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| {{for|the historical development of the concept|Spherical Earth}}
| | I might also prefer to encourage we to consult with your physician before beginning any weight loss system. She or he could give you a fair higher thought on a superior goal for you.<br><br><br><br>Utilizing the metric program, the formula for BMI is weight (kilograms) separated by height (meters) squared. Formula: fat (kg) / [height (m)]2. Imperial system consumers will calculate their BMI by dividing weight in pounds (lbs) by height in inches (in) squared and then increased by a conversion element of 703. Formula: fat (lb) / [height (in)]2 x 703. For those unknown with the formula, many free calories burned walking calculators will be found online.<br><br>If you are a athlete, you aren't the typical particular person. We don't have a very sedentary lifestyle consequently we will commonly have manufactured some aware dietary possibilities simultaneously. You may also consume extra h2o compared to typical person right now, plus drinking water usage is known for aiding fat reduction. Now, because of this the [http://safedietplans.com/calories-burned-walking calories burned calculator] may be challenging for we personally being a runner. It could exhibit which you are "scrawny" when inside truth it is obvious which you just are all slender muscles and filled with energy plus power.<br><br>We can state that weight is a major problem that we are facing now days as folks are thus busy inside their lifetime that they don't get time even to keep their body inside best shape. However, almost all of you feel like putting extra fat from your body. Before we think about throwing mass from your body or placing up mass you need to be aware of the actual calculation which you need to function about. BMI is a source that assists you to learn regarding the actual proportion of weight you ought to take out or put up.<br><br>Weight loss supplements have a variety of elements to get rid of fat. Some supplements focus on improving a person's vitality level. Other supplements target fat burning plus increased metabolism.<br><br>For the concern, the exact technique of acquiring your BMI is that first you must take your fat inside kilograms plus separate it by your height inside meters, squared plus which can be the Body Mass Index. A large amount of people find it really difficult to calculate their BMI Calculating as it is actually a deep procedure and it can be perfectly calculated by a doctor.<br><br>Low-carb diets seems to be efficient in case of overweight teens striving to get rid of fat, according to a latest research by the Cincinnati Children's Hospital Medical Center. |
| '''Earth radius''' is the [[distance]] from [[Earth]]'s center to its surface, about {{convert|6,371|km|mi|sp=us}}. This length is also used as a unit of distance, especially in [[astronomy]] and [[geology]], where it is usually denoted by <math>R_\oplus</math>.
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| This article deals primarily with spherical and [[ellipsoid]]al models of the Earth. See [[Figure of the Earth]] for a more complete discussion of models. The [[Earth]] is only approximately [[Sphere|spherical]], so no single value serves as its natural [[radius]]. Distances from points on the surface to the center range from [[#Notable radii|6,353 km]] to [[#Notable radii|6,384 km]] (3,947–3,968 mi). Several different ways of modeling the Earth as a sphere each yield a [[#Mean radii|mean radius]] of {{convert|6,371|km|mi|sp=us}}.
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| While "radius" normally is a characteristic of perfect spheres, the term as used 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, or of the radius of a sphere whose curvature matches the curvature of the ellipsoidal model of the Earth at a given point.
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| The first scientific estimation of the radius of the earth was given by [[Eratosthenes]] about 240 BC. Estimates of the accuracy of Eratosthenes’s measurement range from within 2% to within 15%.
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| ==Introduction==
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| {{main|Figure of the Earth}}
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| [[Earth]]'s rotation, internal density variations, and external [[tidal force]]s cause it 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 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.
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| 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:
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| * The real surface of the Earth;
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| * 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.</ref>
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| * An ellipsoid: [[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 the 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>
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| * A sphere.
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| 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"''. 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.
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| 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]].
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| ===Physics of Earth's deformation===
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| 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'' <math>a</math> is larger than the ''polar radius'' <math>b</math> by approximately <math>a q</math> where the ''oblateness constant'' <math>q</math> is
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| :::<math>q=\frac{a^3 \omega^2}{GM}\,\!</math>
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| where <math>\omega</math> is the [[angular frequency]], <math>G </math> is the [[gravitational constant]], and <math>M</math> 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]</ref>|group=lower-alpha}} For the Earth {{nowrap|''q''{{sup|−1}} ≈ 289}}, which is close to the measured inverse [[flattening]] {{nowrap|''f''{{sup|−1}} ≈ 298.257}}. 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.<ref>[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>
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| [[Image:Lowresgeoidheight.jpg|400px|right]]
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| The variation in [[density]] and [[Crust (geology)|crustal]] thickness causes gravity to vary on the surface, so that the mean sea level will differ 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 110 m 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>[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>
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| Not all deformations originate within the Earth. The gravity of the Moon and Sun cause the Earth's surface at a given point to undulate by tenths of meters over a nearly 12 hour period (see [[Earth tide]]).
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| ===Radius and local conditions===
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| [[File:Abu Reyhan Biruni-Earth Circumference.svg|thumb|[[Biruni#Geodesy and geography|Biruni]]'s (973–1048) method for calculation of Earth's radius improved accuracy.]] | |
| 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).
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| 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 largest (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 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.
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| 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]], 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]] rose in importance, true global models were developed which, while not as accurate for regional work, best approximate the earth as a whole.
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| ==Fixed radii==
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| The following radii are fixed and do not include a variable location dependence. They are
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| derived from the [[WGS-84]] ellipsoid.<ref name=tr8350_2>http://earth-info.nga.mil/GandG/publications/tr8350.2/wgs84fin.pdf</ref>
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| 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.
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| The radii in this section are for an idealized surface. Even the idealized radii have an uncertainty of ± 2 meters.<ref>http://earth-info.nga.mil/GandG/publications/tr8350.2/tr8350.2-a/Chapter%203.pdf</ref> 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]].
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| The symbol given for the named radius is used in the formulae found in this article.
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| ===Equatorial radius===
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| The Earth's equatorial radius <math>a</math>, or [[semi-major axis]], is the distance from its center to the [[equator]] and equals {{convert|6,378.1370|km|mi|sigfig=8|sp=us}}. The equatorial radius is often used to compare Earth with other [[Planet#Attributes|planets]].
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| ===Polar radius===
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| The Earth's polar radius <math>b</math>, or [[semi-minor axis]], is the distance from its center to the North and South Poles, and equals {{convert|6,356.7523|km|mi|sigfig=8|sp=us}}.
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| ==Radii with location dependence==
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| ===Notable radii===
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| * '''Maximum:''' The summit of [[Chimborazo (volcano)|Chimborazo]] is 6,384.4 km (3,968 mi) from the Earth's center.
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| * '''Minimum:''' The floor of the [[Arctic Ocean]] is ≈{{convert|6,352.8|km|mi|sp=us}} from the Earth's center.<ref>{{cite web|url=http://guam.discover-theworld.com/Country_Guide.aspx?id=96&entry=Mariana+Trench |title=Discover-TheWorld.com - Guam - POINTS OF INTEREST - Don't Miss - Mariana Trench |publisher=Guam.discover-theworld.com |date=1960-01-23 |accessdate=2013-09-16}}</ref>
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| ===Geocentric radius===
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| The distance from the Earth's center to a point on the spheroid surface at geodetic latitude <math>\varphi\,\!</math> is:
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| :<math>R=R(\varphi)=\sqrt{\frac{(a^2\cos\varphi)^2+(b^2\sin\varphi)^2}{(a\cos\varphi)^2+(b\sin\varphi)^2}}\,\!</math>
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| where <math>a</math> and <math>b</math> are the equatorial radius and the polar radius, respectively.
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| ===Radii of curvature===
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| {{see also|Spheroid#Curvature}}
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| [[Eratosthenes#Measurement of the Earth|Eratosthenes]] used two points, one almost exactly north of the other. The points are separated by distance <math> D</math>, and the [[vertical direction]]s at the two points are known to differ by angle of <math>\theta</math>, in radians.
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| A formula used in Eratosthenes' method is
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| :::<math> R= \frac{D}{\theta}\,\!</math>
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| which gives an estimate of radius based on the north-south curvature of the Earth. | |
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| ====Meridional====
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| :In particular the Earth's ''radius of curvature in the (north-south) [[meridian (geography)|meridian]]'' at <math>\varphi\,\!</math> is:
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| :::<math>M=M(\varphi)=\frac{(ab)^2}{((a\cos\varphi)^2+(b\sin\varphi)^2)^{3/2}}\,\!</math>
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| ====Normal====
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| :If one point had appeared due east of the other, one finds the approximate curvature in east-west direction.<ref group=lower-alpha name=curvprim>East-west directions can be misleading. Point ''B'' which appears due East from ''A'' will be closer to the equator than ''A''. Thus the curvature found this way is smaller than the curvature of a circle of constant latitude, except at the equator.
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| West can exchanged for east in this discussion.</ref>
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| :This ''radius of curvature in the [[prime vertical]]'', which is perpendicular, or ''[[orthogonality|normal]]'', to ''M'' at geodetic latitude <math>\varphi\,\!</math> is:<ref group=lower-alpha>''N'' is defined as the radius of curvature in the plane which is normal to both the surface of the ellipsoid at, and the meridian passing through, the specific point of interest.</ref>
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| :::<math>N=N(\varphi)=\frac{a^2}{\sqrt{(a\cos\varphi)^2+(b\sin\varphi)^2}}\,\!</math>
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| Note that ''N=R'' at the equator:
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| [[Image:EarthEllipRadii.jpg|400px|center|Radius at geodetic latitude in black.]]
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| At geodetic latitude 48.46791 degrees (e.g., Lèves, Alsace, France), the radius ''R'' is 20000/π ≈ 6,366.197, namely the radius of a perfect sphere for which the [[meridian arc]] length from the equator to the North Pole is exactly 10000 km, the originally proposed definition of the [[meter]].
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| The Earth's meridional radius of curvature at the equator equals the meridian's [[semi-latus rectum]]:
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| :::<math>\frac{b^2}{a}\,\!</math> {{pad|1em}}={{pad|1em}}6,335.437 km
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| The Earth's polar radius of curvature is:
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| :::<math>\frac{a^2}{b}\,\!</math> {{pad|1em}}={{pad|1em}}6,399.592 km
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| ====Combinations====
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| It is possible to combine the meridional and normal radii of curvature above.
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| The Earth's [[Gaussian curvature|Gaussian radius of curvature]] at latitude <math>\varphi\,\!</math> is:<ref name=Torge>Torge (2001), Geodesy, p.98, eq.(4.23), [http://books.google.com.br/books?id=pFO6VB_czRYC&lpg=PP1&dq=torge%20geodesy&pg=PA98#v=onepage&q=gaussian%20curvature&f=false]</ref>
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| :::<math>R_a=\sqrt{MN}=\frac{a^2b}{(a\cos\varphi)^2+(b\sin\varphi)^2}\,\!</math>
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| The Earth's radius of curvature along a course at an [[azimuth]] (measured clockwise from north) <math>\alpha\,\!</math>, at <math>\varphi\,\!</math> is derived from [[Euler's theorem (differential geometry)|Euler's curvature formula]] as follows:<ref name=Torge>p.97, eq.(4.18)</ref>
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| :::<math>R_c=\frac{{}_{1}}{\frac{\cos^2\alpha}{M}+\frac{\sin^2\alpha}{N}}\,\!</math>
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| The Earth's [[Mean curvature|mean radius of curvature]] at latitude <math>\varphi\,\!</math> is:<ref name=Torge>p.97, eq.(4.19)</ref>
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| :::<math>R_m=\frac{{}_{2}}{\frac{1}{M}+\frac{1}{N}}\,\!</math>
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| ==Mean radii==
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| 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;<ref name=tr8350_2 /> namely,
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| :::<math>\textstyle a = </math> Equatorial radius (6,378.1370 km)
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| :::<math>\textstyle b = </math> Polar radius (6,356.7523 km)
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| 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. | |
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| ===Mean radius===
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| The [[International Union of Geodesy and Geophysics]] (IUGG) defines the mean radius (denoted <math>R_1</math>) to be<ref name="Moritz">Moritz, H. (1980). ''Geodetic Reference System 1980'', by resolution of the XVII General Assembly of the IUGG in Canberra.</ref>
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| :::<math>R_1 = \frac{2a+b}{3}\,\!</math>
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| For Earth, the mean radius is {{convert|6,371.009|km|mi|sp=us}}.<ref name="Moritz2000">{{cite journal |last=Moritz |first=H. |authorlink= |coauthors= |date=March 2000 |title=Geodetic Reference System 1980 |journal=Journal of Geodesy |volume=74 |issue=1 |pages=128–133 |doi=10.1007/s001900050278 |url=http://www.springerlink.com/content/0bgccvjj5bedgdfu/about/ |accessdate= |quote= |bibcode = 2000JGeod..74..128. }}</ref>
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| ===Authalic radius===
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| Earth's authalic ("equal area") radius is the radius of a hypothetical perfect sphere which has the same surface area as the [[reference ellipsoid]]. The [[IUGG]] denotes the authalic radius as <math>R_2</math>.<ref name="Moritz"/>
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| A closed-form solution exists for a spheroid:<ref name="Snyder manual">Snyder, J.P. (1987). ''Map Projections – A Working Manual (US Geological Survey Professional Paper 1395)'' p. 16–17. Washington D.C: United States Government Printing Office.</ref>
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| :::<math>R_2=\sqrt{\frac{a^2+\frac{ab^2}{\sqrt{a^2-b^2}}\ln{\left(\frac{a+\sqrt{a^2-b^2}}b\right)}}{2}}=\sqrt{\frac{a^2}2+\frac{b^2}2\frac{\tanh^{-1}e}e} =\sqrt{\frac{A}{4\pi}}\,\!</math>
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| where <math>e^2=(a^2-b^2)/a^2</math> and <math>A</math> is the surface area of the spheroid.
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| For Earth, the authalic radius is {{convert|6,371.0072|km|mi|sp=us}}.<ref name=Moritz2000/>
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| ===Volumetric radius===
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| 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 <math>R_3</math>.<ref name="Moritz"/>
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| :::<math>R_3=\sqrt[3]{a^2b}\,\!</math>
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| For Earth, the volumetric radius equals {{convert|6,371.0008|km|mi|sp=us}}.<ref name=Moritz2000/>
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| ===Rectifying radius===
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| {{see also|Meridian arc#Polar distance}}
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| Another mean radius is the ''rectifying radius'', giving a sphere with circumference equal to the [[circumference|perimeter]] of the ellipse described by any polar cross section of the ellipsoid. This requires an [[Circumference#Ellipse|elliptic integral]] to find, given the polar and equatorial radii:
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| :::<math>M_r=\frac{2}{\pi}\int_{0}^{\frac{\pi}{2}}\sqrt{{a^2}\cos^2\varphi + {b^2} \sin^2\varphi}\,d\varphi</math>.
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| The rectifying radius is equivalent to the meridional mean, which is defined as the average value of ''M'':<ref name="Snyder manual"/>
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| :::<math>M_r=\frac{2}{\pi}\int_{0}^{\frac{\pi}{2}}\!M(\varphi)\,d\varphi\!</math>
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| 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 {{convert|6,367.4491|km|mi|sp=us}}.
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| The meridional mean is well approximated by the semicubic mean of the two axes:
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| :::<math>M_r\approx\left[\frac{a^{3/2}+b^{3/2}}{2}\right]^{2/3}\,</math>
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| yielding, again, 6,367.4491 km; or less accurately by the [[quadratic mean]] of the two axes:
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| :::<math>M_r\approx\sqrt{\frac{a^2+b^2}{2}}\,\!</math>;
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| about 6,367.454 km; or even just the mean of the two axes:
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| :::<math>M_r\approx\frac{a+b}{2}\,\!</math>;
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| about {{convert|6,367.445|km|mi|sp=us}}.
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| ==Osculating sphere==
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| The best spherical approximation to the ellipsoid in the vicinity of a given point is given by the [[osculating]] sphere. Its radius equals the Gaussian radius of curvature as above, and its radial direction coincides with the ellipsoid [[normal direction]]. This concept aids the interpretation of terrestrial and planetary [[radio occultation]] [[refraction]] measurements.
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| ==See also==
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| * [[Earth mass]]
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| * [[Effective Earth radius]]
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| * [[Geographical distance]]
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| * [[Geodesy]]
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| * [[History of geodesy]]
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| ==Notes==
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| {{notes}}
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| == References ==
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| {{Reflist}}
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| ==External links==
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| * {{cite web|last=Merrifield|first=Michael R.|title=<math>R_\oplus</math> The Earth's Radius (and exoplanets)|url=http://www.sixtysymbols.com/videos/earthradius.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]|year=2010}}
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| {{Units of length used in Astronomy}}
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| {{DEFAULTSORT:Earth Radius}}
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| [[Category:Earth|Radius]]
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| [[Category:Units of length]]
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| [[Category:Human-based units of measurement]]
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| [[Category:Units of measurement in astronomy]]
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| [[Category:Planetary science]]
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| [[Category:Planetary geology]]
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