The ratio of the total angular momentum of the binary star to the angular momentum of star B about the center of mass can be determined by considering the principle of conservation of angular momentum.

Conservation of Angular Momentum:
Angular momentum is a vector quantity defined as the product of the moment of inertia and angular velocity. According to the conservation of angular momentum, the total angular momentum of a system remains constant if no external torques act on it.

In the case of the binary star system, the two stars A and B are rotating about their center of mass. The center of mass is stationary, which means the total angular momentum of the system is conserved.

Angular Momentum of the Binary Star:
The total angular momentum of the binary star can be calculated as the sum of the angular momenta of stars A and B. Let's denote the angular momentum of star A as L(A) and the angular momentum of star B as L(B).

L(total) = L(A) + L(B)

Angular Momentum of Star B about the Center of Mass:
To calculate the angular momentum of star B about the center of mass, we need to consider the mass and the distance of star B from the center of mass.

The distance between the stars A and B is denoted as d. Since the center of mass is stationary, the distance of star B from the center of mass is d/2.

The angular momentum of star B about the center of mass, L(B_cm), can be calculated using the formula:

L(B_cm) = m(B) * v(B_cm) * r(B_cm)

where m(B) is the mass of star B, v(B_cm) is the velocity of star B about the center of mass, and r(B_cm) is the distance of star B from the center of mass.

Since the stars are rotating about their center of mass, the velocity of star B about the center of mass can be calculated as v(B_cm) = ω(B_cm) * r(B_cm), where ω(B_cm) is the angular velocity of star B about the center of mass.

Using the formula for angular momentum of star B about the center of mass, we can rewrite it as:

L(B_cm) = m(B) * ω(B_cm) * (r(B_cm))^2

Ratio of Total Angular Momentum to Angular Momentum of Star B:
To find the ratio of the total angular momentum of the binary star to the angular momentum of star B about the center of mass, we can divide the total angular momentum by the angular momentum of star B:

Ratio = L(total) / L(B_cm)

Substituting the expressions for L(total) and L(B_cm), we get:

Ratio = (L(A) + L(B)) / (m(B) * ω(B_cm) * (r(B_cm))^2)

Since ω(B_cm) and (r(B_cm))^2 are the same for both stars A and B, we can simplify the ratio further:

Ratio = (L(A) / (m(B) * ω(B_cm) * (r(B_cm))^2)) + 1

L(A) / (m(B) * ω(B_cm) * (r(B_cm))^2 represents the ratio of the angular momentum of star A to the angular momentum of star B about the center of mass. Let's denote it as Ratio_AB.

Ratio_AB = L(A) / (m(B) * ω(B_cm