Editing Earth radius (section)

==Global radii{{anchor|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;<ref name=tr8350_2 /> namely,

:''Equatorial radius'': {{mvar|a}} = ({{val|6378.1370|u=km}})
:''Polar radius'': {{mvar|b}} = ({{val|6356.7523|u=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.

===Arithmetic mean radius===
[[File:WGS84_mean_Earth_radius.svg|thumb|Equatorial (''a''), polar (''b'') and arithmetic mean Earth radii as defined in the 1984 [[World Geodetic System]] revision (not to scale)]]

In geophysics, the [[International Union of Geodesy and Geophysics]] (IUGG) defines the ''Earth's [[arithmetic mean]] radius'' (denoted {{math|''R''<sub>1</sub>}}) to be<ref name="Moritz">Moritz, H. (1980). [https://geodesy.geology.ohio-state.edu/course/refpapers/00740128.pdf ''Geodetic Reference System 1980''] {{Webarchive|url=https://web.archive.org/web/20160220054607/https://geodesy.geology.ohio-state.edu/course/refpapers/00740128.pdf |date=2016-02-20 }}, by resolution of the XVII General Assembly of the IUGG in Canberra.</ref>
:<math>R_1 = \frac{2a+b}{3}.</math>
The factor of two accounts for the biaxial symmetry in Earth's spheroid, a specialization of triaxial ellipsoid.<ref name="Moritz2000">{{cite journal |last=Moritz |first=H. |date=March 2000 |title=Geodetic Reference System 1980 |journal=Journal of Geodesy |volume=74 |issue=1 |pages=128–133 |doi=10.1007/s001900050278 |bibcode = 2000JGeod..74..128. |s2cid=195290884 }}</ref>
For Earth, the arithmetic mean radius is published by IUGG and [[National Geospatial-Intelligence Agency|NGA]] as {{convert|6371.0087714|km|mi|abbr=on}}.<ref name="IAU XXIX"/><ref name="NGA"/>

===Authalic radius===
{{see also|Authalic latitude}}
''Earth's authalic radius'' (meaning [[equal-area projection|"equal area"]]) is the radius of a hypothetical perfect sphere that has the same surface area as the [[reference ellipsoid]]. The [[IUGG]] denotes the authalic radius as {{math|''R''<sub>2</sub>}}.<ref name="Moritz"/>
A closed-form solution exists for a spheroid:<ref name="Snyder manual">Snyder, J. P. (1987). ''[https://pubs.usgs.gov/pp/1395/report.pdf Map Projections – A Working Manual (US Geological Survey Professional Paper 1395)]'' p.&nbsp;16–17. Washington D.C: United States Government Printing Office.</ref>
:<math>R_2
=\sqrt{\frac12\left(a^2+\frac{b^2}{e}\ln{\frac{1+e}{b/a}} \right) }
=\sqrt{\frac{a^2}2+\frac{b^2}2\frac{\tanh^{-1}e}e} 
=\sqrt{\frac{A}{4\pi}},</math>
where {{tmath|1=\textstyle e = \sqrt{a^2 - b^2}\big/a }} is the eccentricity, and {{tmath|A}} is the surface area of the spheroid.

For the Earth, the authalic radius is {{convert|6,371.0072|km|mi|abbr=on}}.<ref name=Moritz2000/>

The authalic radius <math>R_2</math> also corresponds to the ''radius of (global) mean curvature'', obtained by averaging the Gaussian curvature, <math>K</math>, over the surface of the ellipsoid. Using the [[Gauss–Bonnet theorem]], this gives
:<math> \frac{\int K \,dA}{A} = \frac{4\pi}{A} = \frac{1}{R_2^2}.</math>

===Volumetric radius===
Another spherical model is defined by the ''Earth's volumetric radius'', which is the radius of a sphere of volume equal to the ellipsoid. The [[IUGG]] denotes the volumetric radius as {{math|''R''<sub>3</sub>}}.<ref name="Moritz"/>
:<math>R_3 = \sqrt[3]{a^2b}.</math>
For Earth, the volumetric radius equals {{convert|6,371.0008|km|mi|abbr=on}}.<ref name=Moritz2000/>

===Rectifying radius===
{{see also|Quarter meridian|Rectifying latitude}}
Another global radius is the ''Earth's 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:
:<math>M_\text{r} = \frac{2}{\pi} \int_0^{\frac{\pi}{2}} \sqrt{a^2 \cos^2\varphi + b^2 \sin^2\varphi} \,d\varphi.</math>

The rectifying radius is equivalent to the meridional mean, which is defined as the average value of {{mvar|M}}:<ref name="Snyder manual"/>
:<math>M_\text{r} = \frac{2}{\pi} \int_0^{\frac{\pi}{2}} M(\varphi) \,d\varphi.</math>

For integration limits of [0,{{sfrac|{{pi}}|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|abbr=on}}.

The meridional mean is well approximated by the semicubic mean of the two axes,{{cn|date=November 2020}}
:<math>M_\text{r} \approx \left(\frac{a^\frac32 + b^\frac32}{2}\right)^\frac23,</math>

which differs from the exact result by less than {{convert|1|um|sigfig=1|abbr=on}}; the mean of the two axes,
:<math>M_\text{r} \approx \frac{a + b}{2},</math>

about {{convert|6,367.445|km|mi|abbr=on}}, can also be used.