## 403(9): Analytical Orbit in the Near Circular Approximation

OK many thanks, this is useful and I will look in to different solutions. This should be straightforward, and the apsidal method is very useful.

Date: Sun, Mar 18, 2018 at 5:55 PM

Subject: Re: Fwd: 403(9): Analytical Orbit in the Near Circular Approximation

To: Myron Evans <myronevans123>

I rechecked the note 403(6). All is o.k. up to eq.(41), first line. Then you replaced 1/r^2 by omega*r. With omega in the denominator, the limit omega –> 0 gives an expression 0/0 which is undefined. Can this be the reason for the wrong limit of delta_phi?

Furthermore, in (44) you assume <drdr> = const. According to the definition of omega in (40), then omega is constant and its derivative vanishes, in contrast to (44). This may be a second problem.

Horst

Am 18.03.2018 um 15:42 schrieb Myron Evans:

I will have another look at this problem of asymptotic behaviour. There must be some minor error or misconception in going from Eq.. (380 to (47) of Note 403(6) because we have already agreed on the basic apsidal method, which you checked to be correct. I will check this tomorrow. In the meantime the numerical solution of Eq. (1) could perhaps be compared with the circular approximation (11) to see if the latter is meaningful,but the numerical integration of Eq. (2) is already sufficient. I think that the Maxima result might be the more reliable. The circular approximation is useful because it gives an analytical orbit. The apsidal method is useful to show the existence of precession for any force law. The apsidal method and your numerical integration are sufficient to prove precession.

Date: Sun, Mar 18, 2018 at 11:11 AM

Subject: Re: 403(9): Analytical Orbit in the Near Circular Approximation

To: Myron Evans <myronevans123>If we assume that the approximation in eq.(10) is justified, the bigger problem seems to be to equate it to the RHS because this term has not the right asymptotic behaviour of delta_phi for epsilon –> 0 as commented for note 8.

The integral in (11) gives a somewhat different result in Maxima, see %o3 of the protocol. Maybe both results are identical when transformed.Horst

Am 17.03.2018 um 14:06 schrieb Myron Evans:

This is given by Eq. (13) from the Wolfram online integrator. This result can be checked with Maxima, which is probably more accurate than Wolfram. At the perihelion, the constant of integration A is adjusted to give exact agreement with the apsidal method, also used in the near circular approximation. This known to give precession at the perihelion. The analytical orbit is given by Eq. (15). The elliptical orbit is given by Eq. (17), which can be compared graphically with Eq. (15). The overall result is that the origin of any precession is vacuum fluctuation, a major advance in cosmology of all kinds.