
    PROOF FOR THE ADVANCE OF MERCURY’S PERIHELION
A. CALCULATING THE CENTER OF MASS OF THE SOLAR SYSTEM 

 
 As it is well known, all celestial bodies of the Solar system, planets, asteroids, comets, etc, as well as the Sun, revolve around the center of mass of our Solar system. Given, however, that every planet P_{i}(i = 1,2,3,…9),
i.e. Mercury, Venus, Earth, … etc, has a period of revolution T_{i} around the Sun, then theoretically at a time t all planets will be found on the same semistraight line oχ´ (or extremely close to it), that is, they will be found on a total general planetary synod, Fig. 1. 
   Let us assume in this case that C is the center of mass of our Solar system. As itis well known, the center of mass of our Solar system remains stable at all times at position C on the straight line χχ´ and this position C is independent of any planetary motion around it.
Taking now the Sun as the measurement principle, distance d between the Sun and the center of mass C of the Solar system is expressed (as it is commonly known) by the following formula: 
    (1) 

   where M_{1}, M_{2}, M_{3}, … M_{9} are the masses of the planets and R_{1}, R_{2}, R_{3}… R_{9} their respective distances from the Sun. M is the mass of the Sun. By substituting into formula (1) the values of M_{1}, M_{2}, M_{3}, … M_{9} and R_{1}, R_{2}, R_{3}… R_{9, }such
as thoseare given in Table 1 (NASA Solar System data), formula (1) yields the following: 
    or 

   which is distance d, separating the Sun from the center of mass C of our Solar system. Moreover, because the Sun’s radius r_{0} is: 
   this signifies that the center of mass of our Solar system is beyond the Sun and in fact it lies at a distance d_{0}, that is: 
   Inotherwords, the center of mass of our Solar system lies at a distance greater than one radius of the Sun, outside its
surface. Note: In the above calculation of the center of mass of our Solar system, the satellites of the planets were not taken into account, since their mass is extremely small in relation to the total mass of our Solar system. 
   
   Consequently, as it has been stated above, because all celestial bodies of our Solar system revolve around the center of mass of the Solar system, then the Sun, also, will revolve around the
center of mass of the Solar system in an approximate circular orbit of radius d = 1,505 10^{9} m. This fact, as we shall see later in this text, plays a decisive role especially in the advance of Mercury’s perihelion, as well as of the perihelia of other planets of our Solar system. .NOTE: Radius d=1,505^{.}10^{9}m is approximately considered to be average value of the radiuses of the sun`s rotation around the center of mass of our solar system, pic.1 (see, http://astro.berkeley.edu/~eliot/Astro7A/Gravity.pdf). 
   In this frame of reference xoy, the Sun moves in a circular orbit C_{0,} with the center being point o and radius d = 1,505 10^{9} m. Furthermore, for the sake of simplicity, we consider that in this frame of reference xoy the orbits C_{i} of all planets P_{i} are circular, where i = 1,2,3,…9 are in sequence Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto.
In this frame of reference xoy, we consider that distance R_{i} between a planet P_{i} and the Sun is the distance R_{i} between two concentric circumferences, i.e. circumference C_{i} of planet P_{i} and circumference C_{0} of the Sun. These distances R_{i} between planets P_{i} and the Sun are those listed in Table 1. A. PLANET MERCURY In frame of reference xoy, with the Sun revolving in its orbitC_{0} having a period T_{0} and with Mercury revolving in its orbit C_{1} having a period T_{1} (T_{0}<<T_{1}), distance R´_{1} separating the Sun from planet Mercury constantly changes (periodically), as a function of time t, R´_{1} = R´_{1}(t).
Distance R´_{1} changes from R´_{1,min} = R_{1} (perihelion) to R´_{1,max} = R_{1 }+ 2d (aphelion). Consequently, in this case (equivalent case), it is as if the Sun is found at point o (i.e. at the center of mass of the Solar system), therefore, distance α between the Sun and Mercury will be: 
 
 B. THE OTHER PLANETS As regards the other planets P_{i}(i = 2,3,…9), i.e. Venus, Earth, Mars,… etc, because distance d=1,505 10^{9}m is extremely small relative to their distanceR_{i} from the Sun (d<< R_{i}), their orbits C_{i}are
not perceptibly affected by the revolution of the Sun around the center of mass of the Solar system, as is the case with planet Mercury. Therefore, for planets P_{i}we accept the accuracy of Kepler’s first law according to which (as it is well known) the Sun coincides with the center of mass of the Solar system. Consequently, in this case, distances R_{i} between planets P_{i}(i = 2,3,…9) and the Sun are
those listed in Table 1. After everything explained above, the calculation of the advance of Mercury’s perihelion assumes the form of the following equivalent problem: THE EQUIVALENT PROBLEM Consider a heliocentric frame of reference xoy, with the Sun being motionless in position o,
where o is the center of mass of our Solar system. Planet Mercury moves in a circular orbit of radius a with the center being point o, according to relation (1), i.e.: 
   All the other planets P_{i}(i = 2,3,…9), i.e. Venus, Earth, Mars,… etc, move in circular orbits C_{i} of radiuses R_{i}(distances from the Sun) stated in Table 1, withthe center being point o.
Note: In relation (2), the exact distance R_{1} between the Sun and Mercury will be calculated here below. PROOF 1. The classical case As it is commonly
known, since the era of Le Verriere to this day, we have been working on the advance of Mercury’s perihelion by considering the Sun as motionless and its center of mass as coinciding with the center of mass of the Solar system, without taking into account its revolution around the center of mass of our Solar system. Therefore, in this classical case, attractive force F_{0} exerted by the Sun on Mercury will be: 
    N (3) 

    (4) 

   where, G = 6,67 10^{11}the universal gravitational constant
m = 3,302 10^{23} Kg the mass of planet Mercury M = 1, 989 10^{30} Kg the mass of the Sun, and F΄_{0} = 1,318 10^{22}Ν.  }
 (5) 

   By substituting the above values (5) into relation (4), we will have: R_{1} = 57,651 10^{9}m. (6)
This is the exact distance between the Sun and Mercury, in a heliocentric system where the Sun is motionless at the center of mass of the Solar system, without taking into account the Sun’s revolution around it. (It is the classical case, known as such to this day). Note: In this classical case, the advance of Mercury’s perihelion equals 531,9´´/century (See Chris Pollock’s paper referred to above).
2 . The case based on the new data and the calculation of the advance of Mercury’s perihelion. On the contrary, according to the new data, i.e. by taking into account the Sun’s revolution around the center of mass of our Solar system, then on the basis of the abovementioned equivalent problem resulting from
relations (2) and (6), distance a between the Sun and Mercury will be: α = R_{1} + 1,505 10^{9} m or α = 57,651 10^{9} m + 1,505 10^{9} m or α = 59,156 10^{9}m (7) After everything explained above, we can now work on the calculation of the advance of Mercury’s perihelion. According to the equivalent problem and on the basis of value (7), force F_{0}exerted by the Sun M on Mercury m will be:

   Moreover, as it is well known, on the basis of the equivalent problem, the advance δφ of Mercury’s perihelion, per Mercury’s period T (See Chris Pollock’s paper mentioned above), is given by
relation (9): 
  
Ψ is the angle of the line of Mercury’s apses and F_{a} is the force exerted by all planets P_{i}(i = 2,3,…9) together on Mercury, which [force] is expressed by the following relation: 
    (11) 

   Relations (10) and (11), yield the following final relation:

    (12) 

   where λ_{i}in relation (12) is the linear masses of planets P_{i}(i = 2,3,…9), i.e. Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune and Pluto. Since the linear mass λ_{i} of a planet is given by: 
    (13) 

   where M_{i} is the mass of planet P_{i}, and R_{i}is
its distance from the Sun, then based on relation (13) and Table 1, we obtain the following values for the linear masses λ_{i} of these planets, namely: LINEAR MASSES OF THE PLANETS 
   Venus : λ_{2} = 7,159 10^{12}Earth :λ_{3} = 6,354 10^{12}Mars :λ_{4} = 4,481 10^{11}Jupiter : λ_{5} = 3,880 10^{14}Saturn: λ_{6} = 6,344 10^{13}Uranus : λ_{7} = 4,815 10^{12}Neptune: λ_{8} = 3,623 10^{12}Pluto : λ_{9} = 350.324.121   (14) 

   Moreover, in relation (12): 
   G = 6,67 10^{11}, the universal gravitational constant m = 3,302 10^{23}, the mass of planet Mercury λ_{i} = the linear masses of planets P_{i}(i = 2,3,…9),
as expressed by relations (14) R_{i} = the distance of planets P_{i}(i = 2,3,…9) from the Sun, as listed in Table 1. α = the distance between the Sun and Mercury as given by relation (7) F_{0} = – 1,252 10^{22} N the force exerted by the Sun
on Mercury as expressed by relation (8)   (15) 

   Note: The minus sign of force F_{0}signifies that force F_{0 }exerted
by the Sun on Mercury is opposite to Force F_{a}exerted by all other planets together on Mercury. From relation (11), based on values given by (14) and (15), it results that force F_{a} is F_{α} = 7,752 10^{15} Ν By substituting now into the final
relation (12) the values provided by (14) and (15), we obtain: 
  
Also, 
    (17) 

 
 From relations (16) and (17), it results that 
    (18) 

   Thus, relations (18) and (9) yield: 
   This, therefore, is the requested value of advanceδ_{φ}of
Mercury’s perihelion, once we take into account the Sun’s revolution around the center of mass of our Solar system. The resulting value of 576,7´´/century greatly converges with the value of 574,8´´/century yielded by astronomic observations about the advanceof Mercury’s perihelion with error εwhich is: 
   that is, ε= 3%_{0 }(per thousand) / century or with difference δ: 
   that is, 
  
        As it can be observed, this difference δ = 1,9´´/century is extremely small and reveals the true causes of the advance of Mercury’s perihelion, which are mainly the following:
a) The perturbative forces exerted by other planets on Mercury, and b) The Sun’s revolution around the center of mass of the Solar system, as demonstrated above. After everything described above, the following basic conclusion can be drawn:
CONCLUSION The difference of 43´´/century with astronomic observations as regards the advance of Mercury’s perihelion is not attributed to the curvature of spacetime around the Sun, as the Theory of Relativity erroneously maintains. The 43´´/century of the advance of Mercury’s
perihelion (as demonstrated above) are due to the revolution of the Sun around the center of mass of our Solar system, a fact that until today has never been taken into account when calculating the advance of Mercury’s perihelion. Finally, after everything discussed in this paper, the Theory of Relativity should be unquestionably deemed erroneous. IMPORTANT REMARK In this paper the advance of Mercury’s perihelion has been calculated with error ε = 3%o_{ }(per thousand)/century or with a difference δ = 1,9´´/century.
Certainly, there are other theoretical methods resting on a different rationale according to which the advance of Mercury’s perihelion can be calculated probably with a smaller error ε and a smaller difference δ.
Should the calculations of these theoretical methods yield an error ε < 3%o_{ }(per thousand)/century and a difference δ < 1,9´´/century (as compared to this paper), it would be very interesting to see them published.
Copyright 2006: Christos A. Tsolkas Christos A. Tsolkas
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