A Back-tracing Code to Study the Evolution of the Magnetosphere Transmission Function for Primary Cosmic Rays
Pavol Bobik1, Matteo Boschini2,3, Davide Grandi3, Massimo Gervasi3,4, Elisabetta Micelotta3 and Pier-Giorgio Rancoita3
______
1Institute of Experimental Physics, Kosice, Slovak Republic
2CILEA, Segrate MI, Italy
3INFN sezione di Milano, Milano, Italy
4Universitá di Milano-Bicocca, Milano, Italy
We developed a code that reconstruct the Cosmic Rays trajectory in the Earth magnetosphere. This code solves the Lorentz equation and propagate a particle backward in time. The total magnetic field is evaluated through the International Geomagnetic Reference Field (IGRF) 2000-2005 adding the external magnetic field Tsyganenko-96. This code has been used both for a simulation of randomly generated inputs and for the analysis of the AMS-01 experiment data taken during the STS-91 Space Shuttle mission in June 1998. By a full spectrum simulation we were able to separate the primary Cosmic Ray component from the secondary one, present at the AMS altitude of 400 km. We built the magnetosphere transmission function for 1998 for several regions with different geomagnetic latitude. The same simulation has been performed also for the magnetic conditions of the year 2005, the starting time of the long-duration AMS-02 data taking. Then we estimate the variation of the transmission function with time and obtain the primary Cosmic Ray flux at 400 km starting from the 1 AU flux predicted by the CREME96 model.
1. INTRODUCTION
The transmission function TF() of the magnetosphere describes the probability that a particle with rigidity , coming from outside, reaches a point inside the magnetosphere. Result depends on particles incoming direction, position and time, besides the rigidity. Models of magnetosphere currently available [Tsyganenko and Stern, 1996] allow to obtain accurate particle tracing inside the magnetosphere, and therefore to get a precise evaluation of the transmission function. The probability approach of the transmission function is suitable particularly to smooth out effect of discreteness during computation. This approach can be applied both to discrete rigidity steps of a single place [Bobik et al., 2001] and to discrete geographical positions on the Earth surface [Bobik et al., 2003a]. In the present paper we use the transmission function (TF hereafter) to describe a probability for a range of rigidities from - to +. In particular we consider the AMS experiment energy bins [Alcaraz et al., 2000a] for the geographic regions covered by AMS orbit.
To evaluate the TF we use the calculation of trajectories in geomagnetic field for a set of points at the AMS orbit excluding South-Atlantic anomaly region. Following the Liouville theorem, if the cosmic ray flux is isotropic outside the magnetosphere, the flux in a random point inside the magnetosphere is the same in all the directions allowed for primaries, while it is zero along all the forbidden directions [Vallarta, 1961]. Back-tracing calculation of the TF is based on the assumption that cosmic rays at 1AU are isotropically distributed in space. Besides this approach applies only to primary particles, i.e. particles coming from outside the magnetosphere. Around the cut-off rigidity we find the transition region of the TF: primary particles with rigidity lower than cut-off can not penetrate the magnetosphere, while trajectories of very high energy particles are unaffected. Besides, trajectories forbidden to primaries could be populated by secondary particles, i.e. particles produced or scattered inside the magnetosphere, in particular in the highest shells of the atmosphere.
Back-tracing method [Smart et al., 2000, Bobik et al., 2004] is based on the inversion of charge sign (Zq) and velocity vector (v) in the equation of motion for a particle with relativistic mass m in a magnetic field B:
m dv/dt = Zq [v B] (1)
In our simulation we have used as external geomagnetic field the Tsyganenko96 model [Tsyganenko and Stern, 1996] ( and as internal geomagnetic field the IGRF model (DGRF 2000 – 2005) [Barton, 1997] (for a full description see web page:
2. THE Transmission Function
2.1. Calculation
We evaluate the TF by back-tracing simulated particles. Starting positions of back-tracing cover a complete sphere at an altitude of 400 km (following the AMS orbit). This grid of points has been built in order to have the same elementary shooting surface. For every position in the grid, starting directions are uniformly distributed in a 2 outgoing hemisphere.
In this way we build transmission functions for the different regions of AMS orbit, obtaining as a set of 10 transmission functions (see table 1). In order to compare results with the AMS-01 experimental data we have reproduced the experimental conditions:
1)geographical polar regions have been excluded (–51.6o latitude +51.6o);
2)South-Atlantic anomaly region has been excluded;
3)Particles incoming directions are restricted to a cone of 32o from the detector axis, pointing at the zenith;
4)Energy bins have been defined equal to the bins used in AMS-01 data analysis.
TABLE 1 : AMS-01 geomagnetic regions.
AMS 01 regionsRegion number / Cosine of
Geomagnetic latitude
Region 1.
Region 2.
Region 3.
Region 4.
Region 5.
Region 6.
Region 7.
Region 8.
Region 9.
Region 10. / 0.2 M
0.2 M 0.3
0.3 M 0.4
0.4 M 0.5
0.5 M 0.6
0.6 M 0.7
0.7 M 0.8
0.8 M 0.9
0.9 M 1.0
1.0 M
For each point of the grid and for every rigidity we evaluated the ratio of the number of allowed trajectories over the number of all the directions. This ratio is the probability for particles with rigidity to reach this point starting from outside the magnetosphere. This ratio is then averaged over every point for each AMS region and for each rigidity bin. This average ratio represents the transmission function for a particle with rigidity inside the actual AMS region. Such a wide region average is able to smooth out the complex structure of penumbra changing from position to position and with rigidity [Bobik et al., 2001].
2.2. Results
In Figure 1 the TF evaluated for some of the geomagnetic regions of AMS orbit is shown. Calculation has been performed taking into account the geomagnetic conditions present during the AMS-01 observations (June 1998). Geomagnetic regions are numbered according to table 1. As we can expect TF extends at lower energy increasing the geomagnetic latitude, i.e. can reach low orbit easily, moving towards the polar regions.
We can also evaluate the TF at the time of AMS-02 long duration flight on ISS. In fact the models of the magnetosphere (IGRF/DGRF and Tsyganenko96) can be extrapolated for the next years. In Figure 2 and Figure 3 we compare the results for the region 10 and region 1 for the time periods of June 1998 (AMS-01 data taking) and October 2005 (the scheduled time of AMS-02 launch on ISS). Variation of the TF is mainly due to the interaction of the Earth magnetic field with the Interplanetary Magnetic Field, at the magnetopause. As expected, the major change occurs for region 10, where low rigidity particles are involved. Minor changes are due to the long term variation of the internal geomagnetic field.
Figure 1. Transmission function evaluated for AMS-01 regions. Calculation parameters are related to STS-91 mission flight time: June 1998.
Figure 2. Transmission function in region 10 of AMS experiment for June 1998 and October 2005 periods.
Figure 3. Transmission function in region 1 of AMS experiment for June 1998 and October 2005 periods.
3. PRIMARY PARTICLES’ SPECTRA
3.1. Primaries in AMS-01 region.
The AMS-01 experiment has measured the spectrum of protons in the 10 geomagnetic regions, obtaining the spectrum of cosmic protons in the energy range: 0.22 - 200 GeV [Alcaraz et al., 2000b]. We evaluate the primary proton spectra at the Space Shuttle orbit combining the measured AMS-01 cosmic proton spectrum (SAMS) with the TF computed for the region ith:
SEarth orbit(i) = SAMS * TF(i) (2)
Results are presented in Figure 4. A comparison of these primary spectra with the measured spectra is shown in Figure 5, where secondary protons are added to primaries.
Figure 4. Primary proton spectra for few of the AMS-01 geomagnetic regions.
Figure 5. AMS-01 measured spectra (label AMS01) compared with the evaluated primary spectra (label TF), for the geomagnetic regions 1, 5, 6, and 10.
From this comparison it is possible to separate the secondary component from the cosmic protons. The low rigidity cut-off of primaries is sharper than corresponding measured spectrum. This is an indication of the presence of secondary protons with energy above the dip.
3.2. AMS-02 Spectra in 2005
Estimations of the proton flux at 1AU for the next years can be obtained using the CREME96 model [Tylka et al., 1997] (web page:
However CREME96 is not taking into account some of the effects of CR propagation in the solar cavity. For this reason we studied a normalization procedure able to reproduce available data within few percent [Bobik et al., 2003b]. Uncertainty for the predicted fluxes is larger at energy lower than few GeV because of the deep solar modulation. For higher energy the accuracy of predicted flux is not an issue.
Using the TF and CREME96 model, both evaluated for October 2005, and equation (2) we obtain expected AMS-02 primary spectrum for 2005 in the several geomagnetic regions, following the prescriptions listed in section 2. Actually the AMS-02 angular sensitivity is extended up to a cone of 45o from the detector axis, but the evaluated TF and proton fluxes do not significantly change.
A comparison of the primary proton spectra in near Earth orbit for the region 10 evaluated for October 2005 and for June 1998 is shown in Figure 6. Predicted intensity of primary protons during AMS-02 flight is significantly lower than intensity during AMS-01 flight for energies higher than 180 MeV. Besides a combination of a higher magnetospheric transmission and a lower intensity of protons outside the magnetosphere in October 2005 than in June 1998 in region 10 produces to a higher proton intensities in AMS-02 spectrum for particles with energy lower than 180 MeV. A comparison of the primary proton spectra in near Earth orbit for the region 1 evaluated for October 2005 and for June 1998 is shown in Figure 7. In region 1, due to the high energy of particles involved, flux changes are extremely restricted.
Figure 6. Primary proton spectra at AMS orbit (region 10) for June 1998 and October 2005.
Figure 7. Primary proton spectra at AMS orbit (region 1) for June 1998 and October 2005.
CONCLUSION
The method of the magnetospheric transmission function in combination with measured (AMS-01) and simulated (CREME96) cosmic protons spectra has been successfully used to obtain the flux of primaries at several geomagnetic regions inside the magnetosphere. As AMS-01 has shown, measured spectra of protons in near Earth orbit are contaminated by a population of secondaries. This method can be used to separate the contribution of primary protons to the measured spectra. Prediction for future experiments are also possible because both geomagnetic model and modulated proton model can be extrapolated.
Moreover this method can be used to recover cosmic ray spectra outside the magnetosphere, starting from measured primary spectra in near Earth orbit, by inverting the relation (2). This procedure is restricted to energies higher than cut-off rigidity, in fact below the cut-off the TF is null and the procedure fails. An independent spectrum of primaries can be obtained from AMS-01 flux data through the separation by back-tracing of the measured particles [Micelotta et al., 2004]. The same procedure can be applied to every measured spectrum of primary Cosmic Rays inside the magnetosphere. In fact the transmission function can be evaluated if the experimental conditions are known: time, position, and attitude of the experiment.
Acknowledgments. Slovak VEGA grant agency (grant 4064) is acknowledged.
REFERENCES
Alcaraz J., et al. (The AMS collaboration), Protons in near Earth orbit, Phys. Lett. B, vol. 472, 215-226, 2000a.
Alcaraz J., et al. (The AMS collaboration), Cosmic Protons, Phys. Lett. B, vol. 490, 27-35, 2000b.
Barton C.E., International Geomagnetic Reference Field: The Seventh Generation, J. Geomag. Geoelectr., vol. 49, 123-148, 1997.
Bobik P., Kudela K., and Usoskin I., Geomagnetic cutoff Penumbra structure: Approach by transmissivity function, Proc. 27th ICRC., Hamburg, Germany, 4056-4059, 2001.
Bobik P., Boschini M., Gervasi M., Grandi D., Kudela K., Micelotta E. andRancoita P.G.,Cosmic ray spectrum at 1 AU: a transmission function approach to the magnetosphere, Proceedings of ICSC 2003,ESA SP-533, 633-636, 2003a.
Bobik P., Boschini M., Gervasi M., Grandi D., Micelotta E. andRancoita P.G.,A normalization procedure for Creme96 spectra, Proceedings of 8th ICATPP 2003, in press (2003b).
Bobik P. et al., A simulation program to reconstruct cosmic rays trajectory in the Earth magnetosphere, in preparation (2004).
Micelotta E. et al. in preparation (2004).
Smart D.F., Shea M.A., and Flückiger E.O., Magnetospheric Models and Trajectory Computations, Space Sci. Rev., vol. 93 (1/2), 305-333, 2000.
Tsyganenko, N.A., and Stern D.P., Modeling the global magnetic field the large-scale Birkeland current systems, J. Geophys. Res., Vol. 101, 27187-27198, 1996.
Tylka A.J., et al., "CREME96: A Revision of the Cosmic Ray Effects on Micro-Electronics Code", IEEE Trans. Nuclear Sci., 44, 2150-2160, 1997.
Vallarta M.S., Theory of the Geomagnetic Effects of Cosmic Radiation, Handbuch der Physik, Vol. XLVI, 1961.
______