With the advancement of space technology, the pace of spacecraft launches continues to increase, leading to a growing density of objects in Earth orbit.  As orbital regimes become increasingly saturated, operators are confronted with more frequent close-approach alerts, requiring a rising number of Collision Avoidance Maneuvers (CAMs) to protect their assets. A key step to ensure mission safety is the evaluation of conjunction risk through the estimation of Probability of Collision (PoC). Existing computation methods are generally tailored to two conjunction regimes: short-term and long-term encounters. While satellite operators rely on well-established heritage methods to assess the criticality of short-term events, PoC computation for long-term encounters remains less mature and more challenging due to the inherent complexity arising from time-dependent uncertainties. Although short-term scenarios constitute the majority of events that operators must address, assessing the risk of long-term encounters is becoming increasingly important due to the rapid deployment of large satellite constellations.
Within this framework, the Flight Dynamics (FD) team at the DLR’s German Space Operations Center (GSOC) is developing a tool to assess the risk of long-term encounters to their assets, thereby enhancing the capabilities of the existing, well-established Collision Avoidance System (CAS).
The methodology employed is a Monte Carlo based approach that uses Differential Algebra (DA) to model the non-linear evolution of the relative motion during the encounter through high-order Taylor expansions, while Automatic Domain Splitting (ADS) subdivides the initial uncertainty set to control accuracy. By sampling the initial relative conditions and evaluating the polynomial expansions over time, the method reduces classical numerical integration of the initial uncertainty to a one-dimensional polynomial evaluation. 
Computational efficiency is crucial for enabling timely and effective execution of potential CAMs. To address this, we present a further advancement of the methodology that incorporates advanced Monte Carlo sampling techniques. Specifically, the approach combines DA for precise nonlinear modeling with the Subset Sampling (SS) algorithm, a rare-event estimation technique particularly effective for small-probability events such as spacecraft collisions. SS works by expressing a rare event as a sequence of intermediate conditional events, each with a higher probability than the original event. Samples are generated iteratively at each conditional level, allowing the probability of the rare event to be estimated as the product of these intermediate probabilities. By doing so, SS dramatically reduces the number of samples required compared to standard Monte Carlo methods, while maintaining statistical reliability. A key advantage of combining these two techniques is that DA provides a high-order Taylor expansion of the system dynamics, allowing the explicit propagation of samples at each conditional level to be replaced by a simple polynomial evaluation, while simultaneously reducing the number of samples required to achieve a given statistical accuracy through the SS algorithm.
The paper outlines the methodology and its implementation, and assesses the performance of both the standalone DA-based approach and the combined DA–SS methodology for long-term conjunctions using representative benchmark scenarios from the literature.