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).
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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.
Step 1: Convert temperatures to Kelvin using the precise offset.
$T_{Sun} = 5727 + 273.15 = 6000.15\ K$ and $T_{Earth} = 27 + 273.15 = 300.15\ K$.
Step 2: Form the temperature ratio first, then raise it to the fourth power.
By the Stefan-Boltzmann law $M \propto T^4$, so \[ \frac{M_{Sun}}{M_{Earth}} = \left(\frac{T_{Sun}}{T_{Earth}}\right)^4 = \left(\frac{6000.15}{300.15}\right)^4 \]
Step 3: Evaluate the temperature ratio.
\[ \frac{6000.15}{300.15} \approx 19.990 \]
Step 4: Square the ratio twice to get the fourth power.
\[ (19.990)^2 \approx 399.60 \] \[ (399.60)^2 \approx 159696 \]
Step 5: Round off and compare.
This gives a ratio of approximately $159696$, essentially the same result as using the simpler $+273$ conversion ($160000$); both fall inside the officially accepted range of $159690$ to $160001$, confirming the Sun radiates roughly $1.6\times10^5$ times more energy per unit area than the Earth.
