A binary star system is observed using a telescope. The angular separation between both the stars is `5 xx 10^(-6)` radians. Calculate the aperture of objective of the telescope if light of wavelength `5,500 Å` is used.

Answer

To solve the problem of calculating the aperture of the objective of the telescope observing a binary star system, we will use the formula for angular resolution of a telescope. The formula is given by:

\[ \theta = \frac{1.22 \lambda}{D} \]

Where:
- \(\theta\) is the angular separation in radians,
- \(\lambda\) is the wavelength of light used (in meters),
- \(D\) is the aperture of the telescope (in meters).

### Step 1: Identify the given values
- Angular separation, \(\theta = 5 \times 10^{-6}\) radians
- Wavelength, \(\lambda = 5500 \, \text{Å} = 5500 \times 10^{-10} \, \text{m}\)

### Step 2: Rearrange the formula to solve for \(D\)
We need to rearrange the formula to find \(D\):

\[ D = \frac{1.22 \lambda}{\theta} \]

### Step 3: Substitute the values into the formula
Now we will substitute the values of \(\lambda\) and \(\theta\) into the rearranged formula:

\[ D = \frac{1.22 \times (5500 \times 10^{-10})}{5 \times 10^{-6}} \]

### Step 4: Calculate \(D\)
Now we will perform the calculation step-by-step:

1. Calculate \(\lambda\):
\[ \lambda = 5500 \times 10^{-10} = 5.5 \times 10^{-7} \, \text{m} \]

2. Substitute \(\lambda\) into the equation:
\[ D = \frac{1.22 \times 5.5 \times 10^{-7}}{5 \times 10^{-6}} \]

3. Calculate the numerator:
\[ 1.22 \times 5.5 = 6.71 \times 10^{-7} \]

4. Now divide by \(\theta\):
\[ D = \frac{6.71 \times 10^{-7}}{5 \times 10^{-6}} = 1.342 \times 10^{-1} = 0.1342 \, \text{m} \]

### Step 5: Convert to centimeters
To convert meters to centimeters:
\[ D = 0.1342 \, \text{m} \times 100 = 13.42 \, \text{cm} \]

### Final Answer
The aperture of the objective of the telescope is \(13.42 \, \text{cm}\).

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