Chapter 33, Problem 9P

To determine

The relative brightness of the Sun as observed from Jupiter in comparison to its brightness from Earth.

Answer to Problem 9P

The relative brightness of the Sun as observed from Jupiter is $0.037$ times the relative brightness of the Sun as observed from Earth.

Explanation of Solution

Given:

Assume the distance between Jupiter and Earth is denoted by r .

The distance between Jupiter and Sun is denoted by 5. 2r .

Formula used:

The expression for the brightness of the solar radiation on a planet is,

  $I=\frac{P}{4\pi r^2}$

Here, P is the Sun’s output power and r is the distance of the planet from the Sun.

Calculation:

The brightness of the solar radiation on Earth is

  $I_{Earth}=\frac{P}{4\pi r^2}$ …… (1)

The brightness of the solar radiation on Jupiter is,

  $\begin{array}{l}{I}_{Jupiter}=\frac{P}{4\pi {\left(5.2r\right)}^2}\\ I_{Jupiter}=\frac{1}{27.04}\left[\frac{P}{4\pi r^2}\right]\end{array}$ …… (2)

Divide equation (2) by equation (1).

  $\begin{array}{l}\frac{I_{Jupiter}}{I_{Earth}}=\frac{\left[\frac{1}{27.04}\left(\frac{P}{4\pi r^2}\right)\right]}{\left(\frac{P}{4\pi r^2}\right)}\\ \frac{I_{Jupiter}}{I_{Earth}}=\frac{1}{27.04}\left(\frac{P}{4\pi r^2}\right)\times \left(\frac{4\pi r^2}{P}\right)\\ \frac{I_{Jupiter}}{I_{Earth}}=\frac{1}{27.04}\\ I_{Jupiter}=0.037I_{Earth}\end{array}$

Conclusion:

Thus, the relative brightness of the Sun as observed from Jupiter is $0.037$ times the relative brightness of the Sun as observed from Earth.