How GPS Works

Every time your phone finds your location to within a few metres, it is silently solving a geometry problem with signals from 24 satellites orbiting 20 000 km above Earth — and applying a correction derived from Einstein's theory of relativity. Without that correction, the error would grow by 10 km every day.

The GPS Constellation

The original US GPS system (now GNSS) maintains a constellation of 31 operational satellites in medium Earth orbit (MEO) at an altitude of ~20 200 km, arranged in 6 orbital planes at 55° inclination. This arrangement ensures that at least 4 satellites are visible from any point on Earth at any time.

Each satellite completes two orbits per day. Their orbits are carefully chosen to be far enough that atmospheric drag is negligible, but close enough that signals reach Earth with adequate power.

Signal power: A GPS satellite transmits ~50 watts on L1 (1575.42 MHz) and L2 (1227.60 MHz) bands. By the time the signal reaches your receiver, its power is about 10⁻¹⁶ watts — one hundred billion billionths of a watt. Your phone needs very sensitive hardware and clever signal processing to extract anything useful from this.

Trilateration: Finding a Point in 3D

Trilateration (not triangulation — which uses angles) is how GPS determines position. Each satellite broadcasts its precise location and the exact time the signal was sent. Your receiver notes when the signal arrives and computes the travel time.

Since the signal travels at the speed of light c ≈ 3 × 10⁸ m/s:

distance = c × (arrival time − transmission time)

One distance measurement places you on a sphere of that radius around the satellite. Two distances give the intersection of two spheres: a circle. Three distances narrow it to two points. A fourth satellite eliminates the ambiguity and also solves for the receiver's clock error (which is much less accurate than the satellite atomic clocks).

Why 4 satellites? Your phone's clock is only accurate to ~1 ms. A 1 ms timing error means a 300 km position error. The fourth satellite allows the receiver to solve for both its 3D position and its clock bias — four unknowns solved with four equations.

Everything Depends on Timing

Light travels 30 cm in 1 nanosecond. For GPS to achieve 1-metre accuracy, timing must be correct to better than 3 nanoseconds.

This is why each GPS satellite carries multiple atomic clocks (caesium and rubidium). These clocks are accurate to about 20–30 nanoseconds per day — roughly 1 part in 10¹³. They are monitored and corrected by ground control stations, and their offset values are uploaded to the satellites and transmitted to receivers.

Special Relativity: Fast Clocks Run Slow

Einstein's 1905 special theory of relativity states that a moving clock ticks slower than a stationary one — time dilation:

Δt' = Δt · √(1 − v²/c²)

GPS satellites orbit at ~3.9 km/s. Substituting into the formula:

Δt_SRT per day = −7.2 μs/day (satellite clocks run slow relative to ground)

7.2 microseconds per day. At 30 cm per nanosecond, this equates to a position error of about 2.16 km per day if uncorrected.

General Relativity: High Clocks Run Fast

Einstein's 1915 general theory of relativity states that clocks tick faster in regions of weaker gravity — gravitational time dilation:

Δt' = Δt · √(1 − 2GM / rc²)

At 20 200 km altitude, gravity is weaker than at Earth's surface. The satellite clocks run fast relative to ground clocks:

Δt_GRT per day = +45.9 μs/day (satellite clocks run fast relative to ground)

This is a much larger effect than special relativity, and in the opposite direction.

The Combined Correction

Effect	Direction	Clock error/day	Position error/day
Special relativity (velocity)	Slow ↓	−7.2 μs	−2.16 km
General relativity (gravity)	Fast ↑	+45.9 μs	+13.77 km
Net effect	Fast ↑	+38.7 μs/day	+11.6 km/day

The net result is that satellite clocks tick 38.7 microseconds faster per day than clocks on the ground. Without correction, GPS position errors would accumulate at about 11.6 km per day.

The engineers solve this by manufacturing the GPS satellite clocks to tick slightly slower than their nominal frequency. On the ground these clocks would be off — but in orbit, after combining both relativistic effects, they tick at exactly the right rate relative to Earth.

Real-world validation: GPS is arguably the best real-world test of general relativity that exists. The system is designed around predictions that would be called nonsense by classical physics — and it works to sub-metre precision globally, 24/7. As physicist Neil Ashby put it: "GPS is a practical application of general relativity."

Other Error Sources

Ionospheric delay: Free electrons in the ionosphere slow the signal (20–30 ns typical, up to 150 ns during solar storms). Dual-frequency receivers (L1 + L2) measure the delay directly to correct for it.
Tropospheric delay: Water vapour causes ~7 ns delay (more at low elevation angles). Corrected by weather models.
Multipath: Signals reflected off buildings arrive after the direct signal, causing errors. Phased-array antennas and signal processing suppress this.
Satellite geometry (PDOP): If all visible satellites cluster in one part of the sky, the trilateration geometry is poor. PDOP (Position Dilution of Precision) quantifies this; PDOP < 2 is excellent.

Beyond GPS: GNSS Systems

GPS (USA) is just one of four global navigation satellite systems (GNSS):

GLONASS (Russia) — 24 satellites at 19 100 km, fully operational
Galileo (EU) — 30 planned satellites at 23 222 km, operational since 2016
BeiDou (China) — 45 satellites (3 orbital layers), global coverage since 2020

Modern smartphones receive signals from all four systems simultaneously, which dramatically improves speed-to-fix and accuracy (especially indoors or in "urban canyons" where sky visibility is limited).

Try It Yourself

The orbital mechanics simulation shows satellite orbits and the geometry of orbital planes — foundation of the GNSS constellation design:

🛰️ Open Orbital Mechanics Simulation →

The solar system simulation demonstrates the N-body gravity physics that satellite trajectory planners must solve:

🌍 Open Solar System Simulation →