The solar constant for a planet is S. The surface temperature of the sun is TK. The sun subtends an angle `theta` at the planet:

Options

A. `S propT^(4)`
B. `S prop T^(theta`
C. `S prop theta^(2)`
D. `S prop theta`

Answer

To solve the problem, we need to establish a relationship between the solar constant \( S \), the surface temperature of the sun \( T_K \), and the angle \( \theta \) subtended by the sun at the planet. We will use Stefan's Law and some basic geometric relationships.

### Step-by-Step Solution:

1. **Understand Stefan's Law**:
Stefan's Law states that the power radiated per unit area of a black body is proportional to the fourth power of its absolute temperature. Mathematically, it is given by:
\[ P = \sigma T^4 \]
where \( P \) is the power, \( \sigma \) is the Stefan-Boltzmann constant, and \( T \) is the temperature in Kelvin.

2. **Calculate Total Power from the Sun**:
The total power radiated by the sun can be expressed as:
\[ P_{\text{total}} = 4 \pi R^2 \sigma T_K^4 \]
where \( R \) is the radius of the sun and \( T_K \) is the surface temperature of the sun.

3. **Power per Unit Area at the Planet**:
The solar constant \( S \) is defined as the power received per unit area at the distance of the planet from the sun. This can be expressed as:
\[ S = \frac{P_{\text{total}}}{4 \pi D^2} \]
where \( D \) is the distance from the sun to the planet.

4. **Substituting Power into the Solar Constant**:
Substitute the expression for \( P_{\text{total}} \) into the equation for \( S \):
\[ S = \frac{4 \pi R^2 \sigma T_K^4}{4 \pi D^2} \]
Simplifying this gives:
\[ S = \frac{R^2 \sigma T_K^4}{D^2} \]

5. **Relate the Angle \( \theta \)**:
The angle \( \theta \) subtended by the sun at the planet can be approximated using the small angle approximation:
\[ \theta = \frac{2R}{D} \]
Rearranging this gives:
\[ D = \frac{2R}{\theta} \]

6. **Substituting \( D \) back into the Equation for \( S \)**:
Substitute \( D \) into the equation for \( S \):
\[ S = \frac{R^2 \sigma T_K^4}{\left(\frac{2R}{\theta}\right)^2} \]
Simplifying this gives:
\[ S = \frac{R^2 \sigma T_K^4 \theta^2}{4R^2} \]
Thus, we have:
\[ S = \frac{\sigma T_K^4 \theta^2}{4} \]

7. **Final Relationship**:
From the above equation, we can conclude that:
\[ S \propto T_K^4 \quad \text{and} \quad S \propto \theta^2 \]

### Conclusion:
The relationship established is that the solar constant \( S \) is directly proportional to the fourth power of the surface temperature of the sun \( T_K \) and the square of the angle \( \theta \) subtended by the sun at the planet.