Ground tracks and coverage

An orbit is a curve in inertial space. A ground track is its projection onto a rotating Earth, and it is what actually determines whether a satellite can see what it needs to see.

Construction and drift

The subsatellite latitude oscillates between $+i$ and $-i$ once per orbit, so inclination sets the highest latitude a satellite ever passes over. A 51.6° orbit never overflies northern Europe; a polar orbit covers everything.

Meanwhile the Earth turns underneath. Each successive pass crosses the equator further west by

with $\omega_\oplus = 7.2921 \times 10^{-5}$ rad/s.

The GEO row is the definition of the regime. The GPS row explains why the GPS ground track repeats daily after exactly two revolutions.

For precision the nodal period should be used rather than the Keplerian one, since $J_2$ makes the equator-to-equator interval slightly different from the orbital period, and the node itself is regressing.

Repeat ground tracks

If the satellite completes exactly $k$ orbits while the Earth turns $j$ times relative to the precessing orbital plane, the ground track closes and repeats indefinitely:

This is a strong constraint. It fixes the semi-major axis to a discrete value rather than a range, which then interacts with the sun-synchronous condition, which already tied inclination to altitude. Two constraints leave very little freedom, and Earth observation orbits are effectively selected from a short list.

The value of an exact repeat is that every pass over a target has the same geometry: same altitude, same viewing angle, same illumination if the orbit is also sun-synchronous. Change detection becomes a matter of differencing two images rather than modelling away the differences between them.

Maintaining the repeat costs a small amount of propellant, because drag constantly perturbs the semi-major axis away from the exact resonant value. The control band is typically a few hundred metres in ground track position.

Access geometry

A satellite is visible from a ground point when it is above the local horizon by at least the elevation mask, the minimum angle set by terrain, obstructions and the point at which atmospheric path loss becomes unacceptable. Typical masks are 5° for a large tracking antenna and 10° to 25° for a small user terminal.

The Earth-central angle to the edge of the access region is

The 400 km rows show how expensive elevation mask is. Raising the mask from 0° to 10° costs 40% of the footprint radius and roughly two thirds of the covered area. This single parameter drives constellation size more than almost anything else, and it is set by the user terminal, not by the space segment.

Pass duration

For a directly overhead pass, contact lasts roughly

At 400 km with a zero-degree mask this gives about 10 minutes; with a 10° mask, about 6 minutes. Most passes are not overhead, so typical usable contacts are shorter, and a low-altitude satellite in view of a single ground station gets on the order of four to six passes per day totalling perhaps 30 minutes.

That number is the origin of the downlink bottleneck. A satellite generating terabytes per day has minutes per day to transmit, which is the argument for edge computing in space, for ground station networks, and for inter-satellite links.

Coverage figures of merit

Coverage is not a single number and specifying it as one is a common source of requirement disputes.

Instantaneous coverage

Fraction of the target region visible right now. Meaningful only for continuous-coverage systems.

Revisit time

Interval between successive accesses to a point. Quote the maximum, not just the mean; the mean hides the gaps that matter.

Response time

Delay from a tasking request to the first opportunity to satisfy it. Includes uplink scheduling, not just geometry.

Availability

Fraction of time a point has access at or above the required elevation. The right metric for communications.

Coverage gap distribution

The full statistics, not a summary. A system with a good mean revisit and one twelve-hour hole is not usable.

Coverage is also strongly latitude dependent. An inclined orbit spends disproportionate time near its extreme latitudes, because the ground track turns there, so revisit at high latitude is often much better than at the equator. For polar and sun-synchronous orbits the poles are covered on every orbit while the equator is covered twice a day. Requirements written as a single global number usually mean the worst latitude, and should say so.

Constellation notation

Walker constellations are specified as $i{:}\,t/p/f$, where $i$ is inclination, $t$ the total satellite count, $p$ the number of equally spaced planes, and $f$ the phasing parameter that sets the relative offset between adjacent planes.

A Walker Delta 53°: 24/3/1 means 24 satellites in three planes at 53° inclination, with each plane offset by one phasing unit from its neighbour. The compactness of this notation is why it is the standard way to state a constellation design. See Constellation design.

References

• Wertz, J. R. Mission Geometry: Orbit and Constellation Design and Management, Microcosm Press, 2001. The standard reference on coverage.
• Walker, J. G. “Satellite Constellations”, Journal of the British Interplanetary Society, 37:559, 1984.
• Vallado, D. A. Fundamentals of Astrodynamics and Applications, 5th ed., Microcosm Press, 2022, chapter 11.
• Capderou, M. Satellites: Orbits and Missions, Springer, 2005, on repeat ground track selection.