Suppose that you can just barely see a certain star that is at a distance of 100 light years. How far away must a star that is 4 times more luminous be barely be seen/how about a star that is 100 times more luminous?

If you need to make a star to appear 4 times more luminous becuase it is because it is barely seen , then you will need to make the star appear 4 times dimmer than it would at 100 l.y. distance. The inverse propostional distance is the square root of 4 which means that you will have to double the distance by multiplying 2(the sq. rt. of 4) with 100 L.Yrs. which is equal to 200 Light Years.

To determine how far away a star must be to be barely seen, we need to consider its luminosity and compare it to the luminosity of the star that is barely visible at a distance of 100 light years.

Luminosity refers to the amount of light energy a star emits per unit of time. It is usually measured in terms of the Sun's luminosity (L☉), which is equivalent to about 3.828 × 10^26 watts.

Given that the star at a distance of 100 light years is barely visible, we can assume it has a luminosity close to the threshold of visibility. Let's call this luminosity L1.

Now, let's consider a star that is 4 times more luminous. We will call its luminosity L2. To calculate how far away this star must be to be barely visible, we need to determine the ratio of the luminosities:

Luminosity ratio (L2/L1) = 4

Since luminosity decreases with distance squared (due to the inverse square law), we can use this ratio and apply it to the formula:

(L2/L1) = (D1/D2)^2

Where D1 is the distance to the original star (100 light years) and D2 is the distance to the star we want to find.

Rearranging the formula, we have:

D2 = √((D1^2) * (L1/L2))

Plugging in the values, we get:

D2 = √((100^2) * (1/4)) = 25 light years

So, a star that is 4 times more luminous must be located about 25 light years away to be barely visible.

Similarly, let's consider a star that is 100 times more luminous. We can follow the same calculations:

Luminosity ratio (L3/L1) = 100

Using the inverse square law equation:

D3 = √((D1^2) * (L1/L3))

Plugging in the values, we get:

D3 = √((100^2) * (1/100)) = 10 light years

Therefore, a star that is 100 times more luminous must be located about 10 light years away to be barely visible.