STAR FORMATION IN MASSIVE CLUSTERS VIA THE WILKINSON MICROWAVE ANISOTROPY PROBE AND THE SPITZER GLIMPSE SURVEY

The star formation rate (SFR) of the Milky Way Galaxy is a fundamental parameter in models of the interstellar medium (ISM) and of Galaxy evolution. The rates at which energy and momentum are supplied by massive stars, which are proportional to the SFR, are the dominant elements driving the evolution of the ISM. The hot gas (∼10⁶ K) component of the ISM is contributed almost exclusively, in the form of shocked stellar winds and supernovae, by massive stars, whose numbers are also proportional to the SFR. Finally, the amount of gas in the ISM is reduced by star formation, as the latter locks up material in stars and eventually in stellar remnants. Since the SFR is of order a solar mass per year, and the gas mass is roughly 10⁹ M_☉, either the gas will be depleted in 10⁹ yr, or it will be replaced by stellar evolution (asymptotic giant branch (AGB) stars), from satellite galaxies, or from the halo surrounding the Milky Way.

Estimates of the SFR generally rely on measuring quantities sensitive to the numbers of massive stars, including recombination line emission (H_α, [N ii]), far-infrared emission from dust (heated primarily by massive stars), and radio free–free emission. Mezger (1978) and Gusten & Mezger (1982) showed that the latter is dominated not by classical radio giant H ii regions, but rather by what Mezger called "extended low density (ELD)" H ii emission. In fact, only ∼10%–20% of the free–free emission comes from classical H ii regions—the bulk comes from the ELD. Free–free emission from H ii regions or the ELD is powered by the absorption of ionizing radiation (photons with energies beyond the Lyman edge, i.e., greater than 13.6  eV). Thus the free–free emission is often characterized by the rate Q, the number of ionizing photons per second needed to power the emission (the conversion from free–free luminosity L_ν to Q is given by Equation (7)). Previous measurements of Q are given in Table 1, along with the value determined in this work. The average of the previous values is Q = 3.2 × 10⁵³  s⁻¹.

The ionizing flux can be estimated from recombination lines as well. Bennett et al. (1994) use observations of the [N ii] 205 μm line and find Q = 3.5 × 10⁵³  s⁻¹; McKee & Williams (1997) use the same observations to estimate Q = 2.6 × 10⁵³  s⁻¹.

The nature of the ELD is uncertain; it may be associated with H ii regions, in which case it is also referred to as extended H ii envelopes (Lockman 1976; Anantharamaiah 1985a, 1985b). The latter author lists the properties of the ELD, based on the emission seen in the H272α line; for Galactic longitudes l < 40° the line is seen in every direction (in the Galactic plane) irrespective of whether there was an H ii region, a supernova remnant, or no point source. The electron densities are in the range 0.5  cm⁻³ < n < 6  cm⁻³; emission measures were in the range 500–3000  pc  cm⁻⁶, with corresponding path lengths 50–200 pc; the filling factor is ∼0.005, and the velocities of the H ii regions, when present, agree well with that of the H272α line velocity.

We note that Taylor & Cordes (1993) model the free electron distribution of the inner Galaxy with two components, one with a mean electron density 〈n_e〉 = 0.1  cm⁻³ and a scale height of 150 pc, and the second, associated with spiral arms, having 〈n_e〉 = 0.08  cm⁻³ and a scale height of 300 pc; both components are reminiscent of the ELD.

We present evidence that the bulk of the ELD is associated with photons emitted from massive clusters not previously identified. We are motivated by the distribution of free–free emission in the Wilkinson Microwave Anisotropy Probe (WMAP) free–free map, shown in Figure 1, and by comparison of higher resolution radio images, e.g., Whiteoak et al. (1994); Cohen & Green (2001) with GLIMPSE (Benjamin et al. 2003; Churchwell et al. 2006, 2007) and the Midcourse Space Experiment (MSX; Price et al. 2001) data. Figure 2 shows the WMAP sources in more detail, while Figure 3 shows the locations of the sources in the plane of the Galaxy.

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In this paper, we determine the SFR in our Galaxy using the free–free flux measured by the WMAP. We describe our data processing and source identification and extraction methods in Section 2. By comparing to catalogs of H ii regions with known distance, we estimate the distance to the WMAP sources in Section 3. The H ii catalogs are known to be biased against H ii regions at large distances; we follow Mezger (1978) and Smith et al. (1978) and crudely account for this by calculating the luminosity of the nearest half of the Galaxy, and then doubling the result to find the total L_ν. In Section 4, we examine GLIMPSE images to solidify our identifications; in this process, we identify 5–75  pc bubbles associated with the bulk (>75%) of the emission. We show that the bubbles and the free–free emission are both powered by massive central star clusters. We derive the ionizing flux Q and the SFR of the Galaxy in Section 5. Half the star formation occurs in the 18 most massive clusters and their retinue; the central clusters have M_* = 4–10 × 10⁴ M_☉. We discuss our results in Section 6. In the Appendix, we describe the machinery needed to convert from ionizing flux Q to SFR $\dot{M}_*$.

The only wavelength range in which free–free dominates the emission from the Galactic plane is in the microwave, between 10 and 100 GHz, placing this in the center of the frequency range of cosmic microwave background (CMB) experiments (Dickinson et al. 2003). Synchrotron radiation and vibrational dust emission are also important contributors in this frequency range. The free–free emission is characterized by a spectral index β, where the antenna temperature T∝ν^−β, and β ≈ 2.1. In contrast, the spectral index for synchrotron radiation is β ≈ 2.7–3.2 and for dust emission β ≈ 1.5–2. In order to isolate the free–free component, some form of the multi-wavelength fitting technique must be used.

In order to optimize the WMAP measurements of cosmological parameters, the galactic foreground emission had to be accurately characterized. This was done using a maximum entropy model (MEM), resulting in maps of the free–free, synchrotron, and dust emission (Bennett et al. 2003a).

These models agree with the observed galactic emission to within 1% overall, with the individual synchrotron and dust emission models matching observations to a few percent. In the case of the free–free map, the correlation to the Hα map is found to be within 12%. This indicates that the MEM process is consistent with Hα where the optical depth is less than 0.5 (Bennett et al. 2003a).

The WMAP free–free model is the only single dish all-sky survey of free–free Galactic emission to date, so it is an attractive database to use to measure the Galactic ionizing photon luminosity and subsequently the Galactic SFR.

2.1. Data Processing

We transformed the WMAP free–free maps from an all-sky HEALPix map to multiple tangential projections centered about the galactic plane. The antenna temperature was converted into flux density using the conversion

$$\begin{equation} F_\nu = \frac{2 k_B \nu ^2}{c^2} \Delta T_A, \end{equation} \tag{ 1 }$$

where ν is the frequency of the WMAP band, k_B is Boltzmann's constant, c is the speed of light, and ΔT_A is the antenna temperature (Bennett et al. 2003b). To determine all-sky flux statistics, an all-sky Cartesian projection of the free–free maps was produced.

The WMAP beam diameter varies from 082 to 021 from the K band to the W band. As part of the map making process, all bands were smoothed to a resolution of 1° (Bennett et al. 2003a). The characteristic size of most H ii regions is of order the smoothed resolution of the foreground maps. Thus we suffer from source confusion from regions with small angular separations. We discuss our method of separating the confused sources in Section 2.2, but argue that in many cases, spatially separate H ii regions are physically associated.

2.2. Source Identification and Extraction

As a first pass at distance determination, we use the source list of Russeil (2003), who lists both giant Molecular Clouds and H ii regions; only the latter are relevant here. In cases where the sources have both a kinematic distance and photometric distance, we use the photometric distance.

Table 3 in Russeil (2003) lists 481 H ii regions; we find 88 sources, with a much higher total flux. It follows that we have likely confused individual sources in comparison to the Russeil (2003) list. Thus, we have initially assumed that each of the 88 sources that we have extracted consists of one or more Russeil sources projected onto the same location in the sky. We use the following procedure to separate these confused sources.

First, in 13 cases we have a source where Russeil has none. In these cases we inspect either MSX or GLIMPSE images to identify likely sources, and use SIMBAD to find any H ii regions at promising locations. For example, we find a source at l = 638, b = +23°, with a flux of 246.5  Jy, having no counterpart in Russeil (2003). We identify this source with the ζ Ophiuchi diffuse cloud, at a distance of 140 pc (Draine 1986), and find Q_ff = 7.4 × 10⁴⁷  s⁻¹ from the free–free emission; we use a subscript to denote the origin of the estimated luminosity (the conversion from L_ν = 4πD²f_ν to Q is given in Equation (7)). This ionizing photon luminosity is reasonably consistent with the estimated stellar rate Q_* = 1.2 × 10⁴⁸  s⁻¹ (Martins et al. 2005), and suggests that ∼35% of the ionizing photons are absorbed by dust grains.

The most outstanding example of a WMAP source with no associated H ii region in Russeil (2003) is that at l = 811, b = 05. This source was, however, mapped by Westerhout (1958), who identified it as part of the Cygnus X region. Examination of the MSX image shows that there are two large bubbles in the region, one centered roughly on Cygnus OB2, and one on Cygnus OB9.

We identify the WMAP source at l = 81°, b = 05 with the southwestern wall of a large bubble in the Cygnus region (see Figure 4). The bubble contains Cyg OB2 (see also Schneider et al. 2006). A second bubble lies further to the southwest, and appears to contain Cyg OB9. The boundary between the two bubbles is a shared wall, which contains Russeil (2003) source 118 at l = 785 b = 00. Her sources 120 and 121 are in the interior of the northern bubble, near the center of Cyg OB2. The southeastern rim of the southern bubble contains Russeil's source 115.

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We assign a distance D = 1.7 kpc (Hanson 2003) to both bubbles (and to the WMAP sources at l = 760, l = 786, and l = 811). We assign the flux from the WMAP source at l = 760 to the southern bubble, and that of the source at l = 811 to the northern bubble. The free–free flux from the wall separating the two bubbles could be powered by photons emitted by clusters inside either bubble. Lacking any further information, we assume that half the ionizing flux comes from clusters located in each of the two bubbles. Split this way, Q_ff = 1.75 × 10⁵¹  s⁻¹ for the northern bubble, and Q_ff = 1.04 × 10⁵¹  s⁻¹ for the southern bubble. We find a total free–free flux in the region of 4033 Jy; Westerhout (1958) finds a total flux of 2520 Jy in "point sources" in the region.

We argue that the free–free flux from the vicinity of the northern bubble can easily be powered by Cyg OB2. Counting only the O stars with spectroscopically determined types listed in Table 5 of Hanson (2003) yields 49 O stars with Q ≈ 5 × 10⁵⁰  s⁻¹. More recently, Negueruela et al. (2008) find 50 O stars, and suggest that there may be as many as 60–70 in the cluster, allowing for some incompleteness due to the strong reddening. This is equal to the number of O stars in the Carina region as tabulated by Smith (2006), who also gives Q_* = 10⁵¹  s⁻¹, which we also adopt for Cyg OB2; the total ionizing flux for the region will be somewhat larger, as there are a number of O and Wolf–Rayet stars with projected locations inside the bubble but outside Cyg OB2.

We suggest that there must be a similar number of O stars in the interior of the southern bubble as well. These stars are difficult to detect since the extinction toward them is large (see the discussion at the end of Section 4).

Returning to the distance determinations, if there is a unique Russeil (2003) source at the location of a WMAP source, we use his distance as a first guess; there are 43 such objects, about half the sample. As in the previous case, we then inspect either MSX or GLIMPSE images at the location of the Russeil source. In some cases, we find sources we believe to be better candidates than the source in the Russeil catalog.

Finally, in 30 cases, we find multiple Russeil (2003) objects in the same direction as our WMAP source. We then assign a portion of our measured flux to each of the Russeil objects. We divide up the WMAP flux using the excitation parameter of each Russeil object. The excitation parameter, U ∝ f_νD², compares the ionizing luminosities of the Russeil objects. Using the distances provided by the catalog, we calculate the free–free luminosity of each Russeil object. The result is a separation of the confused WMAP source into individual H ii regions with flux, distance, and luminosity corresponding to the Russeil (2003) objects.

Using this method, we are able to assign distances to all but 2 of the 88 regions. (One of the original 13 missing regions corresponds to the Large Magellanic Cloud (LMC); we identified 10 using SIMBAD, and their distances are given in Table 2). We assigned the average distance of the known sources to the remaining two unidentified sources.

We list 183 H ii regions in Table 2. For all confused sources, the galactic coordinates, semimajor and semiminor axis sizes are for the WMAP source, not the individual H ii regions. Maps of these regions are presented in Figures 1–3. The distribution of free–free luminosities, dN/dL, of these regions is presented in Figure 5, and will be discussed in Section 4.

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3.1. WMAP Sources, the ELD, and Dispersion Measures

The WMAP free–free sources range in radius (or semimajor axis) from 04 to 10°. The latter is the fitted radius for the nearby H ii region S264 (around λ Orionis) at l = 19505, b = −11995, and D = 400  pc (Fich & Blitz 1984). A visual inspection yields a radius ∼5° or 35 pc, closer to the radius r = 45.5 pc given by Fich & Blitz. As noted above, the effective beam diameter for the free–free map is ∼1°. Six sources have mean angular radii (the geometric mean of the semimajor and semiminor axes) smaller than the effective beam radius; these are likely to be unresolved. The physical radii range from ∼6 pc for ζ Ophiuchi to ∼150 pc, with a mean radius 〈r〉 ∼ 55  pc. We find a filling factor for the WMAP sources (not the diffuse emission) of $\phi _{\rm H\,\mathsc {ii}}\approx 5\times 10^{-3}$, where $\phi _{\rm H\,\mathsc {ii}}$ is the ratio of the (summed) free–free source volume divided by the volume of the galactic disk assuming disk radius R = 8  kpc and scale height H = 200  pc appropriate for the disk between 3–8  kpc, e.g., Binney & Merrifield (1998). Note that the filling factor of the WMAP sources is equal to that of the ELD.

The ionizing luminosities fall in the range 10⁴⁸  s⁻¹ < Q < 1.8 × 10⁵²  s⁻¹, with 〈Q〉 = 2 × 10⁵¹. The median Q = 2.8 × 10⁵⁰  s⁻¹ (all these values are uncorrected for dust absorption).

We can determine the mean electron density for each source from the expression

$$\begin{equation} n_e=\sqrt{3Q\over 4\pi r^3 \alpha (\mbox{H}^+)\phi }, \end{equation} \tag{ 2 }$$

where α(H⁺) = 3.57 × 10⁻¹³  cm³  s⁻¹ is the hydrogen recombination coefficient (Osterbrock 1989) and ϕ is the filling factor of ionized gas in a given WMAP source region. The electron density ranges from n_e ≈ 1ϕ^−1/2  cm⁻³ to n_e ≈ 35ϕ^−1/2  cm⁻³, with a mean n_e = 9ϕ^−1/2  cm⁻³. The density averaged over the disk (i.e., multiplying by the volume filling factor $\phi _{{\rm H\,\mathsc {ii}}}$) is 〈n_e〉 ≈ 0.05  cm⁻³. The typical dispersion measure through a WMAP source is DM ≈ 500  cm⁻³ pc.

The mean mass of ionized gas in a WMAP source is 3 × 10⁵ϕ^1/2 M_☉; the largest sources, with Q ≈ 5 × 10⁵¹  s⁻¹, have an ionized gas mass of ∼10⁶ϕ^1/2 M_☉.

The density-weighted scale height of the WMAP sources is $H_{{\rm H\,\mathsc {ii}}}=145\,\,{\rm pc }$.

Recall that the Taylor & Cordes (1993) model for the inner Galaxy had two components, with scale heights of 150 pc and 300 pc, similar to the scale height we find for WMAP H ii sources. The mean density of the WMAP sources, averaged over the inner Galaxy (i.e., multiplied by the filling factor $\phi _{{\rm H\,\mathsc {ii}}}$) is n_e = 0.05  cm⁻³, compared to the Taylor & Cordes (1993) model values 0.1  cm⁻³ and 0.08  cm⁻³ for the inner annulus and spiral arms, respectively. Following McKee & Williams (1997), we identify the ELD (the sum of the WMAP sources) with the arm and annulus components for the Taylor & Cordes (1993) model.

3.2. Accounting for the H ii Region Distance Bias and for Diffuse Emission

We noted above that catalogs of H ii regions are known to be biased against distant objects, a result apparent in Figure 3. We follow Mezger (1978), Smith et al. (1978), and McKee & Williams (1997), and account for this by doubling the luminosity of sources in our half of the Galaxy, i.e., sources with y ⩾ 0 in Figure 3. This results in

$$\begin{equation} L_{\nu,\rm sources}= 1.2\times 10^{27}\;{\rm erg \; s}^{-1}\;{\rm Hz }^{-1}. \end{equation} \tag{ 3 }$$

There is a selection effect against low-flux sources (less than ∼10 Jy), as mentioned above, due to the source extraction process. The luminosity of a 10 Jy source at 15 kpc is 2.6 × 10²⁴ erg s⁻¹ Hz⁻¹, or Q = 3.5 × 10⁵⁰  s⁻¹, about one fifth that of the ionizing flux of Carina. Since the number counts of free–free sources in ground-based surveys do not increase much with decreasing flux, such sources do not contribute much to the total free–free luminosity of the Galaxy.

On the other hand, there does appear to be a diffuse component to the WMAP free–free sky map (diffuse even compared to the ELD). The total flux over the entire sky is f_ν = 54211.6  Jy, while that in WMAP sources is 36043.0  Jy. We give a rough accounting of this emission by assuming that it arises from gas that has the mean distance of the sources, i.e., we multiply the free–free luminosity emitted by the WMAP sources by the ratio 54211.6/36043 ≈ 1.5 to find our final estimate for the Galactic free–free luminosity,

$$\begin{equation} L_\nu = 1.8\times 10^{27}\pm 2.9\times 10^{26}\;{\rm erg \; s}^{-1}\;{\rm Hz }^{-1}. \end{equation} \tag{ 4 }$$

The error estimate comes from a simple error propagation of errors in the determination of Galactic longitude l (with Δl = 05), the circular velocity of the Galaxy v_c = 220 ±  10  km s⁻¹, and the radial velocities of the associated H ii regions Δv = 5 km s⁻¹. The radial velocity error is by far the largest component.

We show in this section that many of the H ii regions listed in Russeil (2003) and earlier compilations are physically connected. In particular, when several sources appear within ≲1° on the sky, and have radial velocities within Δv_r ≈ ±10  km s⁻¹, examination of Spitzer band 4 GLIMPSE (8 μm) images reveal large (10–100  pc) bubbles, with the H ii regions arrayed around the rim of the bubble. We interpret these bubbles as radiation and H ii gas pressure driven structures powered by a central massive cluster. Here we give one example; more will be presented in a forthcoming paper.

The GLIMPSE images reveal hundreds of small (1–10  pc) bubbles, many listed by Churchwell et al. (2006, 2007). The bubbles we identify are typically larger than 10 pc, ranging up to 100 pc. None of our bubbles are listed in these two references. However, we do find a number of the GLIMPSE team's bubbles contained inside some of our bubbles.

We stress the difference between the WMAP sources we find and the bubbles contained in them. The sources are identified by their free–free emission, and necessarily have sizes comparable to or larger than the WMAP beam (approximately 1°); many of the WMAP sources are resolved, but many are not. In contrast, the ∼5'–30' bubbles we find in the GLIMPSE images, identified by their morphology as revealed by 8 μm polycyclic aromatic hydrocarbon (PAH) emission, are well resolved, and substantially smaller than the surrounding free–free emission regions. We interpret this by asserting that many of the ionizing photons emitted by the clusters associated with the bubbles escape from the bubble to be absorbed by gas in the interior of the WMAP source. These are the photons that produce the ELD (see Section 3.1).

4.1. WMAP Sources are Powered by Massive Star Clusters

There are several arguments that the WMAP sources, and their enclosed, apparently empty large bubbles actually contain the largest star clusters in the Milky Way.

The first is the very large ionizing fluxes found using WMAP, Q ≈ 3–10 ×  10⁵¹  s⁻¹, for the top 20 or so sources. These sources have WMAP-determined radii of order 100 pc, so either there are ∼3–10 Carina size clusters all within 100 pc, and dN_cl/dM is very different than we believe, or there is a single dominant cluster.

The second argument is provided by the shape of the GLIMPSE and MSX 8 μm bubbles inside the WMAP sources. The bubbles are elliptical, with axis ratios one to two or so. This argues for a single massive cluster, which dominates the luminosity of the region.

The third argument is that many of the bubbles show prominent pillars pointing back to a single location in the bubble, again consistent with a single dominant source.

Finally, we present a quantitative argument for the WMAP source G298.3-0.34, showing that there should be a massive cluster M_* ≈ 4 × 10⁴ M_☉ providing the bulk of the ionizing radiation. Along the way we show that the classical giant H ii regions associated with this region are powered by compact star clusters with masses Q ≈ 7 × 10⁵⁰  s⁻¹, and M_cl ≈ 10, 000 M_☉. The total Q ≈ 7.7 × 10⁵¹  s⁻¹ for the region; we show that this most likely arises from a cluster at the location pointed to by the giant pillars in Figure 6, near l = 29866, b = −051.

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Cohen & Green (2001) have shown that the 8 μm emission traces free–free emission reasonably well. This allows us to use the 8 μm images to examine the WMAP sources with much higher resolution.

Figure 6 shows the GLIMPSE image in the direction of the WMAP free–free source G298.4-0.4. SIMBAD lists 7 H ii regions within 05 of the center of the bubble (at l = 2985, b = −0556); we interpret 2MASX J12100188-62500 to be the same source as [GSL2002] 29, and [WMG70] 298.9-00.4 to be the same source as [CH87] 298.868-0.432. The five unique sources are marked by circles in Figure 6 (see Table 3). The H recombination line radial velocities range from +16  km s⁻¹ to +30.3  km s⁻¹, with a mean around +23  km s⁻¹. Given the arrangement of sources around the wall, and the range of radial velocities, we interpret the source as an expanding bubble, with mean r_bubble ≈ 55(D/10  kpc) pc and expansion velocity ∼7 km s⁻¹. We interpret the H ii regions around the rim as triggered star formation. The two largest H ii regions on the rim, G298.227-0.340 and G298.862-0.438, have fluxes f_ν ≈ 47 Jy and 42 Jy, respectively, corresponding to Q ≈ 7.5 × 10⁵⁰(D/10  kpc) s⁻¹ and 6.6 × 10⁵⁰(D/10  kpc) s⁻¹. The total flux from the H ii regions on the rim is 111 Jy, compared to the WMAP flux of 313 Jy. We suggest that there is a massive cluster (Q ≈ 3–5 × 10⁵¹ erg s⁻¹, or M_* ≈ 5 × 10⁴ M_☉) in the interior of the bubble; the pillars point to the location of the cluster.

Table 3. H ii Regions within 05 of l = 2985, b = −0556

Name	Galactic l	Galactic b	R.A.(J2000)	Decl.(J2000)	Flux	vr	Distance	Reference
	(deg)	(deg)	(hh:mm:ss)	(deg:mm:ss)	(Jy)	( km s−1)	( kpc)
[KC97c] G298.2-00.8	298.1869	−0.7821	12:09:03.7	−63:15:46	2.4	+16	10.9	1
GSL2002 29	298.228	−0.3308	12:10:04.0	−62:49:27	47.4	+30.3	12.3	1
KC97c G298.6-00.1	298.5589	−0.1141	12:13:12.6	−62:39:39	2.8	+23	11.7	1
WMG70 298.8-00.3	298.8377	−0.3467	12:15:19.9	−62:55:52	16.0	+25	12.0	2
CH87 298.868-0.432	298.8683	−0.4325	12:15:29.6	−63:01:13	42.4	+25	12.0	1

Note. Fluxes for [GSL2002] 29 and [CH87] 298.868-0.432 are taken from Conti & Crowther (2004); all others are from Caswell & Haynes (1987). References. (1) Caswell & Haynes 1987; (2) Wilson et al. 1970.

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We note that even so-called giant H ii regions are spatially compact, of order a few to 10 pc in radius (e.g., Conti & Crowther 2004); the two classical giant H ii regions, [CH87] 298.868-0.432 (G298.9-0.4 here) and [GSL2002] 29 (G298.2-0.3) are prominent in the 8 μm GLIMPSE image, and have radii 3.8 arcmin, or ∼10(D/10 kpc) pc in 6 cm radio maps (Conti & Crowther 2004). Radial profiles from the centers of the 8 μm sources show the 1/R surface brightness profiles expected from point sources; see Figure 7. These giant H ii regions cannot be responsible for the much more extended 8 μm emission seen in Figure 6 and plotted in Figure 7. Nor can the two giant H ii regions explain the WMAP free–free emission, which has r = 09 ≈ 160(D/10 kpc) pc for G298.

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The surface brightness profile around the large bubble also shows a 1/R shape at large radii (r ≳ 05). Inside the bubble, the surface brightness is generally flat, but with a number of peaks, culminating in the large peak at r ∼ 04, corresponding to the bubble wall. The total 8 μm luminosity is dominated not by the known H ii regions, but by the large-scale emission associated with and surrounding the bubble. Figure 7 shows the surface brightness profiles of the two brightest H ii regions associated with G298; recall that both lie on the rim of the large bubble. Both profiles merge into the background at r ∼ 03 ≈ 50 pc. The figure also shows the azimuthally averaged radial surface brightness profile from the putative location of the massive cluster at l = 29866, b = −0507. In converting from degrees to parsecs, labeled along the top of the figure, we have assumed a distance D = 10 kpc to the object; D = 11.7 kpc for v = +23 km s⁻¹ in this direction.

The figure shows that neither of the classical H ii regions can be responsible for the large-scale (∼200 pc) diffuse emission. We say this because the 1/R scaling for the smaller sources extrapolates to a very low surface brightness at R ≳ 40 pc. It also suggests that a much more luminous source must be embedded in the bubble interior. The surface brightness of the entire region also falls off as 1/r from the point l = 29866, b = −0507, as expected if there is a massive cluster near or at this location. It follows that the total 8 μm luminosity is at least ∼3.5 times that of G298.9-0.4 (the ratio of the surface brightness at large radii in the least-squares fits) and five times that of G298.2-0.3; if the emission associated with the H ii regions does not extend to the edge of the observed 8 μm emission, their contribution to the total flux will be smaller.

The azimuthal averaging leads to an artificially thick bubble wall; surface brightness measurement along radial lines shows that the radial thickness of the bubble wall is Δr ∼ 4(D/10 kpc) pc, about 10% of the bubble radius.

We noted above that the WMAP free–free source G298 has a radius of r ≈ 160(D/10 kpc) pc, similar to the radius 200(D/10 kpc) pc of the 8 μm source we found, once again illustrating the correlation between 8 μm emission and free–free emission.

The total free–free flux in the region is 312 Jy, compared to 47.4 Jy for G298.2-0.3 (∼1/6 of the total) and 42.4 Jy (1/7) for G298.9-0.4; note that these ratios are roughly consistent with the 8 μm flux ratios. We estimate a total flux of ∼110 Jy for all the classical H ii regions in the area, leaving 202 Jy, which we attribute to the massive central cluster. We inferred above that the cluster has an ionizing luminosity Q = 5 × 10⁵¹(D/10 kpc) s⁻¹, and a stellar mass M ≈ 5 × 10⁴(D/10 kpc)M_☉, similar to that of Westerlund 1.

Thus we have a slightly different interpretation of the ELD than Lockman (1976) and Anantharamaiah (1985a, 1985b), at least for our most luminous WMAP sources (recall that these 20 or so sources supply the bulk of the ionizing luminosity of the Galaxy). The cited authors associate the ELD with photons leaking out of classical H ii regions. We argue that the classical H ii regions are not the source of the bulk of the ionizing photons. Rather, in the WMAP sources, the majority of the ionizing flux is produced by a central massive star cluster (M ∼ 5 × 10⁴ M_☉ or larger). These central clusters excite free–free and 8 μm emission out to 50–200 pc. They have also blown ∼10–100 pc bubbles in the surrounding ISM, as seen in Spitzer or MSX images. The rims of the bubbles contain triggered star formation regions, which are younger than the central clusters. Because the triggered clusters are younger, and substantially less luminous (typically by a factor of 5), they have not blown away their natal gas. As a result, they appear as very high surface brightness free–free sources in classical radio emission catalogs (and as bright 8 μm sources); thus Lockman (1976) and Anantharamaiah (1985a, 1985b) attribute the ELD to photons from the smaller, younger, and more embedded clusters, while we attribute them to the larger, less embedded central clusters.

While these young, compact sources are bright and hence easily identified, they are not the source of the ionizing photons in the ELD. Instead, the massive central clusters are the source of the ionizing photons powering the ELD; ionizing photons leak out of the bubbles in all directions, since the bubble walls are far from uniform.

Using the definition of an H ii region (treating all the H ii regions associated with a GLIMPSE bubble as one region) alters the luminosity function somewhat. The new luminosity function is shown in the right panel of Figure 5. At the high-luminosity end, we find dN/dL ∼ N^−1.7±0.2, i.e., most of the luminosity (and stellar mass) is in massive sources (α = 2 corresponds to equal numbers per logarithmic luminosity bin). Half the luminosity due to sources is in the nine most luminous objects, with Q>3.2 × 10⁵⁰  s⁻¹ (not corrected for dust absorption). These sources have luminosities similar to that of the Galactic center, L_ν>3 × 10²⁵ erg Hz⁻¹, or Q>7 × 10⁵¹  s⁻¹. This corresponds to cluster masses M_cl>10⁵ M_☉, ranging up to 2.6 × 10⁵ M_☉.

Kennicutt et al. (1989) survey nearby galaxies and construct Hα luminosity functions; they find a range of values for α between 1.5 and 2.5, with values below 2 being slightly more prevalent. McKee & Williams (1997) refit the data presented in Kennicutt et al. (1989) using truncated power-law fits, and find a lower range, 1.4 < α < 2.3, with a mean α = 1.75 ± 0.23.

We conclude this section with a short discussion of a simple question: why has the central cluster that we suggest powers G298 not been noticed before?

We start by noting that similarly massive (but slightly older) clusters like Westerlund 1 (Figer et al. 1999), red supergiant cluster 1 (RSGC1; Figer et al. 2006), RSGC2 (Davies et al. 2007), and RSGC3 (Clark et al. 2009) escaped detection, or were not understood to be massive, until a few years ago. All these clusters are closer to Earth than G298. The RSGCs, as well as G298, have A_K ∼ 1–2, implying A_v ∼ 10–20; such clusters are difficult to identify using optical data. The RSGCs are easy to find in GLIMPSE images (we found Westerlund 1 and RSGC1 before realizing they were in the literature). In contrast, the central clusters we identify are younger than ∼3.6 Myr, so their most massive stars have not evolved into red supergiants. As a result they are not very prominent in the GLIMPSE images, in contrast to the RSGCs.

Like the RSGCs, the clusters we identify have cleared bubbles around themselves, limiting the IR surface brightness, and even the free–free surface brightness, of the clusters and their surroundings. However, we use these same large bubbles to infer the presence of a massive cluster. A more direct and convincing demonstration of the existence of the large central clusters we identify would be highly desirable.

The emissivity of the free–free flux from an ionizing region is given by

$$\begin{equation} \epsilon ^{\rm ff}_{\nu } = \frac{2^5 \pi e^6}{3 m_e c^3} \left(\frac{2 \pi }{3 k m_e}\right)^{1/2} T^{-1/2} Z^2 n_e n_i e^{-h\nu /kT} g_{{\rm ff}}, \end{equation} \tag{ 5 }$$

where Z is the charge per ion, T is the electron temperature, n_e and n_i are electron and ion densities, respectively, and g_ff is the Gaunt factor. For a fully ionized H ii region, we adopt n_e = n_i and Z = 1. Further, we adopt an electron temperature, T_e = 7000 K for H ii regions, and a Gaunt factor g_ff = 3.3 (Sutherland 1998). At radio frequencies we approximate this as ^ff_ν = ₀n²_e, where ₀ = 2.7 × 10⁻³⁹ g  cm⁵  s⁻³ Hz⁻¹.

To keep an isotropic H ii region ionized, the total number of ionizing photons required is

$$\begin{equation} Q_{\rm tot} = \int n_e^2 \alpha (H^+) dV, \end{equation} \tag{ 6 }$$

where V is the volume of the ionized region.

The total ionizing luminosity (in photons s⁻¹) of a given H ii region is then

$$\begin{equation} Q_{\rm tot} = {\alpha (\mbox{H}^+)\over \epsilon _o} L_\nu \approx 1.33\times 10^{26}L_\nu \,\,{\rm s }^{-1}. \end{equation} \tag{ 7 }$$

Using this expression, we find that the ionizing luminosity of the Galaxy, before correction for absorption by dust, is Q_tot = 2.34 × 10⁵³photons  s⁻¹.

The final step is to correct for the effect of absorption by ionizing photons by dust grains. Following McKee & Williams (1997), we multiply by 1.37, and find

$$\begin{equation} Q_{\rm tot} = 3.2 \times 10^{53}\pm 5.1\times 10^{52} \hbox{ photons} \,\,{\rm s }^{-1}. \end{equation} \tag{ 8 }$$

The error estimate does not take into account any systematic distance bias.

5.1. Star Formation Rate

To estimate the SFR from Q, we follow Mezger (1978) and McKee & Williams (1997), and use the expression

$$\begin{equation} \dot{M}_* = Q {\langle m_*\rangle \over \langle q\rangle } {1\over \langle t_Q\rangle }, \end{equation} \tag{ 9 }$$

where 〈q〉 is the ionizing flux per star averaged over the initial mass function (IMF), and 〈m_*〉 is the mean mass per star, in solar units. The quantity 〈t_Q〉 is the ionization-weighted stellar lifetime, i.e., the time at which the ionizing flux of a star falls to half its maximum value, averaged over the IMF; all the averaged terms are discussed in the Appendix.

All of these averaged quantities depend on the IMF of the stars, in particular on the high-mass slope Γ of the IMF, as discussed in the Appendix; as an example, and to fix notation, the Salpeter (1955) IMF is given by $\xi (m)\equiv mdN/dm={\cal N}m^{-\Gamma }$, with Γ = 1.35.

Using the stellar evolution models of Bressan et al. (1993) we find 〈t_Q〉 = 3.9 × 10⁶ yr (for Γ = 1.35). This is slightly longer than the ionizing flux-weighted main-sequence lifetime 〈t_ms〉 = 3.7 Myr used by McKee & Williams (1997), which is in turn somewhat larger than the 3 Myr used by Mezger (1978). This value is only weakly dependent on Γ.

The WMAP satellite is sensitive to free–free emission, which as we have just indicated is produced by main-sequence O stars with lifetimes of ∼3.9 Myr, i.e., stars with masses in excess of ∼40 M_☉. Most of the stars driving the free–free emission we see have not been identified. We have examined Two Micron All Sky Survey (2MASS) images, and found that in most cases the stars in the bubble interiors have J − K colors indicating that they are heavily reddened (A_K ∼ 2), which is presumably why they have not been identified as O stars, as noted at the end of Section 4.

5.1.1. The Mean Ionizing Flux per Solar Mass

The mean ionizing flux per solar mass, 〈q〉/〈m_*〉, is much more problematic; it depends sensitively on Γ. Figure 8(a) shows 〈q〉/〈m〉 using Q(m) as determined by Martins et al. (2005; the solid line) and as given by Vacca et al. (1996; their evolutionary masses); in making this figure we used the Muench et al. (2002) IMF. The difference between the two estimates for Q(m) results in a difference in 〈q〉/〈m〉 of ∼10%. The filled square represents our favored value,

$$\begin{equation} {\langle q\rangle \over \langle m_*\rangle } =6.3\times 10^{46}\,\,{\rm s }^{-1}\,M_\odot ^{-1}, \end{equation} \tag{ 10 }$$

at Γ = 1.35.

For the Muench et al. (2002) IMF 〈m_*〉 = 0.71 when Γ = 1.35; 〈q〉 = 4.5 × 10⁴⁶  s⁻¹. This is a factor of 5 larger than the value quoted by McKee & Williams (1997), 〈q〉 = 8.9 × 10⁴⁵  s⁻¹; this difference is not primarily a result of our using different expressions for Q(m), since the dashed line uses Vacca et al. (1996), as McKee & Williams (1997) used.

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We show that this factor of 5 arises mostly from the use of a different IMF, with two contributing factors, the use of a different value of Γ, and a different IMF shape, so that McKee & Williams (1997) find fewer massive stars at a fixed value of m, even when Γ is chosen to be the same for the two IMFs; in this comparison, we choose Γ = 1.5 to match their work.

Figure 8(b) shows the mean ionizing flux per solar mass for the Scalo-type IMF used by McKee & Williams (1997; dot-dashed line), the Muench et al. (2002) IMF (solid line), and the Chabrier (2005) IMF (long-dash line), all as a function of Γ. In making this plot, we have used the relation between Q and evolutionary mass given by Vacca et al. (1996), so that the dot-dashed curve goes through the McKee & Williams (1997) result. We note that Scalo (1998) no longer recommends use of the Scalo (1986) IMF.

From this plot we can see that the variation in Γ is responsible for about a factor of 2 out of the total factor 5 difference; the rest comes from the different shapes of the IMF, with the more recent IMFs (Muench et al. 2002; or Chabrier 2005) having many fewer low-mass stars, or alternately, more high-mass stars, even for fixed Γ.

The figure shows that small changes in Γ lead to large changes in the inferred SFR. Recent observations of young massive clusters have suggested that Γ varies from the Salpeter value (Stolte et al. 2002; Harayama et al. 2008); if confirmed, these variations, combined with the results presented here, would lead to large variations in the estimated SFR of the Galaxy.

Using the ionizing flux given by Martins et al. (2005), we can integrate over a Muench et al. (2002) like IMF (Equation (A3)), with Γ as a free parameter. In the Appendix, we find

$$\begin{equation} {\langle q\rangle \over \langle m_*\rangle } \approx 6.3\times 10^{46} \left(m_Q^{1.35-\Gamma }\right) \,\,{\rm s }^{-1}\,M_\odot ^{-1}, \end{equation} \tag{ 11 }$$

where m_Q ≈ 35 M_☉ is the location of the break in a power-law fit to Q(m) (Figure 9).

Finally, we find a SFR for the Milky Way of

$$\begin{equation} \dot{M}_*= 4.1\times 10^{-54}Q=1.3\pm 0.2\,M_\odot \;{\rm yr}^{-1}. \end{equation} \tag{ 12 }$$

Using the McKee & Williams (1997) value of Γ = 1.5 results in $\dot{M}_*=2.2\,M_\odot \;{\rm yr}^{-1}$, lower than their 4.0 M_☉ yr⁻¹ due to the different forms of the IMF (aside from the high-mass slope Γ) and our use of the Martins et al. (2005) temperature scale; as seen in Figure 8, using their IMF and Vacca et al. (1996), we recover $\dot{M}_*\approx 4\,M_\odot \;{\rm yr}^{-1}$. Using the Muench et al. (2002) slope, the result is 0.9 M_☉ yr⁻¹.

5.2. The Magellanic Clouds

We have combined the WMAP free–free map with previous determinations of distances to H ii regions to measure the ionizing flux of the Galaxy. We find Q = 3.2 × 10⁵³ ± 5.1 × 10⁵²  s⁻¹, in agreement with previous determinations. We found 88 sources responsible for a flux of 36043 Jy, out of a total flux of 54211.6 Jy.

The mean WMAP source radius is ∼60 pc. Inspection of Spitzer GLIMPSE images and MSX images shows that diffuse 8 μm emission, which closely tracks the free–free emission, gives sizes consistent with the WMAP sizes, e.g., Figure 7, suggesting that many of the WMAP sources are in fact resolved.

The mean source electron density is ∼9  cm⁻³; hence the mean dispersion measure across a source is DM ≈ 540  cm⁻³ pc; from Figure 1, most of the sources are within ∼60° of the galactic center. Thus we identify these sources with the inner Galaxy and spiral arm components of the free electron model of Taylor & Cordes (1993). The density-weighted scale height of the sources is 114 pc. The total volume filling factor of the sources is ∼0.005. Thus the Galactic mean electron density is 〈n_e〉 ≈ 0.045  cm⁻³.

We used GLIMPSE and MSX images to study the WMAP sources with higher resolution. We found that the bulk of the Galactic star formation (of order half) occurs in ∼20 sources, each with Q ≈ 5 × 10⁵¹  s⁻¹. These sources should be examined in other wave bands, including X-rays. The 8 μm images revealed large bubbles, with r ∼ 20 pc, ranging up to 100 pc, in most of these sources; these bubbles are much larger (up to a factor of 10) than those described by Churchwell et al. (2006, 2007). We showed that classical giant H ii regions associated with the WMAP sources were located in the bubble walls, and interpreted them as triggered star formation. We argued that the bubbles are powered by massive star clusters responsible for the bulk of the ionizing flux in each WMAP source. We estimate that these clusters have masses M_* ≈ 4 × 10⁴ M_☉ or larger.

We note that there are now a number of slightly older (but still young, 10–20 Myr old) Milky Way clusters known to have masses in this range; examples include Westerlund 1 with M ∼ 5 × 10⁴ M_☉ (Brandner et al. 2008), the Arches cluster M ∼ 4 × 10⁴ M_☉ (Figer et al. 1999), and the red supergiant clusters RSGC1 near G25.25-0.15 M ∼ 3 × 10⁴ M_☉ (Figer et al. 2006), RSGC2 (l = 262 b = −006) with M ∼ 4 × 10⁴ M_☉ (Davies et al. 2007), and RSCG3 (l = 292 b = −02) with M ≈ 3 × 10⁴ M_☉ (Clark et al. 2009). In a forthcoming paper, we show that almost all of our high-luminosity WMAP sources, as well as the less-luminous sources, are associated with large bubbles seen in GLIMPSE images, and most have fairly compact clusters in the bubble interior.

Lockman (1976) and Anantharamaiah (1985a, 1985b) suggested that the ELD, which we identify with the WMAP sources, and which accounts for the bulk of the free–free emission in the Galaxy, arises from ionizing photons that leak out of H ii regions. We agree that the ELD is closely associated with giant H ii regions. However, as Figure 7 shows, the bulk of the ionizing flux powering the ELD arises from massive clusters in the centers of large bubbles; the giant H ii regions are due to smaller (but still large) clusters located in the bubble walls. The massive clusters are not readily identified in free–free maps because they have blown away their natal gas, and so do not produce any high surface brightness emission (either free–free, 8 μm, or even far infrared).

Using recent estimates of Q(m) and the IMF ξ(m) of stars, we found a Galactic SFR of $\dot{M}_*=1.3\pm 0.2\,M_\odot \;{\rm yr}^{-1}$. This is somewhat smaller than past determination of the Galactic SFR. We showed that all estimates based on measurements of ionizing radiation are highly sensitive to the slope Γ, where ξ(m) ∼ m^−Γ at high mass. In this case, "high mass" is the critical mass m_Q ≈ 40 M_☉ where stellar luminosities approach the Eddington luminosity. Our quoted value of $\dot{M}_*$ assumes that Γ = 1.35, the Salpeter value.

We thank the anonymous referee for a very thorough and helpful report. We have benefited from ongoing discussions with C. McKee, P. G. Martin, J. Sievers, H. Yee, R. Abraham, and M. Nolta. M.R. would additionally like to thank N. Novikova for support throughout this research. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France, and of NASA's Astrophysics Data System. We acknowledge the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA), SIMBAD, and the GLIMPSE archive. Support for LAMBDA is provided by the NASA Office of Space Science. Part of the research described here was carried out while N.M. was a Visiting Scientist at the Spitzer Science Center, and during a sabbatical supported in part by the Theoretical Astrophysics Center at the University of California, Berkeley. N.M. is supported in part by the Canada Research Chair program and by NSERC of Canada.

Facilities: WMAP - Wilkinson Microwave Anisotropy Probe ()

We collect here the machinery needed to calculate the SFR from observations of free–free radio emission, following Smith et al. (1978) and McKee & Williams (1997).

We use four IMFs, all written in terms of stellar mass in units of the solar mass, m = M/M_☉, with 0.1 ⩽ m ⩽ 120. The first is the Salpeter (1955) IMF,

$$\begin{equation} \xi (m)\equiv mdN/dm={\cal N}(\Gamma)m^{-\Gamma }. \end{equation} \tag{ A1 }$$

Salpeter found Γ = 1.35.

The second IMF is the McKee & Williams (1997) version of Scalo (1986), which at the high-mass end looks like the Salpeter IMF,

$$\begin{equation} mdN/dm={\cal N}m^{-\Gamma }, \end{equation} \tag{ A2 }$$

with ${\cal N}=0.063C_F$; they take C_F = 1.4. McKee & Williams (1997) use Γ = 1.5.

Third, we use a modified Muench et al. (2002) IMF:

$$\begin{eqnarray} && \hspace{-2pc}\xi _{M,m_1}(m) \equiv m{dN\over dm} \nonumber \\ &=& N_0\left\lbrace \begin{array}{@{}ll} m^{-\Gamma } & m_U>m>m_1 \\ m_1^{(0.15-\Gamma)} m^{-0.15} & m_1>m>m_2\\ m_1^{(0.15-\Gamma)} m_2^{(-0.73 - 0.15)} m^{0.73} & m_2>m>m_L. \end{array} \right. \end{eqnarray} \tag{ A3 }$$

Muench et al. (2002) found Γ = 1.21 for the Orion region. As indicated above, m_U = 120 and m_L = 0.1. We use m₁ = 0.6 as the characteristic break mass.

Finally, we use the Chabrier (2005) IMF:

$$\begin{equation} \xi (m)= N_0\left\lbrace \begin{array}{@{}ll} \exp \left\lbrace -{(\log m - \log 0.2)^2\over 2\times (0.55)^2}\right\rbrace & m_L \le m \le 1\\ 0.446m^{-\Gamma } & 1<m \le M_U. \end{array} \right. \end{equation} \tag{ A4 }$$

We use the normalization

$$\begin{equation} \int _{m_L}^{m_U}\xi (m){dm\over m}=1. \end{equation} \tag{ A5 }$$

In that case, the ionizing flux per solar mass is

$$\begin{equation} {\langle q_*\rangle \over \langle m_* \rangle } \equiv \int _{m_L}^{m_U}Q(m)\xi (m){dm\over m} \bigg / \int _{m_L}^{m_U}\xi (m)dm, \end{equation} \tag{ A6 }$$

where

$$\begin{equation} \langle m_* \rangle \equiv \int _{m_L}^{m_U}\xi (m)dm \end{equation} \tag{ A7 }$$

is the mean mass per star.

We use both the Vacca et al. (1996) and Martins et al. (2005) compilations of ionizing fluxes as a function of stellar mass; the ionizing flux Q(m) is given per star by both. Since many clusters harbor stars with mass in excess of 100 M_☉, but neither paper models stars with M>88 M_☉, we have added the result of Martins et al. (2008), who find Q ≈ 10⁵⁰ for each of four WN7-8h stars with log L/L_☉>6.3 (all of which they model by stars with M ≳ 120 M_☉; see their Table 2 and Figure 2). These stars are slightly evolved, but still very young. Figure 9 shows Q(M_*) for both Vacca et al. (1996) and Martins et al. (2005).

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The function Q(m) ∼ m⁴ for 15 < m ≲ m_Q (where m_Q ≈ 40), but Q(m) ∼ m^1.5 for m>m_Q. The integral < Q_*>(m) ∼ mQ(m)ϕ(m) ∼ m^1.65 for m < m_Q, and ∼m^0.15 for larger m, indicating that the bulk of the ionizing flux occurs for stars with mass around m_Q, for all of our IMFs. Doing the integrals on the right-hand side of Equation (A6) from m_L to m_Q gives

$$\begin{equation} {\langle q_*\rangle \over \langle m_* \rangle }\sim m_Q^{-(\Gamma +1)}, \end{equation} \tag{ A8 }$$

which fits the numerical result rather well; it is shown as the dotted line in Figure 8.

The ionizing flux-weighted lifetime of a cluster is given by

$$\begin{equation} \langle t_Q\rangle \equiv \int _{m_L}^{m_U} Q(m) t(m)\xi (m){dm\over m} \bigg / \int _{m_L}^{m_U} Q(m) \xi (m){dm\over m}, \end{equation} \tag{ A9 }$$

where t(m) is the main-sequence lifetime of a star of mass m.