The solar constant is the amount of heat energy received per second per unit area of a perfectly black surface placed at a mean distance of the Earth form the Sun, in the absence of Earth's atmosphere, the surface being held perpendicular to the direction of Sun's rays. Its value is `1388 W//m^(2)`. If the solar constant for the earth is 's'. The surface temperature of the sun is `TK,D` is the diameter of the sun, `R` is the mean distance of the Earth from the sun. The sun subtends a small angle 'theta' at the earth. Then correct options is/are:-

Options

A. `s = sigmaT^(4) ((D)/(R ))^(2)`
B. `s =(sigmaT^(4))/(4)((D)/(R ))^(2)`
C. `s = (sigmaT^(4))/(4)theta^(2)`
D. `s=(sigmaT^(4))/(4)((R)/(D))^(2)`

Answer

To solve the problem regarding the solar constant and the surface temperature of the Sun, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Solar Constant**:
The solar constant \( S \) is defined as the amount of solar energy received per second per unit area on a perfectly black surface placed at a mean distance from the Sun, perpendicular to the Sun's rays. Its value is given as \( S = 1388 \, \text{W/m}^2 \).

2. **Energy Intercept Concept**:
The total energy intercepted by the Earth from the Sun can be expressed as:
\[ \text{Energy Intercept} = S \times A_{\text{Earth}} \]
where \( A_{\text{Earth}} \) is the area of the Earth that receives sunlight.

3. **Area of the Earth**:
The area \( A_{\text{Earth}} \) can be calculated as:
\[ A_{\text{Earth}} = \pi r^2 \]
where \( r \) is the radius of the Earth.

4. **Energy Emitted by the Sun**:
The energy emitted by the Sun can be expressed using the Stefan-Boltzmann law:
\[ \text{Energy Emitted} = \sigma A_{\text{Sun}} T^4 \]
where \( \sigma \) is the Stefan-Boltzmann constant, \( A_{\text{Sun}} \) is the surface area of the Sun, and \( T \) is the surface temperature of the Sun.

5. **Area of the Sun**:
The area \( A_{\text{Sun}} \) is given by:
\[ A_{\text{Sun}} = 4\pi \left(\frac{D}{2}\right)^2 = \pi D^2 \]
where \( D \) is the diameter of the Sun.

6. **Setting Up the Equation**:
The total energy intercepted by the Earth must equal the total energy emitted by the Sun:
\[ S \times \pi r^2 = \sigma \pi D^2 T^4 \]

7. **Simplifying the Equation**:
Canceling \( \pi \) from both sides gives:
\[ S r^2 = \sigma D^2 T^4 \]

8. **Rearranging for Solar Constant**:
Rearranging the equation to express the solar constant \( S \):
\[ S = \frac{\sigma D^2 T^4}{r^2} \]

9. **Conclusion**:
Thus, we have derived that the solar constant \( S \) can be expressed in terms of the surface temperature of the Sun \( T \), the diameter of the Sun \( D \), and the distance from the Sun \( r \).

### Final Expression:
\[ S = \frac{\sigma D^2 T^4}{r^2} \]