# Recombination-Line Sources

Astronomical sources of radio recombination lines are often in local thermodynamic equilibrium (LTE).  The spontaneous emission rate from atomic physics and the laws of radiative transfer for spectral lines can be combined to model sources in LTE.  LTE does not apply to all recombination lines, so departures from LTE must be recognized and treated differently.

Equation 7B7 gives the absorption coefficient at the center frequency $\nu_0$ of the $n \rightarrow n + 1$ electronic transition of hydrogen in an HII region in local thermodynamic equilibrium (LTE) at electron temperature $T_{\rm e}$:
$$\kappa_\nu = {c^2 \over 8 \pi \nu^2_0} {g_{\rm n+1} \over g_{\rm n}} N_{\rm n} A_{\rm n+1,n} \biggl[ 1 - \exp \biggl( - {h \nu_0 \over k T_{\rm e}} \biggr) \biggr] \phi(\nu)~,$$ 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, $n \gg 1$, $h \nu_0 \ll kT_{\rm e}$, and $g_{\rm n+1} \approx g_{\rm n}$. Equation 7A4 gives the spontaneous emission rate $$A_{\rm n+1,n} \approx { 64 \pi^6 m_{\rm e} e^{10} \over 3 c^3 h^6 n^5}$$ and Equation 7A8 parameterizes the normalized line profile $$\phi(\nu_0) \approx \biggl( {\ln 2 \over \pi}\biggr)^{1/2} {2 \over \Delta \nu}~.$$

The number density $N_{\rm n}$ of atoms in the $n$th electronic energy level is given by the Saha equation, a generalization of the Boltzmann equation.  For a derivation of the Saha equation, see Rybicki & Lightman Eq. 9.47).
$$N_{\rm n} = n^2 \biggl( {h^2 \over 2 \pi m_{\rm e} k T_{\rm e}} \biggr)^{3/2} N_{\rm p} N_{\rm e} \exp \biggl({ \chi_{\rm n} \over k T_{\rm e}} \biggr)$$ where $\chi_{\rm n}$ is the ionization potential of the $n$th energy level. For large $n$, $\vert \chi_{\rm n}\vert \ll k T_{\rm e}$ and the exponential term is nearly unity. Thus the opacity coefficient at the line center frequency $\nu_0$ is approximately
$$\kappa_{\nu_0} \approx {c^2 n^2 \over 8 \pi \nu_0^2} \biggl( { h^2 \over 2 \pi m_{\rm e} k T_{\rm e}} \biggr)^{3/2} N_{\rm e}^2 \biggl({ 64 \pi^6 m_{\rm e} e^{10} \over 3 c^3 h^6 n^5}\biggr) { h \nu_0 \over k T_{\rm e} } \biggl[ \biggl( { \ln 2 \over \pi} \biggr)^{1/2} {2 \over \Delta \nu}\biggr]$$ Some algebra reduces this to $$\bbox[border:3px blue solid,7pt]{\kappa_{\nu_0} \approx \biggl( { N_{\rm e}^2 \over T_{\rm e}^{5/2} \Delta \nu} \biggr) \biggl( {4 \pi e^6 h \over 3 m_{\rm e}^{3/2} k^{5/2} c} \biggr) \biggl( {\ln 2 \over 2} \biggr)^{1/2}}\rlap{\quad \rm {(7C1)}}$$ Notice that no explicit dependence on the level $n$ remains; Equation 7C1 and all results derived from it apply to radio recombination lines with any $n \gg 1$. The line optical depth $\tau_{\rm L} = \int \kappa_{\nu_0} ds$ at the 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 contributed by a recombination emission line at its center frequency $\nu_0$ 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 (which occurs at frequency $\nu_0$) 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 is small.

Departures from LTE

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 with large $n$ (hence low $A_{\rm n,n-1}$) in HII regions with high electron densities $N_{\rm e}$. 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 (sometimes called population factors) $b_{\rm n}$ as a function of the electronic energy level $n$ (abscissa) and parameterized by electron density $N_{\rm e}$ in cm$^{-3}$ (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.

Astronomical Applications

Recombination lines can be used to find the electron temperatures of HII regions in LTE. Solving Equation 7C4 explicitly for $T_{\rm e}$ gives the useful formula
$$\bbox[border:3px blue solid,7pt]{\biggl( { T_{\rm e} \over {\rm K}} \biggr) \approx \biggl[ 7.0 \times 10^3 \biggl( { \nu \over {\rm GHz}} \biggr)^{1.1} \, 1.08^{-1} \, \biggl( { \Delta v \over {\rm km~s}^{-1} } \biggr)^{-1} \biggl( {T_{\rm C} \over T_{\rm L}} \biggr) \biggr]^{0.87}}\rlap{\quad \rm {(7C5)}}$$

Example: By mapping the recombination 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 from nonzero radial velocities. With a simple rotational model for the disk of our Galaxy, astronomers can convert radial velocities to distances, albeit with some ambiguities, and map the approximate distribution of HII regions in our Galaxy. They roughly outline the major spiral arms. The spiral pattern of our Galaxy in plan view, 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. Recombination-line observations show that the electron temperatures $T_{\rm e}$ of HII regions increase with distance from the galactic center at the rate of $287\pm46$ K kpc$^{-1}$, probably because metallicity decreases (Quireza et al. 2006, ApJ, 653, 1226).

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. M82 imaged in H92$\alpha$ (contours) and 8.3 GHz continuum (gray scale) (Rodriguez-Rico,  C. A. et al. 2004, ApJ, 616, 783).