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.

 

[Entropia]

First, it's important to note that the precession angle per orbit of perihelion precession of Mercury is not a fixed value, but rather varies depending on the specific model used to calculate it. This is because there are multiple factors that contribute to the precession of Mercury's orbit, such as the gravitational influence of other planets and the shape of the Sun.

In terms of your specific derivation, it's difficult to determine where the discrepancy may lie without seeing your calculations. However, it's possible that your equation may be an approximation or simplification of the more complex equations used to calculate Mercury's precession angle.

As for the intuition behind the approximation, it's likely that it is based on the assumption that the precession angle is small in comparison to the total angle traveled by Mercury in one orbit. This would make sense, as the precession angle is a relatively small deviation from the overall circular orbit of Mercury around the Sun.

In terms of your question about whether the total angle should be less than 2\pi + \Delta\Phi, it's important to remember that the precession angle is not a separate angle that is added on top of the circular orbit, but rather a deviation from it. So while the total angle traveled by Mercury may indeed be greater than 2\pi, the precession angle itself is still a small fraction of that total angle.