the density and $\kappa$ the opacity of the ejecta. Once the ejecta expands
enough to become transparent, it releases the thermal radiation. This occurs
when the diffusion timescale, $t_{\text{diff}}$, becomes comparable to the
dynamical timescale, $t=R/v$. This defines the time at which the maximum of
the luminosity is reached (see e.g. Metzger  _et al._ , 2010b; Fernández and
Metzger, 2016):

$\displaystyle t_{\text{peak}}$ $\displaystyle\approx$
$\displaystyle\left(\frac{\kappa M}{4\pi cv}\right)^{1/2}$
$\displaystyle\approx$ $\displaystyle 1.5\
\text{days}\left(\frac{M}{0.01M_{\odot}}\right)^{1/2}\left(\frac{v}{0.1c}\right)^{-1/2}\left(\frac{\kappa}{\text{cm}^{2}\text{g}^{-1}}\right)^{1/2}$

At timescales beyond the peak time the luminosity can be approximated using
the Arnett’s Law (Arnett, 1980, 1982): $L(t)=M\dot{Q}_{\text{dep}}(t)$.
$\dot{Q}_{\text{dep}}$ is the energy deposition rate corrected by the
thermalization efficiency and can be approximated as
$\dot{Q}_{\text{dep}}\approx\varepsilon\,\times
10^{10}\,(t/\text{day})^{-\alpha}\ \text{erg}\ \text{s}^{-1}\text{g}^{-1}$,
with $\varepsilon<1$ the thermalization efficienty. At peak time the kilonova
luminosity is given by:

$\displaystyle L_{\text{peak}}$ $\displaystyle\approx$ $\displaystyle
1.1\varepsilon\times 10^{41}\ \text{erg}\ \text{s}^{-1}$
$\displaystyle\left(\frac{M}{0.01M_{\odot}}\right)^{1-\alpha/2}\left(\frac{v}{0.1c}\right)^{\alpha/2}\left(\frac{\kappa}{\text{cm}^{2}\text{g}^{-1}}\right)^{-\alpha/2}$

The effective temperature of emission can be obtained from the luminosity
using the Stefan-Boltzmann Law that together with the Wien displacement law
gives the characteristic wavelength of the emission:

$\displaystyle\lambda_{\text{peak}}$ $\displaystyle\approx$ $\displaystyle
514\ \text{nm}$
$\displaystyle\left(\frac{M}{0.01M_{\odot}}\right)^{\alpha/8}\left(\frac{v}{0.1c}\right)^{(2-\alpha)/8}\left(\frac{\kappa}{\text{cm}^{2}\text{g}^{-1}}\right)^{(2+\alpha)/8}$

The above formulas illustrate several characteristic features of kilonova
light curves. Even if the emission mechanism is similar to supernova the
typical ejecta mass is much smaller and the velocity larger. The equations
illustrate the important role played by the opacity that is dominated by
Doppler-broadened atomic line bound-bound transitions (Kasen  _et al._ , 2013;
Fontes  _et al._ , 2015; Tanaka  _et al._ , 2018). Ejecta containing light
r-process elements ($A\lesssim 140$) with $d$-shell valence electrons possess
an opacity $\kappa\lesssim 1$ cm2 g-1. In this case the emission peaks in
around a day in the Blue. This was, indeed, the case for AT 2017gfo (Nicholl
_et al._ , 2017a). If the ejecta contains lanthanide or actinide nuclei
($A\gtrsim 140$), then the optical opacity is very high $\kappa\gtrsim
10\text{--}100$ cm2 g-1 due to the complex structure of $f$-shell valence
electrons for these elements, resulting in a dense forest of lines (Kasen  _et
al._ , 2013; Barnes and Kasen, 2013; Tanaka and Hotokezaka, 2013; Fontes  _et
al._ , 2017). Having a kilonova observation, like in the case of AT 2017gfo,
it is possible to adjust the multi-wavelength evolution of the light curve
using a variation of the model described above and determine the amount of
ejecta, velocity and opacity that is a proxy for the composition. As discussed
in section II.5, to reproduce the AT 2017gfo observations requires at least
two different ejecta components with three-component models being slightly
favored (see e.g. Villar  _et al._ , 2017). This result is consistent with the
existence of several ejecta components in mergers giving rise to different
nucleosynthesis products (see section VI.2). The analysis of sGRB observations
(Wu and MacFadyen, 2018) and kilonova transient (Perego  _et al._ , 2017a)
favors an off-axis viewing angle of $\sim 30^{\circ}\deg$. Hence, the early
blue phase of the kilonova light curve has been suggested to originate from
Lanthanide poor polar ejecta (see e.g. Kasen  _et al._ , 2017), however see
Kawaguchi  _et al._ (2018) for an alternative explanation. This result is
consistent with simulations that predict that weak processes including
electron (anti)neutrino absorption drive the composition to $Y_{e}\gtrsim
0.25$. It may be interpreted as an observational evidence of the important
role of neutrinos in determining the composition of the ejecta. However, there
is a tension between the velocity of the ejecta $v\approx 0.27\ c$ that is
consistent with simulations of dynamical ejecta and the large ejecta mass,
${M}_{\text{ej}}\approx 0.020$ M⊙, that is not. Additional Lanthanide-poor
material is expected to originate from the post-merger neutrino-wind ejecta.
However, its velocity is expected to be smaller, unless the wind is
magnetically accelerated by the strongly magnetized HMNS remnant (Metzger  _et
al._ , 2018). The amount of material and velocity of material involved in the
“purple” and “red” components suggest that they originate from post-merger
outflows from the accretions disk (see e.g. Kasen  _et al._ , 2017).
Simulations predict that the ejecta contain a broad distribution of $Y_{e}$
and is able to produce both light and heavy r-process material including the
Lanthanides/Actinides necessary to account for the high opacity (Just  _et
al._ , 2015a; Wu  _et al._ , 2016).

It is, indeed, the observation of the Lanthanide-rich “red” emission that
provided the first observational evidence that neutron-star mergers produce
r-process nuclei. However, so far no direct spectroscopic evidence has been
obtained pointing to the production of a particular element. The high density
of lines for lanthanides/actinides together with the large velocities of the
ejecta produces line blending and smoothness the spectra (Chornock  _et al._ ,
2017). This aspect has been used to determine the velocity of the ejecta from
spectroscopic information (see e.g. Chornock  _et al._ , 2017). Nevertheless,
the spectra presents peaks that may probe the abundance of individual elements
(see figure 4 of Kasen  _et al._ , 2017). However, uncertainties in current
atomic data hinder a detailed spectral analysis. The lanthanide/actinide
opacities are uncertain because the atomic states and line strengths of these
complex elements are not measured experimentally. Theoretically, such high-Z
atoms represent a challenging problem in many-body quantum mechanics, and
hence are based on statistical models that must be calibrated to experimental
data (Kasen  _et al._ , 2013; Fontes  _et al._ , 2015; Tanaka  _et al._ ,
2018). Beyond identifying the line transitions themselves, there is
considerably uncertainty in how to translate these data into an effective
opacity. The commonly employed “line expansion opacity” formalism (Pinto and
Eastman, 2000a, b), based on the Sobolev approximation and applied to
kilonovae by Barnes and Kasen (2013) and Tanaka and Hotokezaka (2013), may
break down if the line density is sufficiently high that the wavelength
spacing of strong lines becomes comparable to the intrinsic thermal width of
the lines (Kasen  _et al._ , 2013; Fontes  _et al._ , 2015, 2017).

Lacking a direct spectroscopic identification of the abudance of individual
elements, recent work has focused in identifying fingerprints of heavy
elements in kilonova light curves. Particularly promising are late-time
observations as the decay heating can be dominated by a few nuclei (Wu  _et
al._ , 2018). Kasliwal  _et al._ (2018) suggest heavy isotopes (e.g. 140Ba,
143Pr, 147Nd, 156Eu, 191Os, 223Ra, 225Ra, 233Pa, 234Th) with $\beta$-decay
half-live around 14 days. Wu  _et al._ (2018) have shown that at times of
weeks to months, the decay energy input may be dominated by a discrete number
of $\alpha$-decays, 223Ra (half-life $t_{1/2}=11.43$ d), 225Ac ($t_{1/2}=10.0$
d, following the $\beta$-decay of 225Ra with $t_{1/2}=14.9$ d), and the
fissioning isotope 254Cf ($t_{1/2}=60.5$ d) (see also Zhu  _et al._ , 2018),
which liberate more energy per decay and thermalize with greater efficiency
than $\beta$-decay products. Late-time nebular observations of kilonovae which
constrain the radioactive power provide the potential to identify signatures
of these individual isotopes, thus confirming the production of heavy nuclei.
In order to constrain the bolometric light to the required accuracy, multi-
epoch and wide-band observations are required with sensitive instruments like
the James Webb Space Telescope.

An alternative mechanism to probe the in situ production of r-process nuclei
is the identification of X-ray or $\gamma$-ray lines from their decay similar
to the observations of 44Ti $\gamma$-rays in Cas A (Vink  _et al._ , 2001;
Renaud  _et al._ , 2006) and SN 1987A (Grebenev  _et al._ , 2012) remnants.
Qian  _et al._ (1998, 1999) have provided estimates of $\gamma$-ray fluxes for
several r-process nuclei and Ripley  _et al._ (2014) have extended those
estimates to X-ray lines. The predicted fluxes are too low to be detected by
current missions, however improvements in detection techniques may allow to
detect the last merger that took place in our Galaxy.

##  VIII Abundance evolution in the Galaxy and Origin of the r process

After we have summarized in section II the status of abundance observations of
neutron-capture (especially r-process) elements, and passed in the
intermediate sections through the nucleosynthesis working and required nuclear
input, we presented in section VI the most favorable astrophysical sites and
the related abundance predictions. This section addresses their features, in
addition to abundance predictions their occurrence frequency and its time
evolution throughout galactic history, with the aim to provide an
understanding of the impact of these individual sites on the evolution of the
Galaxy. The goal is to identify those astrophysical site(s) responsible for
the total solar r-process abundances as well as the observed features during
galactic evolution.

###  VIII.1 Supernova vs. r-process imprints in early galactic evolution

Based on the nucleosynthesis predictions for (regular) core-collapse and for
type Ia supernovae, plus their occurrence rates, one finds that the early
phase of the evolution of galaxies is dominated by the ejecta of (fast
evolving) massive stars, i.e. those leading to core-collapse supernovae. While
there might exist differences for the ejecta composition of e.g. 13, 15, 20,
25 or 40 M⊙ stars, average production ratios in the interstellar gas will be
found after some time delay when many such explosions and the mixing of their
ejecta with the interstellar medium have taken place. These averaged abundance
ratios reflect integrated ejecta yields over the initial mass function of
stars, i.e. weighted with the probability of finding stars in a certain mass
interval. As type Ia supernovae originate from exploding white dwarfs in
binary systems, i.e. (a) from stars with initially less than 8 M⊙ in order to
become a white dwarf and (b) requiring mass transfer in a binary system before
the type Ia supernova explosion, such events are delayed in comparison to the
explosion of massive single stars. This follows (a) because low and
intermediate mass stars experience longer evolution phases and (b) binary
evolution adds to the delay. For these reasons type Ia supernovae, dominating
the overall production of Fe and Ni (typically 0.5–0.6 M⊙ per event), are only
important at later phases in galactic evolution. As core-collapse supernovae
produce larger amounts of O, Ne, Mg, Si, S, Ar, Ca, Ti (so-called
$\alpha$-elements) than Fe-group nuclei like Fe and Ni (only of the order 0.1
M⊙), their average ratio of $\alpha$/Fe is larger than the corresponding solar
ratio.

Figure 41: Derived [Eu/Fe] abundances for the sample, as a function of
metallicity: r-I stars (green triangles), r-II stars (blue squares), limited-r
(red stars), and non r- process-enhanced stars (black dots); upper limits are
shown with black arrows (Hansen  _et al._ , 2018)

For most stars their surface abundances represent the composition of the
interstellar gas out of which they formed. This is not the case for already
evolved stars which blew off part of their envelope by stellar winds or stars
in binary systems with mass exchange. With these exceptions we can look back
into the early history of the Galaxy via the surface abundances of unevolved
low-mass stars, witnessing the composition of the interstellar medium at the
time of their birth. In Fig. 8 of section II.1 some of these aspects were
displayed, with the ratio [Mg/Fe] plotted as a function of metallicity [Fe/H]
for stars in our Galaxy. For Mg (a typical $\alpha$-element and representative
for many elements from O to Ti) one sees—with a relatively small scatter—a
flat value of [Mg/Fe] between 0.3 and 0.5 up to $\text{[Fe/H]}\leq-1$, which
decreases down to solar values at $\text{[Fe/H]}=0$. This can be explained, as
discussed above, by the early appearance of core-collapse supernovae from fast
evolving massive, single stars, producing on average $\text{[Mg/Fe]}=0.4$ (see
e.g. Woosley  _et al._ , 2002; Woosley and Heger, 2007; Limongi and Chieffi,
2018) before type Ia supernovae set in. The properties of the latter have been
reviewed by Hillebrandt  _et al._ (2013); Maoz  _et al._ (2014); Goldstein and
Kasen (2018); Livio and Mazzali (2018) as well as their nucleosynthesis
features Seitenzahl and Townsley (2017); Nomoto and Leung (2017b). These basic
features of galactic evolution have been understood reasonably well for a
majority of elements (Matteucci and Greggio, 1986; Nomoto  _et al._ , 2013),
while still many open questions exist in stellar evolution and the supernova
explosion mechanisms. (This includes also the question of the role of more
massive stars, probably ending as black holes and related to so-called
hypernovae / long-duration gamma-ray bursts, and even more massive pair
instability supernovae Heger  _et al._ , 2003; Ertl  _et al._ , 2016; Sukhbold
_et al._ , 2016).

Figure 42: Figure taken from Rosswog  _et al._ (2017), indicating the required
r-process ejecta masses as a function of the occurrence frequency of the
production site (for references we refer to the original paper). The figure
shows that on a typical SN frequency of 1/100 yrs about $10^{-4}$ to $10^{-5}$
M⊙ of r-process matter would need to be produced, for binary merger ejecta
with about $10^{-2}$ M⊙ the frequency must be rarer by a factor of 100 to
1000, and if even 1 M⊙ of r-process matter would be ejected in specific
events, the frequency must be again lower by another factor of 100.

The solar abundance of Eu is to more than 90% dominated by those isotopes
which are produced in the r process (Bisterzo  _et al._ , 2015, 2017).
Therefore it is considered as a major r-process indicator. The ratio [Eu/Fe],
already displayed in Fig. 8 of section II.1 and its recent update in Fig. 41
(above), shows a huge scatter by more than two orders of magnitude at low
metallicities, corresonding to very early in galactic evolution. While the
evolution of the average ratio resembles that of the alpha elements, being of
a core-collapse supernova origin and also experiencing a decline to solar
ratios for $\text{[Fe/H]}\geq-1$, it is far more complex to understand Eu than
$\alpha$-elements like Mg. In this section, we will discuss the suggested
origins for the r process and the possibility of their discrimination. A large
scatter seems to indicate a not yet well mixed or averaged interstellar medium
and thus a lower occurrence rate for r-process events, compensated by larger
amounts of their ejecta (see Fig. 42) in order to be consistent with total
solar abundances.

Figure 43: Left: [Ge/H] vs. [Fe/H] ratios from low metallicity observations.
One sees a correlated rise of Ge and Fe, i.e. at these low metallicities
(where type Ia supernovae do not yet contribute significantly) this points to
a joint origin, where Ge and Fe co-produced, i.e. in core-collapse supernovae.
Center: [Ge/Fe] vs. [Eu/Fe]. While Ge/Fe is already almost constant, related
to the ratio displayed in the top panel, Eu/Fe shows large variations, due to
rare (non-supernova?) events which were contributing in a different way.
Right: [Zr/Fe] vs. [Eu/Fe]. One sees a slight/weak correlation, indicating
that Zr is partially co-produced with heavy (strong) r-process elements like
Eu. However, the very weak correlation points also to a combination of
different and joint origins, possibly also a contribution from a weak r
process or other nucleosynthetic origin from core-collapse supernovae (Cowan
_et al._ , 2005).

Also rare events, if consistent with the overall solar r-abundances, would
need to support the averaged [Eu/Fe] ratios at low metallicities, combined
with a scatter of individual observations. The observed approach to the
average [Eu/Fe] ratio with a small scatter occurs only in the interval
$-2\leq\text{[Fe/H]}\leq-1$. This shift in comparison to the behavior of
[Mg/Fe] (see Figs. 8 and 41) is consistent with the fact that the much higher
supernova rate, responsible for Mg (but also e.g. Zn and Ge), leads to a much
earlier approach to average values already at metallicities of about
$\text{[Fe/H]}=-3$.

A further interesting aspect of this analysis is related to the question
whether r-process elements are correlated or not correlated with other
nucleosynthesis products, in order to determine whether they were co-produced
in the same nucleosynthesis site or require a different origin. In Fig. 43 we
show the behavior of Ge, Zr, and Eu with respect to Fe. In case of different
origins of e.g. Fe, Ni, Zn, and Ge (from regular core-collapse SNe) vs.
r-process elements, there should be no correlation between Fe and Eu, as
actually observed (Cowan  _et al._ , 2005).

Combining these considerations, we come to the following preliminary
conclusions for the sites discussed in section VI:

  1. a.

Electron-capture supernovae can possibly produce a weak r process, not a
strong one, and they are probably not rare, if containing stars from the
interval of 8 to 10 M⊙ of the initial mass function.

  2. b.

The neutrino-induced processes in He-shells of low-metallicity massive stars
(Epstein  _et al._ , 1988; Banerjee  _et al._ , 2011, 2016) would also be
frequent events at low metallicities, and not lead to a large scatter of e.g.
[Eu/Fe]. In addition, the related peaks would not be consistent with a strong
r process.

  3. c.

The regular neutrino-driven core-collapse SNe which produce Fe, but at most a
weak r process (e.g. Wanajo  _et al._ , 2011; Martínez-Pinedo  _et al._ ,
2012; Roberts  _et al._ , 2012, for an extended set of references see section
VI.1.1) are excluded as site of a strong r process, because they do not
produce the correct abundance pattern and would also be too frequent, not
permitting a large scatter in [Eu/Fe] at low metallicities.

  4. d.

Quark deconfinement supernovae could be rare, if existent, but present
predictions display not a full strong r process up to the third peak.

  5. e.

Magneto-rotational supernovae, leading to magnetars (i.e. neutron stars with
magnetic fields of $10^{15}$ G) rely still on parameter studies and depend on
assumptions on rotation rates and magnetic field strengths. Due to these
somewhat extreme initial conditions before collapse, they will be rare events,
possibly as infrequent as 1 in 100 to 1 in 1000 of regular core-collapse
supernovae. This would produce a large [Eu/Fe] scatter and also be consistent
with the required total r-process production (see Fig. 42) and abundances
pattern, but this site requires observational confirmation.

  6. f.

Collapsars, if resulting in events as predicted by Siegel  _et al._ (2018),
could also be consistent with observations, if they would occur even rarer
than magneto-rotoational supernovae by a factor of 100. At a first glance they
would contradict the observational finding that r-process sites should not co-
produce Fe (see Fig. 43). However, in solar abundances the ratio of
Fe/r-process elements is of the order 1000\. If collapsars would produce 0.5
M⊙/1 M⊙, this would result in ratios of about 0.5, being off by more than
three orders of magnitude. These negligible Fe contributions in comparison to
r-process matter with respect to solar would not be visible in a plot like
Fig. 43.

  7. g.

Compact binary mergers (NS-NS as well as NS-BH mergers) would lead to a
similar total ejecta mass and (rare) occurrence frequency as discussed in case
(e). Their occurrence and contribution to heavy elements is observationally
proven via gravitational wave, (short) GRB, as well as macronova/kilonova
observations. Abundance predictions are consistent with overall solar
r-abundances.

Summarizing the sites above, we have (of the sites consistent with the
observational constraints of Figs. 42 and 43), as well as the reproduction of
a solar r-process pattern, the following ones remaining: (e) magneto-
rotational supernovae (magnetars), (f) collapsars, and (g) compact binary
mergers. Of these (e) and (f) would belong to massive stars, i.e. occurring
during the earliest instances of galactic evolution. (g) is related to the
coalescence of compact objects produced via the collapse of massive stars and
would possibly experience a delay in their occurrence. Such differences might
be recognizable in galactic evolution modeling.

The following subsection will address the question how rare and frequent
events can be modeled consistently in galactic chemical evolution. Another
aspect is how actually early galactic evolution takes place. There exist
indications that (ultra-faint) dwarf galaxies are the earliest building blocks
of galactic evolution and their merger will finally lead to the evolution of
the early Galaxy as a whole. Due to different gas densities they might
experience different star formation rates and due to a low gravitational pull
they might lose explosive ejecta more easily. This can have an effect on the
point in time (and metallicity) when the first imprints of explosive ejecta
can be observed. These features will be addressed as well.

###  VIII.2 Galactic Chemical Evolution Modelling

####  VIII.2.1 Homogeneous evolution models

Chemical evolution of galaxies has made strong advances since its early days.
Initially all approaches made use of the instantaneous recycling approximation
in the sense, that the ejecta of stellar end stages were utilized without
delay immediately after the corresponding star formation, assuming that the
stellar evolution time scale is short in comparison to galactic evolution. If,
in addition, the instantaneous mixing approximation was applied, i.e. assuming
that the ejecta were instantaneously mixed throughout the galaxy, the whole
galaxy acts as homogeneous box. Neglecting this complete mixing throughout the
Galaxy can explain radial gradients, but would still assume mixing within
large and extended volumes (e.g. radial shells). Further developments included
infall of primordial matter into and outflow of enriched material out of the
Galaxy (for a review of these early investigations see e.g. Audouze and
Tinsley, 1976; Tinsley, 1980). When relaxing the instantaneous recycling
approximation, i.e. taking into account that (explosive) stellar ejecta enter
the interstellar medium (ISM) delayed with respect to the birth of a star by
the duration of its stellar evolution, detailed predictions for the evolution
of element abundances can be made. Based on nucleosynthesis predictions for
stellar deaths, a number of detailed analyses have been performed, from light
elements up to the Fe-group (e.g. Matteucci and Greggio, 1986; Wheeler  _et
al._ , 1989; Timmes  _et al._ , 1995; Matteucci and Chiappini, 2001; Kobayashi
_et al._ , 2006; Pagel, 2009; Matteucci, 2012; Nomoto  _et al._ , 2013). Such
approaches have also been applied to understand the enrichment of heavy
elements in the Galaxy (including r-process contributions) as a function of
time or metallicity [Fe/H] (see e.g. Travaglio  _et al._ , 1999; Ishimaru and
Wanajo, 1999; De Donder and Vanbeveren, 2004; Wanajo and Ishimaru, 2006;
Matteucci  _et al._ , 2014; Ishimaru  _et al._ , 2015; Vangioni  _et al._ ,
2016; Côté  _et al._ , 2017; Hotokezaka  _et al._ , 2018; Côté  _et al._ ,
2018a).

The instantaneous mixing approximation (IMA) simplifies a chemical evolution
model in terms of mass movement. In detail, all event outputs are expected to
cool down and mix with the surrounding ISM instantaneously. Such approaches
always result in an average value of element ratios and do not exhibit a
spread which can be observed at low metallicities. Thus, the greatest problem
of the IMA simplification is: All stars at a given time inherit the same
abundance patterns of elements and therefore it is impossible to reproduce a
scatter in the galactic abundances, which seems to be a crucial aspect,
especially at low metallicities. As a consequence, instead of a spread of
abundance distributions, curves with a single value for each [Fe/H] are
observed. Utilizing the instantaneous mixing approximation leads thus also to
a unique relation between galactic evolution time and metallicity [Fe/H], i.e.
any [Fe/H] can be related to a specific time in the evolution of a galaxy and
for each [Fe/H], due to the instantaneous mixing, only a mean value of [X/Fe]
(X being the element of interest to follow in chemical evolution) is obtained.
This (IMA) approach may be used to get a quick overview of the trends in
chemical evolution with a considerably lower computational effort for such a
model. For a detailed study of especially early chemical evolution, including
the reproduction of spreads in abundance ratios due to local inhomogeneities,
however, this approach is not sufficient.

####  VIII.2.2 Inhomogeneous galactic chemical evolution

Local inhomogeneities can only be produced if only limited amounts of ISM are
polluted by and mixed with the ejecta of each event. The latter effect is of
essential importance especially at low metallicities, where portions of the
ISM are already polluted by stellar winds and supernovae, and others are not.
Inhomogeneous mixing could produce larger element ratios in strongly polluted
areas and smaller values in still less polluted ones. This means that the
scatter in [X/Fe] at low metallicities can be a helpful asset in hinting to
the origin of element X. Inhomogeneous mixing can experience similar [Fe/H]
values in different locations of the Galaxy at different times or different
[Fe/H] values at the same time. In addition, different portions of the ISM are
polluted by different types of events, leading to a scatter at the same
metallicity, which can in fact be utilized as a constraint for these different
stellar ejecta. This is especially the case in the very early galactic
evolution ($\text{[Fe/H]}<-3$), when locally (out of a whole intitial mass
function, IMF) only a few stars might have exploded and imprinted their
stellar neighborhood with their ejecta. Thus, rare events, which produce large
amounts of element X in each event, would cause a large scatter, which could
be used as a very helpful constraint to identify the production site.
Therefore, more advanced chemical evolution studies revoked the instantaneous
mixing approximation (e.g. Chiappini  _et al._ , 2001; Recchi  _et al._ ,
2001; Argast  _et al._ , 2004; Spitoni  _et al._ , 2009; Recchi  _et al._ ,
2009; van de Voort  _et al._ , 2015; Cescutti  _et al._ , 2015; Wehmeyer  _et
al._ , 2015; Shen  _et al._ , 2015; Hirai  _et al._ , 2015, 2017; Haynes and
Kobayashi, 2018). For the reasons summarized above, specially for the origin
of r-process elements like Eu, we think that only such inhomogeneous chemical
evolution models should be utilized.

While some of the models mentioned here are of a more stochastic nature,
Minchev  _et al._ (2014) started with truly chemo “dynamical” galactic
evolution models. These, as well as e.g. van de Voort  _et al._ (2015); Hirai
_et al._ (2015); Shen  _et al._ (2015); Kobayashi (2016); Hirai  _et al._
(2017); Haynes and Kobayashi (2018), are based on SPH simulations. They can
model in a self-consistent way massive mergers of galactic subsystems (treated
as infall in simpler models), energy feedback from stellar explosions causing
outflows (and introduced as such in simpler models), radial migrations in disk
galaxies, mixing and diffusion of matter in the ISM, and the initiation of a
star formation dependence with progenitor mass. Thus, these global SPH
approaches are on the one hand most suitable to model such environments.
However, the mass and smoothing length utilized for the SPH particles will
also determine the resolution and cause an artificial mixing on such scales
which are not neccessarily related to the real mixing mechanisms. These
effects go into the direction of the instantaneous mixing approximation on
such scales. In reality a $10^{51}$ erg supernova explosion mixes via a Sedov
blast wave only with about $5\times 10^{4}$ M⊙ of interstellar medium.
Although different explosion energies and ejecta geometries are encountered,
detailed simulations (Montes  _et al._ , 2016) come to similar results for
neutron star mergers. SPH particle masses or smoothing lengths can exceed
these scales.

Inhomogeneous chem(odynam)ical evolution models for r-process elements, like
Eu, have been provided by Argast  _et al._ (2004), comparing neutron star
mergers and core-collapse supernovae, by Cescutti  _et al._ (2015); Wehmeyer
_et al._ (2015), comparing MHD jet-SNe and regular core-collapse supernovae,
and van de Voort  _et al._ (2015); Shen  _et al._ (2015); Hirai  _et al._
(2015, 2017), only utilizing neutron star mergers. One of the main questions
here is related to the problem of reproducing [Eu/Fe] at low(est)
metallicities. Wehmeyer  _et al._ (2015) treated the galactic chemical
evolution of europium (Eu), iron (Fe) and $\alpha$-elements, like e.g. oxygen
(O), still utilizing a more classical stochastic approach with a parametrized
infall of primordial matter, and a Schmidt law for star formation. This
neglects (large scale turbulent) mixing effects and includes only those
introduced by stellar explosions and the so initiated mixing with the
surrounding ISM according to a Sedov-Taylor blast wave. However, the model
permits inhomogeneities due to a local memory of the addition of stellar
ejecta. This stochastic approach grasps the main features of the impact of the
first stars and their (explosive) ejecta on the evolution of the heavy element
enrichment. Haynes and Kobayashi (2018) did probably perform the most
extensive simulations of all these approaches. Ojima  _et al._ (2018), on the
other hand, utilized a variation of sizes of galactic substructures of the
scale of (ultrafaint) dwarf galaxies within an IMA approach, made use of
related different star formation rates and different outflows according to
their gravity, and added stochastically—within a Monte Carlo scheme—neutron
star mergers in these substructures within the range of possible coalescence
delay times. The merging of these substructures is expected to represent
eventually the early Galaxy as a whole.

One of the major aspects within these approaches for the treatment of compact
binary mergers is the connection of earlier supernova events, producing a
neutron star and Fe ejecta, with the later delayed merger event and its
r-process ejecta. Since explosive events give rise to nucleosynthesis, inside
a supernova remnant bubble (given by a Sedov-Taylor blastwave) the abundances
of metals are higher than outside such a remnant. A star which is born later
_inside_ such a remnant will inherit more metals than a star born _outside_.
Thus, it is of high importance, _where_ a star is born during galactic
evolution, especially in the early phases. If the later merger is occurring
within the supernova remnant bubble, producing large amounts of r-process
matter, the [Fe/H] has been set already by the earlier supernova explosions
and the related [Eu/Fe] ratios will appear at the appropriate [Fe/H]. This is
the main question with respect to having large Eu contributions already at
lowest metallicities. There are ways out of this connection: (a) neutron star
kicks during the supernova explosions act in such a way that the actual
neutron star mergers takes place outside the initial Fe pollution by the
preceding supernovae, (b) neutron star - black hole mergers would have only
experienced the Fe-ejecta of one supernova, (c) large scale turbulent mixing
leads to the dilution of Fe on timescales shorter than the coalescence delay
time of the mergers. Such effects are not (yet) necessarily treated correctly
by present models.

###  VIII.3 Connecting observational constraints on r-process abundances with
different astrophysical sites

In subsection VIII.1, we listed possible production sites for a strong r
process. They need to fulfill the observational constraints: (i) in order to
lead to a large scatter of r/Fe at low metallicities these events must be rare
in comparison to regular core-collapse supernovae (this does not exclude the
latter from being the site for a weak r process), and (ii) if they should be
the dominant site responsible for the solar r-abundances, the combination of
their ejecta mass and occurrence frequency must be able to match this
requirement. Of the listed possible sites (e) magneto-rotational supernovae,
(f) collapsars, and (g) compact binary mergers could possibly fulfill both
criteria. (g) stays for a rare, observed and confirmed event with the ejecta
amounts consistent with reproducing solar abundances. (e) and (f) are
possible, still postulated, events. If the required high rotation rates and
magnetic fields for magneto-rotational supernovae (leading to magnetars) can
exist, they would eject similar amounts of r-process matter as binary mergers,
and if they exist in reality they wil be rare. (f) collapsars, also known as
hypernovae or observed as lGRBs, have been related to high 56Ni ejecta, but
recently also postulated to eject as much as 1 M⊙ of r-process matter (Siegel
and Metzger, 2017; Siegel  _et al._ , 2018; Janiuk, 2018). In a such case,
these events should be very rare, even a factor of 100 rarer than compact
binary mergers.

A further requirement, in addition to the two discussed above (rarity and
reproducing the total amount of solar r-abundances) is that galactic evolution
modeling should reproduce the observed time or metallicity evolution. A number
of inhomogeneous chemical evolution studies, which can follow local variations
of abundances due to the specific contributions by individual explosions, have
implemented this (Argast  _et al._ , 2004; van de Voort  _et al._ , 2015; Shen
_et al._ , 2015; Wehmeyer  _et al._ , 2015; Hirai  _et al._ , 2015; Mennekens
and Vanbeveren, 2016; Hirai  _et al._ , 2017; Haynes and Kobayashi, 2018).
There exists one difference between (e, f) and (g), in the first case the
events result from the final phases of a single massive star, in the latter
one or two prior supernovae (producing a neutron star and Fe-ejecta) are
required, before—with a delay—(the delay or coalescence timescale) the merger
can take place.

Therefore the question arises, whether the scatter of r-process elements like
Eu compared to Fe, [Eu/Fe], at low metallicities (which covers more than two
orders of magnitude, see Figs. 8 and 41, and hints at production sites with a
low event rate) and its time evolution before approaching an average ratio
only in the interval $-2<\text{[Fe/H]}<-1$, can be explained by all scenarios.
The delay of approaching an average value in comparison to [Mg/Fe], where this
takes place at $\text{[Fe/H]}<-3$, can be due to the much lower rate in
comparison to the core-collapse SNe rate, causing the largely different
metallicities where the approach to average values takes place. But the
question is also whether the early appearance of high [Eu/Fe] values can be
consistent with a “delayed” process like compact binary mergers. The
supernovae responsible for producing at least one neutron star produce also
Fe, which can shift the appearance of a typical r-process element like Eu to
higher metallicities [Fe/H] (see Fig. 44). This effect has been discussed in
inhomogeneous galactic evolution models utilizing neutron star mergers, i.e.
events with two prior supernovae and their Fe-ejecta (Argast  _et al._ , 2004;
Cescutti  _et al._ , 2015; Wehmeyer  _et al._ , 2015; Haynes and Kobayashi,
2018). These authors come to the conclusion, also van de Voort  _et al._
(2018), that neutron mergers have problems to explain [Eu/Fe] at lowest
metallicities, while other inhomogenous models claim to do so (van de Voort
_et al._ , 2015; Shen  _et al._ , 2015; Hirai  _et al._ , 2015; Ramirez-Ruiz
_et al._ , 2015; Hirai  _et al._ , 2017). This difference is probably related
to resolution and mixing issues discussed in the previous subsubsection, which
include uncertainties (and also possible solutions), but it should be noted
that the high resolution run of van de Voort  _et al._ (2015) as well as
recent further investigations (van de Voort  _et al._ , 2018) indicates the
rise of [Eu/Fe] to occur at too high metallicities as well. Whether and how
much the use of NS-BH mergers, which explode in an evironment polluted only
with Fe by one prior supernova, improve this situation needs to be seen
(Wehmeyer  _et al._ , 2018). A different issue is the formation of the Galaxy
from small substructures like (ultra-faint) dwarf galaxies, which, due to
different gas densities, experience different star formation rates and, due to
small gravity, the loss of metals from explosive events, both shifting the
occurrence of abundance features to lower metallicities. An early simpler IMA
approach by Ishimaru  _et al._ (2015), recently extended by a stochastic
inclusion of neutron star mergers, which are permitted to vary statistically
with respect to coalescence time scales (Ojima  _et al._ , 2018), seems also
to provide a solution.

Figure 44: top: Influence of coalescence timescale and neutron star merger
probability on Eu-abundances in inhomogeneous galactic chemical evolution.
Magenta stars represent observations. Red dots correspond to model star
abundances as in Argast  _et al._ (2004). The coalescence timescale utilized
is $10^{8}$ years with a typical probability consistent with population
synthesis Wehmeyer  _et al._ (2015). Green dots illustrate the effect on the
abundances if the coalescence timescale is shorter (around $10^{6}$ years).
Blue dots show the abundance change if the probability of neutron star mergers
is increased. bottom: results of a simpler galactic evolution approach with
IMA by Côté  _et al._ (2017) in comparison to observational data (crosses and
circles), utilizing a set of supernova yields and delay time distributions for
the binary merger following a power law $t^{-\gamma}$ in the range $10^{7}$ to
$10^{10}$ years. Population synthesis studies favor $\gamma=1$. Within these
treatments of galactic chemical evolution, none of the binary compact merger
studies would permit a fit with observations of low metallicity stars in the
metallicity range $-4<\text{[Fe/H]}<-2.5$.

The previous investigations utilized coalescence delay times for neutron star
mergers after the formation of the binary neutron star system with a narrow
spread. Population synthesis studies, consistent with the occurrence of short-
duration gamma-ray bursts (sGRBs, related to mergers) indicate that the
possible delay times follow a distribution with a large spread, ranging over
orders of magnitude with a $t^{-1}$ behavior. Based on such a behavior,
studies with simpler IMA modeling of chemical evolution (Côté  _et al._ ,
2017; Hotokezaka  _et al._ , 2018; Côté  _et al._ , 2018a) come to the
conclusion that mergers would not be able to reproduce the early galactic
evolution for metallicities $\text{[Fe/H]}<-2$, and not even the downturn at
$\text{[Fe/H]}=-1$. Thus, problems would remain to explain the strong r
process by NS-mergers alone. But the path to the binary merger is a complex
one. The mass, ejecta, and explosion energy of the second supernova in a
binary system have to be addressed for a full understanding (Müller  _et al._
, 2018). This includes the fact that in the binary evolution neutron star
kicks from supernova explosions could move the neutron star binary far out of
the reach of the initial supernova remnants which polluted the local ISM with
Ni and Fe (Fryer and Kalogera, 1997; Kalogera and Fryer, 1999; Abbott  _et
al._ , 2017b). If in such a way the merger event can be displaced from the
original supernovae, Wehmeyer  _et al._ (2018) finds that mergers could be
made barely consistent with the [Eu/Fe] observations, if such displacement is
taken into account and very short coalescence timescales of $10^{6}$ yr are
used. In addition, the difference between neutron star mergers and neutron
star - black hole mergers, experiencing only the pollution by one prior
supernova, has to be investigated (Wehmeyer  _et al._ , 2018).

After this still somewhat inconclusive result with respect to the question
whether neutron star mergers alone can provide the explanation for a strong r
process throughout and also in the early Galaxy, there exist the other options
(e and f) related to massive stars which can enter their ejecta immediately at
the end of their stellar life and are possibly neglible Fe-producers. Thus,
alternatively, a rare class of core-collapse supernovae (or collapsars?) might
play a significant role, exploding early in galactic evolution. If producing
also a solar r-process pattern, they could be responsible for explaining
observations at low metallicities. Early suggestions, that so-called electron-
capture supernovae (a) in the stellar mass range 8-10 M⊙ (Kitaura  _et al._ ,
2006; Janka  _et al._ , 2008; Wanajo  _et al._ , 2009, 2011) would be able to
produce a strong r process, were never confirmed. They would also not
correspond to rare events. However, other objects driven by strong magnetic
fields and fast rotation (possibly about 1% or less of all core-collapse
supernovae), leaving behind $10^{15}$ Gauss neutron stars (magnetars), could
be important. Such magneto-rotational (MHD)jet-SNe (e) show similar
characteristics in the amount of r-process ejecta and the occurrence frequency
as neutron star mergers, but—because these objects result from massive single
stars—they do not experience the delay of binary evolution (Winteler  _et al._
, 2012; Nishimura  _et al._ , 2015b, 2017; Mösta  _et al._ , 2018; Halevi and
Mösta, 2018). This means that they enter galactic evolution at lowest
metallicities, resulting in a large scatter in r/Fe due to them being rare
events. This can be seen in Fig. 45, which shows the result of a superposition
of magneto-rotational MHD-jet supernovae and neutron star mergers. This way
observations can be matched from lowest metallicities up to present. The very
recent results by Siegel  _et al._ (2018) and also Janiuk (2018) for
collapsars (f) would point in the same direction. One aspect which is not
clear here, is whether hypernovae, collapsars, lGRBs are a homogeneous class
of objects and would all come with huge amounts of 56Ni ejecta. This would in
principle be contradictory to the observational findings of Fig. 43,
indicating that strong r-process sites are not correlated with Fe-group
ejecta, unless the relative overproductions of e.g. Fe vs. Eu with respect to
solar abundances is negligible (which would be the case for 56Ni ejecta of 0.5
M⊙ and r-process ejecta of up to 1 M⊙).

Figure 45: Evolution of Eu-abundances in galactic chemical evolution models
(Wehmeyer  _et al._ , 2015) including both magneto-rotational supernovae and
neutron star mergers as r-process sites. Magenta stars represent observations
whereas green dots represent model stars. The combination of magneto-
rotational MHD-jet supernovae - being of strong importance at low
metallicities - early in the evolution of the Galaxy, and neutron star mergers
permits a perfect fit with observations.

###  VIII.4 r-Process Cosmochronometers

A complete list of isotopes with half-lives in the range $10^{7}-10^{11}$ yr
is given in Table 2. They cover a time span from a lower limit in excess of
the evolution time of massive stars up to (and beyond) the age of the
universe. Such nuclei can be utilized as “chronometers” for processes in
galactic evolution and also serve as a measure for the age of the galaxy (and
thus as a lower limit for the age of the universe, see the earlier discussion
in II.4). The list is not long. Two of the nuclei require predictions for the
production of the ground and isomeric (92Nb, 176Lu). With the exception of
40K, all of the remaining nuclei are heavier than the “Fe-group” and can only
be made via neutron capture.

Table 2: Isotopes with half-lives in the range $10^{7}$-$10^{11}$ yr Isoptope | Half-Life | Isoptope | Half-Life
---|---|---|---
40K |  $1.3\times 10^{9}$ yr |  205Pb |  $1.5\times 10^{7}$ yr
87Rb |  $4.8\times 10^{10}$ yr |  232Th |  $1.4\times 10^{10}$ yr
92Nb |  $3.5\times 10^{7}$ yr |  235U |  $7\times 10^{8}$ yr
129I |  $1.6\times 10^{7}$ yr |  236U |  $2.3\times 10^{7}$ yr
147Sm |  $1.1\times 10^{11}$ yr |  238U |  $4.5\times 10^{9}$ yr
176Lu |  $3.7\times 10^{10}$ yr |  244Pu |  $8\times 10^{7}$ yr
187Re |  $4.4\times 10^{10}$ yr |  247Cm |  $1.6\times 10^{7}$ yr

The nuclei with half-lives comparable to the age of the Galaxy/Universe, 232Th
and 238U, are made in a single nucleosynthesis process (as well as all other
actinide isotopes listed here). These elements, the heaviest ones occurring in
nature beyond Bi, are all products of the r process. The possible
astrophysical settings have been discussed in the previous sections. The
question is how to predict reliable production ratios for these long-lived
isotopes, if (a) not even the site is completely clear, and (b) even for a
given site nuclear uncertainties enter. The abundances of especially these
heaviest nuclei produced in the r process depend on mass models, $\beta$-decay
properties (half-lives, delayed fission, delayed neutron emission), fission
barriers and fragment distributions, and last but not least neutron captures,
especially their rates during the r-process freeze-out, pointing also to the
role direct capture plays in comparison to compound nuclear reactions (see
e.g. Goriely and Martínez-Pinedo, 2015; Eichler  _et al._ , 2015; Mendoza-
Temis  _et al._ , 2015, and the general discussion in sections IV and V).

Among the stars with observed Th and U, there exist a number of so-called
actinide-boost stars with an enhanced ratio of Th/Eu and U/Eu in comparison to
all other r-process enhanced stars (see e.g. Roederer  _et al._ , 2009;
Holmbeck  _et al._ , 2018a), observed especially at low metallicities around
$\text{[Fe/H]}\approx-3$. While most of their element abundances, up to the
third r-process peak, are consistent with solar-system r-abundances, the
deviations occur essentially only for elements beyond the third r-process
peak. This could point to either a different site than the one responsible for
the solar-type r-process abundances or variations of conditions in the same
site, dependent on still unknown aspects (Holmbeck  _et al._ , 2018b).

Nevertheless, for many years such chronometers have been utilzed to attempt
predictions for the age of the Galaxy as a lower limit for the age of the
Universe (for early reviews see Fowler and Hoyle, 1960; Schramm and
Wasserburg, 1970; Cowan  _et al._ , 1991). This has been performed initially
with simplified chemical evolution models via (i) the prediction of 232Th/238U
and 235U/238U ratios in r-process calculations, (ii) applying them in galactic
evolution models, which include assumptions about the histories of star
formation rates and r-process production, and finally (iii) comparing these
ratios with meteoritic data, which provide the Th/U and U/U ratios at the
formation of the solar system. In addition for the early reviews on this
method, mentioned above, see Panov  _et al._ (2017) for recent investigations
along these lines. An advantage (somewhat decreasing the nuclear uncertainties
involved) is that 232Th and 235,238U are populated by $\alpha$-decay chains
und therefore uncertainties in the predictions of individual isotopes average
out to some extent Goriely and Clerbaux (1999). Parametrized, so-called site-
independent fits, based on a superposition of neutron densities which
reproduce all solar r-process abundances from $A=130$ through the actinides,
have been utilized to predict their production ratios (Goriely and Arnould,
2001) (obtained in a similar fashion as those in Fig. 15). Goriely and Janka
(2016) have used adiabatic and steady-state neutrino-driven wind models and
considered the superposition of a large number of contributing components to
study the production of cosmochronometers. Alternatively to the ratio of long-
lived actinide isotopes, also the ratios of e.g. 232Th and 238U to other
(stable) r-process nuclei like Eu can be utilized to check for consistency.
Other options include the ratio of Th and U to stable Pb, which has in
addition to the s-process contribution (i) a direct r-process contribution to
the 206,207,208Pb isotopes (ii) a contribution due to fast $\alpha$\- and
$\beta$-decay chains from unstable nuclei produced in the r process beyond Pb
(decaying within less than $10^{6}$ yr), and finally (iii) a contribution from
the long-lived decay chains originating at 232Th and 235,238U (Roederer  _et
al._ , 2009; Frebel and Kratz, 2009).

Low-metallicity stars have the advantage that one can avoid uncertainties
introduced by chemical evolution modeling. The metallicities of the halo stars
for which neutron-capture element data have become available range from
$\text{[Fe/H]}=-3$ to about $-2$. Typical, still simple (galactic chemical)
evolution calculations suggest roughly the metallicity-age relation
$\text{[Fe/H]}=-1$ at $10^{9}$ yr, $\text{[Fe/H]}=-2$ at $10^{8}$ yr, and
$\text{[Fe/H]}=-3$ at $10^{7}$ yr (if the instantaneous mixing approximation
IMA could be applied at such low metallicities, see, e.g. Tsujimoto  _et al._
, 1997; Chiappini  _et al._ , 2000). Even if these estimates are uncertain by
factors of 2–3, very low metallicity stars most certainly were born when the
Galaxy was only $10^{7}$–$10^{8}$ yr old, a tiny fraction of its present age.
Thus, the neutron-capture elements observed in very low metallicity stars were
generated in only one or at most a few prior nucleosynthesis episodes. If
several events contributed, the time interval between these events had to be
very short in comparison to Th decay ages. Thus, no error is made by simply
adding these contributions, without considering decays, and treating them as a
single r-process abundance distribution which undergoes decay from the time of
its incorporation into a (low metallicity) star until its detection in present
observations. Making use of the earlier mentioned, so-called site-independent,
predictions for r-process production ratios, combined with the presently
observed abundance ratios found in low-metallicity stars, can give an
indication for the decay time of its radioactive isotopes since the star was
born and polluted with an original r-process pattern. Typical results for ages
of most low-metallicity r-process enhanced stars are in the range of 12–14 Gyr
(Cowan  _et al._ , 1999; Schatz  _et al._ , 2002; Kratz  _et al._ , 2004;
Roederer  _et al._ , 2009). This approach assumes that the production ratios
of the site(s) responsible for these observations are consistent with those
reproducing a solar r process.

Actinide boost stars are observed especially at low metallicities around
$\text{[Fe/H]}\approx-3$ (see Fig. 10 from Holmbeck  _et al._ , 2018a). This
could point to either a different site than the one responsible for the solar-
type r-process abundances or variations of conditions in the same site,
dependent on still unknown aspects. An interesting feature, affecting actinide
to Eu ratios is related to the path of the r process and the timing when
fission plays a role. Due to this the r-process results are dependent on the
proton/nucleon ratio $Y_{e}$ in the expanding matter. Holmbeck  _et al._
(2018b) could show, with their nuclear input, that the actinide to Eu ratio is
highest for a $Y_{e}$ in the range $0.15$, while lower $Y_{e}$’s (as expected
from the dynamic tidal ejecta of neutron star mergers) lead to smaller
actinide to Eu ratios, as found in most r-enhanced stars. The Th/U ratio,
reflecting abundances of nuclei relatively close in their mass, seems not that
much affected. Present predictions for magneto-rotational supernovae lean more
to higher $Y_{e}$ values close to 0.15. This is still all very speculative and
strongly affected by uncertainties in site specific conditions as well as in
nuclear properties. However, it is intriguing to check how both types of
abundance patterns can be observed in low-metallicity stars. Improved
predictions for all the most probable main r-process sites, discussed in the
previous section (plus possibly also exotic scenarios Fuller  _et al._ , 2017)
, can hopefully lead to a one-to-one connection between responsible production
sites and observations (although still affected both by astrophysical site as
well as nuclear uncertainties).

When applying the same (decay age) procedure to the actinide boost stars as in
earlier investigations (Cowan  _et al._ , 1999; Schatz  _et al._ , 2002; Kratz
_et al._ , 2004; Roederer  _et al._ , 2009), in order to determine their age
by assuming initial production ratios from the so-called site-independent
analysis responsible for a solar r-process pattern, unreasonably low or even
negative ages result, as can be seen in Fig. 10 on the right scale. Comparable
ages in the range 12–14 Gyr are attained only if the higher Th/Eu (and other)
production ratios are taken from nucleosynthesis conditions with $Y_{e}\approx
0.15$ (see Figs. 16 and 17 in Holmbeck  _et al._ , 2018b). Similar results are
found by Wu  _et al._ (2018) and Eichler  _et al._ (2018b). This is in line
with chemical evolution findings (Wehmeyer  _et al._ , 2015; Cescutti  _et
al._ , 2015; Côté  _et al._ , 2017, 2018b; Hotokezaka  _et al._ , 2018; Haynes
and Kobayashi, 2018) that at low metallicities an additional (early occurring)
r-process site is required before neutron star mergers can take over.

##  IX Summary and Conclusions

Since the postulation of the r process by Burbidge  _et al._ (1957) and
Cameron (1957) many nuclear experiments and astronomical observations improved
tremendously the available input and the constraints for continuous
developments and improvements of the underlying theory and models. Here we
will pass through the important features and the related present situation in
nuclear physics, astrophysical modelling, constraints from astronomical
observations and the understanding of the (chemical, i.e. abundance) evolution
of our Galaxy. With respect to the nuclear physics, which enters decisively in
producing the abundance pattern of the r process, major achievements have been
accomplished. This is related to experimental progress in accessing unstable
nuclei far from stability, combined with a growing theoretical understanding
of their properties. These enter in a number of fundamental ways:

  1. 1.

_nuclear masses_ far from stability and the behavior of shell closures are
responsible for the location and the shape of the r-process peaks and
subpeaks; close to shell closures the correct prediction of the shell gaps and
the on- and offset of deformation plays a major role.

  2. 2.

_$\beta$ -decay properties_, like half-lives, are responsible for the speed
with which the r process moves matter to heavier nuclei, with an impact (in
combination with the path location determined by 1.) on the height of peaks
and subpeaks; $\beta$-delayed neutron emission (affected by nuclear masses via
neutron separation energies) leads, especially in the late phases during decay
back to stability, to a smoothing of the abundance curve. In this final phase
the energy production by $\beta$-decays is also important to understand the
electromanetic emission which can be detected in astronomical observations.

  3. 3.

_$\alpha$ -decays_ are also important in the final phase, especially the
decay-chains originating from actinide nuclei, powering the light emission
detected in astronomical observations and determining the r-process Pb
abundance.

  4. 4.

_fission_ enters strongly in the termination of the r process, causing
fission-cycling which returns matter to lighter nuclei; the onset of fission
(strongest probably near the $N=184$ shell closure), or possible pathways in
the nuclear chart to avoid it, determines the strength of fission cycling, the
ratio of actinide to light and medium-mass r-process nuclei, and also the
possibility to form superheavy nuclei; fission fragment distributions affect
how masses are populated in the vicinity of the $A=130$ peak, especially also
whether substantial fractions can end in nuclei below $A=130$; the amount of
fission-neutrons released during the late phase of the r process, can shape
the final abundance pattern.

  5. 5.

last but not least, _neutron captures_ (and their inverse
photodisintegrations) are the substantial working horse for the rapid neutron
capture process; fortunately, in environments with high neutron densities and
temperatures responsible for the process, they occur on such a fast pace that
a chemical equilibrium between neutron captures and photodisintegrations is
attained, leading to an equilibrium compositon in each isotopic chain with
maxima at the same neutron separation energy throughout the nuclear chart;
this determines the r-process path and makes the connection to 1.; however,
during the final declining phase of neutron densities, the so-called r-process
freeze-out, this equilibrium cannot be maintained, and individual neutron
captures affect the final abundance shape; here it plays an essential role
that direct neutron capture probably wins over compound nucleus prescriptions,
as it leads probably to higher reaction rates than predictions based on a
compound nucleus picture far from stability; therefore, the above mentioned
equilibrium holds probably longer during freeze-out than expected, but
predictions and implementation of direct capture for neutron reactions are
clearly needed (and presently not fully implemented).

In the investigation of all these aspects much progress has occurred, but
major uncertainties are remaining, as experiments have touched nuclei in the
r-process path only at a limited number of locations in the nuclear chart.
However, the purely nuclear structure aspects are not the only ones to
overcome, there exist further nuclear/particle physics, general relativity,
magneto-hydrodynamiscs, atomic physics and numerical/computational challenges
in modelling astrophysical sites responsible for the production of r-process
nuclei:

  1. 1.

most environments expected to be sites of a strong r process involve objects
at highest densities, whether related to core-collapse, compact binary
mergers, or the inner areas of black hole accretion disks; the _nuclear and
supra-nuclear equation of state_ is of highest importance for modelling these
sites, providing the pressure response during dynamic phases, the nuclear and
particle composition, the existence of electron/positron pairs and their
degeneracy, and last but not least determining maximum neutron star masses,
which are important e.g. for answering the question whether in compact binary
mergers a hypermassive neutron star exists intermittantly or even remains as a
final outcome.

  2. 2.

such dense environments have to be modelled in _general relativistic, high
resolution multi-dimensional hydrodynamics_ schemes in order to give provide a
reliable outcome; this is highly challenging and computationally extremely
expensive.

  3. 3.

many of the discussed effects involve the modeling of _magnetic fields_ ,
possibly as a major ingredient to predict jet ejection; a decisive aspect is
whether and how magnetic fields can be enhanced during these events, where the
magneto-rotational instability plays a major role; high resolution magneto-
hydrodynamics modeling is a field only at the brink of getting decent results
for the modelling of complete astrophysical sites.

  4. 4.

weak interactions, including electron/positron captures are essential, but at
least of similar importance are _neutrino interactions with matter_ ; the
correct treatment of multi-D neutrino transport plays a major role in
determining the neutron to nucleon ratio $Y_{e}$, which is probably the key
ingredient for attaining a successful r process; neutrino flavor
transformations, especially via matter-neutrino resonances and fast pairwise
flavor conversion, can play an essential role in compact binary mergers, a
multi-D treatment for such complicated geometries is just at its infancy.

  5. 5.

in addition to neutrino transport, general _radiation transport via photons_
is important to predict the electromagnetic aftermath of explosions in order
to make a connection to observational features like lightcurves and spectra;
this involves still unkown opacities of heavy elements, but also the energy
release of highly unstable radioactive matter after explosive nucleosynthesis
has taken place.

Based on the presently available input for nuclear properties and the present
status with respect to modelling possible r-process sites, three major options
for sites of a strong r process have emerged:

  1. 1.

models of _compact binary mergers_ indicate that they are prolific sites of
r-process nucleosynthesis, with about 10-2 M⊙ of ejected r-process matter in
the dynamic ejecta and a similar amount from accretion disk outflows; when
including all components—dynamic ejecta, neutrino winds and viscous/secular
accretions disk outflows—they produce not only the heaviest r-process nuclei
but also significant amounts of the standard solar r-process abundances for
mass numbers with $A<130$; the first observation of a neutron star merger
(GW170817), accompanied by the AT 2017gfo macronova/kilonova afterglow, makes
this the first proven and confirmed production site of heavy r-process
elements; neutron star - black hole mergers are also promising, but need to be
confirmed.

  2. 2.

there exist observational indications of 1015 Gauss neutron stars (magnetars);
a rare class of _magneto-rotational core-collapse supernovae_ could lead to
such neutron stars with immense magnetic fields and related models produce
r-process matter ejected in polar jets; however, predictions from stellar
evolution about the distribution of magnetic fields and rotation rates before
core-collapse are needed in order to understand the initial conditions,
leading to such events, and investigating the role of the magneto-rotational
instability (MRI) during the collapse/explosion phase is impossible without
high resolution simulations. Thus, this is a promising site, but needs to be
confirmed by observations.

  3. 3.

very recent multi-D MHD simulation for _accretion disk outflows from
collapsars_ , i.e. objects which result from massive star evolution and end in
the formation of a black hole, argue that large amounts of up to ($\approx 1$
M⊙) of r-process material can be ejected; further simulations are needed in
order to understand whether the whole group of massive stars, leading to
hypernovae, collapsars, and long-duration GRBs are a homogeneous class with
the predicted outcome and observational confirmations related to the
composition of ejecta would be extremely helpful.

  4. 4.

while the three above mentioned sites are sites (or candidates) for a strong r
process, producing heavy elements up to the actinides, there exist further
options to produce a so-called _weak r process_ , probably also synthesizing
elements up to Eu, but with a steeper decline as a function of nuclear mass
number than found in the solar system r-abundance pattern; such sites include
electron capture supernovae, regular core-collapse supernovae, possibly also
quark deconfinement supernovae.

All of these models have to confronted with and scrutinzed by astronomical
observations. An interesting aspect is that one of the listed and confirmed
sites is related to binary systems, while the others are resulting from the
evolution of massive single stars. Observations supporting and constraining
r-process sites exist in a number of ways:

  1. 1.

the total amount of _r-process matter existing in the solar system_ (and the
Galaxy) has to be explained together with the overall solar r-abundance
pattern (although this could be a superposition of a multitude of events); it
turns out that all three possible sites are rare events; if the predicted
ejecta masses should be consistent with this requirement, the occurrence
frequency of compact binary mergers as well as magneto-rotational supernovae
should be between 1 event per 100 to 1000 regular supernovae; for mergers this
would be consistent with population synthesis studies; collapsars should be
even less frequent by a factor of about 100, which requires future
investigations.

  2. 2.

_radioactive tracers_ like 244Pu as well as 60Fe are found in _deep sea
sediments_ ; the production of 60Fe in frequent events, related to regular
core-collapse and electron capture supernovae, is supported by the latest
contribution dating back about 2 million years; on the contrary, the amount of
244Pu found in these sediments is lower than expected by about a factor of
100, if a quasi-continuous production of r-process elements is assumed; this
points to substantial decay since the last addition and to much rarer events
than supernovae, underlining the required rarity of events, consistent with
all three possible production sites for a strong r process.

  3. 3.

observations are not only related to explaining the solar abundances, in fact
_observations indicate the presence of neutron-capture elements in halo stars
at lowest metallicities_ ; in some cases these detections indicate a complete
r-process abundance pattern, in other cases only a partial (or incomplete)
r-process pattern is apparent (possibly indicating a weak r process origin),
but it appears as though all of the early stars have some level of r-process
enrichment; these abundance detections indicate nucleosynthesis early in the
history of the Galaxy and provide important clues about the nature of the
earliest stars.

  4. 4.

observations of lowest metallicity stars in our Galaxy and (ultra-faint) dwarf
galaxies show substantial variations in r-process elements, indicating a
production site with a low event rate and consistent high amounts of r-process
ejecta in order to explain solar abundances; this is also underlined by the
_large scatter of Eu/Fe_ (Eu being an r-process element and Fe stemming from
core-collapse supernovae at these low metallicities) _seen in the earliest
stars of the Galaxy_ ; this is explained by a not yet well mixed interstellar
medium with respect to the ratio of products from regular core-collapse
supernobvae and the rare r-process events.

  5. 5.

due to the availability of experimental atomic data and high resolution,
precision observations, r-process abundance determinations have improved much
over time; this permitted also to detect _the presence of long-lived
radioactive nuclei like Th and U_ , making even a _”dating” of stellar ages_
possible; the observed variations in the actinide to intermediate mass
r-process elements like Eu, leading to so-called _actinide-boost stars_ ,
concentrated at lowest metallicities, could even give clues about different
r-process sites.

While the above observations indicate (a) a rare site for the strong r
process, a requirement matched by all three candidates mentioned above, (b)
the observations related to the overall evolution of heavy r-process elements
in comparision to Fe, and especially its large scatter at low metallicities,
require inhomogeneous galactic evolution simulations, which can reproduce this
behavior and might actually point to favored sites:

  1. 1.

A major open question is: _can products of the neutron star merger r process
alone explain the observations_ of a large scatter of Eu/Fe and other
r-process elements seen already at metallicities of $\text{[Fe/H]}\leq-3$? As
the supernovae which produce the neutron stars of a merger already lead to a
substantial floor of Fe, they enhance [Fe/H]. Thus, the high [Eu/Fe] due to
the new ejecta would then be seen first at the metallicity [Fe/H] inherited
from the prior supernovae, if the merger ejecta are mixed with the same
interstellar medium as the prior supernova ejecta. For a typical explosion
energy of $10^{51}$ erg and typical densities of the interstellar medium, this
mixing would occur via a Sedov-Taylor blast wave in a range of about $5\times
10^{4}$ M⊙. Although different energies are encountered in the case of neutron
star mergers, recent investigations indicate that they would mix with a
similar amount of interstellar medium. Stochastic inhomogeneous chemical
evolution calculations, utilizing only this effect alone, show the appearance
of Eu only at about $\text{[Fe/H]}=-2$ for neutron star mergers. It has to be
investigated how this would be affected in the case of neutron star - black
hole mergers, as they would only lead to Fe-ejecta by one prior supernova.

  2. 2.

_Large-scale SPH simulations_ , also addressing these issues, can suffer from
resolution problems via the choice of the mass of the SPH particles involved
as well as smoothing lengths applied. Both of these aspects affect the
resolution and lead therefore (artificially) to more extended mixing than
applied in case 1. above. Such lower resolution blurrs the metallicity
behavior as it implies more extended mixing on the scales of the resolution.
This would mix Fe with larger amounts of interstellar medium and thus also
causes a lower [Fe/H] when the additional pollution by r-process ejecta takes
place. While these resolution issues can be a disadvantage of such
simulations, on the other hand they can also handle substantial turbulent
mixing of interstellar medium matter in the early Galaxy. Some of these
simulations seem to be able to reproduce the r-process behavior of low-
metallicity observations with compact binary mergers, but the most recent ones
also favor an additional site or source at lowest metallicities.

  3. 3.

_Neutron star kicks_ , resulting from a supernova explosion, could have the
binary neutron star system move out of its supernova remnants (polluted with
Fe and Ni), and the merger event could take part in galactic regions
unpolluted by Fe from earlier supernovae in the region of the final merger.
This could permit the ejection of r-process matter in environments with a
lower [Fe/H], also in the case of neutron star mergers. Preliminary
simulations with stochastic inhomogeneous models (as in 1.) are able to
reproduce observations, if coalescence delay times are as short as 1 Myr.

  4. 4.

Another option is that early on, in _galactic substructures of the size of
dwarf galaxies_ , different star formation rates can exist combined with a
loss of nucleosynthesis ejecta out of these galaxies due to smaller gravity.
This can shift the behavior of the [Eu/Fe] ratio as a function of metallicity
[Fe/H] to lower metallicities. When also considering a statistical
distribution of (down to small) coalescence timescales in the individual
substructures, the low-metallicity observations could possibly be matched,
while the merging of these substructures within the early Galaxy at later
times can be made consistent with the [Eu/Fe] decline (similar to alpha
elements) at $\text{[Fe/H]}=-1$.

  5. 5.

In somewhat simpler galactic evolution models, employing the instantaneous
mixing approximation and _coalescence delay time distributions following a
$t^{-1}$ power law_, as expected from population synthesis studies and
statistics of short duration gamma-ray bursts, apparently no model can
reproduce the metallicity dependence of the r-process/Fe abundance ratios with
neutron star mergers alone.

  6. 6.

The discussion above underlines, that it is still inconclusive whether binary
compact mergers alone can explain low metallicity observations, although they
are probably responsible for the dominant amount of r-process products in the
solar system and present Galaxy. _When introducing an additional component
which acts at lowest metallicities, this yields a perfect fit to
observations_. The detection of actinide boost stars, found in particular at
metallicities as low as $\text{[Fe/H]}\approx-3$, supports the argument for
such an additional component with different nucleosynthesis conditions being
active at such low metallicities.

This review of all aspects of the astrophysical r process, from nuclear
physics via stellar (explosive) modeling, astronomical observations, as well
as galactic evolution, has shown that substantial progress has been made since
it was postulated in the 1950s. But it also shows that, despite the very first
observation of an r-process production site (GW170817) in 2017, confirming
neutron star mergers as probably the most important site, many open questions
remain and further progress on all fronts is required in a truly
interdisciplinary effort, in order to answer them.

Existing and upcoming nuclear facilities world-wide (FAIR, FRIB, HIAF, RAON,
RIKEN, SPIRAL) will allow to produce neutron-rich nuclei along the r-process
path and to determine their properties, including, for the first time, nuclei
of the third r-process peak. Relativistic heavy-ion collision experiments
envisioned for FAIR, NICA and RHIC will generate and investigate nuclear
matter at the temperatures and densities as they exist in neutron-star mergers
(and core-collapse supernovae). These exciting experimental prospectives will
constrain and guide advances in global nuclear models, and together they will
decisively reduce the nuclear uncertainties currently hampering r-process
studies.

Observational programs targeting abundances in low-metallicity stars, like
RAVE, Gaia, and APOGEE will be incorporated in future surveys such as SDSS V,
4MOST, and WEAVE. This will provide highest quality information about the
chemical structure of the Galactic disc, the halo, and the bulge.
Multimessenger astronomy with present and future gravitational wave detectors
like LIGO, VIRGO, KAGRA, and IndIGO is expected to detect up to 10 compact
binary mergers per year, which can be followed up with observations of related
gamma-ray bursts and the electromagnetic afterglow. This permits to analyse
the outcome of many sources with different viewing angles and will help our
further understanding of this unique site, producing the heaviest elements in
the Universe.

And finally, the advent of improved theoretical models of the nuclear equation
of state, nuclei far from stability, stellar atmospheres, stellar evolution,
stellar explosions, and state-of-the-art chemo-dynamical models of the Galaxy,
plus by chance observational evidence for an additional (of the discussed or
unexpected) r-process site, will hopefully point us to a clear understanding
of the origin of the heavy elements from Fe to U.

In summary, we are living in exciting times for unravelling the mysteries of
r-process nucleosynthesis.

###### Acknowledgements.

We thank a very large number of colleagues and friends for helpful
collaborations and discussions, but especially Al Cameron and Willy Fowler
should be mentioned here who motivated many of us to explore the topic of this
review. This research was supported in part by: NASA grant HST-GO-14232 (JJC),
NSF grant AST1616040 (CS), NASA grant NNX16AE96G and NSF grant AST-1516182
(JEL), NSF grant Phys-0758100 and the Joint Institute for Nuclear Astrophysics
through NSF Grant Phys-0822648 (AA and MW), NSF award PHY-1430152 (JINA Center
for the Evolution of the Elements AA, JJC, and MW), the Extreme Matter
Institute EMMI (KL and GMP), the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) - Projektnummer 279384907 - SFB 1245 (GMP), the Swiss
SNF, an ERC Advanced Grant (FISH) and COST Actions NewCompstar (FKT) and
ChETEC (FKT and GMP).

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