Normal gravitational field

Proceeding from Equation (2.153) we will derive the expressions for the normal gravitational field and the main attention is paid to the field on the surface of the ellipsoid. To emphasize the fact that we deal with the normal field it is conventional to use the letter y instead of g. Then, at each point outside the mass we have... 

Here C/ is the potential of the field of attraction. Inasmuch as we assume that the earth s surface is equipotential, the vector lines of the normal gravitational field are perpendicular to this surface. This condition can be represented as... 

It is a simple matter, performing a differentiation of the potential U p) and preserving terms proportional to the first and second order of flattening, to derive an expression for the normal gravitational field. As was shown in the previous section, Equation (2.185), it has the form... 

The corresponding international formula for the normal gravitational field is y = 978.0490(1 + 0.0052884sin (p - 0.0000059sin 2(p) Gals The potential on the surface of this ellipsoid is... 

As ealeulations show, when the density inereases with a distance from the earth s surface the parameter I is smaller than 0.4. On the contrary, with a decrease of the density toward the earth s center we have 7 >0.4. Inasmuch as in reality 7 <0.4, we conclude that there is essential concentration of mass in the central part of the earth. In other words, the density increases with depth and this happens mainly due to compression caused by layers situated above, as well as a concentration of heavy components. In conclusion, it may be appropriate to notice the following a. In the last three sections, we demonstrated that the normal gravitational field of the earth is caused by masses of the ellipsoid of rotation and its flattening can be determined from measurements of the gravitational field. 

Here Wq and Uq are the total and normal potentials on the surface of the geoid and on the surface of the reference ellipsoid, respectively. By definition, y — —dUjdz is the magnitude of the normal gravitational field. Thus, Equation (2.292) becomes... 

We see that the gradient of the density and that of the gravitational field are parallel to each other. This means that at each point the field g has a direction along which the maximal rate of a change of density occurs. The same result can be formulated differently. Inasmuch as the gradient of the density is normal to the surfaces where 5 is constant, we conclude that the level surfaces U = constant and 5 — constant have the same shape. For instance, if the density remains constant on the spheroidal surfaces, then the level surfaces of the potential of the gravitational field are also spheroidal. It is obvious that the surface of the fluid Earth is equip-otential otherwise there will be tangential component of the field g, which has to cause a motion of the fluid. But this contradicts the condition of the hydrostatic equilibrium. 

To preserve the earth, the component of the gravitational field along the normal has to be negative and this means that the surface integral satisfies an inequality... 

Earlier we solved the boundary value problem for the spheroid of rotation and found the potential of the gravitational field outside the masses provided that the outer surface is an equipotential surface. Bearing in mind that, we study the distribution of the normal part of the field on the earth s surface, where the position of points is often characterized by spherical coordinates, it is natural also to represent the potential of this field in terms of Legendre s functions. This task can be accomplished in two ways. The first one is based on a solution of the boundary value problem and its expansion into a series of Legendre s functions. We will use the second approach and proceed from the known formula, (Chapter 1) which in fact originated from Legendre s functions... 

The function T p) is called the disturbing potential and it is very small, T U). Now we focus our attention on the space between the geoid and the ellipsoid of rotation and assume the potential of the gravitational field of the geoid, W p), and the normal potential of the spheroid, U q), are equal ... 

Here g is the gravitational field on the physical surface of the earth, y the normal field on the surface S. At the same time, dT/dv and dy/dv have the same values along line V at both surfaces. This is the boundary condition for the disturbing potential and therefore we have to find the harmonic function regular at infinity and satisfying Equation (2.301) on the surface S. In this case, the physical surface of the earth is represented by S formed by normal heights, plotted from the reference ellipsoid. In other words, by leveling the position of the surface S becomes known. 

In the previous chapters our attention was paid mainly to the study of the figure of the earth and with this purpose in mind we represented the total field as a sum of the normal and secondary fields. In this chapter, we will discuss a completely different application of the gravity method, related to exploration geophysics, in which the gravitational field is measured in order to study lateral changes of the density near the earth s surface. By analogy, we also represent the gravitational field as a superposition of two fields ... 

The gravitational field of the Earth is characterized by a potential, , that has a definite value at each point in the field. For all practical purposes this field is independent of the presence of matter in the quantities used in normal thermodynamic systems. Within this approximation the field is independent of the state of a thermodynamic system within it. The potential can be written as... 

Where ip is an arbitrary and unimportant phase. Thus, we see that the normalization of the mode is given by the commutator (7.182) and the normalization of the Lagrangian. The justification of the prefactor of Eq. (7.168) comes from the fact that the Lagrangians for us and ut can be found by varying at second order the Lagrangians for the scalar field and of the gravitational field, respectively. The meaning of the amplitude Uk here is that it corresponds to the variance of the quantities us,ut-... 

These minimum number of variables that determine the state of a system are called the independent variables, and all other variables which can be functions of the independent variables are dependent variables or thermodynamic functions. For a system where no external force fields exists such as an electric field, a magnetic field and a gravitational field, we normally choose as independent variables the combination of pressure-temperature-composition or volume-temperature-composition. 

The normal way to build a house is to acquire a piece of land on planet Earth, and then according to a master plan, to dig and make a foundation, to anchor in this foundation the solid structure allowing to place the walls and floors and at the end to cover the house with a roof. In contrast, theory and virtual reality allow us to proceed in reverse order, starting with a roof somewhere in cosmos and looking for ways to accommodate under it interesting structures that become stable when we add a planet with its gravitational field. 

Colloidal solutions may also be made by dispersing in the solvent j a solid or liquid substance which is normally insoluble, such as gold, ferric oxide, arsenious sulfide, etc. A colloidal solution of this sort consists of very small particles of the dispersed substance, so small that their temperature motion (Brownian movement) prevents them from settling out in the gravitational field of the earth. 

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