Chapter 4, Problem 1P

Pluto has no atmosphere and has an average orbital diameter that is 39.2 times that of the earth. The mean surface temperature is 37 K. Estimate the albedo of Pluto.

To determine

Determine the albedo value of Pluto.

Answer to Problem 1P

The albedo value of Pluto is $\underset{\_}{0.52}$.

Explanation of Solution

Given information:

The average orbital diameter of Pluto is 39.2 times of the earth.

Mean surface temperature $\left(T\right)$ is 37 K.

Pluto has no atmosphere.

Calculation:

The solar flux at the outside of the earth’s atmosphere $\left(S_{\text{earth}}\right)$ is $1.367{\text{kW/m}}^{\text{2}}$.

Find the Insolation of Pluto $\left(S_{\text{Pluto}}\right)$ using the relation given below:

$\begin{array}{c}{\left(r_{\text{Pluto}}\right)}^2{S}_{\text{Pluto}}=S_{\text{earth}}\times {\left(r_{\text{earth}}\right)}^2\\ S_{\text{Pluto}}=1.367{\text{kW/m}}^{\text{2}}\times {\left(\frac{r_{\text{earth}}}{39.2r_{\text{earth}}}\right)}^2\\ =1.367{\text{kW/m}}^{\text{2}}\left(\frac{{10}^3\text{W}}{1\text{kW}}\right)\times {\left(\frac{1}{39.2}\right)}^2\\ =0.89\frac{\text{W}}{{\text{m}}^2}\end{array}$

Let the albedo is a.

The value of Stefan Boltzmann constant is $\sigma =5.67051\times {10}^{-8}\text{W}\cdot {\text{m}}^{-2}\cdot {\text{K}}^{-4}$.

The expression to find the temperature (T) is given below:

$\begin{array}{l}T={\left[\frac{\left(1-a\right)S_{\text{Pluto}}}{4\sigma }\right]}^{1/4}\\ 37={\left[\frac{\left(1-a\right)\left(0.89\right)}{4\left(5.67051\times {10}^{-8}\right)}\right]}^{1/4}\\ {37}^4=\frac{\left(1-a\right)\left(0.89\right)}{4\left(5.67051\times {10}^{-8}\right)}\\ 1-a=0.4776\end{array}$

$a=0.52$

Therefore, the albedo value of Pluto is $\underset{\_}{0.52}$.