Recombination Line Sources

The absorption coefficient at the center frequency $\nu_0$ of the $n \rightarrow n + 1$ electronic transition of hydrogen in an HII region at electron temperature $T_{\rm e}$ is
$$\kappa_\nu = {c^2 \over 8 \pi \nu^2_{\rm n,n+1}} {g_{\rm n+1} \over g_{\rm n}} N_{\rm n} A_{\rm n+1,n} \biggl[ 1 - \exp \biggl( - {h \nu_{\rm n,n+1} \over k T_{\rm e}} \biggr) \biggr] \phi(\nu)$$
where
$$\nu_0 = \nu_{\rm n,n+1} \approx {2 R c \over n^3} = {2 \pi^2 m_{\rm e} e^4 \over h^3 n^3}$$
and
$$g_{\rm n} = 2 n^2~.$$
At radio frequencies, $n \gg 1$, $h \nu_{\rm n,n+1} \ll kT_{\rm e}$, and $g_{\rm n+1} \approx g_{\rm n}$. Recall that
$$A_{\rm n+1,n} \approx { 64 \pi^6 m_{\rm e} e^{10} \over 3 c^3 h^6} {1 \over n^5}$$
and
$$ \phi(\nu_0) \approx \biggl( {\ln 2 \over \pi}\biggr)^{1/2} {2 \over \Delta \nu}~.$$

The last quantity that we need is $N_{\rm n}$, the number density of atoms in the $n$th electronic energy level. This is given by the Saha equation, which we won't derive here (see Rohlfs & Wilson Eq. 13.24).
$$N_{\rm n} = n^2 \biggl( {h^2 \over 2 \pi m_{\rm e} k T_{\rm e}} \biggr)^{3/2} N_{\rm p} N_{\rm e} \exp \biggl({ X_{\rm n} \over k T_{\rm e}} \biggr) $$
where $X_{\rm n}$ is the ionization potential from the $n$th energy level. For large $n$, $\vert X_{\rm n}\vert \ll k T_{\rm e}$ and the exponential term is nearly unity. Thus the opacity coefficient at the line center frequency $\nu_0$ is approximately
$$\kappa_{\nu_0} \approx {c^2 n^2 \over 8 \pi \nu_0^2} \biggl( { h^2 \over 2 \pi m_{\rm e} k T_{\rm e}} \biggr)^{3/2} N_{\rm e}^2 \biggl({ 64 \pi^6 m_{\rm e} e^{10} \over 3 c^3 h^6 n^5}\biggr) { h \nu_0 \over k T_{\rm e} } \biggl[ \biggl( { \ln 2 \over \pi} \biggr)^{1/2} {2 \over \Delta \nu}\biggr]$$
$$ \kappa_{\nu_0} \approx {c^2 n^2 \over 8 \pi} \biggl({h^3 n^3 \over 2 \pi^2 m_{\rm e} e^4}\biggr) { h^3 \over 2 \cdot 2^{1/2} (\pi m_{\rm e} k T_{\rm e})^{3/2} } N_{\rm e}^2 { 64 \pi^6 m_{\rm e} e^{10} \over 3 c^3 h^6 n^5} { h \over k T_{\rm e}} \biggl( { \ln 2 \over \pi }\biggr)^{1/2} {2 \over \Delta \nu}$$
$$\bbox[border:3px blue solid,7pt]{\kappa_{\nu_0} \approx \biggl( { N_{\rm e}^2 \over T_{\rm e}^{5/2} \Delta \nu} \biggr) \biggl( {4 \pi e^6 h \over 3 m_{\rm e}^{3/2} k^{5/2} c} \biggr) \biggl( {\ln 2 \over 2} \biggr)^{1/2}}\rlap{\quad \rm {(7C1)}}$$
Note that no explicit dependence on the level $n$ remains; this equation and results derived from it apply to radio recombination lines with any $n \gg 1$. The optical depth $\tau_{\rm L} = \int \kappa_{\nu_0} ds$ at the line center frequency $\nu_0$ can be expressed in terms of the emission measure $${\rm EM} \equiv \int \biggl( {N_{\rm e}^2 \over {\rm cm}^{-6}} \biggr) \biggl( { d s \over {\rm pc}} \biggr)~;$$
in astronomically convenient units it is
$$\bbox[border:3px blue solid,7pt]{\tau_{\rm L} \approx 1.92 \times 10^3 \biggl( { T_{\rm e} \over {\rm K}} \biggr)^{-5/2} \biggl( { {\rm EM} \over {\rm pc}\,{\rm cm}^{-6} } \biggr) \biggl( { \Delta \nu \over {\rm kHz} } \biggr)^{-1}}\rlap{\quad \rm {(7C2)}}$$

Since $\tau_{\rm L} \ll 1$ in all known HII regions, the brightness temperature at the center of the recombination emission line is
$$\bbox[border:3px blue solid,7pt]{T_{\rm L} \approx T_{\rm e} \tau_{\rm L} \approx 1.92 \times 10^3 \biggl( { T_{\rm e} \over {\rm K}} \biggr)^{-3/2} \biggl( { {\rm EM} \over {\rm pc}\,{\rm cm}^{-6} } \biggr) \biggl( { \Delta \nu \over {\rm kHz} } \biggr)^{-1}}\rlap{\quad \rm {(7C3)}} $$
At frequencies high enough that the free-free continuum is also optically thin, the peak line-to-continuum ratio in LTE is
$$\bbox[border:3px blue solid,7pt]{{T_{\rm L} \over T_{\rm C}} \approx 7.0 \times 10^3 \biggl( { \Delta v \over {\rm km~s}^{-1} } \biggr)^{-1} \biggl( { \nu \over {\rm GHz}} \biggr)^{1.1} \biggl( { T_{\rm e} \over {\rm K} } \biggr)^{-1.15} \biggl[ 1 + { N({\rm He}^+) \over N({\rm H}^+) } \biggr]^{-1}}\rlap{\quad \rm {(7C4)}}$$
where $\Delta v$ is the line FWHM expressed as a velocity and the typical He$^+$/H$^+$ ion ratio is $N({\rm He}^+) / N({\rm H}^+) \approx 0.08$. The term in square brackets is necessary because He$^+$ contributes to the free-free continuum emission but not to the hydrogen recombination line. The line-to-continuum ratio yields an estimate of the electron temperature $T_{\rm e}$ which is independent of the emission measure so long as the frequency is high enough that the continuum optical depth $\tau_{\rm C} \ll 1$.

Detailed calculations show that the assumption of LTE is generally a good one in galactic HII regions. Maintaining LTE requires that collisions, which thermalize the gas, occur more frequently than radiative transitions, which can cause departures from LTE. Thus the assumption of LTE is best for transitions in HII regions with high electron densities $N_{\rm e}$ and large $n$ (hence low $A_{\rm n,n-1}$). The departure from LTE is parameterized by the departure coefficient, $b_{\rm n}$ defined as the ratio of the actual population in the $n$th level to the theoretically expected population in LTE. Detailed calculations give $b_{\rm n}$ values shown in the figure below.

Departure coefficients (aka populations factors) $b_{\rm n}$ as a function of the electronic energy level $n$ (abscissa) and parameterized by electron density $N_{\rm e}$ (Sejnowski, T. J., & Hjellming, R. M. 1969, ApJ, 156, 915).

Hydrogen atoms with very high $n$ are quite large, so collisional broadening smears low-frequency HI recombination lines. The tradeoffs needed to avoid departures from LTE, collisional broadening, and high free-free opacity favor observations of transitions with intermediate $n$-values near $\nu \sim 10$ GHz.

Solving explicitly for the electron temperature $T_{\rm e}$ we get
$$\bbox[border:3px blue solid,7pt]{\biggl( { T_{\rm e} \over {\rm K}} \biggr) \approx \biggl[ 7.0 \times 10^3 \biggl( { \nu \over {\rm GHz}} \biggr)^{1.1} \, 1.08^{-1} \, \biggl( { \Delta v \over {\rm km~s}^{-1} } \biggr)^{-1} \biggl( {T_{\rm C} \over T_{\rm L}} \biggr) \biggr]^{0.87}}\rlap{\quad \rm {(7C5)}}$$

Example: By mapping the recombinantion line-to-continuum ratios $T_{\rm L} / T_{\rm C}$ in a number of H$n\alpha$ transitions, Lockman and Brown determined the temperature distribution in the Orion Nebula, a nearby HII region.

A temperature-distribution model for the Orion Nebula HII region based on the line-to-continuum ratios of hydrogen recombination lines (Lockman, F. J., & Brown, R. L. 1975, ApJ, 201, 134).

Differences between the rest and observed frequencies of radio recombination lines are attributed to Doppler shifts indicating radial velocities. (Frequencies can be measured very accurately because frequency measurements are essentially time measurements, and atomic clocks have extraordinary accuracy, about 1 part in $10^{13}$. In contrast, wavelength measurements are length measurements whose accuracy is limited by the mechanical stability of spectrometer dimensions.) With a simple rotational model for the disk of our Galaxy, radio astronomers can convert radial velocities to distances, albeit with some ambiguities, and map the distribution of HII regions in our Galaxy. They roughly outline the major spiral arms.

The spiral pattern of our Galaxy as traced by H$\alpha$ (circles) and radio recombination lines (squares) (Georgelin, Y. M., & Georgelin, Y. P. 1976, A&A, 49, 57).

A plot showing the observed electron temperatures of galactic HII regions reveals that temperature increases with distance from the galactic center.

The electron temperatures $T_{\rm e}$ of HII regions tend to increase with galactocentric radius $R_{\rm G}$, probably because metallicity decreases (Shaver, P. A. et al. 1983, MNRAS, 204, 53).

The explanation for this trend is the observed decrease in metallicity (relative abundance of elements heavier than helium) with galactocentric distance. Power radiated by emission lines of "metals" is the principal cause of HII region cooling.

Radio recombination line strengths are not affected by dust extinction the way optical lines (e.g., the H$\alpha$ and H$\beta$ lines) are, so they are useful quantitative indicators of the ionization rates and hence star-formation rates in dusty starburst galaxies such as M82.