A planet having surface temperature T, K has a solar constant S. An angle `theta` is subtended by the sun at the planet:

Options

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

Explanation

To solve the problem step by step, we will analyze the relationship between the solar constant \( S \), the surface temperature \( T \) of the planet, and the angle \( \theta \) subtended by the sun at the planet.

### Step 1: Understand the Solar Constant
The solar constant \( S \) is defined as the amount of solar energy received per unit area at the distance of the planet from the sun. It can be expressed using the Stefan-Boltzmann law.

### Step 2: Use the Stefan-Boltzmann Law
According to the Stefan-Boltzmann law, the power emitted by a black body is given by:
\[ P = \sigma A T^4 \]
where \( \sigma \) is the Stefan-Boltzmann constant, \( A \) is the surface area, and \( T \) is the temperature in Kelvin.

### Step 3: Calculate the Power Emitted by the Sun
The surface area \( A \) of the sun can be calculated as:
\[ A = 4 \pi r^2 \]
where \( r \) is the radius of the sun. Thus, the total power \( P \) emitted by the sun is:
\[ P = \sigma (4 \pi r^2) T^4 \]

### Step 4: Calculate the Solar Constant
The solar constant \( S \) is the power received per unit area at the distance \( d \) from the sun:
\[ S = \frac{P}{4 \pi d^2} \]
Substituting the expression for \( P \):
\[ S = \frac{\sigma (4 \pi r^2) T^4}{4 \pi d^2} \]
This simplifies to:
\[ S = \frac{\sigma r^2 T^4}{d^2} \]

### Step 5: Relate the Angle \( \theta \)
The angle \( \theta \) subtended by the sun at the planet can be expressed as:
\[ \theta = \frac{2r}{d} \]
This means that:
\[ d = \frac{2r}{\theta} \]

### Step 6: Substitute \( d \) in the Solar Constant Equation
Substituting \( d \) in the equation for \( S \):
\[ S = \frac{\sigma r^2 T^4}{\left(\frac{2r}{\theta}\right)^2} \]
This simplifies to:
\[ S = \frac{\sigma r^2 T^4 \theta^2}{4r^2} \]
Thus:
\[ S = \frac{\sigma T^4 \theta^2}{4} \]

### Step 7: Final Expression
From the above steps, we can conclude that the solar constant \( S \) varies with the temperature \( T \) and the angle \( \theta \) as follows:
\[ S \propto T^4 \theta^2 \]

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