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An excited-state population lifetime $\tau$ gives a natural linewidth whose Lorentzian full width in ordinary frequency iswhen no extra dephasing broadens it. Additional thermal motion, collisions or inhomogeneity may increase the effective width. Therefore excitation is appreciable over $|D_{\mathrm{in}}{f}_0-f|\lesssim \Gamma /2$, 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 isNear 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 $D_{\mathrm{out}}\Gamma$ 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 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.

Decreasing $q=f_0/f$ increases the blueshift required for excitation. Both characteristic speeds move toward $c$:If $q$ is near one, modest approach speeds can excite the line. If $q\ll 1$, only strongly relativistic motion reaches the line centre. At a fixed qualifying speed, decreasing $q$ moves the resonance toward earlier approach, since a larger $D_{\mathrm{in}}$ is required. It can also move the event outside the illumination window set by $t_e\ge -T$.

At fixed $q$, increasing $\beta$ moves the resonance from the far approaching tail, through closest approach at ${\beta }_{\perp }$, and eventually to the receding side. The received pulse frequency cannot be estimated simply by applying the same blueshift twice: the inward and outward Doppler factors are different, and decay may occur well after excitation. The critical-speed rows of part i make these changes explicit.