𝑃𝑌𝑒
Γ
𝑠𝜌
𝑊 2ℎ

𝑑𝑌𝑒
𝑑𝑟

𝑃
𝑌𝑒

+

−

2
𝑊 2

+

𝐺 𝑀
𝑟𝑐2
𝑠

𝑐2 − 𝑐2
𝑠
𝑐2𝑒2Λ𝑊 2

.

(2.60)

Ideally, we would also include relativistic effects on the wave contribution terms 𝛿 𝑓1 and 𝛿 𝑓2. We do

not for two reasons. First, the derivation of the wave stress contributions to the momentum equation

we follow from Jacques (1977) relies on the eikonal (WKB) approximation. The additional accuracy

to be gained from including relativistic effects would not outweigh the uncertainty introduced by this

21

approximation. Second, the derivation we follow relies on a complicated formalism to differentiate

the waves from the background fluid, and (to our knowledge) no one has yet worked out a general-

relativistic version of this formalism. The final form of the conservation equations is thus

(cid:164)𝑀 = 4𝜋𝑟 2𝑒Λ𝑊𝑣𝜌

( 𝑓1 + 𝛿 𝑓1)

𝑑 ln 𝑣
𝑑 ln 𝑟
𝑑𝑠
𝑣
𝑑𝑟
𝑑𝑌𝑒
𝑑𝑟
𝑑 ln L
𝑑𝑟

𝑣

with

= 𝑓2 + 𝛿 𝑓2
(cid:20)

=

=

1
𝑒Λ𝑊𝑇
1
𝑒Λ𝑊
1
𝑙𝑑

= −

( (cid:164)𝑞𝜈𝑒 − (cid:164)𝑞𝑒+) + ( (cid:164)𝑞 ¯𝜈𝑒 + (cid:164)𝑞𝑒+ − (cid:164)𝑞𝑒− − (cid:164)𝑞𝜈𝑒)𝑌𝑒 +

(cid:21)

L
4𝜋𝑟 2𝜌𝑙𝑑

(cid:2)𝜆𝜈𝑒 + 𝜆𝑒+ − (𝜆 ¯𝜈𝑒 + 𝜆𝑒− + 𝜆𝜈𝑒 + 𝜆𝑒+)𝑌𝑒(cid:3)

(2.61)

(2.62)

(2.63)

(2.64)

(2.65)

𝑓1 = 1 −

𝛿 𝑓1 = −

𝑣2
𝑐2
𝑠
𝑎1L
4𝜋𝑟 2𝜌𝑐𝑠𝑣2
𝑔
(cid:32) 𝑟
Γ𝑃𝑠
𝑃
𝜌𝑌𝑒
𝑠
𝑊 2ℎ
𝜌𝑐2
𝑠

𝑓2 =

𝛿 𝑓2 = −

𝑎1L
4𝜋𝑟 2𝜌𝑐𝑠𝑣2
𝑔

(cid:16) 𝑐𝑠 − 𝑣
𝑣

Γ

𝑐𝑠 𝜌
𝑠𝑌𝑒

(cid:17)

− 2

+

𝑑𝑠
𝑟
𝑌𝑒
𝑑𝑟
(cid:20) 𝑐𝑠 − 𝑣
𝑣𝑔

𝑃𝑌𝑒
Γ
𝑠𝜌
𝑊 2ℎ
(cid:18) 𝑟
𝑠

Γ

(cid:33)

𝑑𝑌𝑒
𝑑𝑟

−

2
𝑊 2

+

𝑐2 − 𝑐2
𝑠
𝑐2𝑒2Λ𝑊 2

𝐺 𝑀
𝑟𝑐2
𝑠
𝑑𝑌𝑒
𝑑𝑟

𝑐𝑠 𝑠
𝜌𝑌𝑒

𝑑𝑠
𝑑𝑟

+

𝑟
𝑌𝑒

𝑐𝑠𝑌𝑒
𝑠𝜌

Γ

− 2Γ

𝑐𝑠 𝜌
𝑠𝑌𝑒

(cid:19)

+

𝑟
𝑙𝑑

(cid:21)

+ 2(1 − 1/𝑎1)

(2.66)

and the 𝜆𝑖 and (cid:164)𝑞𝑖 calculated according to Lippuner & Roberts (2017) assuming a zero chemical

potential Fermi-Dirac spectrum.

2.1.7 Wind Termination Shock

The NDW does not take place in a vacuum—eventually the fast-moving, transonic wind will

impact slower-moving material behind the primary supernova shock. We follow Arcones & Thiele-

mann (2012) in our treatment of this. Once the wind reaches a certain radius 𝑅WT, we employ the

relativistic Rankine-Hugoniot shock conditions:

𝑣0𝜌0𝑊0 = 𝑣1𝜌1𝑊1

𝑊 2
0

ℎ0𝜌0𝑣2

0 + 𝑃0 = 𝑊 2
1

ℎ1𝜌1𝑣2

1 + 𝑃1

𝑊0ℎ0 = 𝑊1ℎ1

22

(2.67)

(2.68)

(2.69)

where the subscript "0" denotes the unshocked wind quantities, and "1" the post-shock quantities.
For the first second post-shock (calculated with Δ𝑡 = ∫ 𝜓2
𝜓1

(𝜕𝑣/𝜕𝑟) −1𝑑𝜓, see section 2.2.3), we

hold the density constant while decreasing velocity as 𝑟 −2. After this, we assume the wind obeys a

purely homologous outflow, with constant velocity and density decreasing as 𝑟 −2 through the end

of the simulation.

2.2 Computational Method

The physics of our model for the NDW is fully contained in the equations above. To obtain

the properties of the steady-state wind, we will integrate a form of equations (2.61)–(2.65) from

the surface of the PNS, where the wind is launched, to a far radius of 1010 cm. There are some

important factors that must be considered, however, in order to translate these equations into a

robust numerical simulation.

2.2.1 EOS Considerations

Our equations are closed by the Helmholtz equation of state (Timmes & Swesty, 2000) for

radiation-dominated material. This equation of state was derived specifically for astrophysical

scenarios like the NDW, in which electrons and positrons are arbitrarily relativistic and degenerate.

This EOS is tabulated, however, with respect to density, temperature, and electron fraction, whereas

the equations we have derived describe the entropy rather than the temperature. To make use of the

tabulated Helmholtz EOS, we must derive an equation for the temperature evolution of the wind.

We begin with

𝑑𝑇
𝑑𝑟

(cid:19)

(cid:18) 𝜕𝑇
𝜕𝑠

=

𝑑𝑠
𝑑𝑟

(cid:19)

(cid:18) 𝜕𝑇
𝜕 𝜌

+

𝑑𝜌
𝑑𝑟

+

(cid:19)

(cid:18) 𝜕𝑇
𝜕𝑌𝑒

𝑠,𝜌

𝑑𝑌𝑒
𝑑𝑟

.

𝑠,𝑌𝑒

𝜌,𝑌𝑒

Substituting from equations (2.61)–(2.65), we have

𝑑𝑇
𝑑𝑟

= Γ𝑇 𝑠
𝜌𝑌𝑒

𝑆𝐸
𝑣𝑒Λ𝑊𝑇

− 𝜌Γ

𝑇 𝜌
𝑠𝑌𝑒

(cid:18)𝑊 2
𝑣

𝑑𝑣
𝑑𝑟

+

2
𝑟

+

(cid:19)

𝐺 𝑀
𝑟 2𝑒2Λ

+ Γ𝑇𝑌𝑒
𝑠𝜌

𝑆𝑌𝑒
𝑣𝑒Λ𝑊

and inserting equation (2.62)

𝑑𝑇
𝑑𝑟

= Γ𝑇 𝑠
𝜌𝑌𝑒

𝑆𝐸
𝑣𝑒Λ𝑊𝑇

− 𝜌Γ

𝑇 𝜌
𝑠𝑌𝑒

(cid:18)𝑊 2
𝑟

𝑓2 + 𝛿 𝑓2
𝑓1 + 𝛿 𝑓1

+

2
𝑟

+

𝐺 𝑀
𝑟 2𝑐2

𝑐2 − 𝑐2
𝑠
𝑒2Λ

(cid:19)

+ Γ𝑇𝑌𝑒
𝑠𝜌

𝑆𝑌𝑒
𝑣𝑒Λ𝑊

.

(2.70)

(2.71)

(2.72)

This will replace equation (2.63) as one of the equations we integrate, and the entropy will be

calculated through the tabulated EOS instead.

23

2.2.2

Initial Conditions

To obtain a solution to the differential equations we have found, we require a set of initial

conditions. Specifically, we require an initial velocity, temperature, electron fraction, and wave

action. The initial velocity we leave as a free parameter, calculated such that the wind becomes

transonic at the critical point (see section 2.2.3 below). Qian & Woosley (1996) argue that at the

surface of the PNS, neutrino heating and cooling will be in equilibrium, i.e.,

(cid:164)𝑞𝜈𝑒 + ( (cid:164)𝑞 ¯𝜈𝑒 + (cid:164)𝑞𝑒+)𝑌𝑒 = (cid:164)𝑞𝑒+ + ( (cid:164)𝑞𝑒− + (cid:164)𝑞𝜈𝑒)𝑌𝑒.

(2.73)

The (cid:164)𝑞𝑖 are temperature dependent, so we employ a rootfinding algorithm to simultaneously calculate

the initial temperature and electron fraction based on this condition. Finally, the initial effective

wave luminosity L is simply equal to the specified wave luminosity 𝐿𝑤.

A number of parameters must also be specified before integrating the equations. These include

the PNS mass and radius, the electron neutrino and antineutrino luminosities and energies, the

location of the wind termination shock, and the wave luminosity and frequency. We can select

these arbitrarily from a chosen parameter space, or extract them from a preexisting data set.

2.2.3 Circumventing the Critical Point

As the evolution equations indicate, a singularity exists when 𝑓1 + 𝛿 𝑓1 = 0, commonly referred

to as the transsonic point. A physical wind solution must become trans-sonic in order to escape

the PNS gravity well, so the singularity must be handled by our code. A singularity will cause

problems for a numerical integrator, so we must find a way to effectively remove or circumvent it

to make the equations tractable. We do so by introducing an auxiliary integration variable 𝜓, and

splitting equation (2.62) into separate differential equations for 𝑣 and 𝑟 in terms of 𝜓:

(2.74)

(2.75)

𝑑 ln 𝑟
𝑑𝜓
𝑑 ln 𝑣
𝑑𝜓

= 𝑓1 + 𝛿 𝑓1

= 𝑓2 + 𝛿 𝑓2.

24

We can similarly recast the other evolution equations:

𝑑 ln 𝑇
𝑑𝜓

=

𝑓1 + 𝛿 𝑓1
𝑣𝑒Λ𝑊𝑇

(cid:18)

Γ𝑇 𝑠
𝜌𝑌𝑒

𝑆𝐸
𝑇

+ Γ𝑇𝑌𝑒

𝑠𝜌 𝑆𝑌𝑒

(cid:19)

−

𝜌
𝑇

( 𝑓1 + 𝛿 𝑓1)Γ

𝑇 𝜌
𝑠𝑌𝑒

(cid:18)𝑊 2
𝑟

𝑓2 + 𝛿 𝑓2
𝑓1 + 𝛿 𝑓1

+

2
𝑟

+

𝐺 𝑀
𝑟 2𝑐2

𝑐2 − 𝑐2
𝑠
𝑒2Λ

(cid:19)

( 𝑓1 + 𝛿 𝑓1)

𝑆𝑌𝑒
𝑒Λ𝑊

( 𝑓1 + 𝛿 𝑓1)

(2.76)

(2.77)

(2.78)

𝑑 ln 𝑌𝑒
𝑑𝜓
𝑑 ln L
𝑑𝜓

=

= −

𝑟
𝑣𝑌𝑒
𝑟
𝑙𝑑

To determine where the waves begin to shock, we also must track the value of the integral in

equation (2.37). We define

so that

and thus

𝐼 =

√︄

∫ 𝑟

0

2𝑐𝑠𝜔N
𝑣𝑔 𝜌

𝑑𝑟′

√︄

𝑑𝐼
𝑑𝑟

=

2𝑐𝑠𝜔N
𝑣𝑔 𝜌

√︄

𝑑𝐼
𝑑𝜓

=

2𝑐𝑠𝜔N
𝑣𝑔 𝜌

𝑓1𝑟.

(2.79)

(2.80)

(2.81)

When 𝐼 is equal to the right-hand side of equation (2.37), shock heating begins in the wind. We

model this via the dissipation length 𝑙𝑑 through an interpolation function:

(cid:16)

𝐽 = 2

1 + 𝑒2𝜋𝑐2

𝑠/𝐼𝜔(𝛾+1)(cid:17) −1

,

so that 𝐽 approaches 1 as 𝐼 approaches the shock condition. We then treat 𝑙𝑑 as

𝑙𝑑 =

(cid:18) 1 − 𝐽
10200

+

𝐽
𝑙𝑑,phys

(cid:19) −1

,

(2.82)

(2.83)

where 𝑙𝑑,phys is given by the expression in equation (2.54), and 10200 is a very large number

representing an infinite dissipation length. At small radii, 𝑙𝑑 is essentially infinite and no shock

heating occurs. Once the wind approaches the point where the waves should begin shocking, 𝑙𝑑

smoothly transitions to its physical value. A smooth transition is helpful for the integrator and is

perhaps more realistic than a sharp transition.

With our equations recast in terms of 𝜓, no infinities will be encountered by the integrator, but

we can still identify the trans-sonic trajectory by checking if 𝑓1 + 𝛿 𝑓1 and 𝑓2 + 𝛿 𝑓2 pass through zero

at the same point.

25

Figure 2.1 The solution families for a simplified wind model without wave contributions. Blue
curves show unphysical double-valued solutions; red curves show breeze solutions that eventually
fall back onto the PNS; and the black curve shows the trans-sonic wind solution.

2.2.4 Numerical Integration Method

To obtain robust profiles for these differential equations, we employ a fourth-order singly-

diagonally implicit Runge-Kutta numerical integration method (Kennedy & Carpenter, 2016) with

an adaptive step size in 𝜓. Solutions to our differential equation set fall into 3 categories (illustrated

in Figure 2.1): unphysical solutions, breeze solutions, and the desired trans-sonic solution. An

overly fast initial velocity leads to an unphysical double-valued solution as the initially positive

𝑑 ln 𝑟/𝑑𝜓 becomes negative before 𝑑 ln 𝑣/𝑑𝜓, curving the solution back to 𝑟 = 0. A slow initial

velocity leads to 𝑑 ln 𝑣/𝑑𝜓 becoming negative first, and the wind slows down and does not become

unbound. Given sufficient time, the material would fall back to the surface of the PNS in a breeze

solution. For a given set of PNS parameters, there exists exactly one initial velocity value for which

the solution becomes transsonic and the wind material becomes unbound from the PNS. We use a

rootfinder to determine this velocity, comparing the values of 𝑑 ln 𝑟/𝑑𝜓 and 𝑑 ln 𝑣/𝑑𝜓 when one

26

100101r/RPNS102101100v/cspasses through zero. When they cross zero simultaneously, because even machine precision is not

sufficiently precise to obtain the transsonic solution, we run the integration taking the absolute values

of 𝑓1 + 𝛿 𝑓1 and 𝑓2 + 𝛿 𝑓2 instead of their signed values. This generates a machine-precision accurate

transsonic solution. We integrate this solution to a maximum radius of 𝑟 = 1010 cm, transitioning

to a homologous outflow when the wind reaches the termination shock radius 𝑅WT. This yields

radius, velocity, temperature, and density profiles describing the wind, and other quantities such as

the entropy can be extracted as well. We also track the various neutrino reaction rates for use in

nucleosynthesis calculations.

2.2.5 Nucleosynthesis Calculations

Our goal in this work is to determine the impact of wave effects on the nucleosynthetic processes

at work in the NDW. To that end, we employ the SkyNet reaction network (Lippuner & Roberts,

2017). SkyNet takes density and temperature profiles as a function of time, and tracks the evolution

of over 7000 different nuclides based on their reaction rates. Reaction rates are taken from the

REACLIB library, and include strong, weak, symmetric fusion, and spontaneous fusion reactions.

Inverse reactions are calculated via detailed balance, and charged-current neutrino reactions on

nucleons can optionally be included as well. To obtain time-dependent temperature and density
profiles from our steady-state models, we take 𝑡 (𝜓) = ∫ 𝜓

(𝜕𝑣/𝜕𝑟) −1. These profiles are then

0

extended to late times (𝑡 = 109 s) with a 𝑡−3 power law to capture nuclear decays. SkyNet takes

these profiles and returns an array of nuclide abundances for each time step it calculates, yielding

both a final abundance distribution for the nuclides formed in the wind as well as a time evolution

of those abundances.

27

CHAPTER 3

PROTO-NEUTRON STAR CONVECTION AND THE R-PROCESS

The contents of this chapter are adapted from Nevins & Roberts (2023), which has been published

in The Monthly Notices of the Royal Astronomical Society.

3.1 Background

The origins of the various chemical elements is one of the fundamental questions of nuclear

astrophysics. In their seminal paper, Burbidge et al. (1957) proposed rapid neutron captures onto

heavy (𝐴 ≥ 12) seed nuclei, followed by 𝛽-decays back toward stability, in a high-energy, high-

neutron density environment as the source of many of the heaviest elements found in the solar

system. The site of this “r-process” has been a topic of much study in recent decades. The visual

counterpart to GW170817 provided exciting evidence of r-processing in the ejecta of a binary

neutron star merger (Drout et al., 2017), but several studies have cast doubt on compact object

mergers as the sole r-process site, with some form of supernovae favored as an additional site (Côté

et al., 2019; Cavallo et al., 2021; Kobayashi et al., 2023; Van der Swaelmen et al., 2023). Many

studies have also noted discrepancies between the abundance distributions of the heaviest (third-

peak) r-process elements and lighter (first-peak) elements, indicating perhaps an additional weak

r-process site that only forms lighter elements (Arcones & Montes, 2011). The neutrino-driven

wind was theorized to be a promising site for r-processing (Woosley & Baron, 1992), and the

simulations of Woosley et al. (1994) catapulted it into the spotlight as a leading candidate for the

primary site of the r-process. These simulations found wind entropies in excess of 400 𝑘B baryon−1

with an electron fraction of 𝑌𝑒 ≈ 0.4, which allowed a robust, solar-like r-process to proceed.

Subsequent work, however, failed to reproduce these optimistic results, consistently yielding

entropies a factor of 2–3 smaller than those initially reported (Witti et al., 1994; Qian & Woosley,

1996; Otsuki et al., 2000; Thompson et al., 2001). Detailed theoretical work by Hoffman et al.

(1997) determined an approximate criterion for strong, third-peak r-processing based on the nearly

constant asymptotic entropy, dynamical timescale, and electron fraction of the wind (see section

28

1.2.1 for the sensitivity of NDW nucleosynthesis to these parameters):

𝑠3
𝜏𝑑𝑌 3
𝑒
evaluated at 𝑇 ≈ 0.5 MeV, indicates a wind likely to undergo a third-peak r-process. No simulated

≳ 1010,

(3.1)

𝜂 ≡

winds met this criterion except in the most extreme conditions. Yet, if a reasonable secondary

heating source could be found to boost the entropy of the wind, Hoffman et al. (1997) hypothesized

that an r-process could become likely.

A number of factors were investigated in the following years that could affect the value of 𝜂 in

the wind. The models derived by Qian & Woosley (1996) and Hoffman et al. (1997) were purely

Newtonian, and did not incorporate corrections from general relativity (though they did provide

estimates of the effects in a post-Newtonian calculation). A number of publications bore out their

predictions that GR corrections would have a positive effect on the entropy (Cardall & Fuller, 1997;

Otsuki et al., 2000; Thompson et al., 2001), but the increase was insufficient. A PNS mass ≳ 2 𝑀⊙

was still required to obtain high enough entropies to predict a strong r-process. Other research

focused on magnetic fields: a strong magnetic field could trap material near the PNS, exposing it

to more neutrino heating and raising the entropy. Again, to obtain a high enough entropy, extreme

conditions—a magnetar-strength field—were required (Thompson, 2003; Suzuki & Nagataki, 2005;

Metzger et al., 2007). Rapid rotation, which accelerates the wind (decreasing 𝜏𝑑) proved helpful,

but high magnetic field strengths were still required. To make matters worse, investigations of PNS

neutrino spectra indicated that, rather than being very neutron rich with 𝑌𝑒 ≲ 0.4, the electron

fraction in the wind was likely 𝑌𝑒 ≳ 0.48, with the wind becoming more proton-rich over time

(Fischer et al., 2010; Hüdepohl et al., 2010; Roberts et al., 2012a; Martínez-Pinedo et al., 2012;

Pllumbi et al., 2015; Xiong et al., 2019). Wanajo (2013) performed a thorough study of NDW

r-processing taking into account many of these factors, and found that unless extremely massive

PNSs are formed, no r-process is predicted. Our base-case models (neglecting wave contributions)

agree with these results, as Figures 3.1 and 3.2 indicate. NDWs in magnetorotational supernovae

with high PNS magnetic fields and rapid rotation are recognized as a plausible r-process site (Desai

et al., 2022; Van der Swaelmen et al., 2023; Prasanna et al., 2024) due to the high wind entropies

29

Figure 3.1 Entropy versus radius profiles for PNSs of varying mass, with a fixed total neutrino
luminosity of 6 × 1051 ergs−1, and 𝐿𝑤 = 0. We find comparable behavior to Wanajo (2013), with
asymptotic entropies well below the 400 𝑘B baryon−1 found by Woosley et al. (1994). The point at
which 𝑇 = 0.5 MeV, roughly where seed formation begins, is marked with a square.

that can attain, but outside these relatively extreme conditions the NDW is generally considered an

unlikely site for the r-process.

There remains at least one physical factor likely to affect the NDW that has not seen much

consideration, and which could thoroughly alter this outlook:

the effects of waves propagating

through the wind. Suzuki & Nagataki (2005) examined magnetic field-driven Alfvén waves, and

found that they could act as an effective heat source to enable r-processing if they carried enough

energy (that is, if the magnetic field were strong enough). Metzger et al. (2007) briefly introduced

PNS convection as another means of generating waves, and predicted that if the luminosity carried

by waves exceeded a fraction of ∼ 10−4 of the neutrino luminosity, the entropy would increase

enough to drive an r-process.

In fact, the predicted luminosity of convection-driven gravito-

acoustic waves is likely to fall in the range of 10−5 to 10−2 𝐿𝜈 (see chapter 1). Vigorous PNS

convection is a common feature of modern 3D supernova simulations (e.g., Nagakura et al., 2020;

Nagakura et al., 2021), so finding strong r-processing in a realistic convective PNS parameter space

would have important ramifications for galactic chemical evolution predictions. The contents of

30

106107Radius (cm)050100150200250300S (kB/baryon)T = 0.5 MeV1.41.51.61.71.81.92.02.1M/MFigure 3.2 Final abundances for the wind profiles of Figure 3.1. We find that no r-processing
takes place for proto-neutron stars of reasonable masses with 𝑌𝑒,eq = 0.48, when wave effects are
excluded.

this chapter (the results of which have been published as Nevins & Roberts, 2023) follow up on

this prediction, carefully examining the impact of convection-generated gravito-acoustic waves on

a possible r-process in a neutron-rich NDW. For the simulations in this section (unless otherwise

noted), we assume an equilibrium electron fraction of 𝑌𝑒,eq = 0.48 with an average electron neutrino

energy of 12 MeV. The average electron antineutrino energy is then calculated to satisfy equation

(1.1). This will yield a reasonable yet still neutron-rich NDW in accord with the studies cited above,

allowing for the possibility of r-processing without artificially enhancing its likelihood.

We focus in this chapter on qualitative analysis of how wave effects alter the wind dynamics and

nucleosynthesis rather than detailed, integrated abundance yields. To that end, the parameters that

are fed into our model (𝑅PNS, 𝑀PNS, 𝐿𝜈, ⟨𝜖𝜈𝑒⟩, 𝑅WT, 𝐿𝑤, 𝜔) are varied independently, allowing us

to freely explore the parameter space. We assume equal neutrino luminosities in all flavors, and the

average electron antineutrino energy is determined by 𝑌𝑒,eq and the neutrino luminosities. Unless

otherwise noted, we use the canonical 𝑅PNS = 10 km, and take 𝜖𝜈𝑒 = 12 MeV, 𝜔 = 2 × 103 s−1, and

𝑅WT = 5 × 108 cm as fiducial values.

31

050100150200250Mass Number1010108106104102Abundance1.41.51.61.71.81.92.02.1M/MFigure 3.3 Radial profiles of the velocity, density, temperature, and entropy in the NDW. Different
lines correspond to different 𝐿𝑤. Other parameters in the wind models were fixed to 𝑀NS = 1.5 𝑀⊙,
𝐿𝜈 = 3 × 1052 erg s−1, and 𝜔 = 2 × 103 s−1. The approximate beginning of seed formation for each
model is marked a square.

3.2 Wind Dynamics

As described in chapter 1, propagating waves affect the NDW in two primary ways: momentum

deposition via mechanical stress, and energy deposition via shock formation. Figure 3.3 shows

these effects for a fiducial PNS parameter set with 𝑀PNS = 1.5 𝑀⊙ and 𝐿𝜈 = 3 × 1052 erg s−1, and

the results are qualitatively similar for more extreme parameter sets. Clearly, above 𝐿𝑤/𝐿𝜈 ≈ 10−5,

the inclusion of wave effects has a significant impact on the dynamics of the wind. Although 𝐿𝑤 in

these models is a relatively small fraction of the total neutrino luminosity, it is a large fraction of

32

1081091010v (cm/s)101102T (GK)Lw=0T = 0.5 MeV106107Radius (cm)0.430.440.450.460.470.480.490.50Ye107108Radius (cm)102S (kB/baryon)105104103102Lw/Lthe neutrino energy that couples to the wind, which goes as

(cid:164)𝑞𝜈
𝐿𝜈,tot

∼ 1.5 × 10−4𝐿2/3
¯𝜈𝑒,51

(cid:18) 𝑅

10 km

(cid:19) 2/3 (cid:18) 1.4 𝑀⊙
𝑀PNS

(cid:19)

.

(3.2)

At small radii, before the waves shock, they accelerate the NDW but do not provide any heating.

This results in increasing velocities with 𝐿𝑤/𝐿𝜈, and therefore lower densities at a given radius

by the relation (cid:164)𝑀PNS = 4𝜋𝑟 2𝑒Λ𝑊 𝜌𝑣. Additionally, since the acceleration of the wind is no longer

provided solely by neutrino heating, the amount of neutrino heating that occurs is lowered, which

results in both lower entropies before the wave-heating activation radius, and in lower electron

fractions at all points in the wind. As the wave contribution increases, fewer neutrino captures are

required to unbind material from the potential well of the PNS and the NDW is accelerated to higher

velocities at smaller radii. Both of these effects work to reduce the number of weak interactions

in the wind and prevent the electron fraction in the wind from reaching 𝑌𝑒,eq, which results in

more neutron-rich conditions at the beginning of nucleosynthesis. The changes in 𝑌𝑒 begin prior

to the waves forming weak shocks, indicating that the wave stress, rather than shock heating, is the

primary contributor. These effects will therefore be present regardless of any uncertainty in the

shock heating mechanism. We observe a spike in 𝑌𝑒 at small radii due to electron-positron capture

when degeneracy is lifted at high temperatures. The electron fraction then relaxes towards 𝑌𝑒,eq,

but may not reach it due to the wave contributions.

The radius at which the waves shock and the rate at which they damp will depend on their fre-

quency content, with the shock formation radius approximately scaling as 𝜔−1 (see equation (2.37))

and the damping length 𝑙𝑑 ∝ 𝜔−1 for a fixed 𝐿𝑤. Therefore, larger wave frequencies will result

in wave heating impacting the thermodynamic conditions of the NDW at smaller radii and higher

temperatures. In Figure 3.4, we show the impact of varying 𝜔 on the entropy of the wind. Clearly,

larger 𝜔 results in a higher entropy at higher temperature, which is potentially more favorable for

later r-processing. The limiting case (𝜔 → ∞) corresponds to instantaneous shock formation in the

wind, but also implies a damping length that goes to zero. Nevertheless, we also show a case with

fixed 𝜔 in 𝑙𝑑 but assuming instantaneous shock formation, as this has been assumed in previous

work looking at secondary heating mechanisms in the NDW (Suzuki & Nagataki, 2005; Metzger

33

Figure 3.4 Early entropy profiles for a 1.5 𝑀⊙ neutron star with 𝐿𝜈 = 3 × 1052 erg s−1 and 𝐿𝑤 =
10−3𝐿𝜈 with varied wave frequencies. For higher frequencies, the shock heating begins to increase
the entropy in the wind earlier and has a larger impact where seed nuclei are formed. The impact
of the shock prescription is illustrated by the black line, which shows the evolution of the entropy
if the waves (with 𝜔 = 2 × 103 s−1) shock immediately instead of when equation (2.37) dictates.

et al., 2007). It is not clear what shock formation radii are favored, given the uncertainty in the

range of frequencies excited by PNS convection and the approximate nature of equation (2.37).

Subsequent to the waves shocking, the entropy rapidly increases in all models. Shock formation

occurs at temperatures between 2 and 10 GK depending on 𝐿𝑤/𝐿𝜈 (and 𝜔, see Figure 3.4). The

extra entropy production provided by (cid:164)𝑞𝑤 is large compared to neutrino heating because of the low

temperatures at which it occurs compared to the temperatures where the bulk of the neutrino heating

takes place (∼ 30 GK in our simulations). For the largest 𝐿𝑤/𝐿𝜈, the entropy can reach asymptotic

values of 300, which is quite large compared to even the largest entropies found for models that do

not experience wave heating. Nevertheless, a significant amount of the entropy production occurs

during or after the temperatures over which seed nuclei for the r-process are produced (∼ 2–8 GK).

Therefore, estimating the likelihood of r-process nucleosynthesis from the 𝜂 metric (see equation

(3.1)) is difficult as 𝑠 is no longer nearly constant while seed production occurs. Before the shock

formation radius, the waves reduce both 𝑠 and 𝜏𝑑 1. This can hinder or abet an alpha-rich freezeout

1We define the dynamical timescale 𝜏𝑑 at a given point in the wind as 𝑇/ (cid:164)𝑇, similar to the 𝑟/ (cid:164)𝑟 used by Hoffman

34

246810Temperature (GK)050100150200250300T = 0.5 MeVInstant heating102103104 (s1)depending on the relative strength of these two effects. After shock formation, 𝑠 is increased relative

to the 𝐿𝑤 = 0 case, but potentially at temperatures that are too low to impact the alpha-richness of

the NDW. Therefore, to better understand the impact of gravito-acoustic wave heating on the wind,

detailed nucleosynthesis calculations are required.

3.3 Nucleosynthesis

We now turn our attention to the nucleosynthesis data obtained from applying the SkyNet

reaction network to the thermodynamic profiles generated in the previous section. First, we

consider the impact of varying 𝐿𝑤/𝐿𝜈 for a fixed 𝜔 = 2 × 103 s−1. The final abundances for NDW

models with 𝑀NS = 1.5 𝑀⊙ and 𝐿𝜈 = 3 × 1052 erg s−1 are shown in Figure 3.5. These correspond

to the NDW models shown in Figure 3.3. In the absence of wave heating, this parameter set only

undergoes an 𝛼-process that terminates with a peak around mass 90 (Woosley & Hoffman, 1992)

and is far from the conditions necessary for producing the third r-process peak. Increasing 𝐿𝑤, we

find that the peak of the abundance distribution increases in mass until 𝐿𝑤/𝐿𝜈 ≈ 10−4. Further

increase of 𝐿𝑤 from this point briefly reduces the mass of the peak of the abundance distribution,

but above 𝐿𝑤/𝐿𝜈 ≈ 10−3 a strong r-process emerges. The final abundances for NDW models

with 𝑀NS = 1.9 𝑀⊙ and 𝐿𝜈 = 6 × 1052 erg s−1 are shown in Figure 3.6. Between 𝐿𝑤/𝐿𝜈 = 10−5

and 𝐿𝑤/𝐿𝜈 = 10−4, these models produce both the second and third r-process peaks, but between

𝐿𝑤/𝐿𝜈 ≈ 10−4 and 𝐿𝑤/𝐿𝜈 ≈ 10−3 production of the third peak is again cutoff and the peak of the

abundance distribution is pushed down to lower mass. As 𝐿𝑤/𝐿𝜈 is increased above 10−3, a strong

r-process re-emerges.

For both sets of parameters, we find the interesting behavior that r-process nucleosynthesis is

inhibited for 𝐿𝑤/𝐿𝜈 in the approximate range of 10−4–10−3. This turnover in the maximum mass

number is due to the competition between the decreasing dynamical timescale (𝜏𝑑) with 𝐿𝑤, which

inhibits seed formation, and the decreasing entropy (𝑠) with 𝐿𝑤, which facilitates seed production

by increasing the density at which alpha recombination occurs (Hoffman et al., 1997). Figure 3.7

illustrates the correlation between the quantity 𝑠3/𝑌 3

𝑒 𝜏𝑑 and the total abundance above mass 150.

et al. (1997).

35

Figure 3.5 Final nucleosynthesis results, using temperature and density profiles for a 1.5 𝑀⊙ neutron
star, with 𝐿𝜈 = 3 × 1052 erg s−1 and using a wave frequency of 2 × 103 s−1. A clear peak around
mass 200 is indicative of a strong r-process taking place.

Figure 3.6 Final nucleosynthesis results, using temperature and density profiles for a 1.9 𝑀⊙ neutron
star, with 𝐿𝜈 = 6 × 1052 erg s−1 and using a wave frequency of 2 × 103 s−1. A clear peak around
mass 200 is indicative of a strong r-process taking place.

36

050100150200250Mass Number1010108106104102AbundanceLw=0105104103102Lw/L050100150200250Mass Number1010108106104102AbundanceLw=0105104103102Lw/LFigure 3.7 Comparison of the total, summed final abundances of all nuclides with mass 𝐴 ≥ 150
(representative of the strength of any r-process taking place) with the quantity 𝑠3/𝑌 3
𝑒 𝜏𝑑 evaluated
𝑒 𝜏𝑑 and 𝑌𝐻 is necessarily approximate
when seed formation begins. The relationship between 𝑠3/𝑌 3
due to the presence of wave heating during seed formation. The relationship found in Hoffman
et al. (1997) was derived under the assumption of constant entropy, which is not generally true in
our models. Nevertheless, we still observe a strong correlation between the two quantities, which
helps to provide a qualitative explanation for the variation in heavy element nucleosynthesis near
𝐿𝑤/𝐿𝜈 = 10−3. These results are for the same parameters as those in Figure 3.6, with a finer grid
in 𝐿𝑤/𝐿𝜈.

Despite entropy no longer being constant during seed formation, we do observe a fairly strong

correlation between r-process strength and this quantity. We find that as the wave luminosity is

increased, 𝜏𝑑 decreases slightly faster than the entropy, but eventually asymptotes to a minimum

value of a few times 10−4 s. The entropy continues to steadily decrease, which creates the trough

in 𝑠3/𝑌 3

𝑒 𝜏𝑑 as a function of 𝐿𝑤 and gives rise to the window of inhibited r-processing we observe

around 𝐿𝑤/𝐿𝜈 = 10−3. At higher 𝐿𝑤, shock heating begins prior to alpha recombination, drastically

increasing the entropy. This, coupled with the reduced electron fraction at high 𝐿𝑤, reinvigorates

a strong r-process.

Second, we consider the impact of varying 𝜔 on gravito-acoustic NDW nucleosynthesis. As

was noted above, increasing 𝜔 results in an earlier activation of shock heating. In Figure 3.8, we

show the final abundances for 𝑀NS = 1.5 𝑀⊙, 𝐿𝜈 = 3 × 1052 erg s−1, and 𝐿𝑤/𝐿𝜈 = 10−3. For

𝜔 < 104 s−1, the nucleosynthesis is similar to the models with 𝐿𝑤/𝐿𝜈 ≈ 10−3 that efficiently form

seed nuclei, as discussed in the preceding paragraphs. Comparing to Figure 3.4, shock heating

begins only after the beginning of seed formation and therefore the resulting increase in entropy

37

105104103102Lw/L1010108106104Total A150 Abundance1010S3Y3e at T = 0.5 MeVYHS3Y3eFigure 3.8 Final abundances for the NDW profiles shown in Figure 3.4. For high frequencies,
the shock heating begins early enough to drive a strong r-process even for a 1.5 𝑀⊙ neutron star.
Instantaneous shock formation is illustrated by the black dashed line, showing the final abundances
for a wind that immediately experiences shock heating from waves with 𝜔 = 2 × 103 s−1.

only has a limited impact on the nucleosynthesis. On the other hand, for the largest frequency

considered (𝜔 = 104 s−1), a full r-process pattern extending through the third peak is produced.

Here, the wave heating due to weak shocks begins before the start of seed formation. Therefore,

the substantial increase in the entropy inhibits seed formation, and leaves a large neutron-to-seed

ratio when alpha capture ends. This is mainly driven by the impact of 𝜔 on the shock heating

activation radius, and less so by the variation in 𝑙𝑑 with 𝜔. This is illustrated by the model shown

in Figure 3.8 that assumes 𝜔 = 2 × 103 s−1 but an instantaneous activation of shock heating. This

results in nucleosynthesis that is very similar to the 𝜔 = 104 s−1 model.

Therefore, as a limiting case given the uncertainty in the shock activation radius and to compare

to previous work (Suzuki & Nagataki, 2005; Metzger et al., 2007), we show in Figure 3.9 final

abundances for varied 𝐿𝑤/𝐿𝜈 for 𝜔 = 2 × 103 s−1, 𝑀NS = 1.5 𝑀⊙, 𝐿𝜈 = 3 × 1052 ergs−1, but with

instantaneous activation of the shock heating. The results are noticeably different than those shown

in Figure 3.5, which shows models with the same parameters but without instantaneous shock

heating. For instantaneous activation, the average mass of the abundance distribution increases

38

050100150200250Mass Number1010108106104102AbundanceInstant heating102103104 (s1)Figure 3.9 Final abundances using the same parameters as in Figure 3.5, but assuming that shock
heating begins instantaneously in the wind. We see that a strong r-process takes place even for
moderate 𝐿𝑤.

monotonically with 𝐿𝑤/𝐿𝜈 and for even moderate wave luminosities is able to produce a full r-

process. This illustrates that uncertainty in the shock formation radius translates into significant

uncertainty in the predicted nucleosynthesis for gravito-acoustic NDWs.

To illustrate the important impact of the reduced electron fraction from the wave contributions,

we show in Figure 3.10 abundance distributions from a wind with 𝑀𝑁 𝑆 = 1.9 𝑀⊙, 𝐿𝜈 = 6 ×

1052 erg s−1, and 𝑌𝑒,eq = 0.52. In the absence of wave effects, the neutrino spectrum used here

should preclude any r-processing whatsoever. The wind would undergo 𝛼-recombination, leaving

only free protons to capture onto seed nuclei. However, with wave effects included, we find

similar r-processing regimes to those obtained with neutrino energies tuned to 𝑌𝑒 = 0.48. In the

wave stress regime, with 5 × 10−4 ≲ 𝐿𝑤/𝐿𝜈 ≲ 5 × 10−3, the change in 𝑌𝑒 is not large enough

to make the wind neutron rich, but the faster outflow caused by the wave stress prevents a robust

𝛼-recombination from occurring. R-process elements are then synthesized from the remaining free

neutrons in the wind, despite the wind being overall proton-rich. This gives rise to the suppressed,

actinide-free r-process patterns in Figure 3.10. Note that charged-current neutrino interactions

are not being considered in these calculations; we could expect to see heavy nucleosynthesis via

39

050100150200250Mass Number1010108106104102AbundanceLw=0105104103102Lw/LFigure 3.10 Final abundances using the same parameters as in Figure 3.6, but with antineutrino
energies tuned to 𝑌𝑒,eq = 0.52. We see r-processing regimes appear, despite a neutrino spectrum
that would otherwise have precluded r-processing entirely.

the 𝜈p-process if they were. Between the r-processing regimes, we again find a region where the

combined entropy and dynamical timescale in the wind favor strong seed formation and thus no

r-processing, regardless of 𝑌𝑒 or an inhibited 𝛼-recombination. At high 𝐿𝑤, the wind becomes

neutron-rich again, and early wave heating suppresses seed formation enough to drive a strong

r-process. Despite a neutrino spectrum predicting a proton-rich wind, a high wave luminosity can

allow a strong r-process to proceed. This will be explored and elaborated further in chapter 4.

3.4 Parameter Study

Our investigation thus far has demonstrated that wave effects can allow for strong NDW r-

processing for certain parameter sets. To better quantify the sensitivity of r-processing to different

parameters, and to determine how common it is likely to be, we survey the likely parameter space

in PNS mass, neutrino luminosity, and wave luminosity. In Figure 3.11, we show the total final

abundance of nuclei with mass number 𝐴 ≥ 150 as a function of 𝐿𝜈 and 𝐿𝑤/𝐿𝜈 for a variety of

PNS masses. Here, we have used 𝑌𝑒,eq = 0.48, 𝜔 = 2 × 103 s−1, and assumed the shock formation

radius is given by equation (2.37). We find the abundance of nuclei with 𝐴 ≥ 150 to be an

effective proxy for the strength of the r-process in the wind (see e.g. Figure 3.6). Two r-processing

40

050100150200250Mass Number1010108106104102AbundanceLw=0105104103102Lw/Lregimes appear. For the highest neutrino and wave luminosities, shock heating begins early enough

in the wind to drive a strong r-process. This shock heating regime is fairly insensitive to PNS

mass but very dependent on wave frequency, which sets how early shock heating can begin in the

wind. The second r-processing regime, driven by acceleration due to the wave stress, is strongly

dependent on mass but insensitive to wave frequency. We see this regime emerge at a PNS mass

of around 1.8 𝑀⊙, and grow to dominate the parameter space for the most massive neutron stars.

The non-monotonic dependence of the average mass number of the final abundances is also visible

here. At higher masses, the wave stress contribution is able to drive strong r-processes even for

very low neutrino and wave luminosities, where shock heating begins too late to strongly affect the

nucleosynthesis. We have also run similar calculations with 𝑌𝑒,eq = 0.45. These show qualitatively

similar behavior to the results shown in Figure 3.11, except that the onset of wave stress-driven

r-process nucleosynthesis is shifted to lower PNS mass.

In order to quantify the impact of the reduced electron fraction due to the wave stress contri-

bution, we show in Figure 3.12 the same parameter set as in Figure 3.11, but with 𝑌𝑒 fixed to a

constant value of 0.48. We find that including a self-consistent 𝑌𝑒 evolution results in a notice-

able broadening of the region in 𝐿𝜈-𝐿𝑤/𝐿𝜈 space where the r-process occurs, especially the wave

stress-dominated regime at lower 𝐿𝑤 and 𝐿𝜈. This is perhaps to be expected, as the change in 𝑌𝑒 is

driven primarily by the wave stress reducing the amount of neutrino heating needed to unbind the

wind material. We also observe generally higher yields of r-process material when 𝑌𝑒 evolution is

included, due to the higher number of free neutrons available.

Finally, in Figure 3.13, we show the impact of instantaneous shock formation on nucleosyn-

thesis across the entire parameter space (once again with 𝑌𝑒,eq = 0.48 and 𝜔 = 2 × 103 s−1, and

self-consistently evolving 𝑌𝑒). In this case, we find third peak r-process production for nearly all

considered neutrino luminosities and PNS masses when 𝐿𝑤/𝐿𝜈 ≳ 2 × 10−4. Although the accel-

eration of the wind due to the wave stress plays a role in determining the nucleosynthesis in these

models, the impact of the waves is mainly driven by the shock heating that they provide.

41

Figure 3.11 A measure of r-process strength across our parameter space, using a wave frequency of
2 × 103 s−1 and 𝑌𝑒,eq = 0.48, and using the shock heating prescription from equation (2.37). Two
regimes of r-process production emerge: a region of high wave and neutrino luminosities across all
masses, driven by shock heating; and a region of moderate wave and neutrino luminosities driven
by the wave stress, which becomes significant at larger neutron star masses.

42

105104103Lw/L10521053L (erg/s)MNS=1.4M105104103Lw/L10521053L (erg/s)MNS=1.5M105104103Lw/L10521053L (erg/s)MNS=1.6M105104103Lw/L10521053L (erg/s)MNS=1.7M105104103Lw/L10521053L (erg/s)MNS=1.8M105104103Lw/L10521053L (erg/s)MNS=1.9M105104103Lw/L10521053L (erg/s)MNS=2.0M105104103Lw/L10521053L (erg/s)MNS=2.1M108107106105104103Total A150 AbundanceFigure 3.12 A measure of r-process strength across our parameter space, using identical parameters
as Figure 3.11 but with 𝑌𝑒 fixed at 0.48. The same two r-processing regimes emerge, but the
wave stress regime is pushed to higher masses and neutrino luminosities by the lowered neutron
abundance.

43

105104103Lw/L10521053L (erg/s)MNS=1.4M105104103Lw/L10521053L (erg/s)MNS=1.5M105104103Lw/L10521053L (erg/s)MNS=1.6M105104103Lw/L10521053L (erg/s)MNS=1.7M105104103Lw/L10521053L (erg/s)MNS=1.8M105104103Lw/L10521053L (erg/s)MNS=1.9M105104103Lw/L10521053L (erg/s)MNS=2.0M105104103Lw/L10521053L (erg/s)MNS=2.1M108107106105104103Total A150 AbundanceFigure 3.13 A measure of r-process strength across our parameter space, using a wave frequency of
2 × 103 s−1 and 𝑌𝑒,eq = 0.48, and assuming the waves immediately shock and begin to deposit heat
into the wind. We see that for higher, but still quite reasonable wave luminosities, r-processing takes
place nearly independent of PNS mass and neutrino luminosity. The r-processing parameter space
broadens to very low wave luminosities at higher masses as the wave stress becomes significant.

44

105104103Lw/L10521053L (erg/s)MNS=1.4M105104103Lw/L10521053L (erg/s)MNS=1.5M105104103Lw/L10521053L (erg/s)MNS=1.6M105104103Lw/L10521053L (erg/s)MNS=1.7M105104103Lw/L10521053L (erg/s)MNS=1.8M105104103Lw/L10521053L (erg/s)MNS=1.9M105104103Lw/L10521053L (erg/s)MNS=2.0M105104103Lw/L10521053L (erg/s)MNS=2.1M108107106105104103Total A150 Abundance3.5 Conclusions

In this chapter we have investigated the impact of gravito-acoustic waves launched by PNS

convection on the dynamics and nucleosynthesis of the neutrino-driven wind. When these waves

propagate through the NDW, they impose additional stresses on the wind and also may shock and

provide an extra source of heating. Using steady-state, spherically symmetric models for the wind

that include the impact of an acoustic wave energy flux, we surveyed the parameter space of the

gravito-acoustic wave luminosity and frequency that is expected to be produced by PNS convection.

The presence of shock heating in the wind precludes reliance upon the common predictive metric

𝑠3/𝑌 3

𝑒 𝜏𝑑, as entropy is no longer nearly constant during seed formation. Therefore, using the results

of our hydrodynamic models, we then performed calculations of nucleosynthesis for the marginally

neutron-rich compositions that may be encountered in some NDWs.

For 𝐿𝑤 ≳ 10−5𝐿𝜈, the waves strongly impact the dynamics of the wind via two mechanisms,

acceleration due to wave stresses and entropy production via wave shock heating. Acceleration

of the NDW by wave stresses reduces the dynamical timescale, but also reduces the entropy and

electron fraction of the wind since a faster wind has less opportunity to undergo neutrino heating.

Depending on 𝐿𝑤/𝐿𝜈, this competition between reduced dynamical timescale and reduced entropy

can make conditions more or less favorable for strong r-process nucleosynthesis.

Similarly to previous work (Suzuki & Nagataki, 2005; Metzger et al., 2007), we find that if

the wave energy is deposited (in our case through shock heating) before r-process seed nucleus

formation begins, the entropy of the wind at seed formation is substantially increased. This in turn

results in limited seed formation and more favorable conditions for producing nuclei in the third

r-process peak. Here, we found that the exact position of shock formation has a strong impact

on the final nucleosynthesis. If wave shock heating begins before a temperature of around 7 GK,

the final nucleosynthesis is strongly impacted and even NDWs with modest wave luminosities and

fiducial PNS masses can produce a solar-like r-process pattern. If wave shock heating begins below

this temperature range, its impacts on nucleosynthesis are muted. For gravito-acoustic waves, the

radius of shock formation depends on their frequency, so higher frequency waves are likely to have

45

a larger impact on nucleosynthesis. For higher PNS masses, wave stress contributions can still

drive a strong r-process even if shock heating begins too late to affect seed formation.

At high wave luminosities (𝐿𝑤/𝐿𝜈 ≥ 10−3), the electron fraction can also be reduced by up

to almost 10% as a result of gravito-acoustic wave acceleration of the NDW. This wave-induced

reduction in 𝑌𝑒 broadens the regions of the parameter space over which the r-process occurs and

can even cause an r-process to be produced (if conditions are otherwise favorable) in winds with

neutrino spectra predicted to result in proton-richness.

46

CHAPTER 4

PROTO-NEUTRON STAR CONVECTION
AND THE 𝜈P-PROCESS

The contents of this chapter are adapted from Nevins & Roberts (2024), which has been published

in The Monthly Notices of the Royal Astronomical Society.

4.1 Background

Historically, the r-process has been the focal point of discussion regarding heavy element

nucleosynthesis. There are, however, other channels by which the heavy elements can be formed.

In neutrino-rich environments, the 𝜈p-process is the most promising of these. The 𝜈p-process

was first proposed by Fröhlich et al. (2006) as a primary nucleosynthetic process that is active in

neutrino-rich environments like the NDW. The 𝜈p-process, similar to the rp-process, forms heavy

nuclei by proton captures onto seed nuclei followed by beta decays toward stability. Unlike the

r-process, which can proceed to the dripline relatively uninhibited regardless of atomic number,

Coulomb repulsion forces the rp-process to proceed closer to stability. This results in longer 𝛽-

decay lifetimes for certain waiting-point nuclei, and when charged-current neutrino reactions are

neglected it is impossible for the classical rp-process to form nuclei heavier than 64Ge in the NDW

due to its long 𝛽 decay lifetime (Pruet et al., 2005). Waiting-point nuclei like 64Ge can be bypassed,

however, by capturing free neutrons created by charged-current antineutrino interactions on free

protons. This modified rp-process, called the 𝜈p-process, potentially allows for formation of heavy

elements up to 𝐴 = 100 and beyond with even a small flux of free neutrons. Since the primary

nucleosynthesis flows will be on the proton-rich side of stability, this process could be the source of

p-nuclei such as 92,94Mo and 96,98Ru. The source of these nuclei and their relative abundances have

been problematic for some time (e.g., Arnould & Goriely, 2003), but a 𝜈p-process in the NDW has

been shown to be a viable source for at least some of these nuclides (Wanajo, 2006; Bliss et al.,

2018b). Figures 4.1 and 4.2 show entropy and final abundance curves for a fiducial proton-rich

wind absent any wave effects. We find, as in the neutron-rich case, the expected dependence of

asymptotic entropy on mass. Our nucleosynthesis calculations are in general agreement with Pruet

47

Figure 4.1 Entropy versus radius for PNSs of varying mass, with the total neutrino luminosity (in
all flavors) fixed to a fiducial value of 3 × 1052 erg s−1, and 𝐿𝑤 = 0.

et al. (2006) and Wanajo et al. (2011), with heavy element synthesis terminating around 𝐴 = 120

for high entropy winds.

While there remains substantial uncertainty in the neutrino spectrum of cooling PNSs, several

recent studies have indicated that the wind becomes increasingly proton-rich at late times (e.g.

Martínez-Pinedo et al., 2014; Fischer et al., 2020; Pascal et al., 2022). Convection in the PNS,

which Nagakura et al. (2021) recently showed to be significant across a broad range of progenitors

for the first few seconds of PNS lifetime, also seems to lead to more proton-rich NDWs (Pascal

et al., 2022). This may be counteracted, however, by the mechanical effects of convection-driven

waves accelerating the NDW, reducing neutrino captures and leading to a more neutron-rich wind

(chapter 3). Nonetheless, the growing consensus over the last decade has been that the NDW

is likely to be either slightly neutron-rich or proton-rich at earlier times, becoming increasingly

proton-rich at late times. A proton-rich NDW naively should not undergo an r-process, but the

𝜈p-process is likely to be active instead and may allow for the formation of heavy nuclei.

A number of parameter studies have explored how the hydrodynamic conditions of the wind

affect a possible 𝜈p-process (e.g. Pruet et al., 2006; Wanajo, 2006; Wanajo et al., 2011; Arcones

et al., 2012; Eichler et al., 2017). As in the case of the r-process, the entropy and dynamical

48

106107Temperature (GK)050100150200250300S (kB/baryon)T = 0.5 MeV1.41.51.61.71.81.92.0M/MFigure 4.2 Final abundances for the wind profiles shown in figure 4.1, in the absence of wave effects
(i.e., 𝐿𝑤 = 0). We find some limited heavy element formation up to about 𝐴 = 130, with modest
PNS mass dependence. Higher PNS masses lead to higher asymptotic entropies in the wind, which
are generally favorable for heavier element nucleosynthesis.

timescale of the wind play key roles in determining the ratio of free nucleons to seeds, which

places a natural limit on how heavy of nuclei can be formed (Wanajo, 2006; Wanajo et al., 2011).

The electron fraction also affects this ratio, both by dictating how many free nucleons are left

after 𝛼 recombination (assuming a complete 𝛼 recombination), and by determining the number of

free protons which can undergo antineutrino capture and transform into neutrons (Wanajo et al.,

2011; Eichler et al., 2017). In chapter 3, we found that the presence of gravito-acoustic waves can

substantially increase the entropy of the wind once the waves begin to shock, possibly allowing

a 𝜈p-process to proceed to higher mass numbers as seed formation is inhibited. However, the

additional heating and momentum flux provided by the waves had the additional effects of reducing

the dynamical timescale of the wind and reducing the electron fraction, driving the wind towards

neutron-richness. The reduced electron fraction will likely inhibit a strong 𝜈p-process by reducing

the potential neutron flux. The effect of a faster outflow is not immediately clear. While it will

reduce the efficiency of seed formation (see chapter 3), it will also reduce the amount of possible

antineutrino captures in the wind, and thus the available neutron flux. A third key factor is the wind

49

050100150200250Mass Number10121010108106104Abundance1.41.51.61.71.81.92.0M/Mtermination shock, when the NDW impacts the slower-moving ejecta behind the main supernova

shock. The 𝜈p-process is most efficient at temperatures between 1.5 and 3 GK, and a termination

shock within this temperature range will prolong efficient 𝜈p-processing.

Our aim in this chapter is to explore the effect of gravito-acoustic waves on heavy element

nucleosynthesis in proton-rich winds. We anticipate this will primarily affect nucleosynthesis via

the entropy and electron fraction, but we will also briefly explore the role of the wind termination

shock in the presence of these waves. We are mainly interested in two aspects of heavy nucle-

osynthesis: how the active nucleosynthetic processes are affected by the presence of these waves;

and what the heaviest elements are that can be formed in a proton-rich wind affected by gravito-

acoustic waves. Understanding the impact of waves on the 𝜈p-process has the potential to impact

our understanding of both the origin of the heavy elements generally, and the origin of the light

p-nuclides 92,94Mo and 96,98Ru specifically. Given the significant impact these waves can have in

a marginally neutron-rich wind, up to driving a full solar-like r-process, we present here a similar

parameter study considering how these waves affect a proton-rich wind and the strength of the

𝜈p-process there. We find that the presence of gravito-acoustic waves in the NDW has a substantial

impact on 𝜈p-process nucleosynthesis and can allow much heavier elements to be formed than

would otherwise be possible.

The model and parameter space explored in this chapter are identical to those in chapter 3,

with two important differences. First, the antineutrino energies in this chapter are tuned to produce

𝑌𝑒,eq = 0.6 to simulate a proton-rich wind (unless otherwise stated), based on simulations such

as Pascal et al. (2022). Second, the SkyNet reaction network is adjusted to include the following

charged-current neutrino and antineutrino interactions on nucleons:

𝜈𝑒 + 𝑛 ←→ 𝑒− + 𝑝

¯𝜈𝑒 + 𝑝 ←→ 𝑒+ + 𝑛.

These were neglected in chapter 3, as they had no noticeable effects on nucleosynthesis in a

neutron-rich wind. They are however necessary to capture a possible 𝜈p-process.

50

Figure 4.3 Radial profiles of the velocity, electron fraction, temperature, and entropy in the NDW.
Different lines correspond to different 𝐿𝑤. Other parameters in the wind models were fixed to
𝑀NS = 1.5 𝑀⊙, 𝐿𝜈 = 3 × 1052 erg s−1, and 𝜔 = 2 × 103 s−1. The approximate beginning of seed
formation at 𝑇 = 0.5 MeV for each model is marked with a square.

4.2 Wind Dynamics

Figure 4.3 shows a set of profiles describing the behavior of the NDW in the presence of

wave effects for a fiducial parameter set. Our results are expectedly similar to those in chapter 3,

with the exception of the increased electron fraction due to the antineutrino energies we assume

here. As the wind is heated and accelerated by high wave luminosities, the number of neutrino

captures required to unbind the material decreases, preventing the electron fraction from reaching

its neutrino capture rate-equilibrium value of 𝑌𝑒,eq = 0.6. At the very highest wave luminosities,

these winds become almost neutron-rich, despite the assumed neutrino spectrum. This reduction

51

1081091010v (cm/s)101102T (GK)Lw=0T = 0.5 MeV106107Radius (cm)0.500.520.540.560.580.600.62Ye107108Radius (cm)102S (kB/baryon)105104103102Lw/Lin 𝑌𝑒 will have a substantial impact on the 𝜈p-process in these models, as fewer free protons will be

available to produce neutrons via antineutrino captures after seed nucleus formation (Wanajo et al.,

2011).

Another important feature to note is the onset of shock heating, visible as a sharp increase

in the entropy. For higher wave luminosities, this takes place near or before the onset of seed

nucleus formation at approximately 6 GK. We also see a clear acceleration of the wind as the

wave luminosity is increased, reducing the dynamical timescale over which seed nuclei can form.

Results for a more extreme parameter set, with higher mass and 𝐿𝜈, are qualitatively similar,

with one notable difference being higher 𝑌𝑒 at large 𝐿𝑤. Increasing the neutrino luminosity can

counteract the reduction in neutrino captures caused by high 𝐿𝑤, pushing the electron fraction closer

to its equilibrium value. A high wave luminosity coupled with a low neutrino luminosity could

result in a neutron-rich wind, even though the neutrino spectrum dictates a proton-rich equilibrium

𝑌𝑒.

4.3 Nucleosynthesis

We now consider the nucleosynthesis resulting from the wind models described in the preceding

section. Figure 4.4a shows the final abundances produced by the fiducial wind profiles shown in

figure 4.3. Two distinct families emerge in the abundance patterns. For low to moderate 𝐿𝑤, we see

a structure that is generally similar to the 𝜈p-process patterns that emerge from the 𝐿𝑤 = 0 case,

with a broad peak near 𝐴 = 100 that shifts towards higher mass numbers with increasing 𝐿𝑤. As in

the neutron-rich case, heavy nucleosynthesis is suppressed in a small window near 𝐿𝑤/𝐿𝜈 = 10−3

due to competition between reduced entropy and reduced dynamical timescale causing increased

seed formation (see chapter 3). At 𝐿𝑤 ≳ 10−3𝐿𝜈, the waves shock prior to seed formation, allowing

again for heavier nucleosynthesis. We see what could be described as a suppressed r-process

emerge, reaching to mass numbers near 𝐴 = 210.

For a more extreme parameter set, with higher mass and 𝐿𝜈, we observe similar abundance

families. Figure 4.4b shows the abundances produced by this parameter set with a higher PNS

mass (𝑀NS = 1.9 𝑀⊙) and neutrino luminosity (𝐿𝜈 = 6 × 1052 erg s−1). Under these conditions, the

52

(a) Fiducial parameter set

(b) Extreme parameter set

Figure 4.4 Final nucleosynthesis patterns for varied wave luminosities, with other parameters held
constant. We assume a wave frequency of 𝜔 = 2 × 103 s−1 for both panels. Left: a fiducial
parameter set with 𝑀NS = 1.5 𝑀⊙ and 𝐿𝜈 = 3 × 1052 erg s−1. Right: a more extreme parameter set
with 𝑀NS = 1.9 𝑀⊙ and 𝐿𝜈 = 6 × 1052 erg s−1. For both parameter sets, we observe two distinct
families of abundance patterns. For 𝐿𝑤/𝐿𝜈 ≲ 2 × 10−4, the nucleosynthesis is characteristic of
a 𝜈p-process, which can proceed to very high mass numbers in the high-entropy environment of
a massive, 1.9 𝑀⊙ PNS. At higher wave luminosities, the nucleosynthesis shifts to a suppressed
r-process pattern reminiscent of that predicted by Meyer (2002).

Figure 4.5 A comparison of the diagnostic quantity 𝑠2/𝜏, evaluated at 𝑇 = 0.5 𝑀𝑒𝑉, with the
maximum atomic mass produced with a final abundance greater than 10−12, for the same PNS
parameters (𝑀NS, 𝐿𝜈, 𝜔) as in figure 4.3. We see that the strength of the 𝜈p-process is correlated
with this quantity, with the fast-outflow r-process seeming to appear once it crosses a certain
threshold value at high 𝐿𝑤.

53

050100150200250Mass Number10121010108106104AbundanceLw=0105104103102Lw/L050100150200250Mass Number10121010108106104AbundanceLw=0105104103102Lw/L105104103102Lw/L100125150175200225250Maximum Atomic Mass1010S3Y3e at T = 0.5 MeVMaximum AS3Y3eFigure 4.6 Time integrated antineutrino capture rates per baryon (𝑛𝜈) from 𝑇 = 3 GK through the
end of the simulation (see Pruet et al., 2006). The black dots correspond to the wind profiles
shown in figure 4.3, which have a wind termination shock imposed at 𝑟ts = 5 × 108 cm, while the
blue squares show 𝑛𝜈 for the same set of models but with a wind termination shock imposed at
5 × 107 cm. In both cases, the additional momentum flux and shock heating from the waves reduces
the number of neutrino captures as the wave luminosity is increased. An earlier wind termination
shock causes the material to remain at smaller radii longer and undergo more neutrino captures.

𝜈p-process is able to proceed all the way to 𝐴 ≈ 200, with the peak shifting to near 𝐴 = 175 for

moderate 𝐿𝑤. Heavy nucleosynthesis is again suppressed near 𝐿𝑤/𝐿𝜈 = 10−3, and revives when

shock heating begins to affect seed formation. The suppressed r-process pattern again emerges at

high 𝐿𝑤, albeit with lower overall abundances.

The diagnostic quantity 𝑠2/𝜏 (similar to 𝑠3𝑌 −3

𝑒 𝜏−1 in neutron-rich winds; see Hoffman et al.,

1997) is helpful in explaining how the strength of the 𝜈p-process varies with 𝐿𝑤, 𝐿𝜈, and mass.

The entropy dependence of seed formation is reduced by one power relative to the neutron-rich

case since seed formation proceeds through the triple-alpha pathway in proton-rich conditions

rather than the effective four-body reaction 4He(𝛼, 𝛾)8Be(𝑛, 𝛾)9Be(𝛼, 𝑛)12C. The 𝜈p-process, just

like the r-process, is limited by the free-nucleon-to-seed-nucleus ratio. A higher 𝑠2/𝜏 implies a

higher proton-to-seed ratio, which will allow heavier nuclei to be formed. Figure 4.5 shows the

correlation, especially for low to moderate 𝐿𝑤, between 𝑠2/𝜏 and the heaviest nuclide formed in

the wind. Although the two quantities are correlated throughout most of the 𝐿𝑤/𝐿𝜈 range, in the

range 2 × 10−4 ≲ 𝐿𝑤/𝐿𝜈 ≲ 10−3, a discrepancy between the two arises. This is driven by two

factors. First, the faster outflow due to wave heating at smaller radii reduces the electron fraction

54

0105104103102Lw/L103102nrts=5×107 cmrts=5×108 cm(a) 𝜈p-process (𝐿𝑤/𝐿 𝜈 = 1 × 10−4)

(b) r-process (𝐿𝑤/𝐿 𝜈 = 5 × 10−3)

Figure 4.7 Final nucleosynthesis results for a fiducial 1.5 𝑀⊙ PNS with 𝐿𝜈 = 3 × 1052 erg s−1,
highlighting the frequency (𝜔) dependence of the two nucleosynthetic processes. Left: 𝐿𝑤/𝐿𝜈 =
1 × 10−4 to produce a 𝜈p-process. The final abundance pattern depends very weakly on wave
frequency. Right: 𝐿𝑤/𝐿𝜈 = 5 × 10−3 to produce a fast-outflow r-process. The fast-outflow r-
process requires a minimum wave frequency of approximately 𝜔 = 103 s−1 to operate, but above
that minimum there is not a strong frequency dependence. For low 𝜔, shock heating occurs after 𝛼
recombination is complete and seed formation is well underway, and thus has a negligible effect on
nucleosynthesis. For a sufficiently high 𝜔, the waves shock early enough to disrupt 𝛼 recombination,
allowing for an r-process.

of the outflow at the beginning of nucleosynthesis (see chapter 3). Second, after seed formation

a shorter dynamical timescale implies there is a shorter period of time for p(𝜈𝑒, 𝑒+)n to produce

neutrons to bypass long-lived waiting point nuclei before the 𝑟 −2 falloff of the neutrino flux shuts

off further captures. Our simulations indicate that this second effect is dominant, and the reduced

electron fraction plays a lesser role. To illustrate this, figure 4.6 shows the reduction in antineutrino

captures (i.e.

the electron antineutrino capture rate integrated from 𝑇 = 3 GK through the end

of the simulation) in the 𝜈p-processing regions of the wind as the wave luminosity is increased.

Above this range of 𝐿𝑤, shock heating begins early enough to be the dominant effect in the wind

and drives suppressed r-processing up to 𝐴 ≳ 200.

A closer analysis of the nucleosynthesis calculations reveals that the abundance patterns we

55

050100150200250Mass Number10121010108106104AbundanceInstant heating102103104 (s1)050100150200250Mass Number10121010108106104AbundanceInstant heating102103104 (s1)Figure 4.8 Final nucleosynthesis results, using temperature and density profiles for a 1.5 𝑀⊙ neutron
star, with 𝐿𝜈 = 3 × 1052 erg s−1, 𝜔 = 2 × 103 s−1, and assuming shock heating begins immediately
in the wind (i.e. 𝑅𝑠 = 𝑅NS). We observe a smoother transition from a 𝜈p-process at low 𝐿𝑤 to a
fast-outflow r-process at high 𝐿𝑤.

observe are the product of several different processes that take place at different temperatures. For

lower wave luminosities, a robust 𝛼-recombination takes place, and we observe a subsequent 𝜈p-

process at temperatures near 3 GK, followed by a neutron capture epoch when the wind enters (𝑛, 𝛾)

- (𝛾, 𝑛) equilibrium, driving the final abundances to the neutron-rich side of stability. This matches

the predictions of Pruet et al. (2006) for winds with enhanced entropy. The 𝜈p-process is very

entropy-dependent, so we do not see nucleosynthesis up to 𝐴 = 200 in the relatively low-entropy

fiducial case. At higher PNS masses (i.e. higher entropies) it can proceed farther, and produces

the peak near 𝐴 = 175 that we see in figure 4.4b. In the range 2 × 10−4 ≲ 𝐿𝑤/𝐿𝜈 ≲ 10−3, we see

the 𝜈p-process is curtailed in both the fiducial and extreme cases. This is due to both the increased

seed formation we previously observed in this region (see chapter 3), as well as a reduction in

neutron production. The additional acceleration from wave stresses reduces the number of neutrino

captures at early times, keeping 𝑌𝑒 closer to neutron richness and reducing the number of free

protons available for neutron production during 𝜈p-processing. Additionally, the faster outflow

reduces the time in which neutron production via antineutrino capture can proceed, decreasing total

56

050100150200250Mass Number10121010108106104AbundanceLw=0105104103102Lw/L(a) 𝜈p-process (𝐿𝑤/𝐿 𝜈 = 1 × 10−4)

(b) r-process (𝐿𝑤/𝐿 𝜈 = 5 × 10−3)

Figure 4.9 Final nucleosynthesis results for a fiducial 1.5 𝑀⊙ PNS with 𝐿𝜈 = 3 × 1052 erg s−1 and
𝜔 = 2 × 103 s−1, highlighting the wind termination shock location (𝑟ts) dependence of the two
nucleosynthetic processes. The 𝑟ts = 5 × 108 cm case used in our other plots is highlighted for
comparison. Left: 𝐿𝑤/𝐿𝜈 = 1 × 10−4 to produce a 𝜈p-process. An earlier shock is beneficial for the
𝜈p-process, keeping the material in the optimal 1.5–3 𝐺𝐾 temperature window that Wanajo et al.
(2011) and Arcones et al. (2012) observed. Right: 𝐿𝑤/𝐿𝜈 = 5 × 10−3 to produce a fast-outflow r-
process. An early shock can delay 𝛼 capture freezeout by keeping the material at high temperatures
longer, leading to rampant seed formation and stifling the r-process. A later shock allows the wind
to cool, "freezing in" the low seed count caused by the additional wave heating and allowing the
fast-outflow r-process to proceed. Kuroda et al. (2008) similarly observed that a later termination
shock is optimal for r-processing.

neutron production during 𝜈p-processing. This reduction is shown in figure 4.6.

At the highest wave luminosities, the early shock heating prevents a robust 𝛼-recombination

phase, leaving a noticeable abundance of both free protons and free neutrons and drastically

reducing seed production. Running the nucleosynthesis calculations without neutrino reactions

shows that these neutrons are not created in-situ, confirming that this is not a 𝜈p-process; the

bulk of the neutrons are left over from an incomplete 𝛼-recombination phase. As the temperature

continues to drop, charged particle reactions freeze out and a neutron capture epoch ensues. In the

fiducial parameter set (figure 4.4a), the lower neutrino luminosity leads to a lower electron fraction,

providing a larger reservoir of neutrons, which results in more pronounced r-process peaks near

57

050100150200250Mass Number10121010108106104Abundancerts=5×108 cm1071081091010Wind termination shock radius (cm)050100150200250Mass Number10121010108106104Abundancerts=5×108 cm1071081091010Wind termination shock radius (cm)𝐴 = 140 and 200. In the more extreme parameter set (figure 4.4b), the higher neutrino luminosity

reduces the number of free neutrons, and thus the r-process pattern is more suppressed. The

abundance patterns for the high 𝐿𝑤 case, especially those in figure 4.4b, agree very closely with

those predicted by Meyer (2002) for r-processing in very fast outflows. It is important to note that,

as figure 4.3 indicates, the r-processing here occurs in proton-rich conditions, with antineutrino

energies that predict an equilibrium 𝑌𝑒 of 0.6.

Incidentally, reviewing the high 𝐿𝑤 nucleosynthesis calculations for neutron-rich winds (e.g.

Figure 3.5 of chapter 3) we find that this same process was operating in those winds. We did not

see the same final abundance patterns because the free neutron abundance was high enough that

the subsequent period of neutron capture washed out any trace of the earlier process, yielding a

standard r-process abundance distribution.

4.3.1 Wave Frequency

The frequency of the waves excited by PNS convection plays an important role in determining

where and how the waves will shock. Though there is substantial uncertainty in the shock mecha-

nism in this context, we expect higher frequencies to shock earlier and deposit their energy faster

than lower frequencies (see section 2.1.4.3). In figures 4.7a and 4.7b, we present a typical 𝜈p- and

fast outflow r-process with varied wave frequencies, as well as one abundance profile from a wind

in which we assume the waves shock instantly in the wind, to illustrate the effect of the uncertainty

in the shock formation prescription. In the case of the 𝜈p-process, we see very little frequency

dependence because of the low wave luminosity, which limits how early the shocks can form in the

main prescription we assume (see chapter 3, section 3.2). Even when the shocks form instantly, the

waves are not carrying enough energy to strongly alter the nucleosynthesis. Since the 𝜈p-process

only operates at lower wave luminosities (𝐿𝑤/𝐿𝜈 ≲ 10−3), we expect this to be consistent across

the parameter space.

The fast-outflow r-process shows a stronger frequency dependence, in the form of what appears

to be a minimum cutoff frequency. Because this r-process only operates at high wave luminosities

(𝐿𝑤/𝐿𝜈 ≳ 10−3), if the waves shock early enough, they will deposit enough energy to strongly

58

affect the resulting nucleosynthesis. At very low frequencies, the waves do not shock until seed

formation has already taken place, so the r-process is stifled. Once above the cutoff frequency,

the waves shock early enough to disrupt 𝛼 recombination, and the final abundance pattern is not

strongly affected by further increasing the frequency.

There is substantial uncertainty in the shock mechanism for the waves we are considering, and

they may shock much earlier than predicted (see figure 10 and the discussion in chapter 3). In figure

4.8, we present abundance profiles assuming the waves shock instantly, so that any heat deposition

will affect every stage of nucleosynthesis. We observe what appears to be a smoother transition

from a 𝜈p-process at low wave luminosities, to a fast-outflow r-process at high wave luminosities,

with higher free neutron abundances altering the abundance patterns as 𝐿𝑤 is increased.

4.3.2 The Role of the Wind Termination Shock

Because of the strong dependence of the 𝜈p-process on the location of the wind termination

shock found in previous work (e.g. Arcones et al., 2012), we now turn our attention there. Figure

4.9a shows how the final abundance pattern for a typical 𝜈p-process varies with the location of

the wind termination shock. Previous studies have found that an early wind termination shock at

temperatures of around 2 GK is optimal for facilitating a strong 𝜈p-process. Our wind models cool

very rapidly, so this optimal temperature range occurs very early on. We can see that, as expected,

the smallest shock radius is favored for the 𝜈p-process. An early shock keeps the wind at higher

temperatures and smaller radii longer, which allows more time for both proton captures and neutron

production via antineutrino capture to proceed (see figure 4.6).

For the fast-outflow r-process, a later shock is optimal. This is in at least qualitative agreement

with Kuroda et al. (2008), who observed less robust r-processes with early shock radii, and little

change in abundance patterns once the radius was increased above 3000 km. We see in figure

4.9b that the r-process cannot proceed for very early termination shocks. A sufficiently early

termination shock can occur at high enough temperatures to restart 𝛼 capture reactions after their

initial freezeout, and the slow-moving shocked material takes much longer to cool than the fast-

moving wind. Seed formation proceeds rapidly in such cases, and we see a robust 𝛼 process

59

Figure 4.10 This plot shows the highest atomic mass produced in a significant amount (𝑌 ≥
1×10−12), giving a measure of how far heavy element nucleosynthesis is able to proceed. We observe
similar heavy nucleosynthesis regimes as in the neutron-rich case, with a mass-dependent wave-
stress driven 𝜈p-process regime at moderate wave luminosities, and a shock-heating driven, mass-
independent r-processing regime at high 𝐿𝑤 and 𝐿𝜈. We do not observe any actinide production in
these models.

emerge, forming large amounts of iron-peak elements and precluding an r-process. Once the shock

is moved beyond approximately 1000 km the r-process can proceed with full vigor, and when the

shock is moved beyond approximately 3000 km the final abundance pattern is no longer sensitive

to its location.

4.4 Parameter Study

In figure 4.10, we show a plot of the maximum atomic mass that is produced with a final

abundance greater than 10−12 for a given parameter set. This gives a helpful measure of how

far the 𝜈p-process is able to proceed as a function of PNS mass, neutrino luminosity, and wave

luminosity as a fraction of neutrino luminosity. For these plots, we have taken 𝜔 = 2000 s−1,

𝑌𝑒,eq = 0.6, 𝑟ts = 5 × 108 cm, and used the shock prescription in equation (2.37). We observe

similar nucleosynthesis regimes as in the neutron-rich case in chapter 3. For high 𝐿𝜈 and 𝐿𝑤,

we see very heavy nuclei being formed as shock heating begins early enough to inhibit seed

60

10510410310521053L (erg/s)MNS=1.4M105104103Lw/LMNS=1.7M105104103MNS=2.0M75100125150175200225250Max Atomic Mass (Y>1e-12)Figure 4.11 This plot is the same as figure 4.10, but assuming that shock heating begins immediately
in the wind. Similar to the neutron-rich case, we see a smoother transition between and blending of
the wave-stress and shock-heating regimes of heavy nucleosynthesis. The high 𝐿𝑤, low 𝐿𝜈 region
of strong heavy element production is revealed to be a full r-process proceeding all the way to the
actinides, as wave effects drive the wind to neutron-richness.

formation, and the high entropy and fast outflow prevent a complete 𝛼-recombination, leading to

a fast-outflow r-process. Because the fast-outflow r-process depends on early shock heating by the

waves disrupting 𝛼-recombination, the sharp cutoff line of this region indicates that the minimum

wave luminosity to obtain such an r-process is of order 1050 erg s−1, and is relatively independent

of neutrino luminosity. We also observe no actinide production in any region of this parameter

space, which could lead to an observable signature of these winds.

We also see a PNS mass-dependent nucleosynthesis regime driven by the wave stress emerge,

as in the neutron-rich case. The nucleosynthesis patterns here are broadly consistent with a 𝜈p-

process, and exhibit the expected correlation between the wave luminosity, the diagnostic quantity

𝑠2/𝜏, and the maximum atomic mass produced shown in figure 4.5. The non-monotonic connection

between wave luminosity and maximum mass produced is very clearly illustrated here. At low

wave luminosities, the effect on 𝑠2/𝜏 dominates, and the reduction in seed formation allows more

massive nuclides to form. For 𝐿𝑤/𝐿𝜈 ≳ 5 × 10−4, unless the neutrino luminosity is high enough

61

10510410310521053L (erg/s)MNS=1.4M105104103Lw/LMNS=1.7M105104103MNS=2.0M75100125150175200225250Max Atomic Mass (Y>1e-12)to make 𝐿𝑤 ≳ 1050 erg s−1 and drive fast-outflow r-processing, the reduction in neutrino captures

caused by the additional acceleration becomes dominant, and the 𝜈p-process is stifled by a lack of

free neutrons. Nucleosynthesis shifts toward the iron peak, and heavy nuclides cannot form. Both

the fast-outflow r-process and the 𝜈p-process benefit from higher asymptotic entropies, so we see

a positive correlation between PNS mass and the maximum atomic mass nuclides formed in the

wind.

Because of the uncertainty surrounding when and where shock formation will take place in the

wind, in figure 4.11 we show the same parameter space as in figure 4.10, but assuming that the waves

instantly shock and deposit their energy into the wind. Just as in the neutron-rich case, we see the

shock-heated heavy nucleosynthesis regime (here the fast-outflow r-process) broadens in both 𝐿𝑤

and 𝐿𝜈, and the wave pressure regime becomes difficult to distinguish as it emerges. Because shock

heating is assumed to begin instantly, 𝛼-recombination is affected even at low wave luminosities,

and we see a smooth transition from 𝜈p-processing to fast-outflow r-processing as in figure 4.8. At

high masses (i.e. higher entropies), the parameter space is dominated by heavy nucleosynthesis of

some kind. Interestingly, we notice a new regime emerge when instant shock heating is assumed:

for low 𝐿𝜈 and high 𝐿𝑤, the wind material becomes neutron-rich, and a full, third-peak, actinide-

producing r-process appears. The rapid acceleration and additional heat provided by the waves,

coupled with a low neutrino luminosity, result in a wind that undergoes comparatively few charged-

current neutrino interactions and maintains a 𝑌𝑒 well below its equilibrium value. We emphasize

again that the neutrino spectra of these proto-neutron stars predicts an equilibrium electron fraction

of 0.6, yet we observe regions of the parameter space (albeit extreme) that predict a full r-process.

4.5 Conclusions

In this chapter, we have continued our exploration of the effects of convection-driven acous-

tic waves on nucleosynthesis in the neutrino-driven wind, now focusing on winds predicted to be

proton-rich and with an eye toward the 𝜈p-process. As in chapter 3, we have surveyed the parameter

space of PNS masses, neutrino luminosities, wave luminosities, and wave frequencies, as well as

considering the impact of the wind termination shock and different shock heating prescriptions.

62

Using a steady-state, one-dimensional wind code and the SkyNet reaction network, we have per-

formed nucleosynthesis calculations to determine the predicted abundance profiles and reaction

flows expected in these winds. For the bulk of our calculations, we assume a PNS neutrino spectrum

that yields an equilibrium 𝑌𝑒 of 0.6, in the range predicted by many recent simulations.

As in the neutron-rich case, we find that wave luminosities of 𝐿𝑤 ≳ 10−5𝐿𝜈 have a substantial

impact on wind dynamics and nucleosynthesis. At modest wave luminosities (𝐿𝑤 ≲ 10−3𝐿𝜈),

the faster expansion caused by the wave stress contribution decreases seed formation efficiency,

leaving a higher ratio of free protons to seed nuclei and enhancing the 𝜈p-process. The resulting

nucleosynthesis flows can produce nuclides up to 204Hg and 203Tl. This is a strong enhancement

from our models that neglect wave effects, which only reach up to around mass 130.

We find these wave driven effects at modest luminosities can have competing impacts on

𝜈p-process nucleosynthesis in the NDW:

1. Gravito-acoustic wave driven acceleration and shock driven energy deposition impacts nu-

clear seed production for the 𝜈p-process. Acceleration of the NDW due to wave stresses

both reduces the dynamical timescale and also reduces the amount of neutrino heating that

occurs (which in turn reduces the asymptotic entropy of the NDW). Additionally, after the

gravito-acoustic waves become non-linear and shock, they deposit heat in the NDW and

increase the entropy. Since seed production in proton-rich winds depends on the parameter

𝑠2/𝜏, with seed production decreasing with increasing values of this parameter, the impact

of waves on seed production can behave non-monotonically with 𝐿𝑤. For 𝐿𝑤 ≲ 2 × 10−4𝐿𝜈,

we find the decreased dynamical timescale dominates and seed production decreases with

increasing 𝐿𝑤. For 2 × 10−4𝐿𝜈 ≲ 𝐿𝑤 ≲ 2 × 10−3, seed production increases with 𝐿𝑤 as the

entropy reduction due to wave stresses dominates. For 𝐿𝑤 ≳ 2 × 10−3𝐿𝜈, wave shock heating

increases both the entropy and dynamical timescale to the extent that even alpha production

is impacted.

2. Acceleration of the NDW by gravito-acoustic waves reduces the time available for p( ¯𝜈𝑒, 𝑒+)n

63

to occur and thus reduces the total number of neutrons that are produced per seed nucleus.

This reduces the maximum nuclear mass to which the 𝜈p-process proceeds, but at lower wave

luminosities the effects on seed formation via the increased entropy and decreased dynamical

timescale dominate. At 2 × 10−4 ≲ 𝐿𝑤/𝐿𝜈 ≲ 10−3, the reduced neutron flux becomes

significant enough to curtail 𝜈p-processing somewhat.

3. For larger gravito-acoustic wave luminosities, the acceleration of the NDW by these waves

can reduce the number of neutrino captures that occur before alpha particle formation. Since

material in the PNS atmosphere is in beta equilibrium and as a result is very neutron rich,

the reduced number of neutrino captures as the outflow starts tends to decrease the proton

richness of the wind. This effect was seen for neutron rich winds, where it helped to make

conditions more favorable for r-process nucleosynthesis, but in proton rich winds where the

𝜈p-process might occur this effect reduces the number of free protons that are available to

undergo p( ¯𝜈𝑒, 𝑒+)n once seeds have been formed. This also contributes to the reduction in

maximum nuclear mass attained by the 𝜈p-process seen for moderate wave luminosities.

At higher wave luminosities (𝐿𝑤 ≳ 10−3𝐿𝜈) and higher neutrino luminosities (𝐿𝜈 ≳ 1052 erg s−1),

we find an interesting result: the early shock heating in this region of the parameter space prevents

a complete 𝛼 recombination phase in the wind, preserving an abundance of free neutrons much

higher than the persistent abundance created by charged-current neutrino interactions. This, cou-

pled with the reduced seed formation due to higher entropy, allows a suppressed r-process to take

place, similar to that predicted by Meyer (2002) for fast-moving winds. Though this r-process does

not reach to the actinides, it can reach beyond the third r-process peak and produce nuclides up

to 210Po1. For a substantial region of the parameter space, we find that winds from a neutron star

with a neutrino spectrum predicted to result in a firmly proton-rich wind can undergo an (albeit

suppressed) r-process. When we consider an alternative prescription for the shock heating, in which

the waves immediately shock upon entering the wind, not only does this region of suppressed r-

1The plots in this chapter show production of elements up to mass number 215. These nuclides should have
decayed by the end of the simulation, but did not due to some missing 𝛼 emission reactions in the version of REACLIB
used here. This issue will be fixed for future publications.

64

processing broaden, but we find an even more extreme possibility: the combination of acceleration

due to wave stresses and additional heating from the shocked waves reduces the amount of neutrino

heating needed to unbind the wind to such a degree that the wind actually becomes neutron-rich,

and undergoes a full, third-peak r-process reaching the actinides. While this only occurs in a small

corner of the parameter space, its mere possibility demonstrates the dramatic changes that can be

caused by the presence of waves in the NDW.

65

CHAPTER 5

INTEGRATED NUCLEOSYNTHESIS FROM CONVECTIVE
PROTO-NEUTRON STARS

5.1

Introduction

Having seen in the preceding chapters the dramatic effects convection-driven waves can have

on the neutrino-driven wind, we now calculate robust predictions of the nucleosynthesis in such

winds. The steady-state approximation is valid for calculating the dynamics of the wind because

the properties of the PNS (mass, neutrino luminosities, convective frequency, etc.) vary slowly in

comparison to the dynamical timescale of the wind. These PNS properties do change with time,

however, and because NDW nucleosynthesis depends heavily on properties such as PNS mass and

neutrino luminosity, the time variation of these properties must be accounted for in order to obtain

accurate nucleosynthesis results.

A number of such studies have been done using parameterized models of the properties of

the PNS, e.g., radius and neutrino luminosity. Arcones & Montes (2011) considered whether the

NDW could explain a certain range of solar system abundances by using an integrated set of simple

steady-state models with a time-varying neutrino luminosity. Wanajo (2013) examined r-process

nucleosynthesis in the NDW using a series of steady-state models, with the PNS mass fixed to a set

value and the neutrino luminosity (in all flavors) given by 𝐿𝜈 (𝑡) = 𝐿𝜈,0(𝑡/𝑡0)−1. The PNS radius was

also varied as a function of the neutrino luminosity. The resulting abundances from these models

were integrated with respect to time and weighted by the mass loss rate (see Equation [2.12]) to

determine the total elemental abundance yields from a wind lasting 10 s post-bounce. Friedland

et al. (2023) used a similar method, by combining a steady-state model with a time-dependent

component and integrating a number of these models for a parameterized 𝐿𝜈 (𝑡). Ekanger et al.

(2022) took a more robust approach and used neutrino luminosities and energies taken from a PNS

simulation done by Pons et al. (1999). Their wind models, however, were highly parameterized,

and they did not appear to consider the time evolution of 𝑅PNS.

Other time-dependent factors such as PNS mass, average neutrino energies, and location of the

66

wind termination shock also play a key role in determining the dynamics and nucleosynthesis of the

wind. Furthermore, the convective parameters examined in this work are also time-varying, and as

we have shown in previous chapters, can have a dramatic impact on nucleosynthesis. Ideally, all this

could be explored in full 3D simulations, and efforts are being made to do so (e.g., Zha et al., 2024;

Wang & Burrows, 2024), but these simulations are limited by extreme computational complexity

and cannot yet explore the full behavior of the wind, which can last ∼ 10 s post-bounce. The most

robust and computationally-feasible approach, at present, is to integrate NDW nucleosynthesis

yields for a full, self-consistent, time-dependent PNS parameter set fed into a robust wind code,

which can easily explore the full wind timescale. Such an investigation was performed by Fischer

et al. (2010), who modeled the core collapse, PNS evolution, and NDW properties for a few different

progenitor stars. We emulate their approach in this chapter. We extract key parameters from a

robust 1D supernova code, STIR, which produces self-consistent explosions and also provides

the PNS convection data we require, and then we compute a number of steady-state models with

time-varying PNS parameters and integrate their yields. This integrated nucleosynthesis from our

models using realistic PNS evolution from STIR provides a much more accurate picture of the

nucleosynthesis in the NDW, and gives us additional insight into the effects of convection-driven

waves.

5.2 Overview of STIR

Though supernovae are fundamentally 3D, the complexity and compute-time requirements of

3D simulations makes 1D, spherically symmetric models an extremely useful tool in parameter

studies. A long-standing problem in 1D, however, is the explodability of the simulated stars.

Historically, it was (almost) impossible to find a stellar model that would explode in 1D without

some ad-hoc method of dumping energy into the core to drive the explosion1. One key reason is

the lack of turbulence in 1D models, which acts as an additional source of pressure to support an

explosion (Couch & Ott, 2015), and also has important effects on the neutrino properties (Roberts

et al., 2012b). To remedy this, Couch et al. (2020) developed a 1D code, Supernova Turbulence

1An exception to this is the z9.6 model, a zero-metallicity 9.6 𝑀⊙ progenitor that explodes self-consistently in 1D,

which we consider in this chapter.

67

In Reduced-dimensionality (STIR), which, like Roberts et al. (2012b), uses mixing-length theory

to simulate the effects of turbulence and convection without the added complexity of increased

dimensionality. This resolves both convection in the main supernova explosion, which allows for

self-consistent explosions in 1D, as well as convection in the PNS. The methods used by Couch et al.

(2020) did not incorporate lepton gradients in their calculations for PNS convection, however, and

the resulting PNS convection did not comport with 3D simulations. Substantial improvements have

now been made to STIR (Couch et al., in prep.). To briefly summarize, the major improvements

are as follows.

1. A full rederivation of the STIR equations was performed, using a Favre average (Pope, 2000)

of the Navier-Stokes equation. STIR is now explicitly energy-conserving, while it was only

approximately so before.

2. A hybrid equation-of-state was implemented that accurately resolves the transition from

NSE abundances to the Helmholtz EOS (Timmes & Swesty, 2000) as the ejecta expand and

cool when the simulation is run to later times. This, combined with the Favre-averaged

rederivation, allow simulations to be run to much later times than were previously possible.

3. Lepton gradients are included in the calculation of the Brunt-Väisälä frequency, which enables

an accurate calculation of convection in the PNS. STIR now accurately reproduces 3D results

for convection in both the explosion and the PNS.

With these improvements, STIR can now provide accurate, self-consistent PNS parameters for use

in our wind code.

5.3

Integrating STIR

A number of parameters need to be extracted from STIR for our neutrino wind model. The

neutrino luminosity 𝐿𝜈 and mean energy ⟨𝜖𝜈⟩ for each flavor are explicitly reported by STIR and

can easily be transferred. The PNS radius and mass are defined, respectively, as the radius and

mass of the region for which the density exceeds 1012 g cm−3. The wind termination shock occurs

where the fast-moving transonic wind collides with slower-moving material at the trailing edge of

68

Figure 5.1 Diagram showing how the PNS radius and wind termination shock radius are determined
from STIR’s output, for the z9.6 progenitor without convection, 3 s post-bounce. The PNS radius
is defined as the boundary of the region having density greater than 1012 g cm−3. The location of
the wind termination shock, where the supersonic wind impacts the trailing edge of the primary
supernova shock, is located where the derivative of the density with respect to radius is maximized.

the primary supernova shock, which can be seen in STIR snapshots as a region of increased density

in the ejecta that moves outward with time. The trailing edge of the primary shock is then easily

detected as the location at which the derivative of density with respect to radius is maximized (see

Figure 5.1).

5.3.1 Convection

STIR reports the convective velocity 𝑣con as a function of radius for each timestep, which allows

us to determine the initial value of 𝐿𝑤 prior to the attenuation experienced when the waves pass

through the evanescent region. Wave generation is concentrated in the outermost pressure scale

height of the convective region (Goldreich & Kumar, 1990), so we take the average of the convective

69

1061071081091010Radius100102104106108101010121014Density (g cm3)4020020406080100120Density Derivative (dimensionless)PNSdlndlnrRWT(a) 𝑙 = 2

(b) 𝑙 = 6

Figure 5.2 Wave branches for a fiducial PNS with 𝑀NS = 1.5 𝑀⊙, 𝐿𝜈 = 3 × 1052 erg s−1, 𝜔 =
2 × 103 s−1, and a low initial wave luminosity 𝐿𝑤/𝐿𝜈 = 10−4 for two modes, 𝑙 = 2 (left) and
𝑙 = 6 (right). Where the wave frequency (black line) is in the red-shaded zone, the waves are
gravitational. In the blue zone, the waves are acoustic, and in the white zones, they are evanescent.
The transmission coefficient as a function of radius is shown in the bottom panels; it decreases
only where the waves are evanescent. The point at which the waves become nonlinear and begin to
deposit heat in the wind is marked with a dot. For lower wave luminosities, which become nonlinear
later, higher angular modes have lower transmission coefficients where wave heating begins.

luminosity, 4𝜋𝑟 2𝜌𝑣3

con, between the edge of the convective zone (𝑟2) and one pressure scale height

below it (𝑟1), multiplied by the convective mach number, and weighted by mass:

∫ 𝑟2
𝑟1

𝐿𝑤 =

4𝜋𝑟 2𝜌𝑣3

con (𝑣con/𝑐𝑠) (4𝜋𝑟 2𝜌)𝑑𝑟
∫ 𝑟2
𝑟1

4𝜋𝑟 2𝜌𝑑𝑟

.

(5.1)

We can similarly calculate the convective frequency, which is equal to the frequency of the waves

(Gossan et al., 2020),

𝜔 =

∫ 𝑟2
𝑟1

𝜋𝑣con
2𝐻 (4𝜋𝑟 2𝜌)𝑑𝑟
∫ 𝑟2
4𝜋𝑟 2𝜌𝑑𝑟
𝑟1

,

(5.2)

where 𝐻 is the pressure scale height. Explicitly calculating the initial wave luminosity and frequency

allows us to improve our wind model as well. Whereas the initial wave luminosity, convective Mach

70

106107108103104Frequency (s1Ll|N|Rs106107108Radius (cm)102101100l106107108103104Frequency (s1Ll|N|Rs106107108Radius (cm)102101100l(a) 𝑙 = 2

(b) 𝑙 = 6

Figure 5.3 Wave branches for a fiducial PNS with 𝑀NS = 1.5 𝑀⊙, 𝐿𝜈 = 3 × 1052 erg s−1, 𝜔 =
2 × 103 s−1, and a high initial wave luminosity 𝐿𝑤/𝐿𝜈 = 10−2 for two modes, 𝑙 = 2 (left) and
𝑙 = 6 (right). Where the wave frequency (black line) is in the red-shaded zone, the waves are
gravitational. In the blue zone, the waves are acoustic, and in the white zones, they are evanescent.
The transmission coefficient as a function of radius is shown in the bottom panels; it decreases
only where the waves are evanescent. The point at which the waves become nonlinear and begin
to deposit heat in the wind is marked with a dot. For higher wave luminosities, the waves become
nonlinear before encountering the evanescent region.

number, and transmission efficiency were previously all combined into a single parameter 𝐿𝑤/𝐿𝜈,

we now have the first two explicitly specified by STIR. The third can be specified as well: instead

of setting the transmission efficiency by hand, we can calculate it in situ by tracking the character

of the waves (gravitational, evanescent, or acoustic) as a function of radius. To do so, we calculate

two frequencies. The first is the Brunt-Väisälä frequency,

𝑁 2 = −

𝑔
𝜌

(cid:20) 𝑑𝜌
𝑑𝑟

−

𝑑𝑃
𝑑𝑟

(cid:18) 𝜕 𝜌
𝜕𝑃

(cid:19)

(cid:21)

.

𝑠

(5.3)

The second is the Lamb frequency, 𝐿2

𝑙 = 𝑙 (𝑙 + 1)𝑐2

𝑠𝑟 −2, which depends on the angular mode 𝑙 of the

waves. If the wave frequency 𝜔 is less than both 𝑁 and 𝐿𝑙, the waves are gravitational in character.

If 𝜔 lies between 𝑁 and 𝐿𝑙, the waves are evanescent, and if 𝜔 is greater than both, the waves are

71

106107108103104Frequency (s1Ll|N|Rs106107108Radius (cm)102101100l106107108103104Frequency (s1Ll|N|Rs106107108Radius (cm)102101100l(a) Asymptotic wave transmission efficiency

(b) Wave transmission efficiency at nonlinearity

Figure 5.4 Wave transmission efficiencies for different angular modes, for a fiducial PNS with
𝑀NS = 1.5 𝑀⊙, 𝐿𝜈 = 3 × 1052 erg s−1, 𝜔 = 2 × 103 s−1, and varied wave luminosities. We confirm
the findings of Gossan et al. (2020) that the asymptotic transmission efficiency (left) peaks at 𝑙 = 2
and 𝑙 = 3 for initial wave luminosities near 𝐿𝑤/𝐿𝜈 = 10−2. The transmission efficiency when the
waves become nonlinear and begin to heat the wind (right) is more relevant for our purposes, and
decreases with 𝑙.

acoustic. The evanescent region is then defined as the region in which the frequency of the waves

is between the Brunt-Väisälä and Lamb frequencies. Equivalently, the evanescent region is defined

as the region for which

𝑘 2
𝑙 =

(cid:0)|𝑁 |2 − 𝜔2(cid:1) (cid:0)𝐿2

𝑙 − 𝜔2(cid:1)

𝜔2𝑐2
𝑠

< 0.

(5.4)

Sample representations of these regions for 𝑙 = 2 and 𝑙 = 6 in a fiducial wind are shown in the upper

panels of Figures 5.2 and 5.3. Knowing where the waves enter and exit the evanescent region allows

us to explicitly calculate the tunneling efficiency T , rather than treating it as a free parameter. For

a given angular mode,

T𝑙 = exp

(cid:20)

−2

∫ 𝑟2

𝑟1

(cid:21)

,

|𝑘𝑙 |𝑑𝑟

(5.5)

where 𝑟1 and 𝑟2 denote the boundaries of the evanescent region. Because the tunneling efficiency

is cumulative across any and all evanescent regions the waves encounter, we can rewrite T𝑙 as

T𝑙 (𝑟) = 𝑒−2𝜙𝑙 (𝑟),

(5.6)

72

246810Angular Mode (l)104103102101100Transmission Coefficient ()105104103102Lw/L246810Angular Mode (l)104103102101100Transmission Coefficient ()105104103102Lw/Lwhere

𝜙𝑙 (𝑟) =

∫ 𝑟

0

|𝑘𝑙 |𝜃 (−𝑘 2

𝑙 )𝑑𝑟′

(5.7)

with 𝜃 (𝑥) denoting the Heaviside step function. The total wave luminosity in a given angular mode

that reaches to a radius 𝑟 in the wind is then

𝑤 (𝑟) = T𝑙 (𝑟)𝐿𝑙
𝐿𝑙

𝑤,0

,

(5.8)

where 𝐿𝑙

𝑤,0 is the initial wave luminosity generated in that angular mode by the PNS convection. The

total effective wave luminosity acting to accelerate and heat the wind at a given radius (neglecting

dissipation for now) is then

L =

𝑙con∑︁

𝑙=1

T𝑙 (𝑟)𝐿𝑙

𝑤,0

.

(5.9)

Examples of the evolution of T𝑙 are shown in the lower panels of Figures 5.2 and 5.3.

Following Gossan et al. (2020), we assume a flat spectrum up to a certain maximum value

𝑙con = 𝑟con/𝐻 (rounded down to the nearest integer), where 𝑟con is the outer radius of the convective
zone of the PNS, and 𝐻 is the pressure scale height at 𝑟con. Then 𝐿𝑙
𝑤,0 = 𝐿𝑤/𝑙con for all angular
modes, where 𝐿𝑤 is the total convective luminosity of the PNS at the surface, given by Equation

(5.1). Since 𝐿𝑙

𝑤,0 is thus independent of 𝑙, the total effective wave luminosity becomes
𝑙con∑︁

L =

𝐿𝑤
𝑙con

𝑙=1

T𝑙 (𝑟).

(5.10)

This effective wave luminosity must also evolve as dissipation occurs in the wind, which Equation

(2.65) accounts for. Since these effects are independent of each other, rather than attempting to

derive a new expression for the evolution of L, we can simply make the change

L →

L
𝑙con

𝑙con∑︁

𝑙=1

𝑒−2𝜙𝑙 (𝑟),

(5.11)

where L appears in the other evolution equations (Equations [2.61]–[2.66]). Equation (5.7) must

also be added to the set of evolution equations for all the angular modes 𝑙 ∈ [1, 𝑙con]. Casting

Equation (5.7) in terms of our computational integration variable 𝜓, we have

𝑑 ln 𝜙𝑙
𝑑𝜓

=

|𝑘𝑙 | 𝑓1𝑟
𝜙𝑙

𝜃 (−𝑘 2

𝑙 ).

73

(5.12)

Gossan et al. (2020) predict 𝑙con ≲ 6, with 𝑙 = 2 and 𝑙 = 3 having the highest tunneling

efficiencies. Considering the higher wave luminosities of 𝐿𝑤 ≈ 10−2𝐿𝜈 they used, we find general

agreement with the asymptotic values of T𝑙 peaking at 𝑙 = 2 and 𝑙 = 3 (see Figure 5.4a). Since we

focus on the inner regions of the supernova rather than the primary shock, however, the transmission

coefficient at the point where the waves become nonlinear and begin to deposit heat is more relevant.

We find that T𝑙 at the point of shock formation monotonically decreases with 𝑙 (Figure 5.4b). As the

initial wave luminosity is increased, the waves become nonlinear earlier, such that they eventually

become nonlinear while still in the gravitational branch. In this situation, T𝑙 = 1 for all modes, and

the full wave luminosity acts to heat the wind. The dissipation mechanism is less clear when the

waves become nonlinear outside of the acoustic branch, but the effect is the same: the energy of

the waves is deposited into the wind as heat.

The parameters extracted from STIR (𝑀PNS, 𝑅PNS, 𝐿𝜈𝑒, 𝐿 ¯𝜈𝑒, ⟨𝜖𝜈𝑒⟩, ⟨𝜖 ¯𝜈𝑒⟩, 𝑅WT, 𝐿𝑤, 𝜔) are used

to assemble an ensemble of steady-state models for integration. We sample the parameters from

STIR every 10 ms, starting a full second post-bounce to allow time for the primary shock to resolve

and a steady-state wind to form, and going through the end of the simulation. With all of the wind

model parameters specified by STIR, the only free parameters in our model are those that feed into

STIR itself: the properties of the progenitor star and the various MLT parameters. STIR has no free

parameters that alter the neutrino physics, and no part of the PNS is excised during the simulation.

The only free parameters in STIR are the various MLT coefficients, which are of order unity and

tuned to match 3D convective profiles.

5.4 Results from STIR: Model z9.6

The first step in analyzing convection-influenced neutrino winds is to compare winds with

convection-driven waves to winds without. One-dimensional supernova models without some

additional energy source, or convection in the case of STIR, are notorious for not exploding.

There are, however, a few progenitor models that do explode without help, including the z9.6

progenitor (A. Heger, private communication) we examine here. This is a 9.6 𝑀⊙ star with zero

metallicity, which explodes in STIR both with and without convection enabled. We ran models for

74

(a) Without convection

(b) With convection

Figure 5.5 Convective velocity reported by STIR as a function of time (post-bounce) and radius,
with the PNS and wind termination radii highlighted, for the non-convective (left) and convective
(right) z9.6 models. Just after 2 s post-bounce, the PNS becomes fully and vigorously convective.

this progenitor, both with and without convection, out to ≈ 6 s post-bounce. We discard the first

second of each model to allow time for a transonic wind to form.

5.4.1 PNS Parameters

Figures 5.5–5.6 show a number of different PNS parameters from STIR, comparing the model

with convection to the model without. The mass of the PNS formed in the explosion does not depend

strongly on the presence or absence of convection, with 𝑀PNS ≈ 1.35 𝑀⊙ in both cases. The radius

and rate of contraction are almost unchanged, as is the expansion rate of the primary shock (Figure

5.5). A close examination shows that in the convective PNS, turbulent pressure causes the PNS

radius to contract more slowly, and the primary shock to move outward more quickly. A more

stark difference is visible in the neutrino luminosities (Figure 5.6). Both the electron neutrino and

electron antineutrino luminosities at early times increase when convection is present, as predicted

by, e.g., Roberts et al. (2012b) and Pascal et al. (2022). The values we obtain for PNS mass, radius,

and (anti)neutrino luminosity in the convective model are in general agreement with the findings of

75

246Time (s)1051061071081091010Radius (cm)RPNSRWT104105106107108109Convective Velocity (cm/s)246Time (s)1051061071081091010Radius (cm)RPNSRWT104105106107108109Convective Velocity (cm/s)(a) Without convection

(b) With convection

Figure 5.6 PNS neutrino luminosities and average energies from approximately 1 to 6 s post-bounce,
both with and without PNS convection. The convective PNS has higher neutrino luminosities at
early times, which accords with prior work. Unlike some prior work, we do not observe the electron
neutrino and antineutrino energies converging at late times, regardless of convection. In the bottom
panels, the predicted equilibrium 𝑌𝑒 in the NDW based on the neutrino parameters is shown. The
neutrino properties we observe predict an increasingly neutron-rich wind, unlike most recent prior
work.

Wang & Burrows (2024), who examined this same progenitor in 3D, though only to approximately

1 s post-bounce. The average neutrino energies from STIR differ significantly, however, from some

prior work, which predicts that the electron neutrino and antineutrino energies should converge at

late times (e.g., Fischer et al., 2010; Mirizzi et al., 2016; Pascal et al., 2022). In both the convective

and non-convective models, we find that the average energies maintain an approximately constant

offset of 5–6 MeV and increase with time. As a result of this neutrino average energy behavior, we

find that 𝑌𝑒,eq moves steadily towards neutron-richness. Both the convective and non-convective

models find equilibrium electron fractions more neutron-rich than the most optimistic predictions

of Roberts et al. (2012a), and at late, instead of early, times.

Examining the convective model more closely (Figures 5.5 and 5.7), we find that the PNS

76

0123456Luminosity (1051 erg/s)246Time (s)0.40.5Ye,eq101214161820Avg Neutrino Energy (MeV)ee0123456Luminosity (1051 erg/s)246Time (s)0.40.5Ye,eq101214161820Avg Neutrino Energy (MeV)eeFigure 5.7 PNS convective frequency and wave luminosity from approximately 1 to 6 s post-bounce.
Luminosities shown are 𝐿𝑤/𝐿𝜈 (solid), 𝐿𝑤 (dashed), and 𝐿trans, the luminosity still present in
the wind when the waves become nonlinear, i.e., equation (5.10) evaluated at 𝑟 = 𝑅𝑠 (dotted).
Convection increases rapidly near 𝑡 = 2 s, becomes most robust at about 2.5 s post-bounce, and then
gradually diminishes. We find lower wave luminosities and frequencies compared to Gossan et al.
(2020), as well as higher angular modes being excited. Comparing to Wang & Burrows (2024),
we find lower 𝐿𝑤, likely due to different calculation methods, but similar values of 𝐿trans at early
times. Our convective frequencies, however, are much higher than the ∼ 100 s−1 they report.

becomes fully and vigorously convective at 2 s post-bounce, with the convective luminosity peaking

at around 2.5 s post-bounce and subsequently tapering off. The wave luminosity as a fraction of

the neutrino luminosity remains above 10−3, which is high enough to substantially impact NDW

nucleosynthesis, through the end of the simulation. Where Gossan et al. (2020) predicted initial

wave luminosities of order 𝐿𝜈 out to 0.6 s post-bounce, we find 𝐿𝑤/𝐿𝜈 ∼ 10−3 at the start of our

model, with a peak value of 𝐿𝑤/𝐿𝜈 ≈ 10−2. The wave luminosity still present in the wind at the

point of nonlinearity (𝐿trans, Figure 5.7), approximately equal to the acoustic luminosity of Wang

& Burrows (2024), is in good agreement with their results at early times, though the absolute wave

77

104103102101100Luminosity123456Time (s)0510lcon102103104Frequency (s1)Lw/LLw (1051 erg/s)Ltrans (1051 erg/s)luminosities (our 𝐿𝑤, their 𝐿PNS) differ substantially, likely due to different calculation methods.

We also find convective frequencies of 𝜔 ≲ 103 s−1 for most of the simulation, slightly lower than

the 𝜔 ≳ 103 s−1 found by Gossan et al. (2020). These frequencies are substantially higher than

the ∼ 100 s−1 found by Wang & Burrows (2024) in 3D. Our models predict the excitation of more

angular modes, up to 𝑙 ≈ 10 at early times when the convective region is small, compared to the

maximum of 𝑙 = 6 predicted by Gossan et al. (2020). We attribute this to a difference in calculating

the maximum value of 𝑙: whereas Gossan et al. (2020) used the full width of the convective region

to determine 𝑙con, we only use the outermost pressure scale height, to maintain consistency with

our calculation of the other convective parameters.

5.4.2 NDW Dynamics

We now turn our attention to the winds launched from these proto-neutron stars. As the

properties of the PNS evolve over time, so too do the dynamics of the NDW. When convection is

neglected, all the relevant properties change very slowly and smoothly, and the NDW follows suit

(Figure 5.8). The entropy of the wind rises as the PNS contracts; the wind becomes more neutron-

rich as the electron antineutrino luminosity remains slightly higher than the electron neutrino

luminosity; and the average electron antineutrino energy remains much higher than that of the

electron neutrinos. Such neutron-rich conditions are not entirely unprecedented: Wang & Burrows

(2024) observed substantial amounts of ejecta with 𝑌𝑒 < 0.5, with trace amounts having 𝑌𝑒 < 0.4.

Given the low acoustic luminosities they report compared to the neutrino luminosities, it is unlikely

that this was caused by wave effects. The velocity profile, and thus the dynamical timescale of the

wind, also varies slowly with time. Even at 6 s post-bounce, the PNS radius remains noticeably

larger (∼ 14 km) than the 10 km assumed in chapter 3. This, coupled with the low PNS mass

of ≈ 1.35 𝑀⊙, results in overall lower wind entropies than we observed in chapter 3, which will

reduce the likelihood of heavy element nucleosynthesis. The likelihood of heavy nucleosynthesis

is quantified in the figures-of-merit shown in Figure 5.9. These figures-of-merit are functions of

entropy, dynamical timescale, and electron fraction calculated at the point of seed nucleus formation

(𝑅seed, where 𝑇 = 0.5 MeV). Hoffman et al. (1997) found that the r-process would likely extend to

78

Figure 5.8 Wind entropy (top left), electron fraction (top right), and velocity (bottom) from the
steady-state models as a function of time (post-bounce) and radius, with the PNS (𝑅PNS), wind
termination (𝑅WT), wave shock formation (𝑅𝑠), and seed formation (𝑅seed) radii highlighted, for
the non-convective z9.6 model. We observe generally low entropies that slowly increase with time,
accompanied by a decreasing electron fraction. The velocity profile does not change significantly
with time, aside from the movement of the wind termination shock.

79

246Time (s)1071081091010Radius (cm)RPNSRWTRseed101102103Entropy (kB/baryon)246Time (s)1071081091010Radius (cm)RPNSRWTRseed0.30.40.50.60.7Electron Fraction (Ye)246Time (s)1071081091010Radius (cm)RPNSRWTRseed1071081091010Velocity (cm/s)(a) Without convection

(b) With convection

Figure 5.9 Figures of merit for both r-processing (𝑠3𝑌 −3
𝑑 , Hoffman et al., 1997) and 𝜈p-processing
(𝑠2𝜏−1), with critical values for heavy, third-peak nucleosynthesis shown with dashed lines. No
heavy nucleosynthesis is expected for winds absent convection. When convection is included, a
window of heavier nucleosynthesis opens from 𝑡 ≈ 2 to 4 s. Though the entropy of the wind is
too low to allow third-peak r-processing, the figure-of-merit becomes high enough to expect some
second-peak production.

𝑒 𝜏−1

Figure 5.10 PNS mass loss rate due to the NDW, both with and without convection. In the absence of
convection, the mass loss rate is approximately monotonic, decreasing with time. When convection
is included, propagating waves drive a significant increase in mass loss, evinced by the jump in (cid:164)𝑀
near 𝑡 = 2 s as the PNS becomes fully convective.

80

246Time (s)104105106107p-process (S2/)1061071081091010r-process (S3/Y3e)246Time (s)104105106107p-process (S2/)1061071081091010r-process (S3/Y3e)246Time (s)105104103102M (M s1)Non-ConvectiveConvectiveFigure 5.11 Wind entropy (top left), electron fraction (top right), and velocity (bottom) from the
steady-state models as a function of time (post-bounce) and radius, with the PNS (𝑅PNS), wind
termination (𝑅WT), wave shock formation (𝑅𝑠), and seed formation (𝑅seed) radii highlighted, for
the convective z9.6 model. As the PNS becomes fully convective (𝑡 ≈ 2 s), the asymptotic entropy
rises, the electron fraction drops, and the wind is strongly accelerated. When convection becomes
vigorous, the waves shock well before seed formation takes place, but the radius of seed formation
moves steadily inward over time until the two are simultaneous near 𝑡 = 5 s. At this point, the shock
heating by the waves will have minimal impact on the nucleosynthesis, but the acceleration due to
the wave stresses will still have a strong effect on the relevant dynamical timescale.

81

246Time (s)1071081091010Radius (cm)RPNSRWTRsRseed101102103Entropy (kB/baryon)246Time (s)1071081091010Radius (cm)RPNSRWTRsRseed0.30.40.50.60.7Electron Fraction (Ye)246Time (s)1071081091010Radius (cm)RPNSRWTRsRseed1071081091010Velocity (cm/s)Figure 5.12 Time evolution of the wave transmission coefficients at the point of nonlinearity for
different angular modes. At early times, wave energy transmission is dominated by low 𝑙 modes.
When the PNS becomes fully convective at ∼ 2 s, the waves (in every mode) become nonlinear while
still in the gravitational branch, resulting in 100% of the initial wave luminosity being deposited in
the wind as heat.

the third peak for

𝑠3
𝑒 𝜏𝑑
𝑌 3
Similarly, the 𝜈p-process for synthesis of isotopes with 𝐴 ≳ 200 is likely to proceed for

≳ 8 × 109.

𝑠2
𝜏𝑑

≳ 2 × 107.

We see in Figure 5.9 that the figures-of-merit at the time of seed formation for both strong r-

processing and strong 𝜈p-processing are well below their threshold values, so we do not expect any

heavy element nucleosynthesis in this simulation. Since the wind is driven only by neutrinos, we

see the PNS mass loss rate due to the wind decreasing steadily with time (Figure 5.10). Absent

convection, a total mass of ≈ 6.9 × 10−5 𝑀⊙ is ejected in the wind.

The inclusion of convection dramatically changes the dynamics of the wind. While the prop-

erties of the PNS still vary slowly compared to the wind’s dynamical timescale, the convective

82

246Time (s)10121010108106104102100Transmission Coefficient ()12345678910Angular Mode (l)luminosity changes significantly as the PNS becomes fully convective at 𝑡 ≈ 2 s, and the wind

dynamics are altered accordingly (Figures 5.11). The vigorous convection drives the launched

waves to become nonlinear while they are still in the gravitational branch, so the full wave lu-

minosity acts to both accelerate and heat the wind (Figure 5.12). The asymptotic entropy of the

wind increases with time, again, as the PNS contracts, but the shape of the steady-state curves

are significantly altered by shock heating from the waves. The entropy at the time of seed for-

mation is highly variable, due to the sensitive balance between rapid cooling of the wind (due

to acceleration from the waves) and reheating of the wind (as the waves form shocks). Though

the asymptotic entropy of the wind is maximized at late times, the wind cools so quickly that the

seed nuclei form before the waves shock, and their energy deposition does not affect the ensuing

nucleosynthesis. With the inclusion of convection, the asymptotic electron fraction is no longer

monotonic in time, but becomes extremely neutron-rich near 𝑡 = 3 s due to the combined effects of

the high average electron antineutrino energy and the acceleration due to wave stresses, and then

slowly rises back toward 𝑌𝑒 = 0.5 as convection tapers off . The wind is strongly accelerated by

the waves very early on, well before either shock formation or seed formation take place, so the

dynamical timescale remains much shorter than the non-convective case after convection becomes

vigorous. The figures-of-merit for r- and 𝜈p-processing at the time of seed formation (Figure 5.9)

are overall much higher when convection is included, with the r-process figure coming within

a factor of 5 of the third-peak threshold. Though this wind is unlikely to produce a third-peak

r-process, we expect to find second-peak nuclides being synthesized. Based on these results, it is

highly likely that a wind from a more massive neutron star would be able to produce third-peak

elements. Both figures-of-merit are maximized from 𝑡 ≈ 2 to 4 s, and the wind is neutron-rich

after 2 s, so any heavy nucleosynthesis will be due to the r-process. Because the waves help to

accelerate the wind material, we see a much higher overall mass loss rate due to the wind (Figure

5.10). The rapid increase in (cid:164)𝑀 near 𝑡 = 2 s is driven by increased wave emission from the PNS

becoming fully convective. The total mass loss in the wind is higher when convection is included,

with ≈ 3.7 × 10−4 𝑀⊙ of material being ejected.

83

(a) Without convection.

(b) With convection.

Figure 5.13 Snapshot abundance profiles of the NDW at different times. The wind absent convection
makes a slow, smooth transition from an iron-peak dominated abundance distribution to a classic
weak r-process distribution, resulting in the integrated abundances seen in Figure 5.15. When
convection is included, the wind initially yields mainly iron-peak nuclei, and transitions to a
stronger second-peak r-process. At late times, an abundance pattern concentrated between mass
numbers 50 and 90 emerges, similar to that found by Bliss et al. (2018a) for low-entropy winds
from low-mass neutron stars. This gives rise to the sudden jump in ¯𝐴 seen in Figure 5.14.

5.4.3 Nucleosynthesis

While inspection of the dynamics of the wind can yield helpful predictions of what nucleosyn-

thesis will take place in the wind, the figures-of-merit were derived assuming a constant entropy

in the wind (Hoffman et al., 1997). When wave contributions are included, the entropy during

nucleosynthesis is no longer approximately constant, so detailed nucleosynthesis calculations are

needed. Following the procedure outlined in section 5.1, we calculate the nucleosynthesis for

each of the steady-state wind profiles generated from the STIR z9.6 models, and sum the resulting

abundance profiles weighted by the mass-loss rate of the corresponding steady-state wind. In the

non-convective case, we find good agreement with prior work (e.g., Arcones & Thielemann, 2012;

Wanajo, 2013): the wind undergoes an 𝛼-process in the slightly neutron-rich conditions found at

84

0255075100125150Mass number108107106105104103102Abundance123456Time (s)0255075100125150Mass number108107106105104103102Abundance123456Time (s)Figure 5.14 Average atomic mass number ¯𝐴 produced in the NDW as a function of time, both with
and without convection. In the absence of convection, ¯𝐴 remains below 20 at all times, increasing
with the figure-of-merit. When convection is included, ¯𝐴 follows the figure-of-merit, staying below
20, until approximately 𝑡 = 4 s. After this point, it rapidly rises to ¯𝐴 ≈ 80 and remains above 40 for
the duration of the wind.

early times, and as the wind becomes more neutron-rich, smoothly transitions to a weak r-process

producing first-peak nuclides (Figure 5.13). The average atomic mass synthesized in the wind

(Figure 5.14) rises smoothly as the wind becomes more neutron-rich with time. The integrated

nucleosynthesis (Figure 5.15) is dominated by the iron peak, with a significant amount of first-peak

r-process elements also being produced.

When convection is included, higher entropies and more neutron-rich winds yield heavier

nucleosynthesis. As Figure 5.13 shows, during the first, proton-rich second of the wind, the yield is

dominated by the 𝛼-process, which produces mainly iron-peak elements with ¯𝐴 ≈ 5 (Figure 5.14).

When convection becomes vigorous, the wind becomes extremely neutron-rich and produces a

second-peak r-process with ¯𝐴 ≈ 15. At late times, as convection tapers off and the wind gradually

becomes less neutron-rich, a peculiar abundance profile emerges with yields concentrated between

𝐴 = 50 and 90, with a much higher average atomic mass ¯𝐴 > 40. Figure 5.16 yields additional

insight into the nucleosynthesis. The two plots show the beginning (𝑇 = 0.5 MeV ≈ 5.8 GK) and

85

246Time (s)020406080100ANon-ConvectiveConvective(a) Without convection.

(b) With convection.

Figure 5.15 Integrated nucleosynthesis in the NDW, both with and without convection. A scaled plot
of the solar r-process abundances (Arnould et al., 2007) is overlaid for comparison. In the absence of
convection, we observe strong iron-peak production with a weak r-process producing only first-peak
elements, in agreement with prior work. When convection is included, nucleosynthesis proceeds
to much heavier elements, producing much larger amounts of both first- and second-peak r-process
material.

end (𝑇 = 2.5 GK) of the 𝛼-process, in which the seed nuclei for any possible r- or 𝜈p-process are

formed, for each timestep, i.e., each steady-state wind for which nucleosynthesis was calculated.

They show the abundances of free protons, free neutrons, 𝛼 particles, and heavy seed nuclei (𝐴 > 4).

When the peculiar abundance profile emerges at 𝑡 ≈ 4 s, we observe no radical change in these

abundances at the start of the 𝛼-process. The most notable change is a slight drop in the 𝛼 abundance,

accompanied by a rise in the seed abundance, which indicates a slightly larger number of seed nuclei

being produced. At the end of the 𝛼-process, however, we see a drastic change in the abundances of

free neutrons and 𝛼 particles. This indicates a strong 𝛼-process in which most of the 𝛼 particles are

being captured into heavier nuclei. Most of these 𝛼 particles, though, are not undergoing a triple-𝛼

reaction to form additional seed nuclei, since the seed abundance is unchanged at the end of the

𝛼-process. Rather, the 𝛼 particles and the free neutrons are capturing onto existing seed nuclei to

form a large abundance of heavier elements. The result is an abundance distribution concentrated

between 𝐴 = 50 and 90, and accompanied by some abundance of surviving 𝛼 particles, depending

86

050100150Mass Number (A)108107106105104103102101100AbundanceIntegrated  windSolar r-process (scaled)050100150Mass Number (A)108107106105104103102101100AbundanceIntegrated  windSolar r-process (scaled)(a) 𝑇 = 0.5 MeV.

(b) 𝑇 = 2.5 GK.
Figure 5.16 Abundances of free protons, free neutrons, 𝛼 particles, and seed nuclei (𝐴 > 4) at
the start (𝑇 = 0.5 MeV) and end (𝑇 = 2.5 GK) of seed nucleus formation in steady-state winds as
a function of time. The late-time 𝐴 = 50–90 abundance pattern at 𝑡 = 4 s arises from a strong
period of 𝛼-capture and neutron-capture reactions before charged-particle reactions freeze out. The
majority of these reactions are not the seed-forming triple-𝛼 but rather 𝛼 captures onto heavier
nuclei; the seed abundance remains approximately constant as the steady-state winds pass through
that temperature regime, while the 𝛼 abundance dramatically decreases. No proper r-process takes
place in these winds, as the free neutrons are almost entirely captured during the 𝛼-process phase.

87

123456Time1010108106104102100AbundanceYpYnYYs123456Time1010108106104102100AbundanceYpYnYYsFigure 5.17 Production factors of various p-nuclides of uncertain origin. Despite the wind being
neutron-rich for most of its duration, both the convective and non-convective models produce
significant abundances of p-nuclides through 94Mo, but produce little to none of the more massive
p-nuclides. The non-convective model, which remains generally closer to proton-richness, naturally
produces more of these nuclides.

on how efficiently they were captured. Bliss et al. (2018a) observed this abundance pattern for fast,

low-entropy, neutron-rich winds, and this matches well with the wind conditions we find at late

times. Though the asymptotic entropy of the late-time wind is fairly high, the entropy at the point of

seed formation is quite low, well below the entropy at seed formation of the non-convective model;

while the outflow speed in the convective case is much higher (Figures 5.8 and 5.11). Overall, the

integrated nucleosynthesis (Figure 5.15) in the convective case is dominated by the longer-lasting

r- and late-time 𝛼-processes, with strong production up to 𝐴 ≈ 130.

While our models do not account for the explosive nucleosynthesis taking place in the rest

of the supernova ejecta, it is of interest to compare our integrated abundance patterns with the

(scaled) solar r-process component (Arnould et al., 2007), which is overlaid in Figure 5.15. In the

88

74Se78Kr84Sr92Mo94Mo96Ru98Ru102Pd106Cd108Cd113In112Sn114Sn115Sn10141011108105102101104Production FactorNon-ConvectiveConvectivenon-convective case, the neutrino-driven wind appears to produce a significant peak near 𝐴 = 90,

which corresponds to a peak in the solar distribution, and synthesize significant amounts of lighter

r-process nuclei. Winds from proto-neutron stars with minimal convection could be a significant

source for these “weak r-process” elements, as others have observed (e.g., Wanajo, 2013; Wang &

Burrows, 2023). Though in both models the wind is neutron-rich for the majority of its lifespan,

the possibility of an incomplete 𝛼-recombination due to high wave luminosities (see chapter 4) and

the brief proton-rich stage of the convective model together allow for the synthesis of p-nuclides

such as 92,94Mo and 96,98Ru. The origin of these nuclei is an outstanding question (Rauscher et al.,

2013), as they cannot be produced by either of the traditional s- or r-processes. NDWs have been

explored as possible sites for their synthesis, and have been found to be plausible sources for the

p-nuclides up to 98Ru (Eichler et al., 2017; Bliss et al., 2018b). Figure 5.17 shows the integrated

production factors for various p-nuclides in both of our models. We find agreement with Eichler

et al. (2017), with significant production of 92,94Mo in both models, but negligible amounts of the

heavier p-nuclides due to the neutron-richness of the winds.

5.5 Conclusions

To circumvent the limitations of steady-state models in predicting nucleosynthesis yields from

realistic, time-dependent neutrino-driven winds, we modeled a suite of steady-state winds for

realistic time-varying PNS parameters, including convection, and integrated the nucleosynthetic

yields. The PNS parameters were sampled from a pair of supernova models run using STIR, a 1D

supernova code that uses MLT to simulate convection both in the primary explosion and in the PNS.

The models used the same progenitor, a zero-metallicity 9.6 𝑀⊙ star that explodes self-consistently

in spherical symmetry, with one model including convective effects and one excluding them. From

the STIR simulations, we find a number of interesting results:

1. In both the convective and non-convective models, the electron neutrino and antineutrino

average energies do not converge at late times. Rather, they maintain an approximately

constant offset of 5–6 MeV throughout the duration of the wind.

89

2. The equilibrium electron fraction in the wind predicted by the neutrino spectrum (Qian &

Woosley, 1996) is increasingly neutron-rich in both models.

3. The PNS becomes fully convective at approximately 2 s post-bounce, later than most 3D

models have explored.

Further STIR simulations will be helpful in determining whether the neutrino properties observed

in these models are typical of CCNSe, or an artifact of this unique progenitor. At the time of writing,

only the z9.6 models were available for analysis, but work is underway both to further improve

STIR and to generate new PNS models for analysis. The results will be detailed in forthcoming

papers (Couch et al., in prep; Nevins et al., in prep).

Using PNS parameters sampled from these models, we ran an ensemble of steady-state winds

and integrated the resulting nucleosynthesis. The neutrino spectra in both cases led to a neutron-

rich wind, and the presence of vigorous PNS convection and highly efficient transmission of the

ensuing gravito-acoustic wave energy to the NDW enhanced this effect.

In the non-convective

case, we observed an increasingly neutron-rich wind that approached 𝑌𝑒 = 0.4 at late times. A low

wind entropy, due to a light, low-compactness PNS, resulted in a classic weak r-process pattern

emerging from this model. Winds from such supernovae, with low-mass remnants and minimal

wave generation, could contribute significantly to the production of first-peak r-process elements

and some of the lighter p-nuclei such as 92,94Mo.

In the convective case, the PNS becoming fully convective at approximately 2 s post-bounce

led to the development of a very neutron-rich wind, with asymptotic electron fractions as low

as 𝑌𝑒 ≈ 0.35. Efficient energy deposition by convection-driven waves generated entropies high

enough to drive a strong second-peak r-process, but were insufficient to reach the third peak. If a

heavier, or more compact, PNS were to form with a similar neutrino spectrum, it is likely that the

higher wind entropy would be sufficient to drive a full, third-peak r-process. Simulations like those

of Wang & Burrows (2024) will prove invaluable in the continued examination of these findings,

especially as they are carried out to later times.

90

In the course of this work, a number of improvements were made to the STIR model as well,

which will be detailed in a forthcoming paper (Couch et al., in prep.). The neutrino spectra that

were observed in this work exhibit notable differences from prior work, especially in the behavior of

the average energies of the electron neutrinos and antineutrinos, which help produce the extremely

neutron-rich conditions found here. We eagerly await further advances in late-time 3D supernova

models to clarify these findings. Regardless, our results show the strong effect convection-driven

gravito-acoustic waves can have on the integrated nucleosynthesis of the NDW, including reduction

of the asymptotic electron fraction by as much as 20% and production of the second r-process peak

in a relatively low-entropy environment.

91

CHAPTER 6

SUMMARY AND FUTURE DIRECTIONS

6.1 Summary

In this work, we have examined gravito-acoustic wave generation by convection in the outer

layers of a proto-neutron star, and the effects resulting from the propagation of these waves into the

neutrino-driven wind. In chapter 1, an overview of core-collapse supernovae, PNS formation, and

the neutrino-driven wind were provided. Within the NDW, as matter freezes out from NSE, there

is a phase of 𝛼-recombination, followed by seed formation and then either an r- or 𝜈p-process. The

waves primarily affect 𝛼-recombination and seed formation, and we discuss in detail the physical

processes involved. The main effects of these waves, we predicted, would be an acceleration of the

wind due to the momentum flux of the waves, and an increase in entropy when the waves began

to shock. We predicted that these twin effects would cause a reduction in neutrino heating and a

consequent decrease in the asymptotic electron fraction of the wind.

In chapter 2, we derived a mathematical model to describe the neutrino-driven wind in the steady-

state, spherically symmetric approximation that is typically used. In this model, we incorporated

all the commonly modelled phenomena affecting the NDW: neutrino heating and its effects on the

electron fraction, corrections from general relativity, and the wind termination shock; as well as

the effects from the waves. We considered both the role of wave stresses in accelerating the wind,

and the entropy deposition that occurs when the waves form shocks. A set of numerical equations

were derived for simulating these winds.

In chapter 3, we used the numerical models developed in chapter 2 to explore the possibility

of r-processing in a slightly neutron-rich wind. For these models, we chose electron neutrino and

antineutrino average energies such that the equilibrium electron fraction in the wind, absent wave

effects, would match the best-case scenarios for neutron-rich winds in recent simulations. We

then examined how the wind dynamics and nucleosynthesis depend on the PNS mass, neutrino

luminosity, wave luminosity, and wave frequency. Our results indicated that a strong r-process

was possible and highly dependent on PNS mass and wave luminosity. If PNS convection were

92

strong, producing a wave luminosity of about 1% the neutrino luminosity, we predicted that a strong

r-process would emerge.

A potential r-process, however, is not the only nucleosynthesis of interest in the NDW. In

chapter 4, we performed a similar analysis of proton-rich winds, which recent simulations favor.

The electron neutrino and antineutrino average energies were tuned to produce a proton-rich

equilibrium electron fraction. Exploring the same PNS parameter space, we found that propagating

waves could have an important impact on the 𝜈p-process and allow nucleosynthesis up to 𝐴 ≳ 200

to take place. As in the neutron-rich case, the nucleosynthesis depends strongly on the PNS mass

and the wave luminosity. Even a modest wave luminosity allowed the 𝜈p-process to proceed to

fairly high mass numbers. At high wave luminosities, of order 1% the neutrino luminosity, we

found an interesting result: the waves formed shocks early enough in the wind to disrupt even the

earliest stages of nucleosynthesis, and a suppressed third-peak r-process took place despite neutrino

energies that predicted a proton-rich wind.

Having determined that propagating waves have a significant impact on nucleosynthesis even

for fiducial PNS parameters, in chapter 5 we sought to explore the time-integrated nucleosynthesis

of a realistic, evolving PNS. To do so, we extracted PNS properties from the supernova code

STIR for a certain progenitor star that explodes reliably in 1D. We used these properties to run

an ensemble of steady-state winds, the nucleosynthesis yields of which we then integrated. This

allowed us to make more robust predictions of the nucleosynthesis likely to result in the inner

ejecta of a core-collapse supernova. We found a curious result: the neutrino spectrum predicted

by STIR gave rise to extremely neutron-rich winds at late times, even when wave effects were

neglected. We also found that the energy transmission by the waves to the wind was far more

efficient than we first expected, so that the energy deposited by the waves approached 10% of the

neutrino luminosity. Even so, we did not observe a third-peak r-process due to the low wind entropy

during the early stages of nucleosynthesis. This was primarily due to the low PNS mass resulting

from this particular progenitor, well below the lowest mass considered in our parameter studies,

and its larger radius. Whereas we had assumed a 10 km PNS radius in our parameter studies, STIR

93

predicts an asymptotic radius close to 14 km. Such a low-compactness PNS resulted in much lower

wind entropy compared to our parameter studies, and we only observed second-peak r-process

production. Nevertheless, our results clearly demonstrate the key role PNS convection and wave

propagation can play in shaping the dynamics and nucleosynthesis of the neutrino-driven wind,

and the importance of well-resolving these effects in future simulations.

6.2 Future Research Directions

Our exploration of time-dependent winds in chapter 5 has opened some interesting pathways

for further study. Though a full r-process did not attain for the PNS we examined, our results

indicate that, if the neutrino spectrum we observed is typical, a full r-process could proceed for a

core-collapse supernova that produces a more massive remnant. Further exploration of integrated,

time-dependent nucleosynthesis from a variety of progenitors is needed: first, to confirm whether

the neutrino spectrum observed here is anomalous; and second, to determine if third-peak r-

processing (or if the neutrino spectrum is anomalous, 𝜈p-processing) is possible or likely in

realistic core-collapse supernovae. Also of interest is the nucleosynthetic yield of winds from other

STIR simulations. As mentioned in chapter 5, there are a number of nuclides of uncertain origin,

for which the NDW is a promising nucleosynthesis site. If proton-rich winds are observed in other

simulations, the strong wave luminosity could drive a robust 𝜈p-process that forms many of these

nuclides.

In the course of this work, STIR has undergone many improvements and is still under active

and vigorous development. We look forward to examining future STIR simulations in the manner

described in chapter 5. We may also consider relaxing the steady-state assumption in our derivation

of the wave effects, and directly including them in STIR. Nucleosynthesis could then be performed

using a tracer particle method to determine the full ejecta composition. We also eagerly await

future 3D exploration of late-time PNS evolution and neutrino wind behavior, which resolves the

wave effects discussed here.

94

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