The production of proton and lepton fluxes in near Earth orbit

Abstract

Substantial fluxes of protons and leptons with energies below the geomagnetic cutoff have been measured by the AMS experiment at altitudes of 370-390 Km, in the latitude interval +/- 51.7 degrees. The production mechanisms of the observed trapped fluxes are investigated in detail by means of the FLUKA Monte Carlo simulation code. All known processes involved in the interaction of the cosmic rays with the atmosphere (detailed descriptions of the magnetic field and atmospheric density, as well as the electromagnetic and nuclear interaction processes) are included in the simulation. The results are presented and compared with the experimental data, indicating good agreement with the observed fluxes. The impact of secondary proton flux on particle production in atmosphere is briefly discussed.

Full text

Page 1

arXiv:astro-ph/0111111v2 17 Jan 2002
The production of proton and lepton fluxes in near Earth orbit
P. Zuccon1, B. Bertucci1, B. Alpat1, R. Battiston1, G. Battistoni2, W.J. Burger1,
G. Esposito1, A. Ferrari2,3, E. Fiandrini1, G. Lamanna1,∗, P.R. Sala2,3
January 17, 2002
1Universit` a and Sezione INFN of Perugia, Italy
2Sezione INFN of Milano, Italy
3CERN Switzerland
Abstract
Substantial fluxes of protons and leptons with energies below the geomagnetic cutoff have been
measuredby the AMS experimentat altitudes of 370-390Km, in the latitude interval ±51.7o. The pro-
duction mechanisms of the observed trapped fluxes are investigated in detail by means of the FLUKA
Monte Carlo simulation code. All known processes involved in the interaction of the cosmic protons
with the atmosphere(detailed descriptions of the magnetic field and atmosphericdensity, as well as the
electromagnetic and nuclear interaction processes) are included in the simulation. The results are pre-
sented and compared with the experimental data, indicating good agreement with the observed fluxes.
The impact of secondary proton flux on particle production in atmosphere is briefly discussed.
1Introduction
Cosmic rays approaching the Earth interact with the atmosphere resulting in a substantial flux of se-
condary particles. The knowledge of composition, intensity and energy spectra of these particles is of
considerable interest, e.g. for the evaluation of background radiation for satellites and the estimate of
the atmospheric neutrino production for neutrino oscillation experiments [1].
The AMS measurements in near earth orbit [2, 3] have allowed, for the first time, to gather accurate
information on the intensity, energy spectra and geographical origin of charged particle fluxes at ener-
gies below the geomagnetic cutoff over a wide range of latitudes and at almost all longitudes. The under
cutoff component of proton fluxes at equatorial latitudes has revealed an unexpected intensity of up to
50% of the primary proton flux, a positron to electron flux ratio has been found in the undercutoff com-
ponent which largely exceeds the cosmic one, differences in residence times and geographical origins
have been reconstructed for positively and negatively charged particles.
A robust interpretation of these and many other characteristics of the undercutoff fluxes in terms
of secondary particles produced in atmosphere requires an accurate description of both the interaction
processes at their origin and of the geomagnetic field effects. Recently, different interpretations of
∗Now at CERN - Switzerland
1

Page 2

the AMS measurements have been proposed [4, 5] based on Monte Carlo simulations using different
approaches on both the generation technique and the interaction model.
In this work, we report results from a fully 3D Monte Carlo simulation based on FLUKA 2000 [6]
for the description of cosmic ray interactions with the atmosphere. The key features of our analysis
are an efficient generation technique for the incoming proton flux and a true microscopic, theory driven
treatment of the interaction processes opposite to empirical parametrization of accelerator data. As a
first attempt the contribution of He and the heavier nuclei, representing (≈ 9%) [7] of the all nuclei
cosmic flux, is neglected.
In the following section we give a detailed description of the basic ingredients of this simulation, the
generation technique and the interaction model. In section 3 we present the results on both protons and
leptons and the comparison with AMS measurements. In section 4 we propose our conclusions.
2The model
An isotropic flux of protons is uniformly generated on a geocentric spherical surface with a radius of
1.07 Earth radii (∼ 500Km a.s.l.) in the kinetic energy range 0.1−170 GeV .
We took the functional form suggested in [8] to describe the proton energy spectrum, the spectral
index and the solar modulation parameter are extracted from a fit to the AMS data [9].
The magnetic field in the Earth’s proximity includes two components: the Earth’s magnetic field,
calculated using a 10 harmonics IGRF [10] implementation, and the external magnetic field, calculated
using the Tsyganenko Model1[11]. To account for the geomagnetic effects, for each primary proton we
back-trace an antiproton of the same energy until one of the following conditions is satisfied:
1. the particle reaches the distance of 10ERfrom the Earth’s center.
2. the particle touches again the production sphere.
3. neither 1 or 2 is satisfied before a time limit is reached.
If condition 1 is satisfied the particle is on an allowed trajectory, while if condition 2 is satisfied the
particle is on a forbidden one. Condition 3 arises for only a small fraction of the events O(10−6).
Particles on allowed trajectories are propagated forward and can reach the Earth’s atmosphere. The
atmosphere around the Earth is simulated up to 120 Km a.s.l. using 60 concentric layers of homoge-
neous density and chemical composition. Data on density and chemical composition are taken from the
standard MSIS model [12]. The Earth is modeled as a solid sphere which absorbs each particle reaching
its surface.
2.1The generation technique
The ideal approach in the generation of the primary cosmic rays spectra would be to start with an
isotropic distribution of particles at a great distance (typically 10ER) from the Earth where the geomag-
netic field introduces negligible distortions on the interstellar flux. However, this computational method
is intrinsically inefficient since most of the particles are generated with trajectories which will not reach
1The external magnetic field is calculated only for distances greater than 2 Earth’s radii (ER) from the Earth’s center . Its
contribution to the total magnetic field is <1% at smaller distances and therefore can be safely neglected.
2

Page 3

the Earth environment. Kinematic cuts can be applied in order to improve the selection efficiency at
generation, however they tend to introduce a bias particles with low rigidity.
A good alternative to this approach is the backtracing method [4],[15] adopted in the present analysis
as outlined in the previous section. In the following, we will shortly discuss the validity of the technique
and report the results of a comparison of the two methods. We recall that this method was applied for
the first time in ref. [8] for the generation of atmospheric neutrino fluxes.
Let us consider first the effects of the geomagnetic field on an incoming flux of charged particles
in the absence of a solid Earth. For the discussion, we start with an isotropic flux of monoenergetic2
protons at large distance, i.e. at infinity, from the origin of a geocentrical reference system. In this
scenario, a negligible fraction of particles, with very particular initial kinematic parameters, will follow
complicated paths and remains confined at a given distance from the origin (semi-bounded trajectories);
for all practical purposes this sample can be ignored. Most of the particles will follow unbounded
trajectories, reaching again infinity after being deflected by the magnetic field.
Unbounded trajectories cross a spherical surface centered in the field source only an even number of
times, as shown in Fig.1: we call legs the trajectory parts connecting the spherical surface to infinity and
loops the parts of the trajectory starting and ending inside the spherical surface.
Since each trajectory can be followed in both directions and no source or sink of particles is contained
within the surface, the incoming and outgoing fluxes are the same. However, the presence of the mag-
netic field breaks the isotropy of the flux “near” the field source, so for a given location there is a flux
dependence due to the direction.
Applying the Liouville Theorem, under the hypothesis of isotropy at infinity, it is straightforward
to prove [18] that the proton flux in a random point is the same as at infinity along a set of directions
(allowed directions), and zero along all the others (forbidden directions).
The pattern of the allowed and forbidden directions depends on both the rigidity and the location and
is known as the geomagnetic cutoff.
With the introduction of a solid Earth, all the trajectories that are crossing the Earth are broken in two
or more pieces (Fig.2): the legs become one-way trajectories and the loops disappear.
The presence of the Earth modifies the flux which exits from the surrounding spherical surface, since
particles are absorbed by the Earth, while it has only a minimal effect on the incoming flux which is
modified only by the absence of certain loops. To generate the flux of particles reaching the Earth’s
atmosphere, it is sufficient to follow the particles along the allowed trajectories corresponding to the
legs, taking care to avoid double or multiple counting.
To respect this prescription we reject all trajectories that are back-traced to the production sphere, this
allow us to correctly consider the cases like the one shown in Fig.3.
We point out that an important difference with respect to the application in the neutrino flux calcu-
lation of [8] is that for the former, the generation sphere coincided with the Earth’s surface, and there-
fore the forbidden trajectories included those which touched again the Earth (plus those who remained
trapped for a long time). In that case there are no problems of double counting.
To check the validity of our technique we made a test comparing the results of the inefficient gen-
eration technique at 10 Earth’s radii distance from the Earth’s center with the backtracing technique
described in this paper.
2The realistic case of an energy spectrum can be treated just as a superposition of monenergetic cases
3

Page 4

5GeV10GeV20GeV30GeV
Mult.
2.770
2.381
2.047
2.840
Particle
p
π+
π−
π0
Mult.
1.983
0.711
0.389
0.638
E frac.
0.409
0.131
0.068
0.114
Mult.
2.676
1.292
0.975
1.601
E frac.
0.337
0.149
0.098
0.169
Mult.
2.744
1.970
1.641
2.378
E frac.
0.307
0.159
0.116
0.175
E frac.
0.294
0.164
0.122
0.177
Table 1: Energy fraction and multiplicity of secondary particles for the proton interactions with atmo-
spheric nuclei in FLUKA 2000. Four typical energies of primary protons are considered.
Fig. 4 shows this comparison for several cha-racteristic distributions, the agreement between the two
methods is good.
2.2The interaction model
We use the software package FLUKA 2000 [6] to transport the particles and describe their interactions
with Earth’s atmosphere. The setup of this simulation is derived from the one used in [1]. This package
contains a tridimensional description of both electro-magnetic and hadronic interactions. This code is
benchmarked against a wide set of data and is already used in many applications, ranging from low
energy nuclear physics to high energy accelerator and cosmic ray physics. For this reason we have
preferred this model with respect to the use of “ad hoc” parametrizations of particle production in the
energy range of our interest.
In FLUKA hadronic interactions are treated in a theory-driven approach, and the models and their
implementations are guided and checked using experimental data. Hadron-nucleon interaction models
are based on resonance production and decay below an energy of few GeV and on the Dual Parton
Model above. The extension from hadron to hadron-nucleus interactions is done in the framework of
a generalized intra-nuclear cascade approach including the Gribov-Glauber multi-collision mechanism
for higher energies followed by equilibrium processes: evaporation, fission, Fermi break-up and γ de-
excitation. The parameters of the models embedded in the FLUKApackage are fixed only by comparing
expectations with data from accelerator experiments.
In fig 5 a) we show the map of the primary proton interaction points in geographical coordinates. The
distribution reflects the influence of the geomagnetic cutoff. Fig 5 b) shows the interactions altitude
profile, the solid histogram is for Ek<30 GeV while the dashed one is for Ek>30 GeV . The mean
interaction altitude depends weakly on the energy.
The cosmic proton impinging in the atmosphere are doing elastic scattering in the 24% of the events
and inelastic interactions in the remaining 76%, in tab.1 we show some characteristic of the inelastic
interactions as simulated by FLUKA 2000.
3Comparison with the AMS data
To compare with the AMS data, we define a detection boundary corresponding to a spherical surface
at the AMS orbit altitude (400Km a.s.l). We record each particle that crosses the detection boundary
4

Page 5

within the AMS field-of-view, defined as a cone with a 32oaperture with respect to the local zenith or
nadir directions.
To obtain the absolute normalization, we take into account the field-of-view, the corresponding AMS
acceptance, and an Equivalent Time Exposure (E.T.E.) corresponding to the number of the generated
primary protons.
Our results are based on a sample of ∼ 6 · 106primary protons generated in the kinetic energy range
of 0.1 − 170 GeV , which corresponds to ∼ 4 10−12s (E.T.E).
3.1Protons
In Fig.6, we show the comparison between the fluxes obtained with the simulation and the measured
AMS downgoing proton flux [2] in nine bins of geomagnetic latitude (θM) [13]. Fig.7 shows the same
comparison for the upgoing proton flux in four selected bins of θM.
As seen in Fig.6, the simulation well reproduces at all latitudes the high energy part of the spec-
trum and the falloff in the primary spectrum due to the geomagnetic cutoff, thus validating the general
approach used for the generation and detection, as well as the tracing technique.
A good agreement among data and simulation is also found in the under-cutoff part of the spectra.
The small and systematic deficit which can be seen in the secondary component of the simulated fluxes
is of the same order of the expected contribution from the interaction of cosmic He and heavier nuclei.
This flux is due to the secondaries produced in the atmosphere and that spiral along the geomagnetic
field lines up to the detection altitude. Therefore it is sensitive to specific aspects of the interaction
model and to the accuracy of the particle transport algorithm.
A correct quantitative prediction of this part of the spectra depends on the quality of low energy
nucleon production both in termsofyield and energy distribution. Thisisinpartdue tothe fragmentation
of the target nucleus, and depends on the details of the nuclear physics algorithms describing excitation
and break up.
From the analysis of the motion of the secondary protons from their production up to their detection,
it can be pointed out that a fraction of the observed flux is due to a multiple counting of the same
particles. Within the formalism of adiabatic invariants [16], it is seen that charged particles trapped in
the geomagnetic field, i.e. the undercutoff protons, move along drift shells which can be associated
with a characteristic residence time3that depends on the fraction of the shell located inside the Earth’s
atmosphere. Thus, particles moving along long-lived shells have a large probability to cross many times
a geocentered spherical detector, while those moving along short-lived shells typically cross the detector
only once.
The drift shells crossing the AMS orbit, at an altitude of 400 km, are in general short-lived, however
in the equatorial region the long-lived shells are present as well [17].
In the following, we will indicate as the real proton flux that one obtained by counting only once each
particle crossing the detector: its intensity is indicated by the dashed distributions in figs. 6 and 7.
A quite relevant effect can be seen in the equatorial region: there the AMS measurement indicates an
important secondary proton flux while the real number of protons crossing the detector is more than one
order of magnitude lower. At high geomagnetic latitudes, the solid and dashed lines tend to merge. The
effect becomes negligible for θM> 0.8.
3The mean time after which a particle is absorbed into the atmosphere. In our case it represents the effective life time of
the particle.
5

Page 6

This can be better seen in Fig.8, where the integral primary proton flux seen by AMS is shown as
a function of geomagnetic latitude. The intensities of the real and measured undercutoff fluxes are
reported in the same plot for comparison and their ratio with the primary component shown in Fig.9. A
minor contribution from the undercutoff proton component can be therefore expected in the atmospheric
shower development and neutrino production.
In Fig.10, the residence time is plotted versus the kinetic energy of the trapped secondary protons for
|θM| < 0.4. In the scatter plot it is possible to distinguish the populations corresponding to long-lived
and short-lived shells similar to those shown in [3] for leptons.
Fig.11 shows the distribution of trapped secondary proton end points for |θM|<0.4, Fig.11a is for a
lifetimes smaller than 0.3 s., while Fig.11b isfor alifetimes greater than 0.3s.. Theend point distribution
agrees with the location of the intersections of the drift shells with the atmosphere as experimentally
verified by [2], and discussed in [17].
3.2Electrons and positrons
In Fig. 12 we show a comparison of the simulated undercutoff electron and positron downgoing fluxes
with the corresponding AMS measured fluxes [3].
We remind that the AMS positron measurement is restricted at energies below few GeVs, with a
dependence of the maximum energy on the geomagnetic latitude which reflects the increasing proton
background with θm.
A comparison of data and simulation in the high energy part of the electron spectra is not possible,
since the cosmic electrons have not been used as an input in the current work. However, their contri-
bution to the cosmic rays reaching the atmosphere is O(10−2) leading to a negligible effect of in the
generation of the undercutoff fluxes.
The simulation well reproduces the general behavior of the undercutoff part of the spectra in terms
of shape and intensity; a similar agreement is observed for the upgoing lepton spectra (not shown). The
real lepton fluxes, corresponding to the real proton flux described earlier, are shown with the dashed
line distribution in Fig. 12. As in the case of protons, a large effect from multiple crossing is present
going toward the equatorial region, more pronounced for the positron component.
As for the undercutoff protons, we would have expected a systematic deficit in the simulated electron
fluxes coming from the missing contribution of helium and heavier nuclei to the CR fluxes. Subcutoff e±
are mainly (97%) coming from decays of pions produced in the proton collisions with the atmospheric
nuclei: charged pions contribute through theπ−µ−echain, while π0through π0→ γ γ with subsequent
e.m. showers. The relative contribution of charged pions to the subcutoff electrons (positron) fluxes at
AMS altitude in our simulation is found to be 37% (47%), while the remaining 60% (50%) appears
to come from π0production. This point deserves some considerations. The level of agreement in
the comparison of data and predictions for e±fluxes turns out to be an important benchmark for the
interaction model in view of a discussion on particle production in atmosphere, since is strictly linked to
the meson production (mostly pions at this energy). This work complements other studies oriented to the
validation of the FLUKA model in terms of particle yields. In particular, the quality of π±generation
in our interaction model has been already checked in [19] through the comparison with muon fluxes
measurements at different depths in atmosphere. The muons are from the charged pions decay chain
and experimental data [20] are well reproduced by the simulation. In the case of e±also π0’s become
relevant. Usually, when parametrized interaction models are used, like in the works of ref. [4, 14], the
6

Page 7

yield of π0is fixed assuming a priori an exact charge symmetry in pion production. In practice, this is
also made necessary by the large errors that affect the scarce existing experimental data on neutral pion
production. Instead, in the case of a microscopic interaction model like FLUKA,there is no constraint of
this type, and the balance of π0vs. π±automatically emerges from the feature of the model. Recently,
it has been pointed out how in FLUKA there exists a significant violation of charge symmetry[21]:
π0’s are in general more (∼20%) of the average of π+and π−. Technically this symmetry violation
emerges in the hadronization of color strings and this normally occurs also in other codes, like JETSET
or PITHYA[22]. We cannot enter here in a detailed discussion of this point, and we limit ourselves to
say that there are reasons to believe that, at least for laboratory energies below 100 GeV, the acceptable
value of charge asymmetry should be lower than that resulting from the present version of the code.
However, the comparisons of predictions to data discussed in this work, combined with the mentioned
work on atmospheric muons[19], already tell that the predicted fraction of π0cannot be significantly
wrong, although no definitive quantitative conclusion can be extracted, since the nuclear component has
not been yet introduced in the primary spectrum.
In Fig.13b-c we show the integrated positron and electron downgoing fluxes for the kinetic energy
range 0.2−1.5 GeV as a function of θM. Their ratio is shown in Fig.13a. One of the most remarkable
features of the AMS measurement is the large value of this ratio, when compared to the natural cosmic
value, and its latitude dependence. In Fig. 12, the contribution from primary protons with Ek> 30 GeV
to the electron and positron fluxes is illustrated by the filled area. We can notice that in the equatorial
region, the electrons are produced essentially by primary protons with Ek> 30 GeV , while for the
positrons lower energy protons contribute as well. This distinction disappears at higher latitudes, where
positron and electrons are produced by the protons in the same energy range.
This behavior reflects the East-West asymmetry of the geomagnetic cutoff on primary protons and the
larger probabilities of escape from atmosphere for secondary electrons(positrons) generated by West-
ward(Eastward) moving protons [15, 14].
Positrons are preferentially injected on drift shells reaching the AMS altitude by eastward moving
protons, which experience a lower rigidity cutoff than westward moving ones. This mechanism is more
effective at the equator, where the cutoff is larger and its asymmetry maximal, resulting in the excess of
undercutoff positrons from low energy protons as indicated by our simulation. The cutoff mechanism
becomes irrelevant at high latitudes, where any difference in positron and electron production should be
given instead by different π+/π−production. Nor the data neither our simulation indicate, within their
uncertainties, a relevant charge asymmetry from this source.
4Conclusions
The interactions of cosmic ray protons with the Earth’s atmosphere have been investigated by means of
a fully 3D Monte Carlo program.
The proton, electron and positron undercutoff flux intensities measured by AMS, as well as their
energy spectra, have been correctly reproduced by our simulation
Geomagnetic effects, and in particular the east-west asymmetry in the cosmic protons rigidity cutoff,
have been confirmed as the mechanism responsible for the measured excess of the positron component.
The main features of the geographical origin and residence time distributions for both protons and
leptons have been replicated and the effect of multiple crossing of the detector by spiraling secondaries
7

Page 8

in the geomagnetic field briefly discussed.
Our results indicate that the intensity of the undercutoff proton flux, when the multiple counting is
taken into account, never exceeds a 10% of the cosmic proton flux, representing a negligible source for
atmospheric production of secondaries. However, this aspect will be object of further and more refined
study in the future.
The analysis on the possible strategies to generate the cosmic rays incoming flux has shown the
validity of a backtracing approach as an accurate and highly efficient technique.
This work provides also additional way to validate the features of the adopted particle production
model. In particular, the study of e±fluxes has revealed to be to be an interesting instrument to check
the meson production in primary interactions, and the results are satisfactory. We believe that our
simulation, validated by the high statistic measurements of AMS, can be used to assess the radiation
environment in near Earth orbit, and represents a valuable tool for a more accurate calculation of
particle fluxes in atmosphere.
This work has been partially supported by the Italian Space Agency (ASI) under contract ARS-98/47.
8

Page 9

References
[1] G. Battistoni et al., Astropart. Phys. 12 (2000) 315;
[2] J.Alcaraz et al., AMS Collaboration, Phys. Lett. B472 (2000) 215;
[3] J.Alcaraz et al., AMS Collaboration, Phys.Lett. B484 (2000) 10;
[4] L. Derome, et al. Phys. Lett. B489 (2000) 1;
[5] V.Plyaskin, Phys.Lett. B516 (2001) 213;
[6] A. Ferrari et al., Physica Medica, VOL XVII, Suppl. 1.
[7] T.K. Gaisser, M. Honda, P. Lipari and T. Stanev, Proc. of the 27th ICRC (Hamburg, 2001), Session OG1.01
[8] M. Honda et al., Phys. Rev. D52 (1995) 4985;
[9] J.Alcaraz et al., AMS Collaboration, Phys. Lett. B490 (2000) 27;
[10] N.A. Tsyganenko Geomagn. and Aeronomy V.26 (1986) 523; N.A. Tsyganenko and M.Peredo, Geopack
Manual, (1992)
[11] N.A. Tsyganenko and D.P. Stern, ISTP newsletter, 6 (1996) 21;
[12] A. E. Hedin, J. Geophys. Res. 96 (1991) 1151;
[13] G.Gustaffson et al., J.Atmos. Terr.Phys 54 (1992) 1609;
[14] L.Derome et al., astro-ph/0103474;
[15] P.Lipari, astro-ph/0101559;
[16] C.E.Mc Ilwain, J. Geophys. Res. 66(1961) 3681;
[17] E. Fiandrini et al. astro-ph/0106241,E.Fiandrini et al, Proceedings of the XXVII ICRC (Hamburg).
[18] M.S.Vallarta, Handbuch der Physik, Springer, Vol. XLVI/1 (1961) 88;
[19] G.Battistoni et al., hep-ph/0107241;
[20] M.Boezio et al., Phys. Rev. D62 (2000) 032007;
[21] P. Lipari, Proc. of the IX Workshop on Neutrino Telescopes, Venice, 2001.
[22] T.Sjostrand, Comp. Phys. Comm. 82 (1994) 74
9

Page 10

leg
loop
leg
Figure 1: Trajectories types crossing a spherical surface around the Earth
leg
loop
leg
EARTH
Figure 2: Trajectories in the presence of a solid Earth
EARTH
A
B
Figure 3: Anexample of multiple counting along atrajectory, this type oftrajectory has to be considered
only at point B.
10

Page 11

Location longitude Location longitude
rad
arbitrary units
Location latitudeLocation latitude
rad
arbitrary units
Angle momentum/zenit directionAngle momentum/zenit direction
rad
arbitrary units
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
-202
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0123
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
1.61.82 2.22.42.62.83
Figure 4: Latitude and longitude of impact points and angle between momentum and zenith directions
for particles generated at a distance of 10 Earth’s radii (solid line) and particles generated at 1.07 Earth’s
radii (shaded histogram).
11

Page 12

geographic longitude (rad)
geographic latitude (rad)
a)
altitude(Km)
arbitra y units
b)
-1.5
-1
-0.5
0
0.5
1
1.5
-3-2-10123
10
10
2
10
3
10
4
10
5
0 20406080100120
Figure 5: a) distribution of primary protons interaction points in geographical coordinates, b) altitude
profile of primary protons interaction points, solid line Ek<30 GeV , dashed line Ek>30 GeV
12

Page 13

0 < |θM| < 0,20,2 < |θM| < 0,3 0,3 < |θM| < 0,4
0,4 < |θM| < 0,50,5 < |θM| < 0,6 0,6 < |θM| < 0,7
0,7 < |θM| < 0,80,8 < |θM| < 0,9 0,9 < |θM| < 1
Kinetic Energy (GeV)
Flux (m2 sr MeV s)-1
10
-4
10
-3
10
-2
10
-1
10
-4
10
-3
10
-2
10
-1
10
-4
10
-3
10
-2
10
-1
1
10
-1
11010210
-1
1 1010
210
-1
110102
Figure 6: Downgoing proton flux, simulation(solid line) and the AMS data (triangles); the dashed lines
are described in the text. ΘMis the geomagnetic latitude in radians.
13

Page 14

0 < |θM| < 0,2 0,3 < |θM| < 0,4
0,5 < |θM| < 0,6 0,7 < |θM| < 0,8
Kinetic Energy (GeV)
Flux (m2 sr MeV s)-1
10
-4
10
-3
10
-2
10
-1
10
-4
10
-3
10
-2
10
-1
10
-1
11010
-1
110
Figure 7: Upgoing proton fluxes, simulation (solid line) and the AMS data (triangles); the dashed lines
are described in the text.
14

Page 15

Geomagnetic?latitude?(rad)
Flux?(m2sr?s)-1
Primary?proton?flux
Secondary?proton?flux
Secondary?real?proton?flux
10
-2
10
-1
1
10
00.10.20.30.40.50.60.70.80.91
Figure 8:
integrated over the kinetic energy range 0.1 − 170 GeV and shown as a function of the geomagnetic
latitude.
Proton fluxes in the AMS field of view as calculated with this simulation. The fluxes are
15