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## Q. 16.6

Because galaxies have been observed with redshifts as great as 10 and quasars with redshifts more than 6, it is useful to have a plot of redshift versus recession velocity. Use Equation (2.33) for the relativistic Doppler shift to calculate such a plot.

Strategy The term redshift used by and physicists refers to $\Delta \lambda / \lambda$. We can use Equation (2.33) to find a relationship between $\Delta \lambda / \lambda \text { and } \beta, \text { where } \beta=v / c$.

$u_{y}=\frac{u_{y}^{\prime}}{\gamma\left[1+\left(v / c^{2}\right) u_{x}^{\prime}\right]}$ (2.23)

## Verified Solution

Equation (2.33) allows us to write

$\frac{\lambda}{\lambda_{0}}=\frac{f_{0}}{f}=\sqrt{\frac{1+\beta}{1-\beta}}$ (16.17)

We determine $\Delta \lambda / \lambda_{0}$ from this equation.

$\frac{\Delta \lambda}{\lambda_{0}}=\frac{\lambda-\lambda_{0}}{\lambda_{0}}=\sqrt{\frac{1+\beta}{1-\beta}}-1$ (16.18)

We show the relationship between the redshift and recession velocity (β) in Figure 16.12. Note that as the redshift increases toward 4, the velocity increases dramatically and approaches the speed of light.