11institutetext: Konkoly Observatory, Research Centre for Astronomy and Earth
Sciences, Eötvös Loránd Research Network (ELKH), Konkoly Thege Miklós út
15-17, H-1121 Budapest, Hungary 22institutetext: CSFK, MTA Centre of
Excellence, Budapest, Konkoly Thege Miklós út 15-17, H-1121, Hungary
33institutetext: INAF – Osservatorio Astronomico di Roma Via Frascati 33,
I-00040, Monteporzio Catone, Italy 44institutetext: NuGrid Collaboration,
http://nugridstars.org 55institutetext: E. A. Milne Centre for Astrophysics,
University of Hull, Hull HU6 7RX, UK 66institutetext: Department of Physics,
North Carolina State University, Raleigh, NC, 27695, USA 77institutetext:
Triangle Universities Nuclear Laboratory, Duke University, Durham, NC, 27710,
USA 88institutetext: Max-Planck Institute for Astrophysics, Postfach 1317,
85741 Garching, Germany 99institutetext: Institute for Nuclear Research
(ATOMKI), H-4001 Debrecen, Hungary 1010institutetext: Eötvös Loránd
University, Institute of Physics, Budapest 1117, Pázmány Péter sérány 1/A,
Hungary 1111institutetext: School of Physics and Astronomy, Monash University,
VIC 3800, Australia

# The $\gamma$–process nucleosynthesis in core-collapse supernovae

I. A novel analysis of $\gamma$–process yields in massive stars

L. Roberti 11223344 M. Pignatari 11225544 A. Psaltis 667744 A. Sieverding 88
P. Mohr 99 Zs. Fülöp 99 M. Lugaro 112210101111

(Received March 31, 2023; accepted June 15, 2023)

###### Abstract

Context. The $\gamma$–process nucleosynthesis in core-collapse supernovae is
generally accepted as a feasible process for the synthesis of neutron-
deficient isotopes beyond iron. However, crucial discrepancies between theory
and observations still exist: the average production of $\gamma$–process
yields from massive stars are too low to reproduce the solar distribution in
galactic chemical evolution calculations, and the yields of the Mo and Ru
isotopes are by a further factor of 10 lower than the yields of the other
$\gamma$–process nuclei.

Aims. We investigate the $\gamma$–process in 5 sets of core-collapse supernova
models published in literature with initial masses 15, 20, and 25 $\rm
M_{\odot}$ at solar metallicity.

Methods. We compared the $\gamma$–process overproduction factors from the
different models. To highlight the possible effect of nuclear physics input,
we also considered 23 ratios of two isotopes close to each other in mass,
relative to their solar values. Further, we investigated the contribution of
C–O shell mergers in the supernova progenitors as an additional site of the
$\gamma$–process.

Results. Our analysis shows that a large scatter among the different models
exists for both the $\gamma$–process integrated yields and the isotopic
ratios. We found only 10 ratios that agree with their solar values, all the
others differ by at least a factor of 3 from the solar values in all the
considered sets of models. The $\gamma$–process within C–O shell mergers
mostly influence the isotopic ratios that involve intermediate and heavy
proton-rich isotopes with $\rm A>100$.

Conclusions. We conclude that there are large discrepancies both among the
different data sets and between the models predictions and the solar abundance
distribution. More calculations are needed particularly updating the nuclear
network, since the majority of the models considered in this work did not use
the latest reaction rates for the $\gamma$–process nucleosynthesis. Moreover,
the role of C–O shell mergers needs further investigation.

###### Key Words.:

$\gamma$–process nucleosynthesis – explosive nucleosynthesis – massive stars

##  1 Introduction

Neutron–capture processes (see, e.g., Käppeler et al., 2011; Cowan et al.,
2019, and references therein) made the majority of the nuclei beyond iron in
the Solar System. However, these processes are not able to make a number of
rare, neutron-deficient, stable isotopes, which constitute a small fraction in
mass of the heavy nuclear species. Already in the pioneering work of Burbidge
et al. (1957) 35 stable proton–rich nuclides111namely: [74]Se, [78]Kr, [84]Sr,
[92,94]Mo, [96,98]Ru, [102]Pd, [106,108]Cd, [112,114,115]Sn, [113]In, [120]Te,
[124,126]Xe, [130,132]Ba, [136,138]Ce, [138]La, [144]Sm, [152]Gd, [156,158]Dy,
[162,164]Er, [168]Yb, [174]Hf, [180]Ta, [180]W, [184]Os, [190]Pt, and [196]Hg.
(the $p$–nuclei) heavier than Fe were identified and the $p$–process was
postulated to explain their existence (see also Cameron, 1957). In the
following years, Arnould (1976) demonstrated that $p$–nuclei can be
synthesised through a sequence of ($\gamma$,$n$), ($\gamma$,$p$) and
($\gamma$,$\alpha$) reactions on pre-existing seed heavy material during the
latest stages of the hydrostatic evolution of massive stars. Woosley & Howard
(1978) showed that the $p$–process material produced during the pre-supernova
evolution is completely reprocessed by the passage of the explosive shock
wave; and that the O/Ne-rich layers during core-collapse supernovae (CCSNe)
explosions provide the conditions to re-make the $p$–nuclei via a chain of
photodisintegrations. They called this the $\gamma$–process. Prantzos et al.
(1990) and then Rayet et al. (1995) performed the first complete computations
of the $\gamma$–process and provided the sets of yields that have been taken
as the reference for the studies of the $p$–nuclei for the following decades
(e.g., Arnould & Goriely, 2003; Rauscher et al., 2013; Pignatari et al.,
2016a, and references therein).

Not all the heavy proton-rich isotopes are made exclusively by the
$\gamma$–process. Branching points, either activated by unstable nuclei or by
stable isotopes that become unstable at stellar temperature, allow neutron
captures to play a role as well. [152]Gd and [164]Er are representative of
this, since they have a potentially dominant $s$–process contribution from
low-mass asymptotic giant branch (AGB) stars due to the branching points at
[151]Sm (unstable in terrestrial conditions) and [163]Dy (stable in
terrestrial conditions), respectively (see, e.g., Arlandini et al., 1999;
Bisterzo et al., 2011, 2015). The production of [113]In and [115]Sn depends on
the $\beta$–decay of the unstable isomers of the stable nuclides [113]Cd and
[115]In, respectively. In this case, however, their origin is still unclear,
with a possible $r$–process contribution (Dillmann et al., 2008). The very
long-lived [180]Ta isomer, with a half-life larger than $\rm 1.2\times
10^{15}yr$, may be efficiently produced both by neutrino-spallation reactions
on [180]Hf (Cumming & Alburger, 1985) and the $s$-process (see, e.g., Rauscher
et al., 2002; Bisterzo et al., 2011). In the latter case, as for [163]Dy,
[179]Hf becomes unstable at stellar temperatures and activates a branching
point on the $s$–process path. Finally, [138]La might receive a substantial
contribution from the neutrino captures on [138]Ba (Goriely, S. & Siess, L.,
2001). Moreover, new processes have been identified in the last decades to
occur in the deepest interior of CCSNe, which may also participate in the
production of the lightest $p$–nuclei between Fe and Pd. These processes range
from the neutrino-driven winds from the forming neutron star (see, e.g.,
Fröhlich et al., 2006; Arcones & Montes, 2011) to the $\alpha$–rich freeze-out
in CCSN ejecta (see, e.g., Woosley & Hoffman, 1992; Pignatari et al., 2016a),
and the $\nu r$-process proposed very recently by Xiong et al. (2023). Modern
multidimensional CCSN simulations (e.g., Harris et al., 2017) and calculations
of the explosion in spherical symmetry with the inclusion of neutrino heating
(e.g., Curtis et al., 2018; Ghosh et al., 2022), point toward the possibility
of ejecting a significant amount of $p$–nuclei yields from the innermost
layers of the star, especially in the Mo - Ru region (see, e.g., Eichler et
al., 2017).

Although the $\gamma$–process nucleosynthesis in CCSNe has been explored for
many decades (see, e.g., the reviews Rauscher et al., 2013; Pignatari et al.,
2016b, and references therein), a complete understanding of the production of
$p$–nuclei in stars is still missing and several discrepancies arise when
comparing theoretical model predictions with the Solar System abundances. The
first fundamental problem is that the average $\gamma$–process yields from
massive star underproduce by about a factor of three the Solar System
abundances. Furthermore, [92,94]Mo and [96,98]Ru are systematically
underproduced by one order of magnitude, compared to the other
$\gamma$–process nuclei (Arnould & Goriely, 2003; Rauscher et al., 2013;
Pignatari et al., 2016a). Several works have shown that nuclear uncertainties
are not the reason of the substantial disagreement between theory and
observations (see, e.g., Rapp et al., 2006; Rauscher et al., 2016; Nishimura
et al., 2018).

The external layers of thermonuclear supernovae (SNIa) resulting from a white
dwarf (WD) accreting material from a stellar binary companion and reaching the
Chandrasekhar mass limit have also been suggested as a site of the
$\gamma$–process (Travaglio et al., 2011, 2014; Battino et al., 2020). First
galactic chemical evolution (GCE) calculations made for $p$–nuclei Travaglio
et al. (2018) confirmed that SNIa scenario would solve the issues related to
the $\gamma$–process in CCSNe. However, the SNIa solution strongly depends on
the number of SNIa generated from Chandrasekhar-mass progenitors within the
single-degenerate scenario (Hillebrandt et al., 2013). According to the
observations, this number is still affected by large uncertainties and could
be much lower than required to solve the missing $\gamma$–process production
(e.g., Woods & Gilfanov, 2013; Battino et al., 2020, and references therein).

Given these still open, significant questions, investigations of potential
solutions to boost the production of $p$–nuclei in massive stars, in
particular in the Mo-Ru region, are still required. For example, Pignatari et
al. (2013) showed that an increase by two orders of magnitude of the
[12]C$+$[12]C reaction rate by Caughlan & Fowler (1988) may lead to a Mo and
Ru $\gamma$–process production up to the level of the other $p$–nuclei, and to
an overall increased $\gamma$–process efficiency. Also, during the latest
stages of the evolution of massive stars, the convective O burning shell may
ingest some C-rich material (Rauscher et al., 2002; Meakin & Arnett, 2006;
Ritter et al., 2018b; Andrassy et al., 2019). This interaction may lead to the
formation of an extended merged convective zone, as well as to asymmetries in
the stellar structure of the progenitor that might facilitate the CCSN
explosion (e.g., Couch & Ott, 2013; Müller et al., 2016). During this ”C–O
shell merger” event, the C shell material is brought down to the base of the O
shell at typical O burning temperature, triggering efficient $\gamma$–process
nucleosynthesis (Rauscher et al., 2002; Ritter et al., 2018a). The mixing of
the $p$–nuclei throughout the extended, mixed C–O shell preserves them from
being fully reprocessed by the CCSN shock wave. Their yields may therefore
increase by orders of magnitude, as compared to models without the C–O shell
merger. It still needs to be investigated, however, if this scenario can solve
some of the puzzles of the missing production of the $p$–nuclei. Finally,
because of the strong sensitivity of the $\gamma$–process yields on the seeds
from the $s$–process (Costa et al., 2000), Choplin, A. et al. (2022) proposed
that an enhanced production of $s$–process seeds in fast rotating massive
stars (”spinstars”) at sub-solar metallicity could increase the efficiency of
$\gamma$–process nucleosynthesis and that this contribution could even
dominate the solar $p$–nuclei abundance distribution. However, the number of
such spinstars as a function of the metallicity is still uncertain and a
further investigation is required.

To shed further light on the mystery of the $\gamma$–process nucleosynthesis
in CCSNe, we started a research program dedicated to implementing the latest
nuclear reaction rates into new calculations of CCSNe for a wide grid of
explosion energies, using the most recent version of the NuGrid
nucleosynthesis codes (e.g., Pignatari et al., 2016b; Ritter et al., 2018b;
Lawson et al., 2022). In this first paper of the series, we present our first
step of the program, corresponding to the analysis of the $\gamma$–process
yields in a number of sets of CCSN models from the literature. The paper is
organized as follows: in Section 2 we present the sets of CCSN models; in
Section 3 we describe our method; in Section 4 we outline our results; in
Section 5 we give a brief overview of the current state of the art in nuclear
physics for the $\gamma$–process; and in Section 6 we discuss the results and
present the conclusions.

##  2 Data sets

We consider yields from five sets of CCSN models (Rauscher et al., 2002;
Pignatari et al., 2016b; Ritter et al., 2018b; Sieverding et al., 2018; Lawson
et al., 2022) with progenitors of initial masses of 15, 20, and 25 $\rm
M_{\odot}$ and solar metallicity. All of the models are based on one-
dimensional (1D) computations using different codes and nuclear networks for
the hydrostatic evolution, the explosion, and the nucleosynthesis. In the
following, we briefly describe the main features of each set.

###  2.1 Rauscher+02 (RAU)

The data from Rauscher et al. (2002) (hereafter labelled as RAU15, RAU20,
RAU25 for the three masses, respectively) are a well-known and widely used
set. They were produced using an adaptive nuclear network including up to a
maximum number of $\sim 2200$ nuclear species. The progenitors of the CCSNe
were calculated by an updated version of the Kepler code (Weaver et al.,
1978), while the supernova explosions were simulated using the piston
technique (Woosley & Weaver, 1995). The explosion energy of the 15 $\rm
M_{\odot}$ model is $1.2\times 10^{51}$ erg, according to the assumption for
SN 1987A (e.g., Woosley & Haxton, 1988), while in the case of 20 and 25 $\rm
M_{\odot}$ models, it is adjusted to obtain the ejection of 0.1 $\rm
M_{\odot}$ of [56]Ni. The initial solar composition is from Anders & Grevesse
(1989) (AG89). The reaction rates involving the nuclei above Fe are mostly
from Rauscher & Thielemann (2000), and from Bao et al. (2000) in the case of
the $\rm(n,\gamma)$ reactions. In case of missing experimental information,
the Hauser-Feshbach rates obtained with the NON-SMOKER code (Rauscher et al.,
1997) were used. In the case of the two reactions, involving $p$–nuclei,
[70]Ge$(\alpha,\gamma)$[74]Se and [144]Sm$(\alpha,\gamma)$[148]Gd, the
experimental rates from Fülöp et al. (1996) and Somorjai et al. (1998) were
implemented, respectively. Moreover, the rates that shared the same
$\alpha$–potentials of the two reactions above were re-computed using those
experimental information. The 20 $\rm M_{\odot}$ model experience a C–O shell
merger about 1 day before the core collapse, forming a single extended mixed
convective zone. As mentioned in the Introduction (Sec. 1), the C–O shell
merger triggers an efficient $\gamma$–process nucleosynthesis during the pre-
supernova evolution of the stars that may dominate over the effect of the
explosive nucleosynthesis during the CCSN.

###  2.2 Pignatari+16 (PGN)

The 15, 20, and 25 $\rm M_{\odot}$ progenitors presented in Pignatari et al.
(2016b) (hereafter PGN15, PGN20, PGN25, respectively) were computed using the
Genec stellar evolutionary code (Eggenberger et al., 2008) and following the
evolution up to and including the central Si burning phase. The further
collapse of the Fe core was not followed and the supernova explosion was
simulated using a semi-analytical approach, with prescription for the mass cut
from Fryer et al. (2012). The adopted initial solar composition is from
Grevesse & Noels (1993) (GN93). The nucleosynthesis during the evolution and
the explosion was computed with the Multi-zone Post-Processing Network –
Parallel code (MPPNP, Pignatari & Herwig, 2012; Pignatari et al., 2016b;
Ritter et al., 2018b), using a dynamical network with a maximum number of 5234
nuclear species. The nuclear network includes 74,313 reactions and above Fe it
uses the experimental rates from KADoNIS compilation (Dillmann et al., 2006)
wherever possible for $\rm(n,\gamma)$ reactions, and the Basel REACLIB
database (Rauscher & Thielemann, 2000) revision 20090121 for the missing
reactions. These models present some interesting features regarding the
$\gamma$–process nucleosynthesis. In the PGN15 model, the mass-cut is deep
enough to allow the ejection of some $\alpha$-rich freeze-out material,
slightly increasing the yields of the lightest $\gamma$–process nuclei. In the
PGN20 model, the shock wave accelerates at the interface between CO and He
core, due to a steep density gradient. This results in an increase of the peak
temperature, which leads to a re-activation of the $\gamma$–process
nucleosynthesis. In the PGN25 model, the $\gamma$–process occurs in a narrow
region of the star that, unfortunately, lacks in spatial resolution. This does
not allow us to distinguish the abundance peaks of all the $p$–nuclei. For
this reason, we decided to exclude this model from the discussion in the next
sections.

###  2.3 Ritter+18 (RIT)

The set of Ritter et al. (2018b) (RIT15, RIT20, RIT25) constitutes a follow-up
work of Pignatari et al. (2016b) except that the stellar progenitors were
calculated with the MESA code (Paxton et al., 2011). The main features of the
RIT models are the same as in the PNG models (initial composition, physics
inputs, explosion mechanism, and nucleosynthesis code), except for parts of
the reaction network database and, as mentioned above, the code used to
compute the stellar progenitors. The RIT models adopt the JINA REACLIB
reaction rate library V1.1 (Cyburt, 2011) instead of the Basel REACLIB
database as the PNG models. The use of a different stellar code led to several
significantly different results. The RIT15 model experiences a C–O shell
merger event, similarly to the RAU20 model. Also in this case, the production
of the $\gamma$–process nuclei greatly increases in the pre-supernova
evolution. The RIT20 model has a large amount of fallback material after the
explosion, such that only the region above explosive Ne burning is ejected,
and ending with a remnant mass of 2.7 $\rm M_{\odot}$. Moreover, in this star,
the convective C–shell is particularly extended and it is able to efficiently
activate the [22]Ne neutron source, triggering the $s$–process
nucleosynthesis. The RIT25 model experience so large fallback that the
explosion ends in a failed supernova, locking all the explosive
nucleosynthesis material into the central remnant of 5.7$\rm M_{\odot}$.

###  2.4 Sieverding+18 (SIE)

The set from Sieverding et al. (2018) (SIE15, SIE20, SIE25) is based on
stellar progenitors calculated using the Kepler code, like the RAU models, and
with the same explosion energy equal to $1.2\times 10^{51}$ erg, but with
different choices for several physical and numerical parameters. The initial
solar composition is from Lodders (2003) (L03). The nuclear network used for
the stellar evolution calculations is not identical to the reaction network
used for the explosion. While the former is mostly identical to the RAU
models, the latter includes the JINA REACLIB reaction rate library V2.2, and
an updated $\nu$–induced reaction rate set. The rate of the neutron–capture
reaction [137]La$(\rm n,\gamma)$[138]La is instead from a previous version of
REACLIB, i.e., Rauscher & Thielemann (2000). The inclusion of $\nu$–induced
reactions is crucial for the production of the two $p$–nuclei [138]La and
[180]Ta, which are directly produced by $\nu_{\rm{e}}$ capture on the
corresponding isobars (Woosley et al., 1990). Sieverding et al. (2018)
presented a parametric study of the explosive yields using several neutrino
energies. Here we include the same set as in den Hartogh et al. (2022), i.e.,
the models with the highest neutrino energy.

###  2.5 Lawson+22 (LAW)

The stellar progenitors of Andrews et al. (2020) and Lawson et al. (2022)
(LAW15, LAW20, LAW25) were computed with a recent version of the Kepler code
(Heger & Woosley, 2010), while for the explosion a 1D hydrodynamic code was
used, mimicking a 3D convective engine (Fryer et al., 1999, 2018). The
hydrostatic nucleosynthesis was calculated via the MPPNP code (as PNG and
RIT), while for the explosive nucleosynthesis the Tracer particle Post-
Processing Network – Parallel code was used (TPPNP, Jones et al., 2019). The
initial solar composition is GN93, as in the RIT models. The stellar yields
for the whole stellar set including the models used in this work are available
online (Andrews et al., 2020). As in RIT20, also in the LAW25 model the
C–shell is particularly extended and has an efficiently active neutron source.
Again similarly to RIT20, also LAW20 has a large fallback and all the
explosive nucleosynthesis is confined within an innermost smaller mass of 2.2
$\rm M_{\odot}$, at the position of the final mass-cut. Therefore, the ejecta
of this model contains only the material ejected from the C shell outward.

##  3 Average overproduction factors

Figure 1: The $p$–nuclide overproduction factors divided by their average (Eq.
1), whose value is indicated in the top right of each panel, together with the
OP factor for 16O ($\rm F_{[}$O16]) for comparison. The blue, orange, and
green symbols represent, respectively: (i) nuclei produced exclusively by the
$\gamma$–process, (ii) nuclei that may have an additional explosive
contribution; and (iii) nuclei that have also $s$–, $r$–process, or neutrino-
capture contributions. Four examples of representative models are shown: the
upper left panel is RAU15, a typical model; the upper right panel is RIT15, a
case where the C–O shell merger occurs; the lower left panel is LAW20, a
failed supernova; and the lower right panel is SIE25, a model where the $\rm
F_{0}$ is the same as for the typical RAU15 model shown upper left panel, but
where this value is driven by two isotopes produced by neutrino captures,
rather than by the $p$–nuclei produced exclusively by the $\gamma$–process
(Sieverding et al., 2018). Note that in the two bottom panels a large number
of $p$–nuclei, including the Mo and Ru isotopes, share the same $\rm
F_{[}$i]/$\rm F_{0}$ $\simeq 0.2-0.3$. However, as shown by the low $\rm
F_{0}$, there is no significant production of these isotopes and their pattern
only reflects the solar abundance distribution used as initial in the models.

The overproduction factor (OP) for each isotope is defined as OP =
$X/X_{\odot}$, i.e., the ratio between its mass fraction $X$ equal to the
total integrated yield divided by the total mass ejected, relative to the
solar mass fraction $X_{\odot}$, for which we use the values of Asplund et al.
(2009)222For the $p$–only isotopes, these values are the same as those given
in the other solar abundance compilations used in the different models as
described in the previous section.. A useful parameter to evaluate the
efficiency of the $\gamma$–process nucleosynthesis is $\rm F_{0}$, defined as
the average of the OP factors of all the 35 $p$–nuclei, i.e.:

$\centering F_{0}=\frac{\sum_{\rm i=1}^{35}F_{\rm i}}{35}\@add@centering$ (1)

where $\rm F_{[}$i] is the OP factor of the $p$–nuclide $i$. The value of $\rm
F_{0}$ has been taken over the years as the representative feature of
$\gamma$–process nucleosynthesis and has been used as a normalisation factor
for the single $p$–nuclei OP factors, i.e., $\rm F_{[}$i]$/$$\rm F_{0}$
measures the deviation of the production of each single isotope $i$ from the
average production. Figure 1 shows the $\rm F_{[}$i]$/$$\rm F_{0}$
distribution for a selection of models. The same figure for each model can be
found in Appendix A, together with a figure showing the OP factors of all the
35 $p$–isotopes.

Two fundamental features are highlighted from the figure and from the
distributions of all the models. First, there are large variations of both the
OP factors and of $\rm F_{0}$ with the initial mass of the stellar progenitor.
Depending on the set of models considered, $\rm F_{0}$ varies by up to a
factor of roughly five, with even larger variation for the OP of single
isotopes. Such a scatter is much larger than what was found in previous
studies (e.g., Rayet et al., 1995), supporting the idea that $\gamma$–process
nucleosynthesis does not depend significantly on the progenitor mass. Second,
large variations are present also when we compare models with the same
progenitor mass but from different sets. This is because the $p$–nuclei
abundances do not depend only on the progenitor mass and the explosion energy.
Rather, they are a complex product of the combination of conditions in the
stellar progenitor layers and the interaction of those layers with the CCSN
shock (e.g., Rauscher et al., 2002; Ritter et al., 2018b).

The $\rm F_{0}$ has been often compared to the OP factor of the nuclide [16]O
(hereafter $\rm F_{[}$O16]), one of the main products of CCSNe. This allows us
to approximately quantify the contribution from massive star to the solar
abundances of the $p$–nuclei without using GCE calculations. If the $p$–nuclei
and [16]O share the same production site, one would expect that $\rm F_{0}$
$\sim$ $\rm F_{[}$O16], however, Rayet et al. (1995) already found an
underproduction of at least of a factor of 2, i.e., $\rm F_{0}$/$\rm
F_{[}$O16] $\simeq$ 0.5. Pignatari et al. (2013) confirmed this result and
pointed out that the $p$–nuclei underproduction is probably more severe, since
at solar metallicity secondary nucleosynthetic products such as
$\gamma$–process nuclei should be synthesised twice as much as primary
isotopes, such as [16]O (see, e.g., Tinsley, 1980). All the models in our sets
have $\rm F_{0}$ $\lesssim$ $\rm F_{[}$O16], except for PGN15, where $\rm
F_{0}$/$\rm F_{[}$O16] $\sim$ 1.05, still far from the factor of roughly two
required to reproduce the solar abundances.

Table 1: Percentages of $p$–nuclei yields in C–O mergers.

isotope | explosive | merger | envelope |  | explosive | merger | envelope
---|---|---|---|---|---|---|---
| ($\%$) | ($\%$) | ($\%$) |  | ($\%$) | ($\%$) | ($\%$)
| RIT15 |  | RAU20
[ 74]Se | 74.1 | 14.9 | 10.9 |  | 47.3 | 49.0 | 3.6
[ 78]Kr | 82.1 | 5.9 | 11.9 |  | 62.1 | 28.8 | 9.1
[ 84]Sr | 70.5 | 11.5 | 17.4 |  | 42.7 | 53.2 | 4.0
[ 92]Mo | 6.4 | 8.4 | 85.2 |  | 10.5 | 15.0 | 74.5
[ 94]Mo | 2.8 | 27.4 | 69.7 |  | 5.9 | 45.6 | 48.3
[ 96]Ru | 8.2 | 4.3 | 87.5 |  | 12.1 | 2.4 | 85.5
[ 98]Ru | 13.1 | 49.6 | 37.2 |  | 16.7 | 37.9 | 45.3
[102]Pd | 36.4 | 40.2 | 23.3 |  | 45.1 | 25.6 | 29.2
[106]Cd | 34.6 | 22.5 | 43.0 |  | 53.5 | 9.2 | 37.3
[108]Cd | 11.4 | 63.4 | 25.1 |  | 19.1 | 46.2 | 34.6
[112]Sn | 34.6 | 35.5 | 29.8 |  | 47.0 | 16.9 | 36.1
[114]Sn | 20.7 | 71.0 | 8.2 |  | 18.0 | 68.7 | 13.0
[115]Sn | 0.3 | 64.9 | 34.8 |  | 1.3 | 61.7 | 36.8
[113]In | 5.4 | 42.5 | 52.0 |  | 8.5 | 46.6 | 44.7
[120]Te | 8.2 | 62.5 | 29.3 |  | 43.0 | 29.6 | 27.3
[124]Xe | 22.0 | 59.9 | 18.1 |  | 70.3 | 8.9 | 20.8
[126]Xe | 7.0 | 87.4 | 5.5 |  | 40.1 | 54.1 | 5.6
[130]Ba | 35.0 | 61.0 | 3.7 |  | 66.6 | 24.1 | 9.3
[132]Ba | 1.8 | 95.9 | 2.2 |  | 18.6 | 78.9 | 2.2
[136]Ce | 40.4 | 51.0 | 8.5 |  | 42.6 | 39.0 | 18.4
[138]Ce | 8.3 | 84.7 | 6.8 |  | 20.7 | 74.4 | 4.6
[138]La | 0.0 | 82.9 | 17.0 |  | 2.0 | 91.6 | 6.2
[144]Sm | 12.6 | 84.0 | 3.2 |  | 16.3 | 80.6 | 2.9
[152]Gd | 0.0 | 3.9 | 96.1 |  | 0.5 | 13.1 | 86.1
[156]Dy | 0.6 | 93.7 | 5.7 |  | 7.7 | 89.5 | 2.3
[158]Dy | 0.0 | 84.4 | 15.5 |  | 1.8 | 95.8 | 1.9
[162]Er | 0.3 | 97.2 | 2.4 |  | 3.6 | 95.0 | 1.0
[164]Er | 7.5 | 39.1 | 53.3 |  | 16.4 | 70.7 | 12.7
[168]Yb | 2.4 | 94.4 | 3.2 |  | 3.8 | 95.1 | 0.7
[174]Hf | 3.7 | 87.5 | 8.6 |  | 6.7 | 92.2 | 1.4
[180]Ta | 0.0 | 98.3 | 1.7 |  | 0.9 | 94.7 | 4.2
[180] W | 18.1 | 70.7 | 10.2 |  | 35.0 | 61.8 | 3.1
[184]Os | 0.9 | 88.9 | 10.2 |  | 0.8 | 96.8 | 2.3
[190]Pt | 1.2 | 58.4 | 40.3 |  | 3.5 | 91.4 | 4.8
[196]Hg | 12.1 | 84.4 | 2.7 |  | 14.9 | 84.2 | 0.6

333Percentages of the total $p$–nuclei yields of the two models with the C–O
merger (RIT15 and RAU20) in three parts of the ejecta: the zone dominated by
the explosive nucleosynthesis (”explosive”), the zone of the mixed C–O shell
(”merger”), and the zone from the He shell outwards (”envelope”).
Contributions are highlighted in bold when larger than 50$\%$.

Relative to a typical model (e.g., top left panel of Figure 1), models with a
C–O shell merger (top right panel) can significantly contribute to the
production of $\gamma$–process nuclei, as evident by comparing the values of
$\rm F_{0}$ and $\rm F_{[}$O16] in these two panels. Nevertheless, also in
these models $\rm F_{0}$ does not exceed $\rm F_{[}$O16]. In both the RIT15
and RAU20 models the effect of the C–O shell merger is dominant on the yields
of most of the $p$-isotopes with $\rm A\gtrsim 110$ (see Table 1). The
production of the lightest isotopes is instead less clear: the RAU20 merger
component is comparable to the explosive component, while the RIT15 merger
component is less relevant. When one element has more than one isotope
produced by the $\gamma$–process, the C–O shell merger favours the synthesis
of the more neutron rich, i.e.: [94]Mo, [98]Ru, [108]Cd, [114]Sn and partially
[115]Sn, [126]Xe, [132]Ba, [138]Ce, partially [158]Dy, but not [164]Er. The
latter, in fact, has a dominant contribution from the $s$–process (Sec. B.9).
These properties indicate that the $\gamma$–process in C–O shell mergers
operates at lower temperatures (those typical of O burning at the bottom of
the convective shell), as compared to the explosion.

Table 2: Definition of average overproduction factors. F | excluded contribution | excluded isotopes
---|---|---
$\rm F_{0}$ | – | –
$\rm F_{0}^{I}$ |  $s$–, $r$–process, |  [74]Se, [78]Kr, [84]Sr, [92,94]Mo,
|  $\nu$–capture, $\nu p$–process, |  [96,98]Ru, [108]Cd, [113]In, [115]Sn,
|  $\alpha$–rich freeze-out |  [138]La, [152]Gd, [164]Er, [180]Ta
$\rm F_{0}^{II}$ |  $\nu p$–process, |  [74]Se, [78]Kr, [84]Sr,
|  $\alpha$–rich freeze-out |  [92,94]Mo, [96,98]Ru
$\rm F_{0}^{III}$ |  $s$–, $r$–process, |  [108]Cd, [113]In, [115]Sn, [138]La,
|  $\nu$–capture |  [152]Gd, [164]Er, [180]Ta
$\rm F_{\gamma}$ |  $s$–, $r$–process, |
|  $\nu$–capture, $\nu p$–process, | all but the 3 most
|  $\alpha$–rich freeze-out, | produced $\gamma$–only nuclei
| partly the $\gamma$–process |

As not all the $p$–nuclei have a $\gamma$–only origin, $\rm F_{0}$ might not
correctly reflect production from the $\gamma$–process. For example, in case
of a failed supernova (such as the LAW20 shown in the bottom left panel of
Figure 1, and RIT25), $\rm F_{0}$ could be dominated by isotopes with
additional nucleosynthetic contributions besides the $\gamma$–process, such as
[180]Ta in LAW20. Neutrino processes producing [138]La and [180]Ta contributed
the most to the $\rm F_{0}$ value in SIE25 (lower right panel of Figure 1).
Therefore, to better constrain the $\gamma$–process itself, we decided to
define the more averages, by excluding different isotope groups, as listed and
described in Table 2. Among the various excluded isotopes, [108]Cd has been
traditionally considered a $\gamma$–only isotope, however, in the models
considered here we found that [108]Cd can also be produced by neutron captures
starting on [107]Ag. This production may be not negligible since the solar
abundance of [107]Ag, used in the models as initial, is more than 15 times
larger than the solar abundance of [108]Cd. For this reason, we included this
isotope among the nuclei with an additional contribution. Our new definitions
help us to identify why some models which models do not produce $p$–nuclei,
and have nevertheless relatively high $\rm F_{0}$ values. Table 3 lists all
the different F–values in our sets of models.

When comparing all these different values to $\rm F_{[}$O16], we found no
significant improvement using $\rm F_{0}^{I}$, $\rm F_{0}^{II}$, or $\rm
F_{0}^{III}$ instead of $\rm F_{0}$. Using $\rm F_{\gamma}$, instead, we find
six models with values larger than $\rm F_{[}$O16], of which four models
(RAU20, PGN20, RIT15 and RIT20) have $\rm F_{\gamma}$/$\rm F_{[}$O16] $>$ 2\.
[196]Hg contributes to the $\rm F_{\gamma}$ of three out of these four models,
and [130]Ba and [168]Yb to two. Overall, we conclude that F–values should be
used carefully as by including many OP factors in the average can hide
information. In fact, different CCSN models may efficiently produce different,
single or groups of, $p$–nuclei, and still result in a low value of $\rm
F_{0}$. Therefore, the use of the F–values only is not enough to investigate
the properties of the $\gamma$–process nucleosynthesis and an accurate
analysis of the production of the single isotopes is required instead.

Table 3: The average overproduction factors in the models.

Model | $\rm F_{0}$ |  $\rm F_{[}$O16] | $\rm F_{0}^{I}$ | $\rm F_{0}^{II}$ | $\rm F_{0}^{III}$ | $\rm F_{\gamma}$ |  $\rm F_{\gamma}$ isotopes
---|---|---|---|---|---|---|---
RAU15 | 4.32E+00 | 1.24E+01 | 5.53E+00 | 4.78E+00 | 4.76E+00 | 1.61E+01 |  [196]Hg, [180]W, [168]Yb
RAU20a | 1.79E+01 | 2.68E+01 | 2.54E+01 | 2.06E+01 | 2.08E+01 | 8.91E+01 |  [168]Yb, [196]Hg, [162]Er
RAU25 | 1.56E+01 | 4.80E+01 | 1.37E+01 | 1.31E+01 | 1.68E+01 | 3.59E+01 |  [196]Hg, [180]W, [130]Ba
PGN15 | 5.03E+00 | 4.72E+00 | 2.93E+00 | 2.95E+00 | 5.54E+00 | 5.64E+00 |  [130]Ba, [132]Ba, [168]Yb
PGN20 | 1.86E+01 | 1.85E+01 | 2.15E+01 | 1.79E+01 | 2.14E+01 | 4.93E+01 |  [196]Hg, [130]Ba, [180]W
RIT15a | 1.01E+01 | 1.39E+01 | 1.43E+01 | 1.17E+01 | 1.17E+01 | 3.63E+01 |  [132]Ba, [162]Er, [168]Yb
RIT20 | 1.66E+01 | 1.94E+01 | 1.40E+01 | 1.34E+01 | 1.78E+01 | 4.34E+01 |  [196]Hg, [130]Ba, [136]Ce
(RIT25) | 2.12E+00 | 1.94E+01 | 6.81E-01 | 2.47E+00 | 6.95E-01 | 8.70E-01 |  [114]Sn, [158]Dy, [168]Yb
SIE15 | 2.78E+00 | 6.09E+00 | 3.05E+00 | 3.02E+00 | 2.75E+00 | 5.89E+00 |  [196]Hg, [180]W, [130]Ba
SIE20 | 3.57E+00 | 1.24E+01 | 2.53E+00 | 3.05E+00 | 3.31E+00 | 6.01E+00 |  [196]Hg, [180]W, [168]Yb
SIE25 | 4.83E+00 | 1.67E+01 | 2.09E+00 | 2.99E+00 | 4.62E+00 | 4.51E+00 |  [184]Os, [168]Yb, [174]Hf
LAW15 | 4.13E+00 | 1.26E+01 | 4.87E+00 | 4.72E+00 | 4.10E+00 | 9.56E+00 |  [190]Pt, [196]Hg, [158]Dy
(LAW20) | 2.80E+00 | 2.41E+01 | 8.88E-01 | 3.27E+00 | 8.98E-01 | 1.40E+00 |  [102]Pd, [158]Dy, [180]W
LAW25 | 1.49E+01 | 5.18E+01 | 1.50E+01 | 1.62E+01 | 1.36E+01 | 3.10E+01 |  [130]Ba, [184]Os, [158]Dy
aThese two models experience the C–O shell merger

444Values of the quantities defined in Table 2 and the reference 16O
overproduction factor $\rm F_{[}$O16] for all the models. The last column
indicates the 3 most produced $\gamma$–only nuclei on which the calculation of
the reported Fγ is based. Values highlighted in bold are larger than $\rm
F_{[}$O16], for each models. The models in parenthesis are not included in our
analysis of the isotopic ratios presented in Sec. 4.

##  4 Isotopic ratios

Table 4: The 23 isotopic ratios.

Isotopic ratio | Subsection | Sub-solar | Solar | Super-solar | Merger | Radiogenic | Additional components
---|---|---|---|---|---|---|---
[74]Se/[78]Kr | B.1 | 0 | 7 | 5 |  |  | $\alpha$
[84]Sr/[78]Kr | B.1 | 0 | 11 | 1 |  |  | $\alpha$
[92]Mo/[94]Mo | B.2 | 0 | 12 | 0 |  |  | $\alpha$
[96]Ru/[98]Ru | B.2 | 0 | 12 | 0 |  |  | $\alpha$
[102]Pd/[108]Cd | B.3 | 0 | 12 | 0 |  |  |
[106]Cd/[108]Cd | B.3 | 0 | 12 | 0 |  |  |
[112]Sn/[114]Sn | B.4 | 1 | 11 | 0 | v |  |  $s$,$r$
[113]In/[114]Sn | B.4 | 4 | 8 | 0 |  | v |  $s$,$r$
[115]Sn/[114]Sn | B.4 | 1 | 11 | 0 | v |  |  $s$,$r$
[120]Te/[126]Xe | B.5 | 2 | 10 | 0 | v | v |
[124]Xe/[126]Xe | B.5 | 2 | 10 | 0 | v |  |
[130]Ba/[132]Ba | B.6 | 1 | 9 | 1 | v | v |
[136]Ce/[138]Ce | B.6 | 1 | 9 | 1 | v | v |
[138]La/[132]Ba | B.7 | 8 | 3 | 1 |  | v | $\nu$
[144]Sm/[132]Ba | B.7 | 0 | 11 | 1 |  | v |
[144]Sm/[152]Gd | B.8 | 4 | 8 | 0 |  |  | $s$
[156]Dy/[152]Gd | B.8 | 6 | 6 | 0 |  | v | $s$
[156]Dy/[158]Dy | B.9 | 2 | 10 | 0 |  | v |
[162]Er/[164]Er | B.9 | 7 | 5 | 0 | v | v | $s$
[168]Yb/[180]Ta | B.10 | 10 | 2 | 0 | v | v |  $s$,$\nu$
[174]Hf/[180]W | B.10 | 0 | 12 | 0 | v | v |
[184]Os/[196]Hg | B.11 | 4 | 8 | 0 |  | v |
[190]Pt/[196]Hg | B.11 | 9 | 3 | 0 |  | v |

555The 23 isotopic ratios are plotted in Figs 7 to 17. The Columns ”Sub-
solar”, ”Solar” and ”Super-solar” identify the number of models that fall
below, within, and above a factor of 3 from the solar $\gamma$–process
component, respectively. The Column ”Merger” reports ”v” if at least one of
the two yields of the isotopic ratio is dominated by the merger (corresponding
to the bold highlighted components in Table 1). The Column ”Radiogenic”
reports ”v” if at least one of the two isotopes of the ratio has a significant
radiogenic contribution. The Column ”Additional components” lists the possible
additional nucleosynthetic processes that may contribute to at least one of
the two isotopes of the ratio; possible additional contributions are: $\alpha$
($\alpha$-rich freeze-out or different explosive component), $s$
($s$–process), $r$ ($r$–process), $\nu$ (neutrino capture).

We analysed the correlations between 12 different couples of OP ratios (for a
total of 23 ratios given that one ratio is used twice). We chose ratios of
isotopes close in mass to study the local properties of the $\gamma$–process
production which are also affected by the impact of the different sets of
reaction rates used in the different sets of models. We avoided including in
each couple of ratios more than one isotope with additional contribution(s)
beyond the $\gamma$–process. While the OP factor quantifies the $absolute$
production of a given isotope relative to solar and therefore indicates if
this production is significant, the ratio of two OP factors quantifies the
overproduction of the two isotopes $relative$ to each other. Therefore, if an
OP ratio agrees with solar, then both isotopes have similar OP factors; if it
does not, then the two isotopes have different OP factors.

Usually, comparisons with solar abundances are made using yields averaged over
the initial mass function (IMF), since the solar composition is the result of
galactic chemical evolution with contributions from many stars. We do not
proceed in this way here because, first, a comparison among different single
star yields allows us to study the $\gamma$–process in different stellar
conditions in detail and to check if there exist best candidates that can
reproduce the solar isotopic ratios. Travaglio et al. (2018) already showed
that, with the current generation of massive star yields, it is not possible
to explain the $\gamma$–process solar distribution using GCE. New GCE
calculations with the yields computed using the updated reaction rates for the
$\gamma$–process is part of our future work. Second, our analysis is also
focused on the C–O shell mergers and at present, the frequency and the
relation with initial mass of such events is unknown. Third, most of the sets
of models available in literature do not include a large enough number of
initial masses and/or metallicities or, if so, the size of the nuclear network
is too limited to produce reliable yields for the $\gamma$–process
nucleosynthesis. This makes the use of IMF averaged yields not particularly
meaningful in the comparison with the solar composition.

The differences in stellar modelling (e.g., different choices of convection,
overshooting, etc.) and in nuclear physics (e.g., number of explicitly
included nuclear species, reaction rate database, etc.), introduce a
significant source of uncertainty and make it difficult to define to which
level a result is in agreement with the solar distribution. To account for
this, we consider models to be in relatively good agreement with solar when
both following conditions are satisfied: (i) they fall within a factor of 3
from the solar ratio and (ii) they have a overproduction at least twice as
solar (OP $>$ 2).

In the following we summarise the main results of our analysis. An extended
discussion for each of the 12 couples of isotopic ratios is presented in
Appendix B. As already mentioned in Sec 2, we did not include the two faint
supernova models (LAW20 and the RIT25), because the $\gamma$–process yields
remain locked in the compact remnant and only the envelope of the star is
ejected in the interstellar medium. We also excluded the PGN25 model due to a
lack of resolution in the $\gamma$–process abundance peaks. Table 4 collects
the main results for each isotopic ratio. We note that the analysis reported
below will need to be complemented by future studies where each of the
isotopic ratios discussed here are carefully investigated in terms of both the
nuclear physics input (see also Sec. 5) and the different modelling
approaches.

There are 10 ratios that satisfy both of our conditions for a good agreement
with the solar ratio for most of the models ([74]Se/[78]Kr, [84]Sr/[78]Kr,
[102]Pd/[108]Cd, [106]Cd/[108]Cd, [120]Te/[126]Xe, [124]Xe/[126]Xe,
[130]Ba/[132]Ba, [136]Ce/[138]Ce, [144]Sm/[132]Ba, and [174]Hf/[180]W). The
other 13 ratios either have low OP factors or be further than a factor of 3
from solar.

We find a general agreement with the solar ratios involving the lightest
$p$–nuclei, except in the case of Mo and Ru isotopes. No model, in fact,
satisfies the condition OP $>$ 2 for [92]Mo, [94]Mo, [96]Ru and [98]Ru. The
production of the isotopes between [74]Se and [108]Cd is mostly dominated by
the $\gamma$–process happening during the explosion, even in the models with
C–O shell mergers.

The majority of the ratios involving heavier isotopes present a significant
contribution from radiogenic species. In some cases, the radiogenic
contribution can be either determinant to reach a good agreement with the
solar ratio (e.g., [120]Te/[126]Xe, [124]Xe/[126]Xe) or conversely to move
away from solar (e.g., [156]Dy/[158]Dy, [184]Os/[196]Hg, [190]Pt/[196]Hg).

As mentioned in Sec. 3, the overproduction of $p$–nuclei with $\rm A\gtrsim
110$ largely increases in models with C–O shell mergers (RIT15 and RAU20).
Moreover, these models have generally larger overproduction of neutron richer
$p$–nuclei. There are 9 isotopic ratios in which the C–O shell mergers stand
out from the other models. In 4 cases ([144]Sm/[152]Gd, [156]Dy/[152]Gd,
[162]Er/[164]Er, and [168]Yb/[180]Ta) the isotopic ratio from the merger is
closer to solar than in the standard models. The opposite occurs in the other
4 cases ([112]Sn/[114]Sn, [120]Te/[126]Xe, [124]Xe/[126]Xe, [130]Ba/[132]Ba,
and [136]Ce/[138]Ce).

Out of the 23 ratios, 8 of them include one isotope with an additional
nucleosynthetic contribution besides the $\gamma$–process (see Sec. 1 and 3
and Table 4). We used the results from the galactic chemical evolution (GCE)
computation of Bisterzo et al. (2014) and from Dillmann et al. (2008), Nemeth
et al. (1994), and Theis et al. (1998) to estimate the neutron-capture
contribution to the $p$–nuclei in the solar composition and derive the
residual component, possibly dominated by the $\gamma$–process. There are
instead no quantitative predictions for the other explosive components (e.g.,
$\alpha$-rich freeze-out, $\nu$$p$–process). The models show a good agreement
with the residual solar ratios only in 4 cases: [112]Sn/[114]Sn,
[113]In/[114]Sn, [115]Sn/[114]Sn, and [144]Sm/[152]Gd. In the other 4 cases
([138]La/[132]Ba, [156]Dy/[152]Gd, [162]Er/[164]Er and [168]Yb/[180]Ta) the
(residual) solar ratio is always underestimated.

##  5 Nuclear Physics Considerations

The models included in our analysis (except for RAU) were computed in the last
decade and the nuclear physics inputs included in these calculations cover a
time span of about 25 years. The RAU models used mostly an early version of
the Basel REACLIB (Rauscher & Thielemann, 2000) compilation and the neutron-
capture rates from Bao et al. (2000). In the case of missing experimental
information, they included the Hauser-Feshbach rates obtained with the NON-
SMOKER code (Rauscher et al., 1997). The PGN models adopted the KADoNIS
compilation (Dillmann et al., 2006) for experimental neutron-capture rates and
the Basel REACLIB (revision 20090121) compilation in all the other cases. The
RIT and LAW models share the same nuclear physics for the $\gamma$–process,
i.e., the KADoNIS compilation for experimental neutron-captures and the JINA
REACLIB, 20120510 V1.1 (Cyburt, 2011) compilation for theoretical neutron-
capture rates and charged-particle reaction rates. Instead, SIE models use a
more recent version of JINA REACLIB, i.e., V2.2, but only for the explosive
nucleosynthesis. The $\gamma$–process nucleosynthesis yields presented in this
work were therefore computed with outdated reaction rates and new calculations
are required to test the impact of the most recent recommendations discussed
below.

Experimental data at energies relevant for $\gamma$–process nucleosynthesis
are rather scarce, and for this reason there is a strong interest in the low-
energy nuclear physics community to measure reaction cross sections that are
related to the synthesis of $p$–nuclei. Typically $\alpha$-induced reaction
cross sections are smaller than their $p$-induced counterparts, due to the
higher Coulomb barrier, and consequently the measurements are more
challenging. Since the latest $\gamma$–process nucleosynthesis review by
Pignatari et al. (2016a), many new measurements on ($\rm p,\gamma$),
($\alpha,\gamma$) and ($\alpha,\rm n$) reactions that produce or destroy
$p$-nuclei have been performed.

In addition to widely used techniques, such as in-beam $\gamma$–spectroscopy
(see Khaliel et al., 2017; Foteinou et al., 2018; Psaltis et al., 2019; Wu et
al., 2020; Heim et al., 2020, for some examples), the activation method (see
Korkulu et al., 2018; Gyürky et al., 2019; Scholz et al., 2020; Cheng et al.,
2021, and references therein), and the total absorption spectroscopy (TAS)/
4$\pi$ summing method (Kelmar et al., 2020; Harissopulos et al., 2021;
Palmisano-Kyle et al., 2022), other few novel methods have been successfully
used recently. Glorius et al. (2019) used the ESR storage ring and silicon
detectors to measure the [124]Xe(p,$\gamma$) reaction. Fallis et al. (2020)
used the DRAGON recoil separator to measure the [76]Se($\alpha$,$\gamma$)
reaction in inverse kinematics. Finally, Lotay et al. (2021) and Williams et
al. (2023) used for the first time a radioactive beam ([83]Rb) and the EMMA
recoil spectrometer to study the [83]Rb(p,$\gamma$)[84]Sr reaction in inverse
kinematics. Efforts for $\alpha$-scattering on exotic isotopes relevant for
the $\gamma$-process are also underway (Galaviz, 2021). These advancements,
along with the already established methods, will enable the low-energy nuclear
physics community to measure key reaction cross sections in the near future.

Despite this experimental progress, it remains impossible for the foreseeable
future to experimentally access all the reaction rates needed in typical
$\gamma$–process networks, which include thousands of nuclei and several
thousands of nuclear reaction rates (Arnould & Goriely, 2003). Consequently,
reaction rates derived from nuclear theory remain crucial. Furthermore,
$\gamma$-induced reactions play a key role in the $\gamma$–process network,
but a measurement of ($\gamma$,$\rm X$) reaction cross sections in the
laboratory cannot provide the ($\gamma$,$\rm X$) reaction rate under stellar
conditions because thermally excited states in the target nucleus play an
important role under stellar conditions, but do not contribute in the
laboratory experiments (for a more detailed discussion see Rauscher et al.,
2013). Instead, the ($\gamma$,$\rm X$) reaction rates are calculated from the
($\rm X$,$\gamma$) capture rates using the detailed balance relation between
forward and backward reaction rates (see e.g., Rauscher & Thielemann, 2000).
For heavy nuclei, the statistical model is appropriate and widely used to
derive reaction rates. In the following paragraphs we discuss very briefly
this approach, the relevant ingredients, and resulting uncertainties for the
reaction rates. Note that, instead, the determination of the astrophysical
reaction rates involving low Z $p$–nuclei may be influenced by effects that
can not be taken into account by the statistical model (Gyürky et al., 2014).
Therefore, in these cases, the experimental determination of the cross section
is required.

In general, the cross section $\sigma$ of a ($\rm X$,$\gamma$) capture
reaction (with $\rm X=\rm n,p,\gamma$) in the statistical model is defined by
the production cross section of the compound nucleus in the entrance channel
and the branching ratio in the exit channel. The production cross section
scales with the transmission $T_{\rm X}$ in the entrance channel. The
branching $b_{\gamma}$ towards the $\gamma$-channel is given by
$T_{\gamma}/\sum_{\rm i}T_{\rm i}$ where the transmissions $T_{\rm i}$ in the
nominator have to take into account all open channels. This leads to the
simple proportionality:

$\centering\sigma(\rm X,\gamma)\sim T_{\rm
X}\times\frac{T_{\gamma}}{\sum_{i}T_{\rm i}}=\frac{T_{\rm
X}T_{\gamma}}{\sum_{\rm i}T_{\rm i}}.\@add@centering$ (2)

This equation allows us to understand the relevance of the essential
ingredients of the statistical model, which are the $\gamma$-strength function
($\gamma$SF), the level density (LD), the nucleon optical model potential
(NOMP), and the $\alpha$-nucleus optical model potential ($\alpha$OMP). Note
that the $T_{\rm i}$ are calculated from global NOMPs and $\alpha$OMPs for the
particle channels and from the $\gamma$SF for the $\gamma$-channel.
Furthermore, the $T_{\rm i}$ are composed of the sum over all bound states in
the respective residual nuclei, which is approximated by the LD above a
certain excitation energy; thus, all $T_{\rm i}$ depend implicitly on the LD.

For ($\rm n$,$\gamma$) neutron capture reactions, typically $T_{\rm n}$
exceeds by far all $T_{\rm i}$ for the other channels. Consequently,
$\sum_{\rm i}T_{\rm i}\approx T_{\rm n}$, and ($\rm n$,$\gamma$) from Eq. (2)
scales mainly with the transmission $T_{\gamma}$ which depends on the
$\gamma$SF and the LD, but is independent of the NOMP and $\alpha$OMP.
Typically, an uncertainty of a factor of two is estimated for the ($\rm
n$,$\gamma$) rates, and this uncertainty was used in recent $\gamma$–process
studies, e.g., Rapp et al. (2006); Rauscher et al. (2016).

For ($\rm p,\gamma$) proton capture reactions on the neutron-deficient nuclei
in the $\gamma$–process, Eq. (2) leads to the following. Because of the
Coulomb barrier, the transmission $T_{\rm p}$ depends sensitively on the
energy. At low energies, corresponding to the Gamow window for the reaction
rates, $T_{\rm p}$ is smaller than $T_{\gamma}$, and the neutron channel is
closed ($T_{\rm n}=0$). Thus, $\sum_{\rm i}T_{\rm i}\approx T_{\gamma}$, and
the ($\rm p,\gamma$) cross section scales with $T_{\rm p}$, which, in turn,
depends mainly on the NOMP and on the LD, but not on the $\gamma$SF. As the
NOMP is well-studied, for ($\rm p,\gamma$) rates a similar uncertainty of a
factor of two as in the ($\rm n$,$\gamma$) case is often used, e.g., Rapp et
al. (2006); Rauscher et al. (2016), although the main origin of the
uncertainty is the NOMP for ($\rm p,\gamma$) rates and the $\gamma$SF and LD
for ($\rm n$,$\gamma$) rates.

The situation for ($\alpha,\gamma$) reactions is similar to the ($\rm
p,\gamma$) reactions. Again, because of the Coulomb barrier, $T_{\alpha}$ is
much smaller than $T_{\gamma}$ at astrophysically relevant energies, and thus
the ($\alpha,\gamma$) cross section scales with $T_{\alpha}$ and becomes
mainly sensitive to the $\alpha$OMP, but is practically independent of the
$\gamma$SF and NOMP. Because of the higher Coulomb barrier, the sensitivity on
the $\alpha$OMP is by far more pronounced than the sensitivity on the NOMP in
the ($\rm p,\gamma$) case. Much larger uncertainties of a factor of ten or
even more were found from the comparison between experimental and calculated
($\alpha,\gamma$) cross sections; e.g., Somorjai et al. (1998). An uncertainty
of a factor of ten was used for ($\alpha,\gamma$) rates in the above mentioned
$\gamma$–process studies (Rapp et al., 2006; Rauscher et al., 2016). It is
interesting to note that also ($\alpha,\rm n$) reaction rates depend
essentially only on the $\alpha$OMP, and a recent study of the weak
$r$–process (Bliss et al., 2020) has also used a factor of ten uncertainty for
the ($\alpha,\rm n$) reaction rates. Recently, the origin of this strong
sensitivity was better understood (Mohr et al., 2020), and updated reaction
rates with an estimated uncertainty of a factor of two using the ATOMKI-V2
$\alpha$OMP were provided in Mohr et al. (2021). This factor uncertainty was
chosen due to the success of ATOMKI-V2 $\alpha$OMP in reproducing
experimentally determined cross sections of $\alpha$-induced reactions (for
example Szegedi et al., 2021). Using these updated ($\alpha,\rm n$) rates,
significant progress in the modelling of the weak $r$–process was achieved
(Psaltis et al., 2022).

Summarising the status of theoretical reaction rates from the statistical
model, nowadays uncertainties of a factor of two are considered for the rates
of the ($\rm n$,$\gamma$), ($\rm p,\gamma$), and ($\alpha,\gamma$) capture
reactions. Because of the relation between forward and reverse reaction rates,
this uncertainty of a factor of two also holds for the ($\gamma$,$\rm n$),
($\gamma,\rm p$), and ($\gamma,\alpha$) photo-disintegration rates. An
experimental confirmation of these claimed uncertainties should become
possible at least in some well selected cases with the increasing availability
of radioactive ion beams and up-to-date detection techniques. Overall, the new
($\alpha,\gamma$) and ($\gamma,\alpha$) reaction rates in Mohr et al. (2021)
are significantly lower than the widely used rates by Rauscher & Thielemann
(2000), in particular for relatively heavier nuclei and lower temperatures.
This clearly calls for the further investigations that we will present in our
future work.

##  6 Discussion and conclusions

We studied the production of $p$–nuclei in 5 different sets of CCSNe. Each set
includes the yields from three massive star progenitors with initial masses of
15, 20, and 25 $\rm M_{\odot}$. We analysed both the overproduction factor of
each isotope (relative to their averages, as defined in several different
ways) and the ratios of $p$–nuclei close to each other in mass, relative to
their solar ratio. This strategy allowed us to investigate the differences
both in the production site and in the nuclear physics among the different
sets of models.

The different sets of CCSN models present several discrepancies both in the
overproduction factors and their ratios. These depend both on the assumptions
on stellar physics implemented in each set, i.e., the adoption of different
criteria for convection, mass-loss prescriptions, the use of overshooting, the
explosion mechanism, etc., and on the different nuclear physics inputs. The
different assumptions in stellar modeling mostly influence the final pre-
supernova structure of the star, as well as the possibility of having merger
events in the advanced phases of the pre-supernova evolution. The variations
in the stellar structure also lead to different modalities of the propagation
of the shock wave during the supernova explosion and hence to different
explosive nucleosynthesis. The use of different nuclear physics inputs leads
to local variations in the proportions of the isotopes of nearby mass, mostly
influencing the isotopic ratios.

In Sec. 3, we discussed that the $\rm F_{0}$ is not appropriate as a general
tool to investigate the properties of the $\gamma$–process nucleosynthesis.
Alternative definitions of this parameter can help us to distinguish the
production of $p$–nuclei from different nucleosynthetic processes, but these
are still not enough to study the details of the $\gamma$–process in the
models. For this reason, we analysed the correlations between 12 different
couples of OP ratios of isotopes close in mass.

Ten of the considered isotopic ratios show a good agreement with the solar
ratio for most of the models. These are [74]Se/[78]Kr and [84]Sr/[78]Kr (Sec
B.1), [102]Pd/[108]Cd and [106]Cd/[108]Cd (Sec B.3), [120]Te/[126]Xe and
[124]Xe/[126]Xe (Sec B.5), [130]Ba/[132]Ba and [136]Ce/[138]Ce (Sec B.6),
[144]Sm/[132]Ba (Sec B.7), and [174]Hf/[180]W (Sec. B.10). The other 13 ratios
show no agreement with solar and/or one or both isotopes have a too low
overproduction factor for their origin to be attributed to the
$\gamma$–process in CCSNe. In particular, the $p$–only Mo and Ru isotopes are
not produced in any of the considered models and their origin remain a
mystery. Nuclei such as [113]In, [115]Sn, [138]La, [152]Gd and [164]Er are not
significantly produced by the $\gamma$–process, which confirm that the bulk of
their abundances in the Solar System is made by other processes. We also
notice that [74]Se, [78]Kr and [84]Sr can be explained by the $\gamma$–process
only, without requiring an $\alpha$–rich freeze-out contribution.

The effect of the C–O shell mergers is mostly to increase the abundances of
the $p$-nuclides heavier than Pd. In the case of the $p$–nuclei lighter than
Pd, i.e., belonging to Se, Kr, Sr, Mo, and Ru, the yield is completely
dominated by the explosion and models with C–O mergers do not significantly
differ from the standard models. Moreover, these events typically favour the
production of the most neutron-rich isotope within $p$-only pairs belonging to
the same element. This is a signature of a $\gamma$–process that occurs at
lower temperature, as compared to the explosive conditions.

In conclusion, we have shown that a large scatter among the existing CCSN
model production exists and no set of models is fully able to reproduce the
distribution of the $p$–nuclei measured in the Solar System. Since most of the
models adopts outdated nuclear reaction rates, our results point toward the
necessity of an upgrade of the $\gamma$–process nucleosynthesis nuclear
networks with the latest results in nuclear physics, as discussed in Sec 5.
Furthermore, it is generally assumed that $\gamma$–process yields from CCSNe
depend only weakly on the energy of the explosion, however, there are no
detailed studies that confirm this. Therefore, we will proceed to improve and
update our $\gamma$–process nuclear network to compute new massive stars and
CCSN models for a wide grid of explosion energies, to further investigate the
$\gamma$–process nucleosynthesis in massive stars.

###### Acknowledgements.

We thank the support from the NKFI via K-project 138031 and the ERC
Consolidator Grant (Hungary) programme (RADIOSTAR, G.A. n. 724560). LR and MP
acknowledge the support to NuGrid from JINA-CEE (NSF Grant PHY-1430152) and
STFC (through the University of Hull’s Consolidated Grant ST/R000840/1), and
ongoing access to viper, the University of Hull High Performance Computing
Facility. LR acknowledges the support from the ChETEC-INFRA – Transnational
Access Project 22102724-ST. MP and ML acknowledges the support from the
”Lendület-2014” Programme of the Hungarian Academy of Sciences (Hungary). AP
acknowledges support from U.S. Department of Energy, Office of Science, Office
of Nuclear Physics, under Award Number DE-SC0017799 and Contract Nos. DE-
FG02-97ER41033 and DE-FG02-97ER41042. AS acknowledges funding by the European
Union’s Framework Programme for Research and Innovation Horizon Europe under
Marie Sklodowska-Curie grant agreement No. 101065891. PM, Gy. Gy., and Zs. F.
acknowledges support from National Research Development and Innovation Office
(NKFIH), Budapest, Hungary (K134197). This work was supported by the European
Union’s Horizon 2020 research and innovation programme (ChETEC-INFRA – Project
no. 101008324), and the IReNA network supported by US NSF AccelNet (Grant No.
OISE-1927130).

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##  Appendix A OP factors

Here we present the overproduction (OP) factors and their $\rm F_{0}$ of the
sets Rauscher et al. (2002); Pignatari et al. (2016b); Ritter et al. (2018b);
Sieverding et al. (2018); Lawson et al. (2022) used in the analysis presented
in this work. Figure 2 3 4 5 6 show the Fi/$\rm F_{0}$ and the isotopic OP
factor distributions for the models from each set. As discussed in Section 3,
we excluded from our analysis the 25 $\rm M_{\odot}$ from Ritter et al.
(2018b) and the 20 $\rm M_{\odot}$ from Lawson et al. (2022), since the
majority of the OP factors are lower than 1, due to the very large remnant
mass obtained after the explosion, and the 25 $\rm M_{\odot}$ from Pignatari
et al. (2016b), due to the low resolution in the $\gamma$–process peak
abundances. The respective $\rm F_{[}$O16] is also reported as a reference.

Figure 2: The Fi/$\rm F_{0}$ of the 15 $\rm M_{\odot}$ (upper left), 20 $\rm
M_{\odot}$ (upper right), 25 $\rm M_{\odot}$ (lower left) and the isotopic OP
factor distributions (lower right) for Rauscher et al. (2002) models. The
horizontal dotted-dashed lines in the lower right panel represent the $\rm
F_{[}$O16] in the 15 (yellow), 20 (green), and 25 $\rm M_{\odot}$ models
(blue).

Figure 3: Same as Figure 2 but for Pignatari et al. (2016b) models.

Figure 4: Same as Figure 2 but for Ritter et al. (2018b) models.

Figure 5: Same as Figure 2 but for Sieverding et al. (2018) models.

Figure 6: Same as Figure 2 but for Lawson et al. (2022) models.

##  Appendix B Isotopic ratios

Here we present an extended and detailed discussion for 12 couples of isotopic
ratios presented in Table 4. The grey regions in the following plots represent
regions of values a factor of two and three from the solar ratios. As already
stated in Sec. 4, we consider models to be in relatively good agreement with
solar when both following conditions are satisfied: (i) they fall into the
grey areas and (ii) they have a overproduction at least twice as solar (OP $>$
2).

###  B.1 [74]Se/[78]Kr vs [84]Sr/[78]Kr (Figure 7)

Figure 7: [74]Se/[78]Kr versus [84]Sr/[78]Kr from the model yields, with both
ratios normalised to their respective solar values. The light and dark grey
shaded areas represent, respectively, values a factor of two and three from
the solar ratio (black filled circle). The different sets of models are
represented by different types of symbols as indicated in the legend on the
right-hand side. The mass of the star is indicated as a small number next to
its corresponding symbol. Filled and empty symbols represent the yields
including or not, respectively, the radiogenic contribution from all unstable
isotopes. The only significant radiogenic contribution comes from [74]Br to
[74]Se.

These two ratios include nuclei that may all have additional channels of
production through explosive nucleosynthesis via the $\alpha$–rich freeze-out
and the neutrino-driven winds (see Sec. 1 and 3). In the models considered for
this work, the bulk of the yields comes from the $\gamma$–process
nucleosynthesis occurring in the explosive O/Ne burning regions. Only the
PGN15 model ejects a significant fraction of material from $\alpha$–rich
freeze-out, however, the total yield is still dominated by the O/Ne explosive
burning region. Also in the two models with the C–O shell mergers (RIT15 and
RAU20), the explosive $\gamma$–process has the most significant contribution
to these three isotopes, in fact the explosion produces about 70–80$\%$ and
40–60$\%$, respectively, of the total yields (Table 1).

The [74]Se/[78]Kr ratio (y-axis) is always super-solar, i.e., OP([74]Se) $>$
OP([78]Kr). The [84]Sr/[78]Kr ratio (x-axis) of three out of the five 15 $\rm
M_{\odot}$ models (RAU15, SIE15, and LAW15) mostly reflects the initial, solar
abundances in the envelope of the stars because both OP([84]Sr) and OP([78]Kr)
are lower than 2. In the other two 15 $\rm M_{\odot}$ models (PGN15 and RIT15)
as well as the RIT20 and SIE25 models, instead, [78]Kr and [84]Sr are
significantly produced during the explosive $\gamma$–process (see Figures in
Appendix A) therefore the values plotted in the figure represent the
$\gamma$–process composition, not the initial composition. Among the remaining
three 20 $\rm M_{\odot}$ models, PGN20 produces one of the highest OP([78]Kr)
of all the models, which it leads to the lowest OP([84]Sr)/OP([78]Kr) ratio.
The last two 20 $\rm M_{\odot}$ models (RAU20 and SIE20) have both a very
similar yield of [78]Kr and [84]Sr. The solar value of [84]Sr/[78]Kr is
instead $\sim 0.7$, which results in OP([84]Sr)/OP([78]Kr) the factor of $\sim
3$ over the solar ratio. In the 25 $\rm M_{\odot}$ case, RAU25 and LAW25 have
different yields and different OP factors, but the ratio between [84]Sr and
[78]Kr yields is the same, therefore, this leads to a similar behaviour of the
ratio of the OP factors.

###  B.2 [92]Mo/[94]Mo vs [96]Ru/[98]Ru (Figure 8)

Figure 8: Same as Figure 7, but for [92]Mo/[94]Mo versus [96]Ru/[98]Ru.

In almost all models, the OPs of these 4 isotopes are significantly lower than
2, which confirms that standard massive star models are unable to produce a
significant amount of these isotopes and represents the reason why most of the
models are in agreement with the solar abundance: they mostly reflect the
envelope composition. All the model shows a sub-solar [96]Ru/[98]Ru ratio
(x-axis), while the [92]Mo/[94]Mo ratio (y-axis) is more variable and there
are 6 models roughly a factor of two different different from solar. In the
case of the two models with C–O shell mergers (RIT15 and RAU20), the
production of the neutron richer isotopes of each element is favoured,
therefore both plotted ratios are sub-solar (Table 1). Out of the other 4
models different from solar, in RIT20 the $\gamma$–process takes place during
explosive Ne burning, in an extended region of $\sim 0.6$ $\rm M_{\odot}$,
with a wide $\gamma$–process peak, in particular of [92]Mo and [98]Ru, which
results in higher yields of these two isotopes compared to the other models.
Instead, in PGN20 there are two distinct $\gamma$–process regions, one in the
typical O/Ne explosive burning zone and one just above the interface between
the ONe and CO cores, where the shock wave accelerates again with a consequent
increase of the peak temperature. This double contribution to the
$\gamma$–process nucleosynthesis increases the OP factors of [92]Mo, [96]Ru
and [98]Ru. However, those features do not affect significantly the overall
distribution of the $p$–nuclei.

###  B.3 [102]Pd/[108]Cd vs [106]Cd/[108]Cd (Figure 9)

Figure 9: Same as Figure 7, but for [102]Pd/[108]Cd versus [106]Cd/[108]Cd.

Most of the models fall within a factor of 2 from the solar ratios. The 4
models RIT15, PGN15, SIE20, and RAU20 have all sub-solar ratios and the same
[102]Pd/[108]Cd (y-axis). In the case of PGN15 and SIE20 the sub-solar
[102]Pd/[108]Cd ratio is due to OP([102]Pd) $<$ 2, while the other two models
(RIT15 and RAU20) are those that experience a C–O shell merger, where the
production of [108]Cd, at denominator in both plotted ratios, is favoured over
that of [106]Cd and [102]Pd. The production of [102]Pd is different between
those two C–O shell merger models: in RIT15 it is dominated by the merger and
in RAU20 by the explosion (see Table 1). Instead, RAU15/25, PGN20 and RIT20
have all super-solar ratios because their OP([102]Pd) and OP([106]Cd) (i.e.,
of the isotopes at numerator) are significantly higher than the other models,
while their OP([108]Cd) is similar. All the other models have super-solar
[102]Pd/[108]Cd and sub-solar [106]Cd/[108]Cd, mostly because OP([108]Cd) and
OP([102]Pd) $\sim$ 2 while OP([106]Cd) $<$ 2\. No model falls into the
quadrant identified by [106]Cd/[108]Cd $>$ solar and [102]Pd/[108]Cd $<$
solar, which means that [102]Pd and [106]Cd are always overproduced together.

###  B.4 [112]Sn/[114]Sn vs [113]In/[114]Sn and [112]Sn/[114]Sn vs
[115]Sn/[114]Sn (Figure 10)

Figure 10: Same as Figure 7, but for [112]Sn/[114]Sn versus [113]In/[114]Sn
(upper panel) and [112]Sn/[114]Sn versus [115]Sn/[114]Sn (lower panel).
[113]In has a radiogenic contribution from [113]Sn. In the lower panel, the
empty circle represents the solar ratio, while the filled circle represents
the solar ratio minus the $s$– and $r$–process contribution to [115]Sn
(1-$s$-$r$). In this plot, the gray shaded areas identify a factor of 2 and 3
from the ratio calculated using the (1-$s$-$r$ ) value.

Of the four isotopes discusses here, the production of [113]In, [114]Sn, and
[115]Sn can also occur via neutron captures because [113]Cd and [115]In have
an unstable isomer (see Sec. 1 and 3). The galactic chemical evolution (GCE)
computation of Bisterzo et al. (2014) found that the $s$–process
nucleosynthesis in low-mass AGB stars provides a negligible contribution to
these nuclei (0, 0.1, and 4.2$\%$, to [113]In, [114]Sn, and [115]Sn,
respectively). The estimated $r$–process contributions to [113]In and [115]Sn,
instead, are between 2 and 16$\%$, and $60\pm 10\%$, respectively (see, e.g.,
Dillmann et al., 2008; Nemeth et al., 1994; Theis et al., 1998, and references
therein). Therefore, the $\gamma$–process nucleosynthesis should have a
significant role in the production of all of these isotopes. In Figure 10 we
accounted for the neutron-capture contribution to [115]Sn only, and subtracted
its $s$\- and $r$-process contributions, because the contributions to the
other isotopes are negligible. In general, the values close to the solar
ratios do not represent the result of $\gamma$–process nucleosynthesis because
the yields are dominated by the initial abundances. The cases that do not
behave like this are discussed in detail below. Specifically, OP([113]In) is
lower than 2 in all the models, therefore [113]In/[114]Sn (x-axis, top panel)
is always lower than solar. The models that effectively activate the
$\gamma$–process are characterised by a radiogenic component to [113]In
dominated by [113]Sn. However, this contribution is never enough to produce
OP([113]In) $>$ 2.

Figure 10 shows that the models that have a $\gamma$–process component
contributing to the isotopes of Sn are typically two to three times away from
the solar value, except for the LAW25 model which has a significant
$\gamma$–process component but the ratios are all close to solar because
[113]In and [114]Sn are similarly overproduced by an $s$–process component in
the C shell, therefore [112]Sn/[114]Sn is slightly sub-solar.

Out of the 5 most extreme models, the C–O shell merger in the RIT15 and RAU20
models favors the production of the Sn neutron richer isotopes, which results
in sub-solar [113]In/[114]Sn and [112]Sn/[114]Sn ratios. The other three most
extreme models (PGN20, RIT20 and RAU25) have the highest OP([112]Sn) and
OP([114]Sn) and therefore their [113]In/[114]Sn ratio (top panel, x-axis) is
strongly sub-solar. In PGN20, the OP([112]Sn) dominates over OP([114]Sn). This
is due to the peculiar structure of this star (see Sec. 2) with two distinct
abundance peaks of $\gamma$–process nucleosynthesis, in both of which the
abundance of [112]Sn is higher than the abundance of [114]Sn. In RIT20, an
additional production of [114]Sn occurs in the C shell, therefore
[112]Sn/[114]Sn decreases to a value below solar. In the RAU25 model
OP([112]Sn) and OP([114]Sn) are similar, therefore their [112]Sn/[114]Sn ratio
is close to one.

In the lower panel of Figure 10 the y-axis is the same as in the top panel, in
the x-axis [115]Sn substitutes [113]In, and the discussion is similar. In
fact, OP([115]Sn) as OP([113]In) is always lower than 2, except in the case of
the C–O shell merger. Most of the models, including those that have a
significant $\gamma$–process production, fall within a factor of 3 of the
value derived by subtracting the neutron-capture contributions from the solar
value.

###  B.5 [120]Te/[126]Xe vs [124]Xe/[126]Xe(Figure 11)

Figure 11: Same as Figure 7, but for [120]Te/[126]Xe versus [124]Xe/[126]Xe
ratio. [126]Xe has a radiogenic contribution from [126]Ba and [126]Cs.

Most of the models move to a sub-solar [120]Te/[126]Xe (y-axis) and solar
[124]Xe/[126]Xe (x-axis) once the radiogenic contribution to [126]Xe is taken
into account. Note that the SIE25 model is close to solar because all the OPs
are close to one, therefore it is non relevant for the $\gamma$–process
analysis (see Figure 1). Three models are instead different from solar: RIT15,
RAU20, and RAU25. Two of these are the models with C–O shell merger (RIT15 and
RAU20), where we found a similar overproduction of [120]Te and [124]Xe (i.e.,
the two isotopes at the numerator), and a relatively larger overproduction of
[126]Xe. This results in both [120]Te/[126]Xe and [124]Xe/[126]Xe) more than
three times lower than solar. Also in the RAU25 model, the overproduction of
[126]Xe is larger than that of [124]Xe and [120]Te, however, in this case
OP([124]Xe) $\sim$ 2OP([120]Te).

###  B.6 [130]Ba/[132]Ba vs [136]Ce/[138]Ce (Figure 12)

Figure 12: Same as Figure 7, but for [130]Ba/[132]Ba versus the
[136]Ce/[138]Ce ratio. [132]Ba and [138]Ce have radiogenic contributions from
[132]Ce, [132]La and [138]Nd, respectively.

The $\gamma$–process nucleosynthesis in the majority of the models reproduces
very closely the solar ratios on both axis (although note that as in the
previous Sec. B.5, the SIE25 model has all the OP factors close to 1), with
two exceptions: RIT20 and RAU20. The RIT20 ratios are both super-solar because
of a combination of the higher OPs ($\sim$30–40) of the more neutron-deficient
isotopes (at numerator) relative to the lower OPs ($\sim$4–8) of the neutron-
richer isotopes (at denominator). The RAU20 ratios are, instead, both sub-
solar due to the effect of the C–O shell merger, which favours the neutron-
richer isotopes. The same effect is present in the other model with C–O shell
merger (RIT15), albeit to a lesser extent because OP([132]Ba) and OP([138]Ce)
are similar in the two models, while OP([130]Ba) and OP([136]Ce) are higher in
RIT15 than in RAU20.

###  B.7 [138]La/[132]Ba vs [144]Sm/[132]Ba (Figure 13)

Figure 13: Same as Figure 7, but for [138]La/[132]Ba versus the
[144]Sm/[132]Ba ratio. [132]Ba has radiogenic contribution from isotope[132]Ce
and [132]La.

The [144]Sm/[132]Ba ratio (x-axis) of all the model falls within a factor of
two of the solar value, except for RIT20, where the [132]Ba is underproduced.
The [138]La/[132]Ba ratio (y-axis), instead, is more scattered and in most of
the models it is more than three times lower than solar, except for the SIE
models. This is because, as mentioned in Sec. 1 and 3, [138]La may have an
additional neutrino capture contribution, which is only included in the
nuclear network of the SIE models. These models also have the lowest
OP([132]Ba), further contributing to higher [138]La/[132]Ba ratio.

###  B.8 [156]Dy/[152]Gd vs [144]Sm/[152]Gd (Figure 14)

Figure 14: Same as Figure 7, but for [156]Dy/[152]Gd versus the
[144]Sm/[152]Gd ratio. The empty circle represents the solar ratio, while the
filled symbol represents the solar ratio minus the $s$–process contribution to
the isotope [152]Gd (1-$s$). Similarly to Figure 10, the gray shaded areas
identify a factor of 2 and 3 from the ratio calculated using the (1-$s$)
value.

Of the three isotopes considered here, as discussed in Sec. 1 and 3, a
fraction of the solar abundance of [152]Gd comes from the $s$–process in AGB
stars. The GCE computation of Bisterzo et al. (2014) reported this fraction to
be $86.6\%$. Therefore, in Figure 14 we represent the value to be matched as
the solar (1-$s$) fraction only, i.e., $13.4\%$. Furthermore, in the models
presented here [152]Gd has a dominant $s$–process contribution from the pre-
supernova He and C shells and the explosive $\gamma$–process component is only
a small fraction of the total yield. [156]Dy and [144]Sm, instead, are
$p$–only nuclei that are produced by $\gamma$–process during the explosion,
and their OPs are significantly higher than that of [152]Gd.

As in the cases of Sec. B.5 and B.6, the SIE25 model is close to solar because
all the OPs are close to one, therefore this is not discussed further. The
[144]Sm/[152]Gd ratio (x-axis) of most of the models falls within a factor of
three of the ratio calculated with the (1-$s$) value. The three SIE15, SIE20,
and PGN15 models, instead, are close to the solar ratio because they have
OP([152]Gd) similar to the other models, but a relatively lower OP([144]Sm).
The [156]Dy/[152]Gd ratio (y-axis) of five models (RIT15, RAU20/25, PGN20 and
LAW25) falls close to the ratio calculated with the (1-$s$) value. Among
those, the models with C–O shell mergers (RIT15 and RAU20) have the best
agreement. All the other models are closer to solar because they have
OP([156]Dy) $<2$, except for RIT20, where OP([156]Dy) $\sim$ 4.

###  B.9 [156]Dy/[158]Dy vs [162]Er/[164]Er (Figure 15)

Figure 15: Same as Figure 7, but for [156]Dy/[158]Dy versus the
[162]Er/[164]Er ratio. As in Figure 14, the empty circle represents the solar
ratio, while the filled symbol represents the solar ratio minus the
$s$–process contribution to the isotope [164]Er (1-$s$). The gray shaded areas
identify a factor of 2 and 3 from the ratio calculated using the (1-$s$)
value.[158]Dy has strong radiogenic contribution from [158]Er in the LAW
models only. [164]Er has a radiogenic contribution from [164]Yb, and a much
less significant contribution from [164]Tm. [162]Er has radiogenic
contribution from [162]Yb.

As in the case of [152]Gd (Sec B.8), also the solar abundance of [164]Er (the
denominator of the x-axis) has an $s$–process component (see Sec. 1 and 3).
Therefore, here we only focus on the possible $\gamma$–process origin of
[164]Er, which corresponds to $16.6\%$ of its solar abundance (Bisterzo et
al., 2014). Five models (PGN15, PGN20, SIE25, plus the two models with C–O
shell mergers RIT15 and RAU20) fall within a factor of 3 of the
[162]Er/[164]Er ratio calculated with the (1-$s$) value. Unlike in the
previous cases (Sec. B.2, B.3, B.4, B.5, and B.6) here the C–O shell merger
favours the production of the neutron-deficient [162]Er relative to [164]Er.

The [156]Dy/[158]Dy ratio (y-axis) is within a factor of 2 from solar, except
in the two LAW models, where the radiogenic contribution to [158]Dy is higher
than in the other models and results in a shift of the ratio to a value more
than 4 times lower than solar. Note that, as in the case of the Er isotopes
discussed above, the C–O shell merger (in RIT15 and RAU20) favours the
production of the neutron-deficient isotope.

###  B.10 [168]Yb/[180]Ta vs [174]Hf/[180]W (Figure 16)

Figure 16: Same as Figure 7, but for [168]Yb/[180]Ta versus the [174]Hf/[180]W
ratio. As in Figure 14, the empty circle represents the solar ratio, while the
filled symbol represents the solar ratio minus the $s$–process contribution to
the isotope [180]Ta (1-$s$). The gray shaded areas identify a factor of 2 and
3 from the ratio calculated using the (1-$s$) value. [168]Yb, [174]Hf, and
[180]W have a radiogenic contribution from [168]Hf, [174]W, and [180]Os,
respectively. In the RAU models the same three isotopes have a further
radiogenic contribution from [168]Lu, [174]Ta, and [180]Re, respectively.

As in the case of [163]Dy (Sec B.9), [179]Hf becomes unstable at stellar
temperatures, activating a $s$–process branching point. Therefore, [180]Ta
(and [180]W to a lesser extent) may show a neutron-capture contribution due to
the chain [179]Hf($\beta^{-}$)[179]Ta(n,$\gamma$)[180]Ta($\beta^{-}$)[180]W.
Therefore, we consider only the [(1-$s$) solar] abundance of [180]Ta, which is
$18.0\%$ (Bisterzo et al., 2014). Note that this residual solar abundance of
[180]Ta is not only due to the $\gamma$–process, since this isotope may also
receive a contribution from neutrino-capture on [180]Hf (Byelikov et al.,
2007; Sieverding et al., 2015). The models mostly populate the region of the
plot where the production of [180]W and [180]Ta dominate, respectively, over
the production of [174]Hf and [168]Yb and they are more than a factor of three
away from [(1-$s$) solar] ratios. The C–O shell mergers (RIT15 and RAU20),
instead, result in the opposite behaviour. Furthermore, these models are the
within a factor of 3 from the [(1-$s$) solar] ratios.

In all the models (except the two LAW models) the radiogenic contributions to
[174]Hf and [180]W shift the [174]Hf/[180]W ratio (x-axis) to lower values. In
the two LAW models, instead, the [180]W yield is much larger than the [174]Hf
yield, due to an efficient production of [180]W via the $s$–process in the
C-shell.

###  B.11 [184]Os/[196]Hg vs [190]Pt/[196]Hg (Figure 17)

Figure 17: Same as Figure 7, but for [184]Os/[196]Hg versus the
[190]Pt/[196]Hg ratio. [196]Hg has a radiogenic contribution from [196]Pb and,
only in RAU models, from [196]Tl. [190]Pt has a radiogenic contribution from
[190]Hg and, only in LAW models, from[190]Au.

The majority of the models overproduce [196]Hg (the denominator in both
ratios), relative to [184]Os and [190]Pt and predict ratios more than three
times lower than solar. The main responsible for this result is the strong
radiogenic contribution from [196]Pb to [196]Hg. In fact, without the
inclusion of this contribution most of the models (except the RAU models) fall
within a factor of three from solar ratio. Specifically, these models have
OP([196]Hg) $>$ OP([184]Os) $>$ OP([190]Pt). It follows that the discrepancy
between the models and the solar ratios is more severe in the case of
[190]Pt/[196]Hg (x-axis), where these models show ratios between a factor of 4
to 20 lower than solar. For the [184]Os/[196]Hg ratio (y-axis), instead, the
most extreme model gives a value a factor of 6 lower than solar.

Of the remaining four models closer to the solar ratios (PGN15, SIE25, LAW15,
and LAW25), PGN15 and SIE25 have OP([190]Pt) $<2$ and OP([184]Os) $>$
OP([196]Hg)$>$ OP([190]Pt), while the LAW models behave differently. This is
because they have OP([190]Pt) significantly higher than the other models, also
due to the radiogenic contribution of [190]Hg and [190]Au. Specifically, in
LAW15 OP([190]Pt) $>$ OP([196]Hg) $>$ OP([184]Os). In this model, the
radiogenic contribution to [190]Pt is twice as large as that to [196]Hg,
therefore, the inclusion of the radiogenic nuclei decreases the ratio by a
factor of two. In LAW25, OP([184]Os) $>$ OP([190]Pt) $>$ OP([196]Hg). In this
model, the radiogenic contribution to [190]Pt and [196]Hg is roughly equal, so
the ratio does not change.