Thanks for the detailed analysis. It will take me some time to digest. My numerical experiments indicate that $a$ and $e$ are indeed constant but the direction of $a$ rotates by a fixed angle each orbit. If I use different values of $K$ then the magnitude of the rotation angle is linearly proportional to $K$. It is prediction of the rotation angle which I am most interested in.

You mean the direction of periapse. Yes, that can be looked at using perturbation theory too.

To be precise I seek the apsidal precession which according to wikipedia is "the first derivative of the argument of periapsis, " en.wikipedia.org/wiki/Apsidal_precession, ideally in terms of other orbital parameters such as period, eccentricity and semi-latus rectum.

A quick trial of your equation for ω (rotation of periapse with angular frequency) for Mercury, with K=3, gives 47 arc seconds per century (109% of the official value 43"). Would you say that error of 9% is reasonable for 1st-order peturbation analysis? Anyway I am greatly impressed with the result, but will check further tomorrow:-).

Hmm, no 9% seems too large an error here, as $v_c\ll c$. So perhaps something is wrong with my algebra or you plugged in slightly wrong numbers (the result depends sensitively on $e$).

I posted details of parameters and values I used for Mercury in UPDATE at end of Question (above). Changing e by +/- 0.0001 still gives 47 "/C. I will check for Mars.

Examining asteroid orbits the error increases exponentially with eccentricity. Oops. I have been using True Anomaly rather than Eccentric Anomaly. I need to re-code.

Correction. It wasnt a matter of using True Anomaly rather than Eccentric Anomaly in my code.

In your last line I think maybe $2+e^2$ needs to be replaced by $2-e^2$. This gives better agreement with GR calculated values, but still error increases exponentially with eccentricity e.

I have corrected the algebra. But I think the formula still disagrees, with the disagreement reduced by a factor 2. I get 44.878879264914915898 arcsec/century.

I agree. I have added more data in Update 2 to main question. Your analysis is considerably beyond my experience so I dont think I will be able to detect anything other than very simple algebraic mistakes. Oops - I just saw your correction retains the $2+e^2$ term by amending the input formula. I will have to Update again.

Added Update 3 to main Question. Nice to see the simpler expression for periapse rotation. But maybe it needs to scale by 1/(2.pi)?

Aha, I see now, Omega is (2.pi/T) i.e. revs/sec not rads/sec. Still the new formula like the others is increasingly divergent as e increases. For example the periapse rotation rate for Icarus (e = 0.8269 ) is 3.16 times the GR-derived rate.

How did you obtain the GR-derived rate?

(a) I used this formula to derive GR rate:- upload.wikimedia.org/math/8/d/a/… which is presented in en.wikipedia.org/wiki/Apsidal_precession. (b) My own software orbit simulator program gives matching apsidal rotation when I apply transvere acceleration of the form originally presented in my Question (At = Ar.K.Vr.Vt/c^2) with K=3.

(Purely as an unjustified fudge at present) I find that if you change the exponent in your expression for little omega from -2 to -1 this gives excellent agreement with GR for all bodies.

Also I find that I do have to scale your expression for little omega by (2.pi)^-1.

I have just realized that if we change the exponent from -2 to -1 in your formula for little w then the formula with K=3 becomes identical to the GR formula. Justification for changing the exponent may come from re-analysing your derivation of Vt from V-Vr which I cannot presently understand algebraically.

That's indeed intriguing. However, I don't think there is anything wrong with the definition of $v_t$. What can't you understand? Are you sure your initial formula was correct (in particular the interpretation of transverse)?

I appreciate something of the origins of the quoted periapse rotation rate (PRR) formula. I'm not sure Einstein was first to use it, just the first to present a widely-accepted model (GR) from which it could be derived. The widely-condemned model of Gerber (1898) also leads to it. My transverse acceleration (TA) formula (At=Ar.3.Vr.Vt/c^2) was discovered in developing an independent physical model and when I plug TA into my orbital simulator it produces “correct” anomalous precession rates (i.e in line with GR and observations). [continued...]

[...] If your peturbation analysis of TA led to the PRR formula that would be a VERY interesting by-product, not one I had envisaged. But my immediate goal is to seek confirmation that TA gives correct anomalous periapse rotation rates. My simulator confirms it and I have good reasons to trust it. But it would be nice to have independent verfication such as through peturbation analysis. [continued …]

[...] I confess that I can't presently understand much of the detail of your analysis because of my very limited experience/skill with celestial mechanics or even vector mechanics. For example (in Appendix: orbit averages) I can see that vectorially Vt = V - Vr but I cannot see how you expressions for V and Vr lead to the expression for Vt. Conscious of the immense help you have given, I will pore over your answer and try to work stuff out for myself as far as possible.