Chapter 10, Problem 9P

The light from a heated atomic gas is shifted in frequency because of the random thermal motion of light-emitting atoms toward or away from an observer. Estimate the fractional Doppler shift $\left(\Delta f/f_0\right)$, assuming that light of frequency f₀ is emitted in the rest frame of each atom, that the light-emitting atoms are iron atoms in a star at temperature 6000 K, and that the atoms are moving relative to an observer with the mean speed

$\bar{\upsilon }=\pm \sqrt{\frac{8k_{\text{B}}T}{\pi m}}$

Must we use the relativistic Doppler shift formulas

$f=f_0\frac{\sqrt{1\pm \upsilon /c}}{1\mp \upsilon /c}$

for this calculation? Such thermal Doppler shifts are measurable and are used to determine stellar surface temperatures.