Earth radius

Earth radius

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Because the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

 is not perfectly spherical
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

, no single value serves as its natural radius
Radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...

. Distances from points on the surface to the center range from 6,353 km to 6,384 km (≈3,947–3,968 mi). Several different ways of modeling the Earth as a sphere each yield a convenient mean radius of 6,371 km (≈3,959 mi).

While "radius" normally is a characteristic of perfect spheres, the term as employed in this article more generally means the distance from some "center" of the Earth to a point on the surface or on an idealized surface that models the Earth. It can also mean some kind of average of such distances. It can also mean the radius of a sphere whose curvature matches the curvature of the ellipsoidal model of the Earth at a given point.

This article deals primarily with spherical and ellipsoidal models of the Earth. See Figure of the Earth
Figure of the Earth
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...

 for a more complete discussion of models.

The first scientific estimation of the radius of the earth was given by Eratosthenes
Eratosthenes
Eratosthenes of Cyrene was a Greek mathematician, poet, athlete, geographer, astronomer, and music theorist.He was the first person to use the word "geography" and invented the discipline of geography as we understand it...

.

Earth radius is also used as a unit of distance, especially in astronomy
Astronomy
Astronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...

 and geology
Geology
Geology is the science comprising the study of solid Earth, the rocks of which it is composed, and the processes by which it evolves. Geology gives insight into the history of the Earth, as it provides the primary evidence for plate tectonics, the evolutionary history of life, and past climates...

. It is usually denoted by .

Radius and models of the earth


Earth's rotation, internal density variations, and external tidal forces cause it to deviate systematically from a perfect sphere. Local topography
Topography
Topography is the study of Earth's surface shape and features or those ofplanets, moons, and asteroids...

 increases the variance, resulting in a surface of unlimited complexity. Our descriptions of the Earth's surface must be simpler than reality in order to be tractable. Hence we create models to approximate the Earth's surface, generally relying on the simplest model that suits the need.

Each of the models in common use come with some notion of "radius". Strictly speaking, spheres are the only solids to have radii, but looser uses of the term "radius" are common in many fields, including those dealing with models of the Earth. Viewing models of the Earth from less to more approximate:
  • The real surface of the Earth;
  • The geoid
    Geoid
    The geoid is that equipotential surface which would coincide exactly with the mean ocean surface of the Earth, if the oceans were in equilibrium, at rest , and extended through the continents . According to C.F...

    , defined by mean sea level at each point on the real surface;
  • An ellipsoid: geocentric to model the entire earth, or else geodetic for regional work;
  • A sphere.


In the case of the geoid and ellipsoids, the fixed distance from any point on the model to the specified center is called "a radius of the Earth" or "the radius of the Earth at that point". It is also common to refer to any mean radius of a spherical model as "the radius of the earth". On the Earth's real surface, on other hand, it is uncommon to refer to a "radius", since there is no practical need. Rather, elevation above or below sea level is useful.

Regardless of model, any radius falls between the polar minimum of about 6,357 km and the equatorial maximum of about 6,378 km (≈3,950 – 3,963 mi). Hence the Earth deviates from a perfect sphere 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". While specific values differ, the concepts in this article generalize to any major planet
Planet
A planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...

.

Physics of Earth's deformation


Rotation of a planet causes it to approximate an oblate ellipsoid
Spheroid
A spheroid, or ellipsoid of revolution 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....

/spheroid
with a bulge at the equator
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....

 and flattening at the North
North Pole
The North Pole, also known as the Geographic North Pole or Terrestrial North Pole, is, subject to the caveats explained below, defined as the point in the northern hemisphere where the Earth's axis of rotation meets its surface...

 and South Pole
South Pole
The South Pole, also known as the Geographic South Pole or Terrestrial South Pole, is one of the two points where the Earth's axis of rotation intersects its surface. It is the southernmost point on the surface of the Earth and lies on the opposite side of the Earth from the North Pole...

s, so that the equatorial radius is larger than the polar radius by approximately where the oblateness constant is

where is the angular frequency
Angular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...

, is the gravitational constant
Gravitational constant
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...

, and is the mass of the planet. For the Earth , which is close to the measured inverse flattening
Flattening
The flattening, ellipticity, or oblateness of an oblate spheroid is a measure of the "squashing" of the spheroid's pole, towards its equator...

 . Additionally, the bulge at the equator shows slow variations. The bulge had been declining, but since 1998 the bulge has increased, possibly due to redistribution of ocean mass via currents.

The variation in density
Density
The mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...

 and crustal
Crust (geology)
In geology, the crust is the outermost solid shell of a rocky planet or natural satellite, which is chemically distinct from the underlying mantle...

 thickness causes gravity to vary on the surface, so that the mean sea level will differ from the ellipsoid. This difference is the geoid
Geoid
The geoid is that equipotential surface which would coincide exactly with the mean ocean surface of the Earth, if the oceans were in equilibrium, at rest , and extended through the continents . According to C.F...

 height
, positive above or outside the ellipsoid, negative below or inside. The geoid height variation is under 110 m on Earth. The geoid height can have abrupt changes due to earthquakes (such as the Sumatra-Andaman earthquake
2004 Indian Ocean earthquake
The 2004 Indian Ocean earthquake was an undersea megathrust earthquake that occurred at 00:58:53 UTC on Sunday, December 26, 2004, with an epicentre off the west coast of Sumatra, Indonesia. The quake itself is known by the scientific community as the Sumatra-Andaman earthquake...

) or reduction in ice masses (such as Greenland
Greenland
Greenland is an autonomous country within the Kingdom of Denmark, located between the Arctic and Atlantic Oceans, east of the Canadian Arctic Archipelago. Though physiographically a part of the continent of North America, Greenland has been politically and culturally associated with Europe for...

).

Not all deformations originate within the earth. Gravity of the Moon and Sun cause Earth's surface to undulate by tenths of meters at a point over a nearly 12 hour period (see Earth tide
Earth tide
Earth tide is the sub-meter motion of the Earth of about 12 hours or longer caused by Moon and Sun gravitation, also called body tide which is the largest contribution globally. The largest body tide contribution is from the semidiurnal constituents, but there are also significant diurnal...

).

Radius and local conditions



Given local and transient influences on surface height, the values defined below are based on a "general purpose" model, refined as globally precisely as possible within 5 m of reference ellipsoid height, and to within 100 m of mean sea level (neglecting geoid height).

Additionally, the radius can be estimated from the curvature of the Earth at a point. Like a torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

 the curvature at a point will be largest (tightest) in one direction (North-South on Earth) and smallest (flattest) perpendicularly (East-West). The corresponding radius of curvature depends on location and direction of measurement from that point. A consequence is that a distance to the true horizon
Horizon
The horizon is the apparent line that separates earth from sky, the line that divides all visible directions into two categories: those that intersect the Earth's surface, and those that do not. At many locations, the true horizon is obscured by trees, buildings, mountains, etc., and the resulting...

 at the equator is slightly shorter in the north/south direction than in the east-west direction.

In summary, local variations in terrain prevent the definition of a single absolutely "precise" radius. One can only adopt an idealized model. Since the estimate by Eratosthenes
Eratosthenes
Eratosthenes of Cyrene was a Greek mathematician, poet, athlete, geographer, astronomer, and music theorist.He was the first person to use the word "geography" and invented the discipline of geography as we understand it...

, many models have been created. Historically these models were based on regional topography, giving the best reference ellipsoid for the area under survey. As satellite remote sensing and especially the Global Positioning System
Global Positioning System
The Global Positioning System is a space-based global navigation satellite system that provides location and time information in all weather, anywhere on or near the Earth, where there is an unobstructed line of sight to four or more GPS satellites...

 rose in importance, true global models were developed which, while not as accurate for regional work, best approximate the earth as a whole.

Fixed radii


The following radii are fixed and do not include a variable location dependence. They are
derived from the WGS-84 ellipsoid.

The value for the equatorial radius is defined to the nearest 0.1 meter in WGS-84. The value for the polar radius in this section has been rounded to the nearest 0.1 meter, which is expected to be adequate for most uses. Please refer to the WGS-84 ellipsoid if a more precise value for its polar radius is needed.

The radii in this section are for an idealized surface. Even the idealized radii have an uncertainty of ± 2 meters. The discrepancy between the ellipsoid radius and the radius to a physical location may be significant. When identifying the position of an observable location, the use of more precise values for WGS-84 radii may not yield a corresponding improvement in accuracy.

The symbol given for the named radius is used in the formulae found in this article.

Equatorial radius


The Earth's equatorial radius , or semi-major axis
Semi-major axis
The major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...

, is the distance from its center to the equator
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....

 and equals 6,378.1370 km
Kilometre
The kilometre is a unit of length in the metric system, equal to one thousand metres and is therefore exactly equal to the distance travelled by light in free space in of a second...

 (≈3,963.191 mi
Mile
A mile is a unit of length, most commonly 5,280 feet . The mile of 5,280 feet is sometimes called the statute mile or land mile to distinguish it from the nautical mile...

; ≈3,443.918 nmi
Nautical mile
The nautical mile is a unit of length that is about one minute of arc of latitude along any meridian, but is approximately one minute of arc of longitude only at the equator...

). The equatorial radius is often used to compare Earth with other planets.

Polar radius


The Earth's polar radius , or semi-minor axis
Semi-minor axis
In geometry, the semi-minor axis is a line segment associated with most conic sections . One end of the segment is the center of the conic section, and it is at right angles with the semi-major axis...

, is the distance from its center to the North and South Poles, and equals 6,356.7523 km (≈3,949.903 mi; ≈3,432.372 nmi).

Notable radii

  • Maximum: The summit of Chimborazo
    Chimborazo (volcano)
    Chimborazo is a currently inactive stratovolcano located in the Cordillera Occidental range of the Andes. Its last known eruption is believed to have occurred around 550 AD....

     is 6,384.4 km (3,968 mi) from the Earth's center.
  • Minimum: The floor of the Arctic Ocean
    Arctic Ocean
    The Arctic Ocean, located in the Northern Hemisphere and mostly in the Arctic north polar region, is the smallest and shallowest of the world's five major oceanic divisions...

     is ≈6,352.8 km (3,947 mi) from the Earth's center.

Radius at a given geodetic latitude


The distance from the Earth's center to a point on the spheroid surface at geodetic latitude is:

Radius of curvature


These are based on an oblate ellipsoid
Spheroid
A spheroid, or ellipsoid of revolution 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....

.

Eratosthenes used two points, one almost exactly north of the other. The points are separated by distance , and the vertical direction
Vertical direction
In astronomy, geography, geometry and related sciences and contexts, a direction passing by a given point is said to be vertical if it is locally aligned with the gradient of the gravity field, i.e., with the direction of the gravitational force at that point...

s at the two points are known to differ by angle of , in radians.
A formula used in Eratosthenes' method is

which gives an estimate of radius based on the north-south curvature of the Earth.

Meridional

In particular the Earth's radius of curvature in the (north-south) meridian
Meridian (geography)
A meridian is an imaginary line on the Earth's surface from the North Pole to the South Pole that connects all locations along it with a given longitude. The position of a point along the meridian is given by its latitude. Each meridian is perpendicular to all circles of latitude...

at is:

Normal

If one point had appeared due east of the other, one finds the approximate curvature in east-west direction.
This radius of curvature in the prime vertical
Prime vertical
In astronomy and astrology, the prime vertical is the vertical circle passing east and west through the zenith, and intersecting the horizon in its east and west points....

, which is perpendicular, or normal
Orthogonality
Orthogonality occurs when two things can vary independently, they are uncorrelated, or they are perpendicular.-Mathematics:In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle...

, to M at geodetic latitude is:

Note that N=R at the equator:
At geodetic latitude 48.46791… degrees (e.g., Lèves, Alsace, France), the radius R is 20000/π ≈ 6,366.1977…, namely the radius of a perfect sphere for which the meridian arc
Meridian arc
In geodesy, a meridian arc measurement is a highly accurate determination of the distance between two points with the same longitude. Two or more such determinations at different locations then specify the shape of the reference ellipsoid which best approximates the shape of the geoid. This...

 length from the equator to the North Pole is exactly 10000 km, the originally proposed definition of the meter.

The Earth's mean radius of curvature (averaging over all directions) at latitude is:


The Earth's radius of curvature along a course at geodetic bearing (measured clockwise from north) , at is derived from Euler's curvature formula
Euler's theorem (differential geometry)
In the mathematical field of differential geometry, Euler's theorem is a result on the curvature of curves on a surface. The theorem establishes the existence of principal curvatures and associated principal directions which give the directions in which the surface curves the most and the least...

 as follows:


The Earth's equatorial radius of curvature in the meridian is:
=6,335.437 km


The Earth's polar radius of curvature is:
=6,399.592 km

Mean radii


The Earth can be modeled as a sphere in many ways. This section describes the common ways. The various radii derived here use the notation and dimensions noted above for the Earth as derived from the WGS-84 ellipsoid; namely,
 Equatorial radius (6,378.1370 km)
 Polar radius (6,356.7523 km)


A sphere being a gross approximation of the spheroid, which itself is an approximation of the geoid, units are given here in kilometers rather than the millimeter resolution appropriate for geodesy.

Mean radius


The International Union of Geodesy and Geophysics
International Union of Geodesy and Geophysics
The International Union of Geodesy and Geophysics is a non-governmental organisation dedicated to the scientific study of the Earth using geophysical and geodesic techniques. The IUGG was established in 1919. Some areas within its scope are environmental preservation, reduction of the effects of...

 (IUGG) defines the mean radius (denoted ) to be

For Earth, the mean radius is 6,371.009 km (≈3,958.761 mi; ≈3,440.069 nmi).

Authalic radius


Earth's authalic ("equal area") radius is the radius of a hypothetical perfect sphere which has the same surface area as the reference ellipsoid
Reference ellipsoid
In geodesy, a reference ellipsoid is a mathematically-defined surface that approximates the geoid, the truer figure of the Earth, or other planetary body....

. The IUGG denotes the authalic radius as .

A closed-form solution exists for a spheroid:

where and is the surface area of the spheroid.

For Earth, the authalic radius is 6,371.0072 km (≈3,958.760 mi; ≈3,440.069 nmi).

Volumetric radius


Another spherical model is defined by the volumetric radius, which is the radius of a sphere of volume equal to the ellipsoid. The IUGG denotes the volumetric radius as .

For Earth, the volumetric radius equals 6,371.0008 km (≈3,958.760 mi; ≈3,440.069 nmi).

Meridional Earth radius


Another mean radius is the rectifying radius, giving a sphere with circumference equal to the perimeter
Circumference
The circumference is the distance around a closed curve. Circumference is a special perimeter.-Circumference of a circle:The circumference of a circle is the length around it....

 of the ellipse described by any polar cross section of the ellipsoid. This requires an elliptic integral to find, given the polar and equatorial radii:
.


The rectifying radius is equivalent to the meridional mean, which is defined as the average value of M:


For integration limits of [0…π/2], the integrals for rectifying radius and mean radius evaluate to the same result, which, for Earth, amounts to 6,367.4491 km (≈3,956.545 mi; ≈3,438.147 nmi).

The meridional mean is well approximated by the semicubic mean of the two axes:


yielding, again, 6,367.4491 km; or less accurately by the quadratic mean of the two axes:
;


about 6,367.454 km; or even just the mean of the two axes:
;


about 6,367.445 km.

See also

  • Effective Earth radius
    Effective Earth radius
    In telecommunication, effective Earth radius is the radius of a hypothetical Earth for which the distance to the radio horizon, assuming rectilinear propagation, is the same as that for the actual Earth with an assumed uniform vertical gradient of atmospheric refractive index.Note: For the...

  • Radius of curvature (applications)
  • Biruni
  • Figure of the Earth
    Figure of the Earth
    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...

  • Geographical distance
    Geographical distance
    Geographical distance is the distance measured along the surface of the earth. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitude.-An abstraction:...

  • Earth tide
    Earth tide
    Earth tide is the sub-meter motion of the Earth of about 12 hours or longer caused by Moon and Sun gravitation, also called body tide which is the largest contribution globally. The largest body tide contribution is from the semidiurnal constituents, but there are also significant diurnal...

  • Geodesy
    Geodesy
    Geodesy , also named geodetics, a branch of earth sciences, is the scientific discipline that deals with the measurement and representation of the Earth, including its gravitational field, in a three-dimensional time-varying space. Geodesists also study geodynamical phenomena such as crustal...

  • History of geodesy
    History of geodesy
    Geodesy ,[1] also named geodetics, is the scientific discipline that deals with the measurement and representation of the Earth.Humanity has always been interested in the Earth...