the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Relativistic Kinematic Effects in the Interaction Time of WhistlerMode Chorus Waves and Electrons in the Outer Radiation Belt
Livia R. Alves
Marcio E. S. Alves
Ligia A. Silva
Vinicius Deggeroni
Paulo R. Jauer
David G. Sibeck
Abstract. Whistlermode chorus waves propagate outside the plasmasphere. As they interact with energetic electrons in the outer radiation belt electrons, the phase space density distribution can change due to energy or pitch angle diffusion. Calculating the waveparticle interaction time is crucial to estimate the particle’s energy or pitch angle change efficiently. Although the wave and particle velocities are a fraction of the speed of light, in calculating the interaction time, the special relativistic effects are often misleading, incomplete, or simply unconsidered. In this work, we derive an equation for the waveparticle interaction time considering the special relativity kinematic effect. We solve the equation considering typical magnetospheric plasma parameters, and compare the results with the nonrelativistic calculations. Besides, we apply the methodology and the equation to calculate the interaction time for one wave cycle in four case studies. We consider waveparticle resonance conditions for chorus waves propagating at any wave normal angle in a dispersive and cold plasma. We use Van Allen Probes for in situ measurements of the relevant wave parameters for the calculation, the ambient magnetic field, and energetic electron flux under quiet and disturbed geomagnetic conditions. Thus, we use a test particle approach to calculate the interaction time for parallel and oblique propagating waves. Also, we evaluate the variation of pitch angle scattering for relativistic electrons interacting with whistlermode chorus waves propagating parallel to the ambient magnetic field. If the relativistic effects are not taken into account, the interaction time can be ∼ 30 % lower for quiet periods and a half lower for disturbed periods. As a consequence, the change in pitch angle is also underestimated. Besides, the longest interaction time occurs at waveparticle interaction with high pitch angle electrons, with energy ∼ 100′ s of keV, interacting with quasiparallel propagating waves. 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 waveparticle interaction. Finally, since we considered only one wavecycle interaction, the average result from some interactions can bring more confident results in the final flux modeling.
Livia R. Alves et al.
Status: open (until 11 Apr 2023)

RC1: 'Comment on angeo20236', Anonymous Referee #1, 28 Mar 2023
reply
General
This draft presents calculations regarding the interaction time for electrons with chorus waves, and the resultant effect on pitch angle transport (“delta alpha”). The main proposed contribution is that the calculation is presented as being consistent with special relativity. Impicitly, the reader is to assume that this is being performed for the first time in the literature.
I believe that the basic principle of the work presented in this draft is sound, and potentially important. Although I do have a major concern regarding the assumption of an inertial satellite frame of reference
Even if that major concern is cleared up, I believe that further major revisions are necessary before publication can be recommended.
These revisions mostly centre around
 Calculations – I have some serious questions regarding some of the calculations. Or clarifications that need to be made regarding the validity of the results
 Data – I do not see what significant value the presented Van Allen Probes data adds, for the manuscript in the current form. More work needs to be done, I would suggest.
 Context – we need much more justification and references to literature. And discussion.
Major comments
Calculations
1. All of the calculations rely on the comparisons of inertial frames of reference. Does a satellite have an inertial frame?
Not only is the satellite orbiting the Earth (and therefore undergoing acceleration), but it is also rotating around its own axis (and as a result so are the devices that measure BuBvBw etc). Please clarify. We need to have absolute clarity on this. Please can the authors give a thorough justification to explain why this is a reasonable approach. How are the inertial frames constructed and justified, given the above mentioned 2 sources of acceleration?
Relatedly, if there is a justification for this, we sould strongly suggest that the authors make a diagram that illustrates what S, S’, and the different relativistic considerations are regarding the wave and the satellite, in order to help the reader understand what is going on
2. e.g. going from equations 14 to 20, and in particular 19 to 20
All of this analysis assumes that
 the forces on the particle are constant during the timescale 0
 It also assumes that T is determined by some version of “T=X/V” (e.g equation 10 from Lakhina et al 2010), or a relativistic version equation 13 in this paper.
Both of these are quite restrictive approximations, although we accept that they are relatively standard methods that are used in the literature e.g. Kennel&Petschek 1966, Lakhina et al 2010 as cited by the authors. The approximations are as follows.
 The presented equations are a big approximation to the real answer, which requires the solution of coupled first order odes for gyrophase, pitch angle, energy, position (e.g. see exact solutions up to second order in Bwave/B0 in Allanson et al 2022 Front. Astron. Space Sci. 8:805699. doi:10.3389/fspas.2021.805699)
 the interaction time could be shorter than this if the particle is scattered away very quickly. Likewise it could be longer if the particle is trapped (e.g. see Bortnik et al 2008 GEOPHYSICAL RESEARCH LETTERS, VOL. 35, L21102, doi:10.1029/2008GL035500, 2008)
Therefore it is important to state these approximations, and to say that the answers could be very different if (i) and (ii) play an important role, e.g. for higher amplitude/nonlinear waves. Conversely, the results in this paper are probably only true for a quasilinear interaction (e.g. Kennel and Engelmann 1966 The Physics of Fluids 9, 2377 (1966).
3. Equation (3) in the manuscript is incorrect. E.g. see
Equation (1) in Lakhina et al 2010
Or equation (13) in Artemyev et al Space Sci Rev (2016) 200:261–355 DOI 10.1007/s1121401602525
or equation 1 in Shprits et al Journal ofAtmosphericandSolarTerrestrialPhysics70(2008)1694–1713
It should only be k_parallel and v_parallel in the doppler shift. The velocity component of interest is
v_parallel = v*cos\alpha
Therefore there are a range of speeds (v) and pitch angles (\alpha) that are resonant with any given wave frequency \omega. E.g.
see equations 151 and 152 from Omura Earth, Planets and Space (2021) 73:95.
Therefore equation 4 not quite correct. This could have serious implications for the rest of the draft? The point is that there are many different values of energy that can resonate with a given wave, and these are a function of pitchangle, e.g. see an example for the n=1 resonance in
Camporeale, E. (2015), Resonant and nonresonant whistlersparticle interaction in the radiation belts, Geophys. Res. Lett., 42, 3114–3121, doi:10.1002/2015GL063874.
Data
4. Case studies 1, 2 and 3 all have magnetic field wave amplitudes >= 1nT (see table 1). With B0 roughly 100nT. That gives Bw/B0 ~ 1/100. These are very intense waves and almost certainly fall into the nonlinear regime (e.g. phase trapping and phase bunching etc) e.g. see
Zhang et al JGR, 2018, 123, 5379–5393, https://doi.org/10.1029/2018JA025390 and
Zhang et al GRL, 2019, 46, 7182–7190. https://doi.org/10.1029/2019GL083833
As per the above comment (2), I question whether interaction time method described in the paper is applicable. Or to put it another way, I would need to be convinced – please!
5. implications of point 3 regarding resonant particles. There is not one single resonant energy. There is a single resonant v_parallel for a given value of n, but this corresponds to many energies and pitchangles. Therefore we do not understand figure 2 as it is currently presented, or any results or discussion that make use of the results from equation 5, and discussions about the given value of resonant energy
6. Lines 208 – 267: There is a lengthy description of some Van Allen Probes data here. But none of it really adds any genuine value to the manuscript in our opinion. From line 268 onwards there is a discussion of some parameters that are used and calculated for the interaction time. But that information is not really linked scientifically to the discussion of the data in lines 208267, except in a very superficial way. What are the implications of one on the other? What is the importance of the results and how does it related to the data? What new inferences can we make because of the data in the table etc? What might this show us in the Van Allen Probes data etc?
Context/References/Discussion
7. ~line 50: The authors claim that the relativistic approach is often simplified or misused, but they provide no references or discussion to support this claim. Please provide references and much more discussion! This is the foundation of the motivation for the paper. We suggest that there needs to be a much more thorough justification and motivation in the introduction of this draft
8. Line 69: Appleton/Hartree equation: please provide a citation and a justification for why this is the correct relation to use
9. Equation (20) How does the presented formula for \Delta\alpha compare to those presented in e.g. papers by Lakhina et al 2010, Tsurutani & Lakhina 1997, Kennel&Petschek66 and Allanson et al 2022? What differences has it made to include relativity, and what differences are observed because the integration was performed approximately and not exactly?
10. Lines 280 onwards: There is no meaningful discussion section. Section 5 has some conclusions. But the authors have not contextualised their work in detail with respect to previous results in the literature. This is an important missing piece of the draft and we would suggest needs to be fixed. Furthermore, now that we have these new estimates, how could they be used in practice by scientists in the future?
Technical
11. line 5 (“wave velocity”): is this group or phase velocity?
12. Line 15 (“interaction time can be ~30% lower for quiet periods”): we are not sure that it is appropriate to characterise things in this way. The authors have performed a small subset of case studies. Also, the details of the interaction time are fundamentally related to the microphysics of the waveparticle interaction, and not really whether it is an active or quiet period. Perhaps it is best not to phrase things in this way.
13. Line 99: The Shprits 2008 paper cited is a paper about radial diffusion. You should replace this with a local diffusion paper by Shprits in the same series in JASTP
14. Line 134: “wellknown formula of the addition of velocities”. What is this formula? Please provide the formula and a citation
15. Equation 18: the authors use \Delta on the LHS but “d” on the RHS for infinitesimal changes. They should either use one or the other, and be consistent on both sides of the equation.
Citation: https://doi.org/10.5194/angeo20236RC1 
RC2: 'Comment on angeo20236', Anonymous Referee #2, 29 Mar 2023
reply
The authors argue that taking into account special relativity effects would alter the waveparticle interactions between chorus and energetic electrons in the radiation belts. Unfortunately their results (most notably shown in Table 1) demonstrate the opposite, that is that relativistic effects are most likely inconsequential. And here is why. Their calculation assumes that electrons interacting with chorus waves will experience weak scattering, and thus nonlinear effects such as phasetrapping (Bortnik et al. 2008 GRL) or physical trapping (Artemyev et al. 2013 PoP, Osmane et al. ApJ 2016) can be neglected. Thus, by assuming that the waves are indeed sufficiently small in amplitude (clearly not the case for some of their events since delta B/B_0 >=1%) and that we are in a quasilinear regime, the transit time they are calculating with or without relativistic effects are comparable, with some minor differences of 0.10.01 ms. What would actually justify their thesis would be to show that the pitchangle diffusion coefficient they can compute from Delta_alpha (Equation 20) would lead to notable differences in the particle scattering. But they do not provide such an analysis and the change in the pitchangle shown in Table 1 (assuming again we are within a quasilinear regime despite the large amplitudes of the waves) are essentially identical for both relativistic and nonlinear relativistic effects with differences of the order 1/10 or 1/100. If we are interested with the nonlinear regime, the transit time they compute only make sense when compared with some nonlinear timescales, such as the trapping time. But since their transit time are essentially identitical I don't see how relativistic effect could turn a linear interaction into a nonlinear one either. I am therefore not certain how the paper could be salvaged since the relativistic effect they study seems to have no impact for both linear and nonlinear waveparticle interactions. If the editors are willing to accept a null result, the authors could perhaps compute the diffusion coefficients with relativistic effects included, and if the difference is marginal publish it as such.
Citation: https://doi.org/10.5194/angeo20236RC2
Livia R. Alves et al.
Livia R. Alves et al.
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