#  The Most Sensitive Radio Recombination Line Measurements Ever Made of the
Galactic Warm Ionized Medium

T. M. Bania Institute for Astrophysical Research, Astronomy Department, Boston
University,
725 Commonwealth Ave., Boston, MA 02215, USA  Dana S. Balser National Radio
Astronomy Observatory, 520 Edgemont Rd., Charlottesville, VA 22903, USA  Trey
V. Wenger NSF Astronomy & Astrophysics Postdoctoral Fellow,
Department of Astronomy, University of Wisconsin–Madison,
Madison, WI, 53706, USA  Spencer J. Ireland Institute for Astrophysical
Research, Astronomy Department, Boston University,
725 Commonwealth Ave., Boston, MA 02215, USA  Department of Physics, New
Mexico Tech, Socorro, NM 87801, USA  L. D. Anderson Department of Physics and
Astronomy, West Virginia University,
Morgantown, WV 26506, USA  Center for Gravitational Waves and Cosmology, West
Virginia University,
Chestnut Ridge Research Building, Morgantown, WV 26505, USA  Green Bank
Observatory, P.O. Box 2, Green Bank, WV 24944, USA  Matteo Luisi Department of
Physics, Westminster College, New Wilmington, PA 16172, USA  Center for
Gravitational Waves and Cosmology, West Virginia University,
Chestnut Ridge Research Building, Morgantown, WV 26505, USA

###### Abstract

Diffuse ionized gas pervades the disk of the Milky Way. We detect extremely
faint emission from this Galactic Warm Ionized Medium (WIM) using the Green
Bank Telescope to make radio recombination line (RRL) observations toward two
Milky Way sight lines: G20, $(\ell,{\it b})$=$(20\,^{\circ},0\,^{\circ})$, and
G45, $(\ell,{\it b})$=$(45\,^{\circ},0\,^{\circ})$. We stack 18 consecutive
Hn$\alpha$ transitions between 4.3–7.1 GHz to derive ${\rm\langle
Hn\alpha\rangle}$ spectra that are sensitive to RRL emission from plasmas with
emission measures EM $\gtrsim$ 10 ${\rm\,cm^{-6}\,pc}$. Each sight line has
two Gaussian shaped spectral components with emission measures that range
between $\sim$100 and $\sim$300 ${\rm\,cm^{-6}\,pc}$. Because there is no
detectable RRL emission at negative LSR velocities the emitting plasma must be
located interior to the Solar orbit. The G20 and G45 emission measures imply
RMS densities of 0.15 and 0.18$\,{\rm cm^{-3}}$, respectively, if these sight
lines are filled with homogeneous plasma. The observed ${\rm\langle
Hn\beta\rangle}$/${\rm\langle Hn\alpha\rangle}$ line ratios are consistent
with LTE excitation for the strongest components. The high velocity component
of G20 has a narrow line width, 13.5$\,{\rm km\,s^{-1}}$, that sets an upper
limit of $\lesssim$ 4,000 K for the plasma electron temperature. This is
inconsistent with the ansatz of a canonically pervasive, low density, $\sim$
10,000 K WIM plasma.

H ii regions — ISM: abundances — radio lines: ISM

Interstellar line emission (844), Milky Way disk (1050), Radio spectroscopy
(1359), Warm ionized medium (1788)

††facilities: GBT††software: TMBIDL (Bania et al., 2016)

##  1 The Milky Way Warm Ionized Medium

Hoyle & Ellis (1963) were the first to suggest the existence of a layer of
ionized gas along the Galactic plane with an electron density of $\sim
0.1$$\,{\rm cm^{-3}}$ and an electron temperature of $\sim 10^{4}$$\,{\rm K}$.
This was based on their Hobart, Tasmania discovery of free-free absorption at
frequencies less than 10$\,{\rm MHz}$ against the synchrotron background
observed at very low radio frequencies (Ellis et al., 1962; Reber & Ellis,
1956). They estimated that the ionizing radiation necessary to produce the
observed free-free absorption was consistent with flux from OB stars. The
dispersion of radio waves from pulsars (Tanenbaum et al., 1968) and the
observation of optical emission lines (Reynolds et al., 1973) from the
interstellar medium (ISM) confirmed the existence of a warm ionized medium
(WIM) in the Milky Way. Detection of H$\alpha$ emission from the diffuse
ionized gas (DIG) in the edge-on spiral galaxy NGC 891 showed that other
galaxies had similar warm plasmas (Dettmar, 1990; Rand et al., 1990). Here, we
follow the convention of using the term “WIM” to refer to the warm ionized
medium in the Milky Way and “DIG” to refer to the diffuse ionized gas in
external galaxies (Haffner et al., 2009).

The WIM has been extensively studied with optical emission lines. H$\alpha$
maps of the Galaxy show emission from almost every direction (e.g., Dennison
et al., 1998; Gaustad et al., 2001; Haffner et al., 2003) with a volume
filling factor larger than 0.2 (Reynolds, 1991a). Sensitive observations of
two emission lines from nitrogen indicate that the WIM is about 2,000$\,{\rm
K}$ warmer than H ii regions (Reynolds et al., 2001). Models of escaping
radiation from H ii regions indicate a harder spectrum for photons capable of
ionizing hydrogen but a suppression of He-ionizing photons (Wood & Mathis,
2004), consistent with observations of helium in the WIM (e.g., Reynolds &
Tufte, 1995).

Although known for 60 years now, a detailed understanding of the WIM’s origin,
distribution, and physical properties remains to be crafted. Because of
attenuation by dust, studies of the WIM using H$\alpha$ are mostly limited to
regions near to the Sun or at high Galactic latitudes. Moreover, the
relatively low spatial and spectral resolution of most H$\alpha$ surveys
cannot separate emission from the WIM with that from discrete, OB-star excited
H ii regions. This is particularly a problem in the inner Galactic disk.

Two recent radio recombination line (RRL) surveys overcome these limitations.
The Green Bank Telescope (GBT) Diffuse Ionized Gas Survey (GDIGS; Anderson et
al., 2021) is a 4-8$\,{\rm GHz}$ RRL survey of the Milky Way disk that probes
the distribution and properties of the WIM in the inner Galaxy ($32\hbox
to0.0pt{.\hss}{{}^{\circ}}3>\ell>-5\,^{\circ},|b|<0\hbox
to0.0pt{.\hss}{{}^{\circ}}5$). GDIGS has a angular resolution of $2\hbox
to0.0pt{.\hss}{{}^{\prime}}6$ and a spectral resolution of 0.5$\,{\rm
km\,s^{-1}}$. A complimentary survey was completed with the Five-hundred-meter
Aperture Spherical radio Telescope (FAST) using RRL transitions spanning
frequencies between 1–1.5$\,{\rm GHz}$ for the Galactic zone
$55\,^{\circ}>\ell>33\,^{\circ},|b|<2\,^{\circ}$(Hou et al., 2022). FAST has
an angular resolution of 3′ and a spectral resolution of 2.2$\,{\rm
km\,s^{-1}}$. Both surveys employ line-stacking of multiple RRL transitions
observed simultaneously to increase the signal to noise ratio.

Using GDIGS data, Luisi et al. (2020) created a WIM-only RRL map of the W43
star formation complex that is devoid of emission from H ii regions. Employing
an empirical model that only accounts for the H ii region locations, angular
sizes, and RRL intensities, they were able to reproduce the observed WIM
emission (also see Belfiore et al., 2022). This supports the notion that UV
photons leaking from the H ii regions surrounding O and B-type stars are the
primary source of the ionization of the WIM.

The main limitation of these new RRL surveys is sensitivity. For example,
GDIGS is sensitive to emission measures, EM $\equiv\ \int\,n_{\rm e}^{2}\,{\rm
d}\ell\ \gtrsim 1,100$${\rm\,cm^{-6}\,pc}$, whereas most H$\alpha$ surveys are
several orders of magnitude more sensitive. GDIGS detects RRL emission from
most directions but not at all possible velocities where H i gas exists. This
may just be a sensitivity issue. For example, the GDIGS spectrum toward G20
shows no RRL emission (see discussion below).

Here, we use the GBT to search for faint RRL emission from two first Galactic
quadrant sight lines: G20 and G45. These directions, at $(\ell,{\it b})$=
$(20\,^{\circ},0\,^{\circ})$, and $(\ell,{\it b})$=
$(45\,^{\circ},0\,^{\circ})$ respectively, were chosen because they contain no
detected discrete H ii regions, show extremely weak radio continuum emission,
and have infrared emission morphologies that do not indicate any incipient
star formation. Any RRL emission detected for these lines of sight would thus
stem from the WIM.

##  2 Observations and Data Analysis

Using the GBT, we made the RRL observations during a series of sessions held
between June and August of 2020 (project GBT/20A–483). Measurements were made
using the C-band (4-8$\,{\rm GHz}$) receiver and the Versatile GBT
Astronomical Spectrometer (VEGAS) back end. Full details of data acquisition
protocols, VEGAS tuning configuration, and calibration are described in our
GDIGS paper (Anderson et al., 2021). GDIGS survey data were acquired using On
the Fly Mapping. Here, however, we employ total power position switching by
observing an off source (Off) position for 6 minutes and then the target (On)
position for 6 minutes, for a total time of 12 minutes per scan. The Off
position lies well outside the Galactic plane because the WIM is expected to
be distributed throughout the Galactic disk with a large volume filling
factor. The Off position is offset from the On by $(\Delta\ell,\Delta{\it
b})=(0\hbox to0.0pt{.\hss}{{}^{\circ}}0,-3\hbox to0.0pt{.\hss}{{}^{\circ}}5)$.

We tuned VEGAS to 64 different frequencies with two orthogonal linear
polarizations each. A single 12 min OffOn spectral pair thus simultaneously
produces 128 independent 8192 channel spectra, each spanning a 23.4375$\,{\rm
MHz}$ bandwidth. The VEGAS tuning consisted of an assortment of hydrogen RRL
transitions that lie in the 4-8$\,{\rm GHz}$ bandpass of the C-band receiver.
These included $\alpha(\Delta\,n=1),\beta(\Delta\,n=2),$ and
$\gamma(\Delta\,n=3)$ transitions, where $n$ is the transition principal
quantum number. For $\alpha$ RRL transitions the frequency extremes of this
tuning are H97$\alpha$ (7.09541$\,{\rm GHz}$ rest frequency) and H115$\alpha$
(4.26814$\,{\rm GHz}$ rest frequency). This large frequency range produces
significant differences in the GBT beam size (the half-power beam width, HPBW)
and spectral resolution. The beam size increases from 107″ for H97$\alpha$ to
177″ for H115$\alpha$ and the spectral velocity resolution decreases from
0.12$\,{\rm km\,s^{-1}}$ per channel for H97$\alpha$ to 0.20$\,{\rm
km\,s^{-1}}$ per channel for H115$\alpha$.

After acquisition, the telescope data were converted into SDFITS format and
then ported to the TMBIDL single dish data analysis software package (Bania et
al., 2016). All subsequent data analysis reported here is done in the TMBIDL
environment.

The intensity scale is determined by injecting signals from a noise diode into
the signal path every other second during observations. The flux density scale
is then established after data acquisition using observations of the quasar 3C
286 which is a primary flux calibrator. For the GBT at C-band frequencies this
quasar is a point source whose intensity has varied by less than 1% over
decades (Ott et al., 1994; Peng et al., 2000). Because the GBT gain at C-band
is 2$\,{\rm K}$/ Jy (Ghigo et al., 2001), the 3C 286 observations allow us to
determine the noise diode fluxes. Overall, this procedure establishes the
intensity scale of our spectra to $\sim$5% accuracy.

To maximize our spectral sensitivity, we derive for each target sight line a
single RRL spectrum for each $\Delta\,n$ transition, $\alpha,\beta,$ and
$\gamma$. We do this by averaging together all the observed, usable RRL
transitions for a given $\Delta\,n$. We call this procedure “stacking” and the
result of this averaging is a “stacked” spectrum: ${\rm\langle
Hn\alpha\rangle}$, ${\rm\langle Hn\beta\rangle}$, and ${\rm\langle
Hn\gamma\rangle}$. The RRL transitions we use to craft these stacked spectra
are compiled in Appendix Table 6. Listed there for both targets are the
transition, the rest frequency, and the system temperatures, $T_{\rm sys}$,
for both linear polarizations (XX and YY).

Stacking is possible because RRLs involve high-$n$ level transitions wherein
adjacent $n$ levels have nearly the same energy. For example, our usable
$\alpha$ transitions span the range from H97$\alpha$ to H115$\alpha$. In LTE
the expected RRL intensities would differ by a factor of $\sim$1.66 between
these two transitions. The expected line profiles, and hence the spectral line
widths, should, however, be similar for all these transitions. Since the
primary goal here is to detect RRL emission, all usable RRL transitions can
therefore be averaged together to increase the signal-to-noise ratio,
producing the ${\rm\langle Hn\alpha\rangle}$, ${\rm\langle Hn\beta\rangle}$,
and ${\rm\langle Hn\gamma\rangle}$ stacked spectra.

Stacking is not a simple average of the different transitions weighted by
their individual integration times and system temperatures, although of course
that must be part of the procedure. Stacking must correct for several other
effects: (1) each transition has a unique rest frequency and spectral velocity
resolution; (2) each transition’s spectrum has a slightly different center
velocity due to our VEGAS tuning; and (3) only one of the RRL transitions is
Doppler tracked due to the limitations of the GBT IF system hardware. Using
the procedure described below we have successfully stacked RRL spectra taken
with the Arecibo telescope/“Interim Correlator” spectrometer (Roshi et al.,
2005) as well as the GBT X-band receiver/ACS spectrometer (Balser, 2006).

To derive the stacked spectra reported here we: (1) inspect for each
transition all the individual OffOn total power pairs and eliminate bad data;
(2) calculate for each transition an average spectrum weighted by integration
time and system temperature; (3) establish a common spectral velocity
resolution for each transition by using a $sin(x)/x$ interpolation (Roshi et
al., 2005) referenced to our poorest resolution, H97$\alpha$, spectrum; (4)
align all the transitions in velocity referenced to the Doppler tracked
H103$\alpha$ spectrum; and (5) obtain a stacked spectrum by averaging all the
individual transition spectra processed in this way weighted by integration
time and system temperature.

Table 1: Stacked Spectra Properties Source | $\ell$ | ${\it b}$ | $\Delta\,n$ | # RRLs | $t_{\rm intg}$ | RMS
---|---|---|---|---|---|---
| (deg) | (deg) |  |  | (hrs) | (mK)
G20 | 20.0 | 0.0 | 1: $\alpha$ | 18 | 243.1 | 0.312
G20 | … | … | 2: $\beta$ | 22 | 302.4 | 0.288
G20 | … | … | 3: $\gamma$ |  07 |  091.0 | 0.483
G45 | 45.0 | 0.0 | 1: $\alpha$ | 18 | 207.0 | 0.343
G45 | … | … | 2: $\beta$ | 22 | 270.8 | 0.314
G45 | … | … | 3: $\gamma$ |  07 |  079.6 | 0.518

The properties of the stacked spectra for the G20 and G45 sight lines are
summarized in Table 1. Listed are the source name, the Galactic position,
$(\ell,{\it b})$, the RRL transition order, $\Delta\,n$, the number of
transitions used to derive the stacked spectrum, the total integration time of
the stacked spectrum, $t_{\rm intg}$, and the root mean square (RMS) noise of
the stacked spectrum after smoothing to 1$\,{\rm km\,s^{-1}}$ resolution. The
integration times for the ${\rm\langle Hn\alpha\rangle}$ spectra exceed
$\sim$200 hrs and the spectral RMS noise attained thereby is
$\sim$300$\rm\,\mu K$. We believe these to be the most sensitive measurements
of WIM RRL emission ever made.

##  3 RRL Emission from WIM Plasma

Figure 1:  Mid-infrared WISE images of G20 (left) and G45 (right) where blue
is 3.4$\,\mu{\rm m}$ (stars), green 12$\,\mu{\rm m}$ (emission from the
photodissociation region; PDR), and red 22$\,\mu{\rm m}$ (dust). The images
are 6′ on a side. The circles show the range of the GBT beam size (HPBW) for
this experiment, from 107″ for H97$\alpha$ to 177″ for H115$\alpha$.

###  3.1 The G20 and G45 Lines of Sight

We detect weak RRL emission from both first Galactic quadrant sight lines: G20
and G45. Images of the mid-infrared emission seen toward these lines of sight
by the Wide-Field Infrared Survey Explorer (Wright et al., 2010, WISE)
satellite are shown in Figure 1. There is clearly no extended IR emission in
these directions consistent with either protostellar activity or high mass
star formation. Accordingly, the WISE Catalog of Galactic H ii Regions
(Anderson et al., 2014a, hereafter “The WISE Catalog”) lists no OB-type star
excited, discrete H ii regions in these fields. (The doughnut shaped object
seen toward $(\ell,{\it b})$ $\sim$ (44.99,+0.04) is a latent image from the
overexposure of a saturated bright star located beyond the field of view shown
in Figure 1.)

The stacked ${\rm\langle Hn\alpha\rangle}$ spectra we derive for G20 and G45
are shown in Figure 2\. Both sight lines show two components of RRL emission
from WIM plasma. In each direction the higher LSR velocity component has the
stronger peak intensity and narrower line width. The G20 components are twice
as strong as their corresponding G45 components. We fit Gaussian functions to
these components in order to quantify their spectral line properties. Table 2
compiles the results of these fits. Listed for each spectral component are the
fit values and fit errors for the LSR velocity, $V_{\rm LSR}$, peak antenna
temperature at the line center, $T_{\rm L}^{\rm A}$, the full width at half
maximum, FWHM, line width, $\Delta\,V$, the area under the component, $W_{\rm
RRL}$, and an estimate of the emission measure, EM. The errors cited for peak
intensity, line area, and emission measure include the 5 % uncertainty in the
intensity scale.

How do these spectra compare with the corresponding GDIGS and FAST
measurements? Figure 3 shows that for G20 our deep integration is much more
sensitive than GDIGS. GDIGS did not observe the G45 sight line but FAST did.
FAST is more EM sensitive than GDIGS. Inspection of their Figure 5, however,
also reveals no RRL emission. The strongest emission component that we measure
toward either of our targets is therefore below the GDIGS and FAST sensitivity
limits.

Figure 2:  G20 and G45 stacked spectra for the ${\rm\langle Hn\alpha\rangle}$
transition. The spectra were smoothed to a 1$\,{\rm km\,s^{-1}}$ velocity
resolution and the instrumental spectral baseline was subtracted. The emission
measure, EM, axes stem from assumptions described in the text.  Figure 3:  G20
${\rm\langle Hn\alpha\rangle}$ spectrum (black) compared with the GDIGS
spectral map pixel for the same direction (red). The ${\rm\langle
Hn\alpha\rangle}$ data were smoothed to 5$\,{\rm km\,s^{-1}}$ resolution. The
$\sim$115$\,{\rm km\,s^{-1}}$ ${\rm\langle Hn\alpha\rangle}$ component is
below the GDIGS sensitivity limit.

###  3.2 RRL Stacked Spectra

For both G20 and G45 RRL emission can be seen at nearly all LSR velocities
between 0$\,{\rm km\,s^{-1}}$ and the terminal velocity produced by Galactic
rotation. The lack of any detected RRL emission at LSR velocities below
0$\,{\rm km\,s^{-1}}$ means that in these directions any RRL emission from WIM
plasma located beyond the Solar orbit about the Galactic Center must be below
the sensitivity limits of our GBT observations.

Table 2: ${\rm\langle Hn\alpha\rangle}$ Emission Properties $Source$ | $V_{\rm LSR}$ | $\sigma_{\rm LSR}$ | $T_{\rm L}^{\rm A}$ | $\sigma_{\rm L}$ | $\Delta\,V$ | $\sigma_{\Delta V}$ | $W_{\rm RRL}$ | $\sigma_{\rm W}$ | $EM$ | $\sigma_{\rm EM}$
---|---|---|---|---|---|---|---|---|---|---
| ($\,{\rm km\,s^{-1}}$) | ($\,{\rm km\,s^{-1}}$) | ($\,{\rm mK}$) | ($\,{\rm mK}$) | ($\,{\rm km\,s^{-1}}$) | ($\,{\rm km\,s^{-1}}$) | ($\,{\rm mK}\,\,{\rm km\,s^{-1}}$) | ($\,{\rm mK}\,\,{\rm km\,s^{-1}}$) | (${\rm\,cm^{-6}\,pc}$) | (${\rm\,cm^{-6}\,pc}$)
G20 |  047.68 | 0.22 | 3.32 | 0.33 | 60.1 | 0.60 | 212.2 | 21.2 |  $<$245.5 | 24.5
G20 | 113.35 | 0.05 | 7.43 | 0.36 | 13.5 | 0.12 | 107.0 |  05.3 |  $<$123.8 |  06.2
G45 |  034.54 | 0.87 | 0.94 | 0.37 | 28.8 | 2.63 |  028.8 | 11.7 |  $<$112.4 | 45.7
G45 |  063.29 | 0.20 | 3.44 | 0.39 | 20.3 | 0.42 |  074.4 |  08.6 |  $<$290.3 | 33.5

We use the observed RRL emission components to estimate the emission measure.
Assuming that the WIM plasma is extended and always fills the GBT beam, the
brightness temperature at the line center of optically thin transitions in
LTE, $T_{\rm L}^{\rm B}$, is [Gordon & Sorochenko (2002), Wilson et al. (2009,
equation 14.28 in the 5th edition), Wenger et al. (2019, equation 16), and
Anderson et al. (2021, equation A4)]:

$\frac{T_{\rm L}^{\rm B}}{\rm K}=3013\left(\frac{T_{e}}{\rm
K}\right)^{-1.5}\left(\frac{\rm EM}{\text{cm${}^{-6}$
pc}}\right)\left(\frac{\nu_{0}}{\rm GHz}\right)^{-1}\left(\frac{\Delta
V}{\text{km s${}^{-1}$}}\right)^{-1}\left(\frac{\Delta n}{n}\right)f_{n+\Delta
n,n}$ (1)

where $T_{\rm e}$ is the plasma electron temperature, $\nu_{0}$ is the rest
frequency, and ${f_{n+\Delta n,n}}$ is the oscillator strength of the
transition between $n+\Delta n$ and $n$. We consider only ${\rm
Hn\alpha}~{}(\Delta n=1)$ transitions and adopt the average rest frequency and
oscillator strength of our ${\rm\langle Hn\alpha\rangle}$ stacked spectra.
Using these values, Equation 1 yields for the emission measure:

$\frac{\rm EM}{\text{cm${}^{-6}$
pc}}=4.57\left(\frac{T_{e}}{10^{4}\,\text{K}}\right)^{1.5}\left(\frac{W_{\rm
RRL}}{\text{mK km s${}^{-1}$}}\right)$ (2)

Here, $W_{\rm RRL}$ [$\,{\rm mK}$$\,{\rm km\,s^{-1}}$], is the spectral area
of an emission component. For a Gaussian line shape, $W_{\rm RRL}$= 1.064
$T_{\rm L}^{\rm B}$ $\Delta\,V$.

The RRL intensity values cited in all tables, however, list antenna
temperature, ${T_{\rm L}^{\rm A}}$, the directly measured quantity, rather
than the brightness temperature, $T_{\rm L}^{\rm B}$. Converting between
antenna temperature to brightness temperature depends on the coupling between
the characteristic angular size, $\Theta_{\rm WIM}$, of the RRL emitting
plasma with the GBT beam. The stacked ${\rm\langle Hn\alpha\rangle}$ spectra
stem from frequencies spanning the 4–8$\,{\rm GHz}$ instantaneous bandwidth of
the C band receiver. For this experiment, the GBT HPBW beam size, $\Theta_{\rm
BEAM}$, thus ranges between 177″ for H115$\alpha$ to 107″ for H97$\alpha$.
Here, we assume that the RRL emitting plasma is spatially extended and fills
the GBT C band beam(s): $\Theta_{\rm WIM}\gg\Theta_{\rm BEAM}$ always. For
this case the brightness temperature is $T_{\rm\,B}=T_{\rm\,A}/\eta_{\rm\,B}$
where the beam efficiency of the GBT at 5.7578 GHz is $\eta_{\rm\,B}=0.93$
(Ghigo et al., 2001; Maddalena, 2010; Balser et al., 2016).

Assuming an electron temperature and FWHM line width, the observed line
antenna temperature, $T_{\rm L}^{\rm A}$, gives an EM of:

$\frac{\rm EM}{\text{cm${}^{-6}$
pc}}=130.71\left(\frac{T_{e}}{10^{4}\,\text{K}}\right)^{1.5}\left(\frac{\Delta
V}{25\,\text{km s${}^{-1}$}}\right)\left(\frac{T_{\rm L}^{\rm A}}{\rm
mK}\right)$ (3)

Here, we use the line peak antenna temperature, $T_{\rm L}^{\rm A}$, for ease
of interpreting the intensities shown in the spectra and listed in the tables.
The needed conversion to brightness temperature has been absorbed into
Equation 3’s numerical constant. The EM axes in Figures 2 and 3 and the values
cited in Table 2 stem from Equation 3 using the observed $T_{\rm L}^{\rm A}$
and $\Delta\,V$ together with an assumed value for the plasma $T_{\rm e}$.

Although the electron temperature can be derived from the RRL line-to-
continuum ratio, we did not measure the radio continuum for G20 and G45 and so
cannot estimate $T_{\rm e}$ from our GBT observations. Studies of the WIM/DIG
in the Milky Way and other galaxies using ${\rm H_{\alpha}}$, [N II], [S II],
and [O III] find that the electron temperature ranges between 6,000 and
11,000$\,{\rm K}$ (Haffner et al., 2009). Here, we use the observed line
widths, $\Delta\,V$, to constrain $T_{\rm e}$. Interpreting the observed line
width as being due solely to thermal broadening, $\Delta\,V$ = $\Delta\,V_{\rm
th}$, sets a firm upper limit on the plasma electron temperature:

$\frac{T_{e}}{\rm K}\leq 21.85\left(\frac{\Delta\,V}{\,{\rm
km\,s^{-1}}}\right)^{2}$ (4)

where $k$ is Boltzmann’s constant and $m_{\rm H}$ is the mass of hydrogen. The
high LSR velocity RRL components seen toward G20 and G45 have the smallest
line width for each sight line. These line widths give $T_{\rm e}$ upper
limits of $\lesssim$ 4,000 $\,{\rm K}$ and $\lesssim$ 9,000 $\,{\rm K}$ for
G20 and G45, respectively. These are the $T_{\rm e}$ values used for the EM
axes in Figures 2 and 3 and for the $EM$ values cited in Table 2.

The ${\rm\langle Hn\alpha\rangle}$ WIM spectra we derive for G20 and G45 are
very deep integrations. The four ${\rm\langle Hn\alpha\rangle}$ emission
components that we detect toward G20 and G45 yield emission measures ranging
between $\sim$ 100 and $\sim$ 300${\rm\,cm^{-6}\,pc}$ (via Equation 2 and
Table 2). All these EM values stem from smoothing the native VEGAS velocity
resolution per channel to 1 $\,{\rm km\,s^{-1}}$. The RMS noise cited in Table
1 is for this velocity resolution. The Figure 2 ${\rm\langle Hn\alpha\rangle}$
spectra are thus sensitive to emission measures $EM\gtrsim$
20${\rm\,cm^{-6}\,pc}$ (the 3 $\sigma$ limit via Equation 3). Below, when we
consider LTE excitation or compare RRL models to our ${\rm\langle
Hn\alpha\rangle}$ spectra, we smooth the per channel velocity resolution to
5$\,{\rm km\,s^{-1}}$. At this resolution these spectra have a 3$\sigma$
sensitivity EM$\gtrsim$ 10${\rm\,cm^{-6}\,pc}$. Nonetheless, even then the
spectra show no hint of additional emission components toward G20 and G45,
especially at negative LSR velocities.

###  3.3 Properties of the Stacked Spectra

Before conducting further analyses of these stacked spectra, we explore how
the RRL emission component line parameters vary as a function of frequency.
After all, the stacked spectra are averages of many RRL transitions spanning a
significant range of rest frequencies. To do this, we create “triad” spectra
by stacking three consecutive Hn$\alpha$ RRL transitions together and
smoothing the resulting spectrum to 5$\,{\rm km\,s^{-1}}$ velocity resolution.
Because our stacked ${\rm\langle Hn\alpha\rangle}$ spectra are comprised of 18
Hn$\alpha$ RRL transitions, there are six of these triad spectra for each
sight line and they have rest frequencies spanning the entire bandwidth of the
C-band receiver. These triad spectra have the sensitivity to enable Gaussian
fits for three of the four RRL emission components present in our sight lines.
Only the weaker, lower velocity component of G45 cannot be analyzed this way.

Figure 4:  G20 sight line RRL emission properties for spectral averages of
three consecutive alpha transitions plotted as a function of frequency. Filled
symbols denote the high velocity, stronger component; open symbols show the
low velocity, weaker component. Horizontal lines flag the component parameters
derived from Gaussian fits to the ${\rm\langle Hn\alpha\rangle}$ spectrum. The
curves shown for $T_{\rm pk}$ and $W_{\rm RRL}$ are not fits to the data,
rather they are notional curves that show the $\nu^{-1}$ frequency dependence
expected for plasma in LTE and a source that fills the telescope beam (see
text).

Figure 5:  G45 sight line RRL emission properties for spectral averages of
three consecutive alpha transitions plotted as a function of frequency. Plot
properties are the same as for Figure 4 except only one RRL component for this
sight line at the higher $\sim$60$\,{\rm km\,s^{-1}}$ velocity is strong
enough to be analyzed in this manner.

This exploration is summarized in Figure 4 for G20 and Figure 5 for G45. These
figures show how the RRL emission component line parameters vary as a function
of frequency. Filled symbols denote the high velocity, stronger component;
open symbols show the low velocity, weaker component. For clarity, in these
plots the frequency of these components has been dithered by $\pm$ 30$\,{\rm
MHz}$. Horizontal lines flag the component parameters derived from Gaussian
fits to the ${\rm\langle Hn\alpha\rangle}$ spectrum (Table 2).

The triad analysis shown in these figures does not reveal any issues that
might compromise the stacking process. Overall, the triad data are consistent
with the Table 2 Gaussian fits to the stacked ${\rm\langle Hn\alpha\rangle}$
spectra. Furthermore, none of the RRL emission component FWHM line widths show
any systematic change with frequency. The strongest G20 emission component
does have a significant LSR velocity gradient as a function of frequency. This
is probably due to the changing GBT beam size sampling different volumes of
plasma. Neither the weaker G20 component nor the G45 sight line component show
any LSR velocity gradient.

For both sight lines, however, the triad component $T_{\rm L}^{\rm A}$ and
$W_{\rm RRL}$ values exhibit significant trends with frequency. The curves
shown for $T_{\rm pk}$ and $W_{\rm RRL}$ are not fits to the data, rather they
are notional curves that show the $\nu^{-1}$ frequency dependence expected for
plasma in LTE and a source that fills the telescope beam (see Equation 1).

###  3.4 Are the G20 and G45 Plasmas in LTE?

Are the ${\rm\langle Hn\alpha\rangle}$, ${\rm\langle Hn\beta\rangle}$, and
${\rm\langle Hn\gamma\rangle}$ stacked spectra consistent with LTE excitation?
Here, we summarize our analysis of this question. The quantum mechanical
details are provided in Appendix B where it shown that for our stacked spectra
the expected LTE $\langle\beta\rangle/\langle\alpha\rangle$ and
$\langle\gamma\rangle/\langle\alpha\rangle$ line intensity ratios should be
0.27565 and 0.10672, respectively (see Table 8). We assess whether LTE
excitation holds for our targets in two ways: (1) we scale the ${\rm\langle
Hn\alpha\rangle}$ spectrum by the Table 8 expected LTE ratios for
$\langle\beta\rangle$ and $\langle\gamma\rangle$ RRLs and compare this in
Figure 6 to our stacked ${\rm\langle Hn\beta\rangle}$ and ${\rm\langle
Hn\gamma\rangle}$ spectra; and (2) we fit Gaussian functions to the RRL
emission components found in the ${\rm\langle Hn\alpha\rangle}$, ${\rm\langle
Hn\beta\rangle}$, and ${\rm\langle Hn\gamma\rangle}$ spectra and compare these
in Table 3 with the expected LTE ratios. In both cases all the stacked spectra
are smoothed to a velocity resolution of 5$\,{\rm km\,s^{-1}}$ to improve
spectral sensitivity.

The first assessment of LTE is shown in Figure 6 where the left hand plots
compare the ${\rm\langle Hn\alpha\rangle}$ and ${\rm\langle Hn\beta\rangle}$
spectra for G20 and G45 with the expected LTE ${\rm\langle Hn\beta\rangle}$
spectrum. The right hand plots compare the ${\rm\langle Hn\beta\rangle}$ and
${\rm\langle Hn\gamma\rangle}$ spectra with the expected LTE intensities. Both
sight lines show ${\rm\langle Hn\beta\rangle}$ RRL emission that is slightly
stronger than what is predicted by LTE. Although the G45 $\sim$35$\,{\rm
km\,s^{-1}}$ component matches the LTE prediction, this ${\rm\langle
Hn\beta\rangle}$ component is only a $\sim$ 1 $\sigma$ signal. Compared with
the other transitions the ${\rm\langle Hn\gamma\rangle}$ spectra are not very
sensitive (see Table 3). Only the $\sim$ 50$\,{\rm km\,s^{-1}}$ emission
component for G20 has any hint of $\gamma$ emission and this is only a
$\gtrsim\,1.5\,\sigma$ signal. For all other ${\rm\langle Hn\gamma\rangle}$
components we can only provide upper limits for the LTE ratio.

Figure 6:  G20 and G45 stacked spectra for the ${\rm\langle Hn\alpha\rangle}$,
${\rm\langle Hn\beta\rangle}$, and ${\rm\langle Hn\gamma\rangle}$ transitions.
The spectra have been smoothed to a 5$\,{\rm km\,s^{-1}}$ velocity resolution.
Left Panels show the ${\rm\langle Hn\alpha\rangle}$ and ${\rm\langle
Hn\beta\rangle}$ spectra. The dashed line spectrum is the expected LTE
intensity of the $\beta$ transition. Right Panels show the ${\rm\langle
Hn\beta\rangle}$ and ${\rm\langle Hn\gamma\rangle}$ spectra. The dashed and
dotted spectra are the expected LTE intensities of the $\beta$ and $\gamma$
transitions, respectively.

The second evaluation of LTE uses Gaussian fits to the individual emission
components found in the stacked $\alpha$, $\beta$, and $\gamma$ spectra. To
make the LTE assessment that is summarized in Table 3, we use the observed
integrated intensity of each component, $W_{\rm RRL}$. (The slight differences
in the reported fits for $\alpha$ components that are seen when comparing
Tables 2 and 3 stem from the different velocity resolutions of the
${\rm\langle Hn\alpha\rangle}$ spectra.)

The “LTE?” column in Table 3 gives the ratio between the observed RRL
transition ratios and the expected LTE ratio. In LTE the value of this ratio
of ratios would be unity. For the strongest WIM components — 113.4$\,{\rm
km\,s^{-1}}$ for G20 and 63.3$\,{\rm km\,s^{-1}}$ for G45 — this ratio is
$1.11\pm 0.11$ and $1.08\pm 0.11$, respectively. We thus find that, within the
errors, our ${\rm\langle Hn\beta\rangle}$/${\rm\langle Hn\alpha\rangle}$
ratios are consistent with LTE excitation for both these emission components.
The lower velocity components in these directions, however, have ${\rm\langle
Hn\beta\rangle}$/${\rm\langle Hn\alpha\rangle}$ ratios that are significantly
larger than unity and thus are not consistent with LTE. Moreover, all ratios
involving the ${\rm\langle Hn\gamma\rangle}$ spectra are compromised by the
much poorer sensitivity of these spectra (see Appendix B).

Table 3: ${\rm\langle Hn\alpha\rangle}$, ${\rm\langle Hn\beta\rangle}$, and ${\rm\langle Hn\gamma\rangle}$ Emission Properties Source |  $\Delta$ n | $V_{\rm LSR}$ | $\sigma$ | $T_{\rm L}^{\rm A}$ | $\sigma$ | $\Delta\,V$ | $\sigma$ | $W_{\rm RRL}$ | $\sigma$ | $W_{\rm RATIO}$ | $\sigma$ | LTE?aa Ratio between observed $W_{\rm RATIO}$ and LTE ratio (Table 8); a plasma in LTE has value unity here.  | $\sigma$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | ($\,{\rm km\,s^{-1}}$) |  | ($\,{\rm mK}$) |  | ($\,{\rm km\,s^{-1}}$) |  | ($\,{\rm mK}\,\,{\rm km\,s^{-1}}$) |  | ($W(\Delta n)\over W(\alpha)$) |  |  |
G20 | 1: $\alpha$ |  047.8 | 0.10 |  +3.33 | 0.17 | 59.4 | 2.97 |  .210.4 | 14.9 |  +1.00 | … |  +… | …
| 2: $\beta$ |  048.5 | 0.21 |  +1.17 | 0.06 | 62.4 | 3.12 |  .077.8 |  05.5 |  +0.37 | 0.04 |  +1.34 | 0.13
| 3: $\gamma$ |  055.3 | 0.74 |  +0.74 | 0.37 | 42.0 | 2.10 |  .033.1 | 16.6 |  +0.16 | 0.08 |  +1.48 | 0.75
G20 | 1: $\alpha$ | 113.4 | 0.02 |  +6.90 | 0.35 | 14.8 | 0.74 |  .108.8 |  02.4 |  +1.00 | … |  +… | …
| 2: $\beta$ | 113.7 | 0.06 |  +2.11 | 0.11 | 14.8 | 0.74 |  .033.3 |  02.4 |  +0.31 | 0.03 |  +1.11 | 0.11
| 3: $\gamma$ | 115.8 | 0.33 |  $<$0.54 | 0.03 | 11.4 | 0.57 |  0$<$6.5 |  0… |  $<$0.06 | … |  $<$0.56 | …
G45 | 1: $\alpha$ |  034.1 | 0.32 |  +0.93 | 0.05 | 27.5 | 1.38 |  .027.2 |  01.9 |  +1.00 | … |  +… | …
| 2: $\beta$ |  037.6 | 2.34 |  +0.32 | 0.02 | 42.8 | 6.41 |  .014.6 |  02.3 |  +0.54 | 0.09 |  +1.95 | 0.34
| 3: $\gamma$ | … | … |  $<$0.50 | … | … | … |  $<$14.9 |  0… |  $<$0.55 | … |  $<$5.14 | …
G45 | 1: $\alpha$ |  063.3 | 0.08 |  +3.35 | 0.17 | 21.2 | 1.06 |  .075.8 |  05.4 |  +1.00 | … |  +… | …
| 2: $\beta$ |  065.3 | 0.18 |  +1.09 | 0.02 | 19.4 | 1.16 |  .022.5 |  01.8 |  +0.30 | 0.03 |  +1.08 | 0.11
| 3: $\gamma$ | … | … |  $<$0.20 | … | … | … |  0$<$4.5 |  0… |  $<$0.06 | … |  $<$0.56 | …

###  3.5 WIM Ionization from Leakage Radiation?

WIM plasma halos envelope the PDRs that surround H ii regions ionized by OB-
type stars. Case studies of individual H ii regions using RRL measurements,
e.g. RCW 120 (Anderson et al., 2015) and NGC 7538 (Luisi et al., 2016), find
that WIM halos are produced by stellar UV photons leaking through the PDRs and
that the fraction of the leaking radio continuum emission escaping into the
WIM is $\sim$25% and $\sim$15%, respectively. Furthermore, optical H$\alpha$
studies also suggest that if OB stars are the source of the WIM ionization,
then the ISM OB-type star excited H ii region distribution must allow at least
$\sim$15–25% of the H-ionizing Lyman continuum photons emitted by the stars to
travel hundreds of parsecs within the Galactic disk (Reynolds, 1991b).

Here, we assess whether the RRL emitting WIM plasma we find toward G20 and G45
is being ionized by leakage radiation from nearby, discrete H ii regions. The
Galactic context of our sight lines is provided by the top panels of Figure 7
which show the locations and LSR velocities of all OB-type star excited H ii
regions in the $(\ell,v)$ zone of the plots. Because we chose our targets to
be devoid of obvious sources of ionizing radiation, as expected, neither sight
line coincides in $(\ell,v)$-space with any major concentration of H ii
regions. This is especially the case for the $\sim$115$\,{\rm km\,s^{-1}}$
component of G20 and the $\sim$35$\,{\rm km\,s^{-1}}$ component of G45:
neither is located on the sky near any significant number of Galactic H ii
regions having their LSR velocities.

Figure 7:  Comparison of G20 and G45 WIM emission with that from known
Galactic H ii regions. Top Plots: $(\ell,v)$ distributions of OB-type star
excited H ii regions near the G20 and G45 sight lines. Black circles are the
LSR velocities of discrete H ii regions. Red triangles and horizontal lines
locate the LSR velocity and FWHM of the four RRL emission components we see
toward G20 and G45. Middle Plots: Comparison of the stacked ${\rm\langle
Hn\alpha\rangle}$ WIM spectra with Galactic H ii regions. Shown are the RRL
$V_{\rm LSR}$ and FWHM line widths of H ii regions from the WISE Catalog
plotted as a function of increasing angular separation of the nebular location
from the G20 and G45 sight lines. The nearest H ii region has the largest
y-axis value. Red points are sources located within 30′ of each sight line;
orange points are those located between 30′ and 100′. Bottom Plots: Predicted
WIM emission due to diffuse ionized gas halos around H ii regions. The solid
red and dashed blue curves are the predictions of the Luisi et al. (2020)
power law and exponential models, respectively. The shaded regions represent
the 95% confidence intervals.

The middle panels of Figure 7 confirm that there are few Galactic H ii regions
that are near both in sky location and LSR velocity to the G20
$\sim$115$\,{\rm km\,s^{-1}}$ and G45 $\sim$35$\,{\rm km\,s^{-1}}$ RRL
emission components. These plots show the $V_{\rm LSR}$ and FWHM of WISE
Catalog H ii regions that are located on the sky within 100′ of the G20 and
G45 sight lines plotted as a function of increasing angular separation of the
nebular location from the G20 and G45 sight lines.

The nearest H ii region has the largest y-axis value. The remaining H ii
regions are plotted in separation sequence with the largest separation having
the smallest y-axis value. (Other than providing a co-ordinate for plotting
the angular separation sequence, these y-axis values have no physical
meaning.) Red points in Figure 7 are sources located within 30′ of each sight
line. There are 17 and 5 WISE Catalog H ii regions located on the sky within
30′ of the G20 and G45 sight lines, respectively. Only 3 nebulae are within
10′ of our sight lines (2 for G20 and 1 for G45) and the closest H ii region
lies 9′ from G45.

Some properties of these H ii regions are compiled in Appendix Table 9. Listed
for each nebula is the angular separation from the fiducial line of sight
(LOS), the IR radius, the Galactic $(\ell,{\it b})$ position, the RRL
parameters (RRL intensity, LSR velocity, and FWHM line width), together with
the measurement errors. Again, the G20 $\sim$115$\,{\rm km\,s^{-1}}$ and G45
$\sim$35$\,{\rm km\,s^{-1}}$ components show very little $(\ell,v)$
correlation with known Galactic H ii regions and there are no H ii regions at
these velocities within 30′ of these sight lines.

Although there are few H ii regions in close $(\ell,v)$ proximity to our sight
lines, we may nonetheless be sensitive to extended WIM halos surrounding these
nebulae. Luisi et al. (2020) develop empirical plasma halo models for the H ii
region complex W43. These models reproduce the observed RRL properties of maps
of the WIM surrounding W43. The models assume that the WIM RRL intensity
depends only on (1) the RRL intensity of nearby H ii regions and (2) the
angular distance to those H ii regions from the fiducial LOS relative to the
angular size of each nebulae. For the W43 complex, they explore both an
exponential and a power law WIM emission distribution model, and they find
that the power law model is better able to reproduce the integrated WIM RRL
intensity.

We apply the Luisi et al. (2020) models to our sight lines in order to explore
the possibility that ionized halos around H ii regions can explain the WIM
emission in these directions. The bottom panels of Figure 7 show the predicted
RRL emission due to WIM halos around all H ii regions within 100′ of each
sight line. The red solid lines show the power law model prediction,

$T_{A,\rm
model}=k\sum_{i}T_{i,\textnormal{H\,{ii}}}\left(\frac{r_{i}}{r_{i,\textnormal{H\,{ii}}}}\right)^{m},$
(5)

and the blue dashed lines show the exponential model prediction,

$T_{A,\rm model}=k\sum_{i}T_{i,\textnormal{H\,{ii}}}\,{\rm
e}^{-m(r_{i}/r_{i,\textnormal{H\,{ii}}})},$ (6)

where $T_{i,\textnormal{H\,{ii}}}$ is the H ii region RRL spectrum, $r_{i}$ is
the angular separation between the sight line and the nominal H ii region
position, $r_{i,\textnormal{H\,{ii}}}$ is the angular IR radius of the H ii
region, and $k$ and $m$ are the free parameters. The H ii region RRL spectrum
is evaluated from the RRL parameters in Appendix Table 9. The sum is taken
over all H ii regions within 100′ of each sight line for which there is a RRL
intensity measurement listed in the table. We adopt $k=0.28\pm 0.08$ and
$m=-1.85\pm 0.12$ for the power law model and $k=0.15\pm 0.04$ and $m=0.33\pm
0.05$ for the exponential model as determined by Luisi et al. (2020) for the
W43 complex.

We note, however, two important differences between Luisi et al. (2020) and
our analysis: (1) they fit their model to the integrated RRL intensity rather
than the RRL spectra, and (2) they parameterize their model in terms of the
average integrated RRL intensity over the H ii region rather than the “peak”
RRL spectrum at the nominal H ii region position. For nebulae comparable in
size to the telescope beam, the “peak” intensity is equal to the average
intensity, but for angularly large nebulae the difference will depend on the
emission morphology. The shaded regions in the bottom panels of Figure 7
represent the 95% confidence intervals determined by Monte Carlo resampling
the H ii region RRL parameters and the model parameters $k$ and $m$.

These empirical models, fit to the W43 complex, may not be applicable to every
Galactic H ii region. By applying such models here, we inherently assume that
the relative distribution of WIM emission around every H ii region is the same
as that around nebulae in the W43 complex. Luisi et al. (2019) find
significant variations in the radial distribution of RRL emission around
several H ii regions. The magnitude of this variation is far greater than the
statistical uncertainties shown in the bottom panels of Figure 7. Furthermore,
the model scaling factor, $k$, is likely related to the fraction of ionizing
photons that leak into the WIM; thus $k$ probably varies from H ii region to H
ii region depending on the environment.

It is thus not surprising that the bottom panels of Figure 7 show clearly that
Luisi et al. (2020) models with $k$ and $m$ parameter values derived for the
W43 complex do not account for the WIM RRL intensities seen toward G20 and
G45. The model intensities are too high by factors of $\sim$4 and $\sim$2,
respectively, for G20 and G45. These sight lines were purposely chosen not to
thread through environments even remotely like those found in massive star
forming regions such as W43.

Finally, it is clear from Figure 7 that the ${\sim}115\,{\rm km\,s^{-1}}$
component toward G20 cannot be due to H ii region leakage radiation. No nearby
H ii region is sufficiently bright or has a sufficiently narrow line width to
explain this feature, as demonstrated by the Luisi et al. (2020) models and
Appendix Table 9. We conclude it to be unlikely that this feature is due to
extended ionized gas halos around H ii regions. It may instead constitute a
heretofore unrecognized phase of the WIM (see Section 5.1). To make further
progress in the exploration of the nature of plasma halos around H ii regions
and this narrow RRL feature we will need sensitive RRL maps of the WIM
surrounding G20 and G45. Such maps will allow us to fit Luisi et al. (2020)
type models to these data.

###  3.6 Galactic ISM Context

Here, we assess the Galactic context of our G20 and G45 sight lines by
comparing spectra for all phases of ISM hydrogen: H+, H i, and H2. This is
done in Figure 8 which shows for each sight line spectra for ${\rm\langle
Hn\alpha\rangle}$, 21$\,{\rm cm}$ H i, and ${\rm{}^{13}CO}$(1$\rightarrow$0)
which is a proxy for H2. These spectra were taken by different telescopes over
a large range of frequencies so their beam sizes sample different LOS volumes.
The ${\rm\langle Hn\alpha\rangle}$ WIM RRL spectra have a HPBW of $2\hbox
to0.0pt{.\hss}{{}^{\prime}}35$. For G20 the H i spectrum is from the HI4PI all
sky survey (HI4PI Collaboration et al., 2016, HPBW=$16\hbox
to0.0pt{.\hss}{{}^{\prime}}2$). The G45 H i spectrum stems from the Boston
University–Arecibo Observatory H i survey (Kuchar, 1992, HPBW=$3\hbox
to0.0pt{.\hss}{{}^{\prime}}2$). The ${\rm{}^{13}CO}$ spectra come from the
Boston University–Five College Radio Observatory ${\rm{}^{13}CO}$ Galactic
Ring Survey (Jackson et al., 2006, HPBW=42″). Despite their different angular
resolutions, comparing these spectra can nonetheless provide useful insights.
For example, the terminal velocity, $V_{\rm T}$, flagged in Figure 8 is
calculated for each sight line using the Clemens (1985) rotation curve. For
both sight lines the H+, H i, and ${\rm{}^{13}CO}$ spectra all have emission
components at or near the terminal velocity.

Figure 8:  Spectra toward G20 and G45 showing all phases of gaseous
interstellar hydrogen: ${\rm\langle Hn\alpha\rangle}$ H i, and
${\rm{}^{13}CO}$ (a tracer of H2). For clarity the H i antenna temperatures in
$\,{\rm K}$ are divided by a factor of 15 and the ${\rm{}^{13}CO}$ $T_{A}^{*}$
antenna temperatures in $\,{\rm K}$ are multiplied by a factor of 6. Vertical
flags mark LSR velocities 0$\,{\rm km\,s^{-1}}$ and the terminal velocity,
$V_{\rm T}$.

Our sight lines probe two important loci in the first quadrant of the Galaxy.
Some key properties of these sight lines, based on an assumed Sun to Galactic
Center distance, $R_{0}$, of 8.5$\,{\rm kpc}$, are summarized in Table 4.
(Other choices for $R_{0}$ would not alter the conclusions we reach here anent
the spatial distribution of the WIM plasma.) Listed in Table 4 for each LOS
are the minimum Galactocentric radius, $R_{\rm min}$=$R_{0}$ sin($\ell$),
tangent point distance from the Sun, $d_{\rm TP}$=$R_{0}$ cos($\ell$), and
total path length inside the Solar orbit about the Galactic Center (GC),
$d_{\rm LOS}$=2$R_{0}$ cos($\ell$).

Table 4: LOS Properties

$LOS$ | $R_{\rm min}$ | $d_{\rm TP}$ | $d_{\rm LOS}$
---|---|---|---
| ($\,{\rm kpc}$) | ($\,{\rm kpc}$) | ($\,{\rm kpc}$)
G20 | 2.9 | 8.0 | 16.0
G45 | 6.0 | 6.0 | 12.0

Note. — $R_{\rm 0}$ = 8.5$\,{\rm kpc}$.

Atomic hydrogen gas is ubiquitously distributed throughout the Galactic ISM.
Because of this, for any line of sight 21$\,{\rm cm}$ H i spectra show
emission at all LSR velocities permitted by Galactic rotation. The velocity
span of these H i spectra defines the maximum velocity spread permitted by
Galactic rotation. For our first quadrant targets, gas at negative velocities
is located in the Outer Galaxy beyond the Solar orbit about the Galactic
Center. The Figure 8 H i spectra have emission at negative LSR velocities that
extends to $\sim-50$$\,{\rm km\,s^{-1}}$ and $\sim-75$$\,{\rm km\,s^{-1}}$ for
G20 and G45, respectively. In contrast, neither of our target sight lines
shows any ${\rm\langle Hn\alpha\rangle}$ RRL emission at negative LSR
velocities at the sensitivity level achieved here. G20 shows ${\rm\langle
Hn\alpha\rangle}$ emission at all velocities between 0$\,{\rm km\,s^{-1}}$ and
the terminal velocity whereas G45 only shows RRL emission between
$\sim$+20$\,{\rm km\,s^{-1}}$ and the terminal velocity. Assuming perfect
circular rotation and no streaming motions, all of the RRL emitting gas we see
toward G20 and G45 must therefore be located in the Inner Galaxy, inside the
Solar orbit. This plasma must reside somewhere along the line of sight paths,
$d_{\rm LOS}$, summarized in Table 4.

Unlike H i, molecular gas in the Milky Way is found in comparatively dense,
discrete clouds. This means that H2/CO spectra will not have molecular
emission spanning all available LSR velocities. As Figure 8 clearly shows, the
molecular gas seen toward our target directions is concentrated into a few
discrete emission components. At the sensitivity of the GRS, G20 shows 5
molecular clouds and G45 has but one. For both sight lines there are
${\rm{}^{13}CO}$ emission components that match the LSR velocities of the
highest velocity RRL components in the ${\rm\langle Hn\alpha\rangle}$ spectra,
albeit the $\sim$ 114$\,{\rm km\,s^{-1}}$ G20 ${\rm{}^{13}CO}$ component is
extremely weak. The GRS spectrometer could not detect emission at negative LSR
velocities so these ${\rm{}^{13}CO}$ spectra provide no Outer Galaxy
information about molecular clouds toward these sight lines.

Cold H i embedded in a molecular cloud produces the absorption dips seen in
the G20 H i spectrum at $V_{\rm LSR}$ = $\sim 5$, $\sim 50$, and
$\sim$70$\,{\rm km\,s^{-1}}$. These H i dips are matched by ${\rm{}^{13}CO}$
emission components. We suspect that all the ${\rm{}^{13}CO}$ components
produce H i absorption but cannot definitively prove this due to a combination
of mismatched angular resolution, insufficient spectral sensitivity, and the
complex structure of Galactic H i spectra.

##  4 WIM RRL Emission Models

Here, using the ${\rm\langle Hn\alpha\rangle}$ emission seen from our targets,
we seek to derive constraints on the line of sight density and temperature
distributions of the RRL emitting plasmas. Complete details of the modeling
are provided in Appendix D. Models for RRL emission from LOS plasmas must
specify the electron density, $n_{\rm e}$, temperature, $T_{\rm e}$, and
velocity dispersion, $\sigma$, at every point along the $d_{\rm LOS}$. Each
model is comprised of one or more plasma “clouds” distributed along the LOS.
All clouds are homogeneous, isothermal plasmas in LTE. Each cloud’s properties
are specified at input. A cloud is defined by: location along the LOS,
$d_{\odot}$, LOS path length size (aka the cloud diameter), $d_{\rm cloud}$,
electron density, electron temperature, and velocity dispersion. We use the
numerical code described in Appendix D to compute synthetic spectra for
H109$\alpha$ RRL emission from the model plasmas. After these model $T_{\rm
B}$($V_{\rm LSR}$) RRL spectra are calculated, we compare them to ${\rm\langle
Hn\alpha\rangle}$ spectra that are converted to brightness temperature and
smoothed to 5$\,{\rm km\,s^{-1}}$ resolution.

We emphasize that the models are only intended to provide estimates of these
quantities so we do not attempt to fully explore a large grid of parameter
choices. Moreover, we do not apply any rigorous numerical metric to evaluate
the “goodness of fit” between a model and the ${\rm\langle Hn\alpha\rangle}$
spectrum. We judge a model’s fit by eye because we can only set limits on the
electron temperature.

From the observed $V_{\rm LSR}$ span of these spectra we know, assuming
circular rotation, that the RRL emitting plasma must be located within the
Solar orbit. The simplest assumption is that a constant density isothermal
plasma fills the entire $d_{\rm LOS}$ in each sight line (see Table 4). For
this case, the observed EM provides an estimate for the rms electron density,
$n_{\rm e}^{\rm rms}$:

$n_{\rm e}^{\rm rms}\ =\left({EM\over d_{\rm los}}\right)^{1/2}$ (7)

Table 2 shows that the total EM from the G20 and G45 emission components is
369.3 and 402.7 ${\rm\,cm^{-6}\,pc}$, respectively. This gives from Equation 7
an $n_{\rm e}^{\rm rms}$ of 0.15 and 0.18$\,{\rm cm^{-3}}$, respectively, for
these sight lines. These $n_{\rm e}^{\rm rms}$ estimates, however, do not
account for gas clumping: there may be significant gaps and/or $n_{\rm e}$
density fluctuations in the plasma distribution along each LOS. Many different
LOS $n_{\rm e}$ distributions can produce identical $n_{\rm e}^{\rm rms}$
values.

Due to the first Galactic quadrant distance ambiguity, we cannot know a priori
where the RRL emitting plasma is located vis a vis the LOS near/far locations.
A series of models seeking to find cloud parameters that produce a synthetic
spectrum matching the ${\rm\langle Hn\alpha\rangle}$ observations confirms
that there is no unique solution. A plethora of models can reproduce the
${\rm\langle Hn\alpha\rangle}$ spectra. Because of this, models for our
target’s emission at LSR velocities having LOS distance ambiguities provide no
meaningful limits for the plasma density and distribution along the LOS.

Table 5: LOS Tangent Point Models  LOS | Model | $R_{\rm gal}$ | $d_{\odot}$ | $d_{\rm cloud}$ | $n_{\rm e}$ | $T_{\rm e}$ | $\sigma$ | $EM$ | $p/k$
---|---|---|---|---|---|---|---|---|---
|  | ($\,{\rm kpc}$) | ($\,{\rm kpc}$) | ($\,{\rm kpc}$) | ($\,{\rm cm^{-3}}$) | ($\,{\rm K}$) | ($\,{\rm km\,s^{-1}}$) | (${\rm\,cm^{-6}\,pc}$) | (${\rm\,cm^{-3}\,K}$)
G20 | A | 2.9 | 8.0 |  02.0 | 0.38 | 4000 | 5.7 | 289 | 1520
| B | 2.9 | 8.0 |  02.0 | 0.29 | 2800 | 5.7 | 168 |  0812
| RMS | 2.9 | 8.0 | 16.0 | 0.15 | 4000 | 5.7 | 360 |  0600
G45 | A | 6.0 | 6.0 |  04.0 | 0.40 | 9000 | 8.6 | 640 | 3600
| B | 6.0 | 6.0 |  04.0 | 0.31 | 6400 | 8.6 | 384 | 1984
| RMS | 6.0 | 6.0 | 12.0 | 0.18 | 9000 | 8.6 | 389 | 1620

Figure 9:  LTE models for RRL emission (solid red curves) from the G20 and G45
high velocity components compared with the observed ${\rm\langle
Hn\alpha\rangle}$ brightness temperature spectrum (black curves). Properties
of the models are summarized in Table 5. Vertical lines flag LSR velocities at
0$\,{\rm km\,s^{-1}}$ and the observed component velocities listed in Table 2.
Model B is not shown because it is indistinguishable from Model A. The dashed
red curve models have plasma filling the entire LOS inside the solar orbit
with the rms density from Equation 7.

###  4.1 LOS Tangent Point Models

Models for RRL emitting gas located at the LOS tangent point distance,
however, can provide useful constraints for the plasma physical properties.
Gas is unambiguously located at the tangent point if it is emitting at the LSR
terminal velocity produced by Galactic rotation for a particular first
quadrant sight line. The high velocity components seen toward G20 and G45 emit
at their terminal velocities and are hence located at their respective tangent
point distances from the Sun.

We model the RRL emission from these components using a single cloud each for
G20 and G45. Each plasma cloud is specified by its LOS distance, $d_{\odot}$,
cloud size, electron density, $n_{\rm e}$, electron temperature, $T_{\rm e}$,
and the velocity dispersion of the emission component, $\sigma$. Each cloud is
located at the tangent point distance (see Table 4). The velocity dispersion
is set by the observed line width (see Table 2 where $\sigma$ =
$\Delta\,V$/2.355 for a Gaussian line shape).

We use the observed RRL line width, $\Delta\,V$, to set the model plasma
electron temperature. This line width stems from a combination of mechanisms
including thermal, $\Delta\,V_{\rm th}$, and non-thermal broadening,
$\Delta\,V_{\rm nt}$. In addition, velocity shear along the line of sight
produced by Galactic rotation, $d\,V_{\rm LSR}/d\,d_{\odot}$, can also broaden
the line emission, $\Delta\,V_{\rm los}$. (Quantum mechanical natural
broadening is insignificant compared with these mechanisms.) The broadening
processes add in quadrature to produce the observed line width:
$\Delta\,V^{2}=\Delta V^{2}_{\rm th}+\Delta V^{2}_{\rm nt}+\Delta V^{2}_{\rm
los}$. By definition the line of sight LSR velocity gradient at any Galactic
tangent point distance, $d_{\rm TP}$, is $d\,V_{\rm LSR}/d\,d_{\rm TP}=0$.
Thus, for plasma located at the tangent point, spectral broadening due to
Galactic rotation, $\Delta\,V_{\rm los}$, is negligible. The WIM plasma is
turbulent, however, so even at the tangent point the observed RRL line width
is an unknown combination of thermal and non-thermal, supersonic turbulent
broadening: $\Delta\,V^{2}=\Delta V^{2}_{\rm th}+\Delta V^{2}_{\rm nt}$.

We estimate $T_{\rm e}$ in two ways, leading to two models each for G20 and
G45. Model A sets $T_{\rm e}$ to be the upper limit derived using Equation 4
and assuming that $\Delta\,V_{\rm obs}$ is entirely due to thermal broadening:
$\Delta\,V$=$\Delta\,V_{\rm th}$. Because the plasma is turbulent, this is a
robust upper limit for $T_{\rm e}$. For Model A, $T_{\rm e}$ is 4,000 $\,{\rm
K}$ and 9,000 $\,{\rm K}$ for G20 and G45, respectively. Model B explores the
effect of turbulent broadening by assuming that the thermal and turbulent
contributions to the line width are equal: $\Delta\,V_{\rm
th}$=$\Delta\,V_{\rm nt}$. The thermal contribution to the observed line width
is then $\Delta\,V_{\rm th}$=$\Delta\,V$/$\sqrt{2}$. For Model B, $T_{\rm e}$
is 2,800 $\,{\rm K}$ and 6,400 $\,{\rm K}$ for G20, and G45, respectively.

Here, we seek to reproduce both the intensity and line shape of the tangent
point emission components. The model cloud parameters that are yet to be
determined are the electron density and cloud size. For an isothermal plasma
the RRL intensity scales linearly with emission measure so $T_{\rm B}\propto
n_{\rm e}^{2}$. The model spectral intensities are very sensitive to $n_{\rm
e}$. We find that changing $n_{\rm e}$ by $\pm$ 0.01$\,{\rm cm^{-3}}$ produces
significant differences in the model peak $T_{\rm B}$ intensity. The cloud
size is constrained by the line shape: if the cloud is too large, Galactic
rotation produces a distinctly non-Gaussian line profile.

To craft the final model parameters we explored a range of choices for $n_{\rm
e}$ and $d_{\rm cloud}$. The models are summarized in Table 5 which lists for
each cloud the minimum LOS Galactocentric distance, $R_{\rm gal}$, location
along the LOS, $d_{\odot}$, LOS path length size (i.e., the cloud diameter),
$d_{\rm cloud}$, electron density, $n_{\rm e}$, electron temperature, $T_{\rm
e}$, velocity dispersion, $\sigma$, emission measure, EM, and the plasma
pressure, $p/k$. The Model A spectra for G20 and G45 are compared in Figure 9
with the observed ${\rm\langle Hn\alpha\rangle}$ emission. The Model B spectra
are not shown because their $n_{\rm e}$ has been adjusted to fit the observed
${\rm\langle Hn\alpha\rangle}$ and so they are indistinguishable from Model A.

The dashed red curve models in Figure 9 have plasma filling the entire LOS
inside the solar orbit with the rms electron density from Equation 7 (see
model “RMS” in Table 5). As expected these models are poor fits to the
observed ${\rm\langle Hn\alpha\rangle}$ spectra because the plasma is surely
not homogeneous and isothermal throughout the $d_{\rm LOS}$ path length. The
models do, however, show the LSR velocity span produced by plasma that fills
the entire $d_{\rm LOS}$. For both sight lines these models produce larger
intensities for $V_{\rm LSR}$’s between $\sim$ 0 and $\sim$ 20$\,{\rm
km\,s^{-1}}$ than what is observed. This may indicate that the plasma density
is rather low near the Sun for both directions.

##  5 Discussion

Here, we compare our results for G20 and G45 with other studies of the WIM.
This is challenging because there is no universally accepted definition of the
WIM. The plasma has been investigated using a variety of spectral observations
over a range of frequencies. The WIM has been studied in the optical, far
infrared (FIR), and radio. Interpreting these spectra requires many different
techniques and assumptions before one derives physical properties such as
density and temperature. Furthermore, because of dust extinction and differing
angular resolutions, these three spectral regimes probe different volumes of
the Milky Way.

Historically, the distribution and physical properties of the WIM in the Milky
Way have primarily been characterized by observations of optical spectral
lines. H$\alpha$ emission studies conclude that the WIM accounts for $\sim
90$% of the ionized gas in the ISM with a scale height of $\sim 1-2$$\,{\rm
kpc}$ (e.g., Reynolds, 1989; Hill et al., 2008). Dispersion measures from
pulsars with known distances, together with H$\alpha$ emission measures,
estimate the range of the average electron density to be $\sim$
0.03-0.10$\,{\rm cm^{-3}}$ with a filling factor, $f\sim$ 0.4-0.2 (e.g.,
Reynolds, 1991a; Taylor & Cordes, 1993; Berkhuijsen et al., 2006; Gaensler et
al., 2008). The ratio of [N ii] $\lambda$5755 and [N ii] $\lambda$6583
provides a direct measure of the electron temperature and indicates the WIM is
about 2,000$\,{\rm K}$ warmer than H ii regions (Reynolds et al., 2001). For a
typical H ii region with $T_{\rm e}=8,000$$\,{\rm K}$, this corresponds to an
electron temperature of $10^{4}$$\,{\rm K}$.

Most of the ionized gas in the Milky Way, however, is located in the inner
Galaxy which is not probed by optical emission lines because of extinction by
dust. Far-infrared collisionally excited lines are less affected by dust and
offer an alternative method to sample the WIM. In particular, [N ii] 122 µm
and [N ii] 205 µm are excellent diagnostics of the WIM since nitrogen has an
ionization potential of 14.5 eV, similar to hydrogen (13.6 eV), and will thus
be associated with H ii.

Using the HIFI instrument on Herschel, Persson et al. (2014) observed [N ii]
205 µm toward four H ii regions. They detected absorption of [N ii] from
foreground gas toward W31C and W49N. This absorbing gas must be located within
the Solar orbit due to the velocity range of the spectral components. Assuming
a N/H abundance ratio and filling factor they estimate mean electron densities
of $\sim 0.1-0.3$$\,{\rm cm^{-3}}$. These results, together with modeling of
N+ with RADEX, are consistent with the optically derived WIM properties,
albeit with somewhat higher electron densities.

Observations of RRLs offer an extinction-free probe of the WIM. Early RRL
observations made with single-dish telescopes toward directions away from OB-
type star excited H ii regions detected weak RRL emission (e.g., Gottesman &
Gordon, 1970; Lockman, 1976; Mezger, 1978; Heiles et al., 1996; Roshi &
Anantharamaiah, 2000). This emission was interpreted by Mezger (1978) as being
associated with the envelopes of H ii regions and called the “extended low
density” (ELD) H ii. Because of their poor angular resolution, however, all
these observations suffer from confusion with the known population of H ii
regions (Anderson et al., 2014b) so any quantitative interpretation of these
data is problematic.

The more recent RRL surveys with the GBT and FAST, however, have sufficient
angular resolution to avoid emission from H ii regions and in principle
produce WIM-only images (Anderson et al., 2021). But there are several reasons
why these WIM-only images may still contain ELD H ii emission and thus not be
truly representative of WIM gas: (1) GDIGS is only sensitive to plasmas with
mean electron densities $\gtrsim 30$$\,{\rm cm^{-3}}$ (for a path length of
1$\,{\rm pc}$) and this is $\sim$ 100 times larger than the density expected
for the WIM (see the Figure 3 sensitivity comparison); (2) since H$\alpha$
studies imply that the WIM fills $\sim$70% of the ISM’s volume, this semi-
ubiquitously distributed plasma ought to emit over the entire range of
velocities that are allowed by Galactic rotation yet for typical sight lines
the GDIGS detected RRL spectra have gaps with no emission at allowed LSR
velocities; and (3) the RRL emission appears to be correlated with the
location of H ii regions (Luisi et al., 2020) and therefore may be part of the
ELD H ii.

For the high sensitivity RRL observations toward G20 and G45, we estimate rms
electron densities of $(n_{\rm e}^{2})^{1/2}\sim 0.15-0.18$$\,{\rm cm^{-3}}$
over a wide range of velocities. For a filling factor of $f=0.3$ this
corresponds to mean electron densities of $\sim 0.3$$\,{\rm cm^{-3}}$. These
electron densities are somewhat higher than expected from the WIM based on
optical data. These higher values may not be surprising since Figure 7
suggests that some of the emission detected toward both G20 and G25 might be
associated with OB-type star H ii regions. But the figure also shows that the
G20 RRL emission above 100$\,{\rm km\,s^{-1}}$ and the G45 emission below
50$\,{\rm km\,s^{-1}}$ have no association with any nearby H ii regions and
are probably tracing the WIM.

###  5.1 G20

The narrow, 13.5$\,{\rm km\,s^{-1}}$, line width from plasma located at the
G20 tangent point distance sets a strong upper limit of $\lesssim$ 4,000 K on
the electron temperature. At this $T_{\rm e}$, Model A with a density of
$n_{\rm e}$=0.38 $\,{\rm cm^{-3}}$ replicates the observed intensity and line
shape of this emission component. The Model A $p/k=n_{\rm e}T_{\rm e}$
pressure is $\lesssim$ 1,520 $\,{\rm cm^{-3}}$$\,{\rm K}$. Models A and B
together span $n_{\rm e}$ between $\sim$ 0.38–0.29 $\,{\rm cm^{-3}}$ and
$T_{\rm e}$ between $\sim$ 4,000–2,800 $\,{\rm K}$ resulting in WIM pressures
ranging from $p/k$ $\sim$ 1,500 to 800 ${\rm\,cm^{-3}\,K}$.

Measuring the pressure in the WIM is challenging because there do not seem to
be any observables from which one can derive both the density and temperature
without bias. Combining the pulsar/H91$\alpha$ density determinations with the
FIR derived electron temperature yields a nominal pressure, $p/k$, of $\sim$
500 to 1,000${\rm\,cm^{-3}\,K}$ for the WIM plasma. But combining these
$n_{\rm e}$ and $T_{\rm e}$ determinations may be problematic. For example,
these estimates for the WIM plasma’s physical parameters are not probing the
same gas volumes. In lieu of robust observational constraints for the WIM
$p/k$, we can seek insight from simulations. The TIGRESS-NCR simulation of the
Galactic ISM (Kim et al., 2023) predicts a WIM pressure $p/k$ ranging between
$\sim$$10^{2}$ and $\sim$$10^{5}$, with a mean value close to
$\sim$$10^{4}$${\rm\,cm^{-3}\,K}$.

Here, we show that a combination of RRL observations with models for the
emission together provide an alternative approach to deriving the WIM
pressure. Using the observed RRL line width to constrain $T_{\rm e}$ and
models to estimate $n_{\rm e}$, we find that the $p/k$ pressure derived here
for the G20 tangent point emission is generally consistent with other
determinations of the WIM pressure that are based on observations. The low
$T_{\rm e}$ limit, however, is not. Too, the electron density is at the high
end for values typically derived for WIM plasma. Altogether, the G20 model for
the tangent point emission challenges our understanding of the WIM as a
canonically pervasive, low density, $\lesssim$ 0.1 $\,{\rm cm^{-3}}$, $\sim$
10,000 $\,{\rm K}$ plasma.

###  5.2 G45

The line width, 20.3$\,{\rm km\,s^{-1}}$, for the G45 tangent point emission
component also sets a strong upper limit for the plasma electron temperature
of $T_{\rm e}$ $\lesssim$ 9,000 $\,{\rm K}$. Model A adopts this $T_{\rm e}$
limit and with a density of $n_{\rm e}$=0.40 $\,{\rm cm^{-3}}$ replicates the
observed intensity and line shape of this emission component. Models A and B
together span $n_{\rm e}$ between $\sim$ 0.40–0.31 $\,{\rm cm^{-3}}$ and
$T_{\rm e}$ between $\sim$ 9,000–6,400 $\,{\rm K}$ resulting in WIM pressures
ranging from $p/k$ $\sim$ 3,600 to 2,000 ${\rm\,cm^{-3}\,K}$.

These model $p/k$ pressures are also consistent with other determinations of
the WIM pressure and are more in line with the TIGRESS-NCR simulation results.
Again, the density is at the high end of values reported for the WIM, but
overall this G45 tangent point emission component’s physical properties are
consistent with WIM plasma.

In sum, the result of these efforts is that the WIM spans a range of
densities. Almost any density up to a $\sim$ few $\,{\rm cm^{-3}}$ may be
consistent with the WIM. Optical studies derive $n_{\rm e}$ $\sim$
0.03–0.1$\,{\rm cm^{-3}}$ (Haffner et al., 2009). FIR observations of fine
structure transitions infer $\sim$ 0.1–0.3$\,{\rm cm^{-3}}$ (Persson et al.,
2014). Our G20 and G45 analysis gives $\sim$ 0.3–0.4$\,{\rm cm^{-3}}$.
Finally, there is evidence from low frequency RRL observations for a more or
less continuous distribution of plasma densities. Several studies have found
volume filling factors inversely proportional to density, so the higher
densities subsume an increasingly smaller fraction of the ISM volume (e.g.,
Anantharamaiah, 1985, 1986; Roshi & Anantharamaiah, 2001).

##  6 Summary

Studies of the Milky Way’s WIM that use H$\alpha$ emission can only probe the
Galactic disk out to distances limited to a few $\,{\rm kpc}$ from the Sun due
to extinction. Because the ISM is optically thin at cm wavelengths radio
recombination lines probe the WIM emission at transgalactic path lengths.
Here, we use RRL emission to study two Galactic sight lines located in the
plane that are devoid of any OB-star produced H ii regions and show no
indication of nearby star formation.

  * •

GBT observations of the WIM made toward Galactic sight lines G20 and G45 show
four Gaussian-shaped emission features in the ${\rm\langle Hn\alpha\rangle}$
stacked RRL spectra. The emission measures of these spectral components range
between $\sim$100 and $\sim$300${\rm\,cm^{-6}\,pc}$.

  * •

For both sight lines, RRL emission can be seen at nearly all LSR velocities
between 0$\,{\rm km\,s^{-1}}$ and the terminal velocity produced by Galactic
rotation. The lack of any significant RRL emission at LSR velocities below
0$\,{\rm km\,s^{-1}}$ means that in these directions RRL emission from any WIM
plasma located beyond the Solar orbit is below the EM $\gtrsim$
10${\rm\,cm^{-6}\,pc}$ (3$\sigma$ limit for 5$\,{\rm km\,s^{-1}}$ resolution)
sensitivity limit of our GBT observations. For G20 and G45, the line of sight
path lengths with RRL emitting WIM plasma, $d_{\rm LOS}$, must thus be no
larger than 12 and 16 $\,{\rm kpc}$, respectively.

  * •

The observed ${\rm\langle Hn\beta\rangle}$/${\rm\langle Hn\alpha\rangle}$
intensity ratios are consistent with LTE excitation for the stronger, higher
velocity components seen toward both sight lines at LSR velocities of
$\sim$113 $\,{\rm km\,s^{-1}}$ and $\sim$63 $\,{\rm km\,s^{-1}}$ for G20 and
G45, respectively. The weaker, lower velocity components at $\sim$48 $\,{\rm
km\,s^{-1}}$ and $\sim$35 $\,{\rm km\,s^{-1}}$, however, show some evidence
for non-LTE excitation.

  * •

Although our sight lines show no H ii regions in their fields, the WIM
emission we observe might originate in extended WIM halos around H ii regions.
Some of the emission detected toward both G20 and G45 occurs at LSR velocities
shared with nearby H ii regions. Empirical models for WIM emission near H ii
regions predict comparable emission to what is seen (see Figure 7), although
the model uncertainties preclude us from making any definitive conclusions.
Despite these uncertainties, it seems unlikely that the G20 RRL emission above
100$\,{\rm km\,s^{-1}}$ is associated with nearby known H ii regions.

  * •

Cloud models with plasma located at the G20 and G45 tangent points reproduce
the observed RRL emission components at the tangent point velocities for these
sight lines. These models have densities $n_{\rm e}$ = $\sim$ 0.29–$\sim$ 0.40
$\,{\rm cm^{-3}}$, temperatures $T_{\rm e}$ = $\sim$ 2,800–$\sim$ 9,000
$\,{\rm K}$, and $p/k$ pressures $\sim$ 800–$\sim$ 3,600 $\,{\rm
cm^{-3}}$$\,{\rm K}$.

  * •

The $p/k$ pressure derived for the G20 tangent point emission is consistent
with other determinations of the WIM pressure. The low, $\lesssim$
4,000$\,{\rm K}$ $T_{\rm e}$ limit, however, is not. The G20 model for the
tangent point emission challenges our understanding of the WIM as a
canonically pervasive, low density, $\lesssim$ 0.1 $\,{\rm cm^{-3}}$, $\sim$
10,000 $\,{\rm K}$ plasma.

We thank the referee for a timely and close reading of our manuscript. We
thank the GBT telescope operators whose diligence and expertise in executing
our observing scripts for project GBT/20A–483 were exceptional. The Green Bank
Telescope is operated by the Green Bank Observatory (GBO). The GBO and the
National Radio Astronomy Observatory (NRAO) are facilities of the National
Science Foundation (NSF) operated under cooperative agreements by Associated
Universities, Inc. This publication makes use of data products from the Wide-
field Infrared Survey Explorer, which is a joint project of the University of
California, Los Angeles, and the Jet Propulsion Laboratory/California
Institute of Technology, funded by the National Aeronautics and Space
Administration. This research was supported by NSF awards AST-1714688 to
T.M.B. and AST-1516021 to L.D.A. T.V.W. is supported by the AST-2202340 NSF
Astronomy and Astrophysics Postdoctoral Fellowship.

##  Appendix A RRL Transitions used for Stacked Spectra

Instrumental effects caused by the C-band receiver, the GBT IF system, and
VEGAS spectrometer compromise some of the transitions we observe. These
effects include unacceptable frequency structure in the spectral baselines and
loss of sensitivity due to high system temperatures. High system temperatures
mostly occur when the C-band receiver performance deteriorates at both the
high and low frequency extremes of the $\sim$4$\,{\rm GHz}$ bandwidth. We
inspected each RRL tuning to evaluate overall quality. The usable RRL
transitions are summarized in Table 6. Listed for the G20 and G45 targets are
the transition, the rest frequency, and the average system temperatures for
both linear polarizations: $\langle T_{XX}\rangle$ and $\langle
T_{YY}\rangle$. For these acceptable tunings system temperatures typically
range between $\sim$20$\,{\rm K}$ and $\sim$40$\,{\rm K}$.

Table 6: RRL Transitions Used for Stacked Spectra Line | Transition | Frequency |  $\langle$_T_ XX$\rangle_{\rm G20}$ |  $\langle$_T_ YY$\rangle_{\rm G20}$ |  $\langle$_T_ XX$\rangle_{\rm G45}$ |  $\langle$_T_ YY$\rangle_{\rm G45}$
---|---|---|---|---|---|---
| $n+\Delta n\rightarrow n$ | (GHz) | (K) | (K) | (K) | (K)
H97$\alpha$ | 98 $\rightarrow$ 97 | 7.09541 | 38.6 | 38.3 | 38.0 | 37.4
H98$\alpha$ | 99 $\rightarrow$ 98 | 6.88149 | 35.4 | 33.4 | 34.6 | 31.9
H99$\alpha$ | 100 $\rightarrow$ 99 | 6.67607 | 35.8 | 27.6 | 35.1 | 26.5
H100$\alpha$ | 101 $\rightarrow$ 100 | 6.47876 | 24.6 | 25.1 | 23.9 | 24.1
H101$\alpha$ | 102 $\rightarrow$ 101 | 6.28914 | 29.8 | 21.1 | 28.5 | 20.2
H102$\alpha$ | 103 $\rightarrow$ 102 | 6.10685 | 28.5 | 21.7 | 27.6 | 20.8
H103$\alpha$ | 104 $\rightarrow$ 103 | 5.93154 | 26.9 | 22.8 | 25.8 | 21.8
H104$\alpha$ | 105 $\rightarrow$ 104 | 5.76288 | 24.2 | 23.4 | 23.0 | 22.4
H105$\alpha$ | 106 $\rightarrow$ 105 | 5.60055 | 23.8 | 24.0 | 22.5 | 22.8
H106$\alpha$ | 107 $\rightarrow$ 106 | 5.44426 | 21.2 | 24.2 | 20.1 | 23.1
H107$\alpha$ | 108 $\rightarrow$ 107 | 5.29373 | 27.1 | 23.7 | 25.6 | 22.5
H108$\alpha$ | 109 $\rightarrow$ 108 | 5.14870 | 24.6 | 26.4 | 23.1 | 25.1
H109$\alpha$ | 110 $\rightarrow$ 109 | 5.00892 | 25.8 | 24.1 | 24.0 | 22.8
H110$\alpha$ | 111 $\rightarrow$ 110 | 4.87416 | 26.0 | 24.0 | 24.2 | 22.8
H111$\alpha$ | 112 $\rightarrow$ 111 | 4.74418 | 27.0 | 23.7 | 24.9 | 22.5
H112$\alpha$ | 113 $\rightarrow$ 112 | 4.61879 | 27.1 | 25.2 | 25.2 | 24.0
H114$\alpha$ | 115 $\rightarrow$ 114 | 4.38095 | 30.2 | 32.4 | 27.6 | 30.7
H115$\alpha$ | 116 $\rightarrow$ 115 | 4.26814 | 38.9 | 39.1 | 36.0 | 36.0
H122$\beta$ | 124 $\rightarrow$ 122 | 7.06872 | 34.1 | 35.0 | 33.3 | 34.8
H123$\beta$ | 125 $\rightarrow$ 123 | 6.89905 | 34.6 | 32.9 | 33.7 | 31.8
H124$\beta$ | 126 $\rightarrow$ 124 | 6.73479 | 34.0 | 31.3 | 33.1 | 29.9
H125$\beta$ | 127 $\rightarrow$ 125 | 6.57570 | 27.7 | 26.4 | 26.7 | 25.1
H126$\beta$ | 128 $\rightarrow$ 126 | 6.42158 | 24.5 | 29.8 | 23.7 | 28.6
H127$\beta$ | 129 $\rightarrow$ 127 | 6.27223 | 30.3 | 22.4 | 29.2 | 21.4
H128$\beta$ | 130 $\rightarrow$ 128 | 6.12748 | 31.0 | 22.2 | 29.4 | 21.2
H129$\beta$ | 131 $\rightarrow$ 129 | 5.98714 | 28.8 | 22.5 | 27.6 | 21.5
H130$\beta$ | 132 $\rightarrow$ 130 | 5.85107 | 25.5 | 22.6 | 24.3 | 21.5
H131$\beta$ | 133 $\rightarrow$ 131 | 5.71909 | 24.3 | 23.1 | 23.1 | 21.9
H132$\beta$ | 134 $\rightarrow$ 132 | 5.59105 | 24.3 | 23.9 | 23.2 | 22.8
H133$\beta$ | 135 $\rightarrow$ 133 | 5.46680 | 22.5 | 23.3 | 21.2 | 22.0
H134$\beta$ | 136 $\rightarrow$ 134 | 5.34619 | 21.7 | 25.7 | 20.5 | 24.3
H135$\beta$ | 137 $\rightarrow$ 135 | 5.22913 | 25.6 | 17.3 | 23.8 | 16.3
H136$\beta$ | 138 $\rightarrow$ 136 | 5.11544 | 26.4 | 22.9 | 24.8 | 21.8
H137$\beta$ | 139 $\rightarrow$ 137 | 5.00502 | 25.4 | 24.1 | 23.8 | 22.8
H138$\beta$ | 140 $\rightarrow$ 138 | 4.89778 | 25.9 | 24.2 | 24.0 | 23.0
H139$\beta$ | 141 $\rightarrow$ 139 | 4.79357 | 26.3 | 23.7 | 24.4 | 22.5
H140$\beta$ | 142 $\rightarrow$ 140 | 4.69229 | 28.2 | 24.8 | 25.9 | 23.3
H141$\beta$ | 143 $\rightarrow$ 141 | 4.59384 | 30.0 | 23.5 | 27.5 | 22.2
H143$\beta$ | 145 $\rightarrow$ 143 | 4.40508 | 29.9 | 30.5 | 27.4 | 28.7
H145$\beta$ | 147 $\rightarrow$ 145 | 4.22650 | 42.4 | 32.8 | 38.3 | 30.8
H139$\gamma$ | 142 $\rightarrow$ 139 | 7.11476 | 40.8 | 42.4 | 39.9 | 41.4
H140$\gamma$ | 143 $\rightarrow$ 140 | 6.96495 | 34.1 | 32.3 | 33.5 | 31.5
H141$\gamma$ | 144 $\rightarrow$ 141 | 6.81933 | 38.5 | 32.8 | 37.5 | 31.6
H143$\gamma$ | 146 $\rightarrow$ 143 | 6.54004 | 26.5 | 26.2 | 25.6 | 25.1
H144$\gamma$ | 147 $\rightarrow$ 144 | 6.40609 | 23.2 | 28.7 | 22.5 | 27.6
H146$\gamma$ | 149 $\rightarrow$ 146 | 6.14898 | 23.9 | 23.5 | 22.9 | 22.4
H147$\gamma$ | 150 $\rightarrow$ 147 | 6.02558 | 27.6 | 23.2 | 26.4 | 22.2

##  Appendix B Radio Recombination Line Excitation Analysis

###  B.1 LTE Excitation

Are the ${\rm\langle Hn\alpha\rangle}$, ${\rm\langle Hn\beta\rangle}$, and
${\rm\langle Hn\gamma\rangle}$ stacked spectra consistent with LTE excitation?
To make this assessment we need to know the expected LTE ratio between a
specific $n+\Delta\,n\rightarrow\,n$ transition and a fiducial
$n+1\rightarrow\,n$ transition. Here, we use ${\rm\langle Hn\alpha\rangle}$ as
the fiducial transition and calculate the LTE ratios expected for ${\rm\langle
Hn\beta\rangle}$ and ${\rm\langle Hn\gamma\rangle}$. First we calculate the
statistical weights, $g_{n}$, and oscillator strengths, $f_{n+\Delta\,n,n}$,
for each RRL transition listed in Appendix Table 6. Since these transitions
were used to craft the stacked spectra, we use the average statistical weight,
$\langle g_{n}\rangle$, and average oscillator strength, $\langle
f_{n+\Delta\,n,n}\rangle$, for ${\rm\langle Hn\alpha\rangle}$, ${\rm\langle
Hn\beta\rangle}$, and ${\rm\langle Hn\gamma\rangle}$ to derive the expected
LTE ratios.

Table 7: Oscillator Strengths Transition | $n$ | $\Delta\,n$ | $g_{\rm n}$ | $f_{\rm n+\Delta\,n,n}$
---|---|---|---|---
H97$\alpha$ |  097 | 1 | 18818 | 18.79132765
H98$\alpha$ |  098 | 1 | 19208 | 18.98210255
H99$\alpha$ |  099 | 1 | 19602 | 19.17287745
H100$\alpha$ | 100 | 1 | 20000 | 19.36365235
H101$\alpha$ | 101 | 1 | 20402 | 19.55442725
H102$\alpha$ | 102 | 1 | 20808 | 19.74520215
H103$\alpha$ | 103 | 1 | 21208 | 19.93597705
H104$\alpha$ | 104 | 1 | 21632 | 20.12675195
H105$\alpha$ | 105 | 1 | 22050 | 20.31752685
H106$\alpha$ | 106 | 1 | 22472 | 20.50830175
H107$\alpha$ | 107 | 1 | 22898 | 20.69907665
H108$\alpha$ | 108 | 1 | 23328 | 20.88985155
H109$\alpha$ | 109 | 1 | 23762 | 21.08062645
H110$\alpha$ | 110 | 1 | 24200 | 21.27140135
H111$\alpha$ | 111 | 1 | 24642 | 21.46217625
H112$\alpha$ | 112 | 1 | 25088 | 21.65295115
H114$\alpha$ | 114 | 1 | 25992 | 22.03450095
H115$\alpha$ | 115 | 1 | 26450 | 22.22527585
H122$\beta$ | 122 | 2 | 29768 |  03.29151250
H123$\beta$ | 123 | 2 | 30258 |  03.31784460
H124$\beta$ | 124 | 2 | 30752 |  03.34417670
H125$\beta$ | 125 | 2 | 31250 |  03.37050880
H126$\beta$ | 126 | 2 | 31752 |  03.39684090
H127$\beta$ | 127 | 2 | 32258 |  03.42317300
H128$\beta$ | 128 | 2 | 32768 |  03.44950510
H129$\beta$ | 129 | 2 | 33282 |  03.47583720
H130$\beta$ | 130 | 2 | 33800 |  03.50216930
H131$\beta$ | 131 | 2 | 34322 |  03.52850140
H132$\beta$ | 132 | 2 | 34848 |  03.55483350
H133$\beta$ | 133 | 2 | 35378 |  03.58116560
H134$\beta$ | 134 | 2 | 35912 |  03.60749770
H135$\beta$ | 135 | 2 | 36450 |  03.63382980
H136$\beta$ | 136 | 2 | 36992 |  03.66016190
H137$\beta$ | 137 | 2 | 37538 |  03.68649400
H138$\beta$ | 138 | 2 | 38088 |  03.71282610
H139$\beta$ | 139 | 2 | 38642 |  03.73915820
H140$\beta$ | 140 | 2 | 39200 |  03.76549030
H141$\beta$ | 141 | 2 | 39762 |  03.79182240
H143$\beta$ | 143 | 2 | 40898 |  03.84448660
H145$\beta$ | 145 | 2 | 42050 |  03.89715080
H139$\gamma$ | 139 | 3 | 38642 |  01.16315647
H140$\gamma$ | 140 | 3 | 39200 |  01.17126209
H141$\gamma$ | 141 | 3 | 39762 |  01.17936771
H143$\gamma$ | 143 | 3 | 40898 |  01.19557895
H144$\gamma$ | 144 | 3 | 41472 |  01.20368457
H146$\gamma$ | 146 | 3 | 42632 |  01.21989581
H147$\gamma$ | 147 | 3 | 43218 |  01.22800143

The quantum properties of the Table 6 transitions are compiled in Table 7.
Listed are the transition, the principle quantum number, $n$, the order of the
transition, $\Delta\,n$, the statistical weight, $g_{n}$, and oscillator
strength, $f_{n+\Delta\,n,n}$. The statistical weight is $g_{n}=2\times n^{2}$
and we use the prescription in Menzel (1968) to derive the oscillator
strengths, $f_{n+\Delta\,n,n}$, for recombination transitions between
principle quantum numbers $n+\Delta\,n\rightarrow\,n$.

The LTE RRL intensity ratio between two optically thin transitions is given by
the ratio of their oscillator strengths,
$f_{n+\Delta\,n,n}/f^{\prime}_{n+\Delta\,n,n}$, and their statistical weights,
$g_{n}/g^{\prime}_{n}$:

${LTE~{}Intensity~{}Ratio}~{}=~{}{{\langle g_{n}\rangle\times\langle
f_{n+\Delta\,n,n}\rangle}\over{\langle g^{\prime}_{n}\rangle\times\langle
f^{\prime}_{n+\Delta\,n,n}\rangle}}.$ (B1)

Here, $\langle f^{\prime}_{n+\Delta\,n,n}\rangle$ and $\langle
g^{\prime}_{n}\rangle$ refer to the fiducial transition, ${\rm\langle
Hn\alpha\rangle}$, and the angle brackets denote the average of the quantum
properties of the transitions used to derive the stacked spectra. The average
quantum properties for the ${\rm\langle Hn\alpha\rangle}$, ${\rm\langle
Hn\beta\rangle}$, and ${\rm\langle Hn\gamma\rangle}$ stacked spectra are
summarized in Table 8 which lists $\langle g_{n}\rangle$, $\langle
f_{n+\Delta\,n,n}\rangle$, and $\langle g_{n}\rangle\times\langle
f_{n+\Delta\,n,n}\rangle$. The last column of Table 8 gives the LTE intensity
ratios for ${\rm\langle Hn\beta\rangle}$ and ${\rm\langle Hn\gamma\rangle}$
relative to ${\rm\langle Hn\alpha\rangle}$.

Table 8: Average Quantum Properties and LTE Ratio

Transition | $\langle g_{n}\rangle$ | $\langle f_{n+\Delta\,n,n}\rangle$ | $\langle g_{n}\rangle\times\langle f_{n+\Delta\,n,n}\rangle$ | LTE Ratio
---|---|---|---|---
${\rm\langle Hn\alpha\rangle}$ | 22365.000 | 20.43411151 | 4.5700890395e+05 | 1.00000
${\rm\langle Hn\beta\rangle}$ | 35271.273 |  03.57159029 | 1.2597453522e+05 | 0.27565
${\rm\langle Hn\gamma\rangle}$ | 40832.000 |  01.19442100 | 4.8770598447e+04 | 0.10672

Note. — Average quantum properties from Table 7. LTE intensity ratio relative
to ${\rm\langle Hn\alpha\rangle}$ from Equation B1.

The “LTE?” column in Table 3 gives the ratio between the observed RRL
transition ratios and the expected LTE ratio. In LTE this ratio of ratios
would be unity. For the strongest WIM components — 113.4$\,{\rm km\,s^{-1}}$
for G20 and 63.3$\,{\rm km\,s^{-1}}$ for G45 — this ratio is $1.11\pm 0.11$
and $1.08\pm 0.11$, respectively. We thus find that, within the errors, our
${\rm\langle Hn\beta\rangle}$/${\rm\langle Hn\alpha\rangle}$ ratios are
consistent with LTE excitation for both sight lines.

The lower velocity components in these directions, however, give values for
the “LTE?” parameter that are significantly larger than one: $1.34\pm 0.13$
and $1.95\pm 0.34$ for G20 and G45, respectively. Moreover, all ratios
involving the ${\rm\langle Hn\gamma\rangle}$ spectra are compromised by the
much poorer sensitivity of these spectra due to their comparatively small
integration times compared with the ${\rm\langle Hn\alpha\rangle}$ and
${\rm\langle Hn\beta\rangle}$ data. These ${\rm\langle Hn\gamma\rangle}$
spectra can only provide upper limits and these limits are not significant.

###  B.2 Non-LTE Excitation

We thus find that the ${\rm\langle Hn\beta\rangle}$ lower velocity components
seen in the G20 and G45 spectra show some evidence for non-LTE excitation in
the WIM gas. These non-LTE effects can be described by the use of departure
coefficients, $b_{\rm n}$. Departure coefficients relate the true level
population, $N_{\rm n}$, to the population level under LTE, $N_{\rm n}^{*}$,
where $b_{\rm n}=(N_{\rm n}/N^{*}_{\rm n})$. Non-LTE effects can alter the RRL
intensities in the following way (see Equation 14.52 in Wilson et al., 2009):

${T_{\rm L}\over T_{\rm L}^{*}}=b_{\rm n}\bigl{(}1-\frac{1}{2}~{}\tau_{\rm
c}~{}\beta_{\rm n}\bigr{)},$ (B2)

where $T_{\rm L}$ is the observed line intensity, $T_{\rm L}^{*}$ is the LTE
intensity, and $\tau_{\rm c}$ is the continuum optical depth. Here,
$\beta_{\rm n}$ is a measure of the gradient of $b_{\rm n}$ with respect to
$n$ (see Equation 14.40 in Wilson et al., 2009):

$\beta_{\rm n}=1-20.836\,\Bigl{(}\frac{T_{\rm e}}{\,{\rm
K}}\Bigr{)}\Bigl{(}\frac{\nu}{\,{\rm GHz}}\Bigr{)}^{-1}\Bigl{(}\frac{{\rm
d\,ln}\,b_{\rm n}}{{\rm d\,n}}\Bigr{)}\,\Delta\,n$ (B3)

The first term in Equation B2 accounts for the effect of non-LTE line
formation whereas the second describes non-LTE line transfer effects which can
include maser amplification of the line radiation. When $\beta$ becomes
negative, $\beta_{\rm n}<0$, maser amplification occurs and the resulting RRL
intensities will depend on radiative transfer details.

For the low densities expected in the WIM, however, the continuum opacity
should be small at cm-wavelengths and thus $\tau_{\rm c}\ll 1$. The main non-
LTE effect will therefore be departures in the level populations and so the
non-LTE line intensity is the LTE intensity times $b_{\rm n}$: $T_{\rm
L}=b_{\rm n}\,T_{\rm L}^{*}$.

As the principle quantum number $n\rightarrow\infty$, $b_{\rm n}$ approaches
unity. Since $n$ increases with $\Delta n$ for RRLs at the same frequency
—e.g., H102$\alpha$, H129$\beta$, and H147$\gamma$ have nearly identical rest
frequencies — we expect the Hn$\beta$ transitions to be closer to LTE than the
Hn$\alpha$ transitions. The observed ${\rm\langle Hn\beta\rangle}$ intensities
should thus be larger than expected when in LTE, consistent with the results
in Figure 6 and Table 3.

In sum, we find that for the strongest WIM components seen in G20 and G45 the
${\rm\langle Hn\beta\rangle}$/${\rm\langle Hn\alpha\rangle}$ ratios are
consistent with LTE excitation. The weaker components, however, show some
evidence for non-LTE excitation. The ${\rm\langle Hn\alpha\rangle}$ intensity
can be weaker than the LTE value because collisions are less effective due the
size of the H atom compared with the larger size of the nearby (in frequency)
${\rm\langle Hn\beta\rangle}$ transition. For the larger ${\rm\langle
Hn\beta\rangle}$ atom collisions are more effective in establishing LTE level
populations and the deviation from LTE is smaller. As a consequence the
${\rm\langle Hn\beta\rangle}$/${\rm\langle Hn\alpha\rangle}$ ratio can exceed
the LTE value.

##  Appendix C H II Regions Located Near the G20 and G45 Sight Lines

Properties of WISE Catalog H ii regions located within 100′ of the G20 and G45
sight lines are compiled in Table 9. Listed for each nebula are the galactic
co-ordinates, $(\ell,{\it b})$, and angular separation from the sight line,
together with the IR radius, $R_{\rm IR}$, RRL intensity, $T_{L}$, LSR
velocity, $V_{\rm LSR}$, FWHM line width, $\Delta\,V$, and their measurement
errors. Some of the entries in the WISE Catalog refer to nebulae that reside
within the same telescope beam. In those cases the Catalog lists identical
$V_{\rm LSR}$ and $\Delta\,V$ values (see Anderson et al., 2014b). When such
confusion within the beam occurs we only list here (and use in Figure 7) a
single nebula, choosing the one with the smallest separation from its fiducial
sight line. Some WISE H ii region spectra have emission at multiple
velocities. Because we have no reason to choose otherwise, in these cases we
plot all the velocities in Figure 7.

Table 9: H ii Regions Near G20 and G45  H ii Region | Separation | $R_{\rm IR}$ | $\ell$ | ${\it b}$ | $T_{L}$ | $\sigma_{T}$ | $V_{\rm LSR}$ | $\sigma_{\rm LSR}$ | $\Delta\,V$ | $\sigma_{\rm\Delta V}$
---|---|---|---|---|---|---|---|---|---|---
| (arcmin) | (arcmin) | (deg) | (deg) | (mK) | (mK) | ($\,{\rm km\,s^{-1}}$) | ($\,{\rm km\,s^{-1}}$) | ($\,{\rm km\,s^{-1}}$) | ($\,{\rm km\,s^{-1}}$)
G020.098$-$00.123 | 9.44 | 0.7 | 20.10 |  $-$0.123 | 32.3 | 0.3 | 43.4 | 0.1 | 20.3 | 0.2
G020.083$-$00.135 | 9.48 | 0.5 | 20.08 |  $-$0.134 | 31.0 | 2.8 | 42.2 | 1.4 | 31.7 | 3.3
G019.818+00.010 | 10.92 | 1.9 | 19.82 | +0.010 | 26.1 | 0.4 | 60.4 | 0.1 | 18.2 | 0.3
G020.227+00.110 | 15.13 | 1.2 | 20.23 | +0.110 | 10.5 | 0.2 | 22.1 | 0.2 | 14.9 | 0.4
G019.728$-$00.113 | 17.67 | 0.7 | 19.73 |  $-$0.113 | 15.4 | 0.4 | 57.3 | 0.3 | 21.7 | 0.7
G019.677$-$00.134 | 20.96 | 1.0 | 19.68 |  $-$0.133 | 48.0 | 5.8 | 55.0 | 1.2 | 20.0 | 2.8
G019.780+00.286 | 21.65 | 1.9 | 19.78 | +0.287 | 21.3 | 0.3 |  $-$9.3 | 0.2 | 21.7 | 0.4
G020.363$-$00.014 | 21.80 | 0.7 | 20.36 |  $-$0.014 | 21.8 | 0.5 | 54.1 | 0.2 | 16.6 | 0.4
G020.150$-$00.335 | 22.03 | 9.4 | 20.15 |  $-$0.335 | 19.0 | 2.9 | 67.3 | 1.5 | 20.3 | 3.6
G019.741+00.280 | 22.89 | 0.7 | 19.74 | +0.280 | 32.2 | 0.4 | 16.3 | 0.1 | 25.5 | 0.3
G019.629$-$00.095 | 22.96 | 5.5 | 19.63 |  $-$0.094 | 54.0 | 3.7 | 58.6 | 0.8 | 22.8 | 1.8
G019.716$-$00.261 | 23.11 | 1.0 | 19.72 |  $-$0.261 | 29.6 | 0.5 | 40.1 | 0.1 | 15.2 | 0.3
G019.675$-$00.226 | 23.69 | 3.5 | 19.68 |  $-$0.225 | 168.5 | 1.7 | 42.5 | 0.1 | 26.5 | 0.4
G019.594+00.024 | 24.40 | 0.7 | 19.59 | +0.024 | 12.8 | 0.3 | 35.3 | 0.4 | 29.0 | 1.0
G019.666$-$00.309 | 27.25 | 1.2 | 19.67 |  $-$0.308 | 33.9 | 0.8 | 44.3 | 0.2 | 19.2 | 0.5
G019.609$-$00.239 | 27.46 | 1.4 | 19.61 |  $-$0.238 | 168.5 | 1.7 | 42.5 | 0.1 | 26.5 | 0.4
G020.457+00.021 | 27.48 | 1.7 | 20.46 | +0.022 | 66.7 | 1.1 | 73.8 | 0.1 | 15.0 | 0.3
G019.554$-$00.248 | 30.60 | 3.2 | 19.55 |  $-$0.248 | 131.0 | 7.4 | 41.0 | 0.8 | 27.7 | 1.8
G020.481+00.168 | 30.63 | 2.6 | 20.48 | +0.169 | 24.0 | 2.3 | 24.1 | 1.0 | 20.5 | 2.2
G019.494$-$00.150 | 31.65 | 1.9 | 19.49 |  $-$0.149 | 34.3 | 0.4 | 32.1 | 0.1 | 16.8 | 0.3
G019.494$-$00.150 | 31.65 | 1.9 | 19.49 |  $-$0.149 | 20.5 | 0.4 | 54.6 | 0.2 | 15.1 | 0.5
G019.489+00.135 | 31.68 | 0.9 | 19.49 | +0.135 | 56.0 | 3.4 | 19.8 | 0.5 | 17.5 | 1.2
G019.504$-$00.193 | 31.88 | 0.9 | 19.50 |  $-$0.193 | 45.4 | 0.4 | 37.8 | 0.1 | 19.9 | 0.2
G019.466+00.168 | 33.58 | 4.8 | 19.47 | +0.168 | 45.0 | 0.5 | 109.3 | 0.1 | 23.2 | 0.3
G019.466+00.168 | 33.58 | 4.8 | 19.47 | +0.168 | 145.0 | 0.6 | 19.9 | 0.0 | 14.9 | 0.1
G019.466+00.168 | 33.58 | 4.8 | 19.47 | +0.168 | 8.2 | 0.6 | 70.8 | 0.6 | 16.3 | 1.4
G020.542$-$00.179 | 34.24 | 3.0 | 20.54 |  $-$0.179 | 40.5 | 0.6 | 48.6 | 0.1 | 16.2 | 0.3
G020.576+00.103 | 35.13 | 1.6 | 20.58 | +0.104 | 12.8 | 0.3 | 28.8 | 0.3 | 25.2 | 0.9
G020.728$-$00.105 | 44.18 | 6.6 | 20.73 |  $-$0.104 | 141.0 | 0.5 | 56.0 | 0.0 | 26.5 | 0.1
G020.727$-$00.259 | 46.34 | 4.1 | 20.73 |  $-$0.259 | 97.8 | 0.3 | 55.2 | 0.0 | 20.4 | 0.1
G019.122$-$00.263 | 54.94 | 1.1 | 19.12 |  $-$0.262 | 32.3 | 0.4 | 56.0 | 0.2 | 35.3 | 0.5
G019.122$-$00.263 | 54.94 | 1.1 | 19.12 |  $-$0.262 | 27.2 | 0.5 | 100.4 | 0.2 | 18.1 | 0.4
G019.064$-$00.282 | 58.63 | 1.3 | 19.06 |  $-$0.282 | 164.8 | 1.8 | 64.4 | 0.1 | 25.2 | 0.3
G019.604$-$00.905 | 59.24 | 1.0 | 19.60 |  $-$0.905 | … | … | 38.7 | 1.2 | 36.7 | 2.8
G020.988+00.092 | 59.54 | 2.2 | 20.99 | +0.092 | 47.0 | 5.6 | 18.6 | 1.3 | 21.4 | 2.9
G021.004$-$00.056 | 60.33 | 0.7 | 21.00 |  $-$0.056 | 18.0 | 0.2 | 28.8 | 0.1 | 19.6 | 0.3
G018.978+00.030 | 61.35 | 4.6 | 18.98 | +0.031 | 26.0 | 2.5 | 52.3 | 1.7 | 36.1 | 3.9
G019.030+00.423 | 63.47 | 1.3 | 19.03 | +0.424 | 8.1 | 0.8 | 25.4 | 0.7 | 14.8 | 1.6
G020.966$-$00.455 | 64.10 | 15.4 | 20.97 |  $-$0.455 | 10.0 | 1.1 | 38.5 | 2.5 | 45.5 | 6.0
G019.045$-$00.588 | 67.27 | 2.0 | 19.05 |  $-$0.588 | 36.0 | 3.7 | 68.2 | 1.0 | 18.6 | 2.2
G018.914$-$00.329 | 68.08 | 12.9 | 18.91 |  $-$0.329 | 217.9 | 0.8 | 68.0 | 0.1 | 23.8 | 0.1
G018.914$-$00.329 | 68.08 | 12.9 | 18.91 |  $-$0.329 | 12.5 | 1.1 | 36.2 | 0.6 | 14.3 | 1.5
G018.832$-$00.300 | 72.35 | 0.7 | 18.83 |  $-$0.300 | 46.1 | 0.4 | 46.4 | 0.1 | 30.0 | 0.3
G018.881$-$00.493 | 73.37 | 1.8 | 18.88 |  $-$0.493 | 92.0 | 5.3 | 65.5 | 0.8 | 27.1 | 1.8
G018.725$-$00.046 | 76.52 | 12.5 | 18.73 |  $-$0.045 | 22.0 | 2.4 | 63.5 | 2.4 | 45.4 | 6.0
G018.741+00.250 | 77.01 | 1.1 | 18.74 | +0.251 | 29.1 | 0.5 | 19.1 | 0.2 | 22.1 | 0.5
G018.710+00.000 | 77.36 | 0.8 | 18.71 | +0.000 | 16.6 | 0.3 | 31.8 | 0.3 | 44.5 | 0.8
G018.677$-$00.236 | 80.63 | 1.2 | 18.68 |  $-$0.236 | 57.0 | 11.3 | 42.6 | 0.6 | 23.2 | 1.3
G018.657$-$00.057 | 80.64 | 1.4 | 18.66 |  $-$0.056 | … | … | 44.1 | 0.9 | 32.4 | 2.2
G018.750$-$00.535 | 81.57 | 1.1 | 18.75 |  $-$0.535 | 18.5 | 0.6 | 66.5 | 0.4 | 25.9 | 1.0
G018.632+00.256 | 83.49 | 1.1 | 18.63 | +0.256 | 22.8 | 0.7 | 18.4 | 0.2 | 12.9 | 0.4
G018.630+00.309 | 84.26 | 0.7 | 18.63 | +0.309 | 25.3 | 0.4 | 14.0 | 0.1 | 17.5 | 0.3
G021.386$-$00.255 | 84.54 | 1.0 | 21.39 |  $-$0.254 | … | … | 92.1 | 1.1 | 32.3 | 2.7
G018.594+00.321 | 86.51 | 3.1 | 18.59 | +0.322 | 44.3 | 0.2 | 13.4 | 0.0 | 17.1 | 0.1
G018.631$-$00.492 | 87.23 | 3.2 | 18.63 |  $-$0.492 | 31.2 | 0.2 | 61.6 | 0.1 | 30.3 | 0.2
G018.584+00.344 | 87.43 | 0.7 | 18.58 | +0.344 | 28.0 | 0.6 | 10.8 | 0.2 | 16.7 | 0.4
G021.426$-$00.546 | 91.64 | 1.0 | 21.43 |  $-$0.546 | … | … | 70.2 | 0.6 | 25.3 | 1.8
G018.461$-$00.003 | 92.33 | 0.5 | 18.46 |  $-$0.003 | … | … | 56.5 | 0.4 | 30.3 | 1.0
G021.560$-$00.108 | 93.84 | 2.9 | 21.56 |  $-$0.108 | 24.9 | 0.3 | 115.1 | 0.2 | 23.0 | 0.5
G021.450$-$00.590 | 93.95 | 2.0 | 21.45 |  $-$0.590 | 38.3 | 0.3 | 72.7 | 0.1 | 20.5 | 0.2
G021.603$-$00.169 | 96.72 | 0.5 | 21.60 |  $-$0.169 | 8.2 | 0.4 |  $-$4.7 | 0.6 | 23.0 | 1.5
G021.634$-$00.003 | 98.05 | 1.9 | 21.63 |  $-$0.002 | 24.5 | 0.3 | 20.2 | 0.1 | 19.8 | 0.3
G045.070+00.132 | 8.99 | 1.0 | 45.07 | +0.132 | … | … | 59.2 | 0.8 | 24.8 | 2.1
G045.121+00.133 | 10.83 | 1.7 | 45.12 | +0.133 | … | … | 56.7 | 0.5 | 44.5 | 1.6
G045.453+00.044 | 27.31 | 4.1 | 45.45 | +0.045 | 742.9 | 1.6 | 54.2 | 0.0 | 27.6 | 0.1
G045.195$-$00.439 | 28.82 | 1.3 | 45.20 |  $-$0.439 | 18.9 | 0.4 | 72.9 | 0.2 | 25.1 | 0.6
G045.475+00.130 | 29.55 | 2.4 | 45.48 | +0.130 | … | … | 55.6 | 0.1 | 29.5 | 0.3
G044.552$-$00.239 | 30.42 | 3.8 | 44.55 |  $-$0.239 | 12.8 | 0.4 | 58.4 | 0.2 | 13.6 | 0.5
G044.811$-$00.492 | 31.59 | 4.6 | 44.81 |  $-$0.492 | 17.0 | 2.5 | 44.8 | 4.0 | 30.0 | 5.5
G045.542$-$00.006 | 32.52 | 0.7 | 45.54 |  $-$0.006 | 42.3 | 0.3 | 55.0 | 0.1 | 26.2 | 0.2
G044.501+00.332 | 35.97 | 0.8 | 44.50 | +0.332 | 48.5 | 0.2 |  $-$43.0 | 0.1 | 22.0 | 0.1
G045.002$-$00.611 | 36.62 | 5.8 | 45.00 |  $-$0.610 | 8.7 | 0.3 | 61.3 | 0.4 | 24.6 | 0.9
G044.521+00.385 | 36.87 | 0.7 | 44.52 | +0.385 | 18.1 | 0.2 |  $-$49.7 | 0.1 | 25.9 | 0.3
G044.375$-$00.076 | 37.76 | 4.2 | 44.38 |  $-$0.076 | 11.4 | 0.1 | 56.7 | 0.1 | 33.5 | 0.3
G045.634$-$00.016 | 38.08 | 1.8 | 45.63 |  $-$0.016 | 3.6 | 0.2 | 9.2 | 0.6 | 19.6 | 1.6
G044.689$-$00.579 | 39.42 | 2.1 | 44.69 |  $-$0.579 | 6.8 | 0.2 | 41.9 | 0.4 | 19.6 | 0.9
G044.379$-$00.327 | 42.11 | 5.0 | 44.38 |  $-$0.327 | 22.8 | 0.5 | 61.1 | 0.2 | 18.6 | 0.5
G045.503+00.495 | 42.38 | 1.9 | 45.50 | +0.496 | 7.4 | 0.4 |  $-$35.9 | 0.7 | 23.8 | 1.9
G045.689$-$00.235 | 43.70 | 4.0 | 45.69 |  $-$0.235 | 8.4 | 0.3 | 19.1 | 0.5 | 34.0 | 1.3
G044.904$-$00.733 | 44.30 | 2.2 | 44.90 |  $-$0.732 | 15.7 | 0.3 | 65.6 | 0.2 | 17.2 | 0.4
G045.197+00.740 | 45.96 | 1.3 | 45.20 | +0.740 | 18.7 | 0.2 |  $-$35.0 | 0.2 | 31.4 | 0.4
G044.224+00.085 | 46.79 | 5.3 | 44.22 | +0.085 | 21.0 | 2.8 | 59.6 | 3.4 | 30.4 | 4.7
G044.418+00.535 | 47.43 | 1.4 | 44.42 | +0.536 | 13.1 | 0.2 |  $-$55.1 | 0.2 | 25.0 | 0.4
G045.391$-$00.725 | 49.39 | 3.2 | 45.39 |  $-$0.724 | 53.0 | 0.5 | 52.5 | 0.1 | 20.6 | 0.2
G045.773$-$00.378 | 51.62 | 1.4 | 45.77 |  $-$0.377 | 11.7 | 0.6 | 51.0 | 0.4 | 15.1 | 0.9
G045.825$-$00.291 | 52.47 | 1.2 | 45.83 |  $-$0.290 | … | … | 61.2 | 0.2 | 26.3 | 0.5
G045.882$-$00.088 | 53.23 | 4.8 | 45.88 |  $-$0.087 | 13.2 | 0.1 | 62.4 | 0.1 | 19.1 | 0.2
G045.882$-$00.088 | 53.23 | 4.8 | 45.88 |  $-$0.087 | 2.2 | 0.1 | 18.2 | 0.5 | 28.9 | 1.3
G044.094$-$00.015 | 54.31 | 1.8 | 44.09 |  $-$0.014 | 12.6 | 0.3 | 66.6 | 0.3 | 23.6 | 0.8
G045.933$-$00.403 | 60.98 | 1.0 | 45.93 |  $-$0.402 | … | … | 63.9 | 0.5 | 21.1 | 1.3
G046.033$-$00.097 | 62.28 | 3.1 | 46.03 |  $-$0.096 | 8.0 | 0.1 | 63.5 | 0.2 | 28.5 | 0.6
G044.331$-$00.837 | 64.26 | 2.8 | 44.33 |  $-$0.837 | 13.9 | 0.4 | 62.5 | 0.2 | 17.0 | 0.6
G046.069+00.216 | 65.47 | 1.1 | 46.07 | +0.216 | 6.2 | 0.3 | 12.4 | 0.4 | 17.6 | 0.9
G046.088+00.254 | 67.05 | 0.4 | 46.09 | +0.255 | 7.6 | 0.3 | 13.4 | 0.4 | 17.0 | 0.9
G043.894+00.197 | 67.37 | 2.8 | 43.89 | +0.198 | 5.7 | 0.2 |  $-$38.1 | 0.4 | 21.3 | 1.0
G043.794$-$00.129 | 72.74 | 0.6 | 43.79 |  $-$0.129 | … | … | 43.3 | 1.1 | 31.9 | 3.0
G043.774+00.057 | 73.64 | 2.5 | 43.77 | +0.058 | 33.1 | 0.3 | 70.5 | 0.1 | 20.8 | 0.2
G043.818+00.395 | 74.78 | 1.8 | 43.82 | +0.395 | 29.6 | 0.2 |  $-$10.5 | 0.1 | 27.6 | 0.2
G043.730+00.114 | 76.49 | 2.1 | 43.73 | +0.115 | 8.3 | 0.3 | 73.1 | 0.4 | 22.4 | 0.9
G046.173+00.533 | 77.36 | 1.0 | 46.17 | +0.533 | 10.2 | 0.2 | 6.3 | 0.2 | 24.8 | 0.6
G046.203+00.532 | 78.95 | 0.7 | 46.20 | +0.532 | 5.5 | 0.2 | 4.6 | 0.6 | 29.9 | 1.4
G046.213+00.547 | 79.88 | 0.6 | 46.21 | +0.548 | 5.5 | 0.2 | 5.8 | 0.6 | 26.3 | 1.3
G043.890$-$00.780 | 81.40 | 1.0 | 43.89 |  $-$0.780 | … | … | 53.4 | 0.4 | 28.4 | 1.0
G043.999+00.978 | 83.97 | 2.3 | 44.00 | +0.979 | 4.4 | 0.2 |  $-$17.1 | 0.6 | 25.4 | 1.5
G043.968+00.993 | 85.92 | 0.8 | 43.97 | +0.993 | 11.1 | 0.5 |  $-$21.6 | 0.6 | 23.8 | 1.5
G046.495$-$00.241 | 90.85 | 5.0 | 46.49 |  $-$0.240 | 257.8 | 1.1 | 57.7 | 0.0 | 18.3 | 0.1
G043.523$-$00.648 | 96.75 | 1.5 | 43.52 |  $-$0.648 | 4.4 | 0.4 | 55.4 | 1.1 | 26.8 | 2.7
G043.432+00.516 | 99.04 | 1.2 | 43.43 | +0.517 | 22.5 | 0.3 |  $-$12.8 | 0.2 | 22.7 | 0.3

##  Appendix D Numerical Modeling of RRL Emission from WIM Plasmas

Models for WIM RRL emission from LOS plasmas must specify the plasma electron
density, $n_{\rm e}$, temperature, $T_{\rm e}$, and velocity dispersion,
$\sigma$, at every point along the $d_{\rm LOS}$. We use a numerical code to
craft synthetic RRL emission spectra for G20 and G45. TMBIDL already includes
code, MODEL_HI, that calculates 21$\,{\rm cm}$ H i spectra for any Galactic
LOS direction and distribution of physical properties. The radiative transfer
calculation is one-dimensional: the LOS is modeled as a series of slabs of
thickness, $dx$, with some total LOS path length. The LOS can have any
arbitrary distribution of density, excitation temperature, and velocity
dispersion.

We modified the TMBIDL MODEL_HI code to calculate instead the LOS radiative
transfer for a pure hydrogen plasma in LTE. The models compute spectra for the
H109$\alpha$ transition whose 5.00$\,{\rm GHz}$ rest frequency is the average
for the ${\rm\langle Hn\alpha\rangle}$ spectra. We follow the Balser et al.
(2021) technique. Assuming the Rayleigh-Jeans limit, $h\nu\ll kT_{\rm e}$, the
brightness temperature as a function of frequency, $\nu$, is given by

$T_{\rm B}(\nu)=T_{\rm e}(1-{\rm e}^{-\tau_{\rm L}(\nu)}),$ (D1)

where $\tau_{\rm L}(\nu)$ is the optical depth, $\tau_{\rm
L}(\nu)=\int\kappa_{\rm L}(\nu)\,d\ell$, and $T_{e}$ is the plasma electron
temperature. Here $\kappa_{\rm L}(\nu)$ is the absorption coefficient and
$d\ell$ is the path length through the ionized gas.

The RRL absorption coefficient (Condon & Ransom, 2016) is:

$\kappa_{\rm L}(\nu)=\frac{c^{2}}{8\pi\nu_{\rm L}^{2}}\frac{g_{\rm u}}{g_{\rm
l}}n_{\rm l}A_{\rm ul}\left[1-{\rm exp}\left(-\frac{h\nu_{\rm L}}{kT_{\rm
e}}\right)\right]\phi(\nu),$ (D2)

where $c$ is the speed of light, $\nu_{\rm L}$ is the frequency of the RRL
transition, ($g_{\rm u}/g_{\rm l}$) is the ratio of statistical weights for
the upper and lower levels, $n_{\rm l}$ is the number density in the lower
state, $A_{\rm ul}$ is the spontaneous emission rate from the upper to lower
state, $h$ is Planck’s constant, $k$ is Boltzmann’s constant, and $\phi(\nu)$
is the normalized line shape. The statistical weights for hydrogen are $g_{\rm
n}=2n^{\rm 2}$. The spontaneous emission rate from the upper to the lower
state, $A_{\rm ul}$, can be approximated (Condon & Ransom, 2016) as:

$A_{\rm n+1,n}=\left(\frac{64\pi^{6}m_{\rm
e}e^{10}}{3c^{3}h^{6}}\right)\frac{1}{n^{5}},$ (D3)

where $m_{\rm e}$ is the electron mass, $e$ is the electric charge, and $n$ is
the principal quantum number. We assume a Gaussian line profile, $\phi(\nu)$:

$\phi(\nu)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\,{\rm
exp}\left[-\frac{(\nu-\nu_{\rm L})^{2}}{2\sigma^{2}}\right].$ (D4)

In order to compare model $T_{\rm B}(\nu)$ spectra with the ${\rm\langle
Hn\alpha\rangle}$ observations we must calculate a $T_{\rm B}$($V_{\rm LSR}$)
model spectrum. To do this each slab must be assigned an LSR velocity derived
from the Galactic LOS direction, $\ell$, location along the LOS, $d_{\odot}$,
and an assumed Galactic rotation curve model. Once this $V_{\rm LSR}$ vs
$d_{\odot}$ relation is established, we use the Doppler equation at the
H109$\alpha$ rest frequency to transform $T_{\rm B}(\nu)$ to $T_{\rm
B}$($V_{\rm LSR}$). TMBIDL has several Galactic rotation curve choices
available. The conclusions we reach here about the WIM do not depend the
choice of any particular rotation curve. Here, for all our models we adopt the
Clemens (1985) rotation curve scaled to $R_{\rm 0}$=8.5$\,{\rm kpc}$.

Each model is comprised of one or more plasma “clouds” distributed along the
LOS. All clouds are homogeneous, isothermal and in LTE. Each cloud’s
properties are specified at input. A cloud is defined by: location along the
LOS, $d_{\odot}$, LOS path length size (aka the cloud diameter), $d_{\rm
cloud}$, electron density, $n_{\rm e}$, plasma electron temperature, $T_{\rm
e}$, and velocity dispersion, $\sigma$.

## References

  * Anantharamaiah (1985) Anantharamaiah, K. R. 1985, Journal of Astrophysics and Astronomy, 6, 203, doi: 10.1007/BF02715006
  * Anantharamaiah (1986) —. 1986, Journal of Astrophysics and Astronomy, 7, 131, doi: 10.1007/BF02714206
  * Anderson et al. (2014a) Anderson, L. D., Bania, T. M., Balser, D. S., et al. 2014a, ApJS, 212, 1, doi: 10.1088/0067-0049/212/1/1
  * Anderson et al. (2014b) —. 2014b, ApJS, 212, 1, doi: 10.1088/0067-0049/212/1/1
  * Anderson et al. (2015) Anderson, L. D., Deharveng, L., Zavagno, A., et al. 2015, ApJ, 800, 101, doi: 10.1088/0004-637X/800/2/101
  * Anderson et al. (2021) Anderson, L. D., Luisi, M., Liu, B., et al. 2021, ApJS, 254, 28, doi: 10.3847/1538-4365/abef65
  * Balser (2006) Balser, D. S. 2006, AJ, 132, 2326, doi: 10.1086/508515
  * Balser et al. (2016) Balser, D. S., Roshi, D. A., Jeyakumar, S., et al. 2016, ApJ, 816, 22, doi: 10.3847/0004-637X/816/1/22
  * Balser et al. (2021) Balser, D. S., Wenger, T. V., Anderson, L. D., et al. 2021, ApJ, 921, 176, doi: 10.3847/1538-4357/ac1db3
  * Bania et al. (2016) Bania, T., Wenger, T., Balser, D., & Anderson, L. 2016, TMBIDL: Single dish radio astronomy data reduction package, Astrophysics Source Code Library, record ascl:1605.005.  http://ascl.net/1605.005
  * Belfiore et al. (2022) Belfiore, F., Santoro, F., Groves, B., et al. 2022, A&A, 659, A26, doi: 10.1051/0004-6361/202141859
  * Berkhuijsen et al. (2006) Berkhuijsen, E. M., Mitra, D., & Mueller, P. 2006, Astronomische Nachrichten, 327, 82, doi: 10.1002/asna.200510488
  * Clemens (1985) Clemens, D. P. 1985, ApJ, 295, 422, doi: 10.1086/163386
  * Condon & Ransom (2016) Condon, J. J., & Ransom, S. M. 2016, Essential Radio Astronomy
  * Dennison et al. (1998) Dennison, B., Simonetti, J. H., & Topasna, G. A. 1998, PASA, 15, 147, doi: 10.1071/AS98147
  * Dettmar (1990) Dettmar, R. J. 1990, A&A, 232, L15
  * Ellis et al. (1962) Ellis, G. R. A., Waterworth, M. D., & Bessell, M. 1962, Nature, 196, 1079, doi: 10.1038/1961079a0
  * Gaensler et al. (2008) Gaensler, B. M., Madsen, G. J., Chatterjee, S., & Mao, S. A. 2008, PASA, 25, 184, doi: 10.1071/AS08004
  * Gaustad et al. (2001) Gaustad, J. E., McCullough, P. R., Rosing, W., & Van Buren, D. 2001, PASP, 113, 1326, doi: 10.1086/323969
  * Ghigo et al. (2001) Ghigo, F., Maddalena, R., Balser, D., & Langston, G. 2001, GBT Commissioning Memo: Gain and Efficiency at S-Band, GBT Commissioning Memo No. 10
  * Gordon & Sorochenko (2002) Gordon, M. A., & Sorochenko, R. L. 2002, Radio Recombination Lines. Their Physics and Astronomical Applications, Vol. 282, doi: 10.1007/978-0-387-09604-9
  * Gottesman & Gordon (1970) Gottesman, S. T., & Gordon, M. A. 1970, ApJ, 162, L93, doi: 10.1086/180631
  * Haffner et al. (2003) Haffner, L. M., Reynolds, R. J., Tufte, S. L., et al. 2003, ApJS, 149, 405, doi: 10.1086/378850
  * Haffner et al. (2009) Haffner, L. M., Dettmar, R. J., Beckman, J. E., et al. 2009, Reviews of Modern Physics, 81, 969, doi: 10.1103/RevModPhys.81.969
  * Heiles et al. (1996) Heiles, C., Reach, W. T., & Koo, B.-C. 1996, ApJ, 466, 191, doi: 10.1086/177503
  * HI4PI Collaboration et al. (2016) HI4PI Collaboration, Ben Bekhti, N., Flöer, L., et al. 2016, A&A, 594, A116, doi: 10.1051/0004-6361/201629178
  * Hill et al. (2008) Hill, A. S., Benjamin, R. A., Kowal, G., et al. 2008, ApJ, 686, 363, doi: 10.1086/590543
  * Hou et al. (2022) Hou, L., Han, J., Hong, T., Gao, X., & Wang, C. 2022, Science China Physics, Mechanics, and Astronomy, 65, 129703, doi: 10.1007/s11433-022-2039-8
  * Hoyle & Ellis (1963) Hoyle, F., & Ellis, G. R. A. 1963, Australian Journal of Physics, 16, 1, doi: 10.1071/PH630001
  * Jackson et al. (2006) Jackson, J. M., Rathborne, J. M., Shah, R. Y., et al. 2006, ApJS, 163, 145, doi: 10.1086/500091
  * Kim et al. (2023) Kim, C.-G., Kim, J.-G., Gong, M., & Ostriker, E. C. 2023, ApJ, 946, 3, doi: 10.3847/1538-4357/acbd3a
  * Kuchar (1992) Kuchar, T. A. 1992, PhD thesis, Boston University, Massachusetts
  * Lockman (1976) Lockman, F. J. 1976, ApJ, 209, 429, doi: 10.1086/154737
  * Luisi et al. (2016) Luisi, M., Anderson, L. D., Balser, D. S., Bania, T. M., & Wenger, T. V. 2016, ApJ, 824, 125, doi: 10.3847/0004-637X/824/2/125
  * Luisi et al. (2019) Luisi, M., Anderson, L. D., Liu, B., Anish Roshi, D., & Churchwell, E. 2019, ApJS, 241, 2, doi: 10.3847/1538-4365/aaf6a5
  * Luisi et al. (2020) Luisi, M., Anderson, L. D., Liu, B., et al. 2020, ApJ, 889, 96, doi: 10.3847/1538-4357/ab643e
  * Maddalena (2010) Maddalena, R. 2010, Theoretical Ratio of Beam Efficiency to Aperture Efficiency, GBT Memo No. 276
  * Menzel (1968) Menzel, D. H. 1968, Nature, 218, 756, doi: 10.1038/218756a0
  * Mezger (1978) Mezger, P. G. 1978, A&A, 70, 565
  * Ott et al. (1994) Ott, M., Witzel, A., Quirrenbach, A., et al. 1994, A&A, 284, 331
  * Peng et al. (2000) Peng, B., Kraus, A., Krichbaum, T. P., & Witzel, A. 2000, A&AS, 145, 1, doi: 10.1051/aas:2000230
  * Persson et al. (2014) Persson, C. M., Gerin, M., Mookerjea, B., et al. 2014, A&A, 568, A37, doi: 10.1051/0004-6361/201423997
  * Rand et al. (1990) Rand, R. J., Kulkarni, S. R., & Hester, J. J. 1990, ApJ, 352, L1, doi: 10.1086/185679
  * Reber & Ellis (1956) Reber, G., & Ellis, G. R. 1956, J. Geophys. Res., 61, 1, doi: 10.1029/JZ061i001p00001
  * Reynolds (1989) Reynolds, R. J. 1989, ApJ, 339, L29, doi: 10.1086/185412
  * Reynolds (1991a) —. 1991a, ApJ, 372, L17, doi: 10.1086/186013
  * Reynolds (1991b) Reynolds, R. J. 1991b, in The Interstellar Disk-Halo Connection in Galaxies, ed. H. Bloemen, Vol. 144, 67
  * Reynolds et al. (1973) Reynolds, R. J., Scherb, F., & Roesler, F. L. 1973, ApJ, 181, L79, doi: 10.1086/181189
  * Reynolds et al. (2001) Reynolds, R. J., Sterling, N. C., Haffner, L. M., & Tufte, S. L. 2001, ApJ, 548, L221, doi: 10.1086/319119
  * Reynolds & Tufte (1995) Reynolds, R. J., & Tufte, S. L. 1995, ApJ, 439, L17, doi: 10.1086/187734
  * Roshi & Anantharamaiah (2000) Roshi, D. A., & Anantharamaiah, K. R. 2000, ApJ, 535, 231, doi: 10.1086/308813
  * Roshi & Anantharamaiah (2001) —. 2001, ApJ, 557, 226, doi: 10.1086/321586
  * Roshi et al. (2005) Roshi, D. A., Balser, D. S., Bania, T. M., Goss, W. M., & De Pree, C. G. 2005, ApJ, 625, 181, doi: 10.1086/429313
  * Tanenbaum et al. (1968) Tanenbaum, B. S., Zeissig, G. A., & Drake, F. D. 1968, Science, 160, 760, doi: 10.1126/science.160.3829.760
  * Taylor & Cordes (1993) Taylor, J. H., & Cordes, J. M. 1993, ApJ, 411, 674, doi: 10.1086/172870
  * Wenger et al. (2019) Wenger, T. V., Balser, D. S., Anderson, L. D., & Bania, T. M. 2019, ApJ, 887, 114, doi: 10.3847/1538-4357/ab53d3
  * Wilson et al. (2009) Wilson, T. L., Rohlfs, K., & Hüttemeister, S. 2009, Tools of Radio Astronomy, doi: 10.1007/978-3-540-85122-6
  * Wood & Mathis (2004) Wood, K., & Mathis, J. S. 2004, MNRAS, 353, 1126, doi: 10.1111/j.1365-2966.2004.07846.x
  * Wright et al. (2010) Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ, 140, 1868, doi: 10.1088/0004-6256/140/6/1868