# Enhanced Conjunction Assessment in LEO: A Hybrid Monte Carlo and Spline-Based Method Using TLE Data
## 3 Physical Characterization and Results
### Case Study Results and Relative Motion Analysis  
This section presents the core results from the evaluated conjunction cases, integrating numerical deviations, statistical behavior, and relative motion analysis. Table 2 summarizes the deviations in time of closest approach (TCA) and miss distance across eight selected events. The multi-stage method consistently reduces temporal error, with spatial accuracy varying depending on encounter geometry and object dynamics. The accuracy of determining the time of closest approach and the corresponding minimum distance depends on precise estimation of satellite and debris orbital states. The evaluation process began by applying the Improved Critical Region Estimation (ICRE) method to identify an approximate TCA within a 120 h prediction window from the TLE epoch, aiming to locate a preliminary point near the expected minimum distance. The next step refined the estimation by narrowing the time interval to 10 min around the TCA estimated by ICRE, enhancing temporal resolution and focusing the search. In the final stage, the multi-stage method applied a 1-millisecond time step over a 3 s window centered on the refined TCA, enabling precise determination of both the TCA and the minimum distance (see Figure 5). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Object A** & \begin{tabular}{c} **Inclination** \\ **Degree** \\ \end{tabular}  & \ \begin{tabular}{c} **No of Detected** \\ **Object** \\ \end{tabular}  & **Object B** \\ \hline STARLINK-1008 & 43 & 5 & Debris Catalog \\ \hline STARLINK-5691 & 54 & 6 &
\begin{tabular}{c} (Irdum33\(\mathrm{\SIUnitSymbolMicro}\)\(\mathrm  
Table 2 presents a detailed comparison between the estimated conjunction results obtained using the proposed multi-stage refinement method and reference values derived from Space-Track data across eight different conjunction cases. For each case, the time of closest approach (TCA) and the minimum separation distance are reported separately for both the proposed method and the comparison data. The deviations in TCA and distance are shown as independent error indicators. No assumption is made that the miss distance deviation results solely from the TCA difference. To account for the difference in the time of epoch between the TLE data and the reference epoch used in the analysis, orbital element propagation was performed to unify the temporal reference frame. This step ensures consistency in the comparison by aligning the orbital states to a common epoch before conjunction assessment. The proposed method achieved an average TCA deviation of \(3.1\times 10^{-2}\) seconds, reflecting consistent temporal precision. The use of a final refinement step with temporal resolution down to \(7\times 10^{-3}\) seconds enables accurate localization of the closest approach within each encounter. Nevertheless, the achievable accuracy remains inherently limited by the quality of the input orbital data, particularly due to the simplified dynamics of the SGP4 propagator, the inherent uncertainty in TLE state vectors, and the absence of formal covariance information. In seven out of eight evaluated cases, the TCA deviation remained below 0.3 s, confirming the effectiveness of the progressive time refinement strategy. The average deviation in minimum separation distance across all cases was 0.688 km. This variability reflects the influence of encounter geometry, numerical sensitivity to propagation conditions, and potential differences between the comparison data source and the TLE-based approach. The results confirm that the structured, multi-stage method enhances the fidelity of temporal estimation, while spatial accuracy remains dependent on orbital dynamics and input fidelity. Figures 6A-D-8A-D presented in this study illustrate key aspects of conjunction analysis and collision risk estimation using the proposed hybrid methodology. Figure 6A-D visualize the offset in distance over time for four selected conjunction events.