Introduction

The inner magnetosphere is a significant and intricate area of magnetospheric research. The dipole region of the magnetosphere close to Earth is called the inner magnetosphere1. It spans from approximately 1 to 10 Earth radii and includes a dipole magnetic field, a co-rotating electric field resulting from Earth’s rotation, and a convection electric field generated by the solar wind2. These fields interact with the plasma present in space and give rise to three large-scale structures: the plasmasphere3,4, the ring current5,6,7,8,9, and the radiation belts10,11, each corresponding to a population of particles having energies within a certain range. Low-energy particles that exit the ionosphere are influenced by the convection and co-rotating electric fields, which form the plasmasphere and can produce waves12,13 that affect signal transmission between Earth and spacecraft14. Medium-energy particles are influenced by the magnetic field and travel around Earth, forming the ring current15, which weakens the Earth’s magnetic field16,17. High-energy particles in the radiation belts could ‘kill’ the instrument of spacecraft18. The inner magnetosphere plays a crucial role in the interactions between fields and plasma in space, energy conversion and matter transport, and serves as a key region in the research of the solar-magnetosphere–ionosphere coupling19,20,21,22.

Medium- and high-energy particles that are consistently injected into the inner magnetosphere from the magnetotail are influenced by the convection electric field and dipole magnetic fields, resulting in two distinct types of paths within the inner magnetosphere23. One is open paths that extend from the magnetotail towards the dayside magnetopause, while the other is closed paths that orbit around the Earth. The boundary between the two types of paths is known as the Alfvén layer. To simulate the Alfvén layer, we adapted a previous analytic model for ring current particles24, while assuming a dipole magnetic field with a uniform dawn-dusk electric field and a corotation electric field and setting the initial kinetic energies of protons and electrons to 6 keV and 2 keV, respectively, as shown in Fig. 1. We found that due to the opposite magnetic drift directions of electrons and ions, the Alfvén layer for electrons and ions is asymmetric and does not overlap completely, resulting in net charge accumulation near the separatrix path for particles of each sign. The phenomenon of charge accumulation in the inner magnetosphere was first proposed by Shield25, but direct observational studies of the accumulation and distribution of charges have been limited due to experimental constraints.

Fig. 1: Illustration of the formation of the Alfvén layer viewed from the Northern Hemisphere with the sun at the top.
figure 1

The protons and electrons are injected into the inner magnetosphere from the plasma sheet on the night side under the action of a dawn–dusk electric field. The red and blue thin lines represent the drift paths of protons and electrons, respectively, in a dipole magnetic field with a uniform dawn–dusk electric field and a corotation electric field. The initial kinetic energies of protons and electrons are set to 6 keV and 2 keV, respectively. The red and blue thick lines denote the proton and electron Alfvén Layer, respectively. The pink area and sky-blue area indicate the regions where the positive and negative charges accumulate due to charge separation.

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The separation of charge generates a shielding electric field that protects the Earth’s surface from harmful effects of the particles and radiation26,27. The charge discharge along magnetic field lines generate Region-2 field-aligned currents (FACs)28 that flow into and out of the ionosphere on the dusk and dawn sides, respectively29. The Region-2 FACs play an important role in space weather phenomena such as magnetic storms and substorms30. Understanding the charge distribution and the relationship between the electric field, charge, and geomagnetic activity is important in predicting and mitigating the impact of space weather.

The recent MMS mission31, consisting of four identical spacecraft, has enabled highly accurate in situ four-point electric-field measurements and thus by Gauss’s law the acquisition of space charge density, which has been further utilized to explore the electric properties of various space physics phenomena such as electron holes32, magnetic reconnection33, and magnetopause34.

In this study, we utilized Gauss’s law and the four-point electric field measurements made by MMS in Earth’s inner magnetosphere to calculate the spatial distribution of the electric charge density there. We present two case studies at the dawn and dusk side respectively to show the features of charge density along the trajectories of the satellites. Furthermore, by using MMS data collected from 2015 through 2018, we perform a statistical investigation on the distribution of charge density in the inner magnetosphere and find that positive (negative) charges accumulate at dusk (dawn) and vary with geomagnetic conditions. The study reveals the presence of charge accumulation in the inner magnetosphere and confirms the existence of the Alfvén layer. The findings provide valuable insights into the plasma dynamics in the inner magnetosphere and can help clarify the magnetosphere-ionosphere energy exchange mechanism.

Results

Event analysis

On 17 December 2015, during a relatively quiet period with a maximum auroral electrojet index of 309 nT, we used data from the MMS mission as it moved from 8.8 Re away from Earth to 4 Re on the dusk side of the inner magnetosphere. As shown in Fig. 2b, the charge was always positive, the charge density (see the Methods section, calculation of charge density calculation) increased with decreasing L-shell, and the maximum charge density was 49 e m−3 at approximately L = 5.91. Additionally, the magnetic field intensity gradually increased from 65 nT to 487 nT, as shown in Fig. 2a. Figure 2e, f delineates spacecraft position at the start \({t}_{0}\) and end \({t}_{1}\) of the observation period.

Fig. 2: Overview of Observations by MMS on 17 December 2015.
figure 2

Overview of field and charge from the MMS constellation on 17 December 2015. a magnetic field intensity B at the center of the constellation, which is the average of the magnetic fields measured by the four spacecraft of MMS; b charge density\(\rho\) distribution; c electric field E at the center of the constellation, which is the average of the electric fields measured by the four spacecraft of MMS; d parallel electric field \({{{{{{\rm{E}}}}}}}_{//}\); e, f the spacecraft positions.

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Figure 3 shows the field and charge distributions on the dawn side on 23 February 2016 during a quiet period (maximum AE index was 174 nT). The satellites moved from 4\({{{{\mathrm{Re}}}}}\) away from Earth to 10, as shown in Fig. 3e, f. As shown in Fig. 3a, the magnetic field intensity decreased gradually from 428 nT to 43 nT. As shown in Fig. 3b, the charge was always negative, and within L = 7, the charge density decreased sharply with increasing L-shell; the maximum negative charge density was −21 e m−3 at L = 4, and between L = 7 and L = 9.8, the charge density remained at approximately −3.47 e m−3. As shown in Fig. 3c, d, the electric field displayed a high-frequency disturbance, and the overall trend was that the electric field intensity decreased with increasing L-value of the drift shell, similar to Fig. 2c, d.

Fig. 3: Overview of Observations by MMS on 23 February, 2016.
figure 3

Overview of field and charge from MMS on 23 February 2016. a magnetic field intensity B at the center of the constellation, which is the average of the magnetic fields measured by the four spacecraft of MMS; b charge density\(\rho\) distribution; c electric field E at the center of the constellation, which is the average of the electric fields measured by the four spacecraft of MMS; d parallel electric field \({{{{{{\rm{E}}}}}}}_{//}\); e, f the spacecraft positions.

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The above comparison between the two cases reveals that positive (negative) charges accumulate at dusk (dawn), and the density decreases with increasing L-shell. To verify this case study and investigate the mechanism of charge accumulation, a statistical study of the net charge distribution is necessary.

Observations of electric field and electric charge density

We project \({E}_{y},{E}_{||}\), and the charge density directly onto the equatorial plane in the solar magnetic (SM) coordinates along the magnetic field lines, assuming that the electric field is constant along a magnetic field line. To carry out a statistical study, first, we separate the data into three groups according to geomagnetic conditions as described by the AE index: (i) quiet times for \({{{{{\rm{AE}}}}}} < 200{{{{{\rm{nT}}}}}}\), (ii) weakly disturbed periods for \({{{{{\rm{200nT}}}}}}\le {{{{{\rm{AE}}}}}} < 500{{{{{\rm{nT}}}}}}\), and (iii) strongly disturbed periods for \({{{{{\rm{AE}}}}}}\ge 500{{{{{\rm{nT}}}}}}\). Then, we separate the space according to L-shell and magnetic local time (MLT) with steps of \(\varDelta L=0.5\) within \(4\le L\le 10\) and \(\varDelta {{{{{\rm{MLT}}}}}}=1{{{{{\rm{h}}}}}}\) in the entire MLT region. Thus, in total, there are 12 × 24 bins in space. Finally, we calculate the average values for the electric field components \({E}_{y}\) and \({E}_{||}\), as well as the electric charge density in each bin. During the quiet periods, the number of paths in each bin varied in the range of 4–69, whereas during the weakly and strongly disturbed periods, the corresponding ranges were much lower (1–32 and 0–16, respectively). The ratio of path numbers for the quiet, weakly disturbed, and strongly disturbed periods occupied 47.40%, 34.79%, and 17.79% of the whole observational duration, respectively.

Figure 4 shows the calculated results for the electric field (\({E}_{y}\), \({E}_{||}\)) and the charge density after being projected and binned onto the equatorial plane in SM coordinates.

Fig. 4: Statistical Results of Electric Fields and Charges.
figure 4

Spatial distributions of electric field and charge density derived from MMS data over all MLT(magnetic local time) and \(4\le L\le 10\) in the inner magnetosphere during three geomagnetic conditions (distinguished by the columns): a–c The y-component of the electric field Ey, d–f charge density\(\rho\) shown by their color scales, and (g–i) parallel electric field E|| shown by its color scale, where red corresponds to a poleward electric field and blue to an equatorward field.

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Figure 4a–c shows the distribution of Ey, where red corresponds to a duskward electric field (Ey > 0) and blue corresponds to a dawnward electric field (Ey < 0). During the quiet periods (i.e., in the early morning sector of 0–4 MLT with L = 6–10 and in the local time sector of 10–22 MLT on the dusk side with L = 5–10) are dawnward electric fields that contain the polarization field produced by the accumulated polarization charges in the Alfvén layer25,26. In the 5–8 MLT sector and the region limited by 22–24-4 MLT and L = 4–6 are duskward electric fields that are enhanced in disturbed periods and spread to cover 5–24 MLT, whereas the early morning sector of 0–6 MLT is dominated by a dawnward electric field. During strongly disturbed periods, Ey is enhanced overall.

Figure 4d–f shows the polarization charge distribution, where red corresponds to positive charges and blue corresponds to negative charges. Some positive charges gathered in 6–22 MLT in a limited L-shell range of 4 < L < 10, and their density increased statistically with increasing geomagnetic disturbance. An azimuthally confined region of 0–6 MLT in quiet periods was occupied mainly by negative charges whose density was much smaller than that of the positive charges on the dusk side. During disturbed periods, the negative charges in this region seemed to fade away. The charge distributions on the dawn and dusk sides were generally consistent with the theoretical charge separation resulting from the grad-B drift and curvature drift of charged particles35.

Next, we analyze the parallel electric field (Fig. 4g–i) that is closely related to the FACs. Because field-aligned electric fields are generated near the equatorial plane and are symmetrical with respect to it36, the parallel components of the electric fields measured on the MMS away from the equator (>5°) in the magnetosphere can be divided into two categories: equatorward (blue) and poleward (red), with the poleward representing the direction from the equator to the South and North poles.

Figure 4d–i clearly shows a distributional accordance in the local time sector of 10–19 MLT between positive charges and the poleward parallel electric field and in 0–6 MLT between negative charges and the equatorward parallel electric field. This accordance can be explained by a direct application of Gauss’s law that electric fields point away from positive charges (thus poleward for 10–19 MLT) and point toward negative charges (thus equatorward for 0–6 MLT). However, at 6–10 MLT at dayside, the accordance was contravened, perhaps because of some unknown physical processes in this local time sector require more-detailed studies in event analyses.

Two distinguished regions

The observations presented above reveal that there were two distinguished MLT sectors: (i) the local time sector limited by 13–16 MLT (sector A), where positive charges were accumulated, and (ii) the sector limited by 2–5 MLT (sector B), where negative charges were accumulated. The observations in these two sectors are consistent with the theoretical predictions for the dusk- and dawn-side parts of the Alfvén layer25,36. Then, we perform further analysis on the two sectors by averaging the charge density over every \(\varDelta L=0.1{{{{{\rm{Re}}}}}}\) (horizontal axis) in Fig. 5a–c (sector A) and Fig. 5d–f (sector B) during different geomagnetic conditions (shown by the columns in the figures).

Fig. 5: Comparative analysis of L-shell charge density distribution.
figure 5

a–c L-shell distribution of charge density in local time sectors 13–16 MLT (magnetic local time) (sector A) and (d–f): 2–5 MLT (sector B) during quiet, weakly disturbed, and strongly disturbed periods.

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As shown in Fig. 5a, the charge density in sector A was positive. During quiet periods, the charge density diminished gradually towards the outer shells; for instance, it was 36.65 e m−3 at L = 4 and 15.39 e m−3 at L = 10; the same trend occurred during disturbed periods, as shown in Fig. 5b, c. By comparing Fig. 5a–c, it was evident that the positive charge accumulations grew with the geomagnetic disturbances. The maximum charge density for all L-shells was 47.97 e m−3 in weakly disturbed periods and 72.02 e m−3 in strongly disturbed periods.

As shown in Fig. 5d, the charge density in sector B during quiet periods was always negative, with an average value of −8.59 e m−3; during disturbed periods, as shown in Fig. 5e, f, its magnitude diminished slightly.

By comparing the charge density distributions in Fig. 5a–f, in sector B on the dawn side, the amount of charge was significantly less than that in sector A on the dusk side. For L = 4–10, the average charge density in sector A was 22.92 e m−3 during quiet periods (approximately three times that in sector B) and 27.17 e m−3 during strongly disturbed periods [six times that in sector B (−4.93 e m−3)]. This finding implied a significant dawn–dusk asymmetry in the net charge distribution of the Alfvén layer in the inner magnetosphere, and this asymmetry become more severe with enhanced geomagnetic disturbances.

Conclusion

Based on the electric field data from the EDP onboard the MMS satellites, we revealed and analyzed the changes in the charge and electric field in the Earth’s inner magnetosphere under different distribution conditions and exhibited, for the first time, the observational features of the Alfvén layer. The findings are summarized as follows. 1) The inner magnetosphere accumulates a positive charge at dusk and a negative charge at dawn. The charge decreases with increasing L-shell value and varies with magnetic activity. 2) The distributions of charge and electric field confirm the existence of charge separation in the Alfvén layer observed by multipoint satellites. 3) The charge distribution in the Alfvén layer has a dawn–dusk asymmetry, with the positive charge density at dusk being much greater than the negative charge density at dawn.

The data used in this study were measured during a solar minimum, and a few samples were collected for events with an AE index greater than 1000 nT. Future work is needed to study the charge distribution during periods of stronger geomagnetic activity.

Discussion

Here, we will explore additional intriguing phenomena that were not previously discussed. For instance, in examining Figs. 4 and 5, we observe a dawn-dusk asymmetry in charge density, contributing to the potential mechanism behind the asymmetry of the Alfvén layer. The current carriers for the Region-2 FACs should be protons and electrons. Given the substantial mass and inertia of protons, a robust field-aligned electric field on the twilight side is necessary to propel them into the ionosphere along magnetic field lines. Conversely, electrons, with their lesser mass and inertia, flow more readily into the ionosphere along magnetic field lines, even in the presence of a relatively weak field-aligned electric field. Figure 6 demonstrates how the Alfvén layers are connected to the polar ionosphere. The ionosphere discharges the Alfvén layers through geomagnetic field lines, creating the Region-2 FACs. The Poynting fluxes point toward the polar regions, facilitating the transfer of electromagnetic energy to the ionosphere. The discharge mechanism of charges into the ionosphere via Region-2 FACs remains uncertain due to the rarity of satellite magnetosynchronization in these regions and the underdeveloped technology for detecting charge densities in the ionosphere. Nevertheless, some evidence supports the significant role of Region-2 FACs in connecting the ionosphere and magnetosphere37,38,39. Notably, the comparison of FAC densities in the ionosphere and magnetosphere using Swarm and Cluster multispacecraft revealed similar latitudinal profiles despite notable differences in FACs transit times40. Observations also indicate a nearly constant current between Arase and AMPERE, implying a connection between the magnetosphere and the ionosphere through FACs at low latitudes, as the ratio of current density along the magnetic lines of force to the total magnetic field remains constant41. Anticipating future events conducive to joint observations with the ionosphere, we aim to gain a comprehensive understanding of the interaction between Region-2 FACs and charge accumulation, contributing to a deeper comprehension of the coupling mechanisms between the magnetosphere and the ionosphere.

Fig. 6
figure 6

The plasmas convection in the inner magnetosphere creates the Alfvén layers, which act as dynamos to drive the Region-2 FACs \({J}_{R2}\) connecting the polar ionosphere. The Poynting fluxes S, which are associated with the magnetic field disturbances \(\delta B\) generated by Region-2 FACs and the electric field E originated from the Alfvén layers, point toward the polar regions and transfer electromagnetic energy to the ionosphere.

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In the realm of magnetospheric physics, the observed charge density in this study appears to be exceptionally high. This may be attributed to a lack of precision in the projection method. Owing to the characteristics of the Earth’s dipole magnetic field, magnetic field lines become denser in the vicinity of the Earth’s polar regions. Given that magnetic field lines represent equipotential lines, the charge density increases as the satellite gets closer to the Earth’s polar region as indicated by the Poisson equation (\(\rho \,=-{\varepsilon }_{0}\,{\nabla }^{2}\varphi \,\)). Consequently, the charge density increases as the L value decreases. When estimating charge density along magnetic field lines, it is essential to scale it based on the density of the magnetic field lines, and the scaling factor is contingent on the corresponding magnetic field line density within the unit volume. Furthermore, in the observations of FACs, it has been noted that the distribution of these currents in the magnetosphere is transiently discontinuous, exhibiting discrete structures. When a satellite observes within the same flux tube, the observed charge density increases significantly. We have also considered the impact of satellite-induced electric fields, corotation electric fields, and satellite motion vorticity on charge density. The results suggest that these effects are minimal, although it is conceivable that other physical processes may counteract some of the impacts of charge separation in the Alfvén layer.

Figure 4 also shows that the electric field distribution and charge distribution have a westward rotation of 45°. It is commonly assumed that the electric current distribution is symmetrical at approximately 0–12 MLT in the magnetosphere and ionosphere41,42,43, and that the convection electric field in the ionosphere is perpendicular to the line of 0–12 MLT44. However, as shown in Fig. 4, the axis of symmetry of the E|| distribution is roughly at 3–15 MLT, and the equatorial electric field in the magnetosphere is mainly a tailward electric field. The phenomenon of westward rotation was also found in previous studies45,46.

The region we observed, which is between 4–10 Re from Earth, partly overlaps with the plasmasphere that contains dense cold plasma with a density ranging from over 104 cm−3 down to about 10 cm−3, which is much greater than the charge density. Although charge compensation may occur, our statistical findings and individual event observations suggest that the charge compensation process in the plasmasphere cannot entirely offset the excess charge resulting from charge separation. It is unfortunate that the plasma data are severely missing when MMS is in the inner magnetosphere, despite being equipped with FPI instruments. Anticipated future advancements in measurement technologies hold the promise of providing more nuanced data for a comprehensive understanding of the inner magnetosphere.

Methods

Charge density calculation

We use the electric- and magnetic-field data from all four MMS spacecraft that orbit Earth in near-equatorial space at low latitude with a perigee of 1.2 \({{{{\mathrm{Re}}}}}\) and an apogee of 12 \({{{{\mathrm{Re}}}}}\). The data were collected three years from September 2015 to September 2018, during which the constellation swept the full range of MLT more than three times. The data from the field instrument suite47,48,49—including electric field measurements—were provided by two sets of biased double-probe sensors (EDP)50 with a temporal resolution of 8−1 s to 32−1 s.

For electric field data products, a quality indicator is provided, ranging from 0 to 3. In this scale, 0 denotes “ very bad data or no data available,” 1 denotes “ bad data, use with caution,” 2 denotes “ fair data, use with caution,” and 3 denotes “ good data.” In our research, we deliberately chose data products with quality indicators of 2 and 3, representing “ok” and “good” data, respectively. In the inner magnetosphere, the magnitude of the electric field is typically less than 100 mVm−1 51,52, so to ensure statistical quality, we have removed any outliers whose values exceed 1000 mVm−1.

Let the position vectors of the four MMS spacecraft be \({{{{{{\bf{r}}}}}}}_{\alpha }\,(\alpha =1,2,\cdot \cdot \cdot ,4)\) in barycentric coordinates53, with \(\frac{1}{4}\mathop{\sum }\nolimits_{\alpha =1}^{4}{{{{{{\bf{r}}}}}}}_{\alpha }=0\). The electric fields measured by the four spacecraft can be expressed as \({{{{{{\bf{E}}}}}}}_{\alpha }={{{{{\bf{E}}}}}}({{{{{{\bf{r}}}}}}}_{\alpha }),\,\alpha =1,2,\cdot \cdot \cdot ,4\), and the linear gradient of component Ei of the electric field can be derived as follows54:

$${\nabla }_{{{{{{\rm{j}}}}}}}{{{{{{\rm{E}}}}}}}_{{{{{{\rm{i}}}}}}}=\frac{1}{4}\mathop{\sum }\limits_{{{{{{\rm{\alpha }}}}}}=1}^{4}{{{{{{\rm{E}}}}}}}_{{{{{{\rm{\alpha }}}}}}{{{{{\rm{i}}}}}}}{{{{{{\rm{r}}}}}}}_{{{{{{\rm{\alpha }}}}}}{{{{{\rm{k}}}}}}}{{{{{{\rm{R}}}}}}}_{{{{{{\rm{kj}}}}}}}^{-1}$$
(1)

where the volumetric tensor is defined as \({{{{{{\rm{R}}}}}}}_{{{{{{\rm{kj}}}}}}}=\frac{1}{4}\mathop{\sum }\nolimits_{{{{{{\rm{\alpha }}}}}}=1}^{4}{{{{{{\rm{r}}}}}}}_{{{{{{\rm{\alpha }}}}}}{{{{{\rm{k}}}}}}}{{{{{{\rm{r}}}}}}}_{{{{{{\rm{\alpha }}}}}}{{{{{\rm{j}}}}}}}\), which characterizes the constellation geometry. The truncation error for the gradient of the electric field is on the order of S/D, where S is the characteristic size of the constellation, and D is the spatial scale of the electric field55. The characteristic scale of MMS is generally S ≈ 20 km, while the spatial scale of the electric field is assumed at the scale of one Earth radius (\({{{{\mathrm{Re}}}}}\) = 6371 km), as indicated by the empirical model of large-scale electric field in the inner magnetosphere56. So the truncation error for the gradient of the electric field is (S/D)2 ≈ 0.0009%57, which implies the high accuracy of the calculation with formula (1). The four satellites are closely spaced so that the measured trends in the electric field are always consistent. This consistency suggests that the non-smoothness observed in the electric field represents a temporal oscillation, thus affirming the feasibility of using the spatial linear approximation58 in charge calculations.

Based on the linear gradient of the electric field measured by each spacecraft, the charge density at the barycentre of the spacecraft constellation is calculated using Gauss’s law as follows:

$${{{{{\rm{\rho }}}}}}\,={{{{{{\rm{\varepsilon }}}}}}}_{0}\,\nabla \cdot E={{{{{{\rm{\varepsilon }}}}}}}_{0}\,\mathop{\sum }\limits_{{{{{{\rm{i}}}}}}=1}^{3}{\nabla }_{{{{{{\rm{i}}}}}}}{{{{{{\rm{E}}}}}}}_{{{{{{\rm{i}}}}}}}$$
(2)

To simulate the actual charge density measurement, we randomly placed a tetrahedron inside a non-uniformly charged sphere. The error in the charge calculation, denoted as \(\varDelta \rho /\rho =(\rho {{\hbox{'}}}-\rho )/\rho\), is defined as the difference between the charge density(\(\rho\)) calculated from the electric field divergence and the actual charge density \(\rho {{\hbox{'}}}\). After running the model 1000 times with random tetrahedron shape inputs, we obtained a maximum error of 7.73% and an average error of 0.1%. These results suggest that the method used for the charge density calculation is reliable.

As shown previously, the observational error in the charge density calculated from MMS electric measurements is \(\delta \rho \, \approx \, {\varepsilon }_{0}(\frac{\delta {E}_{A}}{S}+2\frac{\delta {E}_{S}}{S})\), where the measurement accuracy of the axial electric field \(\delta {E}_{A} < 1{{{{{\rm{mV}}}}}}/{{{{{\rm{m}}}}}}\), and the spin axial electric field \(\delta {E}_{s} < 0.5{{{{{{\rm{mVm}}}}}}}^{-1}\). So that \(\delta \rho \, \approx \, 5.53{{{{{{\rm{em}}}}}}}^{-3}\), where e is the charge of electron, and e = 1.602 176 634 × 10−19 Coulombs. Here, we remove outliers in the charge density (which accounted for less than 0.001% of all the data) whose values exceed 1000 e m−3. Notably, the MMS constellation, at which the electric field is measured, is moving at an orbital velocity of approximately 1 ~ 10 km s−1 relative to Earth. The charge density calculated with Eq. (2) needs to be transformed under a Lorentz transformation59 to obtain the corresponding charge density as viewed in the Earth center frame of reference. However, because the orbital speed of MMS is much less than the speed of light in a vacuum, the change in the charge density under the Lorentz transformation is small and well below the observational error of the charge density (\(\delta \rho\)) given above. Therefore, we neglect the effect of relativity here and assume that the charge density of the plasmas observed in the Earth center frame of reference is the same as that measured in the MMS constellation frame of reference.