With what speed should a galaxy move with respect to us to that the sodium line at `589.0 nm` is observed at `589.6 nm` ?

Options

A. 206 km `s^(-1)`
B. 306 km `s^(-1)`
C. 103 km `s^(-1)`
D. 51 km `s^(-1)`

Explanation

To solve the problem of determining the speed at which a galaxy should move with respect to us for the sodium line at 589.0 nm to be observed at 589.6 nm, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Wavelengths**:
- The original wavelength (λ1) is given as 589.0 nm.
- The observed wavelength (λ2) is given as 589.6 nm.

2. **Calculate the Change in Wavelength (Δλ)**:
- The change in wavelength can be calculated using the formula:
\[ \Delta \lambda = \lambda_2 - \lambda_1 \]
- Substituting the values:
\[ \Delta \lambda = 589.6 \, \text{nm} - 589.0 \, \text{nm} = 0.6 \, \text{nm} \]

3. **Use the Doppler Effect Relation**:
- The relation between the change in wavelength and the speed of the galaxy (V_radial) is given by:
\[ \frac{\Delta \lambda}{\lambda_1} = \frac{V_{\text{radial}}}{c} \]
- Here, \(c\) is the speed of light, approximately \(3 \times 10^8 \, \text{m/s}\).

4. **Rearranging the Equation**:
- We can rearrange the equation to solve for \(V_{\text{radial}}\):
\[ V_{\text{radial}} = \frac{\Delta \lambda}{\lambda_1} \times c \]

5. **Substituting the Values**:
- Now, substituting the values we have:
\[ V_{\text{radial}} = \frac{0.6 \, \text{nm}}{589.0 \, \text{nm}} \times 3 \times 10^8 \, \text{m/s} \]

6. **Calculating the Speed**:
- First, calculate the fraction:
\[ \frac{0.6}{589.0} \approx 0.001018 \]
- Now, multiply by the speed of light:
\[ V_{\text{radial}} \approx 0.001018 \times 3 \times 10^8 \, \text{m/s} \approx 3.054 \times 10^5 \, \text{m/s} \]

7. **Convert to Kilometers per Second**:
- To convert meters per second to kilometers per second, divide by 1000:
\[ V_{\text{radial}} \approx 3.054 \times 10^5 \, \text{m/s} \div 1000 \approx 305.4 \, \text{km/s} \]

8. **Final Answer**:
- The speed at which the galaxy should move with respect to us is approximately:
\[ \boxed{306 \, \text{km/s}} \]