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 $e$).

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)?

Btw, the formula you're quoting originates from Albert Einstein (1915), but didn't use your acceleration formula.

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.