_{1}

^{*}

In this paper it is proposed that the ratio between the apparent radius of any two celestial bodies separated by a very big distance, as seemed each other at a distance, is numerically equal to the ratio of the respective real values of their radius. That relationship is what is called in the paper the Reciprocity Principle. In other words, the apparent size of heavenly bodies, as seemed at a distance, plays a special role in the gravitational interactions. This is so because of some kind of effect over the size due to the very big distances in space. When a couple of celestial bodies interact each other, both of them can be considered as sources of the gravity attraction. In that situation, each body acts upon the other in such a way that it can be considered that there are some degree of reciprocal interaction. In this paper, a formal definition of that reciprocal effect is given.

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 [

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

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

where the

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

In order to obtain

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

Given that A is a number,

The ratio between the apparent radius

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

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

Then, it has that

according to relationship (6).

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

and

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

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

where

That result means that, due to the smallness of the Earth in comparison to the Sun, its escape velocity

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

Thus,

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

It is clear that in the meantime the escape velocity ^{6} km. Let us suppose that along the whole distance, those velocities maintain its values. Then, let the time transit that takes to

In that time

in the meantime

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

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

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 also,

Angel Fierros Palacios, (2016) The Reciprocity Principle in Gravitational Interactions. Open Access Library Journal,03,1-4. doi: 10.4236/oalib.1102412