# Enhanced Conjunction Assessment in LEO: A Hybrid Monte Carlo and Spline-Based Method Using TLE Data
## 3 Physical Characterization and Results
### Collision Probability Estimation

Table 3: _Cont._such, several assumptions were made to balance modeling fidelity with computational efficiency. TLE-based orbital states: All analyses rely on SGP4-propagated TLEs without access to formal covariance matrices. This limits the ability to model correlated uncertainties and dynamic consistency over extended prediction intervals. The performance is therefore most reliable over short timescales (typically \(<5\) days from epoch). Simplified perturbation modeling: The use of SGP4 omits higher-order perturbations such as solar radiation pressure, third-body effects, and atmospheric density variations beyond the standard drag model. This may reduce accuracy for objects in highly elliptical orbits or near the upper LEO boundary (>1200 km). Uncertainty distribution: A diagonal Gaussian or uniform distribution was assumed for perturbing the initial TLE-based state. This simplification neglects correlations and may not fully capture the true shape of the uncertainty region, particularly for objects tracked with limited sensor coverage. Single-sided sampling: Only one object in each conjunction pair was sampled probabilistically, while the other followed a deterministic trajectory. This reduces computational cost but may underestimate risk in dual-uncertainty scenarios. The method is best suited for LEO missions involving short-term risk assessment, mega-constellations (e.g., Starlink, One Web) with dense traffic and limited access to high-fidelity ephemerides, and civilian or academic SSA applications relying on open-source data. Its applicability to MEO, GEO, or highly elliptical orbits may require modification of the propagation model and incorporation of formal covariance data. In such cases, high-fidelity numerical integrators and more complex uncertainty models should be considered. Future extensions of this framework will explore adaptive uncertainty propagation, use of precise ephemerides when available, and application in broader orbital regimes beyond LEO. The use of Monte Carlo sampling allows the propagation of a wide range of initial conditions without assuming linear or Gaussian behavior in the relative motion. Spline-based interpolation of the separation distance enables accurate localization of the time of closest approach (TCA), even when the distance function is non-monotonic or exhibits sharp minima. The hierarchical multi-stage refinement performs brute-force evaluation of distances across increasingly narrow time windows. This avoids relying on gradients or local optimization, which can fail in the presence of oscillatory or non-smooth dynamics. Unlike linearized probabilistic models, the method does not assume a fixed structure of the uncertainty region or linearity in the state transition. Therefore, it remains robust when applied to scenarios with significant orbital curvature, non-planar relative trajectories, or non-Gaussian uncertainty dispersion. However, as the propagation model is based on SGP4, its fidelity under highly non-linear perturbations (e.g., high-altitude eccentric orbits with strong third-body effects) remains limited by the underlying propagator. Future integration of high-order force models could further enhance performance in these regimes. The proposed method can be extended to operate on high-fidelity ephemerides and formal covariance data. This would require replacing the SGP4 propagator with high-order numerical integrators and updating the uncertainty sampling process to incorporate full 6D covariance matrices. Such an enhancement would expand the method's applicability to conjunction scenarios involving high-value or maneuverable assets, and to orbital regimes beyond LEO, such as MEO, GEO, or highly elliptical orbits. While this increase in modeling fidelity is expected to improve accuracy, it would also raise the computational cost per conjunction case.