The tidal field of a planet and the related intrinsic reference systems

Abstract

The tensor describing the tidal gravitational field of the Earth or of a planet is expressed in terms of the curvature parameters of the field.The tensor is then related to the orthonormal reference triads intrinsically defined within the gravity (or gravitational) field of the planet itself, i.e. the only triads recognizable locally with conceivable experiments: the canonical triad, for an observer connected with the Earth, and the tidal triad, for an observer in inertial navigation.Each system of orthonormal triads defines a triple system of mutually orthogonal congruences, to which an anholonomic system of coordinates is attached.Ricci's coefficients of rotation connecting the triads of the frames and the objects of anholonomity of the coordinate's systems are computed together with other geometrical quantities.

Interpreting the direction of the gravity gradient tensor eigenvectors: The main tidal force and its relation to the curvature parameters of the equipotential surface

“…where x1 = x, x2 = y and x3 = z, is the tidal potential used in geodetics (Marussi, 1979). In this case substitution yields…”

“…This allows the eigenvectors of Θ' to be identified as the tidal axes with directions given by (Marussi, 1979):…”

“…The purpose of this paper is to describe the physical meaning of the eigenvectors of the gravity gradient tensor and to show how the direction of the eigenvector associated with the largest eigenvalue can be used in interpretation. In this paper I present the basic, required definitions as proposed by Pedersen and Rasmussen (1990) together with geodetic results from Marussi (1979), illustrate the main properties of the quantity of interest when applied to synthetic data and discuss the application of the method to field data.…”

Microgravitation in space

“…In the same paper (Marussi 1979) it has also been shown that if a steady rotation is superposed to the residual field, the resulting potential W will be written as in which wl, w 2 , w 3 are the components of the rotation vector along the x-,yand z-axes respectively, w2 = w: + 0; + w:, and R the distance of a point P(x, y, z) from the rotation axis drawn through the centre of mass Po. The equipotential surfaces W again constitute a family of concentric homothetic quadrics, under circumstances hyperboloids or ellipsoids, which might degenerate into hyperbolic or elliptical cylinders, or even into a family of parallel planes.…”

“…I have shown (Marussi 1979) that the microgravitational (or tidal) potential T of the residual field sensed by a satellite or in a spacecraft in inertial motion around a planet generating the gravitational potential V, is expressible in the neighbourhood of its centre of mass Po(O,O, 0) by g 2 T.= --(k1x2 t k2y2 -Hz2 t 2 f l~z + 2fZyz).…”