A double star or visual double is a pair of stars that appear close to each other in the sky as seen from Earth when viewed through an optical telescope.
In absence of external force Centre of mass of double star moves like a free particle. In Centre of mass frame both stars moving in a circle about the Centre of mass which is at rest and both star are diametrically opposite to each other.
Thus in our frame of reference, the trajectories of the stars are a combination of (i) uniform motion in a straight line of the Centre of mass and (ii) circular orbits of the stars about the Centre of mass.
As per definition of vector product :-
angular displacement of particle about z axis
differentiate with respect to time