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Earth radius

Because the Earth is not perfectly spherical, no single value serves as its natural radius. Distances from points on the surface to the center range from 6,353 km to 6,384 km (≈3,947–3,968 mi). Several different ways of modeling the Earth as a sphere each yield a convenient mean radius of 6,371 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 the surface or on an idealized surface that models the Earth. It can also mean some kind of average of such distances, or of the radius of a sphere whose curvature matches the curvature of the ellipsoidal model of the Earth at a given point.

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 also 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 $a q$ 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.[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 change abruptly 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. 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 Earth tide).

Radius and local conditions

Biruni's (973-1048) method for calculation of 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 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. 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. Even the idealized radii have an uncertainty of ± 2 meters.[9] The discrepancy between the ellipsoid radius and the radius to a physical location may be significant. 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 symbol given for the named radius is used in the formulae found in this article.

Equatorial radius

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.

Polar radius

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).

Radii with location dependence

Notable radii

• 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 ≈6,352.8 km (3,947 mi) from the Earth's center.[10]

Radius at a given geodetic latitude

The distance from the Earth's center to a point on the spheroid surface 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 an 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 $\theta$, 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 ≈ 6,366.1977…, namely the radius of a perfect sphere for which the meridian arc length 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 derived from Euler's curvature formula as follows:[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:

$\frac{b^2}{a}\,\!$  = 6,335.437 km

The Earth's polar radius of curvature is:

$\frac{a^2}{b}\,\!$  = 6,399.592 km

Mean radii

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)

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

The International Union of Geodesy and Geophysics (IUGG) defines the mean radius (denoted $R_1$) to be[14]

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

For Earth, the mean radius is 6,371.009 km [15] (≈3,958.761 mi; ≈3,440.069 nmi).

Great-circle radius

If the meridional mean and the mean of the two equatorial axes (which, for Earth, are equal) are averaged together, the mean radius approximation of the average great-circle circumference is found:

$Q_r=\sqrt{\frac{\frac{a^2+b^2}{2}+\frac{a^2+a^2}{2}}{2}}=\sqrt{\frac{3a^2+b^2}{4}}\,\!$;
(Average great-circle circumference ≈ 2πQr})

This is the equivalent great-circle radius of an ellipsoid,[16] which for Earth is 6,372.797 km (≈3,959.873 mi; ≈3,441.035 nmi).

Authalic radius

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 $R_2$.[14]

A closed-form solution exists for a spheroid:[17]

$R_2=\sqrt{\frac{a^2+\frac{ab^2}{\sqrt{a^2-b^2}}\ln{\left(\frac{a+\sqrt{a^2-b^2}}b\right)}}{2}}=\sqrt{\frac{a^2}2+\frac{b^2}2\frac{\tanh^{-1}e}e} =\sqrt{\frac{A}{4\pi}}\,\!$

where $e^2=(a^2-b^2)/a^2$ and $A$ is the surface area of the spheroid.

For Earth, the authalic radius is 6,371.0072 km[15] (≈3,958.760 mi; ≈3,440.069 nmi).

Volumetric radius

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 $R_3$.[14]

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

For Earth, the volumetric radius equals 6,371.0008 km[15] (≈3,958.760 mi; ≈3,440.069 nmi).

Meridional Earth radius

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:

$M_r=\frac{2}{\pi}\int_{0}^{\frac{\pi}{2}}\sqrt{{a^2}\cos^2\phi + {b^2} \sin^2\phi}\,d\phi$.

The rectifying radius is equivalent to the meridional mean, which is defined as the average value of M:[17]

$M_r=\frac{2}{\pi}\int_{0}^{\frac{\pi}{2}}\!M(\phi)\,d\phi\!$

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 (≈3,956.545 mi; ≈3,438.147 nmi).

The meridional mean is well approximated by the semicubic mean of the two axes:

$M_r\approx\left[\frac{a^{3/2}+b^{3/2}}{2}\right]^{2/3}\,$

yielding, again, 6,367.4491 km; or less accurately by the quadratic mean of the two axes:

$M_r\approx\sqrt{\frac{a^2+b^2}{2}}\,\!$;

about 6,367.454 km; or even just the mean of the two axes:

$M_r\approx\frac{a+b}{2}\,\!$;

about 6,367.445 km.

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. ^
7. ^
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 c Moritz, H. (March 2000). "Geodetic Reference System 1980". Journal of Geodesy 74 (1): 128–133. Bibcode 2000JGeod..74..128.. DOI:10.1007/s001900050278.
16. ^ Great-circle radius
17. ^ 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.

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