# 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
These plots confirm that the distance follows a clear curve with a distinct minimum point, validating the need for precise temporal resolution. Figure 7A-D provide histograms of relative distances for the same cases. The distribution of distances is concentrated around the minimum value, reflecting the consistency of close approach estimates and supporting probabilistic modeling. Figure 8A-D show the absolute 3D trajectories of the objects during the conjunctions. The visual trajectories confirm proximity and alignment between objects, supporting the physical plausibility of the predicted encounters. Figures 9A-D-11A-D illustrate the anticipated relative motion in the XY, XZ, and YZ planes for four conjunction cases. These projections reveal curved, non-planar trajectories that emphasize the geometric complexity of close approaches. While relative motion in such scenarios is governed by non-linear dynamics, the key limitation to linear or Gaussian assumptions arises from the propagation of uncertainty through inherently non-linear transformations. Although small perturbations can maintain approximate Gaussianity under local linearization, significant deviations and extended propagation intervals typically distort the shape of the uncertainty distribution. This behavior underscores the necessity for sampling-based or non-Gaussian models in high-fidelity conjunction assessment. Figure 12A-D combine the three dimensions into a comprehensive 3D representation of relative motion. This visualization captures the entire interaction trajectory between each pair of objects. The 3D paths exhibit curved, non-planar motion, which justifies the application of spline interpolation and brute-force temporal refinement instead of simplified approximations. Together, the figures confirm that the proposed multi-stage refinement approach enhances TCA estimation and aids in practical collision risk assessment. Figure 6: _Cont._  
Figure 6: (**A**) Distance offsets over time for four separate conjunction events: 24 April 2025. (**B**) Distance offsets over time for four separate conjunction events: 15 May 2025. (**C**) Distance offsets over time for four separate conjunction events: 23 May 2025. (**D**) Distance offsets over time for four separate conjunction events: 23 May 2025. Figure 7: _Cont_. Figure 7: **(A)** Relative distance histogram—24 April 2025 conjunction. **(B)** Relative distance histogram—15 May 2025 conjunction. **(C)** Relative distance histogram—23 May 2025 conjunction. **(D)** Relative distance histogram—24 May 2025 conjunction. Figure 8: _Cont._  
Figure 8: (**A**) Three-dimensional trajectories during conjunction—24 April 2025. (**B**) Three-dimensional trajectories during conjunction—15 May 2025. (**C**) Three-dimensional trajectories during conjunction—23 May 2025. (**D**) Three-dimensional trajectories during conjunction—24 May 2025. Figure 9: _Cont._  
Figure 10: _Cont._  
Figure 9: (**A**) Relative motion in XY plane—24 April 2025. (**B**) Relative motion in XY plane—15 May 2025. (**C**) Relative motion in XY plane—23 May 2025. (**D**) Relative motion in XY plane—24 May 2025. Figure 10: (**A**) Relative motion in XZ plane—24 April 2025. (**B**) Relative motion in XZ plane—15 May 2025. (**C**) Relative motion in XZ plane—23 May 2025. (**D**) Relative motion in XZ plane—24 May 2025. Figure 6:0 _Cont._  
Figure 6:1 (**A**) Relative motion in Z\(Y\) plane—24 April 2025. (**B**) Relative motion in Z\(Y\) plane—15 May 2025. (**C**) Relative motion in Z\(A\) plane—23 May 2025. (**D**) Relative motion in Z\(Y\) plane—24 May 2025. Figure 6:2 (**A**) Full 3D relative motion—24 April 2025. (**B**) Full 3D relative motion—15 May 2025. (**C**) Full 3D relative motion—23 May 2025. (**D**) Full 3D relative motion—24 May 2025.