Question

The Doppler shift in the wavelength of the sodium line is (5890 angstrom) in the light observed from a distant star is 100 angstrom. To determine the speed at which the star is receding.

Answer 1

The change in the received frequency due to the Doppler shift can be written as follows:

$\omega = \frac{\omega_0}{1 + \frac{v}{c}}$ ,

where we take into account the fact that the star is receding from the observer. Hence,

$v=c\left(\frac{\omega_0}{\omega}-1\right)$ .

Utilizing the relationship between the angular velocity and wavelength $\lambda = \frac{2 \pi c}{\omega}$ , we derive:

$v=c\left( \frac{\lambda}{\lambda_0}-1\right)=c \left(\frac{\lambda_0 + \Delta \lambda}{\lambda_0}-1\right) = c \frac{\Delta \lambda}{\lambda_0}$

Substituting the numerical values, we obtain (answer is measured in the number of speed of light):

$v = c \frac{100}{5890} \approx 0.017c$

or substituting the speed of light:

$v \approx 5.1 \times 10^6$ m/s