_{1}

^{*}

The longitude of the perihelion advance of Mercury was calculated for the two and ten-body problem by using a correction to the balance between the force given by the Newton 2
^{nd}
law of motion and the Newton gravitational force. The corresponding system of differential equations was solved numerically. The correction, that expresses the apparent mass variation with the body speed, has a trend that is different from those that usually appear in the electron theory and in the special theory of relativity. The calculated intrinsic precession was ~42.95 arc-sec/cy for the Sun-Mercury system and ~42.98 arc-sec/cy when the difference between the corrected model and the Newtonian model, for the 10-body problem, is taken.

In the scientific literature many papers can be found that deal with alternative theories to the Einstein general theory of relativity (GTR) to model the remarkable observation of Le Verrier with regards to the perihelion precession of Mercury (PPM) which cannot be explained with the influence of other planets. These theories are metric GTR, non-metric GTR, combination of the special theory of relativity (STR) with the Lagrangian (classical or relativistic), etc.

In this work a correction to the balance equation between the force given by Newton 2^{nd} law of motion and Newton gravitational force is introduced to calculate the inherent PPM. The objective of this manuscript is to show that the intrinsic advance of the longitude of the perihelion of Mercury (ALPM) can be accounted for using that correction.

1) Modification of the balance between the Newton 2^{nd} law of motion and the Newton gravitational force

Equating Newton’s 2^{nd} law to the Newton gravitational force, a non-linear ODE is obtained for N point-mass planets in the solar system [

The solution of Equation (1) for N = 1 does not yield an ALPM. Einstein general theory of relativity (GTR) addressed this problem by introducing a curved space-time concept. In this work an empirical approach is used to address the problem.

Let’s modify the l. h. s. of Equation (1) (the l. h. s. is used just for convenience) as

where

(3)

The coefficient of the acceleration for some values of L (assuming

Other equations could be obtained from the Planck balance equation, adapted to a gravitational force:

2) Numerical Solution of the System of Differential Equations

Equation (1) for the heliocentric coordinate system is written as [

k = 0.01720209895 is the Gaussian constant (the Newton gravitational constant expressed in terms of the astronomical unit length, day and taking the Sun mass as 1). Similarly Equation (2) is written as

Assuming

The finite difference method (using a standard two point- finite difference applied to the concept of acceleration and speed to obtain the next value of the speed and the position respectively), with a very small integration step

The longitude of the perihelion,

longitude of the ascending node and

The reciprocal mass and initial conditions

^{4} days (~109.5 years), using a positive

Sgn(L)\L | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

− | −7.20 | −14.34 | −21.51 | −28.65 | −35.82 | −42.99 |

+ | 7.17 | 14.31 | 21.48 | 28.63 | 35.84 | 42.95 |

Note that in this case (the two-body problem) S is not periodic and it is linearly correlated with time and L and that the discrete change is due to the discrete value of L used. Note also that the difference between any two consecutive values of L is about 7, that a linear fit of S with positive L results in a slope of 7.1618, and that the ratio of S_{L}/S_{1} is ~L. The Einstein GTR equation of motion (for m = m_{0}) was also solved numerically, an S = 42.97”/cy was obtained.

It could be worthy to check if L is a constant for the solar planetary system and other bound-orbital-gravita- tional systems or if it represents a state of the moving body. It could also be worthy to assess the potential impact on other gravitational problems as for example on the dark energy/matter problem. It is hoped that a derivation for L = 6 is found.

Note that in this case (the 10-body problem) S is periodically and linearly correlated with time, large fluctuations and periodicities are due to, according to [

Note also that the S (intrinsic to Mercury) difference (for L = 6) between the results of

The longitude of the perihelion advance intrinsic to Mercury was accounted for in the two and 10-body problem by using a correction to the balance between the Newton 2^{nd} law of motion and the Newton gravitational force.

L | 0 | −6 | 6 | Δ: −6 - 0 | Δ: 6 - 0 |
---|---|---|---|---|---|

S ("/cy) | 528.69 | 485.75 | 571.68 | −42.94 | 42.98 |

The correction, that suggests a variation of the mass with the moving body speed, is different from what is usually expected from the special theory of relativity and from the electron theory (L is positive instead of negative).

I would like to thank Dr. M. Krizek for his valuable comments and suggestions.

Barbaro Quintero-Leyva, (2015) On the Intrinsic Precession of the Perihelion of Mercury. Open Access Library Journal,02,1-5. doi: 10.4236/oalib.1102239