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Pressure Distribution along the Radius of the Massive Astronomical Bodies

Author: Nikolay Borislavov Kitov

Content

1.0.0 Introduction

2.0.0 Setup of the problem

3.0.0 Resolve of the problem

4.0.0 Investigations of the result in connection with the Classic Law of Hydrostatic

Balance

5.0.0 Result in integral mode

6.0.0 Investigation of the result in integral mode

7.0.0 Conclusions

8.0.0 Energy of the gravitational body.

* Appendixes

** Bibliography

*** Abbreviations

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1.0.0 Introduction

In the space are seen a row massive bodies – stars, planets, big satellites.

They possess 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 possess 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. 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 toward body’s surface reaches form of the Classic Law of

Hydrostatic Balance.

The finding of such general law is the aim of this work.

To content

2.0.0 Setup of the problem.

2.0.0 Let’s examine the following physic’ object:

Spherical laminated body with volume , consisting of layers, limited by concentric

surfaces with radii respectively of the above limit, and the of the lower limit. Every

layer with volume 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. Has not outside influence.

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Every element is gravitating, and is giving its contribution into the common field of the

corresponding layer , and into the field of the body. In the same time, it is feeling

gravitational impact of the other elements of the body, which integral expression is the

weight of the element . Gravitational interaction is describing by the Newton’s

law, and its 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 weights 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:

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- Mass of the spherical laminated body with radius

- Outer radius of the body

- Arbitrary layer

- Mass of the sphere with radius

- Volume of the layer

- Upper limit of the layer

- Under limit of the layer

- Element of the layer

- Distance of the from the centre of the body

- Density of the layer , and the element

- Average density in the sphere with radius

- Mass of the part of the layer between spheres with radii and

- Gravitational constant

Our goals: Weight of the element

Weight of the layer

Stress and Pressure over the under limit of the layer .

Considerations will be done in spherical coordinates.

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3.0.0 Resolve of the problem.

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The weight is the sum of the components and caused by attraction of the

masses and over .

Let’s express

Spherical coordinates in the frames of the layer

/See APP 1.A/

The weight of the layer will be:

This expression has a sense only in the case , which is the reality for the bodies

investigated with seismic.

The proposed integral is trivial and its result is: /See APP 1.B/

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The layer was chosen in random way /arbitrarily/.

The same kind will be the expression for the weight of 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 limit upon consideration.

The pressure over under limit of the layer will be:

And finally:

This is the basic result for our goals.

Pre boundary case To Conclusions To Example To Content

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4.0.0 Investigations of the result. Connection with classic LHB.

4.1.0 Checking with dimensions.

Obtained result possesses dimension of the pressure.

4.2.0 Boundary behavior toward : /See APP 2.A/

When every another , laying between and , will tend to . Thus

expressions 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 / /

Let’s see what behavior reaches this equation when the layer is very thin, e.g.

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In this particular case the pressure inclines to 0, too. But in before boundary case, when but still , the expression (3) reaches

mode known like LHB. That means, the formula (3) encloses into it LHB like its boundary approximation. 4.3.0 Boundary behavior toward /See APP 2.B/

Here is the big difference from the new expression and the known law!

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4.4.0 The case of a homogenous body. /See APP 3/

Where:

and To 7.0.0. To Content

5.0.0 The result in integral mode. Let’s examine the case, when the body is consisting from unlimited number very thin layers.

In this case: /See APP 4/

And:

In the last formulae, is playing role of the running coordinate, while and are

boundaries of the particular investigation. Respectively:

is analog of the

is analog of the

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For every real body, is constant.

To Content

6.0.0 Investigations of the result in integral mode.

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

7.0.0 Conclusions.

Formulae (3), (4), (7) are generalizations of the LHB.

They are expressions of the central-symmetry force fields.

The LHB is the boundary case (asymptote) of the above formulae.

It is expression of the plane-parallel force fields.

The classic LHB is permissible only in the case .

This is the anthropomorphic experience of the human beings.

Fluidity is a condition of the substance in which gravitational interaction is dominating.

To Content

8.0.0 Energy of the gravitational body.

The result for the gravitational field is obtained after integration the pressure through full

volume of the body:

/See APP 5.A/

For a homogenous body:

/See APP 5.B/

This formula, combined with the Principle of Dirichlet, gives us a physical basis for the

bodies’ expanding.

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In the appendix 5, I have resolved the case only for the homogenous body. But:

1. For the most bodies gravitational energy is dominating.

2. Gravitational force is conservative! Full energy is one and the same for all models!

Example: For the Earth:

by Bullen* by (3)

0 6371 0,000 0,000

33 6338 0,009 0,009

80 6291 0,025 0,025

200 6171 0,071 0,067

400 5971 0,149 0,151

800 5571 0,313 0,337

1200 5171 0,485 0,562

2400 3971 1,056 1,541

2900 3471 1,330 2,270

3600 2771 2,020 4,350

4800 1571 3,060 15,220

5400 971 3,530 ** 40,420

6000 371 3,850 ** 277,300

6371 0 3,920 **

* The data are taken from the book “Таблицы Физических Величин” Edition Москва „Атомиздат” 1976. They are calculated according to Bullen’s second attempt 1936 year. ** These data have a sense only like numerical calculations. Due to a complicated condition of the substance in this depth, every calculation is under question.

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Appendices

Appendix 1: Integral for the weight of the single layer:

A: Weight of the mass element.

Let’s express

Spherical coordinates in the frames of the layer

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Return to the Main Text To Content

B: Integral for the weight of the single layer

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Appendix 2: Boundary Investigations

A: Toward the surface of the body:

When every another , too due to the condition .

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B: Toward the center of the body

For particular , is a real positive number.

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Appendix 3: Homogenous body

For a homogenous 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 4: Transition towards integral mode.

1)

All articles with are omitted.

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Here should play role of the running coordinate .

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Appendix 5: Expressions for the energies

A: Expression for the gravitational energy:

Return to the Main Text To Content

B: Expressions for the Homogenous body:

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Bibliography

1. Хаббард,У. (1987) Внутреннее строение планет. Москва, Мир

Translating from:

Hubbard, William B. (1984 ) Planetary interiors. Van Nostrand Reinhold Company

2. Гибсон, Э. (1977) Спокойное солнце. Москва, Мир

Translating from;

Gibson, Edward G. (1973) The quiet Sun. Washington, Manned Spacecraft Center,

Houston, Texas, Scientific and Technical Information Office, National Aeronautics

and Space Administration

3. Сюняев, Р.А. et al. (1986) Физика космоса: маленькая энциклопедия. Москва,

Советская энциклопедия.

4. Кикоин, И.К. et al. (1976) Таблицы физических величин. Москва, Атомиздат

Table 47.1, p. 993

5. Златев, Иван и Николов, Ангел. (1981) Теоретична механика.

София, Наука и изкуство

6. Уеда, С. (1980) Новы взгляд на Землю. Москва, Мир

Translating from:

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Uyeda, Seya. (1978 ) The new view of the Earth: moving continents and moving

oceans. San Francisco, W.H. Freeman and Company

7. Шкловски, И.С. (1979) Звезди – тяхното раждане, живот и смърт.

София, Наука и изкуство

Translating from:

Шкловский, И.С. (1975) Звезды – их рождение, жизнь, смерть. Москва, Наука

8. Шкловский, И.С. (1976) Сверхновые звезды. Москва, Наука

9. Петков, Ив. Н. (1968) Обща геофизика. София, Наука и изкуство

10. Мельхиор, П. (1975) Физика и динамика планет. Москва, Мир

Translating from:

Melchior, Paul (1975) Physique et dinamique planetaires.

Louven/Bruxelles, Vander-editeur

11. http://imagine.gsfc.nasa.gov/docs/science/know_l2/stars.html

Return to the Main Text

12. http://www.science.org.au/fellows/memoirs/bullen.html

Return to the Main Text

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Abbreviations

LHB – Law of Hydrostatic Balance

CLHB – Classic Law of Hydrostatic Balance

GLHB – General Law of Hydrostatic Balance