Issue 
A&A
Volume 639, July 2020



Article Number  A74  
Number of page(s)  14  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/202038064  
Published online  09 July 2020 
Galactic halo size in the light of recent AMS02 data
^{1}
LPSC, Université Grenoble Alpes, CNRS/IN2P3, 53 avenue des Martyrs, 38026 Grenoble, France
email: david.maurin@lpsc.in2p3.fr
^{2}
Instituto de Física Teórica UAM/CSIC, Calle Nicolás Cabrera 1315, Cantoblanco 28049 Madrid, Spain
^{3}
Niels Bohr International Academy & Discovery Center, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark
^{4}
LUPM, CNRS & Université de Montpellier (UMR5299), Place Eugène Bataillon, 34095 Montpellier Cedex 05, France
email: julien.lavalle@umontpellier.fr
^{5}
LAPTh, Université Savoie Mont Blanc & CNRS, 74941 Annecy Cedex, France
email: pierre.salati@lapth.cnrs.fr
Received:
1
April
2020
Accepted:
14
May
2020
Context. The vertical diffusive halo size of the Galaxy, L, is a key parameter for dark matter indirect searches. It can be better determined thanks to recent AMS02 data.
Aims. We set constraints on L from Be/B and ^{10}Be/Be data, and we performed a consistency check with positron data. We detail the dependence of Be/B and ^{10}Be/Be on L and forecast on which energy range better data would be helpful for future L improvements.
Methods. We used USINE V3.5 for the propagation of nuclei, and e^{+} were calculated with the pinching method.
Results. The current AMS02 Be/B (∼3% precision) and ACECRIS ^{10}Be/Be (∼10% precision) data bring similar and consistent constraints on L. The AMS02 Be/B data alone constrain L = 5_{−2}^{+3} kpc at a 68% confidence level (spanning different benchmark transport configurations), a range for which most models do not overproduce positrons. Future experiments need to deliver percentlevel accuracy on ^{10}Be/^{9}Be anywhere below 10 GV to further constrain L.
Conclusions. Forthcoming AMS02, HELIX, and PAMELA ^{10}Be/^{9}Be results will further test and possibly tighten the limits derived here. Elemental ratios involving radioactive species with different lifetimes (e.g. Al/Mg and Cl/Ar) are also awaited to provide complementary and robuster constraints.
Key words: astroparticle physics / cosmic rays / Galaxy: halo
© N. Weinrich et al. 2020
Open 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.
1. Introduction
Finding the Galactic cosmicray (CR) sources, solving the details of CR transport in the Galaxy, and using the CR as a channel to identify the nature of dark matter (DM) are among the main challenges in CR physics. In the last decade, antimatter data received a lot of scrutiny, owing to the interpretation of the positron fraction rise as a DM signal (Adriani et al. 2009; Bergström et al. 2008; Feng & Zhang 2018), although standard astrophysics explanations are more likely (e.g. Hooper et al. 2009; Delahaye et al. 2010; Serpico 2012; Manconi et al. 2019). Similarly, the presence of a DM contribution in the data is debated in the literature (Reinert & Winkler 2018; Cholis et al. 2019; Cuoco et al. 2019; Boudaud et al. 2020). The latter channel is actually one of the best at constraining weakly interacting massive particles DM candidates in the GeVTeV mass range (e.g. Conrad & Reimer 2017).
DM interpretations for antimatter CRs depend on both the transport and geometry parameters (Donato et al. 2004; Delahaye et al. 2008; Aramaki et al. 2016); the latter is mostly determined via CR radioactive clocks. These clocks have a lifetime of approximately a million years, one order of magnitude shorter than the typical CR propagation time in the Galaxy. Ratios of a secondary (i.e. produced at the propagation stage only) unstable species to their stable counterpart, for instance ^{10}Be/^{9}Be, allow one to break the degeneracy between the diffusion coefficient normalisation and the halo size of the Galaxy (e.g. Donato et al. 2002). However, CR isotopic separation in experiments is challenging. The first measurements were carried out more than forty years ago for ^{10}Be (Webber et al. 1973), ^{36}Cl (Young et al. 1981), ^{26}Al (Webber 1982), and ^{54}Mn (Webber et al. 1979); however, even recent measures (Connell 1998; Yanasak et al. 2001; Hams et al. 2004) are restricted to low energy mostly, that is, below a few hundreds of MeV/n. As an alternative and complementary approach, Webber & Soutoul (1998) proposed to use elemental ratios (e.g. Be/B, Al/Mg) in which the CR clock appears both in the numerator (decayed fraction) and denominator (daughter fed by decaying CR). Elemental ratios have been measured up to hundreds of GeV/n, hence covering an energy range in which ^{10}Be goes from mostly decayed to metastable, with respect to the propagation time, at high energy. While waiting for the AMS02 future release of Be isotope data, the highprecision Be/B ratio already gives useful constraints (Evoli et al. 2020). However, the isotopic fraction of ^{10}Be in B is a few percent only, and the sensitivity of Be/B to L is partly drowned by the dominant presence of the stable nucleus ^{7}Be (Tomassetti 2015a).
The positron data have also been recently shown to provide interesting constraints on L (Lavalle et al. 2014). In particular, for small L, the unavoidable secondary production, mostly from H and He CRs on the interstellar medium (ISM), may overshoot lowenergy data points. Therefore they can be used as a complementary probe to set a lower limit on L.
This study follows up on our previous effort to determine CR transport parameters (Génolini et al. 2019; Weinrich et al. 2020). We performed a joint analysis of Li/C, B/C, and various combinations of ^{10}Be data to determine L. We then drew models from the allowed regions of their parameter space and further checked their consistency with the positron constraint. We stress that in this analysis, as is the case in almost all similar studies, the halo size of the Galaxy is set to be a hard boundary where the CR density goes to zero. This is an effective modelling of a more realistic picture that would probably involve a rapidly growing diffusion coefficient in the halo (Di Bernardo et al. 2013; Tomassetti 2015b). In fact, a modelling of CR, gas, and wave interactions from first principles leads to the picture of a dynamic halo (Breitschwerdt et al. 1991; Zirakashvili et al. 1996). Recent results from Evoli et al. (2018) show, in particular, that the turbulent cascade from CR sources and the selfgeneration of waves by CRs can introduce an effective halo size. Full numerical simulations aimed at accurately computing CRs in a magnetohydrodynamic framework will probably give more insight into this problem, as envisaged in Girichidis et al. (2020). In the meantime, the concept of hard boundary remains a useful benchmark for many CRrelated studies.
The paper is organised as follows: in Sect. 2, we recall the propagation model and configurations used for the analysis. In Sect. 3, we assess the capability of ^{10}Be/^{9}Be, Be/B, or B/C current data to determine L, accounting for various modelling uncertainties; we then provide the resulting constraints on L. In Sect. 4, we discuss other possible observables to determine L, and we look at the flux of secondary positrons, given the Be/B constraint on L. In Sect. 5, we summarise and conclude. Further details of the analysis and additional crosschecks are reported to the appendices: Appendix A provides scaling relations of the transport parameters with L for stable secondary species – they extend, for all transport parameters, those that were given in Appendix C of Génolini et al. (2019); Appendix B highlights the constraints on L set by various ^{10}Be/^{9}Be datasets (lowenergy experiments, ISOMAX, and preliminary PAMELA analysis); Appendix C outlines why ^{7}Be/(^{9}Be+^{10}Be), the most favourable isotopic ratio to extract experimentally, is of no practical use to constrain L.
2. Model and configurations (BIG, SLIM, QUAINT)
The details of the transport equation and approach we follow are detailed in Génolini et al. (2019) and in the companion paper (Weinrich et al. 2020). Here, we only recall the most important features of the model.
We assume the CR density to obey a steadystate diffusionadvection equation. The geometry of the diffusion halo defines the region in which CRs propagate and are confined, here considered an infinite slab of halfthickness L, the parameter we aim at determining in this analysis. In this geometry, CR sources and the gas are pinched in an infinitely thin plan (halfthickness h = 100 pc≪L), where interactions with the gas (spallation, energy gains and losses) are restricted to. A convection term V_{c} is taken to be constant and perpendicular to the disc. We assume isotropic and homogeneous diffusion, and account for diffusion in momentum space: we follow the treatment of Seo et al. (1994), that is K_{pp}(R, x) ∝ 2 h δ(z) (V_{A}p)^{2}/K(R) where V_{A} is the Alfvénic speed of the plasma wave. These assumptions allow one to derive solutions semianalytically (Jones et al. 2001; Maurin et al. 2001). Our calculations are performed with the code USINE V3.5 (Maurin 2020)^{1}. For more details, we refer the reader to Génolini et al. (2019) and Weinrich et al. (2020).
Several theoretical studies have hinted at the possible presence of breaks at low (Ptuskin et al. 2006) or highrigidity (Blasi et al. 2012; Evoli et al. 2018). Actually, spectral breaks are seen in CR data at low (Stone et al. 2013) and highrigidity (Aguilar et al. 2018), and they can be connected to the presence of breaks in the diffusion coefficient at rigidities R_{l} ≈ 4 − 5 GV (Génolini et al. 2019; Vittino et al. 2019; Weinrich et al. 2020) and R_{h} ≈ 250 GV (Génolini et al. 2017, 2019; Reinert & Winkler 2018; Evoli et al. 2019). For these reasons we take
In this study, we fix several parameters whose impact on the results is negligible: the three highrigidity break parameters (R_{h}, δ_{h}, s_{h}) are set to the values reported in Génolini et al. (2019); the smoothness of the lowrigidity break parameter is fixed at s_{l} = 0.04 (fast transition).
We use three benchmark configurations BIG, SLIM, and QUAINT defined in Génolini et al. (2019), whose relevant parameters are collected in Table 1. In this analysis, BIG has 7 free parameters (K_{0}, δ, R_{l}, δ_{l}, V_{c}, V_{A}, L). The configuration SLIM is a special case of BIG, with V_{A} = V_{c} = 0 and η = 1, and it has 5 free parameters (K_{0}, δ, R_{l}, δ_{l}, L). The configuration QUAINT is also a special case of BIG with no lowrigidity break, and it has 6 free parameters (K_{0}, δ, η, V_{c}, V_{A}, L). QUAINT is also an extension (because of the highrigidity break) of older benchmark convection and reacceleration models used for instance in Maurin et al. (2010) and Di Bernardo et al. (2010).
We fitted the model to the data via a χ^{2} minimisation,
where t and q_{t} run over the several flux ratios (e.g. Li/C, Be/C, B/C) measured at different periods, whereas r runs over crosssection reactions. The nondiagonal term allows for ij energy bins correlations in the data (covariance matrix). The and 𝒩_{XS} terms account for Solar modulation and crosssection nuisance parameters, respectively. More details on this procedure are given in Appendix B of Derome et al. (2019). For further details related to the Li/C, Be/C, and B/C data uncertainties (correlation matrix of systematics), and also solar modulation and crosssection priors, we refer the reader to our companion paper (Weinrich et al. 2020).
The minimisation is performed with the MINUIT package (James & Roos 1975), its MINOS algorithm also providing accurate (asymmetric) error bars even if the problem is very nonlinear. In practice, the MINUIT routines are directly called from USINE (Maurin 2020). We also carry out 𝒪(100) minimisations from different starting points to ensure the true minimum is found (Weinrich et al. 2020). All uncertainties reported on the halo size in this study are derived from the profile likelihood method (with MINOS) at the 68% confidence level. As the χ^{2} definition accounts for energy correlations in the data uncertainties and nuisance parameters, the halo size uncertainties also account for them. We stress that uncertainties were derived on log(L) and are mostly symmetric on this parameter. For this reason, we loosely use the notation 1σ in the following, and for instance, 2σ limits can be estimated assuming a lognormal distribution for L.
3. Halo size L from CR clocks
Radioactive secondary species whose lifetime is shorter than escape time decay before experiencing the boundary of the Galaxy. These species are only sensitive to the diffusion coefficient K, whereas stable secondary species can escape and are sensitive to K/L. Any fit combining the information of a stable and radioactive secondary species breaks the K/L degeneracy, allowing for the determination of L (Donato et al. 2002).
Below, we focus on ^{10}Be (t_{1/2} = 1.387 Myr) and related ratios (^{10}Be/^{9}Be, ^{10}Be/Be, and Be/B). To date, ^{10}Be is the only available CR chronometer for which highprecision data exist for the associated elemental flux, Be. The Al, Cl, and Mn fluxes have not been released by the AMS02 collaboration, yet.
3.1. CR datasets and modulation levels
This analysis is based on several datasets. Each dataset is associated to a solar modulation level depending on its data taking period. We use here the simple forcefield approximation (Gleeson & Axford 1967, 1968; CaballeroLopez & Moraal 2004), whose single parameter, ϕ_{FF}, is taken as a nuisance parameter in the analyses. As detailed in Weinrich et al. (2020), the solar modulation level (prior) for each dataset is based on the analysis of neutron monitor data (Maurin et al. 2015; Ghelfi et al. 2016, 2017a,b).
As in Weinrich et al. (2020), the baseline data used to fix the transport parameters are AMS02 Li/C and B/C data (Aguilar et al. 2018) – denoted “base” in the following. The halo size L is then constrained by combining the base with several ratios from different datasets. The largest dataset is that of AMS02 Be/B, covering ∼3 GV to ∼2 TV. We also have several lowenergy ^{10}Be/^{9}Be or ^{10}Be/Be datasets available, retrieved from CRDB^{2} (Maurin et al. 2014). Except for the ISOMAX data reaching ∼2 GeV/n, most of them (ACE, IMP7&8, ISEE3, Ulysses, and Voyager1&2) are at a few hundreds of MeV/n.
The various datasets of interest are listed in Table 2, along with their estimated Solar modulation level and bibliographic reference. Several ratio and dataset combinations are considered, in order to assess and compare their respective impact on L. Combined to “base” (i.e. Li/C and B/C data only), which enables the determination of the transport parameters, at least one ^{10}Berelated dataset is necessary to determine L. In particular, for fits involving ^{10}Be/^{9}Be, we differentiate three groups (see Table 2): the most precise lowenergy data only (“ACE”), lowenergy data without ACE (“LE w/o ACE”), or intermediate energy data only (“ISOMAX”). For consistency for the Solar modulation nuisance parameters, in all fits involving lowenergy isotopic ratios, we consider, if available, the associated similarly modulated Li/C and B/C lowenergy data (in addition to AMS02 data).
Experiments with their datataking periods and associated expected Solar modulation level.
3.2. Expected constraints from B/C, Be/B, and ^{10}Be/Be
Before moving to the fits and results, we wish to understand how strongly L can be constrained by different data combinations of ^{10}Be. Ratios directly involving ^{10}Be should be optimal, but owing to the experimental difficulty of achieving isotopic separation, highprecision elemental ratios like Be/B can be competitive (Webber & Soutoul 1998). In principle, B/C might also lead to some constraint, via the fraction of ^{10}B generated by ^{10}Be decay. The most favourable option would include some information on the ^{10}Be content, highprecision data, and significant dynamic range. For instance, at ∼1 GeV/n, the B/C and Be/C fraction from unstable isotopes are respectively ∼5% and 5 − 10%, with data available from GV to TV at 3 − 5% precision (Aguilar et al. 2018); for ^{10}Be, the fraction is 100% with data in the ∼50 − 200 MeV/n range at ∼10 − 15% precision (Yanasak et al. 2001).
We first pick a reference configuration and L to which other calculations are compared. This configuration is taken from our companion paper, Weinrich et al. (2020), and is based on the simultaneous analysis of AMS02 Li/C, Be/C and B/C data at fixed L = 5 kpc. To see how sensitive to L calculated ratios are, we vary L ensuring that the level of production remains unchanged for secondary stable species–this constraint is satisfied with an appropriate rescaling of the transport parameters (see Appendix A)^{3}. From this setup, we calculate the relative variation of B/C, Be/B, ^{10}Be/Be, and ^{10}Be/^{9}Be with respect to the reference values at L = 5 pc.
The results are shown in Fig. 1, with the dependence upon growing L illustrated via the growing thickness of dotted lines, while different uniformly coloured bands represent different theoretical uncertainties (described and discussed in Sect. 3.3). For B/C (top panel), even for unrealistic variations of L, the maximal impact is a few percent only. On the other hand, varying L from 2.5 to 12 kpc leads to variations of up to 5% for Be/B (second panel) and up to 35% for ^{10}Be/^{9}Be and ^{10}Be/Be (two bottom panels). Secondly, whereas these variations peak strongly around ∼30 GV for Be/B, they are constant for all energy below a few GeV/n for ^{10}Be/^{9}Be and ^{10}Be/Be. We overlay on the plots (hatched regions) the existing data coverage in terms of energy range and total uncertainties. From the comparison with the Ldependent curves (thin to thickdotted lines), we conclude that B/C data alone can only provide very loose upper limits (≲20 − 30 kpc), while Be/B, ^{10}Be/^{9}Be, and ^{10}Be/Be are all expected to constrain L to better than a factor 2.
Fig. 1.
Relative variation of B/C, Be/B, ^{10}Be/Be, and ^{10}Be/^{9}Be ratios (from top to bottom) as a function of rigidity (upper panels) or kinetic energy per nucleon (lower panels). All the panels compare three different effects: (i) the thin to thickdashed lines correspond, at constant grammage, to the impact of L on the calculated ratios (the reference is L = 5 kpc); (ii) hatched boxes show the data relative uncertainties over the data energy coverage (AMS02 data in the two top panels) or for central energy points only (ACE, IMP7&8, ISEE3, ISOMAX, Ulysses, and Voyager1&2 data in the two bottom panels) – see Sect. 3.1 and Table 2 for data references; (iii) the remaining envelopes show 68% CLs on model calculations, including transport, cross sections, and modulation uncertainties separately or combined (“total”). 
3.3. Directions for future experimental efforts
By pursuing further the reasoning, we can also forecast where future measurements could improve the constraint on L. From Fig. 1, energy ranges where L is the most impacting are between 10 and 100 GV for Be/B data, and below a few GeV/n for ^{10}Be/^{9}Be. However, to fairly assess the sensitivity of each ratio, model uncertainties should be taken into account–broadly speaking, model uncertainties originate from transport, crosssection, and solar modulation.
Technically we proceed as follows: starting from the best fit and covariance matrix of the relevant parameters (standard outputs of fits with USINE V3.5), we draw realisations of the parameters of interest, from which new values for the ratios are calculated; we then extract contours and confidence levels on these ratios. Repeating the drawing procedure considering the full covariance matrix of parameters or only block elements of this matrix, we can propagate the model uncertainties all together (e.g. B/C in the top panel of Fig. 1) or separately (all remaining panels). This naturally accounts for the full or partial correlations between the parameters.
We show in the panels of Fig. 1 various model uncertainties (1σ contours) obtained from sampling 10^{4} realisations of the model parameters for SLIM^{4}. Overall, solar modulation uncertainties (≲5%, purple contours) are subdominant in the model error budget; they are also subdominant with respect to data uncertainties (hatched boxes). For isotopic ratios, crosssection uncertainties dominate (∼10 − 20%, green contours), followed by transport uncertainties (≲10%, orange contours). Let us detail separately the conclusions that can be drawn for B/C, Be/B, ^{10}Be/Be, and ^{10}Be/^{9}Be – for completeness, we also show in Appendix C the ^{7}Be/(^{9}Be+^{10}Be) ratio, more easily measured but unfortunately not constraining for L.
Firstly, the status of the B/C ratio (top panel) is different from the other ones, because transport is calibrated on it. For this reason, the associated uncertainties are typically at the level of the data uncertainties. Given the poor sensitivity of B/C to L, at least a factor of ten improvement on data errors would be necessary to bring any improvement on L.
Secondly, for the Be/B ratio (second panel), the optimal region to constrain L is from a few GV to a few tens of GV, exactly in the region where AMS02 data have the smallest errors (∼3%). In this region, crosssection uncertainties (green contours) are at the 5% level, meaning that better nuclear crosssection data could already slightly shrink the allowed L values by ∼30%. Better Be/B data could strengthen the L constraint, but only with crosssections uncertainties brought to par. However, improving CR data systematics below the percent level is very challenging.
Thirdly, for the isotopic ^{10}Be/Be and ^{10}Be/^{9}Be ratios (bottom two panels), we have to slightly change our estimation of crosssection uncertainties. Contrarily to elemental ratios, in which only the overall element production matters (sum of ^{7}Be, ^{9}Be, and ^{10}Be), isotopic ratios directly depend on the associated isotopic production cross sections. To be conservative – though probably too pessimistic –, we draw each isotopic production cross sections (independently) within their expected uncertainty range^{5}. As for Be/B, the crosssection uncertainties are the dominant modelling uncertainties. At variance with Be/B, any region below 10 GeV/n is equally suited to constrain L. In the highenergy end of this interval, current data uncertainties are larger than the model uncertainties, so that there is a small window for improvements on data to improve the constraints on L. Also, CR data on a large energy range should provide a better lever arm to handle crosssection uncertainties.
We can now conclude on the best way to improve the constraints on L in the future. On the short term, forthcoming ^{10}Be/^{9}Be AMS02 and PAMELA data up to 10 GeV/n are the best candidates to improve the constraints on L. On a longer term, the easiest way to improving isotopic CR data would be to focus on GeV/n energies for this same ratio. Improving the precision of Be/B data would be only significant at the subpercent level, but this is likely to remain difficult to achieve, even for future CR projects like HERD (Cattaneo & HERD Collaboration 2019), ALADInO (Adriani et al. 2019), or AMS100 (Schael et al. 2019). In any case, be it for Be/B or isotopic ratios, the common limiting factor to all improvements are crosssection uncertainties.
3.4. Actual constraints on L
In this section, we present the constraints on L from various data combinations. The fitting procedure and free (and nuisance) parameters are as discussed in Sect. 2 – see also our companion paper for more details (Weinrich et al. 2020).
3.4.1. Results
Figure 2 shows the constraints (at 1σ) brought by the ratios discussed in the previous section (B/C, Be/B, ^{10}Be/Be, and ^{10}Be/^{9}Be) for the three configurations BIG (blue circle), SLIM (orange crosses), or QUAINT (green triangles). We first stress that the various transport configurations all give similar constraints on L – we do not show results for the transport parameters as they are available and were abundantly discussed in Weinrich et al. (2020). The behaviour for the different ratios (columns) is in qualitative agreement with the expectations discussed in Sect. 3.2. In the first column, the B/C ratio mostly gives an upper limit L ≲ 15 kpc, whereas in the second and third columns, Be/B AMS02 and ^{10}Be/Be lowenergy data give similar constraints L ≈ 5 ± 3 kpc. The fourth column is based on the results from the same experiments as in the third column, but fitting ^{10}Be/Be instead of ^{10}Be/^{9}Be data, and with the additional use of intermediateenergy ISOMAX data (see Table 2). This gives a slightly lower bestfit value and uncertainties compared to ^{10}Be/Be data, but this apparent improvement is related to a tension between ISOMAX data and all the others. The reduced uncertainties result from an attempt to accommodate all the data at once – Appendix B details results on the brokendown constraints from various lowenergy datasets. The tension with ISOMAX data also reflects in the global fit (last column), which is pushed towards slightly larger L values, also preferred by AMS02 Be/B data (second column).
Fig. 2.
Bestfit halo size and asymmetric uncertainties (from MINOS) for the configurations BIG, SLIM, and QUAINT. From left to right, several data combinations are used. The first column (“Base”) involves AMS02 and lowenergy B/C and Li/C data. The second column is the constraint set from AMS02 data only (Li/C, Be/B, and B/C). The remaining columns combine “Base” data (from first column) to ^{10}Be/Be lowenergy data (third column), to ^{10}Be/^{9}Be low and intermediateenergy (nexttolast column), or combine all the previous cases (last column). 
We gather in Table 3 the bestfit values and 1σ uncertainties on L for the AMSonly analysis (with Be/B, top) and the combined analysis (with Be/B and all isotopic ratios, bottom). In terms of the values, a fair but not perfect agreement is obtained when using AMS02 only data (). An excellent fit is obtained for the isotopic data with , and also when combining elemental and isotopic data with (last column in Fig. 2 or bottom of Table 2); for the latter, lowenergy Li/C, B/C, and also ^{10}Berelated ratios are in good agreement with the constraints set by AMS02 data only and thus merely increases n_{data} without increasing . The last row in Table 3 shows the value of
with and the n_{s} and n_{x} nuisance parameters for solar modulation and cross sections respectively (n_{nui} = n_{s} + n_{x}). As discussed in Weinrich et al. (2020), this quantity gives a direct check that nuisance parameters behave properly. On average, nuisance parameters postfit values should never be more than 1σ away from their prior, that is, , and this is verified for all our fits.
For illustration purposes, we finally show in Figs. 3 and 4 the model calculation and the data for Be/B and isotopic ratios. The parameters are taken from the bestfit to all combined Be/B, ^{10}Be/Be, and ^{10}Be/^{9}Be data (last column in Fig. 2). In both plots, the top panels show the model calculations for the three transport configurations (BIG, SLIM, and QUAINT) along with the data. For SLIM, we also superimpose the 1σ model total uncertainties (contours) as calculated in Sect. 3.3. The second panels illustrate the goodnessoffit to the data via the residuals between the data and the model. For the Be/B case with AMS02 data (Fig. 3), a third panel shows the “rotated” score , as defined in Boudaud et al. (2020) or Weinrich et al. (2020): It suffices to say that this score represents an unbiased visual representation of the distance between the model and the data, accounting for existing rigidity correlations in the systematics of AMS02 data; also, the chisquare is the sum of the squares of these rotated residuals, that is, . The righthand side of the bottom panel is another illustration of the goodness of fit of the model, for the distribution of values is expected to follow a Gaussian distribution of width one.
Fig. 3.
Model prediction (top), residuals (centre), and score (bottom) for Be/B based on the bestfit parameters to B/C, Li/C, ^{10}Be/^{9}Be, ^{10}Be/Be and Be/B data. In the top panel, the contours show the 1σ total model uncertainties for BIG. In the bottom panel, the righthand side shows the distribution of values against a Gaussian with unit width (solid lines). 
Fig. 4.
Model prediction (top) and residuals (bottom) for ^{10}Be/^{9}Be (blue curves and symbols) and ^{10}Be/Be (orange curves and symbols). In the top panel, the 1σ contour corresponds to the total model uncertainties for SLIM (as calculated in Sect. 3.3). The model calculations are based on the parameters from the combined fit of all Li/C, Be/C, Be/B, ^{10}Be/^{9}Be, and ^{10}Be/Be data presented in Table 2 (PAMELA preliminary data excepted). 
3.4.2. Discussion
It is interesting to compare our results to those of previous analyses that considered either ACECRIS ^{10}Be/^{9}Be ratio or Be/B data. Based on the analysis of ^{10}Be/^{9}Be and other radioactive isotopes, and using a diffusion model with δ ≈ 0.3, the GALPROP team found L ∈ [1.5 − 6] kpc (Moskalenko et al. 2001), and later on, using an evolved Bayesian analysis, found L = 5.4 ± 1.4 kpc (Trotta et al. 2011). Actually, the halo size strongly correlates with the diffusion slope δ (Donato et al. 2002; Putze et al. 2010). Using 1D or 2D semianalytical models, our team found L ≈ 5 kpc (Donato et al. 2002) for δ ∼ 0.5, and later on, also in an evolved Bayesian context, found L ≈ 4 ± 1 kpc in a pure diffusion/reacceleration model (Putze et al. 2010). All these values are consistent with the constraints derived here using ACECRIS ^{10}Be/^{9}Be data only (see Fig. 2), that is L ∈ [3 − 8] kpc. Our uncertainties are larger than in previous studies, because we include here production crosssection uncertainties.
Comparatively, less studies focused on elemental ratios. Using HEAO3 Be/B data (and other ratios) in a semianalytical diffusion model with δ = 0.6, a rough range of L ∈ [2 − 4] kpc was found in Webber & Soutoul (1998). A much larger range was found in Putze et al. (2010), with L a few kpc only allowed for δ ≲ 0.3, growing to larger than 10 kpc for δ ≈ 0.5. This tension with ^{10}Be/^{9}Be data was attributed to crosssection uncertainties – their importance in limiting the prediction power of Be/B is detailed in Tomassetti (2015a). Using updated cross sections and propagating their uncertainties, we find here kpc based on AMS02 Be/B data, which is now compatible with values derived from ACECRIS ^{10}Be/^{9}Be data. Similar values were found by the GALPROP team, using both ACECRIS and AMS02 Be fluxes (without ^{10}Be/^{9}Be), with L = 4 ± 0.6 kpc (Boschini et al. 2020). This stronger constraint is somehow in line with the fact that combining ACECRIS and AMS02 data lead to tighter constraints (see Table 3), but as these authors do not use ^{10}Be/^{9}Be, it is most probably attributed to a larger extent to the fact that they do not propagate crosssection uncertainties. In any case, the value of L with a small error inferred in Boschini et al. (2020) is also in mild tension with the recent analysis of AMS02 Be data in Evoli et al. (2020), where a bestfit value of L = 7 kpc but a lower limit L ≳ 5 kpc is found in a semianalytical model similar to the one used here. Their central value, using total or statistical uncertainties only, is always enclosed within our 1σ confidence interval: firstly, the difference with our bestfit value is possibly due to their crosssection parametrisation (Evoli et al. 2019), as illustrated in Fig. 2 of Evoli et al. (2020); secondly, our broader confidence interval is certainly related to our treatment (full propagation) of crosssection uncertainties.
In conclusion, most past and present analyses of ^{10}Be/^{9}Be and Be/B show a preference for L ≈ 4 − 5 kpc, with a currently estimated 1σ uncertainty of about 50%. Several analyses have combined or compared these results to the constraints brought by other ratios of radioactive secondary species (Webber & Soutoul 1998; Moskalenko et al. 2001; Donato et al. 2002; Putze et al. 2010): similar halo sizes were observed, though with a large scatter. Forthcoming AMS02 data on other elemental ratios (e.g. Al/Mg, Cl/Ar...) will allow one to repeat these analyses. Another effect could impact these conclusions. In Donato et al. (2002) and Putze et al. (2010), the impact of a local subdensity, exponentially attenuating the flux of radioactive species, was inspected: slightly larger or much larger uncertainties were observed depending on the transport configuration used (Putze et al. 2010). However, this was calculated assuming a similar diffusion in the local bubble and in the rest of the disc and halo, which may be questionable. Recent γray observations (Abeysekara et al. 2017; Di Mauro et al. 2019) – interpreted as pockets of slow diffusion around pulsars (e.g. Profumo et al. 2018) –, and the indication that the local ISM properties are affected by several SN explosions a few Myr ago (Fields et al. 2019) provide enough motivation to revisit this issue in a future study.
4. Constraints on L from other probes
In this section, we discuss several independent constraints on L. We first review a series of constraints derived from multiwavelength observations of the Milky Way and other spiral galaxies, and then determine explicitly direct constraints induced by lowenergy secondary CR positrons.
4.1. Direct and indirect constraints from radio and gamma’s
Radio emission in the MHz to GHz band from our Galaxy has been used to constrain the magnetised halo thickness L (Bringmann et al. 2012; Orlando & Strong 2013; Di Bernardo et al. 2013, and Biswas & Gupta 2018). This band is dominated by the synchroton emission of the leptonic component of CR in the Galactic magnetic field. The Galactic latitude profile of the radio maps are sensitive to the vertical gradient of CR sourcing the emission, ∝1/L. Although relying on some assumptions (mostly on the magnetic field) these arguments tend to agree in excluding low values of L, typically obtaining L ≳ 2 kpc, with variations within a factor 2 depending on the analysis. They are somewhat less sensitive to large values of L, although upper limits in the range L ≲ 10 − 15 kpc have been derived. Further arguments based on radio observations such as rotation measurements of pulsars also yield results broadly consistent with these constraints (Di Bernardo et al. 2013).
A complementary indirect indication can be derived by looking at the radio emissions of other spiral galaxies seen almost edgeon, which present an average scaleheight of their synchrotron emission of about 1.8 ± 0.2 kpc (Krause 2014). This translates, under the hypothesis of energy equipartition between magnetic field and cosmic ray energy density, in the typical constraint on the magnetised halo size L ≳ 6.2 − 7.8 kpc (Beck 2015). A recent analysis of an indepth view of a spiral galaxy very similar to the Milky Way, NGC 891, reaches similar conclusions as for the vertical extension of the magnetic halo (Schmidt et al. 2019).
In principle, independent constraints can be derived from diffuse Galactic γray data (Stecker & Jones 1977). The advantage is that the bulk of the data in the FermiLAT energy range (GeV) comes from π^{0} decays of hadronic origin. Hence, the γray flux mostly probes the convolution of the hadronic CR flux with the gas density. For instance, let us consider a simple model where CR transport is purely diffusive within a slab (1Dmodel). Assuming the gas is exponentially distributed with typical thickness h, the photon flux originating from the Galactic zenith scales as:
Since we expect h/L ∼ 1%−10%, the factors entering the actual γray flux normalisation should be known to an unrealistically high precision for this method to provide meaningful constraints on L. However, additional inverse Compton contributions from leptonic CR component are also sensitive to L, and cannot be neglected. A global analysis of the diffuse γray emission measured by the FermiLAT satellite actually shows a systematic improvement of the statistical likelihood as L increases up to ∼10 kpc (Ackermann et al. 2012), an effect particularly important when fitting the emission at large longitudes. Conversely, the study of high and intermediatevelocity clouds at few kpcs away from the Galactic plane leads to the constraint L < 6 kpc (Tibaldo et al. 2015). Although this bound can be subject to variations (e.g. from the presence of unaccounted ionised gas in these clouds), the different trends between this constraint and the one obtained from radio observations of distant galaxies might be explained by the radio emission of leptons leaking out beyond the confinement volume. Also, the too shallow γray gradient suggested by FermiLAT data admits alternative explanations, for instance in terms of a physically motivated correlation of the diffusion properties with galactocentric distance (Evoli et al. 2012).
4.2. Constraints from e^{+}
Secondary positrons, usually believed to dominate the local CR positron flux at low energy, can also be used to derive lower limits on the halo size (Lavalle et al. 2014; Boudaud et al. 2017; Reinert & Winkler 2018). Their predicted abundance depends much less on the configuration of the Galactic magnetic field than the radio limits, and their theoretical uncertainties are better under control (see below). Secondary positrons may therefore significantly constrain propagation models, while still on the conservative side since their local flux is known to be dominated by primaries above a few GeV (e.g. Aharonian et al. 1995; Adriani et al. 2009; Hooper et al. 2009; Delahaye et al. 2010).
4.2.1. Positron flux scaling with K_{0} and L and calculation
The steadystate local positron density is expected to scale like the production rate times the minimal propagation timescale involved. An additional dilution factor comes from the fact that the production volume V_{p} is, in most cases, smaller than the diffusion volume V_{λ} (Bulanov et al. 1976). Sticking to a onedimensional picture, we have V_{p}/V_{λ} ≈ h/λ(E). In the latter expression, h is the halfheight of the thin disc (where the ISM gas is confined), and is the positron propagation length scale, featuring the pseudoenergy loss timescale .
The interesting regime to constrain L is, similarly to radioactive species, when h < λ < L, that is, when the vertical boundary does not affect the positron density and does not depend appreciably on K_{0}. This typically happens at energies ≲10 GeV, for which , and the e^{+} flux then scales as . Therefore, since energy loss parameters are fixed independently from the propagation model, the positron flux is a direct probe of the diffusion coefficient normalisation K_{0}. Since the B/C ratio provides constraints on the ratio K_{0}/L, positron measurements enable an indirect probe of L: The lower L in a B/Ccompatible model, the larger the secondary positron flux.
In practice, we calculate the positron flux according to the pinching method introduced in Boudaud et al. (2017) – see also Delahaye et al. (2009) for earlier attempts. For production, we consider incident and target species up to He only, and we take the crosssection parametrisation of Kamae et al. (2006), which accounts for the lowenergy hadronic resonances. Positron fluxes are then compared to AMS02 data (Aguilar et al. 2019a) assuming ϕ_{FF} = 650 MV, as estimated for the corresponding datataking period May 2011–November 2017 (see below for a discussion on Solar modulation level and its uncertainties). There are various sources of uncertainties in the calculation, and we try to list and quantify them below.
4.2.2. Error budget
We consider the uncertainties on the interstellar (IS) flux, and then comment on how the conclusions change for TopofAtmosphere (TOA) fluxes. We mostly focus on results at 1 GeV, because this is the typical energy where our analysis can draw constraints (see next subsection).
Firstly, we consider model uncertainties. Indeed, the method of calculation itself has some limitations. In the low and highenergy regimes, propagation is dominated by energy losses in the disc and in the halo respectively. The pinching method allows to calculate intermediate energies by pinching the halo losses in the disc (Boudaud et al. 2017), ensuring that both the limiting cases are recovered. Further comparisons against full numerical solutions should be carried out to definitively assess the accuracy of the method in the transition zone. Nevertheless, we have checked that the method is robust in the energy range used to define our limits below.
In a broader context, one could question the reliability of 1D models for consistency checks between nuclei and leptons, in the context of spatiallydependent distributions of sources and gas. Given the timescales of various transport parameters and energy losses, both these species originate from a few kpc away at GeV energies (Taillet & Maurin 2003; Maurin & Taillet 2003; Jóhannesson et al. 2016). It means that their production and losses are sensitive to kpcaverages over the gas density properties – of course, this is no longer the case for very high energy leptons. So as long as we focus on the multiGeV energy range, we do not expect strong differences due to gas inhomogeneities between nuclei and positrons, but as would be expected from more refined models, 1D model calculations are sensitive to the absolute value of the averaged gas density. This is at variance with the case of secondary radioactive species, discussed in Sect. 3.4, which could be very sensitive to the local ISM. However, similarly to radioactive nuclei, inhomogeneous spatial diffusion zones around CR sources (Abeysekara et al. 2017; Di Mauro et al. 2019), could also affect primary and secondary lepton spectra in different and very non trivial ways. These complications go beyond the scope of this analysis.
Secondly, we consider uncertainties from the choice of CR projectiles, targets, and cross sections. Any uncertainty on the CR fluxes and production cross sections directly impact the number of secondary positrons. For CR fluxes, we take demodulated proton and helium CR fluxes measured by AMS02 (Aguilar et al. 2015, 2017), and these data typically have uncertainties in the 3 − 10% range. Also, not accounting for the production from heavier CRs and species heavier than He in the ISM underestimates the secondary positron flux. Following the detailed analysis of Boudaud et al. (2020) carried out for antiprotons, we can estimate these effects to be ∼10% and ∼3% respectively.
Concerning production crosssection uncertainties, we recall that we use Kamae et al. (2006) parametrisation. More recent values exist (Kachelrieß et al. 2019) – they are calibrated on more recent collider data and include incident and target species up to Fe –, but they are only valid for incident nucleus energy greater than 4 GeV. Our analysis is mostly sensitive to the lowenergy part, so the latter model is only used to get a rough estimate of the theoretical uncertainties in the production cross sections. The secondary positron flux is 10–20% larger with Kachelrieß et al. (2019) than with Kamae et al. (2006) values.
Thirdly, we consider uncertainties from energy loss modelling. Positrons suffer different energy losses at high, intermediate and lowenergies. Above a few tens of GeV, inverse Compton and synchrotron radiation losses have the shortest timescales. Below a few MeV, ionisation and Coulomb losses dominate, and inbetween, Bremsstrahlung losses dominate. However, some of the positrons measured below 10 GeV have been produced at higher energy and at a more distant place. Hence, there is no onetoone correspondence between the hierarchy in energyloss timescales and impact on the positron flux. This motivates the detailed study of the impact of the various ingredient entering these losses.
Changing the interstellar radiation field (inverse Compton losses) and the magnetic fields (synchrotron losses) according to the values bracketed in Delahaye et al. (2010) have a ∼10% impact at 1 GeV. Coulomb losses on free electrons only dominate at very low energy, and we checked that the uncertainty on n_{e} = 0.033 ± 0.002 cm^{−3} (Nordgren et al. 1992) – see Yao et al. (2017) for an updated model – has a negligible impact on the positron flux (subpercent level) at 1 GeV.
Intermediate energies are dominated by Bremsstrahlung losses on the ISM gas. The same gas density is responsible for the production of secondary positrons. There is a further complication as this very gas density also directly impacts the determination of the transport coefficients, and this is discussed in the next paragraph. The gas density uncertainty is difficult to assess, and it can be probed for instance via the γray emissivity (e.g. Delahaye et al. 2011), especially in the light of FermiLAT data (Ackermann et al. 2012; Casandjian 2015; Acero et al. 2016). The impact of the choice of different density maps was recently investigated in Jóhannesson et al. (2018): between 2D and 3D gas models, variations by a factor two on column density were found – this factor mostly comes from using an outdated value of the Sun’s position in one gas component of the 2D model, and it certainly overestimates the uncertainty on the surface gas density (Σ_{ISM}). More realistically, the H_{I} spin temperature is already responsible for a 10% uncertainty in the gas density (Jóhannesson et al. 2018), and other sources of uncertainties come from the still debated X_{CO} conversion value (Remy et al. 2017) and the dark gas distribution (Grenier et al. 2005). For definiteness, we take a benchmark uncertainty of 50% on Σ_{ISM} in the following. If we consider together the impact on the production and Bremsstrahlung, there should be no net effect in the regime where Bremsstrahlung losses dominate: The gas density cancels out from the integral calculation, as it appears both in the numerator (production) and in the denominator (propagation) – we stress however, that if primary electrons or positrons are considered (no production), their flux now scales with the inverse of the gas density, as found in Cirelli et al. (2013). We checked indeed that there is no impact of the gas density for secondary positrons at 1 GeV. However, at higher energies, in a regime where other energy losses dominate, we also find, as expected, a direct scaling with the gas density.
Fourthly, we consider uncertainties from transport coefficient calibration. The positron flux depends on the transport parameters, calibrated on secondarytoprimary ratios. For instance, assuming a know Σ_{ISM}, the parameter K_{0} is determined with a ∼ ± 12% uncertainty (Weinrich et al. 2020). This leads to a halved uncertainty on the positron flux at very low energy (), but it fully propagates at 1 GeV. However, as discussed in Maurin et al. (2010), any uncertainty on Σ_{ISM} directly translates on the transport parameters K_{0}, V_{c}, and V_{A}, a behaviour also observed in Jóhannesson et al. (2018). A ±50% change on Σ_{ISM} would thus change K_{0} (and V_{A}) accordingly, which, combined with the impact on the positron production and Bremsstrahlung, leads to an overall ±26% uncertainty on secondary positrons at 1 GeV, this number varying with the energy (see above).
Fifthly, we consider uncertainties from Solar modulation. Solar modulating the calculated positron flux also brings uncertainties. Above 100 GeV, CR fluxes are mostly unmodified, and TOA CRs below 1 GeV mostly come from CRs at ∼GeV. As a result, the error budget at 1 GeV applies to lower energies as well for TOA positrons.
Modulation levels for most data in this analysis were taken from Ghelfi et al. (2017b), that is, from averages – over the appropriate CR data taking periods – of time series based on the analysis of neutron monitor data. The fact that postfit values in the LiBeB analyses (see the companion paper, Weinrich et al. 2020) were found to be consistent with the above assumed values further support this choice. In practice, however, Ghelfi et al. (2017b) time series do not extend after 2017. To derive the positron modulation level ϕ_{FF} = 650 MV, we relied on Oulu times series^{6}, derived from Usoskin et al. (2005), and rescaled according to ϕ_{Ghelfi17}(t)≈ϕ_{Usoskin05}(t)+100 MV, as found in Ghelfi et al. (2017b). The overall uncertainties on reconstructed modulation levels from neutron monitors are ±100 MV (Ghelfi et al. 2017b). Those obtained from directly fitting TOA (preAMS) H and He data are in the ±30 MV range (Ghelfi et al. 2016). We choose a very conservative approach below, and for 1 GeV secondary positrons, the ±100 MV (±15%) uncertainty translates into a ∓50% uncertainty. This makes modulation the dominant source of uncertainty for the positron flux calculation.
To conclude this section, we summarise our results on the uncertainties. Although the above analysis does not reach the level of refinement developed for CR nuclei analyses (Derome et al. 2019), we now have a quantitative grasp on the uncertainties on the secondary positron calculations. They are gathered in Table 4, where we also provide a finer view of these uncertainties at three energies (10 MeV, 1 GeV, and 100 GeV). At 1 GeV, which is the energy that is relevant for the analysis below, uncertainties from Solar modulation are the dominant effect, followed by those on production and Σ_{ISM}. Regarding production, we nevertheless stress that the assumptions we make are conservative to derive lower limits on L; they underestimate the positron flux by ∼20 − 30% (10–20% from cross sections and 10% from unaccounted for heavy CR projectiles).
Error budget on the calculation of secondary positrons.
4.2.3. Constraints on L
To check the compatibility with the limits set by AMS02 Be/B data, we carried out calculations for several L values. The latter are taken inside their allowed range (upperhalf of Table 3). To ensure a consistent calculation, the transport parameters are rescaled with L according to Eq. (A.1) and values in Table A.1 (see Appendix A).
We set conservative limits on L by excluding propagation setups for which the prediction of the secondary positron flux exceeds the AMS02 measurements (Aguilar et al. 2019a) by more than 3σ in a single bin. In practice, for all our configurations, the lowestenergy AMS02 data point is the one setting constraints.
In a first step, we check the consistency of the positron flux with the Be/B constraint. In Fig. 5, we report our predictions for the secondary positron fluxes (rescaled by E^{3}) for the BIG, SLIM, and QUAINT propagation models (left, middle, and right panel, respectively) along with the AMS02 data (Aguilar et al. 2019a). These predictions are derived using the bestfit parameters inferred from the combined Li/C, Be/C, and B/C analysis performed in Weinrich et al. (2020), already extensively discussed throughout the paper. For illustration, we have taken the bestfit values obtained for L from the AMS02 Be/B data only (see Table 3). In each panel, the solid curve corresponds to the bestfit value for L, bracketed by its 1σ statistical uncertainty (dashed curve for the lower deviation, and dotdashed curve for the upper deviation).
Fig. 5.
Secondary positron flux predictions using the bestfit transport parameters of the BIG (left panel), SLIM (middle panel), and QUAINT (right panel) propagation models, as inferred from the AMS02 combined Li/C, Be/B, and B/C analysis. The three lines correspond to TOA fluxes, modulated at 650 MV, for the bestfit (solid line) and 1σ upper (dashdotted) and lower (dashed) limits on L (see Table 3) – AMS02 data are from Aguilar et al. (2019a). 
From Fig. 5, we see that the positron constraint on L is only relevant for the QUAINT propagation model. The latter has an effective Alfvèn speed V_{A} ∼ 40 km s^{−1} (see Table A.1) which sets the amplitude of reacceleration^{7}. For the BIG and SLIM models (left and middle panels), both devoid of reacceleration, the nominal predictions undershoot the data. The lowenergy behaviour in these models is furthermore exacerbated by the lowrigidity break in the diffusion coefficient around 3 − 4 GV (Weinrich et al. 2020): it leads to a lower secondary positron flux at low rigidities and hence a decreased constraining power on L. Given the uncertainties on the calculation, only QUAINT can overshoot the data, whereas BIG and SLIM cannot for L values constrained by Be/B.
In a second step, we derive the upperlimit on L in the model QUAINT. Our nominal 3σ constraint for QUAINT is L ≥ 5.54 kpc. However, this number is very sensitive to the model uncertainties. For illustration, we show in Fig. 6 the impact of Solar modulation uncertainties, varying the Fisk potential by a generous ±100 MV (Ghelfi et al. 2017b). The upper value of the modulation, which minimises the modulated positron flux, should be viewed as conservative, weakening the limit to L ≥ 2.9 kpc. The lower value corresponds to a more aggressive constraint (for a lower solar activity) L ≥ 11.3 kpc. These limits do not account for subleading uncertainties discussed above. Without a better handle on the uncertainties, it is difficult to firmly conclude. Overall, we note that the positron constraint on L for the QUAINT model is in agreement with those derived from the Be/B and ^{10}Be/^{9}Be analysis, while being independent from the latter.
Fig. 6.
Secondary positron flux predictions using the bestfit transport parameters of the QUAINT propagation model (also inferred from the AMS02 combined Li/C, Be/B, and B/C analysis). We show the predictions associated with the 3σ limits on L assuming different solar modulation strengths. 
We have only discussed the case of overshooting, which provides a clear situation for excluding part of the parameter space. Given that in the BIG and SLIM models the nominal predictions undershoot the data, one may wonder if their viability is questioned by the positron data. We remark that the various sources of uncertainties (modulation, production, surface gas density), estimated at the ≲50% level at most, seem unable to fully account for the factor of ∼2 mismatch. However, primary positrons, already necessary to explain the highenergy positron fraction in preAMS02 studies, might make up a significant fraction of the positron budget also at low energy – for example, the absence of primary positrons at low energy was raised as an important issue for astrophysical sources like pulsar wind nebulae in Blasi & Amato (2011). Therefore, discussing further the consistency of any propagation model (not only BIG and SLIM) with the positron data should rely on analyses also including primary positrons. This goes beyond the scope of this paper.
5. Conclusions
In the context of recent highprecision AMS02 data, we have revisited the constraints set on the halo size of the Galaxy from radioactive species and positron fluxes.
Using AMS02 Be/B data we find kpc at 1σ, in agreement with Boschini et al. (2020) (but with a larger error) and less than one sigma away from the results of Evoli et al. (2020). Our result holds for several transport configurations (Derome et al. 2019; Weinrich et al. 2020), namely BIG (reacceleration and lowrigidity break), SLIM (pure diffusion with lowrigidity break), and QUAINT (reacceleration and diffusion upturn in the nonrelativistic regime). The constraints are tighter (factor ∼2 reduction in errors) and move to a lower value of L (by 0.3–0.7 kpc) when lowenergy ^{10}Be/^{9}Be data are considered, but this tightening may be related to the fact that ISOMAX data (Hams et al. 2004) prefer a smaller halo size L ≈ 3 kpc and might be indicative of a tension in the data. With the recent release of ^{3}He and ^{4}He data (Aguilar et al. 2019b), AMS02 demonstrated its capabilities for measuring isotopic fluxes. Separating Be isotopes will certainly be even more challenging, but AMS02 will provide a unique picture in the few GeV/n regime, where only ISOMAX and preliminary PAMELA data (Bogomolov & Vasilyev 2019) are available for now, and shed some light on the mutual consistency of these datasets. The balloonborne HELIX, should also provide a complementary view. The instrument is scheduled for a longduration balloon flight in 2020/21, and is expected to achieve a 10% statistical error on ^{10}Be/^{9}Be in the 0.1–10 GeV/n range (Park et al. 2019).
We have also performed a detailed analysis of the modelling uncertainties for both the Be/B and ^{10}Be/^{9}Be ratios to understand whether better data could help improving the estimation on L in the future. Whereas Be/B is maximally sensitive to L at a few tens of GV, ^{10}Be/^{9}Be is sensitive to L over a much larger range (from 100 MeV/n to tens of GeV/n). In terms of possible improvements, Be/B data are already limited by systematics, whereas this is not yet the case for ^{10}Be/^{9}Be. For these reasons, ^{10}Be/^{9}Be seems to be the best target for future experiments. However, in both cases, production crosssections uncertainties dominate the modelling error budget and are already at the level of data uncertainties. Future improvements on L will thus not be possible without improving nuclear data. An alternative strategy to mitigate these uncertainties would be to combine data from different CR clocks (e.g. Al/Mg and Cl/Ar).
In a broader context of multiwavelength and multimessenger observations, we also discussed the constraints set by synchrotron radio (e.g. Di Bernardo et al. 2013) and diffuse γray emissions (e.g. Tibaldo et al. 2015) in the Milkyway (and in some cases in other galaxies). The various observations lead to lower or upper limits, defining a broad range L ∈ [2, 10] kpc compatible with results from radioactive CR nuclei. We also updated the constraints set by positrons (Lavalle et al. 2014). Since BIG and SLIM configurations undershoot the data, the constraints are only significant for the QUAINT model, leading to L ≥ 2.9 − 11.3 kpc depending on the solar modulation, with a nominal central value L ≥ 5.5 kpc; these numbers could shift upwards or downwards depending on uncertainties on the production cross section and the gas surface density. Within the errors, these constraints are also consistent with the ones derived in the main analysis.
While our conclusions appear rather robust within the framework of this analysis, one should keep in mind that these bounds might be altered in presence of inhomogeneities in the local gas density (for radioactive species) and on the diffusion coefficient (for both radioactive species and positrons). These extensions represent interesting and motivated subjects for future studies.
There is a subtlety for B/C, as it contains ^{10}B having decayed from ^{10}Be. Depending how scaling relations are derived, they absorb or not the L dependence of this decayed fraction. We use the theorybased scaling relations to study the sensitivity of B/C to L, but the fitdriven ones for the remaining ratios (see Appendix A).
Configurations having more free parameters (QUAINT and BIG) would provide larger uncertainties; see Fig. 4 in Weinrich et al. (2020).
We follow the NSS prescription (Derome et al. 2019; Weinrich et al. 2020), where we vary the crosssection normalisation and lowenergy behaviour (powerlaw slope) of the dominant production channel for each Be isotope. We use a dispersion σ_{Slope} = 0.15 for these reactions, and σ_{Norm} = 0.25, 0.20, and 0.15 for ^{12}C + H→^{10}Be, ^{12}C + H→^{9}Be, and ^{16}O + H→^{7}Be respectively.
We stress that preAMS02 fit to B/C data (e.g. Maurin et al. 2001) had larger V_{A}. It was found that strong reacceleration gave rise to a prominent bump around 1 GeV in the predicted positron flux, which then easily overshoots the data, especially below a few GeV (Delahaye et al. 2009; Lavalle et al. 2014; Boudaud et al. 2017). However, with milder reacceleration here, we see in the right panel that the flux predictions associated with QUAINT are not in that strong excess with respect to the data, and do not feature any significant bump.
Acknowledgments
We thank A. Marcowith for discussions. This work has been supported by Univ. de Savoie, appel à projets: Diffusion from Galactic HighEnergy Sources to the Earth (DIGHESE), by the Programme National des Hautes Energies of CNRS/INSU with INP and IN2P3, cofunded by CEA and CNES, and by Villum Fonden under project no. 18994. This work is partly supported by the ANR project ANR18CE310006, the national CNRSINSU programmes PNHE and PNCG, and European Union’s Horizon 2020 research and innovation programme under the Marie SkłodowskaCurie grant agreements N° 690575 and N° 674896.
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Appendix A: Scaling relations with L
In Sect. 3.2, we study the sensitivity to L of CR data combinations involving a radioactive species. To do so, the transport parameters obtained from the study of B/C must be known for any L. In Sect. 4.2, we test whether secondary positrons overshoot the data given for various L, given the constraints set from secondarytoprimary ratios and radioactive species. This is a similar but slightly different question than the previous one.
For each model (BIG, SLIM, and QUAINT), we repeat the transport parameter fitting procedure described in Weinrich et al. (2020) at fixed L for several values of L, for either the AMS02 B/C ratio only, or the combined AMS02 Li/C, Be/B, and B/C data. We show for the latter the resulting bestfit transport parameters (symbols) in Fig. A.1. We find that a simple power law is enough to capture the dependence with L (lines in Fig. A.1),
Fig. A.1.
Transport parameters dependence on the halo size L. Colourcoded symbols correspond to the bestfit values on the combined analysis of Li/C, Be/B, and B/C data for different models (BIG, SLIM, and QUAINT). The lines show, for each model and transport parameter, powerlaw fits to these data whose values are reported in the righthand side of Table A.1. 
and we report the bestfit A and B values found for each transport parameter (δ, K_{0}, etc.) in Table A.1 for the two different cases.
Scaling coefficients for the transport parameters, see Eq. (A.1), with B the powerlaw slope and A the transport parameter value for L = 5 kpc.
A.1. Powerlaw behaviour of the rescaling
A few comments are in order: the powerlaw indices, B, for the B/C analysis are very similar to the ones found in Fig. 5 of Putze et al. (2010), that is 1.06 for K_{0} and 0.53 for V_{A} (from the analysis of older B/C data). As argued in Putze et al. (2010), a secondarytoprimary ratio provides constraints degenerated with K_{0}/L and ratios. These small differences (1.06 vs. 1 and 0.53 vs. 0.5) originate in the small fraction of ^{10}B (≲10%) coming from decayed ^{10}Be, which is sensitive to L. Figure A.2 illustrates this sensitivity on the B/C ratio. If the theoretical rescaling is applied (grey lines), the curves with varying L are either above or below the reference, converging to the reference at high rigidity as ^{10}Be halflife grows and ^{10}Be behaves as a stable secondary species. At variance, when the above rescaling is applied, the variation with L is absorbed in the fit, and the highrigidity limit no longer goes to zero. The overall difference is at the few percent level.
Fig. A.2.
Relative variation of B/C ratio (with respect to a reference taken at L = 5 kpc), as a function of rigidity. Thin to thickdashed lines correspond to calculations for various L using either (i) the “theoretical” rescaling which enforce the same level of production for stable secondary species (grey lines), or (ii) the rescaling based on a refit of B/C data for each L as described in the text (red lines). 
A.2. Difference between B/C and combined analysis results
A similar scaling remains for the combined analysis (Li/C, Be/B, B/C), but with some differences, that were all highlighted in Weinrich et al. (2020). First, the diffusion coefficient normalisation K_{0} is smaller than in the B/C only case, and the difference is related to partial degeneracies with production cross sections, that are lifted in the combined analysis of elements. The bestfit values are consistent, though slightly different, from those in Weinrich et al. (2020), and this is mostly attributed to the fact that we fit here the combination Be/B instead of Be/C. Second, whereas for the B/C case BIG parameters are very similar to QUAINT ones, for the combined analysis the BIG parameters are closer to SLIM ones. Last, the lowenergy parameters δ_{l} and η are also different from the B/C analysis only, and correspond to more marked lowrigidity break of the diffusion coefficient; we find here that the latter parameters have almost no dependence on L.
Appendix B: L from several ^{10}Be/^{9}Be datasets
The ^{10}Be/^{9}Be ratio has been measured by different experiments in different energy ranges. To better grasp the respective weight of the data in the final constraint (see Sect. 4), we show the different contributions to the fit. We focus on SLIM, but similar results are observed for BIG and QUAINT.
In Fig. B.1, we show the combined analysis of Li/C, B/C, and different datasets for ^{10}Be/^{9}Be. Fits on lowenergy data only are completely driven by ACE data (compare first and second column). They lead to larger L values than the higherenergy ISOMAX data (third column). The latter have larger uncertainties than lowenergy data (see Fig. 3), so that lowenergy data drive the combined constraint (fourth column).
Fig. B.1.
Constraints on L (top panel) for combined fit of AMS02 Li/C and B/C data, and various datasets for ^{10}Be/^{9}Be. From left to right: lowenergy ACE data, all lowenergy data combined (ACE, IMP7&8, ISEE3, UlyssesHET, and Voyager), higherenergy GeV/n ISOMAX (Hams et al. 2004), LE and ISOMAX combined (fourth column), preliminary PAMELA data (Bogomolov & Vasilyev 2019), and all combined (last column). 
For completeness, we also show the constraints from the preliminary PAMELA data (Bogomolov & Vasilyev 2019). They cover a energy range of 0.1 to 2 GeV/n, combining two independent analyses with the TOF and calorimeter, with a ∼20% uncertainty. They give the best constraints on L (nexttolast column) with smaller error bars, in line with the expectations discussed of Sect. 3.2. Most importantly, they point towards smaller values of the halo size than the other datasets, with L ∼ 2 − 3 pc. If all ^{10}Be/^{9}Be data are combined, the PAMELA data drive the fit (last column).
AMS02 data will cover a similar range as PAMELA preliminary data. With probably slightly smaller uncertainties, they are expected to provide similar or slightly better precision on L. Because of the tension between PAMELA and lowerenergy data, the results from both experiments are crucial to be able to obtain robust results on the central value of the halo size.
Appendix C: Constraints from ^{7}Be/(^{9}Be+^{10}Be)
Owing to the difficulty to achieve isotopic separation in CR experiments, data analyses often start with the most favourable configuration. The Be element is made of ^{7}Be, ^{9}Be, and ^{10}Be. Taking advantage of the ΔA = 2 mass separation between A = 7 and A = 9, 10 isotopes, the ^{7}Be/(^{9}Be+^{10}Be) ratio is experimentally a favourable configuration to analyse.
This ratio has been published by AMS01 (Aguilar et al. 2011) and also by PAMELA (Menn et al. 2018). Following similar steps as the analysis presented in Sect. 3.2, we show in Fig. C.1 prospective limits on L that can be set from using the ^{7}Be/(^{9}Be+^{10}Be) ratio. Because of the subdominant abundance of ^{10}Be in the denominator, this ratio is as sensitive to L as the Be/B ratio. But whereas AMS02 achieves a few percent precision on Be/B, current experiment are at ∼15 − 20% precision for isotopic ratios. For this reason, we conclude that ^{7}Be/(^{9}Be+^{10}Be) is not a competitive target to fix L with current data or with future experiments.
Fig. C.1.
Same as Fig. 1, but for the variation of ^{7}Be/(^{9}Be+^{10}Be) with L (pinkish dashed lines), compared to various model uncertainties (contours) and current data uncertainties (hatched boxes); data from AMS01 (Aguilar et al. 2011) and PAMELA preliminary analysis (Menn et al. 2018). 
All Tables
Experiments with their datataking periods and associated expected Solar modulation level.
Scaling coefficients for the transport parameters, see Eq. (A.1), with B the powerlaw slope and A the transport parameter value for L = 5 kpc.
All Figures
Fig. 1.
Relative variation of B/C, Be/B, ^{10}Be/Be, and ^{10}Be/^{9}Be ratios (from top to bottom) as a function of rigidity (upper panels) or kinetic energy per nucleon (lower panels). All the panels compare three different effects: (i) the thin to thickdashed lines correspond, at constant grammage, to the impact of L on the calculated ratios (the reference is L = 5 kpc); (ii) hatched boxes show the data relative uncertainties over the data energy coverage (AMS02 data in the two top panels) or for central energy points only (ACE, IMP7&8, ISEE3, ISOMAX, Ulysses, and Voyager1&2 data in the two bottom panels) – see Sect. 3.1 and Table 2 for data references; (iii) the remaining envelopes show 68% CLs on model calculations, including transport, cross sections, and modulation uncertainties separately or combined (“total”). 

In the text 
Fig. 2.
Bestfit halo size and asymmetric uncertainties (from MINOS) for the configurations BIG, SLIM, and QUAINT. From left to right, several data combinations are used. The first column (“Base”) involves AMS02 and lowenergy B/C and Li/C data. The second column is the constraint set from AMS02 data only (Li/C, Be/B, and B/C). The remaining columns combine “Base” data (from first column) to ^{10}Be/Be lowenergy data (third column), to ^{10}Be/^{9}Be low and intermediateenergy (nexttolast column), or combine all the previous cases (last column). 

In the text 
Fig. 3.
Model prediction (top), residuals (centre), and score (bottom) for Be/B based on the bestfit parameters to B/C, Li/C, ^{10}Be/^{9}Be, ^{10}Be/Be and Be/B data. In the top panel, the contours show the 1σ total model uncertainties for BIG. In the bottom panel, the righthand side shows the distribution of values against a Gaussian with unit width (solid lines). 

In the text 
Fig. 4.
Model prediction (top) and residuals (bottom) for ^{10}Be/^{9}Be (blue curves and symbols) and ^{10}Be/Be (orange curves and symbols). In the top panel, the 1σ contour corresponds to the total model uncertainties for SLIM (as calculated in Sect. 3.3). The model calculations are based on the parameters from the combined fit of all Li/C, Be/C, Be/B, ^{10}Be/^{9}Be, and ^{10}Be/Be data presented in Table 2 (PAMELA preliminary data excepted). 

In the text 
Fig. 5.
Secondary positron flux predictions using the bestfit transport parameters of the BIG (left panel), SLIM (middle panel), and QUAINT (right panel) propagation models, as inferred from the AMS02 combined Li/C, Be/B, and B/C analysis. The three lines correspond to TOA fluxes, modulated at 650 MV, for the bestfit (solid line) and 1σ upper (dashdotted) and lower (dashed) limits on L (see Table 3) – AMS02 data are from Aguilar et al. (2019a). 

In the text 
Fig. 6.
Secondary positron flux predictions using the bestfit transport parameters of the QUAINT propagation model (also inferred from the AMS02 combined Li/C, Be/B, and B/C analysis). We show the predictions associated with the 3σ limits on L assuming different solar modulation strengths. 

In the text 
Fig. A.1.
Transport parameters dependence on the halo size L. Colourcoded symbols correspond to the bestfit values on the combined analysis of Li/C, Be/B, and B/C data for different models (BIG, SLIM, and QUAINT). The lines show, for each model and transport parameter, powerlaw fits to these data whose values are reported in the righthand side of Table A.1. 

In the text 
Fig. A.2.
Relative variation of B/C ratio (with respect to a reference taken at L = 5 kpc), as a function of rigidity. Thin to thickdashed lines correspond to calculations for various L using either (i) the “theoretical” rescaling which enforce the same level of production for stable secondary species (grey lines), or (ii) the rescaling based on a refit of B/C data for each L as described in the text (red lines). 

In the text 
Fig. B.1.
Constraints on L (top panel) for combined fit of AMS02 Li/C and B/C data, and various datasets for ^{10}Be/^{9}Be. From left to right: lowenergy ACE data, all lowenergy data combined (ACE, IMP7&8, ISEE3, UlyssesHET, and Voyager), higherenergy GeV/n ISOMAX (Hams et al. 2004), LE and ISOMAX combined (fourth column), preliminary PAMELA data (Bogomolov & Vasilyev 2019), and all combined (last column). 

In the text 
Fig. C.1.
Same as Fig. 1, but for the variation of ^{7}Be/(^{9}Be+^{10}Be) with L (pinkish dashed lines), compared to various model uncertainties (contours) and current data uncertainties (hatched boxes); data from AMS01 (Aguilar et al. 2011) and PAMELA preliminary analysis (Menn et al. 2018). 

In the text 
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