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


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.

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.

An impedance probe, which is a kind of RF plasma probe, has been developed as an unique tool to measure the electron number density. The probe impedance in a magnetized plasma has a resonance at the upper hybrid resonance (UHR) frequency fUHR


where fce and fpe 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

We applied a simple sheath model as illustrated in Fig. 1. There are no electrons in the sheath region whose radius is Rs, while ions are distributed uniformly. The equivalent probe capacitance Cpr, which is the output signal from the impedance probe, consists of the sheath capacitance Cs and the equivalent capacitance of the plasma region Cpl


where Zpl represents the impedance of the plasma region. Equation (2) indicates that Cpr takes minimum values when Cpl ↓ 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 Zpl derived by Balmain (1964). It can be seen that Cpr 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, Cpr 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 Cpr Cs.

Fig. 1
figure 1

Applied sheath model. There are no electrons in the sheath region.

Fig. 2
figure 2

Calculated equivalent probe capacitance. Plasma parameters are set as fpe = 3.0 MHz, fce = 1.4 MHz, f ν = 0.05 MHz, and the static magnetic field is perpendicular to the probe axis. The solid line represents calculation including the sheath capacitance of Cs = 30 pF. Dashed line shows equivalent probe capacitance of only the plasma region.

The capacitance of a cylindrical sheath is expressed as


where L and R0 are the probe length and the probe radius, respectively. The relation between the sheath radius Rs and the probe potential ø0 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.

The remaining issue is how to evaluate the electron temperature Te 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 Ie as


where Sp 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 Ii


where Ti, mi and erfc(x) represent the ion temperature, ion mass and the complementary error function, respectively (Chen, 1965; Maassberg and Isensee, 1981). By solving equation Ie= Ii numerically with the acceptable assumption Ti = Te, the floating potential of a cylindrical probe in a Maxwellian plasma øf can be expressed as the following form:


where Ai 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.

Combining Eqs. (4), (5), and (9) gives the following relation:


This equation explicitly shows that the sheath capacitance Cs gives the Debye length λD. The electron temperature is therefore deduced from the observed sheath capacitance if the applied sheath model is valid.

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 N2 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

The impedance probe instrument onboard the S-310-35 sounding rocket was described in Wakabayashi and Ono (2006). A similar impedance probe was installed on the S-520-23 sounding rocket. The capacitance bridge circuit measured the equivalent capacitance of a monopole antenna, whose length and radius were 1.0 m and 6 mm, respectively. Figure 3 shows the block diagram of the impedance probe instrument called NEI (Number density of Electrons by Impedance probe). The sensors of the impedance probes were connected to the electric ground level with a 20 MΩ resistor in order to protect the electric circuit from the high energy particles input due to the electron beam injection as well as the auroral particles precipitation. The capacitance of the variable condenser and the length of the dummy cable were properly adjusted before the flight to eliminate the effects of the stray capacitance.

Fig. 3
figure 3

Block diagram of the impedance probe instrument onboard the sounding rockets S-310-35 and S-520-23.

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.

Figure 4 shows height variations of the sheath capacitance observed in the ascent of the S-520-23 sounding rocket. The black line represents the sheath capacitance ‘measured’ at 300 kHz. A sheath capacitance curve ‘calculated’ from the electron densities and the electron temperatures based on the method described in Section 2 is also plotted in the figure. The measured sheath capacitances showed almost the same profile with that of the calculated curve. However, some differences can be found between the measurements and the calculations below 220 km altitude. In this altitude range, electron plasma frequencies showed lower values than the electron cyclotron frequencies. The difference and validation of the analysis method are discussed in Section 5.1.

Fig. 4
figure 4

Measured sheath capacitance in the ascent of the S-520-23 sounding rocket and the sheath capacitance calculated from the Debye length. In the calculation, we used the factor .

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.

We therefore plotted the equivalent probe capacitance observed at the 300 kHz as a profile of the sheath capacitance (Fig. 5). A notable feature is that measured sheath capacitance indicated sharp enhancements after each Li release. The sheath capacitance observed in the ascending phase showed no such enhancements (see Fig. 4).

Fig. 5
figure 5

Sheath capacitance observed during the descent of the S-520-23 sounding rocket. Dashed lines represent the planned altitudes to start Li release.

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

We have also examined the sheath capacitance observed by the impedance probe onboard the S-310-35 sounding rocket, which was launched into the auroral ionosphere. The electron density and the electron temperature measurements along the rocket trajectory were reported in detail by Wakabayashi and Ono (2006) and Abe 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.

Fig. 6
figure 6

Measured sheath capacitance in the S-310-35 experiment and calculated sheath capacitance.

5. Discussion

5.1 Validation of the calculation method

The first point that should be discussed is the validity of the evaluation method of the sheath capacitance. Figure 7 shows the difference between the measured sheath capacitance and the calculated one in the ascending phase of the S-520-23 sounding rocket. Figure 7(b) shows that most of the sheath capacitances, which were observed in the condition fpe > fce, 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.

Fig. 7
figure 7

Difference between the observed sheath capacitance and the calculated ones in the ascent of the S-520-23 rocket. The difference is defined as (Cs_measuredCs_calculated)/C_calculated × 100.

In contrast, observations with fpe< fce 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 fpe> hce 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 fpe> fce. 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 fpe> fce is suitable for accurate plasma diagnostics via the method described in this paper.

Fig. 8
figure 8

Observed spectra of the equivalent probe capacitance. (a) Measurement at 279.3 km altitude in a condition of fpe = 4.51 MHz and fce = 1.11 MHz, (b) measurement at 150.2 km altitude in a condition of fpe = 0.58 MHz and fce = 1.20 MHz.

We also found a slight difference of the sheath capacitance in the plasma frequency range of more than 4.5 MHz, even though fpe> fce 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 fpe< fce as discussed in Section 5.1.

The difference should be caused by a charge-up effect of the electrode due to the auroral particles (Watanabe, 2000). While the probe potential ø0 in a Maxwellian plasma is equal to øf, the probe potential is predicted to be negatively biased (ø0f) 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 ø0 = 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).

Fig. 9
figure 9

Measured sheath capacitance in the S-310-35 experiment and calculated sheath capacitance. The red line denotes the calculated sheath capacitance with the probe potential of ø0 = 2øf.

Figure 10 enlarges a portion of the Fig. 9(a). The shaded region in the figure shows an altitude range where strong particle precipitation was observed by the Auroral Particle Detector (APD) instrument (Ogasawara et al., 2006). In the region, measured sheath capacitances indicated lower values than the calculated ones with the probe potential ø0 = 2øf. The probe potential of about ø0 = 2.5 − 3.0øf 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.

Fig. 10
figure 10

Extended view of Fig. 9(a). The shaded range shows the region where the auroral arc appeared.

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 fpe> fce 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 fpe < fce, 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.


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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.

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Correspondence to Tomonori Suzuki.

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Suzuki, T., Ono, T., Uemoto, J. et al. Sheath capacitance observed by impedance probes onboard sounding rockets: Its application to ionospheric plasma diagnostics. Earth Planet Sp 62, 579–587 (2010).

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