Ellipsoidal-Sphäroidale Representation of the Gravity Field/Ellipsoidische-sphroidale Darstellung des Schwerefeldes

Abstract

We begin with a chapter on motivation, namely why the Earth cannot be a ball due to Earth rotation which we daily experience. In contrast, the Earth’s gravity field is axially symmetric as a first order approximation, not spherically symmetric. The same axially symmetric gravity field applies to all planets and mini-planets, of course the Moon, the Sun and other space objects which intrinsically rotate. The second chapter is therefore devoted to the definition of ellipsoidal-spheroidal coordinate which allow separation of variables. The mixed elliptic-trigonometric elliptic coordinates are generated by the intersection by a family of confocal, oblate spheroids, a family of confocal half hyperboloids and a family of half planes: in this coordinate system {λ, ϕ, u} we inject to forward transformation of spheroidal coordinates into Cartesian coordinates {x, y, z} and the uniquely inverted ones into the backward transformation {x, y, z}→{λ, ϕ, u}. In such a coordinate system we represent the eigenspace of the potential field in terms of the gravitational field being harmonic as well as the centrifugal potential field being anharmonic. Such an eigenspace is being described by normalized associated Legendre functions of first and second kind. The normalization is based on the global area element of the spheroid $\mathbb {E}_{a,b}^2$. The third chapter is a short introduction into the Somigliana-Pizetti level ellipsoid in terms of its semi-major axis and its semi-minor axis as well as best estimation of the fundamental GeodeticParameters {W₀, GM, J₂, Ω} approximating the Physical Surface of the Planet Earth, namely the Gauss-Listing Geoid. These parameters determine the World Geodetic Datum for a fixed reference epoch. These parameters are called (i) the potential value of the equilibrium figure close to Mean Sea Level, (ii) the gravitational mass, (iii) the second kind, zero order (2, 0) of the gravitational field and finally (iv) the Mean Rotation Speed. These numerical values of the Planet Earth are numerically given. The best estimations of the form parameters derived from two constraints are presented for the Somigliana-Pizzetti Level Ellipsoid. In case of real observations we have to decide whether or not to reduce the constant tide effect. For this reason we have computed the “zero-frequency tidal reference system” and the “tide free reference system” which differ about 40 cm. The radii are {a = 6,378,136.572 m, b = 6,356,751.920 m} for the tide-free Geoid of Reference, but {a = 6,378,136.602 m, b = 6,356,751.860 m} for the zero-frequency tide Geoid of Reference. These results presented in the Datum 2000 differ significantly from the data of the Standard Geodetic Reference System 1980. The geostationary orbit computed in the Chap.4 balances the gravitational force and the centrifugal force to zero, the so-called NullSpace. Its value of 42,164 km distance from the Earth Center has been calculated in the quasi-spherical referenced coordinate system introduced by T. Krarup. This Null Space evaluates the degree/order term (0, 0) of the gravitational field and the degree/order terms (0, 0) and (2, 0) of the centrifugal field. A careful treatment of the axial symmetric gravity field representing this gravitational and centrifugal field of this degree/order amounts to solve a polynomial equation of order ten. The intersection point of these two forces has been calculated with a lot of efforts! Referring to the Somigliana-Pizzetti Reference Gravity Field we compute in all detail Molodenskyheights in Chap.5. In using the World Geodetic Datum 2000 we have presented the Telluroid, telluroid heights and the highlight “Molodensky Heights”. The highlight is our Quasi-geoid Map of East Germany, based on the minimum distance of the Physical Surface of the Earth to the Somigliana-Pizzetti telluroid. The topic of Chap.6 is the famous Mac Cullaghrepresentation of the time-variation of the multipoles of the gravity, generalizing the Newton gravitostatic gravity field to gravito dynamics in terms of Eulerian increments Ω. We build up the theory of the time-varying gravity field of excitation functions of various types: (i) tidal potential, (ii) loading potential, (iii) centrifugal potential and (iv) transverse stress. The mass density variation in time, namely caused by (i) initial mass density and (ii) the divergence of the time displacement vectors, is represented in terms (i) radial, (ii) spheroidal and (iii) toroidal displacement coefficients in terms of the spherical Love-Shida hypothesis. For the various excitation functions we compute those coefficients.

Zusammenfassung

Wie beginnen das erste Kapitel mit dem Argument, dass die Erde auf Grund der Erdrotation keine Kugel mit konstantem Radius sein kann, sondern ein abgeplattetes Ellipsoid ist, im Einklang mit unserer täglicher Erfahrung: die Erde rotiert in etwa 24 Stunden. Wie geben in einführend die Abplattung, die Rotation Periode, den Näherungswert aller terrestrischen Planeten an, ebenso wie die charakteristischen Daten zum Planeten Erde, sowie verschiedene Definitionen über das Jahr, den Monat, der Tag. Das Schwerefeld der Erde besteht aus einem (i) Harmonischen Anteil und einem (ii) anharmonisch Anteil auf Grund der Erdrotation. Die spektrale Bedeutung des ,,geostationären Radius“ von 42.164 km, dem Null-Raum des Schwerefeldes außerhalb der rotierenden Erde. Das zweite Kapitel ist der genauen Definition von elliptischen-sphäroiden Koordinaten einer elliptischenErde gewidmet. Ellipsoid-harmonischen Reihen Entwicklungen des dreidimensionalen Laplace Operator runden das Kapitel ab. Das dritte Kapitel konzentriert sich auf das anharmonische Somigliana-Pizzetti Referenzfeld. Die besten Schätzungen der sog. Form-Parameter der Erde bilden einen ersten Höhepunkt. Das vierte Kapitel erlaubt den geostationären und geosynchronen Satelliten-Radius, den sog. Null-Raum des Schwerefeldes. Das Molodensky-Höhensystem steht im Zentrum des fünften Kapitels, das zentrale Höhensystem in Russland und Europa. Es basiert auf dem Somigliana-Pizzetti Referenz-Schwerefeld. Zentral ist das quasi-Geoid als GPS-Informationssystem der Erdparameterdaten. Die zeitliche Veränderung des terrestrischen Schwe-refeldes, die sog. MacCullagh Darstellung steht zentral im sechsten Kapitel. Den Abschluss bildet unsere Zusammenfassung, und der Ausblick: Geodäsie im 21. Jahrhundert.

Ellipsoidal-Spheroidal representation of the gravity field of a gravitating and rotating Earth, the anharmonic part as well as the harmonic part, zero, first and second derivatives of its potential field, deformable bodies

This contribution is dedicated to the late

Helmut Wolf

Founder of the one and only Institute of Theoretical Geodesy at Bonn University/Germany/ and for his courage to make me the youngest German Professor of Geodetic Sciences as well as

Friedrich W. Hehl

Theoretical Physicist, University of Cologne/Germany/ for his critical accompanying of my research and for his substantial advice.