The solar constant for a planet is `sum`. The surface temperature of the sun is T K. if the sun subtends an angle `theta` at the planet, then

Options

A. `sum propT^(4)`
B. `sumpropT^2`
C. `sumproptheta^(2)`
D. `sumproptheta`

Answer

To solve the problem, we need to derive the relationship between the solar constant (Σ), the surface temperature of the Sun (T), and the angle subtended by the Sun at the planet (θ).

### Step-by-Step Solution:

1. **Understanding the Solar Constant**:
The solar constant (Σ) is the amount of solar energy received per unit area at the distance of the planet from the Sun. It can be expressed as:
\[ \Sigma = \frac{P}{A} \]
where \(P\) is the power emitted by the Sun and \(A\) is the area over which this power is distributed.

2. **Power Emitted by the Sun**:
The power emitted by the Sun can be calculated using the Stefan-Boltzmann law:
\[ P = \sigma A_s T^4 \]
where \(A_s\) is the surface area of the Sun, \(T\) is the surface temperature of the Sun, and \(\sigma\) is the Stefan-Boltzmann constant. The surface area of the Sun is given by:
\[ A_s = 4\pi R_s^2 \]
where \(R_s\) is the radius of the Sun.

3. **Distance from the Sun to the Planet**:
The distance from the Sun to the planet is denoted as \(r\). The solar constant at the planet can then be expressed as:
\[ \Sigma = \frac{P}{4\pi r^2} \]

4. **Combining the Equations**:
Substituting the expression for \(P\) into the equation for Σ, we get:
\[ \Sigma = \frac{\sigma (4\pi R_s^2) T^4}{4\pi r^2} \]
Simplifying this gives:
\[ \Sigma = \frac{\sigma R_s^2 T^4}{r^2} \]

5. **Relation to the Angle θ**:
The angle θ subtended by the Sun at the planet can be related to the radius of the Sun and the distance \(r\). For small angles, we can use the approximation:
\[ \tan\left(\frac{\theta}{2}\right) \approx \frac{R_s}{r} \]
Squaring both sides, we find:
\[ \tan^2\left(\frac{\theta}{2}\right) \approx \frac{R_s^2}{r^2} \]

6. **Final Relation**:
Substituting this back into our expression for Σ, we find:
\[ \Sigma \propto T^4 \tan^2\left(\frac{\theta}{2}\right) \]
For small angles, we can further approximate:
\[ \tan\left(\frac{\theta}{2}\right) \approx \frac{\theta}{2} \]
Therefore, we can express Σ as:
\[ \Sigma \propto T^4 \left(\frac{\theta}{2}\right)^2 \]

### Conclusion:
The solar constant Σ is proportional to \(T^4\) and \(\left(\frac{\theta}{2}\right)^2\).