The plasmasphere tomography (PlaTo) mission concept

Highlights

• A focused radio tomography mission for the plasmasphere is explored,
• This dynamic science investigation can be met with only 14 spacecraft,
• Enables investigations into interchange, drainage plume structuring, and refilling.

1 Introduction

Most of the constituent mass of the Earth’s magnetosphere is located in the plasmasphere (Delzanno et al., 2021), a torus of cold (∼1 eV), dense (∼100–10,000 cm⁻³) plasma primarily of ionospheric origin extending from the topside ionosphere out to a radial distance between 2 and 6 Earth radii (R_E) in the equatorial plane depending on geomagnetic activity (Carpenter and Anderson, 1992; Goldstein and Sandal, 2005]. The plasmasphere can be viewed as cold, high-density plasma co-rotating with the planet. The outer edge of the plasmasphere is defined by the plasmapause, often identified by a steep density drop (>5×) over a spatially narrow (<0.5 R_E) region (Carpenter and Anderson, 1992). During geomagnetically quiet times, up-flowing plasma from the ionosphere accumulates on magnetic flux tubes and co-rotates with Earth. At these times, the plasmasphere has the greatest spatial extent, extending to equatorial radial distances of ∼6 R_E and beyond. During geomagnetically active times, the quasi-steady dawn-to-dusk convective electric field and corresponding sunward ExB drift associated with the Dungey cycle (Dungey, 1961) become enhanced. The region characterized by open drift paths, where the ExB drift associated with sunward convection dominates that associated with corotation, extends earthward. The plasmasphere plasma on formerly closed drift paths is now on open drift paths, allowing that plasma to escape sunward. The result is plasmasphere erosion, with the plasmasphere contracting as the outer plasma is carried sunward in a flow channel known as the plasmasphere drainage plume (henceforth, “plume”). This basic picture of the large-scale plasmasphere dynamics has been known for some time (e.g., Nishida, 1966; Brice, 1967; Grebowsky, 1970; Chappell et al., 1970a; Chappell et al., 1970b). However, a variety of mesoscale structuring and processes embedded in the large global plasmasphere structure play important roles as well, both in terms of controlling the response time of the system to solar storms (Yu et al., 2022) and, as discussed below, processes that are fundamental for controlling other plasma populations, such as the radiation belts, and are, therefore, important for space weather effects.

Plasmaspheric dynamics are a critical component of many other magnetospheric processes. The presence and spatial extent of the high-density regions of plasma, that is, the plasmasphere and plume, determine the distribution of very low frequency (VLF) and ultra low frequency (ULF) waves in the magnetosphere. These waves play a role in the production and loss of energetic particle populations and thus in the magnetosphere radiation environment. The plasmasphere modifies the spatial distribution of whistler-mode waves, with hiss primarily located inside the plasmasphere and dense regions of the plume, and chorus being predominantly outside the plasmasphere (Malaspina et al., 2016). Whistler-mode waves both accelerate relativistic electrons (Horne and Throne 1998; Summers et al., 1998; Horne et al., 2005) and result in their loss (Lyons and Thorne, 1973; Ripoll et al., 2017; Thorne et al., 1973). VLF and other wave–particle interactions depend on the ratio of the plasma to cyclotron frequencies and are thus sensitive to the distribution of plasma density (Agapitov et al., 2020).

While whistler-mode chorus and hiss serve as important sources and loss mechanisms for energetic electrons, respectively, their efficiency is fundamentally governed by the plasmaspheric structure. Specifically, the plasma density regulates wave generation by modulating the ratio of plasma to cyclotron frequencies (Chen et al., 2012; Su et al., 2001; Liu et al., 2023) and affects wave propagation through the modulation of density gradients (Sonwalkar, 2006; Streltsov, 2022; Wu et al., 2021). These combined effects result in regional variations in wave power, wave propagation vectors, and other wave properties. The plasmasphere regulates other kinds of waves which play critical roles in radiation belt dynamics, such as EMIC waves (Chen et al., 2009; de Soria-Santacruz et al., 2013), magnetosonic waves (Ma et al., 2014; Wu et al., 2021; 2023), and ULF waves (Elkington, 2006 and references therein). In the magnetosphere, ULF waves obey magnetohydrodynamics (MHD), and their propagation characteristics are sensitive to the Alfven speed, V A = B / ρ m μ 0, where (for a single ion species with mass m) the mass density is ρ m = m n, n is the plasma number density, and μ 0 is the permeability of free space. As a result, the density gradients associated with the plasma distribution result in refraction and reflection of ULF waves. For example, the plasmapause location (Hartinger et al., 2013; Claudepierre et al., 2016) and plume geometry (Wright and Elsden, 2016; Sandhu et al., 2021) are controlling factors in the distribution of ULF power in the magnetosphere. Effectively, the plume and space between the plasmapause and magnetopause form resonance cavities for these waves. By controlling the spatial distribution of ULF wave power in the magnetosphere, the plasmasphere impacts where drift-bounce resonance between drifting particles and ULF waves is most efficient. The regions where various wave–particle interactions are more or less prevalent and, hence, where energization and loss processes are more or less active depend on the spatial extent of the plasmasphere.

One of the limiting factors in our ability to study the plasmasphere empirically is the spatial and temporal resolution of the observations. A satellite making in situ measurements in the inner magnetosphere typically makes single-point measurements along its trajectory. A typical period for an inner-magnetospheric orbit is ∼8 h, and the time required for the plasmasphere to erode is ∼3–4 h (Goldstein et al., 2005). Thus, while such in situ measurements can capture a density profile in L-shell during the erosion process, the measured profile is evolving considerably over an orbital period. Remote sensing of the plasmasphere, such as observing scattered extreme ultraviolet (EUV) radiation from He⁺ or O⁺, provides maps of line-of-sight intensity corresponding to the plasma density. The EUV measurements thus capture only minor ions, which may behave differently from H⁺, which cannot be imaged using scattered solar photons. The IMAGE satellite EUV camera could capture a full view of the line-of-sight integrated density of the plasmasphere, where the density was >40 cm⁻³ at a 10-min cadence (Goldstein and Sandel, 2005). While studies using space-based remote sensing methods have significantly advanced our understanding of the plasmasphere, they are fundamentally a line-of-sight measurement of a three-dimensional region of space, requiring assumptions to deconvolve observations. The mission proposed here instead measures two-dimensional fields of electron density with mesoscale (≤0.5 R_E) resolution, 5-min cadence, and density accuracy of a few cm⁻³, all of which, as discussed below, are needed to observe various processes and dynamics.

One way to directly observe electron density dynamics is by using radio tomography. Radio tomography with a constellation of spacecraft can create 2-dimensional maps of the plasma density in the spacecraft orbit plane at spatial resolution and cadence depending on the orbits and number of spacecraft used (e.g., Ergun et al., 2000; Zhai and Cummer, 2005; Malaspina et al., 2022). If the transmissions are uncoded, the measurements are made as the spacecraft, one by one and in turn, emit a pair of phased signals of different frequencies, a primary frequency and a reference frequency, for a certain duration. These signals are received by the other spacecraft. Transmissions can be made all at once if the transmissions are coded and therefore distinct from one another. From the phase difference between the two frequencies at each of the receiving spacecraft, the total electron content (TEC) along the ray path between them can be determined. Ambiguities when the phase change is larger than π can be resolved by comparing the delay time with the group velocity difference (Ergun et al., 2000) together with in situ density measurement made at each spacecraft. From the set of all total electron content (TEC) measurements, tomographic inversion can be performed to produce 2D maps of plasma density, similar to computed tomography (CT) scan images used in the medical industry (Zayed 2019). The orbits, number of spacecraft, transmitter frequencies, and cadence of the radio transmitting cycle between all spacecraft all determine the resulting spatial and temporal scales, as well as the accuracy with which the density maps can be derived.

In Figure 1, we show a conceptual depiction of such a spacecraft constellation on two orbits (represented by red dots on the inner orbit and blue dots on the outer orbit), including ray paths (purple dashed lines) between each pair of spacecraft. The 2D maps generated using radio tomography overcome the fundamental limitation of single-point measurements by revealing the density structure simultaneously present over an extended area and show how it evolves in time. The concept of using radio tomography on satellite constellations is not new; it has been discussed for the ionosphere (Fremouw et al., 1992; Bruno et al., 2020) and has been suggested as a way to study the near-Earth plasma sheet and subsolar magnetopause (Ergun et al., 2000) and the plasmasphere (Malaspina et al., 2022). Different constellation configurations require different tomographic techniques.

While multiple radio tomography mission concepts for the advancement of space science have been discussed in the literature, the closest relation to the study presented herein is the PILOT (Plasma Imaging, LOcal Measurement, and Tomographic Experiment) mission concept (Malaspina et al., 2022). The PILOT concept study was designed to provide a comprehensive approach to understand the filling and erosion of the plasmasphere and the impacts on the dynamics of the magnetosphere as a strategic mission concept for consideration by the 2024 Heliophysics Decadal Survey. Many of the science goals from PILOT can be addressed using a focused constellation (e.g., PlaTo). A focused radio tomography spacecraft constellation is an effective and cost-effective tool for such investigations. A small-to-medium science mission concept has therefore been developed to demonstrate the important science objectives that such a mission can address.

This article presents the Plasmasphere Tomography (PlaTo) mission in response to the need to study the plasmasphere cost effectively. Section 2 of this article discusses the science goals and traceability that drive the PlaTo architecture. Section 3 presents an estimation of the number of spacecraft and evaluates the spatial and temporal resolution that can be achieved, related to the number of spacecraft. Section 4 presents the basic properties of a strawman radio payload for the PlaTo mission. Finally, we summarize the results of the investigation and discuss the next steps required to develop a radio tomography mission of the plasmasphere.

2 Science goals and traceability

2.1 Plasmasphere drainage plume structuring

The plume plays an important role in the coupling between the solar wind and magnetosphere. The plume flows to the site of dayside reconnection (Borovsky et al., 1997; Su et al., 2000; Su et al., 2001). By transporting this dense plasma into the reconnection site, the reconnection rate is changed (Borovsky and Denton, 2008; Walsh et al., 2014), and thus the coupling between the solar wind and magnetosphere is modified. Both the density and flow velocity are known to be highly structured (Moldwin et al., 1995; Spasojević et al., 2003; Goldstein et al., 2004; Borovsky and Denton, 2008; Nishimura et al., 2022; Goldstein et al., 2014b). This structuring should play a role in modifying the reconnection rate (Walsh et al., 2014). In addition, Nishimura et al. (2022) found that the plume fine-scale density structures are associated with the modulation of whistler-mode wave intensity and localized electron precipitation. These processes could impact the local conjugate ionosphere conductivity (Nishimura et al., 2022) and thus the coupling of the inner magnetosphere and ionosphere.

This raises the first PlaTo question, namely, What causes the structural variations, or “lumpiness,” in the plume density? Possible sources of the structure include the ExB velocity shear between plume flow and plasmasphere, turbulence, effects arising from coupling timescales between plume plasma and the ionosphere (Borovsky and Denton, 2008), or structuring of the electric field that induces the ExB flow (Goldstein et al., 2004). To examine the source of the structure and the plasma that feeds the plume, observations of the plasmasphere on the dusk side, where the sunward plume separates from the anti-sunward co-rotating plasma, can be examined to see whether the structuring forms in that region and on the expected timescales. The duration of the density structures is ∼2 h at geosynchronous orbital distance around 15 magnetic local time (MLT), with a typical flow speed of ∼12 km/s (Borovsky and Denton, 2008), yielding them a scale size of ∼13 R_E. They have an observed number density of 40 cm⁻³. Projecting these back to L = 3 and L = 5, assuming a uniform electric field responsible for the flow, the magnetic field varying as 1/L³, and assuming the cross-sectional area does not change considerably, a typical plume flow may be ∼430 cm⁻³ and ∼1.1 km/s (at L = 3), and ∼90 cm⁻³ and 5.2 km/s (at L = 5). At these speeds, it would take ∼33 min and ∼7 min for a parcel of plasma to drift across a pixel width of 0.34 R_E. The estimates for L = 5 are consistent with the observations of Nishimura et al. (2022), who examined density irregularities and structures over a range of scale sizes (≲100 km to ≳5000 km) in the magnetosphere with the Van Allen Probes and in the conjugate ionosphere with GNSS receivers. They observed structures with an inner magnetosphere equivalent size of ∼3,000 km and ∼5–8 km/s speed, corresponding to a drift timescale of ∼6–10 min. The normalized plume density variations reported by Borovsky and Denton (2008) were ∼0.57. Therefore, we estimate that the density will vary ∼50 cm⁻³ and ∼250 cm⁻³ at L = 5 and L = 3, respectively. If the base of the plume is the source region of the structuring, the structuring with the density perturbation amplitudes above will be observed to form over a timescale characteristic of the generation mechanism. Following the methodology in Borovsky and Denton (2008) but adjusting for an L ∼ 4, we calculated the timescales for the ExB flow and ionospheric coupling to be 38 min and 18 min, respectively. Resolving on timescales of 7 min will be sufficient to detect these changes.

2.2 Interchange instability

It has long been suggested that the interchange instability may play an important role in magnetosphere dynamics (Gold, 1959; Lemaire, 1974). The interchange instability is expected to play an important role in radial plasma transport at Jupiter (Hill 2006; Bagenal and Dols, 2020) and at Saturn (Goldstein et al., 2014a). The same process may potentially contribute to plasmapause dynamics at Earth (Huang et al., 1990; Lemaire and Gringauz, 1998; Lemaire, 2001). Hence, PlaTo’s second science question: Does the interchange instability operate in Earth’s plasmasphere?

Interchange instability requires that azimuthal mass density gradients in the plasmasphere create divergences in the centrifugal drift current, which must close through Pedersen currents in the ionosphere (Hill, 2006). The associated electric field, mapping between the ionosphere and magnetosphere, results in an ExB drift that amplifies the density gradients at the plasmapause. The interchange instability is expected to become important where the centrifugal force becomes greater than the gravitational force (Lemaire, 1974). During geomagnetic storms, the resulting effective “Roche-limit” (Lemaire, 1974) of the plasmasphere can be <5 R_E on the night side. The growth rate of the interchange instability is dependent on the size of the density structures associated with the azimuthal gradients, the location, radial distance, and geomagnetic activity. An estimate of a typical scale can be made from the analysis by Huang et al. (1990). In the limit where the azimuthal wave number, m, describing the spatial scales of the azimuthal density perturbations is much greater than unity but much less than the ratio of the plasmasphere circumference to the plasmapause thickness, we can estimate the scale size, λ, as λ = 2 π L / m and the growth rate γ ≈ m / 22 h r for L = 5. At L = 5, a wavenumber of m = 80 yields an azimuthal spatial scale of 0.39 R_E and a characteristic growth time of ∼17 min. The density immediately inside the plasmasphere is typically ∼100 cm⁻³, and that outside is ∼<10 cm⁻³; thus, resolutions of density ∼5 cm⁻³ and temporal cadence ∼8 min are sufficient to observe interchange plasma structures in special scales >0.39 R_E. If the interchange instability is present, plasma structuring with these spatial and temporal scales will form at the plasmapause.

2.3 Refilling

Another unresolved question is how plasmasphere refilling occurs. It has been suggested that the refilling rate may depend on the amount of plasma already in flux tubes, scattering the inflowing plasma via Coulomb collisions, resulting in an increase of the refilling rate over time (Su et al., 2001). Hence, the third science question: Is the refilling rate of the plasmasphere uniform over the refilling process or does it change as a function of time? The refilling rate has been observed to change by a factor of 2 or greater at geosynchronous orbit, increasing later in the refilling process (Su et al., 2001). The refilling process could be one process or two (Gallagher and Comfort 2016), depending on how the physical process that leads to ionosphere outflow being trapped on flux tubes changes with the ambient plasma density.

Plasmasphere flux tubes are generally full at L < 2.5 R_E and almost always either refilling or eroding L > 5.0 R_E (Gallagher and Comfort 2016). A range of radial coverage extending from 3.0 to 5.5 R_E provides an ample spatial range in which to observe the refilling dynamics. To determine a spatial scale, we consider that typical densities for this range can be approximated by the Carpenter and Anderson (1992) density profile, n_e = 10^3.9−0.315L, where n_e is in cm⁻³, and L is the L-value. Based on the Carpenter and Anderson density profile, we can infer an equatorial scale size of the plasmasphere density profile of n_e/(dn_e/dL) = 1/(0.315×ln(10)) = 1.38 R_E. A spatial resolution of 0.5 R_E is sufficient to resolve this scale. In addition, at this resolution, the mid-point refilling value is approximately that of the average of the lower and upper L values of the pixel, that is, approximately, the change is linear on those scales. The temporal scale was determined based on fits to observed refilling rates as a function of L reported by Denton et al. (2012), who reported a refilling rate of dn_e/dt = 10^{3.48−0.331L}. At the L values 3.0 and 5.5, the refilling rates are thus ∼307 cm⁻³/day and ∼46 cm⁻³/day, respectively. Over a 2-h observation window, the density should increase by 25 cm⁻³ and 3.8 cm⁻³, respectively, due to refilling. If the refilling rate changes over time, then the observed refilling rates over individual 2-h periods will vary depending on the amount of time between that 2-h interval and the start of the storm recovery. An estimated refilling time of roughly ∼3 days from L 3 to 5.5 was found based on the Denton et al. (2012) refilling rate and the Carpenter and Anderson (1992) density profile. A 3-day period has 36 2-h sub-intervals. While the number of refilling observations is likely to be skewed toward earlier in refilling, we estimate using an equal probability of observing refilling at any stage, 100 observations will provide an average of ∼2.8 observations per 2-h bin. More robust analysis of typical refilling rates must be conducted to develop an actual robust mission architecture.

The results of the above are summarized in the PlaTo science traceability matrix (STM) in Table 1.

TABLE 1. PlaTo science traceability matrix.

The science traceability above specifies the required spatial and temporal resolution, accuracy of the density measurements, density map area, and duration of observations meeting these criteria needed to address the three respective science questions. The goal is to determine the minimum number of spacecraft and their radio payload requirements to answer the three science questions listed above. While the scope of the current study does not allow a complete survey of the available parameter space, a provisional mission concept that achieves science closure, assuming robust density map generation, is identified. Once approximate orbits are known, a reasonable estimate of the mission cost can be determined. While many more mission details remain to be determined, the results presented here establish the basic feasibility of a mission.

3 Number of spacecraft

An initial estimate for the number of spacecraft is calculated based on the number of pixels in the minimum desired area, A_obs, of the density map, and setting that equal to the number of ray paths between all spacecraft. The number of pixels with sides of length Δ x (0.34 R_E for PlaTo) is found as the observation area divided by the area of a pixel, Δ x 2. The number of ray paths, N_rays, between N_sc spacecraft is: N r a y s = N s c N s c − 1 / 2 (Ergun et al., 2000). Each ray yields, or corresponds to, a measure of the total electron content (TEC) between a pair of spacecraft. Each TEC path can be thought of as representing an equation, with TEC equaling the integral of the local plasma density along the path segment, for example, T E C = ∫ n x d x ≈ ∑ n i Δ x i.

To make the density maps, several unknowns, namely, the density in each map pixel, are effectively solved for. Setting the number of equations to the number of unknowns, we can find an expression relating the number of spacecraft, the observational area, and the pixel size: A o b s / Δ x 2 = N s c N s c − 1 / 2. Using the radial (L-value) and MLT extent to estimate the observational area, and the minimum pixel size, the number of spacecraft can be approximated. The desired density map is to have an area of 2.5 R_E × 2 MLT between L of 3 and 5.5, or 5.56 R_E ². The pixels have an area of ∼0.11 R_E ². The result indicates that ∼11 spacecraft should provide sufficient resolution. However, this does not fully determine the number of spacecraft. The required duration over which temporal continuity of the density map is required is another factor controlling the number of spacecraft. There must be enough spacecraft to cover the minimum observation area, but as the spacecraft are moving along their orbital trajectory, the constellation is only periodically in a configuration that allows science measurements. Continuous coverage could be guaranteed if both orbits were fully populated with spacecraft. The observations could be continuous, but this would require an infeasible number of spacecraft.

From the above discussion, we have an estimate of the number of spacecraft. However, additional factors must be considered to ensure mission success. For the mission to be successful, the orbits and number of spacecraft should ensure a minimum number of observations over the needed area, with long enough temporal duration, in the locations where the processes under investigation occur, and under the right geomagnetic conditions for those processes must be made. Many factors contribute to the number of spacecraft: spacecraft separation, the orbit parameters, periods, rate of precession, perigees, apogees, and inclinations. The choice of orbital parameters must result in the apogees allowing the spacecraft to bracket a large enough region of the observation areas for a sufficiently long continuous duration to allow for enough observations over the mission life. The line of apsides for the two orbits must stay reasonably close together so that the tomographic constellation has suitable arrangements for the tomographic inversion technique used. The mission duration and the part of the solar cycle in which the mission is active are critical drivers of whether enough observations of physical processes to be studied will be made. The approach we take here is to increase the estimated number (11) by ∼50% to 16 spacecraft, to compensate for underestimation, and then to “fly” a hypothetical 2-year mission recording the number of observations that meet the science observation criteria (below). We then reduce the number of spacecraft and repeat the experiment to observe how the number of science observations decreases. A science observation interval will be defined to meet the following criteria:

• The fraction of the science area covered by pixels with two or more TEC rays traversing them is equal to or larger than the minimum desired density map area. A robust determination would require further determination of a minimum angle between the rays and the relative arrangement of the voxels.
• The geomagnetic conditions are appropriate for the process under investigation to occur.
• The duration over which density maps with the above two conditions is long enough to observe the dynamics under investigation.

For PlaTo, the identified science areas for the plume, interchange, and refilling are between L of 3.0 and 5.5 and MLT ranges of 14–18 MLT, 1–5 MLT, and 7–17 MLT, respectively. The minimum observation (map) areas are 50% (2 MLT wide), 37.5% (1.5 MLT wide), and 20% (2 MLT wide), respectively. The activity defined by Dst for these three cases is Dst < −30 nT and decreasing, Dst < −30 and decreasing, and Dst > −20 and increasing, respectively. The minimum duration of each science observation interval should be 1 h for the plume and interchange investigations and 2 h for the refilling investigation. The mission duration (2 years) must also result in a sufficient number of usable science observations defined by the criteria above to answer the science questions. For the processes that occur during active conditions, interchange, and plume structuring, we will assume that ten science observations for each study should be sufficient to answer the questions. An initial goal of ten observations of each of the mechanisms under question should allow multiple observations of the processes under investigation if they do occur. As described above, we will require at least 100 observations for refilling. Finally, it is also important to note that a more rigorous evaluation of the angular distribution requirements would also need to be factored in, along with the particular methodology of tomography. For example, a direct inversion of the TEC paths requires an approximately uniform angular coverage. The scenario presented here does not always have uniform angular coverage, as can readily be seen in Figure 1. Radio tomography techniques such as “multiplicative algebraic reconstruction technique” (Na and Sutton, 1994; Bernhardt et al., 1998; Kunitsyn and Tereshchenko, 2003; Hei et al., 2014) have been successfully applied to cases in which the ray paths are mostly vertical.

Using Ansys Government Initiatives’ (AGI’s) Systems Tool Kit (STK) software, the two orbits having the appropriate apogees and similar rates of precession were used to create a 2-year scenario with 16 spacecraft. The parameters used for the spacecraft configuration are listed below and illustrated in Figure 2:

• Outer orbit altitudes of 300 km × 28,702 km (5.5 R_E), 0.01° inclination
• Inner orbit of altitudes of 5684 km × 11,481 km (2.8 R_E), 1.0° inclination
• The 10 outer spacecraft are spaced 5° apart centered around apogee; the six inner spacecraft are spaced 9° apart and similarly centered

To obtain a better time evolution than the two-body equations of motion used to determine the TEC paths, the orbits were propagated in STK using an analytical J4 propagator that incorporates the first- and second-order J2 and first-order J4 secular effects due to the oblateness of Earth. No maintenance maneuvers were included in the simulation to reflect a worst-case spacecraft configuration with no available propulsion, so the orbits were allowed to precess, potentially to unacceptable amounts for taking science, based on the effects of the zonal harmonics.

Two years of position and velocity data, incorporating the Earth’s spherical harmonics, were created and used to calculate the duration and number of pixels in the resulting density map.

The suitability of these orbits with the estimated number of spacecraft is tested against the three criteria (coverage, activity, and duration) with runs incorporating different numbers of spacecraft as follows. The first run of the mission uses all 16 spacecraft. The second run of the mission suppresses two spacecraft on the outer orbit (for a total of 14 spacecraft). The third run suppresses one and three spacecraft on the inner and outer orbits, respectively, for a total of 12 spacecraft. The missions are run for a 2-year duration, and at each 5-min time step, rays are drawn between each pair of spacecraft. Rays passing within 2 R_E of the Earth are suppressed as potentially being distorted due to refraction by the dense inner region of the plasmasphere (discussed more below). For each of the three science regions of interest, the number of pixels within those regions that have more than one ray passing through them is summed every 5 min. We calculate the fraction of the area of each of the three science regions that is covered by these pixels. The result is a time-evolving (with orbit period and precession) of the fractional coverage of the regions of interest. Figure 3 shows an example of the fractional variation in coverage of the interchange region over a period of ∼2 days (Figure 3, top left) and ∼5 months (Figure 3, top right). The variation that occurs on orbital period time scales can be seen in the 2-day panel (Figure 3 left) and longer timescale variations in coverage due to orbital precession in the 6-month timescale (Figure 3 right).

In order to observe the processes to be studied, the level of geomatic activity conducive to that process must be present as well. Thus, a suitable period, or period during which science can be investigated, depends on the constellation being in the right place for a sufficient duration during which the geomagnetic activity is right for the phenomenon in question to be observed. Being in the right place at the right time is probabilistic. To evaluate this, we take historic Dst indices for a 4 year interval, and start the hypothetical 2-year PlaTo mission at some point in the first 2 years of the Dst (2012–2014) and evaluate the number and duration of “events” or, times then the fractional coverage of the region of interest is above the minimum needed and when the Dst index indicates a suitable level of activity. The middle panels in Figure 3 present the Dst index, showing the period of increased geomagnetic activity. The result of sufficient activity and coverage results in the event flag is given a value of 1. The event flag being continuously 1 for a period longer than a minimum duration constitutes a science “observation.” An event is defined as an observation made by the constellation with the three science observation properties listed above.

“Events” as defined above capture the first two criteria; the third criterion is the duration. The example in Figure 3 shows events of a variety of durations. The phasing of the orbital motion determines the coverage. From Figure 3 left, it is clear that if the mission were launched an hour or so later or earlier, the resulting individual coverage segments would be different. In order to account for this, we count the number of “events” of duration longer than the minimum requirement over possible 2-year missions started every hour between September 2012 and September 2014. The hypothetical end times are 2 years later, from September 2014 to September 2016. Note that this period assumes that such a mission would be launched sometime around solar maximum into the declining phase of the solar cycle.

Figure 4 shows the number of usable science observations expected as defined by the three criteria above if the PlaTo mission commenced with apogee in the post-midnight side at the date indicated on the horizontal axis, lasted 2 years, and experienced the geomagnetic activity indicated by the actual Dst index. The results are shown for the three science investigations and for the different numbers of spacecraft in the constellation.

The number of science observations varies significantly with mission start time. For a science investigation on processes that occur during active times, a mission start time and location that will place the apogees in the science region when there are more active conditions will yield more events. For example, in the plume region, the results indicate that if the PlaTo mission were launched near solar maximum with the orbital apogees in the post-midnight sector, the number of events would be ∼10, with some intervals of possible launch times resulting in only a few events. When starting the mission in late Summer 2013 in the post-midnight region, the orbital precession would place the constellation apogees on the post-noon side in late Summer/Fall 2014, then the geomagnetic activity increases, resulting in more science observations. This can also be seen in the interchange investigation, where the event number picks up in the spring and late summer 2014. Mission start times in late 2012 into 2014 were selected because they are near the solar maximum and the early declining phase of the solar cycle. It may be expected that more events will be obtained if the mission commences closer to the start of the declining phase, due to the tendency for an increase in corotating interaction region-driven (CIR) storms, which are typically of longer duration, occurring in the declining phase of the solar cycle (Borovsky and Denton 2006).

In the three scenarios, we have used 12, 14, and 16 spacecraft. This is to explore how the event coverage changes with the number of spacecraft. One of the main goals of the study is to estimate the minimum number of spacecraft needed to complete the mission as defined by the science questions. The reduction in spacecraft does not impact the plume and interchange observations as much as the refilling observations. This is likely due to more spacecraft being required to support the combined requirements of a larger window in MLT and a temporal baseline that is needed for the refilling investigation. The results are summarized in Table 2. For a desired minimum of 10 events, 12 spacecraft with a mission starting anytime in the selected 2-year window near solar maximum will yield, on average, sufficient coverage, as determined by the area coverage identified in the science traceability matrix.

TABLE 2. Influence of the number of spacecraft on event observability.

The analysis presented in Figure 4 and Table 2 indicates that 12 spacecraft will, based on the hypothetical average, record sufficient observations to answer the science questions. A more robust analysis would ensure there was a significantly high chance of making the requisite observations. Based on the analysis here, we note that if we reduce the mean number of observations by one sigma, the 12-spacecraft constellation makes only ∼8 plume observations. The 14-spacecraft constellations make a sufficient number of observations for all three questions.

The investigation using shifting hypothetical start times, historical Dst, and three different constellation sizes suggests that 12 spacecraft are sufficient for all three investigations, but that the time in the solar cycle and initial apogee of the constellation are very important. Starting the constellation in the post-midnight sector near the start of the declining phase will yield, on average, sufficient coverage to address all three science questions. We note that while we varied some parameters such as start time and number of spacecraft, others, such as the orbit parameters and starting MLT of the apogees, were held constant. Furthermore, as outcomes with <10 observations are possible, careful consideration of solar cycle phase and the initial MLT of the orbits is needed. A full exploration of the parameter space would be required for actual mission planning.

While we have explored a narrow range of spacecraft numbers, an idea of how the coverage changes with larger variation in spacecraft numbers can be explored through an analysis determined by the number of spacecraft within the MLT of the observation area. Figure 5 shows an analysis of how the number of spacecraft on the inner and outer orbits affects the total time of coverage over a 30-day period that the constellation is located in the MLT of the science region. This analysis is different in that TEC paths are not used. It is to be thought of in a heuristic sense to illustrate the complex variation of coverage with spacecraft number. Note that the coverage time does not evolve linearly with the number of spacecraft.

3.1 Impact of longevity on the number of spacecraft

The conclusion that a minimum of 12 spacecraft is needed to meet the minimum science goals assumes that all of these spacecraft survive the entire mission duration. Spacecraft with a shorter design life may have a lower reliability as the components and parts may be less robust than those for longer missions. An analysis of the spacecraft design versus actual life was performed by Ferrone et al. (2019). These data are for free-flying, Earth-orbiting unclassified United States military and civilian spacecraft successfully launched between 1 January 1980 and 31 December 2018 with a mass greater than 100 kg and a design life longer than 1 month. The data included no radiation dose information. This analysis will enable general trends to be identified for mission duration.

It is assumed that the spacecraft do not have sufficient fuel to change from the inner to outer orbit, and vice versa, so the orbits are considered independent for this calculation. The spacecraft are identical, so the parallel reliability equation from is used for each orbit: R_O = 1 – (1 – R_A)(1 – R_B)(1 – R_C)…(1 – R_J) for the outer orbit reliability and R_I = 1 – (1 – R_A)(1 – R_B)(1 – R_C)…(1 – R_F) for the inner orbit. To reflect a constellation-level reliability, the series reliability equation is used: R_S = R_OR_I (Wertz and Larson, 1999). Using an individual spacecraft reliability value of 0.8 from the historical data provided by Farrone et al. (2019), the resulting system reliability is shown in Table 3.

TABLE 3. Constellation reliability assuming independent orbits of varying numbers of spacecraft.

Adding one spacecraft to the inner orbit increases the constellation’s reliability by nearly a factor of 4, with only marginal improvement when two to four spacecraft are added.

The above analysis supports the idea that the radio tomography constellation for the focused science mission described will consist of approximately 14 spacecraft. This number allows for redundancy in case of degradation of the constellation during the mission, provided we expect the mean number of observations (Table 2) to be obtained with 12 spacecraft. Or, if constellation degradation is not taken as a concern, 14 spacecraft will increase the statistical likelihood of capturing the desired number of observations, if that is the driving factor. For an actual proposed mission, a more robust handling based on both the expected geomagnetic activity during the mission and the survivability of the spacecraft must be considered.

4 Spacecraft radio payload

The primary spacecraft payload is a radio transmitter and a receiver that measures the phase difference and group delay of the transmitted frequencies. To determine payload requirements, RF frequencies, power, and broadcast duration must be specified. These parameters are tuned to minimize error in the TEC measurements. In the discussion that follows, we use considerations on the RF refraction and noise, analogous to those considered in Ergun et al. (2000), adapted to the plasmasphere.

The transmitter frequencies selected for this investigation depend on the plasma density range and gradients along the ray path line-of-sight, as well as the background radio noise, principally from the Sun. For the outer magnetosphere, Ergun et al. (2000) identified a primary frequency of 1 MHz and a reference frequency of 3 MHz as optimal. For the PILOT mission concept, Malaspina et al. (2022) identified 50 MHz and 150 MHz. However, the specific frequencies or range of frequencies depend on the mission science questions via the requisite spatial resolution and density resolution. Two main error sources act as limiting factors in the choice of frequencies, transmitter power, and transmission time (signal integration time) needed to obtain the density measurements at a desired accuracy, spatial, and temporal resolution. The first limitation is due to the RF signal-to-noise level at a given frequency and the limitations that it places on resolving the phase shift between the two transmitted frequencies and thus the TEC. The second limiting factor is the refraction of the RF signal as it moves through regions of varying plasma density. The tomographic inversion assumes that TEC is measured along a straight path between the transmitting and receiving spacecraft.

Following the discussion of Ergun et al. (2000), it can be shown that the phase difference, Δ Φ 1, between two originally phase-matched frequencies, a primary ω 1 and a reference frequency ω r, is given by Δ Φ 1 = ω 1 e 2 2 ϵ 0 m e c 1 ω 1 2 − 1 ω r 2 ∫ n e d s (Equation 3), where e , ϵ 0 , m e , c , n e , a n d d s are the charge on the electron, permittivity of space, electron mass, the speed of light, electron density, and differential element along the ray path, respectively also see (Zhai and Cummer, 2005). The TEC is the electron density integrated along the ray path, expressed as T E C = ∫ n e d s. If, as in Ergun et al. (2000), we assume a reference frequency as the third harmonic of the fundamental, we can rewrite Equation 3 in terms of the primary frequency (f₁) in Hz, the path length (s) in R_E, and the phase shift in degrees as Δ Φ 1 deg ⁡ . = 2.74 × 10 8 f 1 H z ∫ n e c m − 3 d s R E. The error in TEC due to the error in phase should be less than the minimum TEC to be measured. The minimum TEC, ∼ Δ n e Δ x, to be measured is the smallest change in density (Δ n) over the distance of a pixel width (Δ x); δ T E C < ⁡ min T E C = Δ n Δ x. The error in the measured phase difference, δ ϕ 1, from RF noise depends on the signal-to-noise ratio, S/N, as δ ϕ 1 = tan − 1 S / N − 1 (Ergun et al., 2000). The smallest change, or increment of density, (Δ n), at the spatial resolution (Δ x desired that can be measured due to this error is then Δ n > f 1 δ ϕ 1 Δ x · 2.74 × 10 8 (Equation 4). The effect that a change in density larger than Δ n over Δ x will result in a change in TEC larger than the uncertainty in the TEC measurement. Thus, from Equation 4, a low frequency allows smaller densities and smaller variations in density to be measured. However, the plasmasphere can have large density gradients, especially at the plasmapause, where the density can change by ∼100 cm⁻³ in a few 0.1 R_E in some cases (Carpenter and Anderson, 1992). Density gradients cause refraction of the radio signal so that the RF ray paths are no longer straight lines, which is another source of error. If a true ray path is curved or bent, the actual and assumed propagation paths differ. In that case, the rays will have passed through a different TEC path than they would for a straight line. Quantifying the error in phase difference between the primary and reference frequencies due to the change in the length of the ray path from refraction using Snell’s law is expressed by Equations 10 and 11 in Ergun et al. (2000). The phase error from refraction given by Ergun et al. (2000) is Δ Φ R = ω p 2 8 ω 2 k ∥ 2 k ⊥ 2 Δ Φ 1, where k ∥ and k ⊥ are the components of the RF wave vector parallel and perpendicular to the plasma density gradient respectively. Combined with Equation 3, the lower limit to density resolution per pixel can be determined:

Δ n Δ x > δ T E C = ω p 2 8 ω 2 k ∥ 2 k ⊥ 2 T E C . (3)

A higher frequency thus decreases the error due to refraction. However, a higher frequency also increases the error associated with background RF noise. There is thus some frequency for which the error in the TEC, and hence density, is minimized. To minimize the error, we can start by estimating a frequency that gets close to minimizing the error in TEC. Based on analysis from the PILOT study, we chose a preliminary frequency estimate of ∼100 MHz. Considering a hypothetical case for which strong refraction would occur, in which the ray is incident on the density gradient at an angle of 84° from the normal (choosing an oblique angle corresponding to k ∥ / k ⊥ = 0.1), with the density being a sharp gradient between a plasma of ∼1 cm⁻³ and ∼1500 cm⁻³, we find the ray is only refracted 8 × 10⁻⁵ radians from incidence, resulting in a TEC measurement error of ∼0.02%. The total TEC found integrating the Carpenter and Anderson, (1992) density profile from 2.5 R_E to 5.5 R_E is ∼7.9 × 10¹⁵ m⁻², and the minimum TEC (Δ n Δ x is 1.3 × 10¹² m⁻² or 0.016% of the total. Thus, while indicating 100 MHz is a reasonable starting place, further adjustments of the frequency will be necessary.

Next, the background RF noise must be determined. The main source of environmental noise is the Sun. Terrestrial sources of RF noise in the ∼100 s MHz range are also present but not as significant as those from the Sun (Schroder, 2017). Note that the actual noise value will also include that of spacecraft electronics, which must be evaluated in accordance with the systems chosen. Under active solar conditions, the Sun’s RF spectral irradiance around F ∼100 MHz is ∼10⁻¹⁸ W m⁻² Hz⁻¹ (Ho et al., 2008). For a narrow band over which the noise power spectrum is assumed constant, the intensity of the background RF noise depends on the integration time. I N = ∫ f f + Δ f F N f d f ≈ F N f Δ f. Taking the frequency interval Δ f to be the resolution of the fast Fourier transform (FFT) power spectrum of the received RF signal, the noise intensity in that band is I N ≈ F N / T, where T is the integration time of the signal. The transmitted signal intensity is I s = P T x / 4 π s 2, where P T x is the transmitter power. The ratio of these two intensities forms the signal-to-noise ratio used to find the phase error δ ϕ 1 = tan − 1 I S / I N − 2. Note that by I s I N ∝ P T x T, we observe that increasing the transmitter power or the integration time will decrease the phase error.

The integration time must be long enough to keep the error due to ambient RF background noise sufficiently low, but short enough to allow the constellation to perform a ping cycle and create density maps at the required cadence, or faster. Likewise, the transmitted power must be intense enough to keep the measurement error down, but the spacecraft electronics must be capable of providing sufficient transmit power over the transmit duration. The spacecraft electronics must also be able to sustain the ping cycle for periods >2 h. The time between transmitting intervals of duration T for a constellation of N spacecraft is Δ t = N − 1 T. For N = 14 and a ping cycle of 5 min (to allow margin on the maximum cadence of 7 min), the transmission interval from each spacecraft must be 21.4 s or shorter. For T = 10 s, the time between successive transmissions made by a single spacecraft is 130 s.

Equation 4 and Equation 5 are combined into an expression for the minimum measurable density from the combined two error sources. The total TEC has been estimated between L 2.8 and 5.5 using the density profiles along a radial ray and the Carpenter and Anderson, (1992) plasmasphere model. The resulting error analysis is presented in Figure 6.

Assuming the larger of the two density gradients (a change in density of 1500 cm⁻³ at a sharp spatial distance, which is likely a minority of the cases) and the specification that the error in the minimum density is 20% of the minimum value to be measured (3 cm⁻³) and should be less than 0.6 cm⁻³, we can experiment with different integration periods and transmitter power and determine the frequencies that satisfy the measurement error requirement. This analysis is presented in Figure 6. Note that to develop Figure 6, we applied a constant value for the environmental background noise. The noise level increases at frequencies higher than several GHz and falls off at frequencies lower than ∼30 MHz. We do not explore the higher frequencies, and the error effects for the lower frequency range are dominated by refractive effects as opposed to RF noise. We see that the local measurement error is at a minimum near ∼200 MHz. Smaller errors are found for a gradient with a total change of 100 cm⁻³, which is more representative of a plasmapause at a higher L-value. This suggests broadcast frequencies of 200 MHz and 600 MHz, where 600 MHz is derived using the third harmonic of the probing frequency. Increasing the power or integration time will also reduce the error. In the time Δ t, the transmitting power source (capacitors or batteries) must accumulate 500 J plus extra for loss. The 200 MHz frequency has a wavelength of 1.5 m; a half-wave dipole would have antenna elements of 37.5 cm.

The results displayed in Figure 6 assume an RF signal transmitted at 50 W for 10 s. In this case, the power supplied to the transmitter should be 50 W plus that lost to inefficiencies, maintained for 10 s.

5 Conclusion and future work

This work demonstrates that significant advances in understanding plasmasphere dynamics may be achieved with a small-to-medium-sized tomographic constellation mission. The requirements for a radio tomography mission of the plasmasphere with three science goals capable of being addressed with time-evolving electron density maps in the equatorial plane have been explored at a high level. We have identified a pair of orbits that, when hosting a constellation of 14 spacecraft, provide TEC coverage of sufficient area, duration, and frequency to address three science questions. In addition, we used a simple model of phase error and TEC ray refraction to find an optimal pair of frequencies at prescribed power and transmission interval with which to measure the TEC. An important underlying assumption used was that the density map can be extracted from the TEC measurements with the expected area and durations. We have not yet tested the various density map generation methods, which will be addressed in future work described below.

The selected science questions focus on plume structuring, the importance of the interchange instability, and the refilling rate. These science questions were used to derive measurement criteria and top-level mission requirements. These motivate electron density maps with spatial resolution of 0.34 R_E, cadence of 7 min, and density measurements accurate to <20% for densities >3 cm⁻³. It was found that a ∼2-year mission with 14 spacecraft, and a 10 s, 50 W ping cycle broadcasting at 200 MHz and 600 MHz can meet these requirements. The number of spacecraft (14) was identified by determining the mean number of science observations constellations of three different sizes (12, 14, and 16 spacecraft) would have made, had the mission been launched at different times from 2012 to 2014. Using 14 spacecraft provides a margin for the loss of up to two spacecraft, assuming the mean number of observations based on historic Dst are expected to be obtained. The orbits used were approximately equatorial and had apogees of 2.8 and 5.5 R_E. The radio frequencies, power, and transmit duration were optimized by minimizing the two main error sources, background noise and signal refraction. Mission scientific coverage is sensitive to the initial orbit configuration (MLT of apogee) and the phase of the solar cycle. This study demonstrated potential mission feasibility within the parameters studied. However, mission sensitivity to other parameters, such as orbit procession, has not been explored.

Assuring a robust validation of the provisional mission concept presented in this study involves an extensive future effort, which we detail here. A 2-year simulation of the plasmasphere dynamics using a model that captures a significant amount of detail (i.e., density structures with similar spatial and temporal scales as those of the three science questions), such as MAGE. Even if MAGE does not reproduce all the phenomena of interest, it is still of value to verify that the constellation is capable of collecting TEC data from which density maps capturing similar features can be generated. We would then fly the proposed constellation of 14 spacecraft with the orbits specified through the simulated output. A virtual RF ping cycle with the proposed radio frequencies, power, and transmission intervals would be carried out for the 2-year mission. To verify the assumption that the TEC ray paths may be assumed to be straight, a ray tracing model could be used to propagate the rays. The inclusion of ray tracing would allow detection and quantification of instances of strong refraction and curved ray paths that lead to fundamentally different results than straight-path propagation. Both the ray geometry and the TEC from the ray path and the straight-line approximation could then be compared, as well as results for using both of these in an inversion, data assimilation, or other density map extraction scheme.

As noted in the manuscript, a direct inversion technique usually requires approximately full angular coverage of the region investigated. Thus, we do not expect direct inversion to be a suitable technique in most cases. Instead, radio tomography techniques such as the “multiplicative algebraic reconstruction technique” (Na, 1994; Bernhardt et al., 1998; Kunitsyn and Tereshchenko, 2003; Hei et al., 2014), which have been successfully applied to cases where the ray paths are mostly vertical, will be investigated. In addition, data assimilation techniques using an extended Kalman filter or 3D-VAR approach would also be investigated (Jorgensen et al., 2011). In such an approach, the TEC would be assimilated against a plasmasphere density background model. This density background could be, for example, a simple Carpenter and Anderson (1992) density profile, or a more involved physics-based model. The more realistic dynamic and structured model, such as the MAGE output, would be used as the truth model.

The simulated TEC measurements could be made to include both RF background noise and finite instrument errors based on specific selected hardware, before the assimilated step. The output of the data assimilation would be verified against the truth model (MAGE). We hope to verify that plasmasphere structures of similar density and spatial and temporal scales as those targeted by the science questions are successfully resolved and captured in the assimilative output, and that the number of events in terms of duration and density map area for given geomagnetic conditions are obtained for the 2-year mission. This procedure is effectively an observing system simulation experiment (OSSE) (Shprits et al., 2025; Hsu et al., 2024).

The orbits identified in the current study provide the spatial and temporal coverage needed as determined in the STM, under the assumption that the area covered by the TEC paths yields the density maps expected. As alluded to above, the orbit parameter space must be explored further to determine whether the particular orbits identified in this study are the best ones possible. Here, “best” refers not only to the coverage area and relative rates of precession but also to the ability to deorbit the spacecraft or dispose of it in a graveyard after the end of the mission, the radiation dosage received over the mission life, etc. Optimization of the orbits based on science measurement consideration would be made based on the type of simulation work described above.

Finally, it is worth noting that while the phenomena investigated by the three science questions are not exhaustive of what a PlaTo-like mission could achieve. While we target the structuring of the plume and potential plasmapause structuring due to the interchange instability, these are only two of a larger group of plasmaspheric mesoscale features whose formation mechanisms remain unclear. For example, features such as the dawn side “shoulder,” the nightside “notch,” and “fingers” (Darrouzet et al., 2009) are yet to be fully explained. These features should be resolvable by PlaTo or a similar mission. Plasmaspheric radio tomography thus has potential for investigation processes beyond those directly discussed in this study.

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Footnotes

Edited by: Joseph E Borovsky, Space Science Institute (SSI), United States

Reviewed by: Agnit Mukhopadhyay, MDA Space, Canada

Zhiyong Wu, University of Science and Technology of China, China