Editing Earth radius (section)

==Introduction==
[[File:Earth_oblateness_to_scale.svg|thumb|300px|A scale diagram of the [[Flattening|oblateness]] of the 2003 [[IERS]] [[reference ellipsoid]], with north at the top. The light blue region is a circle.  The outer edge of the dark blue line is an [[ellipse]] with the same [[minor axis]] as the circle and the same [[Eccentricity (mathematics)#Ellipses|eccentricity]] as the Earth. The red line represents the [[Karman line]] {{cvt|100|km|mi}} above [[sea level]], while the yellow area denotes the [[apsis|altitude]] range of the [[International Space Station|ISS]] in [[low Earth orbit]].]]
{{main|Figure of the Earth|Earth ellipsoid|Reference ellipsoid}}

[[Earth's rotation]], internal density variations, and external [[tidal force]]s cause its shape to deviate systematically from a perfect sphere.<ref group=lower-alpha>For details see [[figure of the Earth]], [[geoid]], and [[Earth tide]].</ref> Local [[topography]] increases the variance, resulting in a surface of profound complexity. Our descriptions of Earth's surface must be simpler than reality in order to be tractable. Hence, we create models to approximate characteristics of Earth's surface, generally relying on the simplest model that suits the need.

Each of the models in common use involve some notion of the geometric [[radius]]. Strictly speaking, spheres are the only solids to have radii, but broader uses of the term ''radius'' are common in many fields, including those dealing with models of Earth. The following is a partial list of models of Earth's surface, ordered from exact to more approximate:
* The actual surface of Earth
* The [[geoid]], defined by [[mean sea level]] at each point on the real surface<ref group="lower-alpha">There is no single center to the geoid; it varies according to local [[Geodetic system|geodetic]] conditions. Where there is land, the geoid is generally below ground; it represents where the sea level would be if water could reach it from the ocean via an imaginary canal.</ref>
* A [[spheroid]], also called an [[ellipsoid]] of revolution, [[Geodetic system#Geodetic versus geocentric latitude|geocentric]] to model the entire Earth, or else [[Geodetic system#Geodetic versus geocentric latitude|geodetic]] for regional work<ref group=lower-alpha>In a geocentric ellipsoid, the center of the ellipsoid coincides with some computed center of Earth, and best models the earth as a whole. Geodetic ellipsoids are better suited to regional idiosyncrasies of the geoid. A partial surface of an ellipsoid gets fitted to the region, in which case the center and orientation of the ellipsoid generally do not coincide with the earth's center of mass or axis of rotation.</ref>
* 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"''.<ref group=lower-alpha>The value of the radius is completely dependent upon the latitude in the case of an ellipsoid model, and nearly so on the geoid.</ref> It is also common to refer to any ''[[#Mean radii|mean radius]]'' of a spherical model as ''"the radius of the earth"''. When considering the Earth's real surface, on the other hand, it is uncommon to refer to a "radius", since there is generally no practical need. Rather, elevation above or below sea level is useful.

Regardless of the model, any of these ''geocentric'' radii falls between the polar minimum of about 6,357&nbsp;km and the equatorial maximum of about 6,378&nbsp;km (3,950 to 3,963&nbsp;mi). Hence, the Earth deviates from a perfect sphere by only a third of a percent, which supports the spherical model in most contexts and justifies the term "radius of the Earth". While specific values differ, the concepts in this article generalize to any major [[planet]].

===Physics of Earth's deformation===
{{further|Equatorial bulge}}

Rotation of a planet causes it to approximate an ''[[spheroid|oblate ellipsoid]]/spheroid'' with a bulge at the [[equator]] and flattening at the [[North Pole|North]] and [[South Pole]]s, so that the ''equatorial radius'' {{mvar|a}} is larger than the ''polar radius'' {{mvar|b}} by approximately {{mvar|aq}}. The ''oblateness constant'' {{mvar|q}} is given by
:<math>q=\frac{a^3 \omega^2}{GM},</math>
where {{mvar|ω}} is the [[angular frequency]], {{mvar|G}} is the [[gravitational constant]], and {{mvar|M}} is the mass of the planet.{{refn|This follows from the [[International Astronomical Union]] [[2006 definition of planet|definition]] rule (2): a planet assumes a shape due to [[hydrostatic equilibrium]] where [[gravity]] and [[centrifugal force]]s are nearly balanced.<ref>[http://www.iau2006.org/mirror/www.iau.org/iau0603/index.html IAU 2006 General Assembly: Result of the IAU Resolution votes] {{webarchive|url=https://web.archive.org/web/20061107022302/http://www.iau2006.org/mirror/www.iau.org/iau0603/index.html |date=2006-11-07 }}</ref>|group=lower-alpha}} For the Earth {{math|{{sfrac|1|''q''}} ≈ 289}}, which is close to the measured inverse [[flattening]] {{math|{{sfrac|1|''f''}} ≈ 298.257}}. Additionally, the bulge at the equator shows slow variations. The bulge had been decreasing, but since 1998 the bulge has increased, possibly due to redistribution of ocean mass via currents.<ref>[https://web.archive.org/web/20020810195620/http://www.gsfc.nasa.gov/topstory/20020801gravityfield.html Satellites Reveal A Mystery Of Large Change In Earth's Gravity Field ], Aug. 1, 2002, [[Goddard Space Flight Center]]. </ref>
[[Image:Lowresgeoidheight.jpg|400px|right]]
The variation in [[density]] and [[Crust (geology)|crustal]] thickness causes gravity to vary across the surface and in time, so that the mean sea level differs from the ellipsoid. This difference is the ''[[geoid]] height'', positive above or outside the ellipsoid, negative below or inside. The geoid height variation is under {{convert|110|m|ft|abbr=on}} on Earth. The geoid height can change abruptly due to earthquakes (such as the [[2004 Indian Ocean earthquake|Sumatra-Andaman earthquake]]) or reduction in ice masses (such as [[Greenland]]).<ref>[https://archive.today/20120529015755/http://www.spaceref.com/news/viewpr.html?pid=18567 NASA's Grace Finds Greenland Melting Faster, 'Sees' Sumatra Quake], December 20, 2005, [[Goddard Space Flight Center]].</ref>

Not all deformations originate within the Earth. Gravitational attraction from the Moon or Sun can cause the Earth's surface at a given point to vary by tenths of a meter over a nearly 12-hour period (see [[Earth tide]]).

===Radius and local conditions===
[[File:Abu Reyhan Biruni-Earth Circumference.svg|thumb|[[Al-Biruni#Geodesy and geography|Al-Biruni]]'s (973 – {{c.|1050}}) method for calculation of the Earth's radius simplified measuring the circumference compared to taking measurements from two locations distant from each other.]]
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 {{convert|5|m|ft|abbr=on}} of reference ellipsoid height, and to within {{convert|100|m|ft|abbr=on}} 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]], the curvature at a point will be greatest (tightest) in one direction (north–south on Earth) and smallest (flattest) perpendicularly (east–west). The corresponding [[Radius of curvature (applications)|radius of curvature]] depends on the location and direction of measurement from that point. A consequence is that a distance to the [[horizon|true horizon]] at the equator is slightly shorter in the north–south direction than in the east–west direction.

In summary, local variations in terrain prevent defining a single "precise" radius. One can only adopt an idealized model. Since the estimate by [[Eratosthenes]], many models have been created. Historically, these models were based on regional topography, giving the best [[Figure of the Earth#Historical Earth ellipsoids|reference ellipsoid]] for the area under survey. As satellite [[remote sensing]] and especially the [[Global Positioning System]] gained importance, true global models were developed which, while not as accurate for regional work, best approximate the Earth as a whole.