Field‐Aligned Electron Density Distribution of the Inner Magnetosphere Inferred From Coordinated Observations of Arase and Van Allen Probes

The Radiation Belt Storm Probes (RBSP) and the Arase satellites have different inclinations and sometimes they fly both near the equator and off the equator on the same magnetic field line simultaneously. Such conjunction events give us opportunities to compare the electron density at different latitudes. In this study, we analyzed the plasma waves observed by Arase and RBSP during the three conjunction events during and after the September 7, 2017 storm. The electron number density at the satellite positions was estimated from frequencies of the Upper Hybrid Resonance emissions obtained by the High Frequency Analyzer of the Plasma Wave Experiment onboard the Arase and the Waves instrument onboard the RBSP, respectively. During the three conjunction events, the satellites passed through the plume, inner trough (the narrow region with low electron density between the main body of the plasmasphere and the plume), plasmatrough with variable electron density, and partially refilled plasmasphere. The power‐law index m for the inner trough and plume was inferred to be 4–7 and ∼0, respectively. This is interpreted to mean that the trough was close to collisionless and the plume was relatively near diffusive equilibrium. In the plasmatrough with the varying density, both the high‐density and low‐density regions had m ∼ 0. The low‐density portion of this region may have a different origin from the inner trough, because of the different m indices. For the partially refilled plasmasphere in the storm recovery phase, the power‐law index m showed negative values, meaning that the density in the equatorial plane was higher than at higher latitudes.


Introduction
The plasmasphere is a region in the inner magnetosphere filled with cold, dense plasma from the underlying ionosphere, and the field-aligned profile of its density attracts much interest, because it reflects the physical processes of mass transport from the source region. For example, when the very simple power-law model is used, where n eq is the density on the equatorial plane, L is the magnetic L-shell parameter, R E is Earth radius, R is the geocentric distance to the observation point, and m is the power-law index. The density profile for diffusive equilibrium is closed, having a low-m (m = 0.5-1) power-law distribution, whereas the density profile for collisionless plasma corresponds to high-m (m ∼ 4) power-law distributions (Angerami & Carpenter, 1966;Eviatar et al., 1964;Takahashi et al., 2004).
We have, however, few opportunities to investigate the field-aligned profile of plasma density from the actual observational data. Previous studies have been done using (a) in situ measurements of electron density from plasma wave data by polar orbiting spacecraft (e.g., Denton et al., 2002;Goldstein et al., 2001;Sandhu et al., 2016), (b) passive remote sensing of electron density with whistler waves (e.g., Angerami & Carpenter, 1966), (c) passive remote sensing of plasma mass density with harmonic frequencies of standing Alfven waves in the ultralow frequency (ULF) range (e.g., Denton et al., 2009;Takahashi et al., 2004), and (d) active remote sensing of electron density by the radio plasma imager (RPI) onboard the Imager for Magnetopause-to-Aurora Global Exploration (IMAGE) satellite (Huang et al., 2004;Reinisch et al., 2001). Goldstein et al. (2001) used the electron density measurements of the Polar satellite and obtained a power-law index of m = 0.37 ± 0.8 for high-density regions (n e > 100/cc) and 1.7 ± 1.1 for low-density regions (n e < 100/ cc). Denton et al. (2002) used the same data set as Goldstein et al. (2001) and estimated a power-law index of 1.6-2.1 for the plasmatrough. Takahashi et al. (2004) estimated the power-law index of the plasma mass density and found that m ∼ 0.5 at 4 < L < 6, which is consistent with a diffusive equilibrium solution. On the other hand, no single value of m fit the average observed frequency ratios at 6 < L < 7, and the theoretical solutions indicated that the mass density was peaked at the equator (Takahashi et al., 2004). The subsequent studies by Denton et al. (2009) and Sandhu et al. (2016) analyzed Cluster data and showed that the electron density also had a local maximum in density near the equator in the plasmatrough. As described above, many results have been obtained in which both electrons and ions have had a density distribution close to diffusion equilibrium as long as they have been in the high-density region in the plasmasphere. Also, in high L-shell regions with low density, power-law indices m have had larger values or have not tended to take a specific value, which implied that there may have been a local maximum in density near the equatorial plane.
In recent years, the exploration of the inner magnetosphere has entered the multipoint observation era with satellites flying in formation. In order to obtain a comprehensive understanding of the acceleration, transport, and loss of relativistic electrons in the radiation belts, cross-energy coupling (coupling among the different plasma populations such as the plasmasphere, ring current, and radiation belt) of the inner magnetospheric plasma has been of prime interest (Miyoshi et al., 2016;. To understand the detailed cross-energy coupling processes, it is necessary to know all parameters that contribute to the coupling. Despite the fact that the plasmasphere is the lowest energy population in the inner magnetosphere, it is also important because it dominates the background plasma environment, which controls wave dispersion and resonance conditions. Moreover, the latitudinal distribution of plasma density significantly controls the propagation of plasma waves. In this study, we show simultaneous measurements of the plasma density along the same magnetic field by Arase (Exploration of energization and Radiation in Geospace: ERG) satellite  and Van Allen probes (the Radiation Belt Storm Probes: RBSP, Mauk et al., 2013). We estimate the power-law index of three events during and after the 7 September 2017 storm event using in situ measurements of the electron number density from Arase and RBSP whose orbits have different inclinations. This gives instantaneous information about the density profile along the field lines. Thus, it provides an opportunity to capture dynamic changes of the plasmasphere during magnetic storms.  (Olson & Pfitzer, 1977), and the L-value is McIlwain L-parameter (McIlwain, 1961) for particles with a pitch angle of 90°.

Satellite Orbits
Their orbits seem to cross in the afternoon sector. In fact, the difference of MLT and UT at the same L-values for the two satellites is less than 0.5 hr and 15 min, respectively. Such a conjunction event provides a good opportunity to compare the electron density between the two different latitudes due to the difference of inclinations of Arase and RBSP. As shown in the bottom panel of Figure 1, Arase was ∼10° higher in latitude than RBSP-B.

Evolution of the Plasmasphere During the September 7, 2017 Storm Event
The top and middle panels of Figure 2, respectively, display temporal variations of the Kp and SYM-H indices during the time interval of September 6-10, 2017. They indicate that a magnetic storm with two main phases commenced around 0 UT on 8 September and disturbances continued for approximately 30 hr, followed by a gradual recovery phase. This storm has been considered to be a CME (coronal mass ejection)-driven storm (Redmon et al., 2018). The blue, green, black, and magenta bars in the middle panel indicate the time intervals when the electron density, shown in the bottom panel of this figure, was observed. The event numbers in parentheses are the event numbers defined in Sections 2.3-2.5.
A detailed description of the plasmaspheric dynamics during this storm event is given by Obana et al. (2019); therefore, here we give only a brief description. The bottom panel of Figure 2 shows L-value distribution of electron density obtained by the Arase observations. The exact times are given in the panel. The blue line (0) indicates the plasmasphere, which spreads to L ∼5 before the storm. The green line (1) shows that the plasmasphere has eroded to L = 2.5 during the second main phase. In L > 3 of the green-line profile, the density shows jagged structures. It implies that the satellite was in the plasmatrough with variations in the electron density. In black (2), the plasmasphere has been extremely eroded and the plasmapause reached around L = 1.6-1.7. An isolated increase in density appears to be a plume at L = 3.4-4.3. The magenta line (3) was obtained 2 days after the magnetic storm entered the second recovery phase, and the eroded plasmasphere has partially refilled.

Event #1: 23:20 UT on September 8, 2017
Figure 3 shows the plasma wave observations from Arase and RBSP-B and locations of the satellites during 23:20 UT on 8 September-00:10 UT on September 9, 2017. Figure 3a is the plasma wave spectrogram for the electric field in the frequency range of 10-400 kHz measured by the onboard frequency analyzer (OFA) and the high-frequency analyzer (HFA) of the Plasma Wave Experiment (PWE) Kumamoto et al., 2018) for Arase. The clear emissions of the Upper Hybrid Resonance (UHR) waves are identified by the white line and arrow. The UHR frequency was gradually decreasing from 300 to 100 kHz during 23:20-23:48 UT because Arase was in an outbound pass and then suddenly jumped up to ∼150 kHz at 23:49 UT. This implies In fact, the difference of MLT and UT comparing at the same L-values is less than 0.5 hr and 15 min, respectively. Such a conjunction event gives a significant opportunity to compare the electron density between the two different latitudes due to the difference of inclinations of Arase and RBSP. that the satellite passed the boundary between the low-and high-density regions, which correspond to the inner trough and plasma plume, respectively. Figure 3b shows the plasma wave spectrogram for the electric fields from RBSP-B. These data were measured by the Electric and Magnetic Field Instrument Suite and Integrated Science (EMFISIS) Wave instrument (Kletzing et al., 2013). The UHR waves can be seen as indicated by the white line and arrow and its frequency gradually increased since RBSP-B was in an inbound pass. Its frequency raised in a stepwise fashion at 23:21 UT and decreased around 23:50 UT. This implies that the satellite passed the plume and came into the inner trough around 23:50 UT. differences were less than 0.5 hr. At the moment, the magnetic latitudes of Arase were ∼10° higher than that of RBSP-B due to the difference of their inclinations.
From the upper-limit frequency of the UHR waves (f UHR ), the electron density n e along the spacecraft orbit is estimated using the following equation , 8980 e f f n where f ce is the electron cyclotron frequency. The densities and frequencies are expressed in cubic centimeters and in hertz, respectively. In this study, the local f ce was calculated from the spin average total magnetic field intensity measured by the Magnetic Field Experiment (MGF) instrument (Matsuoka, Teramoto, Nomura, et al., 2018) onboard Arase or the EMFISIS magnetometer onboard RBSP.
The upper panel of Figure 4 shows the electron number density profiles as a function of L. A density jump can be seen around L ∼ 3.45 in the Arase observations (red) and around L ∼ 3.57 in the RBSP-B observations (blue). In both panels, narrow-banded UHR waves are clearly found and indicated by white lines and arrows. In panel 3a, the UHR frequency was gradually decreasing from 300 to 100 kHz during 23:20-23:48 UT because Arase was in an outbound pass and then suddenly jumped up to ∼150 kHz at 23:49 UT. This implies that the satellite passed the boundary between low-and high-density regions, which correspond to the inner trough and plasma plume, respectively. In panel 3b, the observation time is divided into a period showing the high UHR frequency before 23:52 UT and a period showing a low UHR frequency after 23:52 UT. Each corresponds to the times the satellite flying into the high-density plume and the low-density inner trough. The lower three panels show L-value (c), magnetic local time (MLT) (d), and magnetic latitudes (e) of Arase and RBSP-B. They are calculated using the Olson-Pfitzer-Quitet (OPQ 77) magnetic field model (Olson & Pfitzer, 1977). These jumps correspond to the boundary between the inner trough and plume. The difference in the L-values of the boundaries between Arase and RBSP-B may be due to the difference of MLT of the satellites.
Interestingly, the two satellites differed in magnetic latitude by as much as 10°, but in the plume, n e shows similar values at the same L-values. That is, the density of the plume seems to be almost uniform at least in the magnetic latitude range of 10-30°. Whereas in the inner trough, the n e at higher latitudes (Arase) is higher. That is, the n e changes rather steeply along the field line.
When two satellites pass different latitudes of the same flux tube, they, respectively, observe in situ electron densities. Assuming a power-law density distribution as function of geocentric distance, the ratio of the two electron densities is where the subscripts "1" and "2" indicate the values for satellites #1 and #2 and the results shown in the lower panel of Figure 4. For this calculation, moving averages of n e with a 0.05 of window width in L-value were used. The m index is plotted in the panel only if both relative standard deviations (the ratio of the standard deviation to the average) of n e of Arase and RBSP-B in the window are less than 25%. The m indices were 4-7 and −3 to 0 in the inner trough and plume, respectively. These are indicated in the lower panel of Figure 4. The timing of the fluctuations seems to be inconsistent between the two satellites; thus, it is surmised that this change in frequency does not reflect the time change of the electron density, but reflects the spatial variation. As shown in Figure 2, this observation is during the second main phase, with a plasmapause seen at L = 2.5; thus, the fluctuations of f UHR shown in Figure 5 imply the existence of small-scale structures in the plasmatrough.

Event
The results showing this expectation are given in Figure 6. The upper panel of Figure 6 shows the electron number density profiles as a function of the McIlwain's L-parameter calculated using the T04s model during Event #2. At least between L = 3.3 and 3.9, the changes in the electron density observed by Arase and RBSP-B seem to be correlated.
Using ∼10° as the difference in magnetic latitudes (Figure 5e), the m index was calculated as shown in the bottom panel of Figure 6. In both the high-and low-density regions, the derived m index has small values (typically ∼0).
Obviously, the very large negative values for L > 3.9 are meaningless, because the fluctuations in the electron densities observed by Arase and RBSP-B are not correlated to each other in this region.

Event #3: 13:40 UT on September 10, 2017
The third event is a conjunction of Arase and RBSP-A, during the interval of 13:40-14:30 UT on 10 September, in the recovery phase of the storm. Figure 7 (same format as Figure 3) shows a simple picture of emission. The f UHR observed by Arase gradually decreased from 200 to 70 kHz, and those observed by RBSP-A gradually increased from 100 to 200 kHz. This corresponds to the gradual electron density gradient with distance. The time of this event is about 2 days after the commencement of the second recovery phase; thus, the observations depict the density profiles of a partially refilled plasmasphere ( Figure 2). Figure 8 shows the n e obtained by Arase and RBSP-A observations (upper), and the inferred m index (bottom) in the same format as Figure 4. Interestingly, the electron density observed by RBSP-A was higher than that by Arase, despite RBSP-A being at a lower latitude. Because plasmaspheric plasma is supplied from the underlying ionosphere, the density is usually lower near the equator (e.g., Reinisch et al., 2007). However, our results suggest a profile opposite to this. The bottom panel of Figure 8 shows m index, which has negative values due to the higher electron density near the equator.

Accuracy and Coverage of This Study
The three conjunction events shown in this study have <45-min difference in UT and <1-hr difference in MLT when the two satellites cross flux tubes with the same L-value. The L-value includes an uncertainty of 0.05 derived from the window width used for the moving average of n e as indicated in Section 2.3.
Comparing to previous studies, IMAGE-RPI sounding takes only two minutes to make one survey (Song et al., 2005). On the other hand, studies using polar orbiting satellites take several hours until the satellite passes two points of the flux tube with the same L-value (Goldstein et al., 2001). Our surveys have time accuracies intermediate to those studies.
Next, we compare our time accuracies with the timescale of dynamic changes of the plasmasphere. According to Obana et al. (2010), the refilling rates at L = 3.8 were estimated to be 13 amu/cm 3 /hr, so assuming that all the ions are protons, the electron density will increase 10/cm 3 in 45 min. Even in low-density regions (n e ∼ 100/cm 3 ), this will lead to only a 10% increase in density. On the other hand, Goldstein, Sandel, Forrester, and Reiff (2003) investigated images of the He + plasmasphere taken by the IMAGE-EUV camera and reported that the nightside plasmapause had moved inward by ∼2 R E in 3 hr during the erosion. In such a case, the density changes more quickly, and thus, more precise conjunctions (with shorter time differences) will be required.
The difference in MLT must be considered when structures change in both radial and azimuthal directions, like the plume. In fact, as shown in Figure 4, the position of the inner edge of the plume observed by Arase and RBSP was displaced by 0.12 R E . This may be due to the difference in MLT.
The coverage of magnetic latitudes should also be considered. The satellites used in this study explore the magnetic latitudes of <30° (Arase) and <10° (RBSP). Note that the m indices presented in this study do not represent the entire magnetic field line, but only in this range of the magnetic latitude.

Power-Law Index for the Inner Trough and the Plume
In Event #1, the power-law index shows m > 4 at L = 3.3-3.4 (inner trough) and m ∼ 0 at L = 3.6-4.1 (plume). This is interpreted as the inner trough being close to the collisionless and the plume being rather diffusive equilibrium. Previous studies explained that the plume and inner trough are formed by the convection electric field, which causes sunward motion of the plasma, the subsequent corotation, and the subauroral polarization stream (SAPS) electric field. Based on this idea, the plumes originate in the main body of the plasmasphere and the inner troughs originate in the low-density region (so-called outer trough) after the plasmasphere shrank (e.g., Goldstein, Sandel, Hairston, & Reiff, 2003). Our results do not conflict with this model.

Negative Power Law Index in the Partial Refilling Region
In Event #3, power-law index m shows negative values, meaning that the density in the equatorial plane is higher than off the equator. The plasmaspheric density is generally higher closer to Earth. This reflects that the plasma respectively. Bottom: L-value profile of the power-law index m, which is estimated using the differences of electron density and magnetic latitude between Arase and RBSP-B. Between L = 3.3 and 3.9, the changes in electron density observed by two satellites seem to be correlated and thus the m index ∼0 in both the high-and low-density regions.
is of ionospheric origin. However, the electron density in Event #3 shows the opposite features. How can these results be explained?
The period of Event #3 is about 2 days after the commencement of the second recovery phase, and the flux tubes had been partially refilled. When the upper ionosphere comes into the sunlight, the plasma is supplied through photoionization. So, if the length of time after sunrise is different between the flux tubes at Arase and RBSP, the density may be different. However, as shown in Figure 7d, the MLT is almost same for each L-shell, this effect seems unlikely.
Another possibility is the time difference from the end of the main phase. As shown in Figure 7c, the time differences between when the two satellites cross the same L-shells are 0-45 min. According to the estimate in Section 3.1, n e can increase to 0-10/cm 3 for such intervals. This effect is too small to interpret the high n e at the equator. So, we conclude that this difference in density was real, and that there was an increase in density near the equator. Sandhu et al. (2016) statistically analyzed Cluster data and found that the electron density had a localized increase in density near the equator of the plasmatrough. Perhaps even in the partially refilled plasmasphere, the increase in density near the equator may be a common phenomenon. This is an important issue for future research.

Development to Statistical Analysis
In this study, we analyzed three conjunction events between Arase and RBSP. Similar conjunction can occur in different parts (L-value and MLT) of the magnetosphere during the 3 years of 2017-2019, when the two satellite missions overlap. Using the data from such events, statistical analysis of the field-aligned electron density distribution is possible, which is a topic for future research. In particular, statistically examining the structure of the electron density distribution during/after storms is important for understanding the plasma outflow and refilling the process of the plasmasphere.

Application for the Remote Sensing Plasma Mass Density
Standing field-line resonances (FLRs) in the ULF frequency range are ubiquitous in the magnetosphere, and the frequency of FLRs (f FLR ) provides an opportunity to remotely sense the plasma mass density (e.g., Kitamura & Jacobs, 1968;Obayashi & Jacobs, 1958). The procedure for estimating the plasma mass density from the f FLR is to (a) obtain the f FLR from the observation data, (b) choose a suitable magnetic field model, (c) select a suitable model for the variation of the plasma mass density along the field line, and (d) solve the FLR wave equation for the plasma mass density. However, there is very little information about the variation of the density along the field line. Our analysis method makes it possible to obtain this information in the Arase-RBSP era and contributes to the improvement of the accuracy of mass density estimates from the f FLR .

Summary
Using plasma waves observed by Arase and RBSP-A and B, we estimated the power-law index m for the fieldaligned distributions of n e for flux tubes at L = 3.3-4.1. During and after the September 7, 2017 storm event, three conjunction events were studied, in which the differences of MLT and UT for the two satellites were <1 hr and <45 min, respectively. In the plume, n e had comparable values at the higher and the lower latitudes, and the m index was estimated to be ∼0, expecting for rather diffusive equilibrium than collisionless. In the inner trough, n e off the equator was higher than that near the equator, and the m index was estimated as 4-7, corresponding to a collisionless plasma. When in the plasmatrough, n e was variable and the values of n e off the equator and near the equator were similar; the m index was estimated to be ∼0 for both high-and low-density regions. This implies that the plasma transport process might be different between the low-density regions in the variable density structure and the inner trough. In the partially refilled plasmasphere, n e near the equator was higher than off the equator. This suggests the existence of plasma concentration around the equator.

Data Availability Statement
Science data of the ERG (Arase) satellite were obtained from the ERG Science Center operated by ISAS/JAXA and ISEE/Nagoya University (https://ergsc.isee.nagoya-u.ac.jp/index.shtml.en; Miyoshi, Hori, et al., 2018). The present study analyzed PWE-HFA-L2 V01_01   The Van Allen Probes/EMFISIS data are available at http://emfisis.physics.uiowa.edu. The present study used the EMFISIS-L2 v1.6.3 data. We thank the EMFISIS team for providing the EMFISIS data. The solar wind parameters are obtained from the NASA OMNI website (https://omniweb.gsfc.nasa.gov/). We acknowledge the use of NASA/GSFC's Space Physics Data Facility's OMNIWeb service and OMNI data.