Fluorescence from a uniformly moving cloud ($D_{\mathrm{in}}=\gamma (1-\beta s),D_{\mathrm{out}}=[\gamma (1+\beta s){]}^{-1}$)

In this model, let $f_0$ denote the stationary source's emitted frequency and $f$ the cloud's resonance frequency in its own rest frame. Thus $q=f_0/f<1$ means that the source is redder than the cloud's resonance. Excitation requires the incoming Doppler factor to satisfy $D_{\mathrm{in}}{f}_0=f$, or $D_{\mathrm{in}}=1/q$. For the geometry below, the resonant position obeys $s=[1-1/(q\gamma )]/\beta$. At closest approach $s=0$, so $\gamma =1/q$ and $\beta =\sqrt{1-q^2}$. These definitions are essential: reversing the meaning of the frequency ratio would reverse the resonance condition.

For a compact cloud passing a stationary source at distance $L$, put $r(t)=\sqrt{L^2+v^2{t}^2}$ and $s=vt/r$. Incoming and returning rays have opposite Doppler factors. Source return time is $t_R=t+r/c$. If $q=f_0/f<1$, line-centre excitation is possible above $\beta =(1-q^2)/(1+q^2)$ and occurs at closest approach when $\beta =\sqrt{1-q^2}$. In a weak, optically thin, isotropic spontaneous emission model, the excited population obeys ${\dot{N}}_{\ast }=P_{\mathrm{abs}}-N_{\ast }/(\gamma \tau )$ and the received bolometric intensity scales as $D_{\mathrm{out}}^4{N}_{\ast }/(\tau r^2)$. Finite lifetime and line width smear the pulse and its frequency chirp; geometry and source strength are needed for an absolute prediction.