# Sheath capacitance observed by impedance probes onboard sounding rockets: Its application to ionospheric plasma diagnostics

- Tomonori Suzuki
^{1}Email author, - Takayuki Ono
^{1}, - Jyunpei Uemoto
^{2}, - Makoto Wakabayashi
^{3}, - Takumi Abe
^{4}, - Atsushi Kumamoto
^{1}and - Masahide Iizima
^{5}

**62**:620070579

https://doi.org/10.5047/eps.2010.01.003

© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2010

**Received: **18 August 2009

**Accepted: **28 January 2010

**Published: **31 August 2010

## Abstract

Ion sheath which is formed around an electrode significantly affects the impedance of the probe immersed in a plasma. The sheath capacitances obtained from impedance probe measurements were examined for application to plasma diagnoses. We compared analytical calculations of the sheath capacitance with measurements from impedance probes onboard ionospheric sounding rockets. The S-520-23 sounding rocket experiment, which was carried out in mid-latitude, demonstrated that the observed sheath capacitances agreed well with those of the calculations. We concluded that the sheath capacitance measurements allow for estimation of the electron temperature and the electron density of a Maxwellian plasma. On the other hand, the sheath capacitances obtained from the S-310-35 rocket experiment in the auroral ionosphere showed lower values than expected. Auroral particles precipitations should modify the probe potential.

## Key words

## 1. Introduction

In-situ plasma diagnoses are essential for clarifying various aspects of phenomena found in space and laboratory plasmas. A radio frequency (RF) probe technique is a powerful tool for measuring the plasma parameters (e.g., Jackson and Kane, 1959). Over the past several decades, numerous investigations have been conducted on the impedance of an electrode immersed in a plasma in order to achieve accurate plasma diagnoses.

*f*

_{UHR}where

*f*

_{ce}and

*f*

_{pe}denote the electron cyclotron frequency and the electron plasma frequency, respectively. Measurement of frequency dependence of the probe impedance, therefore, allows us to derive the electron density from the UHR frequency. To eliminate measurement errors of the electron density due to the stray capacitance in the electric circuit, Oya and Obayashi (1966) designed the impedance probe to measure the impedance curves by using the capacitance bridge. Accordingly, the impedance probe made it possible to measure the absolute electron density with a high degree of accuracy. Since then, various types of improved impedance probes have been installed on many sounding rockets (e.g., Yamamoto

*et al.*, 1998; Wakabayashi

*et al.*, 2005; Barjatya and Swenson, 2006) and scientific satellites (e.g., Ejiri

*et al.*, 1973; Oya and Morioka, 1975; Takahashi

*et al.*, 1985; Watanabe and Oya, 1986) for measuring electron density.

The observed antenna impedance reflects various physical properties of the ambient plasma as well as the electron density. Evaluations of the observed impedance are actively discussed subjects these days. Tsutsui *et al.* (1997) examined antenna impedance measured by the Geotail spacecraft in the Earth’s magnetosphere, where the electron densities are too low to observe the resonances. Béghin *et al.* (2005) computed the self-impedance and mutual-impedance to model the electric antennas onboard the Cluster satellites. Impedance measurements in a laboratory plasma showed resonances of a long dipole antenna (Blackwell *et al.*, 2007). Miyake *et al.* (2008) developed an analysis tool of antenna impedance via Particle-In-Cell (PIC) simulation. A Plasma-Fluid Finite-Difference Time Domain (PF-FDTD) simulation was applied to estimate the collision frequency in the ionosphere (Ward *et al.*, 2005; Spencer *et al.*, 2008). In addition, unique characteristics of the probe impedance in a thermal magnetized plasma were found by laboratory experiments (Suzuki *et al.*, 2009).

In actual situations, for evaluating the probe impedance it is essential to consider the effect of an ion sheath surrounding the probe. This paper deals with the capacitance of the ion sheath measured by impedance probes. The ion sheath is formed around the probe due to the difference between the thermal velocity of electrons and ions. The sheath capacitance significantly affects the probe impedance at lower frequencies in comparison to that around the UHR frequency. Derivation of the electron temperature from the sheath capacitance was originally proposed by Oya and Aso (1969). Watanabe (2000) reported that sheath thickness was changed with the auroral energetic electron flux. Sheath capacitance has also been studied in terms of its capacitively coupled plasma (e.g., Chen, 2006).

The purpose of this study is to evaluate the sheath capacitance obtained from impedance probe measurements in order to utilize it for ionospheric plasma diagnostics. We applied an analytical formula of the sheath capacitance with a step sheath model. Although our calculation method of the sheath capacitance is fundamentally the same as that of the report by Oya and Aso (1969), we have made some modifications on the evaluation of the probe potential. We examined the sheath capacitance observed from impedance probes onboard the two sounding rockets, S-520-23 and S-310-35. First, we show analysis of the sheath capacitance in a quiet state of the ionospheric plasma to confirm the validity of the analytical model. We also provide observations of the sheath capacitance under disturbed conditions.

## 2. Sheath Capacitance in Impedance Probe Measurements

*R*

_{s}, while ions are distributed uniformly. The equivalent probe capacitance

*C*

_{pr}, which is the output signal from the impedance probe, consists of the sheath capacitance

*C*

_{s}and the equivalent capacitance of the plasma region

*C*

_{pl}where Z

_{pl}represents the impedance of the plasma region. Equation (2) indicates that

*C*

_{pr}takes minimum values when

*C*

_{pl}↓ 0. This indicates that parallel resonance frequency is independent of existence of the sheath. The probe equivalent capacitances of a short cylindrical probe including and excluding the sheath are calculated as shown in Fig. 2. In the calculations, we used the analytical formula of

*Z*

_{pl}derived by Balmain (1964). It can be seen that

*C*

_{pr}takes a minimum value at the UHR frequency. Another parallel resonance appears at the modified plasma resonance (MPR) frequency, which depends on the angle between the probe axis and the static magnetic field (Ejiri

*et al.*, 1968). On the other hand,

*C*

_{pr}shows a peak value not at the cyclotron frequency but at the sheath resonance (SHR) frequency which is higher than the cyclotron frequency (e.g., Oya, 1965; Aso, 1973). It should be noted that the equivalent probe capacitance at the sufficiently lower frequency than the SHR frequency is almost equal to the sheath capacitance

*C*

_{pr}∼

*C*

_{s}.

*L*and

*R*

_{0}are the probe length and the probe radius, respectively. The relation between the sheath radius

*R*s and the probe potential

*ø*

_{0}can be obtained by solving Poisson’s equation (see Jastrow and Pearse, 1957) Note that we set the space potential as the reference potential in this article.

*T*

_{e}from the sheath radius or the probe potential. Some previous works (e.g., Aso, 1973; Steigies

*et al.*, 2000; Wakabayashi and Ono, 2006) assumed that the thickness of the sheath is proportional to the Debye length as However, the value of the coefficient

*α*is arbitrary in the range of about 1–5. We have therefore evaluated the probe potential by applying the Langmuir probe theory to avoid the uncertainty. Maxwellian distribution yields the electron current

*I*

_{e}as where

*S*

_{p}denotes the probe surface area. Since the sheath thickness in the E–F region ionosphere proved to be longer than or comparable to the probe radius, we have adopted the thick sheath approximation to obtain an analytical formula of the ion current

*I*

_{i}where

*T*

_{i},

*m*

_{i}and erfc(

*x*) represent the ion temperature, ion mass and the complementary error function, respectively (Chen, 1965; Maassberg and Isensee, 1981). By solving equation

*I*

_{e}

*= I*

_{i}numerically with the acceptable assumption

*T*

_{i}=

*T*

_{e}, the floating potential of a cylindrical probe in a Maxwellian plasma

*ø*

_{f}can be expressed as the following form: where

*A*

_{i}is a factor depending on the ion species (e.g., ). Note that we have already confirmed that thin sheath approximation showed almost the same results for analysis in Section 4, compared with that for the thick sheath approximation.

## 3. Outline of the Sounding Rocket Experiments and Instruments

### 3.1 Sounding rocket S-520-23 experiment

Wind measurement for Ionized and Neutral atmospheric Dynamics study (WIND) campaign was carried out to investigate the momentum transfer between the ionospheric plasma and the neutral atmosphere. The S-520-23 sounding rocket was launched from the Uchinoura Space Center (31.15°N, 131.04°E in geodetic coordinates), Japan on 2nd September 2007, at 19:20 LT (LT = UT + 9 h). The pay-load reached an apex altitude of 279 km at 268 sec after the launch. The Lithium Ejection System (LES) installed on the S-520-23 rocket was designed to release Li gas three times in the descending phase. The resonantly scattered light of the Li clouds was successfully observed from several ground sites to measure thermospheric neutral wind accurately.

All instruments performed successfully during the rocket flight. Many interesting results were achieved by the neutral wind, the electron density, the electron temperature, the plasma wave, and the electric field measurements. The present paper focuses on the sheath capacitance measured by the impedance probe instrument.

### 3.2 Sounding rocket S-310-35 experiment

The sounding rocket S-310-35 experiment was performed as a part of the Dynamics and Energetics of the Lower Thermosphere in Aurora (DELTA) campaign, whose objectives and results were described in detail in Abe *et al.* (2006a) and references therein. The sounding rocket was launched from the Andøya rocket range (69.29°N, 16.01°E in geodetic coordinates) in Norway on 13th December 2004, at 01:33 LT (LT = UT + 1 h).

While an auroral breakup was occurring simultaneously with the launch, the sounding rocket flew through the auroral active region. There were two auroral active arcs which crossed the rocket trajectory. The second arc around 124–131 km altitude was within the range of sheath analysis. We should also mention that observations by the N_{2} temperature instrument (NTV), which emitted the artificial electron beam (Kurihara *et al.*, 2006), onboard the rocket affected the electron densities and the probe potentials below about 115 km altitude in the ascent (see Wakabayashi and Ono, 2006).

### 3.3 Impedance probe onboard the sounding rockets

While other impedance measurement system found that for sufficiently dense plasma, the parallel circuit resonance appeared at less than the UHR frequency due to the stray capacitance (Kiraga, 2003), the NEI systems from the rocket flights S-520-23 and S-310-35 have not observed the parallel circuit resonance. Pre-flight environmental tests performed in the space plasma simulation chamber at ISAS/JAXA also ensured the reproducibility of the results with well defined, sharp absolute minimum which can be assigned to the upper hybrid frequency. Data pertinent to space ambient plasma show up the same signature of the upper hybrid frequency. The NEI instruments onboard the S-520-23 and S-310-35 rockets therefore realized the accurate measurements of the electron density without the effect of the stray capacitance.

The NEI systems of S-310-35 and S-520-23 are designed to measure the equivalent probe capacitance over the frequency ranges of 300 kHz–10.3 MHz and 300 kHz–12.0 MHz, respectively. The frequency resolution of the impedance probe onboard S-310-35 was 10.0 kHz from (0.3–4.3) MHz and 20 kHz from (4.3–10.3) MHz. In the case of S-520-23, the frequency resolution was 9.4 kHz from (0.3–2.0) MHz, 20.0 kHz from (2.0–4.0) MHz, 50.0 kHz from (4.0–8.0) MHz, and 100.0 kHz from (8.0–12.0) MHz. The time resolution of the impedance probes was about 500 msec.

The observed frequency variations of the equivalent probe capacitance provide the sheath capacitances and the UHR frequencies. In the following analysis, we identified the equivalent probe capacitance measured at around the 300 kHz, which was sufficiently lower than the observed SHR frequencies, as the sheath capacitance. Electron densities along the rocket trajectory were calculated from the observed UHR frequencies and International Geomagnetic Reference Field (IGRF) model. We also used the electron temperature data measured by the fast Langmuir probes onboard the rockets to obtain the Debye length. The Langmuir probe data were running averaged to reduce the effects of the rocket wake.

## 4. Observations of the Sheath Capacitance

### 4.1 Observations during the ascent of the S-520-23 sounding rocket

The sensor of the impedance probe was extended at 56.5 sec after the rocket launch, and measurements were successfully operated above 93.1 km. During the ascending phase of the S-520-23 sounding rocket, measurements were performed under quiet plasma conditions. Although the impedance probe was exposed to the sun during the observations, the effects of photoelectrons on the probe potential are negligible in the ionosphere.

### 4.2 Observations during the descent of the S-520-23 sounding rocket

In the descent of the S-520-23 sounding rocket, LES released Li gas. As the Li releases were strongly disturbing the equivalent probe capacitance, it became difficult to deduce the electron densities by detecting the UHR frequencies after the each Li release. On the other hand, the equivalent probe capacitances measured at lower frequency than SHR frequency were not so fluctuated as capacitance curves around the UHR frequency even after the Li releases.

The most likely interpretation is that Li release caused increases of the plasma density. The observed sheath capacitance variation corresponded to the increase of the electron density of about 1–2 orders of magnitude. The effects of the chemical release on the ionosphere are intriguing (e.g., Szuszczewicz *et al.*, 1996). Uemoto *et al.* (2010) includes detailed observations and discussion of the electron density after the Li releases.

Here, we should emphasize the practicality of the plasma diagnosis from the sheath capacitance. In rocket observations, instruments sometimes fail to measure plasma parameters due to artificial interferences. Analysis of the sheath capacitance in addition to the UHR frequency will contribute to improvement of the success rate of plasma measurements.

### 4.3 Observations in the auroral ionosphere

*et al.*(2006b), respectively. Figure 6 summarizes the analysis of the sheath capacitance in the case of the S-310-35 experiment. It can be clearly seen that the measured sheath capacitances showed lower values than the expected ones. The difference between the measurements and calculations was significantly large in contrast with the results of the S-520-23 ascent. The large fluctuation of the sheath capacitance observed below 115 km altitude in the ascent was caused by the potential change according to the artificial electron beam emitted by the NTV instrument. Periodic modulation of the sheath capacitance, which was remarkable during the descent observation, came from the rocket wake formed on the opposite side of the moving direction.

## 5. Discussion

### 5.1 Validation of the calculation method

*f*

_{pe}>

*f*

_{ce}, were within ±10% different from that of the calculations. The difference is found to be independent from the plasma frequency. It follows that the sheath capacitance, which can be obtained as the probe capacitance at a sufficiently lower frequency than the SHR frequency, allows us to estimate the Debye length. From another point of view, certainty of the electron density and electron temperature measurements was confirmed by the analysis.

*f*

_{pe}

*< f*

_{ce}seemed to make the difference large (Fig. 7(a)). It can be seen that the measured sheath capacitance showed about 10–30% lower values compared with the calculated values, even though the ionospheric plasma was not disturbed. Where does the difference come from? The measurement examples of the equivalent probe capacitance are shown in Fig. 8. The dashed line denotes the sheath capacitance calculated from the observed Debye length. When

*f*

_{pe}

*> h*

_{ce}was satisfied, the probe capacitance curve approached the calculated capacitance at low frequency (Fig. 8(a)). This demonstrated that the equivalent probe capacitance observed at 300 kHz can be recognized as the sheath capacitance. On the other hand, the equivalent probe capacitance at a frequency lower than the SHR frequency was not flat (Fig. 8(b)). There was an additional dip of the equivalent probe capacitance at 0.57 MHz, which corresponded to the plasma frequency. The dip was thus identified as a plasma resonance, which was not clear with a condition of

*f*

_{pe}

*> f*

_{ce}. Due to the existence of the plasma resonance, the equivalent probe capacitances of around the 300 kHz reflected the plasma impedance as well as the sheath capacitance. We concluded that a condition

*f*

_{pe}

*> f*

_{ce}is suitable for accurate plasma diagnostics via the method described in this paper.

We also found a slight difference of the sheath capacitance in the plasma frequency range of more than 4.5 MHz, even though *f*_{pe}*> f*_{ce} was satisfied (see Fig. 7(b)). The difference appeared periodically around the apex of the rocket flight. The periodic variation was caused by the rocket spin. The effect of rocket wake became significant near the apex and descending phase due to the rocket attitude. Since the electron temperature data were running averaged, the calculated sheath capacitances do not reflect the electron temperatures in the wake. There are some physical issues regarding how to evaluate the temperature in the wake. It is beyond the scope of this article to examine the sheath capacitance measured in the wake region.

### 5.2 Effect of the auroral particles precipitation

As shown in Fig. 6, the sheath capacitances measured by the S-310-35 rocket were significantly lower than that of the calculations. Note that the plasma frequencies were higher than the cyclotron frequencies except for the altitudes below 100 km, due to the ionization by auroral particles precipitation (Wakabayashi and Ono, 2006). This indicates that the difference between the observations and calculations was not derived from the condition *f*_{pe}*< f*_{ce} as discussed in Section 5.1.

*ø*

_{0}in a Maxwellian plasma is equal to

*ø*

_{f}, the probe potential is predicted to be negatively biased (

*ø*

_{0}

*<ø*

_{f}) in the auroral ionosphere due to the non-Maxwellian distribution. We therefore recalculated the sheath capacitance with the probe capacitance treated as a free parameter. Consequently, we found that the observed sheath capacitances were well fitted to the calculated sheath capacitances with the probe potential

*ø*

_{0}=

*2ø*

_{f}as shown in Fig. 9. The estimated probe potential was from −1.2to −0.2 volts, which was the same order as that with a rocket potential observation by the fast Langmuir probe as reported in Abe

*et al.*(2006b).

*et al.*, 2006). In the region, measured sheath capacitances indicated lower values than the calculated ones with the probe potential

*ø*

_{0}= 2

*ø*

_{f}. The probe potential of about

*ø*

_{0}= 2.5 − 3.0

*ø*

_{f}was found to be necessary to explain the observed sheath capacitances. Therefore, we concluded that the sheath capacitance reflected the intensity of auroral particle precipitation.

## 6. Conclusions

We examined the sheath capacitance measured by using impedance probe techniques. The sheath capacitance was analytically calculated from the Debye length. Such calculations are a simpler method in comparison with the PIC simulations (Miyake *et al.*, 2008), and as a result, our method does not then require enormous computer facility.

We have compared the sheath capacitances observed by the impedance probes onboard the sounding rockets with the calculated sheath capacitances. When a condition of *f*_{pe}*> f*_{ce} was satisfied, the equivalent probe capacitance observed at a sufficiently lower frequency than the SHR frequency showed a constant value which equaled to the sheath capacitance. The observed sheath capacitances agreed well with the calculations. This indicates that the calculation methodology of the sheath capacitance described in Section 2 is valid; and the Debye length can be estimated from the observed sheath capacitance. The advantages of this technique are summarized as follows: (a) the sheath capacitance in addition to the UHR frequency provides plasma parameters, (b) the electron density, the electron temperature and the probe potential measured via other instruments can be cross-checked with analysis of the sheath capacitance, and (c) the analysis method is simple. Since the detection of the UHR frequencies sometimes becomes hard due to plasma disturbances (e.g., effect of Li release and electron beam emission) in sounding rocket experiments, plasma diagnosis technique from the sheath capacitance is valuable. In future experiments, we propose to design the impedance probe to measure the sheath capacitance with high time resolution. This improvement allows the impedance probe to realize both high-accurate measurements of the electron density from the UHR frequency and high-resolution plasma diagnostics from the sheath capacitance.

In the case of *f*_{pe} < *f*_{ce}, the plasma resonance clearly appeared. As a result, it became difficult to separate the sheath capacitance from the observed the equivalent probe capacitance. For accurate plasma diagnostics from the sheath capacitance obtained by the impedance probe, it is necessary that the plasma frequency is larger than the cyclotron frequency.

Observations in the auroral ionosphere indicated that the probe potential was shifted from the analytical solution of the floating potential due to the non-Maxwellian velocity distribution. Strong precipitation of energetic electrons in the auroral arc should have caused the lower probe potential. Quantitative discussion of the response of the probe potential to the total flux of precipitating auroral particles remains a future issue to examine.

## Declarations

### Acknowledgements

The sounding rocket experiments were conducted by the Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency (ISAS/JAXA) as international projects. We thank all members of the rocket experiments. Pre-flight operation tests of the impedance probes were supported by the Space Plasma Laboratory, ISAS/JAXA. The impedance probes were manufactured by System Keisoku Co., Ltd. This work is supported by the Global COE Program “Global Education and Research Center for Earth and Planetary Dynamics” at Tohoku University.

## Authors’ Affiliations

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