WHAT DRIVES THE EXPANSION OF GIANT H ii REGIONS?: A STUDY OF STELLAR FEEDBACK IN 30 DORADUS

30 Doradus is the most massive and largest H ii region in the Local Group. The primary star cluster powering 30 Doradus is NGC 2070, with 2400 OB stars (Parker 1993), an ionizing photon luminosity of S = 4.5 × 10⁵¹ photons s⁻¹ (Walborn 1991) and a bolometric luminosity of 7.8 × 10⁷ L_☉ (Malumuth & Heap 1994). The IMF of NGC 2070 has masses up to 120 M_☉ (Massey & Hunter 1998), and the stellar population may be the result of several epochs of star formation (Walborn & Blades 1997). At the core of NGC 2070 is R136, the densest concentration of very massive stars known, with a central density of 5.5 × 10⁴ M_☉ pc⁻³ (Hunter et al. 1995); R136 hosts at least 39 O3-type stars and at least 10 WR stars in its core (∼2.5 pc in diameter; Massey & Hunter 1998).

To provide context for how 30 Doradus compares to other local H ii regions, Figure 1 plots Hα luminosity versus H ii region radius for ∼22,000 H ii regions in 70 nearby (distances ≲30 Mpc) galaxies (see references in the figure caption). Morphologically, this galaxy sample is comprised of 13 irregulars/dwarf irregulars and 57 spirals. The black star near the top right denotes 30 Doradus. It is the brightest in Hα of the 613 H ii regions in the irregulars by nearly an order of magnitude. Relative to the H ii regions in spirals (including M33), 30 Doradus has a greater Hα luminosity than ∼99% of that sample.

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The nebula that surrounds the central star cluster has a complex morphology across the electromagnetic spectrum. Figure 2 shows a three-color image of 30 Doradus, with the Spitzer Space Telescope 8 μm IRAC band in red, Hα in green, and soft X-rays (0.5–2.0 keV) in blue (details of these data are given in Section 2). Large- and small-scale structures are evident from thin ionized gas and dust filaments of arcsecond widths to cavities a few arcminutes across filled with hot X-ray gas. The warm ionized gas has several shell-like structures, and many of these are expanding with high velocities (∼100–300 km s⁻¹; Chu & Kennicutt 1994), suggesting that past SN explosions have occurred in the region. In addition to a large ionized gas mass (∼8 × 10⁵ M_☉; Kennicutt 1984), the 30 Doradus nebula also has ∼10⁶ M_☉ of CO (Johansson et al. 1998). The CO(1–0) maps of 30 Doradus have revealed 33 molecular cloud complexes in the H ii region, and in particular, two elongated clouds of CO mass ∼4 × 10⁵ M_☉ that form a "ridge" west and north of R136 (see the CO contours in Figure 2). Estimates of the radius $R_{{\rm H\,\mathsc{ii}}}$ of the nebula range from ∼110 pc (Brandl 2005; using a revised value of D = 50 kpc) to ∼185 pc (Kennicutt 1984). The nearly factor of two uncertainty in $R_{{\rm H\,\mathsc{ii}}}$ arises from the complex shape that precludes accurate determination of the radius. In this paper, we assume ${{\rm H\,\mathsc{ii}}}$ = 150 pc.

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The properties of 30 Doradus described above demonstrate why this H ii region is an ideal candidate for assessing the feedback mechanisms of massive stars. The shear number and energetic output of the OB stars facilitate a detailed study of the effects of radiation, winds, SNe, ionization fronts, etc. Additionally, the proximity of 30 Doradus enables a resolved view of the processes and dynamics associated with starburst activity that was common in the early universe (e.g., Meurer et al. 1997; Shapley et al. 2003). Indeed, the relatively instantaneous formation of the concentrated massive stars in R136 makes 30 Doradus a "mini-starburst" (Leitherer 1997).

We compiled optical photometric data on 30 Doradus from three separate observational programs. For the central 35''× 35'' around R136 (with right ascension α= 05^h38^m455 and declination −69°06'027), we utilize the photometric results of Malumuth & Heap (1994) from Hubble Space Telescope (HST) Planetary Camera (PC) observations. These authors identified over 800 stars within this area and obtained a bolometric luminosity L_bol = 7.8 × 10⁷ L_☉ for their sources.

At larger distances from R136 out to a few arcminutes, we employ the UBV photometric data of Parker (1993). The optical images of 30 Doradus from Parker (1993) were obtained at the 0.9 m telescope at Cerro Tololo Inter-American Observatory (CTIO), with a field of view of 26 × 41 and 049 pixel⁻¹. We followed the analyses of Parker & Garmany (1993) to convert their measured apparent UBV magnitudes to absolute bolometric magnitudes.

For the area outside the field of Parker (1993), we use the UBV data of Selman & Melnick (2005). These observations were taken with the Wide Field Imager on the MPG/ESO 2.2 m telescope at La Silla, out to half a degree away from R136 with 0238 pixel⁻¹. Thus, the three data sets combined provide a full coverage of 30 Doradus in the U, B, and V bands.

To illustrate the H ii region structure, we show the Hα emission of 30 Doradus in Figure 2. This narrowband image (at 6563 Å, with 30 Å full-width at half-maximum) was taken with the University of Michigan/CTIO 61 cm Curtis Schmidt Telescope at CTIO as part of the Magellanic Cloud Emission Line Survey (Smith & MCELS Team 1998). The total integration time was 600 s, and the reduced image has a resolution of 2'' pixel⁻¹.

Infrared images of 30 Doradus were obtained through the Spitzer Space Telescope Legacy project Surveying the Agents of Galaxy Evolution (Meixner et al. 2006) of the LMC. The survey covered an area of ∼7 × 7 degrees of the LMC with the Infrared Array Camera (IRAC; Fazio et al. 2004) and the Multiband Imaging Photometer (MIPS; Rieke et al. 2004). Images were taken in all bands of IRAC (3.6, 4.5, 5.8, and 7.9 μm) and of MIPS (24, 70, and 160 μm) at two epochs in 2005. For our analyses, we used the combined mosaics of both epochs with 12 pixel⁻¹ in the 3.6 and 7.9 μm IRAC images and 249 pixel⁻¹ and 48 pixel⁻¹ in the MIPS 24 μm and 70 μm images, respectively.

30 Doradus was observed with the Australian Telescope Compact Array (ATCA) as part of a 4.8 GHz and 8.64 GHz survey of the LMC (Dickel et al. 2005). This program used two array configurations that provided 19 antenna spacings, and these ATCA observations were combined with the Parkes 64 m telescope data of Haynes et al. (1991) to account for extended structure missed by the interferometric observations. For our analyses, we utilized the resulting ATCA+Parkes 8.64 GHz (3.5 cm) image of 30 Doradus, which had a Gaussian beam of FWHM 22'' and an average rms noise level of 0.5 mJy beam⁻¹. We note that higher resolution ATCA observations of 30 Doradus have been taken by Lazendic et al. (2003), but we have opted to use the ATCA+Parkes image of Dickel et al. (2005) as the latter is more sensitive to the low surface-brightness outskirts of 30 Doradus.

30 Doradus was observed using the Chandra Advanced CCD Imaging Spectrometer (ACIS) in 2006 January for ≈94 ks total (ObsIDs 5906 [13 ks], 7263 [43 ks], and 7264 [38 ks]; PI: L. Townsley) in the timed-exposure VFaint mode. The spatial resolution of the Chandra ACIS images is 0492 pixel⁻¹. Data reduction and analysis was performed using the Chandra Interactive Analysis of Observations (CIAO) Version 4.1. We followed the CIAO data preparation thread to reprocess the Level 2 X-ray data and merge the three observations together. Figure 3 shows the resulting soft X-ray band (0.5–2.0 keV) image following these analyses. Seventy-four point sources were identified in the reprocessed images using the CIAO command wavdetect (a source detection algorithm using wavelet analysis; Freeman et al. 2002); we excluded the identified point sources in our spectral analyses.

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To produce a global X-ray spectrum of 30 Doradus, we extracted Chandra spectra using the CIAO command specextract. Background spectra were also produced from a circular region of radius ≈15'' that is ≈2' east of 30 Doradus, and these were subtracted from the source spectra. Additionally, we removed the counts of the 74 point sources identified above. The resulting spectra were modeled simultaneously as an absorbed, variable-abundance plasma in collisional ionization equilibrium (XSPEC model components phabs and vmekal) in XSPEC Version 12.4.0. Figure 4 gives the spectra with the best-fit model (with χ² = 619 with 396 degrees of freedom (dof)) overplotted. We found a best-fit absorbing column density of N_H = 1.5^+0.3_−0.2 × 10²¹ cm⁻² and an X-ray gas temperature of kT_X = 0.64^+0.03_−0.02 keV. The absorption-corrected soft-band (0.5–2.0 keV) luminosity of the diffuse emission is L_X = 4.5 × 10³⁶ erg s⁻¹.

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Previous Chandra X-ray analysis of 30 Doradus was reported by Townsley et al. (2006a, 2006b) for a different set of observations (ObsIDs 22 and 62520) totalling ∼24 ks. By fitting the X-ray spectra of many diffuse regions across 30 Doradus, they found best-fit absorbing columns of N_H = 1–6 × 10²¹ cm⁻², temperatures of kT_X∼ 0.3–0.9 keV, and absorption-corrected luminosities (0.5–2.0 keV) of log  L_X= 34.2–37.0 erg s⁻¹. Thus, our values are fairly consistent with those of Townsley et al.

The light output by stars produces a direct radiation pressure that is associated with the photons' energy and momentum. The resulting radiation pressure P_dir at some position within the H ii region is related to the bolometric luminosity of each star L_bol and the distance r the light traveled to reach that point:

$$\begin{equation} P_{{\rm dir}}= \sum \frac{L_{{\rm bol}}}{4 \pi r^{2} c}, \end{equation} \tag{ 1 }$$

where the summation is over all the stars in the field. In Section 5.2, we describe an alternative definition of radiation pressure and compare the results from each case.

The above relation assumes that the stellar radiation is not attenuated by dust. In Section 3.2, we calculate separately the radiation pressure associated with the light absorbed by dust using Spitzer IR photometry. Given that our results show that P_dir ≫ P_IR generally (see Section 4), the assumption that the emitted L_bol is unattenuated seems reasonable.

In order to obtain the bolometric luminosities of the massive stars in 30 Doradus, we utilize the UBV photometric data described in Section 2.1. To simplify the calculation, we assume that the bolometric luminosity of R136 obtained by Malumuth & Heap (1994) originates from the point in the middle of the central region marked with the red X in Figure 5. For the stars located outside R136 within a few arcminutes, the Parker (1993) catalog only includes the apparent UBV magnitudes and colors. Therefore, we follow the procedure outlined by Parker & Garmany (1993) to obtain absolute bolometric magnitudes of the 1264 stars in the Parker (1993) catalog that are not in the Selman & Melnick (2005) sample. For the 7697 stars in the Selman & Melnick (2005) catalog that lie outside the field of the Parker (1993) data, we use their published values for the stars' absolute bolometric magnitudes.

Thus, in total, we calculate the bolometric luminosities L_bol of the R136 cluster and 8961 other stars in 30 Doradus. For each of the 441 regions, we sum these 8962 terms in Equation (1), where r corresponds to the projected distance from the 8962 stars' positions to the region center. In this manner, we compute the radiation pressure "felt" by the 441 regions from all of the starlight in 30 Doradus.

Since the stars are viewed in projection, the actual distance r to a star from the R136 center is observed as a projected distance ψ. Therefore, we calculate the direct radiation pressure for two scenarios: one case assuming the stars lie in the same plane (P_dir) and another case where we attempt to "deproject" the stars positions (P_dir,3D). Appendix A outlines the procedure we utilize to obtain the deprojected bolometric luminosity of the stars as a function of r and compares P_dir with P_dir,3D. We find that P_dir,3D is 10%–60% less than P_dir at radii ≲20 pc from 30 Doradus, and the fractional difference between P_dir,3D and P_dir at larger radii is much less (0.1%–3.0%). As these differences do not affect the conclusions of this paper, we will only consider P_dir for the rest of our analyses.

The stars' radiation will be processed by the nearby dust in the region, and an associated pressure is exerted by the resulting infrared radiation field, P_IR. This pressure component could become dynamically important if the expanding H ii shell is optically thick to the IR light, effectively trapping the radiation inside the H ii shell (Krumholz & Matzner 2009). The pressure of the dust-processed radiation field P_IR can be determined by the energy density of the radiation field absorbed by the dust, u_ν (i.e., we assume steady state),

$$\begin{equation} P_{{\rm IR}} = \frac{1}{3} u_{\nu }. \end{equation} \tag{ 2 }$$

To find u_ν in each of our 441 regions, we measure their flux densities F_ν in the IRAC and MIPS images and compare them to the predictions of the dust models of Draine & Li (2007; DL07 hereafter). The DL07 models show how the IR emission spectral energy distribution varies depending on the dust content and the intensity of radiation heating the dust. DL07 assume a mixture of carbonaceous grains and amorphous silicate grains that have a size distribution that reproduces the wavelength-dependent extinction in the local Milky Way (see Weingartner & Draine 2001). One component of this dust mixture is polycyclic aromatic hydrocarbons (PAHs), small-sized carbonaceous grains that produce strong emission features at ∼3–19 μm observed in many galaxies.

Since the infrared emission from dust is relatively insensitive to the spectrum of the incident photons with hν< 13.6 eV, DL07 adopts the spectrum of the local-neighborhood ISM. Then, u_ν is given by

$$\begin{equation} u_{\nu } = U u_{\nu }^{{\rm IRSF}}, \end{equation} \tag{ 3 }$$

where U is a dimensionless scale factor and u^IRSF_ν is the energy density of the hν < 13.6 eV photons in the local ISM, 8.65 × 10⁻¹³ erg cm⁻³ (Mathis et al. 1983). We assume that each region is exposed to a single radiation field because the starlight heating the dust comes largely from NGC 2070. In DL07 parameters, this case corresponds to U_min = U_max and γ = 0, where (1 − γ) is the fraction of the dust mass exposed to the radiation.

For our analyses, we measure the average flux densities F_ν at 8, 24, and 70 μm wavelengths for the 441 regions. We do not consider the 160 μm band because its flux density relative to the 70 μm is much higher than that is consistent with the DL07 models. We suspect that the 160 μm flux is from cold dust that is not associated with 30 Doradus, but is in the sight line to the H ii region.

To ensure that we measure the 8 and 24 μm flux densities only from dust and not starlight, we remove the starlight contribution at these wavelengths based on the 3.6 μm flux density (which is almost entirely from starlight):

$$\begin{eqnarray} F_{\nu }^{\rm {ns}}(8 \;\mu {\rm m}) & = & F_{\nu }(8 \;\mu {\rm m}) - 0.232F_{\nu }(3.6 \;\mu {\rm m}) \end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray} F_{\nu }^{\rm {ns}}(24 \;\mu {\rm m}) & = & F_{\nu }(24 \;\mu {\rm m}) - 0.032F_{\nu }(3.6 \;\mu {\rm m}) \end{eqnarray} \tag{ 5 }$$

where the left-hand sides are the non-stellar flux at the respective wavelengths. The coefficients 0.232 and 0.032 are given in Helou et al. (2004).

To account for the different spatial resolutions of the IR images, we convolved the 3.6, 8, and 24 μm images with kernels to match the point-spread function of the 70 μm image. For this analysis, we employed the convolution kernels and method described in Section 2.3 of Gordon et al. (2008).

Figure 6 shows the resulting average IR flux ratios, 〈νF_ν〉^ns₂₄/〈νF_ν〉₇₀ versus 〈νF_ν〉^ns₈/〈νF_ν〉^ns₂₄, of our regions. Overplotted are the DL07 model predictions for given values of q_PAH, the fraction of dust mass in PAHs, and U. Errors in our flux ratios are ≈2.8% from an ≈2% uncertainty in the Spitzer photometry.

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We interpolate the U–q_PAH grid using Delaunay triangulation, a technique appropriate for a non-uniform grid, to find the U and q_PAH values for our regions. Figure 7 plots the interpolated values of U versus q_PAH. Since the points with the smallest 〈νF_ν〉^ns₈/〈νF_ν〉^ns₂₄ values lie to the left of the U–q_PAH grid in Figure 6, we are only able to set upper limits of q_PAH= 0.47% for them (marked with arrows in Figure 7). Thus, these regions produce the "wall" of points at q_PAH= 0.47% in the U versus q_PAH plot.

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We find that the PAH fraction spans roughly an order of magnitude, with values up to q_PAH= 3.76%. U (and thus u_ν) also varies significantly across 30 Doradus, with U≈91–7640, corresponding to u_ν ≈ 6.6 × 10⁻⁹–7.9 × 10⁻¹¹ erg cm⁻³. These ranges vary radially, with the largest U and smallest q_PAH close to R136. One possible explanation for the q_PAH radial dependence is that PAHs are destroyed more where the radiation field heating the dust is strong (e.g., Guhathakurta & Draine 1989). This result is consistent with the analyses of Peeters et al. (2004), who showed that the ratio of PAH to far-IR (dust continuum) emission in Galactic H ii regions is inversely correlated with the intensity of the UV field absorbed by the dust.

We utilize the interpolated U values and Equation (3) to obtain the energy density u_ν, and thus the pressure of the dust-processed radiation field in the 441 regions.

Next, we consider the pressure associated with both the warm H ii gas and the hot X-ray gas. The warm ionized gas pressure is given by the ideal gas law, $P_{{\rm H\, \mathsc{ii}}} = (n_{{\rm e}} + n_{\rm H} + n_{\rm He}) kT_{{\rm H\,\mathsc{ii}}}$, where n_e, n_H, and n_He are the electron, hydrogen, and helium number densities, respectively, and $T_{{\rm H\,\mathsc{ii}}}$ is the temperature of the H ii gas, which we assume to be the same for electrons and ions. If helium is singly ionized, then n_e + n_H + n_He ≈ 2n_e. The temperature of the H ii gas in 30 Doradus is fairly homogeneous, with $T_{{\rm H\, \mathsc{ii}}} = 10270\pm 140$ K, based on the measurement of [O iii] (λ4959 + λ5007)/λ4363 across 135 positions in the nebula (Krabbe & Copetti 2002); here, we adopt $T_{{\rm H\,\mathsc{ii}}}$ = 10⁴ K. Since $T_{{\rm H\, \mathsc{ii}}}$ is so uniform, the warm gas pressure is determined by the electron number density n_e. We estimate n_e from the average flux density F_ν of the free–free radio emission in each region (Equation (5.14b), Rybicki & Lightman 1979):

$$\begin{equation} n_{\rm e} = \bigg (\frac{6.8 \times 10^{38} 4 \pi D^2 F_{\nu } T_{{\rm H\,\mathsc{ii}}}^{1/2}}{\bar{g}_{\rm ff} V} \bigg)^{1/2}, \end{equation} \tag{ 6 }$$

where we have set the Gaunt factor $\bar{g}_{\rm ff} = 1.2$. In the above relation, D is the distance to 30 Doradus (assumed to be D = 50 kpc) and V is the integrated volume of our regions. For V, we assume a radius of the H ii region R = 150 pc, and we calculate the volume by multiplying the area of our region squares by the path length through the sphere at the region's position. We measure F_ν of our regions in the 3.5 cm ATCA+Parkes image, since bremsstrahlung dominates at that wavelength. Figure 8 shows the resulting map of n_e from these analyses. We find that the central few arcminute area of 30 Doradus has elevated electron densities, with values n_e ≈ 200–500 cm⁻³; the location of these large electron densities corresponds to the two molecular clouds that form the "ridge" in the center of the nebula (Johansson et al. 1998). The area outside the central n_e enhancement has relatively uniform electron density, with n_e≈ 100–200 cm⁻³. In the southwest of 30 Doradus where the supernova remnant (SNR) N157B is located, we obtain elevated 3.5 cm flux densities, possibly because of a non-thermal contribution from that source. Therefore, the actual n_e may be lower than the values we find in that region.

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Our warm gas electron densities are similar to the values obtained by Indebetouw et al. (2009) using Spitzer's Infrared Spectrograph. These authors used the ratio [S iii] λ18.7 μm/ [S iii] λ33.4 μm to map n_e across the central ≈2' of 30 Doradus. They also find enhancements in n_e along the "ridge."

The hot X-ray gas arises from shock heating by stellar winds and SNe, and the associated hot gas pressure is given by the relation P_X = 1.9n_X kT_X, where n_X and T_X are the electron number density and temperature of the X-ray gas, respectively. The factor of 1.9 arises from the assumption that He is fully ionized and that the He mass fraction is 0.3. As in our warm ionized gas calculation, we assume that the electrons and ions have reached equipartition, so a single temperature describes both. Since the hot gas can exist over a range of T_X, we measure both n_X and T_X by modeling the bremsstrahlung spectrum at X-ray wavelengths.

From the three Chandra observations, we extracted Chandra X-ray spectra from each region using the CIAO command specextract. Background spectra were also produced from a circular region of radius ≈15'' that is ≈2' east of 30 Doradus (the cyan circle in Figure 3), and these were subtracted from the source spectra. The resulting spectra were fit using XSPEC Version 12.4.0. Data were grouped such that a minimum of five counts were in each energy bin, and the spectra from the three ACIS observations of a given region were fit simultaneously to improve statistics (i.e., they were fit jointly, with more weight given to the data from the longer integrations). Around the edges of the H ii region, the X-ray signal is weaker, so we combined adjacent regions to achieve sufficient counts for an accurate fit.

Spectra were modeled as an absorbed hot diffuse gas in collisional ionization equilibrium using the XSPEC components phabs and mekal (Mewe et al. 1985, 1986; Liedahl et al. 1995). In this fit, we assumed a metallicity Z ∼ 0.5 Z_☉, the value measured in H ii regions in the LMC (Kurt & Dufour 1998). For regions in the southwest of 30 Doradus with strong emission from the SNR N157B, we added a power-law component to account for the non-thermal emission from the SNR. We obtained good fits statistically, with reduced chi-squared values of 0.80–1.30 with 60–300 dof. If a region's fit had reduced chi-squared values outside this range or less than 60 dof, we combined its spectra with those of adjacent regions to increase the signal. The latter criterion was selected since we found generally that the shape of the bremsstrahlung continuum was not discernable with less than 60 dof.

From our fits, we can estimate the electron number density n_X of each region based on the emission measure EM of our models. Emission measure is defined as EM =∫n²_XdV. Thus,

$$\begin{equation} n_{{\rm X}} = \bigg (\frac{\rm{EM}}{V} \bigg)^{1/2} \end{equation} \tag{ 7 }$$

where V is the integrated volume of our region (the same as used in Equation (6)). Since we are interested in the contribution of the X-ray pressure to the global dynamics, we have divided the $\rm{EM}$ by the integrated volume of a region V in calculating n_X, rather than the volume occupied by the hot gas. In the former case, the density n_X goes as the filling factor f^−1/2; in the latter scenario, n_X ∝ f · f^−1/2 = f^1/2. If the filling factor of the hot gas is small, the thermal pressure of the bubbles may be high internally; however, the hot gas would be insignificant dynamically because it occupies a negligible volume and thus exerts little pressure on the material that bounds the H ii region. Therefore, for our purposes of assessing the dynamical role of the hot gas, it is appropriate to use the integrated volume in calculating n_X.

Figure 9 shows a map of the best-fit temperatures kT_X from the spectral modeling analyses. The X-ray gas temperatures are elevated in several areas, including in the southwest (the bottom right of Figure 9), where the SNR N157B is located, and at the center near R136. Figure 10 gives the map of the hot gas electron density across 30 Doradus from our fits. We find a mean 〈n_X〉 = 0.12 cm⁻³. The hot gas electron density is much less than that of the warm gas since many fewer electrons are heated to X-ray emitting temperatures (∼10⁷ K) than to the moderate temperatures ∼10⁴ K of the warm gas.

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As mentioned previously, the X-ray emission in 30 Doradus arises from the shock heating of gas to temperatures of ∼10⁷ K by stellar winds and SNe. These feedback processes eventually carve out large cavities, called bubbles and superbubbles, filled with diffuse X-ray emission. In Figure 11, we demonstrated that the pressure associated with the hot gas P_X is comparatively low relative to the other pressure components. Here, we explore the implications of this result in regard to the trapping/leakage of the hot gas. For this discussion, we will consider stellar winds only and ignore the contribution by SNe; this assumption is reasonable given that the mechanical energy of one SN is on the order of the amount injected by winds over a single massive star's lifetime (Castor et al. 1975). This assumption is valid at the 0.5 Z_☉ of the LMC: simulations of a 5.5 × 10⁴ M_☉ star cluster in Starburst99 (Leitherer et al. 1999) showed that the total wind luminosity decreased by roughly a factor of two from the solar to half-solar metallicity case.

There are several competing theoretical models to account for the X-ray luminosity in bubbles and superbubbles. The models of Castor et al. (1975) and Weaver et al. (1977) assume that the shock-heated gas is completely confined by a cool shell expanding into a uniform density ISM. An alternative theory proposed by Chevalier & Clegg (1985) ignores the surrounding ISM and employs a steady-state, free-flowing wind. Recently, Harper-Clark & Murray (2009) introduced an intermediate model between these two, whereby the ambient ISM is non-uniform. In this case, only some of the hot gas can escape freely through the holes in the shell.

The fraction of hot gas confined by the shell directly determines the hot gas pressure on the shell as well as the X-ray luminosity within the bubble. If the shell is very porous, the shock-heated gas will escape easily, the wind energy will be lost from the bubble, and the associated pressure and luminosity will be low. By comparison, a more uniform shell will trap the hot gas, retain the wind energy within the bubble, and the corresponding X-ray pressure and luminosity will be much greater. As such, in the latter case, the shocked winds could have a significant role in the dynamics of the H ii region. We note that the warm gas is not able to leak similarly because its sound speed is less than the velocities of the shells (20–200 km⁻¹; Chu & Kennicutt 1994).

To assess whether the hot gas is trapped inside the shell and is dynamically important, we measure the ratio of the hot gas pressure to the direct radiation pressure, f_trap,X = P_X/P_dir, and compare it to what f_trap,X would be if all the wind energy was confined. We can calculate the trapped-wind value using the wind-luminosity relation (Kudritzki et al. 1999; Repolust et al. 2004), which indicates that the momentum flux carried by winds from a star cluster is about half that carried by the radiation field if the cluster samples the entire IMF. Written quantitatively, $0.5\; L_{\rm bol}/c = \dot{M_{\rm w}} v_{\rm w}$, where $\dot{M_{\rm w}}$ is the mass flux from the winds that launched at a velocity v_w. The mechanical energy loss L_w of the winds is then given by

$$\begin{equation} L_{\rm w} = \frac{1}{2} \dot{M_{\rm w}} v_{\rm w}^{2} = \frac{L_{\rm bol}^2}{8 \dot{M_{\rm w}} c^2}, \end{equation} \tag{ 8 }$$

and the mechanical energy of the winds is simply E_w = L_wt, where t is the time since the winds were launched. Putting these relations together, the trapped X-ray gas pressure P_X,T is

$$\begin{equation} P_{\rm X,T} = \frac{2 E_{\rm w}}{3 V_{{\rm H\,\mathsc{ii}}}} = \frac{L_{\rm bol}^2 t}{16 \pi \dot{M_{\rm w}} c^2 R_{{\rm H\,\mathsc{ii}}}^3}, \end{equation} \tag{ 9 }$$

where $V_{{\rm H\,\mathsc{ii}}}$ is the volume of the H ii region.

Given that $P_{\rm dir} = L_{\rm bol}/(4 \pi R_{{\rm H\,\mathsc{ii}}}^2 c)$, then f_trap,X is

$$\begin{equation} f_{\rm trap,X} = \frac{L_{\rm bol} t}{4 \dot{M_{\rm w}} c R_{{\rm H\,\mathsc{ii}}}} = \frac{L_{\rm bol}}{4 \dot{M_{\rm w}} c v_{\rm sh}}, \end{equation} \tag{ 10 }$$

where we have set $R_{{\rm H\,\mathsc{ii}}}/t = v_{\rm sh}$, the velocity of the expanding shell. Finally, we put $\dot{M_{\rm w}}$ in terms of L_bol and v_w, so that Equation (10) reduces to

$$\begin{equation} f_{\rm trap,X} = \frac{v_{\rm w}}{2 v_{\rm sh}}. \end{equation} \tag{ 11 }$$

We use the above equation to obtain an order-of-magnitude estimate of f_trap,X if all the wind energy is confined by the shell. We assume a wind velocity v_w ∼ 1000 km s⁻¹ (the escape velocity from an O6 V star; a reasonable order-of-magnitude estimate, since O3 stellar winds are faster and WR winds would be slower than this value). If we set v_sh∼ 25 km s⁻¹ (the expansion velocity over 30 Doradus given by optical spectroscopy; Chu & Kennicutt 1994), then f_trap,X∼ 20.

We can compare this f_trap,X to our observed values for the regions closest to the shell (the ones along the rim of our 441 squares in Figure 5); Figure 13 shows the histogram of our observed f_trap,X values. We find a mean and median f_trap,X of 0.30 and 0.27, respectively, for our outermost regions. Over 30 Doradus, the highest values of f_trap,X are near the SNR N157B in the southwest corner of 30 Doradus (see Figure 14), where the hot gas is being generated and has not had time to vent. Other locations where f_trap,X is elevated are regions with strong X-ray emission and weak Hα emission. Morphologically, these areas could be where the hot gas is blowing out the 30 Doradus shell.

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The observed f_trap,X values are 1–2 orders of magnitude below what they would be if the wind was fully confined. As a consequence, we find that P_X of our regions is too low to be completely trapped in the H ii region (the Castor et al. model), and the X-ray gas must be leaking through pores in the shell. This result is consistent with the Harper-Clark & Murray model of partial confinement of the hot gas, and the weakness of P_X relative to P_dir suggests that the hot gas does not play a significant role in the dynamics of the H ii region.

We note here that our rim regions in this analysis are ∼70–130 pc from R136, which is less than the estimated radius of $R_{{\rm H\,\mathsc{ii}}} =$110–185 pc. Therefore, our f_trap,X values are lower limits, and the true f_trap,X at the shell may be greater by a factor of a few. Nonetheless, the conclusions would remain the same.

An alternative explanation for the weak X-ray luminosity is that the hot gas mixes with the cool gas, and the hot gas temperature is lowered enough so that the gas can cool efficiently. In that case, the energy is still lost from the system, via radiative cooling instead of the escape of the X-ray emitting material. Far ultraviolet spectroscopy is necessary to determine the level of mixing between the gas components.

In this paper, we have defined the radiation pressure as related to the energy and momentum flux of the light radiated by the stars in 30 Doradus. Alternatively, the radiation pressure could be characterized as the force per unit area exerted by the radiation on matter. The two cases produce divergent results regarding the radial dependence of P_dir. In particular, the former case has large P_dir close to the star cluster and a decline in P_dir with distance from the center. By contrast, the latter predicts small P_dir in the H ii-region interior, and P_dir becomes significant near the neutral shell where the radiation will be absorbed (see Appendix B).

Each definition of radiation pressure reveals distinct information about an H ii region. When considering the global dynamics of expansion of an H ii region, it is necessary to characterize P_dir as the energy density of the radiation field, since that definition reflects the total energy and momentum budget available to drive motion. Alternatively, measurement of the force exerted by radiation on matter facilitates a local estimate of the internal density distribution of an H ii region. As we are interested principally on the dynamical role of radiation pressure, we have adopted the former definition of P_dir in this paper.

In Section 4, we found that the direct radiation pressure P_dir dominates over the ionized gas pressure $P_{{\rm H\,\mathsc{ii}}}$ at radii ≲75 pc, implying that the radiation has played a role in the dynamics in 30 Doradus. Significant radiation pressure alters the properties of an H ii region (e.g., the density profile: Draine 2010) and causes the expansion to proceed differently than in a classical H ii region with ionized gas-driven expansion. In particular, Krumholz & Matzner (2009) demonstrated that radiation-driven expansion imparts more momentum to the shell, accelerating the expansion at early times relative to that of gas-driven expansion. Indeed, an additional force must have dominated at early times in 30 Doradus since the shell velocity v_sh∼ 25 km s⁻¹ (Chu & Kennicutt 1994) is too fast to have been gas-driven alone because the H ii gas sound speed is c_s≈ 10 km s⁻¹.

To determine the characteristic radius r_ch where the H ii region shell transitioned from radiation pressure driven to gas pressure driven, we can set these pressure terms at the shell equal and solve for r_ch. Broadly, the pressures at the shell have different dependences with the shell radius $r_{{\rm H\,\mathsc{ii}}}$: $P_{\rm dir} \propto r_{{\rm H\,\mathsc{ii}}}^{-2}$ and $P_{{\rm H\,\mathsc{ii}}} \propto r_{{\rm H\,\mathsc{ii}}}^{-3/2}$. Setting them equal and solving for r_ch (Equation (4) in Krumholz & Matzner 2009, the "embedded case"), we find

$$\begin{equation} r_{\rm ch} = \frac{\alpha _{\rm B}}{12 \pi \phi } \bigg (\frac{\epsilon _{\rm 0}}{2.2 k_{\rm B} T_{{\rm H\,\mathsc{ii}}}} \bigg)^2 f_{\rm trap,tot}^{2} \frac{\psi ^2 S}{c^{2}}, \end{equation} \tag{ 12 }$$

where α_B is the case-B recombination coefficient and ₀ = 13.6 eV, the photon energy necessary to ionize hydrogen. The dimensionless quantity ϕ accounts for dust absorption of ionizing photons and for free electrons from elements besides hydrogen; ϕ = 0.73 if He is singly ionized and 27% of photons are absorbed by dust (typical for a gas-pressure-dominated H ii region; McKee & Williams 1997). The f_trap,tot represents the factor by which radiation pressure is enhanced by trapping energy in the shell through several mechanisms; the trapped hot gas f_trap,X calculated in Section 5.1 is one component that can contribute to f_trap,tot (as discussed in Section 5.1). Here, we adopt f_trap,tot = 2, as in Krumholz & Matzner (2009). Finally, ψ is the ratio of the bolometric power to the ionizing power in a cluster; we set ψ = 3.2 using the 〈S〉/〈M_*〉 and the 〈L〉/〈M_*〉 relations of Murray & Rahman (2010).

Putting all these terms together, we find r_ch≈ 33 pc. Physically, this result means that early in the expansion before it reached a radius of 33 pc, 30 Doradus dynamics could have been radiation pressure dominated, and it has since become gas pressure dominated. Alternatively, it is possible that the hot gas pressure dominated at early times and has become weaker as the H ii region expands.

The radiation-driven or hot gas-driven expansion at early times in 30 Doradus would have facilitated the expulsion of gas from the central star cluster. In particular, since the warm gas sound speed (∼10 km s⁻¹) is less than the escape velocity of R136 (∼20 km s⁻¹, given a mass M = 5.5 × 10⁴ M_☉ in a radius R = 1 pc; Hunter et al. 1995), an alternative mechanism is necessary to remove the gas and regulate star formation (e.g., Krumholz & Matzner 2009; Fall et al. 2010). We conclude that the radiation pressure or hot gas pressure likely played this role in 30 Doradus, decreasing the available mass to make new stars and slowing star formation in the region.