Relativistic Clock Correction - Revision history

Claudia.Prajanu at 18:54, 8 July 2018

2018-07-08T18:54:22Z

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	The rate of advance of two identical clocks, placed one in the satellite and the other on the terrestrial surface, will differ due to the difference of the gravitational potential (general relativity) and to the relative speed between them (special relativity). This difference can be split into <ref name="Rel">Ashby, N., 2003 [~~http~~://~~relativity~~.~~livingreviews~~.~~org~~/~~Articles~~/~~lrr~~-2003-1~~/download/lrr-2003-1Color~~.pdf Relativity in the Global Positioning System]</ref>:		The rate of advance of two identical clocks, placed one in the satellite and the other on the terrestrial surface, will differ due to the difference of the gravitational potential (general relativity) and to the relative speed between them (special relativity). This difference can be split into <ref name="Rel">Ashby, N., 2003 [https://link.springer.com/content/pdf/10.12942%2Flrr-2003-1.pdf Relativity in the Global Positioning System]</ref>:

Filipe.Pelica: fixed broken link.

2014-09-08T17:33:22Z

fixed broken link.

← Older revision		Revision as of 17:33, 8 September 2014
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	The rate of advance of two identical clocks, placed one in the satellite and the other on the terrestrial surface, will differ due to the difference of the gravitational potential (general relativity) and to the relative speed between them (special relativity). This difference can be split into <ref name="Rel">Ashby, N., 2003 [~~Relativity in the Global Positioning System~~ http://relativity.livingreviews.org/Articles/lrr-2003-1/download/lrr-2003-1Color.pdf]</ref>:		The rate of advance of two identical clocks, placed one in the satellite and the other on the terrestrial surface, will differ due to the difference of the gravitational potential (general relativity) and to the relative speed between them (special relativity). This difference can be split into <ref name="Rel">Ashby, N., 2003 [http://relativity.livingreviews.org/Articles/lrr-2003-1/download/lrr-2003-1Color.pdf Relativity in the Global Positioning System]</ref>:

Filipe.Pelica: fixed broken link.

2014-09-08T17:32:15Z

fixed broken link.

← Older revision		Revision as of 17:32, 8 September 2014
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	The rate of advance of two identical clocks, placed one in the satellite and the other on the terrestrial surface, will differ due to the difference of the gravitational potential (general relativity) and to the relative speed between them (special relativity). This difference can be split into <ref name="Rel"> [Ashby, N., 2003~~] Ashby, N., 2003.~~ Relativity in the Global Positioning System. http://relativity.livingreviews.org/Articles/lrr-2003-1/download/lrr-2003-1Color.pdf</ref>:		The rate of advance of two identical clocks, placed one in the satellite and the other on the terrestrial surface, will differ due to the difference of the gravitational potential (general relativity) and to the relative speed between them (special relativity). This difference can be split into <ref name="Rel">Ashby, N., 2003 [Relativity in the Global Positioning System http://relativity.livingreviews.org/Articles/lrr-2003-1/download/lrr-2003-1Color.pdf]</ref>:

Filipe.Pelica: fixed broken link.

2014-09-08T17:31:12Z

fixed broken link.

← Older revision		Revision as of 17:31, 8 September 2014
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	\|Title={{PAGENAME}}		\|Title={{PAGENAME}}
	}}		}}
	The rate of advance of two identical clocks, placed one in the satellite and the other on the terrestrial surface, will differ due to the difference of the gravitational potential (general relativity) and to the relative speed between them (special relativity). This difference can be split into <ref name="Rel"> [Ashby, N., 2003] Ashby, N., 2003. Relativity in the Global Positioning System. http://relativity.livingreviews.org/Articles/lrr-2003-1/.</ref>:		The rate of advance of two identical clocks, placed one in the satellite and the other on the terrestrial surface, will differ due to the difference of the gravitational potential (general relativity) and to the relative speed between them (special relativity). This difference can be split into <ref name="Rel"> [Ashby, N., 2003] Ashby, N., 2003. Relativity in the Global Positioning System. http://relativity.livingreviews.org/Articles/lrr-2003-1/download/lrr-2003-1Color.pdf</ref>:

Forrest.Voight: word choice

2013-12-12T07:25:32Z

word choice

← Older revision		Revision as of 07:25, 12 December 2013
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	The rate of advance of two identical clocks, placed one in the satellite and the other on the terrestrial surface, will differ due to the difference of the gravitational potential (general relativity) and to the relative speed between them (special relativity). This difference can be ~~splitted~~ into <ref name="Rel"> [Ashby, N., 2003] Ashby, N., 2003. Relativity in the Global Positioning System. http://relativity.livingreviews.org/Articles/lrr-2003-1/.</ref>:		The rate of advance of two identical clocks, placed one in the satellite and the other on the terrestrial surface, will differ due to the difference of the gravitational potential (general relativity) and to the relative speed between them (special relativity). This difference can be split into <ref name="Rel"> [Ashby, N., 2003] Ashby, N., 2003. Relativity in the Global Positioning System. http://relativity.livingreviews.org/Articles/lrr-2003-1/.</ref>:

Carlos.Lopez at 11:41, 23 February 2012

2012-02-23T11:41:11Z

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Jaume.Sanz at 07:54, 30 January 2012

2012-01-30T07:54:05Z

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	::*A constant component that only depends on the nominal value of the semi-major axis of the satellite orbit, which is adjusted modifying (in factory) the clock oscillating frequency of the satellite<ref>Being <math>\displaystyle f_0=10.23 MHz</math>, one has <math>\Delta f_0= 4.464 \cdot 10^{-10}\, f_0=4.57 \cdot 10^{-3} Hz</math> thus the satellite must use <math>f_0'=10.22999999543\,Mhz</math>. Notice that <math>\displaystyle f_0^'</math> is the frequency "emitted" by the satellite and <math>\displaystyle f_0</math> is the one "received" on the terrestrial surface, i.e., an apparent increase of the frequency is of <math>4.57 \cdot 10^{-3} Hz</math>. That is, the clock on satellite appear to run faster (<math>\simeq 38 \mu s/day</math>) than on ground [note: <math>\displaystyle \Delta f/f=\Delta T/T</math>]. This effect is corrected (in factory) decreasing the oscillating frequency of the satellite by this amount <math>4.57 \cdot 10^{-3} Hz</math>.</ref>:		::*A constant component that only depends on the nominal value of the semi-major axis of the satellite orbit, which is adjusted modifying (in factory) the clock oscillating frequency of the satellite<ref group="footnotes">Being <math>\displaystyle f_0=10.23 MHz</math>, one has <math>\Delta f_0= 4.464 \cdot 10^{-10}\, f_0=4.57 \cdot 10^{-3} Hz</math> thus the satellite must use <math>f_0'=10.22999999543\,Mhz</math>. Notice that <math>\displaystyle f_0^'</math> is the frequency "emitted" by the satellite and <math>\displaystyle f_0</math> is the one "received" on the terrestrial surface, i.e., an apparent increase of the frequency is of <math>4.57 \cdot 10^{-3} Hz</math>. That is, the clock on satellite appear to run faster (<math>\simeq 38 \mu s/day</math>) than on ground [note: <math>\displaystyle \Delta f/f=\Delta T/T</math>]. This effect is corrected (in factory) decreasing the oscillating frequency of the satellite by this amount <math>4.57 \cdot 10^{-3} Hz</math>.</ref>:

	::::<math>		::::<math>
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	::where <math>{\mathbf r}^{sat}</math> and <math>{\mathbf v}^{sat}</math> are the satellite position (<math>\displaystyle m</math>) and velocity (<math>\displaystyle m/s</math>) vectors, in an inertial system<ref>Note: Over the osculating orbit, <math>\sqrt{\mu a}\,e\sin E={\mathbf r} \cdot {\mathbf v}</math>. Thence, <math>\Delta_{rel}=- 2\, \frac{\mathbf{r} \cdot \mathbf{v}}{c^2}= -2\,\frac{\sqrt{\mu\, a}}{c^2} \, e\, \sin(E)</math>, where <math>\displaystyle \mu</math> the (geocentric) gravitational constant, <math>\displaystyle c</math> the speed of light in a vacuum, <math>\displaystyle a</math> and <math>\displaystyle e</math> the semi-major axis and eccentricity of the osculating orbit and <math>\displaystyle E</math> the eccentric anomaly.</ref>. The scalar <math>{\mathbf r}^{sat}\cdot{\mathbf v}^{sat}</math> can be evaluated either in a CRS or TRS (i.e., ECEF) system. Notice that in a ECEF system, the earth rotation <math>{\boldsymbol \omega}_E \times {\mathbf r}^{sat}</math> should be discounted from <math>{\mathbf v}^{sat}</math>, but it cancels in the scalar product with <math>{\mathbf r}^{sat}</math>.		::where <math>{\mathbf r}^{sat}</math> and <math>{\mathbf v}^{sat}</math> are the satellite position (<math>\displaystyle m</math>) and velocity (<math>\displaystyle m/s</math>) vectors, in an inertial system<ref group="footnotes">Note: Over the osculating orbit, <math>\sqrt{\mu a}\,e\sin E={\mathbf r} \cdot {\mathbf v}</math>. Thence, <math>\Delta_{rel}=- 2\, \frac{\mathbf{r} \cdot \mathbf{v}}{c^2}= -2\,\frac{\sqrt{\mu\, a}}{c^2} \, e\, \sin(E)</math>, where <math>\displaystyle \mu</math> the (geocentric) gravitational constant, <math>\displaystyle c</math> the speed of light in a vacuum, <math>\displaystyle a</math> and <math>\displaystyle e</math> the semi-major axis and eccentricity of the osculating orbit and <math>\displaystyle E</math> the eccentric anomaly.</ref>. The scalar <math>{\mathbf r}^{sat}\cdot{\mathbf v}^{sat}</math> can be evaluated either in a CRS or TRS (i.e., ECEF) system. Notice that in a ECEF system, the earth rotation <math>{\boldsymbol \omega}_E \times {\mathbf r}^{sat}</math> should be discounted from <math>{\mathbf v}^{sat}</math>, but it cancels in the scalar product with <math>{\mathbf r}^{sat}</math>.

Jaume.Sanz at 07:53, 30 January 2012

2012-01-30T07:53:04Z

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			==Notes==
			<references group="footnotes"/>

	==References==		==References==

Jaume.Sanz at 07:51, 30 January 2012

2012-01-30T07:51:43Z

← Older revision		Revision as of 07:51, 30 January 2012
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	Unlike in GPS, relativistic corrections to GLONASS orbits eccentricity are transmitted within the navigation message into the satellite clock corrections (<math>\displaystyle \tau_n</math>, <math>\gamma_n</math>). Thence, (2) is not needed with such broadcast message <ref name="Rel"/>.		Unlike in GPS, relativistic corrections to GLONASS orbits eccentricity are transmitted within the navigation message into the satellite clock corrections (<math>\displaystyle \tau_n</math>, <math>\displaystyle \gamma_n</math>). Thence, (2) is not needed with such broadcast message <ref name="Rel"/>.

Jaume.Sanz at 07:51, 30 January 2012

2012-01-30T07:51:06Z

← Older revision		Revision as of 07:51, 30 January 2012
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	::*A constant component that only depends on the nominal value of the semi-major axis of the satellite orbit, which is adjusted modifying (in factory) the clock oscillating frequency of the satellite<ref>Being <math>f_0=10.23 MHz</math>, one has <math>\Delta f_0= 4.464 \cdot 10^{-10}\, f_0=4.57 \cdot 10^{-3} Hz</math> thus the satellite must use <math>f_0'=10.22999999543\,Mhz</math>. Notice that <math>\displaystyle f_0^'</math> is the frequency "emitted" by the satellite and <math>\displaystyle f_0</math> is the one "received" on the terrestrial surface, i.e., an apparent increase of the frequency is of <math>4.57 \cdot 10^{-3} Hz</math>. That is, the clock on satellite appear to run faster (<math>\simeq 38 \mu s/day</math>) than on ground [note: <math>\displaystyle \Delta f/f=\Delta T/T</math>]. This effect is corrected (in factory) decreasing the oscillating frequency of the satellite by this amount <math>4.57 \cdot 10^{-3} Hz</math>.</ref>:		::*A constant component that only depends on the nominal value of the semi-major axis of the satellite orbit, which is adjusted modifying (in factory) the clock oscillating frequency of the satellite<ref>Being <math>\displaystyle f_0=10.23 MHz</math>, one has <math>\Delta f_0= 4.464 \cdot 10^{-10}\, f_0=4.57 \cdot 10^{-3} Hz</math> thus the satellite must use <math>f_0'=10.22999999543\,Mhz</math>. Notice that <math>\displaystyle f_0^'</math> is the frequency "emitted" by the satellite and <math>\displaystyle f_0</math> is the one "received" on the terrestrial surface, i.e., an apparent increase of the frequency is of <math>4.57 \cdot 10^{-3} Hz</math>. That is, the clock on satellite appear to run faster (<math>\simeq 38 \mu s/day</math>) than on ground [note: <math>\displaystyle \Delta f/f=\Delta T/T</math>]. This effect is corrected (in factory) decreasing the oscillating frequency of the satellite by this amount <math>4.57 \cdot 10^{-3} Hz</math>.</ref>:

	::::<math>		::::<math>
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	::where <math>{\mathbf r}^{sat}</math> and <math>{\mathbf v}^{sat}</math> are the satellite position (<math>m</math>) and velocity (<math>m/s</math>) vectors, in an inertial system<ref>Note: Over the osculating orbit, <math>\sqrt{\mu a}\,e\sin E={\mathbf r} \cdot {\mathbf v}</math>. Thence, <math>\Delta_{rel}=- 2\, \frac{\mathbf{r} \cdot \mathbf{v}}{c^2}= -2\,\frac{\sqrt{\mu\, a}}{c^2} \, e\, \sin(E)</math>, where <math>\displaystyle \mu</math> the (geocentric) gravitational constant, <math>\displaystyle c</math> the speed of light in a vacuum, <math>\displaystyle a</math> and <math>\displaystyle e</math> the semi-major axis and eccentricity of the osculating orbit and <math>\displaystyle E</math> the eccentric anomaly.</ref>. The scalar <math>{\mathbf r}^{sat}\cdot{\mathbf v}^{sat}</math> can be evaluated either in a CRS or TRS (i.e., ECEF) system. Notice that in a ECEF system, the earth rotation <math>{\boldsymbol \omega}_E \times {\mathbf r}^{sat}</math> should be discounted from <math>{\mathbf v}^{sat}</math>, but it cancels in the scalar product with <math>{\mathbf r}^{sat}</math>.		::where <math>{\mathbf r}^{sat}</math> and <math>{\mathbf v}^{sat}</math> are the satellite position (<math>\displaystyle m</math>) and velocity (<math>\displaystyle m/s</math>) vectors, in an inertial system<ref>Note: Over the osculating orbit, <math>\sqrt{\mu a}\,e\sin E={\mathbf r} \cdot {\mathbf v}</math>. Thence, <math>\Delta_{rel}=- 2\, \frac{\mathbf{r} \cdot \mathbf{v}}{c^2}= -2\,\frac{\sqrt{\mu\, a}}{c^2} \, e\, \sin(E)</math>, where <math>\displaystyle \mu</math> the (geocentric) gravitational constant, <math>\displaystyle c</math> the speed of light in a vacuum, <math>\displaystyle a</math> and <math>\displaystyle e</math> the semi-major axis and eccentricity of the osculating orbit and <math>\displaystyle E</math> the eccentric anomaly.</ref>. The scalar <math>{\mathbf r}^{sat}\cdot{\mathbf v}^{sat}</math> can be evaluated either in a CRS or TRS (i.e., ECEF) system. Notice that in a ECEF system, the earth rotation <math>{\boldsymbol \omega}_E \times {\mathbf r}^{sat}</math> should be discounted from <math>{\mathbf v}^{sat}</math>, but it cancels in the scalar product with <math>{\mathbf r}^{sat}</math>.


	Unlike in GPS, relativistic corrections to GLONASS orbits eccentricity are transmitted within the navigation message into the satellite clock corrections (<math>\tau_n</math>, <math>\gamma_n</math>). Thence, (2) is not needed with such broadcast message <ref name="Rel"/>.		Unlike in GPS, relativistic corrections to GLONASS orbits eccentricity are transmitted within the navigation message into the satellite clock corrections (<math>\displaystyle \tau_n</math>, <math>\gamma_n</math>). Thence, (2) is not needed with such broadcast message <ref name="Rel"/>.