NationMaster - Encyclopedia: Geoid

A geoid is an equipotential surface which (approximately) coincides with the mean ocean surface. It is often referred to as a close representation or physical model of the figure of the Earth. According to C.F. Gauss, it is the "mathematical figure of the Earth", in fact, of the gravity field. It is that equipotential surface (surface of fixed potential value) which coincides on average with mean sea level. Credit ESA as the source of the image. ... Credit ESA as the source of the image. ... ... Earth, also known as Terra, and (mostly in the 19th century) Tellus, is the third-closest planet to the Sun. ... Gravity is the force of attraction between massive particles. ... In fluid mechanics, equipotentials are lines or surfaces of equal head that are in direct relation to pressure. ... The expression figure of the Earth has various meanings in geodesy according to the way it is used and the precision with which the Earths size and shape is to be defined. ... Carl Friedrich Gauss (GauÃŸ) (April 30, 1777 â€“ February 23, 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. ... Gravity is the force of attraction between massive particles. ... For considerations of sea level change, in particular rise associated with possible global warming, see sea level rise. ...

The geoid surface is more irregular than the ellipsoid of revolution often used to approximate the shape of the physical Earth, but considerably smoother than Earth's physical surface. While the latter has excursions of +8,000 m (Mount Everest) and −11,000 m (Mariana Trench), the geoid varies by only about ±100 m about the reference ellipsoid of revolution. 3D rendering of an ellipsoid In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse. ... Mount Everest is the highest mountain on Earth above mean sea level. ... Mariana Trench on Pacific Ocean map The Mariana Trench (or Marianas Trench) is the deepest known submarine trench, and the deepest location in the Earths crust itself. ...

Because the force of gravity is everywhere perpendicular to the geoid (being an equipotential surface), sea water, if left to itself, would assume a surface equal to it—even through the continental land masses if sea water were allowed to freely penetrate them, e.g., by tunnels. In reality it can not, of course; still, geodesists are able to derive the heights of continental points above this imaginary, yet physically defined, surface by a technique called spirit levelling. Spirit levelling is a technique for deterimining differences in height between points on the Earths surface. ...

When travelling by ship, one does not notice the undulations of the geoid; the local vertical is always perpendicular to it, and the local horizon tangential to it. A GPS receiver on board may show the height variations relative to the (mathematically defined) reference ellipsoid, the centre of which coincides with the Earth's centre of mass, the centre of orbital motion of GPS satellites. Over fifty GPS satellites such as this NAVSTAR have been launched since 1978. ... For other uses, please see Satellite (disambiguation) A satellite is an object that orbits another object (known as its primary). ...

Spherical harmonics representation

Spherical harmonics are often used to approximate the shape of the geoid. The current best such set of spherical harmonic coefficients is EGM96 (Earth Gravity Model 1996), determined in an international collaborative project led by NIMA. It contains a full set of coefficients to degree and order 360, describing details in the global geoid as small as 55 km. In mathematics, the spherical harmonics are an orthogonal set of solutions to Laplaces equation represented in a system of spherical coordinates. ...

The mathematical description of this model is

$V=frac{GM}{r}left(1+{sum_{n=2}^{360}}left(frac{a}{r}right){sum_{m=0}^n} overline{P}_{nm}(sinphi)left[overline{C}_{nm}cos mlambda+overline{S}_{nm}sin mlambdaright]right),$

where $phi$ and $lambda$ are geocentric latitude and longitude respectively, $overline{P}_{nm}$ are the fully normalized Legendre functions of degree $n$ and order $m$ , and $overline{C}_{nm}$ and $overline{S}_{nm}$ are the coefficients of the model. One easily counts that there are approx. $begin{matrix} frac{1}{2} end{matrix} n(n+1)approx$ 65,000 different coefficients. The above formula produces the Earth's gravitational potential $V$ at location $phi,;lambda,;r,$ the co-ordinate $r$ being the geocentric radius, i.e, distance from the Earth's centre.

Spherical harmonics representation GA_googleFillSlot("encyclopedia_square");

See also

Spherical harmonics representation