The authors appreciate all suggestions given by Reviewers 1 and 2, which were essential to improve this article. A careful revision was carried out in the manuscript taking into account all the comments and suggestions. The figures and Table 1, mentioned in the response, are available in the attached file (Figures_Table_response_to_reviewers.pdf).

 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?

We add the discussion below in lines 147-154:

To justify the use of an inertial frame associated with the satellite, consider, for instance, the maximum acceleration achieved by the satellite at the perigee (data from satellite orbit can be found e.g. at Mauk et al., 2013), $8.2~{\rm m}{\rm s}^{-2}$. The interaction time between the electron and the wave is of the order of $10^{-3}$ s. Since the period of the satellite is $537.1$ min, its acceleration is nearly constant during the interaction. Therefore, the change in the speed of the satellite in its orbit during one interaction time is around $8.2 \times 10^{-3}~{\rm m}/{\rm s}$ which is 6 orders of magnitude smaller than the speed of the satellite at the perigee 9,8 km/s.  Similarly, the spin period of the satellite is 11 s (Breneman et al., 2022), which leads to a change in angle of about $1.8~{\rm arcmin}$ in one interaction time. Thus, for the purposes of the present article, it is reasonable to consider the satellite as an inertial reference frame during one interaction time. Moreover, it is a standard approach in the literature to consider the satellite as an inertial reference frame.

Breneman, A.W., Wygant, J.R., Tian, S. et al. The Van Allen Probes Electric Field and Waves Instrument: Science Results, Measurements, and Access to Data. Space Sci Rev 218, 69 (2022). https://doi.org/10.1007/s11214-022-00934-y)

Mauk, B.H., Fox, N.J., Kanekal, S.G. et al. Science Objectives and Rationale for the Radiation Belt Storm Probes Mission. Space Sci Rev 179, 3–27 (2013). https://doi.org/10.1007/s11214-012-9908-y

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

Resp.: The diagram was included in the manuscript as Figure 3

1. 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.

1. 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).

Resp.: We thank the referee for this suggestion, and we include the following paragraph in the manuscript (lines 234-241 and 337-348):

The manuscript aims to improve the calculation of the interaction time in the quasilinear wave-particle interaction regime. Thus, we use first-order solutions such as those done by Lakhina and Tsurutani, 2010. The weak turbulence in plasma and nonlinear events occurrence rate is around 10 to 15% considering the average occurrence of whistler-mode chorus waves (Zhang et al., 2018). These events have a solution of the wave-particle interaction equation are based on, at least, second-order terms in wave amplitude, e.g., Allanson et al., 2022; Artemyev et al., 2023; Omura, 2021; Osmane et al., 2016; Bortnik et al., 2008). 

There are several difficulties in calculating the trapping time in non-linear interactions (e.g., eq. (87) Omura, 2021,  Omura et al., 2013, Omura et al., 2008) using in situ measurements. Though, the first-order approach can give approximate solutions to nonlinear regimes, and Equation (13) can be used to estimate the interaction time from in-situ measurements (e.g., Hsieh et al., 2021) since it is important to determine the energy gain of electrons undergoing a wave-particle interaction (Hsieh et al., 2021). However, the interaction time is known to be shorter than the trapping time (Hsieh et al., 2021; Bortnik et al., 2008)

Anton V. Artemyev, Jay M. Albert, Anatoli I. Neishtadt, Didier Mourenas; The effect of wave frequency drift on the electron nonlinear resonant interaction with whistler-mode waves. Physics of Plasmas 1 January 2023; 30 (1): 012901. https://doi.org/10.1063/5.0131297

Omura, Y., Katoh, Y., and Summers, D. (2008), Theory and simulation of the generation of whistler-mode chorus, J. Geophys. Res., 113, A04223, doi:10.1029/2007JA012622.

Osmane, A., Wilson, L. B., Blum, L., Pulkkinen, T. I. (2016), On The Connection Between Microbursts And Nonlinear Electronic Structures In Planetary Radiation Belts. The Astrophysical Journal,  ApJ 816 51, https://dx.doi.org/10.3847/0004-637X/816/2/51

Allanson, O., Thomas, E., Clare, W., Thomas, N. (2022), Weak Turbulence and Quasilinear Diffusion for Relativistic Wave-Particle Interactions Via a Markov Approach. Front. Astron. Space Sci. 8:805699. doi:10.3389/fspas.2021.805699

Bortnik, J., Thorne, R. M., and Inan, U. S. (2008), Nonlinear interaction of energetic electrons with large amplitude chorus, Geophys. Res. Lett., 35, L21102, doi:10.1029/2008GL035500.

Zhang, J., Thorne, R., Artemyev, A., Mourenas, D., Angelopoulos, V., Bortnik, J., Kletzing, C. A., Kurth, W. S., & Hospodarsky, G. B. (2018). Properties of Intense Field-Aligned Lower-Band Chorus Waves: Implications for Nonlinear Wave-Particle Interactions. Journal of Geophysical Research: Space Physics, 123(7), 5379-5393. https://doi.org/10.1029/2018JA025390

1. 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/s11214-016-0252-5

or equation 1 in Shprits et al Journal ofAtmosphericandSolar-TerrestrialPhysics70(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

Resp.: We thank the referee for this point. In the manuscript, we write Equation (3) as a general equation for the Doppler shift in the wave frequency. We explain in lines 113-115 that we are considering the wave vector parallel to the ambient magnetic field for the wave's direction being co-propagating and counter-propagating to the energetic electron. In this case, the equatorial electron pitch angle \alpha used in equation (3) will be the same as done by Lakhina et al., 2020, Artemyev et al., 2016, and Camporeale (2015). 

We call \delta the angle between the wave vector and the electron velocity vector. This correction was included in equation (4) and lines 118-119. Also, Figure 2 was corrected, and the discussion was added in lines 128-131.

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.

Resp.: Yes, we agree, and this is shown by the n (harmonic number) in Eq. (4) in the submitted manuscript.

Therefore equation 4 not quite correct. This could have serious implications for the rest of the draft? 

Resp: Equation (4) was corrected by changing the angle between the electron velocity vector and the wave normal angle as  \delta; both vectors' orientations are defined as related to the ambient magnetic field in a tridimensional coordinate system.  In the manuscript, we solve Eq. (4) for \delta equals \alpha and \pi - \alpha for co-propagating and counter-propagating waves, respectively. In such an approach, both resonant and perpendicular components of the electron velocity vector are considered in the calculation, similar to that done by Summers et al., 2012. 

The point is that there are many different values of energy that can resonate with a given wave, and these are a function of pitch-angle, e.g. see an example for the n=-1 resonance in

Camporeale, E. (2015), Resonant and nonresonant whistlers-particle interaction in the radiation belts, Geophys. Res. Lett., 42, 3114–3121, doi:10.1002/2015GL063874.

Resp.: We agree with the referee that there are many different energies resonating with a given wave frequency, since we have the possibility to match several harmonics with the same wave; besides, for parallel propagating whistler waves, the angle between the wave vector and electron speed vector coincides with the electron's pitch angle, as shown by Camporeale, E. 2015. This information and reference were added in the manuscript in lines 129-131. 

Data

1. 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!

Resp: We thank the referee for this point, and we clarify that the interaction time calculated in this manuscript accounts for the first-order cyclotron effect only. We included the citation of Hsieh et al., 2020 and 2017; Omura, 2021,  Omura et al., 2013; Omura et al., 2008  to show an example in the literature of the interaction time in nonlinear cases. In addition, we changed the case studies presented in Table 1 to apply the derived equations to the linear wave-particle cases. Also, Figures 6, 7, D1, and E1 were updated.

Hsieh, Y.-K., and Omura, Y. (2017), Nonlinear dynamics of electrons interacting with oblique whistler-mode chorus in the magnetosphere, J. Geophys. Res. Space Physics, 122, 675– 694, doi:10.1002/2016JA023255.

1. 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 pitch-angles. 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

Resp.: Figure (2) was corrected for the angle between the wave vector and electron velocity vector. Figure (2) is an example of Eq. (5) for a given ambient plasma parameters, wave normal angle, equatorial electron pitch angle, and one harmonic number. It is shown to give an example of how the resonant electron kinetic energy varies as a function of the wave frequency,  for waves propagating parallel and oblique to the ambient magnetic field. Accordingly, if the harmonic number n is changed, all the curves would be shifted to the higher wave frequency side. We inserted a sentence about this point in line 128.

1. 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 208-267, 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?

Resp.:  We apologize if we were not clear. Section 4 was revised and split to organize the dataset in a subsection (starting in line 246). 

Using Van Allen Probes data, it is possible to identify the periods of the high-energy electron flux dropouts and enhancements in the outer radiation belt concomitant with whistler-mode chorus wave activities for each case study shown in Table 1. It suggests that the resonance between the chorus waves and particles may have locally contributed to this electron flux variability. 

To clarify the importance of the discussions about the PSD in this article, we decided to write a new paragraph before line 208 and rewrote the discussions about Figures 5.1, A1.1 e B1.1, as described below:

Figures 5.1, A1.1 e B1.1 shows the time evolution of the radial PSD profiles at inbound/outbound regions of Van Allen Probe B, which allows the identification of the local relativistic electron loss and/or local low-energy acceleration in the outer radiation belt. The pitch angle diffusion mechanism driven by whistler mode chorus waves can cause this kind of local electron flux loss/acceleration, in which the local (L*), order of magnitude, and energy level are identified at the same period analyzed in previous sections.     

Reference:

Tsyganenko, N. A., & Sitnov, M. I. (2005). Modeling the dynamics of the inner magnetosphere during strong geomagnetic storms. Journal of Geophysical Research, 110, A03208. https://doi.org/10.1029/2004JA010798

Context/References/Discussion

1. ~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

Resp.: We include the discussion and references related to the motivation and justification of the paper in lines 45 - 66:

In the magnetosphere, the kinematics description of the wave-particle interaction for relativistic electrons usually considers the relativistic Doppler shift in the resonance condition (e.g., Thorne et al. 2005, Summers et al. 1998) and the relativistic motion equation (e.g., Omura, 2021). Often, the resonant kinetic energy of the electrons results from the resonance condition and the motion equation, together with the wave group velocity (e.g., Omura 2021, Hsieh et al., 2021, Summers et al., 2012, Glauer et al., 2005, Lyons et al., 1972). The wave-particle interaction time ($T_r$) is a crucial parameter in estimating time-dependent processes as the energy and pitch angle diffusion coefficients \citep{walker1993, lakhina2010, tsurutani2013, hsieh2020, hsieh2022}, however, the relativistic kinematics description mentioned above is incomplete to calculate this parameter. In this paper, we add to the latter approach a complete relativistic description of the problem: the relativistic velocity addition (between the electron and the wave) and the implications of the different reference frames in the estimates of the change in pitch angle and the diffusion coefficient.

We calculate the parameters for four case studies to give a quantitative comparison between the complete relativistic description and a non-relativistic approach (used here as an approximation to calculate the interaction parameters). The interaction time is calculated using the test particle equations \citep{tsurutani1974, lakhina2010, horne2003b, bortnik2008} along with the special relativity theory applied to whistler-mode chorus waves propagating in cold plasma magnetosphere (where group velocity is $~0.3c$ to $0.5 c$) and energetic electrons (with energy $ \sim 0.1$ to $2$ MeV). 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.

We considered parallel propagating whistler-mode chorus waves linearly interacting with relativistic electrons to derive first the group velocity equation, then the resonant relativistic kinetic energy, and finally the interaction time. Thus, we calculate the change pitch angle and the diffusion coefficient rates. We use the Van Allen Probes measurement of wave parameters, ambient magnetic field, density, electron fluxes, and equatorial pitch angle to apply the interaction time equation.

A complete calculation of these parameters can improve relativistic outer radiation belt electron flux variation models.

1. Line 69: Appleton/Hartree equation: please provide a citation and a justification for why this is the correct relation to use

Resp.: We add the following justification and reference in lines 78-81: whistler mode chorus waves which occur in frequencies higher than the ion cyclotron frequency, besides the wave-particle interaction outside the plasmasphere, the dispersion relation for this case is obtained from the solution of the Appleton-Hartree equation \citep{bittencourt2004}.

1. 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?

Resp.: The change in pitch angle for non-relativistic electrons due to wave-particle interaction in a quasi-linear regime is given by equations (3.6) in Kennel and Petschek, 1966, and (11) in Tsurutani and Lakhina, 1997. Lakhina et al., 2010,  derived equation (11) by considering the relativistic resonant condition and the non-relativistic equation of motion. Allanson et al., 2022, show the exact equation for pitch angle scattering and second-order equations for weak turbulence and nonlinear regimes.

In this manuscript, the change in pitch angle is given by equation (20), which is consistent with equations (3.6) in Kennel and Petschek, 1966; (11) in Tsurutani and Lakhina, 1997 and equation (11) in Lakhina et al., 2010. In the limit that \gamma ~ 1, for non-relativistic electrons, Tr equals to \delta t (in Kennel and Petschek, 1966 and Tsurutani and Lakhina, 1997) or to \tau in (Lakhina et al., 2010). Also, Equation (20) in this manuscript is similar to Equation (S3) in Allanson et al., 2021.  

The main difference we observe using a more complete relativity description is that the interaction time is often longer than that calculated with a non-relativistic description, as shown in Table 1. The consequence of underestimating the interaction time is that diffusion coefficients are also underestimated and may deteriorate modeling results.  

These sentences were included in lines (231-248 and in lines 337-341).

1. 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?

Resp.: We revised this paragraph and section 5 to present the main results (lines 322-326) and contextualize them in lines 337-348.

Technical

1. line 5 (“wave velocity”): is this group or phase velocity?

Resp.: We add the wave group velocity in line 5

1. 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 wave-particle interaction, and not really whether it is an active or quiet period. Perhaps it is best not to phrase things in this way.

Resp.: We agree with the referee, and we remove this sentence

1. 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

Resp.: The reference cited in the paper was correct to the following, in line (418):  Yuri Y. Shprits, Dmitriy A. Subbotin, Nigel P. Meredith, Scot R. Elkington, Review of modeling of losses and sources of relativistic electrons in the outer radiation belt II: Local acceleration and loss, Journal of Atmospheric and Solar-Terrestrial Physics, Volume 70, Issue 14, 2008, Pages 1694-1713, ISSN 1364-6826, https://doi.org/10.1016/j.jastp.2008.06.014.

1. Line 134: “well-known formula of the addition of velocities”. What is this formula? Please provide the formula and a citation

Resp.: Equation (7) in the manuscript is the general equation for the relativistic addition of velocities between vg and vgc. In line 133, we add the citation of Jackson, J. D. Classical Electrodynamics, which was included in line 378

1. 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.

Resp.: We made the suggested change in equation (18), line 190

Response to Referee #2

The authors appreciate all suggestions given by Reviewers 1 and 2, which were essential to improve this article. A careful revision was carried out in the manuscript taking into account all the comments and suggestions.

1. The authors argue that taking into account special relativity effects would alter the wave-particle 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 phase-trapping (Bortnik et al. 2008 GRL) or physical trapping (Artemyev et al. 2013 PoP, Osmane et al. ApJ 2016) can be neglected. 

Resp.: We clarify in the manuscript that the derived equations described in this paper are applicable to a quasi-linear regime. 

2. 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 quasi-linear regime, the transit time they are calculating with or without relativistic effects are comparable, with some minor differences of 0.1-0.01 ms. What would actually justify their thesis would be to show that the pitch-angle 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 pitch-angle shown in Table 1 (assuming again we are within a quasi-linear 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. 

Resp.: We thank the referee for this point, which improves the manuscript. As we are interested in showing the relevance of a more complete relativistic description in linear wave-particle events, we choose periods in which linear processes are likely in the previous events and substitute case 2 for a new event. Then we revised the periods selected for each event and chose a linear one. Besides, we calculate the pitch angle diffusion coefficient (Da)  following the same approach as done by Lakhina et al., 2010. The results are summarized in Table 1 (attached). The comparison between non-relativistic and complete relativistic calculations shows that the interaction time is around 4 times higher when the Special Relativity kinematics is considered. Also, the diffusion coefficient rates show notable differences in the pitch angle scattering. 

3. 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 wave-particle 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.

Resp.: Accordingly, we show that for the selected case studies, the relativistic diffusion coefficient rates in Table 1 (attached) can be more than 5 times higher than the non-relativistic diffusion rates.