Problem — Redshift of a galaxy

Problem

A galaxy recedes from us at $v = 0{,}7\,c$. One of its spectral lines, emitted at $\lambda = 486$ nm, is observed at what wavelength? Also calculate the redshift $z = \lambda'/\lambda - 1$.

Solution

Relativistic Doppler effect (recession). For a receding source, the observed wavelength is

$\lambda' = \lambda\sqrt{\frac{1 + \beta}{1 - \beta}}, \qquad \beta = \frac{v}{c} = 0{,}7$

Computing the factor.

$\sqrt{\frac{1 + 0{,}7}{1 - 0{,}7}} = \sqrt{\frac{1{,}7}{0{,}3}} = \sqrt{5{,}67} \approx 2{,}38$

Observed wavelength.

$\lambda' = 486\;\text{nm} \times 2{,}38 \approx 1{,}16\cdot 10^{3}\;\text{nm}$

$\ev{\lambda' \approx 1{,}16\cdot 10^{3}\;\text{nm}}$

Redshift.

$z = \frac{\lambda'}{\lambda} - 1 = 2{,}38 - 1$

$\ev{z \approx 1{,}38}$

The line, emitted in the visible range ($486$ nm, blue-green), is shifted all the way into the near infrared: this is the redshift typical of receding galaxies.

Links

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