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Prepared for submission to JCAP

Secondary cosmic-ray nucleus spectra

disfavor particle transport in the

Galaxy without reacceleration

Qiang Yuana,b,c1, Cheng-Rui Zhua,d, Xiao-Jun Bie,d2, Da-Ming Weia,b

aKey Laboratory of Dark Matter and Space Astronomy, Purple Mountain Observatory,Chinese Academy of Sciences, Nanjing 210008, ChinabSchool of Astronomy and Space Science, University of Science and Technology of China,Hefei, Anhui 230026, ChinacCenter for High Energy Physics, Peking University, Beijing 100871, ChinadUniversity of Chinese Academy of Sciences, Beijing 100049, ChinaeKey Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academyof Sciences, Beijing 100049, China

E-mail: yuanq@pmo.ac.cn, zhucr@pmo.ac.cn, bixj@ihep.ac.cn, dmwei@pmo.ac.cn

Abstract. The precise observations of Galactic cosmic ray fluxes of the secondary family,such as Li, Be, B, are expected to have significant implications on our understanding of thecosmic ray origin and propagation. Here we employ the recent very precise measurementsof those species by the Alpha Magnetic Spectrometer on the International Space Station,together with their parent species (C and O), as well as the data collected by the Voyager-1spacecraft outside the heliosphere and the Advanced Composition Explorer, to investigate thepropagation of cosmic rays in the Milky Way. We consider the diffusion of cosmic rays plusreacceleration or convection effect during the propagation, and find that the reaccelerationmodel can fit the data significantly better than the convection model. We further find that forthe reacceleration model, the spectral hardenings of both the primary and secondary particlescan be well described by the injection hardening without including additional propagationhardening. This is due to that the reacceleration effect results in a steeper secondary-to-primary ratio at low energies, and can thus naturally reproduce the fact that the secondaryspectra harden more than the primary spectra found by AMS-02.

ArXiv ePrint: 1810.03141

1Corresponding author.2Corresponding author.

Contents

1 Introduction 1

2 Methodology 2

3 Results 3

4 Summary and discussion 6

1 Introduction

It has been well established that charged cosmic rays (CRs) propagate diffusively in the MilkyWay, and interact with the interstellar medium, fields, and plasma waves. Such interactionswould leave imprints on their spectra and produce secondary particles and radiation [1].Precise measurements of CR energy spectra, particularly the secondary class such as Li, Be,B or Sc, Ti, V, Cr, Mn are crucial to probing the propagation and interactions of CRs.Using improved data acquired in recent years by, e.g., PAMELA and AMS-02, we do havesignificantly improved constraints on the CR propagation parameters [2–13]. Nevertheless,the big patterns of the CR propagation are still under debate, and those constraints on themodel parameters are model-dependent.

Other than the diffusion in turbulent magnetic fields and the inelastic collisions with theinterstellar gas, CR particles are widely postulated to experience convective propagation [14]which is believed to be the source driving Galactic stellar winds and/or stochastic reaccelera-tion by randomly moving magnetohydrodynamic (MHD) waves [15]. Previous measurementsare not precise enough to discriminate these two classes of model configurations, althoughseveral recent studies showed hints that the reacceleration model was somehow favored com-pared with the convection one [7, 10]. There were also uncertainties from the entanglementof the propagation model with assumptions of the source injection and solar modulation.

Here we investigate this problem with the most recently published data on Li, Be, Band their parent nuclei C and O by AMS-02 [16, 17]. To better constrain the model, thelow-energy measurements of these nuclei by Voyager-1 out of the heliosphere [18], and themedium-energy measurements by the ACE-CRIS instrument at top of the atmosphere arealso used. We keep our studies in a relatively simple framework of particle transportation withuniform, isotropic diffusion of CRs, although spatial variations and/or anisotropic diffusionmay be indicated by some recent observations [19–21]. As for the solar modulation effectof CRs in the heliosphere, we employ the force-field approximation [22]. In spite that morecomplicated model settings may be relevant when confronting all available data such as thediffuse γ-rays, our results show that this simple framework may be enough to account forthe CR data which are largely observed near the Earth. The function form of the injectionspectrum is poorly known. Typically broken power-law forms are assumed to describe theinjection spectrum of accelerated CRs. In this work we adopt a non-parametric, splineinterpolation method to describe the particle’s injection spectrum. This can also minimizethe impact of an improper assumption of the injection spectrum on the determination of thepropagation parameters.

– 1 –

2 Methodology

The propagation of CRs is described by a set of coupled diffusion equations, including theenergy losses and fragmentations due to interactions in the medium, the reacceleration and/orthe convection effects. For details about the propagation equations one can refer to Ref. [1].With proper simplifications the propagation equations can be solved analytically (e.g., [23]).However, for general purposes and realistic astrophysical ingredients, a numerical way isusually necessary [24–27]. We use the numerical code GALPROP to calculate the propagation ofCRs [24, 25]. A cylindrically symmetric geometry of the Galaxy is assumed, with a maximumradius rmax = 20 kpc and a height of 2zh with zh free to be fitted. It has been shown thata three-dimensional configuration of the propagation geometry does have some impacts onthe results of propagation [28–30]. We expect that such effects will affect simultaneously onboth models we are going to compare with, and thus neglect such a complexity.

The diffusion coefficient is parameterized as D(ρ) = βηD0(ρ/ρ0)δ, where ρ is the rigidity

of CR particles, D0 is (approximately) the diffusion coefficient at ρ0 = 4 GV, δ is the rigidity-dependence slope, β is the velocity in unit of light speed, and the βη term is employed toempirically describe possible resonant scatterings of CRs off the MHD waves [31, 32]. We willalso test the case with a high-rigidity break of the diffusion coefficient, which was suggestedto account for the spectral breaks of both the primary and secondary nuclei around 200GV, e.g., [33]. In this case two more parameters of the diffusion coefficient, ρbr and δ′,are added. The reacceleration effect is described by a diffusion in the momentum space,and the Alfven velocity (vA) of the MHD waves is employed to characterize the strength ofreacceleration [15]. The convection is assumed to be perpendicular to the Galactic plane,and the convection velocity is parameterized as a linear function of the vertical height to theGalatic plane, Vc = z · dVc/dz.

To minimize the impact of the parameterization of the injection spectrum of primaryCRs, we adopt a non-parametrized method by means of spline interpolation among a fewchosen rigidity knots [34, 35]. The (logarithmical) flux normalizations at such knots are fittedtogether with other free parameters. In this work we take 7 rigidity knots1 logarithmicallyevenly distributed between 0.2 GV and 2 TV, which covers the energy ranges of observationsfrom Voyager-1 to AMS-02. The abundance ratio between primary nuclei C and O is de-scribed by a constant factor ξO. In addition, there might be uncertainties of fragmentationcross sections to produce secondary nuclei [36], which would degenerate with the propagationparameters. In this work we fix the cross section of the Boron production, and multiply twoconstants2 ξLi and ξBe to the predicted fluxes of Li and Be. Finally, the spatial distributionof CRs is assumed to follow the observed distribution of supernova remnants [2].

To link the ACE-CRIS and AMS-02 measurements near the Earth with the Voyager-1measurements outside the heliosphere, the force-field solar modulation model is employed.The modulation potential is set as a free parameter. Note that we extracted the ACE-CRISdata with the same time period as that by the AMS-02 measurements of nuclei Li, Be, B, C,O, i.e., from May, 2011 to May, 2016 [34].

In total there are 16 or 18 free parameters in the model: 5 or 7 for the modeling ofpropagation (D0, δ, η, zh, vA or dVc/dz), including additionally (ρbr, δ

′) for the case with a

1We also test the fits with different numbers of knots, which result in slightly different values of the fittingmodel parameters and χ2, without affecting all the conclusions in this work.

2The uncertainties of the cross sections may be energy-dependent (e.g., [37]). Here we neglect such com-plexitites due to the lack of precise knowledge about the cross sections.

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break of the diffusion coefficient, 7 for the injection spectrum, 3 for normalizations of Li, Be,and O, and 1 for solar modulation. It is challenging to fit all of those parameters together,due to the degeneracies among many of them. Therefore we adopt an iterative approach. Westart with a given set of propagation parameters (e.g., those given in Ref. [10]), and fit theinjection parameters plus the normalization factors. Then we take the best-fit injection andnormalization parameters as inputs, and re-fit the propagation parameters. We repeat theprocedure until the results get stable. We use the CosRayMC tool which employs the MarkovChain Monte Carlo method to do the fits [38, 39].

3 Results

The data used in this work include the fluxes of Li, Be, B, C, and O measured by AMS-02[16, 17] and Voyager-1 [18], the B, C, and O fluxes measured by the ACE-CRIS [34]. The oldmeasurements of the Beryllium ratios [40–44] are also included to give a loose constraint onthe halo height. We are aware that it is crucial to using the simultaneous measurements inthis study. However, since the new measurements of the Beryllium ratios are not available yet,we can only use those old data at the current stage. For those old data of 10Be/9Be ratios, weapply a force-field solar modulation with a modulation potential of Φ ≈ ΦAMS−02− 0.2 = 0.5GV [7].

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Figure 1. The 1-D and 2-D distributions of the propagation parameters for the convection (left) andreacceleration (right) models. The units of the parameters are given in Table 1.

The fitting results of the main propagation model parameters are given in Table 1. Theone-dimensional (1-D) and two-dimensional (2-D) distributions of the propagation parame-ters are shown in Fig. 1. It is clear that the reacceleration model fits the data significantlybetter than the convection model. For the convection model, the minimum χ2 value is about445.9 for a number of degree-of-freedom (dof) of 403, which gives a p-value of 0.07. As forthe comparison between the reacceleration and convection models, the Akaike informationcriterion (AIC) gives a difference of the AIC values of 140.5, which suggests that the con-vection model is about exp(−140.5/2) = 3.1× 10−31 times as probable as the reaccelerationone. We note that the use of the χ2 values to compare the models or judge the goodness-of-fit of specific models should be cautious, since the reported systematic uncertainties ofthe measurements have been added in quadrature with the statistical errors and the possible

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Table 1. Posterior mean and 68% credible uncertainties of the main model parameters.

Parameter Unit Convection Reacceleration

D0 (1028cm2s−1) 4.10 ± 0.34 7.69 ± 0.60δ 0.477 ± 0.003 0.362 ± 0.004zh (kpc) 4.93 ± 0.53 6.27 ± 0.72vA (km s−1) ... 33.76 ± 0.67

dVc/dz (km s−1 kpc−1) < 0.86† ...η −1.51 ± 0.03 −0.05 ± 0.04

χ2min/dof 445.9/403 305.4/403

ξLi 1.18 ± 0.01 1.18 ± 0.01ξBe 1.00 ± 0.01 1.00 ± 0.01Φ (GV) 0.742 ± 0.010 0.705 ± 0.009

†95% upper limit.

correlation among systematic uncertainties [37, 45] have not been taken into account in thecalculation of the χ2. The chi-squared value of the reacceleration model is 305.4/403, whichseems to be a too good fit. We expect that this may also be related with the correlated sys-tematical uncertainties. A full description of the covariance matrix is currently challengingwithout knowing the details of the analysis.

Fig. 2 shows the comparison between the best-fit model calculated fluxes and the data,for Li, Be, B, C, and O nuclei, respectively. We find that the convection model gives on aver-age larger residuals than the reacceleration model. Furthermore, the value of δ is larger in theconvection model than that in the reacceleration model. It is known that the reaccelerationeffect would make the bump feature of the secondary-to-primary ratio more prominent, andthus δ can be smaller [24]. Therefore the predicted secondary fluxes by the convection modelare generally softer than that by the reacceleration one.

The spectral hardenings above a few hundred GeV/n are found in both the primary andsecondary fluxes [16, 17, 46–50]. A direct fitting to the secondary-to-primary ratios results ina change of the slopes for rigidity intervals of 60.3− 192 and 192− 3300 GV [17], suggestingthat a propagation mechanism is responsible to the spectral hardenings [19, 20, 51]. Similarconclusion was also found in Ref. [33] for a fitting to the B/C data taking into account thepropagation model.

In above we assume a single power-law form of the diffusion coefficient at high rigidities.Clear spectral hardenings can be seen in the fitting results for both primary and secondarynuclei (see Fig. 2), which are expected to be due to the hardening of the injection spectrum.Fig. 3 shows the 95% range of the injection spectra for the convection and reaccelerationmodels. Note that the injection spectra are normalized at ∼ 4.3 GV to show the spectralshapes. These non-parametric injection spectra turn out to be similar with broken power-lawsusually assumed. The spectra experience a softening at ∼GV rigidities and a hardening aboveseveral hundred GV. The physical origin of such spectral shapes would be very important inunderstanding the acceleration and/or confinement of CRs at source. Note that the γ-rayemission from supernova remnants also suggests broken power-law forms of particles aroundGeV energies [52]. Possible physical mechanisms include the strong ion-neutral collisionsnear the shock fronts [53] or the escape of particles from/into finite-size regions [54, 55].The high-energy hardening may be due to the superposition of various sources [56], or thenon-linear acceleration [57].

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Figure 2. Comparison between the best-fit model fluxes and the measurements. The lower sub-panelin each panel shows the residuals (filled for the reaccelerationmodel and open for the convection model)defined as (data−model)/error.

We then check that whether the data require an additional break of the diffusion coef-ficient or not. Two more parameters, the break rigidity ρbr and the high energy slope δhe,have been added in the model3. We find that for the reacceleration model, the additionof the break of the diffusion coefficient improves the fitting very slightly (with a minimumχ2br

value of 303.7). For the convection model, however, a moderate improvement is found(χ2

br = 389.6). Such a difference is anticipated as can be seen from Fig. 2. For the reac-

3Note that the momentum diffusion coefficient depends on the parameter δ [15]. Here the break rigidity isrestricted to be larger than 100 GV, where the reacceleration effect is expected to be small. Therefore onlythe low rigidity slope δ is relevant to the reacceleration.

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Figure 3. Injection spectra for the convection and reacceleration models. For each model we select50 samples within the 95% credible region of the parameter space.

celeration model, the reacceleration effect makes the low energy secondary-to-primary ratiossteeper than that at high energies, which automatically gives the hardening behaviors asfound by AMS-02. Also the δ value of the rigidity-dependence of the diffusion coefficient ofthe reacceleration model is relatively small, which is close to the high energy slope of thesecondary-to-primary ratios. On the other hand, a larger value of δ in the convection modelis required, basically to fit the low energy data. In this case an additional hardening of thediffusion coefficient at high energies is required. In this sense our results are consistent withthat given in Ref. [33], in which a minor reacceleration effect was assumed (with a prior rangeof [0, 10] km s−1 for vA) and a relatively larger value of δ ∼ (0.5 − 0.7) was obtained. Weregard this as a further support of the reacceleration model being a natural explanation ofthe hardenings of the secondary-to-primary ratios revealed by the AMS-02 measurements.

From the fittings we find that a re-normalization factor of ∼ 1.2 is required for the Licomponent. For Be nuclei the re-normalization factor is very close to 1. This is possibly dueto the uncertainties of the fragmentation cross sections of nuclei, either for the productions ofLi or B. Note, however, as pointed out in Ref. [58] it may be unlikely that the cross sectionsfor all the channels to produce Li nuclei are lower by ∼ 20% at the same time. An alternativeexplanation is the existence of primary acceleration sources of Li which are presumably 7Lisources such as novae or type Ia supernovae [58, 59].

4 Summary and discussion

In this work we study the CR propagation models with the newest precise measurementsof both the primary and secondary CR fluxes by AMS-02, ACE-CRIS, and Voyager-1. Theframework as adopted in the GALPROP tool has been employed, which inculdes the diffusion,energy losses, and fragmentations of CRs. We focus on two different propagation models, onewith reacceleration of CRs due to scattering off randomly moving MHD waves in the MilkyWay and the other with global convective propagation from the Galactic disk to the poles.To minimize the effect of the assumption of injection spectrum on the propagation study, weuse a non-parametric method with spline interpolation to describe the injection spectrum.

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With those approaches, we find that the propagation model with reacceleration fits thedata significantly better than that with convection. The best-fit χ2 value of the reaccelera-tion model is smaller by about 140.5 than the convection model, with the same numbers ofdegrees of freedom. This conclusion can only be achieved with significantly improved mea-surements of the data in a very wide energy range. Also the improvement of the analysismethod with a global fitting tool and the elimination of dependence of the source injectionmodels are helpful. Using the PAMELA data of Z ≤ 2 nuclei, Ref. [11] studied the CRpropagation, which also showed that the model with reacceleration fits the data better thanthat of convection, although a combination of reacceleration and convection seems to give thebest fit. In Refs. [13, 37], the authors used the AMS-02 data of secondary-to-primary ratiosto test the propagation models under the semi-analytical model, and found that the modelwith reacceleration and convection or the purely diffusion one can describe fairly equallywell of the data. We note that there are some differences of the model settings and datausage between their works and ours. For example, the propagation geometry is differentbetween two works (one-dimension versus two-dimension). The energy-dependent treatmentof the uncertainties of the fragmentation cross sections in Refs. [13, 37], and the usage ofthe low-energy data in this work may also lead to differences. We expect that further effortsin clarifying effects of those different approaches and reducing the uncertainties of the crosssections will be particularly helpful in understanding the propagation of CRs.

We further find that for the reacceleration model, no high-rigidity (O(102) GV) hard-ening of the diffusion coefficient is necessary to account for the spectral hardenings of boththe primary and secondary nuclei. The observed spectral hardenings can be largely due tothe hardening of the injection spectrum. The reacceleration effect automatially explain thatthe secondary CR spectra harden more than the primary ones through a steepening of thelow-energy part of the spectra. For the convection model, the inclusion of a spectral breakof the diffusion coefficient around hundreds of GV rigidities can improve the fit moderately.Compared with the reacceleration model, the convection model still fits the data much poorereven with such a break of the rigidity dependence of the diffusion coefficient.

Antiprotons can also be used as a diagnostics of the propagation model. We have donefittings to the proton spectra measured by Voyager-1 [18], AMS-02 [49], and DAMPE [60],and the antiproton spectrum by AMS-02 [61] with the convection and reacceleration modelsbased on the mean propagation parameters given in Table 1. The proton injection spectrumis approached with the similar spline interpolation method. An additional re-normalizationparameter of the calculated secondary antiproton spectrum and a force-field solar modulationpotential are employed. The fit of the reacceleration model gives a minimum χ2

p+p = 243.4for a dof of 151, which is better than that of the convection model (with a minimum χ2

p+p =316.1). Both fits are not good enough. Note, however, there are complications from theuncertainties of the propagation parameters, the antiproton production cross section, thecharge-sign dependence of the solar modulation, and/or possible exotic contribution frome.g., the dark matter. More careful dedicated studies will be helpful.

Although there are quite a number of discussions in literature to extend the propagationof CRs with more complicated configurations, such as the spatial variation of the propagationproperties [19, 20] and the anisotropic diffusion with respect to ordered magnetic fields [21],our results show that a simple two-dimensional, isotropic, and uniform propagation scenariocan give quite good description to the locally measured CRs. Better understanding of theCR transport may be achieved with more precise measurements of the CR distribution inthe Galaxy, by e.g., γ-rays.

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Acknowledgments

We thank V. Ptuskin for helpful discussion. This work is supported by the National KeyResearch and Development Program of China (No. 2016YFA0400200), the National NaturalScience Foundation of China (Nos. 11722328, 11851305, U1738205, U1738209), the KeyResearch Program of Frontier Sciences of Chinese Academy of Sciences (No. QYZDJ-SSW-SYS024). QY is also supported by the 100 Talents program of Chinese Academy of Sciencesand the Program for Innovative Talents and Entrepreneur in Jiangsu.

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