Figure of the Earth is a
term of art
Jargon is the specialized terminology associated with a particular field or area of activity. Jargon is normally employed in a particular communicative context and may not be well understood outside that context. The context is usually a particu ...
in
geodesy
Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equivale ...
that refers to the size and shape used to model
Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
. The size and shape it refers to depend on context, including the precision needed for the model. A
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
is a well-known historical approximation of the figure of the Earth that is satisfactory for many purposes. Several models with greater accuracy (including
ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a surface that may be defined as the ...
) have been developed so that
coordinate systems
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
can serve the precise needs of
navigation
Navigation is a field of study that focuses on the process of monitoring and controlling the movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navigation, ...
,
surveying
Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. A land surveying professional is ca ...
,
cadastre
A cadastre or cadaster is a comprehensive recording of the real estate or real property's metes and bounds, metes-and-bounds of a country.Jo Henssen, ''Basic Principles of the Main Cadastral Systems in the World,'/ref>
Often it is represented gra ...
,
land use
Land use involves the management and modification of natural environment or wilderness into built environment such as settlements and semi-natural habitats such as arable fields, pastures, and managed woods. Land use by humans has a long h ...
, and various other concerns.
Motivation
Earth's
topographic
Topography is the study of the forms and features of land surfaces. The topography of an area may refer to the land forms and features themselves, or a description or depiction in maps.
Topography is a field of geoscience and planetary scie ...
surface is apparent with its variety of land forms and water areas. This topographic surface is generally the concern of topographers,
hydrographers, and
geophysicists
Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' so ...
. 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
Pythagorean
Pythagorean, meaning of or pertaining to the ancient Ionian mathematician, philosopher, and music theorist Pythagoras, may refer to:
Philosophy
* Pythagoreanism, the esoteric and metaphysical beliefs purported to have been held by Pythagoras
* Ne ...
concept of a
spherical Earth
Spherical Earth or Earth's curvature refers to the approximation of figure of the Earth as a sphere.
The earliest documented mention of the concept dates from around the 5th century BC, when it appears in the writings of Greek philosophers. I ...
offers a simple surface that is easy to deal with mathematically. Many astronomical and navigational computations use a
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
to model the Earth as a close approximation. However, a more accurate figure is needed for measuring distances and areas on the scale beyond the purely local. Better approximations can be made by modeling the entire surface as an
oblate spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circ ...
, using
spherical harmonic
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Since the spherical harmonics form a ...
s to approximate the
geoid
The geoid () is the shape that the ocean surface would take under the influence of the gravity of Earth, including gravitational attraction and Earth's rotation, if other influences such as winds and tides were absent. This surface is extended ...
, or modeling a region with a best-fit
reference ellipsoid
An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximations ...
.
For surveys of small areas, a planar (flat) model of Earth's surface suffices because the local topography overwhelms the curvature.
Plane-table surveys are made for relatively small areas without considering the size and shape of the entire Earth. A survey of a city, for example, might be conducted this way.
![Earth2014shape SouthAmerica small](https://upload.wikimedia.org/wikipedia/commons/6/69/Earth2014shape_SouthAmerica_small.jpg)
By the late 1600s, serious effort was devoted to modeling the Earth as an ellipsoid, beginning with
Jean Picard
Jean Picard (21 July 1620 – 12 July 1682) was a French astronomer and priest born in La Flèche, where he studied at the Jesuit Collège Royal Henry-Le-Grand.
He is principally notable for his accurate measure of the size of the Earth, base ...
's measurement of a degree of arc along the
Paris meridian
The Paris meridian is a meridian line running through the Paris Observatory in Paris, France – now longitude 2°20′14.02500″ East. It was a long-standing rival to the Greenwich meridian as the prime meridian of the world. The "Paris merid ...
. 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
geosciences
Earth science or geoscience includes all fields of natural science related to the planet Earth. This is a branch of science dealing with the physical, chemical, and biological complex constitutions and synergistic linkages of Earth's four spheres ...
contributed to drastic improvements in the accuracy of the figure of the Earth. The primary utility of this improved accuracy was to provide geographical and gravitational data for the
inertial guidance system
An inertial navigation system (INS) is a navigation device that uses motion sensors (accelerometers), rotation sensors ( gyroscopes) and a computer to continuously calculate by dead reckoning the position, the orientation, and the velocity (dire ...
s of
ballistic missile
A ballistic missile is a type of missile that uses projectile motion to deliver warheads on a target. These weapons are guided only during relatively brief periods—most of the flight is unpowered. Short-range ballistic missiles stay within the ...
s. This funding also drove the expansion of geoscientific disciplines, fostering the creation and growth of various geoscience departments at many universities. These developments benefited many civilian pursuits as well, such as weather and communication
satellite
A satellite or artificial satellite is an object intentionally placed into orbit in outer space. Except for passive satellites, most satellites have an electricity generation system for equipment on board, such as solar panels or radioisotope ...
control and
GPS
The Global Positioning System (GPS), originally Navstar GPS, is a Radionavigation-satellite service, satellite-based radionavigation system owned by the United States government and operated by the United States Space Force. It is one of t ...
location-finding, which would be impossible without highly accurate models for the figure of the Earth.
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
![Horizon, Valencia (Spain)](https://upload.wikimedia.org/wikipedia/commons/1/13/Horizon%2C_Valencia_%28Spain%29.JPG)
The simplest model for the shape of the entire Earth is a sphere. The Earth's
radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
is the
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
from Earth's center to its surface, about . 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, but remained a matter of philosophical speculation until the
3rd century BC
In the Mediterranean Basin, the first few decades of this century were characterized by a balance of power between the Greek Hellenistic kingdoms in the east, and the great mercantile power of Carthage in the west. This balance was shattered ...
. The first scientific estimation of the radius of the Earth was given by
Eratosthenes
Eratosthenes of Cyrene (; grc-gre, Ἐρατοσθένης ; – ) was a Greek polymath: a mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandria ...
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 to . Several different ways of modeling the Earth as a sphere each yield a mean radius of . Regardless of the model, any radius falls between the polar minimum of about and the equatorial maximum of about . The difference correspond to the polar radius being approximately 0.3% shorter than the equatorial radius.
Ellipsoid of revolution
![Earth oblateness to scale](https://upload.wikimedia.org/wikipedia/commons/f/f9/Earth_oblateness_to_scale.svg)
Since the Earth is
flattened at the poles and
bulges at the
Equator
The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can als ...
,
geodesy
Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equivale ...
represents the figure of the Earth as an oblate
spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has cir ...
. The oblate spheroid, or
oblate ellipsoid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circu ...
, is an
ellipsoid of revolution
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has c ...
obtained by rotating an ellipse about its shorter axis. It is the regular geometric shape that most nearly approximates the shape of the Earth. A spheroid describing the figure of the Earth or other
celestial body
An astronomical object, celestial object, stellar object or heavenly body is a naturally occurring physical object, physical entity, association, or structure that exists in the observable universe. In astronomy, the terms ''object'' and ''bod ...
is called a
reference ellipsoid
An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximations ...
. The reference ellipsoid for Earth is called an
Earth ellipsoid
An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximations ...
.
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
Eccentricity or eccentric may refer to:
* Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal"
Mathematics, science and technology Mathematics
* Off-center, in geometry
* Eccentricity (graph theory) of a v ...
;
*
and
flattening
Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening is ...
.
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
The World Geodetic System (WGS) is a standard used in cartography, geodesy, and satellite navigation including GPS. The current version, WGS 84, defines an Earth-centered, Earth-fixed coordinate system and a geodetic datum, and also descri ...
spheroid used by today's GPS systems, the reciprocal of the flattening
is set to be exactly .
The difference between a sphere and a reference ellipsoid for Earth is small, only about one part in 300. Historically, flattening was computed from
grade measurement
Arc measurement, sometimes degree measurement (german: Gradmessung), is the astrogeodetic technique of determining of the radius of Earth – more specifically, the local Earth radius of curvature of the figure of the Earth – by relating th ...
s. Nowadays, geodetic networks and
satellite geodesy
Satellite geodesy is geodesy by means of artificial satellites—the measurement of the form and dimensions of Earth, the location of objects on its surface and the figure of the Earth's gravity field by means of artificial satellite techniques. ...
are used. In practice, many reference ellipsoids have been developed over the centuries from different surveys. The flattening value varies slightly from one reference ellipsoid to another, reflecting local conditions and whether the reference ellipsoid is intended to model the entire Earth or only some portion of it.
A sphere has a single
radius of curvature
In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius o ...
, which is simply the radius of the sphere. More complex surfaces have radii of curvature that vary over the surface. The radius of curvature describes the radius of the sphere that best approximates the surface at that point. Oblate ellipsoids have constant radius of curvature east to west along
parallels, if a
graticule
Graticule may refer to:
* An oscilloscope graticule scale
* The reticle pattern in an optical instrument
* Graticule (cartography), a grid of lines on a map
See also
* Grid (disambiguation)
Grid, The Grid, or GRID may refer to:
Common us ...
is drawn on the surface, but varying curvature in any other direction. For an oblate ellipsoid, the polar radius of curvature
is larger than the equatorial
:
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
![Geoid undulation to scale](https://upload.wikimedia.org/wikipedia/commons/4/48/Geoid_undulation_to_scale.jpg)
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
The geoid () is the shape that the ocean surface would take under the influence of the gravity of Earth, including gravitational attraction and Earth's rotation, if other influences such as winds and tides were absent. This surface is extended ...
. In geodetic surveying, the computation of the
geodetic coordinates
Geodetic coordinates are a type of curvilinear orthogonal coordinate system used in geodesy based on a ''reference ellipsoid''.
They include geodetic latitude (north/south) , ''longitude'' (east/west) , and ellipsoidal height (also known as geo ...
of points is commonly performed on a
reference ellipsoid
An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximations ...
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
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...
) and the centrifugal force of the
Earth's rotation
Earth's rotation or Earth's spin is the rotation of planet Earth around its own Rotation around a fixed axis, axis, as well as changes in the orientation (geometry), orientation of the rotation axis in space. Earth rotates eastward, in retrograd ...
. 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 undulation
The geoid () is the shape that the ocean surface would take under the influence of the gravity of Earth, including gravitational attraction and Earth's rotation, if other influences such as winds and tides were absent. This surface is extended ...
s, 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 (see
equipotential surface
In mathematics and physics, an equipotential or isopotential refers to a region in space where every point is at the same potential. This usually refers to a scalar potential (in that case it is a level set of the potential), although it can ...
). The latter is particularly important because optical instruments containing gravity-reference leveling 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
A plumb bob, plumb bob level, or plummet, is a weight, usually with a pointed tip on the bottom, suspended from a string and used as a vertical reference line, or plumb-line. It is a precursor to the spirit level and used to establish a vertic ...
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
The vertical deflection (VD) or deflection of the vertical (DoV), also known as deflection of the plumb line and astro-geodetic deflection, is a measure of how far the gravity direction at a given point of interest is rotated by local mass anoma ...
. It has two components: an east–west and a north–south component.
[
]
Other shapes
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. Modern technological developments have furnished new and rapid methods for data collection and, since the launch of Sputnik 1
Sputnik 1 (; see § Etymology) was the first artificial Earth satellite. It was launched into an elliptical low Earth orbit by the Soviet Union on 4 October 1957 as part of the Soviet space program. It sent a radio signal back to Earth for t ...
, orbital data have been used to investigate the theory of ellipticity. 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).
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 South Pole
The South Pole, also known as the Geographic South Pole, Terrestrial South Pole or 90th Parallel South, is one of the two points where Earth's axis of rotation intersects its surface. It is the southernmost point on Earth and lies antipod ...
and a bulge of the same degree at the North Pole
The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is the point in the Northern Hemisphere where the Earth's axis of rotation meets its surface. It is called the True North Pole to distinguish from the Mag ...
. This theory contends that the northern middle latitudes are slightly flattened and the southern middle latitudes correspondingly bulged. U.S. Vanguard 1 satellite data from 1958 confirms that the southern equatorial bulge is greater than that of the north, which is corroborated by the South Pole's sea level
Mean sea level (MSL, often shortened to sea level) is an average surface level of one or more among Earth's coastal bodies of water from which heights such as elevation may be measured. The global MSL is a type of vertical datuma standardised g ...
being lower than that of the north. A pear-shaped Earth had first been theorized in 1498 by Christopher Columbus
Christopher Columbus
* lij, Cristoffa C(or)ombo
* es, link=no, Cristóbal Colón
* pt, Cristóvão Colombo
* ca, Cristòfor (or )
* la, Christophorus Columbus. (; born between 25 August and 31 October 1451, died 20 May 1506) was a ...
, based on his incorrect readings of the North Star
Polaris is a star in the northern circumpolar constellation of Ursa Minor. It is designated α Ursae Minoris ( Latinized to ''Alpha Ursae Minoris'') and is commonly called the North Star or Pole Star. With an apparent magnitude that ...
's diurnal motion
Diurnal motion (, ) is an astronomical term referring to the apparent motion of celestial objects (e.g. the Sun and stars) around Earth, or more precisely around the two celestial poles, over the course of one day. It is caused by Earth's rotatio ...
.
John A. O'Keefe and co-authors are credited with the discovery that the Earth had a significant third degree zonal spherical harmonic In the mathematical study of rotational symmetry, the zonal spherical harmonics are special spherical harmonics that are invariant under the rotation through a particular fixed axis. The zonal spherical functions are a broad extension of the notion ...
in its gravitational field
In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
using Vanguard 1 satellite data. Based on further satellite geodesy
Satellite geodesy is geodesy by means of artificial satellites—the measurement of the form and dimensions of Earth, the location of objects on its surface and the figure of the Earth's gravity field by means of artificial satellite techniques. ...
data, Desmond King-Hele
Desmond George King-Hele FRS (3 November 1927 at Seaford in Sussex – 25 December 2019) was a British physicist, poet and author who crossed the divide between the arts and science to write extensively about the life of Erasmus Darwin, whom he ...
refined the estimate to a 45-m difference between north and south polar radii, owing to a 19-m "stem" rising in the North Pole and a 26-m depression in the South Pole. The polar asymmetry is small, though: it is about a thousand times smaller than the earth's flattening and even smaller than the geoidal undulation
The geoid () is the shape that the ocean surface would take under the influence of the gravity of Earth, including gravitational attraction and Earth's rotation, if other influences such as winds and tides were absent. This surface is exten ...
in some regions of the Earth.
Modern geodesy tends to retain the ellipsoid of revolution as a reference ellipsoid
An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximations ...
and treat triaxiality and pear shape as a part of the geoid
The geoid () is the shape that the ocean surface would take under the influence of the gravity of Earth, including gravitational attraction and Earth's rotation, if other influences such as winds and tides were absent. This surface is extended ...
figure: they are represented by the spherical harmonic coefficients and , respectively, corresponding to degree and order numbers 2.2 for the triaxiality and 3.0 for the pear shape.
Local approximations
Simpler local approximations are possible.
Local tangent plane
The local tangent plane
Local tangent plane coordinates (LTP), also known as local ellipsoidal system, local geodetic coordinate system, or local vertical, local horizontal coordinates (LVLH), are a spatial reference system based on the tangent plane defined by the local ...
is appropriate for very small distances.
Osculating sphere
The best local spherical approximation to the ellipsoid in the vicinity of a given point is the ''Earth's osculating sphere''. Its radius equals Earth's Gaussian radius of curvature, and its radial direction coincides with the geodetic normal
The ''n''-vector representation (also called geodetic normal or ellipsoid normal vector) is a three-parameter non-singular representation well-suited for replacing geodetic coordinates (latitude and longitude) for horizontal position representat ...
direction. The center of the osculating sphere is offset from the center of the ellipsoid, but is at the center of curvature
In geometry, the center of curvature of a curve is found at a point that is at a distance from the curve equal to the radius of curvature lying on the normal vector. It is the point at infinity if the curvature is zero. The osculating circl ...
for the given point on the ellipsoid surface. This concept aids the interpretation of terrestrial and planetary radio occultation
Radio occultation (RO) is a remote sensing technique used for measuring the physical properties of a planetary atmosphere or ring system.
Atmospheric radio occultation
Atmospheric radio occultation relies on the detection of a change in a radio ...
refraction
In physics, refraction is the redirection of a wave as it passes from one medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commonly observed phenomeno ...
measurements and in some navigation and surveillance applications.
Earth rotation and Earth's interior
Determining the exact figure of the Earth is not only a geometric
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...
task of geodesy, but also has geophysical
Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' som ...
considerations. According to theoretical arguments by Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the grea ...
, Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
, and others, a body having a uniform density of 5.515 g/cm that rotates like the Earth should have a flattening
Flattening is a measure of the compression of a circle or sphere along a diameter to form an ellipse or an ellipsoid of revolution (spheroid) respectively. Other terms used are ellipticity, or oblateness. The usual notation for flattening is ...
of 1:229. This can be concluded without any information about the composition of Earth's interior
The internal structure of Earth is the solid portion of the Earth, excluding its atmosphere and hydrosphere. The structure consists of an outer silicate solid crust, a highly viscous asthenosphere and solid mantle, a liquid outer core wh ...
.[
] However, the measured flattening is 1:298.25, which is closer to a sphere and a strong argument that Earth's core
The internal structure of Earth is the solid portion of the Earth, excluding its atmosphere and hydrosphere. The structure consists of an outer silicate solid crust, a highly viscous asthenosphere and solid mantle, a liquid outer core whose ...
is extremely compact. Therefore, the density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
must be a function of the depth, ranging from 2.6 g/cm at the surface (rock density of granite
Granite () is a coarse-grained (phaneritic) intrusive igneous rock composed mostly of quartz, alkali feldspar, and plagioclase. It forms from magma with a high content of silica and alkali metal oxides that slowly cools and solidifies undergro ...
, etc.), up to 13 g/cm within the inner core.
Global and regional gravity field
Also with implications for the physical exploration of the Earth's interior is the gravitational field
In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
, which is the net effect of gravitation (due to mass attraction) and centrifugal force (due to rotation). It can be measured very accurately at the surface and remotely by satellites. True vertical
Vertical is a geometric term of location which may refer to:
* Vertical direction, the direction aligned with the direction of the force of gravity, up or down
* Vertical (angles), a pair of angles opposite each other, formed by two intersecting s ...
generally does not correspond to theoretical vertical (deflection
Deflection or deflexion may refer to:
Board games
* Deflection (chess), a tactic that forces an opposing chess piece to leave a square
* Khet (game), formerly ''Deflexion'', an Egyptian-themed chess-like game using lasers
Mechanics
* Deflection ...
ranges up to 50") because topography
Topography is the study of the forms and features of land surfaces. The topography of an area may refer to the land forms and features themselves, or a description or depiction in maps.
Topography is a field of geoscience and planetary sci ...
and all ''geological masses'' disturb the gravitational field. Therefore, the gross structure of the Earth's crust
Earth's crust is Earth's thin outer shell of rock, referring to less than 1% of Earth's radius and volume. It is the top component of the lithosphere, a division of Earth's layers that includes the crust and the upper part of the mantle. The ...
and mantle can be determined by geodetic-geophysical models of the subsurface.
See also
* Clairaut's theorem
Clairaut's theorem characterizes the surface gravity on a viscous rotating ellipsoid in hydrostatic equilibrium under the action of its gravitational field and centrifugal force. It was published in 1743 by Alexis Claude Clairaut in a treatis ...
* EGM96
The Earth Gravitational Models (EGM) are a series of geopotential models of the Earth published by the National Geospatial-Intelligence Agency (NGA). They are used as the geoid reference in the World Geodetic System.
The NGA provides the model ...
* Gravity formula
* Gravity of Earth
The gravity of Earth, denoted by , is the net acceleration that is imparted to objects due to the combined effect of gravitation (from mass distribution within Earth) and the centrifugal force (from the Earth's rotation).
It is a vector quanti ...
*
* Meridian arc
In geodesy and navigation, a meridian arc is the curve between two points on the Earth's surface having the same longitude. The term may refer either to a segment of the meridian, or to its length.
The purpose of measuring meridian arcs is to de ...
* Theoretical gravity In geodesy and geophysics, theoretical gravity or normal gravity is an approximation of the true gravity on Earth's surface by means of a mathematical model representing Earth. The most common model of a smoothed Earth is a rotating Earth ellipsoid ...
; History:
* Pierre Bouguer
Pierre Bouguer () (16 February 1698, Croisic – 15 August 1758, Paris) was a French mathematician, geophysicist, geodesist, and astronomer. He is also known as "the father of naval architecture".
Career
Bouguer's father, Jean Bouguer, one ...
* Earth's circumference#History
* Earth's radius#History
*Flat Earth
The flat-Earth model is an archaic and scientifically disproven conception of Earth's shape as a plane or disk. Many ancient cultures subscribed to a flat-Earth cosmography, including Greece until the classical period (5th century BC), the ...
* Friedrich Robert Helmert
Friedrich Robert Helmert (31 July 1843 – 15 June 1917) was a German geodesist and statistician with important contributions to the theory of errors.
Career
Helmert was born in Freiberg, Kingdom of Saxony. After schooling in Freiberg and D ...
*History of geodesy
The history of geodesy deals with the historical development of measurements and representations of the Earth. The corresponding scientific discipline, '' geodesy'' ( /dʒiːˈɒdɪsi/), began in pre-scientific antiquity and blossomed during th ...
*History of the metre
The history of the metre starts with the Scientific Revolution that is considered to have begun with Nicolaus Copernicus's publication of '' De revolutionibus orbium coelestium'' in 1543. Increasingly accurate measurements were required, and ...
* Meridian arc#History
*Seconds pendulum
A seconds pendulum is a pendulum whose period is precisely two seconds; one second for a swing in one direction and one second for the return swing, a frequency of 0.5 Hz.
Pendulum
A pendulum is a weight suspended from a pivot so that ...
References
;Attribution
Further reading
* Guy Bomford
Brigadier Guy Bomford (1899-1996) was a British geodesist.
He worked at the Survey of India and the Corps of Royal Engineers. In 1947 he was appointed as reader in surveying and geodesy at the University of Oxford, holding this post until his reti ...
, ''Geodesy'', Oxford
Oxford () is a city in England. It is the county town and only city of Oxfordshire. In 2020, its population was estimated at 151,584. It is north-west of London, south-east of Birmingham and north-east of Bristol. The city is home to the ...
1962 and 1880.
* Guy Bomford, ''Determination of the European geoid by means of vertical deflection
The vertical deflection (VD) or deflection of the vertical (DoV), also known as deflection of the plumb line and astro-geodetic deflection, is a measure of how far the gravity direction at a given point of interest is rotated by local mass anoma ...
s''. Rpt of Comm. 14, IUGG
The International Union of Geodesy and Geophysics (IUGG; french: Union géodésique et géophysique internationale, UGGI) is an international non-governmental organization dedicated to the scientific study of Earth and its space environment usi ...
10th Gen. Ass., Rome 1954.
* Karl Ledersteger
Karl Ledersteger (11 November 1900, in Vienna – 24 September 1972, near Vienna) was an important geodesist and geophysicist.
After studies of astronomy, mathematics and geodesy he worked in Germany and later in the National Survey of Austria. ...
and Gottfried Gerstbach, ''Die horizontale Isostasie / Das isostatische Geoid 31. Ordnung''. Geowissenschaftliche Mitteilungen Band 5, TU Wien
TU Wien (TUW; german: Technische Universität Wien; still known in English as the Vienna University of Technology from 1975–2014) is one of the major universities in Vienna, Austria. The university finds high international and domestic recogn ...
1975.
* Helmut Moritz Helmut is a German name. Variants include Hellmut, Helmuth, and Hellmuth.
From old German, the first element deriving from either ''heil'' ("healthy") or ''hiltja'' ("battle"), and the second from ''muot'' ("spirit, mind, mood").
Helmut may refer ...
and Bernhard Hofmann
Bernhard is both a given name and a surname. Notable people with the name include:
Given name
* Bernhard of Saxe-Weimar (1604–1639), Duke of Saxe-Weimar
*Bernhard, Prince of Saxe-Meiningen (1901–1984), head of the House of Saxe-Meiningen 194 ...
, ''Physical Geodesy''. Springer
Springer or springers may refer to:
Publishers
* Springer Science+Business Media, aka Springer International Publishing, a worldwide publishing group founded in 1842 in Germany formerly known as Springer-Verlag.
** Springer Nature, a multinationa ...
, Wien & New York 2005.
* ''Geodesy for the Layman'', Defense Mapping Agency
The National Geospatial-Intelligence Agency (NGA) is a combat support agency within the United States Department of Defense whose primary mission is collecting, analyzing, and distributing geospatial intelligence (GEOINT) in support of national ...
, St. Louis, 1983.
External links
Reference Ellipsoids (PCI Geomatics)
Reference Ellipsoids (ScanEx)
Jos Leys "The shape of Planet Earth"
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Earth
Geodesy
Geophysics