That's Maths:Most hill-walkers can recall an anxious time when, caught on a ridge between steep slopes, they are suddenly enshrouded by dense fog. The immediate question is "Where exactly am I?" Map and compass are vital aids, but they cannot answer that question. A hand-held device about the size of a mobile phone can. How does it do that?
The Global Positioning System (GPS) is a satellite-based navigation system, owned and operated by the United States government, that provides information on location in all weathers, anywhere in the world. It is freely available to anyone with a GPS receiver. The system comprises a constellation of between 24 and 32 satellites, orbiting at about 20,000km above the Earth. Each satellite carries a high-precision atomic clock, accurate to about 1 nanosecond. A nanosecond (ns) is one billionth of a second, the time it takes light to travel one foot.
To compute the position, the GPS receiver uses signals from several satellites, each including the precise time and location of the satellite. The satellites are synchronised so that the signals are transmitted at precisely the same instant. But they arrive at the GPS receiver at slightly different times. Using the known signal speed – the speed of light – the distance to each satellite is determined. These distances are then used to calculate the position of the receiver using trilateration.
Trilateration determines position by using distances to known locations. For example, if you are 110km from Athlone, you are somewhere on a circle of this radius centred at Athlone. If you are also 140km from Belfast, you must be in Dublin or in Garrison, Fermanagh, the points where the two circles intersect. Finally, if you are also 220km from Cork, you can only be in Dublin. Three distances suffice for a unique location.
In three-dimensional space, spheres replace circles and four are needed. The GPS receiver uses signals from four satellites. This provides distances from four known locations, sufficient to pin down the position of the receiver. GPS receivers available today give location to an accuracy of about 10 metres.
Navigation is just one of the applications of GPS. The system is vital for search and rescue, for vehicle tracking, for map-making and surveying, for detecting movements in the Earth’s crust, even monitoring the movements of elephants in Africa. The European Union is developing a system comparable to GPS, called Galileo, which is due to be operational in 2014.
GPS is a striking example of the practical importance of Einstein’s relativity theory. Special Relativity implies that a moving clock ticks slowly relative to a stationary one so, for an observer on Earth, the satellite clocks lose about 7,000ns (7 microseconds) each day. But General Relativity says that these clocks should go about 45,000ns faster, because the Earth’s gravitational pull is weaker higher up. The effect is a speed-up of about 38,000ns per day. To avoid corrections, the clocks are set before launch to compensate for relativistic effects. Without this, GPS would be useless.
Peter Lynch is professor of meteorology at University College Dublin. He blogs at thatsmaths.com
