Which of the following is true for the minimum angular separation of two stars, `Deltatheta_(min)`, , that can be resolved by a telescope? In the following, aperture is the diameter of the objective

Options

A. It decreases with the increase in aperture of the telescope
B. It is independent of the aperture of the telescope
C. It increases linearly with the aperture of the telescope
D. It increases quadratically with the aperture of the telescope

Explanation

To determine the correct statement regarding the minimum angular separation of two stars that can be resolved by a telescope, we need to analyze the relationship between the minimum angular separation (Δθ_min) and the aperture (D) of the telescope.

### Step-by-Step Solution:

1. **Understanding the Formula**:
The minimum angular separation (Δθ_min) that can be resolved by a telescope is given by the formula:
\[ \Delta \theta_{\text{min}} = \frac{1.22 \lambda}{D} \]
where:
- \( \lambda \) is the wavelength of light,
- \( D \) is the diameter of the telescope's aperture.

2. **Analyzing the Relationship**:
From the formula, we can see that Δθ_min is inversely proportional to the diameter of the aperture (D). This means:
- As the diameter (aperture) increases, the minimum angular separation (Δθ_min) decreases.
- Conversely, if the aperture decreases, Δθ_min increases.

3. **Evaluating the Options**:
Now, let's evaluate the provided options based on our understanding:
- **Option 1**: It decreases with an increase in aperture of the telescope. (True)
- **Option 2**: It is independent of the aperture of the telescope. (False)
- **Option 3**: It increases linearly with the aperture of the telescope. (False)
- **Option 4**: It increases quadratically with the aperture of the telescope. (False)

4. **Conclusion**:
The correct statement is that the minimum angular separation of two stars that can be resolved by a telescope decreases with an increase in the aperture of the telescope. Therefore, the correct answer is:
- **Option 1**: It decreases with an increase in aperture of the telescope.