Error analysis for the Global Positioning System: Difference between revisions

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== Relativity ==

The [[Theorytheory of Relativityrelativity]] introduces several effects that need to be taken into account when dealing with precise time measurements. First, accordingAccording to [[special relativity]], time passes differently for objects in relative motion. That is known as "kinetic" [[time dilation]]: in an inertial reference frame, the faster an object moves, the slower its time appears to pass

(as measured by the frame's clocks). [[General relativity]] takes into account also the effects that gravity has on the passage of time. In the context of GPS the most prominent correction introduced by general relativity is [[gravitational time dilation]]: the clocks located deeper in the gravitational potential well (i.e. closer to the attracting body) appear to tick slower.

[[File:Orbit times.svg|thumb|Satellite clocks are slowed by their orbital speed but sped up by their distance out of the Earth's gravitational well.]]

=== [[Special Relativityrelativity]] (SR) ===

SRSpecial relativity predicts that as the velocity of an object increases (in a given frame), its time slows down (as measured in that frame). For instance, the frequency of the atomic clocks moving at GPS orbital speeds will tick more slowly than stationary clocks by a factor of <math>{v^{2}}/{2c^{2}}\approx 10 ^{-10}</math> where the orbital velocity is v = 4&nbsp;km/s and c = the speed of light. The result is an error of about -7.2 μs/day in the satellite. The SRspecial relativistic effect is due to the constant movement of GPS clocks relative to the Earth-centered, non-rotating approximately inertial [[special relativity#Reference frames, coordinates and the Lorentz transformation|reference frame]]. In short, the clocks on the satellites are slowed down by the velocity of the satellite. This [[time dilation]] effect has been measured and verified using the GPS.

=== [[General Relativityrelativity]] (GR) ===

SRSpecial relativity allows tothe comparecomparison of clocks only in a flat [[spacetime]], which neglects gravitational effects on the passage of time. According to GRgeneral relativity, the presence of gravitating bodies (like Earth) curves spacetime, which makes comparing clocks not as straightforward as in SRspecial relativity. However, one can often account for most of the discrepancy by the introduction of [[gravitational time dilation]], the slowing down of time near gravitating bodies. In case of the GPS, the receivers are closer to Earth than the satellites, causing the locks at the altitude of the satellite to be faster by a factor of 5×10⁻¹⁰, or about +45.8 μs/day. This gravitational frequency shift is measurable. During early development some believed that GPS would not be affected by GRgeneral relativistic effects, but the [[Hafele–Keating experiment]] showed that it would be.

=== Combined kinetic and gravitational time dilations ===

Combined, these sources of time dilation cause the clocks on the satellites countto extragain +38.6 microseconds per day, comparedrelative to the clocks on the ground. This is a difference of 4.465 parts in 10¹⁰.<ref>Rizos, Chris. [[University of New South Wales]]. [http://www.gmat.unsw.edu.au/snap/gps/gps_survey/chap3/312.htm GPS Satellite Signals] {{Webarchive|url=https://web.archive.org/web/20100612004027/http://www.gmat.unsw.edu.au/snap/gps/gps_survey/chap3/312.htm |date=2010-06-12}}. 1999.</ref> Without correction, errors of roughly 11.4&nbsp;km/day would accumulate in the position.<ref>{{Cite book |last=Faraoni |first=Valerio |url=https://books.google.com/books?id=NuS9BAAAQBAJ |title=Special Relativity |publisher=Springer Science & Business Media |year=2013 |isbn=978-3-319-01107-3 |edition=illustrated |page=54}} [https://books.google.com/books?id=NuS9BAAAQBAJ&pg=PA54 Extract of page 54]</ref> This initial pseudorange error is corrected in the process of solving the [[GPS#Navigation equations|navigation equations]]. In addition, the elliptical, rather than perfectly circular, satellite orbits cause the time dilation and gravitational frequency shift effects to vary with time. This eccentricity effect causes the clock rate difference between a GPS satellite and a receiver to increase or decrease depending on the altitude of the satellite.

To compensate for the discrepancy, the frequency standard on board each satellite is given a rate offset prior to launch, making it run slightly slower than the desired frequency on Earth; specifically, at 10.22999999543&nbsp;MHz instead of 10.23&nbsp;MHz.<ref name="Nelson">[http://www.aticourses.com/global_positioning_system.htm The Global Positioning System by Robert A. Nelson Via Satellite] {{Webarchive|url=https://web.archive.org/web/20100718150217/http://www.aticourses.com/global_positioning_system.htm |date=2010-07-18 }}, November 1999</ref> Since the atomic clocks on board the GPS satellites are precisely tuned, it makes the system a practical engineering application of the scientific theory of relativity in a real-world environment.<ref>Pogge,{{cite web|last=Pogge|first=Richard W.; [|url=http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html "|title=Real-World Relativity: The GPS Navigation System"]. Retrieved 25 January |access-date=2008.-01-25}}</ref> Placing atomic clocks on artificial satellites to test Einstein's general theory was proposed by [[Friedwardt Winterberg]] in 1955.<ref>{{Cite web |date=1956-08-10 |title=Astronautica Acta II, 25 (1956). |url=http://bourabai.kz/winter/satelliten.htm |access-date=2009-10-23 |archive-date=2014-07-03 |archive-url=https://web.archive.org/web/20140703080406/http://bourabai.kz/winter/satelliten.htm |url-status=dead }}</ref>

{| class="wikitable" style="margin:1em auto;"

|+ SR and GR combined

|-

! Time dilation !! Value !! Notes

|-

| Kinetic || -7.2 μs/day || Clocks slowed in satellites due to Velocity

|-

| Gravitational || +45.8 μs/day || Clocks sped up in satellites due to higher altitude

|-

| Total (Combined) || +38.6 μs/day ||

|}

To compensate for the discrepancy, the frequency standard on board each satellite is given a rate offset prior to launch, making it run slightly slower than the desired frequency on Earth; specifically, at 10.22999999543&nbsp;MHz instead of 10.23&nbsp;MHz.<ref name="Nelson">[http://www.aticourses.com/global_positioning_system.htm The Global Positioning System by Robert A. Nelson Via Satellite] {{Webarchive|url=https://web.archive.org/web/20100718150217/http://www.aticourses.com/global_positioning_system.htm |date=2010-07-18 }}, November 1999</ref> Since the atomic clocks on board the GPS satellites are precisely tuned, it makes the system a practical engineering application of the scientific theory of relativity in a real-world environment.<ref>Pogge, Richard W.; [http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html "Real-World Relativity: The GPS Navigation System"]. Retrieved 25 January 2008.</ref> Placing atomic clocks on artificial satellites to test Einstein's general theory was proposed by [[Friedwardt Winterberg]] in 1955.<ref>{{Cite web |date=1956-08-10 |title=Astronautica Acta II, 25 (1956). |url=http://bourabai.kz/winter/satelliten.htm |access-date=2009-10-23 |archive-date=2014-07-03 |archive-url=https://web.archive.org/web/20140703080406/http://bourabai.kz/winter/satelliten.htm |url-status=dead }}</ref>

=== Calculations ===

To calculate the amount of daily time dilation experienced by GPS satellites relative to Earth we need to separately determine the amounts due to the satellite's velocity and altitude, and add them together.

==== Kinetic time dilation ====

The amount due to velocity will beis determined using the [[Lorentz transformation]]. The time measured by an object moving with velocity <math>v</math> changes by (the inverse of) the [[Lorentz factor]]:

: <math> \frac{1}{\gamma } = \sqrt{1-\frac{v^2}{c^2}} </math>

For small values of ''v/c'' this approximates to:

Line 159 ⟶ 147:

: <math> -8.349\times 10^{-11}\times 60\times 60\times 24\times 10^9\approx -7214 \text{ ns} </math>

That is the satellites' clocks loseare slower than Earth's clocks by 7214 nanoseconds a day due to their velocity.

: Note that this speed of {{val|3874|u=m/s}} is measured relative to Earth's center rather than its surface where the GPS receivers (and users) are. This is because Earth's equipotential makes net time dilation equal across its geodesic surface.<ref>{{Cite web |last=S. P. Drake |date=January 2006 |title=The equivalence principle as a stepping stone from special to general relativity |url=http://www.phys.unsw.edu.au/einsteinlight/jw/2006AJP.pdf |website=Am. J. Phys., Vol. 74, No. 1 |pages=22–25}}</ref> That is, the combination of Special and General effects make the net time dilation at the equator equal to that of the poles, which in turn are at rest relative to the center. Hence we use the center as a reference point to represent the entire surface.

==== Gravitational time dilation ====

The amount of dilation due to gravity will beis determined using the [[gravitational time dilation]] equation:

: <math> \frac{t_r}{t_\infty} =\sqrt{1-\frac{2G M}{r c^2}} </math>

where <math>t_r</math> is the time passed at a distance <math>r</math> from the center of the Earth and <math>t_\infty</math> is the time passed for a far away observer.

Line 186 ⟶ 174:

: 45850 – 7210 = 38640 ns

Hence the satellites' clocks gain approximately 38,640 nanoseconds a day or 38.6 μs per day due to relativityrelativistic effects in total.

In order to compensate for this gain, a GPS clock's frequency needs to be slowed by the fraction: