Attempted to reconcile below derived equation (3) approach with 2-body Newtonian dynamic force equilibrium (4).They were compared in a numeric check, but no luck. Standard notation is used.
$$ \tan \psi =\frac{-2 \omega^2}{\alpha} \,; \,\tan \psi =\frac{-r^3\alpha}{\mu} ; $$
EDIT 1:
{Error in the above (4): $ \tan \psi $ should be perpendicular accln/radial accln.)
In this classic model at ends of major axis for $\psi$ to be $ \pi/2 ,\, \alpha $ should be infinite but that is never so.
What do I miss? Thanks for error trapping help.