# Enhanced Conjunction Assessment in LEO: A Hybrid Monte Carlo and Spline-Based Method Using TLE Data
## 3 Physical Characterization and Results
### Collision Probability Estimation  
Four representative conjunction cases were selected from Table 2, specifically, the ID numbers\" 29798 & 60876, 61278 & 54616, 48826 & 25319, and 41531 & 60251\", to evaluate the accuracy of collision probability estimation using two methods: standard Monte Carlo and Differential Algebra Monte Carlo. The results were compared with reference data obtained from the Space-Track platform. Three test scenarios were defined to examine the performance of both methods under varying initial conditions. The three test cases described in Table 3 were designed as synthetic scenarios to evaluate the performance of collision probability estimation methods under controlled uncertainty configurations. These cases do not directly correspond to the real-world conjunctions listed in Table 2. Instead, they use assumed mean relative positions at the estimated TCA, defined in the local Radial-Transverse-Normal (RTN) frame of the primary object. For example, In Case 1, the mean relative position between the objects was set to [20, 0, 0] meters, with isotropic standard deviations of 50 m in all directions. In Case 2, the mean relative position was increased to [40, 0, 0] meters, and the standard deviations were defined as [100, 80, 60] meters along the x, y, and z axes, respectively. Case 3 involved a mean relative position of [100, 0, 0] meters, with standard deviations of [150, 120, 100] meters along the corresponding axes. In all cases, a uniform collision threshold distance of 5 m was applied. The covariance matrices used in the synthetic test cases are fixed and defined a priori for all Monte Carlo evaluations. They are not adjusted per case to fit reference values but selected to represent typical uncertainty scales consistent with LEO satellite conjunctions derived from TLE data. In the absence of publicly available formal covariance for the involved objects, a diagonal structure is  
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assumed. The same covariance values are applied consistently across all cases to ensure the results' comparability and avoid any tuning that might bias the probability estimates toward agreement with external references. To model orbital uncertainty in the absence of formal covariance data, a diagonal Gaussian distribution was applied to perturb the initial state vector of the primary object in each conjunction scenario. Standard deviations were set to 0.5 km in the radial direction, 1.0 km along-track, and 0.5 km cross-track, consistent with short-term TLE position errors reported in LEO. Monte Carlo sampling using this distribution revealed that typical variations in the time of closest approach (TCA) remained within \(\pm\)2 s, with miss distance deviations generally below \(\pm\)1 km. These values quantify the sensitivity of conjunction geometry to small perturbations in the initial state and justify the probabilistic approach adopted in this study. Table 3 and Figure 13 illustrate the comparison of collision probability estimates using the Monte Carlo (MC) and Differential Algebra Monte Carlo (DAMC) methods for three different cases. Each subplot represents one case, with associated error bars indicating the standard deviation of the computed probabilities. In Case 1 of each conjunction scenario, which represents the closest approach and highest collision risk configuration, the estimated probabilities were the highest among the three tested configurations. The MC method reported average probabilities around 2.45 \(\times\) 10\({}^{-4}\) with a standard deviation of 1.6 \(\times\) 10\({}^{-5}\), while DAMC estimates averaged approximately 2.20 \(\times\) 10\({}^{-4}\) with a lower standard deviation of 1.4 \(\times\) 10\({}^{-5}\). These estimates aligned closely with the real collision probabilities reported from Space-Track, which ranged from 1.11 \(\times\) 10\({}^{-4}\) to 2.73 \(\times\) 10\({}^{-4}\), demonstrating good agreement and validating the methods' applicability in realistic scenarios.