91Ó°ÊÓ

The observed wavelengths of lines in quasar spectrum is $\mathrm{λ}\text{'}=3\mathrm{λ}$

The wavelength of light for observer on Earth is calculated by using the formula for Doppler effect in light and color of light is decided by wavelength of light.

(a)

The relative velocity of recession is given as

$v_s=c\left(\frac{\mathrm{λ}\text{'}}{\mathrm{λ}}-1\right)$

Here, $c$ is the speed of light and its value is $3×{10}^8\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$ .

Substitute all the values in the above equation.

$$\begin{gathered} v_s=c\left(\frac{3\mathrm{λ}}{\mathrm{λ}}-1\right) \\ {v}_s=c\left(3-1\right) \\ {v}_s=2c \\ {v}_s\left(2c\right)>c \end{gathered}$$

Therefore, the relative velocity of recession is greater than speed of light.

(b)

The recession speed of quasar by relativistic Doppler equation is given as:

$u_s=c\left[\frac{1-{\left({\displaystyle \frac{\mathrm{λ}}{\mathrm{λ}\text{'}}}\right)}^2}{1+{\left({\displaystyle \frac{\mathrm{λ}}{\mathrm{λ}\text{'}}}\right)}^2}\right]$

Substitute all the values in the above equation.

$$\begin{gathered} u_s=c\left[\frac{1-{\left({\displaystyle \frac{\mathrm{λ}}{3\mathrm{λ}}}\right)}^2}{1+{\left({\displaystyle \frac{\mathrm{λ}}{3\mathrm{λ}}}\right)}^2}\right] \\ {u}_s=0.8c \end{gathered}$$

Therefore, the recession speed of quasar by relativistic Doppler equation is$0.8c.$ .