Use the Luminosity Distance Formula to find the distance to a star.

Given:
- Luminosity, [tex]\( L = 5.2 \times 10^{23} \)[/tex] watts
- Apparent brightness at Earth, [tex]\( 1.0 \times 10^{-10} \)[/tex] watts/m[tex]\(^2\)[/tex]

Formula:
[tex]\[ \text{Apparent Brightness (AB)} = \frac{\text{Luminosity (L)}}{4 \pi r^2} \][/tex]

Choose the correct distance from the options below:

A. [tex]\( 2.034 \times 10^{-16} \)[/tex] m
B. [tex]\( 3.251 \times 10^{16} \)[/tex] m
C. [tex]\( 2.034 \times 10^{16} \)[/tex] m
D. [tex]\( 3.251 \times 10^{-16} \)[/tex] m



Answer :

To find the distance to a star using the Luminosity Distance Formula, let's follow these steps:

1. Understand the Given Data:
- Luminosity ([tex]\( L \)[/tex]) = [tex]\( 5.2 \times 10^{23} \)[/tex] watts
- Apparent Brightness ([tex]\( AB \)[/tex]) = [tex]\( 1.0 \times 10^{-10} \)[/tex] watts/m[tex]\(^2\)[/tex]
- Formula to use: [tex]\( AB = \frac{L}{4 \pi r^2} \)[/tex]

2. Rearrange the Formula to Solve for Distance ([tex]\( r \)[/tex]):
- The formula [tex]\( AB = \frac{L}{4 \pi r^2} \)[/tex] can be rearranged to solve for [tex]\( r \)[/tex]:
[tex]\[ AB \times 4 \pi r^2 = L \][/tex]
[tex]\[ 4 \pi r^2 = \frac{L}{AB} \][/tex]
[tex]\[ r^2 = \frac{L}{4 \pi AB} \][/tex]
[tex]\[ r = \sqrt{\frac{L}{4 \pi AB}} \][/tex]

3. Plug in the Given Values:
- [tex]\( L = 5.2 \times 10^{23} \)[/tex] watts
- [tex]\( AB = 1.0 \times 10^{-10} \)[/tex] watts/m[tex]\(^2\)[/tex]

4. Calculate the Intermediate Value [tex]\( r^2 \)[/tex]:
[tex]\[ r^2 = \frac{5.2 \times 10^{23}}{4 \pi \times 1.0 \times 10^{-10}} \][/tex]
[tex]\[ r^2 = \frac{5.2 \times 10^{23}}{4 \pi \times 10^{-10}} \][/tex]

Using the fact that [tex]\( 4 \pi \approx 12.5664 \)[/tex], we get:
[tex]\[ r^2 = \frac{5.2 \times 10^{23}}{12.5664 \times 10^{-10}} \][/tex]
[tex]\[ r^2 \approx 4.138 \times 10^{32} \][/tex]

5. Solve for [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{4.138 \times 10^{32}} \][/tex]
[tex]\[ r \approx 2.034 \times 10^{16} \, \text{meters} \][/tex]

Thus, the distance to the star is approximately [tex]\( 2.034 \times 10^{16} \)[/tex] meters.

Given the options:
- [tex]\( 2.034 \times 10^{-16} \)[/tex] m
- [tex]\( 3.251 \times 10^{16} \)[/tex] m
- [tex]\( 2.034 \times 10^{16} \)[/tex] m
- [tex]\( 3.251 \times 10^{-16} \)[/tex] m

The correct answer is [tex]\( 2.034 \times 10^{16} \)[/tex] meters.