Question

With what speed should a galaxy move with respect to us so that the sodium line at 589.0nm is observed at 589.6nm?

Answer 1

To find the speed of the galaxy, first find the change in the wavelength of the sodium line and the wavelength at which it is observed. Then, use the formula for velocity which relates velocity with the speed of light, actual wavelength of the sodium line and the change in wavelengths obtained between the two wavelengths. Substitute the values and calculate the speed at which the galaxy should move.

Complete answer:
Given: Wavelength of sodium line (${\lambda}_{1}$)= 589.0 nm
Wavelength at which sodium line is observed (${\lambda}_{2}$)= 589.6 nm
Change in wavelengths is given by,
$\Delta \lambda = {\lambda}_{2}- {\lambda}_{1}$
Substituting the values in above equation we get,
$\Delta \lambda = 589.6 – 589$
$\Rightarrow \Delta \lambda = 0.6 nm$
Velocity in terms of wavelength is given by,
$v = c \times \dfrac {\Delta \lambda} {{\lambda}_{1}}$
Substituting the values in above equation we get,
$v= 3 \times {10}^{8} \times \dfrac {0.6 \times {10}^{-9}}{589 \times {10}^{-9}}$
$\Rightarrow v =3 \times {10}^{8} \times \dfrac {0.6 }{589}$
$\Rightarrow v = 0.00306 \times {10}^{8}$
$\Rightarrow v = 3.06 \times {10}^{5} {m}/{s}$
So, the speed of the galaxy should be $3.06 \times {10}^{5} {m}/{s}$.

Note:
Students must remember that the greater the speed of the object, shorter will be the wavelength. Astronomers use this shift in the wavelength to find the speed of the object whether moving towards us or moving away from us. If the galaxy is moving away from us, then the spectral lines from different elements will be shifted to a longer wavelength. So, in the given problem the galaxy is moving away from us.