If you wish to contribute or participate in the discussions about articles you are invited to contact the Editor

Relativistic Path Range Effect: Difference between revisions

From Navipedia
Jump to navigation Jump to search
(Created page with "{{Article Infobox2 |Category=Fundamentals |Title={{PAGENAME}} |Authors= J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain. |Level=Advanc...")
 
No edit summary
 
(20 intermediate revisions by 5 users not shown)
Line 1: Line 1:
{{Article Infobox2
{{Article Infobox2
|Category=Fundamentals
|Category=Fundamentals
|Authors=J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
|Level=Intermediate
|YearOfPublication=2011
|Title={{PAGENAME}}
|Title={{PAGENAME}}
|Authors= J. Sanz Subirana, JM. Juan Zornoza and M. Hernandez-Pajares, University of Catalunia, Spain.
|Level=Advanced
|YearOfPublication=2011
|Logo=gAGE
}}
}}
This is a secondary relativistic effect that can be required only for high accuracy positioning. Its net effect on range is less than 2 cm and thence, for most
This is a secondary relativistic effect that can be required only for high accuracy positioning. Its net effect on range is less than 2 cm and thence, for most
purposes it can be neglected.
purposes it can be neglected.




This effect is named the Shapiro signal propagation delay and introduces a general relativistic correction to the geometric range: Due to the space-time curvature produced by the gravitational field, the Euclidean range computed by [[Carrier Phase Ambiguity Fixing|carrier phase measurement]] must be corrected by an amount given by the expression:
This effect is named the Shapiro signal propagation delay and introduces a general relativistic correction to the geometric range: Due to the space-time curvature produced by the gravitational field, the Euclidean range, computed by equation (3) in article [[Code Based Positioning (SPS)]], must be corrected by an amount given by the expression:


<math> \rho_{_{rel}}=\frac{2\,\mu}{c^2}ln\frac{r^{sat}+r_{rcv}+r_{rcv}^{sat}}{r^{sat}+r_{rcv}-r_{rcv}^{sat}} \qquad \mbox{(1)}</math>
::<math>\Delta \rho_{_{rel}}=\frac{2\,\mu}{c^2}ln\frac{r^{sat}+r_{rcv}+r_{rcv}^{sat}}{r^{sat}+r_{rcv}-r_{rcv}^{sat}} \qquad \mbox{(1)}</math>




where <math>r^{sat}, r_{rcv} </math> are the geocentric distances of satellite and receiver and <math> r_{rcv}^{sat} </math> is the distance between them. The constants <math>c = 299792458 m/s </math> and <math>c=299792458\,m/s </math> and <math>\mu=G\,M_\oplus=3.986004418 \cdot 10^{14}\, m^3/s^2 </math> are the sped of light and the earth's gravitational constant, respectively (see [[GNSS Reference Frames]]).
where <math>\displaystyle r^{sat}, r_{rcv} </math> are the geocentric distances of satellite and receiver and <math>\displaystyle r_{rcv}^{sat} </math> is the distance between them. The constants <math>\displaystyle c = 299792458 m/s </math> and <math>\displaystyle \mu=G\,M_\oplus=3.986004418 \cdot 10^{14}\, m^3/s^2 </math> are the speed of light and the Earth's gravitational constant, respectively (see [[Reference Frames in GNSS]]).
 




This correction must be added to the Euclidian distance given by the following equation (2).
This correction must be added to the Euclidian distance given by the following equation (2).


<math> \rho_{rcv}^{sat}=\left\| {r}_{rcv}-{r}^{sat}\right \|=\sqrt{(x_{rcv}-x^{sat})^2+(y_{rcv}-y^{sat})^2+(z_{rcv}-z^{sat})^2} \qquad \mbox{(2)}</math>
:: <math>\rho_{rcv}^{sat}=\left\|{\mathbf r}^{sat}-{\mathbf r}_{rcv}\right \|=\sqrt{(x^{sat}-x_{rcv})^2+(y^{sat}-y_{rcv})^2+(z^{sat}-z_{rcv})^2} \qquad \mbox{(1)}</math>




Next figure 1 illustrates with an example the Shapiro signal propagation delays for the satellites in view from a receiver in Barcelona, Spain (receiver coordinates <math> \lambda\simeq 2^ o $\phi \simeq 41^o</math>).
Next figure 1 illustrates with an example the Shapiro signal propagation delays for the satellites in view from a receiver in Barcelona, Spain (receiver coordinates <math> \lambda\simeq 2^ o</math> <math>\phi \simeq 41^o</math>).


[[File:Geom_Range_Model_Relat_Path_Range_Effect.png|none|thumb|400px|'''''Figure 1:''''' Shapiro correction to the geometric range for the satellites in view from a receiver at <math> \lambda\simeq 2^ o $\phi \simeq 41^o</math> coordinates.]]


A very good review of relativistic effects on GPS can be found in [Ashby, N., 2003], (see http://relativity.livingreviews.org/Articles/lrr-2003-1).
::[[File:Geom_Range_Model_Relat_Path_Range_Effect.png|none|thumb|400px|'''''Figure 1:''''' Shapiro correction to the geometric range for the satellites in view from a receiver at <math> \lambda\simeq 2^ o</math> <math>\phi \simeq 41^o</math> coordinates.]]
 
 
A very good review of relativistic effects on GPS can be found in [Ashby, N., 2003] <ref>[Ashby, N., 2003] Ashby, N., 2003. [https://link.springer.com/content/pdf/10.12942%2Flrr-2003-1.pdf Relativity in the Global Positioning System] </ref> , (see https://link.springer.com/content/pdf/10.12942%2Flrr-2003-1.pdf).




==References==
==References==
[Ashby, N., 2003] Ashby, N., 2003. Relativity in the Global Positioning System. http://relativity.livingreviews.org/articles/lrr-2003-1/.
<references/>
 
[[Category:Fundamentals]]
[[Category:GNSS Measurements Modelling]]

Latest revision as of 14:00, 2 June 2022


FundamentalsFundamentals
Title Relativistic Path Range Effect
Author(s) J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
Level Intermediate
Year of Publication 2011

This is a secondary relativistic effect that can be required only for high accuracy positioning. Its net effect on range is less than 2 cm and thence, for most purposes it can be neglected.


This effect is named the Shapiro signal propagation delay and introduces a general relativistic correction to the geometric range: Due to the space-time curvature produced by the gravitational field, the Euclidean range, computed by equation (3) in article Code Based Positioning (SPS), must be corrected by an amount given by the expression:

[math]\displaystyle{ \Delta \rho_{_{rel}}=\frac{2\,\mu}{c^2}ln\frac{r^{sat}+r_{rcv}+r_{rcv}^{sat}}{r^{sat}+r_{rcv}-r_{rcv}^{sat}} \qquad \mbox{(1)} }[/math]


where [math]\displaystyle{ \displaystyle r^{sat}, r_{rcv} }[/math] are the geocentric distances of satellite and receiver and [math]\displaystyle{ \displaystyle r_{rcv}^{sat} }[/math] is the distance between them. The constants [math]\displaystyle{ \displaystyle c = 299792458 m/s }[/math] and [math]\displaystyle{ \displaystyle \mu=G\,M_\oplus=3.986004418 \cdot 10^{14}\, m^3/s^2 }[/math] are the speed of light and the Earth's gravitational constant, respectively (see Reference Frames in GNSS).


This correction must be added to the Euclidian distance given by the following equation (2).

[math]\displaystyle{ \rho_{rcv}^{sat}=\left\|{\mathbf r}^{sat}-{\mathbf r}_{rcv}\right \|=\sqrt{(x^{sat}-x_{rcv})^2+(y^{sat}-y_{rcv})^2+(z^{sat}-z_{rcv})^2} \qquad \mbox{(1)} }[/math]


Next figure 1 illustrates with an example the Shapiro signal propagation delays for the satellites in view from a receiver in Barcelona, Spain (receiver coordinates [math]\displaystyle{ \lambda\simeq 2^ o }[/math] [math]\displaystyle{ \phi \simeq 41^o }[/math]).


Figure 1: Shapiro correction to the geometric range for the satellites in view from a receiver at [math]\displaystyle{ \lambda\simeq 2^ o }[/math] [math]\displaystyle{ \phi \simeq 41^o }[/math] coordinates.


A very good review of relativistic effects on GPS can be found in [Ashby, N., 2003] [1] , (see https://link.springer.com/content/pdf/10.12942%2Flrr-2003-1.pdf).


References

  1. ^ [Ashby, N., 2003] Ashby, N., 2003. Relativity in the Global Positioning System
Retrieved from "https://gssc.esa.int/navipedia/index.php?title=Relativistic_Path_Range_Effect&oldid=16014"