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## precession angle per orbit of perhelion precession mercury

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]
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