Figure of the Earth
Geodesy |
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In
Motivation
Earth's topographic surface is apparent with its variety of land forms and water areas. This topographic surface is generally the concern of topographers, hydrographers, and geophysicists. While it is the surface on which Earth measurements are made, mathematically modeling it while taking the irregularities into account would be extremely complicated.
The
For surveys of small areas, a planar (flat) model of Earth's surface suffices because the local topography overwhelms the curvature.
By the late 1600s, serious effort was devoted to modeling the Earth as an ellipsoid, beginning with Jean Picard's measurement of a degree of arc along the Paris meridian. Improved maps and better measurement of distances and areas of national territories motivated these early attempts. Surveying instrumentation and techniques improved over the ensuing centuries. Models for the figure of the Earth improved in step.
In the mid- to late 20th century, research across the
Models
The models for the figure of the Earth vary in the way they are used, in their complexity, and in the accuracy with which they represent the size and shape of the Earth.
Sphere
The simplest model for the shape of the entire Earth is a sphere. The Earth's radius is the distance from Earth's center to its surface, about 6,371 km (3,959 mi). While "radius" normally is a characteristic of perfect spheres, the Earth deviates from spherical by only a third of a percent, sufficiently close to treat it as a sphere in many contexts and justifying the term "the radius of the Earth".
The concept of a spherical Earth dates back to around the 6th century BC,[2] but remained a matter of philosophical speculation until the 3rd century BC. The first scientific estimation of the radius of the Earth was given by Eratosthenes about 240 BC, with estimates of the accuracy of Eratosthenes's measurement ranging from −1% to 15%.
The Earth is only approximately spherical, so no single value serves as its natural radius. Distances from points on the surface to the center range from 6,353 km (3,948 mi) to 6,384 km (3,967 mi). Several different ways of modeling the Earth as a sphere each yield a mean radius of 6,371 km (3,959 mi). Regardless of the model, any radius falls between the polar minimum of about 6,357 km (3,950 mi) and the equatorial maximum of about 6,378 km (3,963 mi). The difference 21 km (13 mi) correspond to the polar radius being approximately 0.3% shorter than the equatorial radius.
Ellipsoid of revolution
Since the Earth is
An ellipsoid of revolution is uniquely defined by two quantities. Several conventions for expressing the two quantities are used in geodesy, but they are all equivalent to and convertible with each other:
- Equatorial radius (called semimajor axis), and polar radius (called semiminor axis);
- and eccentricity ;
- and flattening .
Eccentricity and flattening are different ways of expressing how squashed the ellipsoid is. When flattening appears as one of the defining quantities in geodesy, generally it is expressed by its reciprocal. For example, in the WGS 84 spheroid used by today's GPS systems, the reciprocal of the flattening is set to be exactly 298.257223563.
The difference between a sphere and a reference ellipsoid for Earth is small, only about one part in 300. Historically, flattening was computed from
A sphere has a single
because the pole is flattened: the flatter the surface, the larger the sphere must be to approximate it. Conversely, the ellipsoid's north–south radius of curvature at the equator is smaller than the polar
where is the distance from the center of the ellipsoid to the equator (semi-major axis), and is the distance from the center to the pole. (semi-minor axis)
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
The geoid is a surface along which the gravity potential is everywhere equal and to which the direction of gravity is always perpendicular (see
Other shapes
Modern geodesy tends to retain the ellipsoid of revolution as a
Triaxiality (equatorial eccentricity)
The possibility that the Earth's equator is better characterized as an ellipse rather than a circle and therefore that the ellipsoid is triaxial has been a matter of scientific inquiry for many years.[4][5] 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.[3] More recent results indicate a 70 m difference between the two equatorial major and minor axes of inertia, with the larger semidiameter pointing to 15° W longitude (and also 180-degree away).[6][7]
Pear shape
The theory of a slightly pear-shaped Earth arose and gained publicity after the first artificial satellites observed long periodic orbital variations, indicating a depression at the
Local approximations
Simpler local approximations are possible.
Local tangent plane
The
Osculating sphere
The best local spherical approximation to the ellipsoid in the vicinity of a given point is the Earth's
Earth rotation and Earth's interior
Determining the exact figure of the Earth is not only a
Global and regional gravity field
Also with implications for the physical exploration of the Earth's interior is the
See also
- Clairaut's theorem
- EGM96
- Gravity formula
- Gravity of Earth
- Horizon §§ Distance and Curvature
- Meridian arc
- Theoretical gravity
- History
- Pierre Bouguer
- Earth's circumference#History
- Earth's radius#History
- Flat Earth
- Friedrich Robert Helmert
- History of geodesy
- History of the metre
- Meridian arc#History
- Seconds pendulum
References
- .
- ISBN 978-0-8014-0561-7.
- ^ a b c Defense Mapping Agency (1983). Geodesy for the Layman (Report). United States Air Force.
- .
- S2CID 128674427.
- ^ Torge & Müller (2012) Geodesy, De Gruyter, p.100
- ^ Marchenko, A.N. (2009): Current estimation of the Earth’s mechanical and geometrical para meters. In Sideris, M.G., ed. (2009): Observing our changing Earth. IAG Symp. Proceed. 133., pp. 473–481. DOI:10.1007/978-3-540-85426-5_57
- OCLC 70265574.
- OCLC 1154365097.
- ^ O’KEEFE, J. A., ECKEIS, A., & SQUIRES, R. K. (1959). Vanguard Measurements Give Pear-Shaped Component of Earth’s Figure. Science, 129(3348), 565–566. doi:10.1126/science.129.3348.565
- S2CID 4260099.
- ^ King-Hele, D. (1967). The Shape of the Earth. Scientific American, 217(4), 67-80. [1]
- ^ Günter Seeber (2008), Satellite Geodesy, Walter de Gruyter, 608 pages. [2]
- CiteSeerX 10.1.1.594.6212.
- .
- ^
Heine, George (2013). "Euler and the Flattening of the Earth". Math Horizons. 21 (1). Mathematical Association of America: 25–29. S2CID 126412032.
- ISSN 0031-9201
- Attribution
This article incorporates text from this source, which is in the public domain: Defense Mapping Agency (1983). Geodesy for the Layman (Report). United States Air Force.
Further reading
- Guy Bomford, Geodesy, Oxford 1952 and 1980.
- Guy Bomford, Determination of the European geoid by means of IUGG10th 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.
- Geodesy for the Layman, Defense Mapping Agency, St. Louis, 1983.