The WPIA proposed by Fukuhara et al. (2009) uses the three components of observed waveforms and particle velocity vectors. The WPIA quantifies the energy flow by measuring the inner product of the observed instantaneous wave electric field and velocity vectors, E and v, which is the time variation of the kinetic energy of a charged particle and is given by
$$W = \frac{{{\text{d}}K}}{{{\text{d}}t}} = m_{0}\varvec{\upsilon}\cdot \frac{{{\text{d}}(\gamma\varvec{\upsilon})}}{{{\text{d}}t}} = q\varvec{E} \cdot\varvec{\upsilon},$$
(1)
where K = m0c2(γ − 1) is the kinetic energy of a charged particle including relativistic effects, m0 and q are the rest mass and charge of a particle, respectively, c is the speed of light, and γ is the Lorentz factor. According to Katoh et al. (2013), the net variation of the kinetic energy of charged particles, ΔW(r, t), during a time interval Δt is given by
$$\Delta W(\varvec{r},t) = \int_{t}^{t + \Delta t} {} \iiint {q\varvec{E}(\varvec{r},t^{\prime } )} \cdot\varvec{\upsilon}f(\varvec{r},\varvec{\upsilon},t^{\prime } )d\varvec{\upsilon}dt^{\prime } ,$$
(2)
where f is the phase space density of charged particles. Since the measurement of f is performed at discrete times, ΔW(r, t) is discretized as a summation of W(t
i
) = qE(t
i
)·v
i
measured over a time interval Δt, as follows:
$$\Delta W(\varvec{r},t) \simeq \sum\limits_{i = 1}^{N} {q\varvec{E}\text{(}t_{i} ) \cdot\varvec{\upsilon}_{i} = \sum\limits_{i = 1}^{N} {W(t_{i} )} ,}$$
(3)
where \(t \le t_{i} \le t + \Delta t\), N represents the number of particles detected during the time interval Δt, t
i
is the detection time for the i-th particle, E(t
i
) is the wave electric field vector at t
i
, and v
i
is the velocity vector for the i-th particle. Since W(t
i
) represents the gain or the loss of the kinetic energy of the i-th particle, the net amount of the energy exchange in the region of interest is obtained by summing W for all detected particles, where \(W_{\text{int}} = \sum\nolimits_{i = 1}^{N} W(t_i)\). Figure 1 shows a schematic diagram of W and Wint as measured by the S-WPIA for interactions between energetic electrons and whistler-mode waves propagating purely parallel to the background magnetic field (after Katoh et al. 2013), where Ew and Bw are the wave electric and magnetic field vectors, respectively, and v⊥ is the perpendicular component of the velocity vector of a particle. The sign of W is determined by the relative phase angle (θ) between Ew and v⊥ (Fig. 1a, b), and the net energy exchange between particles and waves can be evaluated by summing W for all N particles to obtain Wint (Fig. 1c). By representing the numbers of energetic electrons having positive and negative W by N+ and N−, respectively, it is expected that N+ and N− would be significantly different from each other in the region of efficient wave–particle interactions. Figure 1c indicates the case of an efficient wave–particle interaction resulting in a wave generation, where N− is larger than N+, rendering Wint negative. Alternatively, if the difference (δN) between N+ and N− is negligible, Wint approaches zero and no net energy exchange would occur. Since a finite number of particles are used in the computation of δN and Wint, there is a fluctuation over time. The fluctuation originates from the thermal fluctuation of the distribution of energetic electrons as well as the fluctuation of both wave electric field amplitude and relative phase angle θ. We use the standard deviation σW, which is computed by:
$$\sigma_{W} = \sqrt {\sum\limits_{i = 1}^{N} {(q\varvec{E}_{W} (t_{i} ) \cdot\varvec{\upsilon}_{i} )^{2} - \frac{1}{N}\left( {\sum\limits_{i = 1}^{N} {q\varvec{E}_{W} (t_{i} ) \cdot\varvec{\upsilon}_{i}} } \right)^{2}} } ,$$
(4)
where the first and second terms on the right-hand side correspond to the width and the center of the qEW(t
i
)·v
i
distribution, respectively, to evaluate the statistical significance of the obtained Wint compared to the fluctuation. We can identify an efficient energy exchange between waves and particles when a Wint that is sufficiently larger than the σW is obtained by the S-WPIA. In other words, a sufficient number of particles need to be collected for the computation of Wint, so that the obtained Wint exceeds σW and achieves the required statistical significance, assuming a Gaussian distribution of 1.64 σW for a statistical significance of 90% and 1.96 σW for a 95% scenario. For the case in which a sufficiently large number of particles is expected in the S-WPIA, Wint can be evaluated for different kinetic energy (K) and pitch angle (α) ranges to obtain Wint(K, α). By examining the obtained Wint(K, α), we can identify the specific energy and pitch angle ranges that mostly contribute to the energy exchange through wave–particle interactions. In this case, σW(K, α) should also be computed for the evaluation of the statistical significance of the obtained Wint(K, α).
Specifications of instruments on board the Arase satellite for implementing the S-WPIA
For the WPIA, it is essential to ascertain that the time resolution of t
i
, indicating the detection time for the i-th particle, is shorter than the timescale for the wave–particle interactions. For the S-WPIA on board the Arase satellite, the requirement of the relative time accuracy for each instrument used in the direct measurement of interactions between the chorus and energetic electrons in the inner magnetosphere is studied. The relative phase angle between the electromagnetic field vector for the wave (Ew and Bw) and the velocity vector v⊥ for the energetic electrons should be resolved in order to identify the sign of W correctly for each detected electron. Here, θ represents the relative phase angle between Ew and v⊥ (Fig. 1a, b), and ζ denotes the angle between Bw and v⊥. In addition, identifying of the presence of an electromagnetic electron hole in the velocity phase space is one of the primary goals of the S-WPIA. While the hole is formed in the specific range of ζ (e.g., Omura et al. 2008; Katoh et al. 2013), which rotates in time with the wave period, the wave phase variation needs to be resolved on a timescale that is sufficiently shorter than the wave period. In the inner magnetosphere, chorus emissions appear in a frequency range lower than the electron cyclotron frequency: typically, from 0.2 to 0.5 Ωe0 for the lower band chorus and from 0.5 to 0.8 Ωe0 for the upper band chorus, where Ωe0 is the electron gyrofrequency at the magnetic equator. Assuming 10 kHz as the highest electron cyclotron frequency along the Arase orbit at the equator, the wave period of the chorus is approximately 100 μs. An accuracy greater than 10 μs resolves the wave phase on the order of a few tens of degrees. The same accuracy should be utilized for the synchronization between wave and particle instruments in order to identify θ and ζ correctly.
The instruments on board the Arase satellite meet the requirements for direct measurements of interactions between chorus and energetic electrons by the S-WPIA. Chorus emissions are often observed on the dawn side of the inner magnetosphere and outside the plasmapause. The typical frequency range of chorus emissions is covered by the waveform capture receiver (WFC) of the plasma wave experiments (PWE) on board the Arase satellite (Kasahara et al. 2018a). Furthermore, since the ratio between the plasma frequency (fp) and the electron cyclotron frequency (fce), fp/fce, is typically less than 10, the minimum resonance energy based on the first-order cyclotron resonance condition is estimated to be in the energy range of hundreds of eV to a few keV for the upper band chorus and from a few keV to tens of keV for the lower band chorus, respectively. The resonance energy changes depending on the pitch angle of the resonant electrons and increases to over MeV for large pitch angle ranges. These estimations show that the kinetic energy range of resonant electrons, particularly for the lower band chorus, is covered by the medium-energy particle experiments (MEP-e) (Kasahara et al. 2018b), the high-energy electron instruments (HEP) (Mitani et al. submitted to Earth, Planets and Space), and the extremely high-energy electron experiment (XEP) (Higashio et al. submitted to Earth, Planets and Space) on board the Arase satellite.
Estimation of the required integration time for the S-WPIA
For the direct measurements of wave–particle interactions by the S-WPIA, a certain number of particles detected in the region of interest need to be collected in order to obtain a statistically significant Wint and/or a non-uniform distribution of particles in the wave phase space caused by the presence of an electromagnetic electron hole. Assuming that the distribution of energetic electrons as a function of ζ is changed by 10% from the average due to the presence of an electromagnetic electron hole and that the statistical fluctuation follows a Poisson distribution for which the standard deviation is expressed as N1/2/N for a particle count N, at least more than 100 particles need to be collected in each bin. If the distribution of particles as a function of relative phase angle ζ is analyzed at every 30°, i.e., if 12 bins are assumed for ζ from 0° to 360°, then the collection of 1200 particles would be required to assess each of the kinetic energy and pitch angle ranges.
By referring to the specifications of the MEP-e (Kasahara et al. 2018b), the number of particles required for the S-WPIA required to obtain a statistically significant Wint is estimated. The MEP-e measures electrons in the energy range of 5–80 keV using 16 sensor channels, where each sensor has an angular resolution of 5° for both elevation and azimuth angles. By estimating the expected particle counts for the MEP-e, the observation conditions for the Arase satellite are assumed to be as follows: (1) the background magnetic field is perpendicular to the spin axis of the Arase satellite, and (2) the number of energy steps for the MEP-e is 16, swept four times every second. Since the field-of-view (FOV) for each sensor channel of the MEP-e changes with time due to the satellite spin, the FOV and the corresponding pitch angle for each sensor channel, as well as the energy step for the MEP-e measurement during one spin, are computed as shown in Fig. 2. Figure 2a shows the pitch angle measured by four sensor channels illustrated by colored rectangles in the upper panel, where the same color is used for both lines in Fig. 2a and rectangles indicating the FOV of the corresponding sensor channel. The pitch angle measured by each sensor channel changes in time due to the satellite spin. The coverage of the pitch angle is different depending on the direction of the FOV with respect to the background magnetic field. The energy range measured by each sensor channel also varies in time, as shown in Fig. 2b. Since MEP-e sweeps 16 energy step every 0.25 s, the energy and the pitch angle measured by each sensor vary accordingly. By referring the observation sequence indicated by Fig. 2a, b, we compute the expected count rate during one spin period as a function of both energy and pitch angle of electrons. The flux of incoming energetic electrons to the MEP-e is assumed to be uniform in both time and space during one spin period with a count rate of 5000 counts per second (cps) for each sensor channel. Figure 2c shows the estimated particle count as a function of the energy steps and pitch angle bins, where the width of each pitch angle bin is assumed to be 5°. The estimation shows that a particle count greater than 2000 can be expected in the wide pitch angle range from 60° to 120° in all the kinetic energy range covered by the MEP-e.
The required time interval for the S-WPIA based on the estimation shown in Fig. 2c is evaluated. If the required number of particles is set at 2000 as estimated earlier, the required particle count can be collected by MEP-e within one spin period in the pitch angle range from 60° to 120°. However, additional restrictions and limitations should be taken into account for the S-WPIA. If the number of particles required in order to increase the statistical significance of the obtained results is set at 12,000, the accumulation time should be greater than six spin periods in the pitch angle range from 60° to 120°. In addition to using a large particle count to achieve statistical significance, in order to increase the signal-to-noise ratio for the S-WPIA, the count at the time of the whistler-mode chorus enhancements should be used. We expect that both the net increase of Wint and modulation of the particle distribution as a function of the relative phase angle ζ can only be measured in the presence of chorus emissions. Considering that the statistical fluctuation of the particle count is expressed as N1/2/N, the particle count detected in the absence of chorus emissions only increases the statistical fluctuations without increasing the amount of the modulation due to wave–particle interactions. By selecting the interval of chorus emissions, we expect that the detected count will increase both N1/2/N and the amount of the modulation of the distribution, and therefore, we expect the signal-to-noise ratio to increase. Since chorus elements appear in the spectra intermittently with a timescale of less than 1 s, it can be roughly assumed that one-third of the detected particles are accompanied by chorus elements. Taking these assumptions into account, the required accumulation time for the S-WPIA is estimated to be at least 18 spin periods, corresponding to 144 s. The expected duration of the S-WPIA measurements is more than 3 min in the region of interest, and this expectation is considered in the operation planning for the Arase satellite.
