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A&A
Volume 688, August 2024
Article Number A17
Number of page(s) 18
Section Astrophysical processes
DOI https://doi.org/10.1051/0004-6361/202244660
Published online 30 July 2024

© The Authors 2024

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1. Introduction

The Alpha Magnetic Spectrometer (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. 2021a). In particular, AMS-02 provides elemental fluxes and ratios at percent-level precision from ∼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 into two categories according 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). When releasing their data, the AMS-02 collaboration started with the most abundant species, namely the light primary species H (Aguilar et al. 2015a) and He (Aguilar et al. 2015b), then moved on to C and O (Aguilar et al. 2017), and more recently moved to Ne, Mg, Si (Aguilar et al. 2020), and Fe (Aguilar et al. 2021b). These data are crucial for the modelling of secondary species because the break up of these nuclei populate and contribute, in varying fractions, to the flux of all species lighter than Si. Examples of mixed species are N, Na, and Al (Aguilar et al. 2021c), which receive a roughly similar amount of primary and secondary contributions at a few Giga-Volts (Maurin 2020). Secondary-to-primary ratios are of special interest since they allow the transport parameters to be studied 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), and it was followed by Li/C and Be/C (Aguilar et al. 2018) and by the 3He/4He isotopic ratio (Aguilar et al. 2019).

In this paper, we focus 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. 2017a, 2019; Weinrich et al. 2020a; Maurin et al. 2022a). 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 2012a; Picot-Clémente et al. 2017; Wu & Chen 2019), and the sub-Fe/Fe ratio1 (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, such as the High Energy Astrophysical Observatory (HEAO3; Engelmann et al. 1990) and the Advanced Composition Explorer (ACE; George et al. 2009; Lave et al. 2013), had similar systematics over a wide range of charges, with only ∼ 2 $ {{\sim}\sqrt{2}} $ larger statistical uncertainties for F/Si compared to sub-Fe/Fe (Si/Fe ∼ 1.5 and F/Sub-Fe ∼ 0.8; Engelmann et al. 1990). Nonetheless, the oversight is possibly related to nuclear physics. Until recent times, the community was lacking reliable nuclear production cross sections for F. As we later show, nuclear data for its production are particularly scarce, with only a few relevant measurements in the 1990s and 2000s. Moreover, the production of F involves more progenitors compared to other secondary species, which means that the number of cross sections required for CR calculations is larger. As also discussed below, the five main progenitors of 19F2 (i.e. 20, 22Ne, 24Mg, 28Si, and 56Fe) only make ∼60% of the total production. In comparison (Coste et al. 2012; Génolini et al. 2018; Maurin et al. 2022a), only two progenitors (12C and 16O) make ∼60% of all the Li, Be, and B, and one single progenitor, 4He (resp. Fe), makes ∼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. Whereas the ratios of 3He, Li, Be, B, and sub-Fe to their progenitors are in trace amounts 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 a Giga-electronVolt per nucleon, 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 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 transport equation and describe the propagation model and configuration used in our analysis. In Sect. 3, we inspect the constraints that 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. In Sect. 4, we analyse the behaviour of a combined analysis of the two datasets above. In Sect. 5, we draw constraints on the abundance of F (relative to Si) in CR sources, discussing whether this is consistent with the expectations of the current CR source composition modelling. We conclude in Sect. 6. For readability, we postpone the presentation of crucial but more technical details until the appendices. In Appendix A, we describe the F/Si fitting strategy, which involves priors on solar modulation and nuclear cross-section parameters, and the covariance matrix of uncertainties for AMS-02 F/Si data. In Appendix B, we identify the most important progenitors of F and rescale their production cross sections to recent nuclear data. In Appendix C, we discuss how well our model reproduces the primary CR data and whether the results discussed in the main text are sensitive to the assumption made on the CR source spectral shape.

2. Model and free parameters

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 information. Below, we provide a summary description of the model and of the ingredients relevant for the analysis of F/Si data.

2.1. Transport equation in 1D model

Assuming isotropy, the flux of a CR ion is related to the differential density, dN/dE ≡ N (for short), by ψ = vN/(4π). The differential density Nk for a CR species k is calculated from the transport equation (Berezinskii et al. 1990). For homogeneous and isotropic diffusion (coefficient K) and constant convective transport (Vc) 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)

( − K ∂ 2 ∂ z 2 + V c ∂ ∂ z + 1 γ τ rad k + 2 h δ ( z ) ∑ t ∈ ISM n t v σ inel k + t ) N k + 2 h δ ( z ) ∂ ∂ E ( b k N k − c k ∂ N k ∂ E ) = 2 h δ ( z ) Q k ( E ) + N r γ τ rad r → k + 2 h δ ( z ) ∑ t ∈ ISM ∑ p n t v σ prod p + t → k N p $$ \begin{aligned}&\left( -K \frac{\partial ^2}{\partial z^2} +V_c \frac{\partial }{\partial z} + \frac{1}{\gamma \,\tau ^k_{\rm rad}} + 2h\,\delta (z) \sum _{t\,\in \mathrm {ISM}} n_t\,v\,\sigma ^{k+t}_{\rm inel}\right) N^k \nonumber \\&\quad + 2h\,\delta (z)\frac{\partial }{\partial E}\left( b^k N^k - c^k \frac{\partial N^k}{\partial E}\right)\nonumber \\&\quad = 2h\,\delta (z) Q^k(E) + \frac{N^r}{\gamma \,\tau ^{r\rightarrow k}_{\rm rad}} + 2h\,\delta (z) \sum _{t\,\in \mathrm {ISM}} \sum _p n_t\,v\,\sigma ^{p+t\rightarrow k}_{\rm prod}N^p \end{aligned} $$(1)

with

b ( E ) = 〈 d E d t 〉 ion , coul . − ∇ . V 3 E k ( 2 m + E k m + E k ) + ( 1 + β 2 ) E K pp $$ \begin{aligned}&b(E) = \Big \langle \frac{\mathrm{d}E}{\mathrm{d}t}\Big \rangle _{\rm ion,\,coul.} - \frac{\boldsymbol{\nabla }.\boldsymbol{V}}{3} E_k\left(\frac{2m+E_k}{m+E_k}\right) + \frac{(1+\beta ^2)}{E}\, K_{pp}\end{aligned} $$(2)

c ( E ) = β 2 K pp , $$ \begin{aligned}&c(E) = \beta ^2 \, K_{pp}, \end{aligned} $$(3)

and

K pp × K = 4 3 V a 2 p 2 δ ( 4 − δ 2 ) ( 4 − δ ) · $$ \begin{aligned} K_{pp}\times K= \frac{4}{3}\;V_a^2\;\frac{p^2}{\delta \,(4-\delta ^2)\,(4-\delta )}\cdot \end{aligned} $$(4)

The various quantities appearing in Eq. (1) are: (i) the half-life τrad, either leading to a disappearance if CR species k is unstable, or to a radioactive source term (if r decays into k); (ii) nuclear reaction rates ∑tntv σ on the interstellar medium (ISM) targets t of density nt, either associated with a net loss with a destruction cross section σ inel k + t $ \sigma_{\mathrm{inel}}^{k+t} $, or to a secondary source term, summed over p progenitors (heavier than k) with a production cross section σ prod p + t → k $ \sigma_{\mathrm{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, Qk(E), from astrophysical sources in the thin disc. Finally, Eq. (4) links the diffusion coefficient is momentum (Kpp) and in space (K) from a minimal reacceleration model (Osborne & Ptuskin 1988; Seo & Ptuskin 1994), via the Alfvénic speed Va.

2.2. 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 those of the SS (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 Appendix B.2). For this reason, we also rescale the isotopic abundances in order to match a low-energy ACE-CRIS (Cosmic Ray Isotope Spectrometer) 22Ne/20Ne data point (Binns et al. 2005).

All the results derived in this paper are based on the USINE3 public code (Maurin 2020), which provides a full implementation of the 1D model and its solutions. This model assumes an infinite slab with a thin disc and a thick halo (Jones et al. 2001): the gas and sources (with energy losses and reacceleration) are in the disc, while a spatially independent diffusion and constant convection are enabled in the diffusion halo; CRs freely escape at the boundary L of the halo. In this model, CR fluxes only depend on the vertical coordinate, and the thin disc approximation allows Eq. (1) to be solved analytically in the halo, while the energy part in the disc is solved numerically. Despite its simplicity, this model captures all the essential features of CR transport (Jones et al. 2001), as further illustrated by the occasional use of this model by developers of the numerical code DRAGON (Evoli et al. 2018a) in some of their studies (Evoli et al. 2020; Schroer et al. 2021).

As in our previous publications, we fix the disc half-thickness h = 100 pc and the gas density nISM = 1 cm3 (90% H and 10% He in number) to recover the local gas surface density (Ferrière 2001). The flux of stable species is independent of K0/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. 2020b; Maurin et al. 2022b). For the source term Q(R), we assume a simple power-law in rigidity with a universal slope. 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. 2020a). 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). We check in Appendix C that the use of broken-power laws (instead of a simple power-law), while better fitting the primary flux data, has its own issues. In any case, the results presented below are insensitive to this choice.

2.3. BIG, SLIM, and QUAINT configurations

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

K ( R ) = β η K 0 { 1 + ( R l R ) δ − δ l s l } s l { R R 0 = 1 GV } δ { 1 + ( R R h ) δ − δ h s h } − s h . $$ \begin{aligned} K(R) = {\beta ^\eta } K_{0} \; {\left\{ 1 + \left( \frac{R_l}{R} \right)^{\frac{\delta -\delta _l}{s_l}} \right\} ^{s_l}} {\left\{ \frac{R}{R_0 = 1\,\mathrm{GV}} \right\} ^\delta }\, {\left\{ 1 + \left( \frac{R}{R_h} \right)^{\frac{\delta -\delta _h}{s_h}} \right\} ^{-s_h}}. \end{aligned} $$(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. 2021a) and peculiar microphysics mechanisms (Yan & Lazarian 2004; Ptuskin et al. 2006; Shalchi & Büsching 2010; Blasi et al. 2012; Evoli & Yan 2014; Evoli et al. 2018b; Fornieri et al. 2021).

We re-use here the three propagation scenarios proposed in Génolini et al. (2019)4, namely BIG, SLIM, and QUAINT: (i) BIG enables the “full” transport complexity, that is diffusion (K0,  δ,  Rl,  δl,  sl,  Rh,  δh,  sh, but η = 1), convection (Vc), and reacceleration (Va); (ii) SLIM is a minimal diffusion scenario with η = 1 and Vc = Va = 0; (iii) QUAINT is a more standard diffusion-convection-reacceleration scenario, that is, no low-energy break (Rl = 0) but possible sub-relativistic upturn of the diffusion coefficient via η. 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 (≳10 GV), and they all give a very good fit to light secondary-to-primary data (Weinrich et al. 2020a) and also antiprotons (Boudaud et al. 2020). To limit the number of free parameters in our analysis, we assume a fast transition sl = 0.04 for the low-rigidity break and also fix the three high-rigidity break parameters (Rh, δh, sh) to the values in Génolini et al. (2019); as showed by these authors, this choice only marginally affects the remaining parameters.

2.4. Outline of the F/Si fit procedure

The need for an improved methodology to interpret high-precision AMS-02 data was discussed at length in Derome et al. (2019), and we follow it here. We perform a χ2 analysis (Appendix A.1), accounting for solar modulation and nuclear cross-section uncertainties via nuisance parameters (Appendices A.2 and A.3), in order not to bias the determination of the transport and source parameters; relevant cross sections for the production of F are updated based on recent nuclear data (Appendix B.1). We also account for the covariance matrix of data systematic uncertainties (Appendix A.4).

To ease the reading of the paper and the navigation between the different names and configurations explored in our analyses, they are gathered and described in Table 1. The core of our analyses and results presented below relies on the SLIM propagation configuration (pure diffusion configuration). The configurations QUAINT and BIG (with convection and reacceleration) are used to check that our conclusions are insensitive to this choice. In the following, we present our results based on the updated (for F) production cross-section sets OPT12, OPT22, and OPT12up22 (see Appendix B.1). The latter set is considered to be the most relevant to better describe the nuclear and CR data (see Maurin et al. 2022a, for details).

Table 1.

List of ingredients used for the analysis and their description.

3. 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.

3.1. F/Si versus B/C

Table 2 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 above and in Appendix A). 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 ( χ min 2 / d . o . f . ≲ 1 $ {\chi^{2}_{\mathrm{min}}/\mathrm{d.o.f.}}\lesssim1 $), without using the production cross-section degrees of freedom ( χ nui 2 / n nui ≈ 0 $ {\chi^{2}_{\mathrm{nui}}/n_{\mathrm{nui}}}\approx 0 $). Both ratios lead to consistent values for Rl and δl (low-rigidity break) and give δ ≈ 0.5 consistently (at 2σ 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. 2020a) and Kraichnan-like slope δ (Génolini et al. 2019; Weinrich et al. 2020a; 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 1σ contours of the associated reconstructed K(R): compare the dotted magenta and dash-dotted orange contours for the B/C and F/Si fits respectively in Fig. 1. 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).

thumbnail Fig. 1.

Best-fit and 1σ envelopes for the diffusion coefficient, Eq. (5), for different secondary-to-primary ratio data combinations. The corresponding parameters are gathered in Table 2.

Table 2.

Comparisons of best-fit transport parameters (in SLIM) from various combinations of AMS-02 data.

We note, though, that the best-fit value of the diffusion coefficient normalisation, K0, differs significantly between the F/Si and B/C analyses. However, as highlighted in Weinrich et al. (2020a), considering a single ratio could lead to biased results on K0 because the latter is mostly degenerate with the production cross-section nuisance parameters. We come back to this difference in Sect. 4.

3.2. F production uncertainties and F/Si data systematics

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. 2022a) are consistent with the data. This is illustrated in Fig. 2, 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 rF 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 ≈0.9 to match the F/Si data, of the order of the ∼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.

thumbnail Fig. 2.

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 2). 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 rF (rescaling the overall production of F) to better match the data. A pure secondary origin of fluorine was assumed in both calculations (see text for discussion).

Also highlighted in the legend of Fig. 2 are the χ min 2 / d . o . f . $ \chi^{2}_{\mathrm{min}}/\mathrm{d.o.f.} $ values obtained after the cross-section rescaling. The OPT12up22 set is favoured with χ min 2 / d . o . f . = 1.01 $ {\chi^{2}_{\mathrm{min}}/\mathrm{d.o.f.}}=1.01 $, although the model seems significantly offset from data below 10 GV. We recall that in this range, the “Acc. norm.” data systematics dominates (see top panel of Fig. A.3), with a correlation length of ℓAcc. 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 ≳1σ away from the n low-rigidity points (global normalisation) instead of n × 1σ away (n independent points at 1σ 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. 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 the source abundance of 19F relative to 28Si to be fit, namely as log10(q19F/q28Si). We discuss this Sect. 5.

4.1. Transport parameters

The best-fit values of the fit parameters for the SLIM configuration, OPT12up22 production cross section and for the combined analysis of F/Si and (Li,Be,B)/C are reported in Table 2 (fourth line). For comparison purpose, we also report the results from the (Li,Be,B)/C analysis (Maurin et al. 2022a). The associated K(R) values and contours are shown in Fig. 1, in red and black respectively. From the Table and the figure, we see that the transport parameters are consistent at better than 1σ, 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 three times the number of F/Si data, all having slightly smaller statistical uncertainties. Nevertheless, both the χ min 2 / d . o . f . $ \chi^{2}_{\mathrm{min}}/\mathrm{d.o.f.} $ and χ nui 2 / n nui $ \chi^{2}_{\mathrm{nui}}/n_{\mathrm{nui}} $ values (last two columns in Table 2) show an improvement with the fully combined analysis. The results are consistent with no primary source of F (see next section), 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. (2020a).

These conclusions are generalised in Fig. 3 for the different transport configurations BIG and QUAINT and different cross-section sets (shown as different symbols). We report the results for the (Li,Be,B)/C (Maurin et al. 2022a) 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. (2022a). In particular, for both combination of data, the best χ2/d.o.f. is obtained for OPT12up225.

thumbnail Fig. 3.

Best-fit parameters and uncertainties from the 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. 2022a), 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).

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

In the F/Si-alone analysis, a lower value of K0 was favoured, compared to that obtained in the combined analysis. As detailed in Weinrich et al. (2020a) and Maurin et al. (2022a), only the combined analysis of the secondary-to-primary ratios enables the degeneracy between K0 and the production cross sections to be broken by enforcing unique K0 values for all the ratios. With our methodology (Appendix A), in addition to the best-fit parameters, we also obtained posterior values for the nuisance parameters, that is, we could inspect the values preferred by the fit for the production cross sections. However, this is not direct, as we only used a single proxy reaction per secondary element. Nevertheless, we could calculate the global normalisation factor μ Z (p) $ \mu_Z^{(p)} $ required for each secondary element Z to best-fit the data6. We show them in Fig. 4 for our three production cross-section sets.

thumbnail Fig. 4.

Correlation between log10(K0) and the normalisation factor μ Z (p) $ \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 1σ 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 μZ = 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. (2022a), that is, 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 μZ ≈ 1 (no modification of the production cross sections), but there is a trend for the need to increase the production of B (i.e. μB > 1) and to decrease that of F (i.e. μF < 1). However, the required numbers are of the order of ∼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.

4.3. Best-matching data

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. 5. The middle panel shows the residuals and the bottom panel the z ∼ $ \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 = (model − data)/σ, is similar to the residuals but expressed in unit of the total data uncertainties σtot. 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 z ∼ $ \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, χ 2 = ∑ i z ∼ i 2 $ \chi^2=\sum\nolimits_i \tilde{z}_i^2 $, and the representation of the z ∼ $ \tilde{z} $-score gives an unbiased view of the distance between the model and the data.

thumbnail Fig. 5.

Flux ratios (top), residuals (centre), and z ∼ $ \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) have been 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 z ∼ $ \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 z ∼ $ \tilde{z} $-score values (projected on the y-axis) compared to a 1σ Gaussian distribution (solid black line) (see text for discussion).

In the bottom panel, we also plot on the right-hand side the projected distributions of z ∼ $ \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 χ min 2 $ \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. 2020a).

5. Fluorine source abundance

The combined F/Si+(Li,Be,B)/C analysis also provides constraints on the relative CR source abundance (19F/28Si)CRS, which are shown in the 8-th panel of Fig. 3 and also reported in Table 3. These constraints only depend mildly on the production cross-section sets and the transport configuration selected.

Table 3.

Best-fit values and +1σ upper limits (in parenthesis) on the relative source abundance of F.

From Table 3, we report a best-fit value of ∼10−3. This value is similar to the SS ratio (19F/28Si)SS = 8.7 × 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 1σ upper limit (19F/28Si)CRS ∼ 5 × 10−3.

5.1. 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, Tcond = 734 K, is close to that of S, Tcond = 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 [F/Si]CRS ∼ [S/Si]CRS = −0.59 dex (Tatischeff et al. 2021), where [X/Y] = log10(X/Y) − log10(X/Y)⊙. Thus, in this model we expect that (19F/28Si)CRS ∼ 2.2 × 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 [19F/28Si]CRS ∼ [20Ne/28Si]CRS = −0.76 dex (Tatischeff et al. 2021), and the predicted 19F abundance relative to 28Si is (19F/28Si)CRS ∼ 1.5 × 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 between 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 (19F/28Si)CRS = 2.7 × 10−3. If integrated above 0.1 GV (as discussed in Boschini et al. 2022), the ratio becomes (19F/28Si)CRS = 1.7 × 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 Sect. A.3). We note that, yet another explanation of this discrepancy is proposed in Zhao et al. (2023), where the authors use spatially dependent diffusion.

5.2. Possibility of obtaining better constraints

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

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 6. The fraction grows steadily with the rigidity, which is expected as primary contributions grow Rδ ≈ 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. A.3). For comparison purpose, we report the total uncertainties on AMS-02 F/Si (purple dotted line) in Fig 6: 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 ∼10 − 100 GV range the regime that constrains in practice the fluorine primary fraction.

thumbnail Fig. 6.

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 have been calculated from the best-fit source parameter q19F of the combined analysis of F/Si and (Li,Be,B)/C (see filled symbols in Fig. 3). The thin lines with downward arrows show the 1σ upper limits in the same configuration. For comparison, the purple dotted line shows the total relative uncertainties (errors combined quadratically) of the AMS-02 F/Si ratio (see text for discussion).

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 one 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 ≲0.1% precision on F/Si data at a few Giga-Volt, or a few percent precision above tens of Tera-Volt.

6. 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 have highlighted the importance of accounting for the anomalous isotopic abundance of 22Ne/20Ne in order not to bias the F production (impact ≲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 tens of Giga-Volt, 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 (i.e. no energy correlations in the data) would instead lead to worsened goodness-of-fit values.

We used several propagation setups (with and without convection and reacceleration and two parametrisations of the low-rigidity break) and different production cross sections, but all configurations led 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) and that (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 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 χ2 per degree of freedom is close to one 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); and (iii) 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 a 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 a 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 ∈ [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 (19F/28Si)CRS ∼ 2.2 × 10−4 and ∼1.5 × 10−4, respectively. Unfortunately, owing to the very small primary content in the CR flux of F (≲5% at 1 GV), we could only derive an upper limit at a source of ∼5 × 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 precision of a few percent. This may be within reach of the next generation of CR experiments (Schael et al. 2019; Battiston et al. 2021).

1

In most publications, Sub-Fe=Sc+Ti+V, that is, Z = 21 − 23 elements.

2

We use both F and 19F (only stable isotope for this element).

4

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, 2017; Génolini et al. 2017b), 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.

5

We recall that for the (Li,Be,B)/C subset, the significantly larger-than-one χ2/d.o.f. was caused by the two low-rigidity Be/B points (upturn) that could not be well fitted by the model (Weinrich et al. 2020a).

6

The proxy p for the production reaction of an element Z (see Table A.1) only contributes to a fraction fp of the total. The global bias μ Z (p) $ \mu_Z^{(p)} $ for the production of this element is calculated from ( μ Z (p) $ \mu_Z^{(p)} $ − 1) = (μp − 1)×fp, where μp is the posterior of the normalisation parameter Eq. (A.7). The relevant fp are taken from Génolini et al. (2018); Maurin et al. (2022a), and Fig. B.3, with f16O + H→6Li ≈ 15%, f16O + H→7Be ≈ 19%, f12C + H→11B ≈ 33%, and f22Ne + H→19F ≈ 30%.

7

For instance, nq = 1 for the stand-alone F/Si analysis (Sect. 3) and nq = 4 for the combined analysis of (Li,Be,B)/C and F/Si (Sect. 4).

9

This 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 App. B.1). All in all, σ is difficult to evaluate and anyway, varying its value within reason would not change the conclusions of our analyses.

10

Unfolding is the procedure to estimate ‘true’ rigidities from measured ones, owing to the finite energy resolution of the detector.

12

While we work in this section on the individual production cross sections σ, we stress that, in the rest of the paper, cumulative cross sections σc are considered instead (relevant ones for CR propagation), with σ c ( X → Y ) = σ ( X → Y ) + ∑ G σ ( X → G ) · B r ( G → Y ) $ \sigma^{\mathrm{c}}(X\to Y) = \sigma(X\to Y) +\sum_G\sigma(X\to G)\cdot {\cal B}r\,(G\to Y) $, where G runs over the list of short-lived nuclei for Y with a branching ratio ℬr.

14

Looking at the bottom left panel of Fig. B.1, we see that this originates from a single point (orange hexagon on top of the dashed lines) with very large error bars.

15

As underlined in Génolini et al. (2018), compared to one-step production, the rigidity dependence of n-step channels follows (1/K)n − 1 ∝ R−δ(n − 1), with δ the diffusion slope.

16

Indeed, 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. C.2 and C.1 respectively), the main progenitor of F, is responsible for the growing difference in the F/Si calculation.

Acknowledgments

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.

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Appendix A: χ2 analysis with covariance and nuisance

In order to reduce biases in the model parameter determination, via a minimisation (Sect. A.1), three crucial ingredients are included (Derome et al. 2019): nuisance parameters for the solar modulation level (Sect. A.2) and for nuclear production cross sections (Sect. A.3), and the use of a covariance matrix of data uncertainties (Sect. A.4).

A.1. χ2 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 used in this study the force-field approximation (Gleeson & Axford 1967, 1968), in which

ψ TOA ( E TOA ) = ( p TOA p IS ) 2 ψ IS ( E IS ) , $$ \begin{aligned}&\psi ^\mathrm{TOA}\left(E^\mathrm{TOA}\right) = \left(\frac{p^\mathrm{TOA}}{p^\mathrm{IS}}\right)^2 \psi ^\mathrm{IS}\left(E^\mathrm{IS}\right)\,,\end{aligned} $$(A.1)

E k / n TOA = E k / n IS − Z A ϕ FF . $$ \begin{aligned}&E_{k/n}^\mathrm{TOA}=E_{k/n}^\mathrm{IS}-\frac{Z}{A}\phi _{\rm FF}\,. \end{aligned} $$(A.2)

The parameter ϕFF in Eq. (A.2) 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 χ2 minimisation using MINUIT (James & Roos 1975) with

χ 2 = ∑ q = 1 n q ( D q + ∑ s = 1 n s N s ( q ) ) + ∑ x = 1 n x N x , $$ \begin{aligned}&\chi ^2 = \sum _{q=1}^{n_q} \left( \mathcal{D}^{\,q} + \sum _{s=1}^{n_s} \mathcal{N}^{s(q)}\right) + \sum _{x=1}^{n_x}\mathcal{N}^{\,x},\end{aligned} $$(A.3)

D = ∑ i , j = 1 n E , n E ( y i data − y i model ) ( C − 1 ) ij ( y j data − y j model ) , $$ \begin{aligned}&\mathcal{D} = \sum _{i,j=1}^{n_E,n_E}\left(y^\mathrm{data}_i-y^\mathrm{model}_i\right) \left(\mathcal{C}^{-1}\right)_{ij} \left(y^\mathrm{data}_j-y^\mathrm{model}_j\right),\end{aligned} $$(A.4)

N ( y ) = ( y − μ y ) 2 σ y 2 . $$ \begin{aligned}&\mathcal{N}(y) = \! \frac{\left(y-\mu _y\right)^2}{\sigma _y^2}. \end{aligned} $$(A.5)

In Eq. (A.3), q runs over the nq ratios used in the fit7, for which the quadratic distance 𝒟 between the model and the data, Eq. (A.4), is calculated including energy bin correlations (nE bins in total); these correlations are encoded in the covariance matrix 𝒞 of data uncertainties discussed in Sect. A.4. Gaussian-distributed nuisance parameters 𝒩 of mean μy and variance σ y 2 $ \sigma_y^2 $, Eq. (A.5), are considered on solar modulation and some proxy cross sections (𝒩s and 𝒩x respectively): the indices s and x in Eq. (A.3) run over ns different data taking periods (if applies) and the nx cross-section reactions considered in the error budget respectively. The nuisance terms penalise the χ2 if the associated parameters wander several σ away from their expected value. Although we do not perform a Bayesian analysis, for brevity, we find useful to denote the nuisance parameters values μy and σ y 2 $ \sigma_y^2 $ priors and their post-fit values (found after the minimisation) posteriors.

The minimum χ2 value indicates how good is the fit: χ min 2 $ \chi^2_{\rm min} $/dof ∼ 1 corresponds to an excellent fit, with ndof = ndata − npars − ns − nx. As introduced in Weinrich et al. (2020a), it is also useful to consider

χ nui 2 / n nui ≡ ( ∑ s = 0 n s N s + ∑ x = 0 n x N x ) / ( n s + n x ) , $$ \begin{aligned} {\chi ^{2}_{\rm nui}/n_{\rm nui}} \equiv \left(\sum _{s=0}^{n_s}\mathcal{N}^s + \sum _{x=0}^{n_x}\mathcal{N}^{x}\right)/(n_s+n_x), \end{aligned} $$(A.6)

with nnui = ns + nx. The above quantity indicates whether the post-fit values stay very close to the priors (μpost ∼ μprior) or wander within 1σ of the priors (μpost ≲ μprior ± σ), corresponding to χ nui 2 / n nui ∼ 0 $ {\chi^{2}_{\mathrm{nui}}/n_{\mathrm{nui}}}\sim0 $ or χ nui 2 / n nui ≲ 1 $ {\chi^{2}_{\mathrm{nui}}/n_{\mathrm{nui}}}\lesssim 1 $ respectively.

A.2. Priors for solar modulation

In the Force-Field approximation, Eqs (A.1-A.2), modulation levels can be reconstructed from the analysis of neutron monitor (NM) data, at a precision σϕ ≃ 100 MV (Maurin et al. 2015; Ghelfi et al. 2017). The reconstructed ϕ values are stored in the cosmic-ray database (CRDB8) and can be accessed via the interface in the ‘solar modulation’ tab of CRDB (Maurin et al. 2014, 2020). Figure A.1 shows the modulation levels (averaged over 10 days) retrieved from CRDB from May 19, 2011 to October 30, 2019, for several for the OULU NM station.

thumbnail Fig. A.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 the AMS-02 analysed F/Si data (Aguilar et al. 2021d), and the value in parentheses (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. 3) or their combined analysis with AMS-02 Li/C, Be/C, and B/C data (see Sect. 4). The data taking period of AMS-02 F/Si data (Aguilar et al. 2021d) matches that of Fig. A.1, that is, 8.5 years of AMS-02 data, for which an average modulation level of μϕ = 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. 2021a) are based on a seven-year data taking period (μϕ = 636 MV), but we nevertheless use the previously published dataset based on five years of AMS-02 data (Aguilar et al. 2018) for which μϕ = 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. (2020a), Maurin et al. (2022a), 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 σϕ ≃ 100 MV taken on the solar modulation level. We repeated the above comparison using 10-days slices over the period depicted in Fig. A.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.

A.3. Priors for cross-sections

Production cross-section uncertainties are sizeable (∼5 − 20%), and the propagation equation involves a large network of reactions (≳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 App. B.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 σref:

σ Norm . ( E k / n ) = Norm × σ ref ( E k / n ) , $$ \begin{aligned}&\sigma ^\mathrm{Norm.}(E_{k/n}) = \mathrm{Norm} \times \sigma _{\rm ref}(E_{k/n})\,,\end{aligned} $$(A.7)

σ Scale ( E k / n ) = σ ref ( E k / n × Scale ) , $$ \begin{aligned}&\sigma ^\mathrm{Scale}(E_{k/n}) = \sigma _{\rm ref}\left(E_{k/n} \times \mathrm{Scale} \right)\,,\end{aligned} $$(A.8)

σ Slope ( E k / n ) = { σ ref ( E k / n ) × ( E k / n E k / n thr ) Slope if E k / n ≤ E k / n thresh. , σ ref ( E k / n ) otherwise . $$ \begin{aligned}&\sigma ^\mathrm{Slope}(E_{k/n}) = {\left\{ \begin{array}{ll} \displaystyle \sigma _{\rm ref} (E_{k/n})\times \left(\frac{E_{k/n}}{E_{k/n}^{\text{thr}}}\right)^\text{Slope}\!\!\! \text{ if}\; E_{k/n} \le E_{k/n}^{\text{thresh.}},\\ \sigma _{\rm ref}(E_{k/n})\quad \quad \text{ otherwise}\,. \end{array}\right.} \end{aligned} $$(A.9)

In the above equations, E k / n thresh. $ 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 μ and dispersion σ, to be used as nuisance parameters in Eq. (A.3).

Because we renormalise production cross-section reactions to nuclear data in App. B.1, we enforce μNorm = 1, μScale = 1, and μSlope = 0 for all reactions. To fix σ, as illustrated in Fig. A.2, we visually estimate what transformation law values are needed to encompass at 1σ (dotted grey lines) the different cross-section parametrisations (coloured lines)9. Table A.1 gathers the NSS μ and σ values to use for the production and inelastic reactions listed; σ values for Li, Be, and B production can be found in Table B.1 of Weinrich et al. (2020a). The second column also highlights the typical uncertainties on F/Si observed when using different modelling for the reactions listed in the first column.

thumbnail Fig. A.2.

Illustration of the NSS scheme used for cross-section nuisance parameters (inelastic on the left and production on the right; see Eqs. A.7-A.9). Colour-coded solid lines correspond to existing cross-section parametrisations: inelastic on the left, B94 (Barashenkov & Polanski 1994), W97 (Wellisch & Axen 1996), T99 (Tripathi et al. 1997, 1999), and W03 (Webber et al. 2003); production on the right, 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). The grey lines correspond to the median, 68%, and 95% CLs resulting from varying the NSS parameters. For σNSS of the cross-section nuisance parameters (reported in Table A.1), we chose the NSS values leading to the 68% CLs (dotted grey lines).

Table A.1.

Reaction proxies and values of their nuisance parameters.

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, that is, 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 A.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→19F). These proxies are highlighted in boldface in Table A.1, and compared to the row denoted ‘All’ that shows the impact of the uncertainties from the full network.

A.4. Covariance matrix of uncertainties

An important term in Eq. (A.3) is the covariance matrix of data uncertainties Cij. 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 defined the relative covariance ( C rel α ) ij $ (C_{\rm rel}^\alpha)_{ij} $ between rigidity bin Ri and Rj (and the associated correlation matrix) to be

( C rel α ) ij = σ i α σ j α exp ( − 1 2 ( log ( R i / R j ) 2 ( ℓ α ) 2 ) , $$ \begin{aligned}&(C_{\rm rel}^\alpha )_{ij} = \sigma ^\alpha _i \sigma ^\alpha _j \exp \left(-\frac{1}{2} \frac{(\log (R_i/R_j)^2}{(\ell _\alpha )^2} \right)\,,\end{aligned} $$(A.10)

c ij α ≡ C ij α C ii α × C jj α , $$ \begin{aligned}&\mathrm{c}_{ij}^{\alpha } \equiv \frac{\mathcal{C}_{ij}^{\alpha }}{\sqrt{\mathcal{C}_{ii}^{\alpha } \times \mathcal{C}_{jj}^{\alpha }}}\,, \end{aligned} $$(A.11)

with σ i α $ \sigma_i^\alpha $ the relative uncertainty of error type α at bin i and ℓα the correlation lengths for error type α (in unit of rigidity decade).

The AMS-02 collaboration provides the statistical and systematic uncertainties split in three different components (acceptance, unfolding10, and scale), as displayed in solid lines in the top panel of Fig. A.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 ℓscale = ∞ for the energy scale uncertainty, amounting to a global rigidity shift, and ℓUnfolding = 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 with inelastic cross-section uncertainties (impacting the detector acceptance estimation), with a relatively large correlation length ℓAcc. norm. = 1; (ii) ‘Acc. LE’ is a low-rigidity error associated with the orbit-varying rigidity cut-off of the AMS detector, set to ℓAcc. LE = 0.3; and (iii) ‘Acc. res.’ is a residual error mostly associated with data/Monte Carlo corrections. The latter cannot be associated with a particular physics process and no prescription can be taken for its correlation length.

thumbnail Fig. A.3.

Uncertainties on F AMS-02 data. Top panel: Statistical and Acc., Unf., and Scale systematic uncertainties for F/Si. These data are 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: Model for the correlation matrix c ij tot $ \mathrm{c}_{ij}^{\mathrm{tot}} $, Eq. (A.11), for the combined statistical and systematics uncertainties. These data are colour-coded from no correlation (white, c ij tot = 0 $ \mathrm{c}_{ij}^{\mathrm{tot}}=0 $) to full correlations (blue, c ij tot = 1 $ \mathrm{c}_{ij}^{\mathrm{tot}}=1 $).

As can be seen in the top panel of Fig. A.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. (2020a). As studied on the AMS-02 B/C data in Derome et al. (2019), taking extreme values of the unknown ℓAcc. res. impacts the best-fit χ min 2 / dof $ \chi^{2}_{\mathrm{min}}/\mathrm{dof} $ values (this value increases with ℓAcc. res.), but only marginally the best-fit parameters and thus the best-fit flux ratios. For this reason, the χ min 2 / dof $ \chi^{2}_{\mathrm{min}}/\mathrm{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 ℓAcc. res. = 0.1 (Derome et al. 2019). The bottom panel of Fig. A.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.

Appendix B: Production cross sections and F progenitors

In this Appendix, we focus on the production cross sections. We show first the nuclear parametrisations renormalised on nuclear data points for a selection of important reactions (App. B.1). We then illustrate the importance of correctly setting the source of Ne isotopes for the F/Si calculation (App. B.2). With these updated cross sections and correct CR source settings, we determine the ranking of the most important progenitors of F (App. B.3).

B.1. Rescaling of cross sections on data

Several production cross-section parametrisations are available from the literature. In particular, the widely used GALPROP11 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. These parametrisations were established almost two decades ago, while several nuclear datasets became available since. Indeed, as shown in Maurin et al. (2022a) in the context of Li, Be, and B production, some reactions in these models are at odds with these new nuclear data. In such case, the natural procedure is to rescale these reactions to the new data. We apply this normalisation procedure on 56Fe, 32S, 28Si, 27Al, 24Mg, and 20, 22Ne interacting on H (the most important progenitors of F, see App. B.3), to give either directly 19F (the only stable isotope of Fluorine) or short-lived nuclei ending up their decay chain12 into 19F: the relevant nuclei with ℬr > 5% are (Letaw et al. 1984; Maurin 2001) 19Ne (ℬr = 100%), 19O (ℬr = 100%), 19C (ℬr = 20.9%), 20C (ℬr = 48.6%), 19N (ℬr = 45.4%), and 20N (ℬr = 57%).

In practice, for this nuclear cross-section renormalisation, we (i) start from the OPT12, OPT12up22, and OPT22 datasets (updated for Li, Be, and B production) provided in Maurin et al. (2022a), (ii) extract from the EXFOR database13 (Otuka et al. 2014) all nuclear data relevant for the F production (on H targets), (iii) apply the rescaling procedure of Maurin et al. (2022a) to get updated modelling of the above reaction cross sections, (iv) use the empirical formulae of Ferrando et al. (1988) for σHe/σH 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. B.1 (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 B.1 and discussed below. Our procedure significantly changes the original energy dependence of the cross section for 22Ne+H→19Ne 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 ∼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).

thumbnail Fig. B.1.

Models (OPT12 and OPT22) and data (symbols) for the colour-coded production of 19F and ghosts with ℬr > 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 ≲10−2 mb). ‘Rescaled’ indicates that the isotopes listed have nuclear data on which the model were renormalised. For these isotopes, we do not show the original model values for these reactions (for readability), but we refer the reader to Table B.1, which 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).

Table B.1.

Rescaling factors applied on the main reactions producing F.

To better highlight the difference of our new production sets compared to the original ones, we show in Table B.1 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 ten14.

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.

B.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, 2017a, 2019; Weinrich et al. 2020a; Maurin et al. 2001, 2010, 2022a), elemental abundances were rescaled to match the corresponding elemental CR fluxes, though keeping in the process isotopic abundances fixed to their 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 (22Ne/20Ne)CRS/(22Ne/20Ne)SS ≈ 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 B.2 illustrates the importance of using the correct isotopic abundance for the F/Si calculation. With respect to the calculation based on the SS isotopic abundance (22Ne/20Ne)SS ≃ 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. B.1 where, for the cumulative production of F, we have σc(22Ne + H→19F) = 26 mb and σc(20Ne + H→19F) = 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 ∼3% (see the top panel of Fig. A.3).

thumbnail Fig. B.2.

Impact of changing the isotopic ratio (22Ne/20Ne)CRS at the 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 SS 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.)

It is interesting to compare the (22Ne/20Ne)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 (22Ne/20Ne)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 (22Ne/20Ne)CRS = 0.47 (see Sect. 5). 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.

B.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. (2022a).

We show in Fig. B.3 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 (∼30 − 40%), followed by Mg and Si (∼20%), and then Fe (∼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 ∼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. 2021b), 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).

thumbnail Fig. B.3.

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 one-step and two-step channels (reaction paths linking one isotopic progenitor to a CR isotope). The ‘>2’-step channels (not shown) contribute to a total of ∼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 also show in the bottom panel of Fig. B.3 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 with a unique cross-section reaction (a ranking of the individual cross sections is also possible, but not shown here). In practice, we rank the one-step and two-step channels only. The most important progenitors from direct channels are 20Ne (∼25%), 24Mg and 28Si (both at ∼20%), then 22Ne and 56Fe (both at ∼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 App. B.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 GV15. 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 ∼1% (direct contribution, bottom panel) to ∼10% (direct plus all two-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 one-step channels contribute to ∼70% of the total F production, two-step channels to ∼20%, and ‘> 2-step’ channels to ∼10% of the total (and we recall that multi-step contributions peak at a few GV). We can also tie ∼62% of the F production to 5 direct channels, namely from 20, 22Ne, 24Mg, 28Si, and 56Fe), while ∼25% originates from a few percent-level channels, and ∼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).

Appendix C: Primary fluxes

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. C.1 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 ≲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.

thumbnail Fig. C.1.

Comparison (top panel) and residuals (bottom panel) of the model calculation (lines) and the AMS-02 data (Aguilar et al. 2017, 2020, 2021b,c) 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 parentheses in the legend indicate the fractional contribution of these progenitors to the F production (as read from the top panel of Fig. B.3).

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. C.1 (as taken form the top panel of Fig. B.3). We see that the main progenitors, Ne (30% contribution, black × 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 ≲20%. This makes their individual mismatch on F at the level of ≲30%×20%≲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 ≲30% below these data.

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 ∝ R − α × ∏ k = 0 2 ( 1 + ( R R k ) Δ k / η k ) η k . $$ \begin{aligned} Q \propto R^{-\alpha }\times \prod _{k=0}^2 \left(1+\left(\frac{R}{R_k}\right)^{\Delta _k/\eta _k}\right)^{\eta _k}. \end{aligned} $$(C.1)

The smoothness parameters are fixed to η0 = η2 = 0.18 (with positive values for Δ0 and Δ2 in the fit) and η1 = −0.18 (with negative values for Δ1 in the fit). With more than eight parameters per element—one normalisation per isotope, one slope α and three breaks Δk at three rigidities Rk—, it is no surprise that the model overfits the data ( χ min 2 / dof = 0.282 $ {\chi^{2}_{\mathrm{min}}/\mathrm{dof}}=0.282 $). The fit and the data for the BPL configuration are shown in Fig. C.2; 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.

thumbnail Fig. C.2.

Similar to Fig. C.1 but now fitting the primary fluxes while assuming three breaks in their source spectra (see text). We do not show the residuals, as they all overlap around zero. The model overfits the data with χ min 2 / dof = 0.282 $ {\chi^{2}_{\mathrm{min}}/\mathrm{dof}}=0.282 $.

We can now compare the secondary-to-primary ratios (of interest for this study) obtained with the BPL and PL assumptions in Fig. C.3.

  • High-rigidity: above ∼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.16 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 ±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. 4); (ii) of the order or smaller than the bias on these nuisance parameters (y-axis values of the centre of these ellipses in Fig. 4); (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. 4). 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.

thumbnail Fig. C.3.

Relative difference between secondary-to-primary ratios calculated from the BPL and PL primary spectra, in solid lines. From top to bottom, the four panels show 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.

All Tables

Table 1.

List of ingredients used for the analysis and their description.

In the text
Table 2.

Comparisons of best-fit transport parameters (in SLIM) from various combinations of AMS-02 data.

In the text
Table 3.

Best-fit values and +1σ upper limits (in parenthesis) on the relative source abundance of F.

In the text
Table A.1.

Reaction proxies and values of their nuisance parameters.

In the text
Table B.1.

Rescaling factors applied on the main reactions producing F.

In the text

All Figures

thumbnail Fig. 1.

Best-fit and 1σ envelopes for the diffusion coefficient, Eq. (5), for different secondary-to-primary ratio data combinations. The corresponding parameters are gathered in Table 2.
In the text
thumbnail Fig. 2.

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 2). 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 rF (rescaling the overall production of F) to better match the data. A pure secondary origin of fluorine was assumed in both calculations (see text for discussion).
In the text
thumbnail Fig. 3.

Best-fit parameters and uncertainties from the 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. 2022a), 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).
In the text
thumbnail Fig. 4.

Correlation between log10(K0) and the normalisation factor μ Z (p) $ \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 1σ 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 μZ = 1 (i.e. no modification needed for the production of an element).
In the text
thumbnail Fig. 5.

Flux ratios (top), residuals (centre), and z ∼ $ \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) have been 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 z ∼ $ \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 z ∼ $ \tilde{z} $-score values (projected on the y-axis) compared to a 1σ Gaussian distribution (solid black line) (see text for discussion).
In the text
thumbnail Fig. 6.

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 have been calculated from the best-fit source parameter q19F of the combined analysis of F/Si and (Li,Be,B)/C (see filled symbols in Fig. 3). The thin lines with downward arrows show the 1σ upper limits in the same configuration. For comparison, the purple dotted line shows the total relative uncertainties (errors combined quadratically) of the AMS-02 F/Si ratio (see text for discussion).
In the text
thumbnail Fig. A.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 the AMS-02 analysed F/Si data (Aguilar et al. 2021d), and the value in parentheses (in the legend) gives the average modulation level over the period. Retrieved from CRDB.
In the text
thumbnail Fig. A.2.

Illustration of the NSS scheme used for cross-section nuisance parameters (inelastic on the left and production on the right; see Eqs. A.7-A.9). Colour-coded solid lines correspond to existing cross-section parametrisations: inelastic on the left, B94 (Barashenkov & Polanski 1994), W97 (Wellisch & Axen 1996), T99 (Tripathi et al. 1997, 1999), and W03 (Webber et al. 2003); production on the right, 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). The grey lines correspond to the median, 68%, and 95% CLs resulting from varying the NSS parameters. For σNSS of the cross-section nuisance parameters (reported in Table A.1), we chose the NSS values leading to the 68% CLs (dotted grey lines).
In the text
thumbnail Fig. A.3.

Uncertainties on F AMS-02 data. Top panel: Statistical and Acc., Unf., and Scale systematic uncertainties for F/Si. These data are 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: Model for the correlation matrix c ij tot $ \mathrm{c}_{ij}^{\mathrm{tot}} $, Eq. (A.11), for the combined statistical and systematics uncertainties. These data are colour-coded from no correlation (white, c ij tot = 0 $ \mathrm{c}_{ij}^{\mathrm{tot}}=0 $) to full correlations (blue, c ij tot = 1 $ \mathrm{c}_{ij}^{\mathrm{tot}}=1 $).
In the text
thumbnail Fig. B.1.

Models (OPT12 and OPT22) and data (symbols) for the colour-coded production of 19F and ghosts with ℬr > 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 ≲10−2 mb). ‘Rescaled’ indicates that the isotopes listed have nuclear data on which the model were renormalised. For these isotopes, we do not show the original model values for these reactions (for readability), but we refer the reader to Table B.1, which 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 the text
thumbnail Fig. B.2.

Impact of changing the isotopic ratio (22Ne/20Ne)CRS at the 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 SS 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.)
In the text
thumbnail Fig. B.3.

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 one-step and two-step channels (reaction paths linking one isotopic progenitor to a CR isotope). The ‘>2’-step channels (not shown) contribute to a total of ∼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.
In the text
thumbnail Fig. C.1.

Comparison (top panel) and residuals (bottom panel) of the model calculation (lines) and the AMS-02 data (Aguilar et al. 2017, 2020, 2021b,c) 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 parentheses in the legend indicate the fractional contribution of these progenitors to the F production (as read from the top panel of Fig. B.3).
In the text
thumbnail Fig. C.2.

Similar to Fig. C.1 but now fitting the primary fluxes while assuming three breaks in their source spectra (see text). We do not show the residuals, as they all overlap around zero. The model overfits the data with χ min 2 / dof = 0.282 $ {\chi^{2}_{\mathrm{min}}/\mathrm{dof}}=0.282 $.
In the text
thumbnail Fig. C.3.

Relative difference between secondary-to-primary ratios calculated from the BPL and PL primary spectra, in solid lines. From top to bottom, the four panels show 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.
In the text

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