Whistler Waves and Electron Properties in the Inner Heliosphere: Helios Observations

Histograms in Figure 7 show the distribution of solar wind velocities for all N_tot spectra used in our analysis and for the spectra showing the presence of whistler wave signatures. All the analyzed spectra are shown in gray, slow wind whistlers are shown in green, intermediate wind whistlers are shown in red, and fast wind whistlers are shown in blue, respectively.

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Even though we have observed nearly equal numbers of spectra in the slow and fast solar winds, the majority of the spectra with signatures of whistlers ∼93% (6783) were observed in the slow solar wind (<500 km s⁻¹). This is similar to the study by Lacombe et al. (2014) at 1 au, where the authors suggested that a slow wind speed is one of the necessary conditions for the observation of long-lived whistler waves. Interestingly, this condition at 1 au also seems to hold in the inner heliosphere. However, in our study, we also observe whistler waves in the fast solar wind but they are rare, and represent about 3% (237) of the number of spectra with wave signatures. The remaining 4% are observed in the intermediate wind.

Figure 8 shows the percentage of whistler waves as a function of the wind velocity, i.e., the number of whistlers to the number of spectra available in the corresponding velocity bin. We observe that there is a constant decrease of whistler wave occurrence as the wind velocity is increasing: the higher the velocity, the lower the probability of observing whistler waves.

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Figure 9(a) shows the number of spectra with whistler wave signatures observed at different radial distances from the Sun. We observe that, in the slow wind, whistler waves are present at all distances, from 0.3 to 1 au. In contrast, in the fast wind, whistlers only start to appear for R > 0.6 au, in spite of a large number of spectra available closer to the Sun (see Figure 1).

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Figure 9(b) shows the occurrence of whistlers in the slow and fast solar winds as a function of radial distance between 0.3 and 1 au. Precisely, we show the percentage of spectra with signatures of whistler waves to the spectra analyzed in the corresponding radial distance bin for the slow (red) and fast (blue) solar winds.

Surprisingly, we observe that the percentage of whistler waves increases as we move away from the Sun in the slow and fast winds. However, the occurrence of whistlers in the fast wind is very low when compared to the slow wind. This is the first time that an increase in the percentage of whistlers with the radial distance has been shown.

Using the contribution from the B_y spectral component we have calculated the approximate rms amplitude of the fluctuations that are associated with whistler waves. The amplitude of a whistler wave is estimated as in the Equation (1), where the peak value (PSD_y) of the bump is multiplied with its respective frequency bandwidth (Δf), which gives us the mean square amplitude of the fluctuation. The square root of mean square amplitude can be interpreted as the amplitude of the fluctuation. Note that by this definition we do not have exact amplitudes of the wave, but rather the standard deviation of the associated fluctuating wavefield within 8 s of the integration of the spectrum.

$$\begin{eqnarray}&&\delta {B}_{y}=\sqrt{{\mathrm{PSD}}_{y}\ast {\rm{\Delta }}f}.\end{eqnarray} \tag{ 1 }$$

We have not removed the background turbulence contribution as the amplitudes of the considered whistler waves are large enough that the former represents a minor fraction of the total amplitude (<10%).

Figure 10 shows the radial variation of the mean power spectral densities corresponding to the peak of the spectral bumps for the case of the slow (red) and fast (blue) solar wind. Figure 11(a) shows the radial variation of the mean whistler amplitudes separately for the case of slow (red) and fast (blue) solar wind. We see that the whistler waves have higher amplitudes in fast than in slow wind and that there is no clear radial trend.

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Figure 11(b) shows the radial variation of the normalized whistler amplitudes (δB_y/B) for the slow and fast solar wind. We have normalized the whistler amplitude values with the closest mean magnetic field values. Interestingly we observe that whistler waves have larger relative amplitudes in the fast wind than in the slow wind. We do not observe a clear trend in the radial evolution for both types of the wind. We infer that observed whistler waves may be considered to be in the linear regime (δB/B < 0.1).

We have looked into different plasma parameters such as electron and proton density, electron and proton temperatures, electron core and halo anisotropies, magnetic field, electron and proton beta in order to identify a possible relation with the amplitudes of the observed whistlers. The core electron beta (β_ec) and halo electron beta (β_eh) shows the best possible correlation. For β_ec and β_eh we found a weak but statistically positive correlation of ∼0.40 with the relative amplitudes.

Using the cold plasma theory we have the dispersion relation of the right-hand circularly polarized whistler waves (Bellan 2006),

$$\begin{eqnarray}&&{n}^{2}=\displaystyle \frac{{c}^{2}{k}^{2}}{{\omega }^{2}}\approx 1+\displaystyle \frac{{\omega }_{{pe}}^{2}/{\omega }^{2}}{\left(\tfrac{{\omega }_{{ce}}\cos {\theta }_{{kB}}}{\omega }-1\right)}.\end{eqnarray} \tag{ 2 }$$

For parallel propagating waves with ω ≪ ω_ce we have

$$\begin{eqnarray*}&&\left(\displaystyle \frac{{\omega }_{{ce}}\cos {\theta }_{{kB}}}{\omega }-1\right)\approx \displaystyle \frac{{\omega }_{{ce}}}{\omega },\end{eqnarray*}$$

thus, the dispersion relation becomes:

$$\begin{eqnarray}&&{c}^{2}{k}^{2}\approx {\omega }^{2}+{\omega }_{{pe}}^{2}(\omega /{\omega }_{{ce}}).\end{eqnarray} \tag{ 3 }$$

For ω ≪ ω_pe, we obtain

$$\begin{eqnarray}&&{c}^{2}{k}^{2}\approx {\omega }_{{pe}}^{2}(\omega /{\omega }_{{ce}}).\end{eqnarray} \tag{ 4 }$$

Then, the phase velocity of the quasi-parallel whistler is given by

$$\begin{eqnarray}&&{v}_{\phi }=\displaystyle \frac{\omega }{k}=c\displaystyle \frac{\sqrt{\omega {\omega }_{{ce}}}}{{\omega }_{{pe}}}.\end{eqnarray} \tag{ 5 }$$

The same Equation (5) also holds for antiparallel whistler waves.

Assuming that the spectral bumps correspond to the parallel whistlers and the observed frequency in the satellite frame f is not much influenced by the Doppler shift and taking into account low relative frequency of the observed wave signatures (f ≃ 0.1f_ce), we can apply expression (5) to estimate v_ϕ for our data set.

Figure 12(a) shows the distribution of phase velocities (V_ϕ) determined from the central frequencies of the spectral bumps in the fast wind (blue) and in the slow wind (green). We observe that phase velocities of the slow and fast solar wind whistlers are in the same range 2 × 10² < V_ϕ < 2 × 10³ km s⁻¹ with the median around ≃10³ km s⁻¹. Most of the whistler waves have high phase velocities with respect to V_sw (not shown) and especially with respect to the Alfvén speed, see Figure 12(b) where one observes 9 < V_ϕ/V_A < 21 in the fast wind and 6 < V_ϕ/V_A < 21 in the slow wind. We have also looked into how the phase velocity of whistlers is compared to the electron thermal velocity (V_th,e), we found that 0.5 < V_ϕ/V_th,e < 0.8 (not shown).

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Studies such as Lacombe et al. (2014), Stansby et al. (2016), and Tong et al. (2019b) have suggested that heat flux instability might be acting when the whistlers are observed. Studies by Gary et al. (1999) and Wilson et al. (2013) have shown that A_h > 1 when whistler waves are observed. Recently, Tong et al. (2019b) have shown that if A_h approaches 1 and higher, the plasma becomes unstable for the heat flux instability. The reason for A_h significance will be discussed in detail in the coming sections.

The electron moments on Helios are not always available: there are large gaps. To increase the statistics we have considered the closest electron moment values to the observed whistlers. We kept the constraint of maximum 10 minutes from the nearest observed whistler. In Figure 13 we present (a) the normalized electron heat flux q_e∥/q₀ as a function of the parallel beta of the core of the electron distribution function ${\beta }_{e\parallel c};$ (b) the electron temperature anisotropy of the core A_c as a function of ${\beta }_{e\parallel c}$; and (c) the electron anisotorpy of the halo electrons A_h as a function of ${\beta }_{e\parallel h}$. Green dots correspond to all the electron moments measured during the availability of SCM data; the subset of events for which whistler waves are observed is shown in red.

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From Figure 13(a) we infer that WHF instability (WHF) is probably at work as most of the data follow the marginal stability threshold of $\tfrac{\gamma }{{\omega }_{{ci}}}={10}^{-2}$ (Gary et al. 1999). The q_e∥/q₀ values corresponding to the whistlers are observed close to the threshold. However, some values are below the threshold, which suggests that the WHF instability might not be the sole cause of the whistlers. Other than q_e∥/q₀ values A_h also has an important role as shown by Tong et al. (2019b): A_h close and higher than one may significantly affect the onset of the WHF instability. Indeed, for our data set, we find that for 88% of spectra with signatures of whistlers, A_h > 1. This ascertains the importance of A_h for the whistler wave observations and for their generation through WHF.

Figures 13(b) and (c) show A_c and A_h as functions of β_e∥c and β_e∥h respectively. We can infer that the core and halo whistler anisotropy instability is probably at work as most of the data are well constrained along the instability threshold contour $\tfrac{\gamma }{{\omega }_{{ce}}}={10}^{-2}$. The maximum growth rate contours for the WTA and firehose instability (FH) for the different cases of Kappa (κ = 3, 8) are taken from the work of Lazar et al. (2018), where the authors have considered a bi-Maxwellian core and a bi-Kappa halo of the electron distribution function. All the whistler points are closer to the whistler anisotropy instability threshold than to the firehose instability threshold.

The large number of events located in the vicinity of the WTA threshold supports the role of the core and halo WTA instability in generating whistlers. We notice that when whistlers are observed, for most of the cases we find A_c > 1.0 and A_h > 1.0. We found that nearly 75% of the A_c values corresponding to whistlers satisfied A_c > 1 and as mentioned before 88% of the A_h values satisfied A_h > 1. We can also observe that the higher the β, the lower the instability thresholds, therefore the higher the probability of the instability to develop for high β cases, as in the slow solar wind.

To summarize, Figure 13 suggests that the WHF and WTA instabilities are acting and that the three parameters q_e∥/q₀, A_c, and A_h could be responsible for the observation of whistler wave signatures in the Helios SCM spectra. However, due to the limitations on the data availability we cannot clearly state which type of instability is acting at which time and for how long, whether is it WHF or core WTA or halo WTA instability.

Previously, to have larger statistics we have considered the closest available electron moment values. Now, let us consider the electron properties that correspond to the observed whistler wave signatures. We first look for time intervals in which we observe at least 10 consecutive 8 s spectra with whistlers, i.e., time intervals that are at least 80 s long. Then, we identify the electron moments which are measured in those whistler intervals. By this method we ensure that moments measured are always during the presence of whistler waves, we found only 45 such intervals. Their q_e∥/q₀ values are spread below and above the heat flux instability threshold. However, for all the observed whistler waves, we observe that A_h > 1.01 in agreement with the studies of Gary et al. (1999), Wilson et al. (2013). Our observations also agree with the work of Tong et al. (2019b), who showed the importance of the halo anisotropy (A_h) for the onset of heat flux instability.

Even though we do not have exact values of q_e∥/q₀, A_h and A_c for all the observed spectra with the signatures of whistler waves, we can have a general idea of the conditions around when the whistler waves appear, i.e., usually in the slow wind. A_h and A_c values are relatively higher in the slow solar wind when compared to the adjacent fast solar wind, while q_e∥/q₀ values are lower. A glimpse of this transformation can be seen from an example interval shown in Figure 6.

We have also looked into whether relative amplitudes have any relation to the A_h and A_c values. From Figures 14(a) and (b) we can infer that (i) almost all the whistlers are observed when 0.9 ≤ A_c ≤ 1.2 and 0.9 ≤ A_h ≤ 1.4 and (ii) the relative amplitudes δB_y/B are not dependent on the value of A_c or A_h.

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Different effects can explain the prevalence of whistler waves in the slow solar wind. These can be observational (whistlers are present but cannot be observed), or physical (conditions are not met for generating whistlers, or existing whistlers are damped). Let us first consider the observational ones.

One of the major differences between fast and slow winds is the higher relative fluctuation level of the magnetic field in the former, which may therefore limit the visibility of whistler waves, as noticed by Lacombe et al. (2014). We performed tests to quantify the impact of the fluctuation level, see the Appendix. While this effect plays a role it is not sufficient to explain the much smaller number occurrence probability of whistlers in the fast solar wind. Another effect could be the enhanced Doppler shift in the fast wind. We also quantified this effect, which can be ruled out, see the Appendix. What is then left over are the conditions for whistler waves to be generated in the solar wind.

The two major sources of whistler wave generation in the solar wind are the WHF and WTA instabilities (Gary et al. 1994). Observational studies at 1 au show that whistlers can be generated by the WHF instability (Lacombe et al. 2014; Tong et al. 2019b). We have already shown in Figure 13(a) that the WHF instability conditions are indeed met for some whistlers in the inner heliosphere. Figures 13(b) and (c) show that the WTA instability conditions are also met for core and halo electron populations. From this we conclude that the ratios q_e∥/q₀, A_h, and A_c are important parameters for understanding the presence of whistler waves in the inner heliosphere. In addition, from the studies of Gary & Feldman (1977), Gary et al. (1999), Wilson et al. (2013), and Tong et al. (2019b), we understand that the ratio $\tfrac{{T}_{e\perp h}}{{T}_{e\parallel h}}$ is an important parameter for the onset of the WHF instability.

Gary & Feldman (1977), assuming a two bi-maxwellian velocity distribution function approximation, derived the dispersion relation for the whistler waves, the growth rate is given as

$$\begin{eqnarray}&&\displaystyle \frac{\gamma }{{{\rm{\Omega }}}_{i}}\propto \{({\boldsymbol{k}}\cdot {{\boldsymbol{v}}}_{0H}-{\omega }_{R})\displaystyle \frac{{T}_{e\perp h}}{{T}_{e\parallel h}}+| {{\rm{\Omega }}}_{e}| \left(\displaystyle \frac{{T}_{e\perp h}}{{T}_{e\parallel h}}-1\right)\},\end{eqnarray} \tag{ 6 }$$

where Ω_e is the electron cyclotron frequency, Ω_i is the ion cyclotron frequency, ${{\boldsymbol{v}}}_{0H}$ is the drift velocity for halo electrons, and ω_R is the real frequency.

In Equation (6), the first term on the right-hand side is the contribution of heat flux due to the drift of the electrons and the second term is the contribution of the anisotropy to the growth of the instability. Studies by Gary & Feldman (1977) have suggested that when the halo is nearly isotropic, i.e., when ${A}_{h}\simeq 1$, and the heat flux contribution term is positive, the WHF instability is driven. An example of this has been recently reported by Tong et al. (2019b) using the ARTEMIS data. Gary & Feldman (1977) have also suggested that when A_h > 1, the temperature anisotropy can strongly drive the WTA instability.

We have shown that some of the whistler spectra correspond well to the WHF instability threshold (see Figure 13(a)); however, the majority of them lie below it. In contrast more than 88% of events with the whistlers have A_h > 1 and may contribute positively to the growth of the wave. One should note that whistlers are not always observed when A_h > 1 (see Figure 13(c)). This can be due to several reasons, one of them being that the halo anisotropy alone cannot act as a source for the generation of whistler waves.

Similarly, we have not always observed signatures of whistler waves even if the ratio q_e∥/q₀ is above the WHF instability threshold (see Figure 13(a)). A similar observation was made by Tong et al. (2019a, 2019b). This leads to the conclusion that the heat flux itself might not always be able to generate whistler waves and a more complicated interplay between the heat flux and anisotropy is required.

Figure 15 shows the radial evolution of the normalized heat flux (q_e∥/q₀), calculated by Štverák et al. (2015) for the period of 1975 January 1 to 1979 May 10, separately for the slow and the fast solar wind. Interestingly, we did not observe any correlation to the observed properties in Figures 8 and 9. Contrarily, we have observed that q_e∥/q₀ values are lower in the slow wind as compared to the fast wind. Furthermore as a function of radial distance q_e∥/q₀ values did not show any clear trends in the fast wind, whereas in the slow wind we observe a slight decrease in q_e∥/q₀ with R. These observations might be related to the impact of the whistler waves on the dissipation of the heat flux through the scattering of strahl electrons (Štverák et al. 2015).

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Figures 16 and 17 present the radial evolution of the electron core and halo anisotropies, calculated by Štverák et al. (2009) for the period of 1975 January 1 to 1979 May 10. In Figure 16 we show the variation of the mean electron core and halo anisotropies, while Figure 17 shows the relative proportion of electron core and halo anisotropy values that are greater than one in the slow and fast winds as a function of radial distance. We see that the values of $\langle {A}_{h}\rangle$, $\langle {A}_{c}\rangle$ and also the percentage of electron core and halo anisotropy values that are greater than one (%A_c > 1,%A_h > 1) are higher in the slow solar wind throughout the inner heliosphere as compared to what we see in the fast solar wind. We also observe that as we move away from the Sun the value of $\langle {A}_{h}\rangle$, $\langle {A}_{c}\rangle$ and also the %A_c > 1, %A_h > 1 increases both in the slow and fast solar wind. These observations point to two important conclusions. First, the conditions for the whistler generation are more favorable in the slow wind compared to the fast solar wind. Second, the conditions for whistler generation are improving as we move farther from the Sun.

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In Figure 18 we show the percentages of data with A_c > 1 and A_h > 1 in different wind types as functions of the radial distance. Figure 18 helps in explaining why there is a monotonic decrease in percentage of whistler waves with the wind velocity as we have observed in Figure 8. There is a monotonic decrease of values A_h > 1 and A_c > 1 as the wind velocity increases and consequently we have a decrease in the percentage of whistler waves with velocity. We have also observed a similar monotonic decrease of $\langle {A}_{h}\rangle$ and $\langle {A}_{c}\rangle$ with the increase of wind velocity (not shown). We would also like to mention that we have calculated how the distance from the A_h and A_c values to their respective WTA instability thresholds are varying from 0.3 to 1 au. We found that as we move farther from the Sun, the A_h and A_c values are closer to the WTA instability thresholds, as the β_e∥c and β_e∥h are increasing with the distance and thresholds are lower for higher beta. Therefore, from this analysis too, we can understand that whistler generation conditions are improving as we move farther from the Sun.

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After discussing the observed signatures of whistler wave behaviors with the A_c and A_h values, some other questions can be raised. What can be the reason for the high A_c and A_h values in the slow wind as compared to the fast one? What can be the reason for the increase in A_c and A_h as we move farther from the Sun?

Štverák et al. (2008) showed that the value of A_c increases with the electron collisional age. Collisional age was reported to increase with radial distance and to be anticorrelated to the solar wind speed. This is in line with our observations for A_c. Collisions are effective at low energies (core electrons), but cannot be effective at high energies, such as for halo electron population. We conjecture that the increase we observe in the A_h(R) with R and also the presence of whistler waves are closely related to the broadening (scattering) of the strahl part of the electron distribution. The basis for this idea is related to the observed properties of the strahl of the electron distribution.

Using the Helios electron data, Berčič et al. (2019) has shown that the strahl is in general broader in the slow than in the fast solar wind, and that the width in both cases increases with radial distance. Strahl width was reported to increase with radial distance by Hammond et al. (1996), using data from the Ulysses mission, and by Graham et al. (2017), using data from Cassini.

Studies by Maksimovic et al. (2005) and by Štverák et al. (2009) show that the relative density of the halo and strahl vary oppositely over radial distance: while the strahl population is more pronounced closer to the Sun, the halo density increases farther from the Sun. These observations imply that the strahl is scattered into the halo over the radial distance.

The connection between the occurrence of whistler waves observed and the width of the strahl electrons—the strahl is broader and the occurrence of whistler waves is higher in the slow solar wind and farther from the Sun—suggests that the strahl might be scattered by the whistler waves. Kajdič et al. (2016) in his observational study at 1 au had shown that the strahl is highly scattered in the presence of narrowband whistler waves. In a different context, various numerical studies such as Vocks et al. (2005), Boldyrev & Horaites (2019), and Tang et al. (2020) had shown that the interaction of electron VDF with the whistler wave turbulence could lead to the scattering of energetic strahl electrons and to the formation of an electron halo.

From the observed relation of whistler waves occurrence and the A_h values, we conjecture that the observed changes in A_h(R) might be related to the scattering of the strahl with the whistler waves.

An important and still open question is how the momentum is transferred in this wave–particle interaction, to result in an increase of A_h.

Kajdič et al. (2016) observed a direct link between the scattering of the strahl and the presence of whistler waves. These whistler waves can interact with the strahl electrons through electron cyclotron resonance. In the following, we make rough estimates of the energy ranges in which the whistler waves are able to interact resonantly.

The equation of electron cyclotron resonance is

$$\begin{eqnarray}&&\omega -{\boldsymbol{k}}\cdot {\boldsymbol{v}}={\omega }_{{ce}},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&\omega -{kv}\cos {\theta }_{{kv}}={\omega }_{{ce}},\end{eqnarray} \tag{ 8 }$$

where, ω is the frequency of the wave, k is the wavenumber of the whistler wave, v is the velocity of the resonant electrons, θ_kv is the angle between the whistler wave wavevector and the velocity vector of the resonant electrons and ω_ce is the electron cyclotron frequency.

Let us consider the case when resonant electrons are field aligned, as the strahl electrons (Owens et al. 2017). Most of the whistlers observed to date are quasi-parallel or antiparallel (Lacombe et al. 2014; Stansby et al. 2016; Tong et al. 2019b). The condition for resonance is only satisfied when whistler waves travel opposite to the electrons as ω_ce is always greater than the frequency of the observed whistler waves. Therefore, we can understand that we have resonance when electrons travel in the opposite direction as whistlers. Assuming this condition, i.e., $\cos {\theta }_{{kv}}\approx -1$, we have

$$\begin{eqnarray}&&\omega +{kv}={\omega }_{{ce}},\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{kv}}{\omega }=\displaystyle \frac{\left({\omega }_{{ce}}-\omega \right)}{\omega },\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&v=\left(\displaystyle \frac{{\omega }_{{ce}}}{\omega }-1\right){v}_{\phi }.\end{eqnarray} \tag{ 11 }$$

Equation (11) gives us the velocity of the electrons that can be resonant with the whistler waves (under the assumption of the opposite propagation between the wave and the electrons). The kinetic energy of these resonant electrons,

$$\begin{eqnarray}&&E=\displaystyle \frac{1}{2}{m}_{e}{v}^{2},\end{eqnarray} \tag{ 12 }$$

is shown in Figure 19. Most of the resonant electron energies are above 50 eV, which are the energies representative of the strahl and the halo populations. We conclude that the observed whistler waves might be able to scatter strahl and halo electrons through the electron cyclotron resonance. But how is the energy transferred in this type of interaction?

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Veltri & Zimbardo (1993) have shown that when resonantly interacting whistler waves scatter the electrons, electron energy is transferred from the parallel to the perpendicular direction. Which could be understood as an increase of the temperature perpendicular to the magnetic field direction. On the other hand scattering the beam-like population toward higher pitch-angles could be interpreted as scattering the strahl into the halo component.

We explain this with a feedback mechanism:

• 1.  
  Whistler waves scatter the strahl population.
• 2.  
  Scattered strahl increases the value of A_h.
• 3.  
  Increases in the A_h favors the creation of the whistler waves and on.

Therefore, we can hypothesize that whistler waves could scatter the strahl and create a self-sustaining mechanism that could explain the observed whistler properties.

However, the A_h values cannot keep on increasing without a limit as they are bounded by WTA instability thresholds. This is the reason why we believe that there is a saturation of A_h values in the slow wind farther from the Sun as shown in Figures 16 and 17.

Recently Tang et al. (2020) has shown that whistler wave turbulence interaction with suprathermal electrons can lead to the formation of halo from strahl and could explain the observed opposite variation in the relative density of the halo and strahl with radial distance. In a future paper, we would like to verify our suggested feedback mechanism by modifying the kinetic transport model used in Tang et al. (2020). We would like to treat halo and strahl as separate populations and study the interaction of intermittent narrowband whistler waves with strahl and their influence on halo population.

Even though our explanation appears to be a reasonable one, there are some issues related to it. Our calculations are based on the assumption that $\cos {\theta }_{{kv}}\approx -1$ for the observed whistlers and resonant electrons. This might not always be true. Another case is that the whistler waves could be oblique. Recently, Vasko et al. (2019) using the linear stability analysis has shown that highly oblique whistler waves drive the pitch-angle scattering of strahl electrons, and in turn isotropize the halo and also suppress the heat flux. However, as we discuss in the 1, from the previous observations in the solar wind we know that the oblique whistlers are usually associated with the large scale discontinuities (Breneman et al. 2010). In the free solar wind, narrowband whistlers are mostly quasi-parallel, e.g., (Lacombe et al. 2014; Kajdič et al. 2016; Stansby et al. 2016).