The semimajor axis of the ellipsoid. Figure of the earth

Geoid, quasi-geoid and general terrestrial ellipsoid are three models of the Earth. Let us define them from the point of view modern ideas about the figure of the Earth.

Under figure of the Earth Currently, they understand a figure limited by the physical surface of the Earth, i.e. the surface of its hard shell on land and the undisturbed surface of the seas and oceans.

Land makes up a third of the earth's surface and on average it rises above the water by about 900 meters, which is insignificant compared to the radius of the Earth (6371 km). Therefore, as a first approximation, the geoid is taken as the figure of the Earth.

Let's give two definitions of geoid:

1. Strict: The geoid is the level surface of the Earth's gravity field, passing through the beginning of the height count.

2. Non-strict: The geoid is a figure bounded by the undisturbed surface of the seas and oceans and extended under the continents so that the plumb lines at all its points are perpendicular to it.

For more than a hundred years, that is, from the first half of the last century, surveyors and geophysicists studied the geoid figure and considered this the main scientific task of higher geodesy. In the middle of the last century, the Soviet scientist Molodensky proved that the geoid figure, strictly speaking, is indefinable. He proposed that the main task of higher geodesy be the study of figure real earth and her gravitational field. Molodensky created a theory that allows an accurate determination of the figure of the Earth based on measurements taken on the earth’s surface, without involving any hypotheses about its internal structure.

In Molodensky’s theory, the surface is introduced as an auxiliary one quasigeoid, coinciding with the geoid on the oceans and seas and very little departing from the surface of the geoid on land (less than 2 m) .

Unlike the geoid, the surface of a quasi-geoid can be strictly determined from the results of ground-based observations.

With the concept earth's ellipsoid we have already encountered when considering the main scientific problem of higher geodesy, the surface of the earth's ellipsoid is the mathematically and geometrically simple surface on which geodetic problems of coordinating points on the earth's surface can be solved and which is sufficiently close to the surface of the earth. The Earth's ellipsoid is an ellipsoid of revolution with low polar compression. Its surface can be obtained by rotating the half-ellipse PEP 1 around its minor axis PP 1 (Figure 1.2).

Rice. 1.2. To the concept of the earth's ellipsoid: - semimajor axis; b- minor axis.

The surface of the earth's ellipsoid in geodesy is taken as a reference surface, determining the heights of points on the surface of the studied figure of the earth relative to it.

The shape and dimensions of the earth's ellipsoid are characterized by the major and minor semi-axes and b, and more often by semimajor axis and polar compression

(1.1)

or semimajor axis and eccentricity of the meridian ellipse:

(1.2)

The ellipsoid that has the greatest proximity to the figure of the Earth as a whole is called common earth ellipsoid .

The parameters of the general earth's ellipsoid are determined under the conditions:

1) the center of the ellipsoid must coincide with the center of mass of the Earth, and its minor axis with the axis of rotation of the Earth;

2) the volume of the ellipsoid must be equal to the volume of the geoid (quasi-geoid);

3) the sum of squared deviations in height of the ellipsoid surface from the geoid (quasi-geoid) surface should be minimal.

The parameters of the earth's ellipsoid can be obtained using the so-called degree measurements, which consists in laying out triangulation series in the directions of meridians and parallels at different latitudes with the determination of astronomical latitudes, longitudes and azimuths of the sides at the end points, as well as based on the results of satellite observations.

For a century and a half, scientists from different countries have been determining the parameters of the earth's ellipsoid, using the results of degree measurements available to them. The result of these definitions is the appearance of a number of ellipsoids.

In every country, the ellipsoid that is accepted as a worker is in the best possible way suitable for its territory. In accordance with this criterion, its orientation on the body of the Earth is also performed, i.e. determining the coordinates of the starting point. Such working ellipsoids used in different countries, are called reference - ellipsoids. In the USSR and a number of countries in Eastern Europe, a reference was adopted - Krasovsky’s ellipsoid, 1940. Krasovsky's ellipsoid is the most accurate of all ellipsoids obtained from processing ground-based measurements. Its dimensions are close to the dimensions of the OSE found from satellite observation data.

5. Main sections of higher geodesy; connection of the discipline with other sciences

Higher geodesy is a vast field of knowledge. It consists of a number of large sections, some of which, when examined in detail, are independent disciplines. Let us list the main sections of higher geodesy.

1.Basic geodetic work. This section discusses methods for accurately determining the relative position of points on the earth's surface by performing high-precision angular, linear and leveling measurements (triangulation, polygonometry and leveling); the main coordinate line relative to which these measurements are made is a plumb line.

2. Geodetic gravimetry: examines methods for measuring the acceleration of gravity at points on the earth's surface, as well as methods for taking into account the inhomogeneity of the gravitational field in the results of geodetic measurements.

3. Geodetic astronomy: examines methods for determining latitudes, longitudes and azimuths from observations of celestial bodies.

4. Space or satellite geodesy: solves the same problems as higher geodesy, but with the help of observations of artificial satellites Earth.

5. Spheroidal geodesy: examines methods for solving geodetic problems on the surface of the earth's ellipsoid.

6. Theoretical geodesy: is developing theories and methods for solving the main scientific problem of geodesy - determining the figure and external gravitational field of the Earth - and their changes over time.

In its research, higher geodesy widely uses the latest achievements of physics, mathematics, and astronomy. When developing high-precision measuring equipment - applied optics, precision instrumentation, laser technology, etc. When mathematically processing measurement results, probability theory is used, mathematical statistics, way least squares. All calculations are performed on the latest computers. To solve scientific geodynamic problems, a close relationship between higher geodesy and geology, geotectonics, geophysics, seismology, etc. is necessary.

6.Basic coordinate systems used in higher geodesy. Concept of geodetic and astronomical coordinates and azimuths

In higher geodesy the following coordinate systems are used:

1. Geodetic coordinate system.

2. System of rectangular spatial coordinates.

3. System of plane rectangular coordinates.

4. System of rectangular rectilinear coordinates X, y, related to the plane of the meridian of a given point.

5. Geocentric coordinate system.

6. Coordinate system with reduced latitude and geodetic longitude.

7. System of rectangular spheroidal coordinates.

In the practice of geodetic work, the first three of the listed coordinate systems are most often used, which we will consider in more detail.

E′

Rice. 2.1. Geodetic coordinates IN, L, N points on the earth's surface M.

PE 0 P" -

РmР" - the plane of the local geodetic meridian (drawn through the point M(m) of the terrain).

Мmn is the normal to the ellipsoid, lowered from point M.

Geodetic latitude points M(m) is called acute angle IN between the equatorial plane E and normal (Mmn) to the surface of the ellipsoid at a given point.

Geodetic latitude varies from 0 0 to 90 0. It has a positive sign in the northern hemisphere and a negative sign in the southern hemisphere.

Geodetic longitudeL points M(m) called dihedral angle Рm E 0 between plane PE 0 P" Greenwich (prime) meridian and plane PmP" local geodetic meridian point M(m). Longitudes are measured from the prime meridian and vary from 0 0 to 360 0. In Russia and Belarus from west to east, in some countries it’s the other way around.

Geodetic height points M area is called distance mm this point from the surface of the reference ellipsoid, measured along the normal.

Points lying above the surface of the ellipsoid have positive heights, below – negative.

Geodetic coordinates cannot be directly measured.

Astronomical coordinates characterized by astronomical latitude and astronomical longitude l.

Rice. 2.2. Astronomical coordinates and l points on the earth's surface M.

EE 0 - plane of the earth's equator;

PE 0 E" - plane of Greenwich or prime meridian;

R 1 m P 1 " - the plane of the local astronomical meridian.

M mg is a plumb line passing through point M.

Astronomical latitude points M(m) is called an acute angle between the plane of the earth's equator E and a plumb line Mmg at this point.

Astronomical latitude varies from 0 0 to 90 0. It has a positive sign in the northern hemisphere and a negative sign in the southern hemisphere.

Astronomical longitude points M(m) called the dihedral angle between the plane PE 0 P" Greenwich (zero) meridian and the plane of the astronomical meridian of a given point. Below plane astronomical meridian points understand a plane passing through a plumb line ( Mmg) at a given point and a straight line parallel to the Earth’s rotation axis (in the general case, the plane of the astronomical meridian does not pass through the Earth’s poles).

And named after F.N. Krasovsky.

In any case, the geodetic coordinate system Pulkovo-1942 (SK-42), SK-63, used in Russia and some other countries, as well as the Afgooye and Hanoi 1972 coordinate systems are based on it.

SK-42, by resolution of the Council of Ministers No. 760, was introduced in 1946 to carry out work throughout the entire territory of the USSR. From July 1, 2002, in accordance with the Decree of the Government of the Russian Federation of July 28, 2000 No. 568, new system SK-95, also based on Krasovsky's ellipsoid.

Dimensions of the earth's ellipsoid according to Krasovsky

Links

Morozov V.P. Course of spheroidal geodesy. Ed. 2, revised and additional M., Nedra, 1979, 296 p.

Wikimedia Foundation. 2010.

See what "Krasovsky's ellipsoid" is in other dictionaries:

Earth ellipsoid, introduced in the USA in 1910. Named after the American surveyor John Hayford (1868 1925). The Hayford ellipsoid is also known as the "International ellipsoid 1924" after... Wikipedia

The Earth's ellipsoid, determined from degree measurements in 1940 under the leadership of F. N. Krasovsky. Dimensions of the reference ellipsoid: semimajor axis (equator radius) 6,378,245 m, polar compression 1: 298.3 ... Big Encyclopedic Dictionary

KRASOVSKY ELLIPSOID, earth's ellipsoid, determined from degree measurements in 1940 under the leadership of F. N. Krasovsky. Dimensions of the reference ellipsoid: semimajor axis (equator radius) 6,378,245 m, polar compression 1: 298.3 ... Encyclopedic Dictionary

Krasovsky's ellipsoid is the earth's ellipsoid, determined from degree measurements in 1940 by a group led by F. N. Krasovsky. According to other sources, the definition was completed in 1942 by a group led by surveyor A. A. Izotov and ... Wikipedia

Earth's ellipsoid, the dimensions of which were derived in 1940 by the Central Scientific research institute geodesy, aerial photography and cartography by the Soviet geodesist A. A. Izotov, based on research conducted under the general guidance of F. N ...

Earth's ellipsoid, determined from degree measurements in 1940 at hand. F.N. Krasovsky. Dimensions of the reference ellipsoid: semimajor axis (equator radius) 6378245 m, polar compression 1: 298.3 ... Natural science. Encyclopedic Dictionary

The earth's ellipsoid, determined from degree measurements in 1940 under the leadership of F.N. Krasovsky. Dimensions of the reference ellipsoid: semimajor axis (equator radius) 6378245 m, polar compression 1:298.3 ... Encyclopedic Dictionary

An ellipsoid of revolution, the dimensions of which are selected subject to the best correspondence to the figure of a quasi-geoid for the Earth as a whole (general terrestrial ellipsoid) or its individual parts (reference ellipsoid). Contents 1 Parameters of the earth's ellipsoid 2 ... Wikipedia

Reference ellipsoid is an approximation of the shape of the Earth's surface (or rather, the geoid) by an ellipsoid of rotation, used for the needs of geodesy on a certain area of the earth's surface (the territory of a single country or several countries). In Russia (in the USSR since ... ... Wikipedia

An ellipsoid of rotation that best represents the figure of the Geoid, i.e. the figure of the Earth as a whole. For the best representation of the geoid within the entire Earth, a general geoid is usually introduced. and define it so that: 1) its volume is equal to the volume... Great Soviet Encyclopedia

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

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

There are also other parameters of the ellipsoid:

first eccentricity, $e=\sqrt(\frac(a^2-b^2)(a^2))=\frac(\sqrt(a^2-b^2))(a)$ ;
second eccentricity, $e"=\sqrt(\frac(a^2-b^2)(b^2))=\frac(\sqrt(a^2-b^2))(b)$ .

For the practical implementation of the earth's ellipsoid it is necessary orient in the body of the Earth. At the same time, it extends general condition: orientation must be performed in such a way that the differences between astronomical and geodetic coordinates are minimal.

Reference ellipsoid

The reference ellipsoid figure is best suited for the territory of a single country or several countries. As a rule, reference ellipsoids are adopted for processing geodetic measurements legislatively. In Russia/USSR, Krasovsky's ellipsoid has been used since 1946.

The orientation of the reference ellipsoid in the body of the Earth is subject to the following requirements:

Semi-minor axis of the ellipsoid ( b) must be parallel to the Earth's rotation axis.
The surface of the ellipsoid should be as close as possible to the surface of the geoid within a given region.

To fix the reference ellipsoid in the body of the Earth, it is necessary to set geodetic coordinates B 0, L 0, H0 starting point of the geodetic network and initial azimuth A 0 to the next point. The set of these quantities is called original geodetic dates.

Basic reference ellipsoids and their parameters

Scientist	Year	Country	a, m	1/f
Delambre	1800	France	6 375 653	334,0
Delambre	1810	France	6 376 985	308,6465
Walbeck	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 (before 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	USSR	6 378 210	298,6
Krasovsky	1942	USSR, Soviet republics, east Euro, 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
PZ-90	1990	Russia	6 378 136	298.258

Earth ellipsoid

The general terrestrial ellipsoid must be oriented in the body of the Earth according to the following requirements:

The semi-minor axis must coincide with the Earth's rotation axis.
The center of the ellipsoid must coincide with the center of mass of the Earth.
Heights of the geoid above the ellipsoid h i(the so-called height anomalies) must obey the least squares condition: $\sum_(n=0)^\infty h_i^2 = \min$ .

When orienting a general terrestrial ellipsoid in the body of the Earth (as opposed to a reference ellipsoid), there is no need to enter the original geodetic dates.

Since the requirements for universal ellipsoids in practice are met with certain tolerances, and the latter (3) is impossible to fully implement, geodesy and related sciences can use various implementations of the ellipsoid, the parameters of which are very close, but do not coincide (see below).

Modern global ellipsoids and their parameters

Name	Year	Country/organization	a, m	accuracy m a, m	1/f	accuracy m f	Note
GRS80	1980	IUGG	6 378 137	± 2	298,257 222 101	±0.001	(English) Geodetic Reference System 1980) developed by the International Geodetic and Geophysical Union (eng. International Union of Geodesy and Geophysics ) and recommended for geodetic work
WGS84	1984	USA	6 378 137	± 2	298,257 223 563	±0.001	(English) World Geodetic System 1984) is used in the GPS satellite navigation system
PZ-90	1990	USSR	6 378 136	± 1	298,257 839 303	±0.001	(Earth Parameters 1990) is used in Russia for geodetic support of orbital flights. This ellipsoid is used in the GLONASS satellite navigation system
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

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An excerpt characterizing the Earth's ellipsoid

“Well, in exactly the same way she shuddered, in the same way she came up and smiled timidly then, when it was already happening,” thought Natasha, “and in the same way... I thought that something was missing in her.”
- No, this is the choir from the Water-bearer, do you hear! – And Natasha finished singing the choir’s tune to make it clear to Sonya.
-Where did you go? – Natasha asked.
- Change the water in the glass. I'll finish the pattern now.
“You’re always busy, but I can’t do it,” said Natasha. -Where is Nikolai?
- He seems to be sleeping.
“Sonya, go wake him up,” said Natasha. - Tell him that I call him to sing. “She sat and thought about what it meant, that it all happened, and, without resolving this question and not at all regretting it, again in her imagination she was transported to the time when she was with him, and he looked with loving eyes looked at her.
“Oh, I wish he would come soon. I'm so afraid that this won't happen! And most importantly: I'm getting old, that's what! There will no longer be what is in me now. Or maybe he’ll come today, he’ll come now. Maybe he came and is sitting there in the living room. Maybe he arrived yesterday and I forgot.” She stood up, put down the guitar and went into the living room. All the household, teachers, governesses and guests were already sitting at the tea table. People stood around the table, but Prince Andrei was not there, and life was still the same.
“Oh, here she is,” said Ilya Andreich, seeing Natasha enter. - Well, sit down with me. “But Natasha stopped next to her mother, looking around, as if she was looking for something.
- Mother! - she said. “Give it to me, give it to me, mom, quickly, quickly,” and again she could hardly hold back her sobs.
She sat down at the table and listened to the conversations of the elders and Nikolai, who also came to the table. “Oh my God, my God, the same faces, the same conversations, dad holding the cup in the same way and blowing in the same way!” thought Natasha, feeling with horror the disgust rising in her against everyone at home because they were still the same.
After tea, Nikolai, Sonya and Natasha went to the sofa, to their favorite corner, where their most intimate conversations always began.

“It happens to you,” Natasha said to her brother when they sat down in the sofa, “it happens to you that it seems to you that nothing will happen - nothing; what was all that was good? And not just boring, but sad?
- Yes! - he said. “It happened to me that everything was fine, everyone was cheerful, but it would come to my mind that I was already tired of all this and that everyone needed to die.” Once I didn’t go to the regiment for a walk, but there was music playing there... and so I suddenly became bored...
- Oh, I know that. I know, I know,” Natasha picked up. – I was still little, this happened to me. Do you remember, once I was punished for plums and you all danced, and I sat in the classroom and sobbed, I will never forget: I was sad and I felt sorry for everyone, and for myself, and I felt sorry for everyone. And, most importantly, it wasn’t my fault,” Natasha said, “do you remember?
“I remember,” said Nikolai. “I remember that I came to you later and I wanted to console you and, you know, I was ashamed. We were terribly funny. I had a bobblehead toy then and I wanted to give it to you. Do you remember?
“Do you remember,” Natasha said with a thoughtful smile, how long, long ago, we were still very little, an uncle called us into the office, back in the old house, and it was dark - we came and suddenly there was standing there...
“Arap,” Nikolai finished with a joyful smile, “how can I not remember?” Even now I don’t know that it was a blackamoor, or we saw it in a dream, or we were told.
- He was gray, remember, and had white teeth - he stood and looked at us...
– Do you remember, Sonya? - Nikolai asked...
“Yes, yes, I remember something too,” Sonya answered timidly...
“I asked my father and mother about this blackamoor,” said Natasha. - They say that there was no blackamoor. But you remember!
- Oh, how I remember his teeth now.
- How strange it is, it was like a dream. I love it.
“Do you remember how we were rolling eggs in the hall and suddenly two old women began to spin around on the carpet?” Was it or not? Do you remember how good it was?
- Yes. Do you remember how dad in a blue fur coat fired a gun on the porch? “They turned over, smiling with pleasure, memories, not sad old ones, but poetic youthful memories, those impressions from the most distant past, where dreams merge with reality, and laughed quietly, rejoicing at something.
Sonya, as always, lagged behind them, although their memories were common.
Sonya did not remember much of what they remembered, and what she did remember did not arouse in her the poetic feeling that they experienced. She only enjoyed their joy, trying to imitate it.
She took part only when they remembered Sonya's first visit. Sonya told how she was afraid of Nikolai, because he had strings on his jacket, and the nanny told her that they would sew her into strings too.
“And I remember: they told me that you were born under cabbage,” said Natasha, “and I remember that I didn’t dare not believe it then, but I knew that it wasn’t true, and I was so embarrassed.”
During this conversation, the maid's head poked out of the back door of the sofa room. “Miss, they brought the rooster,” the girl said in a whisper.
“No need, Polya, tell me to carry it,” said Natasha.
In the middle of the conversations going on in the sofa, Dimmler entered the room and approached the harp that stood in the corner. He took off the cloth and the harp made a false sound.

Planet Earth does not have a regular geometric shape. The figure of the Earth is called the geoid. It is generally accepted that the shape of the Earth is close to an ellipsoid, which is obtained by rotating the ellipse around a minor axis (Fig. 1).

The length of the major semi-axis of the earth's ellipsoid is a = 6,378,245 m, minor b = 6,356,863 m. The difference between the semi-axes is 21.4 km. Attitude

called compression of the Earth. These dimensions of the earth's ellipse were established by Prof. N. F. Krasovsky. By Decree of the Council of Ministers of the USSR No. 760 of April 7, 1946, the dimensions of N. F. Krasovsky’s ellipsoid were adopted for all geodetic, topographic and cartographic work in the USSR.

When solving most problems in navigation, the Earth's compression value, which is 0.3%, is neglected and the Earth is taken to be a sphere, the volume of which is equal to the volume of the Earth's ellipsoid. Based on this convention, i.e. that

and by substituting the values a and 6 into this formula, we determine the radius of such a ball R = 6,371,110 m.

Basic points, lines and circles

The imaginary points PN and PS of intersection of the Earth’s rotation axis with its surface are called Earth's poles : northern(Nordic) and southern(south), while the north pole is considered to be the pole from which the Earth’s rotation is directed counterclockwise.

The great circle EABQ (Fig. 2), which is the trace of the intersection of the surface of the globe by a plane perpendicular to the axis of rotation PNPS and passing through its center 0, is called equator. The plane of the equator divides globe into two hemispheres: northern and southern.

Circles of small circles, for example eabq, e1a1b1q1, which are traces of the intersection of the surface of the globe by planes, parallel to the plane equator are called parallels.

Circles large circles, for example, PN aAa1PS and PNbBb1PS, which are traces of the intersection of the surface of the globe by planes passing through the axis of rotation of the Earth (meridial planes), are called meridians.

An unlimited number of parallels and meridians can be drawn, but through one point only one parallel and one meridian can be drawn, which are called, respectively, the parallel of a given point or place and the meridian of a given point or place.

Rice. 2

According to international agreement, it is generally accepted zero or prime meridian meridian passing through the astronomical observatory at Greenwich (near London). He and his opposite divide the globe into two hemispheres: eastern and western.

Parameters of the earth's ellipsoid