# Enhanced Conjunction Assessment in LEO: A Hybrid Monte Carlo and Spline-Based Method Using TLE Data
## 3 Physical Characterization and Results
### Collision Probability Estimation
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Case** & **Method** & **Collision** & **Standard** & **Real** & **TAC** & **Min Distance** (km) \\ \hline \multirow{2}{*}{1} & MC & 0.000243 & 0.000016 & & & \\ \cline{2-6}  & DAMC & 0.000219 & 0.000014 & & & \\ \hline \multirow{2}{*}{2} & MC & 0.000061 & 0.000008 & & 2025-04-25 & \\ \cline{2-6}  & DAMC & 0.000055 & 0.000007 & & 18:56:38.501680 & 0.203779 \\ \hline \multirow{2}{*}{3} & MC & 0.000015 & 0.000004 & & & \\ \cline{2-6}  & DAMC & 0.000014 & 0.000003 & & & \\ \hline \multirow{2}{*}{1} & MC & 0.000246 & 0.000016 & & & \\ \cline{2-6}  & DAMC & 0.000221 & 0.000014 & & & \\ \hline \multirow{2}{*}{2} & MC & 0.000073 & 0.000009 & & & \\ \cline{2-6}  & DAMC & 0.000066 & 0.000008 & & 22:16:14.795 & 0.6264654 \\ \hline \multirow{2}{*}{3} & MC & 0.000015 & 0.000004 & & & \\ \cline{2-6}  & DAMC & 0.000014 & 0.000003 & & & \\ \hline \multirow{2}{*}{1} & MC & 0.000245 & 0.000016 & & & \\ \cline{2-6}  & DAMC & 0.000220 & 0.000014 & & & \\ \hline \multirow{2}{*}{2} & MC & 0.000066 & 0.000008 & & & \\ \cline{2-6}  & DAMC & 0.000059 & 0.000007 & & 07:18:51 & 2.9253064 \\ \hline \multirow{2}{*}{3} & MC & 0.000012 & 0.000003 & & & \\ \cline{2-6}  & DAMC & 0.000011 & 0.000003 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Numerical comparison of collision probability estimates. In Cases 2 and 3, which involved greater mean relative distances and larger standard deviations, the estimated collision probabilities decreased significantly. This inverse relationship between separation distance and collision probability was expected, as increased spatial dispersion reduces the likelihood of intersecting trajectories. In Case 2, the MC method produced probabilities ranging from 5.8 \(\times\) 10\({}^{-5}\) to 7.3 \(\times\) 10\({}^{-5}\), while DAMC values ranged from 5.2 \(\times\) 10\({}^{-5}\) to 6.6 \(\times\) 10\({}^{-5}\). In Case 3, both methods reported probabilities below 1.5 \(\times\) 10\({}^{-5}\), confirming the decreasing trend and the low-risk nature of these configurations. The minimum distances at TCA, ranging from 0.20 km to 2.93 km across the studied cases, further contextualize the probability values. Notably, higher collision probabilities in Case 1 are associated with minimum distances below 0.7 km, while cases with minimum distances above 2 km resulted in negligible probabilities. This correspondence reinforces the dependency of collision likelihood on proximity at closest approach. Methods using TLE-only propagation typically exhibit TCA errors above 1-2 s and spatial errors exceeding 1.5 km over multi-day windows [7; 8]. Gaussian PDF-based approaches are efficient but often underestimate risk in non-linear encounters due to linear assumptions [10]. Full-scale Monte Carlo methods offer higher accuracy but at the cost of significant runtime, often requiring several minutes per case [12]. Compared to these, our hybrid method achieved sub-second TCA accuracy and kilometer-level spatial resolution in under 1 min per case, using standard hardware and real TLE data. The proposed framework was designed for collision risk assessment in low Earth orbit (LEO) using publicly available TLE data. As  
Figure 13: Comparison of collision probability estimates for three conjunction scenarios. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multirow{2}{*}{**Method**} & **Collision** & **Standard** & **Real** & **TAC** & **Min Distance** \\  & & **Probability** & **Deviation** & **Collision** & **TAC** & **(km)** \\ \hline \multirow{2}{*}{1} & MC & 0.000244 & 0.000016 & & & \\ \cline{2-5}  & DAMC & 0.000220 & 0.000014 & & & \\ \hline \multirow{2}{*}{2} & MC & 0.000058 & 0.000008 & & 2025-05-24 & \\ \cline{2-5}  & DAMC & 0.000052 & 0.000007 & & 06:32:54.169398 & 0.295490267 \\ \hline \multirow{2}{*}{3} & MC & 0.000008 & 0.000003 & & & \\ \cline{2-5}  & DAMC & 0.000007 & 0.000003 & & & \\ \hline \hline \end{tabular}
\end{table}