Analogy between electromagnetism and gravity: matter wave model and speed of matter waves calculation with nasa data from horizons

Full analogy between electromagnetism and gravity, with matter wave calculation through solar system simulation sourced with ephemeris data from horizons (nasa). Argument of mercury perihelion precession against general relativity.

More than one year ago I started testing my matter wave model. After trying journal publication (a lot) I finally decided post it here (PDF fileformat) because I have spend more time trying publication than developing this model. I hope you find it useful.

Author: Enrique Dominguez Pinos. © All rights reserved.

Industrial Engineer.

Highlights

• We've proposed a parallel theory to electromagnetism, concluding De Broglie's prediction of a waveform associated with a moving mass.
• We've shown how this mathematical relationship is.
• We've calculated wave speed whose value is less than light speed, according to the astronomical data.
• We finally set out a possible refutation of the theory, through the gravitational repulsion.

Málaga, 8 November 2013

 

Contents

Introduction        3

Start data        4

Planets data        4

Previous results        4

Octave simulations        4

Hardware        4

Tolerances        4

Newton equations (model 32)        4

Measured astronomic data comparison (model 33)        6

MGF model (model 34)        6

Especial relativity (model 45)        8

Remarks        8

Positional vector errors        8

Other planets effect        8

Central force aberration for wave propagation time         8

Matter wave theory implications        9

Appreciable differences        9

Conclusion        9

Append I: Ephemeris data        10

Sun        11

Mercury        12

Venus        13

Earth        14

Mars        15

Jupiter        16

Saturn        17

Uranus        18

Neptune        19

Append II: Sun rotation        20

Append III: Magnetic field equations        21

Append IV: Special relativity        22

Append V: Mass field equations        23

References        24

Introduction: De Broglie duality

The aim of this work is to establish a relationship between a moving mass and its associated wave such as  De Broglie settled. De Broglie states: “every particle of matter with mass m and velocity v a real wave must be 'associated'", without establishing an equation of the particle-associated waves as electromagnetism does.

We'll set an analogy between electric charges and inertial mass. This analogy was previously observed from Heaviside since 1922 (1); we'll show a new point of view which will explain mercury perihelion precession without general relativity introduction. Those analogy between mass and charge will help us to discover a new force law.

We're showing this analogy in Table 1, using the expression of magnetic force for moving charges to search for this new force law.

Mass field	Electromagnetic field
$${\vec{F}}_M^{M\to m}=-G\frac{Mm}{r^2}\widehat{r}$$	$${\vec{F}}_e^{Q\to q}=\frac{1}{{\text{4πϵ}}_o}\frac{Qq}{r^2}\widehat{r}$$
?	$${\vec{F}}_m^{Q\to q}=q{\vec{v}}_q\wedge \frac{{\mu }_o}{\text{4π}}\frac{Q{\vec{v}}_Q\wedge \widehat{r}}{r^2}$$

Table 1: Electromagnetic and mass analogy

In this line of thinking, being the expression of force on moving masses,

$${\vec{F}}_{\mathit{mv}}^{M\to m}=m{\vec{v}}_m\wedge \gamma \frac{M{\vec{v}}_M\wedge \widehat{r}}{r^2}$$

 

We set the constant value that relates the forces known constants, if this quotient is 'cte',

$$\begin{array}{c}\text{cte=}\frac{|{\vec{F}}_e|}{|{\vec{F}}_m|}=\frac{1/\left({\text{4πϵ}}_o\right)}{{\mu }_o/\left(\text{4π}\right)}{\text{=c}}^2=\frac{G}{\gamma },\\ \text{γ=}\frac{G}{c^2}\end{array}$$

 

getting our table with the four forces,

Mass field	Electromagnetic field
$${\vec{F}}_M^{M\to m}=-G\frac{Mm}{r^2}\widehat{r}$$	$${\vec{F}}_e^{Q\to q}=\frac{1}{{\text{4πϵ}}_o}\frac{Qq}{r^2}\widehat{r}$$
$${\vec{F}}_{\mathit{mv}}^{M\to m}=m{\vec{v}}_m\wedge \frac{G}{c^2}\frac{{M\vec{v}}_M\wedge \widehat{r}}{r^2}$$	$${\vec{F}}_m^{Q\to q}=q{\vec{v}}_q\wedge \frac{{\mu }_o}{\text{4π}}\frac{Q{\vec{v}}_Q\wedge \widehat{r}}{r^2}$$

As we told, we'll test the expression through the calculation of the perihelion of Mercury.

Should be noted that the electric field is identified with the gravitational field, likewise that the magnetic field is identified with what we called field mass movement or motion gravitational field. Another papers named it gravitomagnetic field(1).

As a result of the model calibration in the expression of the movement gravitatory force (MGF), the force lacks the minus sign, which has the expression of the gravitational force of Newton.

Start data from nasa ephemeris

Planets data

Eight planets of solar system and the sun are considered for calculations; getting ephemeris data(2) without stellar aberration correction (geometric states).

NOTE: Arbitrary data couldn't be selected because Mars, we set a valid range for all planets.

Initial conditions used for ODEs was showed in Append I: Ephemeris data. The product of (G*m), collected in DE405(5), was used for setting each planet equation. Inestead of the values of horizons constants we have used values of DE405, so we don't need individual data of planets, only (G*m) product.

Previous results

On literature we can find perihelion precession of 574.10±0.65 arc-seconds by century (ICFR reference -mean values-). 532 arcsec of them are from solar system planets influence and 42.98 arcsec must being of general relativity contribution(3),(4).

Calculations referred to sun reference frame (in it's center), so we need to get precessions at this reference frame and set how to perform this calculation.

For precession calculation Mercury position vector when perihelion come at each cycle, starting form first gained vector and determining angle between current and this first vector we get  precessions. For angle between vectors calculation we are using scalar product formula,

Mercury position vector, when perihelion come at each cycle, is needed for precession calculation. Starting form first gained vector and determining angle between current, and this first vector, we get precessions. The angle between the vectors is computed using the formula for the scalar product,

$$\overrightarrow{r_1}\cdot \overrightarrow{r_n}=\left|\overrightarrow{r_1}\right|\left|\overrightarrow{r_n}\right|\cos (\mathrm{\alpha })$$

 

Alpha being angle between vectors, $$\overrightarrow{r_1}$$

being first gained perihelion vector and

$$\overrightarrow{r_n}$$

being position vector of nth perihelion.

 

The angle must be averaged because the actual value fluctuates, so that, we are using cumsum octave function. In an extended period of observation, the value stabilizes and get the real value of the precession. We can't use all simulated data for performing perihelion calculation because stabilization; in each graph will show current promediated years and predicted perihelion precession.

Octave simulations of solar system

Hardware

We're using octave for planets state simulation on an amd64 computer (1GB RAM) and gentoo linux optimized for this architecture.

Tolerances

Getting 1e-4 precision in alpha, needing around 1e4 precision tolerance in the positional vector.

Performing mercury trajectory needing absolute tolerance about 1e-80 for positions and about 1e-100 for speeds, as relative tolerance we are using octave epsilon (~1.49e-8). Other planets getting absolute tolerance in function about it's stellar cycle to mercury cycle ratio.

Newton equations (model 32)

Cumsum results for the perihelion precession in the model of the solar system by Newton's laws are showed in Illustration 1. We got 265 arcsec (simulated one century, last 50 years for getting real tendency)

 

 

Illustration 1: Newton model of solar system (one century)

 

 

We compare positional calculated vector with ephemeris tables in last simulated perihelion for validating this model. We got few minutes of error (ephemeris: 17-Feb-2013 02:25:41; simulated: 17-Feb-2013 02:11:41)

Date	Range
2013-Feb-17 02:15:41.0000	4.600028652353445E+07
2013-Feb-17 02:20:41.0000	4.600028430206786E+07
2013-Feb-17 02:25:41.0000	4.600028324139002E+07
2013-Feb-17 02:30:41.0000	4.600028334150112E+07
2013-Feb-17 02:35:41.0000	4.600028460240115E+07

Table 2: Perihelion calculation date via ephemeris

Relative error in positional vector was about 0.38%-0.11%.

We're tried getting data from another sources (NAIF kernels for example) but can't get all necessary data from there and compared data as sourced by horizons seems a little different.

 

Measured astronomic data comparison (model 33)

Last result was evaluated using ASCII kernel form JPL and ephemutil utility called from octave. We need to get the same position vectors already obtained by the simulation with data from JPL kernel data.

 

Illustration 2: Perihelion precession data shown from EphemUtil

 

 

We got 287arcseg by century. We have to remove a pair of data gained from ephemutil (seems out of range)

So we get a measured reference value that will allow us to perform the calibration of the model in this reference system. Model has to supply 22 arcsec.

MGF model (model 34)

Initially, we get a null effect of this force on the system, consisting of point masses,

$$\begin{array}{c}{\vec{F}}_{\mathit{mv}}^{M\to m}=m{\vec{v}}_m\wedge \vec{g}(\vec{r})\\ \vec{g}(\vec{r})=\frac{G}{c^2}\frac{{M\vec{v}}_M\wedge \widehat{r}}{r^2}\end{array}$$

 

Sun speed is lower (~5km/s) than mercury speed (~59km/s)

But we may consider the force exerted by sun rotating mass; and as in electromagnetic analogy, rotating charges produce a magnetic field and rotating masses produce a moving mass field (and a MGF from sun to mercury)

Rotational data of sun are in Append II: Sun rotation.

Field equation are, (go to Append III: Magnetic field equations for deduction),

$$\begin{array}{c}\vec{g}(\vec{r})=\frac{G\cdot M_S}{c^2{V}_S}\underset{V\text{'}}{\int }\frac{\vec{\mathrm{\omega }}(\mathrm{\varphi }\text{'})\wedge \vec{r}\text{'}\wedge \vec{R}}{R^3}\mathit{dV}\text{'}\\ \vec{R}=\vec{r}-\vec{r}\text{'}\end{array}$$

 

Being Ms y Vs mass and volume of sun. And Rs it's radius.

This equation could be set for ith planet as,

$$\overrightarrow{g_i}(\vec{r})=\frac{G\cdot M_S}{c^2{V}_S}\left(\underset{V\text{'}}{\int }\frac{\vec{\mathrm{\omega }}(\mathrm{\varphi }\text{'})\wedge \vec{r}\text{'}}{R^3}\mathit{dV}\text{'}\right)\wedge \overrightarrow{r_i}$$

 

And,

  $$\overrightarrow{g_i}(\vec{r})=\frac{G\cdot M_S}{c^2{V}_S}\left(\underset{0}{\overset{R_S}{\int }}\underset{0}{\overset{2\mathrm{\pi }}{\int }}\underset{0}{\overset{\mathrm{\pi }}{\int }}\vec{\mathrm{\omega }}(\mathrm{\varphi }\text{'}){\sin }^2(\mathrm{\varphi }\text{'})\frac{(-\widehat{x}\sin (\mathrm{\varphi }\text{'})+\widehat{y}\cos (\mathrm{\varphi }\text{'})){r\text{'}}^3}{R^{3/2}}d\mathrm{\varphi }\text{'}d\mathrm{\theta }\text{'}\mathit{dr}\text{'}\right)\wedge \overrightarrow{r_i}$$

 

With,

$$R={[x_i-r\text{'}\cos (\mathrm{\theta }\text{'})\sin (\mathrm{\varphi }\text{'})]}^2+{[y_i-r\text{'}\sin (\mathrm{\theta }\text{'})\sin (\mathrm{\varphi }\text{'})]}^2+{[z_i-r\text{'}\cos (\mathrm{\varphi }\text{'})]}^2$$

 

Being i coordinates from planet under MGF force.

We have to introduce an additional hypotesis to get an appreciable effect: matter wave propagation speed being less than light propagation speed. From simulations we got cm~3,966.470 km/s (286.61arcsec). Simulation data at Illustration 3.

 

Illustration 3: Limited velocity of matter waves simulation

 

 

Relative error of simulation was about 0.48%.

Especial relativity (model 45)

Analyzing contribution of special relativity as explained in Append IV: Special relativity, we got no appreciable influence (few arcsec, but can't explain perihelion precession of 22 arcsec)

But we can introduce it for accounting it's effect in matter wave speed. In this way, we got 4,687.756 km/s (286.43 arcsec, Illustration 4)

 

Illustration 4: Especial relativity effect consideration

 

 

Remarks

Positional vector errors

We've compared final positions of planets got from horizons ephemeris and simulated ones for checking simulation results. Those data have been checked for one century (error increase as we increase simulation time to two centuries) but wasn't able to get appreciable arcseg deviation in a single direction.

Other planets effect

We've checked MGF wasn't able to produce appreciable effect in another planets. Venus was simulated and effect were negligible.

Central force aberration for wave propagation time

We've seen this effect in (6), Tome II, chap 21, page 21-2, for electromagnetic field. Extrapolation would require us to test the contribution between planets (maybe Venus on Mercury) but we wouldn't consider sun influence, because other planets feeling sun influence as static (they don't see a moving sun) So there is no wave emission.

Matter wave theory implications

This analogy has more implications. Now the gravitational field propagates with movement mass field as electromagnetic waves do, matching a wave equation like Maxwell one, but with cm speed (lower than light speed)

Proposed equations was placed in Append V: Mass field equations.

Appreciable differences

We have to emphasize that the terms of the force will cause, on a body rotating around the sun, in the opposite direction to it, a repulsion force (against the attractive force would predict theory of general relativity) This effect could be used for testing this theory.

Conclusion

We've proposed a parallel theory to electromagnetism, concluding De Broglie's prediction of a waveform associated with a moving mass. We've shown how this mathematical relationship is.

We've checked that mercury precession can't be explained for special relativity and we have calibrated this new theory with astronomical data.

We've calculated wave speed whose value is less than light speed, according to the astronomical data.

We finally set out a possible refutation of the theory, through the gravitational repulsion (because results are against general relativity ones)

 

Append I: Ephemeris data

Ephemeris Type	VECTORS
Coordinate Origin	Sun (body center) [500@10]
Table Settings	Output units=KM-S; quantities code=2; labels=YES; CSV format=YES
Units	Km y km/s
Reference frame	IRCF/J2000.0
Reference plane	Ecliptic and mean equinox of reference epoch

Coordinate system description:

  Ecliptic and Mean Equinox of Reference Epoch

    Reference epoch: J2000.0

    xy-plane: plane of the Earth's orbit at the reference epoch

    x-axis  : out along ascending node of instantaneous plane of the Earth's

              orbit and the Earth's mean equator at the reference epoch

    z-axis  : perpendicular to the xy-plane in the directional (+ or -) sense

              of Earth's north pole at the reference epoch.

Symbol meaning  

    JDCT     Epoch Julian Date, Coordinate Time

      X      x-component of position vector (km)                              

      Y      y-component of position vector (km)                              

      Z      z-component of position vector (km)                              

      VX     x-component of velocity vector (km/sec)                          

      VY     y-component of velocity vector (km/sec)                          

      VZ     z-component of velocity vector (km/sec)                          

 

Geometric states/elements have no aberration corrections applied.

 Computations by ...

     Solar System Dynamics Group, Horizons On-Line Ephemeris System

     4800 Oak Grove Drive, Jet Propulsion Laboratory

     Pasadena, CA  91109   USA

     Information: http://ssd.jpl.nasa.gov/

     Connect    : telnet://ssd.jpl.nasa.gov:6775  (via browser)

                  telnet ssd.jpl.nasa.gov 6775    (via command-line)

     Author     : Jon.Giorgini@jpl.nasa.gov

*******************************************************************************

Sun

Target Body	Sun [Sol] [10]
Coordinate Origin	Sun (body center) [500@10]
Time Span	Start=1912-12-25, Stop=2012-12-25, Step=100 Y

Results

*******************************************************************************

Ephemeris / WWW_USER Fri Dec 14 07:13:25 2012 Pasadena, USA      / Horizons    

*******************************************************************************

Target body name: Sun (10)                        {source: DE405}

Center body name: Sun (10)                        {source: DE405}

Center-site name: BODY CENTER

*******************************************************************************

Start time      : A.D. 1912-Dec-25 00:00:00.0000 CT

Stop  time      : A.D. 2012-Dec-25 00:00:00.0000 CT

Step-size       : 100 calendar years

*******************************************************************************

Center geodetic : 0.00000000,0.00000000,0.0000000 {E-lon(deg),Lat(deg),Alt(km)}

Center cylindric: 0.00000000,0.00000000,0.0000000 {E-lon(deg),Dxy(km),Dz(km)}

Center radii    : 696000.0 x 696000.0 x 696000.0 k{Equator, meridian, pole}    

Output units    : KM-S                                                        

Output format   : 02

Reference frame : ICRF/J2000.0                                                

Output type     : GEOMETRIC cartesian states

Coordinate systm: Ecliptic and Mean Equinox of Reference Epoch                

*******************************************************************************

JDCT ,   , X, Y, Z, VX, VY, VZ,

*******************************************************************************

$$SOE

2419761.500000000, A.D. 1912-Dec-25 00:00:00.0000,  0.000000000000000E+00,  0.000000000000000E+00,  0.000000000000000E+00,  0.000000000000000E+00,  0.000000000000000E+00,  0.000000000000000E+00,

2456286.500000000, A.D. 2012-Dec-25 00:00:00.0000,  0.000000000000000E+00,  0.000000000000000E+00,  0.000000000000000E+00,  0.000000000000000E+00,  0.000000000000000E+00,  0.000000000000000E+00,

$$EOE

*******************************************************************************

Mercury

Target Body	Mercury [199]
Coordinate Origin	Sun (body center) [500@10]
Time Span	Start=1912-12-25, Stop=2012-12-25, Step=100 Y

Results

*******************************************************************************

Ephemeris / WWW_USER Fri Dec 14 06:18:18 2012 Pasadena, USA      / Horizons    

*******************************************************************************

Target body name: Mercury (199)                   {source: DE405}

Center body name: Sun (10)                        {source: DE405}

Center-site name: BODY CENTER

*******************************************************************************

Start time      : A.D. 1912-Dec-25 00:00:00.0000 CT

Stop  time      : A.D. 2012-Dec-25 00:00:00.0000 CT

Step-size       : 100 calendar years

*******************************************************************************

Center geodetic : 0.00000000,0.00000000,0.0000000 {E-lon(deg),Lat(deg),Alt(km)}

Center cylindric: 0.00000000,0.00000000,0.0000000 {E-lon(deg),Dxy(km),Dz(km)}

Center radii    : 696000.0 x 696000.0 x 696000.0 k{Equator, meridian, pole}    

Output units    : KM-S                                                        

Output format   : 02

Reference frame : ICRF/J2000.0                                                

Output type     : GEOMETRIC cartesian states

Coordinate systm: Ecliptic and Mean Equinox of Reference Epoch                

*******************************************************************************

JDCT ,   , X, Y, Z, VX, VY, VZ,

*******************************************************************************

$$SOE

2419761.500000000, A.D. 1912-Dec-25 00:00:00.0000, -5.360665730052117E+07,  1.243397368572870E+07,  5.946675013560415E+06, -2.109596483753719E+01, -4.533775271587504E+01, -1.757353935106772E+00,

2456286.500000000, A.D. 2012-Dec-25 00:00:00.0000, -4.440290472706281E+07, -5.110440497487431E+07, -1.014449602649391E+05,  2.688550693766266E+01, -2.970018707156070E+01, -4.893508405499786E+00,

$$EOE

*******************************************************************************

 

Venus

Target Body	Venus [299]
Coordinate Origin	Sun (body center) [500@10]
Time Span	Start=1912-12-25, Stop=2012-12-25, Step=100 Y

Results

*******************************************************************************

Ephemeris / WWW_USER Fri Dec 14 06:20:44 2012 Pasadena, USA      / Horizons    

*******************************************************************************

Target body name: Venus (299)                     {source: DE405}

Center body name: Sun (10)                        {source: DE405}

Center-site name: BODY CENTER

*******************************************************************************

Start time      : A.D. 1912-Dec-25 00:00:00.0000 CT

Stop  time      : A.D. 2012-Dec-25 00:00:00.0000 CT

Step-size       : 100 calendar years

*******************************************************************************

Center geodetic : 0.00000000,0.00000000,0.0000000 {E-lon(deg),Lat(deg),Alt(km)}

Center cylindric: 0.00000000,0.00000000,0.0000000 {E-lon(deg),Dxy(km),Dz(km)}

Center radii    : 696000.0 x 696000.0 x 696000.0 k{Equator, meridian, pole}    

Output units    : KM-S                                                        

Output format   : 02

Reference frame : ICRF/J2000.0                                                

Output type     : GEOMETRIC cartesian states

Coordinate systm: Ecliptic and Mean Equinox of Reference Epoch                

*******************************************************************************

JDCT ,   , X, Y, Z, VX, VY, VZ,

*******************************************************************************

$$SOE

2419761.500000000, A.D. 1912-Dec-25 00:00:00.0000,  1.018158177530931E+08,  3.703243858802129E+07, -5.386891016958820E+06, -1.208964619890154E+01,  3.275601387483020E+01,  1.138416084527058E+00,

2456286.500000000, A.D. 2012-Dec-25 00:00:00.0000, -8.364393681148392E+07, -6.847969524311861E+07,  3.889032485377138E+06,  2.193837392533697E+01, -2.726409897743777E+01, -1.639716487601144E+00,

$$EOE

*******************************************************************************

 

Earth

Target Body	Earth [Geocenter] [399]
Coordinate Origin	Sun (body center) [500@10]
Time Span	Start=1912-12-25, Stop=2012-12-25, Step=100 Y

Results

*******************************************************************************

Ephemeris / WWW_USER Fri Dec 14 06:22:13 2012 Pasadena, USA      / Horizons    

*******************************************************************************

Target body name: Earth (399)                     {source: DE405}

Center body name: Sun (10)                        {source: DE405}

Center-site name: BODY CENTER

*******************************************************************************

Start time      : A.D. 1912-Dec-25 00:00:00.0000 CT

Stop  time      : A.D. 2012-Dec-25 00:00:00.0000 CT

Step-size       : 100 calendar years

*******************************************************************************

Center geodetic : 0.00000000,0.00000000,0.0000000 {E-lon(deg),Lat(deg),Alt(km)}

Center cylindric: 0.00000000,0.00000000,0.0000000 {E-lon(deg),Dxy(km),Dz(km)}

Center radii    : 696000.0 x 696000.0 x 696000.0 k{Equator, meridian, pole}    

Output units    : KM-S                                                        

Output format   : 02

Reference frame : ICRF/J2000.0                                                

Output type     : GEOMETRIC cartesian states

Coordinate systm: Ecliptic and Mean Equinox of Reference Epoch                

*******************************************************************************

JDCT ,   , X, Y, Z, VX, VY, VZ,

*******************************************************************************

$$SOE

2419761.500000000, A.D. 1912-Dec-25 00:00:00.0000, -1.045637425163494E+07,  1.467385967490118E+08,  2.835631451258451E+04, -3.018886665393219E+01, -2.222940110022601E+00, -8.140263107061256E-04,

2456286.500000000, A.D. 2012-Dec-25 00:00:00.0000, -8.773033517291984E+06,  1.468674792713496E+08, -4.466184044525366E+03, -3.021116715750096E+01, -1.892934819525167E+00,  1.158502276269857E-03,

$$EOE

*******************************************************************************

 

Mars

Target Body	Mars [499]
Coordinate Origin	Sun (body center) [500@10]
Time Span	Start=1912-12-25, Stop=2012-12-25, Step=100 Y

Results

*******************************************************************************

Ephemeris / WWW_USER Fri Dec 14 06:24:47 2012 Pasadena, USA      / Horizons    

*******************************************************************************

Target body name: Mars (499)                      {source: MAR097}

Center body name: Sun (10)                        {source: DE405}

Center-site name: BODY CENTER

*******************************************************************************

Start time      : A.D. 1912-Dec-25 00:00:00.0000 CT

Stop  time      : A.D. 2012-Dec-25 00:00:00.0000 CT

Step-size       : 100 calendar years

*******************************************************************************

Center geodetic : 0.00000000,0.00000000,0.0000000 {E-lon(deg),Lat(deg),Alt(km)}

Center cylindric: 0.00000000,0.00000000,0.0000000 {E-lon(deg),Dxy(km),Dz(km)}

Center radii    : 696000.0 x 696000.0 x 696000.0 k{Equator, meridian, pole}    

Output units    : KM-S                                                        

Output format   : 02

Reference frame : ICRF/J2000.0                                                

Output type     : GEOMETRIC cartesian states

Coordinate systm: Ecliptic and Mean Equinox of Reference Epoch                

*******************************************************************************

JDCT ,   , X, Y, Z, VX, VY, VZ,

*******************************************************************************

$$SOE

2419761.500000000, A.D. 1912-Dec-25 00:00:00.0000, -7.966999905665824E+07, -2.100334861412445E+08, -2.420085859665400E+06,  2.358431013255164E+01, -6.523457368354225E+00, -7.205749072795029E-01,

2456286.500000000, A.D. 2012-Dec-25 00:00:00.0000,  1.514168902923513E+08, -1.419468950342833E+08, -6.691719661214531E+06,  1.749190756063776E+01,  1.974805640407604E+01, -1.571815631304356E-02,

$$EOE

*******************************************************************************

 

Jupiter

Target Body	Jupiter [599]
Coordinate Origin	Sun (body center) [500@10]
Time Span	Start=1912-12-25, Stop=2012-12-25, Step=100 Y

Results

*******************************************************************************

Ephemeris / WWW_USER Fri Dec 14 06:26:01 2012 Pasadena, USA      / Horizons    

*******************************************************************************

Target body name: Jupiter (599)                   {source: JUP230}

Center body name: Sun (10)                        {source: DE405}

Center-site name: BODY CENTER

*******************************************************************************

Start time      : A.D. 1912-Dec-25 00:00:00.0000 CT

Stop  time      : A.D. 2012-Dec-25 00:00:00.0000 CT

Step-size       : 100 calendar years

*******************************************************************************

Center geodetic : 0.00000000,0.00000000,0.0000000 {E-lon(deg),Lat(deg),Alt(km)}

Center cylindric: 0.00000000,0.00000000,0.0000000 {E-lon(deg),Dxy(km),Dz(km)}

Center radii    : 696000.0 x 696000.0 x 696000.0 k{Equator, meridian, pole}    

Output units    : KM-S                                                        

Output format   : 02

Reference frame : ICRF/J2000.0                                                

Output type     : GEOMETRIC cartesian states

Coordinate systm: Ecliptic and Mean Equinox of Reference Epoch                

*******************************************************************************

JDCT ,   , X, Y, Z, VX, VY, VZ,

*******************************************************************************

$$SOE

2419761.500000000, A.D. 1912-Dec-25 00:00:00.0000, -2.325226589284961E+07, -7.864938001501033E+08,  3.721909619674553E+06,  1.291728188825723E+01,  2.232228774062441E-01, -2.905863315990377E-01,

2456286.500000000, A.D. 2012-Dec-25 00:00:00.0000,  2.208900533351294E+08,  7.241860827237822E+08, -7.950563422672395E+06, -1.266535090043849E+01,  4.437528423714215E+00,  2.650446010224173E-01,

$$EOE

*******************************************************************************

 

Saturn

Target Body	Saturn [699]
Coordinate Origin	Sun (body center) [500@10]
Time Span	Start=1912-12-25, Stop=2012-12-25, Step=100 Y

Results

*******************************************************************************

Ephemeris / WWW_USER Fri Dec 14 06:26:55 2012 Pasadena, USA      / Horizons    

*******************************************************************************

Target body name: Saturn (699)                    {source: SAT351}

Center body name: Sun (10)                        {source: DE405}

Center-site name: BODY CENTER

*******************************************************************************

Start time      : A.D. 1912-Dec-25 00:00:00.0000 CT

Stop  time      : A.D. 2012-Dec-25 00:00:00.0000 CT

Step-size       : 100 calendar years

*******************************************************************************

Center geodetic : 0.00000000,0.00000000,0.0000000 {E-lon(deg),Lat(deg),Alt(km)}

Center cylindric: 0.00000000,0.00000000,0.0000000 {E-lon(deg),Dxy(km),Dz(km)}

Center radii    : 696000.0 x 696000.0 x 696000.0 k{Equator, meridian, pole}    

Output units    : KM-S                                                        

Output format   : 02

Reference frame : ICRF/J2000.0                                                

Output type     : GEOMETRIC cartesian states

Coordinate systm: Ecliptic and Mean Equinox of Reference Epoch                

*******************************************************************************

JDCT ,   , X, Y, Z, VX, VY, VZ,

*******************************************************************************

$$SOE

2419761.500000000, A.D. 1912-Dec-25 00:00:00.0000,  6.160627576702619E+08,  1.208915824575386E+09, -4.573515710213210E+07, -9.121766092721035E+00,  4.363134726154207E+00,  2.856552635915862E-01,

2456286.500000000, A.D. 2012-Dec-25 00:00:00.0000, -1.211851731764423E+09, -8.201413680424271E+08,  6.250905459656192E+07,  4.879229156365631E+00, -8.019271442359987E+00, -5.486570783853705E-02,

$$EOE

*******************************************************************************

 

Uranus

Target Body	Uranus [799]
Coordinate Origin	Sun (body center) [500@10]
Time Span	Start=1912-12-25, Stop=2012-12-25, Step=100 Y

Results

*******************************************************************************

Ephemeris / WWW_USER Fri Dec 14 06:27:45 2012 Pasadena, USA      / Horizons    

*******************************************************************************

Target body name: Uranus (799)                    {source: URA095}

Center body name: Sun (10)                        {source: DE405}

Center-site name: BODY CENTER

*******************************************************************************

Start time      : A.D. 1912-Dec-25 00:00:00.0000 CT

Stop  time      : A.D. 2012-Dec-25 00:00:00.0000 CT

Step-size       : 100 calendar years

*******************************************************************************

Center geodetic : 0.00000000,0.00000000,0.0000000 {E-lon(deg),Lat(deg),Alt(km)}

Center cylindric: 0.00000000,0.00000000,0.0000000 {E-lon(deg),Dxy(km),Dz(km)}

Center radii    : 696000.0 x 696000.0 x 696000.0 k{Equator, meridian, pole}    

Output units    : KM-S                                                        

Output format   : 02

Reference frame : ICRF/J2000.0                                                

Output type     : GEOMETRIC cartesian states

Coordinate systm: Ecliptic and Mean Equinox of Reference Epoch                

*******************************************************************************

JDCT ,   , X, Y, Z, VX, VY, VZ,

*******************************************************************************

$$SOE

2419761.500000000, A.D. 1912-Dec-25 00:00:00.0000,  1.672795478653011E+09, -2.443857810328711E+09, -3.084987609136089E+07,  5.577499124725859E+00,  3.531952557946183E+00, -5.938484056216033E-02,

2456286.500000000, A.D. 2012-Dec-25 00:00:00.0000,  2.976113012338417E+09,  3.810762353106240E+08, -3.713403980800174E+07, -9.245322750052556E-01,  6.439861688192399E+00,  3.586407160501116E-02,

$$EOE

*******************************************************************************

 

Neptune

Target Body	Neptune [899]
Coordinate Origin	Sun (body center) [500@10]
Time Span	Start=1912-12-25, Stop=2012-12-25, Step=100 Y

Results

*******************************************************************************

Ephemeris / WWW_USER Fri Dec 14 06:28:42 2012 Pasadena, USA      / Horizons    

*******************************************************************************

Target body name: Neptune (899)                   {source: NEP081}

Center body name: Sun (10)                        {source: DE405}

Center-site name: BODY CENTER

*******************************************************************************

Start time      : A.D. 1912-Dec-25 00:00:00.0000 CT

Stop  time      : A.D. 2012-Dec-25 00:00:00.0000 CT

Step-size       : 100 calendar years

*******************************************************************************

Center geodetic : 0.00000000,0.00000000,0.0000000 {E-lon(deg),Lat(deg),Alt(km)}

Center cylindric: 0.00000000,0.00000000,0.0000000 {E-lon(deg),Dxy(km),Dz(km)}

Center radii    : 696000.0 x 696000.0 x 696000.0 k{Equator, meridian, pole}    

Output units    : KM-S                                                        

Output format   : 02

Reference frame : ICRF/J2000.0                                                

Output type     : GEOMETRIC cartesian states

Coordinate systm: Ecliptic and Mean Equinox of Reference Epoch                

*******************************************************************************

JDCT ,   , X, Y, Z, VX, VY, VZ,

*******************************************************************************

$$SOE

2419761.500000000, A.D. 1912-Dec-25 00:00:00.0000, -1.945382209325346E+09,  4.040882875978233E+09, -3.840489086570295E+07, -4.918386519161444E+00, -2.321455648483695E+00,  1.606763346689215E-01,

2456286.500000000, A.D. 2012-Dec-25 00:00:00.0000,  3.971948263325485E+09, -2.085811431212152E+09, -4.856478539988586E+07,  2.480996673642879E+00,  4.845946535939160E+00, -1.576431831611057E-01,

$$EOE

*******************************************************************************

Append II: Sun rotation

http://en.wikipedia.org/wiki/Solar_rotation

Latitude dependent rotation,

$$\mathrm{\omega }=A+B{\sin }^2(\mathrm{\phi })+C{\sin }^4(\mathrm{\phi })$$

 

constants being,

A= 14.713 deg/day (± 0.0491)

B= –2.396 deg/day (± 0.188)

C= –1.787 deg/day (± 0.253)

and maximum in the equator plane. Equation relating latitude and third spherical coordinate being,

$$\sin (\mathrm{\phi })=\sin (\frac{\mathrm{\pi }}{2}-\mathrm{\varphi })=\cos (\mathrm{\varphi })$$

 

so, rotation module become,

$$\mathrm{\omega }=A+B{\cos }^2(\mathrm{\varphi })+C{\cos }^4(\mathrm{\varphi })$$

 

and constants in rad/s are,

$$A\text{'}=A\frac{\mathrm{\pi }}{180\cdot 24\cdot 3600}$$

 

Getting,

A'= 2.9721098709e-06 rad/sec

B'= –4.84005658308e-07 rad/sec

C'= –3.60984186726e-07 rad/sec

As reference frame is in the sun, z axis match sun rotation axis.

 

 

 

Append III: Magnetic field equations

Starting from magnetic field equations for a point charge, and a volumetric charge (ρ) spinning, with rotation vector $$\vec{\mathrm{\omega }}$$

, and producing a volumetric current

$$\vec{J}=\mathrm{\rho }\vec{\mathrm{\omega }}\wedge \overrightarrow{r\text{'}}$$

,

 

$$\begin{array}{c}\vec{B}=\frac{{\mu }_o}{\text{4π}}\frac{Q{\vec{v}}_Q\wedge \widehat{r}}{r^2},\\ \vec{B}=\frac{{\mu }_o}{\text{4π}}\underset{V\text{'}}{\int }\vec{J}(\vec{r}\text{'})\wedge \frac{\vec{R}}{R}\mathit{dV}\text{'},\end{array}$$

 

where $$\vec{R}=\vec{r}-\vec{r}\text{'}$$

. And making the analogy with the mass field, we get,

 

$$\overrightarrow{B_g}(\vec{r})=\frac{G\cdot M_S}{c^2{V}_S}\underset{V\text{'}}{\int }\frac{\vec{\mathrm{\omega }}(\mathrm{\varphi }\text{'})\wedge \vec{r}\text{'}\wedge \vec{R}}{R^3}\mathit{dV}\text{'}$$

 

Append IV: Special relativity

Followed from (7).

Relativistic equation of a space state system are,

$$\begin{array}{c}\frac{d\vec{x}}{\mathit{dt}}=\vec{v},\\ \frac{d\vec{v}}{\mathit{dt}}=\frac{1}{m}\left(\vec{F}-\frac{(\vec{v}\cdot \vec{F})\vec{v}}{c^2}\right)\sqrt{1-\frac{v^2}{c^2}},\end{array}$$

 

Being $$\vec{F}$$

force executed against the body and being speed vector referenced to selected reference frame.

 

Append V: Mass field equations

Followed from (8).

Maxwell equations set,

$$\begin{array}{c}\nabla \cdot \vec{E}=\frac{\mathrm{\rho }}{{\mathrm{ϵ}}_0}\\ \nabla \cdot \vec{B}=0\\ \nabla \wedge \vec{E}=\frac{-\partial \vec{B}}{\partial t}\\ \nabla \wedge \vec{B}={\mathrm{\mu }}_0\vec{J}+\frac{1}{c}\mathrm{²}\frac{\partial \vec{E}}{\partial t}\end{array}$$

 

and mass field ones become,

$$\begin{array}{c}\nabla \cdot \overrightarrow{E_g}=-4\mathrm{\pi }G{\mathrm{\rho }}_g\\ \nabla \cdot \overrightarrow{B_g}=0\\ \nabla \wedge \overrightarrow{E_g}=\frac{-\partial \overrightarrow{B_g}}{\partial t}\\ \nabla \wedge \overrightarrow{B_g}=\frac{4\mathrm{\pi }G}{c_m\mathrm{²}}\overrightarrow{J_g}+\frac{1}{c_m\mathrm{²}}\frac{\partial \overrightarrow{E_g}}{\partial t}\end{array}$$

 

References

1. (1)Biemond, J., "The gravitomagnetic field of a sphere, Gravity Probe B and the LAGEOS satellites", arXiv:0802.3346v1, 22 Feb 2008.  

2. (2) http://ssd.jpl.nasa.gov/?horizons 

3. (3) http://en.wikipedia.org/wiki/Mercury_%28planet%29 

4. (4) http://www.mathpages.com/rr/s6-02/6-02.htm 

5. (5) http://iau-comm4.jpl.nasa.gov/de405iom/de405iom.pdf 

6. (6)Richard P. Feynman et al. “The Feynman Lectures on Physics”, Addison–Wesley, 1964. 

7. (7) http://en.wikipedia.org/wiki/Relativistic_mechanics 

8. (8) http://en.wikipedia.org/wiki/Gravitoelectromagnetism 

 

 

You can find full development history here:
MGF Model (PDF File)
High resolution Images (eps):
Model 32
Model 33
Model 34
Model 45
This file follows more successful tried steps towards final model.