New method to determine proton trajectories in the equatorial plane of a dipole magnetic field

A parametric description of proton trajectories in the equatorial plane of Earth’s dipole magnetic field has been derived. The exact expression of the angular coordinate contains an integral to be performed numerically. The radial coordinate results from the initial conditions by basic mathematical operations and by using trigonometric functions. With the approximate angular coordinate formula, applicable for a wide variety of cases of protons trapped in Earth’s radiation belts, no numerical integration is needed. The results of exact and approximate expressions were compared for a specific case and small differences were found.