## 5 Results
### User Spacecraft Orbit Determination Results
#### 5.2.2 Case #2
Figure 13: The figure illustrates the uncertainties in the user spacecraft’s position as a function of cumulative errors in the TDRS position. The error in the TDRS position arises from range-only TT&C measurements, as compared to the operational solution that is based on both TT&C and BRTS measurements. The shown error in the user spacecraft and TDRS orbits are obtained solely during the time of DOWD passes. Therefore, they do not represent the implications of long-term TT&C, only TDRS orbital errors on the user spacecraft orbits. Table 6 presents the mean and standard deviation values of the LOF offset estimated for all one-way Doppler passes. In addition to the TT&C-only solutions' values, values obtained from the TT&C and BRTS-based operational solutions are also shown for comparison. All spacecraft estimated values were compatible with operational solutions. This compatibility can be attributed to the fact that, when comparing the TT&C-only DOWD solution with the operational solution, the maximum change in user spacecraft velocity concerning the TDRS satellites resulted in a maximum Doppler shift that was significantly smaller than the mean and standard deviation values of the one-way Doppler residuals (see Table 6). The correlation between Doppler shift and velocity change can be expressed as:  
\[\Delta f\cong\frac{\Delta vf_{r}}{c} \tag{4}\]  
where \(\Delta f\) represents the Doppler shift in Hertz (Hz), \(\Delta v\) is the velocity change in meters/second (m/s), \(f_{r}\) is the S-band reference frequency in Hz, and c is the speed of light in m/s. \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**User Spacecraft**} & \multirow{2}{*}{**Operational Solution**} & \multicolumn{2}{c}{**TT&C Solutions**} & **User-TDRS** \\  & & **Operational Solution** & **Constrained** & **Unconstrained** & **Max. \(\Delta V\)** \\  & **(Hz)** & **(Hz)** & **(Hz)** & **(mm/s)** \\ \hline IXPE & \(-776\pm 79\) & \(-776\pm 79\) & \(-776\pm 80\) & 0.083 \\ LANDSAT-9 & \(251\pm 26\) & \(251\pm 26\) & \(250\pm 23\) & \(1.169\) \\ LANDSAT-8 & \(-96\pm 25\) & \(-97\pm 25\) & \(-99\pm 23\) & \(1.600\) \\ JPSS-2 & \(746\pm 221\) & \(746\pm 221\) & \(746\pm 222\) & 0.012 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The table exhibits the estimated mean and standard deviation values of Local Oscillator Frequency (LOF) drift under different scenarios. Additionally, it also shows the maximum difference in user spacecraft velocity concerning TDRS while comparing the TT&C-only DOWD solution with the operational solution. \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 & 147 & 140 & 690 & 50 \\ LANDSAT-9 & 293 & 249 & 693 & 50 \\ LANDSAT-8 & 273 & 250 & 234 & 18 \\ JPSS-2 & 58 & 42 & 789 & 58 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: 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 TDRS orbit was calculated using a constrained approach that did not factor in any biases present in the TT&C measurements. \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 & 1112 & 1273 & 8519 & 619 \\ LANDSAT-9 & 4988 & 3814 & 18,519 & 1350 \\ LANDSAT-8 & 32,342 & 30,747 & 14,566 & 1056 \\ JPSS-2 & 1340 & 1345 & 15,823 & 1151 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: 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.