# Pressure Distribution Along the Radius of Massive Gravitational Bodies

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1.0.0.   Introduction
2.0.0.   Setup of the problem
3.0.0.   Resolve of the problem
4.0.0.   Investigation of the result
Connection with the Classic Law of Hydrostatic Balance
5.0.0.   Integral mode of the result
Investigations of the result in integral mode
6.0.0.   Conclusions
7.0.0.   Energy of the gravitational bodies
8.0.0.   Appendices
8.1.0.   Abbreviations
8.2.0.   Useful Sites
9.0.0.   Bibliography

1.0.0. Introduction

In the space are seen a row massive bodies - stars, planets, big satellites.
They posses following features:
- Their form is spherical in first approximation.
- Their real form is of the kind of the fluid, rotating in field with central symmetry.
- Their structure is laminated.
- They posses theirs own source of the energy/heat, which is growing with mass of the
bodies in most cases.
Today in the studying of these objects, the Law of Hydrostatic Balance /LHB/ is used,
and it is written in many forms depending from the level and the aim of the particular investigation.
This law is checked on the Earth surface and it is unconditionally right. In this case, and
in the frames of the technical goals, the gravitational field has plane-parallel symmetry.
In the general case gravitational fields have centripetal/central symmetry.

This presupposes availability of the General Law of Hydrostatic Balance /GLHB/,
which is rendering of account the central symmetry of the gravitational fields,
and which in its boundary approximation towards body's surface reaches the
form of the Classic law of Hydrostatic Balance /CLHB/.

The finding of such law is the aim of this work/attempt.

2.0.0. Setup of the problem.
2.1.0. Let's examine the following physics object:

Spherical body with volume    , consisting of     layers, limited by concentric surfaces with
radii respectively    of the above limit, and      of the lower limit. Every layer is with its
own volume      , and is consisting of elements      with density      - constant for the
layer.
Elements are in compact packing - i.e. in the layer have not empty rooms/caverns.
Interaction between elements is only gravitational. All rest interactions are set at naught.
Have not outside influence.

Every element         is gravitating and is giving its contribution into the common field of
the corresponding layer , and into the full field of the body. In the same time it is
feeling gravitational impact from the other elements of the body, which integral expression
is the weight      of this element. Gravitational interaction is describing by the Newton's
law and it's consequences.

The sum of the weights of the elements /in the frames of the layer/ will give the weight
of the full layer    .
The sum of the all layers, laying over spherical surface with radius        and area   will
form the stress over this surface.

The ratio             will form the pressure over this surface.

Let's find            .
2.2.0. Draft and brief writing.

Fig. 1 where:

- Mass of the spherical laminated body

- Outer radius of the body

- Arbitrary layer

- Mas of the sphere with radius

- Radius of the upper limit of the layer

- Radius of the under limit of the layer

- mass element of the layer

- distance from the body's center of the

- Density of the element        , respectively of the layer

- Average density in the sphere with radius
- mass of the part of the layer         between spheres with radii    and

- Gravitational constant

The goals: Weight             of the element
Weight           of the layer
Stress           and pressure         over under limit of the layer

Considerations will be done in spherical coordinates.
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3.0.0. Resolve of the problem

The weight          is the sum of two components             and        caused by attraction of the
masses        and      over       :

Let's express                 :

spherical coordinates in the frames of the layer

and

See Appendix 1

This expression has a sense only in the case                       , which is reality for the bodies
investigated with seismic.

The weight of the full layer will be:

The proposed integral is trivial, and its result is:                      See Appendix 2

The layer       was chosen in random way (arbitrarily).
The same kind will be the expression of the weight                   for the every another
layer .
All layers, from most outer to the layer      including, lay over under limit of the
layer .

Consequently, the sum of the weights of all layers above            , and it including,
will form the stress over the surface upon consideration.

The pressure over under limit of the layer   will be:

And finally:

This is the basic result for our goals                                     Back to Content

4.0.0. Investigations of the result. Connection with Law of Hydrostatic Balance.

4.1.0. Checking with dimensions

Obtained result possesses dimension of the pressure.

4.2.0. Boundary behavior toward                                            See Appendix 3

When           every another     , laying between       and   , will tend to   . Thus
expression in the small brackets will nullify, and with them the full sum will nullify, too.

4.2.1. Let's examine one particular case: The pressure over surface with radius        :

In this case:

Where:          is the surface density of the body
is the average density of the body

What behavior reaches this equation when the layer is very thin - e.g.                     ?
Let's see:

.    .

.    .                .   .          0

In this particular case the pressure incline to    , too.

But in before boundary case, when       , but still     , the expression
reaches mode known like Law of Hydrostatic Balance - LHB.
That means, the formula            encloses into it LHB like its boundary approximation.

4.3.0.Boundary behavior toward                                          See Appendix 4

Here is the big difference between new expression and known law!
4.4.0. The case of homogeneous body                                   See Appendix 5

where:

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5.0.0. Integral mode of the result.
Investigations of the result in integral mode

Let' examine the case, when the body is consisting from unlimited number very thin
layers. In this case:                                     See Appendix 6

All investigations, made in 4.0.0. should be repeated here.
The result is the same.

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6.0.0. Conclusions

Formulas              are generalizations of the LHB.
They are expressions of the central-symmetry force fields.

The LHB is boundary case /asymptote/ of the above formulas.
It is expression of the plane-parallel force fields.

The LHB is permissible only in the case            .
This is anthropomorphic experience of the human beings.
Fluidity is a condition of the substance in which gravitational interaction is dominating.

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7.0.0. Energy of the gravitational bodies
After integration the pressure field, over full volume of the body, should be
found gravitational energy of the body.

Integration of the       over volume of the proposed body gives result:

See appendix 7

For a homogeneous body:

This formula combined with Principle of Dirichlet gives a physics basis for the
bodies expanding.

In Appendix 7.2, I have resolved the case only for the homogeneous body. But:
- For the most bodies gravitational energy is dominating.
- Gravitational force is conservative
The distribution of the density along the radius should change
distribution of the energy along the radius. Full energy of particular
body is one and the same scalar.

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Nikolay Borislavov Kitov
nikolakitov@gmail.com

8.0.0. Appendices

Appendix 1:    Weight of the element from arbitrary layer

spherical coordinates in the frames of the layer

,
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Appendix 2:   Integral for the weight of the single layer
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Appendix 3:      Boundary behavior toward

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Appendix 4:      Boundary behavior toward

For particular   , the sum represents real positive number.

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Appendix 5:    The case of homogeneous body

For a homogeneous body it is no matter how it will be divided by layers.
Consequently     should play role of the running coordinate.

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Appendix 6: Transition toward integral mode of the result

1.

2.

3.
All articles with       are omitted.

Here     should play role of the running coordinate     .           Back to main text

Appendix 7:         Expressions for the gravitational energy.

7.1    General case

The integration over      and    is as the same as in the Appendix 2 .

Back to main text
7.2    Case of the homogeneous body
Back to main text

Abbreviations
LHB - Law of Hydrostatic Balance
CLHB - Classic Law of Hydrostatic Balance
GLHB - General Law of Hydrostatic Balance

Useful Sites

About Law of Hydrostatic Balance - definitions and applying:
- Hydrostatic equilibrium or hydrostatic balance
- What is a star?
- Bullen and Earth's models
- Standard Earth Model
- Why the earth's core is hollow - the standard earth model
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Bibliography

"The Quiet Sun" Edward G. Gibson
Manned Spacecraft Center, Houston, Texas 1977

"The New View of The Earth (Moving Continents and Moving Oceans)
Seiya Uyeda , W. H. Freeman and Company , San Francisco 1980

"Physique et Dinamique Planetaires" Paul Melchior
Louven/Bruxelles Vander-Editeur 1975

"Таблицы физических величин" под редакцией академика Кикоина
Москва, Атомиздат, 1976

"Звезды - их Рождение, Жизнь, Смерть" И. С. Шкловский
Москва, Наука 1975

"Сверхновые Звезды" И. С. Шкловский
Москва, Наука, 1976

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Nikolay Borislavov Kitov
nikolakitov@gmail.com

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Description: Investigation of the behavior of the pressure in the lenths of all radius of the massive gravitational bodies.