Densification Converses for Walker Constellations With Explicit Constants and Reuse Scaling Laws

TL;DR

This paper proves that increasing satellite numbers in Walker LEO constellations under certain conditions causes the signal-to-interference ratio to vanish, establishing explicit bounds and the necessity of frequency reuse for densification.

Contribution

It introduces densification converses for Walker constellations with explicit constants and reuse scaling laws, providing nonasymptotic bounds and visibility-based technical tools.

Findings

Aggregate interference grows at least linearly with satellite count

Coverage probability and spectral efficiency vanish as satellite count increases

Frequency reuse with activity probability q must satisfy qN=O(1) to avoid collapse

Abstract

We establish densification converses for Walker LEO constellations under nearest-visible association in the full-frequency-reuse setting. Performance is evaluated under the invariant (stationary) measure induced by the constellation/Earth dynamics on the user--constellation ``phase state.'' A key Walker-specific feature, absent from unbounded planar models, is that association is restricted to a bounded visible cap determined by Earth geometry. Under power-law path-loss, a two-level antenna-gain model, i.i.d.\ nonnegative fading with unit mean and finite second moment, and nonzero noise, we prove that increasing the total satellite count $N=N_oN_s$ forces the aggregate interference to grow at least linearly in $N$, while the useful signal remains uniformly bounded above. Consequently, the downlink SINR coverage probability at any fixed threshold and the ergodic spectral efficiency both vanish as $N\to\infty$. The key technical ingredient is a deterministic visibility-annulus block lemma, uniform over all sufficiently large constellations and all "phase states", showing that a fixed fraction of visible satellites lies in a distance annulus strictly inside the horizon; this yields explicit finite-$N$ collapse bounds. In particular, we derive nonasymptotic $O(1/N)$ upper bounds on both coverage and ergodic spectral efficiency. Finally, in the case of frequency reuse through independent thinning, with activity probability $q$, we show that avoiding densification collapse necessarily requires $qN=O(1)$, equivalently a reuse factor $\Omega(N)$, and we obtain a corresponding explicit $O(1/(qN))$ upper bound.