Connectivity of LEO Satellite Mega Constellations: An Application of Percolation Theory on a Sphere

2025-02-11

AI Summary

This paper addresses the critical deployment condition required for low Earth orbit (LEO) satellite constellations to achieve global continuous service coverage. We first extend percolation theory to the spherical domain, establishing a spherical percolation model and defining a connectivity phase-transition threshold. By integrating spherical geometry, stereographic projection, and stochastic geometric analysis, we derive a tight closed-form expression for the minimum number of satellites within a single constellation necessary to guarantee global connectivity. Our analysis quantitatively uncovers the coupled influence of orbital altitude and maximum slant range on the connectivity phase transition. The resulting critical thresholds—namely, the minimum satellite count, optimal orbital altitude, and maximum allowable slant range—constitute the first theoretically grounded, quantitative design benchmark for LEO constellation connectivity. This framework significantly enhances both the accuracy and interpretability of large-scale constellation planning.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization

Application Category

• Social Networks and Social Media: Social media analysis through the lenses of networks
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

With the advent of the 6G era, global connectivity has become a common goal in the evolution of communications, aiming to bring Internet services to more unconnected regions. Additionally, the rise of applications such as the Internet of Everything and remote education also requires global connectivity. Non-terrestrial networks (NTN), particularly low earth orbit (LEO) satellites, play a crucial role in this future vision. Although some literature already analyze the coverage performance using stochastic geometry, the ability of generating large-scale continuous service area is still expected to analyze. Therefore, in this paper, we mainly investigate the necessary conditions of LEO satellite deployment for large-scale continuous service coverage on the earth. Firstly, we apply percolation theory to a closed spherical surface and define the percolation on a sphere for the first time. We introduce the sub-critical and super-critical cases to prove the existence of the phase transition of percolation probability. Then, through stereographic projection, we introduce the tight bounds and closed-form expression of the critical number of LEO satellites on the same constellation. In addition, we also investigate how the altitude and maximum slant range of LEO satellites affect percolation probability, and derive the critical values of them. Based on our findings, we provide useful recommendations for companies planning to deploy LEO satellite networks to enhance connectivity.

Problem

• Analyze LEO satellite deployment for global connectivity
• Apply percolation theory to spherical surface first
• Determine critical number and parameters for LEO satellites

Innovation

• Percolation theory application
• LEO satellite deployment analysis
• Stereographic projection utilization