Roles of hot electrons in generating upper-hybrid waves in the earth's radiation belt

The Earth's radiation belt, known as the Van Allen belt, is composed of energetic electrons and ions that circulate around the Earth's dipolar magnetic field lines and persist for extended periods. Actually, the radiation belt consists of at least two belts, and possibly more. In the literature, low-frequency electromagnetic (EM) waves, namely, ultra low frequency waves (ULF), whistler-mode chorus, plasmaspheric hiss, EM ion-cyclotron (EMIC), and magnetosonic (MS) waves known as equatorial noise (and sometimes multiple harmonic electron cyclotron waves) receive much attention as efficient acceleration or loss mechanisms for electrons up to relativistic energies.^1–11 However, these low-frequency waves are usually associated with active magnetospheric conditions such that they do not always occur, and when they do, their occurrence is often limited to spatially localized regions. In contrast, high-frequency quasi-electrostatic (ES) fluctuations in the upper-hybrid frequency range are a constant and pervasive feature in the Earth's radiation belt environment.¹² Such upper hybrid frequency waves had been observed in various regions of the earth or the planetary magnetospheres such as Io torus.^13,14 They suggest that the origin of thermal upper hybrid band might be the hot electrons of $Th∼$ 1 keV in the Io torus.

The twin Van Allen Probes carry identical scientific suites of instruments, which include the Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) suite's Waves instrument that measures plasma waves. The instrument measures three magnetic and three electric field components of plasma waves in the frequency range of 10 Hz to 12 kHz. The Waves instrument also measures the electric field in the higher frequency range of 10 to 500 kHz. The high-frequency instrument is utilized to measure the ES upper-hybrid range fluctuations. In the literature, the ubiquitous upper-hybrid waves are utilized for determining the electron density.¹² This is because one can compute the plasma frequency from the upper hybrid frequency (and the cyclotron frequency), which directly gives the electron density.

The persistent upper-hybrid fluctuations detected by the Waves instrument are analogous to the quasi-thermal noise, or Langmuir fluctuations, in the solar wind,^15–18 except that for the radiation belt environment the assumption of weakly magnetized or unmagnetized plasmas must be relaxed. Strictly speaking, quasi-thermal noise in the solar wind is not exactly the same as Langmuir fluctuations. Langmuir waves are a normal mode satisfying the dispersion relation, i.e., functional relationship between the wave frequency and wave vector, whereas quasi-thermal noise contains contributions from non-eigenmodes as well as the normal mode, that is, from all frequencies and wave vectors not necessarily satisfying the dispersion relation. Typically, Langmuir fluctuations result from an instability caused by a bump-on-tail electron distribution [see, e.g., Ref. 19 while quasi-thermal noise arises from isotropic thermal electron distribution. Nevertheless, they are similar in that the peak intensity occurs at the electron plasma frequency. This is because the quasi-thermal noise is enhanced in the vicinity of the Langmuir wave dispersion relation.

For a magnetized plasma, one can have a quasi-thermal emission at the upper hybrid frequency when the plasma is stable and also an instability of upper hybrid waves caused by an unstable electron distribution such as a loss cone or when the temperature anisotropy exists. The basic theory of quasi-thermal noise is that of spontaneous emission by thermal plasma particles,²⁰ which was extended to magnetized plasmas by Ref. 21, for example. In the recent paper,²² we formulate the theory of ES spontaneous emission in magnetized plasmas, which can be applicable for the radiation belt environment. In the solar wind, quasi-thermal noise spectroscopy, pioneered by Ref. 15, is utilized to diagnose various properties of solar wind electrons including the number density and temperature. By analogy, it appears quite natural that the upper-hybrid fluctuations for the radiation belt could also be employed to unveil the basic properties of underlying electrons, which include not only the number density but also thermal energy. Here, we should caution the readers that our focus is on quasi-thermal emission of upper-hybrid fluctuations by underlying electrons with isotropic distribution, that is, stable situation.

There is a long history of both observational and theoretical studies of non-thermal emissions of harmonic electron cyclotron emissions in the magnetosphere, known in the literature as $(n+1/2)fce$ emissions including the upper-hybrid band. The first detection of intense upper-hybrid waves in the plasmasphere is performed by the authors of Refs. 23–26, who reported that these emissions are related to unstable electron distributions. Some other early works on the magnetospheric upper-hybrid emission and the related $(n+1/2)fce$ bands include.^27–30 However, all these works are related to instability by electrons containing loss cone or temperature anisotropy. The present work is similar to these early papers, but the main difference is that we are interested in quiet-time situations where the electron distribution is quasi-isotropic and no instability mechanism is forthcoming. The methodology to be adopted and discussed in the present paper is similar to that developed by authors of Refs. 13, 14, and 21 in that these authors already developed the theory of spontaneous emission of electrostatic fluctuations in magnetized plasmas and applied the result to the Earth and planetary magnetosphere. Even though the basic methodology is available in principle, the application to the Earth's radiation belt with physical parameters relevant for such an environment has not been made yet.

In the present paper, the theory of thermal fluctuation in magnetized plasmas is applied, combined with data analysis, in order to elucidate the inter-relationship between the measured upper-hybrid fluctuations and the underlying electrons that emit and reabsorb these fluctuations. Upon close comparison with the available results in the literature,^13,14,21 we find that earlier theories and formalisms assume that the contribution from the plasma normal modes, that is, Bernstein modes, dominates the spontaneous emission spectrum. In contrast, the theoretical formalism discussed in Ref. 22 and utilized in the present paper includes both Bernstein modes and fluctuations that do not satisfy the linear dispersion relations, i.e., the non-normal modes.

The electron contents in the Earth's inner magnetosphere may be viewed as being comprised of various electron populations occupying distinct energy ranges. Cold ionospheric electrons (a few eV) mainly contribute to the background electron density. Suprathermal electrons (10–500 eV) are responsible for Landau damping of electrostatic waves and contribute to both the density and temperature. Hot electrons (1–30 keV) are known as the main source for wave generation owing to the thermal and loss-cone anisotropy and also contribute mainly to the temperature. Seed population (10–100 keV) is present mainly during substorm injections, which are accelerated up to relativistic energies. Relativistic electrons (>100 keV) result from particle acceleration processes, but they do not influence average plasma characteristics. Each of these electron populations has different spatial distribution along L-shell and Magnetic Local Time (MLT) and also has characteristic energy spectra, making a complicated mixture of electron distributions over various energy ranges.^31–39 

In this paper, for the sake of simplicity, we simply categorize such various electron species into just three energy ranges: cold electrons (< eV), hot electrons (10–50 keV), and relativistic electrons (100 keV–1 MeV). Of particular interest is the relative contribution of cold electrons characterized by the eV range of thermal energy versus hot electrons with tens keV energy but with typical density, which is of the order of $10−4$ compared to that of the cold electrons. As will be discussed in detail, subsequently, it turns out that hot electrons, even though their number density is very low, might be responsible for the peak wave intensity of upper-hybrid frequency.

This also implies that upper-hybrid fluctuations may be important for maintaining the steady-state energy spectrum of hot electrons, especially during inactive periods. The idea of steady-state or dynamic equilibrium between hot electrons and ES Langmuir fluctuations was recently put forth for the solar wind electrons.^40,41 It has been known since the dawn of space age that the solar wind electrons near 1 AU feature a suprathermal population.^42,43 As mentioned, the solar wind near 1 AU is also replete with the quasi-thermal noise spectrum with peak wave intensity near the plasma frequency. This has been described for decades by Meyer-Vernet and colleagues.^15,17,18 However, the association of the two separate features had not been made until recently when the authors of Refs. 40 and 41 put forth a steady-state model in which quasi-thermal Langmuir wave fluctuations and suprathermal electron population interchange momentum and energy while maintaining an overall dynamical steady state. Quite an analogous model may be envisioned for the quiet-time population of radiation belt electrons undergoing steady-state wave-particle interaction with the upper-hybrid fluctuations.

As a first step in establishing and exploring the possibility of such a mutual relationship between waves and particles, the present paper first analyze the role of cold versus hot electrons in the upper-hybrid fluctuation generation in the earth's radiation belt. In order to isolate the mutual relationship between the upper-hybrid fluctuations and hot electrons, we deliberately choose time intervals where the low-frequency whistler-mode chorus is absent, i.e., we consider relatively inactive and quiet time periods. In the remainder of this paper, we elaborate on the detailed data selection criteria, data analysis methods, comparison between data and theory, and theoretical interpretation.

The organization of the present paper is as follows: In Sec. II, we outline the data selection criteria and procedure. We then proceed to analyze the data in order to extract useful information and reach a physical interpretation. Section III briefly outlines the theory of spontaneous emission of ES harmonic electron cyclotron fluctuations including the peak emission near upper-hybrid frequency. We next interpret observed upper-hybrid fluctuation spectrum in terms of theoretical model. Finally, Sec. IV summarizes the findings and the implications of the present paper.

The Van Allen Probes are two identical satellites and were launched on August 30, 2012. Their orbital inclinations are $10°$ and they cover 500 km to 30 600 km. The orbital period is about 9 h. The two satellites have the same orbit but different transit times.^44,45 We used the Electric and Magnetic Field Instrument and Integrated Science (EMFISIS) investigation onboard the Van Allen Probes. The EMFISIS instrumentation suite provides measurements of wave electric and magnetic fields as well as DC magnetic fields for the twin Van Allen Probes (Radiation Belt Storm Probes, or RBSP) mission. Waves instrument provides a comprehensive set of wave electric and magnetic field measurements covering the frequency range from 10 Hz up to 400 kHz. We analyzed HFR spectral burst Level 2 data measured from a single axis AC electric field spectrum.⁴⁶ The Helium, Oxygen, Proton, and Electron (HOPE) instrument on Van Allen probes,⁴⁷ part of the Energetic particle, Composition, and Thermal plasma (ECT) instrument suite,⁴⁸ is a time of flight mass spectrometer measuring species differentiated ions and electrons from a few eV to 50 keV on alternating 10 s accumulations. Distribution functions and electron densities are computed directly from these data and presented in this paper.

Our emphasis is on the upper-hybrid frequency fluctuations that are persistent in the radiation belt. We reiterate that while the upper-hybrid fluctuations are used just as a passive diagnostics for determining the cold electron density, their possible role in influencing the higher energies' electrons has not been emphasized in the literature. In the present paper, we wish to isolate the effect of upper-hybrid fluctuations from other waves such as whistler-mode chorus and EMIC waves so that we may investigate the inter-relationship between the upper-hybrid waves and electrons. Consequently, we choose quiet (inactive) magnetospheric periods where chorus or EMIC emissions are either absent or minimal.

We selected two cases of quiet periods where the chorus wave is absent. For the sake of comparison with the two selected quiet-time conditions, we also showcase one example of an active period. Of the two quiet cases, we chose one example in which narrowband upper-hybrid fluctuations are accompanied by multiple-harmonic cyclotron waves both below and above the upper-hybrid frequency. The second case shows narrow upper-hybrid frequency emission with weak or no apparent adjacent cyclotron harmonic modes. The third example will be the active period where whistler-mode chorus and all higher-harmonic electron-cyclotron modes are excited up to the upper-hybrid frequency and even higher harmonics.

The momentum distribution function for hot electrons for each case will be also analyzed, as well as the relative density ratio between the hot and cold electron components. Such information will be utilized in order to characterize the dependence of the wave spectrum on the electron properties.

Figure 1(a) (top panel) shows the electric field spectrum for the first case. The data were taken on May, 8, 2013. Figures 1(a) and 1(b) are EMFISIS High Frequency Receiver (HFR) and Waveform Receiver (WFR) spectra from the VAP-A satellite. In this case, upper hybrid waves are detected throughout the entire interval of data collection. The spectrum also features multiple-harmonics electron-cyclotron waves, or $(n+1/2)fce$ emissions near 15:55, as it is commonly known in the literature, which can be seen to take place below the upper-hybrid frequency (indicated by the bold white line). The green patch of broadband wave emission with frequency much higher than the upper-hybrid frequency, f_UH, is probably of remote origin (e.g., Auroral Kilometric Radiation, or AKR). Upon focusing on the wave spectrum in the vicinity of 15:55, we may estimate that the upper-hybrid frequency is roughly given by $fUH∼40$ kHz.

Figure 1(b) (middle panel), which shows the electric field spectrum in the low-frequency band, clearly indicates the absence of whistler-mode chorus wave throughout the entire interval. This event, therefore, is a good example that may reveal the interdependence of high-frequency upper-hybrid waves and hot electrons, since there are no low-frequency waves that may influence the hot electrons. We thus chose this example for the study of how electrons may influence the upper hybrid wave emission. It is well known that the spatial region of the radiation belt slightly overlaps with plasmasphere composed of cold electrons [see, e.g., Ref. 49, but the radiation belt itself is characterized by relativistic electrons (i.e., MeV electrons). Note that one-half of the electron cyclotron frequency is indicated by a solid line in Fig. 1(b). Near 15:55, it is approximately given by $fce/2∼2$ kHz. This implies that the ratio of plasma to electron-cyclotron frequencies is $fpe/fce∼10$⁠.

Figure 1(c) (bottom-left) is a plot of cold electron density profile (blue), which is taken from the EMFISIS instrument, indicated by a bold white line at the top panel, and the hot electron density with energy greater than 10 keV (red). The latter is deduced from the HOPE instrument of VAP-A. The hot electron density with energy greater than 10 keV in Fig. 1(c) is directly derived from the 10–50 keV electron from the HOPE instrument. For an exact calculation of this hot electron density, all electron energy channels of greater than 10 keV from HOPE, MagEIS, and REPT should be considered in principal. But the electron fluxes from REPT and MagEIS are negligible in this low energy range. We found that the electron density in the low energy range of our interest is determined by mostly from HOPE data (not shown here).

Figure 1(d) shows an accompanying momentum distribution function for relativistic electrons, averaged over five minutes near May 8, 2013, 15:55. According to Fig. 1(c), the ratio of hot-to-cold electrons, $Nh/Nc$⁠, for the time period at 15:55, where multiple harmonic electron-cyclotron waves accompany the upper-hybrid frequency waves, is of the order $∼10−3$ to $∼5×10−3$⁠. Here, we focus on the time interval between ∼13:00 and ∼16:00. Figure 1(d) shows the two-dimensional phase space distribution (PSD), or equivalently, momentum distribution function of energetic electrons, obtained from MagEIS from 100 keV to 1 MeV. The vertical axis represents the field-aligned momentum component while the horizontal axis represents the perpendicular momentum normalized by the momentum of a 1 MeV electron. The PSD is averaged in gyrophase. It is interesting to note that the energetic electrons are largely isotropic with only weak indication of the loss cone judging from following the white contour lines. Of course, electron momentum measurement in space is difficult owing to fast time scales involved and relatively slow sampling rates. Consequently, it is possible that the measured electron distribution function may represent a relaxed state of the initial loss cone distribution. However, such an “initial” state is difficult to measure. On the basis of the time averaged PSD, we will simply assume that only a very weak loss cone exists over the time interval of interest to us. Whether such a weak apparent loss cone is capable of driving the instability or has not been examined, however, but given the fact that the hot electron density is much lower than that of the background electrons, $Nh/Nc$ on the order of $10−4$ to $10−3$⁠, it is unlikely that the loss cone driven instability will substantially contribute to the upper-hybrid wave excitation. Of course, if there is a more pronounced “initial” loss cone, then instabilities may be excited. The approach taken in the present paper is to obtain the observed quasi-isotropic electron distribution at face value and perform the spontaneous emission calculation based on thermal distribution of electrons.

Figure 2, which shows the data taken on August 2, 2013, corresponds to the case where a simple single upper hybrid band is manifested, accompanied by weak or no multiple harmonic cyclotron waves. The format is the same as for Fig. 1. Figure 2(a) shows that upper-hybrid emission is a persistent feature with no apparent higher or lower cyclotron harmonic modes. The upper-hybrid frequency in the interval 5:00 to 8:00 is given approximately by $fUH∼30$ kHz or so. In contrast, Fig. 2(b) shows that one-half of the electron cyclotron frequency is on the order of 2 kHz or slightly higher. In this case, therefore, the ratio $fpe/fce$ is as low as 7 or 8, but it can also be higher. Figure 2(b) shows the absence of whistler-mode chorus waves in this interval. Figure 2(c) indicates that for this event, the density ratio between the energetic and background (or relatively cold) electrons is somewhat lower than that of Fig. 1(c), namely, $∼10−4$ or at most $2×10−4$ over the time interval ∼04:00 and ∼08:00. Figure 2(d) shows that the overall particle intensity is substantially lower than Fig. 1(d). For instance, if one focuses on the contour depicting normalized momentum corresponding to 0.5, it can be seen that the intensity of PSD in Fig. 1(d) is on the order of 10⁶⁶. In contrast, for Fig. 2(d), the same is substantially below 10⁶⁶, and is in fact, intermediate to 10⁶⁵ and 10⁶⁶. Notice that the phase space distribution is again characterized by near isotropy with only a weak loss cone signature.

In Sec. III, we will make use of the theory of spontaneous emission for magnetized plasmas, which is fully discussed in the recent paper by Ref. 22, in order to explain the differences between Figs. 1 and 2. Before we do that, however, let us showcase an example of active magnetospheric conditions. By “active period” we mean the existence of wave activities in the low-frequency whistler-mode chorus band. As noted, we are not interested in such a case, since the focus of the present paper is to understand the quiet-time conditions, but for transient events such as sudden acceleration and/or loss of radiation belt electrons by wave-particle interaction, the active conditions are of interest.

A typical active event is shown in Fig. 3. The data are taken on November 20, 2012. Shown in Fig. 3(a) are multiple harmonics of electron-cyclotron waves above and below the upper hybrid wave frequency throughout the entire interval between ∼10:00 and ∼15:00 or so. The white line indicates the upper hybrid band frequency, f_UH. We can consider some possibilities on the causes of the narrow bands above the upper hybrid bands. Where the background changes across the entire band, the instrument might be saturated, and many higher frequency banded emissions are simply a result of clipping of the very large waveforms. Alternatively, the weaker banded emissions above, f_UH, may be narrowband continuum radiation (sometimes called the escaping continuum radiation or myriametric radiation). Another possibility is that it might be the Fq resonances. It is important to note that the active case is not the focus of the present paper. It is shown only for the sake of comparison with quiet time conditions.

The middle panel, Fig. 3(b), displays the excitation of chorus waves. Recall that our criteria of quiet time is the absence of wave activities in whistler chorus wave frequency range. According to this criteria, Figs. 1 and 2 depict quiet times, whereas Fig. 3 is an active period, for which the theory of spontaneous thermal emission probably is not applicable. Upon extrapolating the electric field spectrum in Figs. 3(a) and 3(b), it is quite evident that when the chorus wave is excited, the emission is accompanied by all higher harmonic cyclotron waves up to the upper-hybrid frequency and even above.

For the event on November 20, 2012, Fig. 3(c) shows that the hot electron density is greatly enhanced over the other two cases, such that the ratio $Nh/Nc$ is as high as $∼10−2$⁠. Note that the same ratio is $∼10−3$ in Fig. 1(c) and $∼10−4$ in Fig. 2(c). The enhanced hot electron density may be understood in terms of the particle injection from the nightside during substorms, which excite the chorus wave. The phase space distribution (PSD) or equivalently velocity distribution function (VDF) in Fig. 3(d) shows that while the overall isotropic feature is not too different than the previous two quiet-time examples, one may nevertheless notice a slight enhancement in the perpendicular temperature, and slightly more noticeable loss cone feature. For this case, which represents a typical active condition, the existing theory of $(n+1/2)fce$ emissions including the upper-hybrid waves by instability mechanism may be applicable.^23–30 

Figure 4 shows the comparison of energy spectra between the three cases. Energy spectra are five-minute averages at May 8, 2013, 15:55 (red), August 2, 2013, 05:30 (blue), and November 20, 2012, 12:00 (black). The blue line corresponds to the case shown in Fig. 2 where a single upper-hybrid emission line is present with no accompanying harmonic bands and with the absence of whistler-mode chorus emission. The red curve represents the quiet time condition shown in Fig. 1, where the upper-hybrid line is more intense than in Fig. 2, and where multiple harmonic electron-cyclotron modes are also present. Note that the electron energy spectrum in the case showing multiple-harmonic emissions accompanying the upper-hybrid emission, that is, Fig. 1, has higher intensity than that of Fig. 2, which is an indication that the upper-hybrid and harmonic emissions might be directly related to the presence of hot electrons. For comparison, we also plot the energy spectrum for the active case where chorus wave and all higher-harmonic modes are excited, in the black curve. Note that the active time energy spectrum has higher intensity for the low energy range but the high energy part of the spectrum is actually lower than the two other quiet-time cases. The softening of the energy spectrum is probably related to the injection of tens to hundreds keV electrons during magnetic substorms. We are not interested in analyzing the active case here, as there are many publications that deal with such transient events. Instead, our focus is on understanding the role of quiet-time hot electrons and how they are related to the emission of upper-hybrid fluctuations. The importance of the results shown in Fig. 4, especially the energy spectra for quiet times, is that the thermal spread, or effective temperature, of the hot electrons is comparable for both cases and that only the number density shows variation. This information will be useful in the theoretical analysis to be discussed in Sec. III.

In this section, we briefly summarize the theory of spontaneous thermal emission of upper-hybrid waves in magnetized plasmas. The full details can be found in Ref. 22. Even though the basic formalism is already available in the literature,²¹ the paper of Ref. 22 discusses the basic formulation since the theory of spontaneous thermal emission for magnetized plasmas is generally not very well known in the community. The present section is a brief overview of the result, in order to facilitate the interpretation of the results and to make comparison between the observation and theory of upper-hybrid fluctuations. We defer the systematic discussion to Ref. 22, but the basic theory of spontaneous thermal emission of electrostatic fluctuations is well known. See, for example, Refs. 15 and 17 for unmagnetized plasmas and Ref. 21 for magnetized plasmas. Note that the present formalism is basically the same as that formulated by Ref. 21 except that²¹ eventually focused on the fluctuations from plasma normal modes which satisfy the dispersion relation. In contrast, the present paper discusses fluctuations that contain both plasma normal modes and non-normal modes. For high frequency (higher than characteristic ion frequencies), the electrostatic fluctuation energy density in the spectral form is given by

where the longitudinal dielectric constant for the magnetized plasma is given by

In the above equation, $γ=(1+p2/me2c2)1/2$ is the relativistic Lorentz factor, $ωe=eB0/mec$ is the electron cyclotron frequency, and e, B₀, m_e, and c are the unit charge, ambient magnetic field intensity, electron rest mass, and speed of light in vacuo. In Eqs. (1) and (2), $Jn(x)$ is the Bessel function of the first kind of order n. The total electron velocity distribution function (VDF) may be composed of several components, but in what follows, we consider a dense background cold electrons population and a tenuous but hot population.

Note that the observed energetic electron VDF is characterized by an inverse power-law tail distribution for energies greater than 500 keV, $fe∝p−α$ (see Fig. 4). For the sake of simplicity, however, we will adopt a hot Maxwellian distribution for the energetic electrons. Figures 1–3 show the cold electron density but they do not display the phase space distribution (PSD), or equivalently, velocity distribution function (VDF). We nevertheless adopt the Maxwellian model for the background cold electrons as well. To simplify the matter further, let us also restrict ourselves to non-relativistic formalism (γ = 1), and we ignore small pitch-angle anisotropies, i.e., the weak loss cone feature shown in Figs. 1(d) and 2(d). Under such a set of simplifying approximations, the total electron VDF is modeled by a combination of “cold” background (denoted by c) and “hot” (designated with h) Maxwellians

where $αc2=2Tc/me$ and $αh2=2Th/me$⁠. With this model, we have

where $n0=nc+nh$ is the total electron number density, and the total plasma frequency is defined by $ωpe=(4πn0e2/me)1/2$⁠. The function $In(x)$ is the modified Bessel function of the first kind of order n, and the longitudinal dielectric constant is now explicitly written as

In the above equation, $Z(ζ)$ is the plasma dispersion (Fried-Conte) function.

In terms of dimensionless quantities

the normalized electric field intensity spectrum is given by

As we noted in describing the observations, the parameters relevant to Fig. 1 may be $ρ=ωpe/Ωe∼7$ and $δ=nh/n0∼10−3$⁠, but the temperature ratio $τ=Th/T0$ is uncertain. It turns out, however, upon comparison with observations, we may deduce this parameter, and we will later substantiate the choice of $τ=102$⁠. For the second case displayed in Fig. 2, we choose $ρ∼8$ and $δ∼5×10−5$⁠. Again, the parameter τ is uncertain, but later comparison with observations will show that the choice of $τ∼106$ appears to be appropriate. We will make use of the above input parameters to compare and explain the observations, but before we do that, let us demonstrate the major point of the present paper, namely, it is the hot electrons that might substantially contribute to the upper-hybrid intensity.

In Fig. 5, we choose $ρ=ωpe/Ωe=8, δ=nh/n0=0$ and $δ=10−4$⁠. We also choose τ = 100 for both cases, but since for δ = 0 there are no hot electrons, the choice of τ = 100 is moot. In any case, Fig. 5 shows the electrostatic wave intensity as a function of normalized frequency. The intensity is integrated over q, and we chose $θ=85°$⁠, since the upper hybrid waves are a quasi perpendicular mode. In Fig. 5, the blue curve represents the case without hot electrons, while the red curve depicts the case including the tenuous but hot electrons. The peak intensity is associated with the upper hybrid frequency, and one can also discern multiple harmonic cyclotron modes below the upper hybrid mode. Figure 5 shows that the peak intensities for the harmonic cyclotron modes are largely determined by the dominant background cold electrons (blue and red curves coincide). However, for the intensity of the upper-hybrid frequency, one can see that the presence of hot electrons with $δ=10−4$ and whose temperature is 100 times higher than the background cold electrons clearly dominates. Note that the upper-hybrid intensity depicted by the red curve is an order of magnitude higher than the corresponding blue curve. Note also that for the higher frequency domain characterized by $ω>ωUH=ωpe2+Ωe2$⁠, where ω_UH is the upper-hybrid frequency, the absence of hot electrons causes the intensity to rapidly fall away, while the hot electrons lead to enhanced wave intensity. Such a feature is reminiscent of the quasi-thermal noise spectrum associated with Langmuir fluctuations.^15,17,18

We reiterate that the purpose of Fig. 5 is to theoretically demonstrate that the presence of hot electrons not only leads to enhanced upper-hybrid intensity but also alters the spectral shape such that the emission spectrum above f_UH exhibits a broad extension, reminiscent of the quasithermal noise spectrum.^15,17,18 We now make use the basic theory in order to interpret actual observations. An example of such an analysis is presented in Fig. 6. In Fig. 6, the wave power spectra shown in Figs. 1 and 2 are plotted as a function of frequency. The case corresponding to Fig. 1 (May 8, 2013, 15:55 UT event) is plotted with the blue curve, while the black curve corresponds to that of Fig. 2 (August 2, 2013, 05:30 UT). In Fig. 6, we adopt two-minute averages for each case. For instance, we collected the data during 15:54–15:56 and took an average in order to get the blue line. We did the same for the second case. For May 8, 2013, 15:55 UT event, we chose $ρ=ωpe/Ωe∼7$ and $δ=nh/n0∼10−3$⁠, but since the temperature ratio $τ=Th/T0$ is uncertain, we determined this ratio by trial and error. It turns out that the choice of $τ=102$ produced a reasonable fit. In short, the choice of parameters is

The theoretical curve is superposed over the observed spectrum, by making use of a bold cyan line for the May 8 event. For the second case of August 2, 2013, 05:30 UT, we chose $ρ∼8$ and $δ∼5×10−5$⁠, but again, the parameter τ is uncertain. We thus employed the trial and error method in order to determine that $τ∼106$ is an appropriate choice. To repeat, the choice of parameters for this case is

The theoretic result is depicted by the magenta curve. As the reader may appreciate, a comparison between the theory and observation produces apparently quite reasonable fits. We caution the readers, however, that the fit was achieved by simply employing the thermal equilibrium theory based upon observed quasi-isotropic distribution. We leave open the possibility that the observed distribution may actually represent the relaxed state of initially unstable loss cone distribution. However, such an issue is beyond the scope of the present paper. Our purpose here is to simply adopt the input values from the specific time intervals of interest to us and apply the theory of spontaneous emission in magnetized plasmas such as the Earth' radiation belt. With such caveats, plots such as those in Fig. 6 demonstrate that the upper-hybrid fluctuations may be useful in unveiling the underlying parameter associated with the radiation belt environment, in this case, the temperature ratio between the hot and cold background electrons, $τ=Th/Tc$⁠. We should note, however, that while the cyclotron harmonics below the upper-hybrid frequency peak $f<fuh$ agree quite reasonably, especially for the case represented by cyan curve, the agreement does not appear so well for $f>fuh$⁠. At the present moment, the precise reason for such a discrepancy is not entirely clear.

Electrostatic upper-hybrid fluctuations are a ubiquitous feature in Earth's radiation belt. In the literature, such a signature is used for the purpose of determining the cold electron density, but the present paper shows a possibility that the intensity of upper-hybrid fluctuations might be predominantly emitted by hot electrons. Note, however, that this does not change the frequency of the band; thus, the density measurement is unaffected by the presence of tenuous hot electrons. The importance of this finding may be that for steady-state quiet time conditions, the hot electrons and upper-hybrid waves may form a dynamical steady-state equilibrium solution via constant wave-particle interaction.

As a first step in exploring and understanding such a physical situation, in the present paper, we first selected and analyzed the data from Van Allen Probes, where in two cases, upper-hybrid frequency fluctuations are ubiquitously present while the whistler-mode chorus waves are absent, and in one representative case showing the excitation of whistler-mode chorus and all higher-harmonic waves. We define the quiet time by the absence of whistler-mode chorus activity. The two quiet time examples are theoretically analyzed by means of spontaneous thermal emission theory for magnetized plasmas in order to show that the predominant contribution to the observed wave intensity comes from tenuous hot electrons. The two cases are further analyzed and it was shown that theoretical reconstruction of the observed wave spectra agrees favorably with observations at least in these two events. Note that the active period characterized by the presence of whistler-mode chorus range fluctuations likely involves instabilities such that the spontaneous emission theory in not applicable.

The implication for general astrophysical plasmas is that similar conclusions may be applied to any magnetized planets possessing a radiation belt.^50,51 That is, persistent high-frequency electrostatic fluctuations self generated by energized electrons may undergo a steady-state wave-particle interaction so as to maintain a dynamically steady state. A similar idea was recently put forth in order to model the steady-state velocity distribution of solar wind electrons near 1 AU,⁴¹ where it was shown that high-frequency Langmuir fluctuations and suprathermal electrons maintain a dynamical steady-state. For the Earth's radiation belt where the magnetic field is strong, the analogous process involves upper-hybrid fluctuations.

This work was supported by the basic research funding from KASI. This work was done while P.H.Y. visited KASI. P.H.Y. acknowledges NSF Grant No. AGS1550566 to the University of Maryland, and the BK21 plus program from the National Research Foundation (NRF), Korea, to Kyung Hee University. He also acknowledges the Science Award Grant from the GFT, Inc., to the University of Maryland. The research at the University of Iowa was supported by NASA through Contract No. 131802 with the Applied Physics Laboratory. We are thankful to the Van Allen Probes EMFISIS (http://emfisis.physics.uiowa.edu/Flight/RBSP-A/) and ECT team ftp://stevens.lanl.gov/pub/projects/rbsp) for providing online data access and data analysis tool.