Allowed regions in the problem concerning dynamics of a charged particle in a superposition of dipole and uniform magnetic fields

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The motion of charged particles in the Earth’s magnetic field has been of interest to mathematicians and physicists in connection with the study of the polar aurora and cosmic rays. In 1907, Norwegian mathematician Stromer gave the mathematical formulation of this problem. It became the problem of great importance with the discovery of the Van Allen radiation. As is known, the Earth’s magnetic field can be considered approximately as a superposition of dipole and uniform magnetic fields, and the dipole’s magnetic moment is either parallel or antiparallel to the induction of the uniform field. Thus, the problem concerning the dynamics of the charged particle in the magnetic field of the Earth is reduced to that of charged particle dynamics in the composed field. The paper is devoted to the construction and investigation of the allowed regions in a superposition of dipole and uniform magnetic fields for positive values of Stormer’s constant γ and the same orientation of magnetic moment and uniform field.