Intermediate mass and heavy Galactic cosmic-ray nuclei: the case of new AMS-02 measurements
HHeavy Galactic cosmic-ray nuclei: the case of new AMS-02 measurements
Benedikt Schroer , ∗ Carmelo Evoli , † and Pasquale Blasi ‡ Gran Sasso Science Institute (GSSI), Viale Francesco Crispi 7, 67100 L’Aquila, Italy andINFN-Laboratori Nazionali del Gran Sasso (LNGS), via G. Acitelli 22, 67100 Assergi (AQ), Italy (Dated: February 26, 2021)The recent measurement of the spectra of heavy nuclei carried out with the AMS-02 experimentprovided us with the most complete set of data on cosmic ray fluxes to date, and allowed us to testthe standard model for the transport of these particles through the Galaxy to the finest details.We show that the parameters derived from lighter primary and secondary elements in the cosmicradiation also lead to a good description of the data on heavier nuclei, with no need to invokedifferent injection spectra for such nuclei, provided the whole chain of fragmentation is properlyaccounted for. The only exception to this finding is represented by iron nuclei, which show a veryunusual trend at rigidity (cid:46)
100 GV. This trend reflects in a Fe/O ratio that is at odds with theresults of the standard model of cosmic ray transport, and is in contradiction with data collected byHEAO, ACE-CRIS and Voyager at lower energy. We speculate on possible origins of such findings.
I. INTRODUCTION
The transport of cosmic rays (CRs) in the Galaxy isa crucial piece in the puzzle that leads to understandingthe origin of these high energy particles. The descriptionof CR transport is based on a macroscopic descriptionof the particle motion (advection and diffusion) that isbased on microscopic processes (particle wave resonancesand CR self-generation) and particle physics (radiativelosses and nuclear fragmentation). As such, it representsan endeavour of remarkable proportions, that at best canbe pursued by using effective theories. The advection-diffusion equation engulfs such difficulties in a way thatled to numerous successes and that requires continuousadjustments to reflect pieces of observations that are be-ing acquired.The precision measurements made by AMS-02 in thelast decade or so have affected our understanding of theorigin of CRs in a profound way, by confirming some pil-lar ideas and by revealing anomalies that might be signsof a need for a deep revision of some aspects of the prob-lem: 1) the observation of spectral breaks in both pri-mary and secondary nuclei has driven us towards devel-oping theories that could accommodate such features [1–4]. The fact that the breaks appear also in the ratioof secondary-to-primary nuclei is a clear indication thatthey are most probably due to some new phenomenonoccurring in particle transport rather than in the accel-erators or in the spatial distribution of sources [5, 6]; 2)the AMS-02 measurements showed beyond doubt thatprotons, helium and heavier nuclei need to be injectedat the sources with slightly different spectra, a findingthat is at odds with the basic idea of diffusive shock ac-celeration. 3) The spectrum of positrons in CRs [7, 8] isvery different from what one would expect as secondary ∗ [email protected] † [email protected] ‡ [email protected] products of hadronic interactions, thereby requiring ad-ditional sources of these antimatter particles [9–12]. Thismight also be the case, although less clearly, for antipro-tons [13–15]. 4) The measurement of beryllium [4] pro-vided a firm confirmation that the CR residence time inthe Galaxy requires extended magnetized halos, with size (cid:38) ∼ TV, provides us withthe most complete set of information about CR transportin that such nuclei are the parent particles for lighternuclei through processes of spallation (for stable nuclei)and decay (for unstable isotopes). Such measurementsextend now to iron nuclei, with a predominantly primarynature [19].Provided particles are transported either diffusively oradvectively over distances much larger than the thick-ness of the disc ( h ∼
150 pc), the transport of CRsin the Galaxy is properly described by a weighted slabmodel [20, 21], recently used to derive the basic prop-erties of the CR transport (diffusion coefficient and ad-vection velocity) [22] and the size of the confining halo H (cid:38) a r X i v : . [ a s t r o - ph . H E ] F e b In fact the measurement of the beryllium flux and ofthe Be/B ratio led to a slight readjustment of the pa-rameters due to small production of boron from the de-cay of the unstable Be isotope [16]. As stressed above,the main limitation to the strength of these conclusionsderives from the experimentally poor knowledge of thespallation cross sections, especially the partial cross sec-tions for specific channels of fragmentation. A discussionof these uncertainties was provided in [23–25]. Anothersource of uncertainty, though to a smaller extent, was thefact that the abundances of heavier elements, acting asprimaries for lighter nuclei, were poorly known and hadto be taken from older experiments. This gap has beenfilled by the recent measurements of the fluxes of heavyelements, by AMS-02.In this article we apply the same approach, alreadymastered in [16, 22], to heavy nuclei, up to iron. Weshow that the spectra of all nuclei heavier than Oxy-gen are described within uncertainties with an injectionspectrum at the sources that is the same as for interme-diate mass nuclei (CNO), provided the contribution ofspallation from even heavier nuclei is taken into account.This is in some tension with the statement by the AMS-02 collaboration that Ne, Mg and Si are three primariesthat behave peculiarly, in that they have a different spec-tral index than He, C and O. We maintain that this be-haviour is fully accounted for by properly calculating thesecondary contribution to these nuclei from their heavierparent nuclei, a situation similar to that of nitrogen.The only exception to this general conclusion is thespectrum of iron as measured by AMS-02, which appearsto be at odds with the predictions of the standard model.The main anomaly is a hard shape of the Fe spectrum for R (cid:46)
100 GV, which is not compatible with the grammageinferred from secondary/primary ratios for lighter nuclei.Moreover, the ratio Fe/O measured by AMS-02 appearsto be in tension with previous results from HEAO [26]and Voyager [27]. We present here the results of severaltests proposed to address these problems.The article is structured as follows: in § II we provide abrief summary of the theoretical approach adopted here,based on the weighted slab technique applied to the wholechain of nuclei, from H to Fe. In § III we describe themethodology used to achieve a description of the nuclearspectra, as well as the secondary/primary ratios. In § IVwe describe in detail our findings, starting from nuclei ofintermediate mass (Ne, Mg, Si) and going up to iron. Wesummarize our conclusions in § V. II. PROPAGATION MODEL
The details of the transport equation and the approachwe follow to solve it are detailed in [16, 20–22]. Here,we simply summarize the most important features of themodel.We assume that the CR density of stable and unstablenuclear species obeys a steady-state diffusion-advection equation [28]: − ∂∂z (cid:20) D a ∂f a ∂z (cid:21) + v A ∂f a ∂z − dv A dz p ∂f a ∂p + 1 p ∂∂p (cid:34) p (cid:18) dpdt (cid:19) a, ion f a (cid:35) + µv ( p ) σ a m δ ( z ) f a + f a ˆ τ d,a = 2 h d q ,a ( p ) δ ( z )+ (cid:88) a (cid:48) >a µ v ( p ) σ a (cid:48) → a m δ ( z ) f a (cid:48) + (cid:88) a (cid:48) >a f a (cid:48) ˆ τ d,a (cid:48) , (1)where f a ( p, z ) is the distribution function of species a inphase space, v ( p ) = β ( p ) c is the particles’ velocity, and µ = 2 . is the surface density of the disk.The transport equation in Eq. 1 is obtained assum-ing spatially homogeneous diffusion and that the sourcesof CRs and the target gas are confined in an infinitelythin disc (half-thickness h (cid:39)
100 pc), where interactions(spallation and ionization energy losses) are restricted to.We assume that CRs are confined in a low-density in-finite slab of half-thickness H , extending well beyond thegaseous disc ( H (cid:29) h ). Outside the magnetic halo, theparticles can escape freely into intergalactic space so asto reduce the CR density to f a ( p, z = ± H ) (cid:39) D ( R ) = 2 v A H + βD ( R/ GV) δ [1 + ( R/R b ) ∆ δ/s ] s , (2)where D and δ are fitted to secondary over primaryratios, while the other parameters s , ∆ δ and R b arefixed from observations of primary nuclei and have thefollowing values: s = 0 .
1, ∆ δ = 0 . R b = 312 GV.The change of slope above the rigidity R b in Eq. 2 hasbeen implemented to account for the presence of similarbreaks in the spectra of primaries [29] and in the sec-ondary/primary ratios [4]. As shown in [6], the spectralbreaks seen in CR data at a rigidity of ∼
300 GV mustbe attributed to a change of properties in the Galactictransport rather than in the injection. Theoretical inves-tigations have motivated this functional form in terms ofthe transition from self-generated turbulence to preexist-ing turbulence [21, 30], or of a non-separable spatiallydependent diffusion coefficient in the Galactic halo [23].The plateau at low energies, where advection dominatestransport, was found as a consequence of self-generatedturbulence as described in [31].The advection velocity v A is assumed to be constantabove and below the disk, hence adiabatic energy losses(third term in the LHS of Eq. 1) are determined throughthe expression dv A /dz = 2 v A δ ( z ).The injection of CR nuclei of type a is given as a powerlaw in momentum, having in mind that they are predomi-nantly accelerated at SN shocks [32–34], hence the sourceterm can be written as q ,a ∝ p − γ inj . The slope γ inj is as-sumed to be the same for all nuclei heavier than He, whilethe normalizations are chosen for each individual speciesto match the data. Notice that the assumption of univer-sal injection for all nuclei, although strongly supportedby theoretical grounds, does not apply to light CRs, asit has been shown that the injection of protons(helium)requires a softer(harder) slope than nuclei [22, 35].The other terms on the LHS of Eq. 1 describe the dis-appearance of the species a because of spallation ( σ a )or of radioactive decay (ˆ τ d,a ). In the RHS, one can findthe corresponding terms describing the production of thespecies a by spallation and decay of heavier ( a (cid:48) > a ) el-ements. The inelastic spallation cross sections are com-puted using the parametric fits provided by [36, 37], whilefor the secondary production cross-sections we adoptedthe parametric fits of [22] for the major primary chan-nels of Li, Be and B production, and the approach de-scribed in [25] to compute all other cross-sections (in-cluding the cumulative contribution of ghost nuclei). Weremind that this approach is based on cross-section mea-surements (when available) to set the normalization ofeach spallation channel. To this aim, we make use of theGALPROP cross-section measurement database [38–40]and a collection of more recent measurements [41], in par-ticular for the production of heavy fragments (Ne, Mg,Si, S) by Iron spallation .The quantities ˆ τ d,a = γτ d,a define the Lorentz boosteddecay lifetimes of unstable elements. In [16], some ofus addressed the effect of including the decay of Be,and the corresponding production of B, on the calcu-lation of the Boron-over-Carbon ratio (B/C) and on thedetermination of the transport parameters. Since theCR propagation in the environment of the Galactic discmight take place under different conditions, hostile to dif-fusion, we studied the minimum rigidity above which the Be decay takes place predominantly outside the disc.We found that this condition is well satisfied for rigidities (cid:38) few GV’s even for halos as small as H ∼ H ∼ β − unstable isotopes Al, Cland Mn (see also [43]).The shortest lifetime is the one of Cl, which is afactor ∼ H (cid:38) In order to facilitate the reproducibility of our results we releasethe cumulative cross-sections adopted to compute the secondaryproduction via a dedicated repository [42] . . . . . . . B / C Model PredictionAMS-02 R/[GV] . . . B / O Model PredictionAMS-02
FIG. 1. B/C and B/O ratios as derived by using our calcula-tions (solid line) compared to recent AMS-02 data [4].
Finally, we comment on the fact that we do not modelthe decay modes involving electron capture, since it is im-portant only at energies much smaller than ∼ GeV/n [44].We account for the effect of solar modulation by usingthe force field approximation [45] with a Fisk potential φ . III. METHODOLOGY
In order to get predictions for the fluxes of heavy el-ements, like Ne, Mg, Si, S and Fe, we include in ourcalculation all the stable isotopes from B ( Z = 5) toNi ( Z = 28). Additionally, we propagate the unsta-ble isotopes Be, Al, Cl and Mn as described in § II. This procedure requires fitting of numerous param-eters, that we summarize here. Spatial transport, includ-ing diffusion and advection, comprises 7 free-parameters: D , δ , v A , H , R b , ∆ δ , s . To speed up the analyses, asmentioned before, we fix the halo size to be consistent R / [GV] . . . . . Mg fluxMg primary . × Mg secondary AMS-02 . . . . . Ne fluxNe primary
AMS-02 R / [GV] S flux × primary AMS-02Si fluxSi primary
AMS-02 φ ( R ) · R . / [ m · s r · s · G V ( − . ) ] FIG. 2. Flux of Ne, Mg, Si and S (solid orange line) com-pared to recent AMS-02 data. The shaded area shows theeffect of cross section uncertainties on the predictions. Thedashed lines reflect the fluxes obtained by only including theprimary contribution (no secondary production). The red lineshows the expected flux for the given element, assuming a Mgproduction cross section higher by 30%. with the value derived from the analysis of the Be/Bratio, H = 7 kpc [16], having in mind that secondary-to-primary ratios are degenerate for D /H , to an excel-lent degree of approximation. Moreover, we fix the threehigh-rigidity break parameters ( R b , ∆ δ , s ) to the val-ues of [22], derived from fitting the shape of the protonspectrum. These assumptions reduce the number of freeparameters which describe the transport to 3.The injection efficiencies (cid:15) a of the species C, N, O, Ne,Mg, Si, S and Fe are set in such a way that the total fluxof each species match the AMS-02 data. That amountsto additional 8 parameters. For other primary species,in particular Na, Al, P, Ar, Ca, Cr, Mn and Ni, theTRACER03 [46], CRISIS [47], and HEAO3-C2 [26] mea-surements have been used to fix their abundances and wedo not include these normalizations as free parameters inthe best-fit search. Notice that the efficiency (cid:15) a pertainsto the injection of a given charge, including all isotopes.We distribute the efficiency (cid:15) a among the isotopes in or-der to reproduce the isotopic abundance in the ISM asmeasured in [48].The solar modulation potential, φ , and the injectionslope, γ inj are two additional free parameters, bringingthe total number of free parameters to 13.To fit the propagation parameters against differentdatasets we use the MINUIT package [49]. To ensurethat true minima are found, O (50) minimisations fromdifferent starting points are carried out for all our anal-yses. The quantity we minimise is the χ computed forthe AMS-02 measurements of different datasets. Specifi-cally, the total χ is computed by summing the reduced χ computed over the ratios Be/C, B/C, Be/O, Ne/Mg, . . . . N e / O Model PredictionAMS-02 . . . . M g/ O Model Prediction1 . × Mg secondary AMS-02 R/[GV] . . . . S i / O Model PredictionAMS-02
FIG. 3. Ratios of Ne, Si and Mg over O compared to recentAMS-02 data. The shaded region indicates the effect of crosssection uncertainties on the model. The red line shows thecase with higher Mg production cross sections.
Si/Mg, Ne/O, Mg/O, Si/O, and the absolute fluxes of B,C, N, O, Ne, Mg, Si, and S [4, 18, 29]. To normalize theS primary injection we make use of the AMS-02 prelimi-nary data on the S absolute flux as reported in seminarpresentations . The absolute fluxes are fitted only forrigidities larger than ∼
10 GV in order to minimize theeffect of solar modulation (see also [22]).Finally, for each dataset the total uncertainty iscomputed by adding systematic and statistic errors inquadrature (see also [17] for an attempt to take into ac-count the correlation among systematics). See, e.g., https://indico.gssi.it/event/80/
IV. RESULTSA. Heavy nuclei
The propagation parameters that best fit the data,using the method described in § III are: δ = 0 . γ inj = 4 . D /H = 0 .
35 (in units of 10 cm s − kpc − , v A = 4 . .
49 GV. We notice that the best-fit values are in perfect agreement with the results in [16]where the fit was performed using only the AMS-02 dataon nuclei lighter than Oxygen.Following the procedure first introduced in [22], in or-der to illustrate the impact of cross section uncertain-ties on our results we have repeated our calculations 500times for the best-fit scenario. Each time the spallationcross sections of elements heavier than Oxygen have beenrandomly re-scaled by a factor extracted from a Gaussiandistribution with a width assumed to be ∼
10% for the to-tal cross sections and ∼
30% for the partial cross sections.In all the following figures, the gray-shaded areas repre-sent the 1- σ variance of our calculations associated withthe uncertainty in the cross-section. In fact, the uncer-tainty in the cross sections might also affect the best-fitparameters, in particular the injection efficiencies, hencethe error bands shown in our figures only capture part,though most likely the bigger part, of the effect of uncer-tainties in spallation cross sections.In figure 1 we show how the B/C and B/O ratios arecorrectly reproduced by our best-fit. The variance dueto the uncertainty on the heavy-element cross sections isof the order of ∼
1% at ∼
10 GV (and smaller at higherenergies). This is due to the fact that the secondarycontribution of elements heavier than Oxygen to Boronand Carbon is sub-dominant compared to that of Car-bon, Nitrogen, and Oxygen [25]. This result justifies ourprevious attempts to extract the diffusion parameters byfitting the B/C and B/O even with a poor knowledge ofthe normalization of the flux of heavier elements.More interestingly, our best-fit correctly reproducesthe Ne, Mg, Si and the preliminary S data using thesame injection slope as the intermediate nuclei, C and O.This can be appreciated in Fig. 2. The case of Fe nucleiwill be discussed later. In each panel we show with adashed line the result of our calculations obtained if thecontribution to the flux of that nucleus from spallationof heavier elements were neglected. This allows us to im-mediately identify the species that are mainly primariesfrom those that receive a substantial secondary contribu-tion: for instance, the Si flux is predominantly of primaryorigin (like C and O), while the Ne, Mg and S elementshave mixed origin, resembling the case of Nitrogen. In allthese cases the secondary production fills the low energypart of the spectrum, as one might easily have expected.In Fig. 3 we show the ratios of Ne, Mg and Si overO, as functions of rigidity. The choice of plotting theseratios with respect to Oxygen is due to the fact thatOxygen can be considered (together with protons andIron) pure primary species, being the secondary contri- . . . . . . . N e / M g Model Prediction1 . × Mg secondary AMS-02 R/[GV] . . . . S i / M g Model Prediction1 . × Mg secondary AMS-02
FIG. 4. Ratios of Ne and Si over Mg compared to recentAMS-02 data. The shadowed region indicates the effect ofcross section uncertainties on the model. The red line showsthe case with higher Mg production cross sections. bution from heavier nuclei negligible at all energies. Allof these ratios are in good agreement with measurementsfor R (cid:38)
10 GV.Although Mg/O and Si/O slightly overshoot the dataat very high rigidity, the experimental uncertainties ofthese data points are quite large, and we do not see thisas evidence of a problem at this point. Moreover, for thecase of the Mg/O ratio, we explored the effect of a possi-ble increase by 30% in the production cross section of Mg(solid red line): the increase of the cross section wouldallow to slightly reduce the normalization of the primarycontribution to the flux of Mg, and hence to a somewhatbetter agreement with data points. Similar considera-tions can be made for other heavy nuclei, for which thepartial cross sections for spallation are known with com-parably large uncertainty. In turn, this might be seenas a possible indication that our face values for at leastsome of these cross sections, might be underestimated.The instance of Mg is a good illustration of the pointalready made earlier in this section: the shaded areasonly catch part of the effect due to the uncertainties inthe spallation partial cross sections. Changing these val-ues leads to a change in the normalization of the primarycontribution that is not accounted for in the shaded ar-eas.In figure 4 we also show the ratios of the Ne and Sifluxes over the Mg flux. Again, the agreement with datais remarkable. If to look for weak points, one could iden-tify one in the Ne/Mg ratio, at high rigidities. However,even this ratio is better described in terms of a mild in-crease in the production cross section of Mg, as shown asa solid red line.In all these cases, it remains true that the main limita-tion to uncover the physics of CR transport and acceler-ation is in the poorly known cross sections for productionof elements through spallation reactions.
B. Iron
Iron is as close to a primary nucleus as H and O are,hence the spectrum of Iron can be considered as a reliabletest of the injection spectra derived from the analysis il-lustrated above, and of the grammage traversed by CRs.The latter enters the calculation of the equilibrium spec-trum of Fe through the severe spallation losses, whichare expected to affect the flux of Iron for R (cid:46)
100 GV,making it substantially harder than at higher rigidities.The spectra of Oxygen (top panel) and Iron nuclei(bottom panel) that we derive using our calculations areshown in figure 5, together with data points from AMS-02 and other experiments (see labels on the figure). Afew comments are due: the spectrum of Oxygen is verywell reproduced at all energies, and the spectral break at ∼
300 GV is well visible, because the transport is domi-nated by diffusion in that rigidity range. For Iron nuclei,the situation is different in that no clear evidence for abreak is visible either in the data or the prediction. Thisis expected since the transport of Fe nuclei is dominatedby spallation even at ∼
100 GV. The comparison be-tween predictions of our calculations and AMS-02 dataclearly shows that for R (cid:46)
30 GV, there is a strong dis-agreement, not reconcilable with error bars quoted by theASM-02 collaboration.Following AMS-02 [19], in figure 6 we also show theFe/O ratio as a function of kinetic energy per nucleon,which was also provided by the AMS-02 collaboration.In the same plot we show the same quantity as mea-sured by ACE-CRIS [50], HEAO3 [26] and Voyager [27].The predicted ratio of modulated fluxes as derived usingour calculations is shown as a solid (red) curve. Sincethe plot also contains a data point from Voyager, thatmeasures unmodulated quantities, we also plot the Fe/Oratio calculated with unmodulated fluxes (dashed line).The problem mentioned above at low energies is con-firmed in the Fe/O ratio, which is predicted to be appre- ciably higher than AMS-02 measurements for E kin /n (cid:46)
20 GeV/n. The discrepancy is ∼ −
40% at ∼ the weighted slab model used in our work to describeparticles’ transport neglects the grammage accumulatedby CRs in the halo. This is a good approximation aslong as the density in the halo, n H , is much smaller than n d ( h/H ) ∼ × − cm − . This appears to be a good ap-proximation given that the expected density in the halois thought to satisfy this condition [52]. We actually cal-culated the solution of the transport equation of Fe nucleiin the presence of target gas in the halo and found thatthe low energy solution changes only at the percent levelfor gas density in the halo of order 10 − cm − . Hence,it is unlikely that the discrepancy between predictionsand AMS-02 data may be attributed to oversimplifiedCR transport in the halo. The uncertainties in the spallation cross sectionsare definitely a limiting factor in all these calculations.However most uncertain are the partial cross sections forfragmentation of a nucleus A into a nucleus A (cid:48) , while thetotal cross sections are somewhat better known. Never-theless, for the sake of completeness, we tried to increasethe spallation cross section of Fe by 40%, but even suchdrastic change turned out into a bad fit to the spectrumof Fe nuclei. Despite the lack of physical support to the idea R/GV φ ( R ) · R . / [ m · s r · s · G V ( − . ) ] O fluxAMS-02ACE-CRIS(2009/03-2010/01)CREAM-II(2005/12-2006/01)HEAO3-C2(1979/10-1980/06)TRACER03(2003/12)TRACER06(2006/07)Voyager1(2012/09-2012/12)Voyager1-HET-Aend(2012/12-2015/06)Voyager1-HET-Bend(2012/12-2014/12)Voyager1-LET(2012/12-2015/06) R/GV φ ( R ) · R . / [ m · s r · s · G V ( − . ) ] Fe fluxAMS-02ACE-CRIS(2009/03-2010/01)CREAM-II(2005/12-2006/01)CRISIS(1977/05)HEAO3-C2(1979/10-1980/06)TRACER03(2003/12)TRACER06(2006/07)TRACER99Voyager1-HET-Aend(2012/12-2015/06)Voyager1-HET-Bend(2012/12-2014/12)Voyager1-LET(2012/12-2015/06)
FIG. 5. Spectrum of Oxygen (left) and Iron nuclei (right) as predicted in our calculations (solid line), with an estimate ofthe uncertainties induced by the poor knowledge of cross sections (shaded area). The data points from AMS-02 and otherexperiments are also shown. − E kin /n . . . . F e / O Fe/O unmodulatedFe/O modulatedACE-CRIS(1997/08-1998/04)ACE-CRIS(2009/03-2010/01)HEAO3-C2(1979/10-1980/06)Voyager1-LET(2012/12-2015/06)AMS
FIG. 6. Fe/O ratio of the modulated fluxes (solid line) as afunction of kinetic energy per nucleon, compared with datafrom AMS-02, ACE-CRIS and HEAO. The dashed line showsthe ratio of unmodulated fluxes, compared with the Voyagerdata. that Iron nuclei may be injected with a different injec-tion spectrum compared with other nuclei, we adoptedan agnostic attitude and tried to find a best fit injec-tion spectrum that could improve the fit (after all this iswhat is required for H and He). We find that adopting a harder injection spectrum (as required by the low energyFe AMS-02 data), leads to only a slightly better fit atlower energies, while making the high energy fit worse. Even more aesthetically unappealing is the possibil-ity that the solar modulation may take place in a differentway for Iron than for Oxygen. Imposing this unnaturalsolution, one finds that a good fit would require a Femodulation about 70% stronger than for Oxygen.In a recent article [53], the authors have noticed thesame problem with Fe nuclei when comparing AMS-02data with GALPROP predictions. It was suggested thatin order to fit the AMS-02 Iron spectrum it is necessary tointroduce multiple breaks in the injection (source) spec-trum. Even at high energies, where all nuclei are ex-pected to behave in the same way under the effect ofdiffusion, Iron is assumed to be injected with a spectrumdifferent from that of other nuclei in [53]. Moreover, inorder to accommodate the obvious disagreement of thepredicted flux with Voyager data, they suggest that thecontribution from few local sources could become dom-inant. Clearly it is difficult to disprove such a model,especially because no modeling of such contribution wasdiscussed. We notice however that the contribution oflocal sources at energies of few GeV/n seems unlikely.For an Iron nucleus at that energy the distance trav-eled under the effect of losses is d l = ( Dτ l ) / ∼ τ l is the energy loss timescale including ioiniza-tion and spallation, see also [54]), too large to expectthat something special may be happening. If the authorsof [53] were referring to a fluctuation due to accidentallylocal and recent sources, then this would in general leadto even larger fluctuations at higher energies where theglobal fits illustrated above seem to provide an excellentdescription of the data, with no need for local sources.Finally, we find the assumption of multiple breaks inthe source spectrum in [53] not only poorly based onphysical grounds, but also rather puzzling. In the ab-sence of a more physical model of these phenomena, itis probably fair to simply say that AMS-02 data on theFe spectrum remain at odds with both the present un-derstanding of CR transport and with data collected inprevious experimental endeavours. C. H and He
As discussed in past literature, the situation of H andHe is very peculiar and certainly puzzling. Here we con-firm the conclusion of previous investigations [22] thatAMS-02 data on H and He require different injectionspectra. A similar conclusion was previously reachedbased on PAMELA data [1]. It turns out that H is bestreproduced with a source spectrum with a slope whichis ≈ .
05 softer than other nuclei, similar to what wasfound in [22], while He requires a source spectrum witha slope which is ≈ .
02 harder than that of nuclei.This finding should be a source of concern, in that atpresent no real explanation of this trend exists and cer-tainly this result was not expected. Even models whichsuggest a different injection slope for the two cannot ex-plain the difference in the injection spectra between Heand intermediate mass nuclei, such as C and O [55, 56].Our best-fit to the H and He spectra is shown in figure7, where, as usual, the shaded regions provide an estimateof the effect of uncertainties in the cross sections. Thedashed line shows the primary contribution alone to theHe flux, and its distance from the solid line shows thesecondary contribution (take into account that the fluxplotted in the figure is the sum of He and He, and thelatter is also produced in spallation reactions of He).
V. CONCLUSIONS
The AMS-02 data provide us with the tools for whatcould be the most substantial step forward in our under-standing of the origin of CRs. In the present article wediscussed in detail the implications of these data in termsof CR transport in the Galaxy.The general framework that emerges from our calcula-tions is one in which three slopes are required to describethe source spectra for elements of different masses: asteep injection for protons, one sightly harder spectrumfor nuclei heavier than He and one even harder for Heitself. The difference between the slopes is at the levelof few percents. This might seem a small difference, butthe high precision data now available force us to providea description of physical processes at this level. Taken atface value, this difference, however small, imposes a big
FIG. 7. Fluxes of H and He. The model parameters are thesame of the last section but the injection slope of H is 4.37and of He is 4.31. As in the other plots, the dashed line showsthe predicted flux if the secondary production were neglected. strain on our models of the origin of CRs: all the accel-eration processes that we have devised through the yearsare rigidity dependent, namely nuclei of the same rigid-ity should experience the same acceleration and mani-fest the same spectrum. The situation is somewhat morecomplex, in that the spectrum of particles released intothe ISM is not the same as at the acceleration site (seefor instance [57–59]). But it remains true that nuclei ofdifferent mass but the same rigidity should escape theacceleration region in the same way.It has also been speculated that the integration overtime during a SNR evolution might give rise to a differ-ence in the spectra of H and He [55, 56]. However, asidefrom the details of the model, it predicts that He and R [GV] − . − . − . − . − . s l o p e HCONeMgSiSFe
FIG. 8. Slope of the spectra of nuclei with different mass asa function of rigidity. heavier nuclei should share the same spectrum.It may also be speculated that different types of sourcesmay release particles with slightly different spectra anddifferent relative abundances (for instance some sourcesmay be He-rich), but such solutions all appear very fuzzyat the present time, and it should be admitted that noclear understanding of the origin of the different sourcespectra for H, He and heavier nuclei is yet available.Once it is accepted that three injection slopes are nec-essary to describe observations, we have been able toshow that the spectra of nuclei are all properly accountedfor, with the possible exception of Iron, that will be dis-cussed later. In order to reach such a conclusion it is ofthe utmost importance that all chains of spallation reac-tions and radioactive decays are taken into account. Thecross sections for these processes, or rather the poornesswith which such cross sections are currently known, rep-resent the chief limitation to our capability to extractphysical information from available data. For some nu-clei, for instance Mg, the comparison between observedand predicted spectra would benefit from slight increasesin the production cross sections, compatible with knownuncertainties.The spectra that result from propagation also have atrend in terms of observed spectral shape, as illustratedin figure 8: 1) for increasingly more massive nuclei, thereis a trend towards globally harder spectra, as a result ofthe enhanced role of spallation. 2) At low energies, wherespallation reactions play a more important role the slopedecreases for heavier nuclei. 3) For rigidities R (cid:29) all nuclei heavier than He tend to have the same spectralslope, since at such rigidities CR transport is dominatedby diffusion, which behaves in the same way for all nucleiat given rigidity. Small differences around these trends can be easily identified (and have been discussed above)for nuclei that are substantially contaminated by the con-tribution of spallation from heavier nuclei (for instancethis is the case of N [60]). For such nuclei, deviations arevisible at low rigidities where the secondary contributionbecomes important.The global picture of CR transport that emerges fromour calculations is that, aside from energy losses, the phe-nomenon is well described by diffusion and advection. Afew comments on both ingredients may be useful: thecharacter of particle diffusion is fully reconstructed usingthe proton spectrum and secondary/primary ratios, inaddition to the Be/B ratio that provides an estimate ofthe halo size [22, 35]. Both these pieces of observationssuggest that the spectral breaks observed in spectra at ∼
300 GV are due to a corresponding break in the diffu-sion coefficient. Such a break was proposed to originatefrom a transition from self-generated to pre-existing tur-bulence in the Galaxy [30, 61, 62]. At rigidity R (cid:46) R (cid:46)
30 GV. In this region the discrepancybetween our predictions and the measured spectra are atthe level of ∼ − ACKNOWLEDGEMENTS
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