Feed: Dr. Myron Evans
Posted on: Tuesday, May 14, 2013 11:25 PM
Author: metric345
Subject: Computing Note 242(5)
Many thanks very good work as usual. I would say that both constants of integration A and B have to be used to obtain the Newtonian result. I would suggest going back a little bit and checking eq. (3) of note 242(4) for t against the Newtonian result for t . I will calculate the Newtonian result for t this morning. This procedure determines the constants of integration A and B. Then having determined the constants A and B the true anomaly can be found.
To: EMyrone@aol.com
Sent: 14/05/2013 22:30:05 GMT Daylight Time
Subj: Re: 242(5): The True Anomaly for a Given Distance R between m and M for any Force
I did some calculations for this note. The outer integral is not solveable analytically for the Newton force unless the integration constant is A=0. This solution is %o12 and is plotted in %t14. This does not look like theta(R) for an ellipse.
After that I solved the integral numerically. %t24 shows a plot of the integrands for A>0, A=0, A<0. Only for A<0 the integrandremains finite, in accordance with my earlier findings for the integration constant A.
The numerical results of the integral and the function theta ~ integral / R^2 are plotted in %t35. theta drops like 1/R, also this does not look likely for an ellipse. This needs further checks and study.
The known solution (8) contains a parameter epsilon. I wonder where this comes from in the general solution (2).
Horst
Am 14.05.2013 17:50, schrieb EMyrone
This note gives some more details of the method, notably the inner integral must be an indefinite integral as in eqs. (2) and (4). The outer integral is a definite integral from 0 to R. This note also gives another baseline check of the method in the Newtonian theory, in which eqs. (1) and (9) must be the same. When the baseline checks are complete the theory may be used to compute the perihelion precession for any force and any planar orbit.
242(5).pdf
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