Chapter 5, Problem 43QP

(a)

To determine

The average temperature on the planet.

Answer to Problem 43QP

The average temperature on the planet is $\underset{\_}{271.7\text{K}}$.

Explanation of Solution

The black body temperature of a planet is given by,

    $T={\left[\frac{L_{\text{sun}}\left(1-a\right)}{16\pi \sigma }\right]}^{\frac{1}{4}}\frac{1}{\sqrt{d}}$        (I)

Here, $T$ is the temperature, $L_{\text{sun}}$ is the luminosity of sun, $a$ is the fraction of sunlight that reflects from a planet, $\sigma$ is the Stefan’s constant, $d$ is the distance.

All factors except the albedo are equal so that the change in temperature compared with zero albedo is given by,

    $\begin{array}{c}\frac{T}{T_0}=\frac{{\left[\frac{L_{\text{sun}}\left(1-a\right)}{16\pi \sigma }\right]}^{\frac{1}{4}}\frac{1}{\sqrt{d}}}{{\left[\frac{L_{\text{sun}}\left(1-a_0\right)}{16\pi \sigma }\right]}^{\frac{1}{4}}\frac{1}{\sqrt{d}}}\\ =\frac{{\left(1-a\right)}^{\frac{1}{4}}}{{\left(1-a_0\right)}^{\frac{1}{4}}}\end{array}$        (II)

Conclusion:

Substitute $279\text{K}$ for $T_0$ and $0$ for $a_0$ in equation (II) to find $T$.

    $T=279\sqrt[4]{1-a}$        (III)

Substitute $0.1$ for $a$ in equation (III) to find $T$.

    $\begin{array}{c}T=279\sqrt[4]{1-\left(0.1\right)}\\ =271.7\text{K}\end{array}$

Therefore, the average temperature on the planet is $\underset{\_}{271.7\text{K}}$.

(b)

To determine

The average temperature on the planet if its albedo were $0.9$

Answer to Problem 43QP

The average temperature on the planet is $\underset{\_}{156.9\text{K}}$.

Explanation of Solution

The black body temperature of a planet is  given by,

    $T={\left[\frac{L_{\text{sun}}\left(1-a\right)}{16\pi \sigma }\right]}^{\frac{1}{4}}\frac{1}{\sqrt{d}}$        (I)

Here, $T$ is the temperature, $L_{\text{sun}}$ is the luminosity of sun, $a$ is the fraction of sunlight that reflects from a planet, $\sigma$ is the Stefan’s constant, $d$ is the distance.

All factors except the albedo are equal so that the change in temperature compared with zero albedo is given by,

    $\begin{array}{c}\frac{T}{T_0}=\frac{{\left[\frac{L_{\text{sun}}\left(1-a\right)}{16\pi \sigma }\right]}^{\frac{1}{4}}\frac{1}{\sqrt{d}}}{{\left[\frac{L_{\text{sun}}\left(1-a_0\right)}{16\pi \sigma }\right]}^{\frac{1}{4}}\frac{1}{\sqrt{d}}}\\ =\frac{{\left(1-a\right)}^{\frac{1}{4}}}{{\left(1-a_0\right)}^{\frac{1}{4}}}\end{array}$        (II)

Conclusion:

Substitute $279\text{K}$ for $T_0$ and $0$ for $a_0$ in equation (II) to find $T$.

    $T=279\sqrt[4]{1-a}$        (III)

Substitute $0.9$ for $a$ in equation (III) to find $T$.

    $\begin{array}{c}T=279\sqrt[4]{1-\left(0.9\right)}\\ =156.9\text{K}\end{array}$

Therefore, the average temperature on the planet is $\underset{\_}{156.9\text{K}}$.