| Earth radius | |
|---|---|
 Equatorial (a), polar (b) and arithmetic mean Earth radii as defined in the 1984World Geodetic System revision (not to scale) | |
Other names | terrestrial radius |
Common symbols | R🜨,RE,a,b,aE,bE,ReE,RpE |
| SI unit | meters |
| InSI base units | m |
Behaviour under coord transformation | scalar |
| Dimension | |
| Value | Equatorial radius:a = (6378137.0 m) Polar radius:b = (6356752.3 m) |
| Nominal Earth radius | |
|---|---|
 Cross section of Earth's Interior | |
| General information | |
| Unit system | astronomy,geophysics |
| Unit of | distance |
| Symbol | , , |
| Conversions | |
| 1 in ... | ... is equal to ... |
| SI base unit | 6.3781×106 m[1] |
| Metric system | 6,357 to 6,378 km |
| English units | 3,950 to 3,963 mi |
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Fundamentals | ||||||||||||||||||||||||||
Standards (history)
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Earth radius (denoted asR🜨 orRE) is the distance from the center ofEarth to a point on or near its surface. Approximating thefigure of Earth by anEarth spheroid (anoblate ellipsoid), the radius ranges from a maximum (equatorial radius, denoteda) of nearly 6,378 km (3,963 mi) to a minimum (polar radius, denotedb) of nearly 6,357 km (3,950 mi).
A globally-average value is usually considered to be 6,371 kilometres (3,959 mi) with a 0.3% variability (±10 km) for the following reasons.TheInternational Union of Geodesy and Geophysics (IUGG) provides three reference values: themean radius (R1) of three radii measured at two equator points and a pole; theauthalic radius, which is the radius of a sphere with the same surface area (R2); and thevolumetric radius, which is the radius of a sphere having the same volume as the ellipsoid (R3).[2] All three values are about 6,371 kilometres (3,959 mi).
Other ways to define and measure the Earth's radius involve either the spheroid'sradius of curvature or the actualtopography. A few definitions yield values outside the range between thepolar radius andequatorial radius because they account for localized effects.
Anominal Earth radius (denoted) is sometimes used as aunit of measurement inastronomy andgeophysics, aconversion factor used when expressing planetary properties as multiples or fractions of a constant terrestrial radius; if the choice between equatorial or polar radii is not explicit, the equatorial radius is to be assumed, as recommended by theInternational Astronomical Union (IAU).[1]
Earth's rotation, internal density variations, and externaltidal forces cause its shape to deviate systematically from a perfect sphere.[a] Localtopography 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 geometricradius. Strictly speaking, spheres are the only solids to have radii, but broader uses of the termradius 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:
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".[d] It is also common to refer to anymean 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 thesegeocentric radii falls between the polar minimum of about 6,357 km and the equatorial maximum of about 6,378 km (3,950 to 3,963 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 majorplanet.
Rotation of a planet causes it to approximate anoblate ellipsoid/spheroid with a bulge at theequator and flattening at theNorth andSouth Poles, so that theequatorial radiusa is larger than thepolar radiusb by approximatelyaq. Theoblateness constantq is given by
whereω is theangular frequency,G is thegravitational constant, andM is the mass of the planet.[e] For the Earth1/q ≈ 289, which is close to the measured inverseflattening1/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.[4]
The variation indensity andcrustal thickness causes gravity to vary across the surface and in time, so that the mean sea level differs from the ellipsoid. This difference is thegeoid height, positive above or outside the ellipsoid, negative below or inside. The geoid height variation is under 110 m (360 ft) on Earth. The geoid height can change abruptly due to earthquakes (such as theSumatra-Andaman earthquake) or reduction in ice masses (such asGreenland).[5]
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 (seeEarth tide).
Additionally, the radius can be estimated from the curvature of the Earth at a point. Like atorus, the curvature at a point will be greatest (tightest) in one direction (north–south on Earth) and smallest (flattest) perpendicularly (east–west). The correspondingradius of curvature depends on the location and direction of measurement from that point. A consequence is that a distance to thetrue 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 byEratosthenes, many models have been created. Historically, these models were based on regional topography, giving the bestreference ellipsoid for the area under survey. As satelliteremote sensing and especially theGlobal Positioning System gained importance, true global models were developed which, while not as accurate for regional work, best approximate the Earth as a whole.
The following radii are derived from theWorld Geodetic System 1984 (WGS-84)reference ellipsoid.[6] It is an idealized surface, and the Earth measurements used to calculate it have an uncertainty of ±2 m in both the equatorial and polar dimensions.[7] Additional discrepancies caused by topographical variation at specific locations can 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 inaccuracy.[clarification needed]
The value for the equatorial radius is defined to the nearest 0.1 m in WGS-84. The value for the polar radius in this section has been rounded to the nearest 0.1 m, which is expected to be adequate for most uses. Refer to the WGS-84 ellipsoid if a more precise value for its polar radius is needed.
Thegeocentric radius is the distance from the Earth's center to a point on the spheroid surface atgeodetic latitudeφ, given by the formula[10]
wherea andb are, respectively, the equatorial radius and the polar radius.
The extrema geocentric radii on the ellipsoid coincide with the equatorial and polar radii.They arevertices of the ellipse and also coincide with minimum and maximum radius of curvature.
There are twoprincipal radii of curvature: along the meridional and prime-verticalnormal sections.
In particular, theEarth'smeridional radius of curvature (in the north–south direction) atφ is[11]
where is theeccentricity of the earth. This is the radius thatEratosthenes measured in hisarc measurement.
If one point had appeared due east of the other, one finds the approximate curvature in the east–west direction.[f]
ThisEarth'sprime-vertical radius of curvature, also called theEarth's transverse radius of curvature, is defined perpendicular (orthogonal) toM at geodetic latitudeφ[g] and is[11]
N can also be interpreted geometrically as thenormal distance from the ellipsoid surface to the polar axis.[12]The radius of aparallel of latitude is given by.[13][14]
TheEarth's meridional radius of curvature at the equator equals the meridian'ssemi-latus rectum:
TheEarth's prime-vertical radius of curvature at the equator equals the equatorial radius,
TheEarth's polar radius of curvature (either meridional or prime-vertical) is
Extended content |
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The principal curvatures are the roots of Equation (125) in:[15] where in thefirst fundamental form for a surface (Equation (112) in[15]): E,F, andG are elements of themetric tensor: ,, in thesecond fundamental form for a surface (Equation (123) in[15]): e,f, andg are elements of the shape tensor: is the unit normal to the surface at, and because and are tangents to the surface, is normal to the surface at. With for an oblate spheroid, the curvatures are
and the principal radii of curvature are
The first and second radii of curvature correspond, respectively, to the Earth's meridional and prime-vertical radii of curvature. Geometrically, the second fundamental form gives the distance from to the plane tangent at. |
The Earth'sazimuthal radius of curvature, along anEarth normal section at anazimuth (measured clockwise from north)α and at latitudeφ, is derived fromEuler's curvature formula as follows:[16]: 97
It is possible to combine the principal radii of curvature above in a non-directional manner.
TheEarth'sGaussian radius of curvature at latitudeφ is[16]
whereK is theGaussian curvature,
TheEarth'smean radius of curvature at latitudeφ is[16]: 97
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 theWGS-84 ellipsoid;[6] namely,
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.
In geophysics, theInternational Union of Geodesy and Geophysics (IUGG) defines theEarth'sarithmetic mean radius (denotedR1) to be[2]
The factor of two accounts for the biaxial symmetry in Earth's spheroid, a specialization of triaxial ellipsoid.[17]For Earth, the arithmetic mean radius is published by IUGG andNGA as 6,371.0087714 km (3,958.7613160 mi).[1][7]
Earth's authalic radius (meaning"equal area") is the radius of a hypothetical perfect sphere that has the same surface area as thereference ellipsoid. TheIUGG denotes the authalic radius asR2.[2]A closed-form solution exists for a spheroid:[8]
where is the eccentricity, and is the surface area of the spheroid.
For the Earth, the authalic radius is 6,371.0072 km (3,958.7603 mi).[17]
The authalic radius also corresponds to theradius of (global) mean curvature, obtained by averaging the Gaussian curvature,, over the surface of the ellipsoid. Using theGauss–Bonnet theorem, this gives
Another spherical model is defined by theEarth's volumetric radius, which is the radius of a sphere of volume equal to the ellipsoid. TheIUGG denotes the volumetric radius asR3.[2]
For Earth, the volumetric radius equals 6,371.0008 km (3,958.7564 mi).[17]
Another global radius is theEarth's rectifying radius, giving a sphere with circumference equal to theperimeter of the ellipse described by any polar cross section of the ellipsoid. This requires anelliptic 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 ofM:[8]
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 (3,956.5494 mi).
The meridional mean is well approximated by the semicubic mean of the two axes,[citation needed]
which differs from the exact result by less than 1 μm (4×10−5 in); the mean of the two axes,
about 6,367.445 km (3,956.547 mi), can also be used.
The mathematical expressions above apply over the surface of the ellipsoid.The cases below considers Earth'stopography, above or below areference ellipsoid.As such, they aretopographicalgeocentric distances,Rt, which depends not only on latitude.
Thetopographical mean geocentric distance averages elevations everywhere, resulting in a value230 m larger than theIUGG mean radius, theauthalic radius, or thevolumetric radius. This topographical average is 6,371.230 km (3,958.899 mi) with uncertainty of 10 m (33 ft).[19]
Earth'sdiameter is simply twice Earth's radius; for example,equatorial diameter (2a) andpolar diameter (2b). For the WGS84 ellipsoid, that's respectively:
Earth's circumference equals theperimeter length. Theequatorial circumference is simply thecircle perimeter:Ce = 2πa, in terms of the equatorial radiusa. Thepolar circumference equalsCp = 4mp, four times thequarter meridianmp =aE(e), where the polar radiusb enters via the eccentricitye = (1 −b2/a2)0.5; seeEllipse#Circumference for details.
Arc length of more generalsurface curves, such asmeridian arcs andgeodesics, can also be derived from Earth's equatorial and polar radii.
Likewise forsurface area, either based on amap projection or ageodesic polygon.
Earth's volume, or that of the reference ellipsoid, is Using the parameters fromWGS84 ellipsoid of revolution,a = 6,378.137 km andb =6356.7523142 km,V = 1.08321×1012 km3 (2.5988×1011 cu mi).[20]
In astronomy, theInternational Astronomical Union denotes thenominal equatorial Earth radius as, which is defined to be exactly 6,378.1 km (3,963.2 mi).[1]: 3 Thenominal polar Earth radius is defined exactly as = 6,356.8 km (3,949.9 mi). These values correspond to the zeroEarth tide convention. Equatorial radius is conventionally used as the nominal value unless the polar radius is explicitly required.[1]: 4 The nominal radius serves as aunit of length forastronomy.(The notation is defined such that it can be easily generalized for otherplanets; e.g., for the nominal polarJupiter radius.)
This table summarizes the accepted values of the Earth's radius.
| Agency | Description | Value (in meters) | Ref |
|---|---|---|---|
| IAU | nominal "zero tide" equatorial | 6378100 | [1] |
| IAU | nominal "zero tide" polar | 6356800 | [1] |
| IUGG | equatorial radius | 6378137 | [2] |
| IUGG | semiminor axis (b) | 6356752.3141 | [2] |
| IUGG | polar radius of curvature (c) | 6399593.6259 | [2] |
| IUGG | mean radius (R1) | 6371008.7714 | [2] |
| IUGG | radius of sphere of same surface (R2) | 6371007.1810 | [2] |
| IUGG | radius of sphere of same volume (R3) | 6371000.7900 | [2] |
| NGA | WGS-84 ellipsoid, semi-major axis (a) | 6378137.0 | [6] |
| NGA | WGS-84 ellipsoid, semi-minor axis (b) | 6356752.3142 | [6] |
| NGA | WGS-84 ellipsoid, polar radius of curvature (c) | 6399593.6258 | [6] |
| NGA | WGS-84 ellipsoid, Mean radius of semi-axes (R1) | 6371008.7714 | [6] |
| NGA | WGS-84 ellipsoid, Radius of Sphere of Equal Area (R2) | 6371007.1809 | [6] |
| NGA | WGS-84 ellipsoid, Radius of Sphere of Equal Volume (R3) | 6371000.7900 | [6] |
| GRS 80 semi-major axis (a) | 6378137.0 | ||
| GRS 80 semi-minor axis (b) | ≈6356752.314140 | ||
| Spherical Earth Approx. of Radius (RE) | 6366707.0195 | [21] | |
| meridional radius of curvature at the equator | 6335439 | ||
| Maximum (the summit of Chimborazo) | 6384400 | [18] | |
| Minimum (the floor of the Arctic Ocean) | 6352800 | [18] | |
| Average distance from center to surface | 6371230±10 | [19] |
The first published reference to the Earth's size appeared around 350 BC, whenAristotle reported in his bookOn the Heavens[22] that mathematicians had guessed the circumference of the Earth to be 400,000stadia. Scholars have interpreted Aristotle's figure to be anywhere from highly accurate[23] to almost double the true value.[24] The first known scientific measurement and calculation of the circumference of the Earth was performed byEratosthenes in about 240 BC. Estimates of the error of Eratosthenes's measurement range from 0.5% to 17%.[25] For both Aristotle and Eratosthenes, uncertainty in the accuracy of their estimates is due to modern uncertainty over which stadion length they meant.
Around 100 BC,Posidonius of Apamea recomputed Earth's radius, and found it to be close to that by Eratosthenes,[26] but laterStrabo incorrectly attributed him a value about 3/4 of the actual size.[27]Claudius Ptolemy around 150 AD gave empirical evidence supporting aspherical Earth,[28] but he accepted the lesser value attributed to Posidonius. His highly influential work, theAlmagest,[29] left no doubt among medieval scholars that Earth is spherical, but they were wrong about its size.
By 1490,Christopher Columbus believed that traveling 3,000 miles west from the west coast of theIberian Peninsula would let him reach the eastern coasts ofAsia.[30] However, the 1492 enactment of that voyage brought his fleet to the Americas. TheMagellan expedition (1519–1522), which was the firstcircumnavigation of the World, soundly demonstrated the sphericity of the Earth,[31] and affirmed the original measurement of 40,000 km (25,000 mi) by Eratosthenes.
Around 1690,Isaac Newton andChristiaan Huygens argued that Earth was closer to an oblate spheroid than to a sphere. However, around 1730,Jacques Cassini argued for a prolate spheroid instead, due to different interpretations of theNewtonian mechanics involved.[32] To settle the matter, theFrench Geodesic Mission (1735–1739) measured one degree oflatitude at two locations, one near theArctic Circle and the other near theequator. The expedition found that Newton's conjecture was correct:[33] the Earth is flattened at thepoles due to rotation'scentrifugal force.
