## 3 Simulations and Results
### Orbit Determination Simulation
#### 3.2.2 Effects of the Initial State Covariance  
This subsection is dedicated to the examination of the impact of varying initial state covariances. Two sets of observational data from Case 1 and Case 3 in Section 3.2.1 are used in this subsection for comparison. For each set of observational data, six test cases are defined, each with progressively increasing initial covariance values for position and velocity components. The corresponding diagonal elements of the initial covariance matrix \(\mathbf{P}_{\text{initial}}\) are set as follows:  
* Case A: (100 m)\({}^{2}\) for position, (0.1 m/s)\({}^{2}\) for velocity;
* Case B: (1000 m)\({}^{2}\) for position, (1 m/s)\({}^{2}\) for velocity;
* Case C: (5000 m)\({}^{2}\) for position, (5 m/s)\({}^{2}\) for velocity;
* Case D: (10,000 m)\({}^{2}\) for position, (10 m/s)\({}^{2}\) for velocity;
* Case E: (50,000 m)\({}^{2}\) for position, (50 m/s)\({}^{2}\) for velocity;
* Case F: (100,000 m)\({}^{2}\) for position, (100 m/s)\({}^{2}\) for velocity. These values are assigned without reference to the statistical results shown in Figure 7, as prior knowledge may not be available for most practical cases. In all cases, the measurement noise level is set to be [300 m, 300 arcsec, 300 arcsec], the initial covariance for the along-track acceleration component \(a_{\text{T}}\) is assumed to be (\(10^{-4}\) m/s\({}^{2}\))\({}^{2}\). Figures 10 and 11 illustrate the iterative histories of position errors and the along-track thrust acceleration (\(a_{\text{T}}\)) estimates at the chosen epoch for Cases 1 and 3 with various initial covariance values of position and velocity. As shown in Figures 10 and 11, all tests for both Case 1 and Case 3 converge within 10 iterations. For Cases A-D, the convergence of both position errors and \(a_{\text{T}}\) estimates exhibits oscillatory behavior near their final values. In contrast, Cases E and F with larger initial covariance values for position and velocity demonstrate significantly improved stability after convergence, along with smaller final position errors. The resulting RMS position errors for 1-day OP are shown in Figure 12. Notably, although Cases E and F yield smaller position and velocity errors following OD at the specified epoch, their RMS position errors during OP are actually larger. The cause is the underestimated uncertainty of equivalent along-track acceleration, which is incompatible with the significantly greater uncertainty of position. Additional numerical tests were performed using an initial covariance of (\(10^{-2}\) m/s\({}^{2}\))\({}^{2}\) for \(a_{\text{T}}\), while keeping all other parameters consistent with Case 1-E, Case 1-F, Case 3-E, and Case 3-F. These modified configurations are designated as Case 1-E\({}^{\prime}\), Case 1-F\({}^{\prime}\), Case 3-E\({}^{\prime}\), and Case 3-F\({}^{\prime}\), respectively. The positional errors of 24 h OP after OD using these modified and their corresponding original configurations are presented in Table 3. Clear reductions can be observed in both RMS and the maximum position errors using modified configurations. As evidenced by the results, the RMS position error improvements range from a modest 13% reduction (when comparing Case 1-F relative to 1-F) to a substantial 49% decrease (observed between Case 3-E\({}^{\prime}\) and Case 3-E). Similarly, maximum position error reductions show comparable enhancement, varying between 22% for Case 1-F\({}^{\prime}\) and 48% for Case 3-E\({}^{\prime}\). These results highlight the significant influence that proper state covariance initialization can have on the accuracy of both OD and OP. It seems that a favorable balance between OD stability and OP accuracy can be achieved by setting the uncertainties of position, velocity, and the equivalent along-track acceleration to several tens of kilometers, tens of meters per second, and the order of \(10^{-2}\) m/s\({}^{2}\), respectively.