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5.3.5 Mass distribution of gravitational objects

To examine the mass distribution under the surface of the gravitational object, let us have a look on the relativistic phenomenon (how much the total mass exceeds the rest mass due to its motion with respect to the spacetime structure falling into mass). According to equation (38) we can express the relativistic mass :
(516)

where
m
stands for the relativistic mass
mo
stands for the rest mass
Designating
(517)

as the relativity factor, and applying the equations (170) and (180), we can derive :
(518)

We can derive from the equation (516) that even at the relativity factor kr = 0,1 it will be m = 1,00335 mo. Designating kr = 0,1 as sufficient to apply the equation (516) without the relativistic modification, that is in form
(519)

we can derive from equation (516)
(520)

where
mk
stands for maximum gravitational mass (evenly spread under the respective radius), at which the relativity effect causing mass increase may be neglected, and the equation (519) may be considered as sufficiently precise.
The equation (520) can be modified using the average mass density instead of mass;
(520a)

The curve "average mass density vs radius" acc. to equation (520a) is shown on Fig20, for kr=1,005.
Fig20 The maximum mass mk under radius rlmx that may be considered as not influenced by relativistic mass increase.

Fig20 shows that, to derive the equation of the mass distribution, the relativistic mass increase due to the motion of the mass inside gravitational objects with respect to the falling spacetime structure need not be taken into consideration, since even at the objects of the radius 2*1011 m (higher than the distance Sun - Earth) the average mass density about 6000 kg/m3 (the mass density occurring at planets) is allowable at the relativity factor kr < 1,005. This relativistic mass increas however plays the significant role for the cosmic objects created of the mass of high mass density, like the neutron stars.

Another factor interacting the space flow streaming into a mass, that is also the mass quantity, is the space (spacetime) density, that may influence the mass by the similar way like we derived it in chapter 4.8. The space density inside the gravitational object may be significantly influenced by the mass distribution (depends on mass and radius). However now we are speaking about the space density inside gravitational field of the mass, that is located under the respective radius, not about the space density inside atomic nucleus, nucleon or similar mass particle. In chapter 4.8, respecting the postulate of the constancy of the speed of light, we have derived that the relative speed, with respect to the chosen frame, induces the variation of the space density inside the gravitational field of the mass particle, giving rise to the mass increase (relativity effect). In chapter 5.3.4.2 however, we have derived that any gravitational object is able to influence the space density inside the mass particle in case only, if the object reaches, by its own gravitational field, in position of the mass particle, the space density higher than it is induced by the mass particle itself. This effect may occur in case only, when the gravitational field of the influencing object is extremely intensive, since the space density on the proton's surface reaches the value about

and we can derive that, to reach such high space density in proton, the object of the Earth mass (concentrated in a very small volume) should be situated in a distance from the proton not longer than

This is why, deriving the mass distribution in gravitational object, we need not take into consideration the relativistic mass increase (decrease) due to increase (decrease) of the space density in the gravitational field inside the gravitational object.

Ignoring both the relativistic mass increase due to speed, and the relativistic increase (decrease) due tu change of the space density, the mass distribution under surface of the gravitational object may be considered depending on the mass density dustribution, and on this feature of the space density that is responsible for the mass distribution Note:observing from the chosen reference frame, like from the earth surface, the constant cube vulume, not breathing with the space, we would have to find out that the quantity of mass in this cube differs depending on the position of the cube inside gravitational object, in consequence of the change of the space density.

Between radia rlmx and rlmx+drlmx the mass differential dm may be expected :
(521)

where
stands for the frame intrinsic space density (measured by the instruments breathing with spacetime structure in a respective position inside gravitational object)
In general, we can tell, that under the common radius rlmx the mass m is situated, to which the gravitational radius rx is related according to equation (169):

The gravitational space density :
(522)

Solving the equations (521) and (522) we receive the common equation of the mass distribution vs radius inside gravitational objects :
(523)

5.3.5.1 Mass distribution of gravitational object created of the mass of constant frame intrinsic mass density.

At = o =const
At constant frame intrinsic space density the differential equation (523) gives the solution :
(524)

where
At the boundary condition :
- the equation (524) becomes :
(526)

Substituting for m from equation (526) to equation (522), we recwive the equation defining the gravitational space density :
(527)

Applying the equation (526) we can express the average mass density under respective radius :
(528)

The examination of the equations (526), (527) and (528) leads to the following conclusions :
• The gravitational object can be created of the mass of any frame intrinsic mass density. The object of the mass Mo may be created of the low (high) mass density, since the low (high) mass density is compensated by high (low)space density to reach the requested average mass density.
Note The decision whether the object would be able to exist, made of any frame intrinsic mass density, depends on conditions of stability of the object (like gravitational acceleration,pressure, themperature and many others), and is not discussed in this chapter.
• At k = 3, or, acc. to equation (525) at the frame intrinsic mass density
(529)
• The mass distribution becomes directly proportional to the third power of the radius. See Fig21.

Fig21
• The gravitational space density becomes constant, not depending on radius :
(530)
• The average mass density under radius becomes constant, not depending on the radius :
(531)
• At k < 3, or, acc. to equation (525) at the frame intrinsic mass density
(532)
• The mass becomes directly proportional to the power lower than 3 of the radius.See Fig22

Fig22
• The space density becomes increasing with the decreasing radius. See Fig23.

Fig23
• The average mass density under radius becomes increasing with the decreasing radius. See Fig24.

Fig24
• At k > 3, or, acc. to equation (525) at the frame intrinsic mass density
(533)
• The mass becomes directly proportional to the power higher than 3 of the radius.See Fig25
Fig25
• The space density becomes decreasing with the decreasing radius. See Fig26.
Fig26
• The average mass density under radius becomes decreasing with the decreasing radius. See Fig27.
Fig27

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