Chapter Six
Geometrical Illustration of the Advance
of the Perihelion of Mercury.
6.1 - Conditions Controlling the
Geometrical Shape of an Orbit.
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:
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6.1 |
There are several
measurements showing that this quadratic decrease of the gravitational
field is followed quite accurately in nature. At a distance
RM(o.s.) from M(S)(o.s.), the field is given by:
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6.2 |
where G(o.s.) is the
number of outer space units of the gravitational constant
and M(S)(o.s.) is the number of outer space units of the solar
mass. Equation 6.2 implies that the Sun generates an exact quadratic
gravitational field (in outer space units) in which Mercury is
submerged.
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):
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6.3 |
Since the mass of
Mercury changes with its distance from the Sun, it is incorrect to believe
that the force between the Sun and Mercury still follows an inverse
quadratic function of that distance. Even if the gravitational field
around a central mass decreases exactly as the square of the distance, the
total force between Mercury and the Sun does not decrease at the same rate
as the field. The trajectory of a planet whose mass decreases when it gets
deeper in a gravitational field corresponds exactly to the problem of a
non quadratic force around a central mass. Using classical mechanics we
can calculate the new geometrical shape of the orbit when the force (not
the field) between the Sun and Mercury is non quadratic.
However, when we
consider the proper parameters of the observer moving to different
distances from the Sun, the gravitational field (defined as
the force divided by the proper mass) calculated from equation 5.9 is not
quadratic for the observer traveling between different locations from the
Sun. Consequently, using the parameters existing where Mercury interacts
with the gravitational field leads to an apparent non quadratic field
(since the proper mass of Mercury is constant for a Mercury
observer).
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:
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6.4 |
Equations 6.4 and 4.25
give:
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6.5 |
or:
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6.6 |
Using equation 6.6 in
6.3 gives a force equal to:
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6.7 |
which is equal
to:
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|
6.8 |
Let us define:
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|
6.9 |
Equation 6.8
becomes:
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|
6.10 |
Equation 6.10 shows
that the gravitational force is the difference between a quadratic and a
cubic function. It is known that in a quadratic field, an elliptical orbit
with a small eccentricity (first order expansion) follows the equation r =
a(1 + ecosq) (a is the semi-major axis and e is
the eccentricity). This equation implies two components: a tangential
component of constant radius a and a radial component of amplitude
aecosq. Since Kepler's third law predicts the
same period (first order) for orbits having the same average radius with
or without eccentricity, both the tangential and the radial components
lead to the same period in a quadratic field.
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:
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6.11 |
in which the exact
value of e is:
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6.12 |
A very good
approximation to the first order (with n = 1) gives:
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6.13 |
Combining equations
6.10 and 6.11 gives:
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|
6.14 |
where e is always positive. Equation 6.14 shows that, because
of the decrease of mass due to mass-energy conservation, the
force F between Mercury and the Sun no longer decreases
exactly as the square of the distance. The change of mass of Mercury as a
function of its distance from the Sun is responsible for the change of
power of RM from 2 to 2+e. Therefore even if the gravitational field
affecting Mercury decreases exactly as the inverse of the square
of the distance as written in equation 6.2 (as in a perfect Newtonian
field), the gravitational force is not Newtonian as shown in
equation 6.14. Let us reconsider now the trajectory of bodies submitted to
a force decreasing with a function which is slightly different from
1/R2.
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
RM. The corresponding gravitational potential
VM(o.s.) is obtained by the integral of equation 6.10. This
gives:
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6.15 |
The orbit followed by a
mass submitted to the potential described by equation 6.15 has already
been calculated [2, 3]. Using temporarily Goldstein's notation [2],
the solution of equation 6.15 is a precessing ellipse with a velocity of
precession equal to:
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6.16 |
where W(sec) is in radians per second of time. Transforming
Goldstein's notation into ours, we have m =
M(M)o.s.(o.s.) and h =
(G(o.s.)M(S)(o.s.)M(M)o.s.(o.s.)k1)/2.
t is the period of translation of Mercury around
the Sun. The angular momentum l in equation 6.16 is:
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6.17 |
where dq/dt is the angular velocity. Therefore, from equation
6.9 and the definitions above, we have:
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6.18 |
From equations 6.16,
6.17 and 6.18, we have:
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6.19 |
Let us transform the
precession W(sec) given in radians per second for
radians per circumference W(circ). We
obtain:
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6.20 |
By definition, the
period t equals:
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6.21 |
Equation 6.21 in 6.20
gives:
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6.22 |
Newton's law shows that
the force of gravity FG is equal to the
centrifugal force FC in a circular orbit
(the eccentricity has not yet been taken into account). We have the
fundamental equations:
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6.23 |
Equation 6.23
gives:
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6.24 |
Equations 6.24 and 6.22
give:
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6.25 |
Equation 6.25 gives the
velocity of precession of an ellipse for the case of a perfect quadratic
field in which the orbiting mass changes with its position in the
gravitational potential, due to mass-energy conservation.
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-e2 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:
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6.26 |
The trajectory of a
particle submitted to equation 6.26 is an ellipse as illustrated on
figures 6.1 and 6.2. In figure 6.1, a smooth conic surface is built (in
the Earth gravitational field) in such a way that the height above the
ground increases as the negative of the inverse of the square of the
distance from the central axis. This corresponds to e = 0 in equation 6.26. In this case, the potential
energy of a ball sliding (without friction) on the surface increases
according to the inverse quadratic function from the center. If we throw a
ball on the surface, we can get a circular orbit at various distances from
the center. Using a different initial angular momentum, one can observe a
stationary elliptical orbit as drawn on figure 6.1.
Figure 6.1
Demonstration
of a mass moving in an elliptical orbit in a quadratic potential well
changing as 1/r2. 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".
6.7 - References.
[1] A. Hall, A Suggestion
in the Theory of Mercury, Astr. J. 14, 49-51, 1894.
[2] H. Goldstein, Classical
Physics, Addison-Wesley, Reading, Mass., second Edition, p. 123,
1980.
[3] E. T. Whittaker,
A Treatise on the Analytical Dynamics of Particles and Rigid
Bodies, Cambridge University Press, Fourth Edition, Chapter 4,
1937. (also Dover, New York, 1944).
6.8 - Symbols and
Variables.
|
FM(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 |
|
kgframe |
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 |
|
RM(o.s.) |
number of outer space units for the distance of
Mercury from the Sun |
|
VM(o.s.) |
number of outer space units for the
gravitational potential on
Mercury |
|
