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Hydrostatic Balance General Equation

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					          Equation expressing Law of Hydrostatic Balance is one of the 5
                  basic equations in studying stars and planets.

Now it is known in many forms, and they are used depending of the goals which are
pursued, and the level of studies. Some of these forms are:

                                                   (1)

                                                                                        (1.1)



                                                                                           (1.2)



                                                                                        (1.3)



                 where          is the mass in sphere with radius               (1.4)



                    where      is potential of the conservative force   (1.5)


                                                                                (1.6)


                   basic form                                                              (1.7)

and so on ....
                    (1. ...)

All these forms reflect plane parallel force fields.
Gravitational force fields in the planets and stars are central-symmetrical (centripetal).

Some of given formulae should be expressed in spherical coordinates, but this is not
enough. One parallelogram should be described through spherical coordinates, but it rests
parallelogram. To be in concordance with spherical geometry of the gravitational force
fields, equation for Hydrostatic Balance must to be drown in spherical coordinates from
the beginning.

Spherical form of the equation has to be more general than the plane parallel form, due to, the
plane segments in the gravitational bodies are enough small fragments of the whole.
Thus, the spherical form has to encompass, into it, the plane parallel form like its boundary
approximation or asymptote.
The spherical form has to turn in plane parallel form through limit transition:



Also spherical form should to reflect laminated structure of the astronomical bodies, as this
is proven for the bodies measured directly and is most probable for the bodies with only
models.

Following equation renders these conditions:

                                             (2)



where:
           and

-      ,   are the radii of the lower limits of the layers
-          is the outer radius of the body
-          is the density of the    layer
-          is the average density in the sphere with a radius
-          is the gravitational constant

This expression is suitable for computer work and it is in accordance with the
models of the Earth and the Moon received by seismology.

For simplified model of “homogeneous planets/stars” the expression (2)
 acquires the following form:



(2a)


It is easily ascertain that the expressions (2), (2a) have dimensions of a pressure.
Also in boundary transition             the proposed equations turn into well known form (1.7).

Using particular form of the equation, instead general form, leads to misleads.
Using the general form of the equation does not permit such misleads.




Also, when we use general form of the equation for the massive space bodies:


                       if the mass of the body
Thus should be concluded that, for the massive space bodies expanding is obligatory.
This follows from Principle of Dirichlet, according to which each system aims toward stable
equilibrium at minimum potential energy.
This is the physics basis for the bodies enlarging.

Author: Nikolay Kitov
nikolakitov@gmail.com 

				
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posted:11/17/2012
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Description: List of now known forms of the Law of Hydrostating balance. Generalisation of the Law for the central-symmeric force fields.