 Fluorescence from a uniformly moving cloud

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.