# Enhanced Conjunction Assessment in LEO: A Hybrid Monte Carlo and Spline-Based Method Using TLE Data
## 4 Conclusions  
This study presented a hybrid framework for enhancing the accuracy of collision probability estimation in low Earth orbit, addressing the limitations of traditional TLE-based methods. By integrating Monte Carlo simulations, spline interpolation, and multi-stage temporal refinement, the proposed methodology achieved significant improvements in estimating both the time of closest approach (TCA) and minimum separation distance. Empirical validation using real-world TLE datasets confirmed that the approach consistently reduced TCA errors to below 0.1 s and distance approximation errors to sub-kilometer levels. The framework demonstrated compatibility with standard SSA platforms and maintained computational efficiency, requiring less than one minute per event on conventional hardware. The alignment of the results with authoritative Space-Track data supports the operational viability of the proposed method for large-scale conjunction analysis. The comparative analysis between Monte Carlo and Differential Algebra Monte Carlo techniques further highlighted the latter's potential to balance computational cost with physical fidelity, especially in high-risk configurations. The findings also underscored the dependence of collision probability on both spatial proximity and uncertainty characterization, reinforcing the necessity of non-linear and data-driven modeling strategies. In light of the increasing congestion in orbital environments, this study recommends the adoption of hybrid, high-precision assessment techniques within operational collision avoidance systems. Future work should explore the extension of this framework to incorporate dynamic force models, real-time sensor fusion, and adaptive uncertainty propagation based on improved observation fidelity. While this study focuses on TLE-based data due to its accessibility and widespread use, the proposed framework is designed to be adaptable to higher-fidelity orbital datasets. If precise ephemerides and covariance matrices are available such as those derived from GPS-based tracking or provided in Conjunction Data Messages (CDMs)--the same methodology can be applied with enhanced reliability. The multi-stage TCA refinement algorithm can be integrated with high-order numerical propagators, and the probabilistic models can utilize formal uncertainty descriptions instead of assumed ones. Incorporating accurate force models and validated covariance data would significantly improve the fidelity of \(\mathrm{P_{c}}\) estimates, reduce epistemic uncertainty, and make the method suitable for operational use in high-risk conjunction assessment scenarios. While this study focused on Starlink satellites, the proposed framework is adaptable to satellites of various masses, geometries, and ballistic characteristics. Because the method relies on orbital state uncertainties and relative position sampling rather than on explicit physical parameters such as mass or area, it remains applicable to 1 kg class CubeSats as well as large spacecraft exceeding one metric ton with extended solar arrays. Our proposedmethod is primarily designed for short-term conjunction analysis based on the current or recently updated TLE data. Therefore, in its current form, the method does not predict sudden orbital perturbations in advance, as it assumes nominal propagation from known orbital states. However, if updated TLEs or precise ephemerides reflecting the disturbance are available, the method can still be applied to assess the updated conjunction risk. To handle such scenarios more effectively, it should also be noted that the framework assumes the availability of valid and timely orbital data and does not explicitly predict sudden orbital perturbations due to space weather events or atmospheric disturbances. While our method remains applicable once updated orbital data are available, it does not account for unmodeled perturbations during the propagation window.