= Solution
An excited-state population lifetime $\tau$ gives a <natural linewidth> whose Lorentzian full width in ordinary <frequency> is
$$
\boxed{\Gamma=\frac1{2\pi\tau},}
$$
when no extra dephasing broadens it. Additional thermal motion, collisions or inhomogeneity may increase the effective width. Therefore excitation is appreciable over $|D_{\rm in}f_0-f|\lesssim\Gamma/2$, rather than only at one event. A <Lorentz line shape> has nonzero wings, so $\beta_{\min}$ is a line-centre threshold, not a mathematically sharp cutoff of all signal. Below it, only a weak detuned wing may be excited; at it, finite-width illumination produces a broad weak precursor rather than a delta-function burst at infinity.
For a narrow line with a finite resonance time, a useful local excitation duration is
$$
\Delta t_{\rm exc}\sim\frac{\Gamma}{f_0|\dot D_{\rm in}(t_*)|},\qquad
\dot D_{\rm in}=-\frac{\gamma\beta vL^2}{r^3}.
$$
Near the far-approach threshold this linear estimate becomes large and the full line shape and inverse-square illumination must be retained. After re-emission, the received full width is approximately $D_{\rm out}\Gamma$ at a fixed emission event. Lifetime memory and the changing return <Doppler factor> additionally produce a chirp and a redder tail.
The plots in part <2/c/i> use $\Gamma/f=0.02$ and $c\tau/L=0.4$, consistently corresponding to $fL/c=1/[2\pi(0.4)(0.02)]$ if the width is entirely natural. They are illustrative dimensionless parameters, not an asserted measurement for a specified atomic species.
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