11institutetext:  Kapteyn Astronomical Institute, University of Groningen,
Landleven 12, 9747 AD Groningen, The Netherlands 22institutetext: LPSC,
Université Grenoble Alpes, CNRS/IN2P3, 53 avenue des Martyrs, 38026 Grenoble,
France 33institutetext: LAPTh, Université Savoie Mont Blanc & CNRS, Chemin de
Bellevue, 74941 Annecy Cedex, France 44institutetext: Université Paris-Saclay,
CNRS/IN2P3, IJCLab, F-91405, Orsay, France

# Transport parameters from AMS-02 F/Si data
and fluorine source abundance

E. Ferronato Bueno<EMAIL_ADDRESS>L. Derome 22 Y. Génolini 33 D.
Maurin<EMAIL_ADDRESS>V. Tatischeff 44 M. Vecchi 11

(Received / Accepted)

###### Abstract

Context. The AMS-02 collaboration recently released cosmic-ray F/Si data of
unprecedented accuracy. Cosmic-ray fluorine is predominantly produced by
fragmentation of heavier progenitors, while silicon is mostly accelerated at
source. This ratio is thus maximally sensitive to cosmic-ray propagation.

Aims. We study the compatibility of the transport parameters derived from the
F/Si ratio with those obtained from the lighter Li/C, Be/C, and B/C ratios. We
also inspect the cosmic-ray source abundance of F, one of the few elements
with a high first ionisation potential but only moderately volatile, and a
potentially key element to study the acceleration mechanism of cosmic rays.

Methods. We use the 1D diffusion model implemented in the usine code and
perform $\chi^{2}$ analyses accounting for several systematic effects (energy
correlations in data, nuclear cross sections and solar modulation
uncertainties). We also take advantage of the exfor nuclear database to update
the F production cross sections for its most important progenitors (identified
to be 56Fe, 32S, 28Si, 27Al, 24Mg, 22Ne, and 20Ne).

Results. The transport parameters obtained from AMS-02 F/Si data are
compatible with those obtained from AMS-02 (Li,Be,B)/C data. The combined fit
of all these ratios leads to a $\chi^{2}_{\rm min}/{\rm dof}{}\approx 1.1$,
with $\lesssim 10\%$ adjustments of the B and F production cross sections (the
latter are based on very few nuclear data points, and would strongly benefit
from new measurements). The F/Si ratio is compatible with a pure secondary
origin of F, with a best-fit relative source abundance
${\rm(^{19}F/^{28}Si)}_{\rm CRS}\sim 10^{-3}$ and an upper limit of $\sim
5\times 10^{-3}$. Unfortunately, this limit is not sufficient to test global
acceleration models of cosmic-ray nuclei, for which values at the level of
$\sim 10^{-4}$ are required. Such levels could be attained with F/Si data of a
few percent accuracy at a few tens of TV, possibly within reach of the next
generation of cosmic-ray experiments.

###### Key Words.:

Astroparticle physics – Cosmic rays

##  1 Introduction

The AMS-02 experiment, on the International Space Station since 2011, has the
capability to identify cosmic-ray (CR) elements up to Ni, and it has already
provided a huge body of high-precision CR data (Aguilar et al. 2021b). In
particular, AMS-02 provides elemental fluxes and ratios at percent level
precision from $\sim 1$ GV to a few TV in rigidity ($R=pc/Z$). These data
challenge the standard picture (e.g., Strong et al. 2007; Grenier et al. 2015;
Gabici et al. 2019) of CR transport in the Galaxy.

Broadly speaking, CR nuclei can be separated in two categories, related to the
process making up most of their flux: primary species mostly come from
material synthesised and later on accelerated in astrophysical sources (Meyer
et al. 1997; Ellison et al. 1997; Lingenfelter 2019), while secondary species
originate from nuclear interactions (fragmentation) of the primary species on
the interstellar medium (ISM) (Ginzburg & Syrovatskii 1964). In releasing
their data, the AMS-02 collaboration started from the most abundant species,
namely the light primary species H (Aguilar et al. 2015a) and He (Aguilar et
al. 2015b), then move on to C and O (Aguilar et al. 2017), and more recently,
to Ne, Mg, Si (Aguilar et al. 2020), and also Fe (Aguilar et al. 2021c). These
data are crucial for the modelling of secondary species, because the break-up
of these nuclei populate and contribute, to varying fractions, to the flux of
all species lighter than Si. Examples of mixed species are N, Na and Al
(Aguilar et al. 2021a), which receives a roughly similar amount of primary and
secondary contributions at a few GV (Maurin 2020). Secondary-to-primary ratios
are of special interest, since they allow to study the transport parameters
independently of source ones (e.g., Ginzburg & Syrovatskii 1964; Maurin et al.
2002; Génolini et al. 2015). The B/C ratio was the first secondary-to-primary
ratio released by the AMS-02 collaboration (Aguilar et al. 2016), followed by
Li/C and Be/C (Aguilar et al. 2018) and by the 3He/4He isotopic ratio (Aguilar
et al. 2019).

We focus in this paper on the recently released AMS-02 F/Si ratio (Aguilar et
al. 2021d), extending our previous efforts to interpret AMS-02 secondary-to-
primary ratios (Génolini et al. 2017, 2019; Weinrich et al. 2020b; Maurin et
al. 2022b). While the F/Si ratio can be considered as a secondary-to-primary
ratio (see below), it has not been recognised as such, except in one very
recent study (Boschini et al. 2022). Indeed, the bulk of the CR modelling
literature focused on the B/C ratio (Strong & Moskalenko 1998; Putze et al.
2010; Trotta et al. 2011; Tomassetti & Donato 2012; Cholis & Hooper 2014;
Kappl et al. 2015; Génolini et al. 2019; Evoli et al. 2020), the 3He/4He ratio
(Seo & Ptuskin 1994; Webber & Rockstroh 1997; Coste et al. 2012; Tomassetti
2012b; Picot-Clémente et al. 2017; Wu & Chen 2019), and the sub-Fe/Fe
ratio111In most publications, Sub-Fe=Sc+Ti+V, i.e. $Z=21-23$ elements.
(Garcia-Munoz et al. 1987; Webber et al. 1992; Jones et al. 2001; Maurin et
al. 2001; Evoli et al. 2008). The reason behind this oversight can hardly be
explained by the quality of the data: previous detectors measuring F and Si,
e.g. HEAO3 (Engelmann et al. 1990) and ACE (George et al. 2009; Lave et al.
2013), had similar systematics over a wide range of charges, with only
$\sim\\!\sqrt{2}$ larger statistical uncertainties for F/Si compared to sub-
Fe/Fe (Si/Fe $\sim 1.5$ and F/Sub-Fe $\sim 0.8$, Engelmann et al. 1990). But
it is possibly related to nuclear physics: until recent times, the community
was lacking reliable nuclear production cross sections for F. As we will see,
nuclear data for its production are particularly scarce, with only a few
relevant measurements in the 90’s and 00’s. Moreover, the production of F
involves more progenitors compared to other secondary species, which means
than the number of cross sections required for CR calculations is larger. As
discussed below, the main five progenitors of 19F222We use indifferently F or
19F (only stable isotope for this element). (20,22Ne, 24Mg, 28Si, and 56Fe)
only make $\sim 60\%$ of the total production. In comparison (Coste et al.
2012; Génolini et al. 2018; Maurin et al. 2022b), only two progenitors (12C
and 16O) make $\sim 60\%$ of all the Li, Be, and B, and even one single
progenitor, 4He (resp. Fe), makes $\sim 90\%$ of 3He (resp. sub-Fe).

A possible issue with using F/Si as a probe of CR transport is that F is
expected to have a tiny but non-zero primary abundance. Indeed, whereas the
ratio of 3He, Li, Be, B, and sub-Fe to their progenitors are in trace amount
in the Solar system (SS), F/Si is measured at the percent level (Lodders
2003): this should translate very roughly into a percent level contribution in
the F/Si CR flux ratio at GeV/n, unless some specific mechanism were to
accelerate F more efficiently than Si. Actually, in the range of elements
measured by AMS-02, F might be a key species to study the origin of cosmic
rays (Lingenfelter 2019; Tatischeff et al. 2021). As a matter of fact, two
competing explanations have been advocated to explain the CR source
composition derived from CR data, either from a first ionisation potential
(FIP) bias—similar to that found in solar energetic particles (Meyer et al.
1979; Meyer 1985)—or a volatility bias (Ellison et al. 1997). The two
properties are correlated for most elements, but F is one of the few that
breaks this pattern (Meyer et al. 1997). In the light of the new AMS-02 data,
this makes the study of the F source abundance worth exploring.

The paper is organised as follows: in Sect. 2, we recall the salient
ingredients for the modelling and fitting strategy, highlighting the priors
used on Solar modulation and nuclear cross-section parameters, and building
the covariance matrix of uncertainties for AMS-02 F/Si data. In Sect. 3, we
detail the list of the most important progenitors for the F production, and we
renormalise the associated production cross sections to nuclear data. We
present our results in Sect. 4, where we show that propagation models can
reproduce F/Si and (Li,Be,B)/C ratios together, though it requires a small
bias of the F and B production cross sections. We then draw constraints on the
abundance of F (relative to Si) in CR sources, and discuss whether this is
consistent with the expectations of current CR source composition modelling.
We conclude in Sec. 5. Appendix A illustrates how our model reproduces,
without any fit, the CR fluxes of the progenitors of Li, Be, B, and F.

##  2 Model and methodology

We use here the same model and methodology as used in our previous analyses
(Génolini et al. 2019; Boudaud et al. 2020; Weinrich et al. 2020a, b), and we
refer the reader to these papers for complementary informations. In this
section, we recall the propagation model and its free parameters (Sect. 2.1).
We then recall the minimisation methodology, and exhibit our choices for the
nuisance parameter priors and data uncertainties (Sect. 2.2). For this
analysis, following Maurin et al. (2022b), we also update the most relevant
production cross sections of F (Sect. 3).

###  2.1 Model and free parameters

Assuming isotropy, the flux of a CR ion is related to the differential
density, $dN/dE\equiv N$ (for short), by $\psi=vN/(4\pi)$.

####  2.1.1 Transport equation in 1D model

The differential density $N^{k}$ for a CR species $k$ is calculated from the
transport equation (Berezinskii et al. 1990). For homogeneous and isotropic
diffusion (coefficient $K$) and convective transport ($V_{c}$) taken
perpendicular to the Galactic plane (symmetric w.r.t. the disc), the steady-
state equation in the thin-disc approximation (e.g., Webber et al. 1992)
becomes (Putze et al. 2010)

$\displaystyle\left(-K\frac{\partial^{2}}{\partial
z^{2}}+V_{c}\frac{\partial}{\partial z}+\frac{1}{\gamma\,\tau^{k}_{\rm
rad}}+2h\,\delta(z)\sum_{t\,\in\rm ISM}n_{t}\,v\,\sigma^{k+t}_{\rm
inel}\right)N^{k}$ (1) $\displaystyle\hskip
54.06023pt+\;2h\,\delta(z)\frac{\partial}{\partial
E}\left(b^{k}N^{k}-c^{k}\frac{\partial N^{k}}{\partial E}\right)$
$\displaystyle=$ $\displaystyle
2h\,\delta(z)Q^{k}(E)+\frac{N^{r}}{\gamma\,\tau^{r\to k}_{\rm
rad}}+2h\,\delta(z)\\!\sum_{t\,\in\rm
ISM}\\!\sum_{p}\\!n_{t}\,v\,\sigma^{p+t\to k}_{\rm prod}N^{p}\,,$

with

$\displaystyle b(E)\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\Big{\langle}\frac{dE}{dt}\Big{\rangle}_{\rm
ion,\,coul.}-\frac{\@vec{\nabla}.\@vec{V}}{3}E_{k}\left(\frac{2m+E_{k}}{m+E_{k}}\right)+\frac{(1+\beta^{2})}{E}\,K_{pp},$
(2) $\displaystyle c(E)\\!$ $\displaystyle\\!=\\!$
$\displaystyle\\!\beta^{2}\,K_{pp},$ (3)

and

$K_{pp}\times
K=\frac{4}{3}\;V_{a}^{2}\;\frac{p^{2}}{\delta\,(4-\delta^{2})\,(4-\delta)}\,.$
(4)

The various quantities appearing in Eq. (1) are: (i) the half-life $\tau_{\rm
rad}$, either leading to a disappearance if CR species $k$ is unstable, or to
a radioactive source term (if species $r$ decays into $k$); (ii) nuclear
reaction rates $\sum_{t}n_{t}v\,\sigma$ on the interstellar medium (ISM)
targets $t$ of density $n_{t}$, either associated to a net loss with a
destruction cross section $\sigma_{\rm inel}^{k+t}$, or to a secondary source
term, summed over $p$ progenitors (heavier than $k$) with a production cross
section $\sigma_{\rm prod}^{p+t\to k}$ (straight-ahead approximation is
assumed); (iii) continuous energy losses and gains in the disc only($b$ and
$c$ terms), accounting for ionisation and Coulomb losses (Mannheim &
Schlickeiser 1994; Strong & Moskalenko 1998), adiabatic expansion in the
Galactic wind, and first and second order contribution from reacceleration;
(iv) the so-called primary source term, $Q^{k}(E)$, from astrophysical sources
in the thin disc. Finally, Eq. (4) links the diffusion coefficient is momentum
$(K_{pp})$ and in space $(K)$ from a minimal reacceleration model (Osborne &
Ptuskin 1988; Seo & Ptuskin 1994), via the Alfvénic speed $V_{a}$.

####  2.1.2 Propagation code, setup, and ingredients

The above equation couples about a hundred CR species (for $Z<30$) over a
nuclear network of more than a thousand reactions. To solve this triangular
matrix of equations, one possibility is to start from the heavier nucleus,
which is always assumed to be a primary species, and then to proceed down to
the lightest one. In the process, isotopic source abundances are fixed to
Solar system ones (Lodders 2003), but elemental abundances are normalised to a
high energy CR data point. However, accounting for the anomalous isotopic
overabundance of 22Ne/20Ne at source (e.g., Tatischeff & Gabici 2018) is
mandatory not to bias the F calculation (see below). For this reason, we also
rescale the isotopic abundances in order to match a low-energy ACE-CRIS
22Ne/20Ne data point (Binns et al. 2005).

All the results derived in this paper are based on the
usine333https://lpsc.in2p3.fr/usine public code (Maurin 2020), which provides
a full implementation of the 1D model and its solutions. As in our previous
studies, we fix the disc half-thickness $h=100$ pc and the gas density $n_{\rm
ISM}=1\leavevmode\nobreak\ {\rm cm}^{3}$ ($90\%$ H and $10\%$ He in number) to
recover the local gas surface density (Ferrière 2001). The flux of stable
species is independent of $K_{0}/L$ (e.g., Maurin et al. 2001), so that we can
fix the halo size without loss of generality. We use here $L=5$ kpc, a value
consistent with the latest determination from AMS-02 data (Weinrich et al.
2020a; Maurin et al. 2022a). We assume a simple power-law in rigidity with a
universal slope for the source term $Q(R)$. This was sufficient to give an
excellent match (no fit) to AMS-02 C, N, and O data in our (Li,Be,B)/C
analyses (Génolini et al. 2019; Weinrich et al. 2020b). It also gives a fair
match to the flux of the main progenitors of F, and similarly to the B/C
ratio, this implies that the F/Si ratio becomes independent of its
progenitors’ source spectrum (Maurin et al. 2002; Putze et al. 2011; Génolini
et al. 2015). For consistency, we checked that using broken-power laws (to
better fit the primary flux data) had no impact on the conclusions drawn in
this paper, see App. A.

We stress that there is a rather long list of additional effects or
alternative modelling that may be relevant to interpret CR fluxes, in
particular regarding features at high energy (Vladimirov et al. 2012): energy-
dependence of the nuclear cross sections (Krakau & Schlickeiser 2015), source
spectral breaks or diversity of CR sources (Ptuskin et al. 2013; Bell 2015;
Ohira et al. 2016; Recchia & Gabici 2018), stochasticity of the sources
(Bernard et al. 2012; Thoudam & Hörandel 2012, 2013; Liu et al. 2015; Génolini
et al. 2017; Liu et al. 2017), reacceleration (Wandel et al. 1987; Thoudam &
Hörandel 2014), or secondary production at source (Berezhko et al. 2003; Blasi
& Serpico 2009; Tomassetti & Donato 2012; Cholis & Hooper 2014; Mertsch &
Sarkar 2014; Mertsch et al. 2021), single-source hypothesis (Kachelrieß et al.
2015, 2018), etc. These effects might be relevant to interpret the DAMPE data
(An et al. 2019) at a few tens of TeV (e.g., Fang et al. 2020; Malkov &
Moskalenko 2021). However, none of these affect are considered here, as we
wish to test whether a more standard model can already explain the very
precise AMS-02 data.

####  2.1.3 BIG, SLIM, and QUAINT configurations

Besides $V_{a}$ and $V_{c}$ (for reacceleration and convection), the free
parameters of our analysis are related to the diffusion coefficient,
parametrised as a phenomenologically motivated power-low with a break at both
low-rigidity (Génolini et al. 2019; Vittino et al. 2019; Weinrich et al.
2020b) and high-rigidity (Tomassetti 2012a; Génolini et al. 2017; Reinert &
Winkler 2018; Génolini et al. 2019; Niu & Xue 2020):

$K(R)={\beta^{\eta}}K_{0}\;{\left\\{1\\!+\left(\frac{R_{l}}{R}\right)^{\frac{\delta-\delta_{l}}{s_{l}}}\right\\}^{s_{l}}}{\left\\{\frac{R}{R_{0}\\!=\\!1\,{\rm
GV}}\right\\}^{\delta}}\,{\left\\{1\\!+\left(\frac{R}{R_{h}}\right)^{\frac{\delta-\delta_{h}}{s_{h}}}\right\\}^{-s_{h}}}\\!\\!\\!\\!\\!\\!.$
(5)

As discussed in Génolini et al. (2019), the breaks delineate specific low,
intermediate, and high-rigidity ranges that can be related to both features in
the data (Lave et al. 2013; Cummings et al. 2016; Webber et al. 2017; An et
al. 2019; Adriani et al. 2020; Aguilar et al. 2021b) and peculiar microphysics
mechanisms (Yan & Lazarian 2004; Ptuskin et al. 2006; Shalchi & Büsching 2010;
Blasi et al. 2012; Evoli & Yan 2014; Evoli et al. 2018; Fornieri et al. 2021).

We re-use here the three propagation scenarios proposed in Génolini et al.
(2019), namely BIG, SLIM, and QUAINT: (i) BIG enables the ‘full’ transport
complexity, i.e. diffusion
($K_{0},\,\delta,\,R_{l},\,\delta_{l},\,s_{l},\,R_{h},\,\delta_{h},\,s_{h}$,
but $\eta\,$=1), convection ($V_{c}$), and reacceleration ($V_{a}$); (ii) SLIM
is a minimal diffusion scenario with $\eta\,$=1 and $V_{c}\,$=$\,V_{a}\,$= 0;
(iii) QUAINT is a more standard diffusion/convection/reacceleration scenario,
i.e. no low-energy break ($R_{l}\,$=0) but possible sub-relativistic upturn of
the diffusion coefficient via $\eta$. The latter two scenarios can be viewed
as two different limiting regimes of BIG. All these scenarios lead to similar
predictions in the high-energy regime ($\gtrsim 10$ GV), and they all give a
very good fit to light secondary-to-primary data (Weinrich et al. 2020b) and
also antiprotons (Boudaud et al. 2020).

Minimisations in a large parameter space quickly becomes inefficient. To limit
the number of free parameters in our analysis, we assume a fast transition
$s_{l}$ = 0.04 for the low-rigidity break and also fix the three high-rigidity
break parameters ($R_{h},\delta_{h},s_{h}$) to the values in Génolini et al.
(2019); as showed by these authors, this choice only marginally affects the
remaining parameters. Consequently, our analysis has 4 free transport
parameters for SLIM ($K_{0},\,\delta,\,R_{l},\,\delta_{l})$, 5 for QUAINT
$(K_{0},\,\delta,\,\eta,\,V_{c},\,V_{a})$, and 6 for BIG
$(K_{0},\,\delta,\,R_{l},\,\delta_{l},\,V_{c},\,V_{a})$.

###  2.2 $\chi^{2}$ analysis with covariance and nuisance

The need for an improved methodology to interpret high precision AMS-02 data
was discussed at length in Derome et al. (2019) and is summarised below. In
order to reduce biases in the model parameter determination, via a
minimisation (Sect. 2.2.1), three crucial ingredients are included: nuisance
parameters for the Solar modulation level (Sect. 2.2.2) and for nuclear
production cross sections (Sect. 2.2.3), and the use of a covariance matrix of
data uncertainties (Sect. 2.2.4).

####  2.2.1 $\chi^{2}$ minimisation

Interstellar (IS) fluxes are obtained from the transport equation described in
the previous section. To compare to the top-of-atmosphere (TOA) data, IS
fluxes are Solar modulated. We use in this study the force-field approximation
(Gleeson & Axford 1967, 1968), in which

$\displaystyle\psi^{\rm TOA}\left(E^{\rm TOA}\right)$ $\displaystyle=$
$\displaystyle\left(\frac{p^{\rm TOA}}{p^{\rm IS}}\right)^{2}\psi^{\rm
IS}\left(E^{\rm IS}\right)\,,$ (6) $\displaystyle E_{k/n}^{\rm TOA}$
$\displaystyle=$ $\displaystyle E_{k/n}^{\rm IS}-\frac{Z}{A}\phi_{\rm FF}\,.$
(7)

The parameter $\phi_{\rm FF}$ in Eq. (7) is the Fisk potential, which must be
set appropriately for each data taking period (see below).

The best-fit model parameters and goodness-of-fit of the model to the data are
obtained from a $\chi^{2}$ minimisation using minuit (James & Roos 1975), with

$\displaystyle\chi^{2}$ $\displaystyle=$
$\displaystyle\sum_{q=1}^{n_{q}}\left({\cal D}^{\,q}+\sum_{s=1}^{n_{s}}{\cal
N}^{s(q)}\right)+\sum_{x=1}^{n_{x}}{\cal N}^{\,x},$ (8) $\displaystyle{\cal
D}$ $\displaystyle=$ $\displaystyle\sum_{i,j=1}^{n_{E},n_{E}}\left(y^{\rm
data}_{i}-y^{\rm model}_{i}\right)\left({\cal C}^{-1}\right)_{ij}\left(y^{\rm
data}_{j}-y^{\rm model}_{j}\right),$ (9) $\displaystyle{\cal N}(y)\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\frac{\left(y-\mu_{y}\right)^{2}}{\sigma_{y}^{2}}.$ (10)

In Eq. (8), $q$ runs over the $n_{q}$ ratios used in the fit444For instance,
$n_{q}=1$ for the stand-alone F/Si analysis (Sect. 4.1) and $n_{q}=4$ for the
combined analysis of (Li,Be,B)/C and F/Si (Sect. 4.2)., for which the
quadratic distance ${\cal D}$ between the model and the data, Eq. (9), is
calculated including energy bin correlations ($n_{E}$ bins in total); these
correlations are encoded in the covariance matrix ${\cal C}$ of data
uncertainties discussed in Sect. 2.2.4. Gaussian-distributed nuisance
parameters ${\cal N}$ of mean $\mu_{y}$ and variance $\sigma_{y}^{2}$, Eq.
(10), are considered on Solar modulation and some proxy cross sections (${\cal
N}^{s}$ and ${\cal N}^{x}$ respectively): the indices $s$ and $x$ in Eq. (8)
run over $n_{s}$ different data taking periods (if applies) and the $n_{x}$
cross-section reactions considered in the error budget respectively. The
nuisance terms penalise the $\chi^{2}$ if the associated parameters wander
several $\sigma$ away from their expected value. Although we do not perform a
Bayesian analysis, for brevity, we find useful to denote the nuisance
parameters values $\mu_{y}$ and $\sigma_{y}^{2}$ priors and their post-fit
values (found after the minimisation) posteriors.

The minimum $\chi^{2}$ value indicates how good is the fit: $\chi^{2}_{\rm
min}/{\rm dof}\sim 1$ corresponds to an excellent fit, with $n_{\rm
dof}=n_{\rm data}-n_{\rm pars}-n_{s}-n_{x}$. As introduced in Weinrich et al.
(2020b), it is also useful to consider

$\chi^{2}_{\rm nui}/{n_{\rm nui}}{}\equiv\left(\sum_{s=0}^{n_{s}}{\cal
N}^{s}+\sum_{x=0}^{n_{x}}{\cal N}^{x}\right)/(n_{s}+n_{x}),$ (11)

with $n_{\rm nui}=n_{s}+n_{x}$. The above quantity indicates whether the post-
fit values stay very close to the priors ($\mu^{\rm post}\sim\mu^{\rm prior}$)
or wander within $1\sigma$ of the priors ($\mu^{\rm post}\lesssim\mu^{\rm
prior}\pm\sigma$), corresponding to $\chi^{2}_{\rm nui}/{n_{\rm nui}}{}\sim 0$
or $\chi^{2}_{\rm nui}/{n_{\rm nui}}{}\lesssim 1$ respectively.

####  2.2.2 Priors for Solar modulation

In the Force-Field approximation, Eqs (6-7), modulation levels can be
reconstructed from the analysis of neutron monitor (NM) data, at a precision
$\sigma_{\phi}\simeq 100$ MV (Maurin et al. 2015; Ghelfi et al. 2017). The
reconstructed $\phi$ values are stored in the cosmic-ray database
(crdb555https://lpsc.in2p3.fr/crdb) and can be accessed via the interface in
the ‘Solar modulation’ tab of crdb (Maurin et al. 2014, 2020). Figure 1 shows
the modulation levels (averaged over 10 days) retrieved from crdb from May 19,
2011 to October 30, 2019, for the OULU NM station.

Figure 1: Solar modulation level reconstructed (Ghelfi et al. 2017) from the
OULU neutron monitor station, averaged over a 10 day period. The date range
matches the period of AMS-02 analysed F/Si data (Aguilar et al. 2021d), and
the value in parenthesis (in the legend) gives the average modulation level
over the period. Retrieved from crdb.

The data considered in our analysis are either AMS-02 F/Si data only (see
Sect. 4.1) or their combined analysis with AMS-02 Li/C, Be/C, and B/C data
(see Sect. 4.2). The data taking period of AMS-02 F/Si data (Aguilar et al.
2021d) matches that of Fig. 1, i.e. 8,5 years of AMS-02 data, for which an
average modulation level of $\mu_{\phi}=610$ MV is found (we take as reference
the widely used OULU NM). For Li/C, Be/C, and B/C, the most recent published
data (Aguilar et al. 2021b) are based on a 7 years data taking period
($\mu_{\phi}=636$ MV), but we nevertheless use the previously published data
set based on 5 years of AMS-02 data (Aguilar et al. 2018) for which
$\mu_{\phi}=680$ MV. We do so because we wish to compare and combine the
results of the F/Si analysis to that of our previous (Li,Be,B)/C analysis
Weinrich et al. (2020b); Maurin et al. (2022b), which relies on the smaller
dataset. This choice leads to slightly conservative error bars in our
analyses, as AMS-02 datasets based on longer data taking periods have slightly
smaller uncertainties.

We stress that, in principle, solar modulated fluxes should be calculated as
weighted averages of fluxes modulated on short time intervals (for which the
modulation level is roughly constant), whereas our calculation is based on an
average modulation level over the full data taking period. However, the
difference between the two calculations was shown to be at the few percent
level on the low-energy fluxes, also amounting to a few MV difference on the
average Solar modulation level used (see App. A.2 of Ghelfi et al. 2016). The
latter number is much smaller than the uncertainty $\sigma_{\phi}\simeq 100$
MV taken on the solar modulation level. We repeated the above comparison using
10-days slices over the period depicted in Fig. 1. We reach similar
conclusions for the fluxes and further find a negligible difference on ratios,
at the level of $0.1\%$ in the AMS-02 rigidity range. In our analysis, we thus
use for each dataset the NM-derived average modulation levels, and we recall
that we apply the modulation on each CR isotope separately before forming the
elemental ratios of interest.

####  2.2.3 Priors for cross sections

Production cross-section uncertainties are sizeable ($\sim 5-20\%$), and the
propagation equation involves a large network of reactions ($\gtrsim 1000$).
Owing to the difficulty to assess the uncertainty on a reaction-to-reaction
basis (Génolini et al. 2018), and because different groups of reactions have
similar impact on the quantity of interest (e.g. F/Si), a strategy to
propagate the nuclear uncertainties in the calculation is to pick the most
relevant reactions as proxies for the whole network (Derome et al. 2019). We
first have to identify what these most important reactions are, and then to
parameterise the uncertainties in a way that can be implemented as nuisance
parameters.

In Sect. 3.1, we identify 20Ne, 24Mg, and 28Si as the most impacting
progenitors. To turn the reactions into nuisance parameters, we follow the
methodology detailed in Derome et al. (2019), where transformation laws on
cross sections combine a normalisation, energy scale, and low-energy slope, in
order to modify a reference cross section $\sigma_{\rm ref}$:

$\displaystyle\sigma^{\rm Norm.}(E_{k/n})$
$\displaystyle=\mathrm{Norm}\times\sigma_{\rm ref}(E_{k/n})\,,$ (12)
$\displaystyle\sigma^{\rm Scale}(E_{k/n})$ $\displaystyle=\sigma_{\rm
ref}\left(E_{k/n}\times\mathrm{Scale}\right)\,,$ (13)
$\displaystyle\sigma^{\rm Slope}(E_{k/n})$
$\displaystyle=\begin{cases}\displaystyle\sigma_{\rm
ref}(E_{k/n})\times\left(\frac{E_{k/n}}{E_{k/n}^{\text{thr}}}\right)^{\text{Slope}}\\!\\!\\!\text{if}\;E_{k/n}\leq
E_{k/n}^{\text{thresh.}},\\\ \sigma_{\rm
ref}(E_{k/n})\quad\quad\text{otherwise}\,.\end{cases}$ (14)

In the above equations, $E_{k/n}^{\text{thresh.}}$ is fixed to 1 GeV/n. The
parameters ‘Norm’, ‘Scale’ and ‘Slope’ (NSS) are the ones for which we need to
determine a central value $\mu$ and dispersion $\sigma$, to be used as
nuisance parameters in Eq. (8).

Figure 2: Illustration of the NSS scheme used for cross section nuisance
parameters (inelastic on the left and production on the right), see Eqs.
(12-14). Colour-coded solid lines correspond to existing cross section
parametrisations: on the left (inelastic), B94 (Barashenkov & Polanski 1994),
W97 (Wellisch & Axen 1996), T99 (Tripathi et al. 1997, 1999), and W03 (Webber
et al. 2003); on the right (production, W98 (Webber et al. 1998a, b, c), S01
(A. Soutoul, private communication), W03 (Webber et al. 2003), and G17
(Moskalenko et al. 2001; Moskalenko & Mashnik 2003). Grey lines correspond to
the median, $68\%$, and $95\%$ CLs resulting from varying the NSS parameters:
we choose for $\sigma_{\rm NSS}$ of the cross-section nuisance parameters
(reported in Table 1), the NSS values leading to the 68% CLs (dotted grey
lines).

Because we renormalise production cross-section reactions to nuclear data in
Sect. 3.1, we enforce $\mu_{\rm Norm}=1$, $\mu_{\rm Scale}=1$, and $\mu_{\rm
Slope}=0$ for all reactions. To fix $\sigma$, as illustrated in Fig. 2, we
visually estimate what transformation law values are needed to encompass at
$1\sigma$ (dotted grey lines) the different cross-section parametrisations
(coloured lines)666This crude approach possibly overestimates the uncertainty
seen in nuclear data. However, we recall that we use only a few reactions as
proxies of all reactions involved in the calculation. Moreover, very few
nuclear data points are available, even for the most important reactions (see
Sect. 3.1). All in all, $\sigma$ is difficult to evaluate and anyway, varying
its value within reason would not change the conclusions of our analyses..

Table 1: For the most relevant inelastic (upper half) and production (lower
half) reactions, we show the typical impact nuclear uncertainties have on the
calculated F/Si ratio (col. 2), and the nuisance parameters $\mu$ and $\sigma$
for the NSS transformation laws (cols 3-5), see Eqs (12-14). For the
production reactions, we also show their contributing fraction w.r.t the total
F production (in square brackets, col. 1, as read off Fig. 6). In practice,
for the F/Si $\chi^{2}$ analysis, only two reactions (highlighted in boldface)
are chosen as proxies.

Reaction |  Impact $(\frac{\Delta\sigma}{\sigma})^{\rm XS}$ | Norm. | Scale | Slope
---|---|---|---|---
| on F/Si | $\;\mu\,|\,\sigma$ | $\mu\,|\,\sigma$ | $\mu\,|\,\sigma$
19F+H | 4% | 1.0 — 0.04 | 1.0 — 0.5 | n/a
20Ne+H | 2% | 1.0 — 0.06 | 1.0 — 0.5 | n/a
24Mg+H | 1% | 1.0 — 0.07 | 1.0 — 0.4 | n/a
28Si+H | 4% | 1.0 — 0.07 | 1.0 — 0.4 | n/a
All(a) | 6% |
20Ne +H$\,\mathrel{\vbox{\hbox{\rule[0.0pt]{3.0pt}{0.0pt}}}\mkern-4.0mu\hbox{\char 41\relax}}$19F [30%] | 10% | 1.0 — 0.13 | n/a | 0.0 — 0.01
24Mg+H$\,\mathrel{\vbox{\hbox{\rule[0.0pt]{3.0pt}{0.0pt}}}\mkern-4.0mu\hbox{\char 41\relax}}$19F [15%] | 5% | 1.0 — 0.33 | n/a | 0.0 — 0.04
28Si + H$\,\mathrel{\vbox{\hbox{\rule[0.0pt]{3.0pt}{0.0pt}}}\mkern-4.0mu\hbox{\char 41\relax}}$19F [15%] | 3% | 1.0 — 0.27 | n/a | 0.0 — 0.10
All(a) | 20% |
777 $a$$a$footnotetext: Impact of changing cross sections for all reactions at
once for the F/Si calculation.

Table 1 gathers the NSS $\mu$ and $\sigma$ values to use for the production
and inelastic reactions listed; $\sigma$ values for Li, Be, and B production
can be found in Table B.1 of Weinrich et al. (2020b). The second column also
highlights the typical uncertainties on F/Si observed when using different
modelling for the reactions listed in the first column.

We checked (not shown) that the various production reactions (and inelastic
reactions among themselves) have similar impact on the F/Si shape and mostly
differ on the amplitude of the change. This leads to quasi-degenerate
parameters for the many cross-section reactions, i.e. the same production rate
can be obtained by increasing one reaction cross section and decreasing
another (Derome et al. 2019). As strong degeneracies are difficult to tackle
by minimisation algorithms, the full list of reactions identified in Table 1
is not used in practice (and furthermore, reactions whose impact is smaller
than the data uncertainties are discarded). Consequently, we use a single
proxy for the inelastic cross sections (19F+H) and the production cross
sections (20Ne
+H$\,\mathrel{\vbox{\hbox{\rule[0.0pt]{3.0pt}{0.0pt}}}\mkern-4.0mu\hbox{\char
41\relax}}$19F). These proxies are highlighted in boldface in Table 1, and
compared to the row denoted ‘All’ that shows the impact of the uncertainties
from the full network.

####  2.2.4 Covariance matrix of uncertainties

An important term in Eq. (8) is the covariance matrix of data uncertainties
$C_{ij}$. Because the AMS collaboration does not provide such a matrix, almost
all analyses of their data rely on total uncertainties. However, as shown on
simulated data in Derome et al. (2019), not accounting for the covariance
matrix can significantly bias the determination of the transport parameters;
the crucial role of the covariance matrix was further demonstrated in the
context of analysing AMS-02 antiprotons data (Boudaud et al. 2020; Heisig et
al. 2020).

To remedy this situation, we can build a tentative covariance matrix of
uncertainties based on the information provided in the AMS-02 publications and
supplemental material. Following Derome et al. (2019), we define the relative
covariance $(C_{\rm rel}^{\alpha})_{ij}$ between rigidity bin $R_{i}$ and
$R_{j}$ (and the associated correlation matrix) to be

$\displaystyle(C_{\rm rel}^{\alpha})_{ij}$ $\displaystyle=$
$\displaystyle\sigma^{\alpha}_{i}\sigma^{\alpha}_{j}\exp\left(-\frac{1}{2}\frac{(\log(R_{i}/R_{j})^{2}}{(\ell_{\alpha})^{2}}\right)\,,$
(15) $\displaystyle{\rm c}_{ij}^{\alpha}$ $\displaystyle\equiv$
$\displaystyle\frac{{\cal C}_{ij}^{\alpha}}{\sqrt{{\cal
C}_{ii}^{\alpha}\times{\cal C}_{jj}^{\alpha}}}\,,$ (16)

with $\sigma_{i}^{\alpha}$ the relative uncertainty of error type $\alpha$ at
bin $i$ and $\ell_{\alpha}$ the correlation lengths for error type $\alpha$
(in unit of rigidity decade).

The AMS-02 collaboration provides the statistical and systematic uncertainties
split in three different components (acceptance, unfolding888Unfolding is the
procedure to estimate ‘true’ rigidities from measured ones, owing to the
finite energy resolution of the detector., and scale), as displayed in solid
lines in the top panel of Fig. 3. With the detailed information given in the
supplemental material (Aguilar et al. 2021d), we can estimate the correlation
length associated with these systematic; see Derome et al. (2019) for more
details. We set $\ell_{\rm scale}=\infty$ for the energy scale uncertainty,
amounting to a global rigidity shift, and $\ell_{\rm Unfolding}=0.5$, a mild
correlation resulting from the unfolding procedure. The acceptance uncertainty
combines errors of different origins, which motivates its decomposition in
three extra components (Derome et al. 2019): (i) ‘Acc. norm.’ is associated to
inelastic cross-section uncertainties (impacting the detector acceptance
estimation), with a relatively large correlation length $\ell_{\rm
Acc.\leavevmode\nobreak\ norm.}=1$; (ii) ‘Acc. LE’ is a low-rigidity error
associated to the orbit-varying rigidity cut-off of the AMS detector, set to
$\ell_{\rm Acc.\leavevmode\nobreak\ LE}=0.3$; and (iii) ‘Acc. res.’ is a
residual error mostly associated with data/Monte Carlo corrections. The latter
cannot be associated to a particular physics process and no prescription can
be taken for its correlation length.

As can be seen in the top panel of Fig. 3, the ‘Acc. norm.’ (dashed line) and
‘Acc. res.’ (dash-dotted line) uncertainties dominate the error budget at
intermediate rigidities; a quite similar ranking is seen in (Li,Be,B)/C data,
see Fig. A.1 in Weinrich et al. (2020b). As studied on the AMS-02 B/C data in
Derome et al. (2019), taking extreme values of the unknown $\ell_{\rm
Acc.\leavevmode\nobreak\ res.}$ impacts the best-fit $\chi^{2}_{\rm min}$/dof
values (this value increases with $\ell_{\rm Acc.\leavevmode\nobreak\ res.}$),
but only marginally the best-fit parameters and thus the best-fit flux ratios.
For this reason, the $\chi^{2}_{\rm min}$/dof values we obtain (see Sect. 4)
and their interpretation must be taken with a grain of salt. Otherwise, our
conclusions are expected to be independent of the exact value of this
correlation length, and for definiteness, we set $\ell_{\rm
Acc.\leavevmode\nobreak\ res.}=0.1$ (Derome et al. 2019). The bottom panel of
Fig. 3 shows the corresponding covariance matrix of uncertainties accounting
for all uncertainties: below 10 GV, the data are typically correlated over one
decade in energy (dominated by ‘Acc. norm.’); above, we transition from weakly
correlated (dominated by ‘Acc. res.’ to uncorrelated as statistical
uncertainties take over.

Figure 3: Top panel: statistical and Acc., Unf., and Scale systematic
uncertainties for F/Si, taken from Table S1 of Aguilar et al. (2021d). The
‘Acc. norm.’ (dashed line), ‘Acc. LE’ (dotted line), and ‘Acc. res.’ (dash-
dotted line) systematics are broken down from the ‘Acc.’ systematic as
explained in the text. Bottom panel: correlation matrix ${\rm c}_{ij}^{\rm
tot}$, Eq. (16), for the combined statistical and systematics uncertainties,
colour-coded from no correlation (white, ${\rm c}_{ij}^{\rm tot}=0$) to full
correlations (blue, ${\rm c}_{ij}^{\rm tot}=1$).

##  3 Production cross sections and F progenitors

In this section, we focus on the production cross sections. We show first the
nuclear parametrisations renormalised on nuclear data points for a selection
of important reactions (Sect. 3.1). We then illustrate the importance of
correctly setting the source of Ne isotopes for the F/Si calculation (Sect.
3.2). With these updated cross sections and correct CR source settings, we
determine the ranking of the most important progenitors of F (Sect. 3.3).

###  3.1 Rescaling of cross-sections on data

Several production cross-section parametrisations are available from the
literature. In particular, the widely used
Galprop999https://galprop.stanford.edu/ dataset (Moskalenko et al. 2001;
Moskalenko & Mashnik 2003) is based on systematic fits on existing nuclear
data of parametric models: the underlying models are either the semi-empirical
formulae implemented in the wnew code (Webber et al. 1998a, b, c) or the semi-
analytical formulae implemented in the yieldx code (Silberberg et al. 1998;
Tsao et al. 1998); these models lead to the Galp--opt12 and Galp--opt22
production datasets respectively.

Figure 4: Models (OPT12 and OPT22) and data (symbols) for the colour-coded
production of 19F and ghosts with ${\cal B}r>5\%$ (i.e. 20N, 20C, 19Ne, 19O,
19N, 19C), from 56Fe, 32S 28Si, 27Al, 24Mg, and 20Ne CRs on H. ‘No data’
indicates that no nuclear data were found for the production of the isotopes
listed, hence the model corresponds to the original parametrisation (if not
visible on the plot, it means that the cross-section value is $\lesssim
10^{-2}$ mb). ‘Rescaled’ indicates that the isotopes listed have nuclear data
on which the model were renormalised. For the latter, we do not show the
original model values for these reactions (for readability), but refer the
reader to Table 2 that highlights the rescaling factor between the new and the
original cross sections in the asymptotic regime. The data references (top
right panel) correspond to Villagrasa-Canton et al. (2007), Napolitani et al.
(2004), Webber et al. (1998b, c), Chen et al. (1997), Tull et al. (1993), and
Webber et al. (1990).

In Maurin et al. (2022b), these datasets were updated for Li, Be, and B
production—in particular from recent Fe fragmentation data, unused to build
the original Galprop datasets—, leading to the OPT12, OPT12up22, and OPT22
datasets. We repeat this procedure here (see below for more details) for the
most important progenitors of F, namely ${}^{56}{\rm Fe}$, ${}^{32}{\rm S}$,
${}^{28}{\rm Si}$, ${}^{27}{\rm Al}$, ${}^{24}{\rm Mg}$, and ${}^{20,22}{\rm
Ne}$ (see Sect. 3.3). The fragments to consider are the stable isotopes of F
(only 19F) and short-lived nuclei (dubbed ghosts) ending up their decay chain
into 19F with a branching ratio ${\cal B}r$. While we work in this section on
the individual production cross sections $\sigma$, we stress that, in the rest
of the paper, cumulative cross sections $\sigma^{c}$ are considered instead
(relevant ones for CR propagation), with

$\sigma^{\rm c}(X\to Y)=\sigma(X\to Y)+\sum_{G\in{\rm
ghosts}}\\!\\!\\!\sigma(X\to G)\cdot{\cal B}r\,(G\\!\to\\!Y)\,,$

where $G$ runs over the list of ghost nuclei for $Y$. The relevant ghost
nuclei with ${\cal B}r>5\%$ (that we show here) for 19F are (Letaw et al.
1984; Maurin 2001) ${}^{19}{\rm Ne}$ (${\cal B}{\rm r}=100\%$), ${}^{19}{\rm
O}$ (${\cal B}{\rm r}=100\%$), 19C (${\cal B}{\rm r}=20.9\%$), 20C (${\cal
B}{\rm r}=48.6\%$), 19N (${\cal B}{\rm r}=45.4\%$), and 20N (${\cal B}{\rm
r}=57\%$).

In brief, for the nuclear cross-section update, we (i) start from the OPT12,
OPT12up22, and OPT22 datasets already updated for Li, Be, and B production in
Maurin et al. (2022b), (ii) extract from the exfor
database101010https://www.nndc.bnl.gov/exfor/exfor.htm (Otuka et al. 2014) all
nuclear data relevant for the F production (on H targets), (iii) apply the
rescaling procedure of Maurin et al. (2022b) to get updated modelling of these
cross sections, (iv) use the empirical formulae of Ferrando et al. (1988) for
$\sigma^{\rm He}/\sigma^{\rm H}$ to get the corresponding cross sections on
He, and (v) generate production cross-section sets for the cumulative
reactions, to be used in Sect. 4.

The nuclear data along with the rescaled OPT12 and OPT22 energy-dependent
parametrisations are shown in Fig. 4 (thin and thick lines respectively),
along with the data (symbols). As can be seen, for most reactions, only one or
two nuclear data points (at similar energies) are available. In that case, the
updated cross sections amount to a mere rescaling of the original ones—the
rescaling values are gathered in Table 2 and discussed below. Our procedure
significantly changes the original energy dependence of the cross section for
22Ne+H$\rightarrow^{19}$Ne only (dashed purple line in the bottom left panel):
the asymptotic high-energy value of this reaction is quite uncertain, as it is
fixed by the highest-energy nuclear data point (which suffers large
uncertainties); this reaction would particularly benefit from new data. For
the selection of the most important progenitors of F, we first remark that:
(i) very few data exist and on a very limited energy range; (ii) although they
are expected to play a mostly negligible part in the overall production of F,
most of the associated ghost nuclei (19C, 20C, 19N, and 20N) have no data at
all; (iii) the production into 19O (ghost nucleus) contributes to $\sim 10\%$
of the cumulative cross sections into F, but nuclear data are missing for half
of the progenitors shown. Concerning the models, the parametrisations OPT12
(thin lines) and OPT22 (thick lines) differ at low-energy. This is below the
energy range of AMS-02 F data, but because the renormalisation point is in the
rising regime, the high-energy asymptotic value (relevant for AMS-02 data) can
be significantly different. We stress that OPT22 has a more realistic
behaviour near the production threshold, so it should be favoured. Finally,
for unmeasured reactions, the amplitude of these two parametrisations can also
differ a lot (e.g. 27Al into 19F, compare the thin and thick blue dash-dotted
lines).

Table 2: Rescaling factors ${\cal R}_{\rm model}=\sigma^{\rm new}_{\rm model}/\sigma^{\rm old}_{\rm model}$ at high energy for the production of F (the index model is set to either OPT12 or OPT22). The first column shows the list of progenitors (or projectiles) while the first line shows the list of the main fragments (stable isotope and ghosts with ${\cal B}{\rm r}=100\%$); other significant ghosts of F with ${\cal B}{\rm r}>5\%$, 19C (${\cal B}{\rm r}=20.9\%$), 20C (${\cal B}{\rm r}=48.6\%$), 19N (${\cal B}{\rm r}=45.4\%$), and 20N (${\cal B}{\rm r}=57\%$) are not shown, because no nuclear data are available for any of the projectiles considered. Entries in columns 2-4 show two factors, one for each model, or only one value if $\sigma^{\rm old}_{\tt OPT12}=\sigma^{\rm old}_{\tt OPT22}$. The key $\times$ means no-data available (hence $\sigma^{\rm new}=\sigma^{\rm old}$).  | ${}^{19}{\rm F}$ | ${}^{19}{\rm Ne}$ | ${}^{19}{\rm O}$
---|---|---|---
|  | (${\cal B}{\rm r}=100\%$) | (${\cal B}{\rm r}=100\%$)
${}^{56}{\rm Fe}$ | 5.2—0.6 | 1.92—0.50 | 130—0.7
${}^{32}{\rm S}$ | 0.6—0.6 | 1.04—1.03 | $\times$
${}^{28}{\rm Si}$ | 1 | 0.91—0.90 | $\times$
${}^{27}{\rm Al}$ | 0.98 | 0.87 | $\times$
${}^{24}{\rm Mg}$ | 0.88—0.90 | 1.19 | $\times$
${}^{22}{\rm Ne}$ | 1.16—1.45 | 0.1 | 1.18—1.15
${}^{20}{\rm Ne}$ | 1.03—0.98 | 0.94—0.95 | $\times$

To better highlight the difference of our new production sets compared to the
original ones, we show in Table 2 their ratios above a few GeV/n, in a regime
where cross sections are assumed to be constant. For most progenitors, the new
cross sections are within a few percent below or above the old ones, which was
expected (i.e. no change w.r.t. the original Galprop cross sections). However,
there are three important changes: first, it appears that the production of
19F from 56Fe was significantly underestimated in OPT12 and overestimated in
OPT22 (left and right pipe-separated numbers in the Table); second, the
production of 19F from 32S was also overestimated; third, the production of
the 19Ne ghost from 22Ne is decreased by a factor ten111111Looking at the
bottom left panel of Fig. 4, we see that this originates from a single point
(orange hexagon on top of the dashed lines) with very large error bars..

As obvious from the plots and the Table, gathering new nuclear data for the
production of 19F would be a huge improvement to interpret the high-precision
F/Si (and F) AMS-02 data.

###  3.2 Impact of the anomalous 22Ne/20Ne ratio on F/Si

In all our previous studies dealing with light nuclei, namely (Li,Be,B)/C
(Putze et al. 2010; Génolini et al. 2015, 2017, 2019; Weinrich et al. 2020b;
Maurin et al. 2001, 2010, 2022b), elemental abundances were rescaled to match
the corresponding elemental CR fluxes, though keeping in the process isotopic
abundances fixed to their Solar system (SS) fractions (Lodders 2003). This is
no longer possible here, because Ne, one of the most important progenitors for
F (see next section), has an anomalous isotopic source abundance. Indeed, a
detailed analysis shows that $(^{22}{\rm Ne}/^{20}{\rm Ne})_{\rm
CRS}/(^{22}{\rm Ne}/^{20}{\rm Ne})_{\rm SS}\approx 4$ (Binns et al. 2005),
indicating a different nucleosynthetic origin for 22Ne—possibly from Wolf-
Rayet stars in OB associations. Other isotopic anomalies at source have been
spotted, such as 12C/16O and 58Fe/56Fe (Wiedenbeck & Greiner 1981; Binns et
al. 2005, 2008), but they are smaller than the Ne anomaly and furthermore not
relevant for the F/Si ratio. In passing, we recall that being able to
characterise an anomaly depends on the confidence we have on the production
cross section (see for instance Chen et al. 1997 for an illustration) and to a
lesser extent, to our capability in deconvolving acceleration segregation
mechanisms when comparing the propagated source abundances to CR data (Meyer
et al. 1997; Tatischeff et al. 2021).

Figure 5: Impact of changing the isotopic ratio $(^{22}{\rm Ne}/^{20}{\rm
Ne})_{\rm CRS}$ at source on the propagated F/Si ratio, illustrated for three
equally plausible updates of the F production cross sections (different line
styles). The reference calculation is based on the Solar system value (Lodders
2003), and the various curves (from top to bottom) illustrate that growing
values lead to decreasing F/Si ratios. See text for discussion.

Figure 5 illustrates the importance of using the correct isotopic abundance
for the F/Si calculation. With respect to the calculation based on the Solar
System isotopic abundance $(^{22}{\rm Ne}/^{20}{\rm Ne})_{\rm SS}\simeq 0.07$,
the F/Si ratio decreases when the relative fraction of 22Ne to 20Ne at source
grows. This is understood from the inspection of the bottom plots of Fig. 4
where, for the cumulative production of F, we have
$\sigma^{c}(\mathrm{{}^{22}Ne+H\to^{19}F})=26$ mb and
$\sigma^{c}(\mathrm{{}^{20}Ne+H\to^{19}F})=47$ mb. The typical $3\%$ to $6\%$
difference in F/Si, compared to the case of using the SS value is significant
in the context of F/Si AMS-02 data, for which the best precision is $\sim 3\%$
(see the top panel of Fig. 3).

It is interesting to compare the $\rm(^{22}Ne/^{20}Ne)_{\rm CRS}$ anomaly
obtained from different authors, based on the same ACE-CRIS Ne isotopic data.
In the original analysis of these data, Binns et al. (2005) used a leaky-box
model with Silberberg et al. (1998)’s production cross sections rescaled to
existing nuclear data (that would be similar to our OPT22 set, minus the use
of recent nuclear data). They obtained $\rm(^{22}Ne/^{20}Ne)_{\rm CRS}=0.39$,
to compare to 0.32 reported by the Galprop team using Galp--opt12 production
set (Boschini et al. 2020). At variance, we obtain here
$\rm(^{22}Ne/^{20}Ne)_{\rm CRS}=0.47$ (see Sect. 4). It is beyond the scope of
this paper to investigate further the origin of this difference, which could
possibly be related to the updated production cross sections.

###  3.3 Ranking the most important progenitors

Equipped with the updated production cross sections and using the appropriate
rescaling for the isotopic source abundances of Ne, we can now precisely
determine the most important progenitors of the CR flux of F, assuming all F
is secondary in origin. To do so, we follow the methodology proposed in
Génolini et al. (2018) and Maurin et al. (2022b).

Figure 6: Fractional contributions larger than $1\%$ to the total F production
(modulated at 700 MV) as a function of rigidity. The top panel shows the
ranking for CR element progenitors, while the bottom panel shows more details
via the ranking of the 1-step and 2-step channels (reaction paths linking one
isotopic progenitor to a CR isotope); ‘¿2’-step channels (not shown)
contribute to a total of $\sim 10\%$, with about half of this number
originating from the multi-step fragmentation of the Fe isotopes. In both
plots, contributions starting from the same element share the same colours and
line styles.

We show in Fig. 6 the main F progenitors, from two slightly different
perspectives. The top panel shows how much each CR element contributes to the
total TOA flux of F— the contributions are summed over ISM targets and account
for the production of ghosts nuclei via the cumulative cross sections. The
production from Ne is dominant ($\sim 30-40\%$), followed by Mg and Si ($\sim
20\%$), and then Fe ($\sim 10\%$). The next elements are S and Al,
contributing at the percent level, and we then have a few sub-percent
contributors (Na, Ca, Ar, P, Mn, Cl, and Cr in decreasing order of importance)
whose total contribution reaches $\sim 2\%$. The low-rigidity decrease seen in
the different contributions reflects the properties of the associated
elemental fluxes: ratios of heavier-to-lighter primary fluxes decrease with
decreasing rigidity (Putze et al. 2011; Aguilar et al. 2021c), related to the
growing destruction of CRs with their mass (the heavier the species, the
larger its inelastic cross section). This pattern is seen in the various
progenitors, being more or less marked according to the mass ordering of the
elements (i.e. Ne, Mg, Al, Si, S, Fe).

We also show in the bottom panel of Fig. 6 a more detailed view of the
isotopes of interest. Formally, for any given CR progenitor, one can list all
the (possibly multi-step) reactions that link this progenitor to 19F. We can
then rank the most important reactions (Génolini et al. 2018). We dubbed these
lists of reactions ‘channels’, as they are not associated to a unique cross-
section reaction (a ranking of the individual cross sections is also possible,
but not shown here). In practice, we rank the 1-step and 2-step channels only.
The most important progenitors from direct channels are 20Ne ($\sim 25\%$),
24Mg and 28Si (both at $\sim 20\%$), then 22Ne and 56Fe (both at $\sim 7\%$),
and other ones are at the percent level (25,26Mg and 32S) or below (not
shown). This ranking motivated the choice of the progenitors in Sect. 3.1, for
the update of the most important reactions. Beside the direct channels, two-
step channels only reach the percent level, with a peak at a few GV121212As
underlined in Génolini et al. (2018), compared to 1-step production, the
rigidity dependence of $n$-step channels follows $(1/K)^{n-1}\propto
R^{-\delta(n-1)}$, with $\delta$ the diffusion slope.. Although no multi-step
fragmentation of 56Fe reaches the percent level individually, their multitude
combine in a larger total contribution (also peaking at a few GV): the dashed-
dotted magenta curve at 3 GV moves from $\sim 1\%$ (direct contribution,
bottom panel) to $\sim 10\%$ (direct plus all 2-step contributions, top
panel); the same trend, though with a more moderate impact of multi-step
contributions, can be seen in 24Mg (long-dashed orange lines) and 28Si (dash-
dotted magenta lines).

To conclude this section, at 10 GV, we find that 1-step channels contribute to
$\sim 70\%$ of the total F production, 2-step channels to $\sim 20\%$, and
‘$>$2-step’ channels to $\sim 10\%$ of the total (and we recall that multi-
step contributions peak at a few GV). We can also tie $\sim 62\%$ of the F
production to 5 direct channels, namely from 20,22Ne, 24Mg, 28Si, and 56Fe),
while $\sim 25\%$ originates from a few percent-level channels, and $\sim
13\%$ from hundreds of sub-percent level channels. We checked that these
numbers only marginally depend on the production cross-section set considered
(i.e. OPT12, OPT12up22, and OPT22).

##  4 Results

We can now present our results based on the updated (for F) production cross-
section sets OPT12, OPT22, and OPT12up22 (see Sect. 3.1). The latter set is
considered to be the most relevant to better describe the nuclear and CR data
(see Maurin et al. 2022b for details). In this section, we first inspect the
constraints AMS-02 F/Si data set on the transport parameters (assuming F is a
pure secondary species), and check their compatibility with the constraints
set by the combined AMS-02 (Li,Be,B)/C data (Sect. 4.1). We then perform a
combined analysis of these two datasets (Sect. 4.2), and discuss the
constraints they bring on the source abundance of 19F (Sect. 4.3).

###  4.1 Analysis of F/Si: consistency with Li,Be,B/C?

In this section, we show our results for the propagation configuration SLIM,
whose parameters are listed in Sect. 2.1. Similar conclusions are obtained for
the BIG and QUAINT configurations.

Table 3: Comparisons of best-fit transport parameters (in SLIM) from various combinations of AMS-02 data: F/Si alone with OPT12up22 cross-section set (this analysis), B/C alone (Génolini et al. 2019), combined (Li,Be,B)/C with OPT12up22 cross-section set (Maurin et al. 2022b), and combined (Li,Be,B)/C and F/Si data with updated production cross sections (this paper). See text for discussion. Config. | Ref. | $\delta$ | $\displaystyle\log_{10}\left(\frac{K_{0}}{\mathrm{1\,kpc^{2}\,Myr^{-1}}}\right)$ |  $R_{l}$ [GV] | $\delta_{l}$ | $\displaystyle\log_{10}\left(\frac{q_{{}^{19}\rm F}}{q_{{}^{28}\rm Si}}\right)$ |  $\chi^{2}_{\rm min}$/dof | $\chi^{2}_{\rm nui}/{n_{\rm nui}}$
---|---|---|---|---|---|---|---|---
F/Si ($q_{{}^{19}\rm F}=0$) | this paper | $0.449\pm 0.020$ | $-1.156\pm 0.054$ | $4.21\pm 0.33$ | $-0.45\pm 0.33$ | - | 0.68 | 0.04
B/C | [Gé19] | $0.509\pm 0.017$ | $-1.361\pm 0.047$ | $4.38\pm 0.22$ | $-0.87\pm 0.32$ | - | 0.98 | 0.04
(Li,Be,B)/C | [Ma22] | $0.500\pm 0.010$ | $-1.330\pm 0.025$ | $4.67\pm 0.40$ | $-0.36\pm 0.23$ | - | 1.21 | 0.50
F/Si+(Li,Be,B)/C | this paper | $0.492\pm 0.008$ | $-1.293\pm 0.021$ | $4.55\pm 0.24$ | $-0.39\pm 0.13$ | $-3.0^{+0.6}_{-\infty}$ | 1.12 | 0.4

##### F/Si vs B/C.

Table 3 shows the best-fit parameters obtained from various combinations of
secondary-to-primary data, as obtained from this analysis or from some of our
previous publications (all relying on the methodology recalled in sect. 2.2).
The first two lines show the results obtained from the analysis of a single
secondary-to-primary ratio, namely F/Si (this paper) and the widely used B/C
(Génolini et al. 2019). The last two columns show that both the B/C and F/Si
fits are excellent ($\chi^{2}_{\rm min}/{\rm dof}{}\lesssim 1$), without using
the production cross-section degrees of freedom ($\chi^{2}_{\rm nui}/{n_{\rm
nui}}{}\approx 0$). Both ratios lead to consistent values for $R_{l}$ and
$\delta_{l}$ (low-rigidity break) and give $\delta\approx 0.5$ consistently
(at $2\sigma$ level). This means that considering F as a pure secondary
species is a fair approximation, and that F/Si confirms the trend, in the
diffusion coefficient Eq. (5), for a low-rigidity break (Génolini et al. 2019;
Vittino et al. 2019; Weinrich et al. 2020b) and Kraichnan-like slope $\delta$
(Génolini et al. 2019; Weinrich et al. 2020b; Korsmeier & Cuoco 2021, 2022).
The uncertainties on the parameters are slightly larger for the F/Si analysis
than for the B/C analysis, translating into larger $1\sigma$ contours of the
associated reconstructed $K(R)$: compare the dotted magenta and dash-dotted
orange contours for B/C and F/Si fit respectively in Fig. 7. This difference
is related to the less abundant F and Si fluxes compared to the B and C ones,
with larger statistical uncertainties for the AMS-02 F/Si data compared to B/C
ones (especially at high rigidity).

Figure 7: Best-fit and 1$\sigma$ envelopes for the diffusion coefficient, Eq.
(5), for different secondary-to-primary ratio data combinations. The
corresponding parameters are gathered in Table 3.

We note, though, that the best-fit value of the diffusion coefficient
normalisation, $K_{0}$, differs significantly between the F/Si and B/C
analyses. However, as highlighted in Weinrich et al. (2020b), considering a
single ratio could lead to biased results on $K_{0}$, because the latter is
mostly degenerate with the production cross-section nuisance parameters. We
come back to this difference in Sect. 4.2.

##### F production uncertainties and F/Si data systematics.

Figure 8: Comparison of F/Si data (Aguilar et al. 2021d) with calculations
(SLIM propagation configuration) calibrated on the (Li,Be,B)/C transport
parameters (see Table 3). The thick grey lines show the direct calculation
based on our three production cross-section sets (dashed line for OPT12, solid
line for OPT12up22, and dotted line for OPT22). The thin lines result from the
additional fit of a global factor $r_{\cal F}$ (rescaling the overall
production of F) to better match the data. A pure secondary origin of fluorine
is assumed in both calculations (see text for discussion).

As an alternative view, we inspect here whether the F/Si ratio predicted from
the (Li,Be,B)/C-derived transport parameters (Maurin et al. 2022b) are
consistent with the data. This is illustrated in Fig. 8, where the thick grey
lines show the predicted F/Si ratio (pure secondary hypothesis, not a fit);
the purple symbols show the AMS-02 data. The model calculations clearly
overshoot the data. By allowing an overall rescaling $r_{\rm F}$ of the F
production cross sections, the thin lines shows that the production cross
sections—for any of our production sets—must be rescaled by a factor $\approx
0.9$ to match the F/Si data, i.e. consistent with the typical $\sim 10\%$
errors in the nuclear data. This factor is comparable to the scaling applied
by Boschini et al. (2022), with the Galprop code, to match the F/Si data.

Also highlighted in the legend of Fig. 8 are the $\chi^{2}_{\rm min}$/dof
values obtained after the cross-section rescaling. The OPT12up22 set is
favoured with $\chi^{2}_{\rm min}/{\rm dof}{}=1.01$, although the model seems
significantly offset from data below 10 GV. We recall that in this range, the
‘Acc. norm.’ systematics dominates (see top panel of Fig. 3), with a
correlation length of $\ell_{\rm Acc.\leavevmode\nobreak\ norm.}=1$ decade.
Such a correlation means that if one data point moves, all the others (over
the correlation length) follows. In practice, the model is thus only $\gtrsim
1\sigma$ away from the $n_{\rm pts}$ low-rigidity points (global
normalisation) instead of $n_{\rm pts}\times 1\sigma$ away ($n$ independent
points at $1\sigma$ each). This explains why the fit remains excellent, and
moreover, why it is crucial to account for the covariance matrix of the data
uncertainties in the analyses. It would actually be extremely useful if the
AMS-02 collaboration could directly provide this matrix, as its role is
crucial for the interpretation of their data (Derome et al. 2019; Boudaud et
al. 2020; Heisig et al. 2020).

###  4.2 Combined B/C and F/Si analysis

The next step of the analysis is to perform a combined analysis of (Li,Be,B)/C
and F/Si data. This allows to also fit the relative source abundance of 19F
w.r.t. 28Si, namely $\log_{10}\left(q_{{}^{19}\rm F}/q_{{}^{28}\rm
Si}\right)$.

##### Transport parameters.

The best-fit values of the fit parameters for the SLIM configuration and
OPT12up22 production cross-section set for the combined analysis of F/Si and
(Li,Be,B)/C are reported in in Table 3 (fourth line). For comparison purpose,
we also report the results from the (Li,Be,B)/C analysis (Maurin et al.
2022b); the associated $K(R)$ values and contours are shown in Fig. 7, in red
and black respectively. From the Table and the figure, we see that the
transport parameters are consistent at better than $1\sigma$, and that the
combined analysis of the four ratios slightly decreases the uncertainties
(compared to the (Li,Be,B)/C analysis). The fit values are actually driven by
the subset of (Li,Be,B)/C data, because they combine 3 times the number of
F/Si data, all having slightly smaller statistical uncertainties.
Nevertheless, both the $\chi^{2}_{\rm min}$/dof and $\chi^{2}_{\rm
nui}/{n_{\rm nui}}$ values (last two columns in Table 3) show an improvement
with the fully combined analysis. The results are consistent with no primary
source of F (see below), indicating that $Z=2-5$ and $Z=9$ secondary species
can be explained with the same propagation history; the consistency of 3He
with Li, Be, B data was discussed in Weinrich et al. (2020b).

Figure 9: Best-fit parameters and uncertainties from fit of the SLIM (left),
BIG (middle) and QUAINT (right) models to AMS-02 data using the cross-section
sets OPT12 (downward silver triangles), OPT12up22 (black circles), and OPT22
(upward grey triangles). The empty symbols with thin error bars correspond to
the combined fit to (Li,Be,B)/C data (taken from Maurin et al. 2022b), while
the filled symbols with thick error bars correspond to the combined fit to
(Li,Be,B)/C and F/Si data (this analysis). See text for discussion.

These conclusions are generalised in Fig. 9 for the different transport
configurations BIG and QUAINT (see Sect. 2) and different cross-section sets
(shown as different symbols). We report the results for the (Li,Be,B)/C
(Maurin et al. 2022b) and F/Si+(Li,Be,B)/C analysis (this paper) as empty and
filled symbols respectively. The main differences are related to the choice of
the production cross-section set, as discussed at length in Maurin et al.
(2022b). In particular, for both combination of data, the best $\chi^{2}/$dof
is obtained for OPT12up22131313We recall that for the (Li,Be,B)/C subset, the
significantly larger-than-one $\chi^{2}/$dof was caused by the two low-
rigidity Be/B points (upturn) that could not be well fitted by the model
(Weinrich et al. 2020b)..

##### Posteriors for Li, Be, B, and F production cross sections.

In the F/Si-alone analysis, a lower value of $K_{0}$ was favoured, compared to
that obtained in the combined analysis. As detailed in Weinrich et al. (2020b)
and Maurin et al. (2022b), only the combined analysis of secondary-to-primary
ratios allows to break the degeneracy between $K_{0}$ and the production cross
sections, by enforcing a unique $K_{0}$ values for all the ratios. With our
methodology (Sect. 2.2), in addition to the best-fit parameters, we also get
posterior values for the nuisance parameters, i.e. we can inspect the values
preferred by the fit for the production cross sections. However, this is not
direct as we only use a single proxy reaction per secondary element.
Nevertheless, we can calculate the global normalisation factor $\mu_{Z}^{(p)}$
required for each secondary element $Z$ to best-fit the data141414The proxy
$p$ for the production reaction of an element $Z$ (see Table 1) only
contributes to a fraction $f_{p}$ of the total. The global bias
$\mu_{Z}^{(p)}$ for the production of this element is calculated from
$(\mu_{Z}^{(p)}-1)=(\mu_{p}-1)\times f_{p}$, where $\mu_{p}$ is the posterior
of the normalisation parameter Eq. (12). The relevant $f_{p}$ are taken from
Génolini et al. (2018), Maurin et al. (2022b), and Fig. 6, with
$f_{\mathrm{{}^{16}O+H\to^{6}Li}}\approx 15\%$,
$f_{\mathrm{{}^{16}O+H\to^{7}Be}}\approx 19\%$,
$f_{\mathrm{{}^{12}C+H\to^{11}B}}\approx 33\%$, and
$f_{\mathrm{{}^{22}Ne+H\to^{19}F}}\approx 30\%$.. We show them in Fig. 10 for
our three production cross-section sets.

Figure 10: Correlation between log${}_{10}(K_{0})$ and the normalisation
factor $\mu_{Z}^{(p)}$, which corresponds to the correction factor applied on
the total production cross section (of element $Z$) in order to best-fit
AMS-02 F/Si+(Li,Be,B)/C data. The four elements considered are colour-coded:
$Z=3$ (Li) in orange, $Z=4$ (Be) in green, $Z=5$ (B) in blue, and $Z=9$ (F) in
purple. The $1\sigma$ correlation ellipses are shown for analyses with
different cross section sets (in model SLIM): from left to right OPT12 (filled
downward triangles), OPT12up22 (filled circles), and OPT22 (filled upward
triangles). The horizontal grey dashed line highlights $\mu_{Z}=1$ (i.e. no
modification needed for the production of an element).

For Li, Be, and B production, the results are mostly similar to those
discussed in Fig. 12 of Maurin et al. (2022b), i.e. Li (orange dashed
ellipses) is very sensitive to the selected production set, which is not the
case for Be and B. Also, Be production is consistent with $\mu_{Z}\approx 1$
(no modification of the production cross sections), but there is a trend for
the need to increase the production of B (i.e. $\mu_{\rm B}>1$) and to
decrease that of F (i.e. $\mu_{\rm F}<1$). However, the required numbers are
of the order of $\sim 10\%$ at most, which is within the range of nuclear
uncertainties. New nuclear data are desired in order to confirm this
possibility, or to prove that a tension exists between these different flux
ratios.

##### Best-matching data.

Figure 11: Flux ratios (top), residuals (centre) and $\tilde{z}$-scores
(bottom) for B/C (blue circles), Be/C (green downward triangles), Li/C (orange
squares), and fluorine (purple upward triangles). The models (top panel) are
calculated for the updated OPT12 (dashed grey line), OPT12up22 (solid black
line), and OPT22 (dashed-dotted grey line), from the best-fit transport
parameters of the combined analysis of all three species. In the middle and
bottom panels, the residuals and $\tilde{z}$-score are shown for the OPT12up22
configuration only. The distributions in the right-hand side of the bottom
panel are histograms of the $\tilde{z}$-score values (projected on the
$y$-axis), compared to a $1\sigma$ Gaussian distribution (solid black line).
See text for discussion.

In this global fit, it is also interesting to look at the detailed match of
the model to each ratio, as shown in the top panel of Fig. 11. The middle
panel shows the residuals and the bottom panel the $\tilde{z}$-score. The
latter is a generalisation of the $z$-score in presence of correlations in the
data. We recall that the $z$-score, $z=({\rm model}-{\rm data})/\sigma$, is
similar to the residuals but expressed in unit of the total data uncertainties
$\sigma_{\rm tot}$. As stressed in Boudaud et al. (2020), these quantities
ignore the correlations in the data, leading to a biased view of the goodness-
of-fit between the model and the data. To cure this, Boudaud et al. (2020)
proposed to use the $\tilde{z}$-score, which corresponds to the residuals of
the eigen vectors (data-model) of the total covariance matrix of data
uncertainties (i.e. it accounts for the correlations). By construction,
$\chi^{2}=\sum_{i}\tilde{z}_{i}^{2}$, and the representation of the
$\tilde{z}$-score gives an unbiased view of the distance between the model and
the data.

In the bottom panel, we also plot on the right-hand side the projected
distributions of $\tilde{z}$ ratio by ratio. These distributions are expected
to follow a Gaussian distribution of width one (shown as a solid black line)
for a good match to the data. More directly, the numbers in the legend give
the separate contribution (to the total $\chi^{2}_{\rm min}$) of these various
ratios. We see that they are all in excellent agreement with the data, except
for Be/C; the origin of this discrepancy is understood and related to the
lowest two data points of this ratio (Weinrich et al. 2020b).

###  4.3 Fluorine source abundance

The combined F/Si+(Li,Be,B)/C analysis also provides constraints on the
relative CR source abundance $(^{19}\rm F/{{}^{28}\rm Si})_{\rm CRS}$, which
are shown in the 8-th panel of Fig. 9 and also reported in Table 4. These
constraints only depend mildly on the production cross-section sets and the
transport configuration selected.

From Table 4, we report a best-fit value of $\sim 10^{-3}$. This value is
similar to the Solar system ratio ${\rm(^{19}F/^{28}Si)}_{\rm SS}=8.7\times
10^{-4}$ (Lodders et al. 2009), but should not be compared directly to it (see
below). Our fit is also compatible with a null value (i.e. no primary
contribution necessary to match the data), and we can draw a 1$\sigma$ upper
limit $(^{19}\rm F/{{}^{28}\rm Si})_{\rm CRS}\sim 5\times 10^{-3}$.

Table 4: Best-fit $\log_{10}\left(q_{{}^{19}\rm F}/q_{{}^{28}\rm Si}\right)$ values and $+1\sigma$ upper limits (in parenthesis) from the analysis of AMS-02 F/Si+(Li,Be,B)/C data. Three propagation configurations (columns) and production data sets (rows) are considered: in all cases, the lower limit for the abundance of $q_{{}^{19}\rm F}$ is compatible with zero. | SLIM | BIG | QUAINT
---|---|---|---
OPT12 | -2.7 (-2.3) | -2.7 (-2.3) | -2.8 (-2.4)
OPT12up22 | -3.1 (-2.4) | -3.1 (-2.4) | -3.3 (-2.5)
OPT22 | -3.1 (-2.4) | -3.1 (-2.4) | -3.3 (-2.4)

##### Interpretation.

Fluorine is potentially a key element to distinguish whether the CR source
composition is controlled by atomic effects related to the FIP of the elements
or by their relative concentration into dust in the interstellar medium (Meyer
et al. 1997). It is indeed one of the few elements that has a high FIP, $17.4$
eV, close to that of noble gases, but which is only moderately volatile. The F
fraction in interstellar dust is not measured, but it is found in non-
negligible proportion in some meteorites such as the CI-chondrite Orgueil
(Lodders et al. 2009). The equilibrium condensation temperature of F, $T_{\rm
cond}=734$ K, is close to that of S, $T_{\rm cond}=664$ K (Lodders et al.
2009), so both elements could be incorporated in about the same proportions in
interstellar dust, up to about $20\%$ of their total interstellar abundances
(Tatischeff et al. 2021). In the model of a preferential acceleration of
elements contained in dust (Meyer et al. 1997; Ellison et al. 1997), we can
expect F and S to have similar CR source abundances relative to the solar
system composition, that is ${\rm[F/Si]}_{\rm CRS}\sim{\rm[S/Si]}_{\rm
CRS}=-0.59$ dex (Tatischeff et al. 2021), where
${\rm[X/Y]}=\log_{10}{\rm(X/Y)}-\log_{10}{\rm(X/Y)}_{\odot}$. Thus, in this
model we expect that ${\rm(^{19}F/^{28}Si)}_{\rm CRS}\sim 2.2\times 10^{-4}$.

In the model where the CR source composition is controlled by a FIP bias
similar to that found in solar energetic particles (Meyer et al. 1979; Meyer
1985), the source abundance of 19F can instead be estimated from that of 20Ne,
since the two elements have close masses and FIPs (21.6 eV for Ne). Here, we
assume that the 19F and 20Ne nuclei in the CR source composition come mainly
from the average interstellar medium, and not from the source enriched in
Wolf-Rayet wind material at the origin of the 22Ne excess (Binns et al. 2005).
In the FIP model, we thus expect that ${\rm[^{19}F/^{28}Si]}_{\rm
CRS}\sim{\rm[^{20}Ne/^{28}Si]}_{\rm CRS}=-0.76$ dex (Tatischeff et al. 2021),
and the predicted 19F abundance relative to 28Si is
${\rm(^{19}F/^{28}Si)}_{\rm CRS}\sim 1.5\times 10^{-4}$. Unfortunately, the
upper limit on this ratio derived from AMS-02 data is significantly higher
than the predicted values in both models and does not allow us to distinguish
them.

In their analysis of AMS-02 data, Boschini et al. (2022) found an excess below
10 GV in the F spectrum (after reducing their calculated spectrum by about
$10\%$) and suggest that it could be explained by a primary F component. The
integration of the injection spectra of primary F and 28Si (Boschini et al.
2020) above 1 GV, where the excess is reported, gives
${\rm(^{19}F/^{28}Si)}_{\rm CRS}=2.7\times 10^{-3}$. If integrated above 0.1
GV (as discussed in Boschini et al. 2022), the ratio becomes
${\rm(^{19}F/^{28}Si)}_{\rm CRS}=1.7\times 10^{-3}$. Both values are
consistent with the upper limit obtained in the present work. But they are
about an order of magnitude higher than the values predicted from the FIP and
condensation temperature of F. Such an overabundance would make F very special
compared to all primary and mostly primary CRs from H to Zr, whose abundances
are well explained in the model studied by Tatischeff et al. (2021). It seems
more likely to us that the excess found in the analysis of Boschini et al.
(2022) is related to the non-negligible uncertainties in the 19F production
cross sections (see Section 2.2.3). Note that, yet another explanation of this
discrepancy is proposed in a recent preprint (Zhao et al. 2022), where the
authors use spatially-dependent diffusion.

##### How could we get tighter constraints?

To go further in the interpretation, we need to improve the present
constraints by a factor $\sim 50$.

Figure 12: Primary fraction of 19F as a function of rigidity for the
propagation configuration SLIM and the three production cross-section sets
OPT12 (dashed light-grey lines), OPT12up22 (solid black lines), and OPT22
(dash-dotted grey lines). The thick lines are calculated from the best-fit
source parameter $q_{{}^{19}\rm F}$, from the combined analysis of F/Si and
(Li,Be,B)/C (see filled symbols in Fig. 9). The thin lines with downward
arrows show the $1\sigma$ upper limits in the same configuration. For
comparison, the purple dotted line shows the total relative uncertainties
(errors combined quadratically) of AMS-02 F/Si ratio; see text for discussion.

To understand whether this could be possible, it is interesting to look at the
primary fraction of CR F (w.r.t. its total flux) in Fig 12. The fraction grows
steadily with the rigidity, which is expected as primary contributions grow
$R^{\delta\approx 0.5}$ faster than secondary ones (e.g., Vecchi et al. 2022).
All the curves are parallel because all our propagation configurations assume
the same power-law index value for all primary species (and thus for the
primary F). Were the F/Si data uncertainties constant with rigidity, the high-
rigidity tip would be best to constrain the fluorine relative source
abundance. However, because of decreasing fluxes, statistical uncertainties
dominate at this end (see Fig. 3). For comparison purpose, we report the total
uncertainties on AMS-02 F/Si (purple dotted line) in Fig 12: we see that above
100 GV, the AMS-02 uncertainties are growing faster than the primary fraction;
below this rigidity, the uncertainties are roughly constant, making the $\sim
10-100$ GV range the regime that constrains in practice the fluorine primary
fraction.

Actually, because F is compatible with being a pure secondary (the fit only
gives an upper limit), the primary fraction of F can be directly compared to
the relative uncertainty on the F/Si data (that the model fits): the thin
lines with downward arrows are roughly bounded by the dash-dotted purple line.
The match is not exact and varies from cross-section model to another: this is
because the minimisation procedure allows for a variation of the production
cross sections; moreover the purple dotted line shows the total uncertainties
(statistical and systematic uncertainties added in quadrature), whereas the
analysis accounts for the covariance matrix of data uncertainties. In summary,
to get tight enough constraints on the source abundance of F (to disentangle
between the two source models), the goal of future experiments should be to
reach—owing to the growing primary fraction with rigidity—either a $\lesssim
0.1\%$ precision on F/Si data at a few GV, or a few percent precision above
tens of TV.

##  5 Conclusions

We have studied the AMS-02 F/Si data in a semi-analytical propagation model to
constrain the transport parameters and the source abundance of 19F. We have
updated the production cross sections of F from its main CR progenitors (Ne,
Mg, Si, and Fe). We highlighted the importance of accounting for the anomalous
isotopic abundance of 22Ne/20Ne, in order not to bias the F production (impact
$\lesssim 6\%$). For our analysis, we have propagated the uncertainties on
several important input ingredients (Solar modulation and nuclear production
cross sections), and have accounted for a best-guess covariance matrix of
uncertainties for the F/Si data. The latter is dominated by a systematic with
a significant correlation length (one rigidity decade) below a few ten of GV,
which is equivalent to a global normalisation factor on these low-rigidity
data. This directly impacts the statistical interpretation of the model, since
using the usual quadratic distance estimator instead (i.e, no energy
correlations in the data) would lead to worsened goodness-of-fit values.

We used several propagation setups (with or without convection and
reacceleration, and two parametrisations of the low-rigidity break) and
different production cross sections, but all configurations lead to the same
conclusions. In a first step, we analysed AMS-02 F/Si data alone and found
that (i) the F/Si data can be reproduced by the model assuming F is a pure
secondary species (i.e. no astrophysical source of F); (ii) most of the
transport parameters obtained from the F/Si analysis are consistent with those
derived from the use of the traditional (Li,Be,B)/C ratios. In a second step,
we have performed a combined analysis of F/Si and (Li,Be,B)/C), in order to
break the (partial) degeneracy between the production cross sections and the
normalisation of the diffusion coefficient. We found that (i) the global
$\chi^{2}$ per degree of freedom was close to 1 and significantly better than
that obtained from the (Li,Be,B)/C analysis only; (ii) the transport
parameters were also slightly more constrained when adding F/Si data, as
expected, with a slightly larger diffusion coefficient normalisation (but
within the uncertainties); (iv) the posteriors obtained for the production
cross-section parameters indicate a slight mismatch between the production of
B and F, where a $10\%$ increase of B and $5\%$ decrease of F are needed to
best-fit the data. This is within the typical range of nuclear data
uncertainties, and new or better nuclear data are needed to go further in this
interpretation; in particular, F production data only have a single or pair of
energy points for the whole energy domain, which is certainly not
satisfactory. At this stage, we can nevertheless conclude that all AMS-02
secondary species with $Z\in[2,3,4,5,9]$ can be reproduced in a simple
diffusion model.

Finally, we have obtained an upper limit on the relative source abundance of
19F. Fluorine is potentially a key element to test the underlying CR sources
and their acceleration mechanisms, as it is one of the few light elements
(with Na) for which the correlation between the volatility temperature and FIP
breaks down; these two hypotheses are expected to lead to
${\rm(^{19}F/^{28}Si)}_{\rm CRS}\sim 2.2\times 10^{-4}$ and $\sim 1.5\times
10^{-4}$ respectively. Unfortunately, owing to the very small primary content
in the CR flux of F ($\lesssim 5\%$ at 1 GV), we can only derive an upper
limit at source of $\sim 5\times 10^{-3}$ from AMS-02 data; this limit is set
by the data in the $50-100$ GV rigidity range (where the uncertainties are the
smallest). In order to distinguish between the two above models, at least a
factor $50$ improvement would be needed. This could be achieved by the
challenging measurement of F/Si up to a few tens of TV at a few percent
precision. This may be within reach of the next generation of CR experiments
(Schael et al. 2019; Battiston et al. 2021).

###### Acknowledgements.

We thank our CR colleagues at Annecy and Montpellier for discussions. This
work was supported by the Programme National des Hautes Energies of CNRS/INSU
with INP and IN2P3, co-funded by CEA and CNES. We thank the Center for
Information Technology of the University of Groningen for their support and
for providing access to the Peregrine high-performance computing cluster.

##  Appendix A Primary fluxes

Figure 13: Comparison (top panel) and residuals (bottom panel) between the
model calculation (lines) and the AMS-02 data (Aguilar et al. 2017, 2020,
2021c, 2021a), for the main progenitors of Li, Be, B, and F; we recall that
this is not a fit to the data (see text). The numbers in parenthesis in the
legend indicate the fractional contribution of these progenitors to the F
production (as read from top panel of Fig. 6).

As highlighted in the main text, we do not fit the progenitors of Li, Be, B,
and F in our runs. Instead, we adjust the individual elemental source
abundance (and Ne isotopic source abundance) in order to match the data at a
given rigidity. For AMS-02 data, this rigidity is taken to be the closest data
point to 50 GV. As was shown in Génolini et al. (2019), this procedure gives
an excellent match to the C, N, and O data. The same conclusions are drawn
here in Fig. 13 for CNO (blue, red, and grey lines/symbols), although a few
other elements show some discrepancies. At low rigidity, the model overshoots
the Na and Al data (and to a lesser extent Mg) and undershoots Fe data, at the
level of $\lesssim 30\%$; the Fe undershoot has been observed and discussed in
Schroer et al. (2021) and Boschini et al. (2021). It is beyond the scope of
this paper to quantify this difference (statistically speaking) and to then
discuss the possible origins of these mismatches. We stress that a complete
analysis should account for the covariance matrix of uncertainties in the
data, improving the statistical agreement between the model and the data.

In the context of this analysis, it is enough to focus on the main progenitors
of F, that is Ne, Mg, Si, and Fe. Their contributing fractions to the
production of fluorine are reported in parenthesis in the legend of Fig. 13
(as taken form the top panel of Fig. 6). We see that the main progenitors, Ne
($30\%$ contribution, black $\times$ symbol) and Si ($15\%$ contribution,
empty cyan square), match very well the data. On the other hand, Mg ($15\%$,
orange right-oriented empty triangle) and Fe ($10\%$, empty violet star) are
off by $\lesssim 20\%$. This makes their individual mismatch on F at the level
of $\lesssim 30\%\times 20\%\lesssim 6\%$, but these two contributions cancel
out because of their opposite signs. The remaining contributions from Na and
Al ($1.5\%$) are too small to impact F, even though the model is $\lesssim
30\%$ below these data.

Figure 14: Same as Fig. 13, but now fitting the primary fluxes assuming three
breaks in their source spectra (see text); we do not show the residuals for
legibility. The model overfits the data with $\chi^{2}_{\rm min}/{\rm
dof}{}=0.282$.

To further check this cancellation, we perform a dedicated fit of the primary
source spectrum of C, N, O, Ne, Na, Al, Mg, Si, and Fe on the corresponding
AMS-02 data. Instead of assuming a single power law (PL hereafter) as in the
analysis presented in the main text, we follow the approach of Boschini et al.
(2022) and assume a broken power law (BPL hereafter) with three breaks:

$Q\propto
R^{-\alpha}\times\prod_{k=0}^{2}\left(1+\left(\frac{R}{R_{k}}\right)^{\Delta_{k}/\eta_{k}}\right)^{\eta_{k}}.$
(17)

The smoothness parameters are fixed to $\eta_{0}=\eta_{2}=0.18$ (with positive
values for $\Delta_{0}$ and $\Delta_{2}$ in the fit) and $\eta_{1}=-0.18$
(with negative values for $\Delta_{1}$ in the fit). With more than eight
parameters per element—one normalisation per isotope, one slope $\alpha$ and
three breaks $\Delta_{k}$ at three rigidities $R_{k}$—, it is no surprise that
the model overfits the data ($\chi^{2}_{\rm min}/{\rm dof}{}=0.282$). The fit
and the data for the BPL configuration are shown in Fig. 14; we do not report
the values of the best-fit parameters (obtained assuming total uncertainties
in quadrature on the data) as they are not relevant for the discussion. We
also stress that it is not clear whether the BPL configuration provides more
trustworthy predictions, especially at high rigidity where the last break is
dominated by a few data points (dominated by large statistical uncertainties).
A dedicated analysis to assess the statistical relevance of the various breaks
(in BPL) against some degree or full universality (in PL) in the primary
spectra is necessary, but it goes beyond the scope of the paper. Importantly,
such an analysis would have to include the covariance matrices of
uncertainties on the data.

Figure 15: Relative difference between secondary-to-primary ratios calculated
from the BPL and PL primary spectra, in solid lines (from top to bottom, Li/C,
Be/C, B/C, and F/Si). The hatched area shows a $3\%$ shift related change in
the production cross sections of the secondary in the numerator of these
ratios. The shaded band shows the AMS-02 data relative uncertainties
(systematic and statistical uncertainties added in quadrature). We see that
both areas overlap, showing that the difference between assuming PL or BPL
does not impact our conclusions. We give more details in the discussion in the
expanded text in App. A1.

We can now compare the secondary-to-primary ratios (of interest for this
study) obtained with the BPL and PL assumptions in Fig. 15.

  * •

High-rigidity: above $\sim 100$ GV, the difference between the two
calculations (solid lines) grows. In this regime, the difference (on the
secondary-to-primary ratio calculation) can be interpreted as the propagation
of the primary flux data uncertainties.151515Indeed, with too many parameters
on the model side and large statistical uncertainties on the data side, the
BPL breaks (different breaks per elements) differ from the PL ones (universal
break): for instance, the difference in the Ne flux (black line in Fig. 14 and
13 respectively), the main progenitor of F, is responsible for the growing
difference in the F/Si calculation. In any case, the solid black line curve is
within the secondary-to-primary AMS-02 data uncertainties (shaded band).

  * •

Low-rigidity: below a few tens of GV, the difference between the BPL and PL
calculation (solid lines) is larger than the data uncertainties (shaded area).
However, the discrepancy is reconciled by a minimal change of the secondary
production, shifting the solid lines by $\pm 3\%$, as shown by the hatched
band. This number is (i) of the order of the uncertainties obtained on the
production cross section nuisance parameters (size of the ellipse in Fig. 10);
(ii) of the order or smaller than the bias on these nuisance parameters
(y-axis values of the centre of these ellipses in Fig. 10); (iii) for the Li
case, much smaller than the dispersion related to the choice of the production
cross-section set for the Li case (dispersion between the three orange
ellipses in Fig. 10). This conservative hatched area and shaded one overlap,
showing statistical consistency between the PBL and PL calculations.

For all these reasons, we conclude that while the BPL and PL approaches lead
to few percent differences on the secondary-to-primary ratios (used in our
analysis), these differences are either encompassed in the data uncertainties
or absorbed by a small shift of the production cross sections. This shift
would certainly change the exact numbers provided in the main paper, but would
not change any of our conclusions.

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