Constraining Galactic cosmic-ray parameters with Z<=2 nuclei
aa r X i v : . [ a s t r o - ph . GA ] A ug Astronomy & Astrophysics manuscript no. quartet c (cid:13)
ESO 2018October 2, 2018
Constraining Galactic cosmic-ray parameters with Z ≤ nuclei B. Coste , L. Derome , D. Maurin , and A. Putze Laboratoire de Physique Subatomique et de Cosmologie, Universit´e Joseph Fourier Grenoble 1, CNRS/IN2P3, InstitutPolytechnique de Grenoble, 53 avenue des Martyrs, Grenoble, 38026, France The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, AlbaNova, SE-10691 Stockholm, SwedenReceived / Accepted
ABSTRACT
Context.
The secondary-to-primary B/C ratio is widely used to study Galactic cosmic-ray propagation processes. The H/ He and He/ He ratios probe a different
Z/A regime, therefore testing the ‘universality’ of propagation.
Aims.
We revisit the constraints on diffusion-model parameters set by the quartet ( H, H, He, He), using the mostrecent data as well as updated formulae for the inelastic and production cross-sections.
Methods.
The analysis relies on the USINE propagation package and a Markov Chain Monte Carlo technique to estimatethe probability density functions of the parameters. Simulated data are also used to validate analysis strategies.
Results.
The fragmentation of CNO cosmic rays (resp. NeMgSiFe) on the ISM during their propagation contributes to20% (resp. 20%) of the H and 15% (resp. 10%) of the He flux at high energy. The C to Fe elements are also responsiblefor up to 10% of the He flux measured at 1 GeV/n. The analysis of He/ He (and to a less extent H/ He) data showsthat the transport parameters are consistent with those from the B/C analysis: the diffusion model with δ ∼ . V c ∼
20 km s − (galactic wind), V a ∼
40 km s − (reacceleration) is favoured, but the combination δ ∼ . V c ∼
0, and V a ∼
80 km s − is a close second. The confidence intervals on the parameters show that theconstraints set by the quartet data are competitive with those brought by the B/C data. These constraints are tighterwhen adding the He (or H) flux measurements, and the tightest when further adding the He flux. For the latter, theanalysis of simulated and real data show an increased sensitivity to biases. Using secondary-to-primary ratio along witha loose prior on the source parameters is recommended to get the most robust constraints on the transport parameters.
Conclusions.
Light nuclei should be systematically considered in the analysis of transport parameters. They bringindependent constraints which are competitive with those obtained from the B/C analysis.
Key words.
Astroparticle physics – Methods: statistical – ISM: cosmic rays
1. Introduction
Secondary species in Galactic cosmic rays (GCRs) are pro-duced during the CR journey from the acceleration sitesto the solar neighbourhood, by means of nuclear inter-actions of heavier primary species with the interstellarmedium. Hence, they are tracers of the CR transport inthe Galaxy (e.g., Strong et al. 2007). Studying secondary-to-primary ratios is useful as it factors out the ‘un-known’ source spectrum of the progenitor, leaving H/ He, He/ He, B/C, sub-Fe/Fe—and recently ¯ p/p (Putze et al.2009; di Bernardo et al. 2010)—suitable quantities to con-strain the transport parameters for species Z ≤ A/Z ∼
2, and inthat respect, He/ He is unique since it probes a differentregime and allows to address the issue of the ‘universality’of propagation histories. For instance, in an analysis in theleaky-box model (LBM) framework, Webber (1997) foundthat He/ He data imply a similar propagation history forthe light and heavier species (which was disputed in earlierpapers). Webber also argued that the situation with regardto the H/ He ratio is less clear, because the uncertaintieson the measurements are large (mainly due to instrumen-tal and atmospheric corrections). H and He spectra are themost abundant species in the cosmic radiation, and thus H Send offprint requests to : B. Coste, [email protected] and He are the most abundant secondary species in GCRs.However, achieving a good mass resolution—especially athigh energy—is experimentally challenging. This explainswhy the elemental B/C ratio received more focus both ex-perimentally and theoretically (thanks to its higher preci-sion data w.r.t. to the quartet data).From the modelling side, after the first thor-ough and pioneering studies performed in the 60’s-70’s(Badhwar & Daniel 1963; Ramaty & Lingenfelter 1969;Meyer 1972; Mitler 1972; Ramadurai & Biswas 1974;Mewaldt et al. 1976), the interest for the quartet nucleisomewhat stalled. Several updated analyses of the prop-agation parameters from the quartet were published asnew data became available (see Table A.1 for references).However, very few dedicated studies were carried out in the80’s (Beatty 1986; Webber et al. 1987), likewise in the 90’s(Webber 1990a; Seo & Ptuskin 1994; Webber 1997), andnone in the 00’s. This is certainly related to the very slowpace at which new data became available in this period.Curiously, the most recent published data have not reallybeen properly interpreted, i.e. for H/ He data, IMAX92(de Nolfo et al. 2000) and AMS-01 (Aguilar et al. 2011);and for He/ He data, IMAX92 (Menn et al. 2000), SMILI-II (Ahlen et al. 2000), AMS-01 (Xiong et al. 2003), BESS98(Myers et al. 2003), CAPRICE98 (Mocchiutti et al. 2003).Furthermore, almost all analyses have been performed in
1. Coste et al.: Constraining Galactic cosmic-ray parameters with Z ≤ the successful but simplistic LBM, but in a few studies .At the same time, the analysis of the B/C ratio has beenscrutinised in more details. For instance, to replace the oldusage of matching the data by means of an inefficient man-ual scan of the parameter space (e.g., Jones et al. 2001),more systematic scans were carried out (on the B/C andsub-Fe/Fe ratio) to get best-fit values as well as uncertain-ties on the parameters (Maurin et al. 2001; Lionetto et al.2005; Evoli et al. 2008; di Bernardo et al. 2010). A recentimprovement is the use of Markov Chain Monte Carlo(MCMC) techniques to directly access the probability-density function (PDF) of the GCR transport and sourceparameters (Putze et al. 2009, 2010, 2011; Trotta et al.2011).In this paper, we revisit the constraints set by the quar-tet nuclei and their consistency with the results of heaviernuclei. In the context of the forthcoming PAMELA andAMS-02 data on these ratios, we also discuss the strategyto adopt and intrinsic limitations of the transport parame-ters reconstruction. For that purpose, we take advantage ofthe data taken in the last decade as well as simulated dataof any precision, and analyse them with an MCMC tech-nique implemented in the USINE propagation code. Thisextends and complements analyses of the B/C and primarynuclei (Putze et al. 2010, 2011) in a 1D diffusion model.The paper is organised as follows. In Sect. 2, we brieflyrecall the main ingredients of the 1D diffusion model andthe MCMC analysis. We also list the parameters which areconstrained. The simulated data and their analysis are de-scribed in Sect. 3. The analysis of the real data is givenin Sect. 4. We conclude in Sect. 5. Appendix A gathers thedata sets and the updated cross-sections used in the quartetanalysis.
2. MCMC technique, propagation and parameters
The MCMC technique and its use in the USINE propaga-tion code is detailed in Putze et al. (2009) and summarisedin Putze et al. (2010). The full details regarding the 1Dtransport model can be found in Putze et al. (2010). Below,we only provide a brief description.
The MCMC method, based on Bayesian statistics, is usedto estimate the full distribution (conditional PDF) givensome experimental data and some prior density for theseparameters. Our chains are based on the Metropolis-Hastings algorithm, which ensures that the distribution ofthe chain asymptotically tends to the target PDF.The chain analysis refers to the selection of a subsetof points from the chains (to get a reliable estimate of thePDF). The steps at the beginning of the chain are discarded(burn-in length) if they are too far of the region of interest.Sets of independent samples are obtained by thinning thechain (over the correlation length). The final results of the Seo & Ptuskin (1994) used a 1D diffusion model with reac-celeration whereas Webber & Rockstroh (1997) relied on aMonte Carlo calculation; both studies conclude similarly (con-sistency with the grammage required for heavier species toproduce the light secondaries). A preliminary effort basedon the GALPROP propagation code was also carried out inMoskalenko et al. (2003).
MCMC analysis are the joint and marginalised PDFs. Theyare obtained by counting the number of samples within therelated region of the parameter space.
The Galaxy is modelled to be an infinite thin disc of half-thickness h , which contains the gas and the sources of CRs.The diffusive halo region (where the gas density is assumedto be equal to 0) extends to + L and − L above and belowthe disc. A constant wind V ( r ) = sign( z ) · V c × e z , perpen-dicular to the Galactic plane, is assumed. In this framework,CRs diffuse in the disc and in the halo independently oftheir position. Such semi-analytical models are faster thanfull numerical codes (GALPROP and DRAGON ), whichis an advantage for sampling techniques like MCMC ap-proaches. The differential density N j of the nucleus j is a functionof the total energy E and the position r in the Galaxy.Assuming a steady state, the transport equation can bewritten in a compact form as L j N j + ∂∂E (cid:18) b j N j − c j ∂N j ∂E (cid:19) = S j . (1)The operator L (we omit the superscript j ) describes thediffusion K ( r , E ) and the convection V ( r ) in the Galaxy,but also the decay rate Γ rad ( E ) = 1 / ( γτ ) if the nu-cleus is radioactive, and the destruction rate Γ inel ( r , E ) = P ISM n ISM ( r ) vσ inel ( E ) for collisions with the interstellarmatter (ISM), in the form L ( r , E ) = − ∇ · ( K ∇ ) + ∇ · V + Γ rad + Γ inel . (2)The coefficients b and c in Eq. (1) are respectively firstand second order gains/losses in energy, with b ( r , E ) = (cid:10) dEdt (cid:11) ion , coul . − ∇ . V E k (cid:18) m + E k m + E k (cid:19) (3)+ (1 + β ) E × K pp ,c ( r , E ) = β × K pp . (4)In Eq. (3), the ionisation and Coulomb energy lossesare taken from Mannheim & Schlickeiser (1994) andStrong & Moskalenko (1998). The divergence of theGalactic wind V gives rise to an energy loss term re-lated to the adiabatic expansion of cosmic rays. The lastterm is a first order contribution in energy from reacceler-ation. Equation (4) corresponds to a diffusion in momen-tum space, leading to an energy gain. The associated dif-fusion coefficient K pp (in momentum space) is taken fromthe model of minimal reacceleration by the interstellar tur-bulence (Osborne & Ptuskin 1988; Seo & Ptuskin 1994). Itis related to the spatial diffusion coefficient K by K pp × K = 43 V a p δ (4 − δ ) (4 − δ ) , (5)where V a is the Alfv´enic speed in the medium.We refer the reader to App. A of Putze et al. (2010) forthe solution to Eq. (1) in the 1D geometry. http://galprop.stanford.edu/ ∼ maccione/DRAGON/
2. Coste et al.: Constraining Galactic cosmic-ray parameters with Z ≤ Table 1.
Models tested in the paper.
Model Transport parameters DescriptionII { K , δ, V a } Diffusion + reaccelerationIII { K , δ, V c , V a } Diff. + conv. + reac.
Note 1.
For the sake of consistency, the model identification followsthat of Putze et al. (2009, 2010, 2011) and Maurin et al. (2010).
The exact energy dependence of the source and transportparameters is unknown, but they are expected to be powerlaws of R = pc/Ze (rigidity of the particle).The low-energy diffusion coefficient requires a β = v/c factor that takes into account the inevitable effect of parti-cle velocity on the diffusion rate. However, the recent anal-ysis of the turbulence dissipation effects on the transportcoefficient has shown that this coefficient could increase atlow-energy (Ptuskin et al. 2006; Shalchi & B¨usching 2010).Following Maurin et al. (2010), it is parametrised to be K ( E ) = β η T · K R δ . (6)The default value used for this analysis is η T = 1. The twoother transport parameters are V c , the constant convectivewind perpendicular to the disc, and V a , the Alfv´enic speedregulating the reacceleration strength [see Eq. (5)]. The twomodels considered in this paper are given in Table 1.The low-energy primary source spectrum from accelera-tion models (e.g., Drury 1983; Jones 1994) is also unknown.We parametrise it to be Q E k/n ( E ) ≡ dQdE k/n = q · β η S · R − α , (7)where q is the normalisation. The reference low-energyshape corresponds to η S = − dQ/dp ∝ p − α ,i.e. a pure power-law).The halo size of the Galaxy L cannot be solely deter-mined from secondary-to-primary stable ratios and requiresa radioactive species to lift the degeneracy between K and L . However, the range of allowed values is still very looselyconstrained (e.g., Putze et al. 2010). As the transport andsource parameters can always be rescaled would a differ-ent choice of L assumed (see the scaling relations given inMaurin et al. 2010, where δ is shown not to depend on L ),we fix it to L = 4 kpc. This will also ease the comparison ofthe results obtained in this paper with those of our previousstudies (Putze et al. 2010, 2011).
3. MCMC analysis on artificial data sets
MCMC techniques make the scan of high-dimensional pa-rameter spaces possible, such that a simultaneously es-timation of transport and source parameters is possi-ble (Putze et al. 2009). However, transport parametersare shown to be strongly degenerated for the B/C ra-tio data in the range 0 . −
100 GeV/nuc (Maurin et al.2010), and source and transport parameters are correlated(Putze et al. 2009, 2010). For GCR data in general, thefact that primary fluxes and secondary fluxes are not mea- sured to the same accuracy can bias or prevent an ac-curate determination of these parameters: a simultaneousfit has been observed to be driven by the more accuratelymeasured primary flux (Putze et al. 2011). This, althoughstatistically correct, might not maximise the informationobtained on the transport parameters. Therefore, severalstrategies can be considered when dealing with GCR data: – a combined analysis of secondary-to-primary ratio andprimary flux to constrain simultaneously the source andtransport parameters; – a secondary-to-primary ratio analysis only, either fixingthe source parameters (i.e., using a strong prior), orusing a loose prior. – a primary flux analysis only, either fixing the transportparameters (i.e., using a strong prior), or using a looseprior.In the literature, the strong prior approach has almost al-ways been used to determine the transport or the sourceparameters. The issue we wish to address is how sensi-tive the sought parameters are to various strategies. Thisis the motivation to introduce artificial data, i.e. an idealcase study, as opposed to the case of real data where sev-eral other complications can arise (systematics in the dataand/or the use of the incorrect propagation model or solarmodulation model/level). To be as realistic as possible, we choose models that roughlyreproduce the actual data points (see Fig. 4), but alsomatch the typical energy coverage, number of data points,central value and spread (error bars) of the measurements .To speed-up the calculation and for this section only, we as-sume that all He comes from He (see Sect. 4.1 for all therelevant progenitors). No systematic errors were added al-though they may set a fundamental limitation in recoveringthe cosmic-ray parameters. In practice, the statistical errorsfor the artificial data sets correspond to the sigma of thestandard Gaussian deviations used to randomise the datapoints around their model value: He/ He was generatedwith statistical errors of 10% while He fluxes were generatedwith 1% and 10% errors, to simulate the situation whereprimary fluxes are ‘more accurately’ or ‘equally’ measured(in terms of statistics) than the secondary-to-primary ratio.The parameters of the two models used to simulate thedata are listed in the two italic lines in Table 2, denoted
Model II and
Model III . They correspond to extreme valuesof the diffusion slope δ , but which still roughly fall in therange of values found for instance from the B/C analysis(Putze et al. 2010): for Model II with reacceleration only( V c = 0), δ is generally found to fall between 0 . . Model III with convection and reacceleration, δ is generally found to fall into the 0 . − . Statistical uncertainties are smaller for primary fluxes (moreabundant than secondary fluxes), but the latter a more proneto systematics than ratios (e.g. secondary-to-primary ratios usedto fit transport parameters). The uncertainty on the H and He fluxes is a few percents(for the recent PAMELA data, Adriani et al. 2011) and severaltens of percents for the He/ He ratio. 3. Coste et al.: Constraining Galactic cosmic-ray parameters with Z ≤ Table 2.
Simulated data analysis for several
Models (input parameters in italic ) with L = 4 kpc: each line correspondsto the MCMC-reconstructed values (most-probable value, and relative uncertainties corresponding to the 68% CI) basedon a given data/parameters option (see Sect. 3.2). The last column gives the value of the best χ /d.o.f. configurationfound (corresponding to the curves shown in Fig. 1). Option: data/params η T K × δ V c V a α η S χ / d.o.f.- (kpc Myr − ) - (km s − ) (km s − ) - - - Model II1 10.0 0.2 . . . 70 2.3 1 . . . He/ He+He [1] 10 . +3 . − . . +54% − . . . . 72 . +6 . − . . +0 . − . . +13 . % − . % He/ He+He [1] 10 . +2 . − . . +3 . − . . . . 73 . +2 . − . . +0 . − . . +3 . − . . +3 . − . . +3 . − . . . . 68 . +4 . − . [2.3] [1] 0.973: He/ He [1] 11 . +13 . % − . % . +21% − . . . 39 . +68 . % − . % . +148% − . +326% − . +3 . − . . +5 . − . . . . 69 . +5 . − . [2.3] [1] 0.884: He/ He + src=prior [1] 9 . +11 . % − . % . +15 . % − . % . . . 73 . +5 . − . [1 . , .
5] [ − , +2] 1.08 Model III1.5 0.75 0.7 18 41 2.3 1 . . . He/ He+He . +42 . % − . % . +40 . % − . % . +33 . % − . . +17 . % − . % . +18 . % − . . +0 . − . . +7 . − . % He/ He+He . +2 . − . . +9 . − . . +0 . − . . +3 . − . . +9 . − . . +0 . − . . +4 . − . . +0 . − . . +4 . − . . +1 . − . . +1 . − . . +2 . − . [2.3] [1] 1.013: He/ He 1 . +8 . − . % . +89 . % − . % . +21 . % − . % . +18 . % − . % . +22 . % − . % . +708% − . % − . +50 . % − . % . +29 . % − . . +139% − . % . +24 . % − . % . +6 . − . . +14 . % − . % [2.3] [1] 0.894: He/ He + src=prior 1 . +13 . % − . % . +132% − . % . +29 . % − . . +13 . % − . . +15 . % − . % [1 . , .
5] [ − , +2] 0.96 Model III: analysis with Model II1.5 0.75 0.7 18 41 2.3 1 . . . He/ He+He [1] 13 . +3 . − . . +4 . − . . . . 126 +4 . − . . +0 . − . . +42 . % − . % He/ He+He [1] 11 . +3 . − . . +3 . − . . . . 85 +2 . − . . +0 . − . . +1 . − . He/ He + src=prior [1] 20 . +20 . % − . . +31 . % − . % . . . 107 +5 . − . % [1 . , .
5] [ − , +2] 2.1 Model II: analysis with Model III1 10.0 0.2 0 70 2.3 1 . . . He/ He+He . +5 . − . . +65% − . +24% − . +9 . − . +7 . − . +1 . − . . +12% − He/ He+He . +9 . − . . +19% − . +20% − . . +62% − . +3 . − . . +0 . − . . +7 . − He/ He + src=prior 0 . +193% − . +49% − . +50% − . +12% − . +12% − [1 . , .
5] [ − , +2] 1.05 Note 2.
A value in square brackets corresponds to the fixed value of the parameter for the analysis. An interval in square brackets correspondsto the prior used for the analysis (the posterior PDF obtained is close to the prior).
To test the impact on the reconstruction of the transport ( η T , K , δ , V a , and V c ) and/or source ( α and η S ) param-eters, we test the following combinations (data set | modelparameters) for the analysis. He / He + He data Option 1 ( σ He = 10%): transport + source ;Option 2 ( σ He = 1%): transport + source ;Option 2’ ( σ He = 1%): transport (source = ‘true’ value); He / He ratio only Option 3: transport + source ;Option 3’: transport (source = ‘true’ value); Option 4: transport (source = weak prior).We find that the He data alone cannot constrain thetransport parameters (not shown here), in agreement withPutze et al. (2011) results (strong degeneracy between α and δ , but also with K , V a , and η T ). In a first step, we used the MCMC technique to estimatethe best-fit parameters. The He/ He ratio and He flux areshown for model II (crosses) and the corresponding simu-lated data (plusses) in Fig. 1. When both the He/ He and He data are included in the fit (options 1 and 2, red dottedand red solid lines), the initial flux (crosses) is perfectly re-covered for He, and very well recovered for He/ He. Whenthe fit is only based on He/ He (options 3 and 5, magenta
4. Coste et al.: Constraining Galactic cosmic-ray parameters with Z ≤ [GeV/n] k/n E -2 -1
10 1 10 S ec ond a r y - t o - p r i m a r y r a ti o (I S ) He (simulated) H/ Input model =10%) σ Simulated data (Best-fit from:Option 1Option 2Option 3Option 3’Option 4 [GeV/n] k/n E -1
10 1 10 ] . G e V / n - s r - s - [ m . / n E × I S φ He (simulated) Input model =1%) σ Simulated data (Best-fit from:Option 1Option 2Option 3Option 3’Option 4
Fig. 1.
Analysis of simulated interstellar (IS) data sets forthe He/ He ratio (top panel) and the He flux times E . k/n (bottom panel) based on Model II ( × symbols), the pa-rameters of which are given in Table 2). The best-fit recon-structed curves correspond to the different ‘options’ givenin Sect. 3.2.dashed and blue solid lines), the initial flux is obviously notrecovered (unless the source parameters are set to the truevalue as in option 3’), but the He/ He ratio is consistentwith the data. Unsurprisingly, the associated χ /d.o.f.values (last column of Table 2) are close to 1.The MCMC analysis allows us to go further as it pro-vides the PDF of the parameters, from which the most-probable value and confidence intervals (CIs) are obtained.The results are gathered in Table 2 and Fig. 2. The vari-ous panels of the latter represent the PDFs for transportand source parameters for each ‘option’ for Model II. (Forconcision the correlation plots are not shown.) From theseplots, some arguments are in favour of a simultaneous use ofthe secondary-to-primary ratio and the primary flux (here, He/ He and He), but not all.
A simultaneous analysis ( He/ He + He) gives more strin-gent constraints on the transport parameters than onlyanalysing the secondary-to-primary ratio (compare thePDFs for the red curves and blue curves in Fig. 2 respec-tively, for K , δ , and V c ). This partly comes from the ob-served correlations between transport and source param- K δ
50 60 70 80 9050 60 70 80 90 a V Simulated data
Input param.
He and He data: He/ using = 10%) He σ Option 1 ( = 1%) He σ Option 2 (
He data only: He/ or using Option 3 (src = free)Option 3’ (src = true)Option 4 (src = weak prior) α S η Fig. 2.
Marginalised posterior PDF for the transport andsource parameters on the artificial data for Model II (thevalues for the input model are shown as thick vertical greylines in each panel). The colour code and style correspondto the five ‘options’ described in Sect. 3.2 (and used inFig. 1).eters (e.g., Putze et al. 2009). Option 2’ and option 3’with fixed source parameters show that the better CIs onthe transport parameters come from the information con-tained in primary fluxes . The same conclusions hold truefor Model III ( V c = 0), although with larger relative uncer-tainties due to the two extra transport parameters of themodel ( η T and V c ). As an illustration, we analyse data simulated from modelIII ( V c = 0) with model II ( V c = 0) and vice versa(lower half of Table 2). If we force V c = 0 (while V true c =18 km s − ), the diffusion slope goes to a low value δ ∼ . δ true = 0 . V a ∼
100 km s − ( V true a = 41 km s − ). The larger More stringent constraints on the source parameters (frommore precise data) leads to more stringent constraints on thetransport values: option 1 ( σ He = 10%) vs option 2 ( σ He = 1%). Note that the lack of constraints on the source parametersfor option 4 confirms that the secondary-to-primary ratio is onlymarginally sensitive to the source parameters (e.g., Putze et al.2011). 5. Coste et al.: Constraining Galactic cosmic-ray parameters with Z ≤ χ /d.o.f. value with respect to the one obtained fittingthe correct model easily disfavours this model. The secondtest (simulated with II, analysed with III) indicates whetherallowing for more freedom in the analysis (two additionalfree parameters η T and V c ) affects the recovery of the pa-rameters. The values of δ true = 0 . V true a = 70 km s − are recovered, while the others are systematically offset butless than 3 σ away from their true value. In this simpleexample, adding extra parameters is not an issue as the χ /d.o.f. still favours the minimal model. However, withreal data (see Sect. 4.3 and the B/C analysis of Putze et al.2010; Maurin et al. 2010), in such a situation, it is so farimpossible to conclude whether the correct model is used,due to the possible issue of multimodality and biases fromsystematics (see below). For Model II but even more for Model III (which has morefree transport parameters), a possible worry is the pres-ence of multimodal PDF distributions, which more oftenhappens for the simultaneous analysis. An example of mul-timodality is the analysis with Model II (i.e. V a = 0) ofdata simulated with Model III, which corresponds to a lo-cal minimum of the true Model III parameters. This is ofno consequence for the ideal case, but real data may suf-fer from systematics errors and/or the inappropriate solarmodulation model may be chosen. In that case, the trueminimum can be displaced, or turned into a local mini-mum (and vice versa). Measurements over the last decadesshowed that primary fluxes are more prone to systemat-ics than secondary-to-primary ratios. Primary fluxes arealso more sensitive to solar modulation than ratios. Forthese reasons, the use of secondary-to-primary data only(option 4) for the analysis, although less performant to getstringent limits on the transport parameters, is expected tobe more reliable and robust. The most robust approach to determine the transport pa-rameters (and their CIs) is to analyse the secondary-to-primary ratio using a loose but physically-motivated prioron the source parameters (option 4). This has the advan-tage of taking into account the correlations between thesource and transport parameters. The simultaneous anal-ysis is mandatory to obtain the source parameters (op-tion 1 or 2). It also brings more information on the trans-port parameters, but the primary fluxes can bias their de-termination if it suffers from systematics. We recommendsuch an analysis to be performed in addition to the directsecondary-to-primary ratio analysis, in order to get the fol-lowing diagnosis: if the range of values for the transportparameters from both analyses are – inconsistent , it indicates that the values and CIs ob-tained for the sources parameters are biased or unreli-able; – consistent , the selected propagation model may be thecorrect one, and the source parameters are then themost probable ones for this model. However, the CIs onthe transport parameters are very likely to be underes-timated if the error bars on the ratio are much largerthan the ones on the primary fluxes. Obviously, our analysis does not cover the range of allsystematics when dealing with real data. A more system-atic analysis—e.g. covering a wider family of propagationmodels, several solar modulation models, several sourcesof systematics in the data— goes far beyond the scope ofthis paper. Note that some of these effects are likely to beenergy dependent, complicating even further the analysis.With the successful installation of the AMS-02 detector onthe ISS and its expected high-precision data, these issuesare bound to gain importance.
4. Constraints from the quartet data
We now apply the MCMC technique to the analysis of realdata. We emphasise that for the artificial ones, we assumedthe He to come solely from the He fragmentation, in orderto speed up the calculation. Based on our new compilationfor the cross-section formulae (see App. B), we take intoaccount the contributions from
A > § § § At first order, the contribution to the H and He secondaryproduction from
Z > S j [see Eq. (1)]. For a secondary contribution, thesource term is proportional to the primary flux of the par-ents (which have been measured by many experiments), andto the production cross-section. Normalised to the produc-tion from He, we haveRel P → S ∝ S P S He ∝ Φ P Φ He · γ S P , (8)where P is the CR projectile, S is the secondary fragmentconsidered, and γ SP [see Eq. (B.3)] is the production cross-section relative to the production from He. The fractionalcontribution f P → S for each parent is defined to be f P → S = Rel P → S P P ′ =He ··· Ni Rel P ′ → S . (9)As seen from Table 3, the most important contributionsfrom primary species heavier than He ( Z >
2) are C and O,followed by Mg and Si and finally Fe. The total contributionof these species amounts to ∼
35% for H and ∼
11% for He, but mixed species (such as N) or less abundant speciesalso contribute to ∼ A > ∼
100 GeV/n. The difference can be mostly at-tributed to a preferential destruction of heavier nuclei at
6. Coste et al.: Constraining Galactic cosmic-ray parameters with Z ≤ Table 3.
Estimated fractional contribution of projectile
A > H and He fluxes. The columns are respec-tively the name of the element, atomic number, ratio ofmeasured fluxes, production cross-section ratios for H and He, and the estimated fractional contribution to H and He. P A P Φ P Φ γ HP γ HeP f P → H f P → He % %He 4 1.0 · · · · · · . − . − . − . − . − . − . − . − . − . − < .
1S 32 1 . − . − < . . − . − < . . − . − < . . − . − . − . − . − . − Note 3.
The ratio of measured fluxes is calculated at ∼ ≤ Z ≤ low energy . Note that for H production, the coalescenceof two protons (long-dashed pink curve) contributes up to40% of the total at ∼ Hand He primary fluxes. The peak of contribution occurs atGeV/n as secondary fluxes drop faster than primary fluxeswith energy. Fig. 3 shows this contribution to be .
10% for He. With the high precision measurement from PAMELAand the even better measurements awaited from AMS-02,this will need to be further looked into in the future. Indeed, the primary-to-primary ratios are not constant. Theheavier the nucleus, the larger its destruction cross-section, themore the propagated flux is affected/decreased at low energy,the longer it takes to reach a plateau of maximal contribution athigh energy. The observed trend is consistent with the primary-to-primary ratios shown in Fig. 14 of Putze et al. (2011). [GeV/n] k/n E -1
10 1 10 F r ac ti on a l c on t r i bu ti on → + H/He i CR i=parents ∑ CR parents:He...O...Si...Zn H, He...O...Si H, He...O H, He He, H, He H, H [GeV/n] k/n E -1
10 1 10 F r ac ti on a l c on t r i bu ti on → + H/He i CR i=parents ∑ CR parents:He...O...Si...Zn He...O...Si He...O He [GeV/n] k/n E -1
10 1 10 F r ac ti on a l c on t r i bu ti on CR contributions:Prim + Li...O...Si...ZnPrim + Li...O...SiPrim + Li...O4He (primary) He → + H/He i CR i=parents ∑ + Primary Fig. 3.
Fractional contributions to the propagated Hfluxes (top panel), He fluxes (middle panel) and He fluxes(bottom panel) as a function of E k/n from A > He, the primary contribution is also considered.
Given the accuracy of current data (see Fig. 4), we musttake into account the contribution from all parent nuclei atleast up to Si. In the rest of the analysis, we use PAMELAdata for He (Adriani et al. 2011), as they overcome all oth-ers in the ∼ GeV − TeV range in terms of precision. Beforegiving our final results, and to complement Sect. 3.2, wediscuss the appropriate choice of data to consider here, in
7. Coste et al.: Constraining Galactic cosmic-ray parameters with Z ≤ Table 4.
MCMC analysis of Model III ( V c = 0) with L = 4 kpc: most-probable values and relative uncertainties(corresponding to the 68% CI) for the analysis of various combinations of He/ He, He, and PAMELA He data (+ 3combinations involving the H isotope). The last column gives the χ /d.o.f. for the best-fit model found (which usuallydiffers from the most-probable one). Data K × δ V c V a α η S χ /d.o.f.(kpc Myr − ) - (km s − ) (km s − ) - - - He/ He + He 0 . +10% − . +6 . − . . +1 . − . . +6 . − . . +1 . − . . +59% − He/ He + He ‡ . +27% − . +5 . − . . +3 . − . . +5 . − . . +0 . − . − . +9 . − . He/ He 0 . +84% − . +6 . − . +3 . − . . +34% − [1 . , .
5] [ − , +2] 2.9 He/ He + He 1 . +58% − . . +11 . % − . . +4 . − . . +6 . − . % . +2 . − . − . +33% − He/ He + He ¶ . +35% − . +26 . % − . . +6 . − . . +7 . − . % . +2 . − . − . +18% − He/ He + He + He 0 . +10% − . . +3 . − . . +1 . − . . +4 . − . . +0 . − . . +171% − He/ He + He ¶ + He 0 . +10% − . . +5 . − . . +1 . − . . +5 . − . . +1 . − . . +71% − H/ He + H 15 +13% − . +100% − . +500% − . +52% − . +4 . − . − . +17% − H/ He + H + He 0 . +17% − . +8 . − . . +3 . − . . +8 . − . +1 . − . . +13% − H/ He + H + He ‡ . +55% − . +125% − . +8% − . +39% − . +11% − . +47% − ¶ Including AMS-01 data from the recently published analysis of Aguilar et al. (2011). ‡ Excluding PAMELA He data points below 5 GeV/n and above 183 GeV/n.
Note 4.
An interval in square brackets corresponds to the prior used for the analysis (the posterior PDF obtained is close to the prior). order to get the best balance between robustness and reli-ability for the H and He-related analyses. vs. real data
We start by comparing the results obtained with the simu-lated and the actual data set. To avoid lengthy comparisonsof numbers, we limit ourselves to Model III (where we alsofix η T to its default value, i.e. 1). The obvious differencewith the simulated data is that we no longer have accessto the true source parameters (automatically excluding op-tions 2’ and 4 discussed in Sect. 3.2). For the simultaneousanalysis using He PAMELA data—the precision of which is ∼ χ /d.o.f. value(first row) shows that the model has difficulty to perfectlymatch the high precision PAMELA He data over the wholeenergy range. The analysis of He/ He ratio using a prioron the source parameters (option 4 in Table 2 and third lineof Table 4) gives larger CIs for the transport parameters.The consistency between the results of the latter analysis(third line) and that based on the partial He data (secondline), and their discrepancy with the results of the analysisbased on the full He data set (first line) confirms our sus-picion that high precision measurements for primary fluxescan bias the transport parameters determination. He flux in the analysis Replacing He by He in the simultaneous analysis (4 th and5 th line) further affects the determination of the transportparameters. This is not surprising since He data are notall consistent with one another (see Fig. 4). The bias isstronger when taking into account the recently publishedAMS-01 data (Aguilar et al. 2011). If both He and He aretaken into account , the much better accuracy of the HePAMELA data with respect to the He data amounts to asmaller weight of the latter in the analysis. H / He vs He / HeWe repeat partially the analysis for H in the last 3 rowsof Table 2. The data are so inconsistent with one anotherfor H/ He (see Fig. 4) that we are forced to use at leastthe H flux (whose data points are also markedly incon-sistent with one another). Even so, the results are not re-liable. PAMELA and AMS-02 have the capability to im-prove greatly the situation, but in the meantime, we areforced to include He as well in the analysis (next-to-lastrow in the table). The transport parameter values from the H/ He+ H+He analysis are grossly consistent with thosefrom the He/ He+ He+He analysis, but are likely to sufferfrom similar biases (see the previous paragraph). Reducingthe energy range of He data is not even possible for the Hanalysis (last line in the table), as the results obtained arenot reliable. The simultaneous analysis of He/ He + He + He has notbeen tested in the simulated data since it would have amountedto a double-counting of the He data (appearing in the threequantities). However, real data involve different experiments forthe various quantities (PAMELA for He and other experimentsfor He), and independent measurements are used.8. Coste et al.: Constraining Galactic cosmic-ray parameters with Z ≤ Table 5.
Most-probable values and CIs for Models II and III for our ‘best’ analysis (Sect. 4.3) of the quartet and B/Cdata ( L = 4 kpc). Data K × δ V c V a α η S χ /d.o.f.(kpc Myr − ) - (km s − ) (km s − ) - - - Model II He/ He 15 . +0 . − . . +0 . − . - 116 +11 − [1 . , .
5] [ − , +2] 3.3 He/ He + He + He ‡ . +0 . − . . +0 . − . - 74 +4 − . +0 . − . . +0 . − . H/ He + H + He 14 . +0 . − . . +0 . − . - 44 +5 − . +0 . − . . +0 . − . . +0 . − . . +0 . − . - 73 +2 − α + δ = 2 . − . +0 . − . . +0 . − . - 80 +2 − [1 . , .
5] [ − , +2] 1.5B/C + C (all) [this paper] 6 . +0 . − . . +0 . − . - 57 +2 − . +0 . − . . +0 . − . . +0 . − . . +0 . − . - 78 +1 − . +0 . − . . +0 . − . Model III He/ He 0 . +0 . − . . +0 . − . . +0 . − . +14 − [1 . , .
5] [ − , +2] 2.9 He/ He + He + He ‡ . +0 . − . . +0 . − . . +0 . − . +5 − . +0 . − . . +0 . − . H/ He + H + He 3.2 +0 . − . . +0 . − . . +2 . − . +9 − . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . +2 − α + δ = 2 . − . +0 . − . . +0 . − . . +0 . − . +5 − [1 . , .
5] [ − , +2] 0.9B/C + C (all) [this paper] 0 . +0 . − . . +0 . − . . +0 . − . +1 − . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . +4 − . +0 . − . . +0 . − . ‡ Excluding PAMELA He point below 5 GeV/n and above 183 GeV/n.
Note 5.
An interval in square brackets corresponds to the prior used for the analysis (the posterior PDF obtained is close to the prior).
Note 6.
The B/C results are based on IMP7-8 (Garcia-Munoz et al. 1987), Voyager 1&2 (Lukasiak et al. 1999), ACE-CRIS (George et al.2009), HEA0-3 (Engelmann et al. 1990), Spacelab (Mueller et al. 1991), AMS-01 (Aguilar et al. 2011), and CREAM (Ahn et al. 2008), shownto be the most compatible data for a B/C analysis (Putze et al. 2009).
Taking into account specificities of the actual data (previ-ous section), our ‘best’ analysis is based on the most rele-vant combinations of data for H and He: – the He/ He analysis (with a prior for the sourceparameters) gives robust and conservative resultsfor the transport parameters. The result from the He/ He+ He+He ‡ analysis is more sensitive to biases,but using an energy sub-range for He data is expectedto limit them. – Due to the paucity of H data, the H/ He+ H+He (fullenergy-range for He) analysis is the only reliable option,although it probably suffers from biases.The corresponding most-probable values and CIs are gath-ered in Table 5, and the corresponding envelopes for H/ He, H, He/ He, and He/ He are given in Fig. 4.We also re-analyse the B/C ratio according to our ‘best-analysis’ scheme (B/C alone with a prior for the source pa-rameters or B/C + C). The results are reported in Table 5,where the results obtained in Putze et al. (2010) for fixedsource parameters are also reproduced: we note that thenew strategy gives results in better agreement with thoseof the quartet analysis (e.g., the transport parameters δ and K are shifted by more than 30% for Model II), fur-ther demonstrating its usefulness. If we focus on the transport parameters, we note that com-binations involving the H/ He, He/ He, or B/C ratiogive broadly consistent transport parameter values, be itfor Model II or Model III . Regardless of the actual prop-agation model, we conclude that these results hint at theuniversality of CR transport for all species. Another impor-tant result is that the constraints set by the quartet data onthe transport parameters are competitive with those set bythe B/C ratio, so that the quartet data should be a primetarget for AMS-02. δ ∼ . ) or Model III ( δ ∼ . )? According to Sect. 3.2, comparing the results of thesecondary-to-primary ratio analysis with those of the com-bined analysis (ratio + primary flux) gives an indicationof their robustness. Table 5 show that the results for thediffusion slope δ is very robust, regardless of the modelconsidered. A more detailed comparison shows that for He-related constraints, the transport parameter values forModel II are inconsistent with one another at the 3 σ level,whereas the 68% CIs overlap with one another (but for V c ) The most significant difference is for the H/ He+ H+Heanalysis, which is inconsistent in both models and clearly unre-liable for Model II ( δ ∼ He/ He-related constraints are roughly in the same region but are lo-cated at several σ from each others (they are consistent withone another for Model II). 9. Coste et al.: Constraining Galactic cosmic-ray parameters with Z ≤ [GeV/n] k/n E1 10 ] . G e V / n - s r - s - [ m . / n E × I S φ MASS89Balloon90CAPRICE94IMAX92AMS-01BESS93CAPRICE98Model IIModel III
H (IS) [GeV/n] k/n E -2 -1
10 1 10 S ec ond a r y - t o - p r i m a r y r a ti o ( T OA ) IMP3+4+5 (650 MV)IMP7+Pionneer10 (540 MV)IMP8 (400 MV)ISEE3 (740 MV)MASS89 (1200 MV)IMAX92 (550 MV)BESS93AMS-01 (600 MV)Balloon73 (650 MV)Balloon77+Voyager (400 MV)Balloon84 (650 MV)Balloon90 (1200 MV) = 700 MV) φ Model II ( = 700 MV) φ Model III (
He (TOA) H/ [GeV/n] k/n E1 10 ] . G e V / n - s r - s - [ m . / n E × I S φ MASS89IMAX92BESS93BESS94BESS95BESS97BESS98AMS-01
He (IS) [GeV/n] k/n E -2 -1
10 1 10 S ec ond a r y - t o - p r i m a r y r a ti o ( T OA ) Balloon72 (400 MV)Balloon73 (500 MV)Balloon77+Voyager (400 MV)Balloon89 (1400 MV)IMP7+Pioneer10 (540 MV)ISEE3 (740 MV)ISEE3(HIST) (500 MV) Voyager87 (360 MV)IMAX92 (750 MV)SMILI-I (1200 MV)SMILI-II (1200 MV)AMS-01 (600 MV)BESS98 (700 MV)CAPRICE98 (700 MV) = 700 MV) φ Model II with prior ( = 700 MV) φ Model III with prior ( = 700 MV) φ Model II ( = 700 MV) φ Model III (
He (TOA) He/ Fig. 4.
Left panels: demodulated interstellar H (top) and He (bottom) envelopes at 95% CIs times E . k/n . Right panels:top-of-atmosphere secondary-to-primary ratio H/ He (top) and He/ He (bottom) ratios. The full envelopes correspondto the ‘best’ simultaneous analysis (secondary-to-primary ratio + primary flux) for model II (blue) and III (red). Thehatched envelopes correspond to the He/ He analysis (prior on source parameters). See Table A.1 for references andthe corresponding demodulation (for IS) and modulation (TOA) level φ .for Model III, hence slightly favouring the latter ( δ ∼ . χ /d.o.f. values also tends tofavour model III. Hence, although the value δ ∼ . V c = 0) with δ ∼ .
3. Moreover, as shown in Maurin et al.(2010), many ingredients of the propagation models canlead to a systematic scatter of the transport parameterslarger than the width of their CIs. Data at higher energyfor any secondary-to-primary ratio are mandatory to con-clude on this issue.
The present analysis is more general than that used inPutze et al. (2011), where the transport parameters werefixed. Although it is not the main focus of this paper, weremark that the values of the source slope α from the B/C+ C analysis are consistent with those found in Putze et al.(2011), strengthening the case of a universal source slope α at the ∼
5% level. For the quartet values, α He is broadlyconsistent with Putze et al.’s analysis (based on AMS-01,BESS98 and BESS-TeV data for He). However, the resultsfor the source parameters depend on the choice of data setsand energy-range considered. This indicates that for .
5. Conclusion
We have revisited the constraints set on the transport (andalso the source) parameters by the quartet data, i.e. H, H, He, and He fluxes, but also the secondary-to-primary ra-tios H/ He and He/ He. This extends and complementsa series of studies (Putze et al. 2009, 2010, 2011) carriedout with the USINE propagation code and an MCMC al-gorithm. The three main ingredients on which the analysisrests are: – A minute compilation of the existing quartet data andsurvey of the literature, showing that the most re-cent/precise data (AMS-01, BESS93 →
98, CAPRICE98,IMAX92, and SMILI-II) have not been considered be-fore this analysis. – We have done a systematic survey of the literature forthe cross-sections involved in the production/survivalof the quartet nuclei. This has lead us to propose newempirical production cross-sections of H, H, and He,valid above a few tens of MeV/n, for any projectile onp and He (we also updated inelastic cross-sections). – We have made an extensive use of artificial data setsto assess the reliability of the derived CIs of the GCRtransport and source parameters for various combina-tions of data/parameters analyses.In broad agreement with previous studies, (e.g.Ramaty & Lingenfelter 1969; Beatty 1986), we find thatthe fragmentation of CNO contributes significantly to the
10. Coste et al.: Constraining Galactic cosmic-ray parameters with Z ≤ H flux ( ∼ He fragmen-tation is the dominant channel for the H and He fluxes).Nevertheless, we provide a much finer picture, showing inparticular that heavy nuclei (8 < Z ≤
30) contribute up to10% for He (20% for H) at high energy. We also provide anestimate of the secondary fraction to the He flux. By def-inition, the secondary contribution has a steeper spectrumthan the primary one and therefore becomes quickly negli-gible at high energy. This secondary contribution peaks ata few GeV/n, and it amounts to ∼
10% of the total flux( ∼
7% up to O fragmentation, ∼
2% from elements heavierthan O), which is already a sizeable amount given the ∼ H, the knowledge of the multiplicity of neu-tron and proton produced by the interaction of all elementson the ISM is required to calculate precisely its secondarycontent.Simulated data have allowed us to check several criticalbehaviours. Firstly, the He flux is obviously useful (and re-quired) to constrain the source parameters, but it has alsobeen found to bring significant information on the transportparameters: fitting a secondary-to-primary ratio plus a pri-mary flux brings more constraints than just fitting the ratio(even when source parameters are fixed). Secondly, we havechecked that a model with more free parameters (than theones used to simulate the data) is able to recover the correctvalues. However, our analysis has also strongly hinted at thefact that adding the primary flux He biases the determina-tion of the transport parameters if systematics (which areusually more important in primary fluxes than in ratios) arepresent, and/or if the wrong model is used. For this reason,when dealing with measurements, we recommend to alwayscompare the result from the secondary-to-primary ratio +primary flux analysis to that of the secondary-to-primaryratio using a loose but physically motivated prior on thesource parameters.The analysis of real data has shown that quartet dataslightly favours a model with large δ ∼ . V c ∼
20 km s − and V a ∼
40 km s − ), but that a model withsmall δ ∼ . V c ∼ V a ∼
80 km s − ) can-not be completely ruled out. Better quality data, and espe-cially data at higher energy are required to go further. Theconclusions are similar and the range of transport parame-ters found are consistent with those obtained from the B/Canalysis (Jones et al. 2001; Putze et al. 2010; Maurin et al.2010) . This strongly hints at the the universality of theGCR transport for any all nuclei. Furthermore, we haveshown that the analysis of the light isotopes (and the al-ready very good precision on He) is as constraining as theB/C analysis (similar range of CIs).The several difficulties which have been pointed out inthis analysis could be alleviated by virtue of using bet-ter data. However, it is more likely that the interpretationof future high-precision data will require the developmentof refined models for the source spectra and/or transportand/or solar modulation. For instance, the Force-Field ap-proximation for solar modulation is already too crude tominutely match the PAMELA He data. The forthcoming Note that in this paper, we did not attempt to combinethe results of different secondary-to-primary ratios ( H/ He, He/ He, B/C, sub-Fe/Fe, ¯ p/p ). This is left for a future study,for which a Bayesian evidence could be used to better address(in a Bayesian framework) the crucial issue of model selection.
AMS-02 data at an even better accuracy will definitivelypose interesting new challenges.
Acknowledgements.
D. M. would like to thank A. V. Blinov for kindlyproviding copies of several of his articles. A. P. is grateful for financialsupport from the Swedish Research Council (VR) through the OskarKlein Centre.
Appendix A: Cosmic-ray data
Deuterons and He fluxes are very sensitive to the mod-ulation level, whereas ratios are less affected. The exactvalue for the solar modulation level φ is uncertain. For in-stance, the values given in the seminal papers can differgreatly from those estimated by Casadei & Bindi (2004)(in order to match the electron and positron fluxes of var-ious experiments, see their Table 1), or from those recon-structed from the Neutron monitors (Usoskin et al. 2002).This difference may be attributed to the fact that the lat-ter analysis correctly solves the Fokker-Plank equation ofGCR transport in the Heliosphere, whereas most papersrely on the widely used force-field approximation that isknown to fail for strong modulation level φ & φ necessaryto reproduce the data (as quoted in the seminal paper), butthese values are slightly adjusted in order to give overlap-ping fluxes when all the data are demodulated and plottedtogether. Given the uncertainty on the data, the large un-certainty on φ , and the fact that most-probable region ofparameter space is constrained by the H/ He and He/ Heratio (rather than the best fit to the H and He fluxes),we feel that it is a safe procedure till high precision datafrom PAMELA of AMS-02 are available.The demodulated interstellar (IS) fluxes for H and Heare shown in the left panels of Fig. 4, whereas the yetmodulated top-of-atmosphere (TOA) ratios for H/ He and He/ He are shown in its right panels. The references forthe data are given in Table A.1.
Appendix B: Cross-sections
This appendix summarises the production and destructioncross-sections employed for the quartet nuclei in this paper.
B.1. Elastic and inelastic cross-sections
All reaction cross-sections are taken from the parametri-sations of Tripathi et al. (1999), but for the pH reactioncross-section. The latter is evaluated from σ inelpp = σ totpp − σ elpp ,where the total and elastic cross-sections are fitted to thedata compiled in the PDG . Note also that for He + He,we had to renormalise Tripathi et al. (1999) formulae by afactor 0.9 to match the low-energy data.Our parametrisations (lines) and the data (symbols) areshown in Fig. B.1 for reaction on H and He. Note that werely on Tripathi et al. (1997) for any other inelastic reac-tion. http://pdg.lbl.gov/
11. Coste et al.: Constraining Galactic cosmic-ray parameters with Z ≤ Table A.1.
References for the quartet data.
Exp. φ (MV) Ref. Comment— H —MASS89 9 1989 1200 Webber et al. (1991)Voyager87 7 1987 360 Seo et al. (1994) Voyager is at 23 AUBalloon90 1 1990 1200 Bogomolov et al. (1995)Voyager94 4 1994 150 Seo & McDonald (1995) Voyager is at 56 AUCAPRICE94 14 1994 600 Boezio et al. (1999) Subtraction of H to H from Tab 3.IMAX92 8 1992 550 de Nolfo et al. (2000)AMS-01 10 1998 600 AMS Collaboration (2002)BESS93 5 1993 700 Wang et al. (2002)CAPRICE98 5 1998 700 Papini et al. (2004)— He —MASS89 5 1989 1200 Webber et al. (1991)Voyager87 7 1987 360 Seo et al. (1994) Voyager is at 23 AUVoyager94 4 1994 150 Seo & McDonald (1995) Voyager is at 56 AUIMAX92 24 1992 750 Menn et al. (2000)BESS93 7 1993 700 Wang et al. (2002)BESS94 5 1994 630 Myers et al. (2003) Taken from their Fig. 2BESS95 6 1995 550 Myers et al. (2003) Taken from their Fig. 2BESS97 7 1997 491 Myers et al. (2003) Taken from their Fig. 2BESS98 7 1998 700 Myers et al. (2003) Taken from their Fig. 2AMS-01 5 1998 600 Aguilar et al. (2011)— H/ He —IMP3+4+5 1 65+67+69 650 Hsieh et al. (1971)Balloon73 1 1973 650 Apparao (1973)IMP7+Pioneer10 3 72+73 540 Teegarden et al. (1975)Balloon77+Voyager 14 1977 400 Webber & Yushak (1983)IMP8 1 1977 400 Beatty et al. (1985)ISEE3 1 78-84 740 Kroeger (1986)Balloon84 1 1974 650 Durgaprasad & Kunte (1988) (discarded in the analysis)MASS89 9 1989 1200 Webber et al. (1991)Balloon90 1 1990 1200 Bogomolov et al. (1995)IMAX92 8 1992 550 de Nolfo et al. (2000)AMS-01 4 1998 600 Aguilar et al. (2011)— He/ He —Balloon72 2 1972 400 Webber & Schofield (1975) Re-analysed by Webber et al. (1987)IMP7+Pioneer10 2 72+73 540 Teegarden et al. (1975)Balloon73 2 1973 500 Leech & Ogallagher (1978)Balloon77+Voyager 3 1977 400 Webber & Yushak (1983) Re-analysed by Webber et al. (1987)Balloon81 1 1981 440 Jordan (1985) (discarded, see Webber et al. 1987)ISEE3 2 78-84 740 Kroeger (1986)ISEE3(HIST) 1 78 500 Mewaldt (1986)MASS89 5 1989 1200 Webber et al. (1991)SMILI-I 12 1989 1200 Beatty et al. (1993)Voyager87 1 1987 360 Seo et al. (1994) Voyager is at 23 AUBalloon89 1 1989 1400 Hatano et al. (1995)IMAX92 21 1992 750 Menn et al. (2000)SMILI-II 10 1991 1200 Ahlen et al. (2000)BESS98 7 1998 700 Myers et al. (2003)CAPRICE98 1 1998 700 Mocchiutti et al. (2003)AMS-01 5 1998 600 Aguilar et al. (2011) Supersedes Xiong et al. (2003) data
B.2. Light nuclei production: Nuc + p
The light nuclei He and H are spallative products of cos-mic rays interacting with the interstellar medium (ISM).The total secondary flux is obtained from the combina-tion of production cross-sections and measured primaryfluxes. In principle, all nuclei must be considered, butthe ISM and GCRs are mostly composed of H and He, making the reactions involving these species dom- inant. For heavier species, their decreasing number isbalanced by their higher cross-section. In several stud-ies (e.g. Ramaty & Lingenfelter 1969; Jung et al. 1973b;Beatty 1986), it was found that the CNO CR + H ISM re-actions contribute to ∼
30% of the H flux above GeV/nenergies. The reverse reaction H CR +CNO ISM mostly pro-duces fragments at lower energies, making them irrelevant
12. Coste et al.: Constraining Galactic cosmic-ray parameters with Z ≤ [GeV/n] k/n E -2 -1
10 1 10 [ m b ] i n e l σ He + H He + H H + H H + H Reaction cross-sections (on H) [GeV/n] k/n E -2 -1
10 1 10 [ m b ] i n e l σ He + He He + He H + He Reaction cross-sections (on He)
Fig. B.1.
Total inelastic (reaction) cross-section forthe quartet isotopes. The lines show our parametri-sation (see text), the symbols are data.
Top panel: reaction on H with data from Cairns et al. (1964);Hayakawa et al. (1964); Igo et al. (1967); Palevsky et al.(1967); Griffiths & Harbison (1969); Nicholls et al.(1972); Carlson et al. (1973); Sourkes et al. (1976);Ableev et al. (1977); Klem et al. (1977); Jaros et al.(1978); Velichko et al. (1982); Blinov et al. (1984,1985); Abdullin et al. (1993); Glagolev et al. (1993);Webber (1997).
Bottom panel: reaction on He withdata from Koepke & Brown (1977); Jaros et al. (1978);Tanihata et al. (1985).for CR studies in the regime &
100 MeV/n . Note that H is also produced in these reactions, but it decays in Hewith a life time (12 . He production, but the cross-sections for this fragment areprovided as well below.The energy of the fragments roughly follows a Gaussiandistribution (e.g. Cucinotta et al. 1993). Its impact on thesecondary flux was inspected for the B/C analysis byTsao et al. (1995), where an effect .
10% was found, com-pared to the straight-ahead approximation, in which thekinetic energy per nucleon of the fragment equals that ofthe projectile. The precision sought for the cross-sectionsis driven by the level of precision attained by the CR data Solar modulation also ensures that only species created atenergies & GeV/n matter. to analyse. Given the large errors on the existing data, thestraight-ahead approximation is enough for this analysis.However, future high-precision data (e.g. from the AMS-02experiment) will probably require a refined description.
B.2.1. He + p → H , H , and HeRecent and illustrative reviews on He+H reaction and theproduction of light fragments is given by Bildsten et al.(1990); Cucinotta et al. (1993); Blinov & Chadeyeva(2008). As said earlier, we are only interested in thetotal inclusive production cross-section, not in all thepossible numerous final states (see, e.g., Table 3 ofBlinov & Chadeyeva 2008). We adapt the parametrisationof Cucinotta et al. (1993), which takes into accountseparately the break-up and stripping (for He and H)cross-sections. The former reaction corresponds to the casewhere the helium nucleus breaks up leading to coalescenceof free nucleons into a new nucleus. The latter happensvia the pickup reaction where the incident proton tears aneutron or a proton off the helium nucleus. Both reactionand the total are shown along with the experimental datain Fig. B.2.The most accurate set of data (upward blue emptytriangles) are from the experiments set up in ITEP andLHE JINR (Aladashvill et al. 1981; Glagolev et al. 1993;Abdullin et al. 1994, summarised in Blinov & Chadeyeva2008). Their highest energy data point (Glagolev et al.1993) is a conservative estimate as the more or equalto 6-prong reactions are not detailed (see Table 3 ofBlinov & Chadeyeva 2008 and Table 4 of Glagolev et al.1993). To take into account that possibility, we consider anerror of a few mb in the plots of Fig. B.2. Let us considerin turn each product of interest. He production The stripping cross-section data (d and He in the final state) are well fitted by Eq. (130) ofCucinotta et al. (1993). However, the Griffiths & Harbison(1969) and (Jung et al. 1973a) are ∼
30% below the otherdata. Actually, for the latter (filled stars) the break-upcross-section is above other data, it may be that the endproducts are misreconstructed (in this or the other exper-iments). Nevertheless, the sum of the two—which is theone that matters—is consistent in all data. Note that weslightly modified the break-up cross-section provided byCucinotta et al. (1993) to better fit the high-energy datapoints. For the latter, all the data are consistent withone another, but for the high precision ITEP data at 200MeV/n. H production The stripping cross-section is as for He (dand He in the final state). The high-energy break-up cross-section data (LHE JINR and Webber 1990b) are inconsis-tent. We have decided to rescale the Webber data, to takeinto account the fact that in his preliminary account ofthe results (Webber 1990b), the total inelastic cross-sectionis smaller than that given in a later and updated study(Webber 1997). Still, the agreement between the two setsis not satisfactory. The other high-energy data point is theInnes (1957) experiment, and it suffers large uncertaintiesand maybe systematics (it is for n + He reaction, andthe data point is provided by Meyer (1972) who relied on
13. Coste et al.: Constraining Galactic cosmic-ray parameters with Z ≤ [GeV/n] k/n E -1
10 1 10 [ m b ] p r od σ Breakup
Tannenwald 53[Meyer]Innes 57 [Meyer]Griffiths 69Meyer 72Nicholls 72Jung 73Bizard 77Webber 90Aladashvili 81, Glagolev 93, Abdullin 94
He) Stripping (d+
Tannenwald 53 [Meyer]Rogers 69Griffiths 69Meyer 72Jung 73 He → He + p [GeV/n] k/n E -1
10 1 10 [ m b ] p r od σ Breakup
Innes 57 [Meyer]Cairns 64 [Meyer]Jung 73Webber 90 [rescaled]Aladashvili 81, Glagolev 93, Abdullin 94
He) Stripping (d+
Tannenwald 53 [Meyer]Griffiths 69Rogers 69Meyer 72Jung 73 H → He + p [GeV/n] k/n E -1
10 1 10 [ m b ] p r od σ Breakup
Meyer 72Nicholls 72Jung 73Webber 90Aladashvili 81, Glagolev 93, Abdullin 94 H → He + p Fig. B.2.
Inclusive production cross-sections of He(top), H (centre) and H (bottom) in He+H re-action. The data (see text for details) are fromTannenwald (1953); Innes (1957); Cairns et al. (1964);Rogers et al. (1969); Griffiths & Harbison (1969); Meyer(1972); Jung et al. (1973a); Aladashvill et al. (1981);Webber (1990b); Glagolev et al. (1993); Abdullin et al.(1994).several assumptions to get it). The ITEP/LHE JINR databeing the best available, we have replaced the formula forthe H breakup of Cucinotta et al. (1993) by a form similaras that given for He, but where we changed the parametersto fit the high energy points. [GeV/n] k/n E -2 -1
10 1 10 [ m b ] p r od σ -2 -1
10 110
Griffiths 69Meyer 72Blinov 86, Glagolev 93 H → He + p H → p + p Fig. B.3. H other production channels from the less abun-dant He and the peaked fusion pp reaction (the muchsmaller cross-section is redeemed by a CR flux higher (pinstead of He). The data are from Griffiths & Harbison(1969); Meyer (1972); Blinov et al. (1986); Glagolev et al.(1993). H production There is only break-up for theCucinotta et al. (1993) H production. The data arein broad agreement with one another, but for theNicholls et al. (1972) point (open plus). Again, we haveadapted the Cucinotta et al. (1993) parametrisation tobetter fit the ITEP/LHE JINR data.
B.2.2. He + p → H (breakup) and p + p → H (fusion) There are two other channels for producing H from lightnucleus reactions, and they are shown in Fig. B.3 alongwith the data. The first one is from He (break-up andstripping). The CR flux of the latter is less abundant thanthe He flux. With a ratio of ∼
20% at 1 GeV/n (decreasingat higher energy) and similar production cross-sections ( ∼ −
40 mb), this is expected to contribute by the samefraction at GeV/n energies, and then to become negligible &
10 GeV/n. The second channel is the H coalescence fromtwo protons. The cross-section is non-vanishing only for avery narrow energy range. Even if the cross-section is 10times smaller than for the other channels, the fact that CRprotons are ∼
10 times more numerous than He makes ita significant channel slightly below 1 GeV/n.The fitting curves are taken from Meyer (1972), but weadapted the fit for the He+p channel to match the twohigh-energy ITEP/LHE JINR data points.
B.2.3. Proj ( A> + p → H , H , and HeFor nuclear fragmentation cross-sections of heavier nuclei,the concepts of ‘strong’ or ‘weak’ factorisation relies on thefact that at high energy enough, the branching of the var-ious outgoing particles-production channels becomes inde-pendent of the target. This corresponds to the factorisation σ strong ( P, F, T ) = γ FP γ T or σ weak ( P, F, T ) = γ FP γ PT where σ ( P, F, T ) is the fragmentation cross-section for the projec-tile P incident upon the target T producing the fragmentF. This is discussed, e.g., in Olson et al. (1983), where it
14. Coste et al.: Constraining Galactic cosmic-ray parameters with Z ≤ is concluded that although strong factorisation is proba-bly violated, weak factorisation seems exact (see also, e.g.,Michel et al. 1995). The parametrisation proposed belowtakes advantage of it.Several data exist for the production of light isotopesfrom nuclei A ≥
12 on H (see Table B.1). The most com-plete sets of data in terms of energy coverage are for the pro-jectiles C, N, and O ( h A i = 14), the group Mg, Al, and Si( h A i = 26), and the group Fe and Ni ( h A i = 57). They areplotted in Fig. B.4 (top panels and bottom left panel). Thesolid lines correspond to an adjustment (by eye), rescaledfrom the σ Hep → Hebreakup cross-section (because heavy projectiledo not give A = 3 fragments in the stripping process). As σ prod He ≈ σ prod H , no distinction is made for the fit ( H dataare scarce and do not influence the conclusions drawn fromthese three groups of nuclei). The following parametrisation σ Pp → F ( E k/n , A P ) = γ FP · f ( E k/n , A P ) · σ Hep → Hebreakup ( E k/n ) , (B.1)with f ( E k/n , A P ) = (cid:16) E k/n . / n (cid:17) . · q A P26 if E k/n < . f ( E k/n , A proj ) factor, there is no further energy dependencein the γ FP factor, so that the latter can be determined fromthe data points at any energy. The bottom right panel ofFig. B.4 shows the measured mean value and dispersion as a function of A , from which we obtain: γ HeP = γ HP = 1 . " (cid:18) A P (cid:19) . ,γ HP = 0 . A . . (B.3)The set of formulae (B.1), (B.2), and (B.3) completelydefine the Proj+p production cross-sections for the lightfragments. B.3. Proj ( A ≥ + He → H , H , and He Data for T + He where the target T is heavier than p arescarce. In a compilation of Davis et al. (1995), the authorsfind that the He production scales as A . (based on 4data point with A T ≥ H and H production as well.
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15. Coste et al.: Constraining Galactic cosmic-ray parameters with Z ≤ Table B.1.
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Proj. Frag. E k/n Ref.N, O, Fe H 3 2.2 Fireman (1955)C, N, O, Mg, Al, Fe, Ni, Ag, Sn, Pb H 26 0.45-6.2 Currie (1959); Currie et al. (1956)C H 4 0.225-0.73 Honda & Lal (1960)Mg, Al He 2 0.54 Bieri & Rutsch (1962)C,O,Mg,Al,Si,V,Cr,Mn,Fe,Ni,Cu,Ag,Pb,Bi H, He 38 0.225-5.7 Goebel et al. (1964)CNO H, H, He 14 0.02-7.5 Ramaty & Lingenfelter (1969)C, O, Si H, He 12 0.6-3.0 Kruger & Heymann (1973)Si, Mg He 33 0.02-0.06 Walton et al. (1976)Mg He 6 0.015-0.07 Pulfer (1979)C, O H, H, He 8 1.05-2.1 Olson et al. (1983)Ag He 1 0.48 Green et al. (1984)Mg, Al, Si He 3 0.6 Michel et al. (1989)Mg, Al, Si, Fe, Ni He 21 0.8-2.6 Michel et al. (1995)Mg, Al, Si He 33 0.015-1.6 Leya et al. (1998)C He 3 1.87-3.66 Korejwo et al. (2000, 2002)Pb He 22 0.04-2.6 Leya et al. (2008)Fe, Ni He 53 0.022-1.6 Ammon et al. (2008) [GeV/n] k/n E -2 -1
10 1 10 [ m b ] p r od σ -2 -1 He → C He → N He → O H → C H → N H → O H → C H → N H → O f(E,A) × breakup He → He+p σ × → CNO + p [GeV/n] k/n E -2 -1
10 1 10 [ m b ] p r od σ -2 -1
10 110 He → Mg He → Al He → Si H → Mg H → Al H → Si f(E,A) × breakup He → He+p σ × → MgAlSi + p [GeV/n] k/n E -2 -1
10 1 10 [ m b ] p r od σ -2 -1
10 110 He → Fe He → Ni H → Fe H → Ni H → Fe H → Ni f(E,A) × breakup He → He+p σ ×
6 X → FeNi + p proj A > F P γ < ] /25) proj ) = 1.3 [1 + (A proj (A P γ He: P γ = P γ H: ) = 0.28 A proj (A P γ H: X → Proj + p
Fig. B.4.
Proj ( A> + p → H, H, and He cross-sections for Proj=C,N,O (top left), Proj=Mg,Al,Si (top right), andProj=Fe,Ni (bottom left). The bottom right correspond to the γ FP factor (see Eq. B.1 and text for explanations). Thereferences for the data are gathered in Table B.1. Jung, M., Sakamoto, Y., Suren, J. N., et al. 1973a, Phys. Rev. C, 7,2209Jung, M., Suren, J. N., Sakamoto, Y., et al. 1973b, in InternationalCosmic Ray Conference, Vol. 5, International Cosmic RayConference, 3086Klem, R., Igo, G., Talaga, R., et al. 1977, Physics Letters B, 70, 155 Koepke, J. A. & Brown, R. E. 1977, Phys. Rev. C, 16, 18Korejwo, A., Dzikowski, T., Giller, M., et al. 2000, Journal of PhysicsG Nuclear Physics, 26, 1171Korejwo, A., Giller, M., Dzikowski, T., Perelygin, V. V., & Zarubin,A. V. 2002, Journal of Physics G Nuclear Physics, 28, 1199Kroeger, R. 1986, ApJ, 303, 816