Problem 106 - 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 kW/m\(^2\). The distance from the earth to the sun is \(1.50 \times 10{^1}{^1} m\), and the radius of the sun is \(6.96 \times 10{^8} m\). (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) 6.33 × 10⁷ W/m²; (b) Approximately 5780 K.

Step 1: Calculate Total Power Emitted by the Sun

Using the given radiant energy rate at Earth's upper atmosphere, we first calculate the total power emitted by the Sun. The formula to use is \[ P = I \cdot 4\pi R^2 \]where \( I \) is the solar constant (1.50 kW/m²), and \( R \) is the distance from the Earth to the Sun (\( 1.50 \times 10^{11} \) m).Substituting the values, we get:\[ P = 1.50 \times 10^3 \times 4\pi \times (1.50 \times 10^{11})^2 \]Calculating this, \[ P = 3.96 \times 10^{26} \text{ W} \]where \( P \) is the total power emitted by the Sun.

Step 2: Calculate Rate of Radiation Per Unit Area (Intensity) at Sun's Surface

Having calculated the total power emitted, we now find the intensity at the Sun's surface. The formula is:\[ I_{sun} = \frac{P}{4\pi r^2} \]where \( r \) is the radius of the Sun (\( 6.96 \times 10^8 \) m).Substitute \( P = 3.96 \times 10^{26} \text{ W} \) and \( r \):\[ I_{sun} = \frac{3.96 \times 10^{26}}{4\pi (6.96 \times 10^8)^2} \]Calculating this gives\[ I_{sun} = 6.33 \times 10^7 \text{ W/m}^2 \]This is the rate of radiation of energy per unit area from the Sun's surface.

Step 3: Calculate Surface Temperature of the Sun Using Stefan-Boltzmann Law

If the Sun radiates as a blackbody, we use the Stefan-Boltzmann law:\[ I = \sigma T^4 \]where \( I \) is the intensity, \( \sigma \) is the Stefan-Boltzmann constant (\( 5.67 \times 10^{-8} \text{ W/m}^2\text{K}^4 \)), and \( T \) is the temperature.Rearranging the formula to solve for \( T \):\[ T = \left(\frac{I}{\sigma}\right)^{1/4} \]Substitute \( I = 6.33 \times 10^7 \text{ W/m}^2 \):\[ T = \left(\frac{6.33 \times 10^7}{5.67 \times 10^{-8}}\right)^{1/4} \]Calculating this,\[ T \approx 5780 \text{ K} \]This is the surface temperature of the Sun.

Key Concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is a fundamental concept in the study of thermal radiation. It describes how much energy is emitted by a blackbody in the form of radiation.
This law states that the total energy radiated per unit surface area of a blackbody is proportional to the fourth power of the blackbody's absolute temperature. The formula is given by: \[ I = \sigma T^4 \] where:

• \( I \) is the radiated intensity (energy per unit area).
• \( \sigma \) is the Stefan-Boltzmann constant \((5.67 \times 10^{-8} \text{ W/m}^2\text{K}^4) \).
• \( T \) is the absolute temperature in Kelvin.

The Stefan-Boltzmann Law helps us calculate the energy emitted by the Sun, assuming it behaves like an ideal blackbody.
This law is crucial for understanding how stars, such as our Sun, emit energy and affects calculations involving stellar temperatures.

Blackbody Radiation

Blackbody radiation refers to the theoretical concept of an object that absorbs all incoming radiation, regardless of frequency or angle of incidence.
Such a body is perfect in the sense that it does not reflect or transmit any energy. Instead, it re-radiates energy at a rate determined by its temperature.
The Sun is often approximated as an ideal blackbody, especially when studying its radiation patterns.
This simplification allows astrophysicists to calculate its surface temperature using the Stefan-Boltzmann Law.
Blackbody radiation is characterized by a continuous spectrum, which means it emits radiation at every possible frequency.
Understanding blackbody radiation is key to finding the surface temperature of distant stars and plays an essential role in the study of thermal physics.

Solar Constant

The solar constant is a measure of the rate at which energy from the Sun is received per unit area at the top of Earth's atmosphere.
It is denoted by \( I \) and is approximately 1.50 kW/m². This value can vary slightly due to changes in Earth's orbit and solar activity.
The solar constant is a crucial factor in calculating the energy balance of our planet and plays a significant role in climatic and atmospheric models.
By understanding the solar constant, we are able to estimate the total power output of the Sun. In our exercise, it helps in calculating the total power emitted by the Sun using the formula: \[ P = I \cdot 4\pi R^2 \] where:

• \( P \) is the total power output of the Sun.
• \( R \) is the average distance from the Earth to the Sun.

The solar constant provides an essential baseline for measuring solar energy input and is vital for understanding the Earth's energy dynamics.