Helium Ionization in the Diffuse Ionized Gas Surrounding UCH ii Regions

The existence of a diffuse ionized gas in the Galaxy is evident from a variety of observations (Hoyle & Ellis 1963; see review by Haffner et al. 2009). This gas, referred to as the warm ionized medium (WIM), is now considered to be one of the major components of the interstellar medium (ISM). The WIM has been primarily studied using optical emission lines. These studies show that the local electron density of the WIM is in the range of 0.01–0.1 cm⁻³ and emission measures are typically ≲10 pc cm⁻⁶ (Haffner et al. 2009). In or near the disk of the inner Galaxy, optical lines suffer strong extinction, and hence the distribution of the ionized gas has been studied in the radio frequency regime. In particular, low-frequency (≲2 GHz) radio recombination line (RRL) observations have detected diffuse ionized regions (DIRs) with local density in the range of 1–10 cm⁻³ and emission measure ≲1000 pc cm⁻⁶ (Lockman 1976; Anantharamaiah 1986; Roshi & Anantharamaiah 2000; Alves et al. 2015). It is generally thought that both WIM and DIRs are ionized by massive stars. To maintain ionization, the WIM and the DIR together require about 80% of the ionizing radiation from all OB stars in the Galaxy (Mezger 1978; Murray & Rahman 2010). Thus, the WIM and DIR form an energetically important component of the ISM.

The WIM in the Galaxy has been shown to have low ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ number density ratios from optical line observations (≲0.027; Reynolds & Tufte 1995; see also Haffner et al. 2009). Here ${n}_{{\mathrm{He}}^{+}}$ is the number density of singly ionized helium (He), and ${n}_{{{\rm{H}}}^{+}}$ is that of ionized hydrogen (H). RRL observations toward DIRs have provided an upper limit on ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ of ∼0.013 (Heiles et al. 1996; Roshi et al. 2012). Ionization of both these components of the ISM is thought to be due to UV photons from O6-type or hotter stars that leak out of H ii regions (Mezger 1978; see also Anderson et al. 2011; Luisi et al. 2016, for observational evidence of photon leakage from H ii regions). However, if stars hotter than ∼O6 are the primary ionization sources of the DIR and the WIM, the ratio ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ should be close to that of the actual He/H abundance ratio of ∼0.1 (see Draine 2011, Table 15.1 and Section 15.5). This is because (1) ≤18% of the ionizing photon flux from O6 or hotter stars can fully ionize He in the ionized H region; and (2) the ionization cross sections for photons more energetic than the ionization potentials of H and He decrease with energy proportional to (hν)⁻³, resulting in greater mean-free paths for the highest-energy photons and expected "hardening" of the radiation field with distance from the source of ionization. Thus, the low ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratios in the WIM and DIR are not understood.

There are at least two important issues related to the low helium ionization in the diffuse regions. First, the observed ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratio toward several H ii regions harboring O6 or hotter stars is lower than the cosmic abundance of helium (Shaver et al. 1983). The mean value of the ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratio obtained toward such H ii regions is 0.08, indicating the presence of 20% neutral helium. The possible effects that can produce lower helium ionization include (a) selective absorption by dust (Mezger et al. 1974; see Section 5) and (b) line blanketing in the atmosphere of the O stars (see, e.g., Martins et al. 2005). Second and possibly even more problematic is how the ionizing UV photons escape through the H ii ionization fronts that surround hot O stars and the surrounding DIRs of compact H ii regions. In particular, are the ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratios significantly decreased in their passage to the WIM? This is obviously a function of how clumpy and dusty the ISM is in the neighborhood of massive star formation regions. Although these two are important issues, the main motivation of this investigation is to determine whether the ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratios are systematically decreased in ionized gas surrounding the earliest stages of massive star-forming regions.

Lyman continuum radiation leaking out of H ii regions ionizes gas in its immediate vicinity, as well as at large distances from the H ii region. The collection of ionized gas surrounding compact H ii regions, referred to as envelopes of H ii regions, in the inner Galaxy forms the DIR (Anantharamaiah 1986). The hierarchical structures in the molecular cloud/ISM produce similar morphology at different physical scales and at early stages of star formation. For example, many ultracompact H ii (UCH ii) regions are known to have extended, diffuse ionized gas—referred to as envelopes of UCH ii regions or UCH ii envelopes—associated with them (Garay et al. 1993; Kurtz et al. 1999; Kim & Koo 2001; see also Churchwell 2002). UCH ii regions are ionized by massive stars that are still embedded in their natal molecular cloud, thus representing a very early stage of star formation. The morphology of UCH ii regions and their envelopes is similar to that of compact H ii regions and the DIR. The emission measure of these envelopes is typically ≳ a few times 10³ pc cm⁻⁶, an order of magnitude larger than the emission measure of the DIR. The envelopes of UCH ii regions absorb more than 65% of the ionizing radiation from the embedded stars (Kim & Koo 2001), which is comparable to the percentage of UV photons absorbed by the DIR. Thus, observing He RRLs from higher emission measure diffuse gas surrounding UCH ii regions may provide a clue to resolving the He ionization problem in the DIR. In Section 2 we describe the selection of sources for observation and discuss their properties. The observations and data analysis are discussed in Section 3. Our main results are presented in Section 4, and a discussion of the results is given in Section 5. A summary of the main results is given in Section 6. Appendices A.1–A.3 give the details of the analysis discussed in Section 5.

A systematic study of continuum and RRL emission toward 16 UCH ii regions was done by Kim & Koo (2001). They detected diffuse, extended emission toward 14 UCH ii regions in their sample. The similarity of the LSR velocity of hydrogen RRLs from the UCH ii regions and the diffuse gas suggests that the H ii region and the diffuse component are associated (see Section 4.1 for further discussion on velocity structure based on our data set). This association is also suggested by the continuum morphology of these sources. The continuum emission from UCH ii regions and their envelope was used to estimate the Lyman continuum photon flux required to maintain ionization. This Lyman continuum photon flux is used to estimate the required stellar spectral type. The spectral types for the 14 UCH ii regions range from O4 to O9. We selected three sources (G10.15-0.34, G23.46-0.20, and G29.96-0.02) with embedded stars of type O5.5 or earlier for helium RRL observations. H ii region models with a single ionizing star suggest that the helium and hydrogen Strömgren spheres will overlap for these sources. Hence, helium is expected to be singly ionized in the envelope of the selected UCH ii regions.

VLA 21 cm images of the three selected sources G10.15-0.34, G23.46-0.20, and G29.96-0.02 are shown in Figures 1–3, respectively. These images are from the data obtained by Kim & Koo (2001) and have an angular resolution of ∼40'' × 20''. We show Spitzer three-color images of the three targets in Figures 4–6, with GLIMPSE 3.6 and 8.0 μm data in blue and green (Benjamin et al. 2003; Churchwell et al. 2009) and MIPSGAL 24 μm data in red (Carey et al. 2009). The red MIPSGAL emission is from warm dust grains spatially coincident with the ionized gas in H ii regions. The green GLIMPSE 8.0 μm emission is dominated by polycyclic aromatic hydrocarbons (PAHs) in the photodissociation regions (PDRs). A brief description of the selected sources is given below, and a summary of their properties is given in Table 1.

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Table 1.  Properties of the Selected Sources

Properties	Value	Reference
G10.15-0.34
Distance	3.6 kpc	1
UCH ii regions	G10.15-0.34	2
Angular size of envelope	109 × 67	3
Linear size of envelopea	11.3 pc × 6.9 pc
Flux density of envelope at 1.43 GHz	55.22 Jy	3
Assumed electron temperature of the ionized gas	8000 K
Lyman continuum luminosityb	6.0 × 1049 s−1
Star typec	O3 V
G23.46-0.20
Distance	6 kpc	4, 5
UCH ii regions	G23.455-0.201	2
Angular size of envelope	88 × 58	3
Linear size of envelopea	15.4 pc × 10.1 pc
Flux density of envelope at 1.43 GHz	11.31 Jy	3
Assumed electron temperature of the ionized gas	8000 K
Lyman continuum luminosityb	3.5 × 1049 s−1
Star typec	O4.5 V
G29.96-0.02
Distance	6.2 kpc	6
UCH ii regions	G29.95-0.01, G29.86-0.04	2, 6
Angular size of envelope	63 × 52	3
Linear size of envelopea	11.7 pc × 9.4 pc
Flux density of envelope at 1.43 GHz	12.69 Jy	3
Assumed electron temperature of the ionized gas	8000 K
Lyman continuum luminosityb	4.2 × 1049 s−1
Star typec	O4 V

Notes.

^aLinear size is estimated using the distance give in the table here. ^bLyman continuum luminosity is estimated as described in Rubin (1968) using the flux density at 1.43 GHz, distance to the source, and electron temperature given in the table here. No correction for dust extinction is applied. ^cType of the star from the estimated Lyman continuum luminosity using the results of Martins et al. (2005).

References. (1) Urquhart et al. 2012; (2) Wood & Churchwell 1989; (3) Kim & Koo 2001; (4) Wienen et al. 2015; (5) Brunthaler et al. 2009; (6) Zhang et al. 2014.

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2.1. G10.15-0.34

2.2. G23.46-0.20

2.3. G29.96-0.02

The sizes of the selected sources are listed in Table 1. The typical physical size of the selected diffuse regions surrounding UCH ii regions is ≳9 pc, which corresponds to an angular size ≳5'. We made RRL observations toward 13 positions in the selected sources with the Robert C. Byrd Green Bank Telescope (GBT) near 5 GHz. The FWHM beam width of the telescope near 5 GHz was about 25, allowing us to sample physical scales smaller than the sizes of the diffuse regions in the selected sources. The J2000 R.A. and decl. of the observed positions are listed in Table 2 and are shown in Figures 1–3. For comparison, we also observed three positions (G10.15-0.34a, G23.46-0.20a, G29.96-0.02a; see Figures 1–3 and Table 2) toward peaks in the 21 cm continuum images of the selected regions.

We simultaneously observed eight RRL transitions (104α, 105α, 106α, 109α, 110α, 111α, 112α, and 113α) of hydrogen, helium, and heavy elements. The reference spectra to correct for bandpass shape were obtained by switching the frequency by 8.5 MHz (∼530 km s⁻¹). GBTIDL routines were used to correct for bandpass shape for each RRL transition and calibrate the spectra in units of antenna temperature. Doppler tracking was done using the H110α RRL transition during observation. The residual Doppler correction in other RRL transitions was done offline by shifting them in LSR velocity. The Doppler-corrected spectra for each RRL transition were averaged using GBTIDL routines. The rest of the data analysis was done using routines developed in Matlab. The average spectrum for each transition was examined for radio frequency interference (RFI). After editing RFI-affected spectra and removing spectra that have beta transitions contaminating the helium line, we resampled the subset of RRL spectra to a common velocity resolution. This subset was averaged to obtain the final integrated spectrum. The RRL transitions averaged to get the final spectrum are listed in Table 2. The actual observing time and effective integration time, obtained from the number of transitions averaged, for each position are also included in Table 2. The effective integration time is the actual integration time of the final spectra after editing out the RFI-affected spectra and averaging the data corresponding to the listed transitions in Table 2. A fourth-order polynomial was subtracted from the final spectrum.

Results of Gaussian component analysis of the final spectra are included in Table 3. The source name, peak line antenna temperature in K, FWHM line width in km s⁻¹, LSR velocity of the line in km s⁻¹, and the atom responsible for the line emission are included in Table 3. The final spectra and the Gaussian components are shown in Figures 1–3. The signal-to-noise ratio of the hydrogen line is greater that 10σ in most cases, allowing us to fit multiple Gaussian components to the line profile. We choose the minimum number of Gaussian components for the hydrogen line so that the residuals after subtracting the Gaussian model are consistent with the rms of the spectral noise. Since we did not know a priori which hydrogen line component corresponds to the helium line, we tried several methods to obtain the helium line parameters. We found that the different methods provide a slightly different value for the ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratio (see below), but all these values are consistent within the 1.5σ estimation error of this ratio. We finally adopted the following strategy to get the helium line parameters: we set the LSR velocity of the helium line equal to the central velocity of the strongest hydrogen line while fitting the Gaussian components. The exception is for the position G10.15-0.34d, where the helium line velocity is close to the 8.45 km s⁻¹ hydrogen line component, and hence we used this velocity for the central velocity of helium. No constraints were used to obtain line parameters of atoms heavier than helium. The heavy element is tentatively identified as carbon based on its frequency and its relatively high abundance compared to other heavy elements with ionization potential <13.6 eV.

4.1. LSR Velocity of Hydrogen Line

4.2. Helium-to-hydrogen Line Ratio

The ratio

$$\begin{eqnarray}&&\displaystyle \frac{{n}_{{\mathrm{He}}^{+}}}{{n}_{{{\rm{H}}}^{+}}}=\displaystyle \frac{\int {T}_{L,\mathrm{He}}{dv}}{\int {T}_{L,{\rm{H}}}{dv}}\end{eqnarray} \tag{ 1 }$$

obtained from the RRL data is given in Table 4. Here ${T}_{L,\mathrm{He}}$ and ${T}_{L,{\rm{H}}}$ are the line antenna temperatures of helium and hydrogen, respectively (see Table 3), and the integration is over LSR velocity. The computed values of the ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratio, along with 1σ error and upper limits, are listed in column (2) in Table 4. The velocities of the hydrogen line components used to obtain the ratio are given in column (3). In cases where the helium line is not detected, the upper limit for the ratio is computed using the 1σ value at the line-free region of the spectrum and the net line width of the hydrogen line components listed in column (3) as the expected width of the helium line. The other columns are source name (column (1)), offset distance from the "ionization center" (column (4); see below), 4.875 GHz brightness temperature (column (5); see below) and emission measure (column (6)). The emission measures are obtained from the main beam brightness temperature, estimated from the 4.875 GHz continuum survey of Altenhoff et al. (1979), which has similar angular resolution to that of the GBT observations. The emission measure is obtained using the equation (Mezger & Henderson 1967)

$$\begin{eqnarray}&&\mathrm{EM}=\displaystyle \frac{{T}_{b4.9}}{8.235\times {10}^{-2}{T}_{e}^{-0.35}{\nu }^{-2.1}},\end{eqnarray} \tag{ 2 }$$

where EM has units pc cm⁻⁶, ${T}_{b4.9}$ is the brightness temperature in K at the frequency ν = 4.875 GHz, and T_e is the electron temperature in K (see Table 1).

Table 4.  ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ Ratio toward the Observed Positions

Source	${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ a	VLSR	Offset	${T}_{b4.9}$ b	EM
		(km s−1)	(pc)	(K)	(pc cm−6)
G10.15-0.34a	0.056 (0.009)	13.03	0.9	50.0	4.0 × 105
G10.15-0.34b	≤0.052	16.13	4.7	5.0	4.0 × 104
G10.15-0.34c	≤0.037	12.77, 6.29	3.1	5.0	4.0 × 104
G10.15-0.34d	0.073 (0.018)	8.46	3.9	5.0	4.0 × 104
G10.15-0.34avg	≤0.033	12.83	3.9	⋯	4.0 × 104
G23.46-0.20a	0.048 (0.003)	101.06	2.0	8.0	6.4 × 104
G23.46-0.20b	0.031 (0.005)	99.10	5.7	6.0	4.8 × 104
G23.46-0.20c	0.035 (0.005)	98.10	10.1	3.6	2.9 × 104
G23.46-0.20d	≤0.094	99.94	9.0	2.0	1.6 × 104
G23.46-0.20e	≤0.072	95.61	13.0	1.8	1.4 × 104
G23.46-0.20f	≤0.050	89.20	24.4	1.1	8.7 × 103
G23.46-0.20avg	≤0.051	90.36	15.5	⋯	1.2 × 104
G29.96-0.02a	0.081 (0.005)	101.35	5.2	10.0	7.9 × 104
G29.96-0.02b	0.069 (0.003)	99.28	6.1	8.0	6.4 × 104
G29.96-0.02c	0.053 (0.008)	95.45	5.7	4.0	3.2 × 104
G29.96-0.02d	0.061 (0.008)	96.02	7.8	2.8	2.2 × 104
G29.96-0.02e	0.058 (0.007)	96.90	8.6	2.4	1.9 × 104
G29.96-0.02f	0.061 (0.008)	97.68	11.5	3.0	2.4 × 104

Notes.

^a1σ error in the estimated quantity is given in parentheses. The upper limits are also the 1σ limit. ^bMain-beam brightness temperature at 4.875 GHz at the observed positions obtained from the galactic plane survey of Altenhoff et al. (1979).

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The helium lines are detected toward G10.15-0.34a and d in G10.15-0.34. We use the 24 μm emission in the MIPSGAL data, which has higher angular resolution (∼6'') than the 21 cm images, to identify compact H ii regions (emission shown in red in Figure 4) in the observed positions. This identification can be done because a large part of the 24 μm emission originates from very small dust grains (few nanometers in size) inside H ii regions heated by UV radiation from the star and collision with ionized gas particles (Pavlyuchenkov et al. 2013). The observed correlation between the 24 μm emission and 21 cm flux density of Galactic H ii regions supports this picture (Anderson et al. 2014). Both these positions have bright 24 μm emission, indicating the presence of compact H ii regions. Helium lines are not detected toward G10.15-0.34c and b. No compact H ii regions are present in these directions as inferred from the 24 μm image. We averaged the GBT spectra observed toward these two positions to get a stringent upper limit on helium line emission. The line parameters obtained from the average spectrum are listed in Table 3. The LSR velocity of the hydrogen line (12.8 km s⁻¹) is similar to those observed toward other positions in the source G10.15-0.34. Thus, the positions G10.15-0.34c and b sample the more diffuse region of the UCH ii envelope. The 1σ upper limit for helium line emission and the value of the $\tfrac{{n}_{{\mathrm{He}}^{+}}}{{n}_{{{\rm{H}}}^{+}}}$ toward the diffuse region are ∼2 mK and 0.03, respectively, both obtained from the average spectrum shown in Figure 7.

Helium lines are detected toward positions G23.46-0.20a, b, and c. Examination of the 24 μm image indicates that compact H ii regions are present in these regions (see Figure 5). No helium lines were detected toward positions G23.46-0.20d, e, and f. Toward the position G23.46-0.20e a compact H ii region is present, as inferred from its 24 μm emission. The hydrogen lines toward positions G23.46-0.20d and f have components with LSR velocities 99.9 and 89.2 km s⁻¹, respectively. The strongest hydrogen line components toward positions G23.46-0.20a, b, c, and e show a gradient in LSR velocity with values reducing from 101.1 to 95.6 km s⁻¹. Thus, the diffuse ionized gas toward G23.46-0.20d and G23.46-0.20f may be associated with G23.46-0.20. We averaged the GBT spectra toward G23.46-0.20d and f to get a stringent upper limit on helium line emission (see Figure 7). The upper limit for the value of the $\tfrac{{n}_{{\mathrm{He}}^{+}}}{{n}_{{{\rm{H}}}^{+}}}$ obtained from this average spectrum is 0.05.

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Helium lines are detected toward all the observed positions in G29.96-0.02. All these positions have associated bright 24 μm emission, indicating the presence of compact H ii regions (see Figure 6). The value of $\tfrac{{n}_{{\mathrm{He}}^{+}}}{{n}_{{{\rm{H}}}^{+}}}$ varies between 0.053 and 0.081 at the observed positions. Such variation is also observed toward the sources G10.15-0.34 and G23.46-0.20 at positions where helium lines are detected.

Figure 8 shows the histogram of the values for ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ obtained toward positions where helium lines are detected from all three sources. The values of the ratio from our sample range from 0.033 to 0.081; the mean value is 0.058. The mean value for ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ is consistent with those observed toward compact H ii regions in the Galaxy (Churchwell et al. 1974; Quireza et al. 2006). It may be noted that some of the H ii regions observed by Churchwell et al. (1974) and Quireza et al. (2006) may be ionized by stars later than O6, and hence the mean value of ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ from their data set is biased toward lower values.

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Multiple OB stars are responsible for the ionization of the observed sources. Due to the complexity of the source, however, it is difficult to identify a region, referred to as the "ionization center," where most of the massive stars are located in each source. We assigned the "ionization center" for the region G10.15-0.34 as the center of the NIR emission observed by Blum et al. (2001) (R.A. 18:09:26.71, decl. −20:19:29.7 J2000; see Section 2; shown in Figure 1), which has revealed four O-type stars. The "ionization center" for the region G23.46-0.20 is taken as the UCH ii region G23.455-0.201, and that for the region G29.96-0.02 is taken as the UCH ii region G29.95-0.01. In Figure 9, we plot the values of the ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratio against the estimated distance offset (see Table 4) of the observed positions from the "ionization center." We also plot the measured ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ values against the estimated emission measure (see Table 4) in Figure 10. The upper limits obtained from the average spectra are shown in the plot with filled triangles. No particular trends are evident in these plots, but it is clear that the helium is not uniformly ionized in the observed regions.

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We first examine the observed ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratio toward the continuum bright regions (G10.15-0.34a, G23.46-0.20a, and G29.96-0.02a). The first two positions encompass the UCH ii regions G10.15-0.34 and G23.455-0.201, respectively. The third position is observed near the giant H ii region G29.944-0.042. The mean value of the ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratio obtained from the data toward the three positions is 0.062 ± 0.02. This value is slightly lower than the mean value of ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ (0.08 ± 0.02) observed toward H ii regions ionized by stars of spectral type O6 or hotter star (Shaver et al. 1983). The origin of this lower ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ value may be related to the nonuniform helium ionization observed in the selected sources.

In the diffuse regions in the sources G10.15-0.34 and G23.46-0.20 less than 33% and 51%, respectively, of the helium is singly ionized. The detection of helium lines toward high emission measure regions in the envelopes implies that stars of type O6 or earlier are embedded in these regions. If these stars are responsible for the ionization of the diffuse gas, then helium is expected to be fully ionized. Thus, the situation in the UCH ii envelopes is similar to the WIM and DIR—i.e., stars of type O6 or earlier are required for the ionization of the WIM and DIR, but helium in these regions is observed to be not fully ionized. Observations toward the WIM and DIR probe helium ionization at distances greater than a few tens of parsec from the ionizing stars. Our observations, on the other hand, probe helium ionization in diffuse regions at distances ≲10 pc from newly born massive stars. Below we investigate possible origins for the nonuniform helium ionization.

As mentioned earlier, stars hotter than ∼O6 are the likely primary ionization sources of the DIR and WIM. A "hardening" of the radiation field with distance from the ionizing source is expected since the the ionization cross sections for both H and He decrease with energy proportional to (hν)⁻³. If that is the case, can helium be doubly ionized in regions where He⁺ lines are not detected? The velocity range of RRL spectra obtained in our observation includes 167α, 175α, and 178α recombination line transitions of He⁺⁺. The velocity range of 167α transition overlaps with the H105α line. The 178α transition is affected by a bad baseline. We therefore examined the 175α transition of He⁺⁺ but failed to detect the line from any of the observed positions. We further averaged the spectra from all the observed positions after resampling and shifting them in velocity, but again failed to detect the He⁺⁺ line. The 1σ upper limit we obtained is 0.004 K. Thus, we conclude that helium is not doubly ionized at the observed positions.

The sizes of the helium and hydrogen ionization zones depend on the number of "helium" (q_He; i.e., photon wavelength range 228 Å < λ < 504 Å) and "hydrogen" (q_H; i.e., photon wavelength range 504 Å < λ < 912 Å) Lyman photons available for ionization (Mathis 1971). It is convenient to characterize the Lyman photon spectrum by the ratio $\gamma =\tfrac{{q}_{\mathrm{He}}}{{q}_{{\rm{H}}}}$ (Mathis 1971). To understand physically the dependence of the ionization zone size on γ, to a first approximation, we assume (a) that all the q_He photons ionize helium and (b) that the photons due to the recombination of helium ions ionize hydrogen. The first assumption is justified since the ionization cross section of helium for energies ≥24.6 eV is larger compared to that of hydrogen. The second assumption follows from the energy level of helium and the probability of the paths involved in the recombination process. With the above assumptions and keeping in mind that the helium recombination rate is ∼18% of that of hydrogen in regions where both atoms are (singly) ionized, the ionization equilibrium implies that the size of the helium ionization zone is smaller than the hydrogen ionization zone if γ ≲ 0.2 (see Draine 2011, p. 172, for further reading). A numerical solution to this problem was presented by Mathis (1971), where he derived the relationship between γ and the sizes of the ionization zones of the two atoms (see Appendix A.3). The observed UCH ii envelopes are ionized by multiple OB stars, which may be members of a cluster. In the early stage of star formation, it is likely that the massive members of the cluster are embedded in (locally) ionization-bounded regions and may not be contributing to the ionization of the diffuse regions in the envelopes, which then results in a change in γ of the radiation field. Here we ask, if the diffuse regions in the UCH ii envelopes are ionized by the rest of the stars in the cluster, what upper mass of the cluster member can produce the observed ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratio?

In Appendix A.1, we compute γ as a function of the upper cutoff mass for the cluster. The result is shown in Figure 11. The observed ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratios are used to obtain γ for the radiation field as described in Appendix A.3 (no selective dust absorption is considered here, i.e., a₀ is taken as unity). The measured upper limits to ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ for G10.15-0.34 and G23.46-0.2 are ∼0.03 and 0.05, respectively. The estimated γ for the two sources are, respectively, 0.04 and 0.06. Thus, from Figure 11, it follows that the upper mass of the cluster member that ionizes these UCH ii envelopes has to be ≲30 M_⊙ to explain the observed value of the ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratio. The Lyman continuum emission estimated for G10.15-0.34 and G23.46-0.2 is ∼5 × 10⁴⁹ s⁻¹ (average value for these two sources from Table 1). If all the cluster members are contributing to the ionization, then the mean ionizing flux per solar mass is 6.1 × 10⁴⁶ s⁻¹ M_⊙⁻¹ (see Appendix A.1), implying a cluster mass of ∼800 M_⊙. For this cluster mass, statistical uncertainties in the stellar population of the cluster are expected to be large, especially at the high-mass end. We note here that our analysis does not take into account this statistical uncertainty, and thus the upper mass cutoff estimated above can be somewhat uncertain. Further observations are needed to include the statistical uncertainty in the analysis and will be presented elsewhere.

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The infrared (IR) continuum emission from H ii regions is reprocessed stellar radiation from dust. The bolometric luminosity of H ii regions estimated from the IR emission can thus be used to infer the stellar type. For several H ii regions, the estimated Lyman continuum luminosities, obtained from their radio free–free emission, exceed those expected from their inferred stellar type (Wood & Churchwell 1989; Sánchez-Monge et al. 2013). It has been suggested that this excess ionizing radiation is due to accreting low-mass stars present in the star-forming region (Smith 2014). The "hot" spots on the stellar surface in such accreting stars form the additional source for Lyman continuum emission. We investigate whether the presence of such accreting low-mass stars in our observed sources can contribute to the hydrogen Lyman photons, thus modifying the net γ. The details of the analysis are given in Appendix A.2. We find that the effective temperature of the "hot" spot is ≳35,000 K during a considerable fraction of the accretion phase.⁷ Thus, the contribution from accreting low-mass stars modifies the net γ of the ionizing radiation in such a way that the helium is expected to be ionized (see Figure 11). We conclude that ionizing radiation from accreting low-mass stars cannot be the cause of the low ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratio observed toward the selected sources.

Finally, we investigate whether selective absorption of Lyman photons due to dust in the H ii regions can produce the observed low ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratio, as suggested earlier by Mezger et al. (1974). Appendix A.3 summarizes the analysis. In this process, the dust absorption cross section of helium Lyman photons is larger than that of hydrogen Lyman photons, which changes γ as the photons propagate through the H ii region. This change in γ in turn reduces the size of the helium ionization zone, thus lowering the observed ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratio. To estimate the change in γ, we first express the dust absorption cross sections in terms of the extinction cross section at 13.6 eV provided by the Weingartner & Draine (2001) dust model. The ratio of the helium to hydrogen Lyman photon absorption cross sections, a₀, is then estimated using the observed ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratio and taking the γ due to the stellar cluster to be 0.2 (see Appendix A.3). The results are given in Table 5 for different H ii region filling factors and f_uv. f_uv is a scaling factor that takes into account deviations in the cross section at 13.6 eV from the Weingartner & Draine (2001) value. The estimated a₀ values range from ∼1.8 to 4.4, which are closer to those obtained by Panagia & Smith (1978) but a factor of ∼1.8 smaller than the values obtained by Mezger et al. (1974). In the Weingartner & Draine (2001) dust model, the extinction cross section increases from ∼2 × 10⁻²¹ cm²/H-atom at 13.6 eV to ∼4 × 10⁻²¹ cm²/H-atom at ∼18 eV and then declines below ∼0.2 × 10⁻²¹ cm²/H-atom at 24.6 eV. The extinction and absorption cross sections are related through the dust albedo (see Appendix A.3), which is not known in the relevant photon energy range. The estimated values of a₀ indicate that the absorption cross section needs to be larger by a factor ≥1.8 for photons of energy 24.6 eV compared to that at 13.6 eV. The dust model of Weingartner & Draine (2001) is derived from extinction measurements toward diffuse ISM and consists of silicate and carbonaceous grains with size 4 × 10⁻⁴ μm to ∼0.5 μm. In H ii regions the grains with size ≲10⁻³ μm (mostly PAH molecules) are destroyed in the central region close to the ionizing star (Pavlyuchenkov et al. 2013). The dominant grains are the very small graphite grains, as inferred from the bright 24 μm emission from many H ii regions. Whether a self-consistent modeling of dust emission from H ii regions can provide the above-estimated properties for the absorption cross section of very small graphite grains needs further investigation. Thus, with the current data, we are unable to conclusively establish whether selective absorption is the cause for the low observed ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratio.

Table 5.  Selective Absorption due to Dust

Properties	Value	Note
G10.15-0.34
Path length (pc)	4.4	a
Electron density (cm−3)	135	b
${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$	≤0.033	From Table 4
γ'	0.04c
a0	4.4	ϕ = 0.25d, fuv = 0.5d
	2.4	ϕ = 0.64, fuv = 1.0
	2.9	ϕ = 0.25, fuv = 1.0
	2.2	ϕ = 0.25, fuv = 2.0
G23.46-0.20
Path length (pc)	6.2	a
Electron density (cm−3)	62	b
${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$	≤0.051	From Table 4
γ'	0.06c
a0	4.0	ϕ = 0.25, fuv = 0.5
	2.1	ϕ = 0.64, fuv = 1.0
	2.7	ϕ = 0.25, fuv = 1.0
	2.0	ϕ = 0.25, fuv = 2.0
G29.96-0.02
Path length (pc)	5.2	a
Electron density (cm−3)	90	b
${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$	0.053e	From Table 4
γ'	0.06c
a0	3.4	ϕ = 0.25, fuv = 0.5
	1.9	ϕ = 0.64, fuv = 1.0
	2.3	ϕ = 0.25, fuv = 1.0
	1.8	ϕ = 0.25, fuv = 2.0

Notes.

^aPath length, ${R}_{{{\rm{H}}}^{+}}$, in Equation (11) estimated from the angular size and distance given in Table 1 and assuming a spherical distribution for the ionized gas (Panagia & Walmsley 1978). ^bElectron density, n_e, in Equation (11) estimated using the Lyman continuum luminosities given in Table 1 and ${R}_{{{\rm{H}}}^{+}}.$ ^cγ' is obtained from the listed ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratio as discussed in Appendix A.3. ^dϕ is the H ii region filling factor; f_uv is the ratio of the dust absorption cross section for hydrogen Lyman photons to the extinction cross section at 13.6 eV in the Weingartner & Draine (2001) dust model (see Appendix A.3). ^eMinimum ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratio observed toward G29.96-0.02.

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We observed helium and hydrogen RRLs near 5 GHz toward envelopes of three UCH ii regions—G10.15-0.34, G23.46-0.20, and G29.96-0.02. This data set is used to investigate helium ionization in the UCH ii envelopes. Our main results are as follows:

• 1.  
  Our observations indicated that helium was not uniformly ionized in the UCH ii envelopes. Toward G10.15-0.34 and G23.46-0.20 helium lines were not detected in a few positions, and the upper limits obtained for the ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratio were 0.033 and 0.051, respectively. Our data thus show that helium is not fully ionized in the diffuse regions located at a few parsec from the young massive stars embedded in the observed sources.
• 2.  
  The mean value of the ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratio obtained from the positions nearer to the peak of the radio continuum emission in the observed sources was 0.06 ± 0.02, consistent with the value measured toward compact H ii regions in the Galaxy.
• 3.  
  No He⁺⁺ RRLs were detected toward the observed sources (1σ upper limit is 4 mK), which ruled out the possibility that helium may be doubly ionized in the diffuse regions of the UCH ii envelopes.
• 4.  
  Toward G10.15-0.34 and G23.46-0.20, we investigated the spectrum of the radiation that ionizes the diffuse regions. We considered that the stars of type O6 and earlier were embedded in (locally) ionization-bounded regions and were not contributing to the ionization of the diffuse regions. The observed upper limit on the ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratio then provided an upper mass of ∼30 M_⊙ for the cluster members that ionize the diffuse regions. This upper mass, however, is somewhat uncertain owing to statistical uncertainty in the cluster population.
• 5.  
  Our investigation ruled out accreting low-mass stars (Smith 2014; Cesaroni et al. 2016) as a possible source for additional hydrogen-ionizing photons required to produce a low ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratio in the UCH ii envelopes.
• 6.  
  We also investigated whether selective absorption of Lyman photons by dust was responsible for low helium ionization. We found that the ratio of helium and hydrogen absorption cross section of Lyman photons by dust, a₀, should be in the range of 1.8–4.4 to account for the observed ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ ratio. However, a self-consistent model for dust absorption cross section over a wide wavelength range needs to be developed to conclusively establish the role of selective absorption.

We thank Drs. Kim and Koo for providing their 21 cm images. We acknowledge the critical comments by an anonymous referee that have significantly improved the presentation of the results in the paper.

Facility: Green Bank Telescope - .

Here we summarize the computation of (a) γ, the helium-to-hydrogen photon number ratio, due to a star cluster; (b) γ with emission from low-mass accreting stars in the cluster; and (c) the parameter a₀, which characterizes the selective dust absorption of Lyman photons.

A.1. Expected Helium Ionization due to a Cluster

Computation of hydrogen (q_H) and helium (q_He) Lyman photons from a cluster requires a knowledge of these photon emissions by stars of different mass and the mass function of the cluster. We consider here a modified version of the Muench initial mass function (IMF; Muench et al. 2002) discussed by Murray & Rahman (2010):

$$\begin{eqnarray}\zeta (m) & \equiv & m\ {dN}/{dm}\\ & = & {N}_{0}\ {m}^{-{\rm{\Gamma }}}\qquad \qquad \qquad {m}_{u}\gt m\gt {m}_{1}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&={N}_{0}\ {m}_{1}^{(0.15-{\rm{\Gamma }})}\ {m}^{-0.15}\qquad \qquad \qquad \qquad \ \ \ {m}_{1}\gt m\gt {m}_{2}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&={N}_{0}\ {m}_{1}^{(0.15-{\rm{\Gamma }})}\ {m}_{2}^{(-0.73-0.15)}\ {m}^{0.73}\qquad \qquad \ {m}_{2}\gt m\gt {m}_{l}.\end{eqnarray} \tag{ 5 }$$

We used Γ = 1.35, the slope of the Salpeter IMF, for high-mass stars (Murray & Rahman 2010), m_l = 0.1 M_⊙, m₁ = 0.6 M_⊙ (m₂ = 0.025 M_⊙, which is below m_l). The maximum value for the upper mass cutoff, m_u, is taken as 120 M_⊙. N₀ is fixed through the normalization

$$\begin{eqnarray}&&{\int }_{{m}_{l}}^{{m}_{u}}\zeta (m){dm}/m=1.\end{eqnarray} \tag{ 6 }$$

The Lyman photon emission from stars is taken from the models of Martins et al. (2005). With the above IMF and m_u = 120 M_⊙, we get a mean mass of $\langle m\rangle =0.71$ M_⊙ and mean Lyman continuum photons per s per M_⊙ $\langle q\rangle /\langle m\rangle \,=6.1\times {10}^{46}$ s⁻¹ M_⊙⁻¹. These values are consistent with those obtained by Murray & Rahman (2010). The helium-to-hydrogen Lyman photon number ratio is

$$\begin{eqnarray}&&\gamma \approx \langle {q}_{\mathrm{He}}\rangle /\langle {q}_{{\rm{H}}}\rangle ,\end{eqnarray} \tag{ 7 }$$

where $\langle {q}_{\mathrm{He}}\rangle$ and $\langle {q}_{{\rm{H}}}\rangle$ are the IMF-weighted helium and hydrogen Lyman photon emission per second. In Equation (7), $\langle {q}_{\mathrm{He}}\rangle$ is approximately taken as the Lyman continuum photons with λ < 504 Å (Martins et al. 2005). γ is computed for different m_u in the range 15–120 M_⊙. The result of the computation is plotted in Figure 11.

A.2. The Effect of Accreting Low-mass Stars

Several H ii regions have Lyman continuum emission in excess of what is expected from the star type determined from their IR emission (Wood & Churchwell 1989; Sánchez-Monge et al. 2013). A model that can account for the excess Lyman photon emission in terms of accreting low-mass stars is presented by Smith (2014) (see also Hosokawa et al. 2010). Recent observation of infall tracers toward Lyman excess H ii regions supports the accretion model (Cesaroni et al. 2016). Here we use Figures 2 and 3 of Smith (2014), which gives respectively the stellar radius and accretion luminosity as a function of the mass of the accreting star. We consider the "cold" accretion case as suggested by Smith (2014). The mass–radius and mass–luminosity relationships are given for different accretion rates. To account for the observed Lyman excess in H ii regions, accretion rates in the range of 10⁻⁵ to 10⁻³ M_⊙ yr⁻¹ are required. Calculations here are presented with an accretion rate of 10⁻⁴ M_⊙ yr⁻¹. The excess Lyman emission originates at "hot" spots created on the surface of the star during "cold" accretion. The temperature, T_h, of the hot spot is obtained by assuming that a fraction, f_acc, of the accretion luminosity is converted to thermal energy. Thus,

$$\begin{eqnarray}&&{T}_{h}^{4}=\displaystyle \frac{{f}_{\mathrm{acc}}{L}_{\mathrm{acc}}}{4\pi {R}_{s}^{2}{f}_{h}\sigma },\end{eqnarray} \tag{ 8 }$$

where L_acc is the accretion luminosity, σ is the Stefan-Boltzmann constant, R_s is the radius of the central star, and f_h is the fraction of the surface area covered by the hot spot. Following Smith (2014), we take f_acc = 0.75 and f_h = 0.05. The Lyman continuum photon emission from this hot spot is taken as that due to a star with the same effective temperature in the models of Martins et al. (2005). The fraction of low-mass stars in the cold accretion phase with a hot spot in the cluster is not known; we assume that 1% and 5% of low-mass stars are in such a phase for the computation. We recomputed γ of the radiation from the cluster by considering that 1% and 5% of stars in the mass range 1–6 M_⊙ are in the accretion phase and have hot spots. The result of these computations is shown in Figure 11.

A.3. The Effect of Dust within the H ii Region

We are interested in the effect of dust within the H ii region on the size of the helium ionization region. In Section 5, we presented a physical argument to illustrate that in a dust-free H ii region the size of the ionization zones of helium and hydrogen is determined by the Lyman photon number ratio γ. In the ionization equilibrium equation, the zone sizes are involved to get the total recombination rates of hydrogen and helium, which are volume integrals over the respective ionization regions. The observed helium-to-hydrogen ratio is obtained from the ratio of antenna temperatures (see Equation (1)). The antenna temperatures depend on the line flux densities and hence are proportional to the total recombination rates. Thus,

$$\begin{eqnarray}&&\displaystyle \frac{{n}_{{\mathrm{He}}^{+}}}{{n}_{{{\rm{H}}}^{+}}}=\displaystyle \frac{{\int }_{V({\mathrm{He}}^{+})}{n}_{{\mathrm{He}}^{+}}{n}_{e}{dV}}{{\int }_{V({{\rm{H}}}^{+})}{n}_{{{\rm{H}}}^{+}}{n}_{e}{dV}}\approx {yR},\end{eqnarray} \tag{ 9 }$$

where

$$\begin{eqnarray}&&R\equiv \displaystyle \frac{{\int }_{V({\mathrm{He}}^{+})}{n}_{{{\rm{H}}}^{+}}^{2}{dV}}{{\int }_{V({{\rm{H}}}^{+})}{n}_{{{\rm{H}}}^{+}}^{2}{dV}}\approx \displaystyle \frac{{R}_{{\mathrm{He}}^{+}}^{3}}{{R}_{{{\rm{H}}}^{+}}^{3}},\end{eqnarray} \tag{ 10 }$$

the volume integrals in the numerator and denominator are over the regions, respectively, where helium and hydrogen are ionized, ${R}_{{\mathrm{He}}^{+}}$ and ${R}_{{{\rm{H}}}^{+}}$ are radii of the two zones, if they are spherical in shape, ${n}_{{\mathrm{He}}^{+}}$ is the helium ion density, ${n}_{{{\rm{H}}}^{+}}$ is the proton density, n_e is the electron density, and y is the cosmic abundance of helium. Mathis (1971) derived numerically the relationship between γ and R for y = 0.1 (Figure 3 in Mathis 1971; see also comment by Mezger et al. 1974).

The dust in the H ii region absorbs and scatters Lyman photons. Let σ_H and σ_He be the effective (i.e., weighted by stellar radiation flux density) absorption cross section of hydrogen and helium Lyman photons, respectively, assumed to be constant over the corresponding wavelength range. The absorption optical depths of hydrogen and helium Lyman photons are

$$\begin{eqnarray}{\tau }_{{\rm{H}}} & = & {n}_{{{\rm{H}}}^{+}}{\sigma }_{{\rm{H}}}{x}_{g}{R}_{{{\rm{H}}}^{+}}{\phi }^{1/2}\\ & \approx & 1.8\times {10}^{-21}\displaystyle \frac{{\sigma }_{{\rm{H}}}{x}_{g}}{{\sigma }_{\mathrm{WD}}{x}_{g}/(1-{\rm{\Gamma }})}{n}_{e}{R}_{{{\rm{H}}}^{+}}{\phi }^{1/2}\\ & = & 1.8\times {10}^{-21}{f}_{\mathrm{uv}}{n}_{e}{R}_{{{\rm{H}}}^{+}}{\phi }^{1/2},\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}{\tau }_{\mathrm{He}} & = & {n}_{{{\rm{H}}}^{+}}{\sigma }_{\mathrm{He}}{x}_{g}{R}_{{\mathrm{He}}^{+}}{\phi }^{1/2}\\ & \approx & 1.8\times {10}^{-21}{a}_{0}{f}_{\mathrm{uv}}{n}_{e}{R}_{{{\rm{H}}}^{+}}{\phi }^{1/2},\end{eqnarray} \tag{ 12 }$$

where x_g is the dust-to-gas number density ratio, ϕ is the filling factor, and the parameter ${a}_{0}\equiv {\sigma }_{\mathrm{He}}/{\sigma }_{{\rm{H}}}$. The dust model of Weingartner & Draine (2001), derived for diffuse ISM, provides the extinction cross section, σ_WD, near 13.6 eV as ∼1.8 × 10⁻²¹ cm²/H-atom (their Figure 14, R_V = 3.1 model). In Equations (11) and (12), we expressed the absorption optical depths in terms of the Weingartner & Draine (2001) absorption cross section at 13.6 eV, which is ${\sigma }_{\mathrm{WD}}/(1-{\rm{\Gamma }})$, where Γ is the dust albedo. We assume that Γ is constant over the Lyman continuum. A parameter ${f}_{\mathrm{uv}}\equiv \tfrac{{\sigma }_{{\rm{H}}}{x}_{g}}{{\sigma }_{\mathrm{WD}}{x}_{g}/(1-{\rm{\Gamma }})}$ is defined to take into account deviations from the Weingartner & Draine (2001) dust model. The absorption due to dust changes the Lyman photon rates, which can be expressed as

$$\begin{eqnarray}&&{q}_{{\rm{H}}}^{\prime }={q}_{{\rm{H}}}\exp (-{\tau }_{{\rm{H}}}),\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{q}_{\mathrm{He}}^{\prime }={q}_{\mathrm{He}}\exp (-{\tau }_{\mathrm{He}}).\end{eqnarray} \tag{ 14 }$$

If τ_He is not equal to τ_H, the two photon rates change differently, which results in selective dust absorption proposed by Mezger et al. (1974) (see also Panagia & Smith 1978). Thus, selective absorption modifies the Lyman photon ratio, $\gamma ^{\prime} =\tfrac{{q}_{{\rm{He}}}^{\prime }}{{{\rm{q}}}_{{\rm{H}}}^{\prime }}$, which in turn affects the sizes of the ionization zones of the two atoms (Mathis 1971). γ' is related to the original photon rate γ as (Mezger et al. 1974; Panagia & Smith 1978)

$$\begin{eqnarray}&&\gamma ^{\prime} =\gamma \exp (-({\tau }_{\mathrm{He}}-{\tau }_{{\rm{H}}}))=\gamma \exp (-{\tau }_{{\rm{H}}}({a}_{0}{R}^{1/3}-1)),\end{eqnarray} \tag{ 15 }$$

where the ratio ${R}_{{\mathrm{He}}^{+}}/{R}_{{{\rm{H}}}^{+}}={R}^{1/3}$ for spherically symmetric ionization zones, but is a good approximation for nonuniform H ii regions (Mezger et al. 1974; Panagia & Smith 1978). Note that if the ionization is due to a stellar cluster, then in the above equation γ and hence γ' need to be obtained as given in Equation (7).

Our aim here is to get values for a₀ from the observed ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ values or its upper limit. We use Equation (9) to get R from the observed value ${n}_{{\mathrm{He}}^{+}}/{n}_{{{\rm{H}}}^{+}}$ for y ≈ 0.1. If selective dust absorption is present, then Figure 3 of Mathis (1971) will provide a γ' corresponding to the estimated R. The γ for the cluster radiation is taken as 0.2 (see Figure 11), a reasonable value for the sources we have observed. Equations (15) and (11) can then be used to get a₀ for a given ϕ and f_uv. The radius ${R}_{{{\rm{H}}}^{+}}$ is estimated from the angular size and distance to the sources provided in Table 1, and n_e is obtained using the Lyman continuum luminosity given in Table 1 and ${R}_{{{\rm{H}}}^{+}}$. The estimated a₀ values for a set of ϕ and f_uv are given in Table 5.