fixed the v_c in v_t; added "(for K=3)". why is k_hat component included but e_hat component excluded from final reckoning? What do you mean? To which part of the answer are you refering? The change of the eccentricity vector is in the direction of vector k (the component in direction of e averages to zero), which is exactly the same as a rotation around the vertical (direction h) axis. In the summary, I only refer to the amount of rotation.

Ah, I now understand. I'll add an explanation (that the component in dirction of e_hat vanishes) to the appendix.

@Walter.I failed to stress that direction of extra acceleration is sensitive to sign of Vr. Before apoapsis (Vr>0) the extra acceleration points "forwards" in same direction as Vt. But after apoapsis (Vr<0) and acceleration points "backwards" in opposite direction to Vt. So the (first order) orbit-average extra acceleration is directed towards apoapsis (i.e - e_hat) and has zero k_hat component. Dont know if your analysis takes account.