GPS.

Step 1 of 7

Listening

In the Unit on orbits we saw how a satellite stays in orbit. About thirty GPS satellites circle the Earth about 20,000 km up, each carrying atomic clocks. In the Scene the constellation is frozen at one moment, and the receiver is marked by a red pin on the ground.

Step 2 of 7

Distance from time

Radio waves travel at the speed of light, $c \approx 3 \times 10^8\ \mathrm{m/s}$$c \approx$$3 \times 10^8\ \mathrm{m/s}$⁠. If the receiver knows when a signal left the satellite and when it arrived, it knows how long $t$⁠ the signal was on the way, and so the distance:

$d = c\,t$⁠

The list beside the Scene shows, for each satellite the receiver uses, the signal’s travel time and the range that comes from it.

Step 3 of 7

Spheres

One range says only that the receiver is on a sphere round the satellite. Show the timing spheres is On, and the map beside the Scene shows where each sphere meets the ground. Satellites used is on 2, and Receiver clock error is locked at 0 µs.

Step 4 of 7

The receiver’s clock

GPS satellites carry atomic clocks, but a phone has a cheap quartz crystal, which can easily be off by a millisecond. The receiver times everything on its own clock, so any error in it becomes an error in distance. Satellites used is locked.

Step 5 of 7

Three satellites and a fast clock

With a perfect clock, three satellites would be enough. But the clock here runs fast, and every range is too long by the same amount. Both controls are locked.

Step 6 of 7

The fourth satellite

A position takes three numbers, and the clock one more. Each satellite gives one equation. Receiver clock error is locked at 0.1 ms, and Satellites used is free.

Step 7 of 7

Relativity in your pocket

The satellites’ clocks have a problem of their own. By relativity, a clock in weaker gravity ticks faster, and a clock moving fast ticks slower. In a GPS orbit the weaker gravity wins, and the clock gains on a clock on the ground. The line at the foot of the list beside the Scene spells out both effects.