The wavelength of `H_alpha` line in hydrogen spectrum was found 6563Å in the laboratory. If the wavelength of same line in the spectrum of a milky way is observed to be 6586Å, then the recessional velocity of the milky way will be

Options

A. `0.105 xx 10^6 ms^(-1)`
B. `1.05 xx 10^6 ms^(-1)`
C. `10.5 ms^(-1)`
D. none of these

Answer

To find the recessional velocity of the Milky Way based on the observed shift in the wavelength of the H-alpha line, we can use the formula derived from the Doppler effect. Here’s a step-by-step solution:

### Step 1: Identify the given values
- Wavelength in the laboratory (λ₀) = 6563 Å (angstroms)
- Wavelength observed in the Milky Way (λ) = 6586 Å
- Speed of light (c) = \(3 \times 10^8\) m/s

### Step 2: Calculate the change in wavelength (Δλ)
The change in wavelength (Δλ) can be calculated as:
\[ \Delta \lambda = \lambda - \lambda_0 = 6586 \, \text{Å} - 6563 \, \text{Å} = 23 \, \text{Å} \]

### Step 3: Convert the change in wavelength to meters
Since 1 Å = \(10^{-10}\) m, we convert Δλ to meters:
\[ \Delta \lambda = 23 \, \text{Å} = 23 \times 10^{-10} \, \text{m} = 2.3 \times 10^{-9} \, \text{m} \]

### Step 4: Use the Doppler effect formula
The formula relating the change in wavelength to the recessional velocity is:
\[ \frac{\Delta \lambda}{\lambda_0} = \frac{v}{c} \]
Where:
- \(v\) = recessional velocity
- \(c\) = speed of light

### Step 5: Rearrange the formula to solve for v
Rearranging the formula gives:
\[ v = c \cdot \frac{\Delta \lambda}{\lambda_0} \]

### Step 6: Substitute the values into the equation
Now, substituting the known values:
\[ v = 3 \times 10^8 \, \text{m/s} \cdot \frac{2.3 \times 10^{-9} \, \text{m}}{6563 \times 10^{-10} \, \text{m}} \]

### Step 7: Calculate the recessional velocity
Calculating the fraction:
\[ \frac{2.3 \times 10^{-9}}{6563 \times 10^{-10}} = \frac{2.3}{6563} \approx 0.000350 \]
Now substituting back:
\[ v \approx 3 \times 10^8 \cdot 0.000350 \approx 1.05 \times 10^5 \, \text{m/s} \]

### Final Result
The recessional velocity of the Milky Way is approximately:
\[ v \approx 1.05 \times 10^5 \, \text{m/s} \]