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= \omega₀ + \omega₁

now in one period T, \omega₀ contributes 2\Pi because it is the motion in ecliptic and \omega₁'s contribution is precession per orbit by definition which adds up to angle contributed by real motion.