© D. Maurin et al. 2022

1. Introduction

In the 1950s, Hayakawa et al. (1958) realised that the ¹⁰Be radioactive secondary isotope could be used as a clock to determine the cosmic-ray (CR) age (e.g., Silberberg & Tsao 1990). In modern CR propagation models, the radioactive clocks are used to determine the halo size L of the Galaxy (e.g., Donato et al. 2002). Besides the motivation for a better characterisation of the transport parameters in the Galaxy, the determination of L is also crucial for setting constraints on dark matter from indirect detection of anti-particles (Donato et al. 2004; Génolini et al. 2021).

Many studies have focused on the most abundant and lightest CR clock, ¹⁰Be (Simpson & Garcia-Munoz 1988), either via the isotopic ¹⁰Be/⁹Be ratio (e.g., Moskalenko et al. 2001; Donato et al. 2002) or via the elemental Be/B ratio (O’dell et al. 1975; Webber & Soutoul 1998; Putze et al. 2010). In the former case, the impact of ¹⁰Be decay is maximal in the ratio and, moreover, both the isotopes have roughly the same progenitors and are similarly produced in nuclear reactions. In the latter case, the impact of ¹⁰Be decay is lessened in the numerator by the presence of the more abundantly produced stable isotopes (⁷Be and ⁹Be), even though the presence of the daughter isotope ¹⁰B in the denominator maximises the impact of decay on the ratio; the modelling of the Be/B ratio also involves a larger network or production cross-sections, potentially leading to larger nuclear uncertainties than for the ¹⁰Be/⁹Be calculation. On the experimental side, isotopic separation up to high energies remains very challenging, and thus using the Be/B ratio to set constraints on L is complementary to using the ¹⁰Be/⁹Be ratio.

With the advent of the Alpha Magnetic Spectrometer (AMS-02) experiment on board the International Space Station, data of unprecedented precision have been published for Be/B (Aguilar et al. 2018). These data have been recently analysed by Weinrich et al. (2020a); Evoli et al. (2020), and De La Torre Luque et al. (2021), who all highlighted the impact of cross-section uncertainties on the determination of L, with ΔL  ∼   ± 2.5 kpc. The situation for the ¹⁰Be/⁹Be data is also expected to improve very soon, thanks to the AMS-02 and High Energy Light Isotope eXperiment (HELIX) balloon-borne experiments. In this context, we wish to revisit the uncertainties on L originating from nuclear uncertainties, taking advantage of a recent update on the production cross-sections of Be and B isotopes (Maurin et al. 2022). We also wish to assess the compatibility of the constraints on L brought by Be/B and ¹⁰Be/⁹Be data. In particular, for the latter, nuclear uncertainties may be an issue that will prevent us from fully benefiting from the forthcoming measurements.

The Be isotopes are of secondary origin, that is they are thought to solely originate from the fragmentation of heavier CR nuclei. This makes the calculation of CR flux ratios of these isotopes dependent on: (i) the known half-life of the CR clock (t_1/2 = 1.387 Myr); (ii) the grammage crossed by CRs during their journey through the Galaxy; (iii) the production cross-sections; (iv) the CR fluxes of their progenitors; and finally, (v) the halo size L of the Galaxy. The grammage can be determined from stable secondary-to-primary ratios (e.g., Maurin et al. 2001), while the production cross-sections are set from available nuclear parametrisations and data (Génolini et al. 2018; Maurin et al. 2022), and measured elemental fluxes are available from a wide range of energies from interstellar (IS; Cummings et al. 2016) and top-of-atmosphere (TOA) data (Engelmann et al. 1990; Lave et al. 2013; Aguilar et al. 2021a,b,c,d). We show that the cross-sections and progenitor fluxes can be combined into a single number (denoted ℱ_HE) that can be used as an input to determine L directly from the ¹⁰Be/⁹Be data. This simplification should prove useful for experimentalists willing to constrain L from their data without the need for an underlying propagation model.

This paper is organised as follows. In Sect. 2, we derive a simple analytical formula to calculate the ¹⁰Be/⁹Be ratio and we validate this formula against the full calculation performed with the propagation code USINE; we also introduce the ℱ term and its high-energy limit ℱ_HE, which only depends on the production cross-sections and measured elemental fluxes. In Sect. 3, we compare the limits on L obtained from the analysis of AMS-02 Be/B data or from available ¹⁰Be/⁹Be data; we highlight how the uncertainties on nuclear production cross-sections translate into an uncertainty on ℱ_HE, which dominates the uncertainty on L. We then conclude in Sect. 4.

2. Analytical formula for ¹⁰Be/⁹Be

The framework we consider in this study is the 1D thin disc/thick halo propagation model, which is known to capture all the salient processes of CR transport while remaining simple (e.g., Jones et al. 2001). This framework has been used and shown to reproduce AMS-02 high-precision data in several recent studies (Derome et al. 2019; Génolini et al. 2019; Evoli et al. 2019, 2020; Boudaud et al. 2020; Weinrich et al. 2020a,b; Schroer et al. 2021). The semi-analytical solutions in the 1D model model are implemented in the USINE code (Maurin 2020), which is used here to validate the analytical formula that we propose.

We consider neither convection nor re-acceleration below, as they are not mandatory in order to give an excellent match to AMS-02 Li/C, Be/B, and B/C data (Weinrich et al. 2020b; Maurin et al. 2022). Further omitting energy redistribution (re-acceleration and continuous losses), the diffusion equation for the differential density N^j of a species j reduces to

$$\begin{array}{c}\hfill (-K\frac{{\mathrm{d}}^2}{\mathrm{d}{z}^2}+\frac{1}{\gamma {\tau }_0^j}+2h{n}_{\mathrm{ISM}}v{\sigma }_{\mathrm{inel}}^{j+\mathrm{ISM}}\delta (z))N^j(z)=2h\delta (z){\mathcal{S}}^j(E).\end{array}$$(1)

The various terms appearing in this equation are: the diffusion coefficient K(R); the lifetime for an unstable species ${\tau }_0^j$ (τ₀ → ∞ for a stable species); the disc half-width h pinched in a thin slab δ(z) where the gas and the sources lie; the destruction rate $n_{\mathrm{ISM}}v{\sigma }_{\mathrm{inel}}^{j+\mathrm{ISM}}$ of species j on the interstellar medium (ISM) of density n_ISM (with σ_inel the inelastic cross-section); and a source term 𝒮^j(E) containing a primary component Q^prim(E) for species accelerated at source, and a secondary component $Q^{\sec }(E)=\sum _{\mathrm{p}}{n}_{\mathrm{ISM}}v{\sigma }_{\mathrm{prod}}^{p+\mathrm{ISM}\to j}{N}^{\mathrm{p}}$ from the nuclear production of j from all progenitors p heavier than j.

2.1. Neglecting energy losses

We can provide an analytical formula only if we can neglect energy losses. Figure 1 shows a comparison of calculations performed with USINE, with and without energy losses, based on the best-fit diffusion coefficient and L obtained in Weinrich et al. (2020a,b), for the propagation configuration denoted SLIM. The comparison is shown for IS quantities (blue lines), but also for TOA quantities (orange lines). The difference between the IS and TOA curves indicates that solar modulation (e.g., Potgieter 2013) is an important ingredient (see Sect. 2.3). However, energy losses (a comparison can be made between solid and dotted lines) are not so important. Above a few tens of GeV/n, neglecting energy losses biases the result by a few percent at most for the TOA ¹⁰Be/⁹Be ratio: this is to be compared to the 15 − 20% uncertainties expected in forthcoming AMS-02 (Aguilar et al. 2019) and HELIX (Park et al. 2019) data. This bias only slightly increases as the energy decreases, meaning that this approximation is equally applicable for data going down to tens of MeV/n (see below). We stress, however, that the ∼6% bias gets dangerously close to the best precision current data reach at these energies, with an ∼11% precision for ACE-CRIS data (Yanasak et al. 2001).

Fig. 1. Impact of neglecting energy losses in the 10Be/9Be ratio as a function of the kinetic energy per nucleon (all curves are from USINE runs). Top panel: calculation with energy losses switched on (solid lines) or off (dotted lines) for TOA (blue) or IS (orange) calculations. Bottom panel: relative difference between the ‘no E loss’ and ‘with E losses’ calculations, for the TOA (dotted blue line) and IS (dashed orange line) cases. The range highlighted in yellow corresponds to the region where the AMS-02 and HELIX experiments should measure 10Be/9Be.

This first comparison shows that, to a first approximation, energy losses can be neglected for the calculation of the TOA ¹⁰Be/⁹Be ratio above a few tens of MeV/n. The next step, following Prishchep & Ptuskin (1975) and Ginzburg et al. (1980), is to calculate an analytical formula for this ratio in the 1D model.

2.2. Solutions for stable and radioactive species (and ratio)

We are interested here in the ⁹Be (stable) and ¹⁰Be (unstable) isotopes in the disc (i.e., at z = 0). It is straightforward to solve Eq. (1), and we get (we omit all indices for simplicity):

$$\begin{array}{cc}\hfill N^{\mathrm{stable}}(z=0)& =\frac{\mathcal{S}(E)}{K·{(Lh)}^{-1}+n_{\mathrm{ISM}}v{\sigma }_{\mathrm{inel}}},\hfill \end{array}$$(2)

$$\begin{array}{cc}\hfill N^{\mathrm{rad}}(z=0)& =\frac{\mathcal{S}(E)}{K·{(l_{\mathrm{rad}}h·tanh\left(\frac{L}{l_{\mathrm{rad}}}\right))}^{-1}+n_{\mathrm{ISM}}v{\sigma }_{\mathrm{inel}}},\hfill \end{array}$$(3)

where we have defined $l_{\mathrm{rad}}\equiv \sqrt{\gamma {\tau }_{0}K}$.

To form the ratio of a radioactive to a stable secondary species, it is useful to introduce the diffusion and destruction timescales in the disc, as well as the decay timescale:

$$\begin{array}{cc}& t_{\mathrm{diff}}=\frac{\mathit{Lh}}{K},\hfill \end{array}$$(4)

$$\begin{array}{cc}& t_{\mathrm{inel}}=\frac{1}{nv{\sigma }_{\mathrm{inel}}},\hfill \end{array}$$(5)

$$\begin{array}{cc}& t_{\mathrm{rad}}=\gamma {\tau }_0=\gamma \frac{t_{1/2}}{ln(2)}.\hfill \end{array}$$(6)

Using the superscript 10 and 9 to identify quantities calculated for ¹⁰Be and ⁹Be respectively ($t_{\mathrm{diff}}^{10}$ and $t_{\text{diff}}^9$ are different when calculated at the same E_k/n because the diffusion coefficient depends on the rigidity R = (p/Ze)), we then get

$$\begin{array}{c}\hfill \frac{N^{{}^{10}}}{N^{{}^9}}(E_{k/n})=\mathcal{F}·\frac{t_{\mathrm{diff}}^{{}^{10}}}{t_{\mathrm{diff}}^{{}^9}}·\frac{1+\frac{t_{\mathrm{diff}}^{{}^9}}{t_{\mathrm{inel}}^{{}^9}}}{\sqrt{\frac{L}{h}·\frac{t_{\mathrm{diff}}^{10}}{t_{\mathrm{rad}}^{10}}}·coth\left[\sqrt{\frac{L}{h}·\frac{t_{\mathrm{diff}}^{10}}{t_{\mathrm{rad}}^{10}}}\right]+\frac{t_{\mathrm{diff}}^{10}}{t_{\mathrm{inel}}^{{}^{10}}}},\end{array}$$(7)

with

$$\begin{array}{c}\hfill \mathcal{F}(E_{k/n})\equiv \frac{{\displaystyle \sum _{p\in \mathrm{p}roj}\sum _{t\in \mathrm{I}SM}{n}_{\mathrm{ISM}}^{t}v{\sigma }^{p+t{\to }^{10}\mathrm{Be}}(E_{k/n})\times N^{\mathrm{p}}(E_{k/n})}}{{\sum }_{p\in \mathrm{p}roj}{\sum }_{t\in \mathrm{I}SM}{n}_{\mathrm{ISM}}^{t}v{\sigma }^{p+t{\to }^9\mathrm{Be}}(E_{k/n})\times N^{\mathrm{p}}(E_{k/n})}.\end{array}$$(8)

In this last expression, we put back the summation indices to highlight the fact that ℱ only depends on the fragmentation cross-sections into the Be isotopes and the IS progenitor fluxes ψ^IS; we recall that ψ^IS = Nv/(4π), because CR fluxes are quasi-isotropic.

For all practical calculations in what follows, the index t runs over H and He (90% and 10% in number, respectively), whereas the index p runs over all CR species up to ⁵⁶Fe. We take n_ISM = 1 cm⁻³, but we note that the gas density disappears in ℱ(E_k/n). We come back to Eq. (8) and discuss it in detail in Sect. 2.4.

2.3. Accounting for solar modulation

In practice, the data correspond to TOA quantities, whereas the above formulae correspond to IS quantities. Also, as seen in Fig. 1 (a comparison can be made between the orange and blue lines in the top panel), the solar modulation effect cannot be neglected. Although we cannot directly modulate a ratio, after a bit of tweaking, we can obtain a simple and accurate enough formula.

In the force-field approximation used in our calculations, TOA and IS quantities are related by (Gleeson & Axford 1967, 1968)

$$\begin{array}{cc}\hfill {\psi }^{\mathrm{TOA}}\left(E^{\mathrm{TOA}}\right)& ={\left(\frac{p^{\mathrm{TOA}}}{p^{\mathrm{IS}}}\right)}^2{\psi }^{\mathrm{IS}}\left(E^{\mathrm{IS}}\right),\hfill \end{array}$$(9)

$$\begin{array}{cc}\hfill E_{k/n}^{\mathrm{TOA}}& =E_{k/n}^{\mathrm{IS}}-\frac{Z}{A}\varphi ,\hfill \end{array}$$(10)

where ϕ is the solar modulation level. For our purpose, we need to calculate the flux ratio at the same TOA kinetic energy per nucleon $E_{k/n}^{\mathrm{TOA}}$, whereas solar modulation connects fluxes at total energy. Writing

$$\begin{array}{cc}\hfill E_{k/n,X}^{\mathrm{IS}}& =E_{k/n}^{\mathrm{TOA}}+\frac{Z_X}{A_X}\varphi ,\hfill \\ \hfill {\left(p_X^{\mathrm{TOA}}\right)}^2& =E_{k/n}^{\mathrm{TOA}}{A}_X(E_{k/n}^{\mathrm{TOA}}{A}_X+2m_X),\hfill \\ \hfill {\left(p_X^{\mathrm{IS}}\right)}^2& ={(E_{k/n,X}^{\mathrm{IS}}{A}_X+m_X)}^2-m_X^2,\hfill \end{array}$$

with the subscript X referring to ¹⁰Be or ⁹Be, we get

$$\begin{array}{c}\hfill \frac{{\psi }_{10}^{\mathrm{TOA}}(E_{k/n}^{\mathrm{TOA}})}{{\psi }_9^{\mathrm{TOA}}(E_{k/n}^{\mathrm{TOA}})}={\left(\frac{p_{10}^{\mathrm{TOA}}}{p_{10}^{\mathrm{IS}}}\right)}^2·{\left(\frac{p_9^{\mathrm{IS}}}{p_9^{\mathrm{TOA}}}\right)}^2·\frac{{\psi }_{10}^{\mathrm{IS}}(E_{k/n,10}^{\mathrm{IS}})}{{\psi }_9^{\mathrm{IS}}(E_{k/n,10}^{\mathrm{IS}})}·\frac{{\psi }_9^{\mathrm{IS}}(E_{k/n,10}^{\mathrm{IS}})}{{\psi }_9^{\mathrm{IS}}(E_{k/n,9}^{\mathrm{IS}})}.\end{array}$$(11)

In order to form the ¹⁰Be to ⁹Be flux ratio at the same IS kinetic energy per nucleon (next-to-last term in the equation), we need an extra factor (last term) corresponding to the ratio of the ⁹Be flux calculated at two different IS energies. For a good approximation of this flux, we can take advantage of Voyager data taken outside the solar cavity and TOA data at higher energy. In practice, we fit a log-log polynomial on the IS Be flux models shown in Fig. 4 of Cummings et al. (2016), and defining x ≡ log₁₀[E_k/n/(1 GeV/n)], we have

$$\begin{array}{c}\hfill {log}_{10}\left({\psi }_9^{\mathrm{IS}}\right)=-0.036-1.283x-0.921x^2+0.0078x^3+0.05x^4.\end{array}$$(12)

The normalisation of the flux is not important as our calculation involves ratios of ⁹Be fluxes, and we assume that the energy dependence for ⁹Be is the same as that for Be.

Finally, using the above equations and recalling ψ = Nv/(4π), the formula to calculate the TOA ratio from the IS one is given by

$$\begin{array}{c}\hfill \frac{{\psi }_{10}^{\mathrm{TOA}}(E_{k/n}^{\mathrm{TOA}})}{{\psi }_9^{\mathrm{TOA}}(E_{k/n}^{\mathrm{TOA}})}=\frac{v_{10}}{v_9}\times \frac{N_{10}^{\mathrm{IS}}}{N_9^{\mathrm{IS}}}(E_{k/n}^{\mathrm{TOA}}+\frac{Z_{10}}{A_{10}}\varphi )\times {\mathcal{C}}_{\mathrm{prefact}}\times {\mathcal{C}}_{\mathrm{flux}},\end{array}$$(13)

with

$$\begin{array}{cc}\hfill {\mathcal{C}}_{\mathrm{prefact}}& \equiv \left(\frac{A_{10}(E_{k/n}^{\mathrm{TOA}}{A}_{10}+2m_{10})}{A_9(E_{k/n}^{\mathrm{TOA}}{A}_9+2m_9)}\right)\hfill \\ \hfill & \times \left(\frac{{(E_{k/n}^{\mathrm{TOA}}{A}_9+Z_9\varphi +m_9)}^2-m_9^2}{{(E_{k/n}^{\mathrm{TOA}}{A}_{10}+Z_{10}\varphi +m_{10})}^2-m_{10}^2}\right),\hfill \end{array}$$(14)

$$\begin{array}{cc}\hfill {\mathcal{C}}_{\mathrm{flux}}& \equiv \frac{{\psi }_9^{\mathrm{IS}}(E_{k/n}^{\mathrm{TOA}}+\frac{Z_9}{A_9}\varphi )}{{\psi }_9^{\mathrm{IS}}(E_{k/n}^{\mathrm{TOA}}+\frac{Z_{10}}{A_{10}}\varphi )},\mathrm{with}{\psi }_9^{\mathrm{IS}}=\mathrm{Eq}.(12).\hfill \end{array}$$(15)

We test the accuracy of the approximate formula against the correct TOA modulation of ¹⁰Be/⁹Be in the top panel of Fig. 2. In fact, the figure shows the TOA-to-IS ratio from the exact (solid blue line) and approximate (dashed black line) cases, with the relative difference between the two shown in the bottom panel. The approximation reproduces the exact calculation at the percent level accuracy; the origin of the difference lies in the energy dependence of the IS ⁹Be flux in USINE that is not exactly the same as Eq. (12). In the top panel, we also highlight the various corrective terms appearing in the approximate formula, namely 𝒞_prefact, 𝒞_flux, and ‘shift’ – the TOA ratio must be calculated at IS energy in Eq. (13). The dominant correction is from the ‘shift’ term (dash-dotted grey line), with subdominant but still significant corrections from the 𝒞_prefact (dashed grey line) and 𝒞_flux (dotted grey line) terms.

Fig. 2. Impact of using an approximate solar modulation, illustrated by the calculation of (10Be/9Be)TOA/(10Be/9Be)IS as a function of the kinetic energy per nucleon. Top panel: TOA-to-IS ratio for the exact calculation with USINE (solid blue line) or using the approximate modulation Eqs. (13)–(15) with Eq. (12) for the energy dependence of the 9Be IS flux (dashed black line). The grey lines show the broken-down corrective terms entering Eq. (13). Bottom panel: relative difference between the approximate calculation and the exact calculation. The range highlighted in yellow corresponds to the region where the AMS-02 and HELIX experiments should measure 10Be/9Be.

2.4. ℱ at high energy and ℱ(E_k/n) = ℱ_HE approximation

The ℱ term defined in Eq. (8) is, a priori, an energy-dependent quantity. We recall that it can be directly calculated from the knowledge of measured IS fluxes for ¹⁰Be and ⁹Be progenitors and production cross-sections of these progenitors into the Be isotopes. For IS fluxes, parametrisations for H to Fe elements are available (Shen et al. 2019; Boschini et al. 2020a), based on Voyager data taken at IS energies (Cummings et al. 2016) and ACE (George et al. 2009; Lave et al. 2013), AMS-02 (Aguilar et al. 2021a), and HEAO-3 (Engelmann et al. 1990) data taken at TOA energies. For the production cross-sections, several datasets are also publicly available from the DRAGON¹ (Evoli et al. 2018), GALPROP² (Porter et al. 2022), and USINE³ (Maurin 2020) codes and websites.

Both IS fluxes and production cross-sections are subject to uncertainties that are difficult to quantify. For this reason, we do not provide pre-calculated ℱ terms for the various existing (and still evolving) parametrisations. Rather, we show that taking ℱ to be a constant provides an even simpler framework for analysing (¹⁰Be/⁹Be)^TOA data, while providing meaningful constraints on L.

Asymptotic behaviour at high energy. The ℱ term is expected to become constant above GeV/n. Indeed, the production cross-sections are generally assumed to be constant above a few GeV/n. Moreover, the main progenitors of ¹⁰Be and ⁹Be at high energy are always primary species with the same energy dependence (species from C to Fe are assumed to share the same source slope), so that this energy dependence cancels out of ℱ.

An asymptotically constant ℱ = ℱ_HE term at high energy (HE) means that the ¹⁰Be/⁹Be ratio also becomes a constant – ¹⁰Be behaves like a stable species at high energy (boosted decay time γτ₀). This is illustrated in Fig. 1, where the asymptotic regime is reached above a few hundred GeV/n. Indeed, ℱ_HE can be directly inferred from model calculations of the high-energy (¹⁰Be/⁹Be)_HE ratio, because Eq. (7) reduces to

$$\begin{array}{c}\hfill {\frac{{}^{10}\mathrm{Be}(E_{k/n})}{{}^9\mathrm{Be}(E_{k/n})}|}_{\mathrm{HE}}\equiv {\frac{{\psi }_{10}^{\mathrm{TOA}}(E_{k/n})}{{\psi }_9^{\mathrm{TOA}}(E_{k/n})}|}_{\mathrm{HE}}\approx \frac{t_{\mathrm{diff}}^{10}}{t_{\mathrm{diff}}^9}·{\mathcal{F}}_{\mathrm{HE}}\approx 0.949{\mathcal{F}}_{\mathrm{HE}}.\end{array}$$(16)

For the last equality, we assumed a high-rigidity diffusion coefficient K(R)∝R^δ with δ = 0.5 (Génolini et al. 2019; Weinrich et al. 2020b; Maurin et al. 2022), appropriate for a high-energy (HE) regime, taken to be ≳100 GeV/n here⁴.

Assuming ℱ = ℱ_HE in the analytical model. It is interesting to see whether using ℱ = ℱ_HE in Eq. (7) provides a good enough approximation to further simplify the fit of ¹⁰Be/⁹Be data. Departure of ℱ from a constant happens for two reasons. First, production cross-sections become energy dependent below a few GeV/n. This effect is mitigated by the fact that data are at TOA energies, and thus correspond to cross-sections evaluated at $E_{k/n}^{\mathrm{IS}}\gtrsim E_{k/n}^{\mathrm{TOA}}+0.5$ GeV/n. Second, as observed directly in the AMS-02 data (Aguilar et al. 2021b), the ratio of primary species becomes energy dependent below ≲100 GeV/n. This is because inelastic interactions more strongly impact heavier species, so that all parents have slightly different energy dependences: the flux ratio of heavier-to-lighter primary elements decreases with decreasing energies (Putze et al. 2011; Vecchi et al. 2022). Progenitors of Be isotopes can also be of secondary (e.g., B) or mixed (e.g., N) origin, and thus have different energy dependences compared to purely primary CR fluxes (e.g., O, Si, and Fe). However, to some extent, this variety of energy dependences in the progenitors is mitigated by the fact that the most important parents for ⁹Be and ¹⁰Be are mostly the same species (Génolini et al. 2018).

In Fig. 3 we compare the bias incurred by assuming ℱ(E_k/n) = ℱ_HE in the ¹⁰Be/⁹Be calculation, for two different production cross-section sets: Galp-opt12 (magenta) and OPT12up22 (black). These sets are properly introduced and discussed in Sect. 3.1. The top panel shows the (¹⁰Be/⁹Be)^TOA ratio calculated with USINE (thick lines) and with the analytical model assuming ℱ = ℱ_HE (thin lines); the bottom panel shows the relative difference between the two⁵. For both of the cross-section sets, the analytical model (thick lines) recovers the USINE calculation (thin lines) at a precision better than 1% above ≳4 GeV/n. Below, a non-trivial energy dependence is seen, dependent on the cross-section set: for OPT12up22 (solid black line), the agreement is only a few percent, except for a peak difference of 10% at 300 MeV/n (black line in the bottom panel); for Galp-opt12, the difference grows slowly, and is above 5% at TOA energies below 100 GeV/n.

Fig. 3. Impact of using the approximation ℱ = ℱHE for the calculation of (10Be/9Be)TOA as a function of the kinetic energy per nucleon. Top panel: calculation from USINE (thin lines) and the analytical model (thick lines) for two different production cross-section sets (associated with different transport parameters and L values, as highlighted in the legend). Bottom panel: relative difference between the approximate calculation and the USINE calculation. The range highlighted in yellow corresponds to the region where the AMS-02 and HELIX experiments should measure 10Be/9Be.

2.5. Consequences for the determination of L

Equations (7) and (13), with the full energy dependence accounted for in ℱ, provide an excellent approximation (few percent precision) for the calculation of (¹⁰Be/⁹Be)^TOA from a few tens of MeV/n up to the highest energies. This approximation is less accurate if we assume ℱ = ℱ_HE: differences as large as 10% appear, although in a very narrow energy domain around tens or hundreds of MeV/n. This approximation is also more accurate or less accurate, depending on the cross-section set used.

The advantage of the ℱ = ℱ_HE approximation is that it allows some ingredients entering the calculation to be further separated and identified. Indeed, the main terms in Eq. (13) are now the timescales in Eqs. (4)–(6) and ℱ_HE. First, t_diff corresponds to the confinement time in the disc (or grammage), and it is determined by the analysis of stable secondary-to-primary species (e.g., Weinrich et al. 2020b), that is independently of L. Second, we already stressed that ℱ can be calculated independently of any propagation model but ℱ_HE is even simpler, because it is directly related to (¹⁰Be/⁹Be)_HE via Eq. (16). Third, with t_inel calculated from the inelastic cross-sections, this leaves Eq. (7) exhibiting a single free parameter only, namely the halo size L. This means that L can be directly determined from (¹⁰Be/⁹Be)^TOA data, provided ℱ_HE does not suffer from excessively large uncertainties. In other words, the determination of L will be limited by the precision at which the factor ℱ_HE can be determined, which is mostly related to the precision of the production cross-sections of ⁹Be and ¹⁰Be.

3. Halo size determination: Be/B vs. ¹⁰Be/⁹Be

For all the analyses in this paper, whether we run USINE or use the analytical formula, we rely on inelastic cross-sections from Tripathi et al. (1997, 1999) and an ISM composed of 90% of H and 10% He (in number). The secondary Be/B or ¹⁰Be/⁹Be ratios, and in particular the ℱ term (or ℱ_HE discussed above), also crucially depend on the nuclear production cross-section sets considered. As such, we need to discuss them before setting the constraints on L.

3.1. Production cross-section sets

In this study we consider four different production cross-section sets, including the Galp-opt12 set from the GALPROP team, because it has been the most used set in recent CR analyses (Génolini et al. 2019; Weinrich et al. 2020b; Boschini et al. 2020a,b; Korsmeier & Cuoco 2021; Wang et al. 2022); it is also the set used in our previous effort to determine L from AMS-02 Be/B data (Weinrich et al. 2020a). We also consider the OPT12, OPT12up22, and OPT22 sets, discussed in detail in Maurin et al. (2022), with the most-plausible set being OPT12up22. These sets correspond to various ways to renormalise the original GALPROP cross-section options (Galp-opt12 and Galp-opt22) on the most important production channels (Génolini et al. 2018), taking advantage of recent nuclear data, and also including the misestimated but important contribution of Fe fragmentation into Li, Be, and B isotopes (Maurin et al. 2022).

Other production cross-section sets exist, as derived for instance by Reinert & Winkler (2018), the DRAGON team (Evoli et al. 2018), or from FLUKA (De La Torre Luque et al. 2022). We do not directly use these other sets in this analysis, but we illustrate in Sect. 3.3 how their use would impact the determination of L.

3.2. Updated constraints from AMS-02 Be/B data

In this section, the constraints on the halo size L from AMS-02 Be/B data are derived for the four production cross-section sets presented above. In practice, we repeat the analysis of Weinrich et al. (2020a), that is we fit AMS-02 Li/C, Be/B, and B/C data, accounting for the covariance matrix of uncertainties on the data, and for nuisance parameters on nuclear cross-section parameters and on the solar modulation level. This allows us to determine the transport parameters along with the halo size L – actually log₁₀(L/1 kpc) in the fit–, which is the only parameter we discuss here⁶.

L values. In Table 1 we report the best-fit values (and uncertainties) obtained for L for the various cross-section sets introduced in Sect. 3.1; for Galp-opt12 (first line), we directly reproduce the numbers from Weinrich et al. (2020a). We also report the associated ${\chi }_r^2$ values, confirming that the cross-section set OPT12up22 is the one that best fits the data⁷.

In Table 1 we see that the choice of the production cross-section set strongly impacts the best-fit halo size, but less so their uncertainties (which are at least twice as large than the difference between the best-fit values): the best-fit constraint on L moves from $L_{\mathtt{Galp}-\mathtt{opt}\mathtt{12}}=5.0_{-1.8}^{+3.0}$ (Weinrich et al. 2020a) to $L_{\mathtt{OPT}\mathtt{12}\mathtt{up}\mathtt{22}}=3.8_{-1.6}^{+2.8}$ (this analysis), with a significantly improved χ². These differences arise partly from the fact that the updated cross-section sets (OPT12, OPT12up22, and OPT22) lead to differently enhanced productions of Li (and to a lesser extent Be) with respect to the original Galp-opt12, and thus to a different baseline grammage to reproduce the same secondary-to-primary data (see Fig. 10 of Maurin et al. 2022). However, the main difference is related to the modified cross-section for the production of ¹⁰Be: looking at the columns labelled ⁹Be, ¹⁰Be, and ¹⁰B in Table 1 of Maurin et al. (2022), the numbers correspond to the rescaling applied to obtain OPT12up22 from Galp-opt12 cross-sections, and they vary significantly for the various parents (rows), especially ⁵⁶Fe, ¹⁶O, and ¹²C. Moreover, Table 2 in Maurin et al. (2022) illustrates that, depending on the rescaling procedure used and scatter on nuclear data, the production cross-section for ¹⁰Be can vary by a factor of ≳50% for some reactions (as illustrated in the right panel of their Fig. 6).

Be/B best-fit ratio and 1σ contour. It is interesting to inspect the best-fit values and contours obtained for the Be/B ratio. This is shown in the top panel of Fig. 4, along with the AMS-02 data used for the fit. The difference between the use of the original cross-section set Galp-opt12 (dotted magenta line) and the updated ones (grey and dark lines) is very mild, except above a few hundred GeV/n. To understand the origin of this ≲5% difference, we again have to refer to the underlying cross-section reactions (Maurin et al. 2022): in the updated cross-section sets (OPT12, OPT12up22, and OPT22), the production of Be, and in particular ⁷Be (which is the dominant isotope, see Fig. 13 in Maurin et al. 2022), is increased by the presence of Fe fragmentation compared to the less important extra production of B above ∼10 GV – the importance of Fe in the production of Be and B is shown in their Fig. 8. It is nice, however, to see that the overall shape of the ratio, and the small features seen in the data at low rigidity, seem to be even better reproduced with the updated cross-section sets.

Fig. 4. Best-fit models from the combined analysis of AMS-02 Li/C, Be/B, and B/C data with USINE (see Table 1). The four line styles correspond to the four production cross-section sets discussed in Sect. 3.1 (Galp-opt12, OPT12, OPT12up22, and OPT22). We also show 1σ CLs, but for the OPT12up22 set only (solid black lines), the other sets giving similar contours. Top panel: Be/B as a function of rigidity, along with the AMS-02 data (Aguilar et al. 2018) used for the fit. Bottom panel: 10Be/9Be as a function of kinetic energy per nucleon for the same transport parameters and L as above (i.e., from best-fit to Li/C, Be/B, and B/C data only, no fit to 10Be/9Be). A compilation of data (green symbols) is shown for illustrative purpose.

Resulting ¹⁰Be/⁹Be best-fit ratio and 1σ contour. It is also interesting to look at what these best-fit models predict for the ¹⁰Be/⁹Be ratio. This is shown in the bottom panel of Fig. 4, where we see that all models are in fair agreement with the existing data (not used in the fit in this section). However, there are significant differences between the models, and the origin of these differences is not the same at low and high energy.

At high energy, the ¹⁰Be/⁹Be ratio becomes constant. As already stressed (see Sect. 2), this ratio is independent of L and t_diff, but very sensitive to the production cross-section values of the most-important progenitors via ℱ_HE (see Sect. 2.4). The differences seen between Galp-opt12 (dotted magenta line) and the updated sets OPT12 (dashed grey line), OPT12up22 (solid black line), and OPT22 (dash-dotted grey line), simply reflect the changed proportion of produced ⁹Be and ¹⁰Be – see discussion from the previous paragraph.

At low energy, where decay dominates for ¹⁰Be, we see differences also between the three updated cross-section sets. In this regime, the origin of the differences, as can be read off Eq. (7), is related to both $t_{\mathrm{diff}}^{10}/t_{\mathrm{rad}}^{10}$ and L. Actually, it must be only related to $t_{\mathrm{diff}}^{10}$ and L as all other terms entering the analytical equation remain unchanged from one cross-section set to another. To go into more details, moving from OPT12 to OPT22, there is an increased overall production of Li, Be, and B (Maurin et al. 2022), so that a lower grammage is necessary to produce the same amount of measured secondaries. However, this increase is at the ∼20% level (see Fig. 10 in Maurin et al. 2022), compared to the much larger scatter observed on the L best-fit values (see Table 1); it is not clear which of the previous two effects dominates the scatter on the (¹⁰Be/⁹Be)^TOA isotopic ratio. This scatter is smaller than the width of the 1σ envelope (grey-shaded area) calculated from the covariance matrix of the best-fit parameters (dominated by L uncertainties). At growing energies, the contours shrink as the calculation becomes independent of L and of the transport coefficient.

3.3. Constraints from ¹⁰Be/⁹Be data with the analytical model

We can now move on to the analysis of the constraints set on L by (¹⁰Be/⁹Be)^TOA data, using the analytical model. Below we compare the results obtained using the full energy dependence in ℱ(E_k/n) or the approximation ℱ(E_k/n) = ℱ_HE (i.e., a constant).

L values. To perform the fit on L for a given production set, we calculate ℱ (or ℱ_HE) from USINE IS fluxes and production cross-sections, and t_diff is calculated from the transport parameters given in Maurin et al. (2022)⁸; we report the values for ℱ_HE and t_diff in Table 2. The ¹⁰Be/⁹Be data used are those extracted from CRDB⁹ (Maurin et al. 2014, 2020), namely ACE (Yanasak et al. 2001), IMP7&8 (Garcia-Munoz et al. 1981), ISEE3 (Wiedenbeck & Greiner 1980), ISOMAX (Hams et al. 2004), PAMELA-CALO and TOF (Nozzoli & Cernetti 2021), Ulysses (Connell 1998), and Voyager1&2 (Lukasiak 1999). For each dataset, we apply a different modulation level ϕ, also retrieved from CRDB. The ϕ values are based on the analysis of neutron monitor data (Ghelfi et al. 2017), from a careful calibration of the H and He fluxes (Ghelfi et al. 2016) and neutron monitor response (Maurin et al. 2015). We have ϕ_ACE = 609 MV, ϕ_IMP7&8 = 666 MV, ϕ_ISEE = 741 MV, ϕ_ISOMAX = 597 MV, ϕ_PAMELA = 783 MV, and ϕ_Ulysses = 727 MV; for Voyager1&2, we keep the value ϕ = 460 MV quoted in the original reference (Lukasiak 1999), corresponding to an effective modulation level calculated for the spacecraft at an average distance of 30 AU from the Sun.

We show the best-fit results (and uncertainties) for L in Table 2 using either the approximation ℱ = ℱ_HE or using the full energy dependence in ℱ(E_k/n). Both approaches lead to consistent L values within the uncertainties. The latter originate from the ¹⁰Be/⁹Be data uncertainties, and we have

$$\begin{array}{c}\hfill {(\mathrm{\Delta }L/L)}_{\mathrm{CR}\mathrm{data}}^{{}^{10}\mathrm{Be}{/}^9\mathrm{Be}}\approx 5-10\%.\end{array}$$

We find that the bias on L from the ℱ = ℱ_HE approximation is smaller than (ΔL)_{CR data} and dependent on the cross-section set used. In the rest of the section, we focus on results obtained with the approximation (i.e., using ℱ_HE only). The associated best-fit models (modulated at 700 MV) are shown in Fig. 5 for illustration. They all go nicely through the data for all production sets considered, reflecting the fact that ${\chi }_r^2$ ≈ 0.4 for all these configurations.

Fig. 5. Best-fit model on 10Be/9Be data using the analytical model, for ℱ and tdiff input values estimated from different publications: Galp-opt12 (dotted magenta line) and OPT12up22 (solid black line) are taken from Maurin et al. (2022). The models are shown for ϕ = 700 MV, typically corresponding to the average modulation of all the data (green symbols), whose references are given in the text.

Propagation of cross-section uncertainties on L. As for the Be/B analysis, using the cross-section set OPT12up22 leads to a smaller best-fit value L than using Galp-opt12 (a comparison can be made between the numbers in Table 1 and Table 2). In the context of the analytical model, this behaviour can now be directly linked to the different ℱ_HE values in the two production sets, and more precisely for these sets, to the different production cross-sections of ¹⁰Be and ⁹Be from a few relevant progenitors (¹²C, ¹⁶O, and ⁵⁶Fe). The typical 20% variation in ℱ_HE originating from these cross-section differences is responsible for a variation

$$\begin{array}{c}\hfill {(\mathrm{\Delta }L/L)}_{\mathrm{XS}}^{{}^{10}\mathrm{Be}{/}^9\mathrm{Be}}\approx 40\%,\end{array}$$

which is significantly larger than (ΔL/L)_{CR data}. We further discuss the uncertainties on ℱ_HE and comparisons with recent determination of L in Appendix A.

Finally, we note that the central values for L in the Be/B or ¹⁰Be/⁹Be analyses are similar for Galp-opt12 (5 kpc vs. 4.7 kpc), but slightly different for OPT12up22 (3.8 kpc vs. 2.8 kpc). However, this difference is not significant and could easily be explained by uncertainties on the cross-sections for the production of ¹⁰B and ¹¹B (involved in the Be/B analysis only).

Consequences for future experiments. The impact of production cross-sections for the determination of L is significant, as has been already argued by different teams from Be/B analyses (Weinrich et al. 2020a; Evoli et al. 2020; De La Torre Luque et al. 2021). As demonstrated here with ¹⁰Be/⁹Be, this dependence is directly tied in to the ℱ_HE term, showing that the value of the latter quantity is crucial for a good determination of L. Ideally, future experiments should go up to ≳100 GeV/n in order to get a handle on this crucial ℱ_HE factor. However, AMS-02 and the HELIX project (Park et al. 2019) are expected to reach, at best, energies ≳10 GeV/n. This would already be a great experimental achievement but, unfortunately, these measurements may not be enough to strongly reduce the uncertainties on L. To fully benefit from the high-precision data of these experiments, the alternative is to improve the confidence (and reduce the scatter) that there is in the production cross-sections, in order to better constrain ℱ(E_k/n), and thus L.

4. Conclusions

We have proposed a simple analytical formula to fit the halo size L of the Galaxy from (¹⁰Be/⁹Be)^TOA data, without the need for an underlying propagation model. The minimal ingredients needed are: (i) the grammage (or rather the diffusion timescale), which can be directly taken from existing analyses of secondary-to-primary ratios; (ii) the destruction time, directly calculated from the inelastic cross-sections of ¹⁰Be and ⁹Be; and (iii) a high-energy normalisation constant ℱ_HE ≈ 0.5 (independent of L) or, for an even better accuracy, the energy-dependent ℱ(E_k/n) term calculated from inferred CR IS fluxes and nuclear data, and models for the production cross-sections of ¹⁰Be and ⁹Be.

We have also shown that the constraints set on L from AMS-02 Be/B data using a combined fit and a propagation code are consistent with those obtained from a direct fit of existing (¹⁰Be/⁹Be)^TOA data with the analytical approximation. Moreover, with this approximation, uncertainties on the production cross-sections directly impact ℱ_HE, and from various studies from the literature, we typically have (Δℱ_HE/ℱ_HE)≈20%, which translates into (ΔL/L)≈40%. This uncertainty is clearly a limitation to fully exploiting forthcoming CR data. Fortunately, future data from the NA61/SHINE facility at the CERN SPS (Unger & NA61 SHINE 2019; Amin & NA61 SHINE 2021) should help to improve the situation. In particular, high-energy nuclear data for the production of Be isotopes from ¹⁶O (and to a lesser extent ⁵⁶Fe) are desired.

It would be great if the analytical formula could be extended to other ratios of radioactive clocks. Unfortunately, this may prove difficult. First, the approximate formula is valid for ¹⁰Be/⁹Be, as long as it is expressed as a function of kinetic energy per nucleon (conserved quantity in fragmentation reactions); a similar formula for the ratio evaluated at the same rigidity is expected to be both more complicated and less accurate. Second, it would be nice to be able to extend the approximation for the Be/B ratio calculation. However, compared to the ¹⁰Be/⁹Be case, we checked that a tentative formula for Be/B leads to larger biases, while the formula is to be used on a ratio that shows a weaker dependence on L (¹⁰Be is subdominant in Be). Third, one may think about applying the approximation to heavier CR clocks (e.g., ²⁶Al, ³⁶Cl, and ⁵⁴Mn). However, the stable associated isotopes used to form the ratio of interest (e.g., ²⁷Al in the ²⁶Al/²⁷Al ratio) contain non-negligible primary source terms, which may prevent the derivation of a simple and accurate analytical formula.

Although the analytical model may only apply to the ¹⁰Be/⁹Be ratio, we believe that it will already be very useful in order to interpret the forthcoming data from AMS-02 and HELIX. It should be also useful in order to quickly estimate the benefit that better production cross-sections of ¹⁰Be and ⁹Be can have on the determination of L.

Acknowledgments

D.M. and L.D. thank their AMS colleagues for useful discussions that triggered the development of this project. We thank Y. Génolini for his careful reading of the paper and comments. 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. This work was supported by the Programme National des Hautes Energies of CNRS/INSU with INP and IN2P3, co-funded by CEA and CNES.

References

Appendix A: Comparison with other studies and the impact of re-acceleration

For comparison purposes, we tried to repeat our analysis using ℱ_HE and t_diff values from other publications. The only recent work for which we could retrieve the necessary ingredients is that of De La Torre Luque et al. (2021), hereafter [De21]. In the study, the authors compare the use of different production cross-section sets for the determination of the halo size L.

A.1. ℱ_HE from various cross-section sets

Taking the asymptotic high-energy value of ¹⁰Be/⁹Be from Fig. 8 of [De21] and using Eq. (1), we can calculate ℱ_HE associated with their cross-section sets (Webber, GALPROP, and DRAGON2), with ℱ_HE = 0.35, 0.60, and 0.58 respectively.

The cross-sections sets used in [De21] were introduced, and discussed in detail, in Evoli et al. (2018), and we can compare them to those used in our analysis. The GALPROP set in [De21] and Galp-opt12 used here are one and the same. Although the underlying propagation model (and to some extent, the CR data used) are different in these two studies, we obtain very similar ℱ_HE values (${\mathcal{F}}_{\mathrm{HE}}^{\mathtt{GALPROP}}=0.60$ vs. ${\mathcal{F}}_{\mathrm{HE}}^{\mathtt{Galp}-\mathtt{opt}\mathtt{12}}=0.62$ reported in our Table 2); also similar to the value obtained for DRAGON2 (ℱ_HE = 0.58)¹⁰. These values are also similar to the one obtained directly from the GALPROP team (using their GALPROP cross-section set), as can be calculated from the high-energy ¹⁰Be/⁹Be ratio shown in Fig. 11 of Strong et al. (2007).

However, the very low value ${\mathcal{F}}_{\mathrm{HE}}^{\mathtt{WEBBER}}=0.35$ in [De21] is quite puzzling. As described in Evoli et al. (2018), the so-called Webber set is a mixture of Webber’s 2003 cross-section values (Webber et al. 2003), hereafter W03, and calculations based on the Webber’s 1983 values. The motivation for this hybrid set is not completely straightforward, as W03 should completely supersede Webber’s older calculations. The W03 set was used in Putze et al. (2010)¹¹, where their Fig. 10 allows ℱ_HE = 0.58 to be calculated, a value actually in line with that of the other cross-section sets. We therefore recommend caution when using this hybrid Webber set.

In any case, this comparison shows that the various production cross-section sets used in the literature agree within 10 − 20%. Unfortunately, as highlighted in the text, a 20% difference translates into a ∼40% uncertainty on L.

A.2. Comparison with L values from [De21]

We can try to reproduce the L values obtained in [De21] from our analytical formula, extracting t_diff from their Eq. (2.2) and Table 2.3¹². Using the same data as in [De21], we find L_DRAGON2 = 3.4 kpc and L_WEBBER = 1.1 kpc, to compare to their publication values 6.8 kpc and 2.1 kpc respectively. Despite these significant differences, the strong correlation between L and ℱ_HE remains, and confirms that the uncertainty on ℱ_HE is the dominant source of uncertainty for the determination of L. In addition, we would have expected our analytical formula to lead to similar values for L_DRAGON2 and L_OPT12up22, as ℱ_HE(OPT12up22) ≈ ℱ_HE(DRAGON2), but we only get L_DRAGON2 = 3.4 kpc and L_OPT12up22 = 2.8 kpc.

The likely origin of the above differences is the presence of re-acceleration in [De21]—we checked that varying h or the subset of data used in the fit is not enough to explain them. To test this hypothesis, we calculate, with the USINE code, the ¹⁰Be/⁹Be ratio, varying the re-acceleration, but leaving unchanged the diffusion time. We stress that in the model used in [De21], the re-acceleration, mediated by the Alfvénic speed V_a, is distributed in the full diffusive halo, while it is restricted to the thin disc region in ours (for mathematical reasons, see Maurin 2020)¹³. As a result, V_a ∼ 29.9 km s⁻¹ (with L_Webber = 1.1 kpc) in [De21] corresponds to V_a ≲ 99 km s⁻¹ in our model. In Fig. A.1 we show calculations of ¹⁰Be/⁹Be with and without re-acceleration, considering values up to 150 km s⁻¹. At 100 km s⁻¹ (dotted red line), differences up to 20% are observed, compared to the case with no re-acceleration. This is consistent with the results of Yuan et al. (2020), where the authors find a ∼20% larger halo size when considering re-acceleration (see their Table 1). We do not wish to push further the comparison, but these numbers are in the range necessary to reconcile the difference in L values between [De21] results (with re-acceleration) and those obtained from our analytical formula (without re-acceleration).

Fig. A.1. 10Be/9Be TOA ratio (top) with or without re-acceleration (in the thin disc only, see text) and relative difference with respect to the Va = 0 case (bottom).

All Tables

All Figures

Fig. 1. Impact of neglecting energy losses in the 10Be/9Be ratio as a function of the kinetic energy per nucleon (all curves are from USINE runs). Top panel: calculation with energy losses switched on (solid lines) or off (dotted lines) for TOA (blue) or IS (orange) calculations. Bottom panel: relative difference between the ‘no E loss’ and ‘with E losses’ calculations, for the TOA (dotted blue line) and IS (dashed orange line) cases. The range highlighted in yellow corresponds to the region where the AMS-02 and HELIX experiments should measure 10Be/9Be.

In the text

Fig. 2. Impact of using an approximate solar modulation, illustrated by the calculation of (10Be/9Be)TOA/(10Be/9Be)IS as a function of the kinetic energy per nucleon. Top panel: TOA-to-IS ratio for the exact calculation with USINE (solid blue line) or using the approximate modulation Eqs. (13)–(15) with Eq. (12) for the energy dependence of the 9Be IS flux (dashed black line). The grey lines show the broken-down corrective terms entering Eq. (13). Bottom panel: relative difference between the approximate calculation and the exact calculation. The range highlighted in yellow corresponds to the region where the AMS-02 and HELIX experiments should measure 10Be/9Be.

In the text

Fig. 3. Impact of using the approximation ℱ = ℱHE for the calculation of (10Be/9Be)TOA as a function of the kinetic energy per nucleon. Top panel: calculation from USINE (thin lines) and the analytical model (thick lines) for two different production cross-section sets (associated with different transport parameters and L values, as highlighted in the legend). Bottom panel: relative difference between the approximate calculation and the USINE calculation. The range highlighted in yellow corresponds to the region where the AMS-02 and HELIX experiments should measure 10Be/9Be.

In the text

Fig. 4. Best-fit models from the combined analysis of AMS-02 Li/C, Be/B, and B/C data with USINE (see Table 1). The four line styles correspond to the four production cross-section sets discussed in Sect. 3.1 (Galp-opt12, OPT12, OPT12up22, and OPT22). We also show 1σ CLs, but for the OPT12up22 set only (solid black lines), the other sets giving similar contours. Top panel: Be/B as a function of rigidity, along with the AMS-02 data (Aguilar et al. 2018) used for the fit. Bottom panel: 10Be/9Be as a function of kinetic energy per nucleon for the same transport parameters and L as above (i.e., from best-fit to Li/C, Be/B, and B/C data only, no fit to 10Be/9Be). A compilation of data (green symbols) is shown for illustrative purpose.

In the text

	Fig. 5. Best-fit model on 10Be/9Be data using the analytical model, for ℱ and tdiff input values estimated from different publications: Galp-opt12 (dotted magenta line) and OPT12up22 (solid black line) are taken from Maurin et al. (2022). The models are shown for ϕ = 700 MV, typically corresponding to the average modulation of all the data (green symbols), whose references are given in the text.
In the text

	Fig. A.1. 10Be/9Be TOA ratio (top) with or without re-acceleration (in the thin disc only, see text) and relative difference with respect to the Va = 0 case (bottom).
In the text