5.3.5.2 Mass distribution of gravitational object created of several layers of different frame intrinsic mass density.

1

on its surface, between radia R_o - R₁

2

between radia R₁ - R₂

3

between radia R₂ - R₃

n

between radia R_(n-1) - R_n

• 1st layer

• 	(534)

  
  Solution for R₁ < r_lmx < R_o
  

  	(535)

  
  where
  

  	(536)

  
  Applying the equation (535) we can derive :
• The space density:
  

  	(535a)

  
  The average mass density:
  

  	(535b)

• 2nd layer

• 	(537)

  
  Solution for R₂ < r_lmx < R₁
  

  	(538)

  
  where
  

  	(539)

  
  Applying the equation (538) we can derive :
• The space density:
  

  	(538a)

  
  The average mass density:
  

  	(538b)

• 3rd layer

• 	(540)

  
  Solution for R₃ < r_lmx < R₂
  

  	(541)

  
  where
  

  	(542)

  
  Applying the equation (541) we can derive :
• The space density:
  

  	(541a)

  
  The average mass density:
  

  	(541b)

• n-th layer

• 	(543)

  
  Solution for R_n < r_lmx< R_(n-1)
  

  	(544)

  
  where
  

  	(545)

  
  Applying the equation (544) we can derive :
• The space density:
  

  	(544a)

  
  The average mass density:
  

  	(544b)

• The example
• To calculate the mass distribution and related parameters under Earth surface, on condition that its outer radius R_o = 6,378*10⁶ m, its total mass M_o = 5,977*10²⁴ kg, and, that the mass is created of the following 3 layers :
  1st layer
  Radia : R_o = 6,378*10⁶ m, R₁ = 6,377*10⁶ m
  Frame intrinsic mass density: 1000 kg/m³
  2nd layer
  Radia : R₁ = 6,377*10⁶ m, R₂ = 6,378*10³ m
  Frame intrinsic mass density: 2700 kg/m³
  3rd layer
  Radia : R₂ = 6,377*10³ m, R₃ = 0 m
  Frame intrinsic mass density: 7870 kg/m³
  
  Solution
  Applying the derived equations, the mass distribution, the space density, and the average mass density under radius were calculated in each layer :
  1st layer (See Fig29)
  
  
  
  

  Fig29	Mass distribution, space density and average mass density under radius in 1st layer

  
  
  2nd layer (See Fig30)
  
  
  
  

  Fig30	Mass distribution, space density and average mass density under radius in 2nd layer

  
  
  
  
  3rd layer (See Fig31)
  
  
  
  

  Fig31	Mass distribution, space density and average mass density under radius in 3rd layer

  
  
  
  
  We can see from Fig29, Fig30 and Fig31, that the space density is increasing with the decreasing radius in the upper two layers (1st and 2nd layer), causing, that, in spite of fact that the frame intrinsic density in upper two layers is much less than in the 3rd (inner) layer, the most of the Earth mass is situated in upper two layers. In the 3rd layer the space density becomes decreasing with decreasing radius. At the radius approaching to zero, the space density is comming to zero as well.
  
  Note
  At this low space density must exist the radius r_lmx = R_cr, representing the radius of one spacetime cell (much bigger than the cell on Earth surface, when measured in units of the Earth surface frame). Summarizing the mass between radia R_o and R_cr, we have to come to conclusion, that the summarized mass M_s is less than the total mass M_t of the gravitational object. Under these circumstances we would have to admit, that the mass amount of M_t - M_s must be situated in this very last spacetime cell, or, more likely, in a different zone of the spacetime continuum, spreading from the centre of gravity out to infinity. To determine the radius R_cr (of course it must be very small with respect to R_o, and depending on the frame intrinsic mass density), however, we should have to know the frame intrinsic radius of the spacetime cell.
  
  The average mass density under radius reaches its maximum 2,087*10⁸ kg/m³ at the radius R₂ (on spherical area between the 2nd and 3rd layers). This very high mass concentration arises both due to high frame intrinsic mass density and high space density.