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

TL;DR

The paper addresses global connectivity for LEO satellite megaconstellations by formulating percolation on the Earth’s surface and translating it to plane-based analysis via stereographic projection. It establishes a phase-transition framework with a critical satellite count $N_c$, derived in closed form as $N_c=\frac{\ln 2}{\ln 2-\ln(1+\cos\gamma)}$, along with tight bounds $N_c^{L}$ and $N_c^{U}$ from hexagonal face percolation; it also characterizes how altitude $h$ and maximum slant range $d_m$ influence the percolation threshold. The authors validate the theory with simulations using Starlink, OneWeb, and Kuiper parameters, showing that realistic deployments can achieve large-scale continuous service areas, and they provide deployment guidance and design considerations for achieving global connectivity. Overall, the work offers a rigorous connectivity framework for NTN/Laa satellite systems, bridging stochastic geometry and percolation theory to quantify the existence of giant connected coverage on the sphere with practical design insights.

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.

Figures (13)

• Figure 1: The geometric relationships between coverage angle $\gamma$, nadir angle $\eta$, satellite constellation altitude $h$ and maximum slant range $d_m$.
• Figure 2: The area of spherical cap and the Mercator projection.
• Figure 3: The stereographic projection. The projection plane is on the $xo_zy$ plane, which is tangent to the earth on the South Pole $\textbf{w}_S(\textbf{o}_z)$. The earth's center $\textbf{o}_e$ and North Pole $\textbf{w}_N$ are both on the z-axis. $P'$ is the stereographic projection of $P$. Any circle on the sphere corresponds to a circle on the projection plane. If the spherical cap excludes the North Pole $\textbf{w}_N$, the spherical cap is projected to a finite circular area. Inversely, any finite circular area corresponds to a spherical cap excluding $\textbf{w}_N$.
• Figure 4: Hexagonal lattice on the projected plane. The side length of hexagons is $a$ and $\mathcal{H}_0$ is the hexagon which is centered at the origin.
• Figure 5: Sub-critical case: All satellites are deployed on the same meridian, where neighbour coverage areas are tangent to each other. However, the longest spherical distance inside the coverage areas does not exceed $\pi r_e$ when the number of satellites is not large enough.

Theorems & Definitions (22)

• Lemma 1
• Theorem 1
• Theorem 2
• Lemma 2
• Remark 1
• Definition 1
• Remark 2
• Lemma 3
• Remark 3
• Lemma 4