# Light quark correlators in a mixed-action setup

Fabio Bernardonia, Nicolas Garron 111N.G. acknowledges supported from STFC
grant ST/G000522/1 and EU grant 238353 (STRONGnet). b, Pilar Hernándezc, d,
Carlos Penae
a NIC, DESY, Platanenallee 6, D-15738 Zeuthen, Germany
b SUPA, School of Physics, The University of Edinburgh, Edinburgh EH9 3JZ, UK
c Instituto de Física Corpuscular, CSIC-Universitat de València, Apartado de
Correos 22085, E-46071 Valencia, Spain
d CERN, Physics Departement, 1211 Geneva 23, Switzerland
e Dpto. de Física Téorica and Instituto de Física Téorica UAM/CSIC,
Universidad Autónoma de Madrid, Cantoblanco E-28049 Madrid, Spain E-mail

###### Abstract:

We report our progress in simulating Neuberger valence fermions on $N_{f}=2$
Wilson $O(a)$-improved sea quarks. We compute correlators with valence quark
masses both in the $p$\- and in the $\epsilon$-regime, and we match the
results with the predictions of the Chiral Effective Theory in the mixed
regime. This allows us to extract the Low Energy Couplings (LECs) of the
$N_{f}=2$ theory and to test the validity of the approach.

CERN-PH-TH/2011-229

IFT-UAM/CSIC-11-68

Edinburgh 2011/28

FTUAM-11-55

DESY 11-171

IFIC/11-54

##  1 Introduction

Lattice QCD simulations with different discretizations in the valence and in
the sea sector have been performed by several groups and with different setups
in the last years [1, 2, 3, 4, 5, 6, 7]. This is a viable approach to preserve
exact chiral symmetry for the valence quarks with a “moderate” numerical cost,
which could be particularly advantageous for the computation of weak matrix
elements.

In our work we apply the Neuberger operator on $N_{f}=2$ non-perturbatively
$O(a)$-improved Wilson fermions configurations generated within the CLS common
effort. 222https://twiki.cern.ch/twiki/bin/view/CLS/WebHome In a recent
publication [8] we have reported our results on the topological susceptibility
and the low-lying eigenvalues of the massless Dirac operator, which we used to
test the so-called mixed Chiral Effective Theory approach and finally to
extract the chiral condensate. We now extend our analysis to two- and three-
point functions.

##  2 Left current correlator in Chiral Effective Theory

We consider partially quenched QCD with $N_{s}$ sea flavors with mass $m_{s}$
and $N_{v}$ valence flavors with mass $m_{v}$, and define the two-point
correlators of the left-handed current

${\rm Tr}[T^{a}T^{b}]C(x_{0})\equiv\int d^{3}x\langle
J^{a}_{0}(x)J^{b}_{0}(0)\rangle,\;\;\;\;\;J^{a}_{\mu}\equiv\bar{\psi}T^{a}\gamma_{\mu}P_{-}\psi,$
(1)

where $T^{a}$ is a traceless generator of $SU(N_{v})$ and
$P_{-}=(1-\gamma_{5})/2$ is the left projector. Low energy properties can be
described by the corresponding $SU(N_{s}+N_{v})$ Partially Quenched Chiral
Perturbation Theory (PQChPT); in particular, in a finite volume $V=L^{3}T$ we
consider the following two situations [9]:

  * •

$m_{v/s}\Sigma V\gg 1$: all quarks are in the $p$-regime. In this case the
correlator $C(x_{0})$ at NLO in PQChPT is given by

$\mathcal{C}(x_{0})=\frac{F^{2}_{\rm{eff}}}{4}\frac{M_{vv;\rm{eff}}\cosh\left[M_{vv;\rm{eff}}(T/2-x_{0})\right]}{\sinh\left[M_{vv;\rm{eff}}T/2\right]}-\frac{N_{s}}{2}\frac{dG^{1}(0;M^{2}_{vs})}{dT},$
(2)

with

$\displaystyle F^{2}_{\rm eff}=F^{2}(1+\Delta_{F})$ $\displaystyle=$
$\displaystyle
F^{2}\left\\{1-\frac{N_{s}}{F^{2}}G^{1}(0;M^{2}_{vs})+\frac{8}{F^{2}}(N_{s}M^{2}_{ss}L_{4}+M^{2}_{vv}L_{5})\right\\},$
(3) $\displaystyle M^{2}_{vv;\rm{eff}}=M^{2}_{vv}(1+\Delta_{M})$
$\displaystyle=$ $\displaystyle
M^{2}_{vv}\Bigg{\\{}1+\frac{1}{N_{s}F^{2}}\left[G^{1}(0;M^{2}_{vv})+(M^{2}_{ss}-M^{2}_{vv})G^{2}(0;M^{2}_{vv})\right]$
(4) $\displaystyle-$
$\displaystyle\frac{8}{F^{2}}\left[N_{s}M^{2}_{ss}(L_{4}-2L_{6})+M^{2}_{vv}(L_{5}-2L_{8})\right]\Bigg{\\}},$

with the LO pseudoscalar masses $M_{ij}=(m_{i}+m_{j})B$ and the pseudoscalar
propagators in finite volume

$G^{r}(x;M^{2})=\frac{1}{V}\sum_{p}\frac{e^{ipx}}{(p^{2}+M^{2})^{r}},\;\;\;r\geq
1.$ (5)

  * •

$m_{v}\Sigma V\lesssim 1$, $m_{s}\Sigma V\gg 1$: valence quarks are in the
$\epsilon$-regime, sea quarks are in the $p$-regime. In this case the NLO left
correlator at fixed topology is given by

$\mathcal{C}^{\nu}(x_{0})=\frac{F^{2}_{\epsilon;\rm{eff}}}{2T}\left\\{1+\frac{2T^{2}}{F^{2}V}\mu_{v}\sigma_{\nu}(\mu_{\nu})h_{1}\left(\frac{x_{0}}{T}\right)\right\\}-\frac{N_{s}}{2}\frac{dG^{1}(0;M^{2}_{ss}/2)}{dT},$
(6)

with $\mu_{v}=m_{v}\Sigma V$ and

$\displaystyle F^{2}_{\epsilon;\rm{eff}}$ $\displaystyle=$ $\displaystyle
F^{2}\left(1+\Delta_{F}|_{M_{vv}=0}\right)=F^{2}\left\\{1-\frac{N_{s}}{F^{2}}G^{1}(0;M^{2}_{ss}/2)+\frac{8}{F^{2}}(N_{s}M^{2}_{ss}L_{4})\right\\},$
(7) $\displaystyle h_{1}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\left(\tau-\frac{1}{2}\right)^{2}-\frac{1}{12}\right],$
(8) $\displaystyle\sigma_{\nu}(x)$ $\displaystyle=$ $\displaystyle
x\left[I_{\nu}(x)K_{\nu}(x)+I_{\nu+1}(x)K_{\nu-1}(x)\right]+\frac{\nu}{x},$
(9)

where $I_{\nu}$, $K_{\nu}$ are modified Bessel functions.

A well-established strategy to extract the LO ($F,B$) and the NLO ($L_{i}$)
Low Energy Couplings (LECs) is to match lattice results for the two-point
correlators with the predictions of the ChPT. Depending on the chiral regime,
the systematic uncertainty associated to the matching can vary considerably.
Exploring different regimes in a finite box can then be useful to give
reliable estimates of systematic errors associated to the LECs.

##  3 Lattice results

###  3.1 Setup and results for the Low Energy Couplings

We have carried out our simulations on CLS lattices of size $V=24^{3}\times
48$, with $\beta=5.3$ corresponding to a lattice spacing $a=0.0649(10)$ fm
[10]. We have considered three $p$-regime sea quark masses
$\kappa=0.13620,0.13625,0.13635$, labeling the sets by D4,D5,D6 respectively.
On these configurations we have built the Neuberger-Dirac operator, with
valence quark masses both in the $p$\- and in the $\epsilon$-regime. The
simulations parameters are collected in Table 1, where we report also the
effective sea pion masses in lattice units (see [8] for further details). For
valence quark masses in the $p$-regime, we have collected $\sim 80-160$
measurements. For the $\epsilon$-regime, we have in total $\sim 70-100$
measurements of the left correlator in the topological sectors
$|\nu|=0,1,2,3$.

$\beta=5.3$, $c_{SW}=1.90952$, $V=24^{3}\times 48$
---
lattice | $\kappa$ | $aM_{ss;{\rm eff}}$ |  $am_{v}$ ($\epsilon$-reg.) |  $am_{v}$ ($p$-reg.)
D4 | 0.13620 | 0.1695(14) | 0.002 | 0.012,0.02,0.028
D5 | 0.13625 | 0.1499(15) | 0.002 | 0.012,0.02
D6 | 0.13635 | 0.1183(37) | 0.002 | 0.006,0.015,0.025
Table 1: Simulation parameters

We have computed the two-point left current correlator defined in Eq. (1)
adopting the low-mode averaging technique of [11]. For the renormalization of
the local axial current we used the chiral Ward identity at fixed topological
charge saturated with the zero modes, following the strategy proposed in [12].
In particular we considered

$Z_{A}^{\rm mix}\lim_{m_{v}\rightarrow
0}(m_{v}V)\sum_{\vec{x}}\langle\partial_{0}P^{a}(x)J^{a}_{0}(0)\rangle_{\nu}=\lim_{m_{v}\rightarrow
0}(m_{v}^{2}V)\sum_{\vec{x}}\langle P^{a}(x)P^{a}(0)\rangle_{\nu},$ (10)

where $P^{a}=\bar{\psi}T^{a}\gamma_{5}\psi$ is the pseudoscalar density. The
right-hand side is completely saturated with zero modes and can be directly
computed without chiral extrapolation; on the left-hand side, the chiral limit
has to be taken numerically. The renormalization factor $Z_{A}^{\rm mix}$ can
then be extracted from ratios of correlations functions. For those correlators
we have collected a total of $\sim 70-140$ measurements for the topological
sectors $0<|\nu|<5$. On the left side of Fig. 1 we report the results for the
different sea quark masses corresponding to D4,D5,D6 and for $\nu=1,2,3,4$.
Apart from some fluctuations in the topological sector $|\nu|=1$, the data do
not depend on $\nu$ and on the sea quark masses. From this study we obtained

$Z_{A}^{\rm mix}=1.65(1).$ (11)

Figure 1: _Left_ : the renormalization constant of the local axial current
extracted through the chiral Ward identity at fixed topological charge
saturated with the zero modes. The blue,green and red points refer to the
lattices D4,D5,D6 respectively. _Right_ : $aF_{\epsilon;\rm{eff}}$ extracted
from the left correlator in the mixed $p$/$\epsilon$-regime as a function of
the topological charge. The gray bands represent the average over
$|\nu|=1,2,3$.

We then extracted $M_{vv;\rm{eff}}$, $F_{\rm eff}$, $F_{\epsilon;\rm{eff}}$
from the left correlators, according to Eqs. (2, 6); on the right side of Fig.
1 we show for instance the (bare) pseudoscalar decay constant computed from
the mixed $p$/$\epsilon$-regime correlators at $|\nu|=0,1,2,3$. As it should
be, results do not depend significantly on the topological charge;
$F_{\epsilon;\rm{eff}}$ has been determined by averaging over $|\nu|=1,2,3$
(gray band in the plot). We then performed a global fit of pseudoscalar masses
and decay constants using the NLO predictions of the PQChPT given in Eqs.
(3,4,7). From this fit we obtained the six parameters

$\displaystyle aF^{R}=0.0259(10)(8)$ , $\displaystyle aB^{R}=1.52(11)(9),$
$\displaystyle L_{4}^{R}=0.00054(10)(4)$ , $\displaystyle
L_{5}^{R}=0.00073(15)(2),$ (12)
$\displaystyle(L_{4}-2L_{6})^{R}=0.00033(5)(0)$ ,
$\displaystyle(L_{5}-2L_{8})^{R}=-0.00025(15)(5),$

where “R” means that the couplings are renormalized. The constants $L_{i}^{R}$
are given at the scale $\mu=770$ MeV, while $B^{R}$ is given in the
$\overline{\rm{MS}}$ scheme at $\mu=$ 2 GeV. For the tree-level pseudoscalar
masses we used $M_{ij}=(m_{i}^{R}+m_{j}^{R})B^{R}$; for the sea Wilson quark
masses we used the PCAC masses computed in [13, 10], with renormalization
constants $Z_{A}^{W}=0.78(1)$ [14], $Z_{P}^{W}(\overline{\rm
MS},\;2\;\rm{GeV})=0.543(8)$ [15]; for the overlap valence quarks we have
computed $Z_{S}^{\rm mix}(\overline{\rm MS},\;2\;\rm{GeV})=1.88(15)$ like in
[8]. The first error is statistical, while the second one reflects the
uncertainty on the PCAC quark masses.

Figure 2: Global chiral NLO fit for $M_{vv;\rm{eff}}$, $F_{\rm eff}$. The
squares in the right plot correspond to the $\epsilon$-regime results,
$F_{\epsilon;\rm{eff}}$.

The global fit is shown in Fig. 2. It is important to stress that having
valence quark masses in the $\epsilon$-regime allows us to extract the
pseudoscalar decay constant directly at $am_{v}=0$ and hence to have a good
control over the chiral extrapolation in $m_{v}$.
Appropriate combinations of the $L_{i}$ yield the conventional SU(2) LECs
$\bar{l}_{i}$; in particular we obtain

$\bar{l}_{3}=4.46(30)(14),\;\;\;\;\;\;\;\;\bar{l}_{4}=4.56(10)(4).$ (13)

The FLAG working group [16] gives an estimate $\bar{l}_{3}=3.2(8)$ and quotes
$\bar{l}_{4}=4.66(4)\binom{+4}{-33}$ [17] as best determination of
$\bar{l}_{4}$. Our determinations are in agreement with those results within
the uncertainties. By converting $aF^{R}$ and $aB^{R}$ in physical units we
obtain $F^{R}=79(4)$ MeV and $\Sigma^{R}(\overline{\rm
MS},\;2\;\rm{GeV})=B^{R}(F^{R})^{2}=(306(11)\;{\rm MeV})^{3}$, which also lie
in the right ballpark.

###  3.2 Unitary point

A natural definition of the unitary point corresponds to the matching of the
pseudoscalar masses

$M^{2}_{vv;\rm{eff}}|_{m_{v}=m_{v;\rm{un}}}=M^{2}_{ss;\rm{eff}}.$ (14)

We have extracted $m_{v;\rm{un}}$ through a linear interpolation of
$M^{2}_{vv;\rm{eff}}$. The pseudoscalar decay constants in the full Wilson
theory ($F^{R}_{s;\rm{eff}}$) and in the mixed action theory at
$m_{v;\rm{un}}$ ($F^{R}_{\rm eff}|_{m_{v}=m_{v;\rm{un}}}$) should then be
equal up to discretization effects. In Fig. 3 we show $F^{R}_{\rm
eff}|_{m_{v}=m_{v;\rm{un}}}$ and $F^{R}_{s;\rm{eff}}$ for the three lattices
D4,D5,D6. The Wilson on Wilson data have been taken from [13]. Within our
statistical accuracy, we do not observe large cutoff effects close to the
unitary point.

lat. | $am_{v;\rm{un}}$ | $F^{R}_{\rm eff}|_{m_{v}=m_{v;\rm{un}}}$ | $F^{R}_{s;\rm{eff}}$
---|---|---|---
D4 | 0.0241(5)(4) | 0.0413(6) | 0.0396(7)
D5 | 0.0196(13)(5) | 0.0391(11) | 0.0380(9)
D6 | 0.0088(6)(6) | 0.0320(9) | 0.0328(19)

Figure 3: Effective pseudoscalar decay constant computed in the full Wilson
theory (black points) and in the mixed action theory at the unitary point (red
points) as a function of the pseudoscalar mass. The empty square represents
the decay constant in the chiral limit extracted from the chiral fit, Eq.
(12).

Discretization effects in a mixed action setup can be investigated by means of
Mixed Action PQChPT [18]. If all mass scales are in the $p$-regime and one
assumes a power counting $a\sim m_{s,v}\sim p^{2}$ (_generically small
masses_), it is easy to see that “unitarity violations” of $O(a^{2})$
appearing in $M^{2}_{vs}$ and $(M^{2}_{ss}-M^{2}_{vv})$ affect our observables
only at NNLO, and hence are expected to be small. Our numerical data seem to
confirm this expectation, although a more detailed study is necessary in order
to have a precise estimate of those effects. 333A recent study with Neuberger
valence fermions and Wilson Twisted Mass sea quarks [6] indicates large cutoff
effects close to the unitary point.

##  4 $K\rightarrow\pi\pi$ amplitudes and $\Delta I=1/2$ rule

In [19], a strategy to study the rôle of the charm quark in the $\Delta I=1/2$
rule has been proposed. The basic idea is to start from the GIM limit
$m_{u}=m_{d}=m_{s}=m_{c}$, write down the $\Delta S=1$, CP-even effective weak
Hamiltonian and match it with the effective Hamiltonian in the SU(4) Chiral
Effective Theory. At leading order in ChPT, the ratio of amplitudes
corresponding to $\Delta I=1/2$ and $\Delta I=3/2$ $K\rightarrow\pi\pi$
transitions can be expressed in terms of given LECs $g_{1}^{\pm}$. As usual,
those couplings can be computed by matching suitable correlation functions
computed on the lattice with their analytic expressions obtained in ChPT (see
[19] for details). Quenched results obtained with Neuberger fermions have been
presented in [20]; this strategy, for which exact chiral symmetry at finite
lattice spacing is an essential ingredient, can be implemented also with our
mixed action setup. Preliminary results for the bare ratios $R^{\pm}$ of
three- and two-point functions defined in [20] have been obtained so far for
the lattice D4 (interpolated to the unitary point)

$R^{+}=0.64(6),\;\;\;R^{-}=1.5(2),$ (15)

of the same order as the quenched results for pseudoscalar masses in the same
range.
The second part of the strategy consists in studying the amplitudes ratio as a
function of $m_{c}>m_{u}=m_{d}=m_{s}$ and test if by increasing the charm
quark the $\Delta I=1/2$ enhancement observed experimentally is reproduced.

##  5 Conclusions and outlook

Mixed action simulations with exact chiral symmetry in the valence sector
provide a powerful tool to investigate QCD properties. Given the significant
errors associated to chiral extrapolations, it is important to explore several
chiral regimes where the systematic effects are different, so that such errors
can be properly quantified. In particular, we have considered valence quarks
both in the $p$\- and in the $\epsilon$-regime and matched the lattice results
with the PQChPT predictions at NLO. A global chiral fit of pseudoscalar masses
and decay constants allowed us to determine $F$, $B$, $\bar{l}_{3}$,
$\bar{l}_{4}$ with comparable accuracy with respect to other results present
in the literature. By matching pseudoscalar masses in the valence and in the
sea sector, we did not observed large cutoff effects close to the unitary
point. We stress that a precise determination of the LECs would require an
improvement of the statistical precision and a careful monitoring of
autocorrelations; furthermore, a larger set of volumes, lattice spacings and
quark masses would guarantee a solid estimate of the systematic uncertainties
associated to lattice artefacts, finite-volume effects and chiral
extrapolation.

As next step, we are implementing the strategy proposed in [19] to investigate
the rôle of the charm quark in the $\Delta I=1/2$ rule. This calculation is
more challenging and requires further computational efforts.

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∎

11institutetext:  1Konkoly Observatory, Research Centre for Astronomy and
Earth Sciences, Eötvös Loránd Research Network (ELKH), H-1121 Budapest,
Konkoly Thege M. út 15-17, Hungary
2CSFK, MTA Centre of Excellence, Budapest, Konkoly Thege Miklós út 15-17,
H-1121, Hungary
3ELTE Eötvös Loránd University, Institute of Physics, Budapest 1117, Pázmány
Péter sétány 1/A, Hungary
4School of Physics and Astronomy, Monash University, VIC 3800, Australia
5Institute for Geochemistry and Petrology, ETH Zürich, Zurich, Switzerland
6E. A. Milne Centre for Astrophysics, University of Hull, Cottingham Road,
Kingston upon Hull, HU6 7RX
7Centre for Star and Planet Formation (StarPlan), Globe Institute, Faculty of
Health and Medical Sciences, University of Copenhagen, Øster Voldgade 5-7,
1350 Copenhagen K, Denmark

# Representation of $s$-process abundances for comparison to data from bulk
meteorites

Maria Lugaro1,2,3,4 Mattias Ek5 Mária Pető1,2 Marco Pignatari1,2,6 Georgy V.
Makhatadze7 Isaac J. Onyett7 Maria Schönbächler5

(Received: date / Accepted: date)

###### Abstract

Analysis of bulk meteorite compositions has revealed small isotopic variations
due to the presence of material (e.g., stardust) that preserved the signature
of nuclear reactions occurring in specific stellar sites. The interpretation
of such anomalies provides evidence for the environment of the birth of the
Sun, its accretion process, the evolution of the solar proto-planetary disk,
and the formation of the planets. A crucial element of such interpretation is
the comparison of the observed anomalies to predictions from models of stellar
nucleosynthesis. To date, however, this comparison has been limited to a
handful of model predictions. This is mostly because the calculated stellar
abundances need to be transformed into a specific representation, which
nuclear astrophysicists and stellar nucleosynthesis researchers are not
familiar with. Here, we show in detail that this representation is needed to
account for mass fractionation effects in meteorite data that can be generated
both in nature and during instrumental analysis. We explain the required
internal normalisation to a selected isotopic ratio, describe the motivations
behind such representation more widely, and provide the tools to perform the
calculations. Then, we present some examples considering two elements produced
by the $slow$ neutron-capture ($s$) process: Sr and Mo. We show which specific
representations for the Sr isotopic composition calculated by $s$-process
models better disentangle the nucleosynthetic signatures from stars of
different metallicity. For Mo, the comparison between data and models is
improved due to a recent re-analysis of the 95Mo neutron-capture cross
section.

###### Keywords:

nucleosynthesis meteorites stardust mass fractionation neutron captures

##  1 Introduction

Since its beginning, the study of the $slow$ neutron-capture ($s$) process has
relied on the analysis of meteorites. Thanks to the derivation of the solar
abundances of the $s$-only isotopes111These are isotopes that can only be
produced by the $s$ process, as they are shielded from $r$-process production
by a stable isotope with the same mass but lower atomic number. from analysis
of meteorites suess56  the early $s$-process studies of Don Clayton and
collaborators seeger65 ; clayton61 ; clayton74  and Franz Käppeler and
collaborators kaeppeler82  were able to define the first $s$-process
distributions and identify the weak, main, and strong components. The
reproduction of these components allowed this pioneering research to study in
great detail the characteristics of the $s$-process neutron fluxes in terms of
both their exposure and density. It also provided first constraints on the
$s$-process astrophysical conditions, for example, in terms of the
temperatures that characterises its stellar sites (e.g., kaeppeler93 ). The
meteoritic abundances used in those early works characterise the bulk
composition of the Solar System222Note that the bulk $isotopic$ composition of
meteorites and planetary objects mostly reflects the average abundances
carried by the gas and dust present in the Milky Way interstellar medium at
the place and time of the formation of the Sun. The $elemental$ abundances
instead are also strongly affected by secondary chemical processing.. They
were reproduced initially by parametric models kaeppeler89 , and later by
stellar models, which were taken to be “typical” of the $s$-process sources in
the Galaxy raiteri91 ; straniero95 ; gallino98 ; arlandini99 ; the07 ;
pignatari10 ; bisterzo10PaperI . Today, chemical evolution models of the
Galaxy (e.g., travaglio04 ; prantzos18 ; kobayashi20 ) are employed to
interpret the composition of the Solar System. These models use as input the
ejecta of many $s$-process stellar sources, both massive stars and asymptotic
giant branch (AGB) stars, of different masses and metallicities, to calculate
the evolution of chemical abundances as the Galaxy ages.

On top of this well-mixed, averaged isotopic composition, small
nucleosynthetic isotope variations between the average of bulk meteorite
groups have also been observed. This indicates that large spatial isotopic
heterogeneities were preserved in the early Solar System despite the strong
homogenisation of matter that must have occurred during accretion and in the
proto-planetary disk. For reviews of this complex and far-reaching topic we
direct the reader to dauphas16 ; yokoyama16 ; qin16 ; bermingham20 ; kleine20
; mezger20 . In brief, such heterogeneities are extremely small, with measured
variability of the order of $10^{-4}-10^{-5}$, but they can be measured with
sufficiently high precision, $\sim 10^{-6}$, that they can be interpreted as
significant signals. This improvement in analytical precision is primarily due
to methodological advancements in high-precision multi-collector inductively
coupled mass spectrometry (MC-ICPMS), and in the chemical treatment of
samples. The discovered nucleosynthetic isotope variations could have been
inherited from an originally heterogeneous molecular cloud (including roughly
1% pre-solar dust) and/or represent the product of thermal, gravitational,
and/or chemical processing in the proto-planetary disk itself (e.g.,
trinquier09 ; steele12 ; ek20 ; akram15 ; hutchison22 ). Newly synthesised
stellar material could also have been added (e.g., nanne19 ; lichtenberg21 )
during the accretion and/or onto the proto-planetary disk once it had already
formed, and the accreting material could have varied with time and/or space
according to the composition of the ejecta of potentially nearby stellar
objects, e.g., haba21 .

Possible carriers of the observed nucleosynthetic isotopic variations are
“stardust” grains, i.e., micro-minerals that formed directly in the ejecta of
stars and supernovae and were present in the pre-solar dust inventory of the
original solar molecular cloud (which included both stardust and dust formed
in the interstellar medium). Some of these grains are relatively resistant to
destruction and survived as individual dust grains in the interstellar medium
and in the proto-planetary disk. They were preserved inside primitive
meteorites that never experienced complete melting, and have been identified
and recovered because of their extreme isotopic anomalies zinner14 . Stardust
grains are effectively tiny samples of stellar material and their compositions
can be directly compared to stellar model predictions (e.g., zinner14 ;
nittler16 ; lugaro18grains ; liu19Mo ). For example, the vast majority of
stardust silicon carbide (SiC) grains recovered from meteorites formed in the
external layers of C-rich asymptotic giant branch (AGB) stars and carry the
signature of the $s$ process in the isotopes of many of the elements heavier
that iron, from Sr, Kr, and Zr up to Ba, Nd, and Sm, see, e.g., gallino97 . In
brief, AGB stars are the final phases of the life of low-mass stars (with an
initial mass roughly lower than 10 times the mass of the Sun, see review by
Karakas & Lattanzio karakas14dawes ). They experience H and He burning in
their deep layers and the material processed by nuclear reactions is carried
to the stellar surface via recurrent mixing episodes. In particular, He-
burning is only partially activated at the bottom of the convective thermal
pulses in AGB stars, producing more carbon than oxygen. Therefore, mixing of
this material to the stellar surface allows some of these stars to become
C-rich (C$>$O) and produce C-rich dust such as SiC.

Dauphas et al. dauphas04  found variations in Mo and Ru in meteorites that
clearly follow the pattern of the $s$-process nucleosynthesis that occurs in
AGB stars (Figure 1). Since then, bulk meteorite data has been collected for
many more elements (see, e.g., akram15  and Table 1) and the data shows that
the Earth carries the largest $s$-process excess identified in the Solar
System, for example, for Mo and Ru. This has been interpreted as evidence that
the $s$-process-rich material was present in the innermost part of the disk
mezger20 ; burkhardt21 . The $s$-process signatures may have been generated,
for example, by thermal effects in the proto-planetary disk as a function of
heliocentric distance and/or other disk features. These effects may have
resulted in a distribution of stardust SiC grains from AGB stars different
from the distribution of pre-solar dust particles that condensed in the
interstellar medium. For example, the pre-solar SiC stardust that carried the
$s$-process signature may have survived better closer to the Sun than the
other pre-solar dust ek20 .

While the $s$-process variations appear continuous, other heterogeneities are
discontinuous instead. A compositional “dichotomy” is observed between
different types of meteorites: the non-carbonaceous (NC) and carbonaceous (CC)
meteorites that may have formed in the inner ($<$ 3AU) and outer ($>$ 3AU)
parts of the disk, respectively desch18 . This dichotomy is most apparent in
the neutron-rich isotopes of intermediate-mass and iron-group elements (e.g.,
trinquier07 ; trinquier09 ; leya08 ), such as 48Ca, 50Ti, and 54Cr; elements
heavier than iron affected by neutron captures, such as Mo (Figure 1) and Ru
(e.g., burkhardt11 ); and other isotopes of explosive nucleosynthesis origin,
such as 58Ni (e.g., steele12 ). The dichotomy has been attributed, for
example, to the presence of two dust reservoirs in the proto-planetary disk.
Although the origin of these variations is unclear, the carriers may have
included stardust Cr-rich oxide grains nittler:18  originating in core-
collapse or other types of supernovae jones:19a ; denhartogh22 . Among the
possible mechanisms suggested for creating the two reservoirs are, for
example, the formation of Jupiter kruijer17  and the migration of the snow
line in the proto-planetary disk lichtenberg21 . While all such speculations
seem reasonable, and despite the extensive observational data available, there
is no consensus yet on a full scenario that can explain the origin of the
large length-scale nucleosynthetic variation along the proto-planetary disk
recorded in meteorites.

Figure 1: Mo isotopic variations measured in different meteoritic objects,
shown in the usual representation where isotopic ratios are plotted against
each other. The derivation of the $\epsilon$ quantities reported on each axis
is described in detail in Section 2.1. In brief, they are internally
normalised isotopic ratios (using 98Mo/96Mo) relative to the laboratory
standard used during the measurement (the [0,0] point by definition, close to
the natural terrestrial value, but not necessarily identical). Error bars are
not reported for sake of clarity, with 2$\sigma$ uncertainty for most data
points of the order of $\pm$0.1 to $\pm$0.5. The data are from: (i) the
compilation of burkhardt21  for meteoritic bulk rocks classified in the two NC
and CC groups according to their composition with respect to observed isotopic
dichotomy, with the NC and CC lines from budde19  ; (ii) the compilation of
brennecka20  for calcium-aluminum-rich inclusions (CAIs), which have sizes of
roughly a few cm, are found mostly in primitive meteorites, represent the
first solids to have formed in the Solar System, and show the largest
deviation from the bulk solar composition; and (iii) stephan19  for the the
best fit trend line for meteoritic mainstream (MS) stardust silicon carbide
(SiC) grains that originated from AGB stars, the main sources of the $s$
process in the Galaxy.

Molybdenum is an example of an element for which the measured bulk isotope
composition is available for a wide range of meteorites, showing both the
composition dichotomy between the CC and NC groups and the $s$-process trend
(Figure 1). We do not discuss the nucleosynthetic origin of the Mo dichotomy
here, but note that it is unlikely to originate from the $rapid$ neutron-
capture ($r$) process. This is because $r$-process sites in the late Galaxy
(such as compact mergers, or special types of supernovae, see, e.g., the
review by cowan21 ) are either extremely rare or do not produce the dust
needed to preserve and carry an $r$-process anomaly into the Solar System. The
nucleosynthetic origin of the Mo dichotomy may be instead related to neutron-
capture nucleosynthesis in core-collapse supernovae. These are more common
stellar objects well known to produce dust, as also evidenced by the presence
of such grains in the stardust inventory, e.g.. meyer00MoZr ; pignatari18 ;
liu18SN ). In general, cosmochemists focus on matching the observed variations
using the $s$-, $r$, and $p$-process isotopic abundances predicted for the
bulk Solar System arlandini99 ; bisterzo11PaperII . However, the observed
variations could also derive from sources that are not major contributors to
the Solar System compositions. In fact, Stephan & Davis stephan21  proposed
that distinct $s$ processes from different AGB star sources may be responsible
for the Mo dichotomy. Figure 1 shows that both the NC and CC groups follow
continuous trends on two lines almost parallel to the trend of mainstream (MS)
stardust silicon carbide (SiC) grains (and consistent with an $s$-process
signature, as we will show in Section 3.2). This supports the idea that the
nucleosynthetic variability within the CC and NC groups may have been
controlled by the distribution of SiC grains in the proto-planetary disk ek20
.

To obtain a full picture, it is necessary to compare models of the $s$ process
in AGB stars to meteoritic data for all the available elements, i.e., to add
model predictions onto Figure 1. As presented in detail in the following
methodology section, however, this data-model comparison is far from trivial
and special considerations are required. In brief, models predictions are
given as abundances of each given isotope, while meteoritic data are given as
ratios between two isotopes, normalised to another given ratio with an assumed
value. This is usually referred to as “internal normalisation” and it is
necessary to correct for the ubiquitous presence of mass-dependent
fractionation, which does not reflect nucleosynthethic effects. While such
transformation involves losing some information, also because mass-dependent
effects may mimic nucleosynthetic effects, it is a necessary step as we
demonstrated below. Furthermore, the meteoritic data do not represent pure
stellar nucleosynthetic compositions but only very small deviations towards it
and away from the solar composition. This indicates that the stellar material
was significantly diluted, which points to a relatively small amount of
stardust carriers inside the meteorite relative to the rest of the
isotopically solar material.

A straightforward way to compare stellar models with meteorite data is to
apply the internal normalisation procedure to the stellar models. This makes
it possible to evaluate several stellar models at once in relation to the same
meteorite data. There are two approaches to applying the internal
normalisation procedure to stellar models. The first approach is to construct
a synthetic sample by adding a small amount of stellar abundance to the solar
abundance, for any given isotope:

$C_{SMP}=C_{\odot}+\left(C_{SM}\times x\right)$ (1)

where $C_{SMP}$ is the abundance of the synthetic sample, $C_{\odot}$ is the
solar abundance, $C_{SM}$ is the stellar abundance and $x$ is the dilution
factor of the stellar abundances relative to the solar abundances. The value
of $x$ is such that the magnitude of the internally normalised values of
$C_{SMP}$ are comparable to that of bulk meteorites, e.g., Simon et al.
simon09  and Figure 2. The resulting composition can then be internally
normalised. The second approach is to use a linearised version of the internal
normalisation procedure that can be applied directly to $C_{SM}$, e.g.,
Dauphas et al. dauphas04 . As we show in the Supplementary Materials these two
approaches result in almost identical slopes, when close to the solar
composition, of the mixing lines between the solar composition and the stellar
end-members. Such slopes are the quantity that allows us to compare the models
to the data, as we will describe in details in Section 2.2. Another way to
compare meteorite data with stellar models is to proceed in the reverse way
and renormalise the meteorite data to fit the stellar end-members, e.g.,
Stephan & Davis stephan21 . This approach is most useful if the stellar
components involved are well constrained, as for Mo where abundant stardust
data provide good constraints on the composition of the stellar end-members
stephan21 . We will not discuss this alternative method in further detail here
because we aim to provide guidelines for the stellar modellers and nuclear
astrophysicists to manipulate their model predictions in order to access the
comparison to the meteorite data.

The aim of this paper is to describe and explain the methodology to compare
data and models and highlight possible obstacles and limitations. We will show
the results of the internal normalisation procedure as applied to examples of
$s$-process abundances to demonstrate that it is not intuitive to make even a
qualitative comparison using raw stellar model abundance predictions. In
Section 2 we outline the problem and its solutions, in Section 3 we show some
examples of the effect of the procedure described in Section 2 on $s$-process
abundances, including the models predictions for Figure 1. We will summarize
our methods and results, draw conclusions, and propose future work in Section
4.

##  2 Methodology

The determination of stellar signatures hidden within the isotopic composition
of meteorites via mass spectrometry is a complex task because of the
analytical challenges and because of the presence of effects other than
stellar nucleosynthesis that can affect the isotopic abundances. High
precision isotope analysis generally requires careful chemical separation of
the target element because isobaric interferences333These occur when isotopes
of different elements with nearly identical atomic mass are collected together
with the target isotopes. Resonance ionization mass spectrometry uses highly
specific laser beams and is capable to ionize the target element without
isobaric interference. This method have been used to analyse single stardust
grains, e.g., savina03 ; liu15SrBa , and chondrules from meteorites
trappitsch18Fe60 . must be eliminated for accurate results, and because the
presence of matrix material can affect both the accuracy and precision of the
instrumental analysis rehkamper12 . There are two types of isotopic variation
found in meteorites: mass dependent and mass independent and they are briefly
described below. For a more comprehensive discussion on the different types of
isotope variations found in meteorites and their components we refer the
reader to the reviews by Rehkämper et al. rehkamper12  and Dauphas & Schauble
dauphas16 .

Mass-dependent isotope fractionation (MDF) results in variations of the order
of up to few % and arises from physical, chemical, and geological processes
that may occur in nature and/or during chemical separation and isotope
analysis. As the name indicates, this type of fractionation is a function of
the isotope mass and occurs during kinetic fractionation, where light isotopes
are often enriched in the reaction product, and equilibrium reactions, where
the heavy isotopes are generally enriched in the material with the stiffest
chemical bonds schauble04 . Mass-dependent fractionation occurs during
instrumental analysis (see dauphas16 ; rehkamper12  for reviews) and chemical
separation of the element via ion exchange chromatography, although this can
generally be avoided if more than 95% of the target element is recovered
during the chemical procedure. Correcting for MDF is therefore a vital step in
the processing of isotope data. This correction, and how it may obscure other
isotopic signatures, is explained in the next section. Note that for stardust
the measured isotope ratios can be corrected for instrumental MDF using
external standards and the remaining natural MDF is insignificant relative to
the much larger observed stellar nucleosynthesis anomalies. This means that,
like stellar predictions, stardust compositions cannot be directly compared
against high precision data that has been corrected for MDF.

Mass-independent isotope variations are, for example, the stellar
nucleosynthetic signatures in bulk meteorites, of the order of $\leq 10^{-4}$,
now resolved at the order of $10^{-5}$ to $10^{-6}$. Other examples include
effects from the radioactive decay of unstable isotopes, photochemical
processes, such as the self-shielding during CO photodissociation invoked to
explain O isotope variations in the Solar System, and interaction of material
with galactic cosmic rays, which leads to the production of isotopes via
spallation and/or secondary neutron capture reactions leya13 . The last effect
can be corrected using cosmic neutron dosimeters, i.e., isotopes for which
such production is well characterised (for example, the isotopes of Pt hunt17
; ek20 ). The different size and shape of isotopes can also give rise to
nuclear field shift effects bigeleisen96 , although these typically do not
have a significant impact.

###  2.1 Internal normalisation procedure

When comparing stellar model predictions and stardust compositions to bulk
meteoritic data, it is imperative to follow the same data reduction procedure
that was used for the meteorite isotopic data set considered and is usually
specific to the instrument used to collect the data. In-situ measurements,
such as secondary-ion mass spectrometry SIMS routinely employed for stardust,
are not able to resolve nucleosynthetic differences between average bulk
meteorite groups due to their relatively lower precision. There are two other
major types of instrumentation for high-precision analysis of bulk meteorites:
thermal-ionization mass spectrometry (TIMS) and multi-collector inductively-
coupled-plasma mass spectrometry (MC-ICPMS). These are capable of producing
isotope data of an element with a precision down to a few part per million
(ppm) in a $\sim$1 to 100 mg of sample material dissolved in acidic media. The
MDF effects from such kind of measurements are usually in the permil range for
TIMS and percent range for MC-ICPMS. The exact degree of MDF needs to be
determined for each measurement (see, e.g., rehkamper12 ; schonbachler16a ;
schonbachler16b ) and changes with time during analysis. The TIMS and MC-ICPMS
analyses yield degrees of MDF that are often significantly larger than any
observed natural mass-dependent and mass-independent variation (although in
some instances nucleosynthetic effects in stardust and leachate samples can be
larger), and are routinely corrected for by internal normalisation schemes.

The correction of MDF by internal normalisation assumes that one of the
measured isotope ratios is identical to the known/published value of a chosen
terrestrial standard. The difference between the measured and the reference
value for this particular ratio is then used in the MDF correction procedure.
Any mass-independent variation on the normalisation ratio will result in a
modification of the MDF correction, and this in turn can drastically change
the signature of the resulting mass-independent data, as we will demonstrate
in Figure 2. The exponential law for isotope fractionation described by
Russell et al. russell78  for Ca is commonly applied to isotope studies of
different elements for both TIMS and MC-ICPMS analysis444Note that, in
general, correction of mass-dependent data needs to be carefully made using
the appropriate fractionation law. For example, material that has experienced
equilibrium effects would not follow the exponential law, and assuming such
law in this case would create an artificial mass-independent effect. In many
cases, we do not know which law is the correct one to use. Nevertheless, the
exponential law has been found to adequately account for the MDF in the
majority of studies.. This laws states that the MDF correction applied to the
mass-fractionated measured ratio ($r$)555Note that in Section 3, we will also
use the symbol $r$ in Eq. 2 for stellar model abundances - even though these
ratios do not contain a mass-dependent component. of two isotopes $i$ and $j$
is derived from the corresponding not mass-fractionated ratios ($R$) as an
exponential function of the ratio of their masses $m_{i}$ and $m_{j}$:

$R_{ij}=r_{ij}{\left(\frac{m_{i}}{m_{j}}\right)}^{-\beta},$ (2)

where $\beta$ is a measure of the MDF. For TIMS and MC-ICPMS analyses, if
$m_{i}>m_{j}$, the value of $\beta$ will generally be positive as these
instruments favour increasing the abundance of the heavier relative to the
lighter isotopes666Eq. 2 can also be written as
$r_{ij}=R_{ij}(m_{i}/m_{j})^{\beta}$. Sometimes in literature Eq. 2 is written
as $R_{ij}=r_{ij}(m_{i}/m_{j})^{\beta}$ instead, in which case the value of
$\beta$ changes sign relative to Eq. 2, i.e., a positive $\beta$ value in Eq.
2 would be a negative $\beta$ in this variation..

Since isotopes of the same element have identical chemical properties, all
isotopes during a single measurement follow the same mass fractionation law,
i.e., the value of $\beta$ is the same. The $j$ reference isotope at
denominator is usually chosen such that it has a high abundance and no
isobaric interferences, in order to minimize statistical uncertainties and the
error correlation between $r_{ij}$ and $r_{kj}$, the ratio used for internal
normalisation. If we then consider Eq. 2 but applied to the ratio $kj$ used
for internal normalisation, where the denominator isotope is by convention
always the same, we can extract the exponent ($-\beta$) , equate the two
expressions, and calculate $R_{ij}$ of a sample (SMP) as:

$R^{\rm{SMP}}_{ij}=r^{\rm{SMP}}_{ij}{\left(\frac{R^{\rm{STD}}_{kj}}{r^{\rm{SMP}}_{kj}}\right)}^{\frac{\ln{(m_{i})}-\ln{(m_{j})}}{\ln{(m_{k})}-ln{(m_{j})}}},$
(3)

or, changing the final step of the derivation $e^{ab}=(e^{a})^{b}$ to have the
exponent $=-\beta$, as:

$R^{\rm{SMP}}_{ij}=r^{\rm{SMP}}_{ij}{\left(\frac{m_{i}}{m_{j}}\right)}^{\frac{\ln{(R^{\rm{STD}}_{kj})}-\ln{(r^{\rm{SMP}}_{kj})}}{\ln{(m_{k})}-ln{(m_{j})}}},$
(4)

where it is assumed that $R^{\rm SMP}_{kj}=R^{\rm STD}_{kj}$, i.e., the
reference value used for the internal normalisation, usually the terrestrial
standard (STD). Note that an element needs at least 3 stable, measurable,
isotopes to account for the MDF using this approach. If the final aim is to
derive the nucleosynthetic effect, then the normalising isotopes should also
be free of other mass-independent effects, such as radiogenic decay (see,
e.g., discussion in Sec. 3.1).

Measured deviations from the terrestrial standard ratios are usually so small
that they are not given as percent, i.e., deviation in parts per 100, but as
deviation in parts per 1000, also called permil $\delta$ (and typical for
stardust data), as parts per ten thousands, called $\epsilon$, or parts per
million, called $\mu$. For example:

$\epsilon
R^{\rm{SMP}}_{ij}={\left(\frac{R^{\rm{SMP}}_{ij}}{R^{\rm{STD}}_{ij}}-1\right)}\times
10^{4}.$ (5)

The ratio used for internal normalisation can be indicated in different ways,
for example, by using the last digit of the mass of each isotope in the ratio
within brackets, sometimes separated by a comma. It is also frequently omitted
from the notation if all data presented uses the same internal normalisation
ratio. In the figures presented here we explicitly indicate the normalising
ratio at the top of each plot or in the figure caption (e.g., Fig. 1).
Combining equations 3 and 5, and remembering that $(a/b)^{c}$ is the same as
$(b/a)^{-c}$, gives:

$\epsilon
R^{\rm{SMP}}_{ij}={\left[{\left(\frac{r^{\rm{SMP}}_{ij}}{R^{\rm{STD}}_{ij}}\right)}{\left(\frac{r^{\rm{SMP}}_{kj}}{R^{\rm{STD}}_{kj}}\right)}^{-Q_{i}}-1\right]}\times
10^{4},$ (6)

where $Q_{i}$ is the exponent as given in Eq. 3. We further define
$\epsilon^{\ast}r_{ij}$ as the composition of a sample that has not been
corrected for MDF (i.e., calculated using $r_{ij}$ instead of $R_{ij}$ in Eq.
5).

In Figure 2 we use Eq. 6 to illustrate the effect of internal normalisation
for the example of an $s$-process sample with an added small mass-independent
signal. In the left panel (a) we show the solar abundance lodders09  (solid
line) and the $s$-process component from mainstream SiC grains stephan19
(dashed line). In the middle panel (b) we show a sample created by adding
$10^{-5}$ of the $s$-process component to the solar composition to recreate a
small $s$-process excess (dashed, orange line). Then, we mass-fractionate the
same sample to a degree typical of that seen for MC-ICPMS instruments
($\beta=1.5$), to create a synthetic meteorite isotope data point (solid, blue
line). The variation due to mass-fractionation is two to three orders of
magnitude larger than the nucleosynthetic variation. This highlights the need
of the correction for mass-dependent fractionation to ascertain the
nucleosynthetic composition of the sample, which is often orders of magnitude
smaller. Finally, in the right panel (c) of Figure 2 we show the sample after
it has been internally normalised to the solar abundance of 98Mo/96Mo. The
small negative offset caused by the $s$-process contribution on 98Mo/96Mo
(dashed line in the middle panel of Figure 2) results in a $\beta$ value
calculated from the sample of $1.4988$, instead of $1.5$. This small
difference represents the effect of the nucleosynthetic component and results
in a small under-correction of the mass-dependent component. Note that the
size of this under-correction is independent of the actual mass fractionation
experienced by the sample. This is because the apparent $\beta$ value of the
sample is always equal to the true $\beta$ ($1.5$ in this example) plus the
difference due to the mass-independent component ($-0.0012$ in this example).
Therefore, after internal normalisation (Figure 2c) we obtain the same
abundances starting from both lines in panel b, i.e., we obtain the same
abundances in both cases, if mass fractionation was added or not. The final
result of the internal normalisation procedure is that the $s$-process pattern
is rotated anti-clockwise around the normalising isotope (Figure 2c).

The larger the mass difference between each isotope and the normalising
isotope is, the larger the rotation effect will be, because the applied
correction factor scales with the mass difference between the numerator and
denominator isotopes. While the internally normalised $s$-process composition
is different from the “true” $s$-process pattern, it provides model
predictions in a representation directly comparable to the meteorite data
reported in literature.

Figure 2: Schematic description of the effect of the procedure of internal
normalisation on an example $s$-process sample. In the left panel (a), the
solid line represents the ratios of the Mo isotopes relative to 96Mo using the
solar abundances, the dashed line represents the same ratios calculated using
the $s$-process abundances from mainstream SiC grains stephan19 . In the
middle panel (b), the dashed, orange line (with corresponding values on the
right y-axis) represents the $\epsilon$ values of un-mass-fractionated (UMF)
$s$-process sample obtained by adding $10^{-5}$ of the $s$-process component
to the solar abundances. The solid, blue line (with corresponding values on
the left y-axis) represents the $\epsilon$ values of the sample of the dashed
line, but with some added mass-dependent fractionation (MDF) obtained by
adding a typical MDF effect with $\beta=1.5$. Note the huge difference between
the scale the left and right y-axes. The right panel (c) shows the internally
normalized abundance pattern for both the MDF and UMF samples in the middle
panel. Note that both lines are identical, which highlights the generality of
the internal normalisation method to remove the problem of MDF.

Table 1 reports a list of elements heavier than and including Fe, for which
nucleosynthetic variations are identified in bulk meteorites, together with
the ratios used for the internal normalisation and some recent references. The
normalising ratio is usually chosen to optimise the analytical resolution of
the small isotope anomalies present in meteorites (see, e.g., the discussion
in burkhardt11 ). The normalising ratio can also be chosen to minimize, or to
enhance, the rotational effect resulting from an expected mass-independent
signature. For example, the 98Mo/96Mo ratio commonly used for Mo data
minimises the rotational effect due to $s$-process variations because both
isotopes are significantly produced by the $s$ process. For elements for which
stardust SiC data is available, the normalising isotope used and example
references are also reported in Table 1.

Table 1: Iron and heavier elements that show nucleosynthetic heterogeneity in bulk meteorites and CAIs, the isotopic ratios most commonly used for the internal normalisation, and some example references. When also stardust SiC grains data is available, the isotope at denominator is reported together with example references. | Internal Normalisation | Stardust
---|---|---
Element | Ratio | Value | Reference | Isotope | Reference
Fe | 57/54 | 0.362549 | hopp22  | 56 | trappitsch18
| 57/56 | 0.023095 | hopp22  |  |
Ni | 61/58 | 0.016744 | cook21  | 58 | trappitsch18
| 62/58 | 0.053389 | cook21  |  |
| 62/61 | 3.188500 | steele12  |  |
Zn | 67/64 | 0.082160 | steller22  |  |
| 68/64 | 0.385564 | savage22  |  |
Sr | 86/88 | 0.119400 | charlier19  | 86 | liu15SrBa
Zr | 94/90 | 0.338100 | akram15  | 94 | barzyk07 ; nicolussi97
Mo | 98/96 | 1.453173 | spitzer20  | 96 | nicolussi98
Ru | 99/101 | 0.745075 | fischer17  | 100 | savina04
Pd | 108/105 | 1.188990 | ek20  |  |
Ba | 134/136 | 0.307800 | andreasen07  | 136 | liu14Ba
| 134/138 | 0.033710 | bermingham16  |  |
Nd | 146/144 | 0.721900 | frossard21  | 144 | yin06
Sm | 147/152 | 0.560830 | burkhardt15Nd  | 149 | yin06
Er | 166/168 | 1.241400 | shollenberger18  | 168 | yin06
Yb | 174/172 | 1.477200 | shollenberger18  | 172 | yin06
Hf | 179/177 | 0.732500 | shollenberger18  | 178 | yin06
|  |  |  | 180 | avila12
W | 186/183 | 1.985900 | kruijer17  | 184 | avila12
| 186/184 | 0.927670 | kruijer17  |  |

###  2.2 Calculating mixing lines

There are stark differences between meteorite data and stellar models due to
the fact that the meteorite data do not represent pure stellar compositions
(as in the case of stardust grains), but material in which the stellar
signature was strongly diluted with material carrying the standard solar
composition. In fact, the stellar composition is usually located far outside
the boundaries of a plot as that shown in Figure 1 and needs to be further
converted into the slope of a line that passes through the point representing
material of the standard, terrestrial, or solar composition (usually, but not
necessarily, the zero point). These lines are called “mixing lines” and
represent mixtures between the solar, or terrestrial, isotope composition and
that of the considered stellar source. Points along these lines represent
compositions with different dilution of the stellar source material, with the
stronger the solar dilution, the closer the point to the zero point. When
considering isotopes of the same element, mixing lines are straight in “three-
isotope plots”, i.e., two isotopic ratios with the same denominator plotted
against each other. This is the case, for example, for the comparison between
not internally normalised stardust and stellar model compositions. However, it
is not necessarily the case for mixing lines between the extremely different
internally normalised compositions considered here (stellar and solar). In
this case the mixing line is curved because the magnitude of the internal
normalisation correction scales exponentially, and not linearly, relative to
the difference between the true ratio and the assumed ratio. Therefore, in
this case the slope computed close to the point representing the internally
normalised stellar abundances can be significantly different from the slope
close to the [0,0] point (Figure 3).

To correct the stellar model predictions for MDF and finally derive the slope
of the mixing line that can be compared to that derived from meteorite data we
can reasonably approximate $\epsilon$ from Eq. 6 to the following linear
$\epsilon_{\rm lin}$ formula777This can be done for example by writing
($r^{\rm{SMP}}_{ij}/R^{\rm{STD}}_{ij}$) as
($1+10^{-4}\epsilon^{\ast}r^{\rm{SMP}}_{ij}$) from Eq. 6, and equivalently for
the $kj$ ratios, where the asterisk indicates that the value is calculated
from $r^{\rm{SMP}}_{ij}$ rather than $R^{\rm{SMP}}_{ij}$ as in Eq. 5, and then
using the first two terms of the Taylor expansion $(1+x)^{-Q}=(1-Qx)$ followed
by some simple algebra and the removal of the second order term multiplied by
$10^{-8}$.:

$\epsilon R^{\rm{SMP}}_{ij}\simeq\epsilon_{\rm
lin}R^{\rm{SMP}}_{ij}={\left[{\left(\frac{r^{\rm{SMP}}_{ij}}{R^{\rm{STD}}_{ij}}-1\right)}-Q_{i}{\left(\frac{r^{\rm{SMP}}_{kj}}{R^{\rm{STD}}_{kj}}-1\right)}\right]}\times
10^{4}.$ (7)

The slope is then determined by dividing the $\epsilon_{\rm lin}$ values
calculated for two different isotopes. Note that masses considered in
cosmochemisty are usually not rounded, for example, for 86Sr the accurate
value of 85.9092606 u is used, etc. For the purpose of calculating mixing
lines for comparison with meteorite data, stellar abundances can be used for
the calculation of the slope directly using $\epsilon_{\rm lin}$ from Eq. 7,
without the need to construct a synthetic sample as we did in Figure 2.

As illustrated in Fig. 3, the $\epsilon_{\rm lin}$ values calculated from Eq.
7 are by definition linear approximations and should be used only to derive
the slope close to the [0,0] point representing the solar composition (as done
in Figure 4).

To derive the slope close to the [0,0] point using the $\epsilon$ values
calculated from Eq. 6 instead, it is necessary to create a synthetic sample by
diluting the stellar abundances with the solar abundance using Eq. 1 (see,
e.g., Fig. 2, and Simon et al. simon09 ).

Figure 3: Mixing lines calculated for Mo between the solar composition and the
$s$-process composition, represented by mainstream SiC grains stephan19 ,
using the exponential (Eq. 6; Blue solid line) and linear (Eq. 7; Orange
dashed line) equations. The slopes of the two mixing lines overlap only when
the offset from the solar composition (point [0, 0]) is of the order of a few
100 $\epsilon$ or less. Bulk meteorites exhibit anomalies well within this
range, therefore the linear method is a valid approximation to compare the
models to these data.

We investigated the difference between $\epsilon$ and $\epsilon_{\rm lin}$ for
a synthetic sample created using a selection of stellar models from the FRUITY
database cristallo11FRUITY  in combination with solar compositions from
Lodders et al. lodders09  and Anders & Grevesse anders89 . The values
calculated for the comparisons of Sec. 3 are available in the Supplementary
Material. We find that there is virtually no difference between mixing lines
calculated using $\epsilon$ (Eq. 6) on a synthetic sample (from Eq. 1), or
using $\epsilon_{\rm lin}$ (Eq. 7) directly on the stellar abundances.
Furthermore, the choice of the normalising solar/terrestrial ratios does not
have a large effect on the calculated mixing lines because the stellar
abundance ratios are typically very different from them. However, some
variation between using $\epsilon$ and $\epsilon_{\rm lin}$ are found for
large mass differences (i.e., when $Q$ is large) and for models that produce
variations of two orders of magnitude or more. Generally, to calculate mixing
lines it is sufficient to use the same solar values taken in the model
calculations to represent the initial composition of the star, even if they
are not identical to the reference values in meteorite studies reported in
Table 1. Although, one needs to be careful of artifacts that can appear for
models where no, or little nucleosynthesis takes place if the solar abundances
used in the models as starting compositions differ slightly - even if only in
the fifth digit - from those used in the calculation of $\epsilon$ value. This
will produce spurious values different from zero and therefore erroneous
numbers for the slopes. We also compared $\epsilon_{\rm lin}$ to the
linearised equations in Dauphas et al. dauphas04  frequently used in the
literature to calculate mixing lines. The approach presented here and that of
Dauphas et al. dauphas04  are mathematically identical with the exception that
Dauphas et al. dauphas04  further simplifies the calculation of $Q_{i}$ by
omitting the natural logarithm of the masses, thus approximating $Q_{i}$ as
$(m_{i}-m_{j})/(m_{k}-m_{j})$ (Eq. (3) in dauphas04 ). This simplification
does not significantly affect the mixing lines for most models. The largest
differences occur for the largest mass differences and for the largest
differences in magnitude between the smallest and largest variations. We
recommend using the natural logarithm of the masses in Eq. (3) of dauphas04 ,
which yields identical mixing lines to those calculated using Eq. 7.

A more complex situation arises when isotopes of different elements are
plotted against each other because in this case variations due to chemistry
can also play a role. This is referred to as a four-isotope plot in
geochemistry and the resulting mixing lines are hyperbolas. Here the mixing
relationship not only depends on the isotopic compositions, but also on the
elemental ratio of the elements involved. The more this elemental ratio
differs between Solar System material and the stellar material, the stronger
is the curvature of the hyperbola. There are two ways that the stellar
elemental ratios can differ from solar. First, the nucleosynthesis that
affects the isotopic ratios may also affect the elemental ratios. The
curvature of the hyperbola can be determined directly from the stellar
nucleosynthesis models as the ratio of the over-abundances, relative to solar,
of the two isotopes at denominator. Close to zero, i.e., for high dilutions,
such hyperbola can still be approximated by a straight line and it is possible
to consider this effect by simply multiplying the slope derived from
$\epsilon_{\rm lin}$ (using Eq. 7) by the ratio of the over-abundances,
relative to solar, of the two isotopes at denominator, as in the case of Mo
and Ru discussed by dauphas04 . Second, the chemistry of dust formation may
affect the relative abundance of the different element in a stardust grain. In
this case, a further factor needs to be introduced to account for the fact
that different elements may be incorporated into stardust differently, also
depending on the environment and the type of dust. The difficulty here in
defining the effect of chemical processes is that stardust condenses in
situations of non-equilibrium chemistry; moreover, for many trace elements we
do not know how their condensation works. In these cases, not discussed here
further, we need to be guided by observations and reasonable assumptions. For
example, in the case of the Cr-rich oxide grains, den Hartogh et al.
denhartogh22  used the Al:Mg ratios measured in those grains dauphas:10 . For
heavy $s$-process elements in SiC some data on elemental abundances are
available amari95b , for the case of Pd, however, no quantitative data is
available, and only a qualitative behaviour was considered by Ek et al. ek20
to interpret the data.

###  2.3 Further considerations on the comparison between data and models

For comparison to the bulk meteoritic data, local stellar abundances should be
used at the time and location where the carrier of the observed anomalies
(e.g., stardust) potentially formed. These are different from the total
integrated stellar yields (i.e., the abundances integrated over the total mass
ejected) required by models of the chemical evolution of the Galaxy. For AGB
stars, the final surface abundances are a good approximation to consider for
each stellar model (for example, of a given mass and metallicity). This is
because there is a high degree of both spatial and temporal homogeneity in the
outer regions where dust forms. Spatially, the whole envelope of the star is
convective, and therefore material is well mixed. Temporally, most of the mass
loss occurs in the very final phases of the evolution, therefore, most of the
stardust grains carry such a final composition. Also when considering core-
collapse supernova models, the local abundances as function of the location in
mass within the ejecta should be used, rather than the total yields or the
surface composition. This is because dust formation probably occurs before the
material in the different mass layers of the ejecta is microscopically mixed
with all the layers (see discussion in pignatari13grains ; denhartogh22 ).
Therefore, while for AGB stars, one slope well represents one stellar model,
for core-collapse supernovae, it is necessary to plot the slopes of the mixing
lines derived from the compositions at different mass coordinate steele12 ;
hopp22 . Different mass shells of a given supernova experience different
nucleosynthesis and variable slopes are produced. Total stellar yields are
useful if it is of interest to consider dust formed in the interstellar
medium, after all stellar yields are homogenised. In this case, however, it is
not single star models that should be considered but their implementation into
the global modelling of the chemical evolution of the Galaxy, or at least of a
stellar population. This is because the interstellar medium is built by many
different stellar sources.

Most previous work on the $s$-process elements have considered the $s$-, $r$,
and $p$-component abundances presented by Arlandini et al. arlandini99  and
Bisterzo et al. bisterzo11PaperII . However, these studies were targeted at
matching the bulk $s$-process distribution of the Solar System. The observed
anomalies could derive, instead, from specific astrophysical sources that may
not follow such distribution. In fact, stardust SiC grains from AGB stars are
well known to present $s$-process isotopic features that are not the same as
those of the solar distribution ott90 ; hoppe97 . For example, matching the
grain data requires a lower time-integrated neutron flux - resulting in lower
88Sr/86Sr and 138Ba/136Ba ratios - than that which produced the solar
$s$-process abundance pattern (see, e.g., lugaro18grains ; liu18c13pocket ;
lugaro20  and Table 2). Therefore, specific models of AGB stars of different
masses and metallicities need to be compared to stardust and bulk meteoritic
data because each stardust grain can in principle originate from a different
star. The same consideration in principle applies to the anomalies referred in
the literature as to $r$\- and $p$-process anomalies. The main $r$\- and
$p$-process sites in the Galaxy may be copious and variable over the history
of the Galaxy: for the $r$ process they range from, e.g., neutron stars
mergers to magnetars cowan21 , for the $p$ process they may also include
thermonuclear supernovae travaglio18 . Furthermore, anomalies that mimic the
$r$ and the $p$ processes may be also produced in astrophysical sites that are
not the major contributors of these isotopes in the Galaxy, but that can still
affect relative isotopic abundances and be the site of dust formation. For
example, the neutron bursts (the $n$ process) in the He shell of core-collapse
supernovae and possible explosive neutrino-wind components produce a peculiar
signal that for certain isotopic ratios may look similar to the $r$ process
(e.g., farouqi09 ; pignatari18 ; bliss18 ; kratz19 ).

Finally, we note that for core-collapse supernovae most of the comparisons
available in the literature used the models by Rauscher et al. (rauscher02 ,
although see an exception in den Hartogh et al. denhartogh22 ). It is well
known that core-collapse supernovae models are very uncertain therefore it is
unsatisfactory that typically just one set of models has been employed. More
models, with their detailed composition as function of mass, needs to be made
available for this task. One of the main aims here is to illustrate for the
nuclear astrophysics community the significance of stellar model predictions
within the framework of meteorite data. Working Package 9 of the ChETEC-INFRA
www.chetec-infra.eu project (funded by the European Union’s Horizon 2020
research and innovation programme) has the aim to provide the tools for the
comparison so that more researchers can contribute to the field of
cosmochemistry.

##  3 Examples

To illustrate the effect of the internal normalisation on the $s$-process
predictions from AGB models we present some examples for two selected
elements: Sr and Mo, using the data from the models of lugaro18grains . To
generate the plots we use the open source code available via ChETEC-INFRA at
https://www.chetec-infra.eu/resources/#ToolsMeteorites. Among the models
presented by lugaro18grains , we consider first that of initial mass 3 M⊙ and
solar metallicity (Z=0.014 from asplund09 ) as a typical $s$-process result,
and then compare it to two more models of the same and of higher mass (3 and 4
M⊙) and twice-solar metallicity (Z=0.03). The over-abundances, relative to the
initial values, of the isotopes of interest here for these three models are
reported in Table 2.

Table 2: Over-abundances, i.e., final production factors at the stellar surface relative to the initial values, of the isotopes of interest for the three AGB models considered here. For the Z=0.014 model, the initial values are the solar values given by asplund09 , for the Z=0.03 models, they are the same values multiplied by roughly 2, by definition, which results in lower over-abundances, but does not impact the isotopic ratios. The $p$-only 84Sr and 92Mo, and the $r$-only 100Mo are not included in the network of karakas16  and are assumed here to have production factors equal to unity in all the models, which is accurate within a few percent (see also lugaro18grains  for discussion on 84Sr). Also shown are their ratios relative to another isotope (in brackets), which is the first step for the calculations of $\epsilon$ values.  Isotope | M=3 M⊙; Z=0.014 | M=3 M⊙; Z=0.03 | M=4 M⊙; Z=0.03
---|---|---|---
86Sr (/88Sr) | 26 (0.81) | 20 (1.18) | 10 (1.02)
87Sr (/86Sr, /88Sr) | 25 (1.25, 0.78) | 19 (0.95, 1.12) | 9.5 (0.85, 0.97)
88Sr (/86Sr) | 32 (1.23) | 17 (0.85) | 9.8 (0.98)
94Mo (/96Mo) | 1.28 (0.03) | 1.18 (0.08) | 0.99 (0.13)
95Mo (/96Mo) | 24 (0.63) | 8.3 (0.59) | 5.5 (0.73)
96Mo | 38 | 14 | 7.5
97Mo (/96Mo) | 21 (0.55) | 8.5 (0.61) | 4.2 (0.56)
98Mo (/96Mo) | 30 (0.79) | 11 (0.79) | 5.0 (0.67)

###  3.1 Strontium

Due to mass-independent variations of 87Sr as the result of the radiogenic
decay of the long-lived isotope 87Rb (T1/2=49 Gyr), current bulk meteoritic
data for Sr have limited application for nucleosynthetic studies. However, we
chose this element as a first example because it has peculiar features from
the point of view of the $s$-process nucleosynthesis and only 4 stable
isotopes, at masses 84, 86, 87, 88. This makes Sr a useful element to
demonstrate the effect of the internal normalisation on $s$-process abundance
because for any given internal normalisation there are only two ratios that
can be plotted against each other and therefore only one plot.

In principle, given the 4 isotopes, Sr offers 12 possibilities of internal
normalisation choices, i.e., twice the number of possible combinations of 2
isotopes out of 4. However, it is a general feature that the slope of the line
calculated using the internally normalised ratios remains the same if we
consider one normalising ratio or its reverse (e.g., for 86Sr/87Sr or
87Sr/86Sr, as shown in Figure 4). Therefore, generally speaking the number of
possible combinations given a number of isotopes $n$ is equal to
$n$!/2($n-k$)!, where $k$=2 and $n$ is the number of isotopes ($n$=4 in the
case of Sr).

Of the four isotopes, 84Sr is a pure product of the $p$ process, therefore its
abundance at the surface of AGB stars is unchanged, within a few percent,
relative to the initial abundance. The other 3 isotopes, instead, are mostly
of $s$-process origin, specifically 88Sr has a magic number of neutrons (50)
and represents, with Y and Zr, the first $s$-process peak. 86,87Sr are
produced both in massive stars and in AGB stars (with roughly half and half
contributions to their solar abundances), while 88Sr is most efficiently
produced in AGB stars888A debate about a possible $\sim$20% contribution from
massive stars to the $s$-process first peak elements is ongoing. Such a debate
does not crucially affect our discussion here because we compare data to
single AGB models rather that to the full chemical evolution of the Galaxy.
travaglio04 ; cristallo15LEPP ; prantzos20 .

Strontium is also interesting because the 88Sr/86,87Sr isotopic ratios vary
the most among all the $s$-process elements when changing the metallicity of
the AGB star lugaro20 . Models of metallicity higher than solar are a better
match to the composition of large ($>1\mu$m) SiC grains - especially for Sr
and Zr, but also for Ba - and their composition differ from that of their
solar metallicity counterpart, as discussed at length previously
lugaro18grains ; lugaro20 . This result is also clearly shown in Table 2 where
88Sr is more over-produced than the other Sr isotopes at solar metallicity,
while it has a similar, or even lower, production factor at the higher
metallicity. It is therefore interesting to investigate if these different
$s$-process predictions are still resolved by the different mixing lines
derived within the representation discussed above, and, if so, under which
choices of the internal normalisation these variations are more or less
evident.

Figure 4: Example for calculating the slope of the mixing line in the vicinity
of the Solar System end-member (the [0,0] point) using the $s$-process
predictions at the surface of an AGB star of initial mass 3 M⊙ and solar
metallicity (Z=0.014) at the end of the evolution from Table 2. Solid and
dashed lines represent, respectively, the slopes derived using $\epsilon_{\rm
lin}$ (i.e., the linear approximation Eq. 7) and $\epsilon^{\ast}$ (i.e.,
derived from applying Eq. 5 directly to the stellar abundance ratios). Here we
considered 3 different possible choices of the normalising ratios, each
following the top label “Internal norm. to”. The arrows on each mixing line
indicate the direction of the $s$-process end-member. The numerical values
needed to calculate $\epsilon_{\rm lin}^{\ast}$ are also reported, to support
the clarity of the description in the text. Slopelin and Slope∗ in the legend
of this figure, and following figures, refers to the slope calculated using
$\epsilon_{\rm lin}$ and $\epsilon^{\ast}$, respectively. As the normalising
ratios are indicated separately in each plot, in this and the following
figures $\epsilon$ values are followed only by the isotope at numerator.

Figure 4 presents three example out of the possibilities of internal
normalisation choices mentioned above. The $\epsilon^{\ast}$ and $\epsilon$
values can be easily derived from the values of the over-abundance relative to
solar provided in Table 2 using Eq. 5 and Eq. 7, respectively. The
transformation between $\epsilon^{\ast}$ (derived without internal
normalisation) and $\epsilon_{\rm lin}$ (derived with internal normalisation)
clearly shows that the slopes of the mixing lines can be more or less modified
depending on all the different factors involved. For example, the subtraction
term in Eq. 7 can change sign depending on the values of both the relevant
$\epsilon^{\ast}$ and $Q$ values. Therefore, if the normalising ratio is
86Sr/88Sr (left panel, a), all the $Q$ values are positive, all the
$\epsilon_{\rm lin}$ and $\epsilon^{\ast}$ values remain strongly negative,
and the slope does not change significantly. If the normalising ratio is
88Sr/86Sr (middle panel, b), not all the $\epsilon^{\ast}$ are negative and
not all the $Q$ are positive, which result in a modification of the slope. The
left (a) and middle (b) panels demonstrate that reversing the normalising
ratio does change all the values of $\epsilon_{\rm lin}$ and
$\epsilon^{\ast}$, however, the final slope from the ratio of the
$\epsilon^{\ast}$ values remains unchanged. Note that in these cases the
direction of the $s$-process excess is always towards negative values, as
indicated by the arrows in the figure, and data points located on the line in
the quadrant of positive values would represent material with an $s$-process
deficit. The right panel (c) shows a more dramatic effect from using the
$p$-only 84Sr as the internally normalising isotope and the resulting negative
$Q$ values. In this case the strongly negative $\epsilon^{\ast}$ used for the
normalisation, coupled to the negative $Q$ values, produce an inversion of the
sign of the $\epsilon^{\ast}$ value corresponding to 88Sr. While the examples
presented here are far from exhaustive, they clearly demonstrate that the
representation of the predicted abundances for comparison to meteoritic data
are far from intuitive and need to be accurately calculated for each case
before conclusions can be drawn.

Figure 5: Same as Figure 4 but for the 6 possible different internal
normalisation for Sr (with the corresponding internal normalisation ratio
indicated at the top of each panel) applied to the three AGB models considered
here (M=3 M⊙with Z=0.014 and 0.03, and M=4 with Z=0.03), represented each with
the different colors indicated in the legend.

Figure 5 shows all the possible normalisation for the three models considered
here, where we can identify which normalisation better preserves the large
difference in the relative over-production of the Sr isotopes resulting from
changing the metallicity from 0.014 (blue line) to 0.03 (orange and green
lines). The clearest signature of the different behaviours is shown by the
87Sr/88Sr normalisation, where correlated variations are only appearing for
the Z=0.014 model while for the other models all the lines are almost flat.
Using 84Sr/87,88Sr as normalising ratios also produce significant deviations
between the different models, while with the often used normalisation ratio
86Sr/88Sr, it may be more difficult to pick up the difference between these
specific models, unless the error bars on the data were smaller than the
plotted variation, e.g., less than $\sim\pm$0.10 in $\epsilon$87Sr and for
$\epsilon$84Sr$=-1$.

###  3.2 Molybdenum

Molybdenum has 7 stable isotopes, of which two, the $p$-only 92Mo and the
$r$-only 100Mo are not produced in AGB stars, while all the others are
produced to variable degrees (Table 2). The number of possible normalisation
is 21 and for each given normalisation, 10 different plots can be produced for
all the possible combinations of two isotopic ratios. All these possibilities
are of interest and should be considered in a dedicated study such as that of
stephan21 . Here, we will focus exclusively on the comparison of the model
predictions with the data trends shown in Figure 1.

Figure 6: Slopes predicted using the three different AGB models considered
here (represented each with the color indicated in the legend) for the same Mo
isotopes in the same normalisation as Figure 1. In the right panel the
abundance by number (N) of 95Mo was divided by 1.3 to mimic the effect of the
correspondingly higher neutron-capture cross section reported by koehler22 .
The solid black line represents the mainstream SiC trend line from Figure 1 .

Differently to Sr, the models do not show large variations among each other in
the relative production factors of Mo (see numbers in brackets in Table 2).
They show variations of the order of 10% only, and all in the same direction
relative to the $s$-only 96Mo. This is because the Mo ratios that involve the
isotopes with mass 95, 96, 97, and 98 mostly depend on the ratio of their
neutron-capture reaction rates (although in some conditions the branching
point at 95Zr can be marginally activated, by-passing 95Mo and 96Mo). These
rates have typically a mild temperature dependence because they decrease by
10%, 14%, and 23% between 5 and 30 KeV for 95Mo, 97Mo, and 98Mo, respectively.
The rate of 96Mo instead decreases more, by 50% (see kadonis.org database
dillmann06 ). Some minor production of the classical $p$-only isotope 94Mo may
also occur during the $s$ process, due to neutron captures on the 92Mo and
93Nb initially present in the star.

Figure 6 shows the model predictions for Mo in the representation as used in
Figure 1. The slopes do not vary significantly either when the ratios are
internally normalised, or when different models are considered. All the models
reported here predict slopes between 0.35 and 0.45. Some models of lower mass
(down to 2 M⊙, not discussed here in detail) can reach a slope of 0.5. The
direction of the $s$-process excess in the 98Mo/96Mo normalisation is always
towards negative values, as 96Mo is an $s$-only isotope and the most over-
abundant in AGB models (Table 2).

The slope of the MS SiC grains trend, almost parallel to the bulk meteoritic
slopes, is equal to 0.608 $\pm$ 0.007 (1$\sigma$; Figure 1). Therefore, AGB
models show a discrepancy relative to this value. However, the neutron-capture
rates currently used in the models are based on experiments carried out in
1987 winters87 . A recent analysis for 95Mo koehler22  resulted in a rate with
values highly discrepant from the old values, with the new rate up to 20-30%
higher than the older rate. To mimic the effect of these new rates, we divided
the 95Mo abundances in the models considered here by 1.3 (as a higher rate
results in a lower abundance). In this case, and specifically for the 3 M⊙,
Z=0.03 model, the slope become the same as that of the stardust and bulk
meteoritic data. This modification of the 95Mo neutron-capture cross section
was already predicted by Lugaro et al. lugaro03grains  on the basis of the SiC
data and anticipated by Koehler et al. koehler08 . This example clearly
demonstrates that improvements in the estimates of neutron-capture cross
sections are an essential ingredient to interpret bulk meteoritic data.

##  4 Conclusions

We present the methodology required to transform stellar model predictions
into the representation required to compare them to bulk meteorite data, which
removes the need to accurately know the effect of the ubiquitous process of
mass-dependent fractionation (MDF) in the data. We motivated the necessity of
using such representation as the way by which stellar nucleosynthesis models
can be exploited together with meteorite isotope data to shed light on the
formation of the Solar System and the environment where this occurred. We have
used some examples from the $s$ process as case studies to show in detail the
effect of such normalisation (using Sr as the example) and to stress the fact
that it is not possible to predict a slope, not even qualitatively, without
accurate calculations of the effect of internal normalisation. The
calculations per se are simple enough as various approximations, such as Eq.
7, are valid and can be used to derive the mixing line with an end-member of
stellar composition (see also Supplementary Material). The complications are
due to the large number of possibilities to be explored for each element. In
general, the choice of the isotopic ratios to use for the double normalisation
needs to match that of the data, as determined by analytical limits and
uncertainties. Still, careful consideration of the model predictions is also
pertinent. Depending on the models and the element considered, some results
may become more evident by using one normalising ratio rather than another, as
shown here for the example of the $s$-process Sr (Figure 5). It is possible to
identify the effect of both the properties of the stellar models (e.g., mass
and metallicity) and of the nuclear physics input via a careful analysis of
each isotopic ratio. As illustrative examples, we have shown here that Sr is
strongly affected by stellar metallicity, while Mo is mostly sensitive to the
nuclear physics input. Overall, a huge amount of future work is required to
investigate the significant potential of mapping stellar nucleosynthesis
models onto the representation explained here. Each element, each stellar
model, and each comparison to available data and their interpretation should
be treated in detail, not only for AGB $s$-process predictions but also for
core-collapse supernova models. Within the ChETEC-INFRA www.chetec-infra.eu
project (to run between 2021 and 2025) we are providing the tools and the
models to perform these tasks.

##  5 Supplementary Material

The spreadsheet NormalisationComparison.xlsx contains a comparison between
$\epsilon$ values calculated using the exponential (Eq. 6; “Exponential”), the
linear (Eq. 7; “Linear”) and the linear equation in Dauphas et al. dauphas04
(“Dauphas”) for a selection of $s$-process models from the FRUITY database
cristallo11FRUITY  for all elements listed in Table 1. Mixing lines were
created from these values by dividing the calculated offsets for two different
isotopic ratios. The “Exponential” values were calculated by creating a
synthetic sample (see Eq. 1) where the largest $\epsilon$ offset is always
equal to 1 or -1. The “Linear” and “Dauphas” values were calculated directly
from the stellar abundances and then renormalised such that the largest offset
was equal to 1 or -1, which enables direct comparison between the different
methods. The calculations were performed using the solar values from Lodders
et al. lodders09  (L09) and Anders & Grevesse anders89  (AG89) and the same
values also renormalised to the ratios listed in Table 1. As can be seen in
the figures included in each sheet, there is very little difference in the
slopes calculated with the different methods and solar values.

A Jupyter notebook is available at https://github.com/mattias-
ek/internal_normalisation, which can generate all figures included in the
paper as well as the data included in the spreadsheet discussed above.

###### Acknowledgements.

This paper is based on work carried out as part of Work Package 9 of the
ChETEC-INFRA project funded by the European Union’s Horizon 2020 research and
innovation program under grant agreement No 101008324, the ERC-CoG-2016
RADIOSTAR project (Grant Agreement 724560), and the Hungarian NKFI(OTKA)
K-138031 project. GVM was supported by the European Research Council (Advanced
Grant Agreement 833275—DEEPTIME). IO was supported by the Carlsberg Foundation
(CF20_0209) and the Villum Fonden (00025333). MS and ME acknowledge funding
from the Swiss Science Foundation. We also acknowledge the support of the
NuGrid collaboration, http://www.nugridstars.org, the Joint Institute for
Nuclear Astrophysics - Center for the Evolution of the Elements, USA and the
the US IReNA Accelnet network (Grant No. OISE-1927130). We thank the referees,
Thomas Stephan and Larry Nittler, for their careful reading and for providing
us with many comments, which have helped us to improve the manuscript. We
thank Nicolas Dauphas and Timo Hopp for discussion. ML thanks Evelin Bányai
for help with python and making figures. ML and MP are deeply grateful for the
opportunity to contribute to this volume in memory of Franz Kaeppeler, who has
always been a friend and a mentor to them.

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