Applying the equation (202) we may write the equation acceptable in a linear
geometry :
(211)
where
W_{kr-k}
stands for the difference of the kinetic energy between the orbits at the common radius r_{lmx} and the initial radius r_{k} due to their motion in reference to the spacetime structure
m
mass of the object
v_{m/x}
parameter of the gravitational field: the frame intrinsic speed of the spacetime structure
r_{x}
parameter of the gravitational field: the frame intrinsic radius defining the geometry of the gravitational field
r_{lmx}
common radius in a gravitational field
r_{k}
the initial radius of just observed motion in a gravitational field
v_{zo}
stands for the Z_{ct} orbital speed according to equation (210)
v_{zok}
stands for the initial Z_{ct} orbital speed at the radius r_{k}
v_{zl}
stands for the Z_{ct} free fall speed
v_{zlk}
stands for the initial Z_{ct} free fall speed
Substituting to equation (211) for v_{zo} 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 v_{zok}=0 the equation
(213) is transformed into equation (184).
To make our equations more simplified, we shall consider the initial vertical speed v_{zlk} at a radius r_{k} as zero. Let us find out radia at which the speed v_{zl} 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 v_{zl}=0, we have
(214)
Solving this quadratic equation, we receive:
(215)
(216)
The equations (215) and (216) define two radia, R_{1}=r_{k} (the initial radius) and R_{2}=r_{f} (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 v_{zl}=0 (zero vertical motion) and, at initial radius r_{k}, the object arrives at a final radius r_{f}, which may be smaller, greater, or equal as the initial radius r_{k}, depending on the size of the initial orbital speed v_{zok}. To determine the critical value of the orbital speed v_{zok} = v_{zocr}, at which r_{f} = r_{k}, 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 r_{k}=r_{lmx}, 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 r_{k} (R_{1}), we find what kind of the common motion will follow:
In case v_{zok} > v_{zocr} :
The proceeding vertical speed will be set in opposite direction to the gravitational acceleration. The upward motion will stop at the radius r_{f} > r_{k} .
In case v_{zok} = v_{zocr} :
The motion will continue in a circle of radius r_{k}.
In case v_{zok} < v_{zocr} :
The proceeding vertical speed will be set in direction of the gravitational acceleration. The downward common motion will stop at the radiusr_{f} < r_{k} .
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 Z_{ct} 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*r_{x}).
Consequently we can tell that
at r_{lmx} > r_{x}
the apparent gravitational density is directly proportional to the Z_{ct} orbital speed. At the critical speed v_{zocr} 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 "r_{x}/r_{lmx}" (the natural space density of the gravitational field)
at r_{lmx} < r_{x}
the apparent gravitational density is inversely proportional to the Z_{ct} orbital speed. At the critical speed v_{zocr} it reaches the value "1" and, at the zero orbital speed it reaches the value "r_{lmx}/r_{x}" .
It must be therefore :
at r_{lmx} > r_{x}
(220)
(221)
Hence:
(222)
where k is constant.
Respecting equation (221), we have
(223)
and,
(224)
Substituting for v_{zo} fron equation (210) and for v_{zocr} from equation (219) to the equation (224), we have :
(224-1)
at r_{lmx} < r_{x}
(220a)
(221a)
Hence:
(222a)
where k is constant.
Respecting equation (221), we have
(223a)
and,
(224a)
Substituting for v_{zo} fron equation (210) and for v_{zocr} from equation (219) to the equation (224), we have :
(224-1a)
Orbital and vertical speeds in Z_{cr} frame (in units of a chosen frame)
At r_{lmx} > r_{x} :
At r_{lmx} > r_{x} the circumference of the circle, passed by an object at the common radius r_{lmx}, 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 Z_{cr} orbital speed (v_{oo}) rather than its
Z_{ct} value (v_{zo} or, v_{zok}). The Z_{ct} speed can be derived from the quadratic equation (226). The following equation shows the calculation of the initial Z_{ct} speed
v_{zok} from the initial Z_{cr} speed v_{ook} (at r_{lmx}=r_{k}):
(227)
Applying the equations (185) and (185a), we can derive the equation for the Z_{cr} vertical speed v_{ol} (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 v_{zlk}=0 (at the radius r_{k}), we receive:
(229)
At r_{lmx} < r_{x} :
At r_{lmx} < r_{x} the circumference of the circle, passed by an object at the common radius r_{lmx}, 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 Z_{cr} initial orbital speed (v_{ook}) rather than its Z_{ct} value (v_{zok}). Substituting for v_{oo}=v_{ook} and r_{lmx}=r_{k}, we can derive from the equation (226a):
(227a)
Since the Z_{cr} orbital speed (v_{oo}) goes down with the apparent space density, the Z_{cr} vertical speed (v_{ol}) must go up:
(228a)
(229a)
Confrontation Keppler's laws with the space-flow theory of gravity
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 v_{oo} 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 Z_{ct} orbital speed v_{zo} from the equation (210) instead the Z_{cr}speed v_{oo} into equation (230), we receive
(232)
The law of conservation of angular momentom in spaceflow theory stays valid in Z_{ct} 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 T_{1} and T_{2} of the objects orbiting at critical speeds at the radia r_{1} and r_{2}, we receive