# Inside the box: GPS and relativity

October 9, 2023  -

Image: GPS.gov

Clocks are at the heart of GPS. Advances in space-qualified atomic clocks that kept time to within 10 nanoseconds over a day were a key development that made GPS possible. It turns out that GPS must account for both special relativity and general relativity to deliver position at 1-meter level and time at 100-nanosecond level to its users. We’ll use these round numbers as user expectations from GPS.

In the simple engineering analysis below, we consider the problems that would have arisen if the engineers had ignored relativity in their design of GPS. The issues related to positioning and time transfer are distinct, so we treat them separately.

### GPS is basically a bunch of synchronized, near-perfect clocks in orbit

It’s a mantra worth repeating: To measure ranges to GPS satellites with meter-level accuracy, the clocks on the satellites must keep time with nanosecond-level accuracy.

The clocks aboard GPS satellites are extraordinarily stable, typically to one part in 1013 over a day, which is another way saying that they could gain or lose on average 10-8 seconds, or 10 nanoseconds, over 105 seconds, which is roughly the length of a day. It’s a simple calculation. Suppose you measure a time interval of length with an oscillator advertised to have frequency by counting its periods of oscillation. If the actual frequency is (f + Δf ), you’d measure the time interval as (T + Δt). It is easily shown that:

The fractional frequency stability (f / Δf ) is a key parameter. For an oscillator with stability (f / Δf ) of 10-13  over a day, as noted above, we can limit to 10 nanoseconds on average with data uploads to satellites once a day to re-sync the clocks. An error of 10 nano-seconds in time amounts to an error of about 3 meters in range computation and, speaking roughly, an error of about 3 meters in the position computed by the receiver. We can live with that.

### Gravitational and motional effects on GPS clocks

Our previous calculation of the timekeeping error of a satellite clock would have been fine had we not overlooked an important fact: We pretended as though the clocks were at rest on Earth at mean sea level. So, let’s see what relativity has to say about clocks in 20,000-kilometer-high circular orbits around Earth. The satellite orbits are not perfectly circular, or identical, but for now let’s pretend that they are. We call that modeling. The clocks would move at a rate of about 4 kilometers per second and exist in an environment where Earth’s gravity is only about one-fourth that at sea level.

According to the theory of special relativity, a moving clock ticks more slowly when compared with one that’s stationary at sea level. A clock aboard a GPS satellite will lose about 7 microseconds per day. That is three orders of magnitude larger than our budget for satellite clock error discussed earlier, therefore we can’t simply ignore it.

According to the theory of general relativity, on the other hand, a clock in a weaker gravitational field will tick faster than one that’s stationary at sea level. Apparently, gravity weighs down time, too. A clock aboard a GPS satellite in a medium Earth orbit will gain about 45 microseconds per day over a clock that’s at sea level on the earth.

The net effect: A GPS satellite clock will gain about 38 microseconds per day over a clock at rest at mean sea level. This effect is secular, meaning the time offset will grow from day to day.

So, you ask: Can you show me how you came up with these numbers, 7 micro-seconds and 45 microseconds? No, but I can point you to the references listed below and I can come close using simple mathematical models: (i) Earth’s gravitational potential is complicated and to simplify things we model Earth as homogeneous in composition and spherical in shape with a radius (rE) of 6,400 kilometers; (ii) aGPS satellite orbit is a circle with radius 4 rE; and (iii) the satellites move at the rate of 4 kilometers/second. We saved ourselves a lot of trouble by agreeing on this simple model.

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The calculation of the fractional frequency stability (f / Δf ) due to the relativistic effects is now easy and given in the sidebar. The answers are only approximate, but surprisingly close to the numbers cited above. That’s the beauty of good models. To calculate time gained or lost over a day, multiply by the length of a day in seconds.

As an interesting aside, note that the effects predicted by special relativity and general relativity cancel each other for clocks located at sea level anywhere on Earth. Consider two clocks, one located at the North or South Pole, and the other at the equator. The clock at the equator would tick slower because of its relative speed due to Earth’s spin, but faster because of its greater distance from Earth’s center of mass (about 22 kilometers) due to Earth’s flattening. Because Earth’s spin rate determines its shape, the two effects are not independent, and it’s no coincidence that they cancel exactly.

### What if GPS forgot about relativity?

What would have happened if the engineers responsible for designing GPS had disregarded relativity? If the GPS satellites were in fact in identical, circular or-bits, their clocks would have shown a puzzling, but identical, behavior of gaining time over clocks of the Control Segment on Earth at a steady rate, about 38 microseconds over a day, the combined effect of special and general relativity.

What would that do to range measurements? A GPS receiver would have meas-ured the ranges to all satellites in view as too short by a common amount (up to about 11 kilometers between daily uploads of clock corrections). However, GPS receivers don’t measure ranges. To measure ranges, the receiver clock would have to be synchronized with the satellite clocks, an onerous requirement. The receivers use inexpensive clocks that drift and have frequency stability no bet-ter than . The receivers measure pseudoranges, i.e., ranges with a common bias on account of the receiver clock offset relative to GPS Time. This bias is es-timated by the receiver, along with its three-dimensional position. The price of an inexpensive receiver clock is that we now have four parameters to estimate and need pseudorange measurements from four satellites.

So, what would that do to positioning? The answer is that the common bias introduced by the relativistic effects would get lumped with the typically much larger bias introduced by the offset in the receiver clock. The position estimate would be unaffected.

Now, what about time from GPS? A GPS receiver used for timing is typically stationary with its antenna location carefully surveyed. In principle, a single pseu-dorange measurement can sync it to GPS Time (and UTC). So, if the relativistic effects had been ignored, the timing accuracy would have suffered to the ex-tent of 38 microseconds per day between updates of the clock parameters. That’s a deal-breaker, considering that we expect 100-nanosecond accuracy.

The relativistic effects discussed so far can be compensated for easily by setting the frequency of the satellite clocks lower (by 0.0045674 hertz) in what’s called “factory offset”: The frequency of a satellite clock is set to 10.22999999543 megahertz so that it will tick in orbit at the same rate as a 10.23-megahertz atomic standard at sea level on Earth. What an ingenious solution!

This factory offset would have accounted for the relativistic effects completely if the GPS satellite orbits were perfectly circular and identical. They are not. You can’t control an orbit perfectly.

### So, what about eccentric orbits?

Yes, that’s a complication.

Each orbit is distinct and slightly elliptical. A consequence of this is that the sat-ellite speed is not constant (due to Kepler’s second law): the farther away a sat-ellite gets from Earth in its elliptical orbit, the slower it moves; and the farther away the satellite, the lower is the gravity field. That means the clocks in differ-ent satellites are speeding up and slowing down at different times and at differ-ent rates. The effect for each clock is periodic and quasi-sinusoidal. Averaging the effect over an orbit, we get zero.

For a satellite in an orbit with an eccentricity of 0.02, the net effect is that a clock can be ahead or behind by as much as 45 nanoseconds. The corresponding range error would amount to ± 15 meters. This effect must be accounted for specifically for each orbit. It would require serious bookkeeping on where the satellite has been in its elliptical orbit since the last data upload to sync its clock. It’s a messy business but can be simplified. We’d leave it at that. See ICD-GPS-200C, Section 20.3.3.3.3.1, if you want to see how it is implemented in your GPS receiver.

There is more to relativity than the special theory and general theory. There is the Sagnac effect associated with our rotating reference frames attached to Earth, in which we’d like to determine a position. The principle of constancy of the speed of light cannot be applied in a rotating reference frame, where the paths of the radio rays are not straight lines, but spirals. (Receivers at rest on Earth are moving quite rapidly: 465 meters per second at the equator.) There is also the Shapiro delay associated with the slowing of electromagnetic waves as they near Earth, which amounts to a fraction of a nanosecond. See the refer-ences for more on these topics.

Final thought: Could Einstein have imagined one hundred years ago that a bil-lion people would unknowingly account for the effects of his esoteric theory in their everyday activities?

Refrences

1. Ashby (1993), “Relativity and GPS,” Innovation column in GPS World
2. Ashby (2003), Relativity in the Global Positioning System. Living Reviews in Relativity https://link.springer.com/article/10.12942/lrr-2003-1
3. https://www.gps.gov/technical/icwg/ICD-GPS-200C.pdf