In the previous problem, if `theta` is the angle sbtended by the Sun at the planet, then temperature at which planet radiates energy is proportional to

Options

A. `theta`
B. `theta^(2)`
C. `theta^(1//2)`
D. `theta^(-1//2)`

Answer

To solve the problem, we need to establish the relationship between the temperature at which the planet radiates energy and the angle θ subtended by the Sun at the planet.

### Step-by-Step Solution:

1. **Understand the Concept of Energy Received and Radiated**:
The energy received by the planet from the Sun is equal to the energy radiated by the planet. The energy received can be expressed using the Stefan-Boltzmann law.

2. **Energy Received by the Planet**:
The energy received by the planet (E_received) can be expressed as:
\[ E_{\text{received}} = \sigma T_0^4 \cdot A \]
where \( A \) is the area over which the energy is received. For a planet, the effective area is \( \pi r^2 \), where \( r \) is the radius of the planet.

3. **Energy Radiated by the Planet**:
The energy radiated by the planet (E_radiated) can also be expressed using the Stefan-Boltzmann law:
\[ E_{\text{radiated}} = \sigma T^4 \cdot 4\pi r^2 \]
where \( T \) is the temperature of the planet.

4. **Setting the Energies Equal**:
Since the energy received is equal to the energy radiated, we can set the two equations equal to each other:
\[ \sigma T_0^4 \cdot \pi r^2 = \sigma T^4 \cdot 4\pi r^2 \]

5. **Canceling Common Terms**:
We can cancel \( \sigma \) and \( \pi r^2 \) from both sides:
\[ T_0^4 = 4 T^4 \]

6. **Solving for Temperature**:
Rearranging gives:
\[ T^4 = \frac{T_0^4}{4} \]
Taking the fourth root:
\[ T = T_0 \cdot \left(\frac{1}{2}\right)^{1/4} = T_0 \cdot 2^{-1/4} \]

7. **Relating Temperature to Angle θ**:
The angle θ subtended by the Sun at the planet is related to the distance and the radius of the Sun. The relationship can be expressed as:
\[ \theta = \frac{2R}{d} \]
where \( R \) is the radius of the Sun and \( d \) is the distance from the Sun to the planet.

8. **Expressing Temperature in Terms of θ**:
From the relationship between \( d \) and \( \theta \), we can express \( d \) in terms of θ:
\[ d = \frac{2R}{\theta} \]
Substituting this back into the equation for temperature gives:
\[ T \propto \sqrt{\theta} \]

### Conclusion:
Thus, the temperature at which the planet radiates energy is proportional to \( \sqrt{\theta} \).