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5. SQUARE-LAW DEPENDENCE OF LIGHT SPEED IS NOT REFUTED BY ASTRONOMICAL OBSERVATIONS

5.1. Simulation of the light propagation process in astronomy

Let us consider what changes in astronomical observations of binary stars we shall have if the square-law dependence c_u = c_o (1 + u²/c_o²)^1/2 of light speed upon light source speed exists in nature.

A star maximal speed   u_P (in periastron) at star motion along an ellipse with eccentricity e differs from the same star minimal speed u_A (in apoastron) very sufficiently [ [48]. Duboshin  G. N. Celestial mechanics. Main tasks and methods.  Moscow: Nauka, 1968,  p. 490..]

u_P/u_A = (1 + e)/(1 - e) = m . (5.1)

For example, at e = 0.99 from formula (5.1) we shall have m = 199. But,  in case if square-law dependence c_u = c_o (1 + u²/c_o²)^1/2   exists in nature, the difference between the speed of light emitted by a star, which is in its periastron, and the speed of light emitted by the same star when it is in its apoastron, is very little. For example, at e = 0.99 and   u_A  = 3 ·10⁴ m/s we have

c_P/ c_А - 1 = 2·10^-4, (5.2)

where c_P  is the speed of light emitted by a source moving at a speed of  u_P; c_А is the speed of light emitted by a source moving at a speed of u_A  .

But tremendous astronomical distances from the Earth to binary stars can result  for an observer on the Earth in appearance of unordinary effects even at insignificant changes of light speed. Let us consider for the sake of simplicity an astromentric binary star, i. e. such system of two stars, one of which does not emit light in visible band of wavelengths. Let us assume that A is the apoastron of the star, which radiates light in the visible band of wavelengths, and P is the periastron of this star. Then the light emitted by this star at a time moment  t_A, when the star was in its apoastron, will come to an observer on the Earth, who is at a distance R from this binary star, at a time moment

t_A1 = t_A + R (c_o² + u_A²)^-1/2, (5. 3)

where u_A is the star speed when it is in its apoastron,  and the light emitted by the star at time moment  t_P = t_A + T_o/2 (where T_o is a true period  of star motion along elliptical trajectory, i. e. the period, which will be measured by an observer situated near the binary star under consideration), when the star was  in its periastron, will come to the same observer on the Earth at time moment

t_P = t_P + R (c_o² + u_П²)^-1/2, (5.4)

where u_P is the speed of the star in its periastron (see Fig. 5.1).

Fig. 5.1. Propagation of light emitted by a star  moving along elliptic trajectory if square-law dependence of light speed upon source speed exists in nature.

The light emitted by the star at a time moment t_A2 =  t_A + T_o, i. e. at a time moment, when the star will again be in apoastron, will come to an observer on the Earth at a time moment

t_A2 = t_A2 + R (c_o² + u_A²)^-1/2. (5.5)

In formulas (5.3), (5.4) and (5.5) we for the sake of simplicity consider that  a distance from the Earth to a binary star apoastron is equal to a distance from the Earth to a binary star periastron  and is equal to a distance from the Earth to any point of a binary star elliptic trajectory.

It is true when distance R between a binary star and the Earth is very great, a plane of elliptic trajectory of a binary star is perpendicular to a line connecting the Earth with centre of mass of a binary star and this centre of mass does not move relatively to the Earth.

From expressions (5.3), (5.4) and (5.5) it follows that if for an observer on the Earth an apoastron and a periastron of a binary star do not merge in one point, then this observer will detect that a star moves from an apoastron to a periastron   during a time interval

T₁ = t_P  - t_A1  =  0.5 T_o - 0.5 R (u_P² - u_A²) c_o^-3 , (5.6)

and a star moves back from an periastron to a apoastron during a time interval

T₂ = t_A2 - t_P  =  0.5 T_o +  0.5 R (u_P² - u_A²) c_o^-3 . (5.7)

From formulas (5.6) and (5.7) it follows that the more is the distance between the Earth and a binary star (the more is the value R), the less is the quantity T₁ and the more is the quantity T₂. For example, at R = D_o it will occur that Т₁ = 0, and Т₂ = Т_о. From formulas (5.6) and (5.7) it follows that this will occur, when

D_o= Т_о c_o³ (u_P² - u_A²)^-1. (5.8)

The equality to zero of expression (5.6) means, that a light quantum, which was emmited by a star, when it was in its apoastron, will come to an observer on the Earth simultaneously with a light quantum, which was emitted by a star, when it was in its periastron (see Fig. 5.1).

It can be shown, that a light quanta radiated from any intermediate point of a star elliptic trajectory (between an apoastron А and a periastron P) during accelerated motion of a star from point A to point P will come to an observer on the Earth, which is at a distance of D_о from a binary star, simultaneously with   light quanta from points A and P only in a case, if a star speed at its motion from point A to point P increases according to the formula

u²(t) = u_A² + 2 c_o³ (t - t_A)/D_o . (5.9)

If an observer on the Earth sees ellipse points A and P as separate points, then at star speed changing according to the formula (5.9) an observer on the Earth will see during some time interval all the star trajectory from point A to point P as a luminous arc. But if an observer on the Earth can not resolve points A and P from one another, then, if star speed changes according to formula (5.9), at the moment of coming the light quanta from semiellipse AP an observer on the Earth should see a tremendous flash of  this star.

Indeed, let us suppose that within the period T_о a star  radiates in all directions an energy equal to W. Then, if a star is immovable relatively the Earth and it is at a distance  R   from an observer on the Earth, power flex density P_o of radiation on the Earth will be constant in time and equal to

P_o = W (4 p R² T_о)^-1. (5.10)

But if the energy, which was radiated by a star during half a period of its revolution along elliptic trajectory, is received by an observer on the Earth within a time interval  Dt = T₁<< T_о, then power flex density within the time interval Dt   will be equal (at uniform distribution of power within time interval Dt)

P₁ = 0.5 W (4 p R²Dt)^-1. (5.11)

And the energy radiated by a star within the second half of period, during  which the star  speed is decreasing and the star  is moving from periastron back to apoastron,  will be received by an observer on the Earth within a time interval T₂=T_o - Dt. That is why the power flux density of the star radiation within this time interval will be equal to

P₂ = 0,5 W [4 p R² (T_o - Dt ) ] ^-1. (5.12)

Then

P₁/P₂ = ( T_o - Dt )/Dt  >> 1 . (5.13)

Consequantly, for an observer on the Earth such star shall have periodical flashes (with a period equal to T_o). It is however well known that at Kepler motion there is no direct analytical  dependence of star speed upon time. Therefore star speed changes not in compliance with formula (5.9).

But investigation of this process  using personal computer simulation (see Appendix 2) shows that at Kepler motion of a star grandiose flashes of star with period T_o are observed too, and  the greater is the flash, the more is its period T_o (see Fig. 5.2).

Fig. 5.2. A flash of a binary star (within a period T_o ) conditioned by square-law dependence of light speed upon source speed.

But periodically flashing stars with such attribute are known long before. These are the so called novae, and it was clarified that novae are binary stars and within a time intervaal between two flashes a star radiates an energy, which is aproximately   equal to an energy, which is released during a flash  [ [49]. Pskovsky Ju. P.   Novae and Supernovae.  Moscow: Nauka, 1985, p. 77,  86].

Simulation shows also that if a distance to a binary star is less than a certain value, then an observer sees such a star (if points A and P are still not resolvable for an observer) as cepheid variable - as a star with periodically changing luminosity (see Fig. 5.3). A guess that cepheid variables are binary stars was for the first time stated by A. A. Belopolsky in the beginning of the XX century. But thereat it was difficult to explain the cause of star luminosity changing. And that is why this guess did  not receive recognition from scientific community  [ [50]. Bronshten V. A. Hypotheses about stars and the Universe. Moscow: Nauka, 1974.  pp. 97 - 100].

Fig. 5.3. Smooth change of binary star luminosity (within a period of star revolution) originating from square-law dependence of light speed upon source speed.

Thus, astronomical observations of binary stars moving along elliptic trajectories do not contradict the existence in nature of square-law dependence c_u = c_o(1 + u²/c_o²)^1/2 of light speed upon source speed.  On the contrary,   this observations can be considered as indirect confirmation of this dependence existence in nature.

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