How does GPS work? Satellites, timing and relativity

GPS, the Global Positioning System, works by timing radio signals. Satellites about 20,200 km up carry atomic clocks and broadcast where they are and exactly when each signal left. A receiver, such as the chip in a phone, does the rest: it notes when each signal arrives, multiplies the travel time by the speed of light to get a distance, and works out the one point on Earth that fits the distances to at least four satellites.

In short

• Position comes from distances, and distances come from time: a signal's travel time multiplied by the speed of light.
• A fourth satellite is needed because the receiver's own clock is cheap and wrong; the math solves for the clock error along with the position.
• The satellites' clocks must be corrected for relativity, about 38 microseconds a day, or positions would drift by kilometers.

What is GPS, and who runs it?

GPS is a system of satellites, ground stations and receivers built by the US military and open to everyone free of charge. The first satellite launched in 1978, and the system reached full operation in 1995. After the Soviet Union shot down a Korean airliner that had strayed off course in 1983, President Ronald Reagan announced that GPS would be available for civilian use once it was ready. Through the 1990s, civilian signals were deliberately degraded by a feature called Selective Availability; gps.gov notes that the government ended it in May 2000.

The satellites fly in six orbital planes, each circling Earth twice a day. The US Space Force commits to keeping at least 24 working satellites in orbit, which is enough for at least four to be in view almost anywhere, and it counted 31 operational satellites on 3 July 2023, according to gps.gov's page on the space segment. GPS is one of several such systems: Russia runs GLONASS, the European Union Galileo and China BeiDou, and most modern phones listen to several at once. The general name for all of them is GNSS, global navigation satellite systems.

How does a receiver work out where it is?

The method is called trilateration: finding a point from its distances to known points. GPS receivers only listen; they never transmit anything to the satellites.

Each satellite constantly broadcasts two things: where it is, from orbit data uploaded by ground stations, and the exact time each part of the signal left, by its atomic clock. The receiver compares that departure time with the arrival time by its own clock. A satellite straight overhead is about 20,200 km away, and the signal takes about 0.067 seconds, 67 thousandths of a second, to cover that distance at the speed of light.

One distance puts you somewhere on a giant sphere around that satellite. Two spheres cross in a circle. Three cross in two points, one of which is usually far out in space or deep inside the Earth, so it can be thrown out. So in principle three satellites are enough.

Worked example: why a microsecond matters, and why four satellites

Light travels about 300,000 km a second, which is 300 meters in a microsecond, a millionth of a second. Every timing error becomes a distance error at that rate:

• A clock error of 1 microsecond gives a distance error of about 300 meters.
• An error of 10 nanoseconds, a hundredth of a microsecond, gives about 3 meters.
• A phone's quartz clock can easily be off by a millisecond, a thousandth of a second, which would make every distance about 300 km too long.

The satellites have atomic clocks; the phone doesn't, and that is what the fourth satellite fixes. Suppose the phone's clock runs 1 millisecond fast. Every signal then seems to take 1 millisecond longer than it did, so every distance comes out 300 km too long, by the same amount. With four or more measured distances, the receiver has four unknowns to solve for: its east-west, north-south and up-down position, and its own clock error. Four equations solve four unknowns. The receiver finds the single clock correction that makes all the spheres meet at one point, here 1 millisecond, and it gets the position and the exact time together. That is why GPS is also how cell networks, power grids and banks keep their clocks in step, as gps.gov explains.

Why do GPS clocks need relativity?

The satellites' clocks don't keep the same time as clocks on the ground. Einstein's theories predict two effects. Because the satellites move fast, special relativity slows their clocks by about 7 microseconds a day. Because they sit higher in Earth's gravity, where time runs faster, general relativity speeds them up by about 45 microseconds a day. The net result is that a satellite's clock gains about 38 microseconds a day, as the astronomer Richard Pogge sets out.

At 300 meters per microsecond, 38 microseconds is about 11 km. Left uncorrected, positions would drift by roughly that much every day. So engineers set the satellite clocks to tick slightly slow before launch. Their reference frequency is set to 10.22999999543 MHz instead of 10.23 MHz, so that in orbit they keep pace with the ground. The physicist Neil Ashby's review of relativity in GPS works through every correction the system makes. The physics of both effects is in what the theory of relativity says.

How accurate is GPS, and what goes wrong?

On its own terms, GPS is very good. The US government commits to a daily global average error in the satellites' signals of no more than 2 meters, 95% of the time, which is not the same as a user's accuracy, and gps.gov reports that GPS-enabled smartphones are typically accurate to within about 4.9 meters under open sky. Several things make it worse:

Error	What happens	What helps
The ionosphere	Charged gas 50 to 1,000 km up slows the signals by varying amounts	Receivers that use two frequencies can measure the delay and remove it; many newer phones do
Reflections	In cities, signals bounce off buildings and arrive late, making a satellite seem farther away	Better antennas and software that spots reflected signals
Poor geometry	If the visible satellites are bunched in one part of the sky, the spheres cross at shallow angles and small errors grow	More satellites, from more than one system
Blocked signals	Roofs, tunnels and dense trees block the weak signals; GPS barely works indoors	Wi-Fi and cell positioning

Surveyors and farmers get accuracy down to centimeters by adding corrections from ground stations whose exact positions are known, which cancels most of the shared errors.

Why does a phone also use Wi-Fi and cell towers?

A phone combines several methods because each fails somewhere. GPS needs a clear view of the sky and, from a cold start, can take tens of seconds to collect the satellites' orbit data from their slow broadcast. Phones speed this up with assisted GPS: they download the orbit data over the mobile network, so a first fix takes seconds.

Indoors and in dense cities, phones lean on the radio around them. Cell towers have known locations, so the phone can estimate its position from which towers it hears and how strongly. For Wi-Fi positioning, the phone lists the Wi-Fi networks it can hear and asks a database, built from millions of earlier sightings, where those networks are. That can be accurate to tens of meters with no satellites at all. Motion sensors in the phone fill short gaps, such as a tunnel. This is why a map app's blue dot comes with a circle: the circle shows how unsure the phone is.

The same tiny receivers are now built into watches, cars, cameras, trackers and the connected sensors of the Internet of Things. Signals reach them from satellites far above the International Space Station, which orbits only about 400 km up, and the time those signals carry is as much a product as the position, keeping much of the internet and the financial system in step.

Check yourself

• Without looking back, explain why a receiver needs four satellites and not three.
• Why does an error of one microsecond put a position off by about 300 meters?
• Which two relativity effects act on a GPS satellite's clock, and which one wins?

Sources

• Space Segment — GPS.gov, U.S. government
• GPS Accuracy — GPS.gov, U.S. government
• GPS and Telling Time — GPS.gov, U.S. government
• Real-World Relativity: The GPS Navigation System — Richard W. Pogge, Ohio State University
• Relativity in the Global Positioning System — Neil Ashby, Living Reviews in Relativity, 2003
• How satellite navigation works — European Space Agency