To determine the distance [tex]\( r \)[/tex] to the star given its luminosity [tex]\( L \)[/tex] and its apparent brightness [tex]\( AB \)[/tex], we can use the Luminosity Distance Formula. The formula relates the apparent brightness ([tex]\( AB \)[/tex]), the intrinsic luminosity ([tex]\( L \)[/tex]), and the distance to the star ([tex]\( r \)[/tex]):
[tex]\[ AB = \frac{L}{4 \pi r^2} \][/tex]
Given data:
- Luminosity, [tex]\( L = 3.9 \times 10^{26} \)[/tex] watts
- Apparent brightness, [tex]\( AB = 2.0 \times 10^{-10} \)[/tex] watt/m[tex]\(^2\)[/tex]
We need to solve for the distance [tex]\( r \)[/tex]. Rearrange the formula to isolate [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{ \frac{L}{4 \pi AB} } \][/tex]
1. Plug in the given values:
[tex]\[ L = 3.9 \times 10^{26} \][/tex]
[tex]\[ AB = 2.0 \times 10^{-10} \][/tex]
2. Substitute these values into the formula:
[tex]\[ r = \sqrt{ \frac{3.9 \times 10^{26}}{4 \pi \times 2.0 \times 10^{-10}} } \][/tex]
3. Calculate the denominator:
[tex]\[ 4 \pi \times 2.0 = 8 \pi \][/tex]
4. Now the expression under the square root is:
[tex]\[ r = \sqrt{ \frac{3.9 \times 10^{26}}{8 \pi \times 10^{-10}} } \][/tex]
5. Perform the division inside the square root:
[tex]\[ r = \sqrt{ \frac{3.9 \times 10^{26}}{25.1327412287 \times 10^{-10}} } \][/tex] (using [tex]\( \pi \approx 3.14159 \)[/tex])
6. Simplify the expression:
[tex]\[ r = \sqrt{ \frac{3.9 \times 10^{26}}{25.1327412287 \times 10^{-10}} } \approx \sqrt{ \frac{3.9 \times 10^{26}}{25.1327412287 \times 10^{-10}} } \approx \sqrt{1.55185328066 \times 10^{36}} \][/tex]
7. Simplify the square root:
[tex]\[ r \approx \sqrt{1.55185328066 \times 10^{36}} \approx 3.93923938742745 \times 10^{17} \][/tex]
Therefore, the distance to the star [tex]\( r \)[/tex] is approximately [tex]\( 3.939 \times 10^{17} \)[/tex] meters, which matches one of the provided options.
So, the correct distance to the star is:
[tex]\[ 3.939 \times 10^{17} \, \text{m} \][/tex]
