“One may not consider that aa shell behaves as if all the mass on the shell were concentrated at the shell center.”
Why not? This is simply a geometrical approximation, which becomes more and more relevant, as the distance increases. At the very large distance scale such shell would behave like pin-point object with respect to gravity spreading. Try to draw image below at scale and you'll see, why Newton could use it for derivation of gravity force between Earth and Moon without excessive simplification.
Earth diameter 12756 km reduce to 1
Moon Diameter 3476 km reduced to 3476/12156 = .28
The earth-moon mean distance 384403 / 12156 = 31.622
Moon orbit eccentricity - .0549
Earth constant gravity 980.665 cm/sec^2
Moon 162 cm/sec^2
These numbers will suffer under any approximation
The solar system is not, in general, orbitally planar - the solar system entities all move in a helical trajectory with the sun the axis of the helix. As such interplanetary activity would appear to deserve treatment as the physics is best described - why approximate?
See Flanders take on the speed of gravity forces - effectively infinite.
In a state of pure speculation could Mercury’s anomalous orbit error be corrected with a proper law of gravity applied? Relativity theory and classic physics models both corrected the problem - I suspect fudging on the math and physics by some, but this is only a speculation.
http://metaresearch.org/cosmology/speed_of_gravity.asp [img]
http://www.aetherwavetheory.info/images/physics/gravity/shell_theorem.gif[/img]
Well at least you agree with me that the shell center is not the center of mass attraction between a test particle and the shell. What do you object to by using an accurate system , as opposed to, an 'approximation'? I understand completely your position here and I understand we are, or may be, nit picking rhetorically to a certain degree.
..the shell theorem is described as a "proof" that ..
Nope, it's a geometrical approximation (..a quite trivial and apparent, IMO..), which Newton has used for estimation of gravity force in spherical geometry - no less, no more. In more detailed analysis we should use a more exact approximation and Mr. Newton realized it too a quite well. For more exact derivation of gravity force near sphere please check the article
here and
here. [/quote]
These are the concluding remarks of your links - the language is certainly not in terms of "approximations” or in terms of “averaging”.
FIRST LINK - "This is the desired goal, to show that the force from a thin spherical shell is exactly the same force as if the entire mass M were concentrated at the center of the sphere! Physically, this is a very important result because any spherically symmetric mass distribution can be build up as a series of such shells. This proves that the force from any spherically symmetric mass distribution on a mass outside its radius is the same as if the total mass were a point mass concentrated at the center of the sphere."
I say that the force expression, the result F = gMm/D^2
of the integration in the link merely states the forece on a test mass by a sphere located at a distance D from M.
SECOND LINK - "The net gravitational force on a point mass inside a spherical shell of mass is identically zero! Physically, this is a very important result because any spherically symmetric mass distribution outside the position of the test mass m can be build up as a series of such shells. This proves that the force from any spherically symmetric mass distribution on a mass inside its radius is zero. If a given mass m is inside a spherically symmetric distribution of mass, that part of the mass outside its radius does not contribute to the net force on it. "
As I claim the integration is not complete and we must all recognize that an actual "shell" is an unobserved abstraction only and is being used to describe proof of a flawed system - flawed in the sense that I have been describing.
