Derivation of | |
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by Barry Mingst and Paul Stowe | |
Web Publication by Mountain Man Graphics, Australia | |
Derivation of Newtonian Gravitation from LeSage's Attenuation Concept |
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A long time ago, Lord Kelvin (W. Thompson), Lorentz, Maxwell, and Hemholtz recognized that the behavior of matter had characteristics similar to vortex ring structures in a fluid (the atomic vortex hypothesis). This concept was abandoned in the early 1900's. This abandonment was more philosophical than substantive with the real problem being the math describing the model was, "at the time", intractable. Must more success was being obtained by QM methods. This same model rears up again in modern physics in the form of the mathematical topology of string/super string theory as well as in superconductivity and superfluidity. Penrose's twistor is a vortex ring, as is a magnetic field. It is interesting to note that vortex rings can sustain transverse vibrations (analogous to guitar string vibration), indeed Kelvin proved mathematically that linear disturbances in a saturated 3D vortex fluid (he termed a vortex sponge) would produce propagation of pure transverse waves identical to the equations and properties that describe the propagation of light through space. It was this relationship as well as many others that caused this hypothesis to be considered seriously. It also is interesting to note that Maxwell used this conceptual model as the basis for his derivation of the EM relationships.
The basic gravitational relationship as identified presumes that the ZPE (aether) interacts weakly with the vortex rings that constitute matter. The "macro" effect of a spherical body immersed in this field is derived herein. The derivation below is for any point of evaluation (P) that is external to a spherical material body.
Note that the distance from P to the center of the body will be denoted as R. The radius of the body is denoted as r_o. The tangential angle is denoted as a_o. Any path traversing the sphere may be represented by line. The traverse distance within the sphere along this line is denoted as x. The relationship between x, r_o, and r is given by geometry as:
(x / 2) = (r_o + r)(r_o - r) = (r_o^2 - r^2) {A.1}
The (momentum) flux at point P along any such line will be affected by the interaction of the field within the sphere. This interaction will be the transfer of of some momentum/energy from the field into the ring structures. This general interaction gives rise to a standard thin-shield reduction equation of:
Fee = Fee_o e^(-ux) {A.2}
where Fee is the flux after interaction, Fee_o is the initial flux, u is the attenuation (Transfer) coefficient (in units of inverse length [u = rho u_s, where rho is density, and u_s is mass attenuation coefficient]), and x is the travel length in the sphere along any arbitrary line.
Because of the possibility of scattering (multiple interactions in shields of sufficient thickness), this equation is expanded using a "buildup" term, B. This buildup term corrects the equation for multiple scattering events that will contribute to point P that were not along the original path of the line. The buildup term will depend on the relative importance of each of the three possible interaction modes (removal, scattering, and slowing) in the body, the shape of the body, the size of the body, and the distance of the body from point P. The corrected general removal equation is:
Fee = Fee_o Be^(-ux) {A.3}
Determining the overall directional momentum flux (or current) due to the body at point P, one notes that (in an otherwise isotropic medium) the flux from all directions is identical except for the paths that traverse the body. These interaction paths have their momentum reduced according to the flux attenuation equation. In the isotropic medium, for each of the interaction lines there is another from exactly the opposite direction that has no interaction. The net flux at point P is then given as the sum (integral) of the momentum of all paths from the left and the right of point P. All paths outside of angle a_o are matched exactly by its opposite. The net contribution of paths outside of angle a_o is therefore zero. The contribution of the attenuation within angle a_o can be determined by the difference between the momentum from the left and the momentum from the right:
Delta Fee = (Fee_o - Fee) dOmega = (Fee_o - Fee)(dr / R)((r dTheta) / R) {A.4}
The sum of all such paths is then given by the integral:
Integral [Fee dOmega] = Double Integral [(1 / R^2)(Fee_o - Fee)r dr dTheta {A.5}
which yields:
Fee_net = 1/R^2 Integral(0-2pi) dFee Integral(0-r_o) Fee_o(1-B(-ux)e^-ux)r dr {A.6}
and resolves to:
Fee_net = 2pi Fee_o/R^2 Integral (0-r_o) (1-B(-ux)e^-ux)r dr {A.7}
Noting that x may be replaced by:
L = Sqrt(r_o^2 - r^2)
x = 2L {A.8}
provides the general solution:
Fee_net = 2pi Fee_o/R^2 Integral (0-r_o) (1-B(-2uL)e^-2uL)r dr {A.9}
Fee_net = 2pi Fee_o/R^2 Integral (0-r_o) (1-(1-2uL))r dr {A.10}
simplifying further:
Fee_net = 2pi Fee_o/R^2 Integral (0-r_o) 2uLr dr {A.11}
and:
Fee_net = u4pi Fee_o/R^2 Integral (0-r_o) Lr dr {A.12}
integrating the above equation gives:
Fee_net = u4pi Fee_o/R^2 [L^(3/2)/3] {A.13}
which resolves to:
Fee_net = u4pi Fee_o r^3/3R^2 {A.14}
This equation may be further rearranged to give:
Fee_net = Fee_o/R^2 [u4pir^3/3] {A.15}
It can be seen from the above equation, that for a weak solution, that the bracket term is an alternate form of mass derivation (volume, mass density, and a mass interaction coefficient u_s [u_s = u / mass density]).
This equation can be related to total mass M as:
Fee_net = (Fee_o u_s/R^2)M
Fee_net = 2pi Fee_o/R^2 Integral (0-r_o) r dr {A.17}
integrating:
Fee_net = 2pi Fee_o r_o^2/2R^2 = pi Fee_o r_o^2/R^2 {A.18}
This equation may be rearranged to give:
Fee_net = Fee_o/R^2 [pi r_o^2] {A.19}
where r_o is less than or equal to R.
It can be seen from the above equation that for the strong solution, the "mass" of matter (the field momentum to matter interaction rate) is not important. The gravitational interaction is only proportional to the cross-sectional area seen at point P. The matter mass of the body is irrelevant. Thus there is a maximum gravitational field (the difference between the normal ZPE (aether) field momentum flux on one side and nothing on the other).
If the second body is a weakly interacting body, then it will interact with the net momentum flux (and experience an acceleration) proportional to its interaction constant and volume. The net gravitational interaction (force) will therefore be:
F = Fee_o/R^2 [u_1 4pi r_1^3/3][u_2 4pi r_2^3/3] {A.20}
Since:
u = u_s(rho) {A.21}
this equation resolves to:
F = (Fee_o u_s^2)M_1M_2/R^2 {A.22}
which is the standard gravitational force equation. The experimentally derived constant G can be seen to be:
G = Fee_o u_s^2 {A.23}
Thus we find that A.22 is the familar Newtonian Equation
F = GMm/R^2
From the above equation we see that for the standard form of gravitational force equation, the gravitational force constant is simply the product of the general ZPE (aether) momentum flux and the square of the attenuation coefficient of matter rings with the ZPE (aether) field. The gravitational constant thus contains both the ZPE (aether) momentum flux and its rate of interaction with the vortex ring structures.
In any isotropic particle field, the momentum flux (momentum per unit area per unit time) may be found by:
Fee = rho v^2/3 {A.24}
where "rho" is the density of the medium and
Fee_o = eps_o v^2/3 = eps_o c^2 {A.25}
and substituting equation A.25 into equation A.23 gives:
G = eps_o c^2 u_s^2 {A.26}
a_1 = Fee_o u_s^2 M_2/R^2 = Fee_net u_s {A.27}
Using equation A.23 also gives:
a = eps_o u_s^2 M/R^2 {A.28}
Note that in the case of a weakly interacting body the acceleration resulting from this type of field current is not dependent on the mass of the body. Thus any such matter body responds to a field current in the same manner (regardless of its mass). The important concept here is that field current creates an acceleration independent of mass, the resulting "force" is only a by product of this acceleration. This clearly demonstrates the derivative reason for the postulated "principle of equivalence".
f_d = f_in - f_out
and for the weak solution we find that:
Where u in this case is a linear attenuation coefficient and L is the "effective" travel length through the material body
f_out = f_in[1-2uL]
Thus
f_d = f_in[2uL]
Gravitationally we have related this 2uL to 2GM/c^2r_o^2.
Thus:
f_d = f_in[2GM/c^2r_o]
We see from the above equation that everything is constant except for M and r_o (M is the mass of the body and r_o is the physical radius). Thus we can simply write:
f_d = KM/r_o
where K contains all the constant terms.
So what is K? (Big step here derivation not shown!)
K = (aGe^2/2pi h)/(4pi) = 2.41E-19 m/sec^3
[K can also be back calculated based on a known output {such as the measured anomalous net output of a large astronomical body}]
Where:
The full equation using a lumped heat response model is:
f_d = (KM/r_o^2)(1 - e^-Ht)
and H is:
H = [U(4pi r_o^2)]/[MCp]
Where: U is the OVERALL heat transfer coefficient Cp is the heat capacity at constant pressure t is total time of existence in seconds Here are the results of this equation for known astronomical bodies: Body Mass Radius Net Flux Predicted Flux Difference Moon 7.4E+22 1.74E+06 0.01 .01 -2.5% Earth 6.0E+24 6.37E+06 0.06 .034* ? Jupiter 1.9E+27 7.18E+07 6.60 6.40 3.4% Saturn 5.6E+26 6.03E+07 2.30 2.20 2.7% Uranus 8.6E+25 2.67E+07 0.76 0.78 -2.1% Neptune 1.0E+26 2.49E+07 0.76 0.97 -27.3% * the earth IS NOT in thermal equilibrium! It emits .06 watts/m^2 thus U = .06/delta T. Given delta T is on the order of 10,000 degrees K, U = .06/10,000 or 6.0E-06. H is therefore, H = [(6.0E-06)(4pi)(6.374E+06)^2 ]/[(6.0E+24)(500)] H = 1.02E-18 (1/sec) t = (5E+09)(365.25)(24)(3600) = 1.58E+17 sec Ht = .1612 (1-e^-.1612) = (.149)(.227) = .034 Due to the method employed, this resulting number has a rather high uncertainty and total power is the integration of this flux deposition over the total surface area (which is simply 4pi r_o^2) thus: P_d = 4pi r_o^2 (KM/r_o) = 4pi KMr_o and 4pi K = aGe^2/2pi hHere is the punch line. The individual ZPE (aether) quantum impact the vortex rings, exciting the rings to vibrate just like a guitar pick striking a guitar string (good old string/superstring theory). This "Brownian" excitation WILL create resonances within the masses of rings.
In a perfect medium (bulk viscosity approaches 0), the relationship of fluid particle speed (v) to bulk transverse wave speed (c) is: v = sqrt(3)c where the Sqrt(3) is simply the geometric transform into three equal orthogonal components.
Now consider a small volume element that contains two such vortex rings, one must realize that a vibrational change in one ring couples to the other. The vibration dilates/contracts the ring (resulting in volume/area changes of the ring) and will result in a responding sympathetic distortion of the second ring. This is a result of the fact that volume of the element will tend to remain constant. The resulting coupling factor will consist of square root of 3 (1.73...[the relationship of particle speed to wave speed]), the geometry of the ring (4 PI^2 [the area geometry of a toroidial ring]), and 2 because it is the interaction of the rings. This would lead to numeric constant that would be definitive of these interactions. This term is the inverse of the fine structure constant alpha (a). Alpha (a) then is simply:
1 a = ------------------------- 2[Sqrt(3)(4 PI^2)] which is 1 over 136.76... However. a more basic coupling would be simply 2a = 1 / (Sqrt(3)4PI^2).This ends phase one, the explanation of the basic cause of gravitation,
Barry Mingst and Paul Stowe