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.
