Doppler Effect for Light — Question 7

Question 7

A narrow spread in recession speeds broadens a spectral line

A receding beam emits one rest frequency $f_0$. Its speeds have mean $\beta=0.80$ and small standard deviation $\sigma_\beta=0.003$. Use $D=f/f_0=\sqrt{(1-\beta)/(1+\beta)}$. Intrinsic line width is negligible.

1. Derive the first-order fractional frequency uncertainty.

2. Evaluate the fractional width and invert the relation for an observed width of 0.006 at the same mean speed.

Question 7 - Solution

A narrow spread in recession speeds broadens a spectral line

(a) Method and calculation

$$\frac{d\ln D}{d\beta}=\frac12\left(-\frac1{1-\beta}-\frac1{1+\beta}\right)=-\gamma^2.$$$$\boxed{\sigma_f/f\simeq\gamma^2\sigma_\beta}.$$

(b) Interpretation and further calculation

$$\gamma^2=25/9:\quad\boxed{\sigma_f/f\simeq0.00833333}.$$$$\boxed{\sigma_\beta\simeq(1-\beta^2)(0.006)=0.00216}.$$

Physical check

The widths are standard deviations and the approximation assumes a sufficiently narrow speed distribution.