Fig. 7

Simulation results of wave–particle interactions for electrons under different initial conditions of the initial energy and equatorial pitch angle. Electrons interact with the chorus wave elements under three different initial conditions: (case 1) 500 keV-60°, (case 2) 2.1 MeV-80°, and (case 3) 5.7 MeV-86°. The energy variation shows that electrons in all the three cases gain energy (62.7 keV, 133.5 keV, and 154.4 keV, respectively). The velocity variation shows that the electron in case 1 is accelerated when \( v_{\parallel } \approx V_{R} < 0 \) (the RTA acceleration), whereas those in cases 2 and 3 are accelerated when \( v_{\parallel } \approx V_{R} > 0 \) (the URA acceleration). Energy increase is higher in the URA acceleration (cases 2 and 3) due to a longer passing time through the chorus wave element \( t_{\text{pass}} \)