Past exam of the short options paper shared by Parts A and B of the physics course of the University of Oxford / 2019 / S16 / 2

Population can move between the levels by electron-impact excitation and de-excitation, spontaneous emission, radiative absorption and stimulated emission. Other collision partners and radiative cascades can matter in a fuller atom. In the elementary coronal approximation, the plasma is dilute, optically thin, steady, and weakly illuminated at the transition frequency. Electron impacts predominantly excite the lower level, while spontaneous emission predominantly empties the upper level; collisional de-excitation and photon trapping are negligible. The population balance isA key condition is $n_e{C}_{21}\ll A_{21}$, together with insignificant external pumping and competing channels. A Maxwellian electron Maxwell-Boltzmann distribution lets the tabulated rate coefficient be expressed using one temperature.

At $7000\mathrm{K}$, $k_{B}T=0.603\mathrm{eV}$ andThe $2p$ ground term has fine-structure levels $J=1/2$ and $J=3/2$, with energy-level degeneracy $g_1=2$ and $g_2=4$. Detailed balance gives $C_{21}=C_{12}(g_1/g_2)e^{\Delta E/(k_{B}T)}=7.17\times 10^{-14}{\mathrm{m}}^3{\mathrm{s}}^{-1}$. The critical density is thereforeWith collisions and spontaneous decay both retained, $N_2/N_1=n_e{C}_{12}/(A_{21}+n_e{C}_{21})$. A Boltzmann distribution, $N_2/N_1\to 2e^{-\Delta E/(k_{B}T)}\simeq 1.97$, requires $n_e\gg n_{\mathrm{crit}}$, not merely equality. For example $n_e\gtrsim 3\times 10^8{\mathrm{m}}^{-3}$ brings the ratio within about ten per cent of that limit.

Assume a cold, unmagnetized, weakly collisional plasma, a small fractional bandwidth and $\omega \gg {\omega }_p$. The dispersion relation is ${\omega }^2=c^2{k}^2+{\omega }_p^2$, with ${\omega }_p^2=n_e{e}^2/({ϵ}_0{m}_e)$. The group velocity and propagation time areThe lower frequencies arrive later. Taking the magnitude of the frequency derivative over $\Delta \omega$ givesFor a nonuniform line of sight, replace $n_{e}L$ by the dispersion measure $\int n_{e}dl$. The light travels at the group velocity, not at the phase velocity.

Use the preceding electron density as the line-of-sight mean, $L/c=2000\mathrm{yr}$, $\omega =2\pi (1.4\times 10^9){\mathrm{s}}^{-1}$ and $\Delta \omega =2\pi (4.0\times 10^7){\mathrm{s}}^{-1}$. ThenThis estimate corresponds to dispersion measure about $69.5\mathrm{pc}\mathrm{c}{\mathrm{m}}^{-3}$. A cloud density inferred from absorption need not equal the actual path average, so a real pulsar may differ even at the same distance.