To find the distance to the star given its luminosity [tex]\( L \)[/tex] and apparent brightness [tex]\( AB \)[/tex], we will use the luminosity distance formula:
[tex]\[ AB = \frac{L}{4 \pi r^2} \][/tex]
where:
- [tex]\( L \)[/tex] is the luminosity of the star,
- [tex]\( AB \)[/tex] is the apparent brightness at Earth,
- [tex]\( r \)[/tex] is the distance from the star to Earth.
Given:
- Luminosity [tex]\( L = 2.7 \times 10^{26} \)[/tex] watts,
- Apparent brightness [tex]\( AB = 1.0 \times 10^{-10} \)[/tex] watts/m[tex]\(^2\)[/tex].
First, let's rearrange the formula to solve for [tex]\( r \)[/tex]:
[tex]\[ AB = \frac{L}{4 \pi r^2} \][/tex]
Rearranging to solve for [tex]\( r^2 \)[/tex]:
[tex]\[ 4 \pi r^2 = \frac{L}{AB} \][/tex]
[tex]\[ r^2 = \frac{L}{4 \pi AB} \][/tex]
To find [tex]\( r \)[/tex], we take the square root of both sides:
[tex]\[ r = \sqrt{\frac{L}{4 \pi AB}} \][/tex]
Now, plug in the given values for [tex]\( L \)[/tex] and [tex]\( AB \)[/tex]:
[tex]\[ r = \sqrt{\frac{2.7 \times 10^{26} \text{ W}}{4 \pi \times 1.0 \times 10^{-10} \text{ W/m}^2}} \][/tex]
Next, calculate the denominator [tex]\( 4 \pi \times 1.0 \times 10^{-10} \)[/tex]:
[tex]\[ 4 \pi \times 1.0 \times 10^{-10} = 1.256637061 \times 10^{-9} \][/tex]
Now, perform the division:
[tex]\[ \frac{2.7 \times 10^{26}}{1.256637061 \times 10^{-9}} \approx 2.149999999 \times 10^{35} \][/tex]
Take the square root of the result:
[tex]\[ r = \sqrt{2.149999999 \times 10^{35}} \approx 4.63529 \times 10^{17} \text{ meters} \][/tex]
So, the calculated distance to the star is approximately [tex]\( 4.635 \times 10^{17} \text{ meters} \)[/tex].
Given the options:
- [tex]\( 3.570 \times 10^{-17} \text{ meters} \)[/tex]
- [tex]\( 3.570 \times 10^{17} \text{ meters} \)[/tex]
- [tex]\( 4.635 \times 10^{-17} \text{ meters} \)[/tex]
- [tex]\( 4.635 \times 10^{17} \text{ meters} \)[/tex]
The correct answer is [tex]\( 4.635 \times 10^{17} \text{ meters} \)[/tex].
