Solution (source code)

= Solution

The expanded feature resolves the <Q branch>: each line has $\Delta J=0$ but starts from a different lower rotational level. In the simple <rigid rotor> band model,
$$
\boxed{\widetilde\nu_Q(J)=\widetilde\nu_*+(B'_Q-B'')J(J+1).}
$$
Because the upper <rotational constant> is larger, the lines extend towards higher wavenumber. Their adjacent spacings are
$$
\widetilde\nu_Q(J+1)-\widetilde\nu_Q(J)=2(B'_Q-B'')(J+1),
$$
so they crowd together at the low-frequency edge and become increasingly separated. A strictly resolved bending-band analysis may distinguish the upper parity component sampled by Q lines from that sampled by P/R lines; it is not necessary to impose identical high-resolution constants on every branch in this sketch.

The intensity envelope reflects rotational state populations and the rotational transition strength. For this perpendicular <Q branch>, the usual rotational line-strength factor gives the approximate dependence
$$
\boxed{I_Q(J)\propto(2J+1)
\exp\!\left[-\frac{hcB''J(J+1)}{k_{\rm B}T}\right].}
$$
At small $J$, the increasing magnetic degeneracy makes the line intensity rise; at large $J$, <Boltzmann suppression> dominates. The maximum is therefore at an intermediate $J$. This is a population envelope across distinct transitions, rather than one intrinsically broad vibrational line.