Theoretical severity

The theoretical gravity ( English normal gravity ) is the acceleration due to gravity , which relates to the (theoretical) mean earth ellipsoid ; this hugs the geoid for an average of ± 40 meters. Geodesy and geophysics use theoretical gravity in order to be able to compare the strongly varying gravitational acceleration measured on the earth's surface with a smoothed gravity model . ${\displaystyle g_{\phi }}$

formula

→ Main article : International formula of gravity

Although the stratification and rock density in the earth's interior are not (and never will be) known with absolute accuracy, the theoretical gravity on a level ellipsoid can be described by a relatively simple formula. It is since the pioneering work of the German surveyors Friedrich Robert Helmert as International Heavy formula referred, however, depends on the parameters of the earth ellipsoid used by international conventions are taken every two to three decades to date.

Since 1960

The gravitational formula describes the course of the gravitational acceleration on a level surface approaching sea level as a function of the geographical latitude : ${\displaystyle {\phi }}$

${\displaystyle g_{\phi }=\left(9,780327+0,0516323\cdot \sin ^{2}(\phi )+0,0002269\cdot \sin ^{4}(\phi )\right)\,{\frac {\mathrm {m} }{\mathrm {s} ^{2}}}}$

The value 0.0516323 is called the flattening of gravity and is usually referred to as β in German-speaking  countries . As defined physically, but dimensionless it corresponds to parameters of the (geometric) Earth flattening  f (by English flattening ) of the earth ellipsoid.

Until 1960

Until around 1960, the formula for the Hayford ellipsoid (1924) was mostly used in America , and Helmert's more precise formula in Europe. The equator axis of the global ellipsoid he calculated in 1906 deviates from the modern value by about 70 meters (Hayford by 250 m), but the flattening is almost the same. In contrast to the above, it uses a different angle function of the width, namely the cosine instead of the sine:

${\displaystyle g_{\phi }=\left(9,8061999-0,0259296\cdot \cos(2\phi )+0,0000567\cdot \cos ^{2}(2\phi )\right)\,{\frac {\mathrm {m} }{\mathrm {s} ^{2}}}}$

WGS 84

An ellipsoidal gravity formula that differs only slightly from the first formula is that of the WGS 84 earth model ( World Geodetic System from 1984):

${\displaystyle g_{\phi }=\left(9,7803267714~{\frac {1+0,00193185138639\cdot \sin ^{2}\phi }{\sqrt {1-0,00669437999013\cdot \sin ^{2}\phi }}}\right)\,{\frac {\mathrm {m} }{\mathrm {s} ^{2}}}}$

The difference between the WGS-84 formula and Helmert's equation is below 69 ppb or <6.8 · 10 ⁻⁷ m · s ⁻² .

See also

• Gravity anomaly

literature

• Karl Ledersteger : Astronomical and physical geodesy . Handbook of Surveying Volume 5, 10th edition. Metzler, Stuttgart 1969
• B. Hofmann-Wellenhof, Helmut Moritz : Physical Geodesy , ISBN 3-211-23584-1 , Springer-Verlag Vienna 2006.