From the Doppler ratio to the velocity

Problem

From the Doppler ratio to the velocity. A light source receding from you emits at $\lambda_0=500$ nm and you receive $\lambda=600$ nm. Estimate the recession velocity using the relativistic longitudinal Doppler formula $\lambda/\lambda_0 = \sqrt{(1+\beta)/(1-\beta)}$.

Solution

Ratio of the wavelengths.

$\frac{\lambda}{\lambda_0} = \frac{600}{500} = 1{,}2.$

This is a redshift: the received wavelength is longer, consistent with recession.

Square the Doppler formula.

$\left(\frac{\lambda}{\lambda_0}\right)^2 = \frac{1+\beta}{1-\beta} \quad\Rightarrow\quad 1{,}44 = \frac{1+\beta}{1-\beta}.$

Isolate $\beta$.

$1{,}44\,(1-\beta) = 1+\beta \;\Rightarrow\; 1{,}44 - 1{,}44\,\beta = 1 + \beta \;\Rightarrow\; 0{,}44 = 2{,}44\,\beta,$

$\beta = \frac{0{,}44}{2{,}44} \approx 0{,}18.$

Velocity.

$v = \beta c \approx 0{,}18 \times 3\cdot 10^8 \approx 5{,}4\cdot 10^7\ \text{m/s}.$

$\ev{v \approx 0{,}18\,c \approx 5{,}4\cdot 10^7\ \text{m/s}}$

The source is receding at about $18\%$ of the speed of light.

Links

Topics: Special relativity Concepts: Doppler effect Skills: Symbolic setup