Oops. I just noticed a typo in an equation I wrote in my last post.
In the equation where I wrote the eccentricity e of the Newtonian
ellipse in terms of the parameters E and h I had written:
e = sqrt(1 - |E|*h^2/(2*m*G^2*M^2))...
Of course the correct version of this equation is actually:
e = sqrt(1 - 2*|E|*h^2/(m*G^2*M^2)) .
It seems I accidentally put the factor of 2 in the denominator
rather than the numerator on the 2nd term in the sqrt() function.