Download Universal force-motion equations and solar system implementation

Universal force - motion equations and solar system
implementation
Ahmet YALCIN
R&D Manager
ESER Contracting and Industry Co. Inc. Turan Güneş Blv Cezayir Caddesi 718. Sokak
No: 14 Çankaya, Ankara
E-mail: [email protected]
Abstract
Traditional physics uses Newton’s Universal Law of Gravitation and Einstein’s Theory of
Relativity in order to explain the mechanism of the entire universe. However, these two
approaches are both insensitive about some ordinary observations which somehow have remained
unspoken about to this day for some reason. They cannot answer those three basic questions:
“Why do all celestial bodies generally cluster in the form of a single planar structure?”
“Why do all celestial bodies revolve around the superior one in the same direction?”
“Why do celestial bodies generally rotate around their own axes?”
This article investigates how the celestial bodies’ rotation around their own axes and motion
relative to each other influence their force fields and with three postulates proposed, develops
universal force and motion equations. Afterwards, it discusses to what extent the results correlate
with the observations by applying the equations to a Sun-planet duality. The reason why heavenly
bodies rotate around their own axes and its mathematical explanation will be discussed in another
article soon.
PACS numbers: 04.50.Kd, 04.80.Cc, 95.10.Ce, 95.30.Sf, 96.12.De, 96.12.Fe, 96.30.Bc
Keywords: Universal force equations, universal motion equations, General Relativity, Newtonian
physics, gravitational waves, planetary motion, flat and globular clustering, field relative model of
the Universe, instantaneous remote effect, field propagation rate
1. INTRODUCTION
The Newton’s Laws of Gravitation and motion have been the one and only resource to resolve the
movements of the celestial bodies for two centuries. However, this law of gravitation, as the readers
may as well be aware of, remained inadequate to explain some facts.
At the beginning of the twentieth century, Einstein brought bright solutions to some part of
these problems with the Theory of Relativity. Special and General Theories of Relativity has been the
fundamental router for the science of physics in twentieth century. These theories played an effective
and a fundamental role to solve the problems both in cosmology and inside the atom. With the solution
he brought to the deviation of Mercury’s orbital axis and observational verification of his predictions
on light deflection in the vicinities of stars, Einstein earned great reputation and dignity.
Are those two theoretical approaches exactly enough for explaining the order of the universe?
Let’s try to give an answer to this question with very ordinary observations from the Solar system,
without going too far and mentioning some fundamental problems in cosmology and physics like the
expansion of the universe with increasing acceleration or disconnection between the rules of
operational mechanism in cosmology and quantum world.
Unspoken ordinary observations
The following observations about our solar system are obvious and clear:
Universal force-motion equations and solar system implementation
a. All of the elliptical orbits of the planets are approximately on the same plane. Those orbital
planes are close to the Sun’s equator plane.
b. There exists an Asteroid belt between Mars and Jupiter which is made up of rocks. This belt is
located on the equator plane of the Sun.
c. There is a Kuiper belt beyond the planets which is made up of bigger rocks and distributed on
a wider range of area. This belt is also on the Sun’s equator plane.
d. All giant planets in Solar System have ring like formations which are especially evident on
Saturn. Those rings are all located on each planet’s equator planes.
e. Generally, planets have one or more natural satellites. Those satellites also all revolve almost
around the planets' equator planes.
f. All celestial bodies revolve around the superior one, commonly in the same direction with
its (superior’s) rotation,
g. All celestial bodies, by some means or other, generally rotate around their own axis.
The observations listed above has been ignored, not recognized as a problem and considered
as a natural and inviolable fact by the scientists throughout the history. Newton’s law of gravitation
explains mathematically why planets rotate around on an elliptical orbit; however, there isn’t a theory
that explains why the orbits of all the planets in a system are approximately in the same plane.
Here, since the orbits of all the planets do not exactly overlap, this may be assumed as
coincidence; therefore, it could be said that it’s not right to bring such a rule. As a matter of fact,
clustering of celestial bodies in a flat region around a superior’s body expresses a universal reality.
The above sequence of observations and more others in the universe definitely indicates the existence
of such a rule.
Even though everybody is enchanted by the visual beauty of Saturn’s rings, there isn’t a
satisfactory theoretical study about their formation. The rings mostly consist of individual ice particles
whose sizes range from one to ten meters. The distance of rings to Saturn begins from 70,000 km and
extends up to 213.000 km and their thickness is only around 100 meters. It is needless to say that,
these rings are exactly on the equatorial plane of the Saturn. It is obvious that there should certainly be
repulsive forces both to up and down pushing the pieces of ices to keep them in the equator plane.
If we look at the universe from a wider area, we can observe that this flattened structure
applies to the entire universe. We know that the width of our galaxy Milky Way is about 100-120.000
light years whereas its thickness is only 1000 light years. So, billions of stars in our galaxy are
clustered in a very tiny structure of a disc.
General opinion among the scientists is that this mystery of flattening is originated from the
conservation of angular momentum in the system. It is known that many of the computer simulation
programs made in a great scale support this idea. It should be noted that this explanation is not
satisfactory at all. The reasons can be listed as follows:
1. For the conservation of angular momentum, fixed and individual orbit for each body in the
system is sufficient; they do not necessarily have to share a common plane. Because, the
angular momentum of any component in the systems is totally independent the motion of the
others including the superior’s. If you sum up angular momentums of all components and
make it fixed, you can never get a flattening force. However, above observations claim that a
concrete flattening force is a must.
2. We know that there are a lot more similar computer simulations which failed1. It all depends
on how you start the simulation and what kind of initial conditions you foresee for it. If you
start the simulation with all the dust and gas, which will form the system, to have the tiniest
amount of initial rotation in the same direction, it is likely that you will get the results you
desire.
3. The clumping occurring at the superior’s equator plane in the system is as important as the
flattening itself whereas angular momentum of each component has no concern with the
rotation of the superior in the system.
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Universal force-motion equations and solar system implementation
4. Conservation of angular momentum actually expresses an equilibrium state, therefore does
not imply any force. However an expression saying that “The system acts in a way to
maximize its angular momentum” makes sense. Because the maximizing process includes a
flattening force until the equilibrium is reached. Yet, that process is not a reason for flattening
but a result of it.
Those kind of crude and inelaborate approaches camouflage some bare facts about the
operation of the nature and its cost is heavy. I am not very happy to use these two words “crude” and
“inelaborate” and I hope the readers will justify me in the end. The flattening force is a tangible and
computable quantity which has nothing to do with the initial conditions and the internal dynamics.
Moreover, the issue is not just a problem of flattening. The very significant other observations listed
above make it inevitable to find a new, different and revolutionary approach that will solve all the
problems.
Both Newton's universal law of gravitation and Einstein's General Relativity theory remain
silent in the face of those evident observations listed above. It is clear that the reason for this is that
these two approaches are expressed with “spherical symmetric" force equations. Yet, the universe
shouts out, "our mystery cannot be solved at all using spherically symmetric force equations".
This article takes the heed of this scream.
2. ROTATION EFFECT
Newton's law of gravitation is clear. Any celestial body in space creates a gravitational field around
itself. This field causes spherically symmetric force field towards the body. At every point represented
by a position vector 𝒓 in the effective field of the celestial body, there is a gravitational potential field
expressed by 𝑉(𝑟). In traditional physics, we define the magnitude of this scalar field as follows:
𝐺𝑚
(1)
𝑟
Where 𝑚 is mass of the body, 𝐺 is universal gravitational constant and 𝑟 is the distance from
the point to the center of the celestial body.
𝑉(𝑟) = −
Here, for a particular body, the magnitude of 𝑉(𝑟) only depends on the distance between the
center of the mass and the point in the field together with the mass of the body; it does not matter in
which direction the affected point is due to the spherical symmetry. The dimension of gravitational
potential is the square of the velocity (𝑚⁄𝑠)2 . In this case, there is no reason not to accept the
following postulate logically:
Postulate 1a: Every celestial body in the universe has a scalar speed field around it represented by
the formula given below:
2𝐺𝑚
(2)
𝑣𝑓 (𝑟) = √
𝑟
This scalar quantity, known as the escape velocity in traditional physics, is the lowest
amplitude of the velocity of an object in a gravitational field to escape from the attraction force. In that
case the gravitational potential field 𝑉(𝑟) is equal to the opposite sign of half the square of the scalar
speed field defined above. We will call this quantity as natural, rest or existence speed field of the
source body.
If the celestial body is rotating around its own axis, we will extend the above postulate
assuming that the field of the body is also rotating in the same way:
Postulate 1b: If a celestial body is rotating around its own axis relative to a remote fixed point, in
addition to the scalar speed field, it will also have a velocity field expressed below:
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Universal force-motion equations and solar system implementation
2𝐺𝑚
(𝝎
̂ × 𝒓̂)
𝒗𝒇𝒔 (𝒓) = 𝑘√
𝑟
(3)
̂ and 𝒓̂ are the unit vectors about the axis of rotation of the source body and position vector of
Here, 𝝎
the affected point in the field whereas 𝑘 is a coefficient that scales the magnitude of the rotational
velocity. This coefficient is the measure of, to what extent the velocity field amplitude that the rotation
of the object creates on its equator approaches to the existence speed of the body. Accordingly, for the
rocky type of planets which means that every particle in the planet body has same angular velocity,
this coefficient will be as follows:
𝜋𝑅𝐴 2𝑅𝐴
√
𝑇
𝐺𝑚
Where 𝑅𝐴 is the radius of the planet and 𝑇 is rotational period of the planet.
𝑘=
(4)
It is difficult to identify a firm, rotational velocity for every celestial body. Even though
solid rocky planets have a fixed rotational velocity, it is impossible to tell same thing for Sun and gas
giant planets. We know that the appearing rotational speed of the Sun on its equator slows down
towards the poles. Horizontal lines on the gas giants show that they do not have a solid rotational
velocity. Here, an equivalent average rotational velocity can be mentioned but it is impossible to
determine it exactly with the observational information since their internal structures are unknown.
Equation 4, says that 𝑘 = 0,0415 for Earth. In this case, if the rotational speed of Earth was 1⁄𝑘 =
24 times faster (one revolution per hour) than its velocity field amplitude at the equator would be as
much as its natural speed. The same value is 0,003 if calculated for Sun with its appearing speed on
the equator. The real value for 𝑘 should be much smaller.
The source for the rest of the speed field of celestial bodies will be discussed in detail in the
future article titled "Fundamentals of the field relative theory of the universe". For the time being, it
will be useful to know brief information about it. Particles forming the celestial body are constantly
on the move, and their positions relative to each other continuously change. Therefore, the particles in
the body have no common rotational axis. Thus, the resultant field velocity at every point on the
surface of the body forms a certain equivalent space density. That is what we perceive as the speed
field, and it should naturally be spherically symmetric. If the body is not rotating around its axis, this
spherical symmetry will shape the entire field affected. But if the body is rotating around its axis, due
to the resultant additional movement of the particles, the field of influence will additionally be formed
as axially symmetrical, proportional to this rotation velocity.
The field of influence of a rotating celestial body will now be a speed+velocity field. We will
show the sum of these two fields using a hyper number system called quaternion2 in which the
vector component is a real velocity vector:
(5)
2𝐺𝑚
[1 + 𝑘(𝝎
̂ × 𝒓̂)]
𝒒(𝑟) = √
𝑟
In such a case, the active field potential of this body will be the opposite sign of half of the
quaternion’s norm which is the dot product of it with itself. So:
𝐺𝑚
(6)
[1 + 𝑘(𝝎
̂ × 𝒓̂)] ∙ [1 + 𝑘(𝝎
̂ × 𝒓̂)]
𝑉(𝑟) = −
𝑟
This expression can be written in a reference frame having the body at the center and body’s
equator is in 𝑥 − 𝑦 plane as follows:
𝐺𝑚
(7)
(1 + 𝑘 2 sin2 𝜃)
𝑉(𝑟) = −
𝑟
̂ and 𝒓̂.
Here 𝜃 is the angle between 𝝎
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Universal force-motion equations and solar system implementation
Axiom 1b clearly states that, every celestial body exists with its own field of influence and if it
is rotating or linearly moving, its field is also rotating and moving. Therefore, the rotation speed of the
field is naturally the maximum on the equatorial plane and zero along the axis of rotation.
In that case, since the force field 𝒈 is the gradient of this potential as a conservative field, it can
be expressed as follows:
𝒈 = −𝛁𝑉(𝒓)
𝐺𝑚
(1 + 𝑘 2 sin2 𝜃)]
𝒈 = 𝛁[
𝑟
If we write this expression in spherical coordinates, the force field will now be the function of both
the radial distance and polar angle:
𝐺𝑚
𝐺𝑚
(8)
̂
𝒈(𝑟, 𝜃) = − 2 (1 + 𝑘 2 sin2 𝜃)𝒓̂ + 𝑘 2 2 sin 2𝜃 𝜽
𝑟
𝑟
Results:
1. As seen, the force field has a lateral component along with a radial component. Moreover, the
radial component is no longer spherically symmetric. The radial force field is as large as the
Newton’s gravitational force field in the direction of the axis of rotation but is the largest in
the equator.
2. There is also a polar (𝜃) component for the force field. This component is zero both in the
direction of the rotation axis and in the equatorial plane.
If the equation (7) is considered, it could be said that the mass of the object is perceived
differently relative to where it is located at the field of influence. We can express this as follows:
𝑚𝑓 = 𝑚0 (1 + 𝑘 2 sin2 𝜃)
(9)
Here, 𝑚0 is the rest mass and 𝑚𝑓 appeared mass amplitude. The expression in the equation
𝛾 = (1 + 𝑘 2 sin2 𝜃) is called the mass field factor. This factor determines perceived mass amplitude
in different directions.
Figure 1a shows the change of the additional speed field (sin 𝜃) resulting from the rotation
of the object, the increased radial component of the force field (sin2 𝜃) and the lateral magnitude of
the force field (sin 2𝜃) respectively versus polar angle (𝜃) for k=1.
Figure 1a For 𝑘 = 1 rotation speed field (sin 𝜃), field potential (sin2 𝜃) and polar force field (sin 2𝜃) on the sphere of a
rotating body having one unit of diameter.
Figure 1b visualizes the same magnitudes in an axial section in spherical coordinates and on a
sphere that is one unit distant from the object again for k=1. The figure also shows the additional
radial and polar force field components and their resultant vectors on the sphere of radius 𝑟.
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Universal force-motion equations and solar system implementation
It seems that, the rotation of the object does not create any extra gravitational and lateral force
in axial direction, however, as it moves farther away from the axis, both gravitational field and a new
lateral force field is formed additionally. As angle 𝜃 increases, the polar force increases faster than the
central force, reaches its maximum value at 45 degrees, starts decreasing afterwards and becomes zero
at the equatorial plane again. It is clear that for the angle sin2 𝜃 = sin 2 𝜃 (i.e. 𝜃 = 63,43°) the
magnitudes of radial and lateral field components are equal. For the angles less than this limit value
the polar force is more dominant than the additional radial force.
Any object in the field will be affected by Newton’s gravitational force plus these additional
force fields. The resultant force field lines will no longer be straight lines aiming the source mass of
the field but curved ones. Thus, we can mention that rotating celestial body skews the geometry of the
space. However, this curvature is not a space-time curvature as Einstein designated, so we can easily
visualize it in our minds.
Figure 1b Fields in Figure1a in spherical reference frame. For k=1, field amplitudes on 1 unit of Radius sphere versus 𝜃
polar angle. Vectors are attraction and polar force and their resultants on the specified point. The Radius of green
circles is proportional with the velocity fields.
If the mass, rotation axis and velocity of the affecting body remain unchanged, than its
resultant field amplitude in every affected point also remains unchanged. Thus, an object in the field
will be affected instataneously according to where the object is. So, there is no need for any mediating
particle during the influence process. Therefore, questions about whether remote impact is going
to be instataneously or delayed are pointless.
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Universal force-motion equations and solar system implementation
These results, although considered to provide appropriate and extremely significant
solutions for the observations in cosmological size, at first glance, it can be argued that it is contrary to
the observations on Earth. That’s because “we know that both the coconut in tropical regions near the
equator and the apple in Newton’s family farm in the UK fall with the same speed”. We cannot say,
"Penguins in the Arctic have very small wings, because Arctic has less gravity force field, hence they
do not need bigger wings to fly". Gravitational acceleration is approximately the same in every point
on Earth. In fact, contrary to our assertion, it’s a little more at poles. Then, what does equation (8)
mean? Actually, there is no contradiction. Postulate 1b is valid for a rotation defined “relative to a
point” that is sufficiently distant from the body. Undoubtedly, we mortals on Earth are a part of mass
𝑚 in the equation in Postulate 1b and we rotate with it. In other words, our linear and angular speed
relative to Earth is zero. Therefore, we are insensible to the polarization created by the rotational
motion and are only under the influence of the spherically symmetric gravitational field.
3. MUTUAL INTERACTION
Let’s have two celestial bodies having rest masses 𝑚1 and 𝑚2 located at the positions 𝒓𝟏 (𝑟1 , 𝜃1 , 𝜑1 )
̂ 𝟏 and 𝝎
̂ 𝟐 be unit vectors of their
and 𝒓𝟐 (𝑟2 , 𝜃2 , 𝜑2 ) in a reference frame shown in Figure 3. Let 𝝎
fixed angular rotation axes. In this case, we can express speed-velocity quaternions and field potentials
for each mass where the other mass is located as follows:
2𝐺𝑚1
[1 + 𝑘1 (𝝎
̂ 𝟏 × 𝒓̂𝟏𝟐 )]
𝒒𝟏 = √
|𝒓𝟐 − 𝒓𝟏 |
2𝐺𝑚2
[1 + 𝑘2 (𝝎
̂ 𝟐 × 𝒓̂𝟐𝟏 )]
𝒒𝟐 = √
|𝒓𝟏 − 𝒓𝟐 |
𝑉1 (𝒓𝟐 ) = −
𝑉2 (𝒓𝟏 ) = −
𝐺𝑚1
|𝒓𝟐 − 𝒓𝟏 |
𝐺𝑚2
|𝒓𝟏 − 𝒓𝟐 |
[1 + 𝑘1 (𝝎
̂ 𝟏 × 𝒓̂ 𝟏𝟐 )] ∙ [1 + 𝑘1 (𝝎
̂ 𝟏 × 𝒓̂ 𝟏𝟐 )]
[1 + 𝑘 2 ( 𝝎
̂ 𝟐 × 𝒓̂ 𝟐𝟏 )] ∙ [1 + 𝑘2 (𝝎
̂ 𝟐 × 𝒓̂ 𝟐𝟏 )]
Where 𝒒𝟏 (𝒓𝟐 ) and 𝒒𝟐 (𝒓𝟏 ) are speed-velocity quaternions of 𝑚1 and 𝑚2 at points 𝒓𝟐 and 𝒓𝟏
repectively, 𝑉1 and 𝑉2 are field potentials of the masses in the other mass’ position, 𝒓̂𝟏𝟐 ,and
𝒓̂𝟐𝟏 are unit vectors of 𝒓𝟏𝟐 = 𝒓𝟐 − 𝒓𝟏 and 𝒓𝟐𝟏 = 𝒓𝟏 − 𝒓𝟐 respectively(𝒓̂𝟏𝟐 = −𝒓̂𝟐𝟏 ).
By replacing 𝑟 = |𝒓𝟏 − 𝒓𝟐 | = |𝒓𝟐 − 𝒓𝟏 | and the norms of quaternions as 𝛾1 and 𝛾2 yield:
𝐺𝑚1
𝑉1 (𝒓𝟐 ) = −
𝛾1
𝑟
𝑉2 (𝒓𝟏 ) = −
𝐺𝑚2
𝑟
𝛾2
Those expressions above are independent of speed-velocity quaternions and field potentials for
each of the masses. The norms of the quaternions at the expressions of the potentials determine the
independent potential distribution of each object at their effective fields. The postulate below will be
valid for the field potential that creates the interaction of two objects:
Postulate 2: The distribution of effective field potential causing mutual interaction of two celestial
bodies is equal to the dot product of the velocity-speed quaternions of the objects. So, if 𝛾𝑒 is
the effective field distribution factor:
̂ 𝟏 × 𝒓̂𝟏𝟐 )] ∙ [1 + 𝑘2 (𝝎
̂ 𝟐 × 𝒓̂𝟐𝟏 )]
𝛾𝑒 = [1 + 𝑘1 (𝝎
(10)
̂ 𝟏 × 𝒓̂𝟏𝟐 ) ∙ (𝝎
̂ 𝟐 × 𝒓̂𝟏𝟐 )]
𝛾𝑒 = [1 − 𝑘1 𝑘2 (𝝎
Therefore, the effective field potentials for both masses are:
𝐺𝑚1
𝑉𝑒1 (𝒓𝟐 ) = −
𝛾𝑒
𝑟
𝑉𝑒2 (𝒓𝟏 ) = −
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𝐺𝑚2
𝑟
𝛾𝑒
(11)
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Universal force-motion equations and solar system implementation
The force that acts on each of the objects:
𝐺𝑚1
𝛾
𝑟 𝑒
𝐺𝑚2
𝑭𝟐 = 𝑚1 𝛁
𝛾
𝑟 𝑒
𝑭 𝟏 = 𝑚2 𝛁
Or:
𝛾𝑒
𝑭𝟏 = 𝑭𝟐 = 𝑭 = 𝐺𝑚1 𝑚2 𝛁 ( )
𝑟
(12)
This simple equation is the Universal Force Equation that the two objects apply to each other.
Figure 3 Two celestial bodies being interacted
Some qualifications of the universal force equation can be summarized as follows:
1. If the celestial bodies interacting do not rotate around their own axes, the velocity-speed
quaternions will not have a vector component and the potential field distribution factor will be
equal to one (𝛾𝑖 = 1). In that case, the universal force equation will be reduced to Newton’s
law of gravitation and both of the objects will be under the influence of only the central
gravitational force.
2. If the celestial bodies are rotating around their own axes and both their axes are not in the
same direction, the effective field potential will show a different distribution in all three axes
𝛾
(radial, polar and azimuth). In this case, the expression 𝛁 ( 𝑟𝑒 ) and the force that affects each of
the bodies will have components in all three axes. Thus, the movements of the bodies relative
to each other will not be independent of their rotation axes and directions.
3. Universal force equation is valid only if the celestial bodies stand where they are. In this
context, the Newton’s law of gravitation is rughly valid because the position of Sun and planet
is almost fixed in the radial direction. If they are moving under the influence of a force, than
the speed-velocity quaternion of one body will seem relatively different to the other body and
both will perceive the field potential differently. Because of this, the force that is applied to
the objects will change.
Since there is not only the gravitation force at the interaction anymore, instead of saying gravitational
force or field, we will mention field force or force field.
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Universal force-motion equations and solar system implementation
If we express the gradient at the Universal Force Equation (12) in spherical coordinates, the three
components of the force applied to the bodies will be as follows:
𝑭 = 𝐺𝑚1 𝑚2 [(−
𝛾𝐴 1 ∂𝛾𝑒
1 ∂𝛾𝑒
1 ∂𝛾𝑒
̂+
̂]
+
) 𝐫̂ + 2
𝜽
𝝋
2
2
𝑟
𝑟 ∂r
𝑟 ∂θ
𝑟 sin 𝜃 ∂φ
(13)
We can generalize the universal force equation for a system which is composed of 𝑁 number of
celestial bodies. Since every object will interact with 𝑁 − 1 number of other objects, the forces that
are calculated by means of the other objects’ effective field potentials are summed up. Accordingly,
the total force acting on the 𝑗 𝑡ℎ object -by the help of equation (12)-is:
𝑁
𝑭𝒋 = 𝐺𝑚𝑗 ∑ 𝑚𝑖 𝛁
𝑖=1
𝑖≠𝑗
̂ 𝒊 × 𝒓̂𝒊𝒋 ) ∙ (𝝎
̂ 𝒋 × 𝒓̂𝒊𝒋 )]
[1 − 𝑘𝑖 𝑘𝑗 (𝝎
𝑟𝑖𝑗
This equation can be expressed in matrix form as below:
𝑭 = 𝐺𝑀𝛁(Γ𝑅−1 )
(14)
Here:
𝑭
G
𝑀
Γ
𝑅 −1
: a force column matrix with N components,
: universal gravitational constant,
: symmetric square 𝑚𝑖 𝑚𝑗 mass matrix with 𝑁 × 𝑁 elements,
: effective field distribution matrix for 𝑖 and 𝑗 𝑡ℎ masses with 𝑁 × 𝑁 elements,
: reverse distance matrix (1/𝑟𝑖𝑗 ) with 𝑁 × 𝑁 elements having zero main diagonal elements
(𝑟𝑖𝑖 = 0).
Every constituent body in the system will move under the influence of the resultant force defined
above. To determine the path of the movement, Newton’s second law of motion should be applied.
It is possible to verify the universal force equation (14) without solving it by accepting the observed
orbits of the planets as a solution. In other words, crosschecking can be applied. For that purpose, we
can use the universal force equation with Newton’s second law of motion for the Sun and planet
duality. Namely:
𝑑
𝑭 = 𝐺𝑀𝛁(Γ𝑅−1 ) = (𝑀𝑎 𝑹̇)
𝑑𝑡
Here:
𝑀𝑎
: a row matrix with 𝑁 elements, each having affected masses 𝑚𝑎𝑖 ,
𝑹̇
: a column matrix with 𝑁 elements, each having velocities 𝒓̇ 𝑖 .
We have to open a pharantesis here. The affected masses on the right are not the same as rest masses
on the left. This subject will be examined in the main article titled “Fundamentals of Field Relative
Model of the Universe”. In the model the rest mass of a body is defined as the flux of the force field
created by only the body itself on a closed surface having the body inside. This definition is simly
equivalent with Gauss’ law of gravitation. But this definition of mass is valid only where the body is
not under the influence of any other force field. If the body is moving inside an effective field, and its
velocity is not equivalent with the velocity of the field affecting, than the body will be subject to a
relative mass. The mass of affected body will increase depending on how much its velocity is different
from the field velocity affecting. The “field relative mass” has no relation with the light speed. This is
why the new model of the universe is field relative. Cleraly, the field relative mass will be time
dependent. The right and left hand side of the equations above can be calculated separately and to how
extent they correlate with each other can be found. As it is seen, verifiying the universal force equation
using Newton’s laws of motion is not as simle as it is expected. A simpler and more direct way is to
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Universal force-motion equations and solar system implementation
use a new method called “the universal motion equation” where we will not need to deal with field
relative masses.
For this, we need to do some brainstorming.
Relative Velocities
The interaction of two celestial bodies is a mutually symmetric relationship. The axial symmetrical
field potential, that each of the two rotating celestial bodies creates, gets more complicated at the
mutual interaction. At this interaction both of the objects will affect each other by applying a force to
the other and get affected by it. Therefore, it will be useful to name these objects as active or affecting
and passive or affected bodies.
Effective speed-velocity field quaternions of the celestial bodies are not the only
determining factor in their interactions. We also need to consider their linear velocities relative to each
other. This can be explained as follows:
Let there be an observer or a passive object under the influence of an affecting mass. The
speed-velocity quaternion formed by the affecting mass in a point where the observer is, only valid as
long as the observer does not change its fixed position 𝒓. We know that the amplitude of effective field
quaternion (the scalar and vector parts together) will decrease as moving away. If the angular velocity
of the affecting rotation body does not change (both speed and direction), the speed-velocity
quaternion value will remain unchanged for the observer at fixed position 𝒓. If the bodies change their
positions with a certain velocity, relative to each other, than the effective fields will be perceived to
each as different speed-velocity quaternion.
If both of the celestial bodies have the same linear velocity and direction, that is if their
linear velocities are zero relative to each other, they would not be feeling their linear field velocities,
which is equivalent to both of them stopping. In summary, the perceived speed-velocity quaternion
that the effective body creates at the point where the observer is located is not determined by the
effective body alone. The change in the position of the observer relative to the effective body needs to
be considered. For this reason, the force relationship between the two bodies is not determined by the
absolute position and velocity of these objects relative to a third point but the velocities and positions
of these objects relative to each other.
Let’s simplify the subject even more.
You are being dragged by a flood. If you grasp a branch of a tree to keep your position, you
feel full force of the water acting on you. If your hand and the branch of the tree are strong enough,
you can keep your position by resisting the force of the flood. If the branch of the tree breaks, then you
will start being dragged again. If your speed is slower than flood, due to the friction between your feet
and the riverbed, than you feel less force applied by the flood. But if you and the flood are in the same
speed, then you will never feel of force of the water anymore.
This simple observation can be expressed as follow:
Postulate 3: Things move in a way that cancels forces acting on them.
If the object under the effect of a force can eliminate the force by its motion, than it is
harmonious with the system or is in equilibrium state. The process of the object eliminating or
minimizing this force is called transient response. Postulate 3 is actually nothing more than a
simpler expression of the Newton’s second and third law of motion together.
We feel the gravitational force on Earth so we are moving adherent to the ground. During
free fall this force will not be felt. It is known that Einstein was inspired by this fact the mostly in his
way to form the General Relativity theory.
If the equations (2) and (11) are considered, we can get following relations between the ratio
of the masses and their field speeds and potentials:
𝑣1𝑓
𝑚1
=√
𝑣2𝑓
𝑚2
(15)
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Universal force-motion equations and solar system implementation
𝑉1 𝑚1
=
𝑉2 𝑚2
On the other hand, if these two bodies only move with the forces they apply to each other, the center
of mass of the bodies will remain fixed and both of them will rotate at their own orbitals around that
center of mass. In this case since the total momentum will be zero, the following equation can be
written:
(16)
𝑚1 𝒓̇ 𝟏 = −𝑚2 𝒓̇ 𝟐
Here 𝒓̇ 𝟏 and 𝒓̇ 𝟐 are orbital velocities of 𝑚1 and 𝑚2 bodies respectively.
Considering the equations (15) and (16), the postulate below will be valid:
Postulate 4: Having no external force, in a dual closed system the orbital speed of each of the objects
will be perceived as an additional rotational movement at the point where the other one is located.
Hence, this linear movement will reflect to the point where the other one is located as a velocity field
proportional to the square root of the masses.
We can express this as follows:
𝑚1
𝒗𝟏𝒇 = 𝒓̇ 𝟏 √
𝑚2
(17)
In this equation, it may seem meaningless that the field velocity is dependent on the
magnitude of the objects’ masses. However, it should be remembered that the orbital velocity 𝒓̇ 𝟏 is
dependent to the magnitude of the other object’s mass due to the dual interaction. A planet taking a
full rotation around its axis cannot fully complete its rotation relative to the Sun. This is due to its
movement on the orbital path. For a full tour of the planet around its axis relative to Sun, some more
rotation is needed. The equation (17) is the reflection of this additional rotation to the field.
Figure 4 Two moving bodies by mutual effect in the space
In a two body system, the affected mass will perceive the magnitude of the affecting mass’ speedvelocity quaternion differently due to its velocity. In figure 4, if the velocities of 𝑚1 and 𝑚2 at the
points 𝒓𝟏 and 𝒓𝟐 are 𝒗𝟏 = 𝒓̇ 𝟏 and 𝒗𝟐 = 𝒓̇ 𝟐 , the relative speed-velocity quaternions will be as follws:
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Universal force-motion equations and solar system implementation
2𝐺𝑚1
2𝐺𝑚1
(𝝎
̂ 𝟏 × 𝒓̂𝟏𝟐 ) − 𝒓̇ 𝟐
𝒒𝑹𝟏 = √
+ 𝑘1 √
𝑟12
𝑟12
2𝐺𝑚2
2𝐺𝑚2
(𝝎
̂ 𝟐 × 𝒓̂𝟐𝟏 ) − 𝒓̇ 𝟏
𝒒𝑹𝟐 = √
+ 𝑘2 √
𝑟21
𝑟21
Or if we separate the escape speeds:
2𝐺𝑚1
𝑟12
̂ 𝟏 × 𝒓̂𝟏𝟐 ) − 𝒓̇ 𝟐 √
𝒒𝑹𝟏 = √
[1 + 𝑘1 (𝝎
]
𝑟12
2𝐺𝑚1
(18)
2𝐺𝑚2
𝑟21
̂ 𝟐 × 𝒓̂𝟐𝟏 ) − 𝒓̇ 𝟏 √
𝒒𝑹𝟐 = √
[1 + 𝑘2 (𝝎
]
𝑟21
2𝐺𝑚2
Thus, the relative effective field potential distribution factor will be as follows:
𝑟12
𝑟21
̂ 𝟏 × 𝒓̂𝟏𝟐 ) − 𝒓̇ 𝟐 √
̂ 𝟐 × 𝒓̂𝟐𝟏 ) − 𝒓̇ 𝟏 √
𝛾𝑅𝑒 = [1 + 𝑘1 (𝝎
] ∙ [1 + 𝑘2 (𝝎
]
2𝐺𝑚1
2𝐺𝑚2
(19)
Therefore relative field potentials of each moving body and the force applied each other are as follow:
𝐺𝑚1
𝑉𝑅𝑒1 =
𝛾
(20)
𝑟 𝑅𝑒
𝐺𝑚2
𝑉𝑅𝑒2 =
𝛾
𝑟 𝑅𝑒
𝛾𝑅𝑒
𝑭𝟏 = 𝑭𝟐 = 𝑭 = 𝐺𝑚1 𝑚2 𝛁 ( )
𝑟
Due to Postulate 3, this force should be equal to zero. Thus:
𝛾𝑅𝑒
𝛁( ) = 0
𝑟
(21)
(22)
This equation is the universal motion equation for a two body system. For a system composed of
𝑁 number of bodies, the equation will be generalized as follow:
𝛁(𝛤𝑅 𝑅−1 ) = 0
(23)
Here 𝛤𝑅 effective relative field distribution matrix for 𝑖 and 𝑗𝑡ℎ masses with 𝑁 × 𝑁 elements and 𝑅 −1
reverse distance matrix (1/𝑟𝑖𝑗 ) with 𝑁 × 𝑁 elements having zero main diagonal elements (𝑟𝑖𝑖 = 0).
The universal motion equation explains why the planets are in equilibrium state on their orbits.
The planets are in equilibrium state not because they are under the influence of the Sun’s
gravitational force, but conversly because they are not under the influence of any force. This idea
is exactly equivalent with Einstein’s comments on the orbital stability of the planets. The planets do
not see the Sun; in fact, they have no idea about the Sun. They just travel on a way to make the force
acting on them to be zero. Just like a piece of wood on a river drifting. They also do not know whether
the path they follow is a shortcut or not. Furthermore, for this movement they never need any mediator
particles such as “graviton”.
4. APPLICATION TO THE SOLAR SYSTEM
The Universal Motion Equation is a better and simpler tool to understand the motion of the planets.
As an easiest way we will verify the equation using known orbits of planets by crosschecking.
However, we should not expect that everything to go ferfect at this verification. This expectation is not
correct scientifically. The reasons can be listed as follows:
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Universal force-motion equations and solar system implementation
1. The velocity and acceleration components of the planets at the orbits they travel, are
calculated assuming that they rotate in a smooth elliptical orbit on a pure plane in accordance
with Kepler’s law of motion. Yet, the universal force equation expresses that the force acting
on a planet is not an exact central force but a force having components in all three axes. In that
case, the assumption of the orbits being a smooth ellipse will not be true and this will lead to
wrong results in verification. The solution of the differential equation system in equality (22)
will tell us the exact path of the orbit in three axes.
2. We have to ask ourselves a question: to how extent the quantities (basically the mass and
distance) about the celestial bodies calculated by Newton’s law of gravitation are trustable?
For example, if the masses of the Sun and planets are calculated with respect to different force
relationships, different values can be obtained.
Nevertheless, there is no reason for us not to trust 𝐺 the universal gravitational constant and
the quantities about the Earth at the equations. The gravitational constant has been found
experimentally by Cavendish in 1798 independent of Newton’s law of gravitation. Then, it is
improved with modern measuring methods and its reliability is enhanced. Since we can
measure the dimentions and force field of Earth very sensitively, we will trust its mass
magnitude. The rest masses of the planets will not create much of a problem. Since they will
be on both sides of the equation, they will cancel each other. The field potential distribution
factor in equation (22) is the functions of the magnitudes that can be found by optical
observations like angles and distances. Only relative distances can be measured using optichal
methods rather than absolute ones. Hence, we may expect some little errors arising from
known distances. For now, the greatest source of error may be the Sun’s mass calculated
according to Newton’s law of gravitation.
3. Here, the most distinctive side of the developed universal equations is that it uses the
rotational velocity of the celestial bodies around their axes as a parameter. Proper knowledge
of rotational velocity factor is imperative to reach the correct solution. Yet, it is impossible for
us to exactly know these values except for the rocky planets. This will create an error in the
calculation.
4. The universal motion equation is a differential equation system; therefore it has both transient
and steady state solutions. Verification of the differential equation using the known solution
can only be valid for the steady state solutions. If some of the motions in the Solar system are
still at their transient phase, this will cause an error for the comparison. Since the time
constant is quite long in cosmology, we should consider that this kind of comparisons can be
misleading. For example, in figure 1b it can be seen that the polar angle component of the
force is approaching to zero as getting closer to the equator. Thus it could be said that the
orbital planes of the planets will asymptotically coincide with the Sun’s equatorial plane in an
infinitly long period of time.
Calculations
In our study, for the verification of the universal motion equality for the Solar system, the relative field
potential distribution factor 𝛾𝑅𝑒 at the equation (19) was formed for the Sun-planet duality. The effects
of the other planets were neglected. To simplify the equation, a heliocentric coordinate system with
horizontal x-y plane as the orbital plane of the planet is used. For the initial motion point of the planet,
the vernal equinox for Earth and equivalent points for the other planets were selected. The sampling
with equal orbital angle intervals is used to simplify the calculations for all the planets for their entire
rotation period. In that case, for the calculations in the time domain, Kepler’s law of areas was used.
To adapt the calculation to all planets easily, the orientational data of all planets were transformed to
the same base by using coordinate transformations.
Figure 5 shows the components of the quaternions separately in the used reference frame at
both points in a Sun-planet duality. At the point where the planet is located, the speed and rotational
velocity field of the Sun together with the linear velocity of the planet are present. Sun is fixed at the
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Universal force-motion equations and solar system implementation
reference frame, i.e. it does not have a linear velocity. However, at the origin, the planet has two field
velocity components resulting from its rotation around its axis and linear velocity along with the speed
field as defined with postulate 4. Accordingly, the field quaternions for the Sun and planet in equation
(18), can be written as follows:
2𝐺𝑚𝑆
2𝐺𝑚𝑆
̂ 𝑺 × 𝒓̂)√
𝒒𝑹𝑺 = √
+ 𝑘𝑆 (𝝎
− 𝒓̇ 𝒑
𝑟
𝑟
2𝐺𝑚𝑝
2𝐺𝑚𝑝
𝑚𝑝
̂ 𝒑 × (−𝒓̂)]√
𝒒𝑹𝒑 = √
+ 𝑘𝑝 [𝝎
+ 𝒓̇ 𝒑 √
𝑟
𝑟
𝑚𝑆
Or, by separating the escape velocities:
2𝐺𝑚𝑆
𝑟
̂ 𝑺 × 𝒓̂) − 𝒓̇ 𝒑 √
𝒒𝑹𝑺 = √
[1 + 𝑘𝑆 (𝝎
]
𝑟
2𝐺𝑚𝑆
𝒒𝑹𝒑 = √
(24)
2𝐺𝑚𝑝
𝑟
̂ 𝒑 × 𝒓̂) + 𝒓̇ 𝒑 √
[1 − 𝑘𝑝 (𝝎
]
𝑟
2𝐺𝑚𝑆
Figure 5 Speed and velocity fields in heliocentric coordinate system
Here, the magnitudes with subscripts 𝑆 belongs to the Sun and the ones with 𝑝 belongs to the planet.
Accordingly, the relative effective field distribution factor at equality 19 will be as shown:
𝑟
𝑟
̂ 𝑺 × 𝒓̂) − 𝒓̇ 𝒑 √
̂ 𝒑 × 𝒓̂) + 𝒓̇ 𝒑 √
𝛾𝑅𝑒 = [1 + 𝑘𝑆 (𝝎
] ∙ [1 − 𝑘𝑝 (𝝎
]
2𝐺𝑚𝑆
2𝐺𝑚𝑆
Or:
𝑟
𝑟𝒓̇ ∙ 𝒓̇
̂ 𝒑 × 𝒓̂) ∙ (𝝎
̂ 𝑺 × 𝒓̂) + [𝑘𝑆 (𝝎
̂ 𝑺 × 𝒓̂) ∙ 𝒓̇ + 𝑘𝑝 (𝝎
̂ 𝒑 × 𝒓̂) ∙ 𝒓̇ ]√
𝛾𝑅𝑒 = 1 − 𝑘𝑆 𝑘𝑝 (𝝎
−
2𝐺𝑚𝑆0 2𝐺𝑚𝑆0
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(25)
14
Universal force-motion equations and solar system implementation
If we write the universal motion equation (22) in spherical coordinates, the motion equality in three
axes will be as follows:
𝛾𝑅𝑒 ∂𝛾𝑅𝑒
(−
+
)=0
𝑟
∂r
∂𝛾𝑅𝑒
(26)
=0
∂θ
∂𝛾𝑅𝑒
=0
∂φ
If these expressions are calculated for the heliocentric koordinat system defined above, the following
three equalities for three axes will be obtained on the orbital plane:
The radial component:
1 − 𝑘𝑆 𝑘𝑝 [𝑠𝑖𝑛 𝜃𝑆 𝑠𝑖𝑛 𝜃𝑝 𝑠𝑖𝑛(𝜑 − 𝜑𝑆 ) 𝑐𝑜𝑠 𝜑 + 𝑐𝑜𝑠 𝜃𝑝 𝑐𝑜𝑠 𝜃𝑆 ]
3
𝑟
𝑟 2 |𝒓̇ | 𝜕|𝒓̇ |
+ (𝑘𝑆 𝑐𝑜𝑠 𝜃𝑆 + 𝑘𝑝 𝑐𝑜𝑠 𝜃𝑝 )𝑟𝜑̇ √
+
=0
2
2𝐺𝑚𝑆 𝐺𝑚𝑆 𝜕𝑟
The polar component:
𝑟
−𝑘𝑆 𝑘𝑝 [sin 𝜃𝑆 cos 𝜃𝑝 cos(𝜑 − 𝜑𝑆 ) + sin 𝜃𝑝 cos 𝜃𝑆 sin 𝜑] + 𝑘𝑆 sin 𝜃𝑆 𝑟𝜑̇ √
cos(𝜑 − 𝜑𝑆 )
2𝐺𝑚𝑆
+ 𝑘𝑝 sin 𝜃𝑝 𝑟𝜑̇ √
(27)
𝑟
sin 𝜑 = 0
2𝐺𝑚𝑆
The azimuth component:
−𝑘𝑆 𝑘𝑝 sin 𝜃𝑆 sin 𝜃𝑝 cos(2𝜑 − 𝜑𝑆 ) +
𝑟|𝒓̇ | ∂|𝒓̇ |
=0
𝐺𝑚𝑆 ∂φ
Here, θS , φS, θp and φp are the coordinates of the Sun and the planet’s axial unit vectors
̂ 𝑺 ve 𝝎
̂ 𝒑 ), |𝒓̇ | is the velocity of the planet and 𝜑̇ is the amplitude of the planet’s angular velocity on
(𝝎
the orbit.
In the calculations, for the orbital position vector of the planet the following equation is used3:
𝒓=𝑎
1 − 𝜀2
𝒓̂
1 + 𝜀 cos Φ
Where, 𝑎 is semimajor axis of the planet, 𝜀 is eccentricity of the orbit and Φ is true anomaly
which is the angle between the lines sun-perihelion point and sun-planet.
If the angles of the unit vector pointing the perihelion which is the closest point of the orbit to
sun are θPH and φPH, the true anomaly will be as follow:
Φ = cos−1[sin 𝜃 sin 𝜃𝑃𝐻 cos(𝜑 − 𝜑𝑃𝐻 ) + cos 𝜃 cos 𝜃𝑃𝐻 ]
The verification equations (27) of the universal motion equalities are attained by taking the
partial derivatives of the effective field potential factor 𝛾𝑅𝑒 (25). The calculation of the first and third
equalities is straight forward on the orbital plane. But, since the movement of the planet occurs on the
horizontal plane where 𝜃 = 90° and fixed, the polar component of equations (27) should not nomally
be present. But, we know that the planet is under the influence of a flattening force because its orbital
plane is not the Sun’s equatorial plane. To get these flattening force components in the equalities,
firstly the partial derivative with respect to 𝜃 was taken using general motion equalities, and then its
value was calculated on the orbital plane. Thus, the forces acting on the planet in all three axes can be
calculated.
These equalities (27) also show, what will happen if both celestial bodies do not rotate
(𝑘𝑆 = 𝑘𝑝 = 0) or only one of them is rotating. Clearly, in every equality, the components that create
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Universal force-motion equations and solar system implementation
the force and the reaction components that the movement of the planets creates to cancel out these
forces, are present separately. To test the equalities generally and see the action and reaction forces
altogether, the graph of every component can be observed using different rotation coefficients. Figure
6 shows the action and reaction components in three axes for different rotation coefficients.
Figure 6a For 𝑘𝑆 = 𝑘𝑝 = 0,5 radial action and reaction forces. To be able to see all forces in the same graphic, rotation coefficients
are chase much larger than their estimated values. In fact, rotational repulsive force is proportional with the product of
rotation coefficients hence is very small. The reaction forces are proportional with the coefficients hence they are the source
of orbital axis deviation. Due to the scale selected here, the small variations on the action and reaction forces (except Newton
force) are not visible.
Figure 6b Flattening forces and their reactions. Here rotation coefficients 𝑘𝑆 = 𝑘𝑝 = 1
Figure 6c Azimuth force component of rotating field specifying the revolving direction of the planet and its reaction. As the reaction
force is very small we chase the product of coefficients very small (𝑘𝑆 . 𝑘𝑝 = 0,003).
In universal motion equations (27) we have 2 unknowns but three equalities. Normally,
according to laws of Newton and Kepler, we should be able to find the 𝑘𝑆 and 𝑘𝑝 values that will
cancel out these three equalities since we know the planet’s linear and angular velocities (|𝒓̇ |, 𝜑̇ ) on
the orbit. In accordance with our expectations, we will never be able to find the coefficients that will
make the equalities zero. Because, the planet does not move under the influence of the central
gravitational force only. Therefore, the path it follows along the orbit is not a smooth path. Since every
deviation from zero in the equalities means a force applied to the planet, the planet must follow a wavy
path in accordance with these deflections. We can observe the daily results of this wavy movement,
which may also be called De Broglie waves, from the deviations of axis of Earth throughout the year.
It may be nice work, finding the exact solution of the universal motion equations and determining to
what extent the deviations of the Earth in three axes correlate with the rotation axis deviations.
The first term of the radial component of the universal motion equality (27) i.e. one; expresses
the gravitational force component of the speed field and the last term refers to the reaction force of the
planet to cancel it out. The second and third terms define the interaction force caused by the rotation of
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Universal force-motion equations and solar system implementation
the Sun and the planet and the reaction of the planet to it. The second term here comes from the
general second term expression in equality (25). So:
̂ 𝒑 × 𝒓̂) ∙ (𝝎
̂ 𝑺 × 𝒓̂) = −𝑘𝑆 𝑘𝑝 [𝑠𝑖𝑛 𝜃𝑆 𝑠𝑖𝑛 𝜃𝑝 𝑠𝑖𝑛(𝜑 − 𝜑𝑆 ) 𝑐𝑜𝑠 𝜑 + 𝑐𝑜𝑠 𝜃𝑝 𝑐𝑜𝑠 𝜃𝑆 ]
−𝑘𝑆 𝑘𝑝 (𝝎
This expression defines a new force, originated by the rotating of two bodies around their
axes, and formed in the radial direction along with the Newton’s gravitational force. Since the rotation
coefficients (𝑘𝑖 ) are positive, the sign of this expression will be determined by the direction of rotation
of the objects. If both are rotating in the same direction, it will be positive (attracting); if not, it will be
negative (repulsive). Thus, the rotational directions of the objects relative to each other will work in an
increasing or decreasing way for the gravitational force. The value of the trigonometric expression in
above equation is around one; since the rotation coefficients for the celestial bodies 𝑘𝑖 ≪ 1, their
products will be even smaller. For that reason, this attracting-repulsive effect is negligible. The
physical meaning of this additional force will be discussed in the article “Fundamentals of the field
relative model of the universe” in details. As the readers can guess, the universal motion equation can
also be applicable to atoms; yet this time the neglected components will differ.
While this new attractive-repulsive force is proportional to the product of the coefficients 𝑘𝑖 ,
the reaction of the planet to that is the sum of two terms each propotioal to the coefficients directly as
can be seen from figure 6b. This can be interpreted as more than enough reaction of the planet.
Therefore, it is understood that the deviation of the orbital axis of the planets is inevitable in an
interaction containing at least one celestial body rotating around its own axis. If the field rotation is
taken into account, this would not be hard to visualize in mind. Because the field rotation is very slow,
this can be seen as an observable magnitude only for the orbit of Mercury.
The sum of the four components in the radial direction will not be exactly zero but will wander
in the vicinity of zero. This fact points out that the planet actually will not have a smooth path at its
orbit but will move with a wavy motion in the radial direction.
As seen from (27), the second equality (𝜃 component) is a sum of four different terms. Two of
them are about the flattening forces of the Sun and the planet and the other two are about the reactions
of the movement of the planet to them. Sum of these four terms at every point on the orbit is not
exactly zero. This means that, the path that the planet follows is not an exact plane, in fact, it moves in
a wavy path around the plane Newton-Kepler laws define.
Lastly, the third equation (27) i.e. 𝜑 component shows azimuth component of the force caused by
the rotation of the Sun and the planet and response of the planet to it. Figure 6b indicates that the
motion direction of the planet can not be a coincedance. The sum of the effective rotation force and
planets’ response to it is not exactly zero. Hence, we may expect small deviations from Kepler’s law
of areas.
Results
If we look at the universal motion equations, we can reach the following results:
1. If none of the objects in a system rotate around its own axis, the forces applied each other will
not have polar (𝜃) and azimuth (𝜑) components and all the objects will move under the
influence of a central force only as required by the Newton’s law of gravitation. There will be
no flattening and azimuth forces acting on the planets. In this case, we would have been able
to observe clusters of the globular type and about the half of the objects in the system would
have been rotating in the reverse direction around the superior one.
2. If the superior body in the system, is rotating around but the others not, the rotation of the
main body determines the form of the clustering. In that case, the direction of revolution of the
objects around the superior body must be the same as the rotation direction of it. The satellite
bodies must cluster at the equatorial plane of the ruling body.
3. If the interacting objects are both rotating, the force applied to each on them is determined by
their rotation direction and velocity. Hence, the route they follow is not independent of their
rotation axes.
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Universal force-motion equations and solar system implementation
It is obvious that the Solar system verifies the observations listed above. Mercury and Venus,
whose rotational velocities are almost zero, are travelling closer to the Sun’s equator, while the
Asteroid and Kuiper belts consisting of nonrotation small parts overlap the Sun’s equator. But, the
planets having higher rotational speed and distinct rotational axes from the Sun, wander around on an
angled plane with the Sun’s equator.
The detailed calculation formulas made within the scope of this study and the calculation tables
for all the planets will be brought into use for interested readers in January 2016. Thus, for each
planet, the comparison in all three axes for different 𝑘𝑖 values can be made and to how extent the orbit
that the theory foresees correlates with the Newton-Kepler orbits can be tested by the scientists. For
that reason, the numerical details are not given here.
5. FREE FALL
Freefall is one of the most encountered cases about force relations in our daily lives. The universal
motion equation must have something to say about it.
Free falling objects move with a certain acceleration towards the center of Earth if they do not
encounter any obstacle. We know that the free falling object is insensitive to the rotation of Earth and
have same acceleration everywhere on Earth. It is true for universal motion equation as well, because
the object rotates with earth, i.e. the relative rotation velocity is zero (𝑘𝐸 = 𝑘𝑜 = 0). Therefore, the
components 𝜃 and 𝜑 will not be present in the force relations. Thus, the motion equation (27) will be
as follows:
(28)
𝑟 2 |𝒓̇ | 𝜕|𝒓̇ |
1+
=0
𝐺𝑚𝐸 𝜕𝑟
Where, 𝑚𝐸 is the mass of Earth. If 𝑣𝑜 = |𝒓̇ | is the velocity of the object falling freely than:
𝜕𝑣𝑜
𝐺𝑚𝐸
𝑣𝑜
=− 2
𝜕𝑟
𝑟
In this expression, if we multiply both sides with infinitesimal 𝑑𝑟 and divide them with
infinitesimal 𝑑𝑡:
𝑑𝑣𝑜
𝐺𝑚𝐸 𝑑𝑟
𝑣𝑜
=− 2
𝑑𝑡
𝑟 𝑑𝑡
As, 𝑑𝑟/𝑑𝑡 is the object velocity and 𝑑𝑣𝑜 /𝑑𝑡 is the object acceleration, we get the following
equation for the freefall acceleration as:
𝐺𝑚𝐸
𝑔=− 2
𝑟
6. EQUILIBRIUM and FIELD PROPAGATION RATE
Following very important two points should be emphasized for the universal force-motion equations:
a. Equilibrium
The components of the universal motion equation (26) in three axes can be written as follows:
𝛾𝑅𝑒 1 ∂𝛾𝑅𝑒
(− 2 +
)=0
𝑟
𝑟 ∂r
1 ∂𝛾𝑅𝑒
(29)
=0
𝑟 2 ∂θ
1 ∂𝛾𝑅𝑒
=0
𝑟 2 ∂φ
Here, the distance 𝑟 is shown in order to understand what would happen when departing from
equilibrium i.e. when the equations are not equal to zero because of external disrupting effect.
In these equations the first term in the radial component, decreases faster than the second as
the distance increases. The distance where these two terms are equal is the equilibrium point. At the
distances farther from the equilibrium point, a repulsive force; for closer distances an attractive force
will be effective. In a way, if the distance 𝑟 between the two body increases, the equilibrium will be
destroyed, the repulsive force will be active and the bodies will move further away. However, in this
time, since the azimuth component of the field potential distribution factor will decrease quickly, the
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Universal force-motion equations and solar system implementation
orbital velocity will also decrease. This will cause the velocity term in the end of the radial component
(27) to decrease and the gravitational force to increase thus, balance the system again. The opposite
will be valid for the distances closer than the equilibrium position. On the other hand, taking into
consideration of Figure 1b, we have no doubt about how the system will behave in case of a deviation
on the equilibrium position of the polar component. Similarly, the analysis can be made about how the
disruption of the balance in three axes i.e. distance, angle and velocity; will be fending off to keep
equilibrium. As a result, the planets will remain in stable equilibrium at their orbits.
Taking the radial component of universal motion equations (27) into consideration, I would
like the scientists to think about, the effect of longer distances between the bodies or systems if there is
no equilibrium state in between. I think the comparison of the rotation directions of the galaxies that
are approaching to or diverging from each other (this could be understood from the structure of the
spirals) may lead to very important discoveries about the order of the universe.
b. Field propagation Rate
In universal force equations, the events that are millions of kilometers apart from each other are
associated as if they are happening simultaneously together in the same place. Remote interaction is
actually one of the most controversial issues in both Newtonian physics and Relativity. We made this
issue more complicated here, by adding the remote effects of the motions. Actually if we take up these
equations seriouly, this means we are accepting at least one of the two facts in advance as follow:
i. The propagation velocity of the affecting field in space is infinite. For example, rotation of the
Sun around its own axis will reflect to its entire field instantaneously, or the planet’s linear
velocity on its orbit will be perceived as an instantaneous field velocity at the point where the
Sun is. However, the author of this article believes that the propagation of the field effect is a
propagation of energy just as the waves of robot propeller in the midle of a pool; therefore,
the field velocitiy is finite, even much slower than what’s beieved. Furthermore, this velocity
is not fixed and decreases as it moves away from its source. So, for the universal force
equations to be valid, the second fact mentioned above should be taking place.
ii. In a balanced system with field interactions, there must be certain harmony between the
propagation time of the effective field from source to the affected point and the motion
cycle of the object getting affected from this field. Universal motion equation can only be
valid, if there is an accord between the propagation time of the field and reaction cycle of the
reactant. Otherwise, the erratic changes in the source will apply erratic forces on the object
and there will not be a consistent equilibrium state. This is the theoretical explanation of
why the electrons of an atom are allowed to wander only in certain orbitals. So, is there a
couterpart of this phonemenon in a balanced system in cosmology? Indeed, there is. To
understand this, we need to know the propagation rate of this field energy.
With postulate 1a, we know that every celestial body in space has a speed field in its effective region.
If we choose a volume that is small enough in the effective field, we can assume that it is a
homogeneous part of the space. In that case, it is clear that the velocity field at that point must be the
same in every direction. Therefore, this velocity will be the same as in the direction of the propagation
as well. To find with what velocity it moves away starting from the source of this field, we can write
equation (2) as follows:
𝑑𝑟
2𝐺𝑚
(30)
=√
𝑑𝑡
𝑟
By multiplying both sides with 𝑑𝑡 and and integrating both sides, we can find the following
relation between the distance 𝑟 and the time 𝑡 that the field travels from source to this effective point
i.e. 𝑟 = 0 for initial positon:
(31)
𝑡2
2
=
3
𝑟
9𝐺𝑚
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Universal force-motion equations and solar system implementation
So there will be a fixed ratio between the square of the travel time of the field and the distance
cubed. On the other hand, the third law of Kepler says that “The square of the period of any planet is
proportional to the cube of the semimajor axis of its orbit”. As the semimajor axis is almost sunplanet distance, there should be a certain relation between the period of the planet (𝑇) and travel time
of the field (𝑡) from the Sun to the planet. The constant value of 𝑡 2 ⁄𝑟 3 rate in equation (31) depends
on the dimention of 𝐺 used. If the astronomical unit (𝐴𝑈), kilogram (𝑘𝑔) and the earth year are used
[𝐴𝑈 3 /(𝑘𝑔. 𝑦𝑒𝑎𝑟 2 )], then 𝐺 = 1,9944 10−29. In this case we find as 𝑡 2 ⁄𝑟 3 = 1/180. Since for Earth
𝑇 2 ⁄𝑟 3 = (1𝑦𝑒𝑎𝑟)2 ⁄(1𝐴𝑈)3 = 1, then 𝑇 2 ⁄𝑟 3 : 𝑡 2 ⁄𝑟 3 = 𝑇 2 ⁄𝑡 2 = 180, hence 𝑇⁄𝑡 = 13,416. On the
other hand the period 𝑇 for Earth is 365 days and from equation (31) the travel time (𝑡) of the field
from Sun to Earth is 27,4 days (equals to rotation period of sun, why?). Hence the rate 𝑇⁄𝑡 will be
13,32. There is a little difference between them. This may arise from very small effects of Earth field
and mostly from faulty calculation of Sun mass based on Newton’s law of gravitation. The field
propagation law in (31) is valid for all planets, because Kepler’s third law is valid. Here, we have
arrived at a marvelous result, but we should not forget which problem brought us to this point.
As seen, as long as there is certain equilibrium between the celestial bodies, even if there is a
very large distance between them, there will be no problem in using Universal Force-Motion
equations. Here, the word equilibrium is particularly emphasized. In electro-mechanical systems, the
different components in the system are connected each other with mechanical and/or electrical
fasteners to transfer force or energy amongs the units. For that reason, we can express the force-energy
relations in the system as a whole, adding a delay element when necessary, in the same equation. In
field related systems, the field provides an equivalent connection between interacting bodies. From
this point, equation (31) can be considered as a delay element. A delay element which does not
adumbrate its presence in periodical systems balanced.
The universal motion equation connects the bodies to each other with both the field (existence)
energies they emit and their motions (kinetic). The equilibrium can only occur if there is a
harmony between the movements of the bodies and the propagation time of the field energy
between them. Otherwise there will either be a collission between thebodies or the bodies will
move away from each other.
With the universal force equation, in a force field that is formed by the Sun, the field force
seems to be affecting the planet instantaneously. However, since the field energy spreads out very
slowly, we can notice the sudden disappearance of the Sun after several weeks (27,4 days). Also, the
disappearance of the Earth will be realized by the sun only after 43 years. So in a sense, if the world
disappeared, it would not be noticed by the Sun almost whatsoever. This fact says us that we will
never be able to observe the gravitational waves simultaneously with accompanied radio waves
coming from the same souce. Furthermore the source of any observed gravitational waves
should be rather close to us.
About then years ago, I proposed a method to measure the gravitational field velocity4. The
method aimed to find the field velocity in terms of light velocity. In the method, with a very sensitive
measurement of the force field on Earth in real-time, the Sun’s periodical field effect on Earth’s force
field would be observed. Than the phase difference between the appearing movement of the Sun and
its gravitational effect on Earth would be calculated. This phase difference would give a relation
between the force field velocity and the light velocity. An R&D project was initiated by ESER R&D
Department with the support of TUBITAK the Turkish Scientific Research Council in 2010. However,
due to the low budget, as a result of not being able to use advanced production technology, we could
not maintain the measurement sensitivity we wished for and we could not get satisfactory result. Yet,
with this paper in your hand, it is understood that since Earth is moving through an already formated
force field, it will act like it perceives the Sun’s field effect instantaneously. In that case the method I
proposed can only provide a proof that the light velocity is a universal constant. If a more advanced
project is initiated on this subject, ESER will be happy to contribute with its previous experiences.
So far, we have understood how the universal force and motion equations brought proper
solutions to the obvious observations like flattening and rotation of the celestial bodies in the same
direction. In one sentence, these are due to the rotation of the bodies around their axes which creat
rotating effective fields. But we still did not say anything about why the celestial bodies rotate around
their axes.
© December 2015 ESER Co. Inc. All Rights Reserved
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Universal force-motion equations and solar system implementation
One of the most impressive parts of this theory is that it provides striking and unquestionable
solution for this basic obvious observation. Our research about that issue is ongoing potently and the
work including many stunning aspects of the solution will be presented in a few months.
This work is based on the theoretical study called “Field Relative Model of the Universe”
which I completed basicly in 2012. However, this theory firstly had to prove itself. This was required
notably for me personally and then for the scientists I was going to present. Due to that, I have not
published this base study yet. With this article you have in hand, the theory seems to be validated
perfectly. However, it is understood that I will have to revise the base work that lead to this study
entirely. I aim to complete it in 2016.
Acknowledgment
I would like to thank Mrs Ayşe Gülbeden, Mr Güner Yalçın and my daughter Ceren for thir efforts for
the text in English and Turkish to be simple and easily understandable. This study has been completed
after a long, tiring, exhausting but an exciting process. I am grateful to Prof. Dr. Birol Kılkış for his
advices and encouraging approach through this process. I must thank to some valuable scientists
whom I cannot give names. I tried to work with them insistently, because that way the improvement
process would be much shorter and painless. However, we could not agree on any subject at all and I
had to solve all the complicated problems on the way by myself. That way those scientists made me
have the honor of reaching “Universal Force-Motion Equation” alone. Undoubtedly, I owe the greatest
thanks to the ESER family. There should be a free and peaceful environment to be able to concentrate
myself to this very intensive and troublesome work. The ESER family under the leadership of İlhan
Adiloğlu provided this environment in the broadest sense. Without this friendly working environment
it would not be possible to succeed. During the fırst establishment of the R&D Department of ESER,
Mr Adiloğlu did not provide any constraints on R&D subject. The only criterion was a high grade of
research. I am glad I achived it.
1.
2.
3.
4.
For details on the subject please look at “Properties of galaxies reproduced by a hydrodynamic simulation” M.
Vogelsberger, S. Genel, V. Springel, P. Torrey, D. Sijacki, D. Xu, G. Snyder, S. Bird, D. Nelson, L. Hernquist, 8
May 2014, VOL 509, NATURE”.
Readers who are not familiar with quaternion can find a lot of information about it in advanced Mathematics and
Physics books. A nice informatic section is also present in the 11th section of volumed reference book by Roger
Penrose “The Road to Reality”. Quaternions can be viewed as 4D vectors and the dot product operation can
be applied.
The Astronomical Almanac fort the year 2013, ISBN 978-0-7077-41284 page: E8
For details look at www.realtimegravity.org
References
- Field Analysis and Electromagnetics Masour Javid and Philip Marshall Brown, 1963 by the
McGraw-Hill Book Company, Inc. Catalog Card Number 62-22199
- Theoretical Mechanics with introduction to Lagrange’s Equations and Hamiltonian Theory
by Murray R. Spiegel, 1967 by the McGraw-Hill Book Company, 60232.
- The Astronomical Almanac fort the year 2013, Published by the United Kingdom
Hydrographic Office ISBN 978-0-7077-41284
- Mathematics of Dynamic Systems by H.H. Rosenbrock and C. Storey THOMAS NELSON
and SONS LTD. Great Britain 1970
- Stability Theory of Dynamic Systems by J.L. Willems THOMAS NELSON and SONS
LTD. Great Britain 1970
- Physics for Scientists and Egineers with Modern Physics by Douglas C. Giancoli
PEARSON 2013 ISBH 978-605-4691-34-0
- Philosophical Concepts in Physics The Historical Relations between Philosophy and
Scientific Theories, by James T. Cushing Cambridge University Press 2000.
- Relativity, The Special and General Theory by Albert Einstain 1916, Say Yayıları 8. Baskı
2008
- The Meaning of Relativity by Albert Einstein 1922, ALFA Basım 2014
- Eintein’s Theory of Relativity by Max Born 1962, Evrim Yayınevi 1995.
© December 2015 ESER Co. Inc. All Rights Reserved
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Universal force-motion equations and solar system implementation
-
The Road to Reality A Complete Guide to the Laws of the Universe by Roger Penrose 2004
The Life of the Cosmos by Lee Smolin 1997, ALFA basım 2015
Wikipedia free encyclopedia https://en.wikipedia.org
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