Inverse Square Law of Radiation
        Stars that appear bright to us can either be very bright or very close, and stars that appear dim can either be very dim or very far away.  Far-away stars appear dimmer than nearby ones because the light that they emit is spread out over the vast distance of space before it reaches us  To see how this happens, look at the diagram shown below

As a portion of the light travels away from the star, it covers the small square at one distance unit from the star.   At two distance units from the star, that same amount of light would spread out to cover four of the squares that it covered when it was one unit away.  At a larger distance of three units, the light would have spread out even more and would cover nine of the squares.  At the even large distance of four units, the light would be so spread that it would cover 16 squares.  Notice that the number of squares that the light covers is related to the square of the number of distance units from the source.

    The same amount of light keeps spreading out as it gets farther and farther from its source.  Therefore, as you get farther from the source, there is less light intensity or birhgtness at any one spot.  We can express this mathematically as follows:

                 Brightness at spot 1  =  (distance of spot 2 from light source) 2
             Brightness at spot 2     (distance of spot 1 from light source)

form 2
            Brightness at spot 1 = Brightness at spot 2 x (distance of spot 2 from light source) 2
                                                                                                                                                 (distance of spot 1 from light source)
 
        The distance can be in any units that we  choose, as long as they are both in the same units.  We will express brightness as some number of times as bright as the sun.

            Let's see how we  can use this little piece of mathematics.  Suppose a certain star appears 2 times as bight as the Sun when it is viewed from a distance of 0.5 light years.  How bright would it appears to an observer 10 light years away?  We will call the place that is 0.5 light years away spot 2, and the place that is 10 light years away spot 1.

                Brightness = 2 X (0.5ly)2
                                                                 10ly
                                                   = 2 X 0.052
                                                                    = 0.0005 times as bright as the Sun
 

    As another example, consider how bright the Sun appears to someone viewing it from Mars as compared to someone viewing it from Earth.  Mars is a 1.52 AU from the Sun and Earth is at 1 AU.  We will consider the brightness of the Sun as viewed from Earth to be 1 brightness unit.  The Earth in this case will be spot 2 and mars will be spot 1.

                Brightness = 1 X (1AU)2
                                                                1.52AU
                                                   = 2 X 0.6582
                                                                    = 0.43 as bright as when viewed from Earth

EXERCISE -Inverse Square Law of Radiation
1.  A certain star appears 4 times as bright as the sun when viewed from a distance of 1 light year.  How bright would it appear when seen from a distance of 4 light years?
 
 

2.  A certain star appears 10 times as bright as the Sun when viewed from a distance of 2 light years.  How bight would it appear if viewed from a distance of 1 light year?
 
 

3.  How many times as bright would the sun appear to someone viewing it from Venus at 0.72 AU) as compared to someone viewing it from Earth (at 1 AU)?
 
 

4.  How many times as bright would the Sun appear to someone viewing it from  Jupiter (at 5. 20 AU) as compared to someone viewing it from Earth (at 1 AU)?
 
 

5.  How many times as bright would the Sun appear to someone viewing it from Pluto (at 39.4 AU) as compared to someone vewing it from Earth (at 1 AU)?