Meridian arc

As meridian arc a north-south running is measuring distance on the earth's surface or its mathematical equivalent on the earth ellipsoid referred (see. Meridian ).

The first-mentioned measuring section can be used in the "method of degree measurement " to determine the mean curvature of the earth and thus the earth's radius . To do this, the geographical latitudes of the two route endpoints (φ ₁ , φ ₂ ) must also be measured. These latitudes are determined astronomically by observing the elevation angles of stars.

The route is now reduced to sea level and its length is compared with the difference in geographical latitudes. If the meridional arc has the length B and the difference in width is β = | φ ₁ -φ ₂ |, then the local radius of curvature results with R = B / β. Together with a second meridian arc, the shape of the earth ellipsoid can be derived from this - such as B. 1735–1740 during the famous expeditions of the Paris Academy to Lapland and Peru .

Since around 1900, however, extensive surveying networks have been used in geodesy instead of the meridian method.

Important meridian arcs up to 1900

The first meridian arcs in the history of science served as evidence of the spherical earth figure and its size. According to Eratosthenes' earth measurement , the measurement of the degree by the caliph Al-Ma'mun should be mentioned, a 2 ° long arc near Baghdad , which resulted in 122 km for the meridian degree (today's value depending on the latitude 111–112 km). The measurement of Fernel 1525 north of Paris by means of solar observations and wagon wheel revolution counts has also been handed down. Around 1615, Willebrord van Roijen Snell measured a meridian arc between Alkmaar and Bergen-op-Zoom for the first time with the triangulation of large triangles. In 1670, Jean Picard was the first to determine the length of the meridional arc by triangulation with quadrants , which had telescopes with crosshair eyepieces for aiming at the star. This achieved a level of precision that was previously impossible.

When a noticeable deviation from the spherical shape - i.e. the ellipsoidal earth figure - was to be assumed, several significant degree measurements followed in the 18th and 19th centuries , especially

the French earth measurement in Lapland and Peru
the meridian arc of Kremsmünster in Upper Austria
further arcs in Germany, Italy and Austria-Hungary, v. a. Rome - Rimini ( Roger Boscovich 1751) and Brno - Varasdin ( Vienna Meridian ) by Father Liesganig 1761
the Delambre - Méchain arch from Dunkirk to Barcelona (over 1000 km for the first time), 1792–1798
the Gaußsche Bogen Göttingen - Altona (1821–1823), for the first time using strict mathematics. Adjustment
the 2500 km long Great Arc in India (1802–1841 as part of the Great Trigonometric Survey )
the 3000 km long Struve Arch in Eastern Europe (1816-1852) and
its integration in further arcs for the European Bessel ellipsoid
the meridian arc Grossenhain - Kremsmünster - Pola (~ 1880–1910) from Saxony to the Adriatic
and first degree measurements in the USA for the Hayford ellipsoid .

Important meridian arcs of the 20th century

From around 1900, in addition to astronomical latitude measurements and meridional oriented surveying networks, length determinations (i.e. east-west profiles) were also measured, which gradually gave way to extensive area networks from around 1910 - see astro-geodetic network adjustment .

Of the modern meridian arcs are of particular importance:

the meridian arc of Spitsbergen (measured from Russia & Sweden 1898–1902). With an amplitude of 4.2 ° it is relatively short, but it is the northernmost arc that can be used for earth measurements
the South American meridian arc from Colombia through all of Ecuador to northern Peru, the 1899–1906 remeasurement of the famous French arc from 1735 to 1740
the West European-African arc ( meridian of Paris ) from the Shetland Islands to Laghouat near Algiers. It has 27 ° amplitude (latitude difference) and 38 stations
the North American latitude measurement in the 98 ° meridian from Mexico to the Arctic Ocean (1922)
and other meridional chains of the USA as well as the 23 ° long, oblique arc along the east coast
the African degree measurement in longitude 30 ° East, initiated by Sir D. Gill, but not completed until 1953. Course from Cairo to Cape Town , with 65 ° amplitude the longest measurement profile in classical geodesy.

Mathematical description

A meridian arc on an ellipsoid of revolution has the exact shape of an ellipse . Therefore, its longitude - counted from the equator - can be calculated as an elliptical integral and represented in the form of a series of functions of latitude φ:

{\ displaystyle B = C \ varphi + D \ sin 2 \ varphi + E \ sin 4 \ varphi + F \ sin 6 \ varphi + \ cdots}

etc.

The first coefficient C is related to the mean radius of the earth and is 111.120 km / degree for the Bessel ellipsoid . The second coefficient D is related to the flattening of the earth and is 15.988 km. The values for other ellipsoids differ from the fourth digit.

The development by means of eccentricity e ^{2 was given} by Jean-Baptiste Joseph Delambre in 1799:

{\ displaystyle {\ begin {aligned} B \ approx & \; a (1-e ^ {2}) \ left \ {\ left (1 + {\ frac {3} {4}} e ^ {2} + {\ frac {45} {64}} e ^ {4} + {\ frac {175} {256}} e ^ {6} + {\ frac {11025} {16384}} e ^ {8} \ right) \ varphi \ right. \\ & \ - {\ frac {1} {2}} \ left ({\ frac {3} {4}} e ^ {2} + {\ frac {15} {16}} e ^ {4} + {\ frac {525} {512}} e ^ {6} + {\ frac {2205} {2048}} e ^ {8} \ right) \ sin 2 \ varphi \\ & \ + { \ frac {1} {4}} \ left ({\ frac {15} {64}} e ^ {4} + {\ frac {105} {256}} e ^ {6} + {\ frac {2205} {4096}} e ^ {8} \ right) \ sin 4 \ varphi \\ & \ - {\ frac {1} {6}} \ left ({\ frac {35} {512}} e ^ {6} + {\ frac {315} {2048}} e ^ {8} \ right) \ sin 6 \ varphi \\ & \ + {\ frac {1} {8}} \ left. \ left ({\ frac {315 } {16384}} e ^ {8} \ right) \ sin 8 \ varphi \ right \} \\\ end {aligned}}}

Friedrich Robert Helmert used in 1880: ${\ displaystyle n = {\ frac {1 - {\ sqrt {1-e ^ {2}}}} {1 + {\ sqrt {1-e ^ {2}}}}} \ simeq {\ frac {e ^ {2}} {4}}}$