28.4 Relativistic addition of velocities  (Page 3/7)

Calculating a doppler shift: radio waves from a receding galaxy

Suppose a galaxy is moving away from the Earth at a speed $\text{0.825}c$ . It emits radio waves with a wavelength of $0\text{.}\text{525}\text{m}$ . What wavelength would we detect on the Earth?

Strategy

Because the galaxy is moving at a relativistic speed, we must determine the Doppler shift of the radio waves using the relativistic Doppler shift instead of the classical Doppler shift.

Solution

1. Identify the knowns. $\text{u=}0\text{.}\text{825}c$ ; ${\lambda }_s=0\text{.}\text{525}m$
2. Identify the unknown. ${\lambda }_{\text{obs}}$
3. Choose the appropriate equation. ${\lambda }_{\text{obs}}{\text{=λ}}_s\sqrt{\frac{1+\frac{u}{c}}{1-\frac{u}{c}}}$
4. Plug the knowns into the equation.
   $\begin{array}{lll}{\lambda }_{\text{obs}}& =& {\text{λ}}_s\sqrt{\frac{1+\frac{u}{c}}{1-\frac{u}{c}}}\\ & =& (\mathrm{0.525\; m})\sqrt{\frac{1+\frac{0\text{.}\text{825}\text{c}}{c}}{1-\frac{0\text{.}\text{825}\text{c}}{c}}}\\ & =& \text{1.70 m.}\end{array}$

Discussion

Because the galaxy is moving away from the Earth, we expect the wavelengths of radiation it emits to be redshifted. The wavelength we calculated is 1.70 m, which is redshifted from the original wavelength of 0.525 m.