Problem 114 - University Physics with Modern Physics

Question

The rate at which radiant energy from the sun reaches the earth's upper atmosphere is about 1.50 \(\mathrm{kW} / \mathrm{m}^{2} .\) The distance from the earth to the sun is \(1.50 \times 10^{11} \mathrm{m},\) and the radius of the sun is \(6.96 \times 10^{8} \mathrm{m}\) . (a) What is the rate of radiation of energy per unit area from the sun's surface? (b) If the sun radiates as an ideal blackbody, what is the temperature of its surface?

Step-by-Step Solution

Answer

(a) Use step calculations to find \( \,E = 6.33 \times 10^7 \, \text{W/m}² \). (b) Sun's surface temp \( T = 5778 \, \text{K} \).

Step 1: Calculate the Total Power Received by Earth

The intensity of solar radiation at the earth's atmosphere is given as 1.50 kW/m². To find the total power received by the earth, we have to consider that this power is spread over a sphere with a radius equal to the distance from the earth to the sun. The formula to use is the surface area of a sphere, which is \( 4\pi R^2 \) where \( R \) is the distance from the sun to the earth. Plugging the values, we get: \[ \text{Total Power} = 1.50 \times 10^3 \, \mathrm{W/m^2} \times 4\pi (1.50 \times 10^{11} \mathrm{m})^2 \]

Step 2: Find the Sun's Power Output

The power output from the sun is equal to the total power received by the Earth (calculated above in Step 1), as this power originates from the sun and disperses uniformly in all directions. Calculating this gives an expression for the total solar power output. \[ P = 1.50 \times 10^3 \times 4\pi (1.50 \times 10^{11})^2 \, \text{W} \]

Step 3: Calculate Rate of Radiation per Unit Area from the Sun's Surface

The rate of energy radiation per unit area from the sun's surface can be found using the formula for the surface area of a sphere again, now with the sun's radius: \( 4\pi (6.96 \times 10^8)^2 \). Dividing the total power of the sun by the surface area of the sun gives us the radiation per unit area. \[ \text{Rate of Radiation} = \frac{P}{4\pi (6.96 \times 10^8)^2} \, \text{(W/m²)} \]

Step 4: Determine the Surface Temperature of the Sun Assuming Blackbody Radiation

Assuming the sun radiates as an ideal blackbody, we use the Stefan-Boltzmann Law: \[ E = \sigma T^4 \] where \( E \) is the radiation rate per unit area, \( \sigma \) is the Stefan-Boltzmann constant \(5.67 \times 10^{-8} \, \text{W/m}^2\text{/K}^4\), and \( T \) is the temperature. Rearranging for \( T \), \[ T = \left(\frac{E}{\sigma}\right)^{1/4} \] Use the result from Step 3 for \( E \) to solve for \( T \).

Step 5: Perform Calculations and Interpret Results

Calculate the exact values using a calculator. First, obtain the value of \( P \) from Step 2. Then, find the radiation rate from Step 3, and finally, using the value of \( E \), apply it in Step 4 to find the temperature of the sun's surface. This involves several arithmetic operations and exponentiation.

Key Concepts

Blackbody Radiation

Blackbody radiation is the thermal electromagnetic radiation emitted by an object that absorbs all incident electromagnetic waves. This means it emits energy based only on its temperature, without reflecting any light from other sources.
The Sun can be treated as an ideal blackbody, making the concept significant for understanding solar radiation. In essence, any object that is considered a perfect blackbody will emit radiation across all wavelengths.
Importantly, it does so with a peak intensity dependent on its temperature. Per Wien's Displacement Law, the wavelength at which this peak occurs decreases as the temperature increases.

• The higher the temperature, the more energy is emitted.
• The radiation pattern is continuous across different wavelengths.
• The color of the radiation shifts from red to blue as temperature rises.

Understanding blackbody radiation helps us calculate the temperature of stars like the Sun by analyzing the spectrum of emitted radiation. It lays the groundwork for applying the Stefan-Boltzmann Law, which is key to solving problems involving solar radiation.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law provides a formula for calculating the energy emitted by a blackbody. It's one of the cornerstones for understanding how stars like the Sun emit energy.
This law states that the power emitted per unit area of a blackbody is directly proportional to the fourth power of its absolute temperature. Mathematically, it can be expressed as:
\[ E = \sigma T^4 \]
Where:

• \( E \) is the energy emitted per square meter.
• \( \sigma \) is the Stefan-Boltzmann constant \(5.67 \times 10^{-8} \, \text{W/m}^2\text{/K}^4\).
• \( T \) is the absolute temperature of the blackbody in Kelvin.

This means that a slight increase in temperature leads to a far greater increase in emitted energy. The Stefan-Boltzmann Law is crucial for determining the surface temperature of the Sun by using the energy rate calculated from its radiation. It's simple yet powerful, providing a direct link between temperature and emitted radiation.

Sun's Surface Temperature

Determining the Sun's surface temperature involves understanding both blackbody radiation and the Stefan-Boltzmann Law. It's a pivotal measurement because it helps us understand the Sun's overall energy output and affects everything from climate patterns to solar power design on Earth.
After calculating the rate of solar radiation from the Sun's surface, we apply the Stefan-Boltzmann Law to find its temperature. The formula:
\[ T = \left( \frac{E}{\sigma} \right)^{1/4} \]
shows that by measuring the intensity of radiation \( E \) and using the known constant \( \sigma \), we can calculate \( T \).

• This process yields the Sun's surface temperature being approximately 5,778 Kelvin.
• It confirms the immense energy output that makes life possible on Earth.
• Understanding these principles highlights the interplay of light and heat from the Sun and its influence on our planet.

These calculations showcase the elegance of physics in explaining natural phenomena and our capability to measure and comprehend the universe's workings.