Chapter 16, Problem 56A

To determine

To describe: The unrealistic Doppler shift in terms of wavelength for light from a galaxy moving away from Earth.

Explanation of Solution

Introduction:

The Doppler Effect (or the Doppler shift) is the frequency change of a wave concerning an observer moving relative to the source of the wave. Doppler shift equation tells you that the difference between the apparent wavelength and actual wavelength of light generated by the source is equal to the product of the actual wavelength of the light and the relative speed of the source and the observer divided by the speed of light.

The expression for the apparent wavelength is given by

  ${\lambda }_{\text{app}}-\lambda =\pm \left(\frac{v}{c}\right)\lambda$  ......( (1)

Here, $\lambda$ is the actual wavelength of the light generated by the source, ${\lambda }_{\text{app}}$ is the apparent wavelength of the light observed, $v$ is the relative speed of the source and the observer, and $c$ is the speed of light.

For the galaxy moving away from the earth, the right-hand side is positive.

  ${\lambda }_{\text{app}}-\lambda =\left(\frac{v}{c}\right)\lambda$  ......(2)

Let the source moves with the speed of light $c$ .

Substitute $c$ for $v$ in equation (2)

  $\begin{array}{c}{\lambda }_{\text{app}}-\lambda =\left(\frac{c}{c}\right)\lambda \\ =\lambda \end{array}$

Rewriting the equation for ${\lambda }_{\text{app}}$ ,

  ${\lambda }_{\text{app}}=2\lambda$

Thus, the corresponding apparent wavelength is twice the actual wavelength generated by the source.

As no object can move with a speed equal or greater than the speed of light, the apparent wavelength equal to or greater than twice the actual wavelength is unrealistic.

Conclusion:

Hence, since no object can move at a speed equal to or greater than the speed of light, the apparent wavelength is unrealistic, equal to, or greater than twice the actual wavelength.