An atomic species emits light with a characteristic frequency $f_{0}$ while at a rest state. A diatomic gas of such atomic species has been heated to temperature $T^{\prime}$. Due to their random thermal motions, some atoms move toward or against an onlooker. Assuming that root mean square speed characterizes their thermal motions:

(i) Determine the expression for the fractional variation $\frac{\Delta f}{f_{0}}$ between Doppler shifted frequencies corresponding to those moving straightly toward or against the onlooker, Assume $m c^{2} \gg 3 k T^{\prime}$.

(ii) Compute $\Delta f / f_{0}$ for a gas of oxygen atoms at 300 K.

An alien civilization sends signals toward Earth, and these signals are detected as radio waves with frequencies about 10% lower than those measured on Earth. Estimate the speed of the alien civilization's planet using the Doppler effect for light. How would your answer differ if you used the classical Doppler effect?

A far-off nebula shows a Doppler shift represented by f - f₀ = -12/100 f₀. Estimate its velocity with respect to Earth.