## Gravitation

The lines that intersect all equipotential surfaces orthogonally are not exactly straight but slightly curved (cf. Figure 4.1). They are called lines of gravity force or plumb lines. The gravity vector at any point is tangential to the plumb line. Hence, direction of the gravity vector, vertical and direction of the plumb line are synonymous. As the level surfaces are, so to speak, “horizontal”, i.e., orthogonal to the plumb lines, they play an important part in our daily life (e.g., in civil engineering for the purpose of height determination). Equipotential surfaces of the Earth’s gravity potential W allow, in general, no simple mathematical representation. This is the reason why physical geodesy and geophysics choose a suitable reference surface for modeling the geoid i.e., the equipotential surface at sea level. The reference surface is constructed as an equipotential surface of an artiﬁcial normal gravity potential U . Its gradient ﬁeld, i.e., u = ∇U , is called normal gravity. For reasons of simplicity, physical geodesy usually uses an ellipsoid of revolution in such a way that a good adaption to the Earth’s surface is guaranteed. Closed representations of normal gravity potentials, in consideration of the centrifugal force, can be found

extensively in the geodetic literature (cf. G.G. Stokes [1849], W.A. Heiskanen, H. Moritz [1967], P.A. Meissl [1971], E. Groten [1979], W. Torge [1991], B. Hofmann-Wellenhof, H. Moritz [2005], H. Moritz [2010], E.W. Grafarend et al. [2010]), and the references therein). The deviations of the gravity ﬁeld of the Earth from the normal ﬁeld of such an ellipsoid are small. The remaining parts of the gravity ﬁeld are gathered in a so-called disturbing gravity ﬁeld ∇T corresponding to the disturbing potential T .