Earth radius

Rotation of a planet causes it to approximate an oblate ellipsoid/spheroid with a bulge at the equator and flattening at the North and South Poles, so that the equatorial radius a is larger than the polar radius b by approximately aq. The oblateness constant q is given by

${\displaystyle q={\frac {a^{3}\omega ^{2}}{GM}}\,,}$

where ω is the angular frequency, G is the gravitational constant, and M is the mass of the planet.[lower-alpha 5] For the Earth 1/q ≈ 289, which is close to the measured inverse flattening 1/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 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).[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 (see Earth tide).

Al-Biruni's (973 – c. 1050) method for calculation of the Earth's radius simplified measuring the circumference compared to taking measurements from two locations distant from each other.

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 Earth at a point. Like a torus, the curvature at a point will be greatest (tightest) in one direction (north–south on Earth) and smallest (flattest) perpendicularly (east–west). The corresponding radius of curvature depends on the location and direction of measurement from that point. A consequence is that a distance to the true 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 by Eratosthenes, many models have been created. Historically, these models were based on regional topography, giving the best reference ellipsoid for the area under survey. As satellite remote sensing and especially the Global 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 geocentric radius is the distance from the Earth's center to a point on the spheroid surface at geodetic latitude φ:

${\displaystyle R(\varphi )={\sqrt {\frac {(a^{2}\cos \varphi )^{2}+(b^{2}\sin \varphi )^{2}}{(a\cos \varphi )^{2}+(b\sin \varphi )^{2}}}}}$

where a and b are, respectively, the equatorial radius and the polar radius.

The extrema geocentric radii on the ellipsoid coincide with the equatorial and polar radii. They are vertices of the ellipse and also coincide with minimum and maximum radius of curvature.

There are two principal radii of curvature: along the meridional and prime-vertical normal sections.

In particular, the Earth's meridional radius of curvature (in the north–south direction) at φ is:[9]

${\displaystyle M(\varphi )={\frac {(ab)^{2}}{{\big (}(a\cos \varphi )^{2}+(b\sin \varphi )^{2}{\big )}^{\frac {3}{2}}}}={\frac {a(1-e^{2})}{(1-e^{2}\sin ^{2}\varphi )^{\frac {3}{2}}}}={\frac {1-e^{2}}{a^{2}}}N(\varphi )^{3}\,.}$

where ${\displaystyle e}$ is the eccentricity of the earth. This is the radius that Eratosthenes measured in his arc measurement.

The length PQ, called the prime vertical radius, is ${\displaystyle N(\phi )}$. The length IQ is equal to ${\displaystyle \,e^{2}N(\phi )}$. ${\displaystyle R=(X,\,Y,\,Z)}$.

If one point had appeared due east of the other, one finds the approximate curvature in the east–west direction.[lower-alpha 6]

This Earth's prime-vertical radius of curvature, also called the Earth's transverse radius of curvature, is defined perpendicular (orthogonal) to M at geodetic latitude φ[lower-alpha 7] and is:[9]

${\displaystyle N(\varphi )={\frac {a^{2}}{\sqrt {(a\cos \varphi )^{2}+(b\sin \varphi )^{2}}}}={\frac {a}{\sqrt {1-e^{2}\sin ^{2}\varphi }}}\,.}$

N can also be interpreted geometrically as the normal distance from the ellipsoid surface to the polar axis.[10] The radius of a parallel of latitude is given by ${\displaystyle p=N\cos(\varphi )}$.[11][12]

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 prime-vertical radius of curvature at the equator equals the equatorial radius, N = a.

The Earth's polar radius of curvature (either meridional or prime-vertical) is:

a²/b = 6,399.594 km

The Earth's azimuthal radius of curvature, along an Earth normal section at an azimuth (measured clockwise from north) α and at latitude φ, is derived from Euler's curvature formula as follows:[14]^: 97

${\displaystyle R_{\mathrm {c} }={\frac {1}{{\dfrac {\cos ^{2}\alpha }{M}}+{\dfrac {\sin ^{2}\alpha }{N}}}}\,.}$

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]

${\displaystyle R_{\mathrm {a} }(\varphi )={\frac {1}{\sqrt {K}}}={\frac {1}{2\pi }}\int _{0}^{2\pi }R_{\mathrm {c} }(\alpha )\,d\alpha \,={\sqrt {MN}}={\frac {a^{2}b}{(a\cos \varphi )^{2}+(b\sin \varphi )^{2}}}={\frac {a{\sqrt {1-e^{2}}}}{1-e^{2}\sin ^{2}\varphi }}\,.}$

Where K is the Gaussian curvature, ${\displaystyle K=\kappa _{1}\,\kappa _{2}={\frac {\det \,B}{\det \,A}}}$.

The Earth's mean radius of curvature at latitude φ is:[14]^: 97

${\displaystyle R_{\mathrm {m} }={\frac {2}{{\dfrac {1}{M}}+{\dfrac {1}{N}}}}\,\!}$

In astronomy, the International Astronomical Union denotes the nominal equatorial Earth radius as ${\displaystyle {\mathcal {R}}_{\mathrm {eE} }^{\mathrm {N} }}$, which is defined to be 6,378.1 km (3,963.2 mi).[1]^: 3 The nominal polar Earth radius is defined as ${\displaystyle {\mathcal {R}}_{\mathrm {pE} }^{\mathrm {N} }}$ = 6,356.8 km (3,949.9 mi). These values correspond to the zero Earth 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 a unit of length for astronomy. (The notation is defined such that it can be easily generalized for other planets; e.g., ${\displaystyle {\mathcal {R}}_{\mathrm {pJ} }^{\mathrm {N} }}$ for the nominal polar Jupiter radius.)

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 R₁) to be[2]

${\displaystyle R_{1}={\frac {2a+b}{3}}\,\!}$

The factor of two accounts for the biaxial symmetry in Earth's spheroid, a specialization of triaxial ellipsoid. For Earth, the arithmetic mean radius is 6,371.0088 km (3,958.7613 mi).[15]

Earth's authalic radius (meaning "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 R₂.[2] A closed-form solution exists for a spheroid:[16]

${\displaystyle R_{2}={\sqrt {\frac {a^{2}+{\frac {b^{2}}{e}}\ln {\left({\frac {1+e}{b/a}}\right)}}{2}}}={\sqrt {{\frac {a^{2}}{2}}+{\frac {b^{2}}{2}}{\frac {\tanh ^{-1}e}{e}}}}={\sqrt {\frac {A}{4\pi }}}\,,}$

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).[15]

The authalic radius ${\displaystyle R_{2}}$ also corresponds to the radius of (global) mean curvature, obtained by averaging the Gaussian curvature, ${\displaystyle K}$, over the surface of the ellipsoid. Using the Gauss–Bonnet theorem, this gives

${\displaystyle {\frac {\int K\,dA}{A}}={\frac {4\pi }{A}}={\frac {1}{R_{2}^{2}}}.}$

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 R₃.[2]

${\displaystyle R_{3}={\sqrt[{3}]{a^{2}b}}\,.}$

For Earth, the volumetric radius equals 6,371.0008 km (3,958.7564 mi).[15]

Another global radius is the Earth's 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:

${\displaystyle M_{\mathrm {r} }={\frac {2}{\pi }}\int _{0}^{\frac {\pi }{2}}{\sqrt {{a^{2}}\cos ^{2}\varphi +{b^{2}}\sin ^{2}\varphi }}\,d\varphi \,.}$

The rectifying radius is equivalent to the meridional mean, which is defined as the average value of M:[16]

${\displaystyle M_{\mathrm {r} }={\frac {2}{\pi }}\int _{0}^{\frac {\pi }{2}}\!M(\varphi )\,d\varphi \,.}$

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,

${\displaystyle M_{\mathrm {r} }\approx \left({\frac {a^{\frac {3}{2}}+b^{\frac {3}{2}}}{2}}\right)^{\frac {2}{3}}\,,}$

which differs from the exact result by less than 1 μm (4×10⁻⁵ in); the mean of the two axes,

${\displaystyle M_{\mathrm {r} }\approx {\frac {a+b}{2}}\,,}$

about 6,367.445 km (3,956.547 mi), can also be used.

• Maximum R_t: the summit of Chimborazo is 6,384.4 km (3,967.1 mi) from the Earth's center.
• Minimum R_t: the floor of the Arctic Ocean is 6,352.8 km (3,947.4 mi) from the Earth's center.[17]

The topographical mean geocentric distance averages elevations everywhere, resulting in a value 230 m larger than the IUGG mean radius, the authalic radius, or the volumetric radius. This topographical average is 6,371.230 km (3,958.899 mi) with uncertainty of 10 m (33 ft).[18]