Figure of the Earth

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The expression figure of the Earth has various meanings in geodesy according to the way it is used and the precision with which the Earth's size and shape is to be defined. The actual topographic surface is most apparent with its variety of land forms and water areas. This is, in fact, the surface on which actual Earth measurements are made. It is not suitable, however, for exact mathematical computations because the formulas which would be required to take the irregularities into account would necessitate a prohibitive amount of computations. The topographic surface is generally the concern of topographers and hydrographers.

The Pythagorean concept of a spherical Earth offers a simple surface which is mathematically easy to deal with. Many astronomical and navigational computations use it as a surface representing the Earth. While the sphere is a close approximation of the true figure of the Earth and satisfactory for many purposes, to the geodesists interested in the measurement of long distances—spanning continents and oceans—a more exact figure is necessary. Closer approximations range from modelling the shape of the entire Earth as an oblate spheroid or an oblate ellipsoid to the use of spherical harmonics or local approximations in terms of local reference ellipsoids. The idea of a planar or flat surface for Earth, however, is still acceptable for surveys of small areas as local topography is more important than the curvature. Plane-table surveys are made for relatively small areas and no account is taken of the curvature of the Earth. A survey of a city would likely be computed as though the Earth were a plane surface the size of the city. For such small areas, exact positions can be determined relative to each other without considering the size and shape of the total Earth.

In the mid- to late- 20th century, research across the geosciences contributed to drastic improvements in the accuracy of the Figure of the Earth. The primary utility (and the motivation for funding, mainly from the military) of this improved accuracy was to provide geographical and gravitational data for the inertial guidance systems of ballistic missiles. This funding also drove the expansion of geoscientific disciplines, fostering the creation and growth of various geoscience departments at many universities. [1]

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[edit] Ellipsoid of revolution

Since the Earth is in fact flattened slightly at the poles and bulges somewhat at the equator, the geometrical figure used in geodesy to most nearly approximate the shape of the Earth is an ellipsoid of revolution. The ellipsoid of revolution is the figure which would be obtained by rotating an ellipse about its shorter axis. An ellipsoid of revolution describing the figure of the Earth is called a reference ellipsoid.

An ellipsoid of revolution is uniquely defined by specifying two dimensions. Geodesists, by convention, use the semimajor axis and flattening. The size is represented by the radius at the equator—the semimajor axis—and designated by the letter a. The shape of the ellipsoid is given by the flattening, f, which indicates how closely the ellipsoid approaches a spherical shape. The difference between the reference ellipsoid representing the Earth and a sphere is very small, only one part in 300 approximately.

For such a flattened ellipsoid, the polar radius of curvature is larger than the equatorial

(a2 / b),

even though the Earth's surface is closer to the Earth's centre at the poles than at the equator. Conversely, the equator's vertical radius of curvature is smaller than the polar

(b2 / a).

This circumstance has formed the basis for attempts to determine the flattening of the mean Earth ellipsoid by so-called grade measurements.

[edit] Historical Earth ellipsoids

The reference ellipsoid models listed below have had utility in geodetic work and many are still in use. The older ellipsoids are named for the individual who derived them and the year of development is given. In 1887 the English mathematician Col Alexander Ross Clarke CB FRS RE was awarded the Gold Medal of the Royal Society for his work in determining the figure of the Earth. The international ellipsoid was developed by John Fillmore Hayford in 1910 and adopted by the International Union of Geodesy and Geophysics (IUGG) in 1924, which recommended it for international use.

At the 1967 meeting of the IUGG held in Lucerne, Switzerland, the ellipsoid called GRS-67 (Geodetic Reference System 1967) in the listing was recommended for adoption. The new ellipsoid was not recommended to replace the International Ellipsoid (1924), but was advocated for use where a greater degree of accuracy is required. It became a part of the GRS-67 which was approved and adopted at the 1971 meeting of the IUGG held in Moscow. It is used in Australia for the Australian Geodetic Datum and in South America for the South American Datum 1969.

Reference ellipsoid name Equatorial radius (m) Polar radius (m) Inverse flattening Where used
Modified Everest (Malaya) Revised Kertau 6,377,304.063 6,356,103.038993 300.801699969
Timbalai 6,377,298.56 6,356,097.55 300.801639166
Everest Spheroid 6,377,301.243 6,356,100.228 300.801694993
Maupertuis (1738) 6,397,300 6,363,806.283 191 France
Everest (1830) 6,377,276.345 6,356,075.413 300.801697979 India
Airy (1830) 6,377,563.396 6,356,256.909 299.3249646 Britain
Bessel (1841) 6,377,397.155 6,356,078.963 299.1528128 Europe, Japan
Clarke (1866) 6,378,206.4 6,356,583.8 294.9786982 North America
Clarke (1880) 6,378,249.145 6,356,514.870 293.465 France, Africa
Helmert (1906) 6,378,200 6,356,818.17 298.3
Hayford (1910) 6,378,388 6,356,911.946 297 USA
International (1924) 6,378,388 6,356,911.946 297 Europe
NAD 27 6,378,206.4 6,356,583.800 294.978698208 North America
Krassovsky (1940) 6,378,245 6,356,863.019 298.3 Russia
WGS66 (1966) 6,378,145 6,356,759.769 298.25 USA/DoD
Australian National (1966) 6,378,160 6,356,774.719 298.25 Australia
New International (1967) 6,378,157.5 6,356,772.2 298.24961539
GRS-67 (1967) 6,378,160 6,356,774.516 298.247167427
South American (1969) 6,378,160 6,356,774.719 298.25 South America
WGS-72 (1972) 6,378,135 6,356,750.52 298.26 USA/DoD
GRS-80 (1979) 6,378,137 6,356,752.3141 298.257222101
NAD 83 6,378,137 6,356,752.3 298.257024899 North America
WGS-84 (1984) 6,378,137 6,356,752.3142 298.257223563 Global GPS
IERS (1989) 6,378,136 6,356,751.302 298.257
IERS (2003)[2] 6,378,136.6 6,356,751.9 298.25642 Global ITRS

The GRS-80 (Geodetic Reference System 1980) as approved and adopted by the IUGG at its Canberra, Australia meeting of 1979 is originally defined based on the equatorial radius (semi-major axis of Earth ellipsoid) a, total mass GM, dynamic form factor J2 and angular velocity of rotation ω, making the inverse flattening 1 / f a derived quantity. The minute difference in 1 / f seen between GRS-80 and WGS-84 was produced by inaccurate numerical evaluation from the defining constants...

Some of the above ellipsoid models are actually geodetic datums: e.g., while GRS-80 defines only the geometric shape of its ellipsoid and a normal gravity field formula to go with it, WGS-84 defines a complete geodetic reference system realized in the terrain. Similarly, the older ED-50 (European Datum 1950) is based on the Hayford or International Ellipsoid.

Note that the same ellipsoid may be known by different names, and it is best to mention the values for full identification.

[edit] More complicated figures

The possibility that the Earth's equator is an ellipse rather than a circle and therefore that the ellipsoid is triaxial has been a matter of scientific controversy for many years. Modern technological developments have furnished new and rapid methods for data collection and since the launch of Sputnik 1, orbital data have been used to investigate the theory of ellipticity.

A second theory, more complicated than triaxiality, proposed that observed long periodic orbital variations of the first Earth satellites indicate an additional depression at the south pole accompanied by a bulge of the same degree at the north pole. It is also contended that the northern middle latitudes were slightly flattened and the southern middle latitudes bulged in a similar amount. This concept suggested a slightly pear-shaped Earth and was the subject of much public discussion. Modern geodesy tends to retain the ellipsoid of revolution and treat triaxiality and pear shape as a part of the geoid figure: they are represented by the spherical harmonic coefficients C22,S22 and C30, respectively, corresponding to degree and order numbers 2.2 for the triaxiality and 3.0 for the pear shape.

[edit] Geoid

It was stated earlier that measurements are made on the apparent or topographic surface of the Earth and it has just been explained that computations are performed on an ellipsoid. One other surface is involved in geodetic measurement: the geoid. In geodetic surveying, the computation of the geodetic coordinates of points is commonly performed on a reference ellipsoid closely approximating the size and shape of the Earth in the area of the survey. The actual measurements made on the surface of the Earth with certain instruments are however referred to the geoid. The ellipsoid is a mathematically defined regular surface with specific dimensions. The geoid, on the other hand, coincides with that surface to which the oceans would conform over the entire Earth if free to adjust to the combined effect of the Earth's mass attraction (gravitation) and the centrifugal force of the Earth's rotation. As a result of the uneven distribution of the Earth's mass, the geoidal surface is irregular and, since the ellipsoid is a regular surface, the separations between the two, referred to as geoid undulations, geoid heights, or geoid separations, will be irregular as well.

The geoid is a surface along which the gravity potential is everywhere equal and to which the direction of gravity is always perpendicular. The latter is particularly important because optical instruments containing levelling devices are commonly used to make geodetic measurements. When properly adjusted, the vertical axis of the instrument coincides with the direction of gravity and is, therefore, perpendicular to the geoid. The angle between the plumb line which is perpendicular to the geoid (sometimes called "the vertical") and the perpendicular to the ellipsoid (sometimes called "the ellipsoidal normal") is defined as the deflection of the vertical. It has two components: an east-west and a north-south component.

[edit] Earth rotation and Earth's interior

Determining the exact figure of the Earth is not only a geodetic operation or a task of geometry, but is also related to geophysics. Without any idea of the Earth's interior, we can state a "constant density" of 5.515 g/cm³ and, according to theoretical arguments (see Leonhard Euler, Albert Wangerin, etc.), such a body rotating like the Earth would have an flattening of 1:230.

In fact the measured flattening is 1:298.25, which is more similar to a sphere and a strong argument that the Earth's core is very compact. Therefore the density must be a function of the depth, reaching from about 2.7 g/cm³ at the surface (rock density of granite, limestone etc. — see regional geology) up to approximately 15 within the inner core. Modern seismology yields a value of 16 g/cm³ (iron or hydrogen) at the center of the earth.

[edit] Global and regional gravity field

Another implication to the physical exploration of the Earth's interior is the gravity field which can be measured very exactly at the surface and by satellites. The true vertical does not correspond to the theoretical one (in fact the deflection amounts from 2" to 50") because the topography and all geological masses are slightly disturbing the gravity field. Therefore the gross structure of the earth's crust and mantle can be determined by geodetic-geophysical models of the subsurface.

[edit] Earth's curvature

Since the Earth isn't exactly spherical, its curvature varies with location.

[edit] See also

[edit] External links

[edit] Notes and references

  1. ^ Cloud, John. "Crossing the Olentangy River: The Figure of the Earth and the Military-Industrial-Academic Complex, 1947-1972," Studies in the History and Philosophy of Modern Physics, Vol. 31, No. 3, pp 371-404, 2000.
  2. ^ IERS Conventions (2003) (Chp. 1, page 12)
  • Guy Bomford, Geodesy, Oxford 1962 and 1880.
  • Guy Bomford, Determination of the European geoid by means of vertical deflections. Rpt of Comm. 14, IUGG 10th Gen. Ass., Rome 1954.
  • Karl Ledersteger and Gottfried Gerstbach, Die horizontale Isostasie / Das isostatische Geoid 31. Ordnung. Geowissenschaftliche Mitteilungen Band 5, TU Wien 1975.
  • Helmut Moritz and Bernhard Hofmann, Physical Geodesy. Springer, Wien & New York 2005.