Q. 7.56 - An Introduction to Thermal Physics

Question

The planet Venus is different from the earth in several respects. First, it is only $70\%$ as far from the sun. Second, its thick clouds reflect $77\%$of all incident sunlight. Finally, its atmosphere is much more opaque to infrared light.

(a) Calculate the solar constant at the location of Venus, and estimate what the average surface temperature of Venus would be if it had no atmosphere and did not reflect any sunlight.

(b) Estimate the surface temperature again, taking the reflectivity of the clouds into account.

(c) The opaqueness of Venus's atmosphere at infrared wavelengths is roughly $70$ times that of earth's atmosphere. You can therefore model the atmosphere of Venus as $70$ successive "blankets" of the type considered in the text, with each blanket at a different equilibrium temperature. Use this model to estimate the surface temperature of Venus. (Hint: The temperature of the top layer is what you found in part (b). The next layer down is warmer by a factor of $2^{1/4}$. The next layer down is warmer by a smaller factor. Keep working your way down until you see the pattern.)

Step-by-Step Solution

Answer

(a) $\text{ The solar constant at the location of Venus is }2.8\times {10}^3 \mathrm{W}/{\mathrm{m}}^2 \mathrm{and} \text{ the average temperature of Venus if moon has no atmosphere is }333 \mathrm{K}$

(b) . 

$\text{ The surface temperature of Venus, when reflectivity is taking into account, is }231 \mathrm{K}\text{. }$

(c) .

$\text{ The ground temperature of the Venus is }670 \mathrm{K}\mid \text{. }$

Step 1. Given information

• The amount of solar flux density per unit area is termed as solar constant. and it is given as 

                                                ${\sigma }_{\text{solar }}=\frac{P}{4\pi R^2}$

•  $$\begin{gathered} \text{ From the Stefan's law of radiation we have that } \\ \text{amount of heat radiation is as follows: } \end{gathered}$$                                                                                                            $H=\sigma e\left(\pi r^2\right)T^4$

The above two equations will be used to get the desired result.

Step 2. Calculating the solar constant at the location of Venus .

Solar constant is given as :

${\sigma }_{\text{solar }}=\frac{P}{4\pi R^2}$

Here,$P$ is the solar power and data-custom-editor="chemistry" $\mathrm{R}$ { is the mean distance between the sun and Venus. }

The mean distance between the sun and the Venus is $70\%$of the distance between the earth and the sun.

$R=70\%\left(150\times {10}^9 \mathrm{m}\right)$

$=1.05\times {10}^{11} \mathrm{m}$

Putting $3.9\times {10}^{26} \mathrm{W}\text{ for }P$ and $1.05\times {10}^{11} \mathrm{m}$ $\text{ for }R\text{ in the equation }{\sigma }_{\text{solar }}=\frac{P}{4\pi R^2}$

${\sigma }_{\text{solar }}=\frac{\left(3.9\times {10}^{26} \mathrm{W}\right)}{4\pi {\left[1.05\times {10}^{11} \mathrm{m}\right]}^2}$

          $=2.8\times {10}^3 \mathrm{W}/{\mathrm{m}}^2$

$\text{Hence, the solar constant at the location of Venus is }2.8\times {10}^3 \mathrm{W}/{\mathrm{m}}^2$.

Step 3. Calculating the average temperature of Venus if moon has no atmosphere.

The amount of heat absorbed by planet is 

$H={\sigma }_{\text{solar }}A$

   $={\sigma }_{\text{solar }}\left(4\pi r^2\right)$

$\text{ where, }{\sigma }_{\text{solar }}\text{ is the solar constant and }r\text{ is the radius of the planet. }$

$\text{ From the Stefan's law of radiation, the amount of heat radiation is as follows }$

$H=\sigma e\left(\pi r^2\right)T^4$
$\sigma \text{ is the Stefan's constant and }T\text{ is the absolute temperature. }$

$\text{ Putting }{\sigma }_{\text{solar }}\left(4\pi r^2\right)\text{ for }H$

${\sigma }_{\text{Solar }}\left(4\pi r^2\right)=\sigma e\left(\pi r^2\right)T^4$

$T={\left[\frac{{\sigma }_{\text{Solar }}}{4\sigma }\right]}^{\frac{1}{4}}$

Putting , $2800$ $w/m^2$ for ${\sigma }_{\text{Solar }}$$\text{ and }5.6\times {10}^{-8} \mathrm{W}/{\mathrm{m}}^2·{\mathrm{K}}^4\text{ for }\sigma$.

$T={\left[\frac{\left(2800 \mathrm{W}/{\mathrm{m}}^2\right)}{4\left(5.67\times {10}^{-8} \mathrm{W}/{\mathrm{m}}^2·{\mathrm{K}}^4\right)}\right]}^{\frac{1}{4}}$

$T=333 K$

$\text{Hence, the average temperature of Venus if moon has no atmosphere is }333 \mathrm{K}$

Step 4. Calculating the surface temperature of Venus , when the reflectivity is taking into account .

As the clouds reflect $77\%$ incoming sunlight, then the remaining $23\%$will be absorbed Hence, the new solar constant will be

${\sigma }_{\text{solar }}{}^{\text{'}}=(23\%){\sigma }_{\text{solar }}$

         $=(0.23)\left(2.8\times {10}^3 \mathrm{W}/{\mathrm{m}}^2\right)$

          $=644 \mathrm{W}/{\mathrm{m}}^2$

$\text{ The new surface temperature of the Venus is }$

$T^{\text{'}}={\left[\frac{{\sigma }_{\text{Solar }}}{4\sigma }\right]}^{{]}^{\frac{1}{4}}}$

Putting the value of $644 \mathrm{W}/{\mathrm{m}}^2\text{ for }{\sigma }_{\text{solar }}\text{ 'and }5.6\times {10}^{-8} \mathrm{W}/{\mathrm{m}}^2·{\mathrm{K}}^4$ for $\sigma$.

$T^{\text{'}}={\left[\frac{644 \mathrm{W}/{\mathrm{m}}^2}{4\left(5.67\times {10}^{-8} \mathrm{W}/{\mathrm{m}}^2·{\mathrm{K}}^4\right)}\right]}^{\frac{1}{4}}$

    $=231 \mathrm{K}$

$\text{ Hence, the surface temperature of Venus, when reflectivity is taking into account, is }231 \mathrm{K}\text{. }$

Step 5. As we know that

 The upper blanket  send as much energy downward as it sends upward, so its total emission for each unit of sunlight is two units. 

The equilibrium requires that it also absorb two units of infrared radiation and these two units must come from the lower blanket. Since the lower blanket is radiating twice as much as energy upward as the upper blanket, its temperature must be greater by a factor of $2^{\frac{1}{4}}$.

In the same way, the lower blanket must also send as much energy down as it sends up - in this case, two units, since it emits a total of four units of which one comes from the upper blanket. The other three must come from the ground, as a check note that the ground is absorbed two units from the lower blanket and one from the sun so it must emit three units. In order for the ground to emit three times as much as energy upward as the upper blanket, its temperature must be greater by a factor of  $3^{\frac{1}{4}}$.

Step 6. Calculating the ground temperature of Venus .

Hence, the ${70}^{\text{th }}$ blanket is warmer than first blanket by the order of $(70{)}^{1/4}$. So, the ground is warmer than the order of $(71{)}^{1/4}$.$\text{ The temperature of the Venus is given as }$

$$\begin{gathered} T_{\text{venus }}=(71{)}^{\frac{1}{4}}·T^{\text{'}} \\ {T}_{\text{venus }}=(71{)}^{\frac{1}{4}}(231 \mathrm{K}) \\ =670 \mathrm{K} \end{gathered}$$

$\text{ Hence, the ground temperature of the Venus is }670 \mathrm{K}$