The precession of the perihelion of Mercury explained by Celestial Mechanics of Laplace

. We calculate in this article an exact theoretical value obtained classically for the secular precession of the perihelion of Mercury, followed by the theory of Stockwell, based on planetary theory of Laplace, your Mécanique Céleste : found 5600’’.84 of arc per century for the angular velocity of the longitude of the perihelion of Mercury, d  /dt, adding to the precession of the equinoxes of the Earth relative to the beginning of the year 1850, as calculated by Stockwell. The best known anomaly of the motion of Mercury is the advance of the perihelion precession in relation to the classical theory, discovered by Le Verrier [1], anomaly that which is believed explained by General Relativity [2], [3]. We intend in this article to calculate this precession of the perihelion of Mercury following the Newtonian theory, the Mécanique Céleste of Laplace [4], and show that the theoretical value obtained is in excellent agreement with the observed value. This will lead to the conclusion that General Relativity does not correctly explain the precession of the perihelion of Mercury, unlike the classical theory. First we calculate this precession based on the data of Stockwell [5].

-Mass of planets (M p ) and satellites (M s ) of the solar system in kg and reciprocal of the sum in relation to the mass of the Sun (M S = 1.9891x10 30 kg). is the mass parameter adjustment, satisfying

Planet M planet (kg) M satellites (kg) m -1 = M S / (M p +M s )
where , (2) i.e., the mass of the planet relative to the mass of the Sun. Other invariable elements of the planets, and also required for calculation of variable elements, are shown in table 2 below: where P is the orbital period in days. The values of the precession of the perihelion of Mercury obtained due to the influence of other planets, with and without adjustment masses, with and without satellites, are recorded in table 5 below, rounded to two decimal digits after the decimal point. Will be added to each of these values the precession of the equinoxes on Earth in relation to the apparent ecliptic, which calculation based on Stockwell (for the period 1850-1950) provides   5024''.749 831  5024''.75. The actual calculation of the perihelion advance in relation to classical theory uses as reference the value of 5600''.73 [6]. The values tabulated above correspond to the 100-year period from 1850 to 1950 (January 1), and is noted that the advance of the perihelion for the three cases is less than the value currently  [7]. The longitude  (i) of the perihelion of a planet (i) of the solar system, taking into account only the mutual influence of the planets, and according to the Celestial Mechanics of Laplace [4], is obtained from the arctangent of the ratio between a sum of sines (h (i) ) and a sum of cosines (l (i) ), such that

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where with the index (0) referring to Mercury, (1) to Venus, (2) to the Earth, etc., and ( ) is the eccentricity of the orbit of the planet (i). The solutions to the various h and l must meet the 16 linear ordinary differential equations system of first degree for i equals 0 to 7, corresponding to the eight planets of the solar system (nothing prevents adding up to 8, including Pluto, as it also orbits around the sun and was considered planet, but its contribution would be negligible, and the contributions of other more distant bodies). The following notation is used above: where m is the mass of the planet relative to the Sun's mass, n is the mean motion, is the mean distance from the Sun and We also use with The values of ( ) are positive and ( ) are negative, which is easy to see, while (i, k) and [i, k] have the same sign equal to the sign of n (i) .
As an example, Stockwell obtains the following values for the coefficients of the disturbance suffered by Mercury: With the mass adjustments of  Making the calculation of the second sum we obtain .
For a more accurate calculation of the value above, recalculating the coefficients (0, k) e [0, k] using the actual values of , supposedly constant, found in tables 1, 3 and 4, respectively, we obtain the following results, as shown in table 7. Using the coefficients calculated above and the parameters of eccentricities and longitudes of the perihelion given in table 6, (20) is recalculated as and for (21) we obtain .
The final result for the angular velocity of the perihelion of Mercury is then, according to (18), the difference between (23) and (24), i.e., Within the experimental precision, the theoretical value obtained by the theory of Stockwell, which is the Newtonian theory, the same theory of Laplace, is in agreement with the observed value, then it is not true to say that the classical, Newtonian theory, is not able to explain the advance of secular precession of the perihelion of the planets, and Mercury in particular. Rather, the Newton's gravitation explains with surprising accuracy.
See that our calculations were based on the year 1850, because it is the reference time used by Stockwell. Most likely fixes for the most recent years 1950, 2000, 2014, etc. will reach another total value to this precession, but must proceed in accordance with its observed value of the epoch, their values not differing much from one second of arc per century. The precession of the equinoxes is the largest component in the calculation of the total value of the precession of the perihelion, so it must be the object of careful attention.
If for some reason our calculations were not so surprisingly coincident with the observational result, they would already be able to show the most important: the precession of the perihelion obtained with General Relativity, equal to 43''.03 of arc per century [6], is completely at odds with any hypothetical advance of this precession, because this movement (or deviation, difference, excess) would be much lower, for example, the one obtained with the coefficients of Stockwell, about 31''.05 arc per century (table 5). However, the difference between Newtonian theory and observations obtained here is, essentially, zero (instead 43''.03): theoretical value = value observed, within the measurement accuracy. I.e.: General Relativity does not explain the correct value of the precession of the perihelion of Mercury.
We close this letter clarifying that do not exactly reproduce the calculations of Stockwell, but we rely on it. Our calculations initially used their coefficients and data, we even used all the coefficients obtained for the solution of the system (8), given by sums of sines and cosines, but we used our mass corrections, the masses of the planets added to the masses of the satellites, and the calculations were made by computer programs in C language, using double variables. The average annual motion of the perihelion of Mercury really calculated by Stockwell is 5''.463803 (p. xi, Introduction).
In Laplace was found (13) and (15) into infinite series, recalling the known series expansions of elliptic integrals, while in Stockwell these polynomials in are converted to decimal numbers with up to 7 significant digits; ( ) be a polynomial of degree 30 and ( ) a polynomial of degree 31 in , indicating clearly that the two series are indeed endless.
International Journal of Engineering and Technologies Vol. 3 Laplace tells us that both series only converge for otherwise (and if ≠) we should calculate (k, i) and [k, i] instead of (i, k) and [i, k], using the following relations: Furthermore, the important equation (18) we also find in Laplace only, not in Stockwell. The system (8) becomes unnecessary when what we want is just to calculate the value of the instantaneous temporal variation of  for a single time t, instead of the exact value of  for all time t, and have the values of the various  (k) and e (k) previously tabulated, as the example shown here.