To find the luminosity of the star, we can use the Luminosity Distance Formula. The formula links the apparent brightness (AB) and the distance (r) of a star to its luminosity (L):
[tex]\[ \text{Apparent Brightness (AB)} = \frac{\text{Luminosity (L)}}{4 \pi r^2} \][/tex]
Given:
- Apparent Brightness, [tex]\( AB = 2.4 \times 10^{-10} \ \text{watts/m}^2 \)[/tex]
- Distance, [tex]\( r = 4 \times 10^{17} \ \text{meters} \)[/tex]
We need to solve for the luminosity, [tex]\( L \)[/tex]. Let's rearrange the formula to isolate [tex]\( L \)[/tex]:
[tex]\[ L = AB \times 4 \pi r^2 \][/tex]
Step-by-step solution:
1. Identify the Given Data:
- Apparent Brightness, [tex]\( AB = 2.4 \times 10^{-10} \ \text{watts/m}^2 \)[/tex]
- Distance, [tex]\( r = 4 \times 10^{17} \ \text{meters} \)[/tex]
2. Substitute the Given Data into the Formula:
[tex]\[ L = (2.4 \times 10^{-10} \ \text{watts/m}^2) \times 4 \pi (4 \times 10^{17} \ \text{meters})^2 \][/tex]
3. Calculate the Distance Squared:
[tex]\[ (4 \times 10^{17} \ \text{meters})^2 = 16 \times 10^{34} \ \text{meters}^2 \][/tex]
4. Multiply by [tex]\( 4 \pi \)[/tex]:
[tex]\[ 4 \pi (16 \times 10^{34} \ \text{meters}^2) \][/tex]
[tex]\[ = 64 \pi \times 10^{34} \ \text{meters}^2 \][/tex]
5. Multiply by the Apparent Brightness:
[tex]\[ L = (2.4 \times 10^{-10} \ \text{watts/m}^2) \times (64 \pi \times 10^{34} \ \text{meters}^2) \][/tex]
6. Simplify the Expression:
[tex]\[ L = 2.4 \times 64 \pi \times 10^{-10} \times 10^{34} \][/tex]
[tex]\[ = 153.6 \pi \times 10^{24} \][/tex]
Using [tex]\( \pi \approx 3.141592653589793 \)[/tex]:
7. Perform the Final Calculation:
[tex]\[ L \approx 153.6 \times 3.141592653589793 \times 10^{24} \][/tex]
[tex]\[ \approx 482.5486315913922 \times 10^{24} \][/tex]
[tex]\[ \approx 4.825486315913922 \times 10^{26} \ \text{watts} \][/tex]
Therefore, the luminosity of the star is approximately [tex]\( 4.825 \times 10^{26} \ \text{watts} \)[/tex].
From the given options, the correct answer is [tex]\( 4.825 \times 10^{26} \ \text{watts} \)[/tex].
