Fluid dynamic “Vortex Gravitation” Model — derives Kepler’s Laws for Gravitation

Fluid dynamic “Vortex Gravitation” Model — derives Kepler’s Laws for Gravitation

Since the Gravitational Vortex has been modeled as a fluid dynamic “Gaseous Aether” it would make more since to derive a relation for Kepler’s Laws for Gravitation, using the mechanics and mathematics of a fluid.

Using the mathematical mechanics of Daniel Bernoulli’s Hydrodynamic Theorems, and the concepts of Steven Rado’s Aethro-Kinematic/Dynamic Ideal “Aether Gas” theory, Kepler’s Laws for Gravitation are derived; and  the results predict the “Aether Gaseous” “Gravitational Vortex” that spins or rotates, and has radial, tangential, and orthogonal components.

Below, complements of Wiki: Propagation of a transverse spherical wave in a two (2) dimensional gradient gravitational field grid.

Starting with the Bernoulli Equation for an Inertial Mass body in uniform motion in a rotating fluid is described below.

Bernoulli Total Pressure Conservation Equation

{P_{Pressure-Static}} \,\, + \,\, \frac{1}{2}{\rho_{Net}}\,{|\vec{v}|^2_{CM}} \,\,\, = \,\,\,\,  Constant   \,\,\,\,----> \,\, \frac{kg}{m\,\,s^2}

Where,

Static Pressure of Fluid{P_{Pressure-Static}}   \,\,\,\,----> \,\, \frac{kg}{m\,\,s^2}

Inertial Mass Density{\rho_{Net}} \,\,\, = \,\,\, \frac{m_{Net}}{V_{olume}} \,\,\, = \,\,\, \frac{{m_{1}} + {m_{2}} + {m_{3}} + ...... + {m_{N}}}{\frac{4\pi}{3}\,\,{r^3}}   \,\,\,\,----> \,\, \frac{kg}{m^3}

Average Rectilinear Velocity of Center of Mass of Fluid particles

 {|\vec{v}|_{CM}} \,\,-->\,\, \frac{m}{s}

{|\vec{v}|^2_{CM}} \,\, = \,\, \frac{(\displaystyle\sum_{i=1}^N {m_{i}}{v_{i}})^2}{m_{Net}^2} = \frac{ ({m_{1}}{v_{1}} + {m_{2}}{v_{2}} + {m_{3}}{v_{3}} + ...... + {m_{N}}{v_{N}})^2}{ ({m_{1}} + {m_{2}} + {m_{3}} + ...... + {m_{N}})^2 }\,\,----> \,\, \frac{m^2}{s^2}

Keeping the Inertial Mass Density ({\rho_{Net}}) constant for an isolated fluid system and taking the differential of both sides of the above equation.

Letting the Pressure change ({dP_{Pressure-Static}}) with Velocity changes ({d|\vec{v}|_{CM}}) yields.

{dP_{Pressure-Static}} \,\, + \,\, {\rho_{Net}}\,{|\vec{v}|_{CM}} \,\, {d|\vec{v}|_{CM}} \,\,\, = 0     \,\,\,\,----> \,\, \frac{kg}{m\,\,s^2}

{dP_{Pressure-Static}} \,\, = \,\, {-}{\rho_{Net}}\,{|\vec{v}|_{CM}} \,\, {d|\vec{v}|_{CM}}  \,\,\,\,----> \,\, \frac{kg}{m\,\,s^2}

Next, introducing the”Pressure Gradient” (\frac{dP_{Pressure-Static}}{dr}) of the fluid vortex flow field which is defined as the change in pressure with distance, as described by the equation below:

Pressure Gradient of Vortex Fluid Field

\frac{dP_{Pressure-Static}}{dr} \,\, = \,\, {-}{\rho_{Net}}\,{|\vec{v}|_{CM}} \,\, \frac{d|\vec{v}|_{CM}}{dr}  \,\,\,\,----> \,\, \frac{kg}{m^2\,\,s^2}

Next, introducing Circulation Force ({F_{Force}}) which is equal to the centripetal or centrifugal force which is a constant related to the change in distance with pressure which is the “Pressure Gradient” of the fluid flow field; and likewise is a constant related to the velocity changes with distance.

Below, complements of Wiki: Propagation of a transverse spherical wave in a two (2) dimensional gradient gravitational field grid.

Circulation Force of Vortex Fluid Field

{F_{Force}} \,\,\, = \,\,\, \frac{m_{Net}\,\,{|\vec{v}|^2_{CM}}}{\vec{r}} \,\,\, = \,\,\, {V_{olume}}\,\,\frac{dP_{Pressure-Static}}{dr}  \,\,\,\,----> \,\, \frac{kg\,\,m}{s^2}

{F_{Force}} \,\,\, = \,\,\, \frac{m_{Net}\,\,{|\vec{v}|^2_{CM}}}{\vec{r}} \,\,\, = \,\,\, {-}({V_{olume}} \,\, {\rho_{Net}})\,{|\vec{v}|_{CM}} \,\, \frac{d|\vec{v}|_{CM}}{dr}  \,\,\,\,----> \,\, \frac{kg\,\,m}{s^2}

The Inertial Mass and Density equal, ({m_{Net}} \,\, = \,\, {V_{olume}} \,\, {\rho_{Net}})

Circulation Force of Vortex Fluid Field

{F_{Force}} \,\,\, = \,\,\, \frac{m_{Net}\,\,{|\vec{v}|^2_{CM}}}{\vec{r}} \,\,\, = \,\,\, {-}{m_{Net}} \,{|\vec{v}|_{CM}} \,\, \frac{d|\vec{v}|_{CM}}{dr}  \,\,\,\,----> \,\, \frac{kg\,\,m}{s^2}

Dividing both sides of the above equation by the Net Inertial Mass ({m_{Net}}), and rearranging the above yields,

\frac{d\vec{r}}{\vec{r}} \,\, + \,\, \frac{d|\vec{v}|_{CM}}{|\vec{v}|_{CM}} \,\,\, = \,\,\, 0

Integrating both sides of the above equation yields,

\int{\frac{d\vec{r}}{\vec{r}}} \,\, + \,\, \int{\frac{d|\vec{v}|_{CM}}{|\vec{v}|_{CM}}} \,\,\, = \,\,\, \int{0}

Using the logarithmic law and the Integral of a constant law

\ln{\vec{r}} \,\, + \,\, \ln{|\vec{v}|_{CM}} \,\,\, = \,\, {Constant}

\ln{(\vec{r}} \,\, {|\vec{v}|_{CM})} \,\,\, = \,\, {Constant}

The above produces a new constant,

\vec{r} \,\, {|\vec{v}|_{CM}} \,\,\, = {||\vec{r}||} \,\, {||\vec{v}||_{CM}} \,\,\, = \,\, 2({Constant'})

It follows that, the constant becomes equal to Kepler’s Second Law equation which states that planets orbits in gravitational fields in “Equal Areas in Equal Times.”

Where the Constant equals – ({Constant''} \,\, = \,\, {Constant'}\sqrt{1\,\,\, - \,\,\, e^2})

Kepler’s Second Law – Gravitational Angular Momentum & Aerial Velocity – Circular

{(\frac{dA_{rea}}{dt})_{Gravity}} \,\,\, = \,\,\, \frac{L_{Angular-Momentum}}{2{m_{Mass}}} \,\, = \,\,\, {Constant'} \,\,\, = \,\,\frac{1}{2}\, {||\vec{r}||} \,\, {||\vec{v}||_{CM}} —> \frac{\, m^2}{s}

{(\frac{dA_{rea}}{dt})_{Gravity}} \,\,\, = \,\,\, \frac{1}{2}\sqrt{{K_{Gravity}}\,\,{r}} \,\,\, = \,\,\, \frac{1}{2}\sqrt{({m_{Net}\,G})\,\,{r}}  —-> \frac{m^2}{s}

Kepler’s Second Law – Gravitational Angular Momentum & Aerial Velocity – Elliptical

{(\frac{dA_{rea}}{dt})_{Ellipse-Gravity}} \,\,\, = \,\,\,\frac{L_{Angular-Elliptical}}{2{m_{Mass}}} \,\,\, = \,\,\, {Constant''} \,\,\, = \,\,\, \frac{1}{2}{||\vec{r}||} \,\, {||\vec{v}||_{CM}} \,\, \sqrt{1\,\,\, - \,\,\, e^2} —–> \frac{\, m^2}{s}

{(\frac{dA_{rea}}{dt})_{Ellipse-Gravity} \, = \, \frac{1}{2}\sqrt{{K_{Gravity}}\,\,{(r)}\, (1\,\,\, - \,\,\, e^2)}} = \, \frac{1}{2}\sqrt{({m_{Net}\,G})\,\,{(r)}\,\, (1\,\,\, - \,\,\, e^2)} —–> \frac{m^2}{s}

Therefore the Force now becomes

Circulation Force of Vortex Fluid Field

{F_{Force}} \,\,\, = \,\,\, \frac{m_{Net}\,\,{|\vec{v}|^2_{CM}}}{\vec{r}} \,\,\, = \,\,\, \frac{m_{Net}\,\, {(Constant')}^2} {\vec{r}^3}   \,\,\,\,----> \,\, \frac{kg\,\,m}{s^2}

{F_{Force}} \,\,\, = \,\,\, \frac{m_{Net}\,\,{|\vec{v}|^2_{CM}}}{\vec{r}} \,\,\, = \,\,\, \frac{m_{Net}\,\, {(\frac{dA_{rea}}{dt}_{Gravity})}^2} {\vec{r}^3}  \,\,\,\,----> \,\, \frac{kg\,\,m}{s^2}

{F_{Force}} \,\,\, = \,\,\, \frac{{m_{Net}}\,\,K_{Gravity}}{\vec{r^2}} \,\,\, = \,\,\, \frac{m^2_{Net}\,\,G}{\vec{r^2}}  \,\,\,\,----> \,\, \frac{kg\,\,m}{s^2}

Kepler’s First Law – (Space)

We also know from Kepler’s First law that the planets or bodies in gravitational fields orbit in elliptical and not circular orbits.

Semi-Major Radius – ({r})
Semi-Minor Radius – ({r_{Minor}})
elliptical eccentricity – ({e})

{r_{Minor}} = {r}\,\, \sqrt{1\,\,\, - \,\,\, e^2}  \,\,\,\,----> \,\, {m}

The Area of the Circular Orbit is given by

{A_{rea-Circle}} = \pi \,{r^2}\,\, \,\,\,\,----> \,\, {m^2}

The Area of the Elliptical Orbit is given by

{A_{rea-Ellipse}} = \pi \,{r}{r_{Minor}}\,\, = \pi \,{r^2}\,\, \sqrt{1\,\,\, - \,\,\, e^2} \,\,\,\,----> \,\, {m^2}

Kepler’s First Law & Second Law – (Time)

The Orbital Period of the fluid particles in a “Gravitational Vortex” spin or rotate around the center of the vortex equal areas in equal times.

Orbital “Time” Period – ({T_{Period}}) is defined

{T_{Period}} \,\,\, = \,\,\, \frac{A_{rea-Ellipse}}{(\frac{dA_{rea}}{dt})_{Ellipse-Gravity}}\,\,=\,\,\, \frac{A_{rea-Circle}}{(\frac{dA_{rea}}{dt})_{Gravity}} \,\,  —–> {s}

{T_{Period}} \,\,\,=2\pi \,\,\sqrt{\frac{r^3}{m_{Net}\,G}}\,\,\,=\,\,\, \frac{2\pi \,\,{r}}{{||\vec{v}||_{CM}}} \,\,\,  —–> {s}

Kepler’s Third Law – Evolution Attraction Rate

{K_{Gravity}}= \, 4\pi^2 (\frac{r^3_{Schwarzschild}}{T^2_{Schwarzschild}}) \, = \, 4\pi^2 (\frac{r^3_{1}}{T^2_{1}}) = 4\pi^2 (\frac{r^3_{2}}{T^2_{2}})= ...=4\pi^2 (\frac{r^{3}}{T^{2}_{Period}}) \,

{K_{Gravity}}\,\,\, = \,\,\, \frac{(2{(\frac{dA_{rea}}{dt})_{Gravity}})^2}{r} \,\,\, = \,\,\, \frac{(2{(\frac{dA_{rea}}{dt})_{Ellipse-Gravity}})^2}{{r}\, (1\,\,\, - \,\,\, e^2)} \,\,\,

{K_{Gravity}} \,\, = {m_{Net}}\,G \, \, = ( {m_{1}} + {m_{2}} + {m_{3}} + ...... + {m_{N}})G \,\,   ---> \frac{m^3}{s^2}

Concluding, Kepler’s Laws of the “Gravitational Vortex” using Fluid Mechanics have been derived.

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Citation

Cite this article as:

Robert Louis Kemp; The Super Principia Mathematica – The Rage to Master Conceptual & Mathematical Physics – The General Theory of Relativity – “Fluid dynamic “Vortex Gravitation” Model — derives Kepler’s Laws for Gravitation– Online Volume – ISBN 978-0-9841518-2-0, Volume 3; July 2010

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Author: Robert Louis Kemp

http://www.SuperPrincipia.com

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