Escaping dynamics of relativistic protons in the Earth's magnetosphere

In this work, we study the motion of protons under the influence of the Earth's magnetic field. We investigate the dynamics and topology of invariant spatial regions in two magnetic field models, namely, the dipolar and Luhmann tail configurations. In both cases, we analyze the motion of the protons in the relativistic regime, with kinetic energy intervals in the range 10<E<100MeV. Specifically, we examine situations in which the protons undergo chaotic scattering due to their interaction with the Earth's magnetic field. Additionally, we consider scenarios where the protons, during their interaction, fall onto the Earth's surface. We compare the dipole and Luhmann tail cases by computing the distributions of escape and residence times, the exit basins, and the fractal dimensions of their boundaries. We present a robust symplectic numerical scheme which is suitable for analyzing the impact of the nonlinear effects and the presence of strong sensitivity to initial conditions when the model goes beyond the pure dipolar field. Surprisingly, we uncover a scaling law between the coefficient of the decay law and the energy of the protons. Furthermore, our computation of the fractal dimension of the exit basin boundaries D versus the distance between the protons and the Earth z_{0} reveals a decrease in D as z_{0} increases. We expect this work to be useful for a better understanding of the behavior of protons under the influence of the Earth's magnetic field when they are in the chaotic scattering regime.