 Solution

An additional atomic input is needed: the total Case B recombination coefficient does not specify the number of H-alpha photons per net radiative recombination. Let $y_{\alpha }={\alpha }_{\mathrm{H}\alpha }^{\mathrm{eff}}/{\alpha }_r$, where ${\alpha }_{\mathrm{H}\alpha }^{\mathrm{eff}}$ is the effective recombination coefficient. At the approximately $10^4\mathrm{K}$ temperature implicit in the supplied ${\alpha }_r$, take $y_{\alpha }\simeq 0.45$. The coefficient 2.2 used later in the paper corresponds to $y_{\alpha }=1/2.2\simeq 0.455$, which we use consistently here.

For an unextinguished, fully filled Strömgren sphere with negligible central cavity, the H-alpha luminosity and observed radiative flux satisfySubstitution eliminates $r_S$ and givesConvert the angular diameter to radians: $\theta =56\times 10^{-3}/206265=2.7150\times 10^{-7}$. Also $hc/{\lambda }_{\mathrm{H}\alpha }=3.0268\times 10^{-19}\mathrm{J}$. ThusThe implied hydrogen-ionizing photon production rate is $Q\simeq 1.00\times 10^{39}{\mathrm{s}}^{-1}$. Using $y_{\alpha }=0.45$ instead gives $D\simeq 505\mathrm{pc}$. These values use the printed milliarcseconds and integrated line observed radiative flux. Assuming one H-alpha photon per recombination would give a different, unjustified distance. An appreciable cavity or a nonunit filling factor must also enter the emitting volume.

The Brackett series consists of transitions ending at $n=4$, at infrared wavelengths. They suffer much less interstellar extinction than H-alpha, so their corrected observed radiative fluxes can give a more reliable version of the same inference. At fixed measured angular size and number density, the formula has $D\propto F$: uncorrected interstellar extinction therefore makes this particular distance estimate too small. Infrared lines are weaker and need their own effective recombination coefficients; their advantage is reduced sensitivity to dust.