## 3 Results and Discussion
### Galileo TEC and Ne Measurements  
When the chief satellite soars higher than the deputy satellite, the vertical _Ne_ between the two satellites retrieved from a specific PRN can be computed by the following formula:  
\[Ne^{PRN}=(VTEC^{D}-VTEC^{C})_{PRN}/\triangle h, \tag{2}\]  
where VTEC\({}^{C}\) is the vertical TEC measured by the chief satellite GNSS receiver from a PRN, VTEC\({}^{D}\) is the vertical TEC measured by the deputy satellite GNSS receiver from the same PRN, and \(\triangle h\) is the height difference between the chief and deputy. Using this method, space-born GNSS receivers can be utilized to retrieve localized _Ne_ given a formation flying orbit with non-zero radial offset. If the measurement accuracy is comparable to or outperforms other measurement techniques (e.g., in-situ  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Epoch: 2019.03.08 01:00-00.00 (GPS); Week Number: 2043; GPS Seconds of Week: 435600** \\ \hline
**Orbital Elements** & **Chief Spaeccraft** & **Deputy Spaeccraft** & **Relative State in Body Frame** \\ \hline Semi-major axis \(a\) (m) & 6880540 & 6879040 & Radial offset (m) & \(-\)854.3448 \\ Eccentricity \(e\) & 0.044 & 0.044 & In-track offset (m) & \(-\)1.4978 \(\times\) 10\({}^{4}\) \\ Inclination \(i\) (\({}^{\circ}\)) & 10 & 10 & Cross-track offset (m) & 9.9826 \(\times\) 10\({}^{-5}\) \\ Argument of perigee \(\omega\) (\({}^{\circ}\)) & 180 & 180 & Radial velocity (m/s) & \(-\)0.0328 \\ Long. of ascending node \(\Omega\) (\({}^{\circ}\)) & 67.4893 & 67.479 & In-track velocity (m/s) & 1.7605 \\ Mean anomaly \(M\) (\({}^{\circ}\)) & \(-\)90 & \(-\)90.125 & Cross-track velocity (m/s) & \(-\)1.4944 \(\times\) 10\({}^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Initial orbital states of the satellite fleet.  
Figure 3: Illustration of the Equatorial Spread F (ESF) scenario.  
Langmuir probe or radio-occultation), the design of a satellite mission (e.g., cost, size, and power budget) can be significantly reduced while the same _Ne_ product can still be generated.  
The vertical _Ne_ retrieved from Galileo PRN 30 is plotted against the central height between the two LEO satellites as shown in Figure 5. The TEC result using the E1 and E5a frequency combination is plotted on the left (a), while the result from the E1 and E5b frequency combination is plotted on the right (b). The SR _Ne_ profile is plotted in red, and the measurements using Equation (2) are plotted in blue. Note that, these are not localized (same latitude and longitude) vertical _Ne_ profiles but a vertical projection of the three-dimensional trace of the sounding rocket vertical _Ne_. Due to a high frequency scattering effect from the raw signals, a low pass filter (critical frequency = 0.01 Hz) is applied to process the raw measurements and the filtered results are plotted in black. After filtering, the _Ne_ measurement results from the HIL simulation become much more consistent with the SR model. In Figure 5b, the measurement results above 500 km are relatively deviated from the model. This is due to the inaccurate DCB estimation with respect to PRN 30 of the E1 and E5b frequency combination. Further analysis indicates the scattering features of the _Ne_ raw measurement is a heritage from TEC, which is associated with the particular receiver model and frequency combination of a specific GNSS constellation. For the NovAtel OEM628 receiver, GPS L2P and L2C tracking is aided by GPS L1. While Galileo E5a, E5b, and E5 AltROC tracking is unaided, as is GPS L5. The unaided carrier-phase TEC combinations (e.g., E1 and E5a and E1 and E5b) are noisier than the aided ones (e.g., L1 and L2). Therefore, the vertical _Ne_ retrieved using GPS L1 and L2 TEC by the OEM628 receiver is less fluctuated than the vertical _Ne_ retrieved by Galileo (E1 and E5a or E1 and E5b) TEC in the baseline ESF scenario.  
Figure 4: Hardware-in-the-loop (HIL) simulation results of the ESF scenario: (**a**) radial distance history; (**b**) in-track distance history; (**c**) thrust history.