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: <math> \frac{1}{\gamma } \approx 1-\frac{v^2}{2 c^2} </math>
The GPS satellites move at {{val|3874|u=m/s}} relative to Earth's center.<ref name="Nelson
▲The GPS satellites move at {{val|3874|u=m/s}} relative to Earth's center.<ref name=Nelson>[http://www.aticourses.com/global_positioning_system.htm The Global Positioning System by Robert A. Nelson Via Satellite], November 1999</ref> We thus determine:
: <math> \frac{1}{\gamma } \approx 1-\frac{3874^2}{2 \left(2.998\times 10^8\right)^2} \approx 1-8.349\times 10^{-11} </math>
This difference below 1 of {{val|8.349|e=-11}} represents the fraction by which the satellites' clocks move slower than Earth's. It is then multiplied by the number of nanoseconds in a day:
: <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 lose 7,214 nanoseconds a day due to [[special relativity]] effects.
: 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 | url = http://www.phys.unsw.edu.au/einsteinlight/jw/2006AJP.pdf | title = The equivalence principle as a stepping stone from special to general relativity | author = S. P. Drake | work = Am. J. Phys., Vol. 74, No. 1|pages=
The amount of dilation due to gravity will be determined using the [[gravitational time dilation]] equation:
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: <math> \Delta \left(\frac{1}{\gamma }\right) \approx \frac{G M_{\text{earth}}}{R_{\text{earth}} c^2}-\frac{G M_{\text{earth}}}{R_{\text{gps}} c^2} </math>
Earth has a radius of 6,357
▲Earth has a radius of 6,357 km (at the poles) making ''R<sub>earth</sub>'' = 6,357,000 m and the satellites have an altitude of 20,184 km <ref name=Nelson>[http://www.aticourses.com/global_positioning_system.htm The Global Positioning System by Robert A. Nelson Via Satellite], November 1999</ref> making their orbit radius ''R<sub>gps</sub>'' = 26,541,000 m. Substituting these in the above equation, with ''M<sub>earth</sub>'' = {{val|5.974|e=24}}, ''G'' = {{val|6.674|e=-11}}, and ''c'' = {{val|2.998|e=8}} (all in [[International System of Units|SI]] units), gives:
: <math> \Delta \left(\frac{1}{\gamma }\right) \approx 5.307\times 10^{-10} </math>
This represents the fraction by which the satellites' clocks move faster than Earth's. It is then multiplied by the number of nanoseconds in a day:
: <math> 5.307\times 10^{-10}\times 60\times 60\times 24\times 10^9\approx 45850 \text{ ns} </math>
That is, the satellites' clocks gain 45,850 nanoseconds a day due to [[general relativity]] effects. These effects are added together to give (rounded to 10 ns):
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Hence the satellites' clocks gain approximately 38,640 nanoseconds a day or 38.6 μs per day due to relativity effects in total.
In order to compensate for this gain, a GPS clock's frequency needs to be slowed by the fraction:
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: {{val|5.307|e=-10}} - {{val|8.349|e=-11}} = {{val|4.472|e=-10}}
This fraction is subtracted from 1 and multiplied by the pre-adjusted clock frequency of 10.23
: (1 - {{val|4.472|e=-10}}) × 10.23 = 10.22999999543
That is, we need to slow the clocks down from 10.23
=== Sagnac distortion ===
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Since GPS signals at terrestrial receivers tend to be relatively weak, natural radio signals or scattering of the GPS signals can [[Desensitization (telecommunications)|desensitize]] the receiver, making acquiring and tracking the satellite signals difficult or impossible.
[[Space weather]] degrades GPS operation in two ways, direct interference by solar radio burst noise in the same frequency band<ref>Cerruti, A., P. M. Kintner, D. E. Gary, A. J. Mannucci, R. F. Meyer, P. H. Doherty, and A. J. Coster (2008), Effect of intense December 2006 solar radio bursts on GPS receivers, Space Weather, doi:10.1029/2007SW000375, October 19, 2008</ref> or by scattering of the GPS radio signal in ionospheric irregularities referred to as scintillation.<ref>{{cite journal | author=Aarons, Jules and Basu, Santimay | title=Ionospheric amplitude and phase fluctuations at the GPS frequencies | work=Proceedings of ION GPS | volume=2 | year=1994 | pages=1569–1578}}</ref> Both forms of degradation follow the 11 year [[solar cycle]] and are a maximum at sunspot maximum although they can occur at anytime. Solar radio bursts are associated with [[solar flares]] and Coronal Mass Ejections (CMEs)<ref>S. Mancuso and J. C. Raymond, "Coronal transients and metric type II radio bursts. I. Effects of geometry, 2004, Astronomy and Astrophysics, v.413, p.363-371'</ref> and their impact can affect reception over the half of the Earth facing the sun. Scintillation occurs most frequently at tropical latitudes where it is a night time phenomenon. It occurs less frequently at high latitudes or mid-latitudes where magnetic storms can lead to scintillation.<ref>{{cite journal | author=Ledvina, B. M., J. J. Makela, and P. M. Kintner | year=2002 | title=First observations of intense GPS L1 amplitude scintillations at midlatitude | journal=Geophysical Research Letters | work=Geophys. Res. Lett. | volume=29 | page=1659 | doi=10.1029/2002GL014770 | issue=14 | bibcode=2002GeoRL..29n...4L}}</ref> In addition to producing scintillation, magnetic storms can produce strong ionospheric gradients that degrade the accuracy of SBAS systems.<ref>Tom Diehl, [http://www.faa.gov/about/office_org/headquarters_offices/ato/service_units/techops/navservices/gnss/library/satNav/media/SATNAV_0604.PDF Solar Flares Hit the Earth- WAAS Bends but Does Not Break], SatNav News, volume 23, June 2004</ref>
== Artificial sources of interference ==
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