# Comparison between core-collapse supernova nucleosynthesis and meteoric
stardust grains: investigating magnesium, aluminium, and chromium

Jacqueline den Hartogh Konkoly Observatory, Research Centre for Astronomy and
Earth Sciences, Eötvös Loránd Research Network (ELKH), Konkoly Thege Miklós út
15-17, H-1121 Budapest, Hungary  NuGrid Collaboration, http://nugridstars.org
Maria K. Petö Konkoly Observatory, Research Centre for Astronomy and Earth
Sciences, Eötvös Loránd Research Network (ELKH), Konkoly Thege Miklós út
15-17, H-1121 Budapest, Hungary  Thomas Lawson Konkoly Observatory, Research
Centre for Astronomy and Earth Sciences, Eötvös Loránd Research Network
(ELKH), Konkoly Thege Miklós út 15-17, H-1121 Budapest, Hungary  NuGrid
Collaboration, http://nugridstars.org E. A. Milne Centre for Astrophysics,
Department of Physics and Mathematics, University of Hull, HU6 7RX, United
Kingdom  Joint Institute for Nuclear Astrophysics - Center for the Evolution
of the Elements  Andre Sieverding School of Physics and Astronomy, University
of Minnesota, Minneapolis, MN 55455, USA  Physics Division, Oak Ridge National
Laboratory, P.O. Box 2008, Oak Ridge, TN 37831-6354, USA  Hannah Brinkman
Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Eötvös
Loránd Research Network (ELKH), Konkoly Thege Miklós út 15-17, H-1121
Budapest, Hungary  Graduate School of Physics, University of Szeged, Dom tér
9, Szeged, 6720 Hungary  Marco Pignatari Konkoly Observatory, Research Centre
for Astronomy and Earth Sciences, Eötvös Loránd Research Network (ELKH),
Konkoly Thege Miklós út 15-17, H-1121 Budapest, Hungary  E. A. Milne Centre
for Astrophysics, Department of Physics and Mathematics, University of Hull,
HU6 7RX, United Kingdom  NuGrid Collaboration, http://nugridstars.org Joint
Institute for Nuclear Astrophysics - Center for the Evolution of the Elements
Maria Lugaro Konkoly Observatory, Research Centre for Astronomy and Earth
Sciences, Eötvös Loránd Research Network (ELKH), Konkoly Thege Miklós út
15-17, H-1121 Budapest, Hungary  ELTE Eötvös Loránd University, Institute of
Physics, Budapest 1117, Pázmány Péter sétány 1/A, Hungary  School of Physics
and Astronomy, Monash University, VIC 3800, Australia

(Received June 4, 2021; Revised December 8, 2021; Accepted ecember 21, 2021)

###### Abstract

Isotope variations of nucleosynthetic origin among Solar System’s solid
samples are well documented, yet the origin of these variations is still
uncertain. The observed variability of 54Cr among materials formed in
different regions of the proto-planetary disk has been attributed to variable
amounts of presolar chromium-rich oxide (chromite) grains, which exist within
the meteoritic stardust inventory and most likely originated from some type of
supernova explosions. To investigate if core-collapse supernovae (CCSNe) could
be the site of origin of these grains, we analyse yields of CCSN models of
stars with initial mass 15, 20 and 25 M⊙, and solar metallicity. We present an
extensive abundance data set of the Cr, Mg, and Al isotopes as a function of
enclosed mass. We find cases in which the explosive C-ashes produce a
composition in good agreement with the observed 54Cr/52Cr and 53Cr/52Cr ratios
as well as the 50Cr/52Cr ratios. Taking into account that the signal at atomic
mass 50 could also originate from 50Ti, the ashes of explosive He-burning also
match the observed ratios. Addition of material from the He ashes (enriched in
Al and Cr relative to Mg to simulate the make-up of chromite grains) to the
Solar System composition may reproduce the observed correlation between Mg and
Cr anomalies, while material from the C-ashes does not present significant Mg
anomalies together with Cr isotopic variations. In all cases, non-radiogenic,
stable Mg isotope variations dominate over the variations expected from 26Al.

supernovae: general — nuclear reactions; nucleosynthesis — ISM: meteorites;
meteors; meteoroids — Astrophysics - Solar and Stellar Astrophysics

††journal: ApJ

##  1 Introduction

Isotopic differences of nucleosynthetic origin are observed among meteorite
groups and primitive meteorite components that formed in the Solar System. For
example, spinel-hibonite spherules and ‘normal’ calcium and aluminum rich
inclusions (CAIs) do not show nucleosynthetic variability, while
ultrarefractory platy hybonite crystals and CAIs with fractionation and
unidentified nuclear effects (also known as FUN CAIs) do. This is interpreted
as a record of progressive homogenisation of dust and gas in the inner regions
of the proto-planetary disk via turbulent mixing and thermal heating during
the T-Tauri phase of the Sun (Mishra & Chaussidon, 2014; Pignatale et al.,
2018, 2019; Jacquet et al., 2019). Nucleosynthetic isotope variations are also
observed among bulk compositions of meteorites and planetary objects which
implies that large scale isotopic heterogeneities, inherited from the proto-
solar nebula and/or formed during the evolution of the proto-planetary disk,
have been preserved.

These variations, however, are hard to connect to nucleosynthetic signatures
from specific stellar sources. A number of scenarios have been developed to
explain such connection. These range from isotopic differences inherited from
an inhomogeneous molecular cloud (Burkhardt et al., 2019; Dauphas et al.,
2002; Nanne et al., 2019), late processes acting on a once homogenized
material in the inner regions of the proto-planetary disk (Burkhardt et al.,
2012; Poole et al., 2017; Trinquier et al., 2009; Dauphas et al., 2008;
Regelous et al., 2008), and/or new material added to the proto-planetary disk
after the formation of the Sun (Van Kooten et al., 2016; Schiller et al.,
2018).

Solids from the proto-planetary disk not only display variation in bulk
isotopic compositions, but often also display a discontinuity (gap). For the
isotopes of many elements (e.g. Cr, Ti, Mo, Ru), meteorite types are well
separated into two groups. Because of this compositional gap, nucleosynthetic
isotope variations are often called the “isotopic dichotomy” of the proto-
planetary disk (Warren, 2011). Materials assumed to have formed in the outer
Solar System are associated with enrichment in neutron-rich isotopes of
intermediate-mass and iron group elements, such as 48Ca, 50Ti , 54Cr (see e.g.
Trinquier et al., 2007, 2009; Schiller et al., 2018), neutron-capture affected
isotopes such as those of Mo and Ru (see, e.g. Budde et al., 2016; Kruijer et
al., 2017; Nanne et al., 2019), and other isotopes of explosive
nucleosynthesis origin such as 58Ni (Nanne et al., 2019) and 92Nb (Hibiya et
al., 2019), as compared to materials assumed to have formed in the inner Solar
System111Material from the outer and inner Solar System are represented
respectively by (i) carbonaceous chondrites and “carbonaceous type” iron
meteorites, collectively referred to as CC; and (ii) ordinary chondrites,
lunar and martian samples, “non-carbonaceous” iron meteorites and various
achondrites, collectively referred to as NC., see, e.g., Budde et al. (2016);
Kruijer et al. (2017); Nanne et al. (2019); Hibiya et al. (2019) and the
review by (Kleine et al., 2020).

The nucleosynthetic source of these enrichments has been attributed to
supernovae but the exact origin is still unclear (e.g., Hartmann et al., 1985;
Dauphas et al., 2010). The formation of Jupiter’s core (Helled et al., 2014;
Kruijer et al., 2017), or a pressure maximum in the disk leading to such
formation (Brasser & Mojzsis, 2020) have been invoked as the barrier that kept
these two reservoirs well separated in the early Solar System.

The chromium isotopes are exceptionally useful to deconvolve the origin of
planetary scale nucleosythetic isotope variation in iron group elements
because Cr has four stable isotopes (at atomic mass 50, 52, 53, and 54), which
allows us to obtain two ratios after mass-fractionation effects are removed
with internal normalisation. Furthermore, it appears that the main feature of
the Cr anomaly, i.e., enrichment and depletion of the most neutron rich
isotope (54Cr), is driven by a single, well-identified mineral carrier.
Dauphas et al. (2010) and Qin et al. (2010) identified this carrier phase as
Cr-oxide (with variable structure, but mostly chromium rich Mg-spinel, here
referred to as chromite) and found that variable abundance of such presolar
grains can explain all the variations observed among bulk meteorites. Nittler
et al. (2018) provided high precision Cr data on these presolar chromite,
confirming the previously assumed high 54Cr/52Cr ratios (up to 80 times the
solar ratio). Nittler et al. (2018) compared their data to a limited number of
supernova models and concluded that the observations are better explained by
models of electron capture supernovae (Wanajo et al., 2013) and rare, high
density type Ia SNe (Woosley, 1997) than by models of core collapse supernovae
(CCSNe) by Woosley & Heger (2007)222Also in Asymptotic Giant Branch (AGB)
stars neutron-capture processes can enrich the 54Cr relatively to the other Cr
isotopes. However, the largest anomaly predicted in models of O-rich massive
AGB stars does not exceed values in the order of 40%, based on 6 M⊙ model of
solar metallicity from (Karakas & Lugaro, 2016). Therefore, we can exclude
that neutron captures in AGB stars are sources of presolar chromite with these
anomalies..

Interestingly, 54Cr variations among bulk meteorites and planetary objects may
also correlate with mass independent 26Mg isotope variations (Larsen et al.
2011 and Van Kooten et al. 2016). The observed variation in 26Mg/24Mg stable
isotope ratio can be due to a heterogeneous distribution of the short lived
radionuclide, 26Al (which decays to 26Mg with a half life of 0.717 Myr) along
the proto-planetary disk, or to variations in stable 26Mg and/or 24Mg
abundances, or both. Jacobsen et al. (2008) and Kita et al. (2013) argue for a
homogeneous Mg isotope distribution in the Solar System. Larsen et al. (2011)
proposed that the apparent positive correlation between 54Cr and 26Mg
anomalies among planetary objects is the result of progressive thermal
processing of in-falling 26Al-rich molecular cloud material towards the inner
regions of the disk. This in return results in preferential loss of thermally
unstable and isotopically anomalous dust. Alternatively, Van Kooten et al.
(2016) suggested that the apparent positive correlation between 54Cr and 26Mg
may represent “unmixing” of distinct dust populations with different thermal
properties. Old, thermally processed, presolar, homogeneous dust could mix
with fresh, thermally unprocessed, supernova-derived dust, which formed
shortly before the Solar System. This newly condensed dust is then
preferentially lost from the inner regions of the proto-planetary disk.
Because of the high significance of this apparent correlation and its possible
interpretations, we also make a first attempt to address it here from the
point of view of stellar modelling by exploring the Al and Mg isotopic
composition of the specific CCSN regions that match the nucleosynthesis
anomalies in presolar chromite grains. With simple mixing relations, we
investigate if these CCSN abundances can generate any significant variation in
26Al or stable Mg isotopes among planetary objects. This is a simplified first
attempt to linking stardust data to meteorites and planetary objects because
it assumes that such Al and Mg abundances are carried in the chromite grains
and/or similar carriers enriched in Al. While this is a possible scenario,
there is no evidence for it yet as there are no Mg or O isotope studies on
chrmomite presolar grains.

Here, we compare the predictions from three sets of CCSN models, from stars of
initial mass 15, 20, and 25 M⊙ and solar metallicity, to the chromite data to
evaluate the role of CCSNe as potential sources of chromite grains in the
large scale heterogeneity of the proto-planetary disk. We will also compare
the abundances of the stable isotopes of Cr, Al, and Mg and 26Al in three sets
of CCSN models and evaluate the isotopic abundances and ratios as a function
of the enclosed stellar mass. Our aims are: first, to identify 54Cr production
sites within CCSNe that may match the chromite grains; second, to evaluate the
26Al production and Mg isotope compositions associated with such 54Cr
production sites; and third, to investigate if the Al and Mg abundances of the
CCSN region of potential origin of the chromite grains could produce variation
in 26Al or 26Mg isotopes among planetary objects (under the simple assumption
described above). Furthermore, we compare the 26Al production in the CCSN
models to the 26Al signatures in the presolar grains from CCSN that we found
in the literature, in order to put our analysis of potential Al and Mg
abundances in chromite grains into the wider context of CCSN stardust grains
in general.

The structure of the paper is as follows: in Section 2 we briefly describe the
specifics of the CCSN data sets and outline their differences. The comparison
of the total yields for the nine Al, Mg, and Cr isotopes is presented in
Section 3, and in Section 4 we present the comparison between the CCSN models
and the observed Cr isotopic compositions of stardust grains, as well as the
comparison of the CCSN models to 26Al/27Al in other presolar CCSN grains. In
our discussion in Section 5 we present the effects of uncertainties associated
to neutron-capture reaction rates, an analysis on the Al and Mg isotopic
composition of the 54Cr production sites, and a comparison between modeled
ejecta compositions and the meteoritic data. Our conclusions are presented in
Section 6.

##  2 Methods

Core-collapse supernovae (CCSNe) are explosions associated with the death of
massive stars that process their initial composition through a sequence of
hydrostatic nuclear burning stages until an Fe core is formed (see Langer,
2012, for an extensive review). The self-consistent modeling of the explosion
mechanism is still challenging and requires three-dimensional, high-resolution
simulations, which are currently too expensive to allow us to perform large-
scale surveys for nucleosynthesis studies (for recent reviews see Burrows,
2013; Müller, 2016; Janka et al., 2016). Instead, parameterized, spherically
symmetric simulations have been employed widely to estimate CCSN
nucleosynthesis yields. In such models, the innermost part of the CCSN
progenitor is usually not simulated in detail but replaced with an engine that
artificially drives the explosion, such as a piston (Woosley & Weaver, 1995)
or the injection of thermal energy (Limongi & Chieffi, 2003), which can be
tuned with a few model-specific parameters to yield a desired explosion
energy, measured as the kinetic energy at infinity, and remnant mass, which is
referred to as the mass cut. In addition to the neutron star that is left
behind by the explosion, the mass cut also includes the possibility of
fallback, i.e., material that is initially ejected, but remains
gravitationally bound to the remnant and thus eventually falls back onto it,
possibly leading to the formation of a black hole even after a successful
explosion (Zhang et al., 2008; Fryer, 2009). Recently, models have been
developed that treat the evolution of the stellar core in more detail, instead
of with an engine, and still achieve explosions in spherically symmetric
simulations by different parameterizations (Perego et al., 2015; Sukhbold et
al., 2016; Couch et al., 2020). Such models are promising to improve on the
simple models mentioned above, but remain to be validated by comparison to
multi-dimensional simulations and observations.

We collected three CCSN yield sets for 24,25,26Mg, 26,27Al and 50,52,53,54Cr
(Lawson et al. (submitted), Sieverding et al., 2018; Ritter et al., 2018), for
which we have access to abundance profiles as a function of the stellar mass
coordinate. These yield sets are based on 1D calculations using different
stellar evolution, explosion, and post-processing codes. We include only
models with an initial mass of 15, 20, and 25 M⊙ at solar metallicity, since
higher mass stars are expected to result in the formation of a black hole
without any significant ejection of material processed by explosive nuclear
burning (Heger et al., 2003). We exclude the effects on yields by rotation,
magnetic fields, and binary evolution as these are not known or too uncertain
(Aerts et al., 2019; den Hartogh et al., 2019; Belczynski et al., 2020). The
yield sets are listed in Table 1, together with details on codes used for the
calculations. For each data set we list in the following subsections the codes
used for the calculations and the details of the initial set-ups that are
important for our comparison.

###  2.1 Data set of Lawson et al. (submitted, LAW)

The LAW models are a part of the large data set presented in Fryer et al.
(2018), who performed a parameter study over a broad range for SN explosions.
Andrews et al. (2020) used these models to study the production of radioactive
isotopes relevant for the next generation of facilities for $\gamma-$ray
astronomy, and provided the complete yields for the full stellar set. Jones et
al. (2019b) used the same set to study the production of 60Fe. Here we use the
updated yield set based on the same models, but updated by including a recent
bug fix (Lawson et al., submitted).

The progenitor stellar evolution models were calculated with a recent version
of the Kepler hydrodynamic code (Weaver et al., 1978; Heger & Woosley, 2010),
using initial abundances based on Grevesse & Noels (1993, GN93). The
progenitors were post-processed to obtain the detailed nucleosynthetic results
using MPPNP (Multi-zone Post-Processing Network – Parallel, see Pignatari et
al. 2016 and Ritter et al. 2018). The explosions of the progenitors are
calculated using a 1D code mimicking a 3D convective engine, as described in
Herant et al. (1994) and Fryer et al. (1999). The explosion nucleosynthesis is
calculated using TPPNP (Tracer particle Post-Processing Network – Parallel,
see Jones et al., 2019). The difference between MPPNP and TPPNP is that the
first also performs mixing of mass shells following the mixing as calculated
in the progenitor or explosion model, while the latter does not apply any
mixing and may efficiently streamline the post-processing of trajectories. The
same nuclear reaction package is used from the two post-processing frameworks.

###  2.2 Data set of Sieverding et al. (2018, SIE)

The progenitor models of SIE were calculated with a slightly older version of
the Kepler code than the LAW models. Differences include the neutrino loss
rates as discussed by Sukhbold et al. (2018) and updated photon opacities. Due
to these differences, the SIE models show a less massive C/O core and more
compact structure than the LAW models. The initial abundances for the
progenitors of SIE are based on Lodders (2003, L03). The explosion was
simulated with a piston, as described in Woosley & Weaver (1995). The piston
is put at the mass cut determined by the position where the entropy per baryon
drops below 4 $k_{B}$. The parameters of the piston were adjusted to produce
an explosion energy of $1.2\times 10^{51}\,\mathrm{erg}$. All matter outside
the mass cut is assumed to be ejected, i.e., no additional fallback is
considered.

The explosive nucleosynthesis was post-processed by Sieverding et al. (2018),
who performed a parameter study around the effects of neutrino energies. We
include here the models with the highest neutrino energy.

###  2.3 Data set of Ritter et al. (2018, RIT)

The progenitor models of Ritter et al. (2018) were calculated with the MESA
stellar evolution code (Paxton et al., 2011) with initial abundances based on
GN93. The explosion models were calculated via the semi-analytical approach
using the delayed formalism as described in Pignatari et al. (2016), and using
the mass cuts from Fryer et al. (2012). The detailed nucleosynthesis was
calculated for the progenitor and the explosion with the post-processing code
MPPNP. In the evolution of the 15 M⊙ star the convective O and C shells merge.
This feature can occur during the later phases of stellar evolution, when the
different burning shells are formed close enough to each other to possibly
merge. The shell merger in the 15 M⊙ progenitor model takes place at the end
of the core Si burning phase, see Appendix A for more details. Shell mergers
are often found in 1D and 3D stellar evolution models (see Müller, 2020, for a
recent review), and shell-merger events are often initiated shortly before the
collapse. Collins et al. (2018) found that 40% of their stellar evolution
models with an initial mass between 16 and 26 M⊙ start the core collapse
during an ongoing shell-merger.

###  2.4 Decayed abundances

We present the isotope abundances and isotopic ratios as a function of stellar
mass coordinates for both the progenitor and explosion models. Unless
indicated otherwise, in the following figures we show the abundances obtained
after decaying all the radioactive isotopes created during the explosion into
their respective stable isotope, except for the case of 26Al, as we want to
study its production. For the isotopes of interest here, the most relevant
decay chain is 53Mn($\beta^{+}$)53Cr with a half life of 3.74 Myr. Its effect
on the comparison to the stardust grains will be considered in Section 4.1.

Table 1: Overview of the details of the different yield sets included in this paper. The first three studies are discussed in detail in this work, the last four are only included here and in Section 3.2.  Set | Code for progenitors | Code for explosions | Initial abundances | Solar metallicity
---|---|---|---|---
Lawson et al.(submitted) (LAW) | Kepler | Convective engine | GN93 | Z=0.02
Sieverding et al. (2018) (SIE) | Kepler | Kepler | L03 | Z=0.013
Ritter et al. (2018) (RIT) | MESA | Semi-analytical | GN93 | Z=0.02
Limongi & Chieffi (2018)333This study also investigates the effects of rotation, but we exclude those models in our comparison due to the large uncertainties present in the theory of rotation in stellar evolution (see e.g. Aerts et al., 2019; Belczynski et al., 2020; den Hartogh et al., 2019). (LIM) | FRANEC | FRANEC | AG89444Anders & Grevesse (1989) | Z=0.02
Rauscher et al. (2002) (RAU) | Kepler555Progenitor models from Woosley & Weaver (1995) (all other Kepler progenitor models are more recent) | Kepler | AG894 | Z=0.02
Curtis et al. (2019) (CUR) | Kepler | PUSH | L03 | Z=0.013
Sukhbold et al. (2016)666We include two models (14.9 and 25.2 M⊙) as shown in the paper, other yields can be found in their online data. (SUK) | Kepler | Kepler (W18 engine) | L03 | Z=0.013

###  2.5 Nomenclature

In the following sections we define the regions within the stellar model from
the envelope towards the core in the following way:

  * •

If 4He is the most abundant isotope, the region is called H-ashes (the grey
band located at the highest mass coordinate in the three panels of Figure 1);

  * •

If 12C and 16O are the most abundant isotopes, the region is called He-ashes
(the white band located at the highest mass coordinate in the three panels of
Figure 1);

  * •

If 16O and 20Ne are the most abundant isotopes, the region is called C-ashes
(the grey band located to the left of the He-ashes in the three panels of
Figure 1);

  * •

If 16O and 28Si are the most abundant isotopes, the region is called Ne-ashes
(the white band located to the left of the C-ashes in the three panels of
Figure 1). The shell merger region in the 15 M⊙ RIT model is also labelled as
Ne-ashes;

  * •

If 28Si is the most abundant isotope, the region is called O-ashes (the grey
band located to the left of the Ne-ashes in the three panels of Figure 1);

  * •

If 56Ni and 56Fe are the most abundant isotopes, the region is called Si-ashes
(the white band located to the left of the O-ashes in the three panels of
Figure 1).

This nomenclature represents a simplified structure of the regions within
massive stars before the explosion, and is often used within the massive star
community. The nomenclature of Meyer et al. (1995) is commonly used in the
presolar grain community and, when we compare our version to theirs,
identifies mostly the same zones. The main difference is that they name the
zone based on the most abundant isotopes, while our names refer back to the
main fuel within the region. When putting the two schemes next to each other
we get: our H-ashes are their He/N and He/C zones, our He-ashes are their O/C
zone, our C-ashes are their O/Ne zone, our Ne-ashes are their O/Si zone, our
O-ashes are their Si/S zone, and our Si-ashes are their Ni-zone.

##  3 Yields

We discuss in the first subsection the creation of 24,25,26Mg, 26,27Al and
50,52,53,54Cr in massive stars. Our analysis is focused on these nine
isotopes, and their distribution in CCSN ejecta. The total isotopic yields of
the data sets are compared in the second subsection.

###  3.1 Creation of the Al, Mg, and Cr isotopes in massive stars and CCSNe

We trace the internal structure of the models by plotting the abundance
profiles of the mass fractions of 4He, 12C, 16O, 20Ne, 28Si, 56Ni, and 56Fe as
a function of mass coordinate. In Figure 1 we show the structure plots of the
15 M⊙ models of LAW, SIE, and RIT. In Appendix A we provide figures of all
three initial masses and the three data sets, showing the internal structure
and also the final mass fractions of the Mg, Al, and Cr isotopes of the
progenitor and the explosion model777All the data used to produce the figures
in this paper can be found as Supplemental Data in the online Article Data.

Figure 1: Structure plots showing the three (non-decayed) 15 M⊙ models with grey and white bands to indicate the different regions (labelled as ‘ashes’ in the text). The more compact structure of SIE compared to LAW is visible when comparing the mass coordinates of the bands. Furthermore, the RIT model shows a different internal structure due to the shell-merger. See Appendix A for the Al, Mg, and Cr isotope plots. Table 2: The burning phases in the progenitor and explosion where the isotopes of interest are created and destroyed for the 15 M⊙ LAW model. The burning phases are labelled using the definitions which were set in Section 2.5 (the word ‘ashes’ is not repeated). The dominant nucleosynthesis process responsible for the production or destruction of each isotope are also indicated and explained in more detail in the text. (‘cap’ is short for ‘capture’). This table is specifically for 15 M⊙, which might be where the differences with Table 3 of Woosley et al. (2002) come from. When comparing our table to Curtis et al. (2019), the differences are due to our table excluding radiogenic contributions and including outer layers, which are the exact opposites of the features in the table of Curtis et al. (2019). | Progenitor | Explosion
---|---|---
| Produced in | Via | Destroyed in | Via | Produced in | Via | Destroyed in | Via
24Mg | C, Ne, He |  $\alpha$-cap | - | - | He |  $\alpha$-cap | O, Si | photo-dis
25Mg | C, Ne, He | n-cap | Ne | photo-dis | He | n,$\alpha$-cap | C |  $\alpha$-cap
26Mg | C, Ne, He | n-cap | Ne | photo-dis | He | n-cap | C |  $\alpha$-cap
26Al | C | p-cap | Ne, He | several | C | p-cap | O | photo-dis
27Al | C, Ne | p-cap | O | photo-dis | - | - | O | photo-dis
50Cr | C | p-cap | He | n-cap | O, Si | p-cap | Si | equi
52Cr | - | - | He, C | n-cap | O | equi | Si | equi
53Cr | C | n-cap | He | n-cap | He | n-cap | C | n-cap
54Cr | He | n-cap | - | - | He | n-cap | C | n-cap

Table 2 shows the production and destruction sites of the nine isotopes of
interest, plus their dominant reaction paths for the 15 M⊙ LAW model. Two
reaction paths require explanation: ‘Photo-dis’ stands for photo-
disintegration, the process where an incoming photon removes a neutron,
proton, or an $\alpha$-particle from the nucleus. ‘Equi’ denotes the
production in high temperature equilibrium conditions when most forward- and
backward reaction rates are closely matches (Woosley et al., 1973; Chieffi et
al., 1998). This usually applies to explosive Si and O burning.

In the following we highlight the most important differences between the
models with respect to the production and destruction of the isotopes we are
interested in. The 15 M⊙ SIE progenitor model shows production and destruction
sites (Figure 12) that are comparable to the 15 M⊙ LAW model. The 15 M⊙ RIT
progenitor model, however, experiences a shell-merger, which allows for
C-burning while He-burning is still ongoing. Furthermore, the shell-merger
allows for mixing of Cr-isotopes from the deeper layers outwards. In this
shell-merger region, the 15 M⊙ RIT model shows a higher abundance for 26Mg
than 25Mg (the opposite is visible in the LAW and SIE 15 M⊙ models), and the
presence of 50,52Cr mixed up from deeper layers, which is not taking place in
the LAW and SIE 15 M⊙ models. The 15 M⊙ explosive model of SIE and RIT show
more explosive He burning than the 15 M⊙ LAW model. This allows for the
production of the Mg-isotopes and extra destruction of 50,52Cr in the SIE and
RIT 15 M⊙ models.

The main difference between the 15 and 20 M⊙ LAW progenitor models is that the
mass cut is higher in the 20 M⊙ model (Figure 13), which leads to excluding
the Ne-ashes from the ejecta. The 20 M⊙ SIE model is the only 20 M⊙ model
including the Ne-ashes, where 26Al is produced. The 20 M⊙ RIT model shows a
mass cut similar to the 20 LAW model and a production of 26Al in the C-ashes,
like the 15 M⊙ RIT model. The main difference between the three 20 M⊙ models
is that they show different amounts of explosive nucleosynthesis. The 20 M⊙
LAW explosive model shows no explosive nucleosynthesis involving the nine
isotopes in Table 2. The 20 M⊙ SIE model, however, shows explosive
nucleosynthesis in the inner regions, producing 26Al and 50,52Cr. The 20 M⊙
RIT model shows explosive nucleosynthesis in the whole star, due to its high
temperature compared to the other 20 M⊙ models (Figure 15). The explosive He-
burning in this model is similar to the 15 M⊙ RIT model, while the explosive
C-burning leads to the creation of 26Al and 50,52Cr and the destruction of
27Al and 53,54Cr.

The 25 M⊙ progenitor models (Figure 14) are similar to the 15 M⊙ progenitor
models, except for the high mass cut in the upper C-ashes in the 25 M⊙ RIT
model. The other two models show mass cuts below the Ne-ashes. In the 25 M⊙
explosive models we see explosion nucleosynthesis only in the inner regions of
the LAW and SIE 25 M⊙ models, and not from explosive He-burning. This means
most explosive contributions to the nine isotopes of interest are still
present. In contrast, the 25 M⊙ RIT model only experiences explosive He-
burning as its mass cut is too high to include the other regions.

###  3.2 Comparison of the total yields

Figure 2: Comparison of the isotope yields of the three CCSN data sets
considered in this paper, plus four others found in the literature. The three
models of LAW that are considered in this paper and shown in the Appendix A
are plotted as orange open crosses. The yields of all the models of LAW are
included in the rectangle boxes, where the median value of the models is shown
as orange line. The box plot covers the values between 25% and 75% of all the
data points, while the error bars cover 1.5 times the range of the box plot.
Any points outside the error bars are shown as open circles. The yield sets
are labelled as indicated in Table 1.

Here we present a comparison of the total yields of seven CCSN data sets (not
the net yields, which are calculated as the total yield minus the initial
abundance). An overview of the main characteristics of the models by LAW, SIE,
and RIT is given in Table 1, together with other four sets of CCSN models that
are available in the literature (Rauscher et al., 2002; Sukhbold et al., 2016;
Limongi & Chieffi, 2018; Curtis et al., 2018). The seven sets have been
calculated with different 1D stellar evolution and explosion codes. While this
is not meant to be a comprehensive collection of CCSN yields available, it may
be considered as indicative of the existing abundance variations obtained from
different CCSN models. The comparison of the seven yields sets is presented in
Figure 2. We plot the explosive yields of the nine isotopes: 24,25,26Mg,
26,27Al and 50,52,53,54Cr, for all models grouped together according to their
stellar masses.

We note that the CUR yields are often the lowest yield for the Mg- and Al-
isotopes. The reason for this is that this study only includes the inner
stellar regions in their nucleosynthesis calculations. Parts of the C-ashes
region are cut off, where the Mg and Al isotopes are abundant (see Table 2),
resulting in an apparent reduction of the total yield of the Mg and Al
isotopes compared to other yields. Overall, we find that variations in the
production of the nine isotopes in the seven yield sets are roughly one order
of magnitude at most. The range of yields in the LAW models appears to cover
most other yield sets, thus confirming that the parameter study of Fryer et
al. (2018) well represents the uncertainties within 1D CCSN explosions. We
discuss in the remaining of this section the isotopes that show variations
larger than one order of magnitude in the LAW, SIE, and RIT yields.

The 25,26Mg and 26,27Al yields of LAW are higher than those of SIE in all
panels of Figure 2. This is because according to the LAW models the central
stellar structures are less compact compared to SIE models (as mentioned in
Section 2.2). Thus the Mg- and Al-rich C-ashes of LAW are located at higher
mass coordinates than those of SIE.

Among the 15 M⊙ models (left panel), only the 50,54Cr yields show a spread of
about one order of magnitude (excluding the few outliers of the LAW data set
shown as small circles). The main reason for this is that the RIT model
undergoes a shell-merger. In this region in the 15 M⊙ RIT model creates more
Cr than the other two 15 M⊙ models, as the shell-merger transfers iron group
elements from the deeper layers into the merged region (see Figure 12 and Côté
et al. 2020).

Among the 20 M⊙ models (middle panel) again the Cr-isotope yields show the
largest spread. The lower values of RIT are due to the higher mass cut values
compared to the models of LAW and SIE (Figure 13). This effect is not present
in the Mg and Al isotopes, because these isotopes are produced in regions that
are not affected by the mass cut. The large spreads in the models by LAW are
caused by its large range of values for the mass cuts, see Fryer et al.
(2018).

Also among the yields of the 25 M⊙ models (right panel) the Cr isotopes show
the largest range of variations. The spread in the LAW data set is due to
differences in the explosion energies. The Cr yields of RIT are again lower,
due to its mass cut being higher than in models by LAW and SIE.

In summary, the main differences between the three data sets of LAW, SIE, and
RIT are the structural differences between the progenitors of the LAW and SIE
data sets, the C-O shell merger in the 15 M⊙ RIT model, and the higher mass
cuts in the 20 and 25 M⊙ RIT models.

##  4 Results and comparison with presolar stardust grains

Grains are formed locally within the CCSN ejecta, and thus we cannot use the
total yields as presented in Section 3.2 for the comparison of CCSN yields to
presolar chromite grains. Instead, we compare the high-precision grain data of
Nittler et al. (2018) to the Cr isotopic ratios versus mass coordinate of the
CCSN data sets of LAW, SIE, and RIT (Figures 3-4). The ratios are plotted
against the mass coordinates in Figure 5. Nittler et al. (2018) also
considered the possibility that the signal at atomic mass 50 represents 50Ti
instead of 50Cr and report the 50Ti/48Ti ratios inferred for 5 out of the 19
54Cr-rich grains. Therefore, we also present and discuss here this
possibility, while leaving the extended description of the production of the
Ti isotopes in CCSNe to future work.

Among the models of the LAW data set with different explosion energies, we use
one for each initial mass in this section, which has an explosion energy
closest to the value of $1.2\times 10^{51}\,\mathrm{erg}$ used by Sieverding
et al. (2018). The predicted isotopic ratios are calculated using decayed
stellar abundances to consider the radiogenic contribution to the final
abundances of stable isotopes (as explained in Section 2.4), unless indicated
otherwise. The boxes in Figures 3 and 4 are explained later in this section,
when we precisely locate candidate regions that match the composition of the
chromite grains.

We also explore the predicted Al and Mg isotope profiles of the CCSN models as
a function of mass coordinate. In Figure 6 we show the 26Al/27Al ratio
profiles of the CCSN models in comparison to the highest values determined for
presolar grains of likely CCSN origin, such as SiC type X grains (e.g.,
Groopman et al., 2015) and Group 4 presolar oxides (e.g., Nittler et al.,
2008).

###  4.1 Chromite grains

Figure 3: Comparison between the Cr isotopic ratios as measured by Nittler et
al. (2018) (yellow data points with error bars), and those predicted by LAW,
SIE, and RIT (left, middle, and right panels, respectively). Each point of the
predictions corresponds to the composition of a mass shell, and different
colors and symbols represent different initial masses and ashes, respectively
(as indicated in the legend). Note that the symbols for the models are only
plotted when the mass regions is O-rich, the relevant condition for the
formation of chromite grains

. The solar values are shown as black dashed lines. The black boxes around the
most anomalous grain 2_37 represent a qualitative estimate of nuclear physics
uncertainties as described in the text. The middle row is the same as the top
row, except that the abundance of the radioactive 53Mn is not decayed into
53Cr.

In Figures 3 and 4 we compare the presolar chromite data of Nittler et al.
(2018) to the three data sets of the CCSN model predictions. The predicted Cr
ratios as shown in Figures 3 and 4 vary over orders of magnitude, as the
different Cr isotopes are created and destroyed in different regions of the
progenitor and its explosion (see figures in Appendix A and Table 2). Each
data point of the CCSN data sets in Figures 3 and 4 corresponds to one
numerical zone in a model, and we do not allow for mixing between zones. Only
a few mass shells within each model can reach the stardust data points and are
also O-rich (symbols in the figures), the condition necessary to form the
chromite grains, as opposed to C-rich (not shown in the figures). We focus our
discussion on finding a possible region that has a composition that matches
the grain 2_37 (Nittler et al., 2018), which has the most anomalous 54Cr/52Cr
and 50Cr/52Cr ratios of the grains in this data sets. Less extreme values may
be explained by dilution effects due to mixing with less processed material in
the outer layers of the star, or with material in the interstellar matter
(ISM, see e.g., Zinner, 2014). First, we consider if a possible exact match of
the models to the composition of 2_37 exists, and second, we take into
consideration in the discussion some of the uncertainties due to nuclear
physics. These uncertainties are represented in Figures 3 and 4 by the boxes
around grain 2_37 and are described in detail below.

We start by considering the top and bottom panels of Figures 3. For the three
LAW models, the O-rich regions that can match the 54Cr/52Cr ratio of 2_37 are
the He-ashes and C-ashes (triangles and crosses in the top left panel of
Figure 3). Between these two compositions, the He-ashes are located close to
the required solar value of the 53Cr/52Cr ratio, while the C-ashes provide a
wide range of values for 53Cr/52Cr. The C-ashes, however, match the 50Cr/52Cr
ratio of 2_37, while the 50Cr/52Cr ratio of the He-ashes are at least one
order of magnitude lower than in 2_37 (bottom left panel of Figure 3). Note,
that while the He-ashes of the 20 M⊙ match both the 54Cr/52Cr and 53Cr/52Cr
ratio of the grains, the 50Cr/52Cr is almost three orders of magnitude too
low.

We reach the same conclusions when considering the three SIE models (middle
panels of Figure 3), although the match is slightly worse than for the LAW
models. When the 54Cr/52Cr ratio is matched in the He- and C-ashes, the
53Cr/52Cr ratio is at least 50% higher than the solar ratio. When the
53Cr/52Cr ratio in the C-ashes is equal to the solar ratio, the 54Cr/52Cr
ratio is lower than observed in 2_37. For the 50Cr/52Cr ratio, as in the case
of LAW, the 2_37 data point can only be reached in the C-ashes. The Ne-ashes
of the 20 M⊙ model reach values larger than in 2_37 in the 50Cr/52Cr (3 times
larger than in 2_37) and 53Cr/52Cr ratio (4 times larger than in 2_37).
54Cr/52Cr on the other hand is at least 2 times lower than in 2_37.

The Cr yields of the RIT models differ significantly from LAW and SIE (see
Section 3). The C-, He- and H-ashes of the RIT 25 M⊙ model reach the 2_37
54Cr/52Cr ratio, but only for the He- and H-ashes is the 53Cr/52Cr ratio also
matched. None of the mass shells in this model reach the 50Cr/52Cr ratio of
2_37. The C-ashes of the 20 M⊙ model reach the 2_37 54Cr/52Cr ratio, while the
53Cr/52Cr ratio is two times higher than the solar value observed in the
grain. As for the 50Cr/52Cr ratio the He-ashes within the 20 M⊙ model have a
composition very close to the grains, while the other parts of the ejecta do
not. In the case of the 15 M⊙ model we do not find any region that match the
grain Cr composition.

As mentioned in Section 2.4, in the figures so far we have always presented
results for abundances after radioactive decay. However, to exclude the
radiogenic contributions can make a difference in the case of 53Cr because of
the decay of 53Mn. The time between the CCSN and the formation of grains is
currently unknown (see e.g., Sarangi & Cherchneff, 2015). If the grains are
created long enough after the explosion for all radioactive isotopes to decay
and/or if the radioactive isotopes behave chemically in the same way as their
daughter, then they are incorporated into the grains and contribute therein to
the abundance of the stable isotopes and the decayed results apply. However,
53Mn has a relatively long half life of 3.74 Myr so it might be present in the
grains, Mn is more volatile than Cr (Lodders, 2003), and Mn does not
constitute as a major element in the spinel structure of the refractory Cr-
oxide grains (Dauphas et al., 2010). Therefore, we also show in the middle row
of Figure 3 the predicted isotopic ratios for the case that the dust grains
have formed before 53Mn has decayed888None of the other isotopic ratios
discussed here present an effect due to radiogenic decay, except for the
57Fe/56Fe ratio, which only shows minor differences in the most central O-rich
regions.. When comparing the top and middle panels of Figure 3 we see that the
radiogenic contribution from 53Mn generally does not affect the ratios in
regions that are relevant for the comparison to the grains, except for the Ne-
and C-ashes in all the SIE models and the 20 M⊙ RIT model. In some of the mass
shells within these regions, the non-decayed 53Cr/52Cr ratios are smaller by
factors of a few relative to the decayed values, and closer to 2_37.

Figure 4: Comparison of the Cr isotopic ratio and Ti isotopic ratio as
measured by Nittler et al. (2018) and as predicted isotopic ratios by the
three CCSN data sets. The 50Ti/48Ti ratio is shown instead of 50Cr/52Cr, to
investigate whether the mass 50 measurements are due to 50Ti or 50Cr. Colours
and symbols are as in Figure 3. The yellow 50Ti/48Ti data points of Nittler et
al. (2018) are calculated using the solar value for 50Cr/52Cr.  Figure 5: Cr
and Ti isotopic ratios of all CCSN prediction (from top to bottom: 15 M⊙, 20
M⊙, and 25 M⊙) after the explosion. The red regions are C-rich, and the grey
and white bands indicate the different burning phases, following the
nomenclature of Section 2.5.

An ambiguity in the data of Nittler et al. (2018) is that these authors were
unable to distinguish between 50Cr and 50Ti at atomic mass A=50999The authors
also checked for 50V and found that there is no V present in the grains.
Therefore, it is unclear whether the excess (above the solar ratio) of
50Cr/52Cr ratio in the five grains, is due to 50Cr or 50Ti. For this reason,
we also compare our models in relation to the 50Ti/48Ti ratio, see Figure 4.

None of the three LAW models is able to reach the 50Ti/48Ti ratio as found in
2_37. However, the C-ashes of the 15 M⊙ model and the He-ashes of the 20 M⊙
model are only about a factor of three too low. The 15 M⊙ SIE model reaches
the 2_37 value in its C-rich He-ashes. The other SIE models only approach the
50Ti/48Ti of 2_37, with either their He- or C-ashes. The shell merger region
of the 15 M⊙ RIT model is able to reach the 50Ti/48Ti value of 2_37, however,
the 54Cr/52Cr ratio in that region is at least two orders of magnitude less
than in 2_37. The 20 M⊙ RIT model also approaches the 50Ti/48Ti value of 2_37,
but again the 54Cr/52Cr ratio is too low.

It is clear from the above analysis that it is impossible to identify large
regions within the CCSN models that match three of the four isotopic ratios of
the 2_37 grain shown in Figures 3 and 4. Therefore, we now try to identify
regions that match the grain when taking into consideration uncertainties due
to nuclear physics. In Figures 3 and 4, the grain 2_37 abundances are plotted
within boxes. We defined these boxes as a reasoned qualitative estimate of the
total uncertainty combining both the grain measurement error and the nuclear
physics uncertainties affecting stellar model predictions, as discussed in
more detail in Section 5.1.

Specifically, the different axes of the boxes are set as follows:

  * •

50Cr/52Cr: 50Cr is mainly created in explosive O- and Si-burning (see Table
2), and these burning phases are typically not affected by nuclear physics
uncertainties101010See, for example, the sensitivity study by Parikh et al.
(2013) for Type Ia supernovae, where these processes are active, which shows
that 50Cr in not affected by reaction rates variations. For the 50Cr/52Cr box
we therefore use the measurement of 2_37 and its error bar: 0.317 $\pm$ 0.033.

  * •

50Ti/48Ti: the reaction rate tests in Section 5.1 shows that this ratio can
change from $\sim$4 to $\sim$15 when considering neutron-capture rate
uncertainties (i.e., a factor $\sim$3.7) in the region close to the 2_37 value
(Figure 8). Therefore, we use this factor to extend the lower error bar of the
lowest value of 50Ti/48Ti in 2_37, which is 27. The lower limit of the box is
thus 7.2. For the upper limit of the box we use the upper error bar of the
value of 2_37.

  * •

53Cr/52Cr: we used the same reasoning as for 50Ti/48Ti. The uncertainty factor
resulting from the reaction rate tests in Section 5.1 is $\sim$2 (Figure 7).
We use this factor to extend the upper error bar of 2_37, as the model data is
located at higher values.

  * •

54Cr/52Cr: for this ratio we extended both the upper and lower error bars of
2_37, as the model data is located at both at higher and lower values than the
value of 2_37. The total variation is of a factor of $\sim$2 (Figure 7).

Note that for sake of simplicity we did not consider possible effects on 52Cr.
This isotope is at denominator of all the isotopic ratios, therefore, changing
its abundance would shift all the ratios by the same factor, resulting in a
straight line passing through 2_37, rather than a box.

In Table 3, we report the mass coordinates of the predicted model ratios which
are located within the boxes as shown in Figures 3-4, and using Figure 5 to
identify the mass coordinates. We also list the ashes in which these mass
coordinates are located, for both the decayed or non-decayed cases. Finally,
we list the overlap regions considering mass 50 as either Cr or Ti, and these
regions are indicated in Figure 9 with red dots.

Table 3 shows that for all models we are able to identify a region in which
the predicted ratio can be found within the box of 54Cr/52Cr vs 53Cr/52Cr.
However, not all the models reach one of the other two boxes, which include an
atomic mass 50 isotope. For five models an overlap between the 54Cr/52Cr vs
53Cr/52Cr mass range and at least one of the atomic mass 50 boxes can be
found. The box around 50Ti is larger than the box around 50Cr due to the
stronger sensitivity to reaction rate uncertainties. We find a higher number
of overlap regions for the box around 50Ti (4) than for the box around 50Cr
(2). We note that while an update of the reaction rates that affect the
50Ti/48Ti ratio could lead to smaller boxes and thus a lower number of overlap
regions, the location of these regions would not change. Specifically, the
overlap regions that involve 50Cr/52Cr are always located within the C-ashes,
while for 50Ti/48Ti they are located in the He-ashes and one in the C-ashes.
Based on this analysis, the CCSN He-ashes and the C-ashes are both possible
sites of origin for the Cr-rich grains. In the case of the C-ashes, it would
most likely represent 50Cr. In the case of the He-ashes, the signal at atomic
mass 50 would most likely represent 50Ti. This is in agreement with Table 2,
in which we show that 50Cr is produced in the C-ashes, while it is destroyed
in the He-ashes.

In all three LAW models we have identified overlap regions, as well as in the
15 and 25 M⊙ SIE models. We were unable to do this for the 20 M⊙ SIE model,
likely because the temperature is higher in the region in the 20 M⊙ SIE model,
where the 54Cr/52Cr ratio falls within the box around 2_37, than in the same
region in the 20 M⊙ LAW model. In none of the RIT models were we able to
identify overlap regions. In the 15 M⊙ model the reason for this is that the
shell-merger and the explosive He-burning due to the temperature peak produce
the Cr-isotopes in different ratios than in the other models. These two
processes take place in the regions where the overlap is found in the LAW and
SIE 15 M⊙ models. The 20 M⊙ RIT model experiences higher temperatures in the
C- and He-ashes during the explosion, see Figure 15, also leading to different
Cr-isotopic ratios. For the 25 M⊙ RIT model the main issue is the high mass
cut, which excludes those regions in the ejecta where we find the overlap
regions in the LAW and SIE models.

We also looked at the other models in the data set of Lawson et al
(submitted), as shown as boxplots in Figure 2, which include a variety of
values for the explosion energy and the mass cut. The 15 M⊙ models show little
variability of the relevant isotopic ratios, while the 20 M⊙ models show
differences in all isotopic ratios close to the mass cut. However, this region
does not match the Cr isotopic composition of 2_37. The variations in the 25
M⊙ models larger and present at more regions within the CCSN model. Most
differences between the models, however, are small and fall within the
uncertainty boxes in Figures 3 and 4, and therefore would not lead to more
overlap regions than the ones already listed in Table 3. The exception is that
several 25 M⊙ models with high explosive energies provide a new overlap region
as their 50Ti/48Ti ratio reaches into the 2_37 box within the He-ashes. This
overlap region does not alter our findings that the isotope at atomic mass 50
is likely 50Ti in the He-ashes, and 50Cr in the C-ashes.

Table 3: Mass coordinates (in M⊙) at which the predicted ratio is within the boxes in Figures 3 and 4. The reported ranges cover solutions derived both for the decayed and non-decayed abundances. The overlap is defined as the overlap between 54Cr/52Cr vs 53Cr/52Cr and either 50Cr/52Cr or 50Ti/48Ti vs 53Cr/52Cr. The location of the overlap is labelled by the ashes it is located in, and all overlap regions are indicated in Figure 9. |  54Cr/52Cr vs |  54Cr/52Cr vs |  54Cr/52Cr vs | overlap with | ashes
---|---|---|---|---|---
|  50Cr/52Cr |  53Cr/52Cr |  50Ti/48Ti |  |
LAW |
15 M⊙ | 2.02 | 2.02, 2.57 - 2.61, 2.74 - 2.78 | 2.58- 2.61, 2.77 - 2.78 |  50Cr: 2.02 | C
15 M⊙ |  |  |  |  50Ti: 2.58 - 2.61, 2.77 - 2.78 | He
20 M⊙ | - | 4.05 - 4.50 | 4.05 - 4.19, 4.47 - 4.50 |  50Ti: 4.05 - 4.19, 4.47 - 4.50 | He
25 M⊙ | 3.12 - 3.15 | 5.88 - 7.10 | 6.43-6.44 |  50Ti: 6.43-6.44 | He
SIE |
15 M⊙ | 1.87 - 1.88 | 1.87 - 1.89, 2.23 - 2.26 | 2.36 - 2.38 |  50Cr: 1.87 - 1.88 | C
20 M⊙ | - | 2.28 - 2.34, 2.44 - 3.75 | - | - | -
25 M⊙ | - | 3.13, 4.80 - 5.57 | 4.80 - 5.20 |  50Ti: 4.80 - 5.20 | C (, He)
RIT |
15 M⊙ | - | 3.04 - 3.06, 3.32 - 3.34 | - | - | -
20 M⊙ | 4.86 | 3.24 - 3.29, 5.61 - 5.64 | - | - | -
25 M⊙ | - | 6.54, 6.76 - 7.00 | - | - | -

The analysis above is based on comparison to the most anomalous grain 2_37,
and we justified this choice above by considering that less extreme values may
be explained by invoking some dilution effect due to mixing with less
processed material. However, it is interesting to check if the overall picture
above would significantly change if we aimed at matching the two grains that
are the second and third most anomalous in the 54Cr/52Cr ratios.

In the case of the LAW and SIE models, these grains could be matched by
considering the C-ashes of the 15 and 25 M⊙ models or the He-ashes of the 20
M⊙ models, and the He-ashes 15 M⊙ for the LAW model. In this cases the atomic
mass 50 is always only matched as 50Ti. The only difference between these two
sets of solutions is that in the case of SIE, only the non-decayed abundances
can match the two grains (as otherwise the addition of 53Mn produces too high
abundance at atomic mass 53), while in the case of LAW both the decayed and
the non-decayed predictions match the grains. Finally, these two grains can
also be matched by the composition of the Ne ashes (shell merger) of the RIT
15 M⊙ model, in which case the isotope at atomic mass 50 could be either 50Cr
or 50Ti. We note that all the solutions for all the three most anomalous
grains reported in this section require relatively narrow CCSN mass regions.
Other supernova studies that attempted to explain the presolar Cr-oxide data
share the same problem of localised grain condensation (see e.g. Nittler et
al., 2018; Jones et al., 2019a).

Furthermore, we note that as reported by Nittler et al. (2018) the 57Fe/56Fe
ratio of the grains is compatible within the error bar to the solar value
except for one grain called 2_81. The models can reproduce solar 57Fe/56Fe
ratios but only for very small specific ranges of mass coordinates, for
example, in the O- and C-ashes at mass coordinates 2.2 and 2.9 M⊙ for the LAW
25 M⊙ model. This would make it very difficult for such signature to be
predominant in the grains. However, the error bars on the 57Fe/56Fe ratios are
very large, of the order of the measured anomaly itself, because the overall
abundance of Fe in the grains is very low (Larry Nittler, private
communication). Therefore, we do not consider this as a strong constraint.

###  4.2 The 26Al signature in presolar C-rich and O-rich grains

In Figure 6 we show the 26Al/27Al ratio as predicted in the CCSN models. The
dashed line indicates the highest values of the inferred initial 26Al/27Al
ratios, inferred from the Mg isotope composition of presolar SiC-X and
graphite grains with CCSN origin (Zinner, 2014; Groopman et al., 2015). The
dotted line represents the estimated initial 26Al/27Al ratio of the Group 4
oxides that may also originate from CCSNe (Nittler et al., 2008).

None of the models of LAW in Figure 6 reach the maximum ratio measured in
(Groopman et al., 2015), and only the Ne-ashes (see Figure 5 for
identification of the ashes) of the SIE model with 25 M⊙ initial mass reach an
26Al/27Al ratio higher than the maximum measured in the stardust grains. The
higher ratios are also reached even deeper in the ejecta of the 15 M⊙ and 20
M⊙ SIE models. However, these regions of the ejecta are not C-rich and have a
very low absolute Al abundance. Therefore, including these layers in any
realistic mixing of stellar material coming from different regions of the CCSN
ejecta would not affect the final Al ratio in the resulting mixture. The RIT
models reach the maximum measured ratio in the H-burning ashes that are mildly
C-rich. Typical abundance signatures in C-rich grains from CCSNe, e.g. the
enrichment in 15N and 28Si, and the 44Ca-excess due to the radiogenic
contribution by 44Ti (see e.g., Amari et al., 1992, 1995; Besmehn & Hoppe,
2003) require some degree of mixing with other CCSN layers, where the 26Al
enrichment is lower. It is still a matter of debate which components of the
ejecta shape the mixtures observed in C-rich presolar grains. They could
either undergo extensive mixing with deeper Si-rich regions (e.g., from the so
called Si/S zone Travaglio et al., 1999; Liu et al., 2018a), or more localized
mixing between C-rich layers (e.g., Pignatari et al., 2013, 2015; Xu et al.,
2015). Some degree of contamination or mixing with isotopically normal
material without 26Al has to be expected. More in general, the isotopic
abundances from the RIT models would need to be compared directly with single
presolar grains, to check if the 26Al enrichment can be reproduced along with
other measured isotopic ratios (e.g., Liu et al., 2016; Hoppe et al., 2018).

Pignatari et al. (2015) showed that the ingestion of H in the He-shell of the
massive star progenitor shortly before the onset of the CCSN explosion could
potentially provide enough 26Al to reproduce the most 26Al-rich grains. None
of the models considered in this work have developed late H ingestion events,
and therefore we cannot fully explore the impact of these events in our study.
While H-ingestion in CCSN models has been identified in stellar simulations
since a long time (e.g. Woosley & Weaver, 1995), the quantitative impact of
these events on the nucleosynthesis production is still poorly explored and
there are large uncertainties. This is also due to the intrinsic difficulty of
one-dimensional models to provide robust predictions for these events (see,
e.g., the discussion in Pignatari et al., 2015; Hoppe et al., 2019). We thus
confirm that reproducing the high 26Al/27Al ratios in C-rich grains is a still
a major challenge for modern nuclear astrophysics.

Nevertheless, based on previous works we can qualitatively expect that if
H-ingestion and a following explosive H-burning take place, the neutron burst
in the He-shell material will be mitigated compared to models without
H-ingestion (e.g., Pignatari et al., 2015; Liu et al., 2018b). Therefore, the
isotopes that are created in this region via neutron-captures relevant for
this work, which include 25,26Mg and 53,54Cr (see Table 3) as well as 48,50Ti,
may be produced with smaller efficiency. In this case, the resulting
nucleosynthesis might affect our overlap regions in Table 3. We can speculate
that the reduced 53Cr/52Cr and 54Cr/52Cr ratios could potentially affect the
possibility for an overlap region to exist in the He-shell, depending on the
exact remaining abundance of these isotopes. This aspect will need to be
studied in the future, possibly using a new generation of massive star models
informed by multi-dimensional hydrodynamics simulations of H ingestion (e.g.,
Clarkson & Herwig, 2021).

Nittler et al. (2008) concluded that 4 of their 96 analysed presolar oxide
grains originated from CCSNe. The dotted line in Figure 6 is the maximum of
the 26Al/27Al ratio of those four grains. All nine CCSN models shown in Figure
6 reach this maximum value in an O-rich region. In the LAW models the dotted
line is reached for the 15 M⊙ in the C-ashes. The 20 M⊙ and 25 M⊙ model reach
the dotted line in the He-ashes. The regions of the SIE models that reach the
dotted line are for the 15 M⊙ the He-ashes and the H-ashes, for the 20 M⊙
model the inner C-ashes, and for the 25 M⊙ the Ne-ashes. In the RIT models,
the 15 M⊙ model reaches the limit of Nittler et al. (2008) in the H-ashes, the
20 M⊙ model in the C-ashes, and the 25 M⊙ in the outer He-ashes.

Therefore, in the case of these four presolar oxide grains that are assumed to
have originated in CCSNe, there are extended O-rich regions consistent with
the measured 26Al enrichment. Thus, local or more extended mixing of different
stellar layers may potentially match the observed 26Al/27Al ratio. The O
isotopic ratios reported in Nittler et al. (2008) of these grains, however,
are only reached in the envelope. Further analysis of all isotopic ratios
obtained from these four grains is needed to conclude their region of origin.

Figure 6: The 26Al/27Al ratio is shown for the sets of stellar models, while
the dashed line is the upper limit of SiC-X and Graphite data by Groopman et
al. (2015) and the dotted line the maximum limit of the four group 4 grains of
Nittler et al. (2008). The thick line segments indicate the carbon rich
regions and the thin line segments the oxygen rich regions.

We note that while so far only the Group 4 oxides have been suggested to
originate from CCSNe, recently Hoppe et al. (2021) showed that some silicates
from Group 1 and Group 2 could also be compatible with a CCSN origin, based on
a comparison of their high 25Mg/24Mg ratios to CCSN models affected by the H
ingestion. More investigations are needed to define the full range of 26Al
enrichment and 26Mg abundance signatures in all oxides and silicate grains
with a possible CCSN origin.

##  5 Discussion

###  5.1 Effects of uncertainties in neutron-capture reaction rates on the Cr
and Ti ratios

By considering three different data sets of stellar models, we have derived a
qualitative estimate of the effect of stellar physics uncertainties and
different computational approaches. This, however, does not provide us with a
systematic way to check the effect of nuclear uncertainties. We have
considered these separately and we present them here.

As discussed in Section 3.1, the main channel of production of 53Cr and 54Cr
in regions where the chromite grains potentially originated from, are neutron
captures on other Cr isotopes. The final abundances of 53Cr, 54Cr, 48Ti, and
50Ti after a given neutron flux episode are controlled mostly by their
neutron-capture rates. To test how variations in these rates affect the Cr and
Ti isotopic ratios, we preformed several dedicated tests using the MESA
stellar evolution code, version 10398 (Paxton et al., 2011, 2013, 2015, 2018).
We used the settings for the massive star as described in Brinkman et al.
(2021) and considered models with an initial mass of 20 M⊙ with Z=0.014
evolved up to the core-collapse. We choose this progenitor model for our tests
because the explosion has no significant impact on the abundances in the
regions relevant for our analysis. The supernova explosion was not included in
these tests, which is justified by the fact that the Cr isotopes in the C and
He ashes are more significantly affected by the progenitor evolution than by
the explosion, as shown in Appendix A.

We multiplied the neutron-capture reaction rates of interest by different
constants, as indicated in Table 4. We choose variations in the direction that
would help the models provide a better match to the most anomalous grain and
we varied the rates by up to a factor of 2. This is larger than the up to 50%
uncertainty at 2$\sigma$ reported for the recommended values in the KaDoNiS
database111111See https://kadonis.org/ V0.2 (Dillmann et al., 2006, and
therefore in the JINA reaclib database, which uses KaDoNiS). However, these
reactions were measured several decades ago: these current recommended values
are from Kenny et al. (1977) for the Cr isotopes, from Allen et al. (1977) for
48Ti, and from Sedyshev et al. (1999) for 50Ti. Therefore it is possible that
systematic uncertainties are much higher than the reported uncertainty.

Table 4: Factors used to multiply the indicated reaction rates from their standard values in the 20 M⊙ models considered in this section. |  53Cr(n,$\gamma$)54Cr |  54Cr(n,$\gamma$)55Cr
---|---|---
Model 1121212Using the values of the KaDoNiS database (Dillmann et al., 2006), which produces results very similar to those by LAW and SIE. | 1 | 1
Model 2 | 1.5 | 1
Model 3 | 2 | 1
Model 4 | 1 | 0.5
Model 5 | 2 | 0.5
|  48Ti(n,$\gamma$)49Ti |  50Ti(n,$\gamma$)51Ti
Model 6 | 2 | 1
Model 7 | 1 | 0.5
Model 8 | 2 | 0.5

Figures 7 and 8 show the results for the Cr and Ti isotopic ratios,
respectively, in the C-ashes and He-ashes, which are the two possible sites of
origin for the grains as described in Section 4. In the case of the Cr
isotopic ratios, two expected main trends are visible: (i) in the models with
an enhanced 53Cr(n,$\gamma$)54Cr rate only (Models 2 and 3) the 53Cr/52Cr
ratio decreases relative to the standard Model 1, for example from a maximum
in the He ashes around 0.16 to a minimum of 0.07, i.e., roughly a factor of 2;
(ii) in Model 4, with the reduced 54Cr(n,$\gamma$)55Cr rate, the 54Cr/52Cr
ratio increases relative to Model 1, for example in the C-ashes from $\sim$1
to $\sim$2\. In the combined test (Model 5), the 54Cr/52Cr ratio increases
further to $\sim$3 in the C-ashes compared to Model 1. Although these tests
are only meant to provide a basic estimation of the impact of nuclear
uncertainties, we can already derive that the uncertainties of the neutron-
capture rates of Cr isotopes have a significant impact on stellar
calculations. Therefore, new measurements of these neutron-capture rates are
needed to reduce the uncertainty of the model predictions.

When considering the results of the Ti tests, we find that increasing the
48Ti(n,$\gamma$)49Ti reaction rate only (Model 6) leads to an increase of the
50Ti/48Ti ratio. Decreasing the 50Ti(n,$\gamma$)50Ti reaction rate only (Model
7) does not have a significant effect, because 50Ti is a magic nucleus and
therefore has a very low neutron-capture cross section in both the two nuclear
reaction setups. As a consequence, when both rates are changed in Model 8, the
result is very similar to Model 6. We also tested the case for Ti with the
rates multiplied and divided by 1.5 instead of 2, the results are very similar
to those obtained by using the factor or 2. The result of these reaction rate
tests concerning the Cr and Ti isotopes are used to define the boxes in
Figures 3 and 4 as described in Section 4.1.

We did not test the impact of the nuclear uncertainties affecting the
production of neutrons. In both the He-ashes and the C-ashes the
22Ne($\alpha$,n)25Mg reaction is the main neutron source. The impact of its
present uncertainty on He-burning and C-burning nucleosynthesis is well
studied (e.g., Kaeppeler et al., 1994; Heger et al., 2002; Pignatari et al.,
2010). A more precise definition of the competing $\alpha$-capture rates
22Ne($\alpha$,n)25Mg and 22Ne($\alpha$,$\gamma$)26Mg at relevant stellar
temperatures is an open problem of nuclear astrophysics and an active line of
research for many years (e.g., Longland et al., 2012; Talwar et al., 2016;
Adsley et al., 2021).

Figure 7: Cr-isotopic compositions resulting from the five 20 M⊙ models
calculated using different Cr neutron-capture rates, as listed in the top half
of Table 4. The reference model is Model 1. As in Figure 3 the yellow points
are the grains from Nittler et al. (2018). Figure 8: Same as Figure 7 but for
the 50Ti/48Ti ratio resulting from the four 20 M⊙ models calculated using
different Ti neutron capture rates, and reference model Model 1. The rates
used in these tests are listed in the bottom half of Table 4.

###  5.2 Al and Mg composition of the CCSN regions as candidate sites of
origin of the chromite grains

Figure 9: Ratios of 26Al/27Al, 25Mg/24Mg, and 26Mg/24Mg (all decayed except
for 26Al) for our three CCSN data sets after the explosion as function of the
mass coordinate. The 15, 20, and 25 M⊙ models plotted in the top, middle, and
bottom panels, respectively. The white, grey, and red bands represent the same
regions as in Figure 5. The blue and green dashed lines represent the solar
values for the 25Mg/24Mg and 26Mg/24Mg ratios, respectively (with values from
Bizzarro et al., 2011). The red circles, plotted on the on the dashed lines
for sake of visibility, correspond to the mass coordinates where the Cr and/or
Ti isotopic ratios of the chromite grains by Nittler et al. (2018) match the
ratios predicted by the models, as listed in the “Overlap” column in Table 3.

Here we investigate the link between the Al and Mg isotopic ratios and the
54Cr enrichment in the chromite grains, because of the significance of the
apparent correlation between 54Cr and 26Mg among planetary objects. As
mentioned in the Introduction, our method in this and in the following
subsection is valid only under the assumption that Al and Mg abundances are
carried in the chromite grains and/or similar carriers enriched in Al and
produced in the same region of the chromite grains. While the volatility of Cr
and Mg in an O- and Cr-rich CCSN environment is poorly constrained, at least
under early Solar System conditions they might be comparable: both elements
start condensation in the spinel phase and their major host phases, although
different, have similar 50% condensation temperatures, Lodders (2003).

In Figure 9 we show the 26Al/27Al, 25Mg/24Mg, and 26Mg/24Mg ratios as function
of the mass coordinate of the three CCSN data sets. We also highlight the
overlap regions listed in Table 3 as red dots, which represent the stellar
zones where the Cr-composition of the chromite grains is matched. The Al and
Mg isotopic ratios at the location of red dots in Figure 9 are therefore
expected to reflect the nucleosynthetic signature of these two elements in the
chromite grains. This nucleosynthetic signature may also allow us to determine
if the excess at atomic mass 26 observed in the Solar System material to
accompany the 54Cr excess (Larsen et al., 2011) is due to a 26Al and/or a 26Mg
excess.

As discussed in detail in Section 4.1, the two main regions of interest for
the origin of the chromite grains are the C- and the He-ashes, which is where
the red dots in Figure 9 are located. Specifically, for the LAW set, these are
the He-ashes and the centre of the C-ashes in the 15 M⊙ model, the He-ashes in
the 20 M⊙ model, and the inner C-ashes in the 25 M⊙ model. For the SIE models,
the red dots are location in the centre of the C-ashes in the 15 M⊙ model, and
the inner C-ashes in the 25 M⊙ model. In the RIT models there are no overlap
regions for the most anomalous grain. However, if we consider the second and
third most anomalous grains, the region between 2 and 3 M⊙ for the enclosed
mass in the 15 M⊙ RIT model provides a possible match. The composition of this
region is similar to the C-ashes of the SIE 25 M⊙ model, therefore in the
following we do not discuss it separately.

We remind the reader that these CCSN mass regions appear to be relatively
narrow as we identified them in Section 4.1 by trying to match specifically
the most anomalous observed presolar Cr-oxide grain, without mixing with
material of a different composition. While there is observational evidence
that the composition of the ejecta can be asymmetric (e.g., Höflich, 2004),
mixing within the supernova remnants is still poorly understood. Studies of
high-density graphite grains and SiC grains of Type X suggested that small
scale mixing between different inner and outer region of a supernova must
occur to explain nucleosynthetic signatures typical of the inner layer (such
as the initial presence of radioactive 44Ti and excess in 28Si), together with
signatures from the outer layers, such as the He shell (Travaglio et al.,
1999; Yoshida, 2007). However, Pignatari et al. (2013) matched the grains
without invoking this mixing with the composition that is produced by the
effect of increasing the energy of the explosion on the He-shell. In addition,
Schulte et al. (2020) argue that the CCSN ejecta (especially the material
coming from the inner most regions of the massive star) is too energetic to
condense prior to mixing with the cold interstellar medium. At the location of
the red dots in the He-ashes in Figure 9, the Mg isotopic ratios are roughly a
couple of orders of magnitude higher than their solar values, because 25Mg and
26Mg are produced by the operation of the 22Ne+$\alpha$ reactions. This means
that even if some 26Al is present here, it will not influence the total sum of
26Mg and 26Al. Furthermore, 26Al is mainly destroyed in the He-ashes by the
neutron capture reactions 26Al(n,p)26Mg and 26Al(n,$\alpha$)23Na. In the
C-ashes, the Mg isotopic ratios are typically below their solar values in the
inner part, and above solar in the outer part, with the switch being model
dependent. In the SIE 15 M⊙ model, they are below their solar values in the
whole C-ashes. This is due to the fact that 24Mg is one of the primary
products of C burning, therefore the Mg isotopic ratios 25Mg/24Mg and
26Mg/24Mg decrease towards their solar values. Subsequently, in the inner part
of the C-ashes during the explosion is 24Mg not only strongly produced, but
also 25,26Mg are destroyed via proton captures leading to the production of
26Al. Most of the red dots in the C-ashes are located in the region of the
C-ashes where the Mg isotopic ratios are below their solar values. The
exception is the 25 M⊙ model of SIE where the red dots are located at mass
coordinate 4.8-5.2 M⊙, which corresponds to Mg isotopic ratios a factor of a
few higher than their solar values. We note that these red dots are the only
ones in the C-ashes that match the 50Ti/48Ti ratio.

In summary, the CCSN models predict 54Cr enrichment (as signalled by the
presence of the red dots) together with stable 25Mg and 26Mg excesses in the
He-ashes, while in the C-ashes, both 25Mg/24Mg and 26Mg/24Mg can be either
higher or lower than their solar values. In the next section we compare these
findings to planetary materials. We also take into consideration the possible
radiogenic contribution of 26Al to 26Mg.

###  5.3 Expected isotopic variations in planetary materials

Figure 10: For the LAW 15 M⊙ model we show: Left panel: the $\delta$25Mg
(blue), $\delta$26Mg (black), and $\delta$26Mg* (green) values (where
26Mg*=26Mg+26Al) assuming no mass fractionation of the CCSN ejecta. We also
show the composition of an ejecta enriched in Al, $\delta$26Mg*E (purple
dashed lines), where the Al/Mg ratio is set to 2, comparable to the abundances
measured in colloidal presolar chromite grains (Dauphas et al., 2010)

. The ratios are expressed as $\delta$-values, corresponding to their
deviation to the terrestrial ratio in per mil (see text). Right panel:
$\delta$26Mg’ and $\delta$26Mg*’, which are the internally normalised values
by assuming maximum mass-dependent fractionation of the CCSN ejecta, i.e., the
25Mg/24Mg ratio is set to the terrestrial value making $\delta$25Mg’ equal to
0. This natural mass-dependent fractionation is corrected for by the
exponential mass fractionation law (see, e.g., Bizzarro et al., 2011).

Figure 11: $\mu$54Cr’ and $\mu$26Mg*’ values from meteoritic data (colored
symbols with 2$\sigma$ error bars, see legend, from Larsen et al., 2011) and
predicted trajectories (lines) of $\mu$54Cr’ vs $\mu$26Mg*’ (top panels) and
$\mu$54Cr vs $\mu$26Mg* (bottom panels) obtained by mixing between a solar
component and the CCSN ejecta of the regions identified in Section 4.1 (i.e.,
the red dots) for the LAW and SIE models (left and right panels, respectively)

. The colors of the mixing lines represent models for different stellar masses
as before, i.e, black, blue, and green are used for the 15, 20, and 25 M⊙,
respectively. The C-ashes are indicated by the dotted lines, and the He-ashes
by the solid lines. The dashed and dotted-dashed lines were calculated with
Al- and Cr-enriched composition in the He and C ashes, respectively. Top
panels: The mass independent isotopic composition $\mu$54Cr’ and $\mu$26Mg*’
values of the CCSN ejecta were calculated by setting the 52Cr/50Cr and
25Mg/24Mg ratios to the NIST979 and DSM3 terrestrial standard values for Cr
and Mg, respectively (Bizzarro et al., 2011; Qin et al., 2010). To follow the
data reduction of meteorite measurements, we applied the exponential law to
correct for mass fractionation (Russell et al., 1978). Bottom panels: The CCSN
ejecta are assumed to retain its isotopic composition, i.e., mass-dependent
isotope fractionation is negligible. $\mu$54Cr and $\mu$26Mg* are calculated
by simple deviation of their isotopic ratio values from the terrestrial
standard values in ppm units without internal normalisation.

Here we compare the expected Cr and Mg isotopic compositions of the CCSN
regions whose abundance composition match that of the chromite grains, as
identified in Section 4.1, to the Cr and Mg anomalies identified in planetary
materials. We start with converting the predicted CCSN ejecta into commonly
used variables in cosmochemistry. Then, we present mixing trajectories
calculated with the CCSN ejecta and the solar composition, and compare these
to the meteoritic data.

####  5.3.1 Converting CCSN model data to normalized isotope ratios

In the following we express the ratio of the abundance $N$ of isotope $i$ to
isotope $j$ $(^{i}N/^{j}N)$ from models or measurements as part per mil
($\delta$) or per million ($\mu$) deviations from the terrestrial standard
(TS):

$\delta(\mu)^{i}N=\left(\frac{(^{i}N/^{j}N)_{\rm
sample/model}}{(^{i}N/^{j}N)_{\rm TS}}-1\right)\times 10^{3}\,(\times
10^{6}).$ (1)

Mg isotopic ratios of planetary materials are routinely measured with precise
correction for instrumental mass-dependent fractionation (IMF) using the
standard “bracketing method” (Galy et al., 2001). These IMF corrected values
can be interpreted as the true values of the studied samples. They should
reflect both the original nucleosynthetic mass-independent (which we indicate
as $\delta$26Mg∗) make-up of the analysed materials and all the physical
processes that led to natural (as opposed instrumental) mass-dependent isotope
fractionation of the sample during its chemical history.

Unfortunately, the extent of the natural mass-dependent isotope fractionation,
which we need to remove in order to obtain the original nucleosynthetic
signature $\delta$26Mg∗, is not precisely known (see e.g., Wasserburg et al.,
1977). For meteorites and planetary samples, it is generally assumed that all
the 25Mg/24Mg deviation from the solar values as shown by the IMF corrected
values is caused by natural mass-dependent fractionation. We note that the
deviations from the solar 25Mg/24Mg value are small (on % level). The
normalisation accounts for the maximum possible natural mass fractionation
allowed by the data. The 26Mg/24Mg ratio is therefore corrected for natural
mass-dependent fractionation by using the exponential fractionation law (Galy
et al., 2001) and setting the 25Mg/24Mg ratio to the terrestrial value. This
is referred to as internal normalisation which results in a $\delta$26Mg*’,
identified as the remaining nucleosynthetic mass-independent anomaly. This is
calculated as $\delta$26Mg*’=$\delta$26Mg - $\delta$25Mg/$\beta$, where
$\beta$ is the exponent of mass fractionation (see e.g. Bizzarro et al.,
2011). Finally, we note that the original mass-independent $\delta$26Mg* (and
$\delta$26Mg*’) should reflect both the contribution from the nucleosynthetic
26Mg and the production of radiogenic 26Mg by now extinct 26Al, which is also
produced by nuclear reactions in the star.

We note that in case of Cr, meteoritic and planetary data is obtained via
thermal ionization mass spectrometry using internal normalisation, where
instrumental and natural mass fractionation are corrected together and
therefore cannot be distinguished.

A problem arises when we wish to compare model predictions to meteoritic data
and convert the CCSN yields to internally normalized $\delta$ values. The
issue is that the stellar 25Mg/24Mg or 50Cr/52Cr ratios are almost never equal
to the terrestrial values (see Figure 9), and that some of the nucleosynthetic
sites that can produce the chromite grains, the 25Mg/24Mg ratios differ from
their solar value by up to 3 orders of magnitude. Therefore, if we apply
internal normalisation using the terrestrial 25Mg/24Mg value to obtain the
$\delta$26Mg*’ of the ejecta, we automatically imply that any deviation from
the terrestrial value is due to mass fractionation, which is clearly not the
case. There are two options to consider: (i) we apply internal normalization
in order to treat the data the same way as in case of laboratory measurements
(see Dauphas et al., 2004) or (ii) we take the model results as the true
values of the ejecta and assume no natural mass fractionation, i.e., the
isotope ratio used for normalization is not taken as the terrestrial value.

In Figure 10 we show the $\delta$-value representation of the Al and Mg
isotopic ratios of the LAW 15 M⊙ model (see Figure 9, left top panel) as an
example. We show both $\delta$26Mg (calculated only considering the
contribution of 26Mg at atomic mass 26) and $\delta$26Mg∗ (calculated
considering the contributions of both 26Mg and 26Al at atomic mass 26), to
highlight the impact of the abundance of 26Al on the total mass budget at
atomic mass 26. We show the two options above to convert the CCSN ejecta: the
$\delta$ values are calculated (i) in the right panel, assuming maximum mass
fractionation by setting the 25Mg/24Mg ratio to its terrestrial standard value
(DSM3 standard, see Bizzarro et al., 2011) and (ii) in the left panel,
assuming no mass fractionation of the ejecta.

This figure illustrates how the amplitude of isotope variations changes when
using $\delta$ values instead of simple isotope ratios as in Figure 9. Two
main effects are visible: (i) when the iMg/24Mg ratio is lower than the
terrestrial value, the ratio in Eq. 1 becomes negligible and the
$\delta$-value approaches $-10^{3}$ (the $\delta$-scale is not linear, see Eq.
1), e.g., as in the mass range below 2.1 M⊙ in the left panel, and (ii) the
internally normalised $\delta$iMg’ values (i.e., when setting 25Mg/24Mg to the
terrestrial value) magnify anomalies with respect to the 25Mg abundance (right
panel), as this is again a non-linear transformation of data because of the
exponential fractionation law. Overall, the impact of 26Al at atomic mass 26
is not significant at the location of the nucleosynthetic sites of our
interest (at the red dots, where the black and green lines overlap).

For the 15 M⊙ LAW model, the only location in the star where there is a
difference between $\delta$26Mg’ and $\delta$26Mg*’ is the inner C-ashes,
where the strong depletion of 26Mg accompanied by the enhancement of 26Al
generates a separation between the $\delta$-values calculated using 26Mg only
(green line) or using 26Mg+26Al (black line). However, no red dots are present
in these regions of the 15 M⊙ LAW model, therefore, the 54Cr-rich grains are
not matched here. A similarly strong contribution of the 26Al abundance
relative to the 26Mg abundance at atomic mass 26 in the calculation of the
$\delta$-values shown in Figure 10 only develops in the Ne-ashes, at a mass
coordinate of about 1.8 M⊙. We also checked the behaviour of the other models
and found that the contribution of 26Al to atomic mass 26 also becomes
relevant in the Ne-ashes for the 25 M⊙ LAW model and all SIE models, as well
as in the shell-merger region of 15 M⊙ RIT model, i.e., in regions that did
not produce the composition of the chromite grains. We found one candidate
site in a more central region of the C-ashes in the 25 M⊙ SIE model which
shows 25,26Mg/24Mg ratios higher than the solar value, and while the 26Al
production is ongoing, its abundance relative to 26Mg remains insignificant.

In addition, we show an example of a more likely scenario, where we calculate
an Al-enriched $\delta$26Mg*E (purple dashed lines in Figure 10) using an
Al/Mg=2 ratio, similar to the value reported by Dauphas et al. (2010). This
calculation better represents an ejecta rich in refractory oxide phases. We
find that this enrichment does not play a significant role, except in the case
of the C-ashes when the data is internally normalised (right panel of Figure
10, where the purple dashed line peaks at around 2.05 M⊙). In the regions of
interest here (the red dots), instead, the maximum contribution of 26Al to the
total mass at atomic mass 26 even in this enriched case corresponds to an
increase of at most 50%.

####  5.3.2 Mixing trajectories

In Figure 11 we show the predicted trajectories of two-component mixing
between the particular sites of CCSNe identified in Section 4.1 (i.e., the red
dots, which denote the ejecta whose abundance composition matched Cr-isotopic
signature of the presolar chromite grains) and the solar material with solar
Cr and Mg abundances and terrestrial isotopic composition. For comparison, we
also show the small, correlated mass independent Mg and Cr anomalies reported
in several meteorites as internally normalised $\mu$26Mg*’ vs $\mu$54Cr’
values from Larsen et al. (2011). Note that the data sets on CR chondrules and
CAIs are omitted because these materials are more heterogeneous, showing up to
100 ppm variation in the stable Mg isotopes, and a 5% variation in the initial
26Al abundance (see e.g. Luu et al. 2019 and Larsen et al. 2020 for more
details).

In general, each mixing line is a hyperbola that connects two “end-members”:
the solar isotopic composition, in the origin by definition, and the isotopic
composition of the specific CCSN region fitting the chromite grain
composition. The curvature (K) of the line is determined by the relative
abundance of the normalising isotopes (52Cr and 24Mg) in the ejecta compared
to the Solar System value: K=(52Cr/24Mg)solar/(52Cr/24Mg)CCSN, see Langmuir et
al. (1978); Dauphas et al. (2004). Therefore, the line features are determined
by both the isotopic composition ($\mu$ values in the plots) and the elemental
composition, relative to the solar value of the CCSN ejecta. Note that the
full lines are hyperbolas, but the plots are zoomed into the region of the
meteoritic data, therefore, they appear as linear. It is common practice to
plot the mixing trajectories as symmetric lines going through the
solar/terrestrial value representing not only the addition but also the
subtraction or “unmixing” of a nucleosythetic component. For clarity, here
instead we only plot the mixing trajectories that result from addition of CCSN
material to the solar abundances, to indicate the composition vectors towards
the CCSN composition and to highlight model differences. To calculate the CCSN
end-member we show results with and without mass-dependent fractionation, as
outlined in the previous section. In the top panels of Figure 11 we show the
trajectories derived from internal normalised model data (as in the right
panel of Figure 10), and in the bottom panels of Figure 11 we show the
trajectories derived from no mass fractionation in the CCSN ejecta (as in the
left panel of Figure 10).

The lines in the top and bottom panels are very different from each other
because in both the C ashes and He ashes, the isotopic ratios that we use for
internal normalization, 50Cr/52Cr and 25Mg/24Mg, are very different from the
solar values. For example, in the He ashes, the 50Cr is completely destroyed
and 25Mg is produced (see Appendix A). Therefore, these normalizing isotopic
ratios are at least as anomalous than the isotopic ratios we are investigating
(54Cr/52Cr and 26Mg∗/24Mg). This leads to extreme transformation of the
isotopic space when applying internal normalization, even resulting in a
change of sign in $\delta$ and $\mu$ notation.

All the solid and dotted lines (corresponding to the C- and the He-ashes,
respectively) are horizontal because both in the He- and C-ashes the abundance
of Mg is much higher than the abundance of Cr, therefore, the signature of the
Mg isotopic composition is stronger in this representation relative to that of
the Cr isotopic composition (i.e., K is between 1 and 200). We also note that
the Mg isotopic signature is always dominated by the stable Mg isotopes,
rather than by 26Al.

As mentioned in the Introduction and at the start of Section 5.2, there is no
published study on Mg isotopes in presolar chromite grains that gives evidence
that chromite grains carry anomalies in Mg isotopes. Nevertheless, Dauphas et
al. (2010) reported elemental abundances in chromite grains showing variable
enrichment of Al and Cr relative to Mg. Following this indication, we
considered the possibility of CCSN material enriched in Al and Cr with respect
to Mg, with respect to the CCSN calculated abundances. For simplicity here we
made a test using Al:Mg:Cr=2:1:1 (dashed and dotted-dashed lines in Figure 11
for the He- and C-ashes, respectively), noting that this represents an
enrichment for Al, because the Al/Mg ratios in the CCSN candidate site are
$\simeq$0.15, higher than the solar ratio of 0.06, but still much lower than
2, and for Cr, because the Cr/Mg ratios in the CCSN candidate sites are even
lower than the solar ratio of 0.01. In these enriched cases, the K value
becomes lower than 0.1 and therefore the mixing lines deviate from horizontal.

In these enriched cases for the C-ashes, the mixing lines appear as almost
vertical as they are dominated by the increased relative abundance of 52Cr in
this part of the CCSN ejecta (see Appendix A). None of them match the
correlation displayed by the meteoritic data. This C-ashes material could
still be in agreement with other interpretations of the measured planetary Mg
isotopic data, suggesting an homogeneous proto-planetary disc for Mg isotopes
on a level of a few ppm (e.g., Jacobsen et al., 2008; Kita et al., 2013; Luu
et al., 2019).

In the case of the enriched He-ashes, the 52Cr/24Mg ratio relative to the
solar ratio is less extreme and the highly variable $\mu$ values from the
isotopic composition dominate the mixing trajectories. In the top panels in
Figure 11, where we use internal normalisation the trajectories match the
meteoritic trend only with negative $\mu$ values. This solution, however,
would be inconsistent with our assumption that the observed anomalies are
carried by refractory CCSN grains because such grains would not be
preferentially destroyed in the inner Solar System, relative to less
refractory ISM dust. Therefore, their presence should result in positive
anomalies. In the bottom panels instead, where we assume that the mass
fractionation of the ejecta is negligible, the 15 and 20 M⊙ LAW models may
generate the observed trend via mixing or unmixing of a positive CCSN
component suggested by Larsen et al. (2011). Note that the composition of the
SIE He-ashes is just outside the border of the box defined in Figure 4,
therefore, they are not included in this plot, but if they did they would
behave similarly to the LAW He-ashes.

##  6 Conclusions

We presented a detailed analysis of the production of the Al, Mg, and Cr
isotopes from CCSN models with non-rotating, single star progenitors. We
compared the total isotopic yields between seven CCSN data sets and we
compared the isotopic composition as a function of mass coordinate from three
CCSN sets to the isotopic composition measured in meteoritic stardust grains.
We found potential nucleosynthetic origin sites of the chromite grains
presented in Nittler et al. (2018), and evaluated the contribution at atomic
mass 26 at those potential sites.

Concerning the total CCSN yields, we found that the seven CCSN data sets are
mostly comparable to each other for the nine isotopes of interest: 24,25,26Mg,
26,27Al, and 50,52,53,54Cr. The main differences are due to different mass
cuts (mainly driving variations in the abundances of Cr isotopes), the
occurrence of a shell merger in the 15 M⊙ RIT models, and structural
differences in the progenitors between the LAW and SIE data sets. Based on a
detailed analysis of the production sites of these isotopes in the different
models, we are confident that our findings are representative of most 1D CCSN
models of solar metalicity.

We compared the CCSN models to the composition of the chromite grain most
anomalous in 54Cr: 2_37, including an estimate of the uncertainties due to
neutron capture rates on 53Cr, 54Cr, 48Ti, and 50Ti, based on our sensitivity
tests.

For all models, we were able to identify mass regions within the CCSN ejecta
where the 54Cr/52Cr and 53Cr/52Cr ratios are matched, however, the situation
is more complicated for the ratios including the atomic mass 50 isotopes. Only
in five out of nine models we could find a complete solution and only in small
regions of the CCSN ejecta (see Table 3). These solutions are all located in
either the C-ashes or the He-ashes in all the models of LAW and in the 15 and
25 M⊙ models of SIE. The three RIT models did not show any overlap regions
with grain 2_37. The regions that match the 50Cr/52Cr ratio of 2_37, and
overlap with the regions matching its 54Cr/52Cr and 53Cr/52Cr ratios, are
located in the C-ashes. In contrast, the regions that overlap with the
50Ti/48Ti ratio are located in the He-ashes, with only one case in the
C-ashes.

When we consider the second and third most anomalous grains, we find again
that both the C- and He-ashes from the LAW and SIE models can match these
grains, however, in these cases the signal at atomic mass 50 must always come
from 50Ti. Furthermore, when considering these two grains we also find that
the shell-merger region in the 15 M⊙ RIT model could be a match, in which case
the isotope at atomic mass 50 could be either 50Cr or 50Ti.

We found that adding or not adding the radioactive 53Mn into 53Cr does not
significantly affect the results. This is different from Jones et al. (2019),
who found that for electron-capture supernova ejecta the partial inclusion of
53Mn in 53Cr is crucial to match the values of grain 2_37.

We conclude that the chromite grains analysed by Nittler et al. (2018) could
have originated from CCSNe. We emphasize that CCSNe are the most frequent
stellar events among the production sites considered so far (Nittler et al.,
2018; Jones et al., 2019a), thus making them a likely candidates as the origin
of these grains.

The Al-isotopic data from SiC-X grains are believed to have originated from
CCSNe (Groopman et al., 2015). We confirm, however, that standard CCSN models
do not produce enough 26Al in their C-rich ejecta to match these grains (see
Figure 6). Therefore, mixing of layers within CCSN models and/or a proton
ingestion into the He shell (Pignatari et al., 2015) are needed to match the
grain data. For the few oxide grains of Group IV that also potentially
originate from CCSNe (Nittler et al., 2008), we could match their Al-isotopic
ratios, however, a multi-element isotope analysis is needed to evaluate the
origin of those grains.

In the candidate regions within the He ashes that reproduce the Cr isotopic
ratios of the presolar chromite grains, the iMg/24Mg ratios are orders of
magnitude higher than the solar ratio. Here, 26Al is being destroyed by
neutron capture reactions and has little effect on the total abundance at
atomic mass 26. In the candidate regions within the C-ashes, the iMg/24Mg
ratios of the inner regions are orders of magnitude lower than the solar
values, due to the production of 24Mg by carbon burning and the partial
depletion of 25Mg and 26Mg by neutron capture and proton capture. While the
abundance of 26Al is more significant at these latter site, it never dominates
the production at atomic mass 26. Since presolar chromite grains alone may
drive the variation of Cr isotopes in the proto-planetary disk, we conclude
that the ejecta carrying the chromite grains could have also generated
nucleosynthetic 26Mg isotopic variation in the disk, and such variation would
be dominated by non-radiogenic, stable Mg isotopic anomalies.

We compared Cr and Mg isotopic anomalies measured in meteorites and planetary
materials with our candidate sites from the CCSN models, and derived the
expected mixing trajectories between the solar and the CCSN reservoirs under
the simple assumption that such Al and Mg abundances are carried within the
chromite grains and/or similar carriers enriched in Al. When considering CSSN
refractory material enriched in Al and Cr relative to Mg, as suggested by the
data of Dauphas et al. (2010), we found that the ejecta of the C-ashes does
not generate significant Mg isotopic variations due to the extreme 52Cr
abundances relative to 24Mg. The ejecta of the He-ashes, instead, can generate
a trend similar to the apparent Cr versus Mg isotopic heterogeneity. This
trend is positive, as required under our assumption of refractory carriers,
only if the CCSN material is not double normalised. The validity of such a
comparison method requires further investigation.

Future measurements of chromite grains with Resonant Ionisation Mass
Spectrometry (RIMS, Stephan et al., 2016) are needed to identify if the signal
at atomic mass 50 is related to 50Cr from the C-ashes or 50Ti from the He-
ashes, since this instrument can in principle extract isotopes of a single
element and avoid isobaric interference during the analysis. Furthermore, to
study the link between stardust grains and planetary objects and investigate
the possible 26Al heterogeneity of the proto-planetary disk, we have assumed
here that Al and Mg abundances are also carried in the chromite grains and/or
similar carriers enriched in Al. This assumption needs to be investigated via
future Mg isotope study of the 54Cr-rich chromite grains. New measurements of
the neutron-capture cross sections for the chromium and titanium isotopes are
also required to improve the accuracy of CCSN predictions. Finally, supernova
models based on self-consistent multidimensional simulations are needed to
reduce the uncertainties that result from parameterized mass cuts and
explosion energies.

## Acknowledgements

We thank Larry Nittler for discussion and for sharing unpublished Fe data.
This research is supported by the ERC Consolidator Grant (Hungary) funding
scheme (Project RADIOSTAR, G.A. n. 724560). We thank the ChETEC COST Action
(CA16117), supported by the European Cooperation in Science and Technology,
and the IReNA network supported by NSF AccelNet. This work was supported in
part by the European Union ChETEC-INFRA (project no. 101008324). TL and MP
acknowledge significant support to NuGrid from 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. A.S. acknowledges
support from the U.S. Department of Energy through grant DE-FG02-87ER40328
(UM), Office of Science, Office of Nuclear Physics and Office of Advanced
Scientific Computing Research, Scientific Discovery through Advanced Computing
(SciDAC) program. Research at Oak Ridge National Laboratory is supported under
contract DE-AC05-00OR22725 from the U.S. Department of Energy to UT-Battelle,
LLC. MP thanks support from the National Science Foundation (NSF, USA) under
grant No. PHY-1430152 (JINA Center for the Evolution of the Elements), and
from the ”Lendulet-2014” Program of the Hungarian Academy of Sciences
(Hungary).

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##  Appendix A Figures showing the production and destruction of the Mg, Al,
and Cr isotopes of interest

In this section we show in Figures 12-14 the mass fractions of the three data
sets of Lawson et al.(submitted), Sieverding et al. (2018), and Ritter et al.
(2018) of the progenitor and the CCSN model as a function of mass coordinate.
For each data set we show four figures for the three initial masses, being 15,
20 and 25 M⊙: the top panel shows the stellar structure, and includes seven
isotopes that allow us to identify the various burning phases within the
progenitor and the explosion following the nomenclature as presented in
Section 2.5. The second panel shows the Mg isotopes, the third panel the Al
isotopes, and the fourth panel shows the Cr isotopes as a function of mass
coordinate. In Figure 15 we show the temperatures in the CCSN models, which
helps with identifying the differences in the nucleosynthesis between the nine
CCSN models.

Figure 12: Non-decayed mass fraction profiles of the 15 M⊙ models of the LAW,
SIE, and RIT data sets. The dashed lines are the mass fractions in the
progenitor, the solid lines show the fractions after the CCSN. Figure 13: Non-
decayed mass fraction profiles of the 20 M⊙ models of the LAW, SIE, and RIT
data sets. The dashed lines are the mass fractions in the progenitor, the
solid lines show the fractions after the CCSN. Figure 14: Non-decayed mass
fraction profiles of the 25 M⊙ models of the LAW, SIE, and RIT data sets. The
dashed lines are the mass fractions in the progenitor, the solid lines show
the fractions after the CCSN. Figure 15: Temperature profiles for the
explosions in all three CCSN model sets. The extra peaks in the RIT models
(see e.g., in the 15 M⊙ model at mass coordinate 3.2 M⊙) are due to the
analytic explosion model, which allows the velocity and thus temperature to
increase when the shock decelerates (Pignatari et al., 2016). These peaks
allow for explosive He-burning in the H-ashes.