Editing Geodetic Reference System 1980

{{Short description|Collection of data on Earth's gravity and shape}}
{{More citations needed|date=February 2009}}
{{Geodesy}}
The '''Geodetic Reference System 1980''' ('''GRS80''') consists of a global [[reference ellipsoid]] and a [[normal gravity]] model.<ref name=Moritz1980>{{cite journal |last=Moritz |first=Helmut |authorlink=Helmut Moritz |title=Geodetic Reference System 1980 |journal=[[Bulletin Géodésique]] |volume=54 |issue=3 |date=September 1980 |pages=395–405 |doi=10.1007/BF02521480 |bibcode=1980BGeod..54..395M |s2cid=198209711 |url=http://geoweb.mit.edu/~tah/12.221_2005/grs80_corr.pdf}}</ref><ref name="Moritz1992">{{cite journal | last=Moritz | first=H. |authorlink=Helmut Moritz | title=Geodetic Reference System 1980 | journal=Bulletin Géodésique | publisher=Springer Science and Business Media LLC | volume=66 | issue=2 | year=1992 | issn=0007-4632 | doi=10.1007/bf00989270 | pages=187–192| bibcode=1992BGeod..66..187M | s2cid=122916060 }}</ref><ref name=Moritz2000>{{cite journal |last=Moritz |first=Helmut |authorlink=Helmut Moritz |date=March 2000 |title=Geodetic Reference System 1980 |journal=Journal of Geodesy |volume=74 |issue=1 |pages=128–162 |doi=10.1007/S001900050278 |s2cid=195290884 |url=https://geodesy.geology.ohio-state.edu/course/refpapers/00740128.pdf |access-date=2018-12-15 |archive-date=2016-02-20 |archive-url=https://web.archive.org/web/20160220054607/https://geodesy.geology.ohio-state.edu/course/refpapers/00740128.pdf |url-status=dead }}</ref> The GRS80 gravity model has been followed by the newer more accurate [[Earth Gravitational Model]]s, but the '''GRS80 reference ellipsoid''' is still the most accurate in use for [[coordinate reference system]]s, e.g. for the international [[International Terrestrial Reference System and Frame|ITRS]], the European [[European Terrestrial Reference System 1989|ETRS89]] and (with a 0,1 mm rounding error) for [[World Geodetic System|WGS 84]] used for the American [[Satellite navigation|Global Navigation Satellite System]] ([[Global Positioning System|GPS]]).

==Background==
[[Geodesy]] is the scientific discipline that deals with the measurement and representation of the [[earth]], its [[gravitation]]al field and geodynamic phenomena ([[polar motion]], earth [[tide]]s, and crustal motion) in three-dimensional, time-varying space.

The [[geoid]] is essentially the figure of the Earth abstracted from its topographic features. It is an idealized equilibrium surface of sea water, the mean sea level surface in the absence of currents, air pressure variations etc. and continued under the continental masses. The geoid, unlike the ellipsoid, is irregular and too complicated to serve as the computational surface on which to solve geometrical problems like point positioning. The geometrical separation between it and the reference ellipsoid is called the geoidal [[wiktionary:undulation|undulation]], or more usually the geoid-ellipsoid separation, ''N''. It varies globally between {{val|110|u=m|p=±}}.

A [[reference ellipsoid]], customarily chosen to be the same size (volume) as the geoid, is described by its semi-major axis (equatorial
radius) ''a'' and flattening ''f''. The quantity ''f''&nbsp;= (''a''−''b'')/''a'', where ''b'' is the semi-minor axis (polar radius), is a purely geometrical one. The mechanical ellipticity of the earth (dynamical flattening, symbol ''J''<sub>2</sub>) is determined to high precision by observation of satellite orbit perturbations. Its relationship with the geometric flattening is indirect. The relationship depends on the internal density distribution.

The 1980 Geodetic Reference System (GRS 80) posited a {{val|6378137|u=m}} semi-major axis and a {{frac|298.257222101}} flattening. This system was adopted at the XVII General Assembly of the International Union of Geodesy and Geophysics ([[IUGG]]) in Canberra, Australia, 1979.

The GRS 80 reference system was originally used by the [[WGS 84|World Geodetic System 1984]] (WGS 84). The reference ellipsoid of WGS 84 now differs slightly due to later refinements.{{cn|date=May 2024}}

The numerous other systems which have been used by diverse countries for their maps and charts are gradually dropping out of use as more and more countries move to global, geocentric reference systems using the GRS80 reference ellipsoid.

==Definition==
The reference ellipsoid is usually defined by its [[semi-major axis]] (equatorial
radius) <math>a</math> and either its [[semi-minor axis]] (polar radius) <math>b</math>, [[aspect ratio]] <math>(b/a)</math> or [[flattening]] <math>f</math>, but GRS80 is an exception: ''four'' independent constants are required for a complete definition. GRS80 chooses as these <math>a</math>, <math>GM</math>, <math>J_2</math> and <math>\omega</math>, making the geometrical constant <math>f</math> a derived quantity.

; Defining [[geometrical]] [[constant (mathematics)|constants]]
:Semi-major axis = Equatorial Radius = <math>a = 6\,378\,137\,\mathrm{m}</math>;

; Defining [[nature|physical]] [[constant (mathematics)|constants]]
:[[Geocentric gravitational constant]] determined from the [[gravitational constant]] and the [[earth mass]] with atmosphere <math>GM = 3986005\times10^8\, \mathrm{m^3/s^2}</math>;
:Dynamical form factor <math>J_2 = 108\,263\times10^{-8}</math>;
:Angular velocity of rotation <math>\omega = 7\,292\,115\times10^{-11}\, \mathrm{s^{-1}}</math>;

==Derived quantities==

; Derived geometrical constants (all rounded)
:Flattening = <math>f</math> = 0.003 352 810 681 183 637 418;
:Reciprocal of flattening = <math>1/f</math> = 298.257 222 100 882 711 243;
:Semi-minor axis = Polar Radius = <math>b</math> = 6 356 752.314 140 347 m;
:Aspect ratio = <math>b/a</math> = 0.996 647 189 318 816 363;
:[[Earth radius#Mean radius: R1|Mean radius]] as defined by the [[International Union of Geodesy and Geophysics]] (IUGG): <math>R_1 = (2a+b)/3</math> = 6 371 008.7714 m;
:[[Earth radius#Authalic radius|Authalic mean radius]] = <math>R_2</math> = 6 371 007.1809 m;
:Radius of a sphere of the same volume = <math>R_3 = (a^2b)^{1/3}</math> = 6 371 000.7900 m;
:Linear eccentricity = <math>c = \sqrt{a^2-b^2}</math> = 521 854.0097 m;
:[[Eccentricity (mathematics)|Eccentricity]] of elliptical section through poles = <math>e = \frac{\sqrt{a^2-b^2}}{a}</math> = 0.081 819 191 0428;
:Polar radius of curvature = <math>a^2/b</math> = 6 399 593.6259 m;
:Equatorial radius of curvature for a meridian = <math>b^2/a</math> = 6 335 439.3271 m;
:Meridian quadrant = 10 001 965.7292 m;

; Derived physical constants (rounded)
:Period of rotation ([[sidereal day]]) = <math>2\pi/\omega</math> = 86 164.100 637 s

The formula giving the eccentricity of the GRS80 spheroid is:<ref name=Moritz1980/>

:<math>e^2  =  \frac {a^2 - b^2}{a^2} = 3J_2 + \frac4{15} \frac{\omega^2 a^3}{GM} \frac{e^3}{2q_0},</math>

where

:<math> 2q_0 = \left(1 + \frac3{e'^2}\right) \arctan e' - \frac3{e'}</math>

and <math>e' = \frac{e}{\sqrt{1 - e^2}} </math> (so <math>\arctan e' = \arcsin e</math>).  The equation is solved iteratively to give

:<math>e^2 = 0.00669\,43800\,22903\,41574\,95749\,48586\,28930\,62124\,43890\,\ldots</math>

which gives

:<math>f = 1/298.25722\,21008\,82711\,24316\,28366\,\ldots.</math>

==References==
{{Reflist}}

==External links==
*[https://web.archive.org/web/20051225213554/http://www.gfy.ku.dk/~iag/HB2000/part4/grs80_corr.htm GRS 80 Specification] <!-- The formulas missing in the htm can be found in the original document http://www.iag-aig.org/attach/bc842687a973633aaa20d6617492c5d5/grs80_corr.doc
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[[Category:Geodesy]]