A red star and a green star radiate energy at the same rate which star is bigger.

Options

A. Red
B. Green
C. Both have same size
D. can't be say anything

Explanation

To determine which star is bigger between a red star and a green star that radiate energy at the same rate, we can use the Stefan-Boltzmann law, which states that the power radiated by a star is proportional to its surface area and the fourth power of its temperature.

### Step-by-Step Solution:

1. **Understanding the Stefan-Boltzmann Law**:
The power (P) radiated by a star can be expressed as:
\[ P = \sigma A T^4 \]
where:
- \( P \) is the power radiated,
- \( \sigma \) is the Stefan-Boltzmann constant,
- \( A \) is the surface area of the star,
- \( T \) is the absolute temperature of the star.

2. **Given Information**:
We know that both the red star and the green star radiate energy at the same rate, which means:
\[ P_{red} = P_{green} \]

3. **Setting Up the Equation**:
Since the power is the same for both stars, we can write:
\[ \sigma A_{red} T_{red}^4 = \sigma A_{green} T_{green}^4 \]
The Stefan-Boltzmann constant (\( \sigma \)) cancels out, leading to:
\[ A_{red} T_{red}^4 = A_{green} T_{green}^4 \]

4. **Relating Temperature and Wavelength**:
We know that the temperature of the red star (\( T_{red} \)) is less than the temperature of the green star (\( T_{green} \)):
\[ T_{red} < T_{green} \]
Additionally, we can use the relationship between temperature and wavelength (from Wien's law):
\[ \lambda_{red} T_{red} = \lambda_{green} T_{green} \]
Since \( \lambda_{red} > \lambda_{green} \) (red light has a longer wavelength than green light), it follows that:
\[ T_{red} < T_{green} \]

5. **Analyzing the Area**:
From the equation \( A_{red} T_{red}^4 = A_{green} T_{green}^4 \), since \( T_{red} < T_{green} \), we can infer that:
\[ A_{red} > A_{green} \]
This means that the area of the red star must be greater than that of the green star.

6. **Conclusion**:
Therefore, since the red star has a greater surface area while radiating the same amount of energy, we conclude that:
\[ \text{The red star is bigger than the green star.} \]

### Final Answer:
The red star is bigger than the green star.

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