Implications of current nuclear cross sections on secondary cosmic rays with the upcoming DRAGON2 code
Pedro de la Torre Luque, Mario Nicola Mazziotta, Francesco Loparco, Fabio Gargano, Davide Serini
PPrepared for submission to JCAP
Implications of current nuclearcross sections on secondary cosmicrays with the upcoming
DRAGON2 code
P. De La Torre Luque a,b
M. N. Mazziotta a F. Loparco a,b
F. Gargano a D. Serini a,b a Istituto Nazionale di Fisica Nucleare, Sezione di Bari, via Orabona 4, I-70126 Bari, Italy b Dipartimento di Fisica “M. Merlin" dell’Università e del Politecnico di Bari, via Amendola173, I-70126 Bari, ItalyE-mail: [email protected], [email protected]
Abstract.
Current measurements of cosmic-ray fluxes have reached unprecedented accuracythanks to the new generation of experiments, and in particular the AMS-02 mission. At thesame time, significant progress has been made in the propagation models of galactic cosmicrays. These models include several propagation parameters, which are usually inferred fromthe ratios of secondary to primary cosmic rays, and which depend on the cross sectionsdescribing the collisions among the various species of cosmic-ray nuclei. At present, ourknowledge of these cross sections in the energy range where cosmic-ray interactions occur islimited, and this is a source of uncertainties in the predicted fluxes of secondary cosmic-raynuclei. In this work we study the impact of the cross section uncertainties on the fluxes oflight secondary nuclei (Li, Be, B) using a preliminary version of the upcoming
DRAGON2 code.We first present a detailed comparison of the secondary fluxes computed by implementingdifferent parametrizations for the network of spallation cross sections. Then, we propose forthe first time the use of secondary-over-secondary cosmic-ray flux ratios as a tool to investigatethe consistency of cross sections models and give insight of the overall uncertainties comingfrom the cross sections parametrizations. We show that the uncertainties inferred from thecross section data are enough to explain the discrepancies in the Be and Li fluxes with respectto the AMS-02 data, with no need of a primary component in their spectra. In addition, weshow that the fluxes of B, Be and Li can be simultaneously reproduced by rescaling their crosssections within the experimental uncertainty. Finally, we also revisit the diffusive estimation ofthe halo size, obtaining good agreement with previous works and a best fit value of . ± from the most updated cross sections parametrizations. Keywords:
Cosmic rays, diffusion, propagation, spallation, cross sections, magnetic halo a r X i v : . [ a s t r o - ph . H E ] J a n ontents Cross sections for the main channels
Primary spectra
Theory on secondary-over-secondary flux ratios Propagation of Galactic cosmic rays (CRs) is governed by the magnetic collisionlessinteractions they suffer with the interstellar plasma waves. These interactions make themwander inside the Galaxy following a random walk that can be studied as a diffusivemotion [1, 2]. From the evaluation of the amount of matter traversed by CRs, we knowthat their diffusion is not limited to the disc of the Galaxy [3], but extends up to a few kpc in the so-called magnetic halo.CRs are accelerated inside astrophysical sources (mainly supernova remnants) with anenergy spectrum typically following a power law of the form Q ( E ) = KE − γ . The parameter γ is known as the spectral index. Acceleration of CRs at sources is explained by the diffusiveshock acceleration (DSA) model [4–7], which is based on the first order Fermi mechanism,and correctly reproduces the observed data. CRs produced at these sources are known asprimary CRs and their spectra are usually called “injection spectra” [8].Nevertheless, the spectra of CRs detected at Earth are modified due to their diffusivepropagation, resulting in power laws of the form J ( E ) ∝ E − ( γ + δ ) , where the diffusionparameter δ is related to the time spent by CRs in the Galaxy τ prop ( E ) (propagation time).On the other hand, during their journey, galactic CRs can eventually interact with theinterstellar gas (which consists mainly of hydrogen with about a of helium and traces ofheavier elements [9]) producing lighter nuclei (secondary CRs) that otherwise would be foundin tiny proportions (mainly boron, beryllium, lithium and the so-called sub-Fe nuclei), sincethey are hardly produced in stellar fusion or other thermonuclear processes [10].The spectra of secondary CRs produced from these interactions (spallation reactions)have the form: J k ( E ) ∝ ρ g (cid:80) i J i ( E ) σ i → k ( E ) . Here ρ g is the density of the interstellar medium– 1 –ISM) and σ i → k ( E ) is the inclusive cross section of production of the CR species k from theinteraction of the CR species i with the ISM.The study of secondary CRs can therefore provide valuable information about theinteractions that CRs suffer during their journey. The relevant quantity used to describeCR interactions is the “grammage” X ( E ) = ρ g l ( E ) ∝ ρ g τ prop ( E ) , related to the diffusioncoefficient used to describe the CR diffusive motion ( D ∝ τ − prop ). Here l ( E ) is the effectivelength traversed by CRs, which depends on the interaction cross sections and on the particlekinetic energy per nucleon E .Nonetheless, the current experimental data on inclusive cross sections are based on afew data points that hardly reach energies of tens of GeV / n . These cross sections are crucial,since the best way to constrain the propagation parameters (mainly the diffusion coefficient)is by means of the secondary-over-primary CR flux ratios [11, 12].In this work, we have implemented different publicly available cross sections data setsin the preliminary version of the DRAGON2 cosmic-ray propagation code [13, 14] to study howthe choice of cross sections affects the predictions on the fluxes of the light secondary CRsboron, beryllium and lithium.In section 2 the set-up of the simulations is illustrated. The results of the simulations arepresented in section 3, where the fluxes of these secondary species computed with differentcross sections parametrizations are shown, and some relevant conclusions are drawn. Theeffects of variations on the cross sections are then tested in detail by studying secondary-over-secondary ratios in section 4. These ratios are extremely sensitive to the cross sectionsand, therefore, represent a useful tool to test the validity of the cross sections used in thepropagation code. This allows us to evaluate the uncertainties in the predicted fluxes of B,Be and Li associated to their production cross sections. In this section we also perform asimultaneous fit of the high-energy part of the flux ratios among secondary B, Be and Li byadjusting their production cross sections within the experimental uncertainties. In addition,these secondary-over-secondary ratios will be used to look for possible imprints of any extraprimary source producing them. Then, in section 5 the effect of the halo size on the ratiosinvolving Be is explored with the different cross section models. In this section we also geta more robust estimate of the value of the halo size. Finally, the conclusions are drawn insection 6.
Simulations of CR propagation are performed using the 2-dimensional model of the Galaxyoffered by the DRAGON code [15] , where cylindrical symmetry is assumed. The Galaxyis described as a thin disc with radius R ∼
20 kpc and height h ∼
100 pc with the Sun onthe Galactic plane at a distance of . from the center. The disc is surrounded by thehalo, which is a cylinder with the same radius as the disc and with height H of a few kpc (two-zone model). Gas and CR sources are distributed within the disc. An illustration of themodel is shown in Fig. 1.A diffusion-reacceleration model is used in the CR propagation, without implicitadvection. These models have successfully reproduced the shape of the ratios of secondariesover primaries fluxes at low energy, as for example in refs. [16–18]. The inclusion of stochasticreacceleration seems to be necessary to naturally explain the shapes of these ratios. The software can be downloaded from https://github.com/cosmicrays/DRAGON – 2 – igure 1 . Scheme of the 2D model used for the Galaxy structure. Taken from the lecture https://w3.iihe.ac.be/~aguilar/PHYS-467/PA3.html
The general formulae describing this propagation model are given by the following setof equations: (cid:126) ∇ · ( (cid:126)J i − (cid:126)v ω N i ) + ∂∂p (cid:20) p D pp ∂∂p (cid:18) N i p (cid:19)(cid:21) = Q i + ∂∂p (cid:104) ˙ pN i − p (cid:16) (cid:126) ∇ · (cid:126)v ω N i (cid:17)(cid:105) − N i τ fi + (cid:88) Γ sj → i ( N j ) − N i τ ri + (cid:88) N j τ rj → i (2.1)In the previous equations (cid:126)J i indicates the CR diffusive flux of the i -th species and N i is the density per unit momentum. The term (cid:126)v ω represents the advection speed, whichwe assume to be null. The second term in the left-hand side accounts for the diffusion inmomentum space. The first term in the right-hand side, Q i , represents the distributionand energy spectra of particle sources (injection spectra); the second term describes themomentum losses; finally, the remaining four terms describe losses and gains due to decays andfragmentations. These equations are numerically solved with a new version of the DRAGONcode . More information about the code can be found on refs. [13] and [14].The flux terms (cid:126)J i are related to the densities N i by Fick’s law, (cid:126)J i = − (cid:126)(cid:126)D (cid:126) ∇ N i , where (cid:126)(cid:126)D is the spatial diffusion tensor [13]. In this study, we assume that the spatial diffusioncoefficient parallel to the regular magnetic field lines vanishes because of the azimuthalsymmetry adopted for the Galaxy structure [15] and we used the most general form of thespatial diffusion coefficient perpendicular to the magnetic field lines, without including anyspatial dependence (i.e. homogeneous spatial coefficient): D ( R ) = D β η (cid:18) RR (cid:19) δ (2.2)Here β = v/c , R is the rigidity of the particle, D is the constant diffusion coefficient at thereference rigidity R (it is set to ). The η parameter describes the complex physicaleffects that may play a major role at low energy, as the dissipation of Alfven waves [19].These phenomenological formulas for the diffusion coefficient have been widely employed,usually setting η = 0 or η = 1 [16, 20]. In this work η will be treated as a free parameter, Now available in https://github.com/cosmicrays/DRAGON2-Beta_version – 3 –ince also negative values of this parameter seem to be allowed although they are yetunexplored [19].Finally, the relation between the spatial diffusion coefficient D ( R ) and the diffusioncoefficient in momentum space D pp depends on the Alfven velocity v A and, following refs. [21]and [22], it is given by: D pp = 43 1 δ (4 − δ )(4 − δ ) v A p D ( R ) (2.3)Hereafter, when speaking of “diffusion parameters”, we will refer to the set of parametersin the spatial diffusion coefficient and to the Alfven velocity.The injection spectra of primary nuclei are parametrized with a doubly broken powerlaw: Q = K × (cid:16) RR (cid:17) γ for R < R K × (cid:16) RR (cid:17) γ for R < R < R K × (cid:16) RR (cid:17) γ for R > R (2.4)where Q is the differential energy flux in units of m − s − sr − GV − , R , are the rigiditybreaks, γ , , are the logarithmic slopes below and above each break and the parameters K i set the normalization of the flux. The low-energy break was set to R = 8 GV for all nuclei,while the high-energy break was set to R = 335 GV for protons and R = 200 GV forthe heavier nuclei. In the present work we injected H, He, C, N, O, Ne, Mg and Si as primary nuclei with the spatial distribution of sources following the model in ref.[23].The injection spectra are tuned such that these primary CRs reproduce the AMS-02 spectra(Figure 11).In the last years CR experiments have reached unprecedented precision in the fluxmeasurements of CRs, making possible the study of several features unexplored in the past.The accuracy showed in the last experimental results of the AMS-02 collaboration is of theorder of − (for the main nuclei involved in the creation of light secondary CRs, as C,N and O). However, the uncertainties on the cross sections reach levels of − in somechannels (see [24]), which makes clear the need of new cross sections data with better accuracyto improve the precision of the predictions on CR fluxes. Even the exact determination ofthe uncertainties is complicated for most of the channels, since sometimes data from differentexperiments are difficult to reconcile. In addition, there are many channels with no data orjust with a few data at low energies (below
10 GeV / n ), which makes the parametrizations athigh energies not straightforward at all. Also the so-called ghost-nuclei (i.e. those unstablenuclei whose lifetime is so small that they are not propagated and their contribution is directlyadded up to their daughter nuclei) have a sizeable effect in the estimation of the productionof secondary nuclei.Currently there are several parametrizations of the nuclei cross sections publiclyavailable. The first parametrizations came from the early measurements of B. Webber,published in a series of papers during the 80s (see [25] and references therein) that ultimatelylead to the semi-empirical WNEW code [26] with a last update in 2003 [27]. Then, therewere important efforts to expand the known experimental measurements to other channels,turning out in the semi-empirical parametrizations by Silberberg and Tsao [28–30] thatconverged in the YIELDX code [31, 32]. More recently, the GALPROP team developed a https://galprop.standford.edu – 4 –et of routines combining semi-empirical formulae [33–35]. Nevertheless, although theseroutines have remained the state of the art, it is well known that with the current availablecross section data they show serious shortcomings that limit the precision of CR studies [24].A couple of years ago, a new set of cross sections derived from a different parametrizationof data for all individual channels was presented as the default option for the incoming DRAGON2 code [14]. They have been successfully used in other studies as [36] and are fully available inthe github repository of the new DRAGON version .Due to uncertainties in cross section, the discrepancies found when comparing thepredicted secondary CR fluxes to data might drive to misleading conclusions. It is thereforeclear the need of comparing different cross section data sets and their performance inreproducing the measurements.In the present simulation we are propagating nuclei up to Z = 14 , which implies we canfully describe the generation of secondary nitrogen and boron (which is considered to be fullysecondary, as Be and Li). Nevertheless, the missing iron (mainly F e , but also its isotopes)and, in very low proportion, S make us underestimate the total amount of Li and Be ina 3.22% and 3.7% in average, respectively (see tables IV and V in [24]). In order to savecomputational time for the computations required for the rest of nuclei (up to Z = 14 ), wecompensate for the missing source terms (primary Fe and S) by adding just this extra ∼ to the Li and Be spectra.Finally, another important parameter affecting the low-energy region of the CR spectrais the solar modulation, which is modeled using the Parker equations and the force-fieldapproximation [37]. This parameter depends on the solar activity and, therefore, on theepoch in which the experimental data were taken.The variable used to characterize the solar modulation is the Fisk potential φ . The fluxof CRs reaching a detector at Earth is related to that in the Local Interstellar medium (LIS)by the following equation: Φ obs ( E obs ) = (cid:18) mE obs + E obs mE LIS + E LIS (cid:19) Φ LIS ( E LIS ) (2.5)with E LIS = E obs + eφ | Z | /A . In eq. 2.5, E LIS and E obs are the kinetic energies per nucleonin the LIS and at Earth, respectively, while m indicates the proton mass.In this work we have chosen for the solar modulation potential the value of .
61 GV ,consistent with the data from the NEWK neutron monitor experiment (see [38, 39]), whichallows us to reproduce the Voyager-1 [40, 41] and AMS-02 data in the period 2011-2016 andis similar to the value found in previous studies [17]. In this section we show and discuss the predicted fluxes of B, Be and Li obtained in ouranalyses. These predictions are performed using the cross sections parametrization from theupcoming
DRAGON2 code,
GALPROP and a combination of the WNEW and YIELDX (only forthe Li production) called here
Webber , and selecting the set of diffusion parameters thatreproduce the boron-over-carbon data (see Fig. 12 in appendix B) of AMS-02 [42] and usingthe optimal value for the halo size found from the procedure explained in Sec. 5. Theseparameters are summarized in Table 1. Additionally, we adjust the parameters of the source Publicly available at https://dmaurin.gitlab.io/USINE/input_xs_data.html – 5 – ebber GALPROP DRAGON2 D ( cm s − ) 2.3 6.65 7.1 v A ( km/s ) 29.9 25.5 27.7 η -0.25 -0.55 -0.6 δ kpc ) 2.07 6.93 6.76 Table 1 . Diffusion parameters used in the CR propagation with the different cross section parametrizations.These values have been obtained from a fit of the B/C data from AMS-02 [42], [43] and of the Be/ Be datafrom various experiments (see sec. 7). spectra in Eq. 2.4 to reproduce the fluxes of individual CR species measured by AMS-02[43, 44]. As an example, Figure 11 in appendix B shows the observed spectra at Earth ofthe main primary CR nuclei (from carbon to silicon) compared with the predictions from oursimulation with the
DRAGON2 set of cross sections. We remark here the importance of addingthe new released data of AMS-02 for the Mg, Si and Ne fluxes for testing the production ofsecondary CRs (mostly Be and Li). We then proceed to study the spectra of secondary CRsand compare their fluxes to the available experimental data.Very few works have studied the spectra of Li and Be in order to determine the diffusionparameters (see [45] for a recent work), while nearly all authors just use boron and its ratiosto a primary CR species (usually C) to develop their models. This is due to the fact that,before AMS-02, the experimental data on Li and Be fluxes were poor and the uncertainty onthe predicted fluxes from the cross sections parametrizations in the B channels is expected tobe the smallest [46]. Typical uncertainties on the predicted Li and Be fluxes around − and − respectively are usually quoted [46], while the uncertainties on the predicted Bflux are around [24]. The reason is that the production of B is mainly regulated by the Cand O reaction channels, while other very poorly constrained cross sections channels (mainlythose of Mg, Ne and Si) become important for the production of Li and Be, contributing to ∼ of their flux.Figure 2 shows the spectra of B, Be and Li for the three cross section parametrizationsadopted in the present work. However, in the very high-energy region, above
200 GeV / n , wesee that the measured fluxes of Li, Be and B are slightly higher (harder) than the predictedones for all the three cross section parametrizations. This feature suggests the need ofintroducing a break in the energy dependence of the diffusion coefficient (see, e.g., ref. [50])rather than in the CR injection spectra, but this does not affect the conclusions of this article.In the case of the Webber cross sections, the Li and Be fluxes follow a similar trend, beingoverestimated with respect to the AMS-02 data with residuals generally below . In turn,for the
GALPROP parametrizations, we see an opposite behaviour for Li, with the simulationunderestimating the experimental data of about , while in the case of Be the predictionis closer to the experimental data, with discrepancies less than above
10 GeV / n . Theshape of the residuals found for Be with the Webber and
GALPROP parametrizations may alsobe related to the adjustment of the halo size value (since the flux of the unstable isotope Be strongly depends on this parameter, as explained in sec.5). Finally, the
DRAGON2 defaultcross sections seem to reproduce all the secondary CR fluxes at the same time within discrepancies in the full energy range, with differences just in their normalization.Comparing the fluxes of Li, Be and B shown in the panels of Fig. 2, we see that the main– 6 – E . J ( E )(( G e V / n ) . s m s r ) B - Webber cross sectionsModulated spectrum, =0.61Unmodulated spectrumB AMS-02 dataPamela(2006-2008)VOYAGER-1(2012-2015) Energy (GeV/n) R e s i d u a l s B spectrum - Webber cross sections E . J ( E )(( G e V / n ) . s m s r ) Be - Webber cross sectionsModulated spectrum, =0.61Unmodulated spectrumBe AMS-02 dataACE-CRIS(1998/01-1999/01)VOYAGER-1 Energy (GeV/n) R e s i d u a l s Be spectrum - Webber cross sections E . J ( E )(( G e V / n ) . s m s r ) Li - Webber cross sectionsModulated spectrum, =0.61Unmodulated spectrumLi AMS-02 dataACE-CRIS(1998/01-1999/01)VOYAGER-1 Energy (GeV/n) R e s i d u a l s Li spectrum - Webber cross sections E . J ( E )(( G e V / n ) . s m s r ) B - GALPROP cross sectionsModulated spectrum, =0.61Unmodulated spectrumB AMS-02 dataPamela(2006-2008)VOYAGER-1(2012-2015) Energy (GeV/n) R e s i d u a l s B spectrum - GALPROP cross sections E . J ( E )(( G e V / n ) . s m s r ) Be - GALPROP cross sectionsModulated spectrum, =0.61Unmodulated spectrumBe AMS-02 dataACE-CRIS(1998/01-1999/01)VOYAGER-1 Energy (GeV/n) R e s i d u a l s Be spectrum - GALPROP cross sections E . J ( E )(( G e V / n ) . s m s r ) Li - GALPROP cross sectionsModulated spectrum, =0.61Unmodulated spectrumLi AMS-02 dataACE-CRIS(1998/01-1999/01)VOYAGER-1 Energy (GeV/n) R e s i d u a l s Li spectrum - GALPROP cross sections E . J ( E )(( G e V / n ) . s m s r ) B - DRAGON2 cross sectionsModulated spectrum, =0.61Unmodulated spectrumB AMS-02 dataPamela(2006-2008)VOYAGER-1(2012-2015) Energy (GeV/n) R e s i d u a l s B spectrum - DRAGON2 cross sections E . J ( E )(( G e V / n ) . s m s r ) Be - DRAGON2 cross sectionsModulated spectrum, =0.61Unmodulated spectrumBe AMS-02 dataACE-CRIS(1998/01-1999/01)VOYAGER-1 Energy (GeV/n) R e s i d u a l s Be spectrum - DRAGON2 cross sections E . J ( E )(( G e V / n ) . s m s r ) Li - DRAGON2 cross sectionsModulated spectrum, =0.61Unmodulated spectrumLi AMS-02 dataACE-CRIS(1998/01-1999/01)VOYAGER-1 Energy (GeV/n) R e s i d u a l s Li spectrum - DRAGON2 cross sections
Figure 2 . Spectra of the light secondary nuclei obtained with the diffusion parameters fitting the B/Cspectrum using the
Webber cross sections (upper row), the
GALPROP parametrizations (middle row) and the
DRAGON2 model (bottom row). The residuals (defined as (model-data)/model throughout all the paper)are also shown to have an idea about how large discrepancies may be for different cross sections models.Experimental data from CR experiments were taken from //https://lpsc.in2p3.fr/crdb/ [47, 48] and https://tools.ssdc.asi.it/CosmicRays/ [49]. difference arise in the predicted Li flux, which is the nucleus that suffers more from missingcross sections data and poorly known reaction channels. Nevertheless, the differences amongthe predictions obtained with the three cross section parametrizations are subject to thechoice of diffusion coefficient used, limiting a direct study of their production cross sections.The fluxes of individual secondary CRs, therefore, do not allow a clear discrimination amongthe various cross section models. On the other hand, as we will discuss in the next partof this section, the flux ratios among secondary CRs are almost unaffected by the diffusioncoefficient, making this observable much more sensitive to the spallation cross sections thanthe fluxes of individual species.The employed cross sections parametrizations for the channels of production of B, Beand Li from C and O projectiles (the main channels for their production) are compared tothe available data in appendix A (see ref. [24] for other rarer channels).– 7 – R a t i o H=2kpcH=4kpcH=8kpc H=16kpcfit: H= 3.96 +/- 0.45 KpcBe/B AMS-02 data Energy (GeV/n) R e s i d u a l s Be/B spectrum - Webber R a t i o H=2kpcH=4kpcH=8kpc H=16kpcfit: H= 3.96 +/- 0.45 KpcLi/B AMS-02 data Energy (GeV/n) R e s i d u a l s Li/B spectrum - Webber R a t i o H=2kpcH=4kpcH=8kpc H=16kpcfit: H= 3.96 +/- 0.45 KpcLi/Be AMS-02 data Energy (GeV/n) R e s i d u a l s Li/Be spectrum - Webber R a t i o H=2kpcH=4kpcH=8kpc H=16kpcfit: H= 7.55 +/- 1.69 KpcBe/B AMS-02 data Energy (GeV/n) R e s i d u a l s Be/B spectrum - GALPROP R a t i o H=2kpcH=4kpcH=8kpc H=16kpcfit: H= 7.55 +/- 1.69 KpcLi/B AMS-02 data Energy (GeV/n) R e s i d u a l s Li/B spectrum - GALPROP R a t i o H=2kpcH=4kpcH=8kpc H=16kpcfit: H= 7.55 +/- 1.69 KpcLi/Be AMS-02 data Energy (GeV/n) R e s i d u a l s Li/Be spectrum - GALPROP R a t i o H=2kpcH=4kpcH=8kpc H=16kpcfit: H= 7.69 +/- 1.68 KpcBe/B AMS-02 data Energy (GeV/n) R e s i d u a l s Be/B spectrum - DRAGON2 R a t i o H=2kpcH=4kpcH=8kpc H=16kpcfit: H= 7.69 +/- 1.68 KpcLi/B AMS-02 data Energy (GeV/n) R e s i d u a l s Li/B spectrum - DRAGON2 R a t i o H=2kpcH=4kpcH=8kpc H=16kpcfit: H= 7.69 +/- 1.68 KpcLi/Be AMS-02 data Energy (GeV/n) R e s i d u a l s Li/Be spectrum - DRAGON2
Figure 3 . Secondary-to-secondary ratios of the light secondary CRs for the
Webber , GALPROP and
DRAGON2 cross sections models. The residuals are also shown to better illustrate how large are the discrepancies betweensimulations and data. These plots include the simulated spectra for various halo sizes, since the presence of Be and its beta decay to B modify the shape of the spectra at low energies. The simulated spectrum forthe halo size that best fits the flux ratios of Be isotopes, as explained in section 5, is also included. Data takenfrom //https://lpsc.in2p3.fr/crdb/ [47, 48] and https://tools.ssdc.asi.it/CosmicRays/ [49].
Figure 3 shows the Be/B, Li/B and Li/Be flux ratios predicted using the three crosssection parametrizations, assuming the halo sizes reported in Table 1, compared to theexperimental data in the energy range from
500 MeV / n up to / n . We see that thedifferences among the predictions obtained with the three parametrizations are quite large, inparticular for the ratios involving Li, as expected. In appendix C we show that ratios amongsecondary CRs are roughly unaffected by the parametrization of the diffusion coefficient in theenergy region above a few tens of GeV/n, while they are mainly dependent on their productioncross sections and the spectra of primary CRs (set always to fit the AMS-02 data).On top of this, another parameter which can significantly affect the fluxes of secondaryCRs and their ratios, at low energy, is the size of the galactic halo. To show the effect of thehalo size on these ratios, we have performed an additional set of simulations changing the– 8 –alo size from to
16 kpc . From Figure 3 we see that in the case of the Li/B and Li/Beratios, variations of the halo size do not yield large changes at any energies (the variations ofthe ratios are less than ). On the other hand, in the case of the Be/B ratio, variations ofthe halo size yield variations in the spectra up to in the low energy region. As mentionedabove, different unstable CR species have different decay lengths and, depending on the pathlength they travel until reaching the Earth, different fractions of unstable nuclei can decay,thus influencing the secondary fluxes and their ratios. In particular, the Be/B ratio is highlysensitive to the halo size due to the presence of the radioactive isotope Be, which can decayinto B (see [36], where the authors discuss the halo size repercussions on the Be/B ratioat low energy). This dependence is particularly evident below
10 GeV / n , given the shortlifetime of this isotope at low energies.Finally, we point out that, as the Be decay length at low energies is of the order of a fewhundred pc , the Be flux at low energies can also depend on the gas density distribution in theGalaxy. Changing this distribution could therefore change the predictions on the secondaryflux ratios involving Be. However, a detailed study of this effect is beyond the goal of thepresent paper. Other uncertainties in the secondary CR fluxes are related to total inelasticcross sections and are expected to have negligible effects in comparison ( O (2%) [12]).In conclusion, we find that the secondary-over-secondary spectra are mainly related totheir production cross sections, and above ∼ −
20 GeV / n they have very little dependenceon all the other discussed effects. Assuming the current uncertainties on spallation crosssections, the predictions on the secondary-over-secondary ratios can be as uncertain as ∼ and higher at energies below
10 GeV / n because of their dependence on the diffusionparameters and halo size at these energies. Hence, these ratios represent an extremely usefultool to constrain the parametrizations of the inclusive cross sections used in CR propagationcodes.From Figure 3, we can see that the largest residuals are usually found at low energies,as expected. The shape of the flux ratios at low energy is not well reproduced when usingthe Webber cross sections, with residuals up to ∼ . On the other hand, the shapes ofthe flux ratios obtained from the GALPROP and
DRAGON2 parametrizations are very similar,with different normalizations. The smaller residuals with respect to data are found with the
DRAGON2 parametrization and are less than above / n .In the next section we will use the flux ratios among B, Be and Li in order to evaluatethe uncertainties associated to their production cross sections and achieve a simultaneous fitfrom a rescaling on their production cross sections within the experimental uncertainties. Given the different predictions from the different parametrizations, each cross section modelmay lead to a different interpretation of the CR data. As an example, from Fig. 2 we seethat the
Webber cross section parametrization yields a ∼ excess in the Be and Li fluxesin comparison to the B flux. One could correct this discrepancy either by adding a primarycomponent of boron (injecting boron from the source) and reducing the total grammagetraversed by CRs (in order to fit the B/C ratio) or by rescaling the cross sections of boronproduction (which would have the same effect on the grammage necessary to fit the B/Cratio), such that the three fluxes will reproduce AMS-02 data at the same time. On the otherhand, the GALPROP parametrization yields a ∼ deficit in the Li flux. As a consequence ofthis result, one could add an additional component of Li generated at the source (see ref. [51]).– 9 –evertheless, the GALPROP parametrizations could also be tuned to reproduce the flux ratiosby a proper rescaling of the production cross sections (i.e. renormalizing some channels),within the experimental uncertainties.To investigate the effects of the uncertainties on the cross section parametrizations wehave defined two bracketing models from the
GALPROP parametrization, in which the spallationcross sections for each interaction channel with C and O as projectiles have been shiftedup or down using a scaling factor corresponding to the average uncertainties on the crosssections experimental data at ± σ . This is motivated by the fact that the energy dependenceof the cross sections is supposed to be well known, while their normalization is not preciselydetermined (see ref. [52]). A couple of channels were differently rescaled, to better contain thecross sections data. In the case of the Li production channels from O and C, extra shiftsof and respectively were applied to the upper model, since a few data points exhibitlarger excursions with respect to the nominal model. In the channels of O producing Beand Be the shift was taken to be half of the experimental uncertainties.The bracketing models are shown in figures 4 and 5, where the shifts, typicallycorresponding to a ± variation from the cross sections normalization, are shown in thelegends for each channel. We see that the Be channels are those with smaller uncertaintiesand more data points, while the Li channels are those with the higher uncertainties and lessdata points. We also see that the channels coming from the spallation of O exhibit largerexperimental errors and less data points than those from the spallation of C. The original
GALPROP cross sections are also shown to illustrate the relative changes.Figures 4 and 5 also show the cross sections obtained from the fit of the secondary-over-secondary flux ratios above
10 GeV / n , which will be discussed in Section 4.1 (see figure 6).These cross sections are scaled from the original GALPROP parametrizations and allow us tosimultaneously reproduce the flux ratios involving Li, Be and B.We point out here that we are not changing the cross sections from the spallationreactions of other nuclei than C and O (the main channels). Most of the channelswith minor importance have no or very few data measurements, what originates most of theuncertainties on the total production cross sections of the isotopes we are studying. Althoughthe individual contribution to the secondary CR fluxes from each of these channels is small,the sum of all their contributions is relevant ( ∼ for B and ∼ for Li and Be) and thevery recent AMS-02 data for Mg, Si and Ne are very important to accurately constrain thesesecondary CRs.The next step here consists on the evaluation of the spectra of secondary CRs for theseupper and lower models. According to ref. [24], variations of the cross section just in the mainchannels considered in this work can lead to variations ∼ of the B flux, ∼ of the Beflux and ∼ of the Li flux. To show the effects of the cross sections uncertainties on the secondary-over-secondary fluxratios, we have derived these ratios using the two bracketing models for the cross sections,demonstrating that the AMS-02 data lie between these two models. The results are shownin Figure 6, where the yellow bands represent the values of the flux ratios within the twolimiting models.The upper bound of each band corresponds to the situations in which the numerator istaken from the upper bracketing model and the denominator from the lower bracketing model– 10 – Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) Galprop v54Upper model - shift: 12.9%Lower model - shift: 12.9%Ratios-derivedData compilation
Direct C + H B Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) Galprop v54Upper model - shift: 24.4%Lower model - shift: 24.4%Ratios-derivedData compilation
Direct C + H B Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) Galprop v54Upper model - shift: 21.0%Lower model - shift: 21.0%Ratios-derivedData compilation
Direct C + H Be Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) Galprop v54Upper model - shift: 19.4%Lower model - shift: 19.4%Ratios-derivedData compilation
Direct C + H Be Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) Galprop v54Upper model - shift: 15.6%Lower model - shift: 15.6%Ratios-derivedData compilation
Direct C + H Be Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) Galprop v54Upper model - shift: 24.9%Lower model - shift: 19.9%Ratios-derivedData compilation
Direct C + H Li Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) Galprop v54Upper model - shift: 25.6%Lower model - shift: 20.6%Ratios-derivedData compilation
Direct C + H Li Figure 4 . Cross sections describing the production of secondary nuclei from C. The upper and lowerbracketing cross sections are shown and the percentage of renormalization is indicated in the legends. Thecross sections derived by the fit of the secondary-over-secondary flux ratios are also shown and compared withthe
GALPROP cross sections. (thus maximizing the ratio) and vice versa for the lower bound of the band.We remark here again that, unlike the fluxes of individual secondary CR species, the fluxratios exhibit only a weak dependence on the diffusion parameters and therefore this bandrepresents a solid prediction for the effects of the CR spallation cross sections uncertainties.In Figure 6 the expected full uncertainty bands on the flux ratios are also representedby black dashed lines. These bands are evaluated taking into account also the contributionsfrom other channels than C and O. From this figure, we see that the full uncertainty bandfor the Li/Be ratio is twice larger than that evaluated taking into account the uncertaintyassociated to their main production channels (i.e. with the two bracketing models), whilefor the other ratios it is around larger. In any case, we see that almost all the AMS-02data lie within the bands obtained just varying the main production channels. These resultsconfirms that primary components of B, Be or Li are not needed to explain the experimentaldata (although they could still be present).On the other hand, it should be stressed that the use of these secondary-over-secondary– 11 – Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) Galprop v54Upper model - shift: 27.7%Lower model - shift: 27.7%Ratios-derivedData compilation
Direct O + H B Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) Galprop v54Upper model - shift: 24.2%Lower model - shift: 24.2%Ratios-derivedData compilation
Direct O + H B Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) Galprop v54Upper model - shift: 23.4%Lower model - shift: 23.4%Ratios-derivedData compilation
Direct O + H Be Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) Galprop v54Upper model - shift: 21.2%Lower model - shift: 21.2%Ratios-derivedData compilation
Direct O + H Be Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) Galprop v54Upper model - shift: 24.8%Lower model - shift: 24.8%Ratios-derivedData compilation
Direct O + H Be Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) Galprop v54Upper model - shift: 34.6%Lower model - shift: 26.6%Ratios-derivedData compilation
Direct O + H Li Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) Galprop v54Upper model - shift: 40.7%Lower model - shift: 32.7%Ratios-derivedData compilation
Direct O + H Li Figure 5 . Cross sections describing the production of secondary nuclei from O. The upper and lowerbracketing cross sections are shown and the percentage of renormalization is indicated in the legends. Thecross sections derived by the fit of the secondary-over-secondary flux ratios are also shown and compared withthe
GALPROP cross sections. flux ratios can help improving, not only the determination of the diffusion coefficient, but alsothe spallation cross sections used in the CR propagation codes. This strategy has been alreadyused for improving the estimation of the production of antiprotons from CR interactions withthe interstellar gas in [53] obtaining better results than previous analyses.In the analysis of the
GALPROP model, from figure 3, it is obvious that an increase of Licross sections is needed; however, just a change of this cross sections does not account for thediscrepancies in the Be/B and Li/B flux ratios. This means that making variations just in theBe and Li cross sections (or whichever pair of secondaries) independently will never reproducethe three ratios at the same time, implying that a simultaneous adjustment of all the threefluxes is needed. Nevertheless, the simultaneous adjustment of the studied ratios does nothave an exact solution (there can be degenerate solutions), so that finding the correct relationbetween the cross sections and the ratios is not straightforward.Moreover, in this work, we have obtained a set of cross sections from a fit of the highenergy part of the secondary-over-secondary flux ratios. The strategy followed here consists– 12 – Energy (GeV/n) R a t i o B e / B Expected total uncertaintiessDerived model Main channels uncertaintiesBe/B AMS-02 data
Be/B spectrum Energy (GeV/n) R a t i o L i / B Expected total uncertaintiessDerived model Main channels uncertaintiesLi/B AMS-02 data
Li/B spectrum Energy (GeV/n) R a t i o L i / B e Expected total uncertaintiessDerived model Main channels uncertaintiesLi/Be AMS-02 data
Li/Be spectrum E . J ( E )(( G e V / n ) . s m s r ) B - Derived cross sectionsModulated spectrum, =0.61Unmodulated spectrumB AMS-02 dataPamela(2006-2008)VOYAGER-1(2012-2015) Energy (GeV/n) R e s i d u a l s B spectrum - Derived cross sections E . J ( E )(( G e V / n ) . s m s r ) Be - Derived cross sectionsModulated spectrum, =0.61Unmodulated spectrumBe AMS-02 dataACE-CRIS(1998/01-1999/01)VOYAGER-1 Energy (GeV/n) R e s i d u a l s Be spectrum - Derived cross sections E . J ( E )(( G e V / n ) . s m s r ) Li - Derived cross sectionsModulated spectrum, =0.61Unmodulated spectrumLi AMS-02 dataACE-CRIS(1998/01-1999/01)VOYAGER-1 Energy (GeV/n) R e s i d u a l s Li spectrum - Derived cross sections
Figure 6 . In the top panels the secondary-over-secondary flux ratios involving Li, Be and B are shown.The yellow bands correspond to the uncertainties obtained using the bracketing cross section models forthe main production channels. The bands between the black dashed lines correspond to the expected totaluncertainties, obtained adding the contributions from all the minor production channels. The blue lines areobtained by simultaneously fitting the AMS-02 data, using the diffusion parameters obtained from the fitof the B/C ratio. In the bottom panels the B, Be and Li fluxes obtained with the cross sections derivedfrom the fit procedure are compared with the AMS-02 data. The residuals are also shown. Data taken from //https://lpsc.in2p3.fr/crdb/ [47, 48] and https://tools.ssdc.asi.it/CosmicRays/ [49]. of a progressive shift of the cross sections from the original parametrization for each of thechannels, until we find a configuration that matches the three ratios at the same time. Indeed,in this case we see that the fit convergence is reached when Li and Be cross sections arevery close to touch the bracketing limits (Upper model for Li and Lower for Be), whichmeans that there is very narrow margin to find other degenerate solutions. The fit yieldeda renormalization of the cross sections of ∼ down for the B flux (constant for all thechannels), of ∼ down for the Be flux ( ∼ in the C channels and ∼ in the Ochannels) and of up for the Li flux ( ∼ for C channels and ∼ O channels).The uncertainties on the scaling factors obtained for this derived model are of above
10 GeV / n , as discussed in appendix C.As a final remark, from the lower plots of Figure 6 we see that the predicted fluxes ofthese three light secondary CRs reproduce the AMS-02 data at the same time in a broadenergy region. The common discrepancy at high energies is due to the choice of the diffusionparameters, as already mentioned, which can be also the reason for the discrepancy of theLi flux below / n (although it is also related to the cross sections parametrization).The low-energy part of the Be spectrum is extremely influenced by the halo size, which mayexplain its deviation from experimental data as commented in ref. [42].As expected, the scaling needed in the cross sections of the the B channels with respectto the original GALPROP parametrization is very small, while the Li and Be main channelsneed larger shifts, in agreement with the uncertainties that one would expect, as the impact– 13 –f a change of these minor channels (very poorly known) for B production is very small. Inturn, the impact of the main channels for Li and Be fluxes is approximately the same as thatof the rest of the minor channels, which means larger level of uncertainty. This combinationof flux ratios among secondary CRs may also be considered as a strategy to take care of thedeficiencies on the extrapolations typically used for the rest of the minor channels.In conclusion, we have demonstrated that we can reproduce the fluxes of B, Be and Li atthe same time within the experimental cross sections uncertainties, with no need of includingany primary extra source and that we can tune the spallation cross sections of production ofsecondary species to reproduce the secondary-over-secondary ratios and overcome the lack ofknowledge we have in the normalization of the cross sections parametrizations. This is crucialin order to determine the diffusion parameters with better accuracy. In fact, with a correctbalance of the secondary CRs, we could in principle also use Li and Be data in addition to Bdata to determine the diffusion parameters. This will be explored in a next paper.
The height of the galactic halo H plays a relevant role for the study of secondary-to-primaryratios involving leptons (the radiative energy loss rates are of the same order of magnitude asthe reciprocal of the diffusion time), antiprotons and, as discussed above, unstable isotopeslike Be.The usual way to constrain the halo size is by means of the study of the ratios of Beto the total Be flux or to the Be flux [54, 55]. The spectrum of the isotope Be depends onan interplay between the diffusion time of the primary CRs ( τ ∝ E − δ ) and the decay time ofthis isotope (which is around . , as determined in ref. [56]). Other methods have beenused to set constraints on H from radio observations of lepton synchrotron emission [57], fromX-ray and gamma-ray studies [58], from studies on CR leptons and other heavy nuclei [59]and even antiprotons [60].One of the obvious consequences of using secondary isotopes to determine any featureof the propagation is that the results will be highly influenced by the cross section modelused. We have therefore decided to evaluate the halo size with four different cross sectionmodels: the DRAGON2 , Webber and
GALPROP parametrizations and the cross sections derivedin section 4.1. For each model, we have implemented a fit of the halo size using the availabledata on the Be/ Be flux ratios. We have simulated different halo sizes from to
16 kpc and, for each halo size, we have evaluated the Be/ Be flux ratios. As mentionedabove, in each of simulation (i.e. for every halo size value) the diffusion parameters have beenchosen to fit the B/C spectrum of the AMS-02 experiment. The flux ratios with halo sizesdifferent from the tabulated values have been evaluated using a 2D interpolation with the tool
RegularGridInterpolator . The error introduced by the interpolation is smaller than forevery energy bin. The fit is performed with the curve _ f it package from the scipy.optimize library.Figure 7 shows the fit results for all the cross section models. For each cross sectionmodel the experimental data are compared with the predicted Be/ Be flux ratios obtainedfor different halo sizes. As expected, the differences among the various predictions are largerat low energies, while the curves tend converge above
10 GeV / n . The curve correspondingto the halo sizes which yields the best fit is also shown. https://docs.scipy.org/doc/scipy/reference/generated/scipy.interpolate.RegularGridInterpolator.html – 14 – Energy (GeV/n) R a t i o ACE-CRIS(1997/08-1999/07)ACE-SIS(1997/08-1999/07)Balloon(1973/08)Balloon(1977/05)Balloon(1977/09)IMP7&8(1972/09-1975/09) IMP7&8(1974/01-1980/05)ISEE3-HKH(1978/08-1979/08)ISOMAX(1998/08)Ulysses-HET(1990/10-1997/12)Voyager1&2(1977/01-1991/12)Voyager1&2(1977/01-1998/12) Be/ Be spectrum - Webber
Simulated Be/ Be H=1 KpcSimulated Be/ Be H=4 KpcSimulated Be/ Be H=8 KpcSimulated Be/ Be H=16 Kpcfit: H= 2.07 +/- 0.34 Kpc Energy (GeV/n) R a t i o ACE-CRIS(1997/08-1999/07)ACE-SIS(1997/08-1999/07)Balloon(1973/08)Balloon(1977/05)Balloon(1977/09)IMP7&8(1972/09-1975/09) IMP7&8(1974/01-1980/05)ISEE3-HKH(1978/08-1979/08)ISOMAX(1998/08)Ulysses-HET(1990/10-1997/12)Voyager1&2(1977/01-1991/12)Voyager1&2(1977/01-1998/12) Be/ Be spectrum - GALPROP
Simulated Be/ Be H=1 KpcSimulated Be/ Be H=4 KpcSimulated Be/ Be H=8 KpcSimulated Be/ Be H=16 Kpcfit: H= 6.93 +/- 0.98 Kpc Energy (GeV/n) R a t i o ACE-CRIS(1997/08-1999/07)ACE-SIS(1997/08-1999/07)Balloon(1973/08)Balloon(1977/05)Balloon(1977/09)IMP7&8(1972/09-1975/09) IMP7&8(1974/01-1980/05)ISEE3-HKH(1978/08-1979/08)ISOMAX(1998/08)Ulysses-HET(1990/10-1997/12)Voyager1&2(1977/01-1991/12)Voyager1&2(1977/01-1998/12) Be/ Be spectrum - Derived
Simulated Be/ Be H=1 KpcSimulated Be/ Be H=4 KpcSimulated Be/ Be H=8 KpcSimulated Be/ Be H=16 Kpcfit: H= 6.62 +/- 1.10 Kpc Energy (GeV/n) R a t i o ACE-CRIS(1997/08-1999/07)ACE-SIS(1997/08-1999/07)Balloon(1973/08)Balloon(1977/05)Balloon(1977/09)IMP7&8(1972/09-1975/09) IMP7&8(1974/01-1980/05)ISEE3-HKH(1978/08-1979/08)ISOMAX(1998/08)Ulysses-HET(1990/10-1997/12)Voyager1&2(1977/01-1991/12)Voyager1&2(1977/01-1998/12) Be/ Be spectrum - DRAGON2
Simulated Be/ Be H=1 KpcSimulated Be/ Be H=4 KpcSimulated Be/ Be H=8 KpcSimulated Be/ Be H=16 Kpcfit: H= 6.76 +/- 1.00 Kpc
Figure 7 . Be/ Be predicted flux ratios compared to all experimental data available and for everycross section parametrization studied here. For each parametrization, various simulations with differenthalo sizes are shown, along with the simulation yielding the best fit value. Data taken from //https://lpsc.in2p3.fr/crdb/ [47, 48].
The value of the halo size that best fits the experimental data for each model isaccompanied by the uncertainties coming from its determination. The fit results aresummarized in Figure 8 together with a line indicating the mean halo size value. The errorbar associated to the halo size fitted with the Webber cross section model is smaller thanthose associated to the halo sizes fitted with the other cross section models. This is due tothe fact that, as can be seen from the curves in Figure 7, a change in the value of H leads tochanges in the flux ratios which become smaller as H increases. Therefore, the smaller errorbar foundWe see that the derived cross sections cast a value of the halo size very similar tothe
DRAGON2 and
GALPROP predictions, of around . (very similar to the value foundin refs. [36, 61]). The small value found for the Webber parametrizations can be explainedhaving a look to the cross sections around hundreds
MeV . The most important channelsgenerally show a good agreement with cross sections data, but in the case of the C → Bechannel, the discrepancy is considerable. Therefore, from this underestimation of the crosssection, the halo size prediction is expected to be also underestimated (less Be productionimplies a lower halo size to reduce the fraction of decaying Be nuclei).In any case, the value obtained using the
Webber parametrizations seems to be out of thestandard picture of a halo size between and
10 kpc height found in measurements of diffusegamma-ray background [62] and with other radio observations [63–65]. The mean of thesebest fit values (including the determination of the halo size using the
Webber parametrization)– 15 –
Halo size (kpc)
DRAGON2GALPROPWebberDerived
Halo sizes summary
Figure 8 . Summary of the results obtained for the fits of halo sizes to the Be/ Be experimental data.Also the values obtained with the derived cross sections are shown for having a broader comparison. Errorbars reflect only statistical uncertainties, as explained above. The dashed blue line represents the mean valueof the halo best fit values and the red band the uncertainties of the mean ( σ ), calculated as the mean of thehalo best fit values ± σ . is exactly . +0 . − . kpc , and the value obtained using only the most updated cross sections isaround . ± . We are living in a very exciting time for CR physics thanks to the high quality data providedby the AMS-02 experiment, which allows making precise studies on the nature of CRs andtheir propagation throughout the Galaxy.However, while the astrophysical information seems to be accurate enough to face newphysical challenges, the information about the spallation cross section, which is the otherkey element in the CR puzzle, is still largely incomplete. Since the study of secondary-over-primary CRs is the best tool to test the diffusion coefficient parametrizations, we need to havehigh precision on the secondary CRs fluxes and in their production (inclusive) cross sections.The amount of channels that matter in the spallation network and the difficulty toperform the measurements lead to parametrizations that extend the energy range of theactual measurements and expand to all important nuclei (even those for which experimentaldata are missing). While we have some insight in the shape of cross sections as a function ofenergy, the normalization is still somewhat uncertain.In this paper we have deeply discussed different cross sections parametrizations and wehave investigated their direct impact in the fluxes of Li, Be and B. We have highlighted theimmediate relation of the secondary-over-secondary ratios with the spallation cross sectionsused. We have shown that predictions from different cross sections models can differ in theamount of Be and Li fluxes at a level from to more than . We also emphasize that theuncertainties in the determination of secondary CR fluxes are in general quite large, althoughthose on the B flux seem to be remarkably smaller, since its production is essentially due tothe contribution of C and O channels for which more experimental data are available.In addition, we have demonstrated the good performance achieved in the simulationswhen using the default
DRAGON2 cross sections model, being able to reproduce every observablewithin very small errors. – 16 –t has been shown that there is no need of invoking primary sources of secondary CRs,as the cross sections uncertainties can largely account for the discrepancies found with respectto data. To better show this, two models of cross sections have been considered as limitingcases. These bracketing models represent minimum and maximum credible renormalizationsof the
GALPROP model of cross sections, maintaining the same energy dependence. Wehave investigated how the secondary-over-secondary flux ratios would behave under thecombination of the bracketing models to test whether, between these two limiting cases,there is a space of cross sections that can reproduce all secondary CRs above
10 GeV / n atthe same time. A set of cross sections has been derived by matching these flux ratios at highenergy, demonstrating that they are an excellent tool to constrain and even adjust the crosssections parametrizations. This derived cross sections model balances the deficiencies on thedescription of those channels which are very poorly known and gives some insight on howmuch our cross sections network is biased for every secondary species. We showed that, whilethe B production channels should not be renormalized by more than to reproduce theratios, the beryllium and lithium ones needed an overall renormalization of around and (in average), respectively.In this way we have been able to combine the information arising from secondary CRsto get rid of systematic uncertainties related to spallation cross sections. We argue thatthis combined tuning of the normalization on a cross sections parametrization to reproducethe secondary-over-secondary flux ratios serves to improve the determination of the diffusioncoefficient parameters as well, and it is important to obtain predictions which are consistentwith all the observables at the same time.Finally, a study of the effects on the halo size on the secondary-over-secondary fluxratios for each cross section model has been performed. In particular, the ratio Be/ Be wasstudied for each of the models, as the Be isotope has a lifetime of the order of the diffusiontime of CRs in the Galaxy. This allowed the discussion on the wide repercussion of the crosssections on the determination of the halo size too. The determination of the halo size fromthe most updated parametrizations gives a mean value around . ± . , which is inagreement with most of the values obtained in other works.In a next work, the diffusion parameters will be studied in a combined analysisof the secondary-over-primary and secondary-over-secondary flux ratios of B, Be and Liincluding nuisance factors to allow adjustments on the normalization of the cross sectionsparametrizations studied. Acknowledgments
We remark the crucial help of Daniele Gaggero in the development of the paper and hisexaminations during the evolution of the ideas commented here and also in the implementationof the preliminary version of the
DRAGON2 code.Special acknowledgements to Carmelo Evoli for his invaluable advice and commentsduring all the work process regarding the
DRAGON2 preliminary version used here and themanuscript elaboration.Many thanks to the instituto de física teórica (IFT) in Madrid for hosting Pedro De laTorre for a long stay there and specially to the DAMASCO (DArk MAtter AStroparticlesand COsmology) group for their support and valuable conversations related to this work.– 17 –his work has been carried out using the RECAS computing infrastructure in Bari( ). A particular acknowledgment goes to G.Donvito and A. Italiano for their valuable support.
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In this appendix we are showing a comparison between the different cross section models usedin the study and the experimental data in the most important reaction channels for the lightsecondary CRs Li, B, Be. Experimental data are taken from various experiments and authors:some can be found in EXFOR (Experimental Nuclear Reaction Data) others in the GALPROP database of cross sections ( isotope _ cs.dat ) and the rest come from various publicationsand experiments (Bodemann1993, Davids1970, Fontes1977, Korejwo1999, Korejwo2002,Moyle1979, Olson1983, Radin1979, Read1984, Roche1976, W90, W98a and Zeitlin2011). Thisdata is available upon request. More information about the references used can be found insection 5 of ref. [14] and mainly in the appendix of ref. [52]. Secondary nuclei from C Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) DRAGON2WNEW03Galprop v54Data compilation
Direct C + H B Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) DRAGON2WNEW03Galprop v54Data compilation
Direct C + H B Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) DRAGON2WNEW03Galprop v54Data compilation
Direct C + H Be Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) DRAGON2WNEW03Galprop v54Data compilation
Direct C + H Be Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) DRAGON2WNEW03Galprop v54Data compilation
Direct C + H Be Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) DRAGON2TS98Galprop v54Data compilation
Direct C + H Li Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) DRAGON2TS98Galprop v54Data compilation
Direct C + H Li Figure 9 . Comparison on the public cross sections to the available experimental data for various channelscoming from C. – 21 – econdary nuclei from O Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) DRAGON2WNEW03Galprop v54Data compilation
Direct O + H B Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) DRAGON2WNEW03Galprop v54Data compilation
Direct O + H B Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) DRAGON2WNEW03Galprop v54Data compilation
Direct O + H Be Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) DRAGON2WNEW03Galprop v54Data compilation
Direct O + H Be Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) DRAGON2WNEW03Galprop v54Data compilation
Direct O + H Be Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) DRAGON2TS98Galprop v54Data compilation
Direct O + H Li Projectile energy (GeV/n) Sp a ll a t i o n ( m b ) DRAGON2TS98Galprop v54Data compilation
Direct O + H Li Figure 10 . Comparison on the public cross sections to the available experimental data for various channelscoming from O. – 22 – Primary spectra
In this section we show the spectra of the main primary CRs involved in the generation of thesecondary CRs Li, Be and B. A good description of them is necessary for a correct assessmentof the secondary ratios. E . F l u x (( G e V / n ) . s m s r ) Modulated spectrumUnmodulated spectrumO AMS dataO Voyager data10 Energy (GeV/n) R e s i d u a l s Oxygen spectrum E . F l u x (( G e V / n ) . s m s r ) Modulated spectrumUnmodulated spectrumC AMS dataC Voyager data Energy (GeV/n) R e s i d u a l s Carbon spectrum E . F l u x (( G e V / n ) . s m s r ) Modulated spectrumUnmodulated spectrumN AMS dataN Voyager data Energy (GeV/n) R e s i d u a l s Nitrogen spectrum E . F l u x (( G e V / n ) . s m s r ) Modulated spectrumModulated spectrumUnmodulated spectrumNe AMS dataNe Voyager data Energy (GeV/n) R e s i d u a l s Neon spectrum E . F l u x (( G e V / n ) . s m s r ) Modulated spectrumUnmodulated spectrumMg AMS dataMg Voyager data Energy (GeV/n) R e s i d u a l s Magnesium spectrum E . F l u x (( G e V / n ) . s m s r ) Modulated spectrumUnmodulated spectrumSi AMS dataSi Voyager data Energy (GeV/n) R e s i d u a l s Silicon spectrum
Figure 11 . Fits of the primary CR spectra used in the simulations, using the
DRAGON2 cross sections. Datafrom the Voyager-1 mission and AMS-02 are plotted, including the very recent Ne, Mg and Si data from AMS-02. Voyager-1 is outside the heliosphere, which means no solar modulation effects, while the Fisk potentialvalue used for to fit AMS-02 data is φ = 0 .
61 GV . Data taken from //https://lpsc.in2p3.fr/crdb/ [47, 48]and https://tools.ssdc.asi.it/CosmicRays/ [49].
In addition, the fits of the boron-over-carbon spectra with the three tested cross sectionsare shown in fig. 12, for the unmodulated and modulated predictions, in comparison to theexperimental data of the Voyager-1, PAMELA and AMS-02 experiments.– 23 – ebber GALPROP DRAGON2
Energy (GeV/n) - -
10 1 10 B / C VOYAGER 1 (2012-2015)AMS-02 (2011-2016)PAMELA (2006-2008)=0.61 F Webber XSecs, Webber XSecs, unmodulated
Energy (GeV/n) - -
10 1 10 B / C VOYAGER 1 (2012-2015)AMS-02 (2011-2016)PAMELA (2006-2008)=0.61 F DRAGON2 XSecs, DRAGON2 XSecs, unmodulated
Energy (GeV/n) - -
10 1 10 B / C VOYAGER 1 (2012-2015)AMS-02 (2011-2016)PAMELA (2006-2008)=0.61 F Galprop XSecs, Galprop XSecs, unmodulated
Figure 12 . Fits of the boron-over-carbon ratios for the three compared cross sections models. Thesemodels are set to a Fisk potential of φ = 0 .
61 GV in the time period corresponding to the AMS-02data taking. A line with a solar modulation φ = 0 . is added for completeness. Data taken from //https://lpsc.in2p3.fr/crdb/ [47, 48] and https://tools.ssdc.asi.it/CosmicRays/ [49]. C Theory on secondary-over-secondary flux ratios
In this appendix, the basic theory about the secondary-over-secondary flux ratios is illustrated.Here we demonstrate that they are mainly dependent on the cross sections of secondary CRproduction and on the injection spectra used to describe primary CRs. Roughly speaking,if one considers that the flux of a primary CR species at Earth is J α ∝ q α × τ prop ( E ) and τ prop ( E ) = H /D ( E ) , it will take the form J α ∼ C α E − γ α − δ at high energies (more precisely,around few GeVs, when /σ inelα ( E ) (cid:29) n g hH cτ prop ( E )) ). The secondary CR fluxes take theform J i ∝ (cid:80) α J α cn g σ α → i × τ prop ( E ) whose ratio leaves the expression: J k J j (cid:16) E (cid:17) ∝ (cid:80) α → k J α ( E ) σ α → k ( E ) (cid:80) α → j J α ( E ) σ α → j ( E ) high energies −−−−→ ∼ (cid:80) α → k C α E − γ α σ α → k ( E ) (cid:80) α → j C α E − γ α σ α → j ( E ) (C.1)Notice that the term E − δ is common in the summation terms and therefore is common innumerator and denominator, cancelling out. With the equation C.1 one appreciates that theseratios have direct dependence on the local spectrum of primary nuclei ( α ) and on the overallspallation cross sections. Nonetheless, the local primary spectrum may have some dependenceon the diffusion parameters at low energies and reacceleration can slightly contribute. Thetotal repercussion of these parameters in the low energy region can be estimated to be at alevel smaller than . Then, due to the presence of the radioactive Be, the most importantcontributor to the low energy uncertainties is the halo size being able to introduce variationsof more than , respectively. At the end, these ratios are mainly dependent on the sourceterm of CRs and on their spallation cross sections at high energies. As AMS-02 data for thelocal spectrum of primary CRs are highly precise, the largest uncertainty at high energy lies onthe spallation cross sections, and is this what makes them so suitable for adjusting the overallcross sections (not individual channels) with high precision (AMS-02 precision, indeed). Thenext figure shows the effect of these changes on the Li/B ratio (since the importance of thehalo size and gas profile is negligible):As we see, at
10 GeV / n the maximum difference between the predictions is around , while at
30 GeV / n it goes down to for variations of the diffusion parameters largerthan the usual uncertainties related to their determination. This largest variation between– 24 – R a t i o =0.05V_A=18.5D_0=7.65=0.40Best fit modelLi/B AMS-02 data Energy (GeV/n) R e s i d u a l s Li/B spectrum
Figure 13 . Simulated ratios with same primary source term (except for the simulation with change in thediffusion spectral index, δ , in which the source term was changed to hold same α + δ value) and changingdiffusion parameters. The best fit model has been shown in table 1 and the simulations change by ∆ η ∼ . , ∆ V A ∼ / s , ∆ D ∼ × cm / s , ∆ δ ∼ . , respectively, much higher than the usual uncertaintiesquoted in the best fits of these parameters. predictions corresponds to the ∼ change in the D parameter, which translates into ∼ variation in the fluxes of secondary species, while changes in the other parametersdo not involve variations larger than . Nonetheless, the differences at high energy are notdue to the change on the diffusion parameters by itself, but are mostly due to the subtledifferences in the primary fluxes used in each simulation and to the change in the amountof secondary C (roughly / of its flux is secondary) and N ( ∼ of it is secondary at
10 GeV / n ), which leads to sizeable changes in the secondary CRs. This may also meanthat the ∼ − uncertainties in the AMS-02 primary fluxes imply a maximum of ∼4%