Earth radius
From Free net encyclopedia
The radius of Earth (or any planet) is the distance from its center to a point on its surface at mean sea level. Like most planets, Earth is not a perfect sphere, but instead is somewhat flattened at the North and South Poles, and bulges at the equator, which means that its radius and corresponding radius of curvature, or arcradius, differs depending on where you measure it (and, in the case of the arcradius, even which direction is being faced!). This elliptic shape is known as an oblate spheroid/ellipsoid.
Given all of the advancements in measuring technology (including satellites) and tailoring to regional topography, many different reference ellipsoid models have made their way into general usage over the years, providing slightly different values.
However, local variations in terrain negate any chance of pronouncing an absolutely "precise" radius/arcradius—one can only find a mathematically precise value based on a given model (with the plethora of—some seemingly outdated—models accommodating regional terrain and accumulated data found from them).
Therefore, the values defined below are based on a "general purpose" model, refined as globally precise as possible.
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Equatorial radius: <math>a</math>
The Earth's equatorial radius, or semi-major axis, is the distance from its centre to the equator and equals 6,378.135 km (≈3,963.189 mi; ≈3,443.917 nmi).
Polar radius: <math>b</math>
The Earth's polar radius, or semi-minor axis, is the distance from its center to the North and South Poles, and equals 6,356.750 km (≈3,949.901 mi; ≈3,432.370 nmi).
Radius at a given geocentric latitude
The Earth's radius at geocentric latitude <math>l</math> is:
- <math>R_l = \frac{a b}{\sqrt{a^2-(a^2-b^2)cos^2 l}}</math>
(the latitude is <math>\frac{\pi}{2}</math> at the north pole, 0 at the equator, and <math>-\frac{\pi}{2}</math> at the south pole).
Radius of curvature
The Earth's equatorial radius of curvature in the meridian is:
- <math>\frac{b^2}{a}</math>= 6335.437 km.
The Earth's polar radius of curvature is:
- <math>\frac{a^2}{b}</math>= 6399.592 km.
The Earth's radius of curvature in the meridian (north-south) (meridional radius of curvature) at geodetic latitude <math>l</math> is:
- <math>R_m = a^2 b^2 (a^2 cos^2 l + b^2 sin^2 l )^{- \frac{3}{2}}</math>
The Earth's radius of curvature in the prime vertical (east-west) (transverse radius of curvature) at geodetic latitude <math>l</math> is:
- <math>R_t = \frac{a^2}{\sqrt{a^2 cos^2 l + b^2 sin^2 l}}</math>
The Earth's radius of curvature along a course at bearing β at geodetic latitude <math>l</math> is:
- <math>R_c = \frac{R_m R_t}{R_m sin^2 \beta + R_t cos^2 \beta}</math>
The Earth's mean radius of curvature (averaging over all directions) at geodetic latitude <math>l</math> is:
- <math>R_a = \sqrt{R_m R_t} = \frac{a^2 b}{a^2 cos^2 l + b^2 sin^2 l}</math>
Quadratic mean radius: <math>Q_r</math>
The ellipsoidal quadratic mean radius provides the best approximation of Earth's average transverse meridional arcradius and radius:
- <math>Q_r = \sqrt{\frac{3a^2 + b^2}{4}}</math>
It is this radius that would be used to approximate the ellipsoid's average great ellipse (i.e., this is the equivalent spherical "great-circle" radius of the ellipsoid).</br> For Earth, <math>Q_r</math> equals 6,372.795477598 km (≈3,959.871 mi; ≈3,441.034 nmi).
Authalic mean radius: <math>A_r</math>
Earth's authalic ("equal area") mean radius is approximately 6,371.005076123 km (≈3,958.759 mi; ≈3,440.067 nmi). This number is derived by square rooting the average (latitudinally cosine corrected) geometric mean of the meridional and transverse equatorial, or "normal" (i.e., perpendicular), arcradii of all surface points on the spheroid, which can be reduced to a closed-form solution:
- <math>A_r = \sqrt{\frac{a^2+\frac{ab^2}{\sqrt{a^2-b^2}}\ln{(\frac{a+\sqrt{a^2-b^2}}b)}}{2}}= \sqrt{\frac{A}{4\pi}}</math>
- (where <math>A</math> is the authalic surface area of Earth. This
- would be the radius of a hypothetical perfect sphere
- which has the same, geometric mean oriented surface
- area as the spheroid.)
Volumetric radius: <math>V_r</math>
Another, less utilized, sphericalization is that of the volumetric radius, which is the radius of a sphere of equal volume:
- <math>V_r = \sqrt[3]{a^2b}</math>
For Earth, the volumetric radius equals 6,370.998685023 km (≈3,958.755 mi; ≈3,440.064 nmi).
- (Note: Earth radius is sometimes used as a unit of distance, especially in astronomy and geology. It is usually denoted by <math>R_E</math>.)