## The Approximate Redshift Equation

The redshift
$z$
of a galaxy is a measure of the speed
$v$
with which is moving away from us - redshift - or towards us - blueshift. Because of the expansion of the Universe, the light from most galaxies is redshifted, so that the wavelength of light emitted by stars in a galaxy is longer when it is received at the Earth then when it was emitted. If the speed of the galaxy is
$v m/s$
away from us then
$\frac{\lambda_{RECEIVED LIGHT} - \lambda_{EMITTED LIGHT}}{\lambda_{EMITTED LIGHT}} = \sqrt{\frac{1+ v/c}{1-v/c}} -1$
.
We can find an approximation for small
$v/c$
using the binomial expansion.
$\sqrt{1+\frac{v}{c}}= (1+ \frac{v}{c})^{{1/2}} = 1+\frac{v}{2c} - \frac{v^"}{8c^2} + higher \, powers \, of \frac{v}{c}$

$\frac{1}{\sqrt{1-\frac{v}{c}}} = (1+ \frac{v}{c})^{-{1/2}} = 1+\frac{v}{2c} + \frac{v^"}{8c^2} + higher \, powers \, of \frac{v}{c}$

Then
\begin{equation} \begin{aligned} \frac{\lambda_{RECEIVED LIGHT} - \lambda_{EMITTED LIGHT}}{\lambda_{EMITTED LIGHT}} &= \sqrt{\frac{1+ v/c}{1-v/c}} -1\\ &= (1+ \frac{v}{c})^{{1/2}} (1- \frac{v}{c})^{-{1/2}} -1 \\ &= (1+\frac{v}{2c} - \frac{v^"}{8c^2} + higher \, powers \, of \frac{v}{c})(1+\frac{v}{2c} + \frac{v^"}{8c^2} + higher \, powers \, of \frac{v}{c}) -1\\ &= 1 + \frac{v}{c} + higher \, powers \, of \frac{v}{c} -1 \\ &= \frac{v}{c}\end{aligned} \end{equation} 