Field of attraction

Thus, the field of attraction caused by volume, surface, linear, and point masses is... 

We have found that the volume density tends to infinity differently near point, linear, and surface masses, and this fact influences the field behavior in the vicinity of such places. Of course, Equation (1.6) always allows us to calculate the field of attraction g. At the same time, in many cases the use of Equation (1.15) greatly simplifies this procedure. 

This means that any solutions of Poisson s equation, for instance U ip) and U2(p), can differ from each other at every point of the volume Fby a constant only, if their normal derivatives coincide on the boundary surface S. Thus, this boundary value problem defines also uniquely the field of attraction, and it can be written as... 

Unlike the previous case. Equation (1.87) define the potential only to within a constant, but of course the field of attraction is determined uniquely. 

Four types of boundary conditions have been formulated and each of them uniquely defines the field of attraction within the volume V. 

Fig. 1.12. (a) The field of attraction and its potential inside mass, (b) spherical shell of finite thickness, (c) field of attraction due to shell masses, (d) field inside a shell, (e) illustration of Equation (1.137). 

It is also instructive to consider components of the field gs along and perpendicular to the direction of the field of attraction. From Equation (2.13) we have... 

As was shown in the previous seetion the gravitational field of the earth consists of two parts. One of these describes the field of attraction but the other is caused by rotation, and 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... 

In Section 2.4 we have studied the behavior of the gravitational field of the spheroid outside of masses. Now let us focus our attention on the field of attraction inside masses. It may be proper to notice that the determination of the field caused by masses in the spheroid and, in general, by an ellipsoid, was a subject of classical works performed by Maclaurin, Lagrange, Laplace, Poisson, and others. As is well known, the equation of the ellipsoid, when the major axes are directed along coordinate lines is... 

Here p is the pressure, ga the field of attraction, 5 the density of the fluid, and r the vector directed away from the axis of rotation and it is equal in magnitude to the distance between a particle and this axis. The first two terms of Equation (2.332) characterize the real forces acting on the particle, namely the surface and attraction ones. At the same time the last term is a centrifugal force, and it is introduced because we consider a non-inertial frame of reference. It is convenient to represent Equation (2.332) as... 

Here ga a) is the field of attraction on the sphere surface with radius a, and gTae the field at points of the spheroid equator. After multiplication of Equations (2.349-2.351), we obtain... 

If there is a difference between the measured and calculated fields, all parameters of the first approximation or some of them are changed in such a way that a better fit to these fields is achieved. Thus, we obtain a second approximation of mass distribution. Of course, in those cases when even the new set of parameters does not provide a satisfactory match of these fields, this process of calculation has to be continued. As we see from this process, every step of the interpretation requires application of Newton s law. Let us recall that this procedure, based on the use of Newton s law, is often called the solution of the forward problem of the field of attraction, and by definition we have... 

SOLUTION OF THE FORWARD PROBLEM (A CAECUEATION OF THE FIELD OF ATTRACTION)... 

Inasmuch as determination of the field of attraction is an important element of interpretation of gravitational data, let us derive some equations allowing us to simplify the calculation of the useful signal. As follows from the Newton s law of... 

Similar expressions can be written for horizontal components of the field of attraction. Thus, instead of the volume integral, we have derived an expression for the field that requires integration only over the surface. The formulas described in this section allow us, in many cases, to simplify the solution of the forward problem. 

In an alkali metal atom such as sodium, the 3s electron penetrates the neon core, i.e., it moves into the field of attraction of the nucleus, being only partially screened by the K and L shells. In an excited sodium atom the electron in a 3p orbital penetrates the electron cloud to a lesser extent, and the electron raised to a 3d orbital is practically non-penetrating. Thus the 3s, 3p, and 3d orbital in a many-electron atom have different energies whereas these orbitals in hydrogen atom have the same energy. 

Kovalev, B. H., V. V. Stan, T. K. Antoch, V. P. Konyukhov, and S. F. Nedopekina Synthesis in field of attractants (sexual attractants) of insects. III. Easy method for preparation of higher dialkylacetylenes. Synthesis of muscalure, an indoor fly attractant for Musca domestica. Zh. Org. Khim. 13, 2049—2052 (1977). 

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