Does the Sun's spin have effect on the orbit of the Earth?
Does the Sun's spin have affect on the orbit of the Earth?
It has a very, very small effect. The Sun would have a larger equatorial bulge if it rotated faster, and a smaller equatorial bulge if it rotated slower. An equatorial bulge increases gravitational acceleration near the plane of the body's equator and decreases gravitational acceleration above below the body's poles.
Since the Sun is already rotating rather slowly, it doesn't have much of an equatorial bulge. Moreover, the effects of oblateness drop off as 1/R^4, as opposed to the 1/R^2 drop-off in Newton's law of gravitation for a point mass.
The combination of small oblateness, 1/R^4 drop-off, and large distances makes the effects of the Sun's oblateness on the planet's orbits very small.
The effect is very, very small, about three orders of magnitude smaller than the general relativistic effect on Mercury's precession (which is already very small, about 43 arc seconds per century). The primary effect of the Sun's oblateness is to make Mercury precess by 0.0286 arc seconds per century. That said, the extremely precise observations of Mercury's orbit about the Sun do make that tiny precession statistically important because the uncertainty in Mercury's precession rate is down to 0.0015 arc seconds per century, according to Park, Ryan S., et al. "Precession of Mercury’s Perihelion from Ranging to the MESSENGER Spacecraft." The Astronomical Journal 153.3 (2017): 121.