How does the brightness of a star depend on its distance from us?
The apparent brightness of a star is proportional to 1 divided by its distance squared. That is, if you took a star and moved it twice as far away, it would appear 1/4 as bright; if you moved it four times the distance, it would appear 1/16 as bright.
The reason this happens is simple. Imagine that you are the star, spitting out imaginary light rays in all directions. Now take each of your hands and touch your thumb to your forefinger to make a circle that you can look through. Hold both hands up in front of your face and imagine that each circle is a "telescope" that someone is using to "look" at you (in other words, to collect the light rays that you are spitting out).
Now move one hand twice as far away from your eyes as the other and look at the faraway "telescope" through the nearby one. You should see that it is only about 1/4 the size. (What has happened is that each of its two dimensions has shrunk by 1/2, so when you multiply them together to get the area it has gone down by 1/4.) This means that the second circle would only collect 1/4 as many of the light rays that you're throwing at it, so it would measure you (the star) to only be 1/4 as bright.
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