Error analysis for the Global Positioning System: Difference between revisions

Revision as of 00:36, 24 May 2022

Revision as of 00:41, 24 May 2022

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A number of sources of error exist due to [[Theory of relativity|relativistic]] effects<ref>Webb (2004), p. 32.</ref> that would render the system useless if uncorrected. Three relativistic effects are time dilation, gravitational frequency shift, and eccentricity effects. Examples include the relativistic time ''slowing'' due to the speed of the satellite of about 1 part in 10¹⁰, the gravitational time dilation that makes a satellite run about 5 parts in 10¹⁰ ''faster'' than an Earth-based clock, and the [[Sagnac effect]] due to rotation relative to receivers on Earth. These topics are examined below, one at a time.

=== Special and generalGeneral relativityRelativity ===

[[Special relativity]] (SR) and [[General Relativity]] (GR) are two separate and distinct theories behind the [[theory of relativity]]. SR and GR make different (opposite) predictions when it comes to the clocks on-board GPS satellites. Note the opposite signs (plus and minus) due to the different effects.

SR ([[Special relativityRelativity]]) predicts that the frequency of the atomic clocks moving at GPS orbital speeds will tick more slowly than stationary ground clocks by a factor of <math>{v^{2}}/{2c^{2}}\approx 10 ^{-10}</math>, or result in a delay of about -7 μs/day, where the orbital velocity is v = 4&nbsp;km/s, and c = the speed of light. The SR effect is to their constant movement and height 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.

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Combining SR and GR, the discrepancy is about +38 microseconds per day. 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.

<CENTER>

{| class="wikitable"

|+ SR and GR combined

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| Total (Combined) || +38.6 μs/day || GR is larger effect than SR

|}

</CENTER>

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}}</ref> The conclusion is that the GPS satellites must compensate for GR, the physics of [[black holes]] and extreme gravity.