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GPS receivers need four satellites to determine their position and time. They provide a receiver with data that is fed into four equations. Three equations are solved (simultaneous with the fourth) to get position. The fourth equation solves for the exact time. The best explanation I've found for the solution of time is here (Paragraph starting with "The Global...").

If your GPS receiver had an onboard atomic clock (imagine a \$50,600 cell phone, \$600 for the iPhone and \$50,000 for the atomic clock), could you do away with the fourth satellite with the atomic clock on board the receiver? You would directly know the exact time of flight from each satellite. This would mean you also know your distance from each of the three satellites directly (no system of linear equations or need to solve for time).

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    $\begingroup$ Pretty much. I'm going for an idealized geometrical understanding. No relativity, no ionosphere, no echos off buildings etc. Most of which can be accounted for with additional math. $\endgroup$ – chessofnerd Jun 24 '17 at 16:53
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    $\begingroup$ The onboard atomic clock would help only if it was synchronized to the GPS atomic clock before. $\endgroup$ – Uwe Jun 24 '17 at 18:01
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    $\begingroup$ In general I believe the answer is yes. In your specific example, no--atomic clocks are way too big, heavy and power hungry to put in your cell phone. $\endgroup$ – Loren Pechtel Jun 24 '17 at 20:25
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    $\begingroup$ @anthony, this is primarily a could-it-work/deeper-understanding question. There are very narrow niche applications where this might be worth it too. The problem with a GPS satellite is it literally tells an adversary where it is at any time making it inherently easy to attack and destroy. An advantage with the on-board atomic time piece approach is one could work with a minimal GPS constellation and not be giving away one's position. I suppose the government could go back to S/A but they claim they never will. $\endgroup$ – chessofnerd Jun 25 '17 at 16:27
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    $\begingroup$ @chessofnerd War is often a matter of economics and practicality. To degrade the GPS constellation to the point that it is only useful to a much more sophisticated (and heavy/bulky/expensive) receiver could be a rather expensive and logistically prohibitive challenge. If a satellite is under threat, it would likely be detected in advance. The satellite could conceivably go offline, make a small orbital adjustment, and perhaps temporarily go silent or transmit false positioning data so that an inbound threat can no longer effectively target it. $\endgroup$ – Anthony X Jun 25 '17 at 16:38
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If your GPS receiver had an onboard atomic clock (imagine a \$50,600 cell phone, \$600 for the iPhone and \$50,000 for the atomic clock), could you do away with the fourth satellite with the atomic clock on board the receiver?

Yes, in theory. Actually in normal GPS, four signals will normally not give a unique solution, even with zero error bars, but usually only one solution will be close enough to the surface of the earth to make sense.

It seems that chip-scale atomic clocks (CSAC) are now down to about \$1500. However, their precision is not good enough for them to work for long periods of time without recalibration, and that recalibration would normally require that you get four GPS satellites again. The spec sheet for a current model says "<9E-10 /mo Aging Rate (Typical)," which suggests that if you ran the thing for a month without reacquiring signals from four satellites, you would be off by the end of the month by a few milliseconds, which is an error of hundreds of kilometers.

To go one step further, if you assume you are in the surface of the earth I think you might be able to get away with only TWO satellites (although in this case there are two solutions on the surface of the earth that are both reasonable and choosing the correct one is harder). This wouldn't work for a plane and would require a very precise knowledge of the Earth's topography.

I don't think this works very well. The intersection of three spheres is typically two points. With normal GPS, you get multiple solutions with 4 satellites, but you can usually tell which one is real because it's near the earth's surface. With only two satellites plus the CSAC, you would get two solutions with no obvious way to tell which was correct.

The problem with a GPS satellite is it literally tells an adversary where it is at any time making it inherently easy to attack and destroy. An advantage with the on-board atomic time piece approach is one could work with a minimal GPS constellation [...] you could determine a receivers position with only three entities revealing their position as opposed to the conventional four.

I don't really see how this helps. A GPS satellite that doesn't reveal its position isn't functioning as a GPS satellite, so it might as well have been put out of commission. Also, the most likely method that's been discussed for destroying GPS satellites is as a result of a nuclear explosion in space. You can get EMP or the trapping of high-energy charged particles in the earth's magnetic field. The latter has already happened unintentionally in 1962 with the Starfish Prime nuclear test. In this type of scenario, an attacker doesn't need to know the satellite's position to high precision -- they may not need to know its position at all.

Maybe a more compelling application is that you don't want to lose GPS tracking if you're in a deep canyon where you only have a limited view of the sky. This happens to me fairly often when I'm hiking. If a limited view of the sky cuts you from 4 satellites to 3 for 15 minutes, it would be nice to still be able to use GPS. However, we now have so many different satellites up there (GPS, Galileo, Beidou, and GLONASS) that with a multi-constellation receiver you should probably always have plenty of satellites visible. I think in the future consumer-grade GPS units will all start to be multi-constellation.

As Aron pointed out in a comment, it may be easier to jam a GPS signal than to destroy a GPS satellite.

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  • $\begingroup$ Regarding the 2 satellite on the earth issue, I think the two terrestrial (ie. Reasonable) solutions are on opposite sides of the great circle formed between the earth and the two satellites. If you are at a position far from the great circle, I think you could reasonably pick the correct one. For instance, if the two satellites are on the equator (I recognize that in practice they aren't, just work with me!) and you know you are in the Northern hemisphere, you could make two satellites work. $\endgroup$ – chessofnerd Jun 27 '17 at 0:36
  • $\begingroup$ Might work especially well for Mariners where the topography of the ocean is well known. $\endgroup$ – chessofnerd Jun 27 '17 at 0:37
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    $\begingroup$ Destroying the GPS satellites is such an overkill and of such economic damage to both sides that no state would ever do it. The simple fact is that it is far easier to jam GPS, since the GPS signal is sooo weak. $\endgroup$ – Aron Jun 27 '17 at 9:23
  • $\begingroup$ I also have to note how you conflated and genericized the term GPS. A quick Google with Bing would tell you that correct generic term is "global navigation satellite system" (GNSS). $\endgroup$ – Aron Jun 27 '17 at 9:26
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Yes

There was in fact the belief that future GPS receivers would contain a tiny atomic clock.

https://www.wired.com/insights/2013/09/how-chip-scale-atomic-clocks-will-change-the-future-of-time/

However, the opposite occured. Atomic clocks became redundant in many applications as the number of Navigation satilites increased.

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    $\begingroup$ The article mentions atomic clocks, and it mentions GPS, but it does not really answer the question "Would a GPS receiver with an onboard atomic clock only need 3 satellites to determine position?" There's no mention of the minimum number of satellites that are needed either with or without an atomic clock. $\endgroup$ – uhoh Jun 26 '17 at 13:03

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