Earth radius

The earth radius is the radius of the approximated spherical shape of the earth (geoid), the “earth globe”. It is an astronomical unit of measurement and also a fundamental quantity for many fields of knowledge - especially for geosciences - and for technology . It is on average 6,371 kilometers. The earth's diameter is twice the earth's radius, on average around 12,742 kilometers.

Depending on the application, different, more precise earth radii are used that are not based on a spherical shape, but on that of an ellipsoid of revolution or other approximations.

Frequently used values

In a simple approximation, the earth is an ellipsoid of revolution. (Representation with the north pole of the earth at the top.) The ellipsoid of revolution has smaller semiaxes in the polar direction and larger semiaxes towards the equator. (The radii of curvature (in the NS direction) behave the other way around.) If there is an intermediate value, the radius (center distance and at the same time radius of curvature) of the sphere of equal volume lies.

Since the earth or the sea level ( geoid ) does not have an exact spherical shape , but is flattened at the poles by around 21 km (0.335 percent) , the term " globe " must be defined in more detail for more precise details of its radius . The following values are most commonly used:

Equator radius R _A = 6,378,137 m of the mean earth ellipsoid (internationally defined value of the major semi-axis a of the GRS 80 )
Equator radius R _A = 6,378,388 m of the (older) Hayford ellipsoid from 1924
Average radius R ₀ = 6,371,000.785 m (sphere of equal volume, cube root of a a b , the semi-axes of the GRS 80 ellipsoid)
- Rounded value R = 6,371.0 km to the top or
- the older value 6,371.2 km (Hayford ellipsoid 1924 )
Arithmetic mean R = (2a + b) / 3 = 6,371,008.767 m or
- Sphere of equal area 6,371,007.176 m (GRS 80 ellipsoid)

Radii of some important earth ellipsoids

Ellipsoid	year	Equatorial radius a	Pole radius b	Mean value R ₀
GRS 80 , WGS 84	1979	6,378,137.0 m	6,356,752.314 m	6,371,000.8 m
WGS 72	1972	6,378,135.0 m	6,356,750.5 m	6,370,998.9 m
Boarding school Ellipsoid	1967	6,378,165.0 m	6,356,779.702 m	6,371,028.6 m
Hayford ellipsoid	1910/24	6,378,388.0 m	6,356,911.946 m	6,371,221 m
Bessel ellipsoid	1841	6,377,397.155 m	6,356,078.962 m	6,370,283 m
Peru / Lapland	1740	6,379,500 m	6,349,800 m	6,369,600 m

history

From antiquity to Columbus

The idea that the earth was spherical appeared as early as 600 BC. Chr. In the ionic nature philosophy ( Thales of Miletus , Anaximander ) and v 4th century. BC Aristotle gave three astronomical proofs of this fact.

The first determination of the circumference of the earth is from Eratosthenes (around 240 BC ), the inventor of the degree measurement method. He compared the angular heights of the highest point of the sun in Egypt between Alexandria and Syene (today's Aswan), which differ by 1/50 of a full circle. This resulted in the circumference of the earth as 50 times the distance from Alexandria to Aswan, in today's units 835 km by 50 = 41,750 km. The radius can be calculated from the circumference. Eratosthenes calculated in stages ; For the accuracy of his determination of the earth's radius, however, the unit of length used does not play a role: Eratosthenes then came to an earth radius of approx. 6,645 km and thus a value that is 4.2 percent above today's.

In the early Middle Ages, the Arabs determined the length of a degree to be 56 2/3 Arab miles. Since this is to be equated with approx. 2,000 m, the radius of the earth's body is R = 6,500 km, which deviates 2 percent from today's value. In 1023, the mathematician Al-Biruni determined the radius of the globe to be 6,339.6 km using a new measuring method he had invented.

In the 15th century these values were certainly known in Europe, but the Arabic values were sometimes assigned the 25 percent shorter Italian mile. On this basis and at the same time overestimating the length of Asia, Columbus finally came to the incorrect conclusion that one would have to get to East Asia in a few weeks on a western course.

Ferdinand Magellan began a circumnavigation of the world in August 1519 . When the fleet reached the Philippines and thus demonstrably Asian waters in 1520 , the final proof of the spherical shape of the earth was provided, the earth's circumference, which had long been underestimated, was now correctly recognized.

Earth measurement in modern times

The actual size of the earth was only known with a few percent accuracy at the end of the period of discovery . French scientists of the 17th century determined their deviation from the spherical shape by measuring degrees over a few hundred kilometers, but this was still uncertain and in some cases even led to an elongated pole radius. In contrast, calculated Isaac Newton that the Earth's rotation (falsely: centrifugal force) due to the inertia of a flattening would cause the earth.

This question was clarified by the earth measurements carried out by the French Academy with its two expeditions to Lapland and Peru ( 1736 to 1741 ). They also served to define meters (postulated circumference of the earth over the poles = 40,000 km) and provided an accuracy of 0.02 percent or 1.5 km ( meridian quadrant = 10,002,250 m, mean earth radius R ₀ = 6,369.6 km).

Since then, the accuracy with which the mathematical figure of the earth is known has doubled every 50 years. Around 1965, satellite geodesy brought about an enormous increase in accuracy to 20 meters and is now advancing into the centimeter range. Newly developed gradiometric satellites such as GRACE ( 2004 ) and GOCE even aim at changes in the shape of the earth that are assumed to be in the range of a few millimeters per year.

Regional and local details of the earth's shape

The sea is part of the geoid; the elevations of the continents do not

The deviations of the earth from the spherical shape would not yet be visible on an ideal globe , but the high mountains could be felt by their "roughness". The flattening of the earth (flattening at the poles by 21 km or 0.3 percent), on the other hand, must be taken into account in every precise map , often also the typical course of the earth's curvature of each continent ("waves" in the geoid up to ± 100 meters). The radius of curvature can vary regionally between about 6,330 km and 6,400 km, locally even between 5,000 and 8,000 km.

Overall, the regional variability of the earth's radius means a change in the scale of maps and computer models up to a few kilometers to 1,000 km and must be taken into account in almost all applications. When measuring today's technical projects with millimeter precision , these effects already have an effect at a distance of 100 meters.

The altitude or shape of the terrain , on the other hand, is not added to the earth's radius, but - in geographic information systems , for example - added to the databases as an attribute. For tasks with a physical background, the variability of the gravitational acceleration must also be taken into account, which on the earth's surface assumes values of 9.76 to 9.94 m / s².

For tasks in astronomy or space travel , the distance of a point from the center of the earth is often required, the so-called geocentric radius. It can be calculated from the earth ellipsoid used and is, for example, 6,365 to 6,368 km in Central Europe, plus the sea level of the point. However, the radius of curvature of level surfaces and height measurements , which can be up to 30 km larger, must be distinguished from this.

Physical influences

The earth's radius and its variation is not only a fundamental variable in geometric tasks, but also in physics and various geosciences. Here it appears as a distance from the center of the earth or from the earth's axis (R · cos (width)), as a radius of curvature in movements or in measuring beams , as a Gaussian measure of curvature (1 / R²) or in the effect of gradients of different forces.

The mean gravity on the earth's surface is also related to the radius and the earth's mass , as is the mean density of the earth's body. Its value of 5.52 g / cm³ gives geophysics a clear indication that the density of the earth's interior must be much higher than the usual rock densities of 2.5–2.8 g / cm³. The inner shell structure of the earth has been researched for over 100 years using gravimetry , mathematical and seismic models, among other things .

Résumé

The exact figure of the earth is already known to a few centimeters today, although its elevation varies by 10 to 15 km on both sides:

Mean (equal volume) earth radius = 6,371 km
geocentric mean, variation = 6,368 km ± 11 km
Semi- axes of the earth ellipsoid = (equatorial :) 6,378.1 or (polar :) 6,356.7 km
continental radii of curvature (north-south) = (equatorial :) 6,330 to (polar :) 6,400 km
"Earth globe" is therefore only sufficient up to 0.5% accuracy.

It is often unknown that not only does the earth's radius vary due to the flattening of the earth, but that there is also a "latitude problem": the geographical and geocentric latitudes differ by up to 0.19 ° or 22 kilometers. Therefore, by reason of additional local variations of the shape of the Earth by a bullet was for the land surveying the earth's surface by locally best matching reference ellipsoid approximates of which are over a hundred different in use worldwide. The specification of geographical coordinates of a place always relates to a specific reference system ( geodetic datum ).

This means that a plumb bob , thought to be extended under our feet, passes by up to 20 km from the center of the earth . Subjects such as earth measurements , geophysics and satellite geodesy have to deal with the related facts on a daily basis.

Circumference of the earth

Is approximately adopted a spherical shape for the earth's shape, the circumference of the earth can by means of the circumference calculation for a circle approximated by the radius of the earth are calculated . With an earth radius of 6,371 km, this results in a circumference of around 40,030 km. ${\ displaystyle U}$ ${\ displaystyle r}$ ${\ displaystyle U = 2 \ pi \ cdot r}$

Because of the flattening of the earth, the circumference is greatest at the equator at around 40,075 km. The distance between the poles and the equator is about 10,002 km, which corresponds to a circumference of the earth along a longitude of about 40,008 km. This value is remarkably close to the round value of 40,000 km. The reason for this is that, according to an early definition, the meter should be defined as the 10 millionth part of an earth quadrant .

In contrast to the length of the longitudes, the length of the parallels is not uniform and decreases from the equator to the poles. If the earth flattening is neglected, its length can be approximately calculated , whereby the geographical latitude means. The length of the parallel 50th parallel running through Mainz is accordingly about . ${\ displaystyle U (x) = 2 \ pi \ cdot r \ cdot \ cos (x)}$ ${\ displaystyle x}$ ${\ displaystyle U (50 ^ {\ circ}) = 2 \ pi \ cdot 6378 \, \ mathrm {km} \ cdot \ cos (50 ^ {\ circ}) \ approx 25759 \, \ mathrm {km}}$

literature

Wolfgang Torge : Geodesy. 3. completely revised and extended edition, ISBN 3-11-017072-8 . De Gruyter-Verlag, Berlin 2001

Individual evidence

↑ https://blogabissl.blogspot.com/2019/10/wie-biruni-1023-den-erdradius-auf-26.html accessed October 19, 2019
↑ Ludwig Bergmann: Bergmann-Schaefer textbook of experimental physics : Vol. 1. Mechanics, acoustics, heat. 10th edition. de Gruyter, Berlin 1990, p. 3

[1] ttps://blogabissl.blogspot.com/2019/10/wie-biruni-1023-den-erdradius-auf-26.html accessed October 19, 2019

[2] Ludwig Bergmann: Bergmann-Schaefer textbook of experimental physics : Vol. 1. Mechanics, acoustics, heat. 10th edition. de Gruyter, Berlin 1990, p. 3