Past exam of the symmetry and relativity course of the University of Oxford / 2019 / 2 / c / ii / Solution

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.