Figure: Explication of the Solar System center of mass motion relative to the Sun

Caption: The motion of the Solar System center of mass relative to the Sun, 1945--1995.

Features:

1. The "nucleus of Sun" is the Solar core (radius ≅ 0.2*( solar radius R_☉ = 6.957*10**5 km = 109.1 R_eq_⊕ = 4.650*10**(-3) AU)) and the "limb of Sun" in this context is the solar photosphere (solar radius R_☉ = 6.957*10**5 km = 109.1 R_eq_⊕ = 4.650*10**(-3) AU).

   The term limb (i.e., astronomical limb) in astronomy is the curved edge of the apparent disk of a astronomical object: e.g., lunar limb. Recall in astronomy, "apparent" means as seen from the Earth.

2. The Solar System center of mass motion is determined only by the gravitational field EXTERNAL to the Solar System. The EXTERNAL gravitational field is due to all the rest of the observable universe, but the main effective component of this EXTERNAL gravitational field for motion relative to the Milky Way center (AKA Galactic center of mass): distance = 8.122(31) kpc = 26.490(100) kly is due to the Milky Way itself.

3. The tidal forces due to the EXTERNAL gravitational field are completely negligible for Solar System (which also true for almost all other planetary systems), and thus all motions relative to the Solar System center of mass are determined by INTERNAL forces overwhelmingly dominated by the INTERNAL gravitational field.

4. The Solar System center of mass is, in fact, the origin for the center-of-mass inertial frame of the Solar System (i.e., the most relevant inertial frame for the INTERNAL motions of the Solar System: i.e., the motions relative to the Solar System center of mass).

5. Since the Sun contains 99.86 % of the Solar System's known mass (Wikipedia: Solar System: Composition), the Sun's center is to 1st order at the Solar System center of mass. However, as the image shows, the Solar System center of mass moves around depending on the overall arrangement of Solar System astro-bodies and is only sometimes inside the Sun.