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#### 5.3.3.8 Analysis of the common motion in gravitational field

Analyzing the equations (213), we are coming to conclusion that the free fall speed reaches zero under the following circumstances:
1. In case when vzo = vzocr according to equation (218) (the critical or, the 1st cosmic speed is reached), and the free fall speed reaches zero (vzlk = 0) at the radius rk. In this case the examined object becomes circulating around the gravitational object in a circle of the radius rk.
2. In a common case, the examined object does not circulate in the circle, but in the ellipse-like, egg-like or pear-like curve. In this case the vertical speed reaches zero in two points (radia):
1. At the upper initial radius rk, provided that the horizontal (orbital) initial speed fulfills the condition:
(261)
2. And, at the lower final radius rf,
(262)

Remark :
Of course, the lower culminating radius rf may be reached in case only, if it exceeds the radius of the surface of gravitational body.
1. Or, in a similar way:
1. At the lower initial radius rk, provided that the initial orbital speed fulfills the condition:
(263)
2. And, at the upper final radius rf according to the equation (262). The upper culminating point however exists on condition only, that :
(264)

If the equation (264) is not fulfilled, the object escapes from the gravitational field. The equation (264) defines the orbital escaping speed for the case, when the extent of the gravitational field is not restricted by another (superior) gravitational field.

#### The shift of culminating points

We can see from Tab13b-1, Tab13b-2, Fig13d-1 and Fig13e-1 that at the same time when the object makes one turn from the upper to the lower culminating point and back to the upper culminating point, the radius vector of the object turns by the angle , differring from 360 degrees by . The angle would be expected of 360 degrees in a common geometry. General relativity derives slight advance (that is > 0) of culminating points
Observations have revealed that the major axis of the Mercury's orbit rotates in space with respect to stars, covering angle of about 16 degrees every 10000 years, that is 1,38758 seconds of arc within one turn of Mercury's rotation around Sun (Perihelion advance of Mercury). The Einstein,s theory of general relativity predicts perihelion advance of Mercury of 0,10358 seconds of arc within one turn. The example of the calculation of the Mercury's orbital trajectory in Tab13b-1 shows that accordind to space-flow theory the whole observed value of the perihelion advance of Mercury might be explained by the relativistic phenomena and the related deformation of the gravitational field. In this example the perihelion advance of 3,8537*10-4 degrees per one turn is calculated, corresponding to 1,38733 seconds of arc per one turn. So, the perihelion advance by space-flow theory is 13,396 times greater than the perihelion advance by the theory of general relativity for planet Mercury.

It has become of common knowledge that the seasons on Earth and, probably on any planet, are caused by the tilt of the earth's rotation, that is, due to the deflection 23,4 degrees (for the Earth) of the axis from a direction perpendicular to the earth's orbital plane. The direction of the axis stays nearly the same with regard to stars, even as the Earth revolves around the Sun. As a result, when the Earth is at a certain place in its orbit, the northern hemisphere is tilted towards the Sun, and there is summer on the northern hemisphere. Half a year later the northern hemisphere is tilted away from the Sun and there is winter on northern hemisphere. Two days during year, when the earth's axis is most directly toward or away from the Sun are named the solstices and they are also the longest and shortest days of the year.

The earth's orbit however is not a perfact circle. It is somewhat elliptical and the distance between the Earth and the Sun varies during the year. Since the variation of the distance is relatively small, this effect is too weak to cause seasons, but it might have some influence over their intensity.

The Earth reaches perihelion - the point when it is closest to the Sun on January 4th in this era. The date of perihelion does not remain fixed. As we can see in Tab13c-1, the perihelion is constantly delayed, it will be reached gradually on all days of the year, and reaches again the january 4th within about 7139 years. Even if this period cannot be considered as verified (tha data in Tab13c-1 should be considered as the results of calculation examples), the real time of Earth per 360 degrees perihelion advance would not probably differ much from this value. This discovery might be significant for the evaluation of the global development of weather, since the balance of conditions determining weather is pretty subtle. Up to now scientific sources determine the period of the 360 degrees perihelion advance of Earht to more than 50000 years.

#### The energy of free object in free motion

The unit kinetic energy (kinetic energy on unit of mass) of the object in free motion in Zct reference frame, on condition that the relativistic influence is ignored, is defined (see equation 211):
(265)

Substituting for rx and vm/x from equations 169 and 170, we can derive:
(266)

This is the same energy as it was derived by Sir Isaac Newton:
(267)

In a space-flow theory energy can be expressed yet in another two reference frames, Zcr and intrinsic, that is in frames differring from the Zct frame by different length or time units. Respective equations are rather complicated and that is why they are not included in this paper.

The relativistic influence caused by increasing speed of the space flow in gravitational field (defined by mass of gravitational body) has no consequences on energy. Say if we apply parameters rx from the equation (244) and vm/x from the equation (246), which have already been modified regarding the relativistic influence mrntioned, the equation (267) stays unchanged. So, the unit kinetic energy is not influenced by relativistic phenomena. The energy of the object however varier with the relativistic increase of its mass, as it is was defined by the equation (192).

#### How long does it take sun's light to reach earth

To determine the time interval within it the photon from the Sun passes over the distance Sun - Earth, it is necessary to consider the following entries :
• initial Zct speed of light c = 2,997*108 m/sec
• distance Sun - Earth : 1,496*1011 m
• radius of the sun's surface rk = 6,959*108 m
• period of sun's rotation: 2,12544*106 secs
• corresponding Zcr radial speed on sun's surface vook = 2057,206 m/sec
• basic parameters of the sun's gravitational field:
rx = 4,43776*108 m
vm/x = 7,73412*105 m/sec
In case the photon starts from the centre of the sun's disc
Applying the equation 227 for the vook = 2057,206 m/sec we can calculate the Zct initial orbital speed vzok on sun's surface:
vzok = 3,212548*103 m/sec
Since it must be
(268)

where vzlk stands for the initial vertical speed of the photon, we have:
(269)

Substituting for vzlk in equation 213, we have for the Zct vertical speed:
(270)

Applying equations 228 and 224-1, we have :
(271)

Ignoring motion of the photon in earth's gravitational field (we can afford it, since earth's gravitational field is significantly weaker than the sun's gravitational field), we can calculate (numerically) time interval Tse within it sun's light reaches the Earth:
(272)

In case the photon starts from the limb of the sun's disc
By reason that now photon starts from the sun's surface horizontally, must be:
vzok = c = 2,997*108 m/sec
vzlk = 0
Applying the equation 229, we have :
(273)

and
(274)

Up to now calculations, based on the idea of the space-time of constant space density (homogenous and heterogenous space) and, on the idea of constant speed of light, result in an explicit value of the time interval Tse = 8 minutes: 18 seconds. According to space-flow theory the time interval Tse varies with distance of the place of the photon's launch from the centre of sun's disc, reaching 52,45 seconds for the photon starting from the centre of sun's disc and 11 hours: 9 minutes: 57 seconds for the photon starting from the limb of sun's disc. The dependence Tse vs distance from the centre os sun's disc is not directly proportional. The greatest growth of the time Tse comes from photons starting nearby the sun's limb.

#### The dark matter

By measuring Doppler shifts in our galaxy (Milky Way) scientists determined orbital speeds of stars. This in turn allows them to calculate the mass of galaxy. However the mass calculated in such a way is much larger than the mass of all stars and gas, that can be seen in galaxy. That is how scientists came to a conclusion that at least 90% of the mass of any galaxy, and perhaps 99% of the mass of the universe do exist in the form of matter that cannot be seen (dark matter).
Space-flow theory can significantly contribute to the solution of this great cosmological problem, especially in a disc-shaped galaxies. The unit space flow (space flow per unit space angle) in a narrow space angle close to the plane of galactic disc is much greater than the unit space flow in the remaining space angle. The space flow in the narrow disc angle therefore reaches much higher value than it would correspond to this angle in case of uniform distribution of the mass. Since space flow is directly proportional to the mass, this implies conclusion that gravitational field in this space angle is as strong as if it was created by the such much times greater gravitational mass how much times is the unit space flow in this narrow space angle greater than it would be there in case of the uniform distribution of the mass.
Orbital speeds of the stars in galaxies are approaching their critical values according to equation (218), since the trajectories of the stars are approximately circular. By substitution for rx and vm/x from equations (169) and (170) to equation (218) we can derive:
(275)

where are
M
stands for the mass of galaxy under radius rlmx at which orbits the respective star.
vzo
orbital speed of the star in galaxy
rlmx
distance of the star from the centre of galaxy
kapa
gravitational constant
In case the unit space flow in a narrow disc angle is 10 times greater than in a uniform distribution, corresponding mass of (imaginary) gravitational body will be also 10 times greater, and the orbital speed will have to be:
(276)

Resume:
Too high orbital speed observed at the stars in galaxies can be explained by unequal distribution of the mass in galaxy. There is no need to look for dark matter for that reason.
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