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Magnetospheric Models and Trajectory Computations


The calculation of particle trajectories in the Earth's magnetic field has been a subject of interest since the time of Störmer. The fundamental problem is that the trajectory-tracing process involves using mathematical equations that have `no solution in closed form'. This difficulty has forced researchers to use the `brute force' technique of numerical integration of many individual trajectories to ascertain the behavior of trajectory families or groups. As the power of computers has improved over the decades, the numerical integration procedure has grown more tractable and while the problem is still formidable, thousands of trajectories can be computed without the expenditure of excessive resources. As particle trajectories are computed and the characteristics analyzed we can determine the cutoff rigidity of a specific location and viewing direction and direction and deduce the direction in space of various cosmic ray anisotropies. Unfortunately, cutoff rigidities are not simple parameters due to the chaotic behavior of the cosmic-ray trajectories in the cosmic ray penumbral region. As the computational problem becomes more manageable, there is still the issue of the accuracy of the magnetic field models. Over the decades, magnetic field models of increasing complexity have been developed and utilized. The accuracy of trajectory calculations employing contemporary magnetic field models is sufficient that cosmic ray experiments can be designed on the basis of trajectory calculations. However, the Earth's magnetosphere is dynamic and the most widely used magnetospheric models currently available are static. This means that the greatest uncertainly in the application of charged particle trajectories occurs at low energies.

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  1. Berthelier, A.: 1993, in J. Hruska, M. A. Shea, D. F. Smart, and G. Heckman (eds.), 'The Geomagnetic Indices: Derivation, Meaning and Use in Solar-Terrestrial Physics', Solar-Terrestrial Predictions IV(3), 3-20, NOAA, ERL, Boulder CO, USA.

  2. Bieber, J. W. and Evenson, P.: 1995, 'Spaceship Earth -- An Optimized Network of Neutron Monitors', Proc. 24th Int. Cosmic Ray Conf. 4, 1316–1319.

  3. Bieber, J. W., Clem, J., and Evenson, P.: 1997, 'Efficient Computation of Apparent Cutoffs', Proc. 25th Int. Cosmic Ray Conf. 2, 389–391.

  4. Boberg, P. R., Tylka, A. J., Adams, J. H., Flückiger, E. O., and Kobel, E.: 1995, 'Geomagnetic Transmission of Solar Energetic Protons During the Geomagnetic Disturbances of October 1989', Geophys. Res. Lett. 22, 1133–1136.

  5. Brunberg, E.-A.: 1953, 'Experimental Determinations of Electron Orbits in the Field of a Magnetic Dipole', Tellus V, 135–156.

  6. Brunberg, E.-A.: 1956, 'Cosmic Rays in the Terrestrial Magnetic Dipole Field', Tellus VII, 215–232.

  7. Brunberg, E.-A. and Dattner, A.: 1953, 'Experimental Determinations of Electron Orbits in the Field of a Magnetic Dipole II', Tellus V, 269–292.

  8. Byrnak, B.: 1979, 'A Helix Predictor-Corrector Method for Cosmic-Ray Tracing', Nucl. Inst. Meth. 161, 303–309.

  9. Byrnak, B., Lund, N., Rasmussen, I. L., and Petrou, N.: 1981, 'The Isotopic Composition of Cosmic Ray Nuclei at 0.6, 3 and 7 GeV/n', Proc. 17th Int. Cosmic Ray Conf. 2, 18–11.

  10. Bütikofer, R., Flückiger, E.O., Smart, D.F., and Shea, M.A.: 1995,'Effects of the Magnetosheath on Cosmic Ray Particle Propagation in Near-Earth Space', Proc. 24th Int. Cosmic Ray Conf. 4, 1070–1073.

  11. Clem, J. M., Bieber, J. W., Duldig, M., Evenson, P., Hall, D., and Humble, J.: 1997, 'Contribution of Obliquely Incident Particles to NM Counting Rate', J. Geophys. Res. 102, 26 919–26 926.

  12. Cooke, D. J., Humble, J. E., Shea, M. A., Smart, D. F., Lund, N., Rasmussen, I. L., Byrnak, B., Goret, P., and Petrou, N.: 1991, 'On Cosmic-Ray Cutoff Terminology', Il Nuovo Cimento 14C, 213–234.

  13. Copenhagen-Saclay: 1981, 'The HEAO-3 French-Danish Cosmic Ray Spectrometer: Prelim. Results of the Elemental Abundances of Cosmic Ray Nuclei in the Iron Peak', Adv. Space Res. 1, 173–184.

  14. Cramp, J. L., Duldig, M., and Humble, J. E.: 1995, 'Neutron Monitor Response to Highly Anisotropic Ground Level Enhancements', Proc. 24th Int. Cosmic Ray Conf. 4, 248–251.

  15. Danilova, O. A. and Tyasto, M.: 1991, 'Variations of Cosmic Ray Cutoff Rigidities at Mid-latitude Stations due to Asymmetric Magnetosphere', Proc. 21st Int. Cosmic Ray Conf. 7, 6–9.

  16. Dryer, R. and Meyer, P.: 1975, 'Isotopic Composition of Cosmic-Ray Nitrogen at 1.5 GeV/amu', Phys. Rev. Lett. 35, 601–604.

  17. Fanselow, J. L. and Stone, E. C.: 1972, 'Geomagnetic Cutoff for Cosmic-Ray Protons for Seven Energy Intervals Between 1.2 and 39 MeV', J. Geophys. Res. 77, 3999–4009.

  18. Fermi, E.: 1950, 'Nuclear Physics', compiled by Orear, J., Rosenfeld, A. H., and Schultes, R. A., University of Chicago Press, Chicago IL, USA.

  19. Flückiger, E. O. and Kobel, E.: 1990, 'Aspects of Combining Models of the Earth's Internal and External Magnetic Fields', J. Geomag. Geoelectr. 42, 1123–1136.

  20. Gall, R. and Bravo, S.: 1973, 'Role of the Neutral Sheet in the Illumination of Polar Caps by Solar Protons', J. Geophys. Res. 78, 6773–6776.

  21. Gall, R., Jiménez, J., and Camacho, L.: 1968, 'Arrival of Low-Energy Cosmic Rays Via theMagnetospheric Tail', J. Geophys. Res. 73, 1593–1605.

  22. Gall, R., Jiménez, J., and Orozco, A: 1969, 'Directions of Approach of Cosmic Rays for High Latitude Stations', J. Geophys. Res. 74, 3529–3540.

  23. Gall, R., Smart, D. F., and Shea, M. A.: 1971a, 'The Direct Mode of Propagation of Cosmic Rays to Geostationary Satellites', Planetary Space Sci. 19, 1419–1430.

  24. Gall, R., Smart, D. F., and Shea, M. A.: 1971b, in K. Ya. Kondratyev, M. J. Rycroft, and C. Sagan (eds.), 'The Daily Variation of Cosmic-Ray Cutoff at the Altitude of a Geostationary Satellite', Space Research XI 2, 1259–1264, Akademie-Verlag, Berlin.

  25. Golightly, M. J. and Weyland, M.: 1997, 'Modeling Exposures Aboard the Space Shuttle from the August 1989 Solar Particle Event', Paper No. 13, Impact of Solar Energetic Particle Events on Design of Human Missions, Center for Advanced Space Studies, Houston TX, U.S.A.

  26. Gussenhoven, M. S., Brautigam, D. H., and Mullen, E. G.: 1988, 'Characterizing Solar Flare High Energy Particles in Near-Earth Orbits', IEEE Trans. Nuclear Sci. 44 (6), 1412–1419.

  27. Gustafesson, G., Papitashvilli, N. E., and Papitashvilli, V. O.: 1992, 'A Revised Corrected Geomagnetic Coordinate System for Epochs 1985 and 1990', J. Terrest. Phys. 54, 1609–1631.

  28. Hapgood, M.A.: 1992,'Space Physics Transformations: A User Guide', Planetary Space Sci. 40, 711–717.

  29. IGRF: 1992, 'IGRF, 1991 Revision', EOS, Trans. American Geophys. Union 73 (16), 182.

  30. Jory, F. S.: 1956, 'Selected Cosmic-Ray Orbits in the Earth's Magnetic Field', Phys. Rev. 103, 1068–1075.

  31. Kasper, J. K.: 1959, 'The Earth's Simple Shadow Effect on Cosmic Radiation', Il Nuovo Cimento (Supplemento) XI, 1–26.

  32. Klecker, B., McNab, M. C., Blake, J. B., Hamilton, D. C., Hovestadt, D., Kästle, H., Looper, M. D., Mason, G. M., Mazur, J. E., and Scholer, M.: 1995, 'Charge State of Anomalous Cosmic-Ray Nitrogen, Oxygen and Neon: SAMPEX Observations', Astrophys. J. 442, L69–L72.

  33. Kobel, E.: 1990, 'Bestimmung der Grenzsteifigkeiten und der Asymptotischen Richtungen der Kosmischen Strahlung für das Solare Protonenereignis von 7./8.Dez. 1982 unter Berücksichtigung der Einflüsse der gestörten Erdmagnetosphäre', Lizentiatsarbeit, Physikalisches Institut, Universität Bern.

  34. Langel, R. A., Kerridge, D. R., Barraclough, D. R., and Mailn, R. C.: 1986, 'Geomagnetic Temporal Change: 1903–1982', J. Geomag. Geoelectr. 38, 573–597.

  35. Lemaitre, G. and Vallarta, M. S.: 1936a, 'On the Geomagnetic Analysis of Cosmic Radiation', Phys. Rev. 49, 719–726.

  36. Lemaitre, G. and Vallarta, M. S.: 1936b, 'On the Allowed Cone of Cosmic Radiation', Phys. Rev. 50, 493–504.

  37. Leske, R. A., Cummings, J. R., Mewaldt, R. A., and Stone, E. C.: 1995, 'Measurement of the Ionic Charge State of Solar Energetic Particles Using the Geomagnetic Field', Astrophys. J. 452, L149–L152.

  38. Leske, R. A., Mewaldt, R. A., Stone, E. C., and von Rosenvinge, T. T.: 1996, 'The Isotopic Composition of Anomalous Cosmic Rays from SAMPEX', Space Sci. Rev. 78, 149–154.

  39. Leske, R. A., Mewaldt, R. A., Stone, E. C., and von Rosenvinge, T. T.: 1997, 'Geomagnetic Cutoff Variations During Solar Energetic Particle Events – Implications for the Space Station', Proc. 25th Int. Cosmic Ray Conf. 2, 381–384.

  40. Lin, Z., Bieber, J., and Evenson, P.: 1995, 'Electron Trajectories in a Model Magnetosphere: Simulation and Observations under Active Conditions', J. Geophys. Res. 100, 23 543–23 549.

  41. Lund, N., Peters, B., Cossick, R., and Pal, Y.: 1970, 'On the Isotopic Composition of Primary Cosmic Ray Nuclei', Phys. Lett. 31B, 553–556.

  42. Lund, N., Ramsussen, I. L., and Peters, B.: 1971, 'A Method for Determining the Mean Atomic Charge of the Elements in the Primary Cosmic Radiation Throughout the Latitude Sensitivity Part of the Spectrum', Proc. 12th Int. Cosmic Ray Conf. 1, 130–134.

  43. Lust, R.: 1957, 'Impact Zones for Solar Cosmic Ray Particles', Phys. Rev. 105, 1827–1839.

  44. McCracken, K. G. and Ness, N.: 1966, 'The Collimation of Cosmic Rays by the Interplanetary Magnetic Field', J. Geophys. Res. 77, 3315–3318.

  45. McCracken, K. G., Rao, U. R., and Shea, M. A.: 1962, 'The Trajectories of Cosmic Rays in a High Degree Simulation of the Geomagnetic Field', Massachusetts Institute of Technology, Laboratory for Nuclear Science, Technical Report 77 (NYO-2670), Cambridge MA, USA.

  46. McCracken, K. G., Rao, U. R., Fowler, B. C., Shea, M. A., and Smart, D. F.: 1965, 'Cosmic Ray Tables (Asymptotic Directions, Variational Coefficients and Cutoff Rigidities', IQSY Instruction Manual No. 10, IQSY Committee, London, U.K.

  47. McCracken, K. G., Rao, U. R., and Bukata, R. P.: 1967, 'Cosmic Ray Propagation Process, 1. A Study of the Cosmic-Ray Flare Effect', J. Geophys. Res. 72, 423–434.

  48. McCracken, K. G., Rao, U. R., Fowler, B. C., Shea, M. A., and Smart, D. F.: 1968, 'Cosmic Ray Tables (Asymptotic Directions, etc.)', Annals of the IQSY 1, Chapter 14, 198–214, MIT Press, Cambridge MA, U.S.A.

  49. McIlwain, C. E.: 1961, 'Coordinates for Mapping the Distribution of Trapped Particles', J. Geophys. Res. 66, 3681–3691.

  50. Macmillan, S.: 1996, 'A Geomagnetic Jerk for the Early 1990s', Earth Planetary. Sci. Lett. 13, 189–192.

  51. Mischke, C.F.W., Raubenheimer, B. C., Stoker, P. H., van der Walt, A. J., Shea, M. A., and Smart, D. F.: 1979, 'Experimental Observations of Secular Changes in the Vertical Cutoff Rigidity', Proc. 16th Int. Conf. Cosmic Ray 4, 279–284.

  52. Mitra, B., Biswas, S., Durgaprasad, N., Singh, R. K., Vahia, M. N., Dutta, A., and Goswami, J. N.: 1989, 'Studies of Anomalous Cosmic Radiation Oxygen Ions in Space and their Ionization States in ANURADHA Experiment in Spacelab-3', Adv. Space Res. 9, 17–20.

  53. Morfill, G. and Quenby, J. J.: 1971, 'The Entry of Solar Particles Over the Polar Cap', Planetary. Space. Sci. 19, 1541–1577.

  54. Morfill, G. and Scholer, M.: 1972a, 'Solar Proton Intensity Structures in the Magnetosphere During Interplanetary Anisotropies', Planetary Space. Sci. 20, 2113–2123.

  55. Morfill, G. and Scholer, M.: 1972b, 'Reconnection of the Geomagnetic Tail Deducted from Solar-Particle Observations', J. Geophys. Res. 77, 4021–4026.

  56. Morfill, G. and Scholer, M.: 1973, 'Study of the Magnetosphere Using Energetic Solar Particles', Space Sci. Rev. 15, 267–353.

  57. Nagashima, G. and Fujimoto, K.: 1994, 'Interplanetary Magnetic Field Colliminated Cosmic Ray Flow Across Magnetic Shock From Inside of Forbush Decrease, Observed as Local-Time-Dependant Precursory Decrease on the Ground', J. Geophys. Res. 99, 21 419–21 427.

  58. Orloff, S.: 1998, 'A Computational Investigation of Solar Energetic Particle Trajectories in Model Magnetospheres', PhD Thesis, Rice University.

  59. Orloff, S. and Freeman, J.: 1999, 'Anomalous Trapped Energetic Electron Orbits', Proc. 6th Spacecraft Charging Techn. Conf., Air Force Research Laboratory, Hancom AFB, Bedford MA, USA.

  60. Ostapenko, A. A. and Maltsev, Y. P.: 1997, 'Relation of the Magnetic Field in the Magnetosphere to the Geomagnetic and Solar Wind Activity', J. Geophys. Res. 102, 17 467–17 473.

  61. Paulikas, G. A.: 1974, 'Tracing of High-Latitude Magnetic Field Lines by Solar Particles', Rev. Geophys. Space Sci. 12, 117–128.

  62. Peters, B.: 1974, 'Isotropic Analysis of High Energy Cosmic Ray Nuclei', Nucl. Inst. Meth. 12, 205–238.

  63. Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T.: 1989, 'Numerical Recipes', Cambridge University Press, London, U.K.

  64. Ralston, A. and Wilf, S. H. (eds.): 1960, 'Mathematical Methods for Digital Computers', John Wiley and Sons, New York NY, USA.

  65. Rao, U. R., McCracken, K. G., and Venkatesan, D.: 1963, 'Asymptotic Cones of Acceptance and Their use in the Study of the Daily Variation of Cosmic Radiation', J. Geophys. Res. 68, 345–369.

  66. Sabka, T. J., Langel, R. A., Baldwin, R. T., and Conrad, J. A.: 1997, 'The Geomagnetic Field, 1900–1995, Including the Large Scale Fields from Magnetospheric Sources and NASA Candidate Models for the 1995 Revision of the IGRF', J. Geomag. Geoelectr. 49, 157–206.

  67. Shea, M. A. and Smart, D. F.: 1970, 'Secular Variation in Cosmic Ray Cutoff Rigidities', J. Geophys. Res. 75, 3921–3922.

  68. Shea, M. A. and Smart, D. F.: 1983, 'AWorld Grid of Calculated Vertical Cutoff Rigidities for 1980.', Proc. 18th Int. Cosmic Ray Conf. 3, 415–418.

  69. Shea, M. A. and Smart, D. F.: 1990, 'The Influence of the Changing Geomagnetic Field on Cosmic Ray Cutoff Rigidities', J. Geomag. Geoelectr. 42, 1107–1121.

  70. Shea, M. A., Smart, D. F., and McCracken, K. G.: 1965, 'A Study of Vertical Cutoff Rigidities Using Sixth Degree Simulations of the Geomagnetic Field', J. Geophys. Res. 70, 4117–4130.

  71. Shea, M. A., Smart, D. F., and Gentile, L. C.: 1987, 'Vertical Cutoff Rigidities Calculated From the Estimated 1985 Geomagnetic Field Coefficients', Proc. 20th Int. Cosmic Ray Conf. 4, 205–207.

  72. Smart, D. F. and Shea, M. A.: 1967, 'A Study of the Effectiveness of the McIlwain Coordinates in Estimating Cosmic-Ray Vertical Cutoff Rigidities', J. Geophys. Res. 72, 3447–3454.

  73. Smart, D. F. and Shea, M. A.: 1972, 'Daily Variation of Electron and Proton Geomagnetic Cutoffs Calculated for Fort Churchill, Canada', J. Geophys. Res. 77, 4595–4601.

  74. Smart, D. F. and Shea, M. A.: 1981a, 'Optimum Step Length Control for Cosmic Ray Trajectory Calculations', Proc. 17th Int. Cosmic Ray Conf. 4, 255–258.

  75. Smart, D. F. and Shea, M. A.: 1981b, 'Muon Trajectories from the Batavia Accelerator N-E Beam Dump', Proc. 17th Int. Cosmic Ray Conf. 5, 6–9.

  76. Smart, D. F. and Shea, M. A.: 1994, 'Geomagnetic Cutoffs: a Review for Space Dosimetry Applications', Adv. Space Res. 14(10), 787–796.

  77. Smart, D. F. and Shea, M. A.: 1997a, 'World Grid of Cosmic Ray Vertical Cutoff Rigidities for Epoch 1990.0', Proc. 25th Int. Cosmic Ray Conf. 2, 401–404.

  78. Smart, D. F. and Shea, M. A.: 1997b, 'Calculated Cosmic Ray Cutoff Rigidities at 450 km for Epoch 1990.0', Proc. 25th Int. Cosmic Ray Conf. 2, 397–400.

  79. Smart, D. F., Shea, M. A., and Gall, R.: 1969, 'The Daily Variation of Trajectory-Derived High-Latitude Cutoff Rigidities in a Model Magnetosphere', J. Geophys. Res. 74, 4731–4738.

  80. Smart, D. F., Shea, M. A., and Flückiger, E. O.: 1999a, 'Calculated Vertical Cutoff Rigidities for the Int. Space Station During Magnetically Quiet Times', Proc. 26th Int. Cosmic Ray Conf. 7, 394–397.

  81. Smart, D. F., Shea, M. A., Flückiger, E. O., Tylka, A. J., and Boberg, P. R.: 1999b, 'Calculated Vertical Cutoff Rigidities for the International Space Station DuringMagnetically Active Times', Proc. 26th Int. Cosmic Ray Conf. 7, 398–401.

  82. Smart, D. F., Shea, M. A., Flückiger, E. O., Tylka, A. J., and Boberg, P. R.: 1999c, 'Changes in Calculated Vertical Cutoff Rigidities at the Altitude of the International Space Station as a Function of Magnetic Activity', Proc. 26th Int. Cosmic Ray Conf. 7, 337–340.

  83. Soutoul, A., Engelmann, J. J., Goret, P., Juliusson, E., Koch-Miramond, L., Masse, P., Petrou, N., Rio, Y., and Risho, T.: 1981, 'Isotopic Analysis Using the Geomagnetic Method', Proc. 17th Int. Cosmic Ray Conf. 9, 105–108.

  84. Stoer, J. and Bulirsch, R.: 1980; Introduction to Numerical Analysis, Chapter 7, Springer-Verlag, New York NY, U.S.A.

  85. Störmer, C.: 1930, 'Periodische Elektronenbabahnen im Feld lines Elementramagneton und ihre Anwendung auf Bruches Modellverauche und auf Eschenhagens Elementarwellen des Erdmagnetismus'. Zeits. f. Astrophys. 1, 237–274.

  86. Störmer, C.: 1950, 'The Polar Aurora' Oxford University Press, London.

  87. Thomas, G. R., Willis, D. M., and Pratt, R. J.: 1974, 'Solar Proton Entry to the Magnetosphere on 18 Nov. 1986 and 25 Feb. 1969-I. Interpretation of Satellite Data Using Trajectory Computations in a Model Magnetosphere', J. Atmospheric Terrest. Phys. 36, 995–1017.

  88. Tsyganenko, N. A.: 1989, 'A Magnetospheric Magnetic Field Model With a Warped Tail Current Sheet', Planetary Space Sci. 37, 5–20.

  89. Tsyganenko, N. A. and Usmanov, A. V.: 1982, 'Determination of the Magnetospheric Current System Parameters and Development of Experimental Geomagnetic Field Models Based on Data from IMP and HEOS Satellites', Planetary Space Sci. 30, 985–998.

  90. Tylka, A. J., Boberg, P. R., Adams, J. H., and Beahm, L. P.: 1995, 'The Mean Ionic Charge State of Solar Energetic Fe Ions Above 200 MeV per Nucleon', Astrophys. J. 444, L109–L113.

  91. Vallarta, M. S.: 1938, 'An Outline of the Theory of the Allowed Cone of Cosmic Radiation', University of Toronto Series, Appl. Math, Series No. 3, The University of Toronto Press, Toronto, Ontario, Canada.

  92. Vallarta, M. S.: 1961, 'Theory of the Geomagnetic Effects of Cosmic Radiation', in: Handbuch der Physik XLVI/I, 88–129, Springer-Verlag, New York NY, USA.

  93. Vallarta, M. S.: 1978, 'Manuel Sandoval Vallarta, Obra Cientifica', Universidad Nacional Autonoma de Mexico, Instituto Nacional de Energia Nuclear, Mexico City, Mexico.

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Smart, D., Shea, M. & Flückiger, E. Magnetospheric Models and Trajectory Computations. Space Science Reviews 93, 305–333 (2000).

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  • Neutron Monitor
  • Solar Energetic Particle Event
  • Trajectory Calculation
  • Asymptotic Cone
  • Cutoff Rigidity