Contents 1 Introduction 1.1 Physics of Earth's deformation
1.2
2 Fixed radius 2.1 Equatorial radius 2.2 Polar radius 3 Location-dependent radii 3.1 Geocentric radius 3.1.1 Notable geocentric radii 3.2 Radii of curvature 3.2.1 Principal sections 3.2.1.1 Meridional 3.2.1.2 Prime vertical 3.2.2 Directional 3.2.3 Combinations 4 Global average radii 4.1 Mean radius
4.2 Authalic radius
4.3 Volumetric radius
4.4 Rectifying radius
4.5
5 Osculating sphere 6 See also 7 Notes 8 References 9 External links Introduction[edit] Scale drawing of the oblateness of the 2003 IERS reference ellipsoid.
The outer edge of the dark blue line is an ellipse with the same
eccentricity as that of the Earth, with north at the top. For
comparison, the outer edge of light blue area is a circle of diameter
equal to the minor axis. The red line denotes the
Main article: Figure of Earth Earth's rotation, internal density variations, and external tidal forces cause its shape to deviate systematically from a perfect sphere.[b] Local topography increases the variance, resulting in a surface of profound complexity. Our descriptions of the Earth's surface must be simpler than reality in order to be tractable. Hence, we create models to approximate characteristics of the 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 the Earth. Here 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;[c] An ellipsoid, geocentric to model the entire Earth, or else geodetic for regional work;[d] 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
q = a 3 ω 2 G M , displaystyle q= frac a^ 3 omega ^ 2 GM ,, where ω is the angular frequency, G is the gravitational constant,
and M is the mass of the planet.[f] For the
The variation in density and 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 110 m (360 ft) on Earth. The geoid
height can change abruptly due to earthquakes (such as the
Sumatra-Andaman earthquake) or reduction in ice masses (such as
Greenland).[9]
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
Al-Biruni's (973–1048) method for calculation of the 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 (16 ft) of
reference ellipsoid height, and to within 100 m (330 ft) of
mean sea level (neglecting geoid height).
Additionally, the radius can be estimated from the curvature of the
R ( φ ) = ( a 2 cos φ ) 2 + ( b 2 sin φ ) 2 ( a cos φ ) 2 + ( b sin φ ) 2 displaystyle R(varphi )= sqrt frac (a^ 2 cos varphi )^ 2 +(b^ 2 sin varphi )^ 2 (acos varphi )^ 2 +(bsin varphi )^ 2 where a and b are, respectively, the equatorial radius and the polar radius. Notable geocentric radii[edit] Maximum: The summit of Chimborazo is 6,384.4 km (3,967.1 mi)
from the Earth's center.
Minimum: The floor of the
Radii of curvature[edit]
See also:
M ( φ ) = ( a b ) 2 ( ( a cos φ ) 2 + ( b sin φ ) 2 ) 3 2 . displaystyle M(varphi )= frac (ab)^ 2 big ( (acos varphi )^ 2 +(bsin varphi )^ 2 big ) ^ frac 3 2 ,. This is the radius that
N ( φ ) = a 2 ( a cos φ ) 2 + ( b sin φ ) 2 . displaystyle N(varphi )= frac a^ 2 sqrt (acos varphi )^ 2 +(bsin varphi )^ 2 ,. This radius is also called the transverse radius of curvature. At the equator, N = R. Three different radii as a function of Earth's latitude. R is the geocentric radius; M is the meridional radius of curvature; and N is the prime vertical radius of curvature. The Earth's meridional radius of curvature at the equator equals the meridian's semi-latus rectum: b2/a = 6,335.439 km The Earth's polar radius of curvature is: a2/b = 6,399.594 km Directional[edit] The Earth's radius of curvature along a course at an azimuth (measured clockwise from north) α at φ is derived from Euler's curvature formula as follows:[14]:97 R c = 1 cos 2 α M + sin 2 α N . displaystyle R_ mathrm c = frac 1 dfrac cos ^ 2 alpha M + dfrac sin ^ 2 alpha N ,. Combinations[edit] It is possible to combine the principal radii of curvature above in a non-directional manner. The Earth's Gaussian radius of curvature at latitude φ is:[14] R a = M N = a 2 b ( a cos φ ) 2 + ( b sin φ ) 2 . displaystyle R_ mathrm a = sqrt MN = frac a^ 2 b (acos varphi )^ 2 +(bsin varphi )^ 2 ,. The Earth's mean radius of curvature at latitude φ is:[14]:97 R m = 2 1 M + 1 N displaystyle R_ mathrm m = frac 2 dfrac 1 M + dfrac 1 N ,! Global average radii[edit]
The
a = Equatorial radius (7006637813700000000♠6378.1370 km) b = Polar radius (7006635675230000000♠6356.7523 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. Mean radius[edit] Equatorial (a), polar (b) and mean
In geophysics, the International Union of
R 1 = 2 a + b 3 displaystyle R_ 1 = frac 2a+b 3 ,! For Earth, the mean radius is 6,371.0088 km
(3,958.7613 mi).[16]
In astronomy, the
R e E N displaystyle mathcal R _ mathrm eE ^ mathrm N , which is defined to be 6,378.1 km (3,963.2 mi).[1]:3 The
nominal polar
R p E N displaystyle mathcal R _ mathrm pE ^ mathrm N = 6,356.8 km (3,949.9 mi). These values correspond to the
zero tide radii. Equatorial radius is conventionally used as the
nominal value unless the polar radius is explicitly required.[1]:4
Authalic radius[edit]
See also: Authalic latitude
Earth's authalic ("equal area") radius is the radius of a hypothetical
perfect sphere that has the same surface area as the reference
ellipsoid. The
R 2 = a 2 + a b 2 a 2 − b 2 ln ( a + a 2 − b 2 b ) 2 = a 2 2 + b 2 2 tanh − 1 e e = A 4 π , displaystyle 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 4pi ,, where e2 = a2 − b2/a2 and A is the surface area of the spheroid. For the Earth, the authalic radius is 6,371.0072 km (3,958.7603 mi).[16] Volumetric radius[edit] 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 R3.[15] R 3 = a 2 b 3 . displaystyle R_ 3 = sqrt[ 3 ] a^ 2 b ,. For Earth, the volumetric radius equals 6,371.0008 km (3,958.7564 mi).[16] Rectifying radius[edit] See also: Meridian arc § Polar distance, and Rectifying latitude Another mean radius is the rectifying radius, giving a sphere with circumference equal to the perimeter of the ellipse described by any polar cross section of the ellipsoid. This requires an elliptic integral to find, given the polar and equatorial radii: M r = 2 π ∫ 0 π 2 a 2 cos 2 φ + b 2 sin 2 φ d φ . displaystyle M_ mathrm r = frac 2 pi int _ 0 ^ frac pi 2 sqrt a^ 2 cos ^ 2 varphi + b^ 2 sin ^ 2 varphi ,dvarphi ,. The rectifying radius is equivalent to the meridional mean, which is defined as the average value of M:[17] M r = 2 π ∫ 0 π 2 M ( φ ) d φ . displaystyle M_ mathrm r = frac 2 pi int _ 0 ^ frac pi 2 !M(varphi ),dvarphi ,. 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: M r ≈ ( a 3 2 + b 3 2 2 ) 2 3 , displaystyle M_ mathrm r approx left( frac a^ frac 3 2 +b^ frac 3 2 2 right)^ frac 2 3 ,, yielding, again, 6,367.4491 km; or less accurately by the quadratic mean of the two axes: M r ≈ a 2 + b 2 2 ; displaystyle M_ mathrm r approx sqrt frac a^ 2 +b^ 2 2 ,; about 6,367.454 km; or even just the mean of the two axes: M r ≈ a + b 2 ; displaystyle M_ mathrm r approx frac a+b 2 ,; about 6,367.445 km (3,956.547 mi).
Notes[edit] ^ See Figure of
References[edit] ^ a b c Mamajek, E. E; Prsa, A; Torres, G; et al. (2015). "IAU 2015
Resolution B3 on Recommended Nominal Conversion Constants for Selected
Solar and Planetary Properties". arXiv:1510.07674
[astro-ph.SR].
^ a b Frédéric Chambat; Bernard Valette (2001). "Mean radius, mass,
and inertia for reference
External links[edit] Merrifield, Michael R. (2010). " R ⊕ displaystyle R_ oplus The Earth's
v t e Units of length used in Astronomy Astronomical system of units
See also Cosmic distance ladder Orders of magnitude (length) Convers |