Doppler Effect for Light — Question 8

Question 8

The classical longitudinal approximation misses a second-order correction

For recession, the exact frequency factor is $D=\sqrt{(1-\beta)/(1+\beta)}$. A low-speed estimate uses $D_{\mathrm{lin}}=1-\beta$.

1. Expand D through second order in $\beta$.

2. At $\beta=0.10$, find the exact factor and the fractional error of the linear estimate relative to the exact factor.

Question 8 - Solution

The classical longitudinal approximation misses a second-order correction

(a) Method and calculation

$$D=(1-\beta)(1-\beta^2)^{-1/2}.$$$$\boxed{D=1-\beta+\tfrac12\beta^2+O(\beta^3)}.$$

(b) Interpretation and further calculation

$$\boxed{D=\sqrt{9/11}=0.904534},\qquad D_{\mathrm{lin}}=0.900000.$$$$\boxed{\frac{D-D_{\mathrm{lin}}}D=1-\sqrt{1-\beta^2}=0.00501256}.$$

Physical check

The linear estimate is too low for positive recession speed. Its relative error starts at one half beta squared.