Gravity in View of the Theory of Orbiting Binary Stars

In this paper, we investigate orbiting of two stars having equal masses. We consider two models: with a circular orbit and with two elliptical orbits having a common center of a mass located in a common focal point. In the case of the circular orbit, we applied the notion of the instantaneous complex frequency. The paper is illustrated with numerous formulas, derivations and discussion of results.


Introduction
Three years ago, the author presented a paper describing the gravitational forces as a result of anisotropic energy exchange between baryonic matter and quantum vacuum [1]. Here, we try to show that the theory of circulation of double stars around a common center of mass yields arguments in favor of the above theory. Our goal can be achieved by investigating orbiting of two stars having equal masses. We present two such models: the first one with a circular orbit and the second one with two elliptical orbits with a common center of a mass located in a common focal point. The presented mathematical descriptions of the above models are derived by the author and certainly only the methods of derivations are new. Most of the results belong to the existing knowledge. As regards the circular orbit, we applied the notion of the instantaneous complex frequency. We introduce the following notations:

A Model of a Double Star of Equal Mass Orbiting on a Circular Orbit
We get the following time-independent relations between the angular velocity ω 0 and the radius ρ 0 : ( The orbital tangential velocity is 0 0 0 v ω ρ = . Therefore, the kinetic energy of the system is 2 0 k E mv = and the potential energy energy is negative. Its value equals twice the kinetic energy. Therefore, the total energy of the system is negative and time independent. Several authors derived formulae for calculation of the power of gravitational waves emitted by the system of Figure 1. ( In the book of Gasperini [3], we find [ ]

Instantaneous Complex Frequency Description of the Circular Binary System
We have to show that due to the emission of gravitational waves, the trajectory of the stars is not circular since the instantaneous radius decrease in time and the angular velocity and tangential velocity increase in time. The stars are orbiting along spirals ( Figure 2). A convenient method of description of this phenomenon is the notion of instantaneous complex frequency. The phasor representing the first star has the form and the second one α (t) is called instantaneous radial frequency and ω(t)-instantaneous angular frequency. The instantaneous radius is In the is a line connecting the mass centers through the origin (0, 0) and defines the direction of gravitation force. Differently, the direction of the centrifugal force F c is defined by the curvature radius ρ c . The geometry of the addition of the two forces is presented in Figure 3 (with large rate of inspiral). In the case ( ) 0 t α α = − , the angle γ is given by the formulae (see Ap-  The inspiral orbit is defined by the equation However, there is a tangential force (see Figure 3) However, for the quasi-circular orbit, the tangential force is extremely small.
This force induces acceleration of the mass m given by ( ) In consequence, the instantaneous angular frequency is increasing in time where T(t) is a decreasing instantaneous period. The energies of the system also increase in time. The instantaneous tangential velocity of the stars is The curvature radius is (see Appendix 2) The instantaneous kinetic energy of both stars is total energy of the system is negative. The power of the gravitational waves emitted by the system given by Equation (3) is [ ] 23 6.523698 10 W P = × .
1) Estimation of the value of the radial frequency α 0 The decrease of the radius of the circular model in one period T 0 is We have an increase of the negative value of the potential energy ( ) Therefore, we get the increase ( ) Assuming arbitrary that this increase should be equal to the energy emitted by gravitational waves during one period we get ( ) Therefore, 2) The increase of the angular frequency (or decrease of the period T 0 ) during the inspiral Taylor and Hulse have measured that the period of the PSR system decreases by 76.5 μs per year [6]. Let us derive a formula for this decrease for the circular system. We insert in Equation (2 It is more than one order of magnitude smaller in comparison to 76, 5 μs per year of the PSR system. Therefore, the circular model cannot be applied to describe the properties of the PSR elliptical system. 3) The increase of the negative value of the potential energy The potential energy at the moment t = 0 is i.e., exactly the value defined by Equation (3) which represents the power of the emitted gravitational waves. This result validates the correctness of Equation (17) defining α 0 and Equation (19) defining the delay per year. The negative sign of this power is applied in the book of Gasperini [3] with no comment. We found that the authors of reference [9] derived a formula with a negative sign of the gravitational "Poynting vector" also with no comment.

4) The inspiral time
The main goal of this paper is to validate the explanation of the nature of gravity presented in [1]. The instantaneous radius is The decrease of the radius during a year is

The Theoretical Model of the Binary Pulsar PSRB1913+16
The PSR system differs considerably from the above described circular system.
The two stars are orbiting along elliptical orbits (see Figure 4) around a common center of mass located in the focus. We consider again equal masses The local velocity of the stars by orbiting from periastrone to apastron is (deceleration)  The mean velocity (in terms of φ) is the same for both directions The local tangential velocity is alternatively defined as Where ρ c is the local curvature radius (see Figure 5). The local velocity is shown in Figure 6. The maximum value equals 448.172 [km/s] and the minimum 106.287 (compare with Appendix 1).
Note that in this model, the maxima and minima of the velocity are located near the periastrone and apastrone (not exactly at these locations). The mean value in terms of φ (as in Equation (38)

Final Conclusions
o Both orbits, the circular and the elliptical, are defined by two forces of opposite directions: the centrifugal force and a term of the gravitational force (see Figure 3). It is logical to assume that both forces have the same physical explanation: the anisotropic energy exchange as described in reference [1].
Here, both anisotropies of radiation cancel. The other part of the gravitational force responsible for tangential acceleration or deceleration is the result of the tangential anisotropy of radiation. o The cancelation of the two forces shows that gravity and inertia have the same physical origin. They are recoil forces of radiation. The radiation pattern should be symmetric w.r.t. the tangent of the orbit. Differently, the pattern is asymmetric w.r.t. the line perpendicular to the orbit resulting in a recoil force of radiation.
In a word, it is logical to assume that all described here forces are recoil forces of radiation. The radiation pattern is symmetric w.r.t. the tangent of the orbit (cancellation of gravitation and inertia) and asymmetric w.r.t. the line perpendicular to the orbit, i.e., the direction of the curvature radius.

Appendix 1
This paper is illustrated by the properties of the binary pulsar PSR1913+16, a system of two binary neutron stars discovered and measured during many years by Taylor and Hulse [7] [8]. This great achievement of radioastronomy and also of time-frequency metrology was awarded by the Nobel Prize in physics in 1993.
Let us repeat here the data compiled by Robert Johnston [8].

Appendix 2: The Derivation of the Curvature Radius of the in Spiral Orbit
Let us define the in spiral orbit by the equation