Measurement of boron and carbon fluxes in cosmic rays with the PAMELA experiment
O. Adriani, G. C. Barbarino, G. A. Bazilevskaya, R. Bellotti, M. Boezio, E. A. Bogomolov, M. Bongi, V. Bonvicini, S. Bottai, A. Bruno, F. Cafagna, D. Campana, R. Carbone, P. Carlson, M. Casolino, G. Castellini, I. A. Danilchenko, C. De Donato, C. De Santis, N. De Simone, V. Di Felice, V. Formato, A. M. Galper, A. V. Karelin, S. V. Koldashov, S. Koldobskiy, S. Y. Krutkov, A. N. Kvashnin, A. Leonov, V. Malakhov, L. Marcelli, M. Martucci, A. G. Mayorov, W. Menn, M. Mergé, V. V. Mikhailov, E. Mocchiutti, A. Monaco, N. Mori, R. Munini, G. Osteria, F. Palma, B. Panico, P. Papini, M. Pearce, P. Picozza, C. Pizzolotto, M. Ricci, S. B. Ricciarini, L. Rossetto, R. Sarkar, V. Scotti, M. Simon, R. Sparvoli, P. Spillantini, Y. I. Stozhkov, A. Vacchi, E. Vannuccini, G. I. Vasilyev, S. A. Voronov, Y. T. Yurkin, G. Zampa, N. Zampa, V. G. Zverev
aa r X i v : . [ a s t r o - ph . H E ] J u l Measurement of boron and carbon fluxes in cosmic rays with thePAMELA experiment.
O. Adriani , , G. C. Barbarino , , G. A. Bazilevskaya , R. Bellotti , , M. Boezio ,E. A. Bogomolov , M. Bongi , , V. Bonvicini , S. Bottai , A. Bruno , , F. Cafagna ,D. Campana , R. Carbone , P. Carlson , M. Casolino , , G. Castellini ,I. A. Danilchenko , C. De Donato , , C. De Santis , , N. De Simone , V. Di Felice , ,V. Formato , , A. M. Galper , A. V. Karelin , S. V. Koldashov , S. Koldobskiy ,S. Y. Krutkov , A. N. Kvashnin , A. Leonov , V. Malakhov , L. Marcelli , ,M. Martucci , , A. G. Mayorov , W. Menn , M. Merg´e , , V. V. Mikhailov ,E. Mocchiutti , A. Monaco , , N. Mori , , R. Munini , , G. Osteria , F. Palma , ,B. Panico , P. Papini , M. Pearce , P. Picozza , , C. Pizzolotto , ∗ , M. Ricci ,S. B. Ricciarini , , L. Rossetto , R. Sarkar ∗ , V. Scotti , , M. Simon , R. Sparvoli , ,P. Spillantini , , Y. I. Stozhkov , A. Vacchi , E. Vannuccini , G. I. Vasilyev ,S. A. Voronov , Y. T. Yurkin , G. Zampa , N. Zampa , V. G. Zverev University of Florence, Department of Physics and Astronomy, I-50019 Sesto Fiorentino,Florence, Italy INFN, Sezione di Florence, I-50019 Sesto Fiorentino, Florence, Italy University of Naples “Federico II”, Department of Physics, I-80126 Naples, Italy INFN, Sezione di Naples, I-80126 Naples, Italy Lebedev Physical Institute, RU-119991, Moscow, Russia University of Bari, Department of Physics, I-70126 Bari, Italy INFN, Sezione di Bari, I-70126 Bari, Italy INFN, Sezione di Trieste, I-34149 Trieste, Italy Ioffe Physical Technical Institute, RU-194021 St. Petersburg, Russia INFN, Sezione di Rome “Tor Vergata”, I-00133 Rome, Italy University of Rome “Tor Vergata”, Department of Physics, I-00133 Rome, Italy 2 – National Research Nuclear University MEPhI, RU-115409 Moscow KTH, Department of Physics, and the Oskar Klein Centre for Cosmoparticle Physics,AlbaNova University Centre, SE-10691 Stockholm, Sweden RIKEN, Advanced Science Institute, Wako-shi, Saitama, Japan IFAC, I-50019 Sesto Fiorentino, Florence, Italy University of Trieste, Department of Physics, I-34147 Trieste, Italy Universit¨at Siegen, Department of Physics, D-57068 Siegen, Germany INFN, Sezione di Perugia, I-06123 Perugia, Italy Agenzia Spaziale Italiana (ASI) Science Data Center, Via del Politecnico snc I-00133Rome, Italy INFN, Laboratori Nazionali di Frascati, Via Enrico Fermi 40, I-00044 Frascati, Italy Centro Siciliano di Fisica Nucleare e Struttura della Materia (CSFNSM), Viale A. Doria6, I-95125 Catania, Italy Indian Centre for Space Physics, 43, Chalantika, Garia Station Road, Kolkata 700 084,West Bengal, India ∗ Previously at INFN, Sezione di Trieste, I-34149 Trieste, ItalyReceived ; accepted 3 –
ABSTRACT
The propagation of cosmic rays inside our galaxy plays a fundamental rolein shaping their injection spectra into those observed at Earth. One of the besttools to investigate this issue is the ratio of fluxes for secondary and primaryspecies. The boron-to-carbon (B/C) ratio, in particular, is a sensitive probe toinvestigate propagation mechanisms. This paper presents new measurements ofthe absolute fluxes of boron and carbon nuclei, as well as the B/C ratio, fromthe PAMELA space experiment. The results span the range 0.44 - 129 GeV/nin kinetic energy for data taken in the period July 2006 - March 2008.
1. Introduction
Propagation in the interstellar medium (ISM) significantly affects the spectrum ofgalactic cosmic rays. After being accelerated by high-energy astrophysical processes suchas supernovae explosions, cosmic rays are injected into the interstellar space, propagatethrough it and eventually reach the Earth where they are detected. The multitude ofphysical processes that cosmic rays undergo during propagation (e.g. diffusion, spallation,emission of synchrotron radiation etc.) shape the injection spectra and chemical compositioninto the observed values. A detailed knowledge of these processes is therefore needed inorder to interpret the experimental data in terms of source parameters, or in estimating theexpected background when searching for contributions from new sources.There is still a relatively high degree of uncertainty regarding the physical processesrelevant to propagation of cosmic rays and the impact of experimental uncertainties on thedetermination of propagation parameters (see Maurin et al. (2010) and references therein).The propagation is usually modelled in terms of a diffusive transport equation Ptuskin 4 –(2012). The equation contains terms which account for diffusion in the irregular galacticmagnetic field, convection due to the galactic wind, energy losses, re-acceleration (modelledas diffusion in momentum space), spallation and radioactive decay, and source terms.Some parameters of the equation are simply related to directly measurable quantitiesunrelated to cosmic rays, and thus they can be obtained from independent measurements(e.g. the density of atomic hydrogen in the ISM, which is needed in order to estimate thespallation rate, can be measured by means of 21 cm radio surveys). Other parameters areobtained by fitting distributions derived from numerical propagation models like GALPROPStrong & Moskalenko (1998); Vladimirov (2012) or DRAGON Gaggero et al. (2013) todirect cosmic ray measurements.In order to test and tune the propagation models, a particularly useful measurablequantity is the secondary to primary flux ratio. Primary nuclei are those accelerated bycosmic ray sources such as supernova remnants, whereas secondaries are those producedin interactions of primaries with the ISM during propagation. The boron to carbon fluxratio (B/C) has been widely studied. Since boron is produced in negligible quantities bystellar nucleosynthesis processes Bethe (1939), almost all of the observed boron is believedto be from spallation reactions of CNO primaries on atomic and molecular H and Hepresent in the ISM. The B/C flux ratio is therefore a clean and direct probe of propagationmechanisms, and it is considered as the “standard tool” for studying propagation modelsStrong et al. (2007); Obermeier et al. (2012).The B/C flux ratio, as well as the absolute boron and carbon fluxes, have beenmeasured by balloon-borne Freier et al. (1959); Panov et al. (2007); Ahn et al. (2008);Obermeier et al. (2011) and by space-based experiments Engelmann et al. (1990);Swordy et al. (1990); Webber et al. (2002); Aguilar et al. (2010); Lave et al. (2013);Oliva et al. (2013), with different techniques and spanning various energy ranges from about 5 –80 MeV/n up to a few TeV/n. Even if the spread in the measurements and their associatederrors makes it difficult to clearly discriminate between the various models or to tightlyconstrain model parameters, there is a general consensus about several points. The relativeabundance of the light elements Li, Be and B in cosmic rays is significantly higher thanin the solar system de Nolfo et al. (2006). This supports the idea of creation by spallationreactions in ISM. The B/C flux ratio has a peak value at ∼
2. The PAMELA detector
A schematic view of the PAMELA detector system Picozza et al. (2007) is shown inFigure 1. The design was chosen to meet the main scientific goal of precisely measuring thelight components of the cosmic ray spectrum in the energy range starting from tens of MeVup to 1 TeV (depending on particle species), with a particular focus on antimatter. Tothis end, the design is optimized for | Z | = 1 particles and to provide a high lepton-hadrondiscrimination power. The core of the instrument is a magnetic spectrometer Adriani et al.(2007) made by six double-sided silicon microstrip tracking layers placed in the bore of a 6 –permanent magnet. The read-out pitch of the silicon sensors is 51 µ m in the X (bending)view and 66.5 µ m in the Y view. The spectrometer provides information about the magneticrigidity ρ = pc/ ( Ze ) of the particle (where p and Z are the particle momentum and theelectric charge, respectively). Six layers of plastic scintillator paddles arranged in three X-Yplanes (S1, S2 and S3 in Figure 1) placed above and below the magnetic cavity constitutethe Time-Of-Flight (TOF) system Barbarino et al. (2008); Osteria & Russo (2008). Theflight time of particles is measured with a time resolution of 250 ps for | Z | = 1 particlesand about 70 ps for boron and carbon nuclei Campana et al. (2009). This allows albedoparticles to be rejected and, in combination with the track length information obtainedfrom the tracking system, precise measurement of the particle velocity, β = v/c . TheTOF scintillators can identify the absolute particle charge up to oxygen by means of sixindependent ionisation measurements. The tracking system and the upper TOF system areshielded by an anticoincidence system (AC) Pearce et al. (2003) made of plastic scintillatorsand arranged in three sections (CARD, CAT and CAS in Figure 1), which allows spurioustriggers generated by secondary particles to be rejected during offline data analysis. Asampling electromagnetic calorimeter Boezio et al. (2002); Bonvicini et al. (2009) is placedbelow S3. It consists of 22 modules, each comprising a tungsten converter layer placedbetween two layers equipped with single-sided silicon strip detectors with orthogonalread-out strips. The total depth of the calorimeter is 16 . X , while the readout pitch ofthe strips is 2.44 mm. The calorimeter measures the energy of electrons and positrons, andgives a lepton/hadron rejection power of ∼ by means of topological shower analysis,thanks to its fine lateral and longitudinal segmentation. A tail-catcher scintillating detector(S4) and a neutron detector placed below the calorimeter help to further improve therejection power.The geometric factor of the apparatus, defined by the magnetic cavity, is energydependent because of the track curvature induced by the magnetic field, and increases as 7 –the energy of the particle increases. However, for rigidities above 1 GV it varies only by afew per mil, reaching the value of 21.6 cm sr at the highest rigidity.The PAMELA apparatus was launched on June 15th 2006, and has been continuouslytaking data since then. It is hosted as a piggyback payload on the Russian satelliteResurs-DK1, which executes a 70 ° semi-polar orbit. The orbit was elliptical with variableheight between 350 and 620 km up to 2010, after which it was converted to the currentcircular orbit with height about 600 km. Neutrondetector Anti-coincidence m a g n e t B XZ Y
Proton AntiprotonScintill. S4CalorimeterTrackingsystem(6 planes) GeometricacceptanceCATCASCARDSpectrometerTOF (S1)TOF (S2)TOF (S3)
Fig. 1.— Schematic view of the PAMELA apparatus. 8 –
3. Data analysis3.1. Data processing
The event reconstruction routines require silicon strips to be gathered into clusters.A “seed” strip is defined as a strip with a signal to noise ratio (S/N) greater than 7; it isgrouped with its neighbouring “signal” strips with S/N > | Z | ∼ | Z | ∼ √
12, which translates to ∼ µ m for the X (bending) view and ∼ µ m for the Y view.The associated MDR (Maximum Detectable Rigidity ) is ∼
250 GV.Prior to event reconstruction, the clusters with an associated energy release less than5 MIP have been removed. This helps to eliminate clusters associated with delta raysand light secondary particles, e.g. backscattered particles from the calorimeter. There isa twofold effect: the tracking efficiency is increased since the tracking algorithm has less The MDR is defined as the rigidity with an associated 100% error due to the finitespatial resolution of the spectrometer | Z | = 1 minimum ionisingparticle 9 –clusters to deal with, and the energy dependence of the tracking efficiency is reduced at highenergies by removing backscattering clusters, which are mainly produced by high energyprimaries interacting in the calorimeter. In order to be able to reliably measure the magnetic rigidity, events with a single trackin the spectrometer containing at least 4 hits in the X view and 3 hits in the Y view havebeen selected. A good χ value for the fitted track was required. The χ distribution isenergy dependent and thus the selection criterion has been calibrated in order to obtain aconstant efficiency of about 90% over the whole energy range, in particular at low energieswhere multiple scattering leads to generally higher χ values. Reconstructed tracks wererequired to lie entirely inside a fiducial volume with bounding surfaces 0.15 cm fromthe magnet walls. Galactic events were selected by imposing that the lower edge of therigidity bin to which the event belongs exceeds the critical rigidity, ρ c , defined as 1.3times the cutoff rigidity ρ SV C computed in the St¨ormer vertical approximation Shea et al.(1987) as ρ SV C = 14 . /L , where L is the McIlwain L -shell parameter McIlwain (1961)obtained by using the Resurs-DK1 orbital information and the IGRF magnetic field modelMacMillan & Maus (2005). The South Atlantic Anomaly region has been included in theanalysis. Reconstructed particle trajectories were required to be down-going according tothe TOF. No selections on the hit pattern in the TOF paddles or AC were made, since thiscan lead to very low efficiencies due to the production of delta rays in the aluminum domeof the pressurized vessel in which PAMELA is hosted. This introduces a contaminationfrom secondaries produced in hadronic interactions of primaries in the dome. This effecthas been accounted for using Monte Carlo simulations.Boron and carbon events have been selected by means of ionisation energy losses 10 –in the TOF system. Charge consistency has been required between S12 and h S2 i and h S3 i (the arithmetic mean of the ionisations for the two layers constituting S2 and S3,respectively). Requiring charge consistency above and below the tracking system rejectedevents interacting in the silicon layers. The selection bands as functions of the rigiditymeasured by the spectrometer are shown in Figure 2.In order to assess the presence of possible contamination in the selected samples, theabove selection cuts have been applied to boron and carbon samples independently selectedby means of S11 (the upper layer of S1) and the first silicon layer of the calorimeter. Theprobabilities of misidentifying a carbon nucleus as boron and vice versa are about 3 · − and 10 − , respectively, over the whole energy range considered in this analysis. Stricteranalysis criteria were imposed by narrowing the selection bands. When properly correctedby the selection efficiency (see Section 3.3), the event counts showed no statisticallysignificant deviation from that obtained using the standard selection. The contaminationis therefore assumed to be negligible. Selected events have been binned according to therigidity measured by the magnetic spectrometer. The tracking efficiency has been evaluated with flight data and Monte Carlo simulationsusing a methodology similar to that described in Adriani et al. (2013). Two samples ofboron and carbon were selected by means of a β dependent requirement on ionisationenergy losses in the TOF system. Fiducial containment was verified using calorimeterinformation. Firstly, non-interacting events penetrating deeply into the calorimeter were S12 is the lowest of the two layers constituting S1; the upper layer S11 was used forefficiency measurement as explained in Section 3.3 11 – [GV] ρ d E / d x [ M I P ] CB ρ vs S12
TOF dE/dX [GV] ρ d E / d x [ M I P ] CB ρ vs
Fig. 2.— Charge selection bands for S12, h S2 i and h S3 i as a function of rigidity. The redvertical dotted lines denote the upper and lower rigidity limits of this analysis. The absenceof relativistic protons in this sample is due to the 5 MIP cluster selection described in Section3.1. 12 –identified, and a straight track fitted. Then, the rigidity of the nucleus was derived from the β measured by the TOF and used to back-propagate the track through the spectrometermagnetic field up to the top of the apparatus. The containment criteria were applied tothis back-extrapolated track. The tracking efficiency was determined for this sample ofnon-interacting nuclei as a function of the rigidity derived from β . The 70 ps resolution ofthe TOF system for carbon leads to ∆ β/β ∼
2% at β = 0 . β . To account forthese effects, a simulation of the PAMELA apparatus based on GEANT4 Agostinelli et al.(2003); Allison et al. (2006) has been used to estimate the isotropic, rigidity-dependenttracking efficiency which is subsequently divided by a Monte Carlo efficiency obtained usingthe same procedure as the experimental efficiency. The resulting ratio, which has an almostconstant value of about 0.97, has been used as the correction factor for the experimentalefficiency. The constancy of the ratio results from an isotropic efficiency that is also almostconstant above 10 GV because of the data processing procedures described in Section 3.1.The efficiencies for the selection of down-going particles and for charge selectionhave been estimated using flight data exclusively. The down-going requirement is 100%efficient due to the 70 ps resolution of the TOF system. To evaluate the charge selectionefficiency, the redundancy of the PAMELA subdetectors has been exploited. Two samplesof boron and carbon have been tagged requiring charge consistency on S11 and on the firstsilicon layer of the calorimeter. These two detectors are placed at the two extrema of theapparatus, so this selection rejects interactions which change the reconstructed charge of 13 –the incident particle. The resulting efficiencies have a peak value of ∼
75% at 3 GV andthen decrease at high energies towards an almost constant value of about 50% for boronand 60% for carbon above some tens of GV.The tracking and the charge selection efficiencies are shown in Figure 3 together withthe total selection efficiency. (GV) ρ ε Carbon
Tracking efficiencyCharge sel. efficiencyTotal efficiency
Carbon (GV) ρ ε Boron
Tracking efficiencyCharge sel. efficiencyTotal efficiency
Boron
Fig. 3.— Selection efficiencies as functions of rigidity. The dashed line is a fit of the chargeselection efficiency above 3 GV with a power law at low rigidities and a constant value athigh rigidities. The slope, the break point and the normalization are free parameters of thefit. The fitted charge selection efficiency is used to compute the total efficiency above 3GeV/n (about 7.6 GV for C and B and 8.4 GV for B) in order to smooth the statisticalfluctuations.The measurement of the charge selection efficiency sets the lower rigidity limit forfluxes to 2 GV, corresponding to about 0.44 GeV/n for B and C. Below this thresholdcharge confusion in the calorimeter selection becomes too large to be able to reliably tag 14 –pure boron and carbon samples for an efficiency measurement.The effects of a possible contamination in the efficiency samples tagged with S11 andthe calorimeter (S11+CALO tag) have been investigated by considering a single TOF layerand measuring the charge selection efficiency both on the event set tagged with S11+CALOand on a purer sample obtained by adding the other TOF planes to the S11+CALO tag.The two efficiencies were found to be consistent within statistical errors for each layer. Noeffect due to contamination in the S11+CALO tagged set was observed.
The selected boron and carbon samples are contaminated by secondaries producedduring fragmentation processes occurring in the aluminum dome on top of the pressurizedvessel hosting PAMELA. This effect has been studied with a Monte Carlo calculation basedon the FLUKA code Battistoni et al. (2007) by simulating the cosmic spectra for C and O,which are the main contributors to the contamination. The resulting contamination is ofthe order of 10 − for carbon, whereas for boron it ranges from about 5% at some GV up toabout 20% at ∼
200 GV, coming mainly from spallation of carbon.After subtracting the contamination, the rigidity distributions of boron and carbonevents have been corrected for folding effects using a Bayesian procedure D’Agostini (1995),in order to obtain the distributions at the top of payload. These effects include possiblerigidity displacements at high energies due to the finite position resolution of the silicontracking layers and the energy loss of low-energy nuclei traversing the apparatus. Thesmearing matrix was derived using the GEANT4 simulations.Interactions with the aluminum dome also remove primaries from the selected samples.Elastic scattering processes can remove primaries from the instrument acceptance or slow 15 –them down so that they are swept out by the magnetic field; inelastic scattering can destroythe primary. A correction factor for these effects has been evaluated using the FLUKAsimulations, and applied to the unfolded event count. The correction is almost flat above10 GV and amounts to 15% for carbon and 14% for boron, increasing at lower energiesbecause of energy loss. These numbers have been treated as corrections to the geometricalfactor for the two nuclear species. The resulting geometrical factors are shown in Figure 4. (GV) ρ s r) G ( c m Fig. 4.— Effective geometrical factors including the fiducial containment criterion and thecorrection for interactions of primary particles above the tracker. The dashed lines are fitsused to obtain asymptotic values at high energy.Energy loss in the apparatus may lower the measured rigidity below the critical rigidity,leading to rejection of galactic nuclei with initial rigidity above the critical one. A “cutoffcorrection factor” for each nuclear species was computed by assigning a random cutoffvalue (distributed as observed for in-flight values) to events simulated with GEANT4 andderiving the fraction of rejected events. This correction factor rises from about 0.97 at 2 16 –GV to unity (i.e. no correction) at 3 GV and above.
The live time of the apparatus is measured by on-board clocks and has been evaluatedas a function of the vertical cutoff as the time spent in regions where the critical rigidityis below the lower limit of the rigidity bin. The total live time is constant at a valueof ∼ . × s for rigidities above 20 GV and decreases at lower rigidities because ofthe shorter time spent by the satellite in high latitude (i.e. low cutoff) regions down to ∼ . × s at 2 GV. The overall error on live time determination is less than 0.2%, andhas therefore been neglected. Due to the requirement of track containment inside a fiducial volume (see Section 3.2),the effective geometrical factor turns out to be lower than the nominal one, and assumesa constant value of 19.9 cm sr above 1 GV. This value has been cross-checked using twodifferent numerical methods. The first one is a numerical computation of the integraldefining the geometrical factor Sullivan (1971), taking into account the curvature of thetrack due to the magnetic field, while the second method relies on a Monte Carlo simulationSullivan (1971). The two methods yield results differing by less than 0.1%. This error hasalso been neglected. 17 – The fluxes have been computed both as functions of rigidity and as functions of kineticenergy per nucleon. For each bin i , the event count ∆ N ′ i corrected for the effects describedin Section 3.4 was divided by the live time ∆ T i , the effective geometrical factor ˜ G i , thetotal selection efficiency ǫ i and the bin width ∆ ρ i or ∆ E i . The flux expressed as a functionof rigidity is computed as: φ ( ρ i ) = ∆ N ′ i ∆ T i ˜ G i ǫ i ∆ ρ i , while as a function of kinetic energy per nucleon: φ ( E i ) = ∆ N ′ i ∆ T i ˜ G i ǫ i ∆ E i . For boron, the latter formula needs to properly account for isotopic composition, asexplained in the next section.
In cosmic rays, both the isotopes B and B are present in comparable quantities.Since the event selection did not distinguish between them and since the events arebinned according to their rigidity, a given value for the isotopic composition of boronmust be assumed in order to perform the measurements as functions of kinetic energy pernucleon. Large uncertainties plague the available estimates of the isotopic compositionof boron. Direct measurements are available only at relatively low energies Ahlen et al.(2000); Hams et al. (2004); Aguilar et al. (2011). Galactic propagation models predict ahigh-energy value for the B fraction (i.e., B/( B + B)) which is weakly dependent onkinetic energy per nucleon and whose consensus value is ˜ F B = 0 . ± .
15. This value hasbeen used in this analysis for the whole energy range. 18 –The boron flux has been evaluated considering two different hypotheses: pure B andpure B. Assuming a binning in kinetic energy per nucleon, the corresponding binningin rigidity for each of the two hypotheses has been derived. Event selection, efficiencymeasurements, flux computation and corrections have then been performed in the sameway for the two binnings. The two boron fluxes are combined to obtain the final flux,considering that each bin of each flux distribution contains B and B events with thesame rigidity but different energy due to the different masses. Consequently, in each binthe isotopic fraction does not resemble the usual fraction expressed as a function of kineticenergy per nucleon, and a simple bin-by-bin linear combination of the two fluxes using˜ F B as the weight would lead to an incorrect result. A fraction F B ( ρ ) has been derived bymeans of Monte Carlo simulations and used as a weight in order to linearly combine thetwo boron fluxes bin by bin and obtain a final flux. A detailed description of the calculationis presented in Appendix B.
4. Results
The observed number of selected boron and carbon events, the absolute fluxes and theB/C flux ratio are reported in Tables 1 and 2. The quoted systematic uncertainties arediscussed in detail in Appendix A. The fluxes and the B/C ratio are also shown in Figures 5and 6 along with measurements from other experiments and a theoretical calculation basedon GALPROP. The details of the calculation are described in Section 5. The mean kineticenergy < E > and the mean rigidity < ρ > for each bin have been computed accordingto Lafferty & Wyatt (1995) using an iterative procedure starting from the middle point ofeach bin. The resulting mean energies and rigidities for boron and carbon differ by lessthan 1%, and have been considered to be equal.The discrepancies with other experiments at low energies can be reasonably ascribed 19 –to solar modulation effects. The data used for this analysis were taken by PAMELAduring an unusually quiet solar minimum period, resulting in an enhanced flux of galacticcosmic rays at low energies in the heliosphere, which has already been observed for protonsAdriani et al. (2013) and nuclei Mewaldt et al. (2010). Above 6 GeV/n the fluxes are inoverall agreement with the other available measurements, especially with those from HEAOand CREAM. A power-law fit above 20 GeV/n results in a spectral index γ B = 3 . ± . γ C = 2 . ± .
06 for carbon.
5. Discussion
A comprehensive and detailed study of the results presented above is beyond the scopeof this paper. The following discussion is intentionally limited to a single propagation modelin order to compute an estimate of the most significant propagation parameters from thePAMELA boron and carbon data. Results may vary when considering different models orpropagation software packages.The data presented in the previous section as a function of kinetic energy per nucleonhas been fitted with a diffusive cosmic ray propagation model using the GALPROP codeinterfaced with the MIGRAD minimizer in the MINUIT2 minimization package distributedwithin the ROOT framework Brun & Rademakers (1997). Only a few parameters havebeen left free because of the high computation time required for multiple GALPROPruns. The values for the other parameters have been taken from Vladimirov (2012).The diffusion coefficient is found to have a fitted slope value of δ = 0 . ± .
007 and anormalization factor D = (4 . ± . · cm /s. Other fitted parameters are the solarmodulation parameter in the force-field approximation Φ = (0 . ± .
01) GV and the overallnormalization of the fluxes N = 1 . ± .
03. The result of the fit is shown in Figures 5 and6. A contour plot of the confidence intervals for δ and D is shown in Figure 7. 20 – E(GeV/n)0.4 1 2 3 4 5 6 7 8 910 20 30 40 50 100 . ( G e V / n ) - s s r) ( m . E × F l ux -1 C B
PAMELAGalpropCREAMTRACERATIC-2HEAOCRN
E(GeV/n)0.4 1 2 3 4 5 6 7 8 10 20 30 40 100 B / C PAMELAAMS-02 (preliminary)GalpropCREAMTRACERATIC-2HEAOAMS-01
Fig. 5.— Absolute boron and carbon fluxes multiplied by E . (upper panel) and B/C fluxratio (lower panel) as measured by PAMELA, together with results from other experiments(AMS02 Oliva et al. (2013), CREAM Ahn et al. (2008), TRACER Obermeier et al. (2011),ATIC-2 Panov et al. (2007), HEAO Engelmann et al. (1990), AMS01 Aguilar et al. (2010),CRN Swordy et al. (1990)) and a theoretical calculation based on GALPROP (see Section5), as functions of kinetic energy per nucleon. For PAMELA data the error bars representthe statistical error and the shaded area is the overall systematic uncertainty summarized inAppendix A. 21 – (GV) ρ . ( GV ) - s s r) ( m . ρ × F l ux C B
PAMELAGalprop (GV) ρ B / C PAMELAGalprop
Fig. 6.— Absolute boron and carbon fluxes multiplied by ρ . (upper panel) and B/C fluxratio (lower panel) as measured by PAMELA, together with a theoretical calculation basedon GALPROP (see Section 5), as functions of rigidity. The error bars represent the statisticalerror and the shaded area is the overall systematic uncertainty summarized in Appendix A,except for the boron mixing error which does not affect the rigidity-dependent boron flux. Kinetic energy h E i C events C flux B events B events B flux B/Cat top of payload value ± stat. ± syst. value ± stat. ± syst. value ± stat. ± syst.(GeV/n) (GeV/n) (GeV/n m s sr) − (GeV/n m s sr) − . ± . ± .
26) 1566 1795 (1 . ± . +0 . − . ) (3 . ± . +0 . − . ) · − . ± . ± .
21) 1955 2092 (1 . ± . +0 . − . ) (3 . ± . +0 . − . ) · − . ± . ± .
16) 2300 2320 (1 . ± . +0 . − . ) (3 . ± . +0 . − . ) · − . ± . ± .
12) 2351 2248 (7 . ± . +0 . − . ) · − (3 . ± . +0 . − . ) · − . ± . ± . . ± . +0 . − . ) · − (3 . ± . +0 . − . ) · − . ± . ± . . ± . +0 . − . ) · − (2 . ± . +0 . − . ) · − . ± . ± . · − . ± . +0 . − . ) · − (2 . ± . +0 . − . ) · − . ± . ± . · − . ± . ± . · − (2 . ± . +0 . − . ) · − . ± . ± . · − . ± . ± . · − (2 . ± . +0 . − . ) · − . ± . ± . · −
811 704 (3 . ± . +0 . − . ) · − (2 . ± . +0 . − . ) · − . ± . ± . · −
612 540 (1 . ± . ± . · − (2 . ± . +0 . − . ) · − . ± . ± . · −
454 369 (7 . ± . ± . · − (2 . ± . ± . · − . ± . ± . · −
253 217 (3 . ± . ± . · − (1 . ± . +0 . − . ) · − . ± . ± . · −
149 121 (1 . ± . +0 . − . ) · − (1 . ± . ± . · − . ± . ± . · −
85 69 (6 . ± . ± . · − (1 . ± . ± . · − . ± . ± . · −
79 65 (2 . ± . ± . · − (1 . ± . ± . · − . ± . ± . · −
31 24 (4 . ± . ± . · − (1 . ± . ± . · − . ± . ± . · − . ± . ± . · − (10 ± ± . · − Table 1: Observed number of events, absolute fluxes and the B/C flux ratio as function of kinetic energy per nucleon.Both the event counts for pure B and pure B hypotheses are reported.
Rigidity h ρ i C events C flux B events B flux B/Cat top of payload value ± stat. ± syst. value ± stat. ± syst. value ± stat. ± syst.(GV) (GV) (GV m s sr) − (GV m s sr) − . ± . ± .
10) 1566 (6 . ± . ± . · − (3 . ± . ± . · − . ± . ± .
08) 1955 (5 . ± . ± . · − (3 . ± . ± . · − . ± . ± . . ± . ± . · − (3 . ± . ± . · − . ± . ± . . ± . ± . · − (3 . ± . ± . · − . ± . ± . · − . ± . ± . · − (3 . ± . ± . · − . ± . ± . · − . ± . ± . · − (3 . ± . ± . · − . ± . ± . · − . ± . ± . · − (3 . ± . ± . · − . ± . ± . · − . ± . ± . · − (2 . ± . ± . · − . ± . ± . · − . ± . ± . · − (2 . ± . ± . · − . ± . ± . · −
811 (1 . ± . ± . · − (2 . ± . ± . · − . ± . ± . · −
612 (8 . ± . ± . · − (2 . ± . ± . · − . ± . ± . · −
454 (4 . ± . ± . · − (2 . ± . ± . · − . ± . ± . · −
253 (2 . ± . ± . · − (2 . ± . ± . · − . ± . ± . · −
149 (8 . ± . ± . · − (1 . ± . ± . · − . ± . ± . · −
85 (3 . ± . ± . · − (1 . ± . ± . · − . ± . ± . · −
79 (1 . ± . ± . · − (1 . ± . ± . · − . ± . ± . · −
31 (2 . ± . ± . · − (1 . ± . ± . · −
152 - 260 193 86 (4 . ± . ± . · − . ± . ± . · − (1 . ± . ± . · − Table 2: Observed number of events, absolute fluxes and the B/C flux ratio as function of rigidity. 24 – δ / s ) ( c m D × Fig. 7.— Contour plot of the 1-, 2- and 3-sigma confidence levels for δ and D .The fitted value for δ falls between the predicted values for Kolmogorov ( δ = 1 /
3) andKraichnan ( δ = 1 /
2) diffusion types, thus the PAMELA data cannot distinguish betweenthese two types.
6. Acknowledgements
We acknowledge support from The Italian Space Agency (ASI), Deutsches Zentrum f¨urLuft- und Raumfahrt (DLR), The Swedish National Space Board, The Swedish ResearchCouncil, The Russian Space Agency (Roscosmos) and The Russian Science Foundation.
A. Systematic uncertainties
The following contributions to the systematic uncertainty have been considered: 25 – • Selection efficiencies : the measurement of the tracking and charge selection efficienciesfrom flight data is performed using samples of finite size. The associated statisticalerror has been propagated to the flux as a systematic uncertainty. • Fiducial containment : the finite tracking resolution of the calorimeter can lead toa contamination of the tracking efficiency sample by events coming from outsidethe fiducial acceptance, and possibly also crossing the magnet walls. These canin principle be eliminated by further restricting the fiducial volume for both eventselection and efficiency measurement, but this would significantly reduce the samplesizes. The chosen approach is to use protons from both flight and simulated data tomeasure the tracking efficiency for both the fiducial volume defined in Section 3.2and a more restrictive one. Their relative difference is taken as an estimate of thesystematic uncertainty, which is about 2%. Monte Carlo simulations give results forboron and carbon which are consistent with the one obtained with protons. Theuncertainty is propagated to the final flux. • Monte Carlo correction factor for the tracking efficiency : this correction factor shouldintroduce only relatively small errors, since it is computed as the ratio of two MonteCarlo efficiencies. Systematic effects should largely cancel out. The correction factoris constant at 0.97 for both boron and carbon. That this factor remains constant athigh rigidity is due to the isotropic efficiency being constant at relativistic rigidities. Aconservative factor of 3% has been taken as an estimate of the systematic uncertaintyon the flux because of this correction factor. • Residual coherent misalignment of the spectrometer : the spectrometer alignmentprocedure results in a residual coherent misalignment producing a systematic shift inthe measured rigidity. The error estimation procedure is described in the SupportingOnline Material of Adriani et al. (2011) . This error has been propagated to the 26 –measured flux. It is negligible at low energy and increases up to about 2% at 250 GV. • Cutoff, contamination and geometrical factor corrections : all these factors have beenevaluated on finite-size samples, so they are affected by a statistical error which hasbeen propagated to the flux as a systematic uncertainty. • Unfolding : the unfolding error has been assessed by means of the procedure describedin the Supporting Online Material of Adriani et al. (2011), comparing a given initialspectrum and an unfolded Monte Carlo simulation. The two were found to be inagreement within 3%, so this value has been taken as the unfolding contribution tothe flux error. • Isotopic composition of boron : the uncertainty associated with this poorly knownparameter has been propagated to the flux by assuming the extreme values of 0.2and 0.5 for the B fraction and taking the difference between these fluxes and theone obtained with ˜ F B = 0 .
35 as the estimated upper and lower errors on the flux.This error affects only the measurement expressed as a function of kinetic energy pernucleon.The overall uncertainty has been estimated as the quadratic sum of the above terms in thehypothesis of uncorrelated errors. A summary plot is shown in Figure 8.
B. Isotopic composition of boron
In this analysis the events have been binned according to their rigidity as measured bythe magnetic spectrometer. Given that the event selection does not distinguish between thetwo isotopes B and B, each bin is populated by B and B events with approximatelythe same rigidity (within the bin limits) but different kinetic energy per nucleon because of 27 –
E (GeV/n)1 10 S y s t . un c . % Boron
Charge selectionTrack selectionsContaminationUnfoldingGeometrical factorCutoff correctionIsotopic CompositionTrack efficiency correctionsCoherent misalignmentTotal
Boron
E (GeV/n)1 10 S y s t . un c . % Carbon
Charge selectionTrack selectionsContaminationUnfoldingGeometrical factorCutoff correctionTrack efficiency correctionsCoherent misalignmentTotal
Carbon
Fig. 8.— Systematic uncertainties for absolute fluxes. The total contribution is computedas the quadratic sum of the individual terms. The track selections term is the quadratic sumof the contributions from statistics and from fiducial containment. The contributions of thetrack efficiency correction and of the unfolding have been slightly shifted apart from their3% value for the sake of readability. 28 –the different mass numbers. Consequently, the isotopic composition in a given bin is notdescribed by the B fraction F B expressed as a function of kinetic energy per nucleon E : F B ( E ) = φ B ( E ) φ B ( E ) + φ B ( E ) , (B1)where φ B ( E ) and φ B ( E ) are the fluxes of B and B respectively. A fraction expressedas a function of rigidity must then be derived in order to correctly account for the isotopiccomposition in each bin: F B ( ρ ) = φ B ( ρ ) φ B ( ρ ) + φ B ( ρ ) . (B2)Using rigidity bins of finite size leads to: F B ( ρ i ) = ∆ N B ( ρ i )∆ N B ( ρ i ) + ∆ N B ( ρ i ) , (B3)where F B ( ρ i ) is the B fraction for the i -th rigidity bin centered at ρ i , while ∆ N B ( ρ i )and ∆ N B ( ρ i ) are the B and B event count for the same bin, respectively. ∆ N B ( ρ i )can be rewritten using the B fraction in kinetic energy:∆ N B ( ρ i ) = ∆ N B ( E i ) = 1 − F B ( E i ) F B ( E i ) ∆ N B ( E i ) . (B4)Here ∆ N B ( E i ) denotes the B event count in a bin in kinetic energy per nucleon whoselimits are obtained by converting the limits in rigidity of the i -th bin to kinetic energyassuming the mass and the charge of B. E i is the kinetic energy per nucleon of a Bnucleus of rigidity ρ i . Then, by construction, the first equality in the above equation follows.The second equality follows from the definition of F B ( E i ) which is the equivalent of eq. B3for kinetic energy bins. Note that:∆ N B ( ρ i ) = ∆ N B ( E i ) , (B5)since the limits of the energy and rigidity bins do not correspond for B. Converting the binlimits in energy back to rigidity but assuming now the mass and the charge of B yields:∆ N B ( E i ) = ∆ N B ( ρ ′ i ) . (B6) 29 – ρ ′ i is then the rigidity of a B nucleus having the same kinetic energy per nucleon E i of a B nucleus of rigidity ρ i (the same relation holds between the limits of the bins centered in ρ ′ i and ρ i ). To obtain the explicit relationship between ρ ′ i and ρ , write E i as: ρ ′ i = A Z q ( E i ) + 2 m p E i , (B7)where Z is the atomic number of boron, A is the mass number of B and m p is the protonmass, and then E i as a function of ρ i : E i = s Z A ρ i + m p − m p , (B8)with A the mass number of B. It follows that: ρ ′ i = A A ρ i . (B9)The final form of the rigidity-dependent B fraction is then: F B ( ρ i ) ≈ ∆ N B ( ρ i )∆ N B ( ρ i ) + − ˜ F B ˜ F B ∆ N B ( ρ ′ i ) , (B10)where the approximated energy-independent value F B ( E ) ≈ ˜ F B has been used.Generally speaking, the fraction expressed as a function of rigidity is not constant anddepends on the spectral shape. To account for this a toy Monte Carlo simulation of realistic B and B spectra taken from a galactic propagation model has been set up, the resultingevent counts have been trimmed to reproduce ˜ F B = 0 .
35 and finally the events have beenbinned according to their rigidity for both the pure B and pure B hypotheses. Knowingthe fraction in each rigidity bin of the two binnings one can express the final boron flux inthe i -th energy bin as: φ B ( E i ) = F B i φ ( E i ) + (1 − F B i ) φ ( E i ) , (B11)where F B i and F B i are the B fraction obtained from the toy Monte Carlo in the i -thrigidity bin for pure B and pure B hypotheses respectively, and φ ( E i ) and φ ( E i ) are 30 –the experimental fluxes for the pure B and pure B hypotheses respectively (see Section3.8).To assess the difference between the B fraction as a function of kinetic energy pernucleon and as a function of rigidity, eq. B10 can be computed at high energies. Above fewGeV/n, where the spectrum can be well described with a power-law function with index γ ,eq. B10 gives a B fraction F B ( ρ i ) ≈
11 + − ˜ F B ˜ F B ( A /A ) − γ ≃ . , (B12)which is in agreement with the value obtained from the toy Monte Carlo and differs from˜ F B = 0 .
35 by about 18%. 31 –