Question

In the redshift of radiation from a distant galaxy, certain radiation, known to have a wavelength of 434 nm when observed in the laboratory, has a wavelength of 462 nm. (a) What is the radial speed of the galaxy relative to Earth? (b) Is the galaxy approaching or receding from Earth?

Short answer

(a) The radial speed of the galaxy is 0.062c or $1.86\times {10}^{7}m/s$.

(b) The observed wavelength is longer than the proper wavelength; hence, the galaxy is receding from us.

Step-by-step solution

Doppler effect

(a)

In astronomy applications, the velocities of galaxies are estimated using Doppler shifts. Doppler shift is the difference between the observed and proper wavelength of light.The wavelength measured in the rest frame of the source is called proper wavelength ${\lambda }_o$. And the detected wavelength $\lambda$is related to the proper wavelength as

$\lambda ={\lambda }_o\sqrt{\frac{1+\beta }{1-\beta }}$

Where, $\beta$is the speed parameter.

Inserting the given values in the above equations

$$\begin{gathered} \frac{\lambda }{{\lambda }_o}=\sqrt{\frac{1+\beta }{1-\beta }} \\ \beta =\frac{1-{\left({\displaystyle \frac{{\lambda }_0}{\lambda }}\right)}^2}{1+{\left(\frac{{\lambda }_0}{\lambda }\right)}^2} \\ =\frac{1-{\left(0.94\right)}^2}{1+{\left(0.94\right)}^2} \\ =0.062 \end{gathered}$$

Hence the radial velocity of the galaxy is 0.062c or $1.86\times {10}^{7}m/s$.

Redshift or blue shift.

(b)

If the observed wavelength is longer than the proper wavelength, it implies that the galaxy is moving away from us and is said to be red-shifted.

If the observed wavelength is shorter than the proper wavelength, it implies that the galaxy is moving toward us and is said to be blue-shifted.

Here, the observed wavelength is longer than the proper wavelength; hence, the galaxy is receding from us.