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Why Do GPS Satellite Clocks Run Fast, and How Is It Fixed?

Category: Space & Astronomy

Seen from the ground, a clock on a GPS satellite runs fast. It gains about 38 millionths of a second (38 microseconds) every day. That sounds like nothing, but GPS works by timing radio signals that travel at the speed of light, and at that speed 38 microseconds is more than 11 kilometers (about 7 miles) of travel [1].

The gain comes from two effects of Einstein's relativity that pull in opposite directions, and engineers cancel most of it before the satellite ever leaves the ground. This article follows the numbers, then asks what would have gone wrong if nobody had noticed.

1. Why GPS Needs Clocks Accurate to Billionths of a Second

GPS satellites fly in medium Earth orbit, about 20,200 km (12,550 miles) up, and each one circles the Earth twice a day [4]. Every satellite broadcasts its position along with the time the signal was sent. A receiver compares that time with its own clock and turns the delay into a distance (the four-satellite article explains the details).

Light covers about 30 centimeters (1 foot) in one billionth of a second (a nanosecond). So the satellite clocks have to be right to nanoseconds. An engineering article in GPS World puts it this way: an error of 10 nanoseconds in a satellite clock becomes an error of roughly 3 meters (10 feet) in the measured range [1]. The atomic clocks in the satellites are built to be that steady, typically to a frequency stability of about one part in 1013 over a day [1].

That is why a clock that is off by 38,000 nanoseconds a day is a serious problem, not a rounding error. The question is where the 38,000 comes from.

2. Why a Fast-Moving Satellite Clock Loses About 7 Microseconds a Day

A GPS satellite travels at roughly 4 kilometers per second (about 14,000 km/h, or 8,700 mph) [2]. According to special relativity, a clock that moves relative to you ticks more slowly than a clock that stays with you. Seen from the ground, a satellite clock therefore falls behind by about 7 microseconds each day [2] [1].

Everyday speeds never show this. It only becomes visible because the satellite is fast, its clock is extremely steady, and the time is added up over many hours.

3. Why Weaker Gravity Makes the Same Clock Gain About 45 Microseconds a Day

General relativity adds a second effect, and this one goes the other way. A clock deeper in a gravitational field ticks more slowly than a clock higher up. The satellite is 20,200 km above the surface, where Earth's gravity is much weaker, so its clock ticks faster than a clock on the ground. The gain is about 45 microseconds per day [2] [1].

Ashby, a physicist who studied this problem for decades, calculated the gravity effect alone as a fractional rate difference of about 5.3 parts in 1010. If it were ignored, it would build up to a position error of about 13.7 km in one day [3].

There is a neat side note. Clocks on the ground are not all equal either. A clock on the equator moves faster because of Earth's spin, which slows it, but it also sits farther from Earth's center because the planet is slightly flattened, which speeds it up. At mean sea level (the surface called the geoid) the two effects cancel almost exactly, so clocks at the equator and the poles tick at the same rate [1] [3]. That is what makes "a clock on the ground" a fair thing to compare against.

4. What the Two Effects Add Up To, and What Would Go Wrong Without a Fix

The speed effect (about 7 microseconds slow) and the gravity effect (about 45 microseconds fast) do not cancel. The gravity effect wins, and the satellite clock ends up gaining about 38 microseconds a day: 45 − 7 = 38 [2] [1]. The offset also grows steadily from day to day, so waiting does not help [1].

Sources describe the damage a little differently, so here is the short version:

  • A simplified teaching estimate. Richard Pogge of Ohio State University says that without the corrections a GPS fix would be wrong within about 2 minutes, and position errors would pile up at roughly 10 km per day [2].
  • A more detailed engineering analysis. The GPS World analysis points out that a receiver already estimates its own clock error, so a common bias shared by all satellites mostly hides inside that estimate. The ranges would look too short by up to about 11 km, yet the position itself would be mostly unaffected. Timing is another matter. Anyone using GPS as a time source would be off by 38 microseconds a day, far outside the 100-nanosecond accuracy users expect [1].

Sources disagree on how the failure would look, so this article does not give a single dramatic number. What both agree on is the size of the clock offset (about 38 microseconds a day) and that the system could not meet its accuracy goals if the offset were left alone.

5. How Engineers Cancel the Offset Before the Satellite Is Launched

The fix is surprisingly simple. The satellite clock is set to tick slightly slow on the ground, so that in orbit the relativity effects speed it up to exactly the right rate. A reference atomic clock at sea level runs at 10.23 megahertz. The satellite clock is set to 10.22999999543 megahertz, which is lower by about 0.0046 hertz. Once it reaches its orbit, it ticks at the same rate as the 10.23 MHz standard on the ground [1]. This is called the "factory offset."

Ashby's calculation of the constant part of the effect, about 4.46 parts in 1010, matches that offset. He adds that this pre-launch adjustment was used on the older satellites, while for newer rubidium clocks, whose frequency can shift during launch, engineers measure the rate after the satellite reaches orbit and send the needed correction to receivers in the navigation message [3].

Orbits are never perfect circles, so a small leftover remains. A satellite on a slightly stretched orbit speeds up and slows down as it moves, and its altitude changes too, which makes its clock drift ahead and behind by tens of nanoseconds over each orbit. A receiver applies a calculated correction for this "eccentricity effect" [1] [3].

One historical detail is worth knowing. The test satellite NTS-2, launched on June 23, 1977, was part of the first demonstration phase of the NAVSTAR GPS program. Its cesium clock was paired with a "relativistic frequency synthesizer" that adjusted the clock's rate toward the rate of the Naval Observatory master clock [5]. The fuller story of how engineers tested the prediction in orbit is told in Ashby's long review, which I could not open, so I leave it out here.

6. Try It: How Far Does Light Travel in 38 Microseconds?

You can check your work with a calculator. Ask an adult if you would like help.

  • Light's speed: about 299,792 km per second. Multiply by 0.000038 seconds. What do you get? Compare your answer with the "more than 11 km" in the lead.
  • Nanoseconds: light goes about 30 cm in one nanosecond. Look at a 30 cm (12 inch) ruler and imagine light traveling that far in one nanosecond. How many such rulers fit into 11 km?
  • Ask a question: if a clock in a plane flying at 900 km/h ticks slightly slower than a clock on the ground, would you ever notice? What would you need to measure it?

If this made you curious, Ashby's article Relativity in the Global Positioning System (Living Reviews in Relativity) goes much deeper. The version cited below is a shorter teaching paper by the same author.

Sources

Facts, numbers and years in this article were checked against the sources below. Where reading was limited, the note says so.

  1. "Inside the box: GPS and relativity," GPS World (trade magazine). https://www.gpsworld.com/inside-the-box-gps-and-relativity/ (Read in full. Source for the 7, 45 and 38 microsecond figures, the factory offset of 10.22999999543 MHz, the 11 km range error, timing versus positioning, and the eccentricity effect.)
  2. Richard Pogge, "Real-World Relativity: The GPS Navigation System," Ohio State University. https://www.astronomy.ohio-state.edu/pogge.1/Ast162/Unit5/gps.html (Read in full. Source for orbit speed, the 7/45/38 microsecond figures, and the "2 minutes, about 10 km per day" claim, which is a simplified teaching statement.)
  3. Neil Ashby, "Relativistic Effects in the Global Positioning System," teaching paper (University of Colorado Boulder / NIST), 2006. https://www.aapt.org/doorway/tgru/articles/ashbyarticle.pdf (Read in full. Source for the gravity effect of 13.7 km/day if ignored, the cancellation at sea level, the 4.4647 × 10-10 factory offset, newer rubidium clocks, and the eccentricity correction. His longer review in Living Reviews in Relativity could not be opened.)
  4. "Space Segment," GPS.gov (U.S. government). https://www.gps.gov/space-segment (Read in full. Source for the altitude of about 20,200 km and two orbits a day.)
  5. Stability analysis of the NTS-2 cesium frequency standard, Defense Technical Information Center report ADA082797. https://apps.dtic.mil/sti/tr/pdf/ADA082797.pdf (The page itself returned an access error. The launch date and the relativistic frequency synthesizer are known only from the abstract as shown in a search summary, so treat that sentence as less certain.)