A-level Physics (Advancing Physics)/Doppler Effect/Worked Solutions

1. M31 (the Andromeda galaxy) is approaching us at about 120kms⁻¹. What is its red-shift?

$z=\frac{v_s}{c}=\frac{-120000}{300000000}=-0.4\times 10^{-3}$

The minus sign is important! Andromeda is blue-shifted!

2. Some light from M31 reaches us with a wavelength of 590 nm. What is its wavelength, relative to M31?

$-0.0004=\frac{\mathrm{\Delta }\lambda }{{\lambda }_0}=\frac{\lambda -{\lambda }_0}{{\lambda }_0}=\frac{\lambda }{{\lambda }_0}-1=\frac{590\times 10^{-9}}{{\lambda }_0}-1$

$0.9996=\frac{590\times 10^{-9}}{{\lambda }_0}$

${\lambda }_0=\frac{590\times 10^{-9}}{0.9996}=590.23\text{ nm}$

3. Some light has a wavelength, relative to M31, of 480 nm. What is its wavelength, relative to us?

$0.9996=\frac{\lambda }{{\lambda }_0}=\frac{\lambda }{480\times 10^{-9}}$

$\lambda =0.9996\times 480\times 10^{-9}=479.808\text{ nm}$

4. A quasar emits electromagnetic radiation at a wavelength of 121.6 nm. If, relative to us, this wavelength is red-shifted 0.2 nm, what is the velocity of recession of the quasar?

$\frac{v_s}{c}=\frac{\mathrm{\Delta }\lambda }{{\lambda }_0}$

$\frac{v_s}{3\times 10^8}=\frac{0.2}{121.6}=0.00164$

$v_s=3\times 10^8\times 0.00164=493{\text{ kms}}^{-1}$

However, this is about as high a velocity as we can use the classical Doppler effect for.