Earth ellipsoid

Earth ellipsoid - an ellipsoid of revolution , the dimensions of which are selected subject to the best fit to the quasi-geoid figure for the Earth as a whole (a common earth ellipsoid) or its individual parts ( reference ellipsoid ).

Content

1 Earth Ellipsoid Parameters
2 Reference ellipsoid
- 2.1 Basic reference ellipsoids and their parameters
3 Earth Ellipsoid
- 3.1 Modern terrestrial ellipsoids and their parameters
4 See also
5 Links

Earth Ellipsoid Parameters

The Earth ellipsoid has three main parameters, any two of which uniquely determine its shape:

semimajor axis (equatorial radius) of the ellipsoid, a ;
minor axis (polar radius), b ;
geometric (polar) compression, $f={\frac {a-b}{a}}$ ${\ displaystyle f = {\ frac {ab} {a}}}$ .

There are also other parameters of the ellipsoid:

first eccentricity $e={\sqrt {\frac {a^{2}-b^{2}}{a^{2}}}}={\frac {\sqrt {a^{2}-b^{2}}}{a}}$ ${\ displaystyle e = {\ sqrt {\ frac {a ^ {2} -b ^ {2}} {a ^ {2}}}} = {\ frac {\ sqrt {a ^ {2} -b ^ { 2}}} {a}}}$ ;
second eccentricity $e'={\sqrt {\frac {a^{2}-b^{2}}{b^{2}}}}={\frac {\sqrt {a^{2}-b^{2}}}{b}}$ ${\ displaystyle e '= {\ sqrt {\ frac {a ^ {2} -b ^ {2}} {b ^ {2}}}} = {\ frac {\ sqrt {a ^ {2} -b ^ {2}}} {b}}}$ .

For the practical implementation of the Earth's ellipsoid, it is necessary to orient in the body of the Earth . In this case, a general condition is put forward: orientation must be performed in such a way that the differences between the astronomical and geodetic coordinates are minimal.

Reference Ellipsoid

The reference ellipsoid shape is best suited for the territory of a single country or several countries. As a rule, reference ellipsoids are accepted for the processing of geodetic measurements by law . In Russia / USSR from 1946 to 2012, 3 basic coordinate systems based on the Krasovsky ellipsoid (SK-42, SK-63 and SK-95) were used. Starting from 2012, the RF PP 1463 and 1240 dated December 28, 2012 and November 24, 2016, respectively, the use of SK-42 and SK-95 is allowed until January 1, 2021).

- ellipsoids GSK-2011 and PZ-90.11. The ellipsoids SK-95 and Krasovsky are no longer in use by January 1, 2021. In the USA, the reference ellipsoid WGS-84 is common.

Orientation of the reference ellipsoid in the body of the Earth obeys the following requirements:

The minor axis of the ellipsoid ( b ) should be parallel to the axis of rotation of the Earth.
The surface of the ellipsoid should be as close as possible to the surface of the geoid within the region.

To fix the reference ellipsoid in the Earth’s body, it is necessary to set the geodetic coordinates B ₀ , L ₀ , H _{0 of the} starting point of the geodetic network and the initial azimuth A ₀ to the neighboring point. The totality of these quantities is called the original geodetic dates .

Basic reference ellipsoids and their parameters

Scientist	Year	A country	a, m	1 / f
Delambre	1800	France	6 375 653	334.0
Delambre	1810	France	6 376 985	308.6465
Valbek	1819	Finland, Russian Empire	6 376 896	302.8
Airy	1830		6 377 563.4	299.324 964 6
Everest	1830	India, Pakistan, Nepal, Sri Lanka	6 377 276,345	300.801 7
Bessel	1841	Germany, Russia (until 1942)	6 377 397,155	299.152 815 4
Tenner	1844	Russia	6 377 096	302.5
Clark	1866	USA, Canada, Lat. and Center. America	6 378 206.4	294.978 698 2
Clark	1880	France, South Africa	6 377 365	289.0
Listing	1880		6 378 249	293.5
Helmert	1907		6,378,200	298.3
Hayford	1910	Europe, Asia, South America, Antarctica	6 378 388	297.0
Heiskanen	1929		6,378,400	298.2
Krasovsky	1936	the USSR	6 378 210	298.6
Krasovsky	1942	USSR, Soviet republics, Eastern Europe, Antarctica	6 378 245	298.3
Everest	1956	India, Nepal	6 377 301,243	300.801 7
IAG-67	1967		6 378 160	298.247 167
WGS-72	1972		6 378 135	298.26
IAU-76	1976		6 378 140	298.257

Earth Ellipsoid

Earth-wide ellipsoid should be oriented in the body of the Earth according to the following requirements:

The minor axis should coincide with the axis of rotation of the Earth.
The center of the ellipsoid should coincide with the center of mass of the Earth.
The heights of the geoid above the ellipsoid h _i (the so-called height anomalies) must obey the least squares condition: $\sum _{n=0}^{\infty }h_{i}^{2}=\min$ ${\ displaystyle \ sum _ {n = 0} ^ {\ infty} h_ {i} ^ {2} = \ min}$ .

When orienting a common terrestrial ellipsoid in the body of the Earth (unlike a reference ellipsoid), it is not necessary to enter the initial geodetic dates.

Since the requirements for terrestrial ellipsoids in practice are satisfied with some tolerances, and the fulfillment of the latter (3) is not possible in full, then in geodesy and related sciences various implementations of the ellipsoid can be used, the parameters of which are very close, but do not coincide (see below).

Modern Earth-Ellipsoids and Their Parameters

Title	Year	Country / Organization	a, m	accuracy m _a , m	1 / f	accuracy m _f	Note
GRS80	1980	MAGG (IUGG)	6 378 137	± 2	298,257 222 101	± 0.001	( Eng. Geodetic Reference System 1980) was developed by the International Geodetic and Geophysical Union ( Eng. International Union of Geodesy and Geophysics ) and recommended for geodetic works
WGS 84	1984	USA	6 378 137	± 2	298,257 223 563	± 0.001	( Eng. World Geodetic System 1984) used in GPS satellite navigation system
PZ-90	1990	the USSR	6 378 136	± 1	298,257 839 303	± 0.001	(Earth parameters 1990) is used on the territory of Russia for geodetic support of orbital flights. This ellipsoid is used in the GLONASS satellite navigation system.
MSVZ (IERS)	1996	Iers	6 378 136.49	-	298,256 45	-	( English International Earth Rotation Service 1996 ) recommended by the International Earth Rotation Service for processing VLBI observations

Links

V.L. Panteleev. Earth Theory (lecture course)
Web site of the International Geodetic and Geophysical Union
A brief biography of Walbeck on the website of the University of Helsinki
Le procès des étoiles 1735-1771 ASIN: B0000DTZN6
Le Procès des étoiles ASIN: B0014LXB6O
Le procès des étoiles 1735—1771 ISBN 978-2-232-11862-3