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 centripetal force and gravitational force.

### Step-by-Step Solution:

1. **Identify the given data:**
- Radius of star alpha's orbit, \( R_{\alpha} = 1.00 \times 10^9 \text{ km} \)
- Radius of star beta's orbit, \( R_{\beta} = 5.00 \times 10^8 \text{ km} \)

2. **Use the relationship between the masses and the radii:**
In a binary star system, the centripetal force acting on each star due to the gravitational attraction between them can be expressed as:
\[ M_{\alpha} \cdot R_{\alpha} = M_{\beta} \cdot R_{\beta} \]
where \( M_{\alpha} \) is the mass of star alpha and \( M_{\beta} \) is the mass of star beta.

3. **Rearranging the equation to find the mass ratio:**
We can rearrange the equation to find the ratio of the masses:
\[ \frac{M_{\beta}}{M_{\alpha}} = \frac{R_{\alpha}}{R_{\beta}} \]

4. **Substituting the values of the radii:**
Substitute 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. **Simplifying the ratio:**
Simplifying the fraction:
\[ \frac{M_{\beta}}{M_{\alpha}} = \frac{1.00}{5.00} \times \frac{10^9}{10^8} = \frac{1}{5} \times 10 = 2 \]

6. **Conclusion:**
Therefore, the ratio of the masses of star beta to star alpha is:
\[ \frac{M_{\beta}}{M_{\alpha}} = 2 \]

### Final Answer:
The ratio of the masses of star beta to star alpha is \( 2:1 \).