Relativistic kinematic effects in the interaction time of whistler-mode chorus waves and electrons in the outer radiation belt - L. R. Alves; M. E. S. Alves; M. E. S. Alves; L. A. da Silva; L. A. da Silva; V. Deggeroni; P. R. Jauer; P. R. Jauer; D. G. Sibeck (Annales Geophysicae, 2023)
Whistler-mode chorus waves propagate outside the plasmasphere, interacting with energetic electrons in the outer radiation belt. This leads to local changes in the phase space density distribution due to energy or pitch angle diffusion. The wave–particle interaction time (Tr) is crucial in estimating time-dependent processes such as the energy and pitch angle diffusion. Although the wave group and particle velocities are a fraction of the speed of light, the kinematics description of the wave–particle interaction for relativistic electrons usually considers the relativistic Doppler shift in the resonance condition and relativistic motion equation. This relativistic kinematics description is incomplete. In this paper, to the literature we add a complete relativistic description of the problem that relies on the relativistic velocity addition (between the electron and the wave) and the implications of the different reference frames for the estimates of the interaction time. We use quasi-linear test particle equations and the special relativity theory applied to whistler-mode chorus waves parallel propagating in cold-plasma magnetosphere interaction with relativistic electrons. Also, we consider that the resonance occurs in the electron's reference frame. At the same time, the result of such interaction and their parameters are measured in the local inertial reference frame of the satellite. The change pitch angle and the average diffusion coefficient rates are then calculated from the relativistic interaction time. The interaction time equation is consistent with previous works in the limit of non-relativistic interactions (Tnr). For the sake of application, we provide the interaction time and average diffusion coefficient Daa for four case studies observed during the Van Allen Probes era. Our results show that the interaction time is generally longer when applying the complete relativistic approach, considering a non-relativistic calculation. From the four case studies, the ratio Tr/Tnr<svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="29pt" height="14pt" class="svg-formula" dspmath="mathimg" md5hash="4f6d000e2d68b0aa0773f3c8b3697fc8"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="angeo-41-429-2023-ie00001.svg" width="29pt" height="14pt" src="angeo-41-429-2023-ie00001.png"/></svg:svg> varies in the range 1.7–3.0 and Daa/<msubsup>Daanr</msubsup><svg:svg xmlns:svg="http://www.w3.org/2000/svg" width="42pt" height="14pt" class="svg-formula" dspmath="mathimg" md5hash="abb2437cf734bfe5586c9af979fcd4c0"><svg:image xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="angeo-41-429-2023-ie00002.svg" width="42pt" height="14pt" src="angeo-41-429-2023-ie00002.png"/></svg:svg> in the range 1.9–5.4. Accurately calculating the interaction time with full consideration of special relativity can enhance the modeling of the electron flux in Earth's outer radiation belt. Additionally, the change in pitch angle depends on the time of interaction, and similar discrepancies can be found when the time is calculated with no special relativity consideration. The results described here have several implications for modeling relativistic outer-radiation-belt electron flux resulting from the wave–particle interaction. Finally, since we considered only one wave cycle interaction, the average result from some interactions can bring more reliable results in the final flux modeling.