A binary star consists of two stars A (mass $2.2M_{s})$ and B (mass $11M_{s}),$ where $M_{s}$ is the mass of the sun. They are separated by distance d and are rotating about their centre of mass, which is stationary. The ratio of the total angular momentum of the binary star to the angular momentum of star B about the centre of mass is

Step 1: Identify the given parameters.

The masses of the two stars and their separation distance are given.

$$m_A = 2.2 M_s$$ $$m_B = 11 M_s$$ $$d = r_A + r_B$$

Step 2: Relate the orbital radii to the center of mass.

Since the stars rotate about their stationary center of mass, the position of the center of mass satisfies the balancing condition.

$$m_A r_A = m_B r_B$$ $$\frac{r_A}{r_B} = \frac{m_B}{m_A}$$

Step 3: Express the orbital radii in terms of the separation distance.

Using the total distance $d = r_A + r_B$, we find the individual radii.

$$r_A = \frac{m_B}{m_A + m_B} d = \frac{11}{13.2} d = \frac{5}{6} d$$ $$r_B = \frac{m_A}{m_A + m_B} d = \frac{2.2}{13.2} d = \frac{1}{6} d$$

Step 4: Write the expressions for the angular momenta.

Both stars rotate with the same angular velocity $\omega$ about the center of mass.

$$L_B = m_B r_B^2 \omega$$ $$L_A = m_A r_A^2 \omega$$ $$L_{\text{total}} = L_A + L_B = \left(m_A r_A^2 + m_B r_B^2\right) \omega$$

Step 5: Calculate the ratio of the total angular momentum to the angular momentum of star B.

Divide the total angular momentum by the angular momentum of star B.

$$\frac{L_{\text{total}}}{L_B} = 1 + \frac{m_A r_A^2}{m_B r_B^2}$$

Substitute the ratio $\frac{r_A}{r_B} = \frac{m_B}{m_A}$ into the expression.

$$\frac{m_A r_A^2}{m_B r_B^2} = \frac{m_A}{m_B} \left(\frac{m_B}{m_A}\right)^2 = \frac{m_B}{m_A}$$ $$\frac{L_{\text{total}}}{L_B} = 1 + \frac{m_B}{m_A}$$

Step 6: Substitute the given mass values to find the final ratio.

Substitute $m_A = 2.2 M_s$ and $m_B = 11 M_s$.

$$\frac{L_{\text{total}}}{L_B} = 1 + \frac{11}{2.2}$$ $$\frac{L_{\text{total}}}{L_B} = 1 + 5 = 6$$

Answer: The ratio of the total angular momentum of the binary star to the angular momentum of star B about the centre of mass is 6.