The gravitational field

In order to study the attraction of masses of the earth which moves around the axis of rotation, it seems appropriate to use the field g, which depends on the distribution of masses and the angular velocity, as well as coordinates of the point. Besides, it has a physical meaning of the reaction force per unit mass. However, it has one very serious shortcoming, namely, unlike the attraction force it is directed outward. In other words, it differs strongly from the attraction field, in spite of the fact that the contribution of rotation is extremely small. To overcome this problem we introduce the gravitational field g which differs from the reaction field in direction only  

---

An interesting historical application of the Boltzmann equation involves examination of the number density of very small spherical globules of latex suspended in water. The particles are dishibuted in the potential gradient of the gravitational field. If an arbitrary point in the suspension is selected, the number of particles N at height h pm (1 pm= 10 m) above the reference point can be counted with a magnifying lens. In one series of measurements, the number of particles per unit volume of the suspension as a function of h was as shown in Table 3-3. 

To determine the size of a particle having a velocity, w, in the gravitational field, both sides of equation 63 are multiplied by the complex Re/ArK ... 

For example, at n = 1200 rpm = 20 liter/sec and r = 0.5 m, the settling velocity in the centrifuge is almost 28 times greater than that in free settling. Note that the above expressions are applicable only for Re > 500. For small particles. Re < 2, migration toward the wall is laminar. The proper settling velocity expression for the gravitational field is... 

Our most important insight into the connection between thermodynamics and black holes comes from a celebrated result obtained by Bardeen, Carter and Hawking [bard73], that the four laws of black hole physics can be obtained by replacing, in the first and second laws of thermodynamics, the entropy and temperature of a thermodynamical system by the black hole event horizon (or boundary of the black hole) and surface gravity (which measures the strength of the gravitational field at the black hole s surface). 

As we saw in Section A, we can think of a field as a region of influence, like the gravitational field of the Earth. 

Equation (1.84) form Dirichlet s boundary value problem, which can be either exterior or internal one. Fig. 1.8a, and it has several important applications in the theory of the gravitational field of the earth. It is worth to notice that in accordance with Equation (1.83) we can say that along any direction tangential to the boundary surface, the component of the field is also known, since = dU/dt. Consequently, the boundary value problem can be written in terms of the field as... 

Certainly, Equations (1.127 and 1.128) can be derived directly from Equation (1.6), but it requires a rather cumbersome integration. This example also allows us to illustrate the fact that the gravitational field has a finite value inside any mass. With this purpose in mind, imagine that the observation point p is located at the center of a small and homogeneous sphere. Fig. 1.12a. Then, the total field can be represented as a sum ... 

In essence, Equation (1.162) is an example, when the function is presented as a sum of partial solutions of Laplace s equation. Further, we will use Equation (1.170) to describe the gravitational field of the earth. 

FORCES ACTING ON AN ELEMENTARY VOLUME OF THE ROTATING EARTH AND THE GRAVITATIONAL FIELD... 

Later we will demonstrate that the gravitational field g has the same magnitude as the field of the surface forces, g, but the opposite direction. Until now we have... 

After a division by a mass m we again obtain for the gravitational field (Equation... 

Then, the gravitational field for a moving particle has the form ... 

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... 

By definition, at each point p the gravitational field is tangent to the vector line and its equation is... 

This field of the centrifugal force, unlike the attraction field, is fictitious, and correspondingly, we observe a volume distribution of fictitious sources with a density proportional to co. A summation of the first and second Equations (2.62 and 2.63) gives the system of equations of the gravitational field at regular points... 

As was shown earlier the gravitational field is a superposition of the attraction and centrifugal fields ... 

Certainly, the expression for the potential is much simpler than that for the field, and this is a very important reason why we pay special attention to the behavior of this function U(p). As follows from the behavior of the gravitational field, the potential U has a maximum at the earth s center and with an increase of the distance from this point it becomes smaller, since the first derivative in the radial direction, that is, the component of the gravitational field, is negative. At very large distances from the earth the function U has a minimum and then it starts to increase, but this range is beyond our interest. In the first chapter we demonstrated that the potential of the attraction field obeys Poisson s and Laplace s equations inside and outside the earth, respectively ... 

Here Uq is the value of the potential on the surface. It is proper to notice that potentials of the attraction field and the centrifugal force usually vary on the level surface of the gravitational field. Changing the value of the constant, Uq, we obtain different level surfaces, including one which coincides in the ocean with the free undisturbed surface of the water and, as was pointed out earlier, this is called the geoid. As follows from Equation (2.73) the projection of the field g on any direction / is related to the potential U by... 

This relationship between this geometrical concept and the gravitational field plays a very important role for determination of heights. 

Now we express this curvature in terms of the gravitational field and potential. Since the coordinate z depends on v, differentiation of U with respect to v gives at the point p... 

Now we express the curvature of the plumb line in terms of the gravitational field, and with this purpose in mind consider its elementary displacement [Pg.81]

Thus, the potential of the gravitational field and pressure differ by a constant multiplier, and the pressure plays the role of the potential even though its meaning is completely different. 

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... 

Now we will start to apply the theory of the potential U(p) and its field g(p) to study a gravitational field caused by masses of the earth. Earlier, it was pointed out that the behavior of the gravitational field on the earth s surface has mainly a regular character, while the irregular part is very small, less than 0.1%. Correspondingly, it is natural to divide the mass of the earth into two parts ... 

---