Gravity flattening

As a theoretical gravity in which is Geophysics the dependence of the theoretical acceleration due to gravity (gravity) of the latitude indicated. It is calculated from the gravity at the equator of the earth's ellipsoid and the gravity at the poles: ${\displaystyle \beta }$ ${\displaystyle \gamma }$ ${\displaystyle \phi }$ ${\displaystyle \gamma _{a}}$ ${\displaystyle \gamma _{b}}$

${\displaystyle \beta ={\frac {\gamma _{b}-\gamma _{a}}{\gamma _{a}}}={\frac {\gamma _{b}}{\gamma _{a}}}-1.}$

It is a consequence of the earth's rotation and the resulting flattening of the earth : ${\displaystyle f}$

${\displaystyle f={\frac {a-b}{a}}=1-{\frac {b}{a}},}$

where and are the semi-axes of the earth's ellipsoid. ${\displaystyle a}$ ${\displaystyle b}$

The exact values ​​of and depend u. a. on the earth ellipsoid used. ${\displaystyle f}$ ${\displaystyle \beta }$

In the international GRS80 earth model, the following applies:

${\displaystyle a_{\text{GRS80}}=6\,378\,137{,}0\;{\text{m}}}$ (Semi-axis at the equator)

${\displaystyle b_{\text{GRS80}}=6\,356\,752{,}3141\;{\text{m}}}$ (Semi-axis at the poles)

${\displaystyle \Rightarrow f_{\text{GRS80}}={\frac {1}{298{,}257}}=0{,}00335281}$

On a global average:

${\displaystyle \gamma _{a}=9{,}7805\;\mathrm {m/s^{2}} }$ (Acceleration due to gravity at the equator)

${\displaystyle \gamma _{b}=9{,}8322\;\mathrm {m/s^{2}} }$ (Gravitational acceleration at the poles)

${\displaystyle \Rightarrow \beta =0{,}005163\approx 0{,}52\,\%}$

The increase in gravitational acceleration from the Earth's equator towards the poles is 0.52 percent. B. makes clearly noticeable in the length of the seconds pendulum .

The (physical) flattening of gravity is much stronger than the (geometric) flattening of the earth . This expresses the noticeable difference between the two gravity values ​​at the equator and at the poles, which are the smallest and greatest theoretical gravity at sea level. ${\displaystyle \beta }$ ${\displaystyle f}$

The flattening of gravity is one of the reasons why the level surfaces of the earth's gravitational field are not completely parallel to sea ​​level .

History

While Newton's geometric flattening of the earth was postulated as early as 1680 and was only empirically confirmed in 1742 after the Peru-Lapland expeditions of the Paris Academy , the opposite was true for the flattening of gravity: Jean Richer clearly set a position in Cayenne (French Guiana, South America) in 1673 Shorter seconds pendulum than fixed in Paris, but the theory for this was created in 1743 by Alexis-Claude Clairaut ( theory of the shape of the earth according to the laws of hydrostatics ). It could, of course, serve as an additional safeguard for the definition of the Paris original meter (1793).