Consequences of a Generalized Newtonian Gravity with an Exponential Factor

The central interaction of bodies is investigated, which enhances the Newtonian interaction by the exponential factor. As a consequence, it has been shown that Black Holes are subordinate to this enhanced interaction. All Black Holes can be systematized in accordance with their mass, the radius of the event horizon and the gravitational field intensity exponent, created by the Black Hole.


Introduction
In order to build an acceptable theory of the solar system planetary motion, humanity has passed a long, centuries-old path. In the III century B.C. the representative of Pythagorean school of Greece Aristarcus of Samos (Aristarcus of Samos, 310-230 B.C.) put forward the heliocentric system of the planetary motion, but it was rejected by the ancient astronomers, as in their opinion it was baseless. Later on, the most famous one was the geocentric system of Ptolomey (Claudius Ptolomey, 90-168 A.D.), who lived in Alexandria in the II century A.D. From the observations of the starry sky, the ancients concluded that it goes around our Earth, which was considered to be motionless and in the center of the Universe. In the system of Ptolemey everything is explained with the help of circles and circular motions. But this system, which existed over a thousand years, turned out to be very complicated and regularly came into conflict with the data of the astronomic observations, which became intensive after the invention of the telescope by Galileo Galilei (Galileo Galilei, 1564-1642). proportional to the square of the distance between the masses and is directed along the line, connecting the centers of the masses. Newton's another important achievement was that he proved that the orbit of the bodies moving around the Sun, may be any of the curves of the conical sections family (circle, ellipse, parabola, hyperbola). In the next decades and centuries Newton's gravity law has received a lot of convincing and vivid confirmations. We note some of them.  by both theories one and the same value of the gravitational radius g r is obtained. And the opponents of GRT affirm that this theory is not applicable, as the solution of its equations has singularity, unacceptable when describing natural phenomena.
In this connection, we give an opinion of Peter Bergman, Albert Einstein's student and fellow campaigner: "In the solar system and even in our entire Galaxy, the relative motions of the bodies entering these formations are so slow that it is practically indifferent whether to choose the Galaxy as the Lorentz's system or the system in which is resting the center of inertia of the solar's system. The neglected smallness of relativistic effects may explain why the use of Newtonian methods of calculation leads to extremely satisfacting results" [1].
In the papers [5] [6] a new type of central interaction is determined, generalizing Newtonian interaction of gravity. This interaction at short distances is stronger than Newtonian interaction and practically coincides with it at long distances. This confirms the assumption of the famous astronomer Newcomb. In the given paper some important properties of the potential field, created by generalized-Newtonian gravity force [6] are described, the problem of Black Holes ("Dark Bodies") is interpreted in a new way.

Generalized Newtonian Interaction of World-Wide Gravitation
Let us have bodies with mass , M m . We place the beginning of the polar coordinates ( ) where the constant of integration h is determined from the initial condition: at Trajectory of the body motion with mass m is conic section [6] ( ) where constant C is equal to the moment value of the initial velocity, relative to the center of gravity. The parameters of the conical section are determined by formulas: the movement trajectory will be an ellipse, if the semi-axis of the ellipse are determined by formulas   (1) the escape velocity is more than the classical one. Summarizing, we can state that there is a central interaction of bodies, such as (1), which at short distances is more powerful than the Newtonian interaction. This confirms prediction of Newcomb, made on the problem of perihelium of planet Mercury. In the framework of interaction (1), this problem requires a separate detailed consideration, in accordance with the approaches outlined in the Roseveare's monograph [7].
We determine the gravitational radius ( g R ), which corresponds to the interaction (1). The body with mass M will be "Black Hole" ("Dark", invisible), if any body with mass m and initial velocity, even equal to velocity of light "c" cannot overcome the field of gravity of the mass M. The initial conditions of the problem will be: at 0 * , g r R v c = = . Substituting these conditions into (11) "Black Holes" (in our opinion term "Dark Bodies" is more suitable) will differ from each other by mass, gravitational radius g R and gravitational field intensity exponent γ . According to graph of function g g R r (see Figure 1), arbitrarily many "Black Holes" may exist. The astronomers and astrophysics proved this [8]- [13], as in any galaxy there is a center (supposedly "Black Hole"), round which as well turn stars [14] [15] [16]. The mass of such a body comparing with the Sun mass (  ),

On Horizon of Events
The horizon of events is called the boundary (surface) of the space area, the gravity of which is so great, that even the objects, moving with velocity of light (c) cannot leave it. Usually this area is considered to be sphere, the radius of which coincides with the gravitational radius. The radius determined from the relation of the area surface S of this sphere to ( 4π ), i.e. 1942-2018) the horizon of events is made of light, which is not able to leave the Black Hole and that is why "soars" on this horizon [21]. By Newton theory and by GRT gravitational radius The Earth can become a Black Hole if in some way its whole mass is succeeded to be put into a ball with radius of 9 mm, for the Sun 3 km, which is difficult to imagine. In the presence of interaction (1) the gravitational radius g R is determined by formula (12) and its value depends as on the mass, as well as on the gravitational field intensity exponent γ . Above we showed that g R comparing with Newtonian g r can be arbitrarily large. I.e. by interaction (14) the Black Hole will have real dimensions, which is principally observed in reality for massive Black Holes  (Table 1), calculated from the relation (12). International Journal of Astronomy and Astrophysics

On Density of Black Hole
By Newtonian theory of gravity the medium density of the Black Hole is much more than the volume of the Black Hole by Newtonian theory.
The above results allow to conclude that the real Black Holes will have a density much lower than the density and the volume much larger than the volume according to Newton's theory.

On Accelerations of Bodies in the Force Field of Gravity (1)
The force field created by Newtonian force of gravity ( ) has a remarkable property, that all the bodies, being at the same distance from the center of gravity, independent from their dimensions and mass, get from field of the same acceleration [1]. We shall prove that the field, created by force interac-tion has this property (1), as well.
Let the body with the mass M be the center of gravity by (1) and we have bodies with the masses 1 2 , m m , being at the distance 0 r from the center of gravity.
Then according to (2) According to Newton second law of Mechanics where 1 2 , w w are the accelerations of the bodies with masses 1 2 , m m . From here it follows Matching (21) with (19) we have, 1 2 1 w w = i.e. two bodies, having arbitrary masses 1 2 , m m and dimensions, being at the same distance from the center of gravity, under the action of the field get the same accelerations.

On Gravity Force
Let us imagine that the body of mass m is located on the surface of the Black Hole. Let's find out what its weight will be. The force of gravity (14) where N g acceleration of gravity according to Newton's theory. From (23) it follows that the weight of the body ( ) P mg = , located on the surface of the Black Hole, can be arbitrarily big, depending on the value of the field intensity exponent γ , created by the Black Hole.

Discussion and Conclusions
A new type of central interaction of bodies has been established, generalizing the Newtonian interaction of world-wide gravitation. The prediction of famous astronomer Newcomb is verified. It is proved that with such an interaction, the trajectory of the body is a conical section, however, the escape speed is much more than the speed at Newtonian interaction. The close relationship of this in-teraction with the Black Holes problem is shown.
The gravitational field created by the Black Hole obeys the generalized-Newtonian interaction: . Could exist any number of Black Holes, which will differ by mass, radius of the event horizon (Schwarzchild's gravitational radius) and gravitational field intensity exponent γ ? The Schwarzchild's gravitational radius g R can be arbitrarily larger than the Newtonian's gravitational radius g r . At generalized-Newtonian interaction (14), the density of Black Holes is significantly lower than the density at Newtonian interaction. All Black Holes can be systematized according to their mass M, Schwarzchild's gravitational radius g R , gravitational field intensity exponent γ .
The solution corresponding to the Generalized-Newtonian Force of World-Wide Gravitation does not contain the singularity.
Newton's theory of gravity also sheds light on the problem of Black Holes. For this should use the generalized-Newtonian interaction (14).

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.