Sun Heat Flux Calculation Using Black Body Radiation

This problem requires us to calculate the heat flux from the Sun's surface, assuming it behaves like a perfect black body. We are given the wavelength where the Sun's radiation output is at its peak intensity ($\lambda_{max} = 0.49 \, \mu\text{m}$). To find the heat flux, we will use two essential laws governing black body radiation: Wien's Displacement Law and the Stefan-Boltzmann Law.

Black Body Radiation Principles Explained

A black body is a theoretical object that absorbs all electromagnetic radiation falling on it and emits thermal radiation based solely on its temperature. The characteristics of this emitted radiation, like its peak wavelength and total intensity, are determined by its temperature.

Temperature Calculation Using Wien's Law

First, we determine the Sun's surface temperature ($T$) using Wien's Displacement Law. This law establishes an inverse relationship between the peak emission wavelength ($\lambda_{max}$) and the temperature ($T$) of a black body:

$$ \lambda_{max} T = b $$

Key constants and given values:

• Peak wavelength, $\lambda_{max} = 0.49 \, \mu\text{m}$
• Wien's displacement constant, $b \approx 2.898 \times 10^{-3} \, \text{m}\cdot\text{K} $

Before calculation, we convert the wavelength from micrometers ($\mu\text{m}$) to meters (m):

$$ \lambda_{max} = 0.49 \, \mu\text{m} = 0.49 \times 10^{-6} \, \text{m} $$

Rearranging Wien's Law to solve for $T$:

$$ T = \frac{b}{\lambda_{max}} $$

Plugging in the values:

$$ T = \frac{2.898 \times 10^{-3} \, \text{m}\cdot\text{K}}{0.49 \times 10^{-6} \, \text{m}} $$

$$ T \approx 5914.29 \, \text{K} $$

This calculation gives us the Sun's approximate surface temperature as $5914.29 \, \text{K}$.

Heat Flux Calculation Using Stefan-Boltzmann Law

Next, we calculate the heat flux ($E$), also known as the emissive power, using the Stefan-Boltzmann Law. This law relates the total energy radiated per unit surface area of a black body to the fourth power of its absolute temperature:

$$ E = \sigma T^4 $$

Where:

• $E$ = Heat flux ($W/m^2$)
• $\sigma$ = Stefan-Boltzmann constant $\approx 5.670 \times 10^{-8} \, \text{W}\cdot\text{m}^{-2}\cdot\text{K}^{-4}$
• $T$ = Temperature (K)

Using the calculated temperature $T \approx 5914.29 \, \text{K}$:

$$ E = (5.670 \times 10^{-8} \, \text{W}\cdot\text{m}^{-2}\cdot\text{K}^{-4}) \times (5914.29 \, \text{K})^4 $$

First, calculate $T^4$:

$$ T^4 \approx (5914.29)^4 \approx 1.2146 \times 10^{15} \, \text{K}^4 $$

Now, compute the heat flux $E$:

$$ E \approx (5.670 \times 10^{-8}) \times (1.2146 \times 10^{15}) $$

$$ E \approx 6.887 \times 10^{7} \, \text{W/m}^2 $$

Final Answer Analysis

Our calculation yields a heat flux of approximately $6.887 \times 10^{7} \, \text{W/m}^2$. Let's compare this to the options provided:

• Option 1: $5.9 \times 10^{7} \, \text{W/m}^2$
• Option 2: $6.93 \times 10^{7} \, \text{W/m}^2$
• Option 3: $4.93 \times 10^{7} \, \text{W/m}^2$
• Option 4: $7.93 \times 10^{3} \, \text{W/m}^2$

The calculated value, $6.887 \times 10^{7} \, \text{W/m}^2$, is very close to Option 2 ($6.93 \times 10^{7} \, \text{W/m}^2$). Minor discrepancies often arise from using slightly different values for physical constants or intermediate rounding. Based on our calculations, Option 2 is the most accurate choice.