Plate 41 · Information

How a Phone Knows Where It Is

Far above the clouds, thirty-odd satellites circle the Earth twice a day. Each carries atomic clocks and says, over and over: this is the time, and this is where I am. Your phone never answers. It only listens, and from how late each message arrives it works out where it is standing, to within a few metres.

Try it

Chapters · 7

01Six Rings of Satellites
02A Message at the Speed of Light
03One Distance, One Sphere
04The Fourth Satellite
05Einstein in Orbit
06Not Only GPS
07Your Turn

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01

Six Rings of Satellites

GPS, the Global Positioning System, is a fleet of satellites run by the United States: 32 were in service on the day these orbits were recorded. Each flies about 20,200 km up and goes round the Earth twice a day.

They are spread over six tilted rings, so that from almost anywhere on Earth, at any hour, at least four are above the horizon.

The GPS satellites on 12 October 2026, placed from their published orbits · satellites not to scale

02

A Message at the Speed of Light

Every satellite carries atomic clocks and broadcasts, without pause, two things: the exact moment each message left, and where the satellite was.

Radio waves travel at the speed of light, about 300,000 km a second, and even so the trip takes time: about 0.067 s from a satellite overhead, about 0.086 s from one low on the horizon. Multiply the delay by the speed of light and you have your distance from that satellite.

03

One Distance, One Sphere

Knowing you are 21,000 km from one satellite puts you somewhere on a giant sphere around it. Where that sphere meets the ground is a circle, and you are somewhere on it.

A second satellite draws a second circle; the two cross at two points. A third circle passes through only one of them. That is where you are.

04

The Fourth Satellite

There is a catch. To time the delay, your phone needs a clock as good as the satellites’, and it doesn’t have one. Phone clocks are often off by a good fraction of a second: in 6.4 million checks against an internet time server, half were off by more than 0.7 seconds. In a thousandth of a second, light travels 300 km.

So every distance is wrong by the same amount, and the circles no longer meet. A fourth satellite settles it: the phone looks for the one clock correction that makes all four circles meet at a single point. That gives it its position and the exact time at once.

05

Einstein in Orbit

The satellites’ clocks don’t keep time quite like ours. Moving at 3.9 km a second, they run slow by about 7 millionths of a second a day, as Einstein’s special relativity predicts; up high, where gravity is weaker, they run fast by about 46, as his general relativity predicts.

Together they gain about 38 millionths of a second a day, the time light takes to cover more than 11 km. So before launch, each clock is set to tick slightly slow: 10,229,999.99543 times a second instead of 10,230,000. In orbit, it keeps time with clocks on the ground.

06

Not Only GPS

GPS is not the only system of its kind. China’s BeiDou, Europe’s Galileo and Russia’s GLONASS work the same way, and many phones can receive more than one of them.

More satellites in view mean more circles, and a sharper fix between tall buildings, where some satellites are hidden. BeiDou also has satellites much farther out, a few of which hang over one spot above the equator, south of Asia.

07

Your Turn

Choose where you stand. Set the phone’s clock wrong and see where three satellites put you, then let the fourth correct it. Switch to bunched satellites and watch the circles cross at shallow angles and smear the fix.

Try thisSet the phone clock 1 ms wrong and use only three satellites: how far off are you?

What the model leaves out

Satellite positions come from published mean orbital elements (CelesTrak), carried forward with Kepler’s laws and the main effect of the Earth’s equatorial bulge; that is good to tens of kilometres, while real receivers use far more precise orbits broadcast by the satellites themselves. The Earth here is a perfect sphere, the satellites are drawn hugely enlarged, and their signals are slowed down. The circles on the ground assume you are standing on the ground; a real receiver also works out its height, which is one reason it needs four satellites. In space, three spheres meet at two points, but one of them lies far out in space and is easily ruled out. Real receivers also correct for the signal’s delay through the upper atmosphere. The bands drawn for uncertainty in the lab are exaggerated: under open sky, phones are typically good to about 5 m.

A common misreading

Your phone sends nothing to the GPS satellites, and the satellites have no idea you exist: the signals go one way, from space to the ground, and any number of receivers can listen at once. When a map app shares your location, it does so over the phone network or Wi-Fi, not through GPS. Nor is the relativity correction a small fudge: without it, the satellite clocks would run ahead of ground clocks by about 38 millionths of a second more every day.

Sources

Data

Orbits of the GPS, BeiDou, Galileo and GLONASS satellites: CelesTrak (GP data from the 18th Space Defense Squadron), fetched 12 October 2026. Earth: NASA Earth Observatory, Blue Marble Next Generation, October 2004 (Reto Stöckli); city lights, NASA Earth Observatory Black Marble 2016 (Joshua Stevens, Miguel Román, Suomi NPP VIIRS); clouds, NASA Blue Marble 2002 (Reto Stöckli, NASA GSFC). Stars: Yale Bright Star Catalogue, 5th revised edition (Hoffleit & Warren 1991), via CDS VizieR. Milky Way: NASA/Goddard Space Flight Center Scientific Visualization Studio, Deep Star Maps 2020; Gaia DR2: ESA/Gaia/DPAC.