Back-Tracing and Flux Reconstruction for Solar Events with PAMELA
A. Bruno, O. Adriani, G. C. Barbarino, G. A. Bazilevskaya, R. Bellotti, M. Boezio, E. A. Bogomolov, M. Bongi, V. Bonvicini, S. Bottai, U. Bravar, F. Cafagna, D. Campana, R. Carbone, P. Carlson, M. Casolino, G. Castellini, E. C. Christian, C. De Donato, G. A. de Nolfo, 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, M. Lee, 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, M. Ricci, S. B. Ricciarini, J. M. Ryan, R. Sarkar, V. Scotti, M. Simon, R. Sparvoli, P. Spillantini, S. Stochaj, 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 . I M ] N ov Back-Tracing and Flux Reconstruction for Solar Events withPAMELA
A. Bruno , ∗ , O. Adriani , , G. C. Barbarino , , G. A. Bazilevskaya , R. Bellotti , ,M. Boezio , E. A. Bogomolov , M. Bongi , , V. Bonvicini , S. Bottai , U. Bravar ,F. Cafagna , D. Campana , R. Carbone , P. Carlson , M. Casolino , , G. Castellini ,E. C. Christian , C. De Donato , , G. A. de Nolfo , 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 , M. Lee , 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 , , M. Ricci ,S. B. Ricciarini , , J. M. Ryan , R. Sarkar , , V. Scotti , , M. Simon , R. Sparvoli , ,P. Spillantini , , S. Stochaj , Y. I. Stozhkov , A. Vacchi , E. Vannuccini , G. I. Vasilyev ,S. A. Voronov , Y. T. Yurkin , G. Zampa , N. Zampa , and V. G. Zverev . Department of Physics, University of Bari, I-70126 Bari, Italy. Department of Physics and Astronomy, University of Florence, I-50019 Sesto Fiorentino,Florence, Italy. INFN, Sezione di Florence, I-50019 Sesto Fiorentino, Florence, Italy. Department of Physics, University of Naples “Federico II”, I-80126 Naples, Italy. INFN, Sezione di Naples, I-80126 Naples, Italy. Lebedev Physical Institute, RU-119991 Moscow, Russia. 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. Space Science Center, University of New Hampshire, Durham, NH, USA. KTH, Department of Physics, and the Oskar Klein Centre for Cosmoparticle Physics,AlbaNova University Centre, SE-10691 Stockholm, Sweden. 2 – INFN, Sezione di Rome “Tor Vergata”, I-00133 Rome, Italy. RIKEN, Advanced Science Institute, Wako-shi, Saitama, Japan. IFAC, I-50019 Sesto Fiorentino, Florence, Italy. Heliophysics Division, NASA Goddard Space Flight Ctr, Greenbelt, MD, USA. National Research Nuclear University MEPhI, RU-115409 Moscow, Russia. Department of Physics, University of Rome “Tor Vergata”, I-00133 Rome, Italy. Agenzia Spaziale Italiana (ASI) Science Data Center, I-00133 Rome, Italy. Department of Physics, University of Trieste, I-34147 Trieste, Italy. INFN, Laboratori Nazionali di Frascati, I-00044 Frascati, Italy. Department of Physics, Universit¨at Siegen, D-57068 Siegen, Germany. Indian Centre for Space Physics, 43 Chalantika, Garia Station Road, Kolkata 700084,West Bengal, India. Previously at INFN, Sezione di Trieste, I-34149 Trieste, Italy. Electrical and Computer Engineering, New Mexico State University, Las Cruces, NM,USA.Received ; accepted * Corresponding author. E-mail address: [email protected]. 3 –
ABSTRACT
The PAMELA satellite-borne experiment is providing first direct measure-ments of Solar Energetic Particles (SEPs) with energies from ∼
80 MeV to severalGeV in near-Earth space. Its unique observational capabilities include the possi-bility of measuring the flux angular distribution and thus investigating possibleanisotropies related to SEP events. This paper focuses on the analysis methodsdeveloped to estimate SEP energy spectra as a function of the particle pitchangle with respect to the Interplanetary Magnetic Field (IMF). The crucial in-gredient is provided by an accurate simulation of the asymptotic exposition of thePAMELA apparatus, based on a realistic reconstruction of particle trajectoriesin the Earth’s magnetosphere. As case study, the results of the calculation forthe May 17, 2012 event are reported.
1. Introduction
Solar Energetic Particles (SEPs) are high energy particles associated with explosivephenomena occurring in the solar atmosphere, such as solar flares and Coronal MassEjections (CMEs). They can significantly perturb the Earth’s magnetosphere producing asudden increase in particle fluxes and, consequently, in the radiation exposure experiencedby spacecrafts and their possible crew. SEPs constitute a sample of solar material andprovide important information about the sources of the particle populations, and theirangular distribution can be used to investigate details of the particle transport in theinterplanetary medium.SEP measurements are performed both by in-situ detectors on spacecrafts and byground-based Neutron Monitors (NMs). While the former are able to measure SEPs with 4 –energies below some tens of MeV, the latter can only register the highest energy SEPs ( & ∼
80 MeV up to several GeV, hence bridging the low energydata by space-based instruments and the GLE data by the worldwide network of NMs. Inaddition, PAMELA is sensitive to the particle composition and it is able to reconstructthe flux angular distribution, thus enabling a clearer and more complete view of the SEPevents.This paper reports the analysis methods developed for the estimate of SEP energyspectra with the PAMELA apparatus, as a function of the particle asymptotic directionof arrival. As case study, the results used in the analysis of the May 17, 2012 solar event(Adriani et al. 2015b) are presented.
2. The PAMELA Experiment
PAMELA is a space-based experiment designed for a precise measurement of thecharged cosmic radiation in the kinetic energy range from some tens of MeV up to severalhundreds of GeV (Picozza et al. 2007; Adriani et al. 2014). The Resurs-DK1 satellite,which hosts the apparatus, was launched into a semi-polar (70 deg inclination) andelliptical (350–610 km altitude) orbit on June the 15 th .
3. Geomagnetic Field Models
The analysis of SEP events described in this work relies on the IGRF-11 (Finlay et al.2010) and the TS07D (Tsyganenko & Sitnov 2007; Sitnov et al. 2008) models for thedescription of the internal and external geomagnetic field, respectively: the former employsa global spherical harmonic implementation of the main magnetic field; the latter is a highresolution dynamical model of the storm-time geomagnetic field in the inner magnetosphere,based on recent satellite measurements. Consistent with the dataset coverage, the modelis valid within the region delimited by the magnetopause (based on Shue et al. 1998)and by a spherical surface with a radius of 30 Earth’s radii (Re). Solar Wind (SW)and IMF parameters are obtained from the high resolution (5-min) Omniweb database(King & Papitashvili 2004). The TS07D model is more flexible and accurate with respectto all past empirical models in reconstructing the distribution of storm-scale currents, so itis particularly adequate for the study of SEP events.
4. Back-tracing Techniques
Cosmic Ray (CR) cutoff rigidities and asymptotic arrival directions (i.e. the directionsof approach before encountering the Earth’s magnetosphere) are commonly evaluated bysimulations, accounting for the effect of the geomagnetic field on the particle transport (seee.g. Smart et al. 2000 and references therein). Using spacecraft ephemeris data (position,orientation, time), and the particle rigidity ( R = momentum/charge) and direction providedby the tracking system, trajectories of all detected protons are reconstructed by means of a 6 –tracing program based on numerical integration methods (Smart & Shea 2000, 2005), andimplementing the afore-mentioned geomagnetic field models.To reduce the computational time, geomagnetically trapped (Adriani et al. 2015a)and most albedo (Adriani et al. 2015c) particles (originated from the interactions of CRswith the Earth’s atmosphere) are discarded by selecting only protons with rigidities R >R min = 10 /L − . L is the McIlwain’s parameter (McIlwain 1966). Then, eachtrajectory is back propagated from the measurement location and traced, with no constraintlimiting the total path-length or tracing time, until one of the two following conditions issatisfied:1. it reaches the model magnetosphere boundaries (see Section 3);2. or it reaches an altitude of 40 km.The two categories correspond to “allowed” and “forbidden” trajectories, respectively: theformer includes contributions from Solar and Galactic CRs (hereafter SCRs and GCRs),while events satisfying the latter condition, including albedo particles with rigidities greaterthan R min , are excluded from the analysis.
5. Asymptotic Arrival Directions
The asymptotic arrival directions are evaluated with respect to the IMF direction,with polar angles α and β denoting the particle pitch and gyro-phase angle, respectively.Both Geographic (GEO) and Geocentric Solar Ecliptic (GSE) coordinates are used. Figure Such a value approximately corresponds to the mean production altitude for albedoprotons. 7 –Fig. 1.— Sample CR proton trajectories reconstructed in the magnetosphere. The totalmagnetic field intensities [nT] are also shown (color coding) for the X-Y (top), the X-Z(middle) and the Y-Z (bottom) GSE planes. See the text for details. 8 –Fig. 2.— Estimated asymptotic arrival directions for protons detected during the May 17,2012 SEP event. See the text for details.1 reports some sample trajectories reconstructed in the GSE coordinate system. The totalmagnetic field intensities obtained with the IGRF-11 + TS07D models ( ≤ R E ) are alsoshown (color coding) for the X-Y (top), the X-Z (middle) and the Y-Z (bottom) GSE planes.The crosses denote the estimated position of the bow shock nose (King & Papitashvili2004).The trajectory analysis allows a deeper understanding of SEP events. To improve 9 –the interpretation of results, the directions of approach and the entry points at the modelmagnetosphere boundaries can be visualized as a function of particle rigidity and orbitalposition (Bruno et al. 2014). As an example, Figure 2 reports the proton results ( R . rd solar cycle. Only the first PAMELA polar pass which registered the event isincluded, corresponding to the interval 01:57 – 02:20 UT.Left panels show the reconstructed asymptotic directions for the selected proton sample(counts) in terms of GEO (top and middle) and GSE (bottom) coordinates; in the top panelcolors denote the number of proton counts in each bin, while in middle and bottom panelsthey refer to the particle rigidity. Distributions are integrated over the polar pass. Thespacecraft position is indicated by the grey curve. The contour curves represent values ofconstant pitch angle with respect to the IMF direction, denoted with crosses. As PAMELAis moving (eastward) and changing its orientation along the orbit, observed asymptoticdirections rapidly vary performing a (clockwise) loop over the region above Brazil (seemiddle panel).Right panels in the same figure display the distributions of asymptotic directions as afunction of UT, and particle rigidity (top and middle) and pitch-angle (bottom); colors inthe top panel denote the number of proton counts in each bin while, in middle and bottompanel, they correspond to particle pitch-angle and rigidity respectively. Solid curves denotethe estimated St¨ormer vertical cutoff (St¨ormer 1950; Shea et al. 1965) for the PAMELAepoch ( ∼ . /L GV).Since PAMELA aperture is about 20 deg, the observable pitch-angle range at agiven rigidity is quite small (a few deg) except in the penumbral regions around the localgeomagnetic cutoff, where particle trajectories become complex (chaotic trajectories) andboth allowed and forbidden bands of CR trajectories are present (Cooke et al. 1991). 10 –Corresponding asymptotic directions rapidly change with particle rigidity and lookingdirection. Conservatively, these regions are excluded from the analysis.
6. Flux Evaluation6.1. Apparatus Gathering Power
The factor of proportionality between fluxes and counting rates, corrected for selectionefficiencies, is by definition the gathering power Γ ( cm sr ) of the apparatus:Γ = Z Ω dωF ( ω ) Z S dσ · ˆ r = Z Ω dωF ( ω ) A ( ω ) , (1)where Ω is the solid angle domain limited by the instrument geometry, S is the detectorsurface area, ˆ r d σ is the effective element of area looking into ω , F ( ω ) is the flux angulardistribution (varying between 0 and 1) and A ( ω ) is the directional response function(Sullivan 1971).In the case of the PAMELA instrument, Γ is rigidity dependent due to the spectrometerbending effect on particle trajectories: it decreases with decreasing rigidity R since particleswith lower rigidity are more and more deflected by the magnetic field toward the lateralwalls of the magnetic cavity, being absorbed before reaching the lowest plane of the Timeof Flight system, which provides the event trigger (see Figure 3).In terms of the zenith θ and the azimuth φ angles describing downward-going particle 11 –Fig. 3.— Schematic view of the PAMELA apparatus, including only parts constrainingthe field of view: the Time of Flight system, denoted with S1, S2 and S3; the magneticspectrometer with first and last tracking system planes (denoted with T1 and T6) and themagnetic cavity. The PAMELA reference frame is also reported.directions in the PAMELA frame :Γ( R ) = Z − dcosθ Z π dφ F ( R, θ, φ ) S ( R, θ, φ ) | cosθ | , (2)where S ( R, θ, φ ) is the apparatus response function in units of area ( cm ), and the cos θ factor accounts for the trajectory inclination with respect to the instrument axis.For non-ideal detectors, it is necessary to account for the effect of elastic and inelasticparticle interactions inside the apparatus (especially with its mechanical parts), on the The PAMELA reference system has the origin in the center of the spectrometer cavity;the Z axis is directed along the main axis of the apparatus, toward the incoming particles; theY axis is directed opposite to the main direction of the magnetic field inside the spectrometer;the X axis completes a right-handed system. 12 –measured coincidence rate. Consequently, Γ is different for each particle species (protons,electrons, ions, etc.).
For an isotropic particle flux, the gathering power does not depend on looking direction(i.e. F ( ω ) = 1), and it is usually called the geometrical factor G . A technically simple but efficient solution for the calculation of the geometrical factorof the apparatus is provided by Monte Carlo methods (Sullivan 1971). The solid angle issubdivided into a large number of (∆ cosθ, ∆ φ ) bins, with the angular domain limited todownward-going directions. For each bin: • particles are produced with random position on a plane generation surface with area S gen , placed just above the apparatus opening aperture, so that a large number ofevents corresponds to the intensity incident on the instrument. • For each particle, a random (i.e. random cosθ and φ ) direction is chosen in theselected (∆ cosθ, ∆ φ ) bin. • Trajectories are propagated through the apparatus and tested at successively deeperlayers: only particles satisfying all the detector geometrical constraints defining theapparatus Field of View (FoV) are selected. 13 –The procedure is repeated until the desired statistical precision (see below) is achieved.Then, for each rigidity value, the geometrical factor is obtained as: G ( R ) ≃ S gen ∆ cosθ ∆ φ X cosθ X φ k ( R, θ, φ ) | cosθ | (3)with k ( R, θ, φ ) = n sel ( R, θ, φ ) n tot ( R, θ, φ ) , (4)where n sel ( R, θ, φ ) and n tot ( R, θ, φ ) are the numbers of selected and generated trajectoriesin each angular bin, and the cosθ factor accounts for the probability of a particle trajectorywith direction ( cosθ, φ ): P ( θ, φ ) dcosθ dφ = | cosθ | dcosθ dφ. (5)The statistical uncertainty on G ( R ) can be evaluated with binomial methods:∆ G ( R ) ≃ S gen ∆ cosθ ∆ φ vuutX cosθ X φ (cid:20) k ( R, θ, φ )(1 − k ( R, θ, φ )) n tot ( R, θ, φ ) (cid:21) cos θ. (6)Advantage is taken of the angular subdivision, by varying directions over the ∆ cosθ ∆ φ solid angle rather than a full 2 π hemisphere. In particular, a number n tot ( R, θ, φ ) of incidenttrajectories proportional to cos − θ c (with θ c taken at ∆ cosθ bin center) is chosen in orderto minimize the statistical error associated to the angular bins with a small k ( R, θ, φ ).The dependence of the instrument response on particle rigidity is studied by performingthe calculation for 30 rigidities in the range 0.39–10 GV, and the value of G ( R ) atintermediate R is evaluated through interpolation methods. An accurate estimate of thePAMELA geometrical factor based on the Monte Carlo approach can be found in Bruno(2008).As an example, Figure 4 reports the k ( R, θ, φ ) ratio (color codes) over the PAMELAFoV as a function of local φ and cosθ , for 0.39 GV (top panel) and 4.09 GV (bottom panel) A more rigorous treatment is provided by Bayesian approaches. 14 –Fig. 4.— The k ( R, θ, φ ) = n sel ( R, θ, φ ) /n tot ( R, θ, φ ) ratio (color codes) over the PAMELAfield of view, as a function of local polar coordinates φ and cosθ (apparatus reference frame).Results for 0.39 GV (top) and 4.09 GV (bottom) protons are displayed. See the text fordetails.protons. The four peaks structure reflects the rectangular section of the apparatus; thedifferences in the FoV distributions at different rigidities are due to the bending effect ofthe magnetic spectrometer. 15 – However, in presence of an anisotropic flux exposition ( F = const ), the gathering powerdepends also on the flux angular distribution. SCR fluxes can be conveniently expressed interms of asymptotic polar angles α (pitch angle) and β (gyro-phase angle) with respect tothe IMF direction: F = F ( R, α, β ). The corresponding gathering power can be written as:Γ( R ) = Z π sinα dα Z π dβ F ( R, α, β ) S ( R, θ, φ ) | cosθ | , (7)with θ = θ ( R, α, β ) and φ = φ ( R, α, β ). The flux angular distribution F ( R, α, β ) is unknown.
For simplicity, we assume that SCR fluxes depend only on particle rigidity R andasymptotic pitch angle α , estimating an apparatus effective area (cm ) as: H ( R, α ) = sinα π Z π dβ S ( R, θ, φ ) | cosθ | , (8)by averaging the directional response function over the β angle. The method is valid alsofor isotropic fluxes (independent on α ): in this case, the effective area is related to thegeometrical factor G ( R ) by: G ( R ) = 2 π Z π dα H ( R, α ) . (9)The approach is analogous to the one developed for the measurement of geomagneticallytrapped protons (Bruno et al. 2015), with α and β denoting the polar angles with respectto the local geomagnetic field direction. But while local angles can be calculated from ( θ, φ )by means of basic rotation matrices of the FoV involving only the spacecraft orientationwith respect to the local magnetic field, the conversion to asymptotic angles depends on theparticle propagation in the magnetosphere, so trajectory tracing methods are needed. 16 – The effective area definition given in Equation 8 is based on the assumption ofapproximately isotropic fluxes within small pitch-angle bins. Consequently, H ( R, α ) can bederived from Equation 3 by integrating the directional response function over the ( cosθ, φ )directions corresponding to pitch angles within the interval α ± ∆ α/ π Z ∆ α dα H ( R, α ) ≃ S gen ∆ cosθ ∆ φ X θ,φ → α k ( R, θ, φ ) | cosθ | . (10)The approach accuracy depends on the number of (∆ cosθ, ∆ φ ) bins used in the angularpartitioning (see Section 6.2.1), while the width of the ∆ α bins is chosen accounting for thedetector angular resolution.To convert local ( cosθ, φ ) into asymptotic ( α, β ) directions and apply Equation 10, alarge number of trajectories N , uniformly distributed inside PAMELA field of view, has tobe reconstructed in the magnetosphere, for each rigidity value and each orbital position. Inorder to assure a high resolution, the calculation is performed for time steps with a 1-secwidth, back-tracing about 2800 trajectories for each rigidity bin, for a total of more than10 trajectories for each polar pass ( ∼
23 min). At a later stage, results are extended overthe full field of view of PAMELA through a bilinear interpolation.The procedure is illustrated in Figures 5-6 for 0.39 and 4.09 GV protons respectively,at a sample orbital position (May 17, 2012, 02:07 UT). Top panels report the distributionof reconstructed directions in the PAMELA field of view, with each point associated to agiven asymptotic direction ( α , β ); middle panels show the calculated (after interpolation)pitch-angle coverage ; bottom panels illustrate the estimated effective area as a function ofthe explored pitch-angle range. Final calculation results for 22 rigidity values between 0.39 The β calculation can be neglected under the gyro-tropic approximation. 17 –Fig. 5.— Top: simulated directions (red points) inside PAMELA field of view. Middle:pitch-angle coverage (color axis, deg). Bottom: the apparatus effective area as function ofpitch-angle; minimum and maximum observable pitch-angles are reported, along with thevalue corresponding to the vertical direction. Results correspond to 0.39 GV protons for asample orbital position (May 17, 2012, 02:07:00 UT).and 4.09 GV (see the color code) are displayed in Figure 7: the peaks in the distributionscorrespond to vertically incident protons.Figure 8 reports the asymptotic cones of acceptance of the PAMELA apparatusevaluated for the first polar pass (01:57–02:20 UT) during the May 17, 2012 SEP event(Adriani et al. 2015b). Results for sample rigidity values (see the color code) are shown as 18 –Fig. 6.— Same than Figure 5 but for 4.09 GV protons.a function of GEO (top panel) and GSE (middle panel) coordinates; grey points denote thespacecraft position, while crosses indicate the IMF direction. The pitch-angle coverage asa function of orbital position is shown in the bottom panel. While moving (and rotating)along the orbit, PAMELA covers a large pitch-angle interval, approximately ranging from0 to 145 deg. In particular, PAMELA is looking at the IMF direction between 02:14 and02:18 UT, depending on the proton rigidity. 19 – Pitch-angle [deg]
90 95 100 105 110 115 120 ] E ff . a r ea [ c m -4 -3 -2 -1 Fig. 7.— The PAMELA effective area at a sample orbital position (May 17, 2012, 02:07:00UT) as function of pitch-angle, for different values of particle rigidity.
Differential directional fluxes are obtained at each orbital position t as:Φ( R, α, t ) = N tot ( R, α, t )2 π R ∆ R dR R ∆ α dα R ∆ t dtH ( R, α, t ) , (11)where N tot ( R, α, t ) is the number of proton counts in the bin (
R, α, t ), corrected by theselection efficiencies, and the denominator represents the asymptotic exposition of theapparatus integrated over the selected rigidity bin ∆ R .Averaged fluxes over the polar pass T = P ∆ t are evaluated as:Φ( R, α ) = N tot ( R, α )2 π R ∆ R dR R ∆ α dα R T dtH ( R, α, t ) , (12)where N tot ( R, α ) = P T N tot ( R, α, t ) and the exposition is derived by weighting each effectivearea contribution by the corresponding lifetime spent by PAMELA at the same orbitalposition. 20 –Fig. 8.— Asymptotic cones of acceptance of the PAMELA apparatus for sample rigidityvalues (see labels), evaluated in GEO (top) and GSE (middle) coordinates, and as a functionof UT and pitch-angle (bottom). Grey points denote the spacecraft position, while crossesindicate the IMF direction. Calculations refer to the first PAMELA polar pass (01:57–02:20UT) during the May 17, 2012 SEP event. 21 –
Since it is not possible to discriminate between SCR and GCR signals, solar fluxintensities are obtained by subtracting the GCR contribution from the total measured flux.The GCR component is evaluated by estimating proton fluxes during two days prior thearrival of SEPs. We found that the GCR flux is isotropic with respect to the IMF directionwithin experimental uncertainties. Consequently, the same flux Φ
GCR ( R ) is subtracted forall pitch angle bins: Φ SCR ( R, α ) = Φ tot ( R, α ) − Φ GCR ( R ) == N tot ( R, α ) − N GCR ( R, α )2 π R ∆ R dR R ∆ α dα R T dtH ( R, α, t ) . (13) Flux statistical errors can be obtained by evaluating 68.27% C.L. intervals for apoissonian signal N tot ( R, α ) in presence of a background N GCR ( R, α ) (Feldman & Cousins1998). Systematic uncertainties related to the reconstruction of asymptotic directions areevaluated by introducing a bias in the particle direction measurement from the trackingsystem, according to a gaussian distribution with a variance equal to the experimentalangular resolution.
7. Conclusions
This paper reports the analysis methods developed for the estimate of SEP energyspectra as a function of the particle asymptotic direction of arrival. The exposition ofthe PAMELA apparatus is evaluated by means of accurate back-tracing simulations basedon a realistic description of the Earth’s magnetosphere. As case study, the results of the 22 –calculation for the May 17, 2012 event are discussed. The developed trajectory analysisenables the investigation of flux anisotropies, providing fundamental information for thecharacterization of SEPs. It will prove to be a vital ingredient for the interpretation of solarevents observed by PAMELA during solar cycles 23 and 24.
Acknowledgements
We acknowledge support from The Italian Space Agency (ASI), Deutsches Zentrumf¨ u r Luftund Raumfahrt (DLR), The Swedish National Space Board, The Swedish ResearchCouncil, The Russian Space Agency (Roscosmos) and The Russian Scientific Foundation.We gratefully thank N. Tsyganenko for helpful discussions, and M. I. Sitnov and G. K.Stephens for their assistance and support in the use of the TS07D model. 23 – REFERENCES