# Enhanced Conjunction Assessment in LEO: A Hybrid Monte Carlo and Spline-Based Method Using TLE Data
## 1 Introduction  
The rapid proliferation of artificial satellites and orbital debris in low Earth orbit (LEO) has significantly increased the probability of in-orbit collisions [1]. Mega-constellations such as Starlink and One Web have accelerated congestion in critical orbital regions, raising concerns regarding the long-term sustainability of near-Earth space [2]. While the 750-850 km sun-synchronous orbit (SSO) region remains the most collision-prone zone due to the accumulation of long-lived debris and inactive objects, recent large-scale deployments of mega-constellations such as Starlink (~550 km) and One Web (~1200 km) have substantially increased the number of active satellites across multiple altitude bands in low Earth orbit (LEO). Although these constellations do not operate within the SSO range, their presence contributes to growing orbital congestion and increases the frequency of potential conjunction events. Cross-altitudeinteractions, influenced by orbital decay, maneuvering activities, and propagation uncertainties, further elevate the complexity of space traffic management across LEO. Increasing solar activity and the rapid growth of mega-constellations have intensified concerns about orbital stability in low Earth orbit (LEO). Periods of heightened geomagnetic activity such as those recorded during solar cycle 25 have caused significant perturbations in satellite orbits due to enhanced atmospheric drag. These disturbances shorten orbital lifetimes and complicate conjunction risk forecasting, particularly for low-mass spacecraft. The dynamic and congested nature of LEO now demands more resilient and adaptive approaches to conjunction assessment that can operate under uncertain and time-varying orbital conditions [3] The total number of tracked space objects in low Earth orbit (LEO) larger than 10 cm, including active satellites, defunct spacecraft, and debris fragments, has surpassed 63,774, as reported by CelesTrak statistical summaries. As of May 2025, over 11,000 of these are active satellites. However, publicly available Two-Line Element (TLE) data [4], which serves as the primary input for this study, encompasses only a subset of these tracked objects. At the time of analysis, approximately 20,314 cataloged objects had valid TLE entries in LEO. This discrepancy with the reported 42,000 objects occurs because not all cataloged debris is assigned or published with corresponding TLEs, as illustrated in Figure 1. The analytical framework of this study relies solely on TLE-available entries, increasing the risk of chain-reaction collisions, known as the Kessler Syndrome [5]. Reliable collision probability estimation is essential for conjunction assessment and mitigation planning [6]. TLE-based approaches remain widely used due to their availability and ease of integration, particularly for academic and civil space situational awareness efforts [7]. However, they are constrained by limited accuracy, infrequent updates, and lack of formal uncertainty representation, which affects conjunction prediction reliability [8]. Two-Line Element sets remain the most widely accessible source for orbital data due to their global availability and ease of integration. However, TLEs suffer from known limitations, including infrequent updates, simplified drag modeling, and a lack of formal covariance information. These constraints limit their predictive accuracy, especially for high-drag or maneuvering objects. Despite this, TLEs are still widely used in civil and academic space situational awareness due to the absence of alternatives for many cataloged objects [9]. Existing collision assessment models often linearize orbital dynamics and assume Gaussian distributed uncertainties. For instance, the widely used Gaussian probability density function (PDF) method by [PERSON] assumes linear propagation and symmetric error ellipsoids [10], which are not well-suited for non-linear relative motion scenarios [11].