1108 Cosmogony 

Tools 1110 
1109 * Derivation Of The Inverse Square Law Of Gravity
By The Johnson Cosmological Model
Created Dec 313, 2003 Stavropol, RU.
Observe Porpoise vortices!

Observe a source of basic Insight! View a saved copy. 
(With Laws And Possibilities)
Universe  All space, matter and motion regarded as a whole.
Logos  the source of fundamental order of the universe (the eternal laws that govern physical and behavioral
reality).
Aether  The medium that fills all space and supports the propagation of electromagnetic waves which are constructed from vortex
rings formed in and of Aether.
Cosmos  the universe regarded as an orderly, harmonious whole.
Chaos  The background random matter in translational equilibrium (an ideal fluid in thermal equilibrium with local fluctuations).
Random background motion of elements of matter in space.
The Physical Universe is the Space, Matter and Motion in which and from which our visible and invisible physical Universe is formed. Aether (matter) is the one and only formless substance of the Cosmos from which and in which all material bodies are formed and exist. The background Matter is in constant motion with collisions from all directions transferring momentum in all directions by the properties of space occupation, lack of cohesiveness, probability of location and conservation of momentum.
universe (y 'nəv?s') The totality of matter, energy, and space, including the Solar System, the galaxies, and the contents of the space between the galaxies. Current theories of cosmology suggest that the universe is constantly expanding. 
The American Heritage? Science Dictionary Copyright ? 2002. Published by Houghton Mifflin. All rights reserved.
Cite This Source
Ae?ther
n. Greek Mythology
The poetic personification of the clear upper air breathed by the Olympians.
[Latin Aethr, from Greek aithr, upper air.] 
The American Heritage? Dictionary of the English Language, Fourth Edition copyright ?2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved.


See : 1191 Michelson  Morley experiment What is so hard to understand? To the extent that the Aether Chaos is an inviscus, inelastic, infinitely divisable fluid, the Aether vortex rings pass through the Aether chaos with out effect.
To the extent that the rings interact with the chaos we find the cause of the background red shift! 
*Logos (From Wikipedia, the free encyclopedia) Logos in ancient Greek philosophy, mathematics, rhetoric, theology, and Christianity.Logos (pronounced /ˈloʊɡɒs/ or /ˈlɒgɒs/; Greek λόγος Logos) is an important term. Heraclitus (ca. 535?475 BCE) established the term as meaning the source and fundamental order of the cosmos. John 1:15 In the beginning was the Logos, and the Logos was with God, and the Logos was God. The same was in the beginning with God. All things were made by him; and without him was not any thing made that was made. In him was life; and the life was the light of men. And the light shineth in darkness; and the darkness comprehended it not. 
cos?mos
(kŏz'məs, mŏs',
mōs')
n.
1. The universe regarded as an orderly, harmonious whole. [Middle English, from Greek kosmos , order .] 
chaos
(kā'ŏs') The behavior of systems that follow deterministic laws but appear random and unpredictable. Chaotic systems very are sensitive to initial conditions; small changes in those conditions can lead to quite different outcomes. One example of chaotic behavior is the flow of air in conditions of turbulence. See more at fractal. 
For a founding statement see 1.1.3 Postulates Of Physical Reality  A Foundation Statement
The extensive property of the Cosmos exists and is called Space! Space is continuous in three linear, independent, infinite dimensions.
Note: a vector is an arrow going from a starting point to an end. A vector can be a location (given an arbitrary starting point) or displacement . An example of the use of terms would be the corner stone in a building as the origin. Locations are vectors from the origin to points. Displacements are vectors from one point to another. Displacements subtract out reference to the origin.
In the Figure below R_{a}_{a} and R_{b}_{b} are locations and r is a displacement. The vector r gives no information about the location of its beginning R_{a}_{a} or its end R_{b}_{b}.
Defining an arbitrary initial, orthogonal Cartesian coordinate system starting point of R_{0}_{0} as the origin with three perpendicular directions (_{1},_{2},_{3}).
We define location R in physical space in terms of its components
R_{0} ≡ R_{01}_{1} + R_{02}_{2} + R_{03}_{3}.
We define displacement r_{ab} in physical space in terms of its components
r_{ab} ≡ R_{b}_{b
} R_{a}_{a}.
≡ R_{b1}_{1}  R_{a1}_{1} + R_{b2}_{2}  R_{a2}_{2} + R_{b3}_{3}  R_{a3}_{3}
≡ (R_{b1}  R_{a1})_{1} + (R_{b2}  R_{a2})2 + (R_{b3}  R_{a3})_{3}
≡ r_{ab1}_{1} + r_{ab2}_{2} + r_{ab3}_{3}
Location r in an
orthogonal Cartesian coordinate system.
Notice that displacement subtracts out reference to R_{0}_{f} and R_{i}_{i} so r_{fi} has magnitude and direction but not location.
We define area A_{12} as the cross product
A_{12}_{12} ≡ r_{1}_{1}
x r_{2}_{2}.
We define volume V as the dot product of area and extension
V_{123 }≡ A_{12}_{12}
r_{3}_{3},
= r_{1}_{1} x r_{2}_{2} r_{3}_{3}.
The inertial property of the Cosmos Aether called Matter exists! Matter is not created or destroyed. Each element of Matter is infinitely divisible. Each
element of Matter occupies its own space uniquely. Basic Matter is a nonviscous, inelastic fluid. Out of the Aether Matter all material things formed and in
the unformed chaos the move and have their being. The inertial property of Aether is called mass m.
The Aether concept was previously rejected through error in understanding 
See : 1191 Michelson  Morley experiment
Aether Matter is not just a medium in which electromagnetic radiation propagates and is effected (like the imagined Aether or Dark Matter). The radiation and all material objects are forms created of the Aether Matter and swimming in the Aether Matter.
Each point of Aether Matter m_{i} occupies it's own unique location R_{i}_{i} in space.
R_{i}_{i} ≡ R_{i1}_{i1} + R_{i2}_{i2} + R_{i3}_{i3}.
We define density of a volume of space as the mass divided by the volume of space containing it.
_{x} ≡ m_{x}/V_{x}
Given two elements of Matter m_{a} and m_{b} the displacement between them r_{ab}_{ab}. Relative motion (velocity) v_{ab}_{ab} between them exists!
Given two elements of Matter in space separating at a constant velocity v_{ab}_{ab}
and the associated displacement r_{ab}_{ab} , we operationally define a
clock with time t by
t_{ab} ≡ r_{ab}_{ab} / v_{ab}_{ab} , where r_{ab }≡ r_{ab}_{ab} and v_{ab}
≡ / v_{ab}_{ab}.
With a clock defined we can graph the position of an element of Matter as a function of time. Collisions in the Matter change the position of an element of Matter instantaneously so the graph is jagged as represented in one dimension below.
Position as a function of time.
This kind of graph has no well defined slope at the points of collision. In calculus terms r(t) is not a differentiable function of time.
With a clock to measure time and a length standard to measure distance we operationally define the velocity vector in a given direction v_{a} for a period of time in space where there are not any collisions as
v_{a} ≡ r_{a}_{a}/t_{a}.
In the inertial frame of reference the net flow in a given direction is equal to the net flow in the opposite direction because as with the one dimensional physics toy motion flow is possible simultaneously in all directions (meaning + and  directions along each axis).
The magnitude of velocity (speed v_{a}) of any element of matter m_{a }in space exists with an average speed v_{0}. The average speed v_{0} is uniform through out background matter. From this we conclude there exists an inertial frame of reference R_{0} where the average speed v_{0} is zero.
Given the random distribution of background matter, with a large enough volume V, we observe the effects of chaotic motion matter is in equilibrium giving an
average background density_{0} that is constant.
The average background density of
matter _{c} is constant.
The dimensionless saturation is the ration of the volume of mass V_{m} in a volume
of space V_{s}
≡ V_{m} / V_{s}
By Observation 1 we define a universal constant the saturation of the chaos _{c}
for the ratio of a volume of matter V_{mc} and the volume of confining space V_{sc}.
_{c} ≡ V_{mc} / V_{sc}
With space, density, and motion we can define momentum passing through a given area or impinging on a given surface. The definition of momentum as mass m times
its velocity v needs some insight when talking about a fluid like the Aether Matter. What mass are we talking about and what velocity are we talking about?
Source of basic Insight.
Basic matter is not viscous and not compressible:
This means that if we had two pieces of pure basic matter with no internal motion, in a collision the momentum of one could be transmitted through the other without friction. For example see the diagram below showing a collision.
Basic matter is not viscous and not compressible.
In Figure 3.3 we observe the mechanism for keeping a form in the chaos.
In collision momentum is conserved but instantaneously transferred ahead with a space discontinuity in the path of the center of mass of momentum. In the chaos the shape depends on the shapes and mass distributions in the collision. Since the shape is chaotic so is the outcome.
Momentum can move in opposite directions at the same time in the same pure matter. Below is a model of a Physics Toy with hanging steel balls.
Physics Toy with hanging steel balls.
In Figure 3.3 we observe the mechanism for keeping a form in the chaos.
Matter deposited on one side of a stream is removed by collisions on the opposite side!
When we consider that the pure matter is not viscous and not compressible, we see that momentum freely moves in all directions at the same time.
Question: In background matter, what is the mass moving in a given direction?
Considering uniform random motion (similar to a gas) where half the matter has a component moving in any given direction and half the matter has a component
moving in the opposite direction.
Elements of matter moving through random collisions.
Answer: At any given time, half the mass has a motion component in any direction and the other half of the mass has a component of motion in the opposite direction. With the infinitesimal sizes in the Aether Matter, this is on a scale large enough that random fluctuations are averaged out.
The expression of the half of the mass m_{i}_{i+}
moving in a given direction _{i+} in terms of the density (by Observation 1) _{o} ≡ m_{x}/V_{x} is
m_{i+} = (1/2)m_{i}
=(1/2)_{0} V_{i+} , ( i+ is direction on the i axis.)
Question: What is the average component in any given direction at any given time of the average speed?
The average speed associated with any momentum is distributed in all directions equally, so we need to find the average component of the average speed in any
given direction (note only one direction along the axis, both ways average to zero).
Answer: The velocity v_{k+}_{k+} in a given direction associated with
the mass m_{ik+} is the average velocity component of the average speed in any given direction. This ranges from 0 to v_{o}. An
approximate average of this in any direction is half the average speed.
v_{k+}_{k+ } (1/2)v_{o}_{k+,} (where k+ indicates the + direction on the k axis.)
The definition of momentum of a mass element m_{i} is m_{i} times its velocity v_{i}_{k+}
p_{i+}_{k+} ≡
m_{i+}v_{k+}_{k+}
For a component of momentum passing through a given area in a given direction, we use Equation 2 and Equation
3 in Definition 9 to get:
p_{i+}_{k+} m_{i+}(1/2)v_{o}_{k+} , Equation 3 in Definition 9
(1/2)_{0} V_{k} (1/2)v_{o}_{k+} , Equation 2 in Definition 9
(?)_{0} V_{k} v_{o}_{k+}
(The speed of a longitudinal wave as a function of density)
Matter transfers its momentum in collision. The collisions are of incompressible fluid matter. Consider a set of billiard balls in a row. An element of matter traveling with the average background speed *v_{o}=dl/dt (*one dimensional).
Billiard Balls.
From Figure 4.1, momentum is transmitted the distance L_{t} in time t_{t}.
L_{t} = the distance momentum is transmitted.
n = the numver of elements in length L_{t}.
d = the diameter of an element.
The length of matter the momentum traveled through instantaneously was
L_{m} ≡ nd.
The length of space the momentum traveled through at *v_{o}
L_{f} ≡ L_{t}  L_{m} , or L_{m}= L_{t} L_{f} .
The observed time t_{t} for the momentum to be transmitted distance L_{t} is
t_{t} = observed time to transmit momentum distance L_{t} .
Matter traveled the free space distance L_{f} in time t_{t}. By Definition 6 t_{t} ≡ L_{t}
/v_{t} , the speed of the free matter is
v_{f} = L_{f} /t_{t}.
From Observable 4.1 L_{t} and Observable 4.4 t_{t}
the speed of momentum v_{p} is
v_{p}=L_{t} /t_{t}
From Equation 4.1 v_{f} =L_{f} /t_{t} and Equation 4.2 v_{p}=L_{t} /t_{t} the ratio of the speed of momentum transmission v_{m} to the average background
free space speed v_{f} is
v_{p} /v_{f} =( L_{t} /t_{t})/(L_{f} /t_{t})
= L_{t} /L_{f} .
From Equation 4.3 v_{p} goes to infinity as L_{f} goes to zero. That is as the density increases the speed of transmission
of momentum increases while the speed of free motion is unchanged.
By Definition 5 _{i} ≡ m_{i}/V_{i}
the density of pure matter is its mass divided by its volume
_{0m} =m_{i}/V_{m}.
We have Observation 1 _{0} = m_{x}/V_{s}
but here we are varying the background volume to explore the effect on speed of momentum transmission in the Chaos
_{b} = m_{x}/V_{b}
By taking the ratio of Equation 4.4 _{0m}=m_{x}/V_{m}
and Equation 4.5 _{b} = m_{x}/V_{b}
_{0m}/_{b} = (m_{x}/V_{m})/(m_{x}/V_{b}) = V_{b} / V_{m} =
L_{m}^{3}/L_{t}^{3}
Given the average background free space speed v_{o}, consider average speed as a function of density. We look at this
system for an arbitrary unit of distance (large enough that the averages are not invalidated). We use 1000 units of total space and very the free space 1000 to
0.01.
The Table uses a 1000 units total distance Lt and varies the free space Lf 1000 to 0.01.
The graph below represents the data table above.
Graph as the density approaches 100% the speed approaches infinity!
Consider the improbable event of streams forming in the chaos.
A stream forming in the chaos.
Our ideal matter has its own characteristics. To have intuitive insight into behavior of a stream on the chaos we need to have the reality experiences that reflect these characteristics.
A stream is a group organization out of the chaotic matter. A steam travels with a group velocity with respect to the background which averages to zero.
From Observation 3 says that the stream shape will tend to be maintained while Observation 2 says there is a difference in the speed of any element and the external speed of momentum as the momentum advances through transfer. An unverified thought is that the average speed of an element of matter participating in the stream is the same as the ambient average speed. This would mean that there is a direct relationship between that average ambient speed and the speed of any stream.
The next level of organization is a stream forming a vortex. Consider the improbable event of a vortex forming from streams. Below we consider a section of a vortex about the axis of rotation.
Figure 5.2 A vortex forming in a stream.
What is the difference in effect of ambient chaos on the inner vortex boundary and the outer vortex boundary?
The need here is to discover the mechanics that give identity and multiplicity there is some condition (probably density dependent) that gives a size constraint. The spiral form of solar systems and galaxies may also be an outcome of the same mechanics on a larger scale. Here I am not talking about some abstract statistical formulation but a concrete visible mechanics. What is happening differently on the inside surface and the outside surface interaction with the chaos is a focus worth attention.
A vortex cross section.
Consider the improbable event of a vortex ring (toroid) forming from the vortices.
A toroid cut out.
With vortex rings and multiring objects formed from them in the Chaos, we can find objects and observe their position as a function of time. The position as a function of time is smoother than collisions in the Chaos. An example would be throwing a ball. up as shown below.
Position as a function of time.
Using the limit as t 0 in Definition 7 operationally defines the threedimensional velocity of and
object as the slope of the displacement time graphs.
v_{i}_{i }≡ Limit
(r_{i}/t_{i})_{i
}t
0
= Limit (r_{i1}/t)_{i1}+ (r_{i2}/t)_{i2 }+ (r_{i3}/t)_{i3
}t
0
= (dr_{i1}/dt)_{i1}+ (dr_{i2}/dt)_{i2 }+ (dr_{i3}/dt)_{i3
}= v_{i1}_{i1}+ v_{i2}_{i2 }+ v_{i3}_{i3}
Velocity as a function of time.
We operationally define the threedimensional acceleration of and object as the slope of the velocity time graphs
a_{i}_{i} ≡
Limit (v_{i}/t_{i})_{i
}t 0
= Limit (v_{i1}/t)_{i1}+ (v_{i2}/t)_{i2 }+ (v_{i3}/t)_{i3 }t 0
= (dv_{i1}/dt)_{i1}+ (dv_{i2}/dt)_{i2 }+ (dv_{i3}/dt)_{i3}._{
}= a_{i1}_{i1}+ a_{i2}_{i2 }+ a_{i3}_{i3}
We can define the three dimensional force F acting on an element of mass m_{i} with cross section area A_{i} in the Chaos by
time t rate of change matter r_{0} V_{i} in the impacting volume moving at average background speed v_{oi} .
F_{i}_{i}
≡ dp_{i}_{i} /dt
= dp_{1}_{i1}
/dt + dp_{2}_{i2} /dt + dp_{3}_{i3} /dt
Now we need to recognize the infinite divisibility of the Chaos. This means that when two particles of matter collide the momentum of the contacting parts is
not lost like in the crash of to cars but as with the physics toy continues in exchange and jumps ahead into the forward element in both directions. Since Chaos
is ubiquitous (everywhere like a fog) this continuation of momentum and in fact jumping ahead was accounted for in the average speed in any given direction.
We have the net force F_{ir+} acting in the r+ direction on an element of mass m_{ir+}. The mass is in a volume
with cross section area A_{ir}. The average momentum p_{ir+}_{r+} of the ith element moving in a given direction (r+) is from Equation 4 p_{ir+}_{r+} (1/4)_{o}V_{ir+}v_{o}_{r+}. The ambient force F_{ir+} in any direction _{r+} on an arbitrary area of obstruction A_{ir} is
F_{ir+}_{r+} ≡ d(p_{ir+}_{r+}) /dt ,
d((1/4)_{0} V_{ir+} v_{o})/dt_{r+ }, note (1/4)(_{0} A_{ir} v_{o}) is not a function of time
= (1/4)(_{0}
A_{ir} v_{o})dr_{i}/dt_{r+} , by Definition 7 dr_{i}/dt_{r+} = v_{ir+}_{r+}
= (1/4)(_{0}
A_{ir} v_{o}) v_{ir}_{+r+} , by Equation 3 v_{ir+}_{r+}= (1/2)v_{o}_{r+}
= (1/4)(_{0}
A_{ir} v_{o})(1/2)v_{o}_{r+} ,
= (1/8) _{0}
A_{ir} v^{2}_{o}_{r+} .
We can define the average pressure on an element of mass m_{i} in a given direction r_{r} by dividing the average force F_{i ave} over the area A_{r}
.
P ≡ F_{ir+ave}_{r+}●A_{ir}_{r+} / (A_{ir}_{r+} ● A_{ir}_{r+} )
= F_{ir+} / A_{ir} ,_{
}
Using Equation 5 F_{ir+}_{r+} = (1/8)_{o}A_{ir} v^{2}_{o}_{r+} or F_{ir+} = (1/8) _{o} A_{ir} v^{2}_{o} in Definition 13 P = F_{ir+}/A_{ir}
gives
P = (1/8) _{o}A_{ir}
v^{2}_{o }/ A_{ir}
= (1/8) _{o}
v^{2}_{o} ,
Note that in the primitive (no vortexes) Chaos this is universal and for a vortex it is the external pressure.
For uniform circular motion we study the diagram below.
A point in uniform circular motion.
We have a point moving at a constant speed v_{c}. At time t_{1} the point is at r_{1}. At time t_{2} the point is at r_{2}. In the change of time is dt, the change in position is dr and the change in velocity is dv.
We have similar triangles that give equal ratios for dr/r and dv/v
dr/r = dv/v.
Solving Definition 10 v=dr/dt for dt gives:
dt = dr/v
Rearranging Observation 6 dr/r = dv/v gives vdr/r = dv.
Then dividing both sides by Equation 6.1 dt = dr/v gives (vdr/r)/(dr/v)=dv/dt=a then reducing gives centripetal acceleration
a=v^{2}/r.
Consider a vortex section shown below.
A vortex cross section.
The pressure on the surface area exerts the force that is equal to the centripetal (inward) force of rotation. Note that momentum in a stable vortex behaves
like an element of matter m_{i} and momentum is associated with the same element as it moves.
We apply Newton's second law to a segment of the vortex cut out
F_{i }= m_{i}a_{i}.
Using Equation 6.2 a_{i}=v_{i}^{2}/r_{i} in Equation 6.3 F=ma
gives
F_{i} = m_{i}v_{i}^{2}/r_{i}.
Using Equation 4.5 m_{i} = m_{i}/V_{mi} in Equation 6.4 F_{i} =m_{i}v_{i}^{2}/r_{i} gives
F_{i} =_{mi}V_{mi}
v_{i}^{2}/r_{i}.
Looking at the segment in the diagram above we have the volume V_{i}
V_{i} = d_{l}A_{i}
Using Equation 6.6 V_{i}=d_{l}A_{i} in Equation 6.5 F_{i}
=_{mi}V_{mi} v_{i}^{2}/r_{i} gives
F_{i} =_{m}d_{l}A_{i}
v_{i}^{2}/r_{i}.
We have the pressure from the Chaos by Equation 5.2 P_{i} = (1/8) _{0} v^{2}_{o} .
Using this in Definition 13 P_{i} = F_{i }/A_{i} gives the external force as
F_{ext} = PA =(1/8)_{0}v^{2}_{o}A
We apply Newton's third law (to every force there is an equal and opposite force)
F_{r}= F_{r+} .
Using Definition 5 Density _{x}
≡ m_{x}/V_{x}^{ }in^{ }Equation 6.4 F_{c} =mv^{2}/r gives
F_{c} = _{m}d_{l}A_{i}v_{i}^{2} /r_{i} .
Using Equation 6.8 F_{ext} =PA =(1/8)_{o}v^{2}_{o}A
and Equation 6.10 F_{c} = _{m}d_{l}A_{i}v_{i}^{2}
/r_{i} in Equation 6.9 Fr=Fr+ gives
_{m}d_{l}A_{i}v_{i}^{2}
/r_{i} = (1/8)_{0}v_{o}^{2}A_{i} ,
_{m}d_{l} v_{i}^{2} /r_{i} = (1/8)_{0}v_{o}^{2} ,
d_{l}v_{i}^{2} /r = (1/8)(_{0}/_{m})v_{o}^{2
}.
If we assume that the driving speed of the Chaos v_{o}^{2} is the same as the tangential speed of the vortex v_{i}^{2}
we have d_{l}/r =(1/8)( _{o}/_{m}), or:
r = 8d_{l}(_{m}/_{o})
This Equation 6.12 r = 8d_{l}(_{m}/_{o}) would mean that the ratio of the wall radius and the wall thickness would be a
constant. Since vortex rings are the building blocks of magnetic fields, this could explain how they can range so much in major radius. That is that as the
major radius expands conservation of energy is accomplished by the shrinking the minor radius and wall thickness.
The question remains, what limits the decrease in the radius of the vortex ring? We could speculate on the ratio of the densities, say if
the pure density is equal that of the chaos the ratio of r = 8d_{l}.
If the pure density _{m} was twice that of the chaos the ratio of r = 16d_{l}.
This is a limit. the minor radius r (the minor radius of the vortex must be smaller than 8d_{l} (the thickness of the vortex wall). We have:
r > 8d_{l} .
The internal energy is kinetic. The energy of a segment is (1/2)mv^{2}, m= 2rd_{l}_{m} ,
K.E._{R}/z =rd_{l}_{m}v^{2}.
We start with a toroid cut out shown below. We assume this is a closed system in the sense the the mass is constant and the internal energy is constant.
A toroid cut out.
The total length of z (the axis of the toroid segments) is the center circumference 2R
z = 2R.
Using Equation 7.1 z = 2R in Equation 6.14 K.E._{R}/z =rd_{l}_{m}v^{2}, the total energy K.E._{R} would be
K.E._{R} = rd_{l}_{m}v^{2}(z) , solving Equation 6.14 for K.E._{R
}= rd_{l}_{m}v^{2}(2R)
= 2^{2}Rrd_{l}_{m}v^{2}.
Using Equation 6.12 r =8d_{l}(_{m}/_{o}) in Equation 7.2 K.E = 2^{2}Rrd_{l}_{m}v^{2}
gives
K.E=2^{2}R {8d_{l}(_{m}/_{o})} d_{l}_{m}v^{2 }= 16^{2}Rd^{2}_{l}v^{2}^{2}_{m} /_{o}
This show us the variables while energy is conserved is such large change as the major diameter changes in magnetic fields.
Consider two groups of vortex rings (making up visible matter) as depicted in the diagram below. The center and top spots represents the cross sectional area of a spherical object.
Two groups (center and top) of vortex rings (making up visible matter).
The idea here is that the pressure of the chaos is ubiquitous but clusters of vortex rings deflect the incoming momentum causing a shadow effect. This shadow effect causes the shadowed surface to experience decrease in momentum impacting on the shadowed side. This decrease in pressure between the objects causes them to experience a net external force toward each other. Groups of rings (objects) are forced toward each other rather than the false concept of "action at a distance" mediated by some imaginary thing called a field.
Given the average cross section area of a single vortex ring is a_{o}. The average cross sectional area for the n center rings is
A_{center} = n a_{o} .
We assume that the chaos is fine enough that the cross section area of every ring is seen.
The free vortex rings at the center experience pressure from the momentum of the chaos impacting giving a force F_{center} from all
directions. We use Equation 8.1 A_{center} = n a_{o} to multiply the pressure in Equation
6 P_{i} = (1/8) _{o} v^{2}_{o} get the force
impinging on the surface of the center group from any direction. The ambient force on the center object is
F_{o center} = P A_{center }= (1/8) _{o} v^{2}_{o} n_{center }a_{o}
What part of that force is missing due to the shielding or shadow effect of the object at the top?
Given the shielding and its shadow the decrease in force in that direction is proportional to the ration of the area of the obstruction divided by the area of the sphere on which it lies with the shadowed object at the center of the sphere and the shadowing object at the top of the sphere.
When object at the top has k vortex rings it casts a shadow of
A_{top} = ka_{o} .
For a sphere of projection with radius R_{s} we have the projected spherical surface area of 4R^{2}_{s} or
A_{sphere} = 4R^{2}.
Where the center object has n rings of crosssectional area of a_{o}. The shadow effect causes reduction in pressure from that direction. The net force pushing the two groups together is proportional to the ratio of the area of the cross section of a single ring a_{o} times the number k of rings and the area of the sphere of projection radius R_{s} and projected surface area of 4R^{2}_{s} or
A_{top} / A_{sphere} or ka_{o} /(4R^{2} ) .
Given the mass m_{o} of a single ring (constant mass of a single ring). The mass of the top object with n rings is
m_{center }= nm_{o} .
The mass of the center object with k rings is
m_{top}= km_{o} .
So the net force from the shielding imbalance is
F_{center net} = F_{o center} A_{top} / A_{sphere} , shielding
from Equation 8.4
= (1/8) _{o}
v^{2}_{o} A_{center} A_{top} / A_{sphere} , using Equation 8.2
= (1/8) _{o}
v^{2}_{o }ka_{o }na_{o} /(4R^{2} ) , using Equation 7 ,8.5 , and 8.6
= (1/32) _{o} v^{2}_{o }a^{2}_{o }k_{ }n / R^{2} ,
simplifying
= (1/32) _{o} v^{2}_{o }a^{2}_{o }(m_{top}/m_{o})
(m_{center}/m_{o}) / R^{2} ,
= (1/32^{} m^{2}_{o}) _{o} v^{2}_{o} a^{2}_{o}
m_{top} m_{center} / R^{2} .
In Equation 8.7 we define the Universal Gravitational Constant
G ≡ (1/32^{} m^{2}_{o}) _{o} v^{2}_{o} a^{2}_{o} .
Using Definition 14 in Equation 8.7 gives
F_{center net} = G m_{top} m_{center} / R^{2} .
This is then a derivation Newton's general law of gravitation!
(a direction in Space) 
Postulate 1 Space R_{0} Exists! Definition 1 Location R_{0} Postulate 2 Matter m Exists! Definition 5 Density_{x} Postulate 3 Motion v of Matter Exists! Definition 6 Time t 
Figure 1.1 Random background motion of Matter in space. Figure 2.1 Location r in an orthogonal Cartesian coordinate system. Figure 3.1 Position as a function of time. Figure 3.2 Basic Matter is not viscous and not compressible. Figure 3.3 Physics Toy with hanging steel balls Figure 3.4 Elements of Matter moving through random collisions. Figure 4.1 Billiard Balls Figure 4.2 As the density approaches 100% speed approaches infinity! Figure 5.1 A stream forming in the chaos. Figure 5.2 A vortex forming in a stream. Figure 5.3 A vortex cross section. Figure 5.4 A toroid cut out. Figure 5.5 Position as a function of time. Figure 5.6 Velocity as a function of time. Figure 6.1 A point in uniform circular motion. Figure 6.2 A vortex cross section. Figure 7.1 A toroid cut out. Figure 8.1 Two groups of vortex rings (making up visible Matter) 
Postulate 1 Space R_{0} Exists! 
Universe, Aether, Logos,
Cosmos, Chaos 
Equation 1 R_{i}_{i} ≡ R_{i1}_{i1} + R_{i2}_{i2} + R_{i3}_{i3}. Equation 2 m_{i+} =(1/2)_{0} V_{i+} , ( i+ is direction on the i axis.) Equation 3 v_{k+}_{k+}=(1/2)v_{o}_{k+} (where k+ indicates +direction on k axis.) Equation 3.1 saturation _{c} ≡ V_{mc} / V_{sc} Equation 4 p_{i+}_{k+} = (?)_{0} V_{k} v_{o}_{k+} Equation 4.1 velocity in free space v_{f} = L_{f} /t_{t}. Equation 4.2 velocity of momentum in the chaos v_{p}=L_{t} /t_{t} Equation 4.3 speed of momentum / speed in free space v_{p} /v_{f} = L_{t} /L_{f} . Equation 4.4 density of pure matter _{0m} =m_{i}/V_{m}. Equation 4.5 varying the background volume _{b} = m_{x}/V_{b} Equation 4.6 ratio of constant and varied density _{0m}/_{b} = L_{m}^{3}/L_{t}^{3} Equation 5 Force F_{ir+}_{r+} ≡ (1/8) _{0} A_{ir} v^{2}_{o}_{r+} . Equation 6 Pressure P_{i} = (1/8) _{0} v^{2}_{o} , Equation 6.1 dt = dr/v Equation 6.2 centripetal acceleration a=v^{2}/r. Equation 6.3 Newton's second law F_{i}=m_{i}a_{i}. Equation 7 surface area of a sphere of radius R A_{sphere} = 4R^{2}. Equation 7.1 Major diameter of ring z_{total} = 2R. Equation 8.1 A_{center} = n a_{o} . Equation 8.2 F_{o center} = P A_{center}= (1/8) _{o} v^{2}_{o} n_{center }a_{o} Equation 8.3 A_{top} = ka_{o} . Equation 8.4 Ratio of view obstruction A_{top} / A_{sphere} or ka_{o} /(4R^{2} ) . Equation 8.5 center object m_{center }= nm_{o} . Equation 8.6 top object m_{top}= km_{o} . Equation 8.7 Force F_{center net} = (1/32^{} m^{2}_{o}) _{o} v^{2}_{o} a^{2}_{o} m_{top} m_{center} / R^{2} . Equation 8.8 Force F_{center net} = G m_{top} m_{center} / R^{2} 