## 5 Results
### User Spacecraft Orbit Determination Results
#### 5.2.2 Case #2  
Insufficient knowledge about the TT&C bias and its covariance can lead to significant errors in the orbits of the TDRS satellites. As a result, the user spacecraft's orbit determination accuracy can be severely affected. Any error in the TDRS orbit can propagate to the user spacecraft orbit and cause substantial errors in the determination of its position and velocity. To illustrate this impact, Figure 13 demonstrates the impact of the TDRS orbit error on the user spacecraft's orbit. The plot shows the relationship between the error in the user spacecraft's orbit and the cumulative error in the TDRS orbit. Each point on the plot represents a unique solution obtained using the TDRS orbits described in Section 5.1.3. Remarkably, despite being in different orbital configurations, the errors in all user spacecraft's orbits were linearly related to the errors in the TDRS orbits. However, the slope of linearity varies for each spacecraft. Several factors could have contributed to this variance, including the relative position of the TDRS with respect to the user spacecraft during the acquisition of DOWD measurements, the number of DOWD passes acquired, and dynamic errors in the user spacecraft's orbit. For instance, the LANDSAT-8 spacecraft exhibited a significant error, even for similar cumulative errors in the TDRS orbit. This can be attributed to the limited number of usable DOWD passes. Only three passes were available due to a TDRS maneuver, and one of these was sigma edited out by GTDS because of increasing orbit uncertainties. DOWD generally requires a minimum of four passes for a reasonable orbit solution, as short arcs of data are typically insufficient to provide reliable estimates. As a result of the limited DOWD passes, \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**User Spacecraft**} & \multicolumn{2}{c}{**User Spacecraft Orbit Error**} & \multicolumn{2}{c}{**Cumulative TDRS Orbit Error**} \\  & **Position (m)** & **Velocity (mm/s)** & **Position (m)** & **Velocity (mm/s)** \\ \hline IXPE & 16 & 16 & 37 & 2 \\ LANDSAT-9 & 50 & 45 & 86 & 6 \\ LANDSAT-8 & 31 & 29 & 97 & 7 \\ JPSS-2 & 39 & 26 & 48 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The table indicates the largest error observed between the user spacecraft’s orbit and the operational solution, along with the total errors estimated in the TDRS satellite orbits. The operational TT&C bias values and their respective covariance was utilized to determine the TDRS orbit. the estimated LANDSAT-8 solutions were not highly accurate. This situation represents the worst-case scenario in terms of the quality of the solutions obtained. On the other hand, JPSS-2 exhibited the gentlest slope in the linear relationship with TDRS orbit errors. This can be attributed to the wealth of DOWD passes (16 in total) available for JPSS-2. Tables 4 and 5 provide a summary of the impact of using constrained TT&C bias and unconstrained TT&C bias TDRS solutions on the accuracy of the user spacecraft's orbit. These tables present uncertainties in position and velocity compared to the operational solution. The constrained bias TDRS solutions were observed to provide significantly better user spacecraft orbit precision than the unconstrained bias TDRS solutions. For instance, the maximum position error in the LANDSAT-8 orbits using a constrained TDRS solution was approximately 293 m, whereas it was \(\approx\)33,000 m when using an unconstrained TDRS solution. The findings suggest that when there is a lack of adequate understanding of the TT&C measurement bias and its covariance, a constrained TDRS solution could offer a more dependable user spacecraft orbit. In these scenarios, a constrained TDRS solution has the potential to mitigate the spread of any errors originating from measurements in the TDRS orbit to the user spacecraft orbit. This leads to more precise determinations of position and velocity compared to an unconstrained solution.