The two stars in a certain binary star system move in circular orbits The first star `alpha` moves in an orbit of radius `1.00 xx 10^(9)km` The other star `beta` moves in an orbit of radius `5.00 xx 10^(8)km` What is the ratio of masses of star `beta` to the star `alpha` ? .

Answer

To find the ratio of the masses of star beta to star alpha in a binary star system, we can use the concept of the center of mass. The two stars revolve around their common center of mass, and the distances from the center of mass are inversely proportional to their masses.

### Step-by-Step Solution:

1. **Identify the Radii of Orbits**:
- The radius of the orbit of star alpha, \( r_{\alpha} = 1.00 \times 10^9 \, \text{km} \)
- The radius of the orbit of star beta, \( r_{\beta} = 5.00 \times 10^8 \, \text{km} \)

2. **Use the Center of Mass Concept**:
- For two objects in a binary system, the relationship between their masses and their distances from the center of mass is given by:
\[ m_{\alpha} \cdot r_{\alpha} = m_{\beta} \cdot r_{\beta} \]

3. **Rearrange the Equation**:
- We want to find the ratio of the masses \( \frac{m_{\beta}}{m_{\alpha}} \):
\[ \frac{m_{\beta}}{m_{\alpha}} = \frac{r_{\alpha}}{r_{\beta}} \]

4. **Substitute the Values**:
- Plug in the values of \( r_{\alpha} \) and \( r_{\beta} \):
\[ \frac{m_{\beta}}{m_{\alpha}} = \frac{1.00 \times 10^9 \, \text{km}}{5.00 \times 10^8 \, \text{km}} \]

5. **Calculate the Ratio**:
- Simplifying the ratio:
\[ \frac{m_{\beta}}{m_{\alpha}} = \frac{1.00}{0.50} = 2 \]

6. **Final Result**:
- The ratio of the masses of star beta to star alpha is:
\[ \frac{m_{\beta}}{m_{\alpha}} = 2 \]

### Summary:
The ratio of the masses of star beta to star alpha is 2.

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