Walker-delta pattern i : T / P / F. T satellites in P equally spaced circular orbital planes (right ascension 2πp/P), S = T/P per plane, equally spaced in argument of latitude; plane p is shifted by F x 360°/T. Position in the inertial frame: r = a [cos u cosΩ - sin u cos i sinΩ, cos u sinΩ + sin u cos i cosΩ, sin u sin i], u = u0 + n t.
a = R + h n = sqrt(mu / a^3) T_orb = 2 pi sqrt(a^3 / mu)
mu = 398600.44 km^3/s^2, R = 6371 km (h = 550 km gives about 95.5 min, 7.59 km/s)
Footprint. A user sees a satellite when its elevation is at least the mask e. The Earth-central half angle of the visible cap is lambda = arccos(R cos e / (R + h)) - e, the cap covers (1 - cos lambda)/2 of the sphere, so the average number of satellites above the mask anywhere on a uniform sphere is exactly T (1 - cos lambda)/2. The slant range at elevation e is sqrt((R+h)^2 - R^2 cos^2 e) - R sin e.
Why not equal everywhere. Satellites linger near the turning latitude of their orbit (≈ inclination), so the visible count peaks at mid latitudes and there is no coverage poleward of about i + lambda for an inclined shell. That is the profile chart above.
Latency. Light in vacuum travels at c = 299,792 km/s; in silica fibre at c/1.468 (about 0.68 c) and cables are longer than the great circle (route factor). A satellite path is user-to-satellite slant range + laser links between satellites + satellite-to-far-end range, all at c. The laser mesh is idealised: the great circle between the two serving satellites at orbit radius, split into straight chords that stay above 80 km altitude, multiplied by a stretch factor for grid routing, plus a switching delay per hop. Over short distances fibre wins (the satellite path pays two slant ranges, about 1.1 to 2 ms each way); the crossover chart shows where the longer laser path through near-vacuum catches up. Only propagation is modelled: no queueing, ground-network, or processing delay.
Doppler. f_D = -f0 (dr/dt)/c, with dr/dt the range rate between satellite and the (rotating) ground point. For a circular orbit seen from a ground point at elevation e the range rate is v R cos e / (R + h) at the horizon-most point of a pass, so the shift is bounded by f0 v R cos e/((R+h) c): about 250 to 290 kHz at 11.7 GHz for a 25° mask. Real terminals track and pre-compensate it.
Handover. A pass above a 25° mask lasts at most a few minutes, so the serving satellite has to change. Each switch is assumed to cost a fixed outage you set; no-coverage time (no satellite above the mask) is counted too.
Simplifications. Spherical Earth, circular two-body orbits (no J2 drift, drag, or station-keeping), perfect clear sky, one beam per user, no capacity or congestion model, no satellite pointing limits, no interference avoidance. Cities are generic marker points; coordinates are rounded.
Independent educational model, not affiliated with or endorsed by any company named here; trademarks belong to their owners. The default 550 km, 53 degree, 72 x 22 shell and the 25 degree mask are publicly described illustrative values for large low-orbit broadband systems; all numbers here are textbook-level approximations.