On the phase difference of ECH waves obtained from the interferometry observation by the Arase satellite

Abstract

We analyzed electrostatic electron cyclotron harmonic waves observed by the interferometry observation mode of the Arase satellite. It is found that the magnitude of the phase difference varies with the satellite spin. The spin dependence of this phase difference was investigated by examining the trend of the spin dependence for the 84 events of interferometry observation of ECH waves. We found that they are divided into two categories. One is that the phase difference tends to show sinusoidal variations as a function of the angle $\gamma _B$ between the ambient magnetic field projected on the spin plane and the electric field sensor. The other is that the phase difference is close to zero and does not depend on $\gamma _B$. A numerical model of interferometry observation of single plane wave is constructed to explain the observed phase differences. We performed the numerical calculations when the background magnetic field was oriented in the direction often observed in the Arase satellite. The result of the calculations shows the wave vector direction relates to the spin angle with the maximum phase difference. Using this relation, we show that it may be possible to estimate the wave vector direction of ECH waves from one-dimensional interferometry data. This is expected to enable more accurate estimates of phase velocity.

Graphical Abstract

1 Introduction

Electrostatic electron cyclotron harmonic (ECH) waves are one type of plasma waves observed in the magnetosphere. These waves exhibit frequency harmonic structures close to $(n+1/2)$ times of the electron cyclotron frequency, $f_{\textrm{ce}}$. Strong emissions are also observed near the upper hybrid resonance (UHR) frequency. ECH waves are electrostatic in nature, with a wave vector nearly perpendicular to the ambient magnetic field. The electric field oscillation of ECH waves is parallel to the wave vector due to their electrostatic properties. ECH waves were first discovered by the OGO-5 satellite (Kennel et al. 1970). They are excited by a loss cone distribution of hot electrons coexisting with a cold core electron population (Ashour-Abdalla and Kennel 1978). ECH waves have been observed by various satellites in the Earth’s magnetosphere, and are frequently observed near the magnetic equator (Oya et al. 1990; Shinbori et al. 2007). ECH waves are also observed in the magnetosphere of Jupiter and Saturn (Menietti et al. 2012, and references therein).

ECH waves have been reported to have large amplitudes on the order of approximately 10 mV/m (Kennel et al. 1970; Fredricks and Scarf 1973), suggesting their significant impact on the plasma environment. ECH waves are known to cause electron pitch angle scattering, leading to diffuse aurora (Meredith et al. 2009; Li et al. 2012; Ni et al. 2011a). THEMIS observations have shown evidence of pitch angle scattering of several keV-level electrons due to ECH waves (Kurita et al. 2014; Zhang and Angelopoulos 2014). Simultaneous observations with the Arase satellite and ground-based optical imagers have revealed a correlation between the intensity of ECH waves and pulsating auroras, suggesting the possibility that electrons scattered by ECH waves in the loss cone contribute to the generation of diffuse auroras (Fukizawa et al. 2018).

ECH waves have been analyzed for diagnosing plasma properties. Hubbard and Birmingham (1978) classified the frequency structure of ECH waves observed by the ISEE satellite into several categories. In order to understand the excitation conditions of ECH waves in each category, calculations of the linear growth rate were also performed by varying the plasma properties. Hubbard and Birmingham (1978) proposed that the frequency structure of ECH waves can be used to diagnose the density and temperature of cold electrons, and this method was utilized in Hubbard et al. (1979). Moncuquet et al. (1995) experimentally derived the dispersion relation of ECH waves through a comparison between spin modulation of the observed waves by the Ulysses spacecraft and theoretically predicted waves. The experimental dispersion curve was fitted to the theoretical curve, enabling the determination of the density and temperature of cold electrons.

Using interferometric techniques to estimate the wavelength of waves from satellite measurements, it is possible to experimentally derive the dispersion relation (e.g., Graham et al. 2016). The Arase satellite (Miyoshi et al. 2018a) has the capability to perform interferometry observations, allowing for the determination of the dispersion relation of observed ECH waves. To evaluate the wavelength of waves from the interferometry observation, it is essential to derive the phase difference of the waves from the observation. In this study, events in which ECH waves are observed during interferometric measurements by the Arase satellite are investigated. This study shows that the phase difference of ECH waves obtained from the interferometry observation by the Arase satellite can be categorized into two groups, which are followed by simple numerical experiments to explain the phase difference of ECH waves observed by the Arase satellite.

2 Instruments and analysis methods

In this study, we use the data obtained by the plasma wave experiment (Kasahara et al. 2018) (PWE) and the magnetic field experiment (Matsuoka et al. 2018) (MGF) onboard the Arase satellite (Miyoshi et al. 2018a). The Arase satellite is placed in an elliptical orbit with a perigee altitude of $\sim$460 km and an apogee altitude of 32110 km. The satellite is spin-stabilized with a spin period of 8 s. The spin axis of the satellite is roughly directed toward the Sun. PWE consists of two pairs of wire probe antenna (Kasaba et al. 2017) (WPT) in the spin plane and three-axis magnetic search coil (Ozaki et al. 2018) (MSC), and three different receivers. Signals obtained from WPT and MSC in the frequency range from a few Hz to 20 kHz are processed by onboard frequency analyzer/waveform capture (Matsuda et al. 2018) with the nominal sampling frequency of 65 kHz. Because of the large amount of waveform data, three magnetic and two electric field waveforms are obtained during the burst mode operation (Kasahara et al. 2018). MGF measures ambient magnetic field vectors and the data are used to determine the relative angles of WPT antennas to the ambient magnetic fields. The intensity of the ambient magnetic field is also determined, which is used to compute local electron gyrofrequency. The four probes of WPT are labeled U1, U2, V1, and V2, respectively, and opposite probe pairs are operated as dipoles to measure the electric fields (Eu and Ev) during the nominal operation. The V1 and V2 probes can be operated as monopoles during the special operation, measuring the potential difference between themselves and the spacecraft body. The individual potential from the opposite sensor pair can be used to perform the interferometry observation. Note that, during the special operation, signals from U1 and U2 are not available.

In the interferometry observation, the phase difference of the observed waveforms can be calculated (Graham et al. 2016). The electric fields measured by the V1 and V2 probes (hereafter $E_{\textrm{V1}}$ and $E_{\textrm{V2}}$) are computed by dividing the potential obtained from each sensor by the physical length of the antenna (15.6 m). The phase difference $\Delta \theta$ between the $E_{\textrm{V1}}$ and $E_{\textrm{V2}}$ in the frequency domain is evaluated from the cross spectrum of the complex Fourier transforms of the waveforms:

$$\begin{aligned} W_j (f)={\mathcal {F}}[E_{{\textrm{V}j}} (t)]\;\;\;(j=1,2), \end{aligned}$$

(1)

where $W_1$ and $W_2$ are the complex Fourier transform of $E_{\textrm{V1}}$ and $E_{\textrm{V2}}$, respectively. $W_1$ and $W_2$ can be represented using their amplitudes and phases as:

$$\begin{aligned}{} & {} W_1 = |W_1| \exp \left( \textrm{i} \theta _1 \right) , \end{aligned}$$

(2)

$$\begin{aligned}{} & {} W_2 = |W_2| \exp \left( \textrm{i} \theta _2 \right) , \end{aligned}$$

(3)

where $\theta _1$ and $\theta _2$ are the phase of the complex Fourier transforms $W_1$ and $W_2$, respectively. The cross spectrum of $W_1$ and $W_2$ is given as:

$$\begin{aligned} W_1 W_2^* = |W_1||W_2| \exp \left( \textrm{i} \left( \theta _1 - \theta _2 \right) \right) , \end{aligned}$$

(4)

where the asterisk denotes the complex conjugate. From Eq. (4), the phase difference $\Delta \theta$ can be obtained by computing the phase of $W_1 W_2^*$ as

$$\begin{aligned} \Delta \theta = \arctan \frac{\textrm{Im} \left( \langle W_1 W_2^* \rangle \right) }{\textrm{Re} \left( \langle W_1 W_2^* \rangle \right) }, \end{aligned}$$

(5)

where the angle brackets represent the ensemble average, which are adopted so as to increase the signal to noise ratio in the case of actual observation.

The observed phase difference can be expressed using the wave frequency f and phase velocity $v_{\textrm{ph}}$:

$$\begin{aligned}{} & {} \Delta \theta =2\pi f\Delta t, \end{aligned}$$

(6)

$$\begin{aligned}{} & {} \Delta t=\frac{l}{v_{\textrm{ph}}}, \end{aligned}$$

(7)

where l is the distance between observation points. Assuming that the waveforms are observed at the midpoint of the antennas, $l = 15.6$ m in the case of Arase/WPT. When the phase velocity of the observed wave is very slow relative to the sampling period, the phase difference between the two waveforms will be large. When the phase difference becomes larger than $2\pi$, it is impossible to obtain an accurate phase difference.

3 Phase difference calculated by interferometry observation of ECH waves

3.1 Event analysis

We show three examples of ECH waves observed by the interferometry observation mode of the Arase satellite. The first event is shown in Fig. 1, which starts at 00:00:10 on 8 September 2019 with a duration of 60 s. The Arase satellite was located at a radial distance of 5.9 Re, a magnetic latitude of $-0.8$ degrees, and a magnetic local time of 5.7 MLT. The $f_{\textrm{ce}}$ of this event is 3585  Hz. Figure 1a and b shows frequency–time spectrograms computed from electric and magnetic field waveforms observed by antenna V1 and one of the magnetic search coil sensors, $B_{\beta }$, respectively. The multiple harmonic emissions are only seen in the spectrogram of the electric field, which corresponds to the ECH waves. Electromagnetic waves, which appear below the electron gyrofrequency shown in Fig. 1a, correspond to whistler mode waves. Figure 1c represents the coherence between $E_{V1}$ and $E_{V2}$, which is calculated using the following equation:

$$\begin{aligned} Coh(f)=\sqrt{\frac{|\langle W_1(f)W_2^*(f) \rangle |^2 }{\langle |W_1| \rangle \langle |W_2| \rangle } } \end{aligned}$$

(8)

The ensemble average is calculated using a total of 3 and 3 points in the time and frequency domain, respectively. The coherence is used as a confidence indicator of results obtained from the interferometry technique. The coherence threshold of 0.9 is used to discriminate valid signals from unreliable ones. In addition, we removed data points with spectral intensities less than $2.0\times 10^{-3}$ mV/m from the analysis. The frequency–time spectrogram of the phase differences is shown in Fig. 1d. It can be seen that the phase difference is larger at low frequencies and smaller at high frequencies in each frequency band of the harmonic structure of the ECH waves. This tendency is clear, especially in the frequency range from $f_{\textrm{ce}}$ to $2f_{\textrm{ce}}$. Figure 1e shows the angle $\gamma _B$ between the V1 antenna and the ambient magnetic field projected on the spin plane and Fig. 1f shows the elevation angle of the background magnetic field from the spin plane. It is found that the phase difference between the two antennas is maximized at the timing of $\gamma _B$ close to $\pm 90$  degrees, which corresponds to the timing that the amplitude of the ECH waves is maximized during the satellite spin. These analyses are applied to the other events.

Fig. 1

ECH events near 00:00:20 on September 8, 2019; for each part, from top left: a spectral intensity of $E_{\textrm{V1}}$ at each angular frequency, where the red line represents the cyclotron frequency and the white lines represent half and integer multiples of the cyclotron frequency, b spectral intensity of the magnetic field in the $\beta$ direction, c coherence between $E_{\textrm{V1}}$ and $E_{\textrm{V2}}$, d phase difference between $E_{\textrm{V1}}$ and $E_{\textrm{V2}}$ at times and frequencies with high coherence, e angle between the background magnetic field and antenna V1 when projected onto the spin plane, f elevation angle of background magnetic field relative to the spin plane

To understand the dependence of the phase difference on $\gamma _B$ in detail, average profiles of the phase difference as a function of $\gamma _B$ are evaluated using the data shown in Fig. 1d. The wave amplitude of ECH waves as a function of frequency is averaged over the time interval shown in Fig. 1 and the frequency bin with the maximum amplitude in each harmonic band of the ECH waves is chosen to analyze the $\gamma _B$ dependence of the phase difference. For the event shown in Fig. 1, five frequency bins of 5696 Hz, 9024 Hz, 11776 Hz, 15040 Hz, and 18304 Hz are selected.

The average profiles of the phase differences at the selected frequency bands are shown in Fig. 2 together with standard deviations. The averages and standard deviations are computed every 5 degrees of $\gamma _B$. It is found that the phase difference tends to have large deviations when $\gamma _B$ is close to 0 degrees and $\pm 180$  degrees, and the phase difference formed two peaks with $\gamma _B$ around $\pm 90$ degrees.

Fig. 2

Phase difference at the events near 00:00:20 on September 8, 2019; Phase difference between $E_{\textrm{V1}}$ and $E_{\textrm{V2}}$ relative to the spin angle $\gamma _B$ at the chosen frequencies between integer multiples of the cyclotron frequency. The orange line represents the result fitted with a sine function. Phase difference has a sinusoidal shape

In an ideal case of ECH wave observation, when $\gamma _B$ is 0 degrees and ±180 degrees, the antenna is nearly perpendicular to both the wave vector and electric field of ECH waves. Since the electric field is hardly picked up by the antennas in this configuration, it is expected that the phase difference computed in this case would be dominated by noises and does not represent the true phase difference of ECH waves. When $\gamma _B$ is ±90 degrees, the antennas lie in the plane where the electric field fluctuations are present. This allows us to obtain the phase difference with maximum intensity and good signal-to-noise ratio. It is also important to note that the distance between wavefronts measured by two antennas can be the largest when $\gamma _B$ is $\pm 90$ degrees, which results in a large phase difference between antenna V1 and V2.

The second event is shown in Fig. 3, which starts at 23:52:35 on 7 September 2019 with a duration of 60 s. The Arase satellite was located at a radial distance of 5.9 Re, a magnetic latitude of $-0.2$ degrees, and a magnetic local time of 5.6 MLT. The $f_{\textrm{ce}}$ of this event is 3633  Hz. The analysis method applied to the first event is used for the second event. Applying the same procedure as in the analysis of the first event, the average profile of the phase difference as a function of $\gamma _B$ was obtained at frequencies of 5632 Hz, 9344 Hz, 11712 Hz, 15360 Hz and 18688 Hz. The results are shown in Fig. 4. At 9344 Hz, 11712 Hz, 15360 Hz, and 18688 Hz, the phase difference is close to zero and independent of $\gamma _B$. On the other hand, at the 5632 Hz, the same sinusoidal shape as shown in Fig. 2 was observed.

Fig. 3

ECH event near 23:52:35 on September 7, 2019; each item in the figure represents the same parameters as those shown in Fig. 1

Fig. 4

Phase difference at the event near 23:52:35 on September 7, 2019; Phase difference between $E_{\textrm{V1}}$ and $E_{\textrm{V2}}$ relative to the spin angle $\gamma _B$ at the chosen frequencies between integer multiples of the cyclotron frequency. Phase difference is almost constant around 0 at limited frequencies and at the other frequencies, the same sinusoidal shape as shown in Fig. 2

The third event is shown in Fig. 5, which starts at 23:29:50 on 7 September 2019 with a duration of 60 s. The Arase satellite was located at a radial distance of 5.8 Re, a magnetic latitude of 1.5 degrees, and a magnetic local time of 5.4 MLT. The $f_{\textrm{ce}}$ of this event is 3823  Hz. The analysis method applied to the first event is used for the third event. Applying the same procedure as in the analysis of the first event, the average profile of the phase difference as a function of $\gamma _B$ was obtained at frequencies of 4096 Hz, 9856 Hz, 14592 Hz, 15872 Hz and 22848 Hz. The results are shown in Fig. 6. There is no intense signal in the fundamental frequency band, and waves are observed in the higher frequency bands. In all frequency bands where waves are observed, the phase difference is close to zero regardless of $\gamma _B$.

Fig. 5

ECH event near 23:29:50 on September 7, 2019; Each item in the figure represents the same parameters as those shown in Fig. 1

Fig. 6

Phase difference at the event near 23:29:50 on September 7, 2019; Phase difference between $E_{\textrm{V1}}$ and $E_{\textrm{V2}}$ relative to the spin angle $\gamma _B$ at the chosen frequencies between integer multiples of the cyclotron frequency. Phase difference is almost constant around 0

Fig. 7

Parameter $\gamma$ set to represent the rotation of the antenna in DSI coordinate system, and the spin angle $\gamma _B$ with respect to the background magnetic field projected onto the spin plane

3.2 Statistical investigation of the dependence of the averaged phase difference on $\gamma _B$

From the event studies of the phase difference of the ECH waves computed from the interferometry observation performed by the Arase satellite, two patterns are seen in the dependence of the averaged phase difference on $\gamma _B$. To understand the generality of the patterns, we performed a statistical analysis of the $\gamma _B$ dependence of the averaged phase difference of ECH waves using the interferometry observation by the Arase satellite from August to September 2019. During this period, ECH waves were the target of the interferometry observation by the Arase satellite. A total of 160 interferometry observation events were acquired during the period.

We first selected the interferometry observation events during which ECH waves were observed. The phase difference of ECH waves between two antennas is analyzed using the same method as shown in the previous section. The average profiles of phase difference as a function of $\gamma _B$ are computed in each harmonic band of the ECH waves. Of the 160 events, we find 84 interferometry observations of ECH waves.

We classified the 84 observed ECH events into four types based on their trends. Of the observed ECH events, 47 events show the $\gamma _B$ dependence of the averaged phase difference similar to the event shown in Fig. 2. There are 24 events with no $\gamma _B$ dependence of the phase differences in the entire ECH frequency band. In these cases, the calculated phase differences are close to zero. Most events with this characteristic have waves in the high frequency band rather than the fundamental frequency band, as shown in Fig. 6. 9 events show a mixture of the ECH frequency bands with $\gamma _B$-dependent phase difference and the phase difference close to zero in all $\gamma _B$ as shown in Fig. 4. At last, there are 4 events in which the phase difference fluctuates and shows no trend related to $\gamma _B$.

4 Numerical calculation of phase difference using model waves

We construct a simple model to understand the results from the interferometry observation performed by the Arase satellite. In the model, the properties of ECH waves are considered to construct model waves.

Numerical calculations are performed in the Despun Sun sector Inertia (DSI) coordinate system used on the Arase satellite. The Z-axis of the DSI coordinate system is parallel to the spin axis, and the X-axis is defined with respect to the sun direction determined by the sun sensor onboard the Arase satellite. The Y-axis in DSI is defined to complete the right-hand coordinate system. In the DSI coordinate system, the two antennas V1 and V2 rotate in the XY plane with a period of 8 s according to the satellite spin. As shown in Fig. 7, $\gamma$ is the angle between the antenna V1 and the DSI-X axis, and $\gamma$ is expressed as $\gamma =2\pi t/8$. Assuming that the midpoint of each antenna is the observation point, the coordinates of the two observation points $r_1$ and $r_2$ are, respectively, expressed as $(0.5 l \cos \gamma , 0.5 l \sin \gamma , 0)$ and $(-0.5l \cos \gamma , -0.5l \sin \gamma , 0)$ in the DSI coordinate system. As shown in Fig. 8, we define the angle between the DSI-Z axis and the background magnetic field as $\psi$, and the angle between the background magnetic field projected in the DSI-XY plane and the X axis as $\phi$.

Fig. 8

The direction of the background magnetic field, denoted by the zenith angle $\psi$ and azimuth angle $\phi$, and the k vector, denoted by the angle $\alpha$ from $X_B$ in the coordinate system along the magnetic field

Fig. 9

a Outline of the observed model wave of ECH in the spin period b Calculated phase difference

ECH waves are longitudinal waves oscillating almost perpendicular to the background magnetic field. We set the coordinate system $X_B Y_B Z_B$ shown in Figure 8, where the $Z_B$ axis is the direction of the background magnetic field. The $Y_B$ direction is the outer product of the Z direction and the $B_0$ direction projected in the XY plane. The cross product of the $Y_B$ and $Z_B$ axes is the $X_B$ axis.

The DSI-XYZ can be converted into $X_B Y_B Z_B$ coordinate system using the transformation matrix T:

$$\begin{aligned} T= \left( \begin{array}{ccc} \cos \psi \cos \phi &{} \cos \psi \sin \phi &{} -\sin \psi \\ -\sin \phi &{} \cos \phi &{} 0 \\ \sin \psi \cos \phi &{} \sin \psi \sin \phi &{} \cos \psi \end{array} \right) . \end{aligned}$$

(9)

In this numerical calculation, the propagation direction of the ECH wave is perpendicular to the ambient magnetic field for simplicity. The wave vector is represented as $\varvec{k}=k(\cos \alpha , \sin \alpha , 0)$ in the $X_B Y_B Z_B$ coordinate system, where k is the absolute value of $\varvec{k}$ and $\alpha$ is the angle between the wave vector and the $X_B$ axis in the $X_B Y_B$ plane.

Using this, the electric field of ECH waves at the observation point $\varvec{r_1}$ can be expressed as

$$\begin{aligned} E(\varvec{r_{B1}},t)= A \cos ( \varvec{k} \cdot \varvec{r_{B1}} -\omega t), \end{aligned}$$

(10)

where A represents the electric field amplitude, which was calculated as 1 and $\varvec{r_{B1}}$ is the observation point $\varvec{r_1}$ in the $X_B Y_B Z_B$ coordinate, and it is obtained as

$$\begin{aligned} \varvec{r_{B1}}=T \varvec{r_1}. \end{aligned}$$

(11)

We assumed the phase velocity of ECH waves $v_{\textrm{ph}}$ of 990 km/s at frequency f of 5570 Hz, which gives $k \sim 0.03$. These parameters are quite similar to those of ECH waves observed in the inner magnetosphere(Zhou at al. 2017). Due to the directivity of the antenna, only the electric field in the direction of the antenna is received. The observed waveform $E_1$ is expressed as

$$\begin{aligned} E_1(t)=T^{-1} \frac{\varvec{k}}{k} \cdot \left( \begin{array}{c} \cos \gamma \\ \sin \gamma \\ 0 \end{array} \right) E(\varvec{r_{B1}},t). \end{aligned}$$

(12)

The observed amplitude in Equation (12) is computed from the inner product of the wave vector $\varvec{k}$ and the antenna direction.

Figure 9a shows the waveforms obtained from the calculation with $\psi =90$ degrees, $\phi =0$ degrees and $\alpha =90$ degrees. Figure 9b shows the phase difference between $E_1$ and $E_2$.

In this case, the antenna direction is perpendicular to the wave vector of the ECH wave at the initial timing ($t=0$ s) and 4 s. As the antenna rotates, the antenna direction is parallel to the wave vector at $t=2$ s, and anti-parallel to the wave vector at $t=6$ s.

The amplitude of observed waveforms depends on the angle between the antenna and electric field fluctuations. The amplitude becomes the maximum when the antenna is parallel to the electric field fluctuations which is parallel to the wave vector. When the wave vector is parallel or anti-parallel to the antenna direction ($t=2$ s and 6 s), the amplitude is maximum due to the directivity of the antenna, and the absolute value of phase difference between $E_1$ and $E_2$ is also maximum. At $t= 0$ s and 4 s, when the antenna direction is perpendicular to the wave vector, the observed amplitude is zero and the phase difference is zero because the antenna direction is parallel to the wavefront at these timings.

Next, the phase difference between $E_1$ and $E_2$ is examined by changing the direction of the ambient magnetic field which corresponds to the change in the direction of the wave vector. The Arase satellite observes most ECH waves at low latitudes, where the angle between the background magnetic field and the observation plane is about 0 degrees to 15 degrees, and accordingly this numerical calculation is also performed for the case where $\psi$ is close to 90 degrees.

Figure 10 shows the calculated phase difference for $\psi =90$ degrees. The horizontal axis is the angle $\gamma _B$ between the antenna and the background magnetic field in the spin plane, and the vertical axis shows the phase difference. $\gamma _B$ is calculated as $\gamma - \phi$. Comparing three cases with $\alpha =90$ degrees and $\phi$ of 0, 45, and 90 degrees, the dependence of the phase difference on $\gamma _B$ remains unchanged when $\phi$ is changed. This means that $\phi$ varying in the range of 0 degrees to 360 degrees does not cause any change in phase difference when $\psi =90$ degrees. Next, comparing the six cases where $\phi =0$ and $\alpha$ is varied from 0 degrees to 90 degrees every 15 degrees, the maximum phase difference becomes smaller as $\alpha$ becomes smaller. When $\alpha =90$ degrees, both the wave vector and the amplitude of the electric field are in the $Y_B(=Y)$ direction. As $\alpha$ becomes smaller, the wave vector rotates to the $X_B(=-Z)$ direction, so the component perpendicular to the observation plane increases. At $\alpha =0$ degrees, the wave vector is perfectly perpendicular to the observation plane in the $-Z$ axis direction in the DSI coordinate system, and the amplitude of the observed waveform is zero. The $\alpha$ varies in the range from 0 degrees to 360 degrees, which affects the maximum phase difference in one satellite spin.

Figure 11 shows the results for $\psi =80$ degrees. In this case, both the maximum value of the phase difference and its timing change when $\phi$ and $\alpha$ change, respectively. When $\alpha$ changes, the maximum value of the phase difference changes as in the case of $\psi =90$ degrees and $\gamma _B$ with the maximum phase difference also changes. At $\psi =80$ degrees, the change by $\phi$ is small, and the change $\alpha$ is relatively large. For any value of $\alpha$, the phase difference is consistently expressed as a sine function, and the magnitude of the phase difference becomes 0 and the maximum value occurs every 90 degrees.

We reproduced the observed phase difference of the ECH waves using this numerical model. In the case of the phase difference of ECH wave at 5696 Hz shown in Fig. 2, the maximum phase difference is 9.89 degrees at $\gamma _B=$82.5 degrees. The black line with error bars in Fig. 12 shows the phase difference at 5696 Hz in Fig. 2. Since the mean of phase differences is calculated every 5 degrees, there is an error of $\pm 2.5$ degrees in $\gamma _B$ which takes the maximum value. The standard deviation of the phase difference is 0.75 degrees at the maximum. The red line in Fig. 12 is the phase difference reproduced by this numerical model. The used parameters are $\psi =$83.1 degrees, $\phi =$0 degrees, $\alpha =$222 degrees, $f=$5696 Hz and $v_{\textrm{ph}}=$2309  km/s. The maximum phase difference of 9.89 degrees appears at $\gamma _B=82.3$ degrees in our numerical model. Table 1 lists the parameters to reproduce for the remaining four frequencies in Fig. 2 and one frequency shown in Fig. 4 for which the phase difference has a sinusoidal shape. Our model can be used to estimate possible parameters of ECH wave vectors by comparison of the observed phase difference with the numerical model (Fig. 13).

Fig. 10

Phase difference of the ECH waves with respect to the angle $\gamma _B$, which represents the satellite spin, when $\psi =90$ degrees and $\phi$ and $\alpha$ are varied, respectively

Fig. 11

Phase difference of the ECH waves with respect to the angle $\gamma _B$, which represents the satellite spin, when $\psi =80$ degrees and $\phi$ and $\alpha$ are varied, respectively

Fig. 12

The black line shows the phase difference relative to $\gamma _B$ at 5696 Hz in Fig. 2, and the red line is the phase difference reproduced by this numerical model at $\psi =83.1$ degrees, $\phi =0$ degrees, $\alpha =222$ degrees, $f=5696$ Hz and $v_{\textrm{ph}}=2309$ km/s

Fig. 13

Phase difference relative to $\alpha$ and $\gamma _B$ at $\psi =80$ degrees and $\phi =0$ degrees where the black dots indicate $\gamma _B$ with the largest phase difference in each $\alpha$ and the red dots indicate $\gamma _B$ with the smallest phase difference

Table 1 Parameters that reproduce the phase difference in frequencies shown in Fig. 2 in the upper panel and Fig. 4 in the lower panel

5 Discussion and conclusions

We calculated the phase difference of the ECH waves observed by the Arase satellite and found that the majority of them had a sine-shape phase difference depending on the spin period. The sine-type phase difference is also shown in the numerical results, and 47 events, including the event shown in Fig. 1, can be explained by the numerical results. The analyzed phase difference represented by the sine function has a maximum value around $\gamma _B=90$ degrees and 270 degrees. From the numerical calculations in the previous section, $\gamma _B$ at the peak will differ from 90 degrees or 270 degrees in the case that the wave vector has a large component perpendicular to the observation plane. Assuming that the ECH waves are plane waves propagating in one direction, the wave vector of observed ECH waves is approximately in the observation plane. If $\gamma _B$ values that take the peak can be calculated accurately, it is possible to correctly obtain the ECH wave vector from a single dimensional electric field waveform observation.

The events where the phase difference was 0 for any $\gamma _B$ are the second most frequently observed. This result can also be explained by the numerical experiments. The numerical calculations show that the phase difference becomes zero when the ECH wavefront coincides with the antenna. In this case, the observed amplitude is also zero, which differs from the Arase satellite observations showing that significant ECH wave intensity is observed. However, even when the antenna and wavefront do not perfectly coincide, the observed phase difference might be near zero if the apparent phase velocity is very fast.

According to this numerical calculation, there are two possible reasons why two different phase difference trends are observed in one event, as shown in Fig. 4. One is the possibility that the phase velocity is different at each frequency, and the other is the possibility that the wave vector direction is different at each frequency. Since the growth rate of ECH waves depends on frequency and wave normal angle, it is theoretically possible for ECH waves to have different wave vector directions in different frequency bands(Horne et al. 2003; Ni et al. 2011b).

The statistical investigation shows that the spin-dependent phase difference with the sinusoidal shape is more often observed compared with the phase difference close to zero without spin dependence. Numerical calculations show that changing the value of $\alpha$ does not result in a linear change in the amplitude of the phase difference. When $\alpha$ is small in the range of 0 to 90 degrees, both the amplitude and the $\gamma _B$ of the peak phase difference change significantly, whereas when $\alpha$ is larger than 45 degrees, this change is small. Figure 13 shows the phase difference for $\alpha$ and $\gamma _B$ at $\psi = 80$ degrees. $\gamma _B$ with the largest phase difference for each $\alpha$ is indicated by the white dot and $\gamma _B$ with the smallest phase difference for each $\alpha$ is indicated by the pink dot. In many $\alpha$, the peak in the phase difference is close to $\gamma _B =90$ degrees. In the case of $\alpha$ near 0 and 180 degrees, the peak changes greatly and the amplitude of the phase difference become small. Almost all of the events used in the statistical analysis have a peak phase difference with $\gamma _B \sim 90$ degrees, and this result is consistent with the numerical results for $\psi =80$ degrees.

Although we constructed numerical model to compute the phase difference, the analytic expression of the phase difference is also derived. The waveforms at the observation point $\varvec{r_1}$ and $\varvec{r_2}$ can be expressed as:

$$\begin{aligned} E_j (t) =A' \cos \left( \varvec{k} \cdot T \varvec{r_j} -\omega t \right) \,\,\, (j=1,2), \end{aligned}$$

(13)

where $A'$ represents the amplitude that takes into account the direction difference between the antennas and electric field vector. The phase difference between the observed points can be expressed as

$$\begin{aligned}{} & {} \Delta \theta = \varvec{k}\cdot T \varvec{r_1} - \varvec{k}\cdot T \varvec{r_2}. \end{aligned}$$

(14)

Substituting $\varvec{k}$, T, $\varvec{r_1}$, and $\varvec{r_2}$ in Equation (14), $\Delta \theta$ is given as

$$\begin{aligned}{} & {} \Delta \theta = 2k\times 0.5l \left( \cos \alpha , \sin \alpha , 0 \right) \cdot \left( \begin{array}{ccc} \cos \psi \cos \phi &{} \cos \psi \sin \phi &{} -\sin \psi \\ -\sin \phi &{} \cos \phi &{} 0 \\ \sin \psi \cos \phi &{} \sin \psi \sin \phi &{} \cos \psi \end{array} \right) \left( \begin{array}{c} \cos \gamma \\ \sin \gamma \\ 0 \end{array} \right) ,\end{aligned}$$

(15)

$$\begin{aligned}{} & {} \Delta \theta = kl \left( \cos \psi \cos \phi \cos \gamma \cos \alpha + \cos \psi \sin \phi \sin \gamma \cos \alpha - \sin \phi \cos \gamma \sin \alpha + \cos \phi \sin \gamma \sin \alpha \right) . \end{aligned}$$

(16)

From the relationship of $\gamma _B = \gamma -\phi$, following equations are available:

$$\begin{aligned}{} & {} \sin \gamma _B = \sin \gamma \cos \phi - \cos \gamma \sin \phi , \end{aligned}$$

(17)

$$\begin{aligned}{} & {} \cos \gamma _B = \cos \gamma \cos \phi + \sin \gamma \sin \phi . \end{aligned}$$

(18)

Thus, $\Delta \theta$ is expressed by the function of k, $\psi$, $\alpha$, and $\gamma _B$ as

$$\begin{aligned} \Delta \theta =kl (\cos \psi \cos \alpha \cos \gamma _B +\sin \alpha \sin \gamma _B). \end{aligned}$$

(19)

The wave vector can be estimated from the $\gamma _B$ that takes the peak of the phase difference. The Arase satellite acquires only a one-dimensional electric field when it performs the interferometry observation. In such a case, it is very difficult to estimate wave vector direction because of the lack of complete electric field vector information. However, if $\gamma _B$ at the peak of the phase difference can be accurately determined from the interferometry observation, it is expected that the accuracy of the wave vector direction improves. We will attempt to accurately estimate wave vectors in future work, which is important to understand the excitation and propagation of ECH waves and their effects on the plasma environment.

Availability of data and materials

The datasets analyzed during the current study are available in the ERG Science Center repository operated by ISAS/JAXA and ISEE/Nagoya University, https://ergsc.isee.nagoya-u.ac.jp/index.shtml.en (Miyoshi et al. 2018b) The present study analyzed level 2 MGF v03.03 (DOI:10.34515/DATA.ERG-06001), level 2 PWE v01.01 (DOI: 10.34515/DATA.ERG-09000, 10.34515/DATA.ERG-09001), and orbit L2 v02 (DOI:10.34515/DATA.ERG-12000) data.