# Quasi Thermal Noise Spectroscopy for Van Allen Probes

## Abstract

Quasi thermal fluctuations in the Langmuir/upper-hybrid frequency range are pervasively observed in space plasmas including the radiation belt and the ring current region of inner magnetosphere as well as the solar wind. The quasi thermal noise spectroscopy may be employed in order to determine the electron density and temperature as well as to diagnose the properties of energetic electrons when direct measurements are not available. However, when employing the technique, one must carefully take the spacecraft orientation into account. The present paper takes the upper-hybrid and multiple harmonic—or (*n* + 1/2)*f*_{ce}—emissions measured by the Van Allen Probes as an example in order to illustrate how the spacecraft antenna geometrical factor can be incorporated into the theoretical interpretation. This method can in principle be applied to other spacecraft, including the Parker Solar Probe.

## Key Points

- Quasi thermal noise can be used as a diagnostic tool for space plasmas
- Upper-hybrid quasi thermal noise detected by the Van Allen Probes are analyzed with spacecraft orientation taken into account
- The present method may be extended for the Parker Solar Probe mission

## 1 Introduction

The quasi thermal noise (QTN) represents the spontaneously emitted electromagnetic fluctuations that take place in a plasma, which may be close to but not necessarily at thermal equilibrium state (Sitenko, 1967). A theoretical tool to analyze QTN is known as the QTN “spectroscopy,” which was developed by Meyer-Vernet (1979) in the context of solar wind. The QTN spectroscopy is a versatile research tool that can be employed to indirectly determine not only the total electron density and temperature (Maksimovic et al., 1995; Meyer-Vernet et al., 1986) but also the properties of energetic and often nonthermal electrons. It is well known that the solar wind contains tenuous but energetic nonthermal electron population. These electrons may contribute significantly to the energy transport in interplanetary space, but their detection is difficult owing to the spacecraft charging and photoelectrons. However, QTN spectroscopy is a passive technique, which is thought to be less susceptible to such limitations. In this regard, Zouganelis (2008) employed the QTN method in order to study the nonthermal electron velocity distribution measured by the Ulysses spacecraft.

The Parker Solar Probe (PSP), whose science operation is just beginning, will eventually reach the unprecedented close vicinity of the Sun (as close as 9 or 10 solar radii). One of its objectives is to study the properties of nonthermal electrons of coronal origin, which includes the highly field-aligned *strahl* electrons. When PSP is near its closest approach, however, direct measurements of strahl electrons are not possible, since the largely radial solar magnetic field geometry and the spacecraft heat shield prevent the charged particles moving along the field to reach the detector. On the other hand, QTN can be measured with the *FIELDS* instrument (Bale et al., 2016), which subsequently may be analyzed in order to unveil the nonthermal electron properties. In anticipation of such a scientific endeavor Lazar et al. (2018) analyzed the QTN spectrum for plasma conditions to be encountered by PSP. However, in order to fully utilize the information gathered by the spacecraft, one must carefully take the spacecraft orientation and antenna geometry into consideration. The original QTN spectroscopy does indeed take the antenna response into account, but it is for unmagnetized solar wind plasma (Meyer-Vernet & Perche, 1989). Later extension to magnetized plasmas (Moncuquet et al., 1993) is relevant to the present investigation, but our discussion complements the existing method, as will be explained.

## 2 Theoretical Considerations

*F*(

**k**) (Meyer & Vernet, 1974; Meyer-Vernet et al., 2017),

*k*

_{⊥}and

*k*

_{‖}are wave vector components perpendicular and parallel with respect to the linear antenna, respectively, and

*a*is the antenna radius. For the solar wind plasma near Earth orbit, which is practically unmagnetized, we may simply take the formula for spontaneously emitted electrostatic fluctuations in unmagnetized plasma (Sitenko, 1967),

*a*,

*Z*(

*ζ*) represents the plasma dispersion function (Fried & Conte, 1961),

*T*is the equilibrium plasma temperature, denotes the square of plasma frequency defined with respect to species labeled

*a*(

*a*=

*e*,

*i*for electrons and ions, respectively),

*e*

_{a}= −

*e*for electrons and

*e*

_{a}=

*e*for protons,

*m*

_{a}designates the particle mass, and

*n*

_{0}designates the ambient plasma density.

*ω*

_{pe}/Ω

_{e}, for the radiation belt environment can be on the order of . Here Ω

_{e}=

*eB*

_{0}/

*m*

_{e}

*c*stands for the electron cyclotron (or gyro) frequency,

*B*

_{0}being the ambient magnetic field intensity, and

*c*denoting the speed of light

*in vacuo*. The parameter regime of is expected to be similar for the case of solar wind near 9 or 10 solar radii, which will eventually be surveyed by the PSP. For such cases, one must instead employ the formula appropriate for magnetized plasmas (Moncuquet et al., 1993; Sentman, 1982), for which the dielectric constant in 2 must be replaced by

_{n}(

*x*) =

*I*

_{n}(

*x*)

*e*

^{−x}, with

*I*

_{n}(

*x*) being the modified Bessel function of the first kind of order

*n*. Yoon et al. (2017) and Hwang et al. (2017) made use of the magnetized version of spontaneous emission theory in order to analyze the VAP probe data, but they did so without including the antenna response function. Instead, they fixed the propagation angle, , and simply integrated over

*k*space in order to make specific comparisons against observed data. That is, they simply chose angle

*θ*that produced the best fit to data.

The outstanding issue is obviously that theoretical justification for the choice of *θ* was not sufficiently provided. Thermal plasmas are supposed to emit fluctuations in all directions, but the linear antenna is generally expected to best respond to electric field along the antenna length. Consequently, it is reasonable to expect that the choice of *θ* must be related to the antenna orientation with respect to the local magnetic field. However, Yoon et al. (2017) and Hwang et al. (2017) did not address this issue in a quantitative manner. To address this problem, one must possess the knowledge of spacecraft orientation with respect to the local magnetic field in conjunction with the antenna response. It is the purpose of the present investigation to revisit this problem and address these outstanding issues.

The twin spacecraft VAP spin on each axis, which is directed along the Sun-Earth axis. The electric field antenna lies in a plane orthogonal to the spin axis. As the spacecraft executes elliptical orbits around the Earth, it surveys the local geomagnetic field vector that is generally tilted with respect to the spin axis, see Figure 1a, where the local magnetic field vector is displayed to be tilted with respect to the spin axis by angle *α*. The linear antenna spins so that the cartoon representation is but a snapshot when the antenna orientation is vertical. For a general configuration that involves the spinning spacecraft, the geometrical consideration is substantially more complicated such that we choose to analyze the singular case when the antenna happens to be vertically oriented.

For VAP spacecraft all the electric and magnetic field instruments are aligned with coordinate system known as the UVW science coordinate, where *W* axis is coaligned with the spin axis, while *U* and *V* are two axes orthogonal to *W* axis. By taking advantage of the measured magnetic field strengths in UVW coordinate, it is possible to determine the angle *α*. Specifically, by making use of three components of magnetic field vector, *B*_{V}, *B*_{U}, and *B*_{W}, we have
. If *α* is close to 0°, then it is likely that the electric field antenna will pick up QTN associated with quasi perpendicular propagation angle *θ* close to 90°. On the other hand, if *α* is close to 90°, then the antenna is expected to pick up quasi parallel signals associated with QTN. Thus, we expect a rough relationship *α*≈*π*/2 − *θ*. Quasi parallel QTN in magnetized plasmas is characterized by a single peak in the vicinity of plasma frequency, whereas quasi perpendicular QTN is generally associated with the upper-hybrid peak, with the accompanying multiple harmonic electron cyclotron harmonic, or (*n* + 1/2)*f*_{ce} emissions. The upper-hybrid frequency is, of course, given by
.

Hwang et al. (2017) analyzed the 8 May 2013 event between UTC 12:00 and 18:00. Of this time interval, (*n* + 1/2)*f*_{ce} emissions occurred in the limited time range around UTC 16:00. If angle *α* for this time period is relatively low, then the interpretation of QTN being associated with quasi perpendicular angle *θ* is justified. We will thus revisit this event and reanalyze the data with the spacecraft orientation taken into consideration. Before we do that, however, let us examine the antenna response function associated with the spacecraft, *F*(**k**), defined in 1. It should be kept in mind that the perpendicular and parallel wave vector components, *k*_{⊥} and *k*_{‖}, which appear in 3 are different from the same quantities defined in 1. Whereas *k*_{⊥} and *k*_{‖} that appear within the definition of the antenna response function refer to wave vector components defined with respect to the linear antenna geometry, the same quantities in 3 refer to wave vector components defined with respect to the ambient magnetic field vector. For unmagnetized plasmas, where the spontaneously emitted electric field fluctuations are isotropic, such an ambiguity does not exist. However, for magnetized plasmas, one must exercise caution when dealing with the antenna response function. Figure 1b graphically depicts the interrelationship between the wave vector defined with respect to the local magnetic field vector (i.e.,
and
) and the wave vector components defined with respect to the antenna direction, which we denote as
and
. The graphical representation in Figure 1b is for a singular situation when the linear antenna happens to be vertically oriented. Of course, the antenna spins in the plane orthogonal to the spin axis, so that the geometrical orientation depicted in Figure 1b happens only once per rotation. However, this singular case serves the purpose of illustrating the role of antenna response function in determining the angle *θ*.

*F*(

**k**) in terms of and

*k*

_{‖}and

*k*

_{⊥}by and . Consequently, we have

This result is essentially equivalent to the formula derived by Moncuquet et al. (1993), see the discussion in section 3.2 of their paper, who considered the relative geometrical configuration between the wave vector **k** and antenna direction via angle *φ*−*θ* as well as the relative angle *α* between the spacecraft and local magnetic field, if we interpret their result using our notation. However, they made one crucial assumption that the most important wave propagation angle is quasi perpendicular, |*k*_{‖}| ≪ *k*_{⊥}. This implies that in their theory, the wave propagation is assumed to be predominantly close to 90° with respect to the ambient magnetic field. In contrast, our purpose is to allow for an arbitrary angle *θ* to start with, but to show that only when *θ* is close to 90°, or equivalently, the angle *α* is close to 0°, via the relationship *α* = *π*/2 − *θ*, that the antenna will pick up (*n* + 1/2)*f*_{ce} type of emissions.

*F*(

*x*,

*y*) as a function of

*x*for three different values of

*y*. Note that

*F*rapidly reduces for finite value of

*y*and maximizes for

*y*= 0. If

*θ*=

*φ*, then

*y*= 0 so that

*J*

_{0}(

*y*= 0)→1, which maximizes the antenna response, and

*F*(

**k**) becomes simply the function of

*kL*,

*θ*−

*φ*is finite, then the Bessel function factor reduces the antenna response. This implies that the antenna response is most effective when

*θ*=

*φ*. In other words, the antenna will predominantly pick up QTN fluctuation signal, whose wave propagation angle

*θ*corresponds to the antenna orientation angle itself, namely,

*φ*. In short, the above theoretical discussion provides the necessary theoretical justification for choosing

*θ*as a fixed input parameter, while integrating the fluctuation formula over

*k*space, as was done by Hwang et al. (2017) and Yoon et al. (2017). Note that the relationship,

*φ*=

*π*/2 −

*α*, is valid only if the antenna happens to be vertical, as shown in Figure 1c, which is realized only once per rotation. Nevertheless, the above discussion is sufficient to illustrate the essential point that if

*α*happens to be close to 0°, then it is likely that the signal detected by antenna will bear the feature of (

*n*+ 1/2)

*f*

_{ce}emissions, while for

*α*close to 90°, it is more likely that QTN spectrum will peak in the vicinity of plasma frequency, with little or no multiple harmonic structure. In short, we have established that the general relationship,

*φ*by

*θ*.

## 3 Reinterpretation of Data

The outstanding issue to be addressed is to check whether during the 8 May 2013 event, the VAP spacecraft orientation angle *α* was indeed quasi parallel or not. In Figure 3 upper three panels, we display the results of renewed analysis for the same event between UTC 14:00 and 18:00. The first panel shows the high-frequency electric field dynamic spectrum in frequency range 10 to 40 kHz. The first panel shows the band of persistent upper-hybrid/Langmuir frequency range fluctuations throughout the entire interval. (The broad emissions that take place at frequencies much higher than the upper-hybrid/Langmuir band are most likely the Auroral Kilometric Radiation from remote sources.) We show in white curves the multiple harmonic electron cyclotron frequencies. For most of the time interval, the upper-hybrid/Langmuir frequency line is generally much higher than the electron gyrofrequency such that the ratio of plasma to electron cyclotron frequency *ω*_{pe}/Ω_{e} = *f*_{pe}/*f*_{ce} is of the order of ∼15 or higher. However, around UTC 16:00 or so, the frequency ratio drops to *ω*_{pe}/Ω_{e}∼7 for a brief period, which is also accompanied by excitation of multiple electron gyrofrequency modes, or equivalently, (*n* + 1/2)*f*_{ce} emissions. Upon close examinations, however, faint but distinguishable multiple harmonic emissions actually start around UTC 15:00 or so.

The second panel of Figure 3 shows the electron energy spectrum in the range of 10 eV to 10 keV. For most of the duration, the electron spectrum remains more or less constant, indicating that the radiation belt is in a relatively inactive period. However, around the time when the frequency ratio drops to *ω*_{pe}/Ω_{e}∼7 when excitation of (*n* + 1/2)*f*_{ce} emissions take place, it is seen that the ∼100 eV electron population undergoes an appreciable increase in intensity.

In the third panel, we plot the time variation of angle *α*. Before UTC 15:00 where the excitation of multiple harmonic cyclotron emissions is beginning to manifest themselves, the spacecraft orientation is quasi perpendicular, with *α* close to 90°. Indeed, *α* starts out with ∼100° value at MTL 14:00 but is gradually reduced until around UTC 15:00; *α* is now close to 40°. Around UTC 16:00, *α* is ∼20° and stays around this (or slightly lower) value beyond this point in time. This analysis thus shows for the first time that the (*n* + 1/2)*f*_{ce} emissions are generally related to the antenna orientation with respect to local magnetic field.

*n*+ 1/2)

*f*

_{ce}emissions when the frequency ratio

*ω*

_{pe}/Ω

_{e}drops to ∼7 or so. To understand the cause of this feature, we have constructed a theoretical QTN spectrum by integrating the electric field fluctuation spectrum over the wave number while fixing the propagation angle

*θ*via the relationship

*θ*=

*π*/2 −

*α*. We have artificially varied the frequency ratio

*ω*

_{pe}/Ω

_{e}from the asymptotic value of

*ω*

_{pe}/Ω

_{e}= 15 but allowing it to sharply drop to the value

*ω*

_{pe}/Ω

_{e}∼7 and back up, via an adiabatic temporal dependence,

*T*from −5 to 5. We have also exploited the fact that there is an accompanying increase in the ∼100 eV electron intensity. In order to reflect this increase, we have considered an overall increase in the background electron temperature, , where

*ε*= 0.1. The basic theoretical formulae are discussed in (Yoon et al., 2017) and (Hwang et al., 2017) so will not be repeated here; see, for example, equations (25) and (26) of (Yoon et al., 2017) or, equivalently, equations (6) and (7) of (Hwang et al., 2017). We have numerically integrated the theoretical spectrum over

*k*space, as was done by Hwang et al. (2017) and Yoon et al. (2017).

Figure 3 fourth panel is the result of theoretical QTN spectrum as a function of frequency (vertical axis) and artificial “time” *T* (horizontal axis). We may interpret *T* as representing the observed time period as shown in the first panel of Figure 3. The theoretical construction appears to be in excellent agreement with observation in that the brightening of multiple harmonic cyclotron emission lines coincides with the sudden drop in the background electron density (and small increase in the background temperature). High ratios of *ω*_{pe}/Ω_{e} imply weakly magnetized plasma, so that QTN spectrum features are dominated by the plasma line (*f*∼*f*_{pe}) with weak harmonic structure. In contrast, for lower values of *ω*_{pe}/Ω_{e} the plasma is strongly magnetized so that the peak frequency coincides with the upper-hybrid line (*f*∼*f*_{uh}) and multiple harmonic emissions become more apparent.

## 4 Conclusions

In the present study we have revisited the recent analysis by Hwang et al. (2017) and Yoon et al. (2017), on the upper-hybrid frequency range fluctuations pervasively observed by the VAP, in terms of the spontaneously emitted electrostatic fluctuations in thermal magnetized plasma. In their interpretation of the multiple harmonic (*n* + 1/2)*f*_{ce} emissions, which sometimes accompany the upper-hybrid range fluctuations, however, the authors simply chose the angle *θ* between the wave vector and ambient magnetic field vector that produced the best fit with the data. In the present paper, we have taken the spacecraft orientation into account and calculated the relative angle between the local magnetic field vector and the spacecraft spin axis, *α*. We have then analyzed the antenna response function, given by Moncuquet et al. (1993) and Meyer-Vernet et al. (2017), and showed that the relationship *α*≈*π*/2 − *θ* exists.

We have validated this prediction against the same data analyzed before. In order to further interpret the observation, which shows enhanced (*n* + 1/2)*f*_{ce} emissions when the local value of *f*_{pe}/*f*_{ce} drops, we also calculated the theoretical electric fluctuation spectrum, by artificially varying the frequency ratio *f*_{pe}/*f*_{ce}. The result shows an excellent agreement with observation in that, for *f*_{pe}/*f*_{ce} close to ∼7, cyclotron emissions are enhanced, while for higher values of *f*_{pe}/*f*_{ce}, the emission characteristics become similar to the unmagnetized case.

Making use of the spacecraft coordinate defined with respect to the local magnetic field in order to interpret the data of the QTN analysis, as carried out in the present paper for the first time, is a new concept, and we have demonstrated that such a piece of information can be very useful in accurately interpreting the electric and magnetic field data. Note that Moncuquet et al. (1993) and Meyer-Vernet et al. (2017) have already carried out a similar analysis as ours by taking the relative angle between the antenna and wave vector, *θ* − *φ*, as well as the relative angle between the local magnetic field vector and antenna, namely, *α*. However, one crucial difference is that in their calculation, they already presupposed that *θ* is close to *π*/2. Our purpose had been to provide the justification of why *θ* has to be close to such a value in the first place. We have shown that such a configuration does not always occur, but only when the spacecraft is oriented in certain directions in relation to the local magnetic field that it does occur. Such an identification technique may be useful not only for interpreting existing spacecraft data but also for new and future missions, such as the PSP (Bale et al., 2016).

As we have noted in section 1, the QTN, which will be detected by the FIELDS instrument onboard the PSP spacecraft, may be very useful for an indirect diagnosis of the incoming solar wind strahl electron population, since such particles cannot be detected when the spacecraft reaches its closest approach to the Sun, owing to the largely radial nature of the solar wind magnetic field and the spacecraft heat shield. However, as discussed by Zouganelis (2008)—in the case of unmagnetized plasmas—and Lazar et al. (2018)—in the case for magnetized plasmas, QTN theory can be employed to uncover the underlying properties of energetic nonthermal electrons. For PSP situation, the analysis by Lazar et al. (2018) becomes relevant, since for distances of 9 or 10 solar radii from the Sun, the plasma must be considered to be magnetized, *ω*_{pe}/Ω_{e} being
or so. In the analysis by Lazar et al. (2018), the choice of angle *θ* becomes an issue, just as in the works by Hwang et al. (2017) and Yoon et al. (2017). The present analysis thus becomes relevant for PSP mission as well. As the data from PSP are just beginning to become available, we believe that it is timely to disseminate the present findings to the wider space physics community. Note that our finding may also apply to the Solar Orbiter (Müller et al., 2013) mission but to a lesser degree since the radial distance from the Sun where Solar Orbiter will operate will largely be characterized by the unmagnetized plasma condition, *ω*_{pe}/Ω_{e} being much higher than
.

## Acknowledgments

Authors acknowledge the Van Allen Probes EMFISIS (http://emfisis.physics.uiowa.edu/Flight/RBSP-A/) and ECT team (ftp://stevens.lanl.gov/pub/projects/rbsp) for providing online data access and data analysis tool. This work was supported by the basic research funding from KASI. P. H. Y. acknowledges the Science Award grant from the GFT Charity, Inc., and NSF grant AGS1550566 to the University of Maryland. He also acknowledges the BK21 plus program from the National Research Foundation (NRF), Korea, to Kyung Hee University. S. J. acknowledges support by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B03036181).