Maths – The Driving Force Behind GPS
‘If these calculations are error free, then the spheres should intersect at an exact point, the point where the receiver is’
Andrew Youngs | 11 November 2016

GPS is something we all use so regularly that we rarely question it – in our phones, our sat navs – yet how does a simple handheld device manage to find our location to within just a few metres of accuracy?

It starts with 6 orbits of satellites, each containing 4 satellites each, making a total of 24 satellites whizzing over our heads every day at an altitude of about 22,000 kilometres. The orbits are evenly spaced every 60° around the earth at planes inclined at 55° from the equator, and they have a time period of 12 hours. For those who are less au fait with physics jargon, that is to say that every 12 hours each satellite completes one full orbit. This network of satellites all fit together very handily meaning that, at any moment in time, there are several visible satellites from any given place on the earth.

These satellites transmit signals at radio frequencies of 1227.60mHz and 1575.42mHz, giving the position and the exact time of transmission. The receiver then records the difference between the time the signal was transmitted by the satellite, and the time it arrived at the receiver. Since the speed of radio waves is constant (a value of 3 x 108m/s), the receiver can use the relationship between speed, distance and time to calculate the distance which the signal has travelled. This tells the receiver that the receiving point is on the edge of a huge sphere which has a radius equal to the distance the signal has travelled, with the centre of the sphere being the point of transmission, i.e. the satellite.

It is here that the network of satellites comes in useful. Since at any point on the earth you can see multiple satellites, signals are sent from all the satellites which are in view of the receiver, creating as many of these ‘spheres’ as there are satellites, indicating the position of the receiver. If the calculations are error free, then the spheres should intersect at an exact point, the point where the receiver is. Simple.

Sadly, it doesn’t work out quite this perfectly due to a few snags. The layers of gas in the earth’s atmosphere slow down the radio waves, and therefore affect the calculated distance. This is partially solved by the receivers automatically altering results in accordance to the mean (average) atmospheric density. Unfortunately, atmospheric density is very variable, so even altering our value does not necessarily give a precise distance value.

Yet despite this, GPS is still incredibly accurate, and can tell us our position to a high degree of accuracy, all calculated using basic mechanics.


Original Image by Andrew Youngs

James Routledge 2016