Curvature of the earth

Italy from space and the curvature of the earth as seen from an altitude of 400 km ( ISS )

The curvature of the earth is understood as the fact that the shape of the earth corresponds roughly to a sphere and therefore deviates from a tangential plane over short distances . Because of the curvature of the earth, height measurements must be corrected accordingly.

That the Earth is approximately spherical is, was ionic scientists already v to 600th Known.

calculation

Calculated with a mean earth radius of 6371 km (for more details see), the ideal earth surface deviates from a tangential plane downwards (radial, towards the center of the earth) as follows:

0.8 mm at 100 m

20 mm at 500 m

78 mm over 1 km

1.96 m over 5 km

7.85 m over 10 km

As a simple approximation, precisely for small distances L, the formula can be used, where the distance, the earth's radius is 6,371,000 meters and the deviation is in meters. ${\ displaystyle y = {\ tfrac {L ^ {2}} {2R}}}$ ${\ displaystyle L}$ ${\ displaystyle R}$ ${\ displaystyle y}$

An example to illustrate this: Two people stand 10,000 m (= ) apart. In order for them to have visual contact, either both must be at eye level of 1.96 m (each = 5000 m to the level in the middle at 0 m height) or one person on the level at eye level of zero meters and the 10,000 m distant person at eye level of 7.85 m. ${\ displaystyle 2L}$ ${\ displaystyle L}$

Graphic to illustrate the curvature of the earth; the two points marked in black are 1000 km apart and are both at an altitude of 19.6 km

With a somewhat more precise approximation formula with = earth radius, = distance and = lowering, that is the height that disappears under the tangential plane with "straight view " (see also geodetic visibility ), the following values result from with a given (calculated with ): ${\ displaystyle y = {\ sqrt {L ^ {2} + R ^ {2}}} - R}$ ${\ displaystyle R}$ ${\ displaystyle L}$ ${\ displaystyle y}$ ${\ displaystyle y}$ ${\ displaystyle L}$ ${\ displaystyle R = 6378 \, {\ rm {km}}}$

0000.31 m at 02 km

0001.96 m at 05 km

0007.85 m at 10 km

0031 m at 020 km

0196 m at 050 km

0784 m at 100 km

1764 m at 150 km

3135 m at 200 km

4898 m at 250 km

The correct height measurements because of the curvature of the earth is thus already on short distances vital and growing square with distance. When measuring the location , the curvature of the earth only has an effect at a greater distance and led to the distinction between “ lower ” and “ higher geodesy ”.

Historical representation of the theoretical visibility from the peaks of Mont Blanc and Monte Venda

In a practical example, the determination of the elevation angle of mountains in the mountains, arithmetically the curvature of the earth z. B. for Mont Blanc at 4810 m altitude, depending on the distance, the following elevation angles (assuming a viewpoint at sea level, the values in brackets without the curvature of the earth):

at 050 km + 5.27 ° (5.49 °)

at 100 km + 2.30 ° (2.75 °)

at 150 km + 1.16 ° (1.83 °)

at 200 km + 0.48 ° (1.38 °)

at 250 km −0.02 °

The 250 km value means that at this distance the top of Mont Blanc is below the "horizon line". For observation points above sea level, the calculated elevation angle increases because the "horizon line" moves away from the observer and only the earth curvature component becomes effective beyond it. In practice, terrestrial refraction also plays a role. They refract the light rays in the direction of the curvature of the earth, so that the elevation angles are slightly increased. It can be interpreted to mean that the subsidence caused by the curvature of the earth is reduced by 5 to 15%, depending on meteorological conditions. If z. B. the influence of the refraction were 15%, then in the latter case an elevation angle of 0.04 ° would result.

Photographic documentation

The curvature of the earth can, for example, be documented with telephoto recordings of distant ships on water or of mountains with great visibility. Objects that are far away not only appear smaller due to the shortened perspective , but also lie deeper in the image due to the curvature of the earth than would be the case on a geometric plane. The lower areas of the motif are covered by the horizon. The size of the effect is subject to some fluctuations, which are mainly attributable to terrestrial refraction .

Departing boat disappears behind the horizon.
Due to the curvature of the earth, the base of the wind turbines in the Thorntonbank offshore wind farm is more or less hidden, depending on the distance.
Distant view from Fluchthorn (3399 m) to Rheinwaldhorn (3402 m) and Güferhorn (3379 m), about 100 km away. In between, the cathedral (4545 m), 202 km away, appears lower in the background .

Recordings with non- distortion- free wide-angle lenses from low heights are not suitable . The curved horizon line does not show the curvature of the earth, but an aberration of the lens . The error increases in the direction of the image edges and can be avoided if the horizon runs through the center of the lens ( optical axis ). The curvature of the earth can be technically proven on wide-angle photos from normal cruising altitudes of around 10.5 km, but the curved horizon line is only clearly visible from heights of around 15 km.

Image aberration : as a result of a barrel-shaped distortion caused by the lens , the kink line is shown curved

Image taken from the ISS from a height of 400 km, no significant aberrations: position of the horizon near the center of the image and straight horizontal comparison lines in the foreground

literature

Heribert Kahmen : Surveying Volume 1. De Gruyter Berlin, New York 1988. ISBN 3-11-011759-2 .
Karl Ledersteger : Astronomical and Physical Geodesy . Volume V of the Jordan-Eggert- Kneissl series of books , Handbook of Surveying ; 871 pp. (Pp. 79–155, 455 ff., 705 ff.), Verlag JB Metzler, Stuttgart 1969.
Günter Petrahn (2000): Basics of surveying technology, pocket book survey. Cornelsen-Verlag Scriptor.

Web links

Commons : Curvature of the earth - collection of images, videos, and audio files

Remarks

↑ The natural philosophers knew three proofs that Aristotle later adopted in his writings: 1) Different star skies depending on the latitude , 2) Sinking visibility of ships after leaving, 3) Circular earth shadow during lunar eclipses.
↑ In fact, the earth has a flattening of 0.3 percent; the axes of the mean earth ellipsoid therefore differ between the equator and the poles by 21 km (6378 km and 6357 km, respectively). The minimum and maximum radius of curvature are 6334 km and 6400 km, respectively.
^ David K. Lynch: Visually discerning the curvature of the Earth . In: Applied Optics . tape 47 , no. 34 , December 2008, p. H39–43 (English, thulescientific.com [PDF; 4.4 MB ; accessed on August 17, 2018]).
↑ Andrea Schorsch: From where do you see the curvature of the earth? In: NTV.de. May 8, 2018. Retrieved August 23, 2018 .

[1] The natural philosophers knew three proofs that Aristotle later adopted in his writings: 1) Different star skies depending on the latitude , 2) Sinking visibility of ships after leaving, 3) Circular earth shadow during lunar eclipses.

[2] In fact, the earth has a flattening of 0.3 percent; the axes of the mean earth ellipsoid therefore differ between the equator and the poles by 21 km (6378 km and 6357 km, respectively). The minimum and maximum radius of curvature are 6334 km and 6400 km, respectively.

[3] David K. Lynch: Visually discerning the curvature of the Earth . In: Applied Optics . tape 47 , no. 34 , December 2008, p. H39–43 (English, thulescientific.com [PDF; 4.4 MB ; accessed on August 17, 2018]).

[4] Andrea Schorsch: From where do you see the curvature of the earth? In: NTV.de. May 8, 2018. Retrieved August 23, 2018 .