# Constraints on the frequency and mass content of r-process events derived
from turbulent mixing in galactic disks

Anne Noer Kolborg Niels Bohr Institute, University of Copenhagen, Blegdamsvej
17, DK-2100 Copenhagen, Denmark  Department of Astronomy & Astrophysics,
University of California, Santa Cruz, CA 95064, USA  Enrico Ramirez-Ruiz
Department of Astronomy & Astrophysics, University of California, Santa Cruz,
CA 95064, USA  Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17,
DK-2100 Copenhagen, Denmark  Davide Martizzi Department of Astronomy &
Astrophysics, University of California, Santa Cruz, CA 95064, USA  Phil Macias
Department of Astronomy & Astrophysics, University of California, Santa Cruz,
CA 95064, USA  Melinda Soares-Furtado NASA Hubble Postdoctoral Fellow
Department of Astronomy, University of Wisconsin-Madison, 475 N. Charter
Street, Madison, WI 53703, USA

###### Abstract

Metal-poor stars in the Milky Way (MW) halo display large star-to-star
dispersion in their r-process abundance relative to lighter elements. This
suggests a chemically diverse and unmixed interstellar medium (ISM) in the
early Universe. This study aims to help shed light on the impact of turbulent
mixing, driven by core collapse supernovae (cc-SNe), on the r-process
abundance dispersal in galactic disks. To this end, we conduct a series of
simulations of small-scale galaxy patches which resolve metal mixing
mechanisms at parsec scales. Our set-up includes cc-SNe feedback and
enrichment from r-process sources. We find that the relative rate of the
r-process events to cc-SNe is directly imprinted on the shape of the r-process
distribution in the ISM with more frequent events causing more centrally
peaked distributions. We consider also the fraction of metals that is lost on
galactic winds and find that cc-SNe are able to efficiently launch highly
enriched winds, especially in smaller galaxy models. This result suggests that
smaller systems, e.g. dwarf galaxies, may require higher levels of enrichment
in order to achieve similar mean r-process abundances as MW-like progenitors
systems. Finally, we are able to place novel constraints on the production
rate of r-process elements in the MW, $6\times 10^{-7}{M_{\odot}/\rm
yr}\lesssim\dot{m}_{\rm rp}\ll 4.7\times 10^{-4}{M_{\odot}/\rm yr}$, imposed
by accurately reproducing the mean and dispersion of [Eu/Fe] in metal-poor
stars. Our results are consistent with independent estimates from alternate
methods and constitute a significant reduction in the permitted parameter
space.

††journal: ApJ

tablenum SIXtablenum

##  1 Introduction

The detailed physical ingredients required for r-process nucleosynthesis to
take place were identified in the pioneering works of Burbidge et al. (1957)
and Cameron (1957). Despite that, the dominant astrophysical site for
r-process production in the early Universe remains highly debatable (Cowan et
al., 2021), even after the landmark discovery of the kilonova associated with
GW170817 (Abbott et al., 2017; Coulter et al., 2017; Kasen et al., 2017;
Watson et al., 2019). To this end, metal-poor stars in the galactic halo can
be used as unique probes of r-process element synthesis in the early Universe
and could help elucidate the astrophysical assets of the dominant progenitor
system (Sneden et al., 2008; Thielemann et al., 2017).

Abundance similarities observed among metal-poor halo stars with ages
discrepant by billions of years but with a distribution that is representative
of the solar system (Sneden et al., 2003; McWilliam et al., 1995; Sneden et
al., 2008; Roederer et al., 2014) suggest that nuclear pathways responsible
for r-process elements are rather robust and have been operating coherently
over extensive periods of time in the assembly history of the Milky Way (MW).
From the large star-to-star chemical abundance dispersion in r-process
elements (such as Eu), with respect to the $\alpha$ elements (such as Mg), we
can infer that the injection of r-process elements occurs at a dramatically
diminished rate when compared to core-collapse supernovae (cc-SNe)(Wasserburg
& Qian, 2000; Fields et al., 2002). Additionally, the largest enhancements in
[Eu/Fe] observed in metal poor stars imply that individual r-process sites
need to synthesize a minimum of roughly $10^{-3}M_{\odot}$ r-process material
(Macias & Ramirez-Ruiz, 2018).

A central goal of this paper is to use a series of idealized hydrodynamic
simulations of the response of galactic disks to SN feedback and metal mixing
(Kolborg et al., 2022) to provide a deeper physical interpretation of the
star-to-star r-process scatter observed in the MW in the metallicity extent
[Fe/H] of approximately -3.0 to -1.5, where Fe enrichment is driven primarily
by cc-SNe (Hotokezaka et al., 2018; Côté et al., 2019; Wanajo et al., 2021).
The constraints derived from the abundance pattern of r-process elements in
the MW may ultimately help decipher the dominant production mechanism in the
early Universe.

The numerical modeling aimed at addressing the inhomogeneous enrichment of
r-process elements in the MW has been primarily performed in a cosmological
context (Shen et al., 2015; van de Voort et al., 2015; Naiman et al., 2018;
Haynes & Kobayashi, 2019; van de Voort et al., 2020, 2022). These studies are
limited by resolutions of a few tens to hundreds of parsecs, and they
inevitably involve “sub-grid” conjectures for SN feedback, star formation and
turbulent mixing. As such, large uncertainties remain at scales at which metal
injection and turbulent metal mixing are taking place. In preference, in this
paper we propose to model the gas interactions in small patches of a galaxy
disk simulations in order to effectively isolate the small-scale mixing
effects of metals at sub parsec scales from cc-SNe and r-process producing
events (such as neutron star mergers (NSM)) into the interstellar medium
(ISM). Besides being computationally workable (Kolborg et al., 2022),
investigating the distribution of r-process elements in these simulations and
comparing them extensively with observational data of metal poor stars can, in
turn, help constrain the frequency of events and the mass content of r-process
per event.

This paper is structured as follows. In Section 2, we summarize the simulation
setups and introduce the key parameters relevant for metal production in a
wide range of galaxy models. Sections 3 through 6 present the results of our
investigation. Initially, Section 3 shows the results of the turbulent mixing
study in a given galaxy potential. This section serves to introduce the reader
to the salient concepts and build an understanding of the key mixing processes
from which we build in subsequent sections. In Section 4, we present the
mixing results across the three galaxy potentials studied, while in Section 5
we investigate how changing the allowed height of r-process events influences
the mixing and ejection of r-process mass. Section 6 investigates the
production rate of r-process elements by studying how the spread of elements
changes in response to variations in the mass per event (Section 6.1) and the
relative rate of events (Section 6.2). This section concludes with a
comparison between simulations and observations of [Eu/Fe] abundance of metal
poor halo stars (Section 6.3). Section 7 gives a discussion of the
implications of our simulation results, while Section 8 provides a final
summary of our findings.

##  2 Numerical Methods

We use the hydrodynamic code RAMSES (Teyssier, 2002) to simulate the evolution
of gas in patches of galactic disks over timescales of hundreds of Myr. The
fundamental simulation setup is the same as that used in Martizzi et al.
(2016) and Kolborg et al. (2022), with the inclusion of a passive scalar field
in order to trace the enrichment of r-process elements. In this section, we
describe the salient aspects of the models as they pertain to this project.
For additional details, we refer the interested reader to Martizzi et al.
(2016) and Kolborg et al. (2022). The full suite of simulations used in this
study are listed in Table 1.

Galaxy model | $\Sigma_{\mathrm{SFR}}$ | $\rho_{0}$ | $z_{\mathrm{eff}}$ | $z_{\mathrm{SNe}}$ | $L$ | $dx$ | $T_{\rm end}$ | $\kappa$ | $z_{\text{NSM}}$ | $f_{\rm rp}$ | $m_{\rm rp}$
---|---|---|---|---|---|---|---|---|---|---|---
| M⊙ / kpc2 / Myr | g/cm3 | pc | pc | pc | pc | Myr | pc km/s | $z_{\text{SNe}}$ |  | $M_{\odot}$
MW progenitor, high SFR | $3\text{\times}{10}^{4}$ | $3.47\text{\times}{10}^{-23}$ | 40 | 100 | 1000 | 3.9 | 120 | 147 | 1.33 | $1\text{\times}{10}^{-3}$ | $1\text{\times}{10}^{-2}$
|  |  |  |  |  |  |  |  |  |  | $1\text{\times}{10}^{-1}$
|  |  |  |  |  |  |  |  |  | $1\text{\times}{10}^{-2}$ | $1\text{\times}{10}^{-2}$
|  |  |  |  |  |  |  |  |  | $1\text{\times}{10}^{-2}$ | $1\text{\times}{10}^{-3}$
|  |  |  |  |  |  |  |  | 1.0 |  | $1\text{\times}{10}^{-2}$
|  |  |  |  |  |  |  |  | 2.0 |  |
MW progenitor, low SFR | $1\text{\times}{10}^{3}$ | $2.08\text{\times}{10}^{-24}$ | 80 | 160 | 1000 | 3.9 | 250 | 213 | 1.33 | $1\text{\times}{10}^{-3}$ | $1\text{\times}{10}^{-2}$
Satellite | $8\text{\times}{10}^{2}$ | $4.67\text{\times}{10}^{-25}$ | 365 | 800 | 4000 | 15.6 | 500 | 1460 | 1.33 | $1\text{\times}{10}^{-3}$ | $1\text{\times}{10}^{-2}$
|  |  |  |  |  |  |  |  |  | $1\text{\times}{10}^{-1}$ |
|  |  |  |  |  |  |  |  |  | $1\text{\times}{10}^{-2}$ | $1\text{\times}{10}^{-2}$
|  |  |  |  |  |  |  |  |  | $1\text{\times}{10}^{-2}$ | $1\text{\times}{10}^{-3}$
Table 1: Overview of all the simulations presented in this project. The
columns are: name of the galaxy patch model, surface density of star formation
rate, mid-plane density initial condition, effective scale height of the
gaseous disk, maximum vertical height of a cc-SNe, box side length, cell size,
total evolution time, turbulent diffusion coefficient (Kolborg et al., 2022),
maximum vertical height of an r-process event (as a fraction of the maximum
cc-SNe altitude), relative rate of r-process to cc-SNe events, and mass of
r-process material per r-process event.

###  2.1 Galaxy models

The simulations assume a static gravitational potential produced by gas,
stars, and dark matter as described by Kuijken & Gilmore (1989). The
parameters of this gravitational potential are varied to emulate three
different types of galaxies (Kolborg et al., 2022).

The first model mimics an early MW progenitor with a high star formation rate
(SFR). The second model simulates a MW progenitor with a more modest SFR. This
second model is motivated by the recent findings of Wang et al. (2021), who
argue that the presence of the Large Magellanic Cloud satellite suggests a MW
progenitor with a less active SFR. The third model mimics a weak gravitational
potential with a high gas fraction and low SFR, emulating the expected
properties of a classical dwarf galaxy.

We employ cubic boxes111RAMSES does not yet allow tall box simulations with
periodic boundary conditions on the four edges perpendicular to the disk and
outflow boundary conditions on the two parallel ones. The resolution of each
simulation box is chosen such that the evolution of individual supernova
remnants (SNR) are always well resolved. Specifically the cooling radius is
resolved by at least 5 cells for least 94% of all remnants, in all galaxy
models (Kolborg et al., 2022).

These various setups of galaxy disk patches at relatively high resolution,
which capture the momentum injection of individual SNe, are able to
effectively replicate the local conditions of galactic environments that might
be representative of metal-poor stars assembled in the early MW and within
accreted dwarf satellites.

###  2.2 Core collapse supernovae and neutron star mergers

cc-SNe and r-process events are seeded randomly with a constant rate (see the
following section) and a flat distribution in space within fixed maximum
allowed heights from the center of the galactic disk.

We model cc-SNe using the sub-grid model for SN feedback implemented by
Martizzi et al. (2015). In this model, each cc-SN event has an ejecta mass
equal to the initial mass function (IMF) weighted average
($M_{\mathrm{ej}}=$6.8\text{\,}\,${M_{\odot}}$) and energy
$E=$1\text{\times}{10}^{51}\text{\,}\,$\mathrm{erg}$. Metals are introduced
into the gas by individual cc-SNe, which are assumed to be chemically
identical.

We apply the same sub-grid model of injection of mass and energy to the
r-process producing events, however, we selected $M_{\rm
ej}=$1\text{\times}{10}^{-2}\text{\,}\,${M_{\odot}}$ and $E=
$1\text{\times}{10}^{51}\text{\,}\,$\mathrm{erg}$. These values reflect the
properties of the ejecta inferred in the gravitational-wave triggered NSM
event GW170717 (Kasen et al., 2017).

Although the morphology of a NSM remnant is highly asymmetric at early times
(Rosswog & Ramirez-Ruiz, 2002; Ramirez-Ruiz & MacFadyen, 2010; Roberts et al.,
2011), the subsequent radiative evolution is notably analogous to that of a
SNR with similar total energy (Montes et al., 2016). The shell formation
epoch, which occurs when the remnant becomes radiative, takes place by the
time the mass of swept-up material reaches $M_{\rm
c}\approx{10}^{3}{({n}_{{\rm{H}}}/1\,{\mathrm{cm}}^{-3})}^{-2/7}{M}_{\odot}$
(Karpov et al., 2020; Macias & Ramirez-Ruiz, 2018; Cioffi et al., 1988;
Thornton et al., 1998; Martizzi et al., 2015). The implications of this are
twofold. First, sub-grid SN feedback models, like the ones used in this study,
can be effectively used to resolve the key evolutionary phases of NSM remnants
(Montes et al., 2016). Second, our r-process injection models can be applied
whether enrichment has occurred via extremely rare cc-SNe (e.g., Cowan &
Thielemann, 2004; Winteler et al., 2012; Nishimura et al., 2015; Mösta et al.,
2018; Siegel et al., 2019), or through NSMs (e.g., Metzger et al., 2010;
Roberts et al., 2011), provided that any contribution of other freshly-
synthesized metals (i.e., non r-process) is less than those contained in the
swept-up ISM mass by the time the blast wave reaches the cooling phase (Macias
& Ramirez-Ruiz, 2019). For the purposes of this study we focus our attention
on NSMs as the source of r-process elements.

The maximum allowed height of standard cc-SNe events ($z_{\rm SNe}$) is fixed
to twice the scale height of the gaseous disk. Fixing the allowed height of
the cc-SNe to the gaseous scale height is motivated by the typical short
distances traveled by massive stars between the time of their birth and the
SNe. The r-process events, on the other hand, are allowed to occur within a
region defined by the parameter $z_{\rm NSM}$. As a starting point, we set
this value to $z_{\rm NSM}=1.33z_{\rm SNe}$. In Section 5, we explore how
changes to this scale height impact the turbulent mixing process, and we
discuss the limitations of our galaxy patch simulations.

####  2.2.1 Event and metal injection rates

The surface density rate of cc-SNe, $\Gamma$, is set by the rate of star
formation in the galaxy patch, such that $\Gamma=\frac{\dot{\Sigma}_{\rm
SFR}}{$100\text{\,}\mathrm{M}_{\odot}$}$. The star formation rate (and hence
the cc-SNe event rate) is assumed to be constant. Due to the relatively short
time span and the physical extent of the galaxy patches modelled, one can
think of this as modelling a short star formation burst early in the history
of the galaxy.
The rate of NSM events is set as a rate relative to standard cc-SNe,
$\Gamma_{\rm NSM}=f_{\rm rp}\Gamma$. In this study we explore relative rates
of $f_{\rm rp}$ in the range [$10^{-3},10^{-2}$], which is equivalent to 1 NSM
event per [100,1000] cc-SNe.

For the purposes of this project, we are interested in studying the spread of
r-process elements relative to elements produced by cc-SN. We select Fe as our
representative cc-SNe element, with each cc-SNe yielding $M_{\rm
Fe}=$8\text{\times}{10}^{-2}\text{\,}\,${M_{\odot}}$ (Kolborg et al., 2022).
We model r-process events as NSMs and assume that the entire ejecta is
predominately comprised of r-process elements with a total mass $m_{\rm
rp}=M_{\rm ej,NSM}$. That is, r-process events are assumed not to contribute
to Fe enrichment, in our simulations.

As commonly carried out by the community, we designate Eu as the
representative r-process element. We calculate the mass of Eu by applying the
solar r-process abundance pattern (Asplund et al., 2009; Sneden et al., 2008)
and calculating the mass fraction of Eu to all the r-process elements.

This is particularly useful when comparing with observations of metal-poor
stars (Sneden et al., 2008). For the purpose of this study, we consider atomic
mass number 69 as the strict lower limit for neutron capture element
production ($A_{\rm min}=69$). In Section 7.1, we discuss the implications of
this choice and the consequences for the results if $A_{\rm min}=90$, which
corresponds to the second and third r-process peaks (Sneden et al., 2008).

$\alpha$, iron peak and r-process elements are tracked as individual passive
scalar fields. For the purposes of this study only iron and r-process elements
are of interest and the individual scalar fields allow us to make adjustments
to $M_{\rm Fe}$ and $m_{\rm rp}$ in post-processing. This approach allows us
to simulate a wide range of r-process masses per event, $m_{\rm
rp}=[10^{-3},10^{-1}]{\,M_{\odot}}$, studying their effects on the metal
enrichment of the ISM.

In Kolborg et al. (2022) we showed that the metals injected into the ISM by
these cataclysmic events are mixed through the galactic disk by turbulent
diffusion, which is in turn driven by the energy and momentum deposition from
the most common events. The turbulent diffusion coefficient sets the timescale
for metal mixing, which depends mainly on the scale height of the disk and the
turbulent velocity dispersion (Kolborg et al., 2022). Martizzi et al. (2016)
conducted resolution tests on the velocity dispersion in the MW progenitor
with high SFR set-up and found that the turbulent diffusion coefficient is
effectively converged at the resolution employed in this work. Furthermore,
the turbulent driving scales in the MW progenitor models are
$\gtrsim$100\text{\,}\mathrm{p}\mathrm{c}$$ in both galaxy set-ups (Martizzi
et al., 2016). This characteristic scale is nearly two orders of magnitudes
greater than the resolution size scale of our models and, as a result, we
expect the turbulent cascade to be well resolved.
The advantage of galaxy patch simulations is that they allow us to effectively
capture the driving of the turbulence by cc-SNe, as well as, accurately
studying its effects on the mixing of other freshly synthesized metals that
are produced by much rarer cataclysmic events. The simulations are designed to
isolate the mixing driven by cc-SNe feedback, which is primarily driven by
turbulence and by galactic wind launching. By design, other important sources
of mixing in galaxies, such as gas inflow, recycling of gas in galactic
fountains, and disc shearing are not present in these simulations.

###  2.3 Steady State

All simulations are initialized with the gas in hydrostatic equilibrium with
the static gravitational potential. As cooling and SNe feedback turn on
simultaneously some of the thermal pressure support falls and turbulent
pressure support increases. Hence, at early times, the disk is primarily
supported by thermal pressure, while at later times, the disk is supported by
both thermal and turbulent pressure driven by cc-SNe. Since information needs
to be effectively communicated to all regions before turbulent motions in the
bulk of the disk reach a statistical steady state, an initially transient
phase is produced. The duration of this phase is usually well described by the
characteristic relaxation time in our disk models, $t_{r}=4z_{\rm
eff}{\langle\sigma_{v}\rangle}^{-1}$, where $\langle\sigma_{v}\rangle$ is the
time averaged, mass-weighted velocity dispersion of the gas (Kolborg et al.,
2022) . In what follows, we neglect this initial phase, which lasts
$\approx{1\over 5}T_{\rm end}$ in all of our simulations (Table 1). In this
work, we only discuss the results from the steady state of the simulations.

##  3 Metal Mixing in high SFR environments

In this section, we examine the mixing of elements in the MW progenitor model
with high SFR. For simplicity, we limit our attention to a single instance of
this model where r-process events take place at a rate relative to cc-SNe of
$f_{\rm rp}=10^{-3}$ and each event yields a mass of r-process elements of
$m_{\rm rp}=10^{-2}\,M_{\odot}$. This assumes that cc-SNe occur in a region of
$z_{\rm SNe}=$100\text{\,}\mathrm{p}\mathrm{c}$$, while the r-process events
occur in the region defined by $|z|\leq 1.33z_{\rm SNe}$. We use this
representative model to present a detailed account of the metal mixing
features that may be present in the interstellar gas. These salient features
can result naturally when one examines metal turbulent mixing by events
occurring at disparate rates, yet their exact properties depend on the
specific galaxy model and more sensitively on $f_{\rm rp}$ and $m_{\rm rp}$.

###  3.1 The Tomography of a Galaxy Patch

Figure 1: Density-weighted projection plots of the thermodynamic proprieties
of the gas in the MW progenitor with high SFR (left-hand column: density;
right-hand column: temperature). The top row presents an edge-on view of the
galaxy disk, while the bottom presents a face-on view. The SNe drive turbulent
mixing in the disk and a galactic wind. Individual SNR are clearly visible in
the plane of the disk. Figure 2: Density weighted projection plots of the
metal mixing properties of the gas in the galaxy patch simulation shown in
Figure 1. The top row presents an edge-on view of the galaxy disk, while the
bottom presents a face-on view. The panels show the abundances of [Fe/H]
(left), [Eu/H] (middle) and [Eu/Fe] (right). Both Fe and Eu are efficiently
ejected from the galactic disk and drastically increase the metal abundance in
the galactic wind region. The effects of the relative rate of metal injection
on the metal mixing of Fe and Eu are clearly seen; the less frequently
injected Eu is less evenly distributed than Fe, which is injected more
frequently.

In Figure 1 we show the density-weighted projections of the thermodynamic
properties of the gas along two axes in the simulation box. The density (left-
hand column) and temperature (right-hand column) projections along an axis
perpendicular to the galactic disk (top row) clearly illustrate how the
momentum deposition from cc-SNe ejects gas from the disk in the form of a hot,
rarefied galactic wind. From another point of view, the projection along the
galactic disk plane (similar to a face-on view of the galaxy, bottom row)
distinctively shows individual cc-SN and r-process remnants. They can be
observed as nearly spherical regions of lower density and higher temperature
gas, which are embedded in the denser, cooler, star forming gas.

In addition to momentum and energy, the individual events inject new metals:
iron peak elements in the case of cc-SNe and heavy metals in the case of
r-process events. These metals are mixed into the surrounding ISM by the
turbulent diffusion, which is driven primarily by cc-SNe. Figure 2 shows the
density-weighted projections of the metal abundances in the gas for both Fe
and Eu enrichment.

Figure 3: Temporal evolution of the volume weighted mean and $1\sigma$ spread
of the abundances of [Fe/H] (left), [Eu/H] (middle) and [Eu/Fe] (right) for
the MW progenitor with high SFR. The global values refer to the mean and
spread when considering all the gas in the simulation volume, while the cold,
dense, disk material is defined by gas located at $|z|\leq z_{\rm eff}$ and
with the following thermodynamic properties:
$T\leq$1\text{\times}{10}^{4}\text{\,}\,$\mathrm{K}$ and
$\rho\geq\rho_{4}/4\approx$1\text{\times}{10}^{-23}\text{\,}\,$\mathrm{g\,cm^{-3}}$.
The spread in [Fe/H] is always significantly less than the spread of [Eu/X]
due to cc-SN metals being injected at much higher rates.

Several key aspects can be noted by examining the maps shown in Figure 2.
First, the rarefied galactic wind is highly enriched in Fe and Eu when
compared to cold, dense gas in the mid-plane. This is primarily due to the low
density of the hot wind within this region, which implies that even the mixing
of a small amounts of metals can lead to a comparatively high enrichment.

Second, the abundance of [Fe/H] is relatively uniform across the disk and
within the galactic wind region. The full span of [Fe/H] variations is
generally small. Across the disk, we observe localized [Fe/H] variations where
neighboring cc-SN remnants remain compact and fail to overlap (at scales of a
few tens of pc). These localized Fe enhancements are then subsequently
smoothed out as the enriched gas is given more time to expand and mix
throughout the disk (Kolborg et al., 2022). Yet, individual remnants always
expand to characteristic lengths that are smaller than the disk scale height.

Lastly, [Eu/H] is inspected in the middle panels of Figure 2 to exhibit
significantly larger scatter in abundance concentrations than [Fe/H] in the
left panels. This originates from the comparative rarity of r-process metal
injection, which naturally produces a chemically inhomogeneous and unmixed ISM
at these early epochs. As more events are injected and r-process material
diffuses and mixes, we can expect these metal concentrations to be gradually
smoothed out.

###  3.2 Metal mixing as a function of time

From visual inspection of the abundance maps alone, we can gain a clear
insight regarding the differential distribution of Eu and Fe in the galactic
disk. In this section, we present a quantitative description of these
differences.

Figure 3 depicts the volume weighted mean and $1\sigma$ spread of the gas
abundances when considering all the gas in the simulation volume (global) and
the cold, dense gas in the disk. For reference, the snapshots depicted in
Figures 1 and 2 correspond to a simulation time $57.4{\rm Myr}\approx
0.475T_{\rm end}$.

Our scheme does not explicitly include star formation so, for the purpose of
making an informed matching, we isolate the cold, dense gas within the disk
($|z|\leq z_{\rm eff}$), that is the most apt to form stars. We impose a
temperature boundary of $T\leq 10^{4}\,\mathrm{K}$, which is related to the
cooling function temperature floor, and a density limit of
$\rho\gtrsim\rho_{0}/4$ (see Table 1).

As the simulation evolves, the [Fe/H] within the disk is gradually enriched as
ensuing cc-SNe increase the mean Fe metallicity of the gas. At the same time,
cc-SNe feedback drive outflows which cause Fe to spread from the disk onto the
rarefied wind. The volume weighted average tends to emphasize the abundance of
the hot, rarefied gas, which has a large volume filling factor. This becomes
evident when examining the underlying distributions at fixed times shown in
Appendix A. The global gas distribution of [Fe/H] has a very narrow peak at
high metallicities (which drives the mean value) and a fairly long tail at
lower metallicities (which influences the spread calculation). By contrast,
the global and disk [Fe/H] distributions are rather similar when considering
the mass weighted distributions. A similar behavior is seen for the [Eu/H]
distributions, albeit with much broader ”peaks” given the lesser degree of
metal mixing which results from less frequent injections.

Initially, the metal distributions of both [Fe/H] and [Eu/H] are highly
inhomogeneous because there are significant regions that contain unmixed
material. Due to its much more frequent injection, [Fe/H] is more uniformly
distributed than [Eu/H]. The impact of higher metal injection rates on the
abundance of [Fe/H] is twofold. First, the spread in the distribution is much
narrower than the one seen for [Eu/H] in both the global and the cold, dense
gas at all times in the simulation. In other words, there is a higher degree
of homogenization of Fe than Eu as cc-SN products migrate and mix across the
disk more effectively. Second, the evolution of the mean metallicity with
time, which is caused by the rate of metal injection, is much swifter for
[Fe/H] than for [Eu/H]. As time evolves, the localized inhomogeneities in both
[Fe/H] and [Eu/H] are smoothed out and the evolution of the mean abundance
becomes more gradual as each new injection of metals contributes progressively
less to the total metal content. Yet r-process products migrate and mix
throughout the box much more slowly. This is because large amount of freshly
synthesized r-process material are injected at a rate that is usually faster
than the the rate at which turbulent mixing smooths them down.

###  3.3 Mass loading factor

Through injection of energy and momentum, cc-SNe drive material out of the
disk and launch galactic winds. In this section, we consider the mass loading
factor of the wind, which is commonly defined as (e.g. Martizzi et al., 2016;
Li & Bryan, 2020):

$\eta(z)=\frac{\dot{M}_{\mathrm{out}(z)}}{\mathrm{SFR}},$ (1)

$\dot{M}_{\mathrm{out}}$ denotes the rate at which mass is leaving the galaxy.

Analogously, the metal mass loading factor is defined as the ratio of the mass
in metals leaving the galaxy to the ratio of mass of metals that are injected
(Li & Bryan, 2020):

$\displaystyle\eta_{Z_{i}}(z)$
$\displaystyle=\frac{\dot{M}_{Z_{i},\mathrm{out}}(z)}{\dot{M}_{Z_{i},\mathrm{inj}}}.$
(2)

Here we calculate the rate of metal injection as

$\displaystyle\dot{M}_{Z_{i},\mathrm{inj}}=\dot{n}_{\mathrm{SNe}}M_{\mathrm{ej}}y_{Z_{i}}f_{Z_{i}}.$
(3)

$y_{Z_{i}}$ is the fractional yield of element $Z_{i}$ produced by a
particular event ejecting a mass $M_{\rm ej}$, so that $M(Z_{i})=M_{\rm
ej}y_{Z_{i}}$. The parameter $f_{Z_{i}}$ denotes the fraction of events that
produce metal $Z_{i}$ relative to the rate of cc-SNe. For cc-SN elements
$f_{Z_{i}}=1.0$, while for r-process elements $f_{Z_{i}}=f_{\rm rp}$.

The galaxy patch set-up is uniquely equipped to study the launching of
galactic winds, which develop naturally from SNe feedback (Martizzi et al.,
2016) and require no sub-grid model. We estimated the rate at which mass
leaves the disk by measuring the mass flux that streams through a surface
parallel to the disk with an out-flowing z-velocity. We measure the outflow
rate at three different heights. In order of increasing height from the disk
midplane, the chosen distances are as follows: $|z|=3z_{\rm eff}$,
$|z|=2z_{\rm SNe}$, and $|z|=\frac{L}{2}-dx$.The last measurement height is
one cell size away from the edge of the simulation domain. The physical size
of the relative heights are listed in Table 1 for each galaxy model.

It is important to note that the local boxes we consider in this study do not
have a clearly defined escape velocity (Martizzi et al., 2016) and the wind
mass loading factor is observed to decline with increasing box height.
Martizzi et al. (2016) studied SNe feedback in the galaxy patch simulations
and argue that the structure of the galactic winds are not always well
captured by these boxes. However, in the case of the MW progenitor with high
SFR simulation (equivalent to their model FX-ULTRA-MW-L8), they found that the
global wind properties are well modelled within
$|z|\lesssim$200\text{\,}\mathrm{p}\mathrm{c}$$, which is equivalent to
$2z_{\rm SNe}$ for that model. Therefore, we proceeded with the analysis of
the wind loading, focusing on results from heights comparable to this value.
Moreover, Li & Bryan (2020) compared loading factors from a wide range of
simulations, including the work by Martizzi et al. (2016), and found similar
values reported across different global and local studies. This lends further
credence to the robustness of the derived loading factors in this study.

Figure 4 shows the evolution of the loading factors relating to the total
mass, cc-SN elements, and r-process elements during the steady state of the
simulation. The mass loading factor has a nearly constant value throughout the
evolution of the simulation, with values that are consistent with the results
of Martizzi et al. (2016). The evolution of loading factor of Fe closely
traces $\eta_{M}$. This seems natural to expect for a galactic wind driven
primarily by momentum injection from the same cc-SNe that are also injecting
Fe. Interestingly, $\eta_{Z_{\rm iron}}$, indicates that only about $\approx
10$% of the injected metals are incorporated into the galactic winds. The
evolution of $\eta_{Z_{\rm rp}}$, on the other hand, is much less smooth and
contains large metal outbursts. The location of these outbursts closely follow
the injection timing of r-process events, which are marked by dark vertical
lines in Figure 4.

The rise of r-process material in the wind takes place near the local
injection sites, which produce large amounts of heavy elements. This freshly
synthesized r-process material does not have time to mix effectively within
the disk before being expelled. The smoothing of the individual outburst with
$z$ is caused by r-process metals further mixing with wind material. It is
noteworthy that the loading factors of cc-SN and r-process elements are very
similar to each other at all times in the simulation, suggesting that
r-process metals are carried out of the disk by winds driven by cc-SNe only
after mixing significantly with the ISM. On average, the mass loading factor
of iron elements is $\approx 7.5$%, while the mass loading factor of r-process
elements is $\approx 11$% (both measured at $|z|=2z_{\rm SNe}$). The slightly
higher mass loading factor implies that r-process elements are retained
slightly less effective than cc-SN elements. This is closely related to the
r-process metal outbursts captured in Figure 2, which occurs because
individual r-process events do not have time to mix effectively with the
surrounding ISM before being expelled from the disk.

Figure 4: Evolution of the loading factors of the total, iron and r-process
mass (see legend) as a function of time in the MW progenitor with high SFR.
The loading factor is shown for two different heights in the potential:
$|z|=3z_{\rm eff}$ (solid lines) and $|z|=2z_{\rm SNe}$ (dashed lines). The
dark vertical lines indicate the timing of r-process injections in the disk.
The iron loading factor evolves very similarly to the total mass loading
factor albeit with a different normalization. The r-process loading factor is
strongly correlated with the injection of new r-process events. Interestingly,
the loading factors of iron and r-process elements are comparable to each
other, suggesting that both metal groups are carried away at similar rates
irrespective of their very different injection rates.  Figure 5: The temporal
evolution of the volume weighted mean and $1\sigma$ of [Fe/H] (left), [Eu/H]
(middle), and [Eu/Fe] (right) in each of the three galaxy potentials. From top
to bottom we present the results for the MW progenitor with high SFR, the MW
progenitor with low SFR, and the satellite model. The abundances were
calculated for all the gas in the box (global) and for the cold, dense gas in
the disk as done in Figure 3. Overall, the evolution of the abundances are
similar in all three galaxy potentials. Yet, the difference between the global
and the cold, dense gas mean values is smaller for the weaker galactic
potentials, implying larger volume filling factors of enriched gas in all gas
phases.

##  4 Metal Mixing in different galaxy types

In this section, we consider the influence of the galaxy potential on the
turbulent mixing of the freshly synthesized metals. To facilitate comparison,
we assume standard values for all parameters that do not relate to the galaxy
type. More explicitly, the relative rate of NSM to cc-SN events is set to
$f_{\rm rp}=10^{-3}$, the mass per event is set to $m_{\rm
rp}=10^{-2}\,M_{\odot}$, and the scale height of NSM is set to $z_{\rm
NSM}=1.33z_{\rm SNe}$.

Figure 6: Loading factors of total, iron and r-process mass (see legend)
measured at $|z|=2z_{\rm SNe}$ for three different galaxy potentials: MW
progenitor with high SFR (left); MW progenitor with low SFR; and satellite
galaxy (right). The dark vertical lines along the bottom of each panel denote
the times at which r-process events take place. The times are normalized to
the run times of the simulations (see Table 1). The initial phase (before
steady-state is reached) is indicated by lower opacity color. The loading
factors are generally larger for weaker gravitational potentials. The weaker
gravitational potentials also display smaller differences in loading factors
at greater distances from the disk (see Appendix B), indicating the material
streams from the disk more easily.

###  4.1 Abundance evolution

Figure 5 illustrates the evolution of the (volume-weighted) mean and spread of
the [Fe/H], [Eu/H], and [Eu/Fe] abundances as a function of time in each of
the three galaxy potentials. Similar to Figure 3, the mean and spread are
shown for all the gas in the box (global) and the cold, dense gas within the
disk. For all galaxy types, the global spread is observed to be larger than
the spread of the cold, dense gas. We also see a significant offset between
the mean abundance of the global gas and the cold, dense gas for all galaxy
models. This offset is the smallest for the satellite model and is related to
the lower gas density in the disk, which naturally leads to a higher volume
filling factor of the enriched material in both the wind and the disk regions.

As a whole, galaxy models with lower SFRs take longer to achieve the same mean
[Fe/H] metallicity. As such, Fe is given more time to migrate throughout the
disk. This is because the lower-density gas associated with a low SFR allows
each cc-SN remnant to reach larger physical sizes than remnants in higher-
density environments. As such, galactic disks with lower SFRs show less spread
in abundances at a comparable average [Fe/H] metallicity. As anticipated, the
spread of [Eu/H] is generally larger than that of [Fe/H], particularly in the
cold, dense gas phase.

In contrast with [Fe/H], the [Eu/H] spread is primarily driven by the relative
rate of r-process to cc-SN events (which is the same in all simulations) and
is less dependent on the SFR. It is compelling to note that the evolution of
[Eu/H] for the MW progenitor with low SFR is rather odd (see middle panel in
Figure 5). This transpires because the SFR is so low in this model that the
galaxy patch is host to only two r-process events over its entire evolution.
As a result, very metal poor stars in this model will not be enriched with any
r-process material until after the first event takes place. This behavior
clearly illustrates that galaxies with very low global SFRs might not be
enriched with r-process material, as it is commonly expected to be the case
for ultra faint dwarf (UFD) galaxies in the MW halo. Most of these systems are
very metal poor and exhibiting low r-process enhancements (Cowan et al.,
2021). A notable exception is Reticulum-II, which has been shown to contain
several highly enriched stars (Ji et al., 2016a, b, 2022; Roederer et al.,
2016). In this context, it is interesting to consider the likelihood that any
one small system hosts an r-process event given the very low SFRs associated
with these systems.

###  4.2 Mass loading factors

One of our goals in this study is to understand the effects that turbulence
driven by cc-SNe in galactic disks has on metal mixing. cc-SNe increase the
mean metallicity of the gas in the disk and drive turbulent mixing, which
prompts the metals to diffuse across the disk. At the same time, the momentum
and energy that goes into driving turbulence launches galactic winds and
drives metals out of the disk. Figure 6 shows the temporal evolution of the
loading factors of mass, iron, and r-process metals for the three different
galaxy potentials. The loading factors are all measured at $|z|=2z_{\rm SNe}$
(see Table 1). As expected, the mass loading factor is significantly larger
for weaker galaxy potentials, highlighting the greater relative importance of
cc-SN energetics in these systems. This result is consistent with what is
frequently ascertained in dwarf galaxy simulations (e.g., Fielding et al.,
2017).

During the simulations, the satellite galaxy model and the MW progenitor with
low SFR release $\approx 35$% and $\approx 10$% of their initial mass to cc-
SNe-driven winds, respectively. In contrast, the MW progenitor model with high
SFR loses $\lesssim$5% of its initial mass (Kolborg et al., 2022). This mass
loss is naturally explained by the mass loading factors observed for these
less massive systems. The associated larger mass loading factors in low mass
systems are, for example, necessary in order to interpret the observed galaxy
stellar mass function (e.g., Li & Bryan, 2020).

Across the different galaxy potentials, the loading factor of iron and
r-process elements are remarkably similar to each other, with a major caveat
being that the MW progenitor with low SFR undergoes very low r-process mass
loading before the first event occurs at $t\approx 0.6T_{\rm end}$.

Models with fewer cc-SNe give the freshly synthesized r-process metals more
time to diffuse and travel across the disk and, as a result, show less
prominent r-process outbursts when compare with the MW progenitor with high
SFR. The average mass loading factor of iron and r-process elements in
satellite galaxy model (MW progenitor with low SFR), at $|z|=2z_{\rm SNe}$,
are $\approx 70$ % ($\approx 30$ %) and $\approx 89$% ($\approx 22$ %),
indicating significantly less retention of metals in the ISM than the MW
progenitor with high SFR. In essence, higher mass injection rates of r-process
material are necessary in a satellite galaxy in order to average [Eu/H]
abundance comparable to a more massive system.

##  5 Varying the scale heights of r-process injection and its effect on metal
mixing

Here we examine the impact of altering the scale height of r-process metal
injection sites on metal mixing. The galaxy patch setup does not lend itself
well to studying large offsets from the disk (e.g., Rosswog et al., 2003;
Zheng & Ramirez-Ruiz, 2007; Kelley et al., 2010; Behroozi et al., 2014; Zevin
et al., 2022), as would be expected for systems with long merger timescales
($t_{\rm delay}$) and large velocity kicks ($v_{\rm kick}$). This setup can,
however, be used to probe the impact of modest event offsets, $\bar{z}\approx
140(t_{\rm delay}/10{\rm Myr})(v_{\rm kick}/20{\rm km\,s^{-1}})$ pc, on the
subsequent mixing efficacy, as could be envisaged for r-process production in
fast-merging double neutron stars (Ramirez-Ruiz et al., 2015; Safarzadeh et
al., 2019) or rare cc-SNe (Winteler et al., 2012; Nishimura et al., 2015;
Mösta et al., 2018). These prompt channels are commonly argued to be more
effective at enriching very metal poor stars with r-process products (e.g.,
Siegel et al., 2019; Cowan & Sneden, 2006), which is the central focus of this
study.

In this section, we present results for three different simulations. For these
runs, we use the same galaxy type. We select the MW progenitor with high SFR
in view of the fact that, as we argued in Section 4, r-process products in
this model are given the smallest amount of time to migrate throughout the
disk. As such, the effects of varying the scale height of metal injection
should be particularly obvious for this model. The relative rate and mass per
event of r-process events remain unchanged. That is, $f_{\rm rp}=10^{-3}$ and
$m_{\rm rp}=10^{-2}\,M_{\odot}$. Yet, the maximum allowed height of NSM is
varied between the following three scale heights: $z_{\rm NSM}=z_{\rm SNe}$,
$z_{\rm NSM}=1.33z_{\rm SNe}$, $z_{\rm NSM}=2z_{\rm SNe}$. To facilitate
comparison, the injection procedure for cc-SNe remains the same in all
simulations.

###  5.1 Mass loading of r-process elements

It is natural to expect that the event scale height will influence the rate at
which r-process material is lost via galactic winds. We explore this
assumption here by examining the consequences of increasing $z_{\rm NSM}$ on
$\eta_{Z_{\rm rp}}$.

We expect $\eta_{M}$ and $\eta_{Z_{\rm iron}}$ to remain unchanged in these
models as cc-SNe, whose injection properties are the same in all models,
dominate the energy and momentum injection in the disk. In essence, cc-SNe are
primarily responsible for driving both the turbulent mixing and the resulting
galactic wind in these models. This is confirmed by our simulations which show
that the average $\eta_{M}$ and $\eta_{Z_{\rm iron}}$ are similar to within a
few percent in all simulations.

In contrast, the average mass loading factor of r-process elements vary
perceptibly with $z_{\rm NSM}$ (see Figure 7). More precisely, the values of
$\eta_{Z_{\rm rp}}$ measured at $|z|=2z_{\rm SNe}$, are $\approx 8.2$ %
($z_{\rm NSM}=z_{\rm SNe}$), $\approx 12.3$ % ($z_{\rm NSM}=1.33z_{\rm
SNe}$)222There is a small difference between the average reported here and in
section 3.3. The number reported in this section includes only the time that
is common among the three simulations, this shorter time span leads to small
differences in the reported figures. and $\approx 12.6$ % ($z_{\rm
NSM}=2z_{\rm SNe}$). As expected, the model where r-process events are
confined to a region that is significantly larger than the one for cc-SNe
shows the largest $\eta_{Z_{\rm rp}}$, indicating slightly less effective
retention of r-process metals in the ISM.

Figure 7: The temporal evolution of the metal r-process loading factor,
$\eta_{Z_{\rm rp}}$, in units of the characteristic relaxation time, $t_{r}$,
for varying $z_{\mathrm{NSM}}$ in the MW progenitor high SFR model.
$\eta_{Z_{\rm rp}}$ is measured here at $|z|=2z_{\rm SNe}$. The dark vertical
lines denote the times at which r-process events transpire. The r-process
retention is slightly altered by changes in $z_{\mathrm{NSM}}$, with more
prominent metal outbursts observed when injection takes place at higher scale
heights.

###  5.2 Abundance evolution

Figure 8: The temporal evolution of the volume weighted mean and $1\sigma$ of
[Eu/Fe] as a function of the global mean [Fe/H] abundance in MW progenitor
with high SFR simulations with varying $z_{\mathrm{NSM}}$ (see legend). The
[Eu/Fe] abundance is shown in the left panel for the entire simulation box and
in the right panel for the cold, dense gas in the disk (defined as for Figures
3 and 5). We note no discernible difference in the mean abundance of the gas,
suggesting that the offset of events does not strongly influence the observed
abundances of r-process elements within the limits examined here.

Having found only a small increase in $\eta_{Z_{\rm rp}}$ with r-process event
offsets, we now consider if the location of events are capable of altering the
metal abundance of the ISM. Figure 8 shows the volume-weighted mean and
$1\sigma$ of [Eu/Fe] as a function of the mean [Fe/H] for the global (left)
and cold, dense, disk gas (right) in the three simulations. The definition of
cold, dense disk gas remains unchanged from Sections 3.2 and 4.1.

While there is a trend of slightly higher mean [Eu/Fe] abundances with
increasing event offsets in the global measurement, we note no discernible
difference when considering only the cold, dense gas. This is to be expected
as the retention of r-process material in these models is $\gtrsim$ 90%. Both
the mass loading factors and abundances are found to be similar across models
with different allowable scale heights of r-process events. Thus, we conclude
that the range of offsets examined here do not produce measurable changes in
observed galaxy properties.

We caution the reader that global disk galaxy simulations are necessary in
order to study the mixing effects of r-process metals deposited by NSM with
large kicks and long merger times. This will require a large range of scales
to be resolved simultaneously, which is exceedingly challenging given the
complexity of the interplay between the various galaxy assembly mechanisms at
all scales. Considering the somewhat emerging stage of modeling in the field,
this study amounts to a sizable improvement within the stated limitations in
our understanding of how the location of metal injection within the galaxy
influences the inhomogeneous enrichment of the ISM and the mixing of r-process
elements.

##  6 On the repercussions of the production rate of r-process material on
metal mixing

The purpose of this paper is to help isolate some of the key mechanisms that
regulate turbulent mixing of r-process elements in galactic disks and, in
particular, how the statistics of abundance variations in stars can help
constrain astrophysical r-process synthesis models. It seems reasonable to
expect that most astrophysical r-process production models would have
r-process mass per event, $m_{\rm rp}$, and (relative) rate of events, $f_{\rm
rp}$, as essential parameters. While a given r-process production mechanism
may display a correlation or even interdependence between these two
parameters, it is instructive to consider these two independently for
understanding the mixing properties. Motivated by this, in this section, we
examine more closely how the shape of the distribution of [Eu/Fe] abundances
in the ISM changes in response to shifts in $m_{\rm rp}$ and $f_{\rm rp}$.

First, we consider changes in $m_{\rm rp}$, keeping all other parameters
fixed. Next, we consider changes in $f_{\rm rp}$ while keeping $m_{\rm rp}$
constant. Finally, we address how the statistics of variations of metal poor
halo stars in chemical space can provide valuable constraints on $m_{\rm rp}$
and $f_{\rm rp}$. Throughout this analysis, we focus on the results derived
from the MW progenitor with high SFR model and the satellite galaxy model.

###  6.1 The role of $m_{\rm rp}$

The key to using galaxy patch models productively is to isolate the role of
key parameters and then to analyse the simulations so that we can learn the
role of these model ingredients on metal mixing. In this analysis we first
begin by isolating the role of $m_{\rm rp}$.

Figure 9 shows the stacked one-dimensional distributions of [Eu/Fe] within the
cold, dense gas at each average [Fe/H] metallicity. The color bar notes the
fractional mass in each bin. The filled-in contours represent the standard
enrichment model (i.e., $f_{\rm rp}=10^{-3}$ and $m_{\rm
rp}=10^{-2}\,M_{\odot}$), while the line contours denote the model with
$m_{\rm rp}$ increased by a factor of 10 (i.e., $m_{\rm
rp}=10^{-1}\,M_{\odot}$). The left-hand panel shows our results from the MW
progenitor with high SFR model, while the right-hand panel reveals our
outcomes from the satellite galaxy model.

In both galaxy models, as expected, the shape of the contours is unchanged by
altering $m_{\rm rp}$. The main difference is an increase in the mean
abundance of [Eu/Fe], as the mass of r-process injected over time is augmented
in the box. The metal dispersion, on the other hand, stays the same as the
relative rate of metal mixing, which is controlled by the cc-SN rate, remains
unchanged.

Figure 9: Stacked 1d distributions of [Eu/Fe] within the cold, dense gas at
each average [Fe/H] metallicity. The truncation at higher [Fe/H] abundance is
caused by the different time scales for enrichment in the two galaxy patch
models. Contours show the fractional mass per bin in this abundance plane and
the effects of altering the mass per event while keeping the relative rate
($f_{\rm rp}=10^{-3}$) constant. Filled in contours represent the simulations
with $m_{\rm rp}=$1\text{\times}{10}^{-2}\text{\,}\mathrm{M}_{\odot}$$, while
line contours represent $m_{\rm
rp}=$1\text{\times}{10}^{-1}\text{\,}\mathrm{M}_{\odot}$$. Results are shown
for two galaxy models: the MW progenitor high SFR model (left panel) and
satellite galaxy model (right panel). Increasing the mass per event shifts the
abundance distributions to higher mean [Eu/Fe] at constant mean [Fe/H], while
the abundance spread, which is driven by turbulent mixing, remains unchanged.

###  6.2 The role of $f_{\rm rp}$

Our findings in Section 6.1 provide supporting evidence that while the average
[Eu/Fe] abundance can be altered by changes in $m_{\rm rp}$, the dispersion of
the [Eu/Fe] abundance can only be modified by varying $f_{\rm rp}$.

It is to this issue that we now turn our attention. Figure 10 shows how the
stacked one-dimensional distributions of [Eu/Fe] as a function of the average
[Fe/H] metallicity within the cold, dense gas are altered when $f_{\rm rp}$ is
modified. This result should be directly compared with those presented in
Figure 9. For both of the models considered in Figure 10, the relative rate of
r-process events is changed from $f_{\rm rp}=10^{-3}$ to $f_{\rm rp}=10^{-2}$
while keeping all other inputs constant.

Some conspicuous points should be underscored from Figure 10. First, it is
important to note that the [Fe/H] abundance is determined solely by the cc-
SNe, whose rate remains unchanged in all simulations. Thus, changes in $f_{\rm
rp}$ lead simply to adding more r-process material, thereby increasing the
mean [Eu/Fe] abundance but producing very little change in the overall Fe
mass. Second, the relative injection rate, $f_{\rm rp}$, alters both the
evolution of the mean [Eu/Fe] abundance, as well as, the [Eu/Fe] spread. This
is because the total amount of metals injected is significantly increased with
time but at the expense of augmenting the number of injection sites and not
the total mass per event as was done in in Section 6.1. As such, the mean
separation between injection sites is substantially decreased, which helps the
effectiveness of turbulent mixing in smoothing metal inhomogeneities. As
expected, this diminishes the [Eu/Fe] abundance spread. Finally, the mean
[Eu/Fe] abundance increases more swiftly when $f_{\rm rp}$ is augmented,
causing the model to spend more time at higher relative abundances in the
plane depicted in Figure 10. By its very nature, a mass build up at higher
relative abundances is naturally produced in these models.

Motivated by the results presented in Sections 6.1 and 6.2, we direct our
efforts to comparing our models to observations under the assumption that the
selected cold gas within the disk is the most likely to form stars.

Figure 10: Stacked 1d distributions of [Eu/Fe] within the cold, dense gas at
each average [Fe/H] metallicity. The truncation at higher [Fe/H] abundance is
caused by the different time scales for enrichment in the two galaxy patches,
as well as, slightly different evolution times between different simulations
of the same galaxy patch. Results are shown for the same two galaxy models as
in Figure 9. Color indicates the fractional mass per bin in this abundance
plane. Filled in contours represent the simulations with $f_{\rm rp}=10^{-3}$,
while line contours represent $f_{\rm rm}=10^{-2}$, in both cases $m_{\rm rp}
=$1\text{\times}{10}^{-2}\text{\,}\mathrm{M}_{\odot}$$. Changing the relative
rate of events from $f_{\rm rp}=10^{-3}$ to $f_{\rm rp}=10^{-2}$ reduces the
spread of abundances and leads to more cold, dense gas at higher [Eu/Fe]
abundances.

###  6.3 Constrains from observations of metal poor halo stars

Figure 11: Stacked 1d distributions of [Eu/Fe] in the cold, dense gas at each
average [Fe/H] metallicity in the MW progenitor with high SFR model (left
panel) and satellite galaxy model (right panel). The color indicates the
fractional mass in each bin, the white solid (dashed) lines indicate the
$1\sigma$ ($2\sigma$) spread of the distribution. In both models, $f_{\rm
rp}=10^{-2}$ and $m_{\rm rp}=$5\text{\times}{10}^{-3}\text{\,}\,${M_{\odot}}$.
We consider the cold, dense gas within the disk as the most likely to form
stars. The black symbols indicate the abundances of metal poor halo stars with
uncertainties as reported by Roederer et al. (2014). The mean [Eu/Fe]
abundance in high SFR model is slightly elevated when compared to the
satellite galaxy model as expected from the ability of this model to more
effectively retain r-process material. The truncation at higher [Fe/H]
abundance is caused by the different time scales for enrichment in the two
galaxy patch models.

Having considered the influence of $m_{\rm rp}$ and $f_{\rm rp}$ on the mean
and spread of [Eu/Fe] as a function of the mean [Fe/H] abundance, we dedicate
ourselves to comparing the simulated abundance distributions with the observed
stellar abundances in the MW halo. We do this in order to provide useful
constraints on the operating conditions of r-process production models in the
early Universe.

The metal-poor halo stars for this analysis are taken from Roederer et al.
(2014). Our selection has a twofold rationale. First, this data set
constitutes the largest homogeneously reduced sample of its kind (about 300
metal-poor halo stars). And, second, the authors have systematically and
scrupulously documented the uncertainty of their measurements. This allows us
to use the star-to-star [Eu/Fe] scatter and mean as a function of [Fe/H]
derived by Roederer et al. (2014) as a constraint for r-process production
models in the [$m_{\rm rp}$,$f_{\rm rp}$] plane.

In Figure 11, we show the distributions of [Eu/Fe] as a function of the
average abundance of [Fe/H] within the cold, dense gas in the disk for both
the MW progenitor with high SFR (left panel) and the satellite galaxy model
(right panel). The color indicates the fractional mass in a given bin. Also
shown are the abundances and measurement uncertainties of the MW halo stars,
as reported by Roederer et al. (2014). The parameters of the r-process events
have been adjusted to achieve a reasonable agreement with the observations for
both models. Shown is a representative model that effectively recovers both
the mean and the spread in [Eu/Fe] as a function of [Fe/H] for the sample of
MW halo stars that we selected. The representative values used to generate the
models in Figure 11 are $f_{\rm rp}=10^{-2}$ and $m_{\rm
rp}=$5\text{\times}{10}^{-3}\text{\,}\,${M_{\odot}}$, respectively. Given that
the MW progenitor with high SFR model retains the r-process products more
effectively, the mean [Eu/Fe] is slightly more elevated compared to the
satellite model, albeit within the uncertainties of the data and the range of
applicability of the simulations (e.g., results are only shown after steady
state is reached).

From the simple model comparison presented in Figure 11, we find clear
evidence that the properties of the metal-poor halo stars are consistent with
stellar birth sites in a MW progenitor with a high SFR or a satellite galaxy.
It is, however, important to note that model comparison is hindered by the
fact that the observational sample used in this analysis is impaired by
incompleteness and selection effects, which are most evident at low [Eu/Fe]
abundances.
In the preceding sections we outlined how the abundance distributions of
[Eu/Fe] are influenced by changes in $m_{\rm rp}$ and $f_{\rm rp}$. In what
follows, we briefly consider how these results may be extrapolated beyond the
parameter space explored in this paper. The results presented in Section 6.1
indicate that that impact of altering $m_{\rm rp}$ while keeping $f_{\rm rp}$
unaltered is straight forward. Changing $m_{\rm rp}$ only leads to a change in
normalization, with an increase of mass per event leading to a larger
abundance normalization.

Extrapolating $f_{\rm rp}$ is more complicated due the intricate influence of
this parameter on the shape of the [Eu/Fe] distribution. The results presented
in section 6.2 clearly indicate that lower relative rate leads to later
r-process enrichment. At $f_{\rm rp}=10^{-3}$ the simulations struggle to
explain the high levels of r-process pollution of low metallicity stars. For
this reason, lower relative rates are unlikely to produce reasonable fits to
the observed values. Higher relative rates, i.e. more frequent r-process
events, seem more likely to reproduce the observational data at low
metallicities. On the other hand, as we have shown here, higher relative rates
also lead to more efficient metal mixing, quickly reducing the width of the
abundance distribution at higher metallicity values. This makes high relative
rates to be unable to reproduce the spread observed in [Eu/Fe]. We expand on
these arguments in the following section.

##  7 Discussion

Observations of [Eu/Fe] in metal-poor stars suggest relative yields and
variations of yields in r-process production events. Such inherent
fluctuations are evident when r-process events inject metals at rates that are
significantly reduced when compared to cc-SN rates. These fluctuations are
smoothed out by turbulent diffusion driven by cc-SNe, which sets the rate of
metal mixing. As we have contended in this paper, the degree of fluctuations
along with the mean abundances of [Eu/Fe] are sensitive to the relative mass
injection rate of r-process and, to a lesser extent, to the type of galaxy
environment.

###  7.1 Production rate of r-process elements

Figure 12: The constraints derived for r-process production events in the
[$m_{\rm rp}$, $R_{\rm MW}$] plane, which has been adapted from Hotokezaka et
al. (2018). The left panel includes a compilation of the production rate of
r-process element constraints in the MW: the constraints from GW170818, limits
on rates from short gamma-ray burst observations, the mass limits from
extremely metal-poor stars (Macias & Ramirez-Ruiz, 2018), and the mass limits
from macronova candidates. The derived relationships between the mean and
spread of abundances when the rate or mass per event are altered are
highlighted in the left panel. On the right-hand panel we show the constraints
derived from turbulent mixing simulations. The red symbols indicate
simulations that do not successfully explain the data while the blue symbols
are for simulations that give a reasonable description of the [Eu/Fe]
abundances (see also Appendix C). The lavender shaded region highlights the
parameter space for which simulations provide a reasonable description of the
data. The upper limit for the mass production rate (purple line) arise
naturally from the highest [Eu/Fe] abundance measurement (see text). Models
above this region are stringently ruled out given that there are very few
selection effects against uncovering the highest [Eu/Fe] metal poor stars.

The total r-process mass in the MW is estimated to be $M_{\rm
tot,rp}\approx$2.3\text{\times}{10}^{4}\text{\,}\,${M_{\odot}}$ (Hotokezaka et
al., 2018). Combining this number with an estimate of the age of the MW
($t_{\rm MW}\approx$1\text{\times}{10}^{10}\text{\,}\,${\rm yr}$), one can
calculate an average production rate of r-process elements in the galaxy as

$\dot{m}_{\rm rp} \approx\frac{M_{\rm tot,rp}}{t_{\rm
MW}}\approx$2.3\text{\times}{10}^{-6}\text{\,}\mathrm{M}_{\odot}\mathrm{/}\mathrm{y}\mathrm{r}$.$
(4)

Hotokezaka et al. (2018) used this relationship to estimate the mass
production rate of r-process events in the MW as

$R_{\mathrm{MW}}\approx$230\text{\,}\mathrm{/}\mathrm{M}\mathrm{y}\mathrm{r}$\left(\frac{m_{\mathrm{rp}}}{$0.01\text{\,}\mathrm{M}_{\odot}$}\right)^{-1}.$
(5)

This relationship is represented by the thick green line in Figure 12 along
which the production rate of r-process elements is unchanged. The shaded
region below the line is limited by the same relationship by for a lower total
r-process mass in the MW. This relationship is set by $\dot{m}_{\rm
rp}\approx$1\text{\times}{10}^{-7}\text{\,}\mathrm{M}_{\odot}\mathrm{/}\mathrm{y}\mathrm{r}$$,
which is thought to give a strict lower limit to $\dot{m}_{\rm rp}$ based on
the minimum amount of r-process mass in stars in the MW (Kasen et al., 2017;
Hotokezaka et al., 2018).

The shaded-pink region in Figure 12 shows the range of values in the [$m_{\rm
rp}$, $R_{\rm MW}$] plane derived from observations of GW170817 (Kasen et al.,
2017; Kilpatrick et al., 2017; Waxman et al., 2018; Hotokezaka et al., 2018).
We also show in this plane the estimates of the rates of short gamma-ray
bursts ($R_{\rm MW}$), which are believed to be an observational signature of
NSM (Eichler et al., 1989; Lee et al., 2005; Lee & Ramirez-Ruiz, 2007) and the
mass constraints ($m_{\rm rp}$) from observations of afterglows of sGRBs
(Roberts et al., 2011; Hotokezaka et al., 2013; Tanvir et al., 2013; Berger et
al., 2013; Yang et al., 2015; Jin et al., 2016; Kasen et al., 2015; Kasliwal
et al., 2017; Kilpatrick et al., 2017; Murguia-Berthier et al., 2017; Ascenzi
et al., 2019), referred to in Figure 12 as macronova candidates. Finally, the
vertical black line marks the lower limit on r-process mass per event as
deduced by Macias & Ramirez-Ruiz (2018) based on observations of metal poor
halo stars in the MW.

In the left panel of Figure 12 we call attention to the results from our
turbulent mixing study, which explores how the abundance distribution of
[Eu/Fe] is altered by $m_{\rm rp}$ and $f_{\rm rp}$ (i.e., $R_{\rm MW}$). As
highlighted by the horizontal arrow, changing $m_{\rm rp}$ while keeping
$f_{\rm rp}$ constant shifts the mean abundance at a fixed [Fe/H] abundance
but does not alter the scatter. This implies that the observed spread of
abundances can not be used to constraint $m_{\rm rp}$. In contrast, as
underscored by the vertical arrow, $f_{\rm rp}$ changes both the spread and
the mean of the [Eu/Fe] abundance distributions at a fixed [Fe/H] abundance.

The results of the galaxy patch simulations of the MW progenitor with high SFR
model are shown in the right panel of Figure 12. To accurately position our
simulations in the [$m_{\rm rp}$, $R_{\rm MW}$] plane, we follow Beniamini et
al. (2016) who estimates an average rate of cc-SNe in the MW to be

$\tilde{n}_{\rm
SNe,MW}\approx$6.4\text{\times}{10}^{4}\text{\,}\mathrm{M}\mathrm{y}\mathrm{r}^{-1}$.$
(6)

This allows us to chart the relative rate of r-process to cc-SN events from
the local boxes to standard MW values by assuming constant relative rates
across the galaxy. Then it follows that

$R_{\rm MW,rp}=f_{\rm rp}\tilde{n}_{\rm SNe,MW}.$ (7)

From Figures 9-11 we know that the SFR does not strongly impact the shape of
the distribution. As such, we expect that that conclusions based on assuming a
constant SFR everywhere in the disk will not significantly be altered if the
SFR changes across the disk.
We divide the simulations broadly into two distinct groups by how well they
describe the observed abundances of MW halo stars (Roederer et al., 2014). We
consider both the mean and the spread of [Eu/Fe] abundance in bins of [Fe/H],
as well as, the range of [Fe/H] abundance over which there is rough agreement
when making these comparisons. In Appendix C we present the results of the
simulations with varying $m_{\rm rp}$ and $f_{\rm rp}$ and how they compare to
observations. The blue symbols indicate reasonable agreement between
observations and simulations, while the red symbols indicate little to no
overlap between the abundances predicted by the simulations and those observed
in the comparison sample.

Obviously, the comparison presented in Figure 12 is only cursory and should be
taken as an order of magnitude estimate at present. Having said this, the
results of our metal mixing study allow us to clearly define a model
confidence region within the [$m_{\rm rp}$, $R_{\rm MW}$] plane, which is
specified by the lavender shaded region in the left panel. The lavender region
is restrained by a strict upper limit (purple line). Such a limit can be
derived by the underlying fact that simulations above this boundary will
naturally produce stars with a mean abundance of [Eu/Fe] = 1.4, which is
higher than the abundance of the most r-process enhanced star in the Roederer
et al. (2014) sample: [Eu/Fe] = 1.37. The slope of the line is derived from
the mass ratios of Eu and Fe production leading to [Eu/Fe] = 1.4. This value
can be written as a production rate with $\dot{m}_{\rm
rp,max}\approx$4.7\text{\times}{10}^{-4}\text{\,}\mathrm{M}_{\odot}\mathrm{/}\mathrm{y}\mathrm{r}$$.
The lavender region presented in the left panel of Figure 12 is in remarkable
good agreement with previous independent constraints derived both
theoretically and empirically and provide a significant reduction in the
permissible parameter space.

###  7.2 Co-production of light and heavy r-process elements?

Several astrophysical processes might be required to explain the solar
r-process abundance pattern of both the lighter neutron capture elements
(between the first and second r-process peaks at $A\approx 80$ and $A\approx
130$, respectively) and heavier nuclei such as Eu (Cowan et al., 2021). A
fascinating characteristic that emerges when comparing the relative abundances
of metal poor halo stars is the robustness of the pattern for elements with
$A\geqslant 137$ (Sneden et al., 2008). In this study we have selected to
model the r-process mass contribution as that comprised by all elements with
$A\geq 69$. If, for example, we change the minimum atomic number from $A_{\rm
min}=69$ to $A_{\rm min}=90$ the average abundances of [Eu/Fe] shift by
$+0.68$ dex with no other changes to the results presented in Figure 12.
Appendix C includes the simulated distributions for $m_{\rm rp}$ calculated
with and without this shifting (i.e., with $A_{\rm min}=69$ and $A_{\rm
min}=90$). This simply implies that constraints on $\dot{m}_{\rm rp}$ are
sensitive to whether r-process sites, which are assumed in this paper to be
standard, are responsible for producing all r-process elements or only the
heavier ones. If Eu is produced by sources that only produce heavier r-process
elements ($A\geq 90$), this implies that the derived constraints on
$\dot{m}_{\rm rp}$ should be lower by $0.68$ dex when compared to those
illustrated in Figure 12.

###  7.3 On the origin of the metal poor halo stars

The MW halo is expected to have been assembled from stars originally residing
in disrupted satellites (e.g. Naidu et al., 2021; Santistevan et al., 2021).
The presence of r-process enhanced stars (Beers & Christlieb, 2005; Holmbeck
et al., 2020) in several dwarf galaxy systems and the orbital properties of
r-process enhanced stars in the halo(Roederer et al., 2018) indicate that such
stars likely were accreted from disrupted satellites. From a theoretical
perspective, Hirai et al. (2022) recently performed hydrodynamic zoom-in
cosmological simulations, focusing on a MW-like galaxies. They found that the
vast majority (90%) of r-process enhanced stars ([Eu/Fe] $>0.7$) are formed
early in the evolution of a galaxy and concluded that the majority of these
r-process enhanced stars were accreted from disrupted satellite galaxies.

Our results support the hypothesis that both MW-like and dwarf-like galaxy
systems are capable of producing stars with r-process abundances similar to
those observed in the MW halo. It is, however, important to note that
retention of r-process elements in satellite galaxies is found to be smaller
than in MW-like progenitors (Section 4). This implies that viable mass
production rates in satellite galaxies should be larger (by a factor of a few)
than those presented in Figure 12 for a MW-like progenitor. Retention of
r-process elements might also be significantly reduced for r-process events
with large offsets from the star forming disk, such as expected for NSM with
long delay times and large kick velocities. With that said, systems with large
displacements are thought to be much less effective at polluting low
metallicity gas with r-process material in the early Universe (e.g., Siegel et
al., 2019).

##  8 Summary and conclusions

In this paper we study the patchy enrichment of the ISM using numerical
simulations at kpc scales that are able to resolve the mixing of metals by cc-
SNe-driven turbulence. By investigating the statistics of variations of cc-SN
and r-process products in these simulations, we are able to derive constraints
on the allowed range of the production rate of r-process elements in the MW.
By systematically varying the model parameters, we were able to identify some
of the physical process that we believe are most relevant to explain the mean
and dispersion of [Eu/Fe] abundances in metal-poor stars. Our salient findings
are:

  * •

cc-SNe inject freshly synthesized metals and drive turbulent mixing which
causes metals to diffuse across the disk (Figure 1). By virtue of their much
more frequent injection, cc-SNe products (for example, Fe) are more evenly
distributed than r-process elements (for example, Eu) synthesized in rarer
events. This difference naturally produces an ISM with r-process elements that
are nonuniform and highly undiluted at early epochs (Figure 2). These
congenital variations are most notable when r-process sources inject metals at
rates that are considerably decreased compared to cc-SN rates (Sections 3 and
4).

  * •

The momentum and energy that goes into driving turbulence in the disk also
launches galactic winds and flings metals out of the disk (Figure 4). The
rarefied galactic wind is highly enriched in both Fe and Eu compared to the
star forming gas in the disk (Figure 2). This result implies that a
considerable mass of r-process elements might reside in the hot inter galactic
medium (Sections 3 and 4).

  * •

The metal mass loading factors of cc-SN and r-process products are not exactly
the same, suggesting that r-process metals are launched by winds before they
are able to be efficiently mixed with the ISM (Figure 2). However, models with
fewer cc-SNe give r-process elements more time to diffuse and mix in the disk
and, as a result, show winds with less prominent Eu-enriched outbursts
(Sections 3 and 4).

  * •

Across different galaxy models, the metal mass loading factors of iron and
r-process elements are rather similar. However, the magnitude of the loading
factors depend on the specific galaxy potential, with larger mass loading
factors found in less massive galaxies (Figure 6). Implicitly, higher mass
injection rates of r-process material are required in a satellite galaxy in
order to achieve an average [Eu/H] abundance comparable with a MW-like
progenitor galaxy (Section 4).

  * •

The r-process metal mass loading factor shows a measurable increase with
increasing allowed height of r-process events (Figure 7). Despite this, the
[Eu/Fe] abundances, especially in the cold, dense gas phase, are found to be
very similar across models (Figure 8). Thus, we conclude that no measurable
changes in observed galaxy properties are expected for the range of offsets we
are able to probe in these simulations (Section 5).

  * •

The degree of fluctuations in and the mean of [Eu/Fe] abundances in the cold,
dense gas are found to be highly responsive to the mass injection rate of
r-process (Section 6) and, to a lesser extent, to the type of galaxy (Section
4). Concretely, increasing the r-process mass per event increases the mean
[Eu/Fe] abundance (Figure 9), while increasing the rate of r-process events
relative to cc-SNe increases the mean [Eu/Fe] abundance and reduces the
[Eu/Fe] spread (Figure 10).

  * •

Observations of [Eu/Fe] in metal-poor stars are used to derive constraints on
the mass per event and the event rate of r-process sources (Section 7). We
find a production rate of $6\times 10^{-7}{M_{\odot}/\rm
yr}\lesssim\dot{m}_{\rm rp}\ll 4.7\times 10^{-4}{M_{\odot}/\rm yr}$ best
explains the data. The constraints presented are in notable agreement with
other independently derived confinements and produce a marked reduction in the
permitted parameter range (Figure 12).

  * •

Our findings give credence to the idea that stars with r-process abundances
similar to those observed in the MW halo can be manufactured by both MW-like
and dwarf-like galaxy progenitors (Figure 11). Although we note that r-process
mass retention in satellite galaxies is found to be smaller than in MW-like
progenitors and, as such, the viable mass production rates should be
appropriately higher (Section 6).

We thank the referee for helpful comments which improved the quality of this
paper. We thank C. Sakari, K. Ruiz-Rocha, A. Ji, D. Kasen, N. Imara and L.
Kenoly for insightful discussions. This work made use of an HPC facility
funded by a grant from VILLUM FONDEN (project number 16599). A.N.K. and
E.R.-R. acknowledge support by the Heising-Simons Foundation, the Danish
National Research Foundation (DNRF132) and NSF (AST-2206243, AST-1911206 and
AST-1852393). PM gratefully acknowledges support from NASA grant
14-WPS14-0048. M.S.F gratefully acknowledges support provided by NASA through
Hubble Fellowship grant HST-HF2-51493.001-A awarded by the Space Telescope
Science Institute, which is operated by the Association of Universities for
Research in Astronomy, In., for NASA, under the contract NAS 5-26555.

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##  Appendix A Mass and volume weighted distributions

In Section 3.2 we discussed the evolution of the volume weighted mean and
spread of [Fe/H], [Eu/H] and [Eu/Fe] in all the gas in the simulation volume,
as well as, in the cold, dense gas in the disk. We showed that the global mean
abundances are always higher than those of the cold, dense gas, and we argued
that this is due to the larger volume filling factor of the gas in the wind
region. We present here data to support this claim.

In the top row of Figure 13 we reproduce Figure 3, adding vertical lines to
indicate three specific times in the simulation. These times are selected to
more closely examine the structures of the volume and mass weighted abundance
distributions. The subsequent three rows in the figure correspond to each of
these time slices. In these rows the solid lines outline the volume weighted
distributions (the color of the lines are the same as in the top row with
darker color indicating cold, dense, disk gas while lighter color refers to
the global gas values). We also note the mean value of these abundances with
an arrow. The dashed lines show the equivalent distributions but weighted by
gas mass rather than volume.

We first investigate the [Fe/H] distributions. The volume weighted
distribution of all the gas shows a sharp peak at relatively high abundances,
where the mean of the distribution lies. Furthermore, this distribution has a
long tail on the lower abundance side, at all time slices considered. The
volume weighted distribution of the cold, dense gas is much more symmetric and
peaks at lower metallicity values. Thus, there are considerable differences
between the two distributions when considering the volume weighted
distributions. However, the mass weighted distributions are rather similar
(dashed lines in Figure 13). Hence, we conclude that the difference in mean
abundance between the global and the cold, dense gas is driven by the higher
abundance of the hot, diffuse gas in the disk winds being emphasized in the
volume weighted distributions of the global gas.

We now turn our attention to the [Eu/X] distributions and see many similar
behaviors as seen in the [Fe/H] distributions. First, the volume weighted
distribution of the global gas abundance is off-set at high metallicity when
compared to all other distributions. Second, there is a nearly perfect overlap
between the volume weighted distribution of the cold, dense gas and the mass
weighted distributions of both the global and the cold, dense gas phases.
Finally, the volume weighted global distribution features a tail at the low
enrichment side. The distributions [Eu/X] are generally much broader than
those of [Fe/H] indicating that the r-process elements are less well mixed in
all gas phases than Fe is. As time increases in the simulation the offset
between the mean of the global and the cold, dense disk gas slowly decreases.
This is caused by gas mixing in the ISM.

Figure 13: The abundance distributions at three distinct times in the metal
mixing evolution of the MW progenitor with high SFR simulation. The top row is
the same as Figure 3. The subsequent three rows show the distributions of
[Fe/H], [Eu/H] and [Eu/Fe] at each of the three times indicated by vertical,
black lines in the top row. The colors indicate the gas phase; lighter color
for global values and darker for cold, dense disk gas. The solid lines show
the volume weighted distributions while the dashed lines give the mass
weighted distributions. The arrows show the mean value of the volume weighted
distributions.

##  Appendix B Loading in all galaxy potentials

In Section 4 we presented the loading factors of the total, iron and r-process
mass in three galaxy potentials but we limited our attention to results for
$|z|=2z_{\rm SNe}$. For completeness we present here the results for these
loading factors at varying heights. In the middle row of Figure 14 we present
an expanded version of Figure 6. The top and bottom rows of the figure show
the same corresponding loading factors at $|z|=3z_{\rm eff}$ and $|z|={L\over
2}-dx$, respectively. Across all galaxy potentials, the loading factors are
larger when measured closest to the disk and decrease as we approach the edge
of the box. This effect is caused (partly) by circulation of material in the
wind whereby some of the ejected material falls back. The drop in loading
factor is strongest in the MW progenitor with high SFR and smallest in the
satellite galaxy model indicating that material is more effeciently launched
when residing in a weaker galaxy potential.

Figure 14: Loading factors for the total, iron and r-process mass (see legend)
in each of the three galaxy patch models: MW progenitor with high SFR (left
hand column), MW progenitor with low SFR (middle column) and satellite galaxy
model (right hand column). Each row corresponds to mass outflow at increasing
distance from the disk mid-plane: $3z_{\rm eff}$ (top row), $2z_{\rm
SNe}$(middle row) and ${L\over 2}-dx$ (bottom row). The middle row of this
Figure is equivalent to Figure 6 although it has been extended to a slightly
lower $\eta$ values.

##  Appendix C R-process production rate in the Milky Way

###  C.1 Light and heavy r-process elements

We expand here on the presentation of our results in Sections 6.3, 7.1 and
7.2, and show how the abundances derived from various simulations compare with
observations. Figure 15 shows the 50% and 90% contours of the distributions of
[Eu/Fe] abundance as a function of the median [Fe/H] abundance of the cold,
dense, disk gas. In the left (right) hand column we present the results when
taking $A=69$ ($A=90$) as a lower limit for r-process elements. The r-process
mass per event increases with each row, while the relative rate of r-process
events is indicated by the color (see legend). We also include the 50% and 90%
widths of the corresponding abundances of the metal-poor halo stars (Roederer
et al., 2014), which are calculated in 0.23 dex wide [Fe/H] bins.

In Section 6.3 we showed the comparison between the best fitting r-process
parameters and the observations. In Figure 15 we show how models with other
parameter combinations fit the data. When choosing the best fit we take into
account how well the simulations reproduce the mean of the observations, as
well as, the abundance spread. This is done only for the range of metallicity
over which the model is in steady state. We strongly disfavor models that
predict a large number of stars with greater [Eu/Fe] abundances than those
observed.

In Section 7.2 we discussed the dependence on the constraints of $\dot{m}_{\rm
rp}$ on the assumed range of r-process elements. Figure 15 shows how the
distributions of [Eu/Fe] change as a function of the average [Fe/H] when we
change $A_{\rm min}$. When an r-process pattern that includes only the heavy
r-process elements ($A_{\rm min}=90$) is assumed, the relative mass of Eu to
all the r-process elements increases and thus, the distribution of [Eu/Fe]
shifts by $+0.68$ dex.

Figure 15: The simulated 50% and 90 % contours of the abundance distribution
in the MW progenitor with high SFR when $m_{\rm rp}$ and $f_{\rm rp}$ are
altered. The black and gray bars indicate the 50% and 90% spreads of the
metal-poor MW halo stars (Roederer et al., 2014) in bins of [Fe/H]. The left
hand columns use the Eu mass fraction to all r-process elements, while the
right hand uses the fraction relative only to the heavy r-process elements
(defined here as $A>90$).

###  C.2 Independence of $m_{\mathrm{rp}}$ and $f_{\rm rp}$

In Figure 16 we explore the possible interdependence between $f_{\mathrm{rp}}$
and $m_{\rm rp}$. Along the top row of the figure the relative rate, $f_{\rm
rp}=10^{-2}$, is constant and the mass per event increases from left to right.
Within each column the mass per event is constant but the relative rate
decreases from the top to the bottom row such that the bottom rows have,
$f_{\rm rp}=10^{-3}$. The color of the shaded areas show the achieved average
production rate of r-process elements per unit time in the simulation.

Figure 16: The same contours of the abundance distributions as the left hand
column in Figure 15 (omitting the $m_{\rm rp}=5\times 10^{-3}M_{\odot}$ case).
Here the distributions are organized by the relative rate (constant along the
rows) and mass per event (constant within a column) and color coded by the
total r-process production rate.