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#### 5.3.3.2 Equations of common free motion

Applying the equation (202) we may write the equation acceptable in a linear geometry :
(211)

where
Wkr-k
stands for the difference of the kinetic energy between the orbits at the common radius rlmx and the initial radius rk due to their motion in reference to the spacetime structure
m
mass of the object
vm/x
parameter of the gravitational field: the frame intrinsic speed of the spacetime structure
rx
parameter of the gravitational field: the frame intrinsic radius defining the geometry of the gravitational field
rlmx
common radius in a gravitational field
rk
the initial radius of just observed motion in a gravitational field
vzo
stands for the Zct orbital speed according to equation (210)
vzok
stands for the initial Zct orbital speed at the radius rk
vzl
stands for the Zct free fall speed
vzlk
stands for the initial Zct free fall speed
Substituting to equation (211) for vzo from equation (210), we receive :
(212)

Thus :
(213)

The equation (213) shows the free fall speed on condition that the time flow is not influenced by the orbital speed (at the orbital speed approaching zero). We can see, that substituting vzok=0 the equation (213) is transformed into equation (184).

To make our equations more simplified, we shall consider the initial vertical speed vzlk at a radius rk as zero. Let us find out radia at which the speed vzl reaches the zero value (the experience shows us that at common motion the zero value of the vertical speed is reached at two radia):
Under condition vzl=0, we have
(214)

(215)

(216)

The equations (215) and (216) define two radia, R1=rk (the initial radius) and R2=rf (the final radius), between which takes place common motion in a gravitational field.
The excentricity of the object's trajectory in a gravitational field is defined as :
(216a)

Starting its motion at vzl=0 (zero vertical motion) and, at initial radius rk, the object arrives at a final radius rf, which may be smaller, greater, or equal as the initial radius rk, depending on the size of the initial orbital speed vzok. To determine the critical value of the orbital speed vzok = vzocr, at which rf = rk, that is at which the object revolves in the circle around the centre of gravity, we have to expect, that:
(217)

(218)

In a common case rk=rlmx, since of course, each radius may become the initial one for our start of the common motion.
Thus
(219)
Hence, comparing the initial orbital speed (at vertical speed = 0) with the critical orbital speed at the radius rk (R1), we find what kind of the common motion will follow:
In case vzok > vzocr :
The proceeding vertical speed will be set in opposite direction to the gravitational acceleration. The upward motion will stop at the radius rf > rk .
In case vzok = vzocr :
The motion will continue in a circle of radius rk.
In case vzok < vzocr :
The proceeding vertical speed will be set in direction of the gravitational acceleration. The downward common motion will stop at the radius rf < rk .

#### The apparent change of the space density due to orbital motion

Looking newly at Fig15a we can come to the following significant conclusion:
The greater (less) is the Zct orbital speed of the object in a common motion, the more (less) times the circumference of the imaginary circle, that might be created by one revolution of the object's trajectory on condition that its instantaneous orbital speed became constant and its vertical speed became zero, might encircle the circumference of the circle in the chosen frame (2*PI*rx).

Consequently we can tell that
at rlmx > rx
the apparent gravitational density is directly proportional to the Zct orbital speed. At the critical speed vzocr it reaches the value "1" (we already know that the motion in a circle is possible at the critical speed only) and, at the zero orbital speed it reaches the value "rx/rlmx" (the natural space density of the gravitational field)
at rlmx < rx
the apparent gravitational density is inversely proportional to the Zct orbital speed. At the critical speed vzocr it reaches the value "1" and, at the zero orbital speed it reaches the value "rlmx/rx" .
It must be therefore :
at rlmx > rx
(220)

(221)

Hence:
(222)

where k is constant. Respecting equation (221), we have
(223)

and,
(224)

Substituting for vzo fron equation (210) and for vzocr from equation (219) to the equation (224), we have :
(224-1)
at rlmx < rx
(220a)

(221a)

Hence:
(222a)

where k is constant. Respecting equation (221), we have
(223a)

and,
(224a)

Substituting for vzo fron equation (210) and for vzocr from equation (219) to the equation (224), we have :
(224-1a)

#### Orbital and vertical speeds in Zcr frame (in units of a chosen frame)

At rlmx > rx :
At rlmx > rx the circumference of the circle, passed by an object at the common radius rlmx, is longer (in units of the chosen frame) than the corresponding circumference in the chosen frame. Therefore it must be:
(225)

Respecting the equations (224) and (225), we have:
(226)

Angular speed
(226b)

Observing the common motion, we can detect the Zcr orbital speed (voo) rather than its Zct value (vzo or, vzok). The Zct speed can be derived from the quadratic equation (226). The following equation shows the calculation of the initial Zct speed vzok from the initial Zcr speed vook (at rlmx=rk):
(227)

Applying the equations (185) and (185a), we can derive the equation for the Zcr vertical speed vol (the vertical speed of the object in a common motion, expressed in units of a chosen frame):
(228)

Substituting from equations (210), (213) and (224) and, presuming the initial speed vzlk=0 (at the radius rk), we receive:
(229)
At rlmx < rx :
At rlmx < rx the circumference of the circle, passed by an object at the common radius rlmx, is shorter (in units of the chosen frame) than the corresponding circumference in the chosen frame. Therefore it must be:
(225a)

Respecting the equations (224) and (225), we have:
(226a)

Again, observing the common motion, we can detect the Zcr initial orbital speed (vook) rather than its Zct value (vzok). Substituting for voo=vook and rlmx=rk, we can derive from the equation (226a):
(227a)

Since the Zcr orbital speed (voo) goes down with the apparent space density, the Zcr vertical speed (vol) must go up:
(228a)

(229a)

#### Confrontation Keppler's laws with the space-flow theory of gravity

First Keppler's law
Statutory text: The shape of the orbit of any body orbiting the sun is an ellipse with the sun at one of the foci.
Comparison with space-flow theory: This law does not conform to a space-flow theory, even if at the orbital speeds not differring much with the critical orbital speeds the shapes of orbits are very similar to the ellipses.
Derivation: The law was derived for the motion in a spacetime structure of a constant space density. This condition can be considered as approximately fulfilled in case only of the orbital speeds not differring much with the critical orbital speeds.
Second Keppler's law
Statutory text: The line joining the planet to the sun sweeps out equal areas in equal times.
Comparison with space-flow theory: This law does not conform to a space-flow theory.
Derivation:The law was derived as a consequence of the conservation of angular momentum. We have :
(230)

where
L...stands for angular momentum
I...stands for moment of inertia
Omega...stands for angular speed
m...stands for object's mass
Substituting for voo from equation (226), we have:
(231)

We can see from equation (231) that the angular momentum is not constant, but varies with radius in a common motion.
Note: However substituting the Zct orbital speed vzo from the equation (210) instead the Zcrspeed voo into equation (230), we receive
(232)

The law of conservation of angular momentom in spaceflow theory stays valid in Zct frames only (that is, in frames in which the length units are expressed in units of the spacetime structure, where the motion takes place, and time units are expressed in a chosen frame).
Third Keppler's law
Statutory text: The square of the sidereal orbital periods of the planets are proportional to the cube of the planets' mean distance from the sun.
Comparison with space-flow theory: The law conforms to the spaceflow theory in case only when the objects (planets) are orbiting around the sun at critical speeds (in a circular trajectories). The more the initial orbital speed differs from the critical value, the more the law is incorrect.
Derivation: For the sidereal orbital period of the object orbiting around the centre of gravity at the critical initial orbital speed, we have
(233)

By comparison of the sidereal periods T1 and T2 of the objects orbiting at critical speeds at the radia r1 and r2, we receive
(234)

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