The Reciprocity Principle in Gravitational Interactions

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**Subject Areas:** **Classical Mechanics**

1. Introduction

Let us consider two heavenly bodies having masses m_{1} and m_{2}, and radius r_{1} and r_{2}, respectively, separated by a certain distance, and in gravitational attraction acting along the line joining them. In that situation, the values of their escape velocities due to its apparent size are dependent on their masses, and critically on their apparent radius [1] . When those celestial bodies interact each other, both of them must be considered as sources of gravity force. Then, along that dynamical process, as it was said before, there exist some degree of reciprocity [1] . By definition of escape velocity it has that [2]

(1)

where G is the Gravitational Constant, M the mass of the source of the gravity force, and R its radius [1] .

2. The Reciprocity Principle

When the heavenly bodies before mentioned, are in gravitational interaction, it is easy to see that from Equation (1) the following relationship can be obtained

(2)

where the are the apparent radius of the bodies.

Let’s consider one of the bodies as the source of the gravity attraction; and be that body the one marked with the index 2. Its mass is m_{2}, and its apparent radius is. So that, the equation (1) takes the following form [1] .

(3)

In order to obtain, and also it can be use the following procedure. It is well known from Optics that the ratio of the image size q, to the object size p is the Magnification A, that is to say

(4)

To obtain the apparent size of that body, it can be used some optical astronomical instrument that has a magnification A; in order to get the apparent image, and also the apparent size, given by the following formula

(5)

Given that A is a number, is equal to p/2. To calculate is enough to introduce into Equation (2) the result obtained. However, in that equation it has the unknown quantity. In order to known its value, it is pro- posed that:

The ratio between the apparent radius keep the same proportion among them as the ratio of the respectives real values of their radius; that is to say

(6)

where r_{1} and r_{2} are the values of the real radius. The proposition before given is the Reciprocity Principle.

3. The Sun-Earth System

As an illustrative example, let’s consider the case of the gravitational interaction between the Sun and our planet. Therefore, Equation (2) becomes

(7)

Then, it has that

(8)

according to relationship (6).

Taking into account the values that appears in the Appendix, it is obtained that

(9)

and

(10)

in such a way that in Equation (8) it has that

(11)

Substituting this result into Equation (8) the following value for the Earth’s escape velocity due to its apparent size is obtained; that is to say

(12)

where = 13c, and c is the speed of light in vacuum [1] . Then

(13)

That result means that, due to the smallness of the Earth in comparison to the Sun, its escape velocity is orders of magnitude smaller than. Hence, when both of those velocities meets in some point of the space, they pull each other, because they are the carriers of the respective gravity force.

Finally, from Equation (10) it is easy to obtain that

(14)

Thus,

(15)

But, this is the same numerical value given in (10).

4. The Point of Meet

It is clear that in the meantime the escape velocity travels 68,371 km, the escape velocity travels a distance equal to 3.9 ´ 10^{6} km. Let us suppose that along the whole distance, those velocities maintain its values. Then, let the time transit that takes to to travel the given distance, be the unit of measure; that is to say

In that time travels the distance

(16)

in the meantime travels the distance

(17)

Those results indicate that the point of meet occurs near the Earth

5. Conclusions

It is proposed that in Gravitational Interactions, the Reciprocity Principle is valid for any couple of heavenly bodies that interact each other. Also, it is a useful concept, and a powerful tool to calculate the escape velocities and radius due to the apparent size of each body. In other words, its validity can be extended to the whole Universe. On the other hand, it is possible to prove that the ratios and for two celestial bodies 1 and 2, respectively, are numerically equal. The same is valid for the ratios and; in such a way that from this point of view; it is always true that

; and

where the prime refers the escape velocity of these bodies due to their apparent size.

In the case of the Sun-Earth System, it has that

; and

and also,

; and.

Appendix