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# Encyclopedia

Because the Earth is not perfectly spherical, no single value serves as its natural radius. Instead, being nearly spherical, a range of values from 6,357 km to 6,378 km (≈3,950 – 3,963 mi) spans all proposed radii according to need, and several different ways of modeling the Earth as a sphere all yield a convenient mean radius of 6371 km (≈3,959 mi). While "radius" normally is a characteristic of perfect spheres, the term as employed in this article more generally means the distance from some "center" of the Earth to a point on some idealized surface that models the Earth. That distance can vary over the idealized surface. This article deals primarily with spherical and ellipsoidal models of the Earth. See Figure of the Earth for a more complete discussion of models.

The first scientific estimation of the radius of the earth was given by Eratosthenes.

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

## Introduction

### Radius and models of the earth

Earth's rotation, internal density variations, and external tidal forces cause it to deviate systematically from a perfect sphere.[1] Local topography increases the variance, resulting in a surface of unlimited complexity. Our descriptions of the Earth's surface must be simpler than reality in order to be tractable. Hence we create models to approximate the Earth's surface, generally relying on the simplest model that suits the need.

Each of the models in common use come with some notion of "radius". Strictly speaking, spheres are the only solids to have radii, but looser uses of the term "radius" are common in many fields, including those dealing with models of the Earth. Viewing models of the Earth from less to more approximate:

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 Earth at that point".[4] It is also common to refer to any mean radius of a spherical model as "the radius of the earth". On the Earth's real surface, on other hand, it is uncommon to refer to a "radius", since there is no practical need. Rather, elevation above or below sea level is useful.

Regardless of model, any radius falls between the polar minimum of about 6,357 km and the equatorial maximum of about 6,378 km (≈3,950 – 3,963 mi). Hence the Earth deviates from a perfect sphere by only a third of a percent, sufficiently close to treat it as a sphere in many contexts and justifying the term "the radius of the Earth". While specific values differ, the concepts in this article generalize to any major planet.

### Physics of Earth's deformation

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 ω is the angular frequency, G is the gravitational constant, and M is the mass of the planet.[5] For the Earth q−1 ≈ 289, which is close to the measured inverse flattening f−1 ≈ 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.[6]

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).[7]

Not all deformations originate within the earth. Gravity of the Moon and Sun cause Earth's surface to undulate by tenths of meters at a point over a nearly 12 hour period (see Earth tide).

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 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.

The following radii are fixed and do not include a variable location dependence. They are derived from the WGS-84 ellipsoid.[8]

The value for the equatorial radius is defined to the nearest 0.1 meter in WGS-84. The value for the polar radius in this section has been rounded to the nearest 0.1 meter, which is expected to be adequate for most uses. Please refer to the WGS-84 ellipsoid if a more precise value for its polar radius is needed.

The radii in this section are for an idealized surface. The distance from the center of the Earth to an observable location may differ from the WGS-84 radius by ± 2 meters.[9] The uncertainty may be greater. When identifying the position of an observable location, the use of more precise values for WGS-84 radii may not yield a corresponding improvement in accuracy.

The Earth's equatorial radius a, or semi-major axis, is the distance from its center to the equator and equals 6,378.1370 km (≈3,963.191 mi; ≈3,443.918 nmi). The equatorial radius is often used to compare Earth with other planets.

The Earth's polar radius b, 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).

• Maximum: The summit of Chimborazo is 6,384.4 km (3,968 mi) from the Earth's center.
• Minimum: The floor of the Arctic Ocean is ≈6352.8 km (3,947 mi) from the Earth's center.[10]

### Radius at a given geodetic latitude

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

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

These are based on a oblate ellipsoid.

Eratosthenes used two points, one almost 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 θ, in radians. A formula used in 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.[11]
This radius of curvature in the prime vertical, which is perpendicular, or normal, to M at geodetic latitude $\phi\,\!$ is:[12]
$N=N(\phi)=\frac{a^2}{\sqrt{(a\cos(\phi))^2+(b\sin(\phi))^2}}\,\!$

Note that N=R at the equator:

At geodetic latitude 48.46791… degrees (e.g., Lèves, Alsace, France), 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 meter.

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

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

The Earth's radius of curvature along a course at geodetic bearing (measured clockwise from north) $\alpha\,\!$, at $\phi\,\!$ is:[13]

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

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

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;[8] namely,

$\textstyle a =$ Equatorial radius (6,378.1370 km)
$\textstyle b =$ Polar radius (6,356.7523 km)

The International Union of Geodesy and Geophysics (IUGG) defines the mean radius (denoted R1) to be[14]

$R_1 = \frac{2a+b}{3}\,\!$

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

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 R2.[14]

A closed-form solution exists for a spheroid:[15]

$R_2=\sqrt{\frac{a^2+\frac{ab^2}{\sqrt{a^2-b^2}}\ln{\left(\frac{a+\sqrt{a^2-b^2}}b\right)}}{2}}=\sqrt{\frac{A}{4\pi}}\,\!$
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).

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 R3.[14]

$R_3=\sqrt[3]{a^2b}\,\!$

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

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 computing the average value of M:[15]

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

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

## Notes and references

1. ^ For details see Figure of the Earth, Geoid, and Earth tide.
2. ^ There is no single center to the geoid; it varies according to local geodetic conditions.
3. ^ In a geocentric ellipsoid, the center of the ellipsoid coincides with some computed center of the earth, and best models the earth as a whole. Geodetic ellipsoids are better suited to regional idiosyncrasies of the geoid. A partial surface of an ellipsoid gets fitted to the region, in which case the center and orientation of the ellipsoid generally do not coincide with the earth's center of mass or axis of rotation.
4. ^ The value of the radius is completely dependent upon the latitude in the case of an ellipsoid model, and nearly so on the geoid.
5. ^ This follows from the International Astronomical Union definition rule (2): a planet assumes a shape due to hydrostatic equilibrium where gravity and centrifugal forces are nearly balanced. IAU 2006 General Assembly: Result of the IAU Resolution votes
6. ^ Satellites Reveal A Mystery Of Large Change In Earth's Gravity Field, Aug. 1, 2002, Goddard Space Flight Center.
7. ^ NASA's Grace Finds Greenland Melting Faster, 'Sees' Sumatra Quake, December 20, 2005, Goddard Space Flight Center.
8. ^ a b http://earth-info.nga.mil/GandG/publications/tr8350.2/wgs84fin.pdf
9. ^ http://earth-info.nga.mil/GandG/publications/tr8350.2/tr8350.2-a/Chapter%203.pdf
10. ^ http://guam.discover-theworld.com/Country_Guide.aspx?id=96&entry=Mariana+Trench
11. ^ East-west directions can be misleading. Point B which appears due East from A will be closer to the equator than A. Thus the curvature found this way is smaller than the curvature of a circle of constant latitude, except at the equator. West can exchanged for east in this discussion.
12. ^ N is defined as the radius of curvature in the plane which is normal to both the surface of the ellipsoid at, and the meridian passing through, the specific point of interest.
13. ^ A related application of M and N: if two nearby points have the difference in latitude of $d\phi\,\!$ and longitude of $d\lambda\,\!$ (in radians) with $M\,\!$ and $N\,\!$ calculated at mean latitude $\phi_m\,\!$, then the distance D between them can be found loxodromically:
${\color{white}\frac{\big|}{}}dH=\sqrt{(d\phi)^2+(\cos(\phi_m)d\lambda)^2}=\sqrt{(dH\cos(\alpha))^2+(dH\sin(\alpha))^2}\,\!$
\begin{align}{\color{white}\frac{\big|}{}} dD&=\sqrt{(M(\phi_m)d\phi)^2+(N(\phi_m)\cos(\phi)d\lambda)^2}\ &=\sqrt{(dD\cos(\alpha))^2+(dD\sin(\alpha))^2},\ &=\sqrt{(M(\phi_m)\cos(\alpha))^2+(N(\phi_m)\sin(\alpha))^2}\cdot{dH}\ &=\overset{{}_{\smile}}{R}\cdot{dH}\end{align}\,\!
Thus $\overset{{}_{\smile}}{R}\,\!$ is the radius of arc, or arcradius, and $d\phi\,\!$ and $d\lambda\,\!$ can be estimated from D, M, and N.
14. ^ a b c Moritz, H. (1980). Geodetic Reference System 1980, by resolution of the XVII General Assembly of the IUGG in Canberra.
15. ^ a b Snyder, J.P. (1987). Map Projections—A Working Manual (US Geological Survey Professional Paper 1395) p. 16–17. Washington D.C: United States Government Printing Office.