Chapter 4 · Satellites, clocks and relativity, from 1978 to today
GPS and satellite navigation
A constellation in orbit
GPS, the Global Positioning System, is built and run by the United States. Its first satellite was launched in 1978 and the system reached full operation in 1995. About 31 working satellites orbit at 20,200 km, spread over six orbital planes, and each circles the Earth twice a day. The layout guarantees that at least four satellites stand above the horizon almost anywhere. The orbit is high enough for each satellite to see a large part of the planet, yet low enough for the signal to stay strong. Using it is free. In the animation the satellites sweep around a blue globe.
A shout with a timestamp
Every satellite carries atomic clocks and broadcasts a radio message that says, in effect: I am satellite number so-and-so, here is my orbit, and at this very instant the time is exactly this. Radio travels at 300,000 km/s, so from a satellite straight overhead, 20,200 km away, the signal needs 20,200 / 300,000 ≈ 0.067 s, about 67 ms. Your receiver compares the time written in the signal with its own clock. The difference multiplied by the speed of light is the distance to that satellite.
Why nanoseconds matter
Because light is so fast, tiny timing errors become huge distances. An error of one microsecond, a millionth of a second, means 300 metres. To be right within 3 metres, a receiver has to measure time to about 10 nanoseconds. That is why every GPS satellite carries atomic clocks, which keep time to within a few billionths of a second per day. The satellites are checked by ground stations and receive corrections several times a day, so that the whole sky agrees on what time it is.
Spheres that meet
One satellite says: you are somewhere on a huge sphere around me, with a radius equal to the distance. A second satellite adds another sphere, and the two spheres cut each other in a circle. A third sphere cuts that circle in two points, and one of them lies far out in space, so the Earth's surface tells you which is real. This method, finding a position from distances alone, is called trilateration. It uses lengths, not angles. Try moving the receiver in the sandbox and watch the four delays change.
The fourth satellite
Your phone has no atomic clock. Its own clock may be wrong by a few milliseconds, which would put you hundreds of kilometres off. So the clock error becomes a fourth unknown beside x, y and z, and a fourth satellite is needed to solve for all four. The result: GPS.gov reports that phones are typically accurate to about 5 metres under open sky. Tall buildings bounce signals and add error in cities. Move the receiver and the sandbox recomputes the four delays, then finds the position back from them.
Einstein rides along
Clocks in orbit tick differently from clocks on the ground, for two reasons. By special relativity the satellite's speed, about 3.9 km/s, slows its clock by around 7 microseconds a day. By general relativity gravity is weaker at 20,200 km, which speeds the clock up by around 45 microseconds a day. The net gain is about 38 microseconds a day. Left uncorrected, 38 µs × 300,000 km/s is about 11 km of error piling up every day. Engineers set the clocks slightly slow before launch, so GPS works as a daily test of Einstein's theories.
Four systems, one sky
GPS is one of several global navigation satellite systems, GNSS for short. Russia's GLONASS began launching in 1982 and was restored to full strength in 2011. Europe's Galileo started its first services in December 2016. China's BeiDou declared global service in July 2020. A modern phone listens to several systems at once, which means well over a hundred satellites to choose from. More satellites in view mean a better fix between tall buildings, and a backup if one system has a problem.
In numbers
- 20,200 km
- the altitude of GPS satellites
- +38 µs
- net daily gain of a satellite clock (relativity)
- 300 m
- distance error from one microsecond of timing