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The advance of the perihelion of Mercury given in equation 5.46 was calculated using perturbations of individual parameters. This advance can also be illustrated using geometrical considerations. Newton stated the universal law of gravitation which predicts an exact quadratic gravitational field around a mass. Newton has shown that in the gravitational field around a central body, all masses move in elliptical orbits independently of the mass of the orbiting body. According to classical mechanics, the necessary condition to get an exact elliptical orbit is for the mass to move in a gravitational field whose intensity decreases exactly as the inverse of the square of the distance R from the central mass:

6.1 |

6.2 |

Although the inverse quadratic law is generally accepted, a very slight deviation of that law was first suggested by Aseph Hall in 1894 [1]. Since we have seen that the mass of a body changes when it is moved into a gravitational potential, we can show that such a slight change of mass leads to an effect equivalent to the slight change of the quadratic function suggested by Hall.

Classical mechanics shows that a massive body travels in an elliptical orbit when the force F rather than the field between the central mass and the orbiting mass decreases as the square of the distance. Let us consider Newton's equation (written in a correct way, contrary to equation 5.1):

6.3 |

However, when we consider the proper parameters of the observer moving to different distances from the Sun, the gravitational

Using either the non quadratic force as seen by an outer space observer that takes into account the change of mass of Mercury or the apparent non quadratic force given by equation 5.9 (with constant proper mass) leads to a similar advance of the perihelion of Mercury. However, these calculations are incomplete because other fundamental phenomena, like the change of mass as a function of the velocity of Mercury on its orbit, are not taken into account. Changes of length and clock rate due to Mercury's velocity and gravitational potential should also be taken into account.

Since we have already calculated the total precession in equation 5.46, we will limit our demonstration here to the change of one parameter using only the change of mass of Mercury as a function of its distance from the Sun. We will use only the perturbation of this parameter and show that it is one of the contributions to the geometrical precession of the ellipse which can be illustrated in a classical experiment that can be done in a laboratory using a simple apparatus.

**6.2 - The Change of
Mass of Mercury.**

Let us
consider the change of force on Mercury due to its change of
mass as a function of its distance from the Sun. Equation 4.25
shows how the absolute mass of a kilogram decreases when getting
closer to the Sun. Consequently, the total mass of Mercury
decreases by the same ratio. From equations 4.39, 4.40 and 4.41,
the mass of Mercury (in outer space units) follows the
relationship:

6.4 |

6.5 |

6.6 |

6.7 |

6.8 |

6.9 |

6.10 |

However, in the case of a non quadratic field (cubic term in equation 6.10), the period of oscillation of the radial component becomes longer than the period of the circular tangential component. Of course, a circular component does not 'feel' the field gradient. Because the cubic radial component of oscillation has a longer period, there is a continual shift of phase between the periods of the tangential and of the radial components.

Consequently, the cubic term in equation 6.10 which does not follow Kepler's quadratic gradient of force, is responsible for the precession of the ellipse because the radial component, having a longer period, becomes out of phase with the circular component. It is the difference of period between the tangential and the radial components of motion that produces the precession of the ellipse. We also notice that it is the radial component of oscillation which is most affected by the change of parameters resulting from mass-energy conservation.

Let us examine the bracket on the right hand side of equation 6.10. Using a series expansion, we can show that it is mathematically equivalent to a simple exponential form given by:

6.11 |

6.12 |

6.13 |

6.14 |

**6.3 - Orbital Shapes
and Gravitational Force Gradients.**

We have
calculated in equation 6.10 the force on Mercury as a function
of the distance R_{M}. The corresponding gravitational
potential V_{M}(o.s.) is obtained by the integral of
equation 6.10. This gives:

6.15 |

6.16 |

6.17 |

6.18 |

6.19 |

6.20 |

6.21 |

6.22 |

6.23 |

6.24 |

6.25 |

**6.4 - Identity of
Mathematical Forms.**

We find
that the advance of the perihelion of Mercury obtained with the
perturbation method used by Einstein and by us in equation 5.46,
has the same mathematical form as equation 6.25 which clearly
corresponds to the precession of an elliptical orbit. There are
two obvious differences. Since we have not taken into account
the eccentricity of the orbit, the term 1-e^{2} is naturally missing in equation 6.25 as explained in
section 5.10. Other similar parameters are ignored here since we
do not take into account the perturbations explained in section
6.1. If we take into account these perturbations, other similar
terms will be added and the full precession will be found as
obtained in chapter five. The aim of the present demonstration
is only to illustrate the reality of the classical precession of
the ellipse in the case of a non quadratic force.

**6.5 - Illustration of
Trajectories in Potential Wells.**

When
the force on a planet moving around the Sun decreases as the
square of its distance from the Sun, it travels on a perfect
ellipse. However, due to mass-energy conservation, the exact
intensity of the force does not decrease as the square of the
distance. As seen in equation 6.14 the force follows the
relationship:

6.26 |

**Figure 6.1**

Demonstration of a mass moving in an elliptical
orbit in a quadratic potential well changing as 1/r^{2}.

However, if the shape of the cone is different (see figure 6.2) so that the potential increases more rapidly than the inverse square of the distance (corresponding to equation 6.26 with e ¹ 0), after throwing a ball, we see that the axis of the elliptical orbit precesses just as observed for Mercury in its orbit around the Sun. The cause of that classical precession on that apparatus is (in part) the same as the cause of the precession of 43 arcsec per century of Mercury. Of course, this demonstration assumes that the friction and the rotation of the ball are negligible.

**Figure 6.2**

Demonstration of the precessing orbit of a mass
moving in a potential well changing as 1/r^{(}^{2+e}).

This shows that the advance of the perihelion of Mercury is not caused by space or time distortion. It is simply a beautiful demonstration of classical mechanics that predicts precessing orbits giving the shape of a rosette.

**6.6 - Validity of the
Classical Model.**

We have
found above that there is a perfect mathematical agreement
between the result calculated in equation 5.46 and the result
predicted using Einstein's mathematics. Moreover, those results
are in perfect agreement with the observations of the advance of
the perihelion of Mercury.

In
order to arrive to his equation, Einstein, needed several new
hypotheses called Einstein's relativity principles. Let us
compare the hypotheses used by Einstein with the ones used in
this book to find the Lorentz transformations and the equation
for the advance of the perihelion of Mercury. This comparison is
important if we wish to apply Occam's razor which gives a
preference to the theory that requires the minimum number of
hypotheses. Einstein's theory requires many new hypotheses, for
example:

1) the
reciprocity principle which is not compatible with mass-energy
conservation as showed in section 3.9;

2) the
hypothesis that the acceleration produced by a change of
velocity is undistinguishable from the acceleration due to
gravity (see chapter ten);

3) the
non conservation of mass-energy in general relativity.

Einstein then arrived at the consequences that space and time
can be distorted, contracted and dilated. In fact, Einstein's
model not only requires new physical hypotheses, it also
requires "new logic" which is not compatible with the natural
understanding of nature. Classical logic can no longer be
applied in relativity. In this book, we use the Bohr model of
the atom which is so familiar everywhere in physics. We also
find that using proper values, the physical relationships are
valid in all frames as in Einstein's relativity. At the same
time, a rational explanation is given. No time nor space
distortion is required and the new interpretation is compatible
with classical logic. There is certainly an extremely strong
preference in favor of this new model when we apply Occam's
razor.

**Important Note:**

After the writing of that
book, a complete detailed description of the advance of the
perihelion of Mercury entitled:

*"A Detailed Classical Description of the
Advance of the Perihelion of Mercury".*

**This new paper appears: At this location**

**6.7 - References.**

[1] A. Hall, ** A
Suggestion in the Theory of Mercury,** Astr. J. 14,
49-51, 1894.

[2] H. Goldstein,

[3] E. T. Whittaker,

F _{M}(o.s.)number of outer space newtons for the gravitational force on Mercury G(o.s.) number of outer space units for the gravitational constant kg _{frame}mass of the local kilogram in absolute units M( M)_{M}(o.s.)number of outer space kilograms for Mercury at Mercury location M( M)_{o.s.}(o.s.)number of outer space kilograms for Mercury in outer space m( M)_{M}[o.s.]mass of Mercury in outer space units at Mercury location m( M)_{o.s.}[o.s.]mass of Mercury in outer space units in outer space M( S)(o.s.)number of outer space units for the mass of the Sun R _{M}(o.s.)number of outer space units for the distance of Mercury from the Sun V _{M}(o.s.)number of outer space units for the gravitational potential on Mercury

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