I wonder if coronal mass ejections (CMEs) effect the movement of the Sun at all? Newton’s Law: for every action there is an equal and opposite reaction.
Those are tiny in mass in comparison with the Sun, but velocity also matters, so there’s some kind of movement of the whole works. OTOH, CMEs aren’t confined to one side, so they probably cancel out. And standing on Mercury, one couldn’t even use ‘em to make smores. ;’)