108 lines
4.2 KiB
Markdown
108 lines
4.2 KiB
Markdown
---
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title: Constellations
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tags: constellation, shell, planes, phasing, coverage
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summary: Why a constellation is an arithmetic problem about planes, phasing and shell, and what the arithmetic forces the design to be.
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---
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# Constellations
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A constellation is a set of satellites in planes chosen so that their
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combined coverage is what the service needs. The parameters that matter are
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few, the constraints are many, and the design is the intersection.
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## Definition
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A constellation is described by:
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- The number of **planes** and their **inclination**.
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- The number of satellites **per plane** and their phasing within it.
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- The **shell**, meaning the altitude or semi-major axis.
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- The **walk**, the relative motion between adjacent planes, which keeps the
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pattern from sitting still relative to the Earth.
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Coverage is the fraction of the surface that sees at least one satellite at
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a given elevation, at a given instant or averaged over a period. The two are
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different numbers and a design that meets the instantaneous figure can be
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poorly covered at any given place and time.
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## Mathematics
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### The geometry of a shell
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At altitude $h$, the Earth subtends a half angle from the satellite, and the
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ground area visible above an elevation angle `e` is
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$$
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A(e) = 2 \pi R^2 \left( 1 + \cos e \right)
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$$
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For $e = 0$, the visible cap is a hemisphere. The fraction of the globe a
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satellite can see at once is therefore fixed by its altitude alone, and no
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amount of satellites improves it beyond covering more of that hemisphere at
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the same time.
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### Cost per unit coverage
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The cost of a constellation scales with the number of satellites, and the
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number needed for a given coverage scales roughly with the area to be
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covered divided by the area one satellite covers at that elevation. Two
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consequences follow:
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- A high shell covers more per satellite and costs more per satellite to
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place, and the balance sets the altitude.
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- The expensive part of a shell is launch, and launches carry a limited
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number of satellites, which is what makes plane count a launch-rate
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decision as much as a coverage one.
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### Phasing and the seam
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Satellites in a plane spread along it at equal intervals. If the phasing is
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not chosen well, the satellites align over the poles or over the equator in
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a way that leaves a seam of poor coverage. Walker delta describes the
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phasing; a delta that is not coprime with the satellite count produces
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repeats.
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### Walk
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A relative motion between planes rotates the whole pattern. A sun-synchronous
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orbit has a nodal drift of about one degree per day, which is what keeps the
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planes aligned with the terminator. The walk is therefore not decoration:
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it is a service requirement, and it costs propellant.
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## Unit conventions
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- Altitude above the equatorial radius, kilometres.
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- Inclination in degrees when describing a constellation, because that is how
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shells are quoted, and in radians inside any calculation.
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- Phasing as an integer offset.
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## Applicability
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The design point is set by three competing constraints: coverage at low
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elevation pushes to a high shell, which pushes to fewer satellites; revisit
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time for the ground segment pushes to more planes, which pushes to more
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launches; and the propagation medium of the band in use pushes some of it
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below the rain-limited altitudes.
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The hard constraint is often the physical one rather than the coverage one: a
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satellite cannot occupy a shell that the link budget or the propagation
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environment forbids, and a design that ignores this is a design that
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degrades in the worst weather.
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Coverage figures are also easy to misread. Averaged over the service area,
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a constellation may have excellent coverage; the failure is that a
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particular user at a particular time is below the elevation angle, which is
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an availability question rather than a coverage one.
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## Standards
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- Constellation filings for the geostationary and non-geostationary regimes:
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the Radio Regulations, Annex 4
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- Phase and coverage conventions are not standardised; they are engineering
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practice.
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## See also
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- [[access-and-availability]] for the question coverage does not answer
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- [[solar-radiation-pressure]] for why a plane drifts
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- [[link-budget]] for which shells are usable |