Low Earth Orbit (LEO) Satellite Mega-Constellations: Performance Metrics and Design Insights

Last updated: June 10, 2025

Certainly! The following is a meticulously edited and fact-faithful academic article synthesizing the findings, analytical methods, and practical design insights ° for Low Earth Orbit ° (LEO °) Satellite Mega-Constellations, strictly based on (Okati et al., 2020 ° ).

Abstract

This work presents an analytical framework ° for the performance assessment of Low Earth Orbit (LEO) satellite mega-constellations, focusing on coverage probability ° and data rate for inclined (non-uniform) satellite deployments. Unlike traditional analyses that depend on exact satellite positions or pure simulation, the methodology introduces tractable analytic expressions that incorporate real-world constellation geometry via a novel effective number of satellites parameter. The framework supports systematic optimization of key constellation parameters, enabling efficient design and accurate evaluation across a wide range of deployment scenarios.

1. Analytical Performance Framework

The framework is engineered to compute downlink and uplink performance metrics for large-scale LEO constellations, considering general fading and arbitrary constellation parameters. The methodology comprises two central phases:

1. Uniform Distribution ° Abstraction: The constellation is first abstracted as a set of $N$ satellites uniformly distributed over a spherical shell, allowing closed-form analysis of user coverage and data rate.
2. Latitude-Dependent Correction: The effective number of satellites parameter, $N_{\mathrm{eff}}$, corrects the uniform model to accurately reflect the latitude-dependent, non-uniform density ° of realistic inclined constellations, ensuring model fidelity at every user location.

2. Analytical Expressions for Coverage Probability and Data Rate

Coverage Probability

Coverage probability $c(T)$ is defined as the probability that the signal-to-noise ratio (SNR °) at a given user position exceeds a threshold $T$:

$c(T) = \mathbb{P}(\mathrm{SNR} > T)$

The serving satellite is the nearest visible satellite above a minimum elevation, with noise-limited assumption and general fading (e.g., Rician fading as a worked example). For $N$ uniformly distributed satellites, the coverage probability is given by:

[ c(T) = \frac{N}{2(R_E + h)^2} \int_{r_{\min}}^{r_{\max}} \left[1 - F_{G_0}\left(\frac{T r_0^\alpha \sigma^{2}{P_s}\right)\right]} \left(1 - \frac{r_0² - r_{\min}^2}{4(R_E + h)^{2}\right){N-1}} r_0 \, dr_0 ]

where:

• $R_E$ is the Earth's radius,
• $h$ is satellite altitude,
• $r_0$ is user-to-satellite distance ($r_0 \in [r_{\min}, r_{\max}]$, $r_{\min} = h$),
• $F_{G_0}$ is the CDF ° of channel fading ° gain,
• $\alpha$ is path loss ° exponent,
• $\sigma^2$ is noise power,
• $P_s$ is satellite transmit power.

The maximum coverage ° (when $T=0$) is:

$c_{\max} = 1 - (1 - V)^N$

with $V$ the visibility probability for a single satellite.

Average Achievable Data Rate

The average achievable spectrum-normalized capacity is:

$\bar{C} = \mathbb{E}\left[\log_2(1 + \mathrm{SNR})\right]$

Main analytical formula:

[ \bar{C} = \frac{N}{2 \ln 2 \cdot (R_E + h)^2} \int_{r_{\min}}^{r_{\max}} \int_{0}^{\infty} \ln \left[1 + \frac{P_s g_0 r_0^{{-\alpha}}{\sigma2}\right]} f_{G_0}(g_0) \left(1 - \frac{r_0² - r_{\min}^2}{4(R_E + h)^{2}\right){N-1}} r_0 \, dg_0\, dr_0 ]

where $f_{G_0}$ is the PDF ° of the channel gain °.

3. The Effective Number of Satellites Parameter

Real inclined constellations do not yield uniform satellite visibility across latitudes; satellites concentrate near inclination-edge regions and are sparse near the equator.

To correct for this, the effective number of satellites ($N_{\mathrm{eff}}$) is defined as:

$N_{\mathrm{eff}} = \frac{2 f_{\phi_s}(\varphi)}{\cos \varphi} \, N_{\text{act}}$

where:

• $N_{\text{act}}$ is the actual constellation size,
• $\varphi$ is user latitude,
• $f_{\phi_s}(\varphi)$ is the satellite-latitude distribution (PDF), which—given inclination $\iota$—is:

[ f_{\phi_s}(\varphi) = \frac{\sqrt{2}}{\pi} \cdot \frac{\cos \varphi}{\sqrt{\cos(2\varphi) - \cos(2\iota)}} \quad \text{for } \varphi \in [-\iota, \iota] ]

This parameter ensures the analytical formulas, when plugged with $N_{\mathrm{eff}}$, accurately predict real-world performance at each latitude.

• Polar Orbits ($\iota=90^\circ$): $N_{\mathrm{eff}}$ at the equator ≈ 0.64 $N_{\text{act}}$.
• High Latitudes: $N_{\mathrm{eff}}$ can exceed $N_{\text{act}}$, indicating frequent satellite crossings.

Replacing $N$ with $N_{\mathrm{eff}}$ in all analytical expressions enables accurate, location-dependent predictions for design and planning.

4. Design Insights for LEO Constellation Parameters

The framework provides actionable insights to guide satellite network design:

(A) Total Number of Satellites ($N$)

• More satellites increase coverage probability (especially at higher elevations).
• Diminishing returns: Coverage improvement per satellite levels off as $N$ increases.
• Latitudinal non-uniformity: Additional satellites disproportionately benefit equatorial regions.

(B) Altitude ($h$)

• Higher $h$ enhances individual satellite visibility (larger cap), raising maximum coverage at low $N$.
• Trade-off: Higher $h$ incurs greater path loss, reducing SNR and data rate. There exists an optimal altitude maximizing either coverage or data rate for each set of parameters.

(C) Inclination Angle ($\iota$)

• Larger inclination increases coverage at high latitudes but reduces equatorial density.
• Design can be tailored through inclination for target market coverage (e.g., polar vs. equatorial regions).

(D) Minimum Elevation Angle ($\theta_{\min}$)

• Higher $\theta_{\min}$ (i.e., requiring satellites to be higher above horizon) reduces the visible area but increases link quality.
• Tuning $\theta_{\min}$ is a trade-off between coverage probability and link robustness.

Performance predictions for coverage and data rate, including optimal configuration points for any parameter, can be made by directly evaluating the above analytic expressions using simulation-backed $N_{\mathrm{eff}}$ for each latitude.

5. Numerical Validation

• Monte Carlo simulations closely match the analytic predictions when $N_{\mathrm{eff}}$ is used.
• Validation spans varying satellite numbers ($N$), orbital altitudes ($h$), inclination angles ° ($\iota$), and elevation angle thresholds ($\theta_{\min}$).
• The framework identifies optimal values for altitude, inclination, and satellite count for various traffic and region requirements.

6. Summary Table of Key Formulas

7. Conclusion

This analytical methodology delivers a robust, generalizable tool for evaluating and optimizing the coverage and data rate of LEO mega-constellation networks—regardless of constellation geometry, fading, or user location. By introducing and validating the effective number of satellites parameter, the approach permits existing uniform-analytical models to be adapted, with high accuracy, to real-world satellite constellations °. This enables satellite system designers to efficiently explore trade-offs in satellite count, altitude, inclination, and coverage requirements, leading to better-informed technical and commercial deployment strategies °.