Question

Earth is about 150 million kilometers from the Sun, and the apparent brightness of the Sun in our sky is about1300 watts/m2. Use these two facts and the inverse square law for light to answer the questions.

1. Determine the apparent brightness that we would measure for the Sun if we were located half Earth’s distance from the Sun.

2. Determine the apparent brightness that we would measure for the Sun if we were located twice Earth’s distance from the Sun.

3. Determine the apparent brightness that we would measure for the Sun if we were located 7 times Earth’s distance from the Sun.

Answer 1

1.
`I=5200(W)/(m^2)`

2.
`I=325(W)/(m^2)`

3.
`I=26.53(W)/(m^2)`

Step-by-step explanation:

The apparent brightness is inversally proportional to the square of distance, and it can be written as:

`(I_(1))/(I_(2)) =(r_(2)^2)/(r_(1)^2)`

Where, for instance, state 1 is the Earth (given, known values) and state 2 is the unknown value. Knowing a new distance leads to a known apparent brightness, and viceversa.

This is a very simple proportionallity law, so, rearranging the equation:

`I_(2)=I_(1)(r_(1)^2)/(r_(2)^2)=I_(1)((r_(1))/(r_(2)))^2`

1. For a distance of half the distance from the Sun (i.e., 0.5 AU, regardless its actual value since we are using AU multiples):

`I_(2)=1300\cdot2^2=5200(W)/(m^2)`

2. Similarly, for r2=2AU:

`I_(2)=1300\cdot0.5^2=325(W)/(m^2)`

3. Finally, for r2=7AU:

`I_(2)=1300\cdot(1)/(7^2)=26.53(W)/(m^2)`