Answer to Question #206548 in Mechanics | Relativity for Safi Ullah

Question

A certain line in the spectrum of the light from a nebula has a wavelength of

656 nm instead of the 434 nm measured in the laboratory. If the nebula is moving

radially, what is its speed relative to the Earth? Ans.: It is moving away from the

Earth with a speed of V ¼ 0:391c.

Answer 1

The shift is quite large, so we should use the formula of relativistic Doppler effect (https://en.wikipedia.org/wiki/Relativistic_Doppler_effect).

Let $\beta = \dfrac{v}{c},$ $\lambda_r$ is received wavelength, $\lambda_s$ is emitted wavelength.

Therefore, $\tau=\dfrac{\lambda_r}{\lambda_s} =\sqrt{\dfrac{1+\beta}{1-\beta}}$ , $\tau = \dfrac{656}{434} \approx 1.512,$

so $\dfrac{1+\beta}{1-\beta} = \tau^2, \; \beta = \dfrac{\tau^2-1}{\tau^2+1} = 0.391, \; v = 0.391c.$