= 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 $y_\alpha=\alpha_{\rm H\alpha}^{\rm eff}/\alpha_r$, where $\alpha_{\rm H\alpha}^{\rm eff}$ is the <effective recombination coefficient>. At the approximately $10^4\ \mathrm K$ <temperature> implicit in the supplied $\alpha_r$, take $y_\alpha\simeq0.45$. The coefficient 2.2 used later in the paper corresponds to $y_\alpha=1/2.2\simeq0.455$, which we use consistently here.
For an unextinguished, fully filled <Strömgren sphere> with negligible central cavity, the <H-alpha> <luminosity> and <observed radiative flux> satisfy
$$
4\pi D^2F_{\rm H\alpha}=y_\alpha\frac{hc}{\lambda_{\rm H\alpha}}\alpha_rn_e^2\frac{4\pi r_S^3}{3},\qquad
r_S=\frac{\theta D}{2}.
$$
Substitution eliminates $r_S$ and gives
$$
\boxed{D=\frac{24F_{\rm H\alpha}\lambda_{\rm H\alpha}}
{y_\alpha hc\alpha_rn_e^2\theta^3}.}
$$
Convert the angular diameter to <radians>: $\theta=56\times10^{-3}/206265=2.7150\times10^{-7}$. Also $hc/\lambda_{\rm H\alpha}=3.0268\times10^{-19}\ \mathrm J$. Thus
$$
\boxed{D=1.5422\times10^{19}\ \mathrm m\simeq500\ \mathrm{pc},\qquad
r_S\simeq2.09\times10^{12}\ \mathrm m\simeq14.0\ \mathrm{au}.}
$$
The implied <hydrogen-ionizing photon production rate> is $Q\simeq1.00\times10^{39}\ \mathrm{s^{-1}}$. Using $y_\alpha=0.45$ instead gives $D\simeq505\ \mathrm{pc}$. 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.
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