Ionization Profiles of Galactic H ii Regions

H ii regions, first described by Strömgren (1939), are regions of ionized gas surrounding O- and B-type stars. Only these short-lived and massive stars emit a sufficient number of high-energy photons (>13.6 eV) to ionize their surroundings fully. H ii regions are among the most luminous objects in our Galaxy at radio wavelengths and can be studied using radio recombination line (RRL) and radio free–free continuum emission. Compared to observations at near- to mid-infrared (IR) wavelengths, these radio observations are faint, but have the benefit of being essentially free from extinction.

Between the H ii region and the ambient diffuse interstellar medium (ISM) lies the H i front, which is the boundary between fully ionized hydrogen within the region and neutral hydrogen outside, followed by a photodissociation region (PDR). In PDRs, hydrogen is predominantly neutral, but carbon and other species with ionization potentials lower than that of hydrogen are mostly ionized. PDRs can be studied using numerous atomic or molecular transitions, e.g., IR emission from polycyclic aromatic hydrocarbons (PAHs) or C⁺ emission at 158 μm.

Low-density diffuse gas known as the "diffuse ionized gas" or "warm ionized medium" (WIM) is a major component of the ISM of our Galaxy. The WIM, with electron temperatures ranging from 6000 to 10,000 K, accounts for over 90% of all ionized hydrogen in the ISM (Haffner et al. 2009). It has a scale height of ∼1 kpc and was found to be in a lower ionization state than gas in H ii regions (Haffner et al. 1999; Madsen et al. 2006). The "extended low-density medium" (ELDM), another diffuse component of the ISM (see Gottesman & Gordon 1970; Mezger 1978), has a smaller scale height of only ∼100 pc and has been observed to be spatially correlated with the locations of discrete H ii regions (Alves et al. 2012).

While it is still not fully known how the WIM maintains its ionization (Haffner et al. 2009), it is believed that O stars are the most likely source of ionizing photons since all other possible ionization mechanisms (e.g., supernova explosions) cannot fulfill the energy requirements (Domgörgen & Mathis 1994; Hoopes & Walterbos 2003). However given the distribution of the WIM, it remains unclear precisely how the radiation from O stars within H ii regions is able to propagate through their surrounding PDRs and across kiloparsec-size scales into the ISM. While Wood et al. (2010) argue that a supernova-driven turbulent ISM has low-density paths that would allow ionizing photons to reach and ionize gas several kiloparsecs above the midplane, it is unknown whether this scenario could explain the WIM distribution in the Galactic plane.

Observations have shown that a significant amount of ionizing radiation is leaking from individual H ii regions. While most of these analyses focus on H ii regions in external galaxies (e.g., Oey & Kennicutt 1997; Zurita et al. 2002; Giammanco et al. 2005; Pellegrini et al. 2012), a few studies were performed on Milky Way H ii regions. Using H-alpha emission data, Anderson et al. (2015) showed that the Galactic H ii region RCW 120 is leaking ∼25% of its ionizing radiation into the ISM. They further showed that the PDR is clumpy at 8.0 μm and that photons preferentially escape through low-density pathways into the ISM. We performed a similar analysis on the compact Galactic H ii region NCG 7538 (Luisi et al. 2016, hereafter L16) using Green Bank Telescope (GBT) RRL and radio continuum data to understand better how a single H ii region may contribute to the ionization of the WIM. We computed an ionizing leaking fraction of 15% ± 5% and found that, unlike giant H ii region complexes, the radiation leaking from NCG 7538 seems to affect only the local ambient medium.

It is not well understood how emission from the WIM is affected by the presence of H ii regions, their sizes, and their morphologies. We showed in L16 that the hydrogen RRL intensity around the compact H ii region NGC 7538 decreases rapidly with its distance from the central region, whereas the RRL emission decrease in the giant H ii region complex W43 is much less steep. This result implies that giant H ii regions may have a much larger effect on maintaining the ionization of the WIM compared to compact H ii regions. In fact, Zurita et al. (2002) suggest that essentially all ionizing radiation escapes from H ii regions with the highest luminosities. Using a model from Beckman et al. (2000), Zurita et al. (2000) suggest that radiation leaking from luminous clusters of H ii regions may be sufficient to ionize the diffuse gas. As a result, a large escape fraction of less luminous regions may not necessarily be required to maintain the ionization of the WIM.

The spectrum of the radiation field provides additional information on the physical processes within H ii regions and the effect of leaking radiation on the WIM. Within an H ii region, the radiation field depends on the temperature of the ionizing star(s). The ratio of emitted helium (E > 24.6 eV) to hydrogen (E > 13.6 eV) ionizing photons, Q₁/Q₀, can be estimated indirectly by measuring the N(⁴He⁺)/N(H⁺) ionic abundance ratio. While Q₁/Q₀ is determined by the temperature of the ionizing star(s), its value is only ∼0.25 even for the hottest O3V stars (see Martins et al. 2005; Draine 2011). For an O9V star, Q₁/Q₀ is reduced to 0.015. As the radiation travels through the stellar atmosphere, metals will selectively absorb more energetic photons in a process known as line blanketing. While dependent on the metallicity of the star, this process will generally result in a decrease in the ratio of He-to-H–ionizing photons (e.g., Pankonin et al. 1980; Afflerbach et al. 1997). The He-to-H–ionizing photon ratio is also affected by dust, which causes selective attenuation of ultraviolet (UV) photons.

As radiation propagates through an H ii region, its spectrum changes further due to absorption and re-emission processes. Wood & Mathis (2004) suggested that these interaction processes preferentially result in a hardening of the H-ionizing continuum and a suppression of He-ionizing photons. Recently, Weber et al. (2019) showed that for O stars with low effective temperatures (<35,000 K) nearly the entire He-ionizing radiation is absorbed within the H ii regions. The radiation hardening has been demonstrated by Osterbrock (1989) based on the dependence of the absorption cross-section on frequency: ionizing photons with energies E ≳ 13.6 eV are preferentially absorbed by hydrogen compared to photons with much higher energies. Since the ionization cross-section of He is much greater than that of H, a large fraction of He-ionizing photons is absorbed well within the ionization front of the H ii region, resulting in a depletion of He-ionizing photons outside the PDR.

The suppression of He-ionizing photons implies a reduced N(⁴He⁺)/N(H⁺) ionic abundance ratio outside the H ii region, as there will be a fewer number of photons with sufficient energy to ionize He compared to H. Such H ii regions may be density bounded beyond the He⁺ zone, but within the H⁺ zone (see Reynolds et al. 1995). This effect was observed by Pankonin et al. (1980), who show that the ionized helium abundance in the Orion Nebula decreases with its distance from the exciting star. We also indirectly confirmed these results in L16 for the compact H ii region NCG 7538 by observing a decrease in the N(⁴He⁺)/N(H⁺) ratio with an increasing distance from the region's central position. However, it is unknown whether these findings are applicable to Galactic H ii regions in general and their relation to the age or geometry of the region.

Previous observational work and simulations confirm that the ratio of He-to-H–ionizing photons in the WIM is lower than that found in H ii regions. In a study of optical emission lines toward faint Hα-emitting regions in the Milky Way, Madsen et al. (2006) show that the He I/Hα line ratio is suppressed compared to that of H ii regions, indicating a softer radiation field. Using Monte Carlo photoionization simulations, Wood & Mathis (2004) suggest that this line ratio depends strongly on the H ii region leaking fraction. They find that He I/Hα is significantly reduced only for low escape fractions (∼15%). This result is in disagreement with Roshi et al. (2012), who observed an N(⁴He⁺)/N(H⁺) upper limit of only 0.024 in the diffuse gas near the H ii region G49, despite an apparent escape fraction of ∼63% and N(⁴He⁺)/N(H⁺) ratios of >0.066 within the H ii region (Churchwell et al. 1974; Lichten et al. 1979; Thum et al. 1980; McGee & Newton 1981; Mehringer 1994; Bell et al. 2011). Clearly, a larger sample size is required for further study.

The goal of this study is to observe a variety of H ii regions to determine the role of leaking radiation from H ii regions in maintaining the ionization of the WIM. Our observed sample includes H ii regions of different sizes and morphologies for which the PDR boundary can be identified and for which the spectral class of its central star(s) is known. The properties of our observed H ii regions are summarized in Table 1, which lists the source name, the Galactic longitude and latitude, the radius of the H ii region, its Galactocentric radius, the distance to the Sun, and the spectral type of the ionizing source(s).

The GBT RRL observations are described in Section 2 of this paper, and we outline the process of defining PDR boundaries in Section 3. In Section 4 and 5, we analyze the hydrogen RRL emission around the observed H ii regions and derive N(⁴He⁺)/N(H⁺) ionic abundance ratios for the observed directions, respectively. Physical properties of the ionized gas, including electron temperatures, emission measures, and electron densities, are derived in Section 6. In Section 7, a line profile analysis is performed as an alternative method to derive electron temperatures. We analyze carbon and doubly ionized helium emission in Sections 8 and 9, respectively, and conclude in Section 10.

We observed eight H ii regions with the GBT from 2017 February to 2017 May. The properties of the observed H ii regions are given in Table 1, which lists the source, the Galactic longitude and latitude, the radius of the H ii regions as defined in Section 3, the Galactocentric radius and heliocentric distance given by the WISE Catalog of Galactic H ii Regions (Anderson et al. 2014), and the spectral type of the ionizing source(s). For each targeted H ii region, several positions within and outside of the region's PDR were observed (see Section 3 for a description of how the PDR boundaries were determined). We employed total-power position-switching observations with variable off-source and on-source integration times, ranging from 3 minutes to 36 minutes, depending on the expected brightness of the position. For integration times exceeding 6 minutes, the observation was split up into blocks of 6 minutes each to reduce the overall impact of radio-frequency interference (RFI).

The individual pointings for each observed H ii region lie along an imaginary line on the plane of the sky intersecting the region center. The angle of the line was chosen such that there is as little confusion by other radio sources (e.g., nearby H ii regions) as possible. For each targeted H ii region, between 7 and 17 positions were observed. The goal was to include as many positions as possible within the regions' PDR boundaries, separated by as little as the average GBT half-power beamwidth (HPBW) of 123''. Outside the PDR boundaries, our pointings are spaced apart by one to four GBT HPBWs, depending on the spatial extent of the source and our total available observing time. For the largest regions, we sample up to a maximum distance of 47' from the center. The locations of the observed positions for each source are shown in Figure 1. The locations are labeled according to their predominant direction from the region center in Galactic coordinates (N ... north, S ... south, E ... east, W ... west) and their distance from the center of the regions in multiples of 2 × HPBW. The coordinates of all observed directions are given in Table 3, which lists the source, the source coordinates, and the location of the observed direction with respect to the PDR boundary (see Section 3).

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The GBT C-band instantaneous bandpass in combination with the Versatile GBT Astronomical Spectrometer (VEGAS) backend includes 22 Hnα lines, 19 Hnβ lines, and 8 Hnγ lines between 4.054 and 7.793 GHz. In our setup, the α lines range from n = 95 to n = 117, the β lines range from n = 120 to n = 146, and the γ lines range from n = 138 to n = 147 (see Luisi et al. 2018). In addition, we tuned to seven HeInα lines and eight molecular lines, including formaldehyde and methanol. To achieve a higher S/N, we average together all Hnα, Hnβ, and Hnγ lines, respectively, using TMBIDL⁷ after first regridding and shifting the spectra so that they are aligned in velocity (see Balser 2006). Spectra affected by RFI were discarded before averaging. The averaged spectra were smoothed to a resolution of 1.86 km s⁻¹ and a fourth-order polynomial baseline was subtracted.

Gaussian models were fit to the H and He profiles for which the signal-to-noise ratio (S/N) is at least 5, as defined by the method given by Lenz & Ayres (1992),

$$\begin{eqnarray}&&{\rm{S}}/{\rm{N}}=0.7\left(\displaystyle \frac{{T}_{{\rm{L}}}}{\mathrm{rms}}\right){\left(\displaystyle \frac{{\rm{\Delta }}V}{{\rm{\Delta }}{V}_{{\rm{k}}}}\right)}^{0.5},\end{eqnarray} \tag{ 1 }$$

where T_L is the peak line intensity, rms is the root mean square spectral noise, ΔV is the FWHM of the line, and ΔV_k = 1.86 km s⁻¹ is the FWHM of the Gaussian smoothing kernel. We also fit carbon lines with an S/N of at least 3. Here, the narrow width of the carbon lines allows for a lower S/N threshold as they are less likely to be confused with baseline fluctuations. In the few cases where more than one hydrogen line velocity component is detected, we assume that the brightest component is due to the H ii region.

The continuum antenna temperature, T_C, was derived for all observed positions from the continuum background of our averaged spectral line data by removing all lines above the 3σ level. We also remove 20% of all channels toward each end of the bandpass, since the bandpass edges are prone to instabilities, and average the antenna temperature over the remaining baseline to find T_C. Since the baseline stability is the dominant uncertainty contribution, the error in T_C is estimated by averaging over individual segments of each baseline, 1000 channels in width, and by computing the standard deviation between these segments. All continuum antenna temperatures are given in the Appendix.

Also shown in the Appendix are the averaged Hnα, Hnβ, and Hnγ spectra, as well as the corresponding RRL parameters.

Defining the PDR boundaries of the observed H ii regions is a crucial step in determining the region properties. Since PAHs are abundant in PDRs where they emit strongly in the 8 μm, 12 μm, and 24 μm bands (e.g., Hollenbach & Tielens 1997), enhancements in the 12 μm Wide-field Infrared Survey Explorer (WISE) emission can be used to trace the PDR itself. We define the PDR boundary for all observed regions by following the enhanced 12 μm emission surrounding the H ii region by hand. We estimate the inner and outer PDR boundary for each region such that the width of the PDR is the FWHM of the enhanced emission. Although this characterization of the PDR structure is by no means unique, results from L16 show that the above method appears to trace the PDR around compact H ii regions reliably.

The PDR boundaries are determined using the above method for each of the eight observed H ii regions and the results are shown in Figure 1. The term "strong" refers to a PDR boundary for which the enhanced 12 μm emission shows the largest contrast to the surrounding medium, whereas a "weak" PDR is barely distinguishable from the 12 μm background. Several PDR boundaries are asymmetrical or incomplete in the associated 12 μm emission. For example, in the case of G45, the western PDR boundary is much weaker and further away from the region center than the eastern boundary. For M16, the PDR shows discontinuities to the east and west, but is strong toward the north (the sampled directions). Similarly, the PDR boundary of Orion appears to be strongest toward the east and west and less so toward the south where it is at the largest distance from the center. Defining the PDR boundary of G29 is perhaps the most challenging of all of the observed regions due to its very weak 12 μm emission, many discontinuities, and proximity to several nearby continuum sources.

For M16 and Orion, several PDR components are visible in a given direction. In these cases, we define the strongest PDR boundary as the "main" PDR. For a number of regions, the sharp decrease in 12 μm intensity makes it difficult to locate the PDR boundary unambiguously toward some directions.

Using our GBT data, we test the hypothesis that high-luminosity H ii regions have a greater effect on maintaining the ionization of the WIM compared to compact H ii regions, possibly because they allow a larger fraction of ionizing radiation to escape into the ISM. In Figure 2, the hydrogen RRL intensity is shown, averaged over the Hnα transitions given in Section 2, and the ionic abundance ratio (see Section 5) is shown as a function of the distance from the region center for all observed H ii regions. As expected, the RRL intensity decreases with its increasing distance for all regions. Regions with more extended PDR boundaries (e.g., M16, M17, and Orion) have more extended emission than the observed compact regions, possibly because they are in a later evolutionary stage. In Figure 3, the hydrogen RRL emission is shown for all observed regions with the distance normalized by the radius of the PDR boundary along the given direction. It is striking that all observed H ii regions, except for Orion and perhaps M17, exhibit roughly the same hydrogen RRL emission gradient. We also show exponential fits of the form I = a × exp(b r_PDR) (where I is the hydrogen RRL intensity, r_PDR is the normalized distance from the region center, and a and b are the fit parameters) to the data to highlight the similarity between the gradients.

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The inverse correlation between the H ii region size and the slope of the hydrogen RRL intensity may indicate that larger regions allow more ionizing photons to escape through their PDRs, which in turn maintain the ionization of the surrounding WIM. Our results suggest that the total amount of escaping ionizing radiation is fundamentally correlated with the radius of the PDR boundary of that region. It is possible that the PDR boundaries surrounding large and luminous H ii regions are generally weaker or more inhomogeneous than those surrounding more compact regions. In Section 3, we show that the PDRs around M16 and Orion are not as well defined as the PDR boundaries around the more compact regions in our sample. For M16, this result could be related to the fact that large, high-luminosity H ii regions are more likely to be density bounded rather than radiation bounded (Beckman et al. 1998); however, the PDR boundaries may be less well defined for regions with multiple sources of ionization.

For a number of sources (M16, G29, N49) the intensity does not continue to decrease with the distance from the center but rather flattens out beyond a certain radius. We hypothesize that this emission far from the region center is not due to the H ii region itself but from the WIM. The observed hydrogen line intensities of ∼10–50 mK are not uncommon for WIM emission (see Anderson et al. 2015; Luisi et al. 2017). It is noteworthy that all of these regions are at relatively low Galactic latitudes where presumably emission from the WIM is the strongest (Alves et al. 2012).

Several regions have more than one hydrogen line component. The southern positions of G29 exhibit a second component approximately −40 km s⁻¹ offset from the velocity of the H ii region itself. There is also a second hydrogen line visible in N49 and in the southernmost positions of M16. Due to their low RRL intensities and since the existence of an additional H ii region along the line of sight is unlikely, these additional line components are probably due to emission from the WIM (see Anderson et al. 2015). The second hydrogen component in the central position of M17 is too strong and too spatially constrained to be caused by the WIM. It may instead be due to expansion processes within the H ii region itself.

It has been suggested that He-ionizing photons are suppressed as UV photons escape from H ii regions (Hoopes & Walterbos 2003; Wood & Mathis 2004). The complex absorption and re-emission processes in the surrounding gas, however, have never been observed in detail, and it is unclear whether this result is applicable to the Galactic H ii region population as a whole. The hardness of the interstellar radiation field for our region sample can be constrained by deriving the y⁺ = N(⁴He⁺)/N(H⁺) ionic abundance ratio using our GBT data. Since helium (with an ionization potential of ∼24.6 eV) is ionized by harder radiation compared to hydrogen (∼13.6 eV), a larger value of y⁺ indicates a more energetic radiation field.

We calculate y⁺ using

$$\begin{eqnarray}&&{y}^{+}=\displaystyle \frac{{T}_{{\rm{L}}}{(}^{4}{{\rm{He}}}^{+}){\rm{\Delta }}V{(}^{4}{{\rm{He}}}^{+})}{{T}_{{\rm{L}}}({{\rm{H}}}^{+}){\rm{\Delta }}V({{\rm{H}}}^{+})},\end{eqnarray} \tag{ 2 }$$

where T_L(⁴He⁺) and T_L(H⁺) are the line temperatures of helium and hydrogen, respectively, and ΔV (⁴He⁺) and ΔV (H⁺) are the corresponding FWHM line widths (Peimbert et al. 1992). For positions with hydrogen but no helium detections, we use upper limits of T_L(⁴He⁺) = 3 × rms and ${\rm{\Delta }}V{(}^{4}{{\rm{He}}}^{+})=\bar{x}\,{\rm{\Delta }}V({{\rm{H}}}^{+})$, where $\bar{x}$ is the average line width ratio ΔV(⁴He⁺)/ΔV (H⁺) for the observed region. Our values for $\bar{x}$ range from 0.59 to 0.87, which is consistent with previous studies by L16 ($\bar{x}=0.84$) and (Wenger et al. 2013, $\bar{x}=0.77\pm 0.25$). Since the atomic mass of hydrogen is approximately one-fourth that of helium, $\bar{x}$ should be equal to ∼0.5 in the absence of turbulence. The above values for $\bar{x}$, therefore, indicate that turbulence plays a significant role in broadening the observed line widths of our positions (see Section 7).

y⁺ is shown for each individual H ii region in Figure 2 and for all observed regions in Figure 4. We observe a decrease in y⁺ with the distance from the center for most regions. While previous results indicated an approximately constant value of y⁺ within the region and a decrease outside the PDR boundary (L16; see also Balser et al. 2001), our sample shows a relatively steady decrease with the angular offset, regardless of the sampled location with respect to the PDR. We fit a linear profile, y⁺ = a + b × r (where r is the distance to the center of the H ii region in degrees, and a and b are the fitting parameters), to the ionic abundance ratios of each region separately for each direction from the region center. We also calculate Spearman's rank correlation coefficient, ρ, both for each direction separately and for each H ii region as a whole. We give the fitting parameters and Spearman's ρ in Table 2. Our results support the hypothesis that a large fraction of He-ionizing photons is being absorbed well within the H ii region boundary. We note that the measured y⁺ gradient may also partly be due to the geometries of the observed H ii regions since we implicitly assume in the derivation of y⁺ that both H⁺ and He⁺ fill the beam. We expect to find environments near ionization fronts where hydrogen exists in a predominantly ionized form, but where helium remains mostly neutral (see Pankonin et al. 1980). Depending on the geometry of the region, the telescope beam may intersect several ionization fronts, and for these lines of sight the value of y⁺ would be lower than expected.

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Two regions do not follow the general trend of decreasing y⁺: S206 and M17. S206 shows an increase of y⁺ with the distance, and for M17 the results are inconclusive. It is unclear why these two regions exhibit such different behavior than the rest of our sample. S206 is a relatively compact H ii region without much extended emission, whereas M17 is one of the largest and brightest H ii regions in the Milky Way. This suggests that the measured increase in y⁺ may not be related to the size or morphology of these two regions.

6.1. LTE Electron Temperatures

The electron temperature, T_e, is a proxy for the metallicity of an H ii region (e.g., Rubin 1985) and can be used to study its intrinsic heating and cooling processes. Previous studies have found conflicting results regarding the relationship between T_e inside and outside the PDR boundaries of H ii regions. Most research shows a relatively constant electron temperature distribution within H ii regions (e.g., Roelfsema et al. 1992; Adler et al. 1996; Rubin et al. 2003), but there has been evidence for a decrease of T_e with an increasing distance from Orion A (Wilson et al. 2015). Under the assumption of LTE T_e can be derived by

$$\begin{eqnarray}\begin{array}{rcl}\left(\displaystyle \frac{{T}_{{\rm{e}}}^{* }}{{\rm{K}}}\right) & = & \left\{7103.3{\left(\displaystyle \frac{{\nu }_{{\rm{L}}}}{{\rm{GHz}}}\right)}^{1.1}\left[\displaystyle \frac{{T}_{{\rm{C}}}}{{T}_{{\rm{L}}}({{\rm{H}}}^{+})}\right]\right.\\ & & {\left.\times {\left[\displaystyle \frac{{\rm{\Delta }}V({{\rm{H}}}^{+})}{{\mathrm{km}{\rm{s}}}^{-1}}\right]}^{-1}\times {\left[1+{y}^{+}\right]}^{-1}\right\}}^{0.87},\end{array}\end{eqnarray} \tag{ 3 }$$

where ν_L = 6 GHz is the average frequency of our Hnα recombination lines, T_C is the continuum antenna temperature, T_L is the H line antenna temperature, ${\rm{\Delta }}V({{\rm{H}}}^{+})$ is the FWHM line width, and y⁺ is the ionic abundance ratio from Equation (2) (Mezger & Ellis 1968; Quireza et al. 2006).

The derived ${T}_{{\rm{e}}}^{* }$ are shown in Figure 5 for all observed positions where the He line could be detected. While the average electron temperature for each H ii region is slightly different, there are no large variations of ${T}_{{\rm{e}}}^{* }$ with the distance for any of the observed regions. For G45, our derived electron temperature of 1240 ± 1080 K at a normalized distance of r_PDR = 1.42 is abnormally low. At this position, our calculated continuum antenna temperature used to derive ${T}_{{\rm{e}}}^{* }$ is affected by severe baseline instability in our the spectra. Therefore, we argue that here our value of ${T}_{{\rm{e}}}^{* }$ does not reflect the actual electron temperature. This assumption is supported by our line profile analysis (Section 7), which yields an estimated electron temperature of ∼6000 K for this direction. We disregard this position for all further analyses.

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6.2. Electron Densities

Assuming that the ionized gas is in LTE we are able to constrain the emission measure and the rms electron number density for each observed direction. The emission measure, EM, is defined as the integral of the electron density squared, ${n}_{{\rm{e}}}^{2}$, along the line of sight. Because the emission measure is proportional to the optical depth at the line center, τ_L, the brightness temperature of a recombination emission line can be estimated by

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{T}_{{\rm{b}}}}{{\rm{K}}} & \approx & {T}_{{\rm{e}}}{\tau }_{{\rm{L}}}\approx 1.92\,\times \,{10}^{3}{\left(\displaystyle \frac{{T}_{{\rm{e}}}}{{\rm{K}}}\right)}^{-3/2}\\ & & \times \,\left(\displaystyle \frac{\mathrm{EM}}{\mathrm{pc}\,{\mathrm{cm}}^{-6}}\right){\left(\displaystyle \frac{{\rm{\Delta }}\nu }{\mathrm{kHz}}\right)}^{-1},\end{array}\end{eqnarray} \tag{ 4 }$$

where T_b is the brightness temperature at the line center and Δν is the line FWHM (Condon & Ransom 2016). Assuming that the RRL emitting region is extended evenly across the GBT beam and using a GBT main beam efficiency of 0.94 at C-band (Maddalena 2010, 2012), the emission measure can be expressed as

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\mathrm{EM}}{\mathrm{pc}\,{\mathrm{cm}}^{-6}} & \approx & 1.109\times {10}^{-2}\left(\displaystyle \frac{{T}_{{\rm{L}}}({{\rm{H}}}^{+})}{{\rm{K}}}\right)\\ & & \times \,\left(\displaystyle \frac{{\rm{\Delta }}V({{\rm{H}}}^{+})}{{\mathrm{km}{\rm{s}}}^{-1}}\right){\left(\displaystyle \frac{{T}_{{\rm{e}}}}{{\rm{K}}}\right)}^{3/2}.\end{array}\end{eqnarray} \tag{ 5 }$$

We use our LTE electron temperatures, ${T}_{{\rm{e}}}^{* }$, to calculate the EM for all positions for which an He line was detected. The resulting EM values range from 270 to 2.4 × 10⁶ pc cm⁻⁶, with the largest values found toward the central directions of Orion and M17 (see Figure 6, top panel). All observed H ii regions show a decrease in EM with the distance from the center, with the exception of M16 whose EM remains roughly constant within the PDR boundaries. The derived EM values are given in Table 5.

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The rms electron number density, ${\bar{n}}_{{\rm{e}}}$, can be estimated from the derived emission measures, assuming the H ii region geometry is approximated by a slab of the constant line of sight path length and a uniform density. We further assume that the path length for each H ii region is twice its radius, R_PDR, as given in Table 1. By definition,

$$\begin{eqnarray}&&\left(\displaystyle \frac{\mathrm{EM}}{\mathrm{pc}\,{\mathrm{cm}}^{-6}}\right)={\int }_{\mathrm{los}}{\left(\displaystyle \frac{{n}_{{\rm{e}}}}{{\mathrm{cm}}^{-3}}\right)}^{2}d\left(\displaystyle \frac{s}{\mathrm{pc}}\right),\end{eqnarray} \tag{ 6 }$$

such that

$$\begin{eqnarray}&&\displaystyle \frac{{\bar{n}}_{{\rm{e}}}}{{\mathrm{cm}}^{-3}}\approx 0.707{\left(\displaystyle \frac{\mathrm{EM}}{\mathrm{pc}{\mathrm{cm}}^{-6}}\right)}^{1/2}{\left(\displaystyle \frac{{R}_{\mathrm{PDR}}}{\mathrm{pc}}\right)}^{-1/2}.\end{eqnarray} \tag{ 7 }$$

We calculate ${\bar{n}}_{{\rm{e}}}$ for all positions with an He line detection and show the calculated ${\bar{n}}_{{\rm{e}}}$ values in the bottom panel of Figure 6. Due to our assumption of constant path length for each region ${\bar{n}}_{{\rm{e}}}$ follows the same trends as EM.

6.3. Non-LTE Analysis

The observed line widths of RRLs within and near H ii regions predominantly depend on two variables: the temperature of the plasma (thermal line broadening) and the amount of turbulence in the local ISM (turbulent line broadening). Other mechanisms that affect the observed line widths, such as natural broadening or pressure broadening, are thought to be negligible given our observed frequencies and electron densities (Hoang-Binh 1972).

Here, we assume that the RRL widths are only affected by thermal broadening and turbulent broadening. Therefore,

$$\begin{eqnarray}&&{\rm{\Delta }}{V}_{{\rm{H}}}=2.355{\left(\displaystyle \frac{{{kT}}_{{\rm{e}}}}{{m}_{{\rm{H}}}}+{V}_{\mathrm{turb}}^{2}\right)}^{1/2},\end{eqnarray} \tag{ 8 }$$

where ΔV_H is the observed FWHM of the hydrogen RRL, T_e is the electron temperature, m_H is the mass of the hydrogen atom, and V_turb is the velocity contribution due to turbulence. The first term in the parentheses thus corresponds to the thermal contribution, while the constant of 2.355 accounts for the conversion from the one-dimensional velocity dispersion to FWHM. Since we can measure ΔV_H directly, only two unknowns remain: T_e and V_turb. However, these can be expressed in terms of each other by observing the helium RRL toward the same direction, as long as T_e and V_turb are unchanged between the two species. After accounting for natural constants and atomic masses,

$$\begin{eqnarray}&&\left(\displaystyle \frac{{T}_{{\rm{e}}}}{{\rm{K}}}\right)=29.22\left[{\left(\displaystyle \frac{{\rm{\Delta }}{V}_{{\rm{H}}}}{\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{2}-{\left(\displaystyle \frac{{\rm{\Delta }}{V}_{\mathrm{He}}}{\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{2}\right],\end{eqnarray} \tag{ 9 }$$

where ΔV_He is the observed FWHM of the helium RRL.

Our results for T_e are shown in Figure 7. As for ${T}_{{\rm{e}}}^{* }$, only directions for which the helium line was detected are included. The average electron temperature is T_e = 7940 ± 4720 K outside the H ii region PDRs and T_e = 9180 ± 3950 K within. As for ${T}_{{\rm{e}}}^{* }$, this difference is not statistically significant. The larger deviation between individual values of T_e suggests that this method of calculation is less robust than that described in Section 6.1. This may be due to the strong dependence of T_e on the width of the observed hydrogen and helium RRLs and the fact that ΔV_He may have larger uncertainties than those given in Table 4 because of its low intensity and sensitivity to baseline variations. Due to the large deviation between individual values of T_e we use T_e only as a consistency check in this work.

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Previous results suggest that carbon RRL emission is often observed from H ii region PDRs (e.g., Hollenbach & Tielens 1999, L16). This is possibly due to its first ionization potential of only ∼11.3 eV, which is lower than that of hydrogen or helium. The carbon in H ii region PDRs may, therefore, be ionized by soft-UV photons (E < 13.6 eV) that pass through the H ii region essentially undisturbed, aside from being attenuated by dust.

We do not observe enhanced carbon RRL emission near the PDR boundaries for most H ii regions in our sample (see the top panel of Figure 8). Instead, the emission is strongest within the H ii regions and decreases steadily with the distance from the regions. While it is likely that a fraction of the PDR is contained within the telescope beam along the line of sight, this may also suggest that a large number of soft-UV photons are attenuated by dust present within the H ii regions. S104 is the only observed source for which the line emission may be increased near the PDR. However, the large number of nondetections for S104 casts doubt on the statistical significance of this interpretation.

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At low frequencies, carbon can be observed in either absorption lines or amplified emission lines due to stimulated emission from inverted populations. Observations of Cas A at 26 MHz show spectral features consistent with the detection of a carbon-α absorption line at n ∼ 630 (Konovalenko & Sodin 1981; Walmsley & Watson 1982). In an Ooty Radio Telescope study of RRLs near 327 MHz, Roshi et al. (2002) found evidence of stimulated emission, resulting in a strong correlation between carbon RRL intensity and continuum emission.

To test whether carbon RRL emission is amplified by stimulated emission at our higher average observing frequency of ∼6 GHz we show the carbon RRL intensity as a function of continuum intensity in the bottom panel of Figure 8. While there is a correlation between the carbon emission and the continuum intensity, the spread in our data is quite large. However, most of the spread is caused by only two sources, Orion and M17, which show stronger carbon RRL emission than expected given their background continuum intensities. These two sources are among the largest and most luminous H ii regions in our sample. This suggests that much of the carbon emission may be amplified by stimulated emission. At large distance offsets, a possible lack of amplification of the carbon lines by stimulated emission may explain our carbon nondetections where the continuum emission is weak. It is also possible that a low carbon abundance in the diffuse medium at these positions is responsible for the nondetections.

The second ionization potential of helium at ∼54.4 eV exceeds helium's first ionization potential by over a factor of two, and thus He⁺ is ionized by only the most energetic radiation fields within H ii regions. Previous GBT observations by Roshi et al. (2017) failed to detect emission from He⁺⁺ in the diffuse gas surrounding ultracompact H ii (UCH ii) regions. Their combined 1σ upper limit for their setup is 4 mK; however, they only observed 3 UCH ii regions with a total of 18 independent pointings.

Although many of our sampled positions are cospatial with luminous H ii regions, we did not detect the He⁺⁺ line. The rms values for our individual positions range from 3 to 20 mK, corresponding to upper limits for the He⁺⁺ line of 9 to 60 mK. Our best upper limit for the He⁺⁺/He⁺ line ratio is 0.064, sampled at the central position of Orion. We average together all positions for which the line intensity of singly ionized He is at least 30 mK after shifting them in velocity and again fail to detect the He⁺⁺ line. The rms of our combined, averaged spectrum is 0.6 mK, corresponding to a 3σ upper limit for the He⁺⁺ line of 1.8 mK.

Despite our recent studies (Anderson et al. 2015; L16), radiation leaking from Galactic H ii regions is still not well understood. Models and observations of external galaxies suggest that 30%–70% of the emitted ionizing radiation escapes from H ii regions into the ISM (Oey & Kennicutt 1997; Zurita et al. 2002; Giammanco et al. 2005; Pellegrini et al. 2012). However, due to observational constraints, these studies tend to be biased toward the largest and most luminous H ii regions, which only make up a small fraction of the total number of H ii regions in a given galaxy. Here, we observe a sample of Galactic H ii regions of various sizes, geometries, and luminosities, including several compact (radius < 5 pc) H ii regions whose extragalactic counterparts may have been missed in previous studies.

If ionizing radiation leaking from an individual H ii region is responsible for maintaining the ionization of the WIM around that region, we expect a decrease in RRL intensity with distance from the region, for which the slope of the decrease is believed to constrain the amount of leaking photons roughly (see L16). We observe a power-law decrease in the hydrogen RRL emission intensity outside the PDR of all observed H ii regions (see Figure 3). We showed in Section 4 that the hydrogen line intensity decreases with roughly the same slope for all of our targets, except Orion, when normalizing the angular offset from the region center by the radius of the PDR. This suggests that the slope in the decrease of the hydrogen RRL intensity with the distance is directly related to the size of the region. The case of Orion is further complicated by its highly asymmetrical geometry and the presence of several PDR boundaries toward the observed directions. In fact, when defining the innermost visible enhancement in 12 μm emission as the main PDR boundary, its slope is similar to that of the other sources, suggesting that the power-law decrease of the hydrogen RRL intensity with the distance and region size may fundamentally be the same for all Galactic H ii regions. The physical implications of this result are unclear, although it may be related to a transition from ionization-bounded to density-bounded behavior (e.g., Rozas et al. 1998; Zurita et al. 2002).

Using the ionic abundance ratio, y⁺, the hardness of the interstellar radiation field can be constrained for our region sample. We observe a general trend of a decreasing y⁺ with the distance from the H ii regions. This is indicative of absorption of He-ionizing photons within the ionization front of the H ii region. It is unclear why two of the observed regions, S206 and M17, do not follow this trend. The difference in their physical properties, such as M17 being ionized by more than one O star, makes it unlikely that this is due to the region's size, luminosity, or morphology.

The electron temperature, T_e, is believed to remain relatively constant within H ii regions (Roelfsema et al. 1992; Adler et al. 1996; Rubin et al. 2003), although recently Wilson et al. (2015) reported on a decrease in T_e with an increasing distance from Orion A. We calculate the electron temperature for our positions with He detections using two independent methods. The first method assumes that LTE is satisfied for all observed directions (see also Mezger & Ellis 1968; Quireza et al. 2006). We find no general trends in the derived electron temperatures with distances from the H ii regions and no statistically significant difference between T_e within and outside the H ii region PDRs. The second method is based on a line profile analysis and assumes that T_e and the amount of turbulent line broadening remain the same between hydrogen and helium. Although LTE is not required, this method appears to be less robust than the first, possibly due to the larger contribution of line width uncertainties on the overall uncertainty in T_e. We, again, do not detect a significant difference between electron temperatures within and outside the H ii region PDRs.

Our H ii region sample spans a wide range of EMs and rms electron densities, ${\bar{n}}_{{\rm{e}}}$. While most observed directions have ${10}^{2}\lesssim {\bar{n}}_{{\rm{e}}}\lesssim {10}^{3}\,{\mathrm{cm}}^{-3}$, the electron densities at the central locations of M17 and Orion are as large as 258 cm⁻³ and 596 cm⁻³, respectively. These values are significantly lower than those found by optical line tracers (e.g., Danks 1970; Kitchin 1987), possibly because they are based on ${T}_{{\rm{e}}}^{* }$ averaged along the line of sight and over the HPBW. With an average departure coefficient of b_n = 0.86 ± 0.04 we do not find strong departure from LTE in our sample. We note that b_n is largest toward the high-${\bar{n}}_{{\rm{e}}}$ regions at the central locations of our H ii regions sample where the level populations are dominated by collisions. As expected, b_n decreases with an increasing distance from all observed H ii regions, suggesting that non-LTE effects become more significant in the lower-density envelopes surrounding Galactic H ii regions.

Finally, we do not find enhanced carbon RRL emission near the PDR boundaries, as has been observed previously (Hollenbach & Tielens 1999; L16). This may indicate selective attenuation of soft-UV photons by dust within the H ii regions. However, a correlation exists between the carbon RRL intensity and the continuum intensity. This suggests that the carbon emission is amplified by stimulated emission.

We thank the referee for insightful comments that improved the clarity of this manuscript. We thank Zachary Tallman for his help in identifying the PDR boundaries used in this work. This work is supported by NSF grant AST1516021 to LDA. This research has made use of the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.

Facilities: Green Bank Telescope - , WISE - .

Software: TMBIDL (Bania et al. 2014).

Here, we show the observed RRL spectra and give the RRL parameters and derived source properties. The averaged Hnα, Hnβ, and Hnγ spectra are shown in Figures 9, 10, and 11, respectively. The derived RRL parameters are given in Table 4, which lists the source, the transition, the element, the line intensity, the FWHM line width, the LSR velocity, the rms noise in the spectrum, and the total on-source integration time for each position, including the corresponding 1σ uncertainties of the Gaussian fits. All continuum antenna temperatures are given in Table 5, which lists the source, TC, the local thermodynamic equilibrium (LTE) electron temperatures (Te*), and the emission measures (EM), including all 1σ uncertainties.

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