Precession angle per orbit of perhelion precession mercury

[fakrudeen]

Many derivations of mercury precession angle per orbit $$\Delta$$$$\Phi$$, equate this to extra angle per orbit $$\delta$$ in excess of $$2\pi$$

that is $$\delta$$ = $$\Delta$$$$\Phi$$

But I am getting this relation

$$\sqrt{4\pi\delta}$$= $$\Delta$$$$\Phi$$

from arc length of helix and circular orbit.
What am I doing wrong?
What is the intution for above approximation? Shouldn't the total angle be less than $$2\pi$$ + $$\Delta$$$$\Phi$$
because planet takes short cut [analagous to hypotenuse length being less than sum of other two sides]

 

[fakrudeen]

I was trying to approximate perihelion precession as helix and its arc length.

It turns out the approximation used is simple angular velocity addition

if angular velocity about ecliptic is w₀ and angular velocity of precession is w₁,
then their vector sum is the real angular velocity
$$\omega$$= $$\omega$$₀ + $$\omega$$₁

now in one period T, $$\omega$$₀ contributes 2$$\Pi$$ because it is the motion in ecliptic and $$\omega$$₁'s contribution is precession per orbit by definition which adds up to angle contributed by real motion.