Alpha Institute for Advanced Studies

Many thanks again, I plan to use this note as the main part of UFT403, Sections 1 and 2. Your check is very interesting, and I will include it in the final paper.. I will rework Note 403(7) it is probably power 3/2 in the denominator, a minor error of hand calculation. Use of the computer to mend these errors is obviously of key importance. Maxima may have an analytical or numerical solution for the correction function delta phi, giving the complete orbit phi + delta phi as well as the precession of the perihelion for orbits of small eccentricity from Note 403(6). Einstein’s famous or infamous result for the precession of the perihelion did not attempt to give the orbit. The major advance in knowledge is that precession is due to vacuum fluctuations.

Subject: note 403(6)
To: Myron Evans <myronevans123>

The Schwarzschild radius is 3*10^3 m, not 3*10^5 m, a small typo.
In eq. (34) there is a sign typo but it has been corrected in (35).

A check for the correctness of the result (37) is as follows:
assume
omega = a/r
as in previous papers. Then eq.(37) gives

psi = pi (1+a).

For a small a, a<<1, we have psi ~ pi which is the result for vanishing precession.

Horst

OK thanks, there may be an analytical solution from Wolfram of the correction function, and there will also be a numerical solution.

This note shows that the Maxima solution of the ECE2 force equation, Eq. (2), is a small perturbation of an elliptical orbit. From the previous note it is known that this perturbation is a precession in the small eccentricity limit. That is easy to show with the apsidal method of Note 403(6). The solution for phi is given by Eq. (22), which can be worked out if the integral in Eq. (21) can be evaluated by Maxima, either analytically or numerically.. The solution (2) can also be worked out numerically with Maxima.

a403rdpapernotes7.pdf

OK thanks again, I will write a note to show that Eq. (8) of the protocol with positive sign is the same as Eq. (7.38) of Marion and Thornton if the spin connection is zero. In this case its solution is Eq. (7.39), a conic section with half right magnitude (7.40a) and eccentricity (7.40b). The conic section is Eq.(7.41) The spin connection is very small, so an analytical approximation can proceed on that basis. A numerical integration is the best way forward.

Fwd: 403(5): Solution of the ECE2 Force Equation for Small Eccentricity (Planetary Systems)
To: Myron Evans <myronevans123>

Eq.(8) of the protocol is a kind of implicit solution of(7) because it is not u(phi) but an integral form of the inverse function phi(u). This is obviously what we need here.
The bracket in the upper line of Eq. (8) marks that there are two solutions, written as a vector in brackets. So both lines are a solution, differering only by the sign of the integral.
The integral can be solved analytically in case of two alternative simplifications:

1. The log function is omitted. Then the result does no more depend on omega_u. It is essentially a arcsin function, see eq. (9) of the augmented protocol.
2. The linear and quadratic terms of u are omitted. Then an error function expression is obtained ( see (10)).

If one of these simplifications can be justified, I can graph the result. Otherwise we could try a numerical solution of the integral.

Horst

Am 14.03.2018 um 10:56 schrieb Myron Evans:

Thanks again. This is very useful as usual. Eq. (8) of the protocol is the needed analytical solution and it is a precessing ellipse. As the spin connection goes to zero, the integral gives an ellipse as in Marion and Thornton chapter seven of the third edition. So the integral in Eq (8) of the protocol can be evaluated numerically or perhaps with an analytical approximation that does not restrict the range of phi.. It is similar to the integral of the Einstein theory, except that one term is replaced by a log term. It is known from the apsidal method for small eccentricities (note 403(6)) that the ellipse will be a precessing ellipse. So I think that Eq. (8) of the protocol should be used in UFT403 together with Note 403(6), and the rest of the notes mentioned but not used. I have one question, what is the bracket in the upper line of Eq. (8) of the protocol/ I assume that the lower line of Eq. (8) is the answer. From the apsidal method, any force that is not inverse square will give a precession. However, the force of the ECE2 equation is derived as part of a unified field theory rigorously based on geometry as you know. The major discovery is that precession is due to vacuum fluctuations An entirely new cosmology can be developed from the theory of vacuum fluctuations of the same type as used in Lamb shift theory. As shown in UFT399, infinite energy from the vacuum can also be based on vacuum fluctuations.

Date: Tue, Mar 13, 2018 at 6:43 PM
Subject: Re: 403(5): Solution of the ECE2 Force Equation for Small Eccentricity (Planetary Systems)
To: Myron Evans <myronevans123>

This is which solutions Maxima gives for eqs. (1) and (2). It has to be assumed that omega_r is constant, otherwise there is no analytical solution. omega is written omega_u because the variable u=1/r has to be used in the equation solver. Then eq.(1) has an analytical solution, see eq. o6 in the protocol. By precondition, this solution is only valid in the range r ~ alpha, as far as I understand this. It can be seen that this solution comprises a homogeneous part (with constants %k1 and %k2) and an inhomogeneous part.

Eq. (2) of the note only gives an incomplete solution (with integrals) because u appears in the denominator:

r*omega_r = omega_u / u.

The approximatin of eq. (5) leads to the analytical solution o10, again with parts from the homogeneous and inhomogeneous diff. equation. This seems to differ from what the Worlfram solver puts out. However, o10 seems to be more or less identical with eq.(10) of the note.
Since we have r ~ alpha, this solution seems only to be valid in the range phi ~ pi/2. Using this (and setting %k1=0) in o10 gives the result o11 which depends on omega_u, in contrast to your eq.(14). Where did omega_r go in your calculation?

Horst

Am 09.03.2018 um 13:39 schrieb Myron Evans:

This is given by Eq. (7) in the approximation of Eq. (5), and by Eq. (10) in the next approximation, the equation following equation (8). These equations can be used to describe any precessing orbit with precision and the ECE2 equation is clearly preferred to the Einstein equation because the latter has been refuted in nearly a hundred different ways in the UFT series and refuted in many other ways by many authors for many years. It is used only by dogmatic ostriches who hope that ECE2 will go away, but it’s here to stay.

403(5).pdf

Thanks again. This is very useful as usual. Eq. (8) of the protocol is the needed analytical solution and it is a precessing ellipse. As the spin connection goes to zero, the integral gives an ellipse as in Marion and Thornton chapter seven of the third edition. So the integral in Eq (8) of the protocol can be evaluated numerically or perhaps with an analytical approximation that does not restrict the range of phi.. It is similar to the integral of the Einstein theory, except that one term is replaced by a log term. It is known from the apsidal method for small eccentricities (note 403(6)) that the ellipse will be a precessing ellipse. So I think that Eq. (8) of the protocol should be used in UFT403 together with Note 403(6), and the rest of the notes mentioned but not used. I have one question, what is the bracket in the upper line of Eq. (8) of the protocol/ I assume that the lower line of Eq. (8) is the answer. From the apsidal method, any force that is not inverse square will give a precession. However, the force of the ECE2 equation is derived as part of a unified field theory rigorously based on geometry as you know. The major discovery is that precession is due to vacuum fluctuations An entirely new cosmology can be developed from the theory of vacuum fluctuations of the same type as used in Lamb shift theory. As shown in UFT399, infinite energy from the vacuum can also be based on vacuum fluctuations.

Date: Tue, Mar 13, 2018 at 6:43 PM
Subject: Re: 403(5): Solution of the ECE2 Force Equation for Small Eccentricity (Planetary Systems)
To: Myron Evans <myronevans123>

This is which solutions Maxima gives for eqs. (1) and (2). It has to be assumed that omega_r is constant, otherwise there is no analytical solution. omega is written omega_u because the variable u=1/r has to be used in the equation solver. Then eq.(1) has an analytical solution, see eq. o6 in the protocol. By precondition, this solution is only valid in the range r ~ alpha, as far as I understand this. It can be seen that this solution comprises a homogeneous part (with constants %k1 and %k2) and an inhomogeneous part.

Eq. (2) of the note only gives an incomplete solution (with integrals) because u appears in the denominator:

r*omega_r = omega_u / u.

The approximatin of eq. (5) leads to the analytical solution o10, again with parts from the homogeneous and inhomogeneous diff. equation. This seems to differ from what the Worlfram solver puts out. However, o10 seems to be more or less identical with eq.(10) of the note.
Since we have r ~ alpha, this solution seems only to be valid in the range phi ~ pi/2. Using this (and setting %k1=0) in o10 gives the result o11 which depends on omega_u, in contrast to your eq.(14). Where did omega_r go in your calculation?

Horst

Am 09.03.2018 um 13:39 schrieb Myron Evans:

This is given by Eq. (7) in the approximation of Eq. (5), and by Eq. (10) in the next approximation, the equation following equation (8). These equations can be used to describe any precessing orbit with precision and the ECE2 equation is clearly preferred to the Einstein equation because the latter has been refuted in nearly a hundred different ways in the UFT series and refuted in many other ways by many authors for many years. It is used only by dogmatic ostriches who hope that ECE2 will go away, but it’s here to stay.

403(5).pdf

Much appreciated as ever. I agree that dr / dt is not zero at the angle phi = pi / 2 that defines the half right latitude alpha, but any elliptical orbit is characterized by a constant alpha and a constant eccentricity epsilon. So defining alpha as a constant, d alpha / dt = 0. So r has been defined as the constant alpha at phi = pi / 2.

Date: Tue, Mar 13, 2018 at 4:48 PM
Subject: Re: 403(1): Analytical Solution for the Precession Angle of the ECE2 Covariant Orbit
To: Myron Evans <myronevans123>

The relation
d^2u/dphi^2 = 0
follows directly from the assumption r=alpha. At this point the radial velocity is not zero, so I would avoid arguments like eqs. (12-14).

That the second derivative d^2u/dphi^2 vanishes does not mean that the first derivative du/dphi vanishes. This is not the case here. Therefore eq.(11) seems not justified to me. Furthermore, eq.(11) does only hold for r=alpha, therefore it should be set r=alpha at the RHS. Unfortunately the subsequent development then becomes obsolete. Maybe I am overlooking something here.

Horst

Am 03.03.2018 um 14:33 schrieb Myron Evans:

This is given by the formula (29) which is valid for small precessions. Precise agreement with experimental data can be found by adjusting the relativistic angular momentum L

Many thanks for this meticulous checking, this is hard work and is of key importance for the accuracy and impact of the work.

The Relativistic Binet Equation of Orbits
To: Myron Evans <myronevans123>

I checked the equations of the notes. They are correct. In note 7 there is a small sign typo in eq.(24). There is no analytical solution for eq.(41) but I think you found one in paper 403. I will write up section 3 with some example graphics.

Horst

Am 25.02.2018 um 15:08 schrieb Myron Evans:

This given by Eq. (25) using the results of Note 402(5). It is the same result as derived in earlier UFT papers, giving a proof of overall self consistency. It is developed to give the precise orbital equation (36), which is approximated by Eq. (42), a second order non linear differential equation. It is known from numerical integration in UFT401 that the starting equation, the relativistic Newton equation, gives precessing ellipses. Eq. (36) gives the precise analytical result for the same orbit, so Eq. (36) must give precession.

This note makes straightforward use of the well known apsidal equation (1) to give orbital precessions and trace their origin to vacuum fluctuations. Firstly, for the sake of reference only, the orbital precession of the obsolete Einstein theory is obtained very simply. It is then shown that EGR is a special case of ECE2, one in which the isotropically averaged vacuum fluctuation is given by Eq. (26). So vacuum fluctuation is the origin of Einsteinian general relativity. However that theory is totally wrong and has been refuted many times by the famous UFT series. ECE2 is based on the correct geometry and is much simpler and powerful than EGR. Finally the apsidal method is used to give the ECE2 precession at the perihelion, Eq. (48). Any precession is given by Eq. (47) in the limit of small eccentricity. For eccentric orbits such as those of comets, the ECE2 equation must be integrated numerically. Vacuum fluctuations give rise to relativity itself in one way of thinking.

a403rdpapernotes6.pdf

If F = – mMG / r squared + omega phi then the precession at the perihelion of a precessing elliptical orbit is approximately delta psi = pi (omega r + r squared partial omega / partial r) / 2 where r = a (1 – eps) at the perihelion. If the sign of omega is reversed the direction of precession is reversed and teh theory gives forward or retrograde precessions. The apsidal method for low eccentricity orbits is simple and very useful for calculating precessions. So ECE2 theory has progressed far in advance of the standard model, and there have been no objections for over a decade. I will give the dtails of teh calculation tomorrow and then write up Sections 1 and 2 of UFT403, hich gives three or four methods ofproving rpecession from the ECE2 force equation. Tis verifies HOsrts’ excellent numerical work. The numerical method to machineprcissiion is the most accurate method bu the analytical approximations are every useful.

This is given by Eq. (7) in the approximation of Eq. (5), and by Eq. (10) in the next approximation, the equation following equation (8). These equations can be used to describe any precessing orbit with precision and the ECE2 equation is clearly preferred to the Einstein equation because the latter has been refuted in nearly a hundred different ways in the UFT series and refuted in many other ways by many authors for many years. It is used only by dogmatic ostriches who hope that ECE2 will go away, but it’s here to stay.

a403rdpapernotes5.pdf