# Geometry of satellite constellations

Can someone point me to a site explaining how they determine the optimum orbits for satellites that have to provide continuous ground coverage, such as GPS satellites? How you provide guaranteed coverage with the minimum number of satellites, minimize times where most of the satellites will be “bunched up” over one hemisphere of the Earth, and so forth?

It’s around 26k miles. At that height they orbit at the same speed the earth rotates.

So you just spread them out.

Do a google on geosychronous orbits

GPS satellites are not in geosynchronous orbit.

GPS and satellite phone systems like Iridium do not use geosynchronous satellites because the distance to them would be too great for the low powered receivers and transmitters those devices have. They use Low Earth Orbits (LEO), typically a few hundred miles up.

At that distance, a satellite traverses the sky over any given location in just a few minutes, which is why a larger number of them is needed to assure continuous coverage. IIRC, GPS requires a minimum of four satellites “visible” to establish a fix.

The Iridium system was so named because it was originally planned to have 77 satellites. (The atomic number of iridium is 77.) As it happened, they found they could get by with the 66 they have now. (For some strange reason, they chose not to rename the system “Dysprosium.”)

The OP is asking about the geometry of such constellations of LEO satellites. Unfortunately, while I know what he is talking about, I don’t know any more than him. I’ll be interested in seeing if anyone has any interesting links about this subject.

Iridium satellites are in LEO (~480 miles), but GPS hangs at 12,100 miles.
JPL on GPS:

Here’s a great long academic overview of GPS.

At approximately half the geosynchronous height, GPS satellites need about 25% of the transmit power to achieve a given signal strength at the earth’s surface. Non-synchronous orbits also offer the advantage that the satellites can be controlled from a single command station.

There’s also the point that the demand for geosynchronous slots is high - adding a bunch of satellites there might lead to overcrowding.

I think it’s too complex an issue to have one generic solution. The optimal orbit depends on many factors, such as:
[ul]
[li]Distance requirements. If the satellite needs to receive transmissions from the ground, the closer the better. For broadcasts it doesn’t matter too much, geostationary orbit is fine.[/li][li]How high in the sky they need to be to be usable. Elevation of 10 degrees is enough to contact a purpose-built antenna (built on top of mountains and tall buildings), but useless for communicating with car-mounted or handled devices.[/li][li]How many satellites need to be in view. A satellite phone needs to contact just one satellite, while a GPS receiver needs to see 2 or 3 satellites simultaneously - and more the better.[/li][li]What exactly you mean by “global coverage”. There really are very few users in polar regions and oceans, and you need to consider whether 100% coverage is worth the cost or settle for 90% coverage.[/li][li]Whether you want uniform global coverage or not. In some applications you want the satellites to be concentrated above populated areas, and get away with fewer satellites above oceans and polar regions. [/li][/ul]

Other people have mentioned most of the other points… I believe that ‘bunching’ is generally not a problem for global satellite networks because they’re placed in orbit at the same nearly-circular orbital distance, spaced out evenly so that they form regular pentagons of some sort. If there’s a need to sweep north-to-south this is probably also done in some sort of balanced and symmetric way, though that’s more of a WAG.

Bunching would only be an issue if all the satellites were at diferent orbital distances, like the planets of the solar system are with respect to the sun.

GPS satellites are also not geosynchronous because geosynchronous orbits are necessarily over the equator. GPS works much better with ‘non-collinear’ satellites.

Bunching’s not an issue because the satellites see themselves as travelling with straight acceleration - the sine wave pattern is from the earth’s perspective, and is a result of the earth turning underneath.

The question in the OP is mostly dependent on how much ‘sky’ you can see from your point on the ground, and you want 4 satellites (3’s OK, especially when you’re on the ground, but they try for 4) in view.

As others have said, it’s dependent on how high the orbit - there’s more sky you can see from one point the higher you go.

Selection of satellite orbits involves, as one can imagine, a number of tradeoffs to achieve an optimal solution, including altitude, orbital plane, eccentricity, and number of satellites in the constellation.

For altitude, higher means more earth-coverage, but more transmit power to achieve the same received signal strength (or worse ground resolution for some kind of imaging application). It’s also more expensive to loft the payload higher, although in LEO you may have some drag-makeup burns to do (there still being some atmospheric drag down in 150-200 mile orbits). If you get to GEO, then the satellite stays parked in the same spot in the sky (nice for pointing a high-gain antenna at it), but it’s over the equator so you may not have good reception for high-northerly/southerly locations as well as easier obscuration in urban canyons or due to terrain. If you are trying to provide service off the equator, then you may need an inclined orbit, which means that each satellite is only (relatively) overhead part of the time – leading to multi-satellite constellations for uninterrupted coverage. For example, the Soviets went to tundra or Molniya orbits for some of their satellites, where a highly elliptical orbit kept the vehicles in the northern sky for much of their orbits, and then they would speedily whip around through the southern latitudes. It’s also energy-intensive to expeditiously change orbital planes, so when you are launching multiple satellites off of one booster it’s nice if they can then inhabit the same plane.

For GPS, it’s actually the case that you need 4 satellites visible to achieve a position fix (3 spatial degrees of freedom plus solving for the receiver time offset). [It can be done with fewer satellites visible, but those are for specialized applications – I can describe further if there’s an interest.] It is desirable to have good geometric diversity to reduce the error in the position estimate – particularly you’d like lots of high-elevation satellites to reign in the vertical estimation error for airborne applications. Spatial diversity of the constellation, adequate satellite visibility in most parts of the world (in addition to 1-out redundancy), and practical transmit power limitations are all among the reasons that the GPS satellite constellation is formed the way it is.

(I’m full-time GPS, so may be able to answer some further questions in that regard. Hope that helps.)

Correction: geostationary orbits must be over the equator. Geosynchronous means a 24-hour period, but not necessarily over equator. Geostationary orbits are a subset of geosynchronous orbites.

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