Question

Assuming Sun's temperature to be 5727 °C and Earth's temperature to be 27 °C, the ratio between the total radiant exitance of Sun and that of Earth is ________ (Rounded off to the nearest integer).

Hint:

Apply the Stefan-Boltzmann law, M equal to sigma times T to the fourth power, after converting both temperatures to Kelvin, and take the ratio of the fourth powers.

Answer 1

Correct Answer: 159690

Step 1: Recall the Stefan-Boltzmann law.
The total radiant exitance of a blackbody is proportional to the fourth power of its absolute temperature: \[ M = \sigma T^4 \] where \(\sigma\) is the Stefan-Boltzmann constant and \(T\) is the temperature in Kelvin.

Step 2: Convert the given temperatures to Kelvin.
\[ T_{Sun} = 5727 + 273 = 6000\ K, \qquad T_{Earth} = 27 + 273 = 300\ K \]

Step 3: Write the ratio of exitances.
Since \(\sigma\) is a universal constant common to both bodies, it cancels out in the ratio: \[ \frac{M_{Sun}}{M_{Earth}} = \frac{\sigma T_{Sun}^4}{\sigma T_{Earth}^4} = \left(\frac{T_{Sun}}{T_{Earth}}\right)^4 \]

Step 4: Substitute the temperatures.
\[ \frac{M_{Sun}}{M_{Earth}} = \left(\frac{6000}{300}\right)^4 = (20)^4 \]

Step 5: Evaluate and round off.
\[ (20)^4 = 20\times20\times20\times20 = 160000 \] Rounded to the nearest integer, the ratio is \(160000\), which lies in the accepted range of \(159690\) to \(160001\).

\[ \boxed{\dfrac{M_{Sun}}{M_{Earth}} \approx 160000} \]