Determining distance using luminosity

[bobo1455]

I have a question where I am asked to determine the apparent distance in light years using brightness and luminosity: With a very large telescope on a very dark night we can detect a star like our Sun that appears just 10^−20× as bright as our Sun does during the day. Use this information to determine how far that very faint sun is from Earth.

The answer choices are: (A) 1600 ly, (B) 6000 ly, (C) 16,000 ly, (D) 60,000 ly, (E) 160,000 ly

The answer I calculated was closest to (D), my exact answer that I got is 49,125.553 ly and here is how I got that:

Using the formula d^2 = L / 4*pi*B
Where L = luminosity in watts, B = apparent brightness and d^2 is the distance in meters

so the question said bright as the Sun, so I used the Sun's luminosity which is 3.8 x 10^26 Watts
and for brightness B, I calculated it using the apparent brightness of the Sun, which I searched was 1.4 × 10^3 W/m^2 and then multiplied it by 10^-20, because the question said that the brightness of the star we are looking at has a brightness of 10^-20 x brightness of the Sun.

So then I plugged in those values and ended up with an answer in meters and converted to light years and got approx. 50,000 ly.

That's my shot at the answer, but I still don't fully understand how I did this question.

 

[mfb]

You made this more complicated than necessary, and the result looks wrong. All you need are relative values and the distance between Earth and sun.

A star at the same distance as our sun appears as bright as our sun, a star at 10 times the distance appears 10^-2 as bright as our sun and so on. The result is very close (within 2%) of one of the given answers.

 

[bobo1455]

so I can get the answer without using the formula I have?

 

[mfb]

Yes.

 

[bobo1455]

Ok, so I converted distance between Earth and Sun (1 AU) to light years and got 149,600,000 km.

So if this star is 10^-20 as bright as the Sun, then it would be 10^20 times as farther?

How do know if a star is 10 times the distance than its brightness is 10^-2?

 

[mfb]

Ok, so I converted distance between Earth and Sun (1 AU) to light years and got 149,600,000 km.

That is not converted to light years, it is converted to km.

So if this star is 10^-20 as bright as the Sun, then it would be 10^20 times as farther?

No.

How do know if a star is 10 times the distance than its brightness is 10^-2?

See the square in d^2 = L / (4*pi*B): If you increase d by a factor of 10, B goes down by a factor of 100.
Alternatively: In 10 times the distance, the area (that gets light from the star) increases by a factor of 10²=100, so the light per area goes down by the same factor.

 

[bobo1455]

distance between Earth and Sun is 1 AU, which is equal to 1.58128451 × 10^-5 light years.

Since I'm given the brightness times sun's brightness, I need to find the n times the distance using 10^-20 and then multiply 1.58128451 × 10^-5 light years by n, right?

Also in my textbook, there is a formula for light-collecting area of a telescope that is d^2, where d is diameter and then when you square it, it gives you what the increase in brightness is in comparison to the other telescope.

Would I be able to use that here? Say, d^2 = 10^-20 and then square root both sides and get 10^-10 and multiply that by the distance 1.58128451 × 10^-5 still gives me a wrong answer.

 

[bobo1455]

got the answer lol, thanks for the help mfb