The two-dimensional probability of collision (2D PoC) formulation remains one of the most widely used tools for operational conjunction assessment due to its closed-form representation and computational simplicity. Despite its popularity, extensive operational experience and multiple studies have highlighted that the 2D PoC may exhibit substantial inaccuracies for specific classes of conjunctions, particularly when its key assumptions - high relative velocity and negligible velocity uncertainty, rectilinear relative motion over the encounter interval, and Gaussian nature of the relative position distribution - are challenged. In contrast, high-fidelity Monte Carlo (MC) methods have evolved into reliable reference techniques, though their computational burden remains prohibitive for operational use and their outcome may depend on implementation settings [Hall 2018]. Between these two extremes lie several intermediate approaches, among which Coppola’s long-term encounter formulation (3D PoC) occupies a fundamental role.
What is currently lacking, however, is a robust set of indicators reflecting the domain of validity of relevant PoC methods. This shortcoming has limited both the interpretability and reliability of operational PoC estimates across agencies. The present study establishes a systematic classification of conjunction types through a detailed numerical exploration, enabling the identification of domains of validity for different methods, and the diagnosis of assumption failures responsible for inaccurate PoC predictions. Instead of classical fast-versus-slow encounter dichotomy, we demonstrate that conjunctions between Earth-orbiting objects span a considerably richer spectrum of behaviors. This complexity necessitates a multidimensional set of quantitative metrics to assess the validity of simplifying assumptions, and to guide the appropriate selection of PoC algorithms.
 
The analysis integrates multiple datasets and methodological frameworks, including the extensive CNES database of more than 40,000 conjunctions, a curated subset of roughly 100 representative cases, and selected scenarios from the NASA–CARA repository. Across these datasets, PoC estimates are computed using the classical 2D PoC, a CNES/GMV Monte Carlo approach initiated at TCA with samples drawn either from Cartesian (MCCar) or equinoctial coordinates (MCeq) and 3D PoC [Coppola 2012]. Coppola’s short-term encounter time-span validity is also revisited.
 
A reference classification of encounters is defined based on the above computations and function of several readily accessible indicators including the offset, extended and inaccuracy indicators of 2D PoC usage violation [Hall 2023], offset-from-TCA [Hall 2019] miss distance, Mahalanobis distance, 2D PoC and instantaneous PoC. The relative total variation of the offset-from-TCA 2D PoC function and the indicator V', measuring the amplitude of the offset-from-TCA 2D PoC variations are then introduced and recalled.
 
A central contribution of this work is an in-depth comparison of the two Monte Carlo PoC estimates MCCar and MCeq. A striking observation is that these two MC approaches can yield noticeably different PoC values. This means that MC-based PoC values must be interpreted with care, as differences may arise from modeling conventions rather than physical characteristics of the encounter. Moreover, for a large subclass of these cases, we observe that MCCar and 3D PoC agree on one side when MCeq and 2D PoC are very close on the other side.  Across the full database, however, a large variety of combinations of agreement and disagreement appear. In parallel, an automated clustering approach was developed to detect recurring geometric and uncertainty-structure patterns across the dataset. These findings further reinforce that no single indicator, nor any single computational method, can reliably predict PoC accuracy across all encounter scenarios.
 
References:
[Coppola 2012] V. T. Coppola. Including Velocity Uncertainty in the Probability of Collision between Space Objects. Advances in the Astronautical Sciences, 143, 2012.
[Hall 2018] D. T. Hall, S. J. Casali, L. C. Johnson, B. B. Skehart, L. G. Baars. High fidelity collision probabilities estimated using brute force Monte Carlo simulations. AAS/AIAA Astrodynamics Specialist Conference, Number AAS 18-244, Snowbird, UT, USA, August 2018.
 
[Hall 2019] D. T. Hall. Implementations recommendations and usage boundaries for the two-dimensional probability of collision calculation. AAS/AIAA Astrodynamics Specialist Conference, Number AAS 19-632, Portland, Oregon, USA, August 2019.
[Hall 2023] D. T. Hall, L. G. Baars, and S. J. Casali. A multistep probability of collision computational algorithm. AAS/AIAA Astrodynamics Specialist Conference, Big Sky, MT USA, August 2023.