Earth radius

The Earth's shape, like that of all major planets, may be approximated by a sphere. A sphere has a constant radius, but it is usual when speaking of the radius of the Earth to refer to various fixed distances from surface to "center", and to various mean radii, discussed below. The distance from the mean sea level at each point on the surface to the center (the radius of the Earth at that point) varies with latitude. This distance ranges from 6,356.750 km to 6,378.135 km (≈3,949.901 – 3,963.189 mi), which are the polar radius and the equatorial radius, respectively. Variations in radius due to topography, such as at mountain peaks, are not addressed in this article. The various values given in this article for radius of the earth differ by less than one percent.

For all planets, including Earth, the systematic causes of the distortion from true sphericity are rotation, variation of mass density within the planet, and tidal forces.

Note: Earth radius is sometimes used as a unit of distance, especially in astronomy and geology. It is usually denoted by $R_{\oplus }$.

Introduction

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 $aq$ where the oblateness constant $q$ is

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

where $\omega$ is the angular frequency, $G$ is the gravitational constant, and $M$ is the mass of the planet. For the Earth $q^{-1}\approx 289$, which is close to the measured inverse flattening $f^{-1}\approx 298.257$. 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.

The variation in density and 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 have abrupt changes due to earthquakes (such as the Sumatra-Andaman earthquake) or reduction in ice masses (such as Greenland).

The tides from the gravity of the Moon and Sun cause the surface of the Earth to rise and fall by tenths of meters at a point over a nearly 12 hr period.

Therefore, 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).

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 depends on 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 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 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.

Fixed radii

The following radii are fixed and do not include a variable location dependence.

Equatorial radius:  $a$

The Earth's equatorial radius, or semi-major axis, is the distance from its center to the equator and equals 6,378.137 km (≈3,963.191 mi; ≈3,443.918 nmi). At -0°N 121.83°E / -0; 121.83, the geoid height rises to 63.42 m above the reference ellipsoid (WGS-84), giving a total radius of 6,378.200 km. The equatorial radius is often used to compare Earth with other planets.

Polar radius:  $b$

The Earth's polar radius, or semi-minor axis, is the distance from its center to the North and South Poles, and equals 6,356.7523 km (≈3,949.903 mi; ≈3,432.372 nmi). The geoid height (WGS-84) at the North Pole is 13.6 m above the reference ellipsoid, and at the South Pole 29.5 m below the reference, giving the more exact 6,356.766 km and 6,356.723 km, respectively.

Radii with location dependence

Radius at a given geodetic latitude

The Earth's radius at geodetic latitude, $\phi \,\!$, is:

$R=R(\phi )={\sqrt {\frac {(a^{2}\cos(\phi ))^{2}+(b^{2}\sin(\phi ))^{2}}{(a\cos(\phi ))^{2}+(b\sin(\phi ))^{2}}}};\,\!$

Radius of curvature

These are based on a oblate ellipsoid.

Eratosthenes used two points, one exactly north of the other. The points are separated by distance $D$, and the vertical directions at the two points are known to differ by angle of $\theta$, in radians. A formula based on Eratosthenes method is

$R={\frac {D}{\theta }};\,\!$

which gives an estimate of radius based on the north-south curvature of the Earth.

Meridional

In particular the Earth's radius of curvature in the (north-south) meridian at $\phi \,\!$ is:

$M=M(\phi )={\frac {(ab)^{2}}{((a\cos(\phi ))^{2}+(b\sin(\phi ))^{2})^{3/2}}};\,\!$

Normal

If one point had appeared due east of the other, one finds the approximate curvature in east-west direction.

This radius of curvature in the prime vertical, which is perpendicular, or normal, to M at geodetic latitude $\phi \,\!$ is:

$N=N(\phi )={\frac {a^{2}}{\sqrt {(a\cos(\phi ))^{2}+(b\sin(\phi ))^{2}}}};\,\!$

Note that N=R at the equator:

Radius at geodetic latitude in black.

At geodetic latitude 48.46791… degrees, e.g. Lèves, Alsace, the radius R is 20000/π ≈ 6366.1977…, namely the radius of a perfect sphere for which the distance from the equator to the North Pole is exactly 10000 km, the originally proposed definition of the metre.

The Earth's mean radius of curvature (averaging over all directions) at latitude $\phi \,\!$ is:

$R_{a}={\sqrt {MN}}={\frac {a^{2}b}{(a\cos(\phi ))^{2}+(b\sin(\phi ))^{2}}};\,\!$

The Earth's radius of curvature along a course at geodetic bearing (measured clockwise from north) $\alpha \,\!$, at $\phi \,\!$ is:

$R_{c}={\frac {{}_{1}}{{\frac {\cos(\alpha )^{2}}{M}}+{\frac {\sin(\alpha )^{2}}{N}}}}.\,\!$

The Earth's equatorial radius of curvature in the meridian is:

${\frac {b^{2}}{a}}\,\!$= 6335.437 km

The Earth's polar radius of curvature is:

${\frac {a^{2}}{b}}\,\!$= 6399.592 km

Mean radii

The various radii explained below use the notation and dimensions noted above for the Earth as derived from WGS (E2008); namely,

$\textstyle a=$ Equatorial radius (6,378,137.0 m)

$\textstyle b=$ Polar radius (6,356,752.3 m)

Mean radius: $R_{1}$

The International Union of Geodesy and Geophysics (IUGG) defines the mean radius (denoted $R_{1}$) to be

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

For Earth, the mean radius is 6,371.009 km (˜3,958.761 mi; ˜3,440.069 nmi).

Authalic radius: $R_{2}$

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 $R_{2}$.

A closed-form solution exists for a spheroid:

$R_{2}={\sqrt {\frac {a^{2}+{\frac {ab^{2}}{\sqrt {a^{2}-b^{2}}}}\ln {\left({\frac {a+{\sqrt {a^{2}-b^{2}}}}{b}}\right)}}{2}}}={\sqrt {\frac {A}{4\pi }}};\,\!$

where $A\,\!$ is the surface area of the spheroid

For Earth, the authalic radius is 6,371.0072 km (˜3,958.760 mi; ˜3,440.069 nmi).

Volumetric radius: $R_{3}$

Another, less utilized, sphericalization is that of the volumetric radius, which is the radius of a sphere of equal volume. The IUGG denotes the volumetric radius as $R_{3}$.

$R_{3}={\sqrt[{3}]{a^{2}b}};\,\!$

For Earth, the volumetric radius equals 6,371.0008 km (˜3,958.760 mi; ˜3,440.069 nmi).

Meridional Earth radius

Another radius mean is the meridional mean, which equals the radius used in finding the perimeter of an ellipse. It can also be found by just finding the average value of M:

$M_{r}={\frac {2}{\pi }}\int _{0}^{90^{\circ }}\!M(\phi )\,d\phi \;\approx \left[{\frac {a^{1.5}+b^{1.5}}{2}}\right]^{1/1.5};\,\!$

For Earth, this works out to 6367.4491 km (≈3,956.545 mi; ≈3,438.147 nmi).

See also

• Effective Earth radius
• Radius of curvature
• Figure of the Earth
• Geographical distance
• Earth tide