OB STARS AND STELLAR BOW SHOCKS IN CYGNUS-X: A NOVEL LABORATORY ESTIMATING STELLAR MASS LOSS RATES

We retrieved data from the Spitzer archive obtained as part of the Spitzer Legacy Survey of the Cygnus-X Complex (Hora et al. 2008) using data from the Infrared Array Camera (IRAC) at 3.6, 4.5, 5.8, and 8.0 μm (Fazio et al. 2004) and the Multi-band Infrared Photometer for Spitzer (MIPS) bandpasses at 24 & 70 μm (Rieke et al. 2004). One of us (H.A.K.) visually examined mosaiced single-band and multi-color images of the ∼24 deg² survey area to identify BS candidates on the basis of symmetric arc-shaped structures less than a few arcminutes in size enclosing a symmetrically placed stellar source. The complex ISM structure in this region, coupled with the large dynamic range of features, renders BS identification a subjective and imprecise task. In practice, the 8 μm band (sensitive to emission from polycyclic aromatic hydrocarbon (PAH) features excited by non-ionizing photons) and the 24 μm band (sensitive to emission from warm dust) were the principal images used to locate candidate BSs. While a plethora of arc-like structures appear in both bands, very few exhibited a high degree of symmetry and included a point source near the apsis. In total, ten objects were deemed to be BS candidates and were retained for further investigation. Table 1 lists the equatorial and Galactic coordinates and Two Micron All Sky Survey (2MASS) JHK, and Spitzer IRAC photometry of the central stars for each object. The final column provides common stellar cross-identifications from other works, where applicable.

Figure 1 is a three-color image ∼2° square centered on Cygnus OB2 (ℓ = 803, b = +10) showing the location of the first seven candidate BSs. Blue, green, and red represent the 4.5, 8.0, and 24 μm images, respectively. Objects 8–10 have Galactic latitudes b < 015 and lie off the lower edge of Figure 1. Figures 2–11 show enlarged views of each BS candidate using the same color scheme as Figure 1. We discuss each object in detail in Section 3.

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We obtained optical spectra for several of the stellar sources located near the apsides of candidate BSs. We observed five of the stars (Stars 1, 2, 3, 5, and 8) at the Wyoming Infrared Observatory's (WIRO) 2.3 m telescope equipped with the Longslit spectrograph on 2008 June 23–30, 2009 May 20, and 2009 September 16. The 1800 l mm⁻¹ grating was used in first order. The spectral coverage was 5210–6680 Å. Exposure time varied from 360 s for Star 3 (HD 195229) to 3600 s for Stars 2 and 5, yielding signal-to-noise ratios ranging from 50:1 for Star 5 to 450:1 for Star 2. Copper–argon lamp exposures were taken after each stellar spectrum for wavelength calibration.

The spectra were reduced, extracted, wavelength calibrated, and Doppler corrected to the heliocentric frame using standard IRAF⁴ tasks in the KPNOSLIT package. The absolute radial velocities were checked against the standards HD161096, HD 161797, HD131156, HD171391, and HD188512. Table 1 lists the heliocentric radial velocities, which are typically accurate to 6 km s⁻¹ rms.

We also obtained spectra of BD+43°3654 and HD195229 during an observing run at the WIYN⁵ 3.5 m telescope on 2008 June 11–16 using the Hydra multispectrograph. We used the Red camera and 3'' blue fiber cables with the 1200 l mm⁻¹ grating in second order. Exposure times were 600 s and the spectral coverage was 3820–4500 Å. Copper–argon lamp exposures were taken to wavelength calibrate the spectra. The spectra were reduced, extracted, wavelength calibrated, and Doppler corrected to the heliocentric frame using standard IRAF tasks in the TWODSPEC and RVCOR packages. Radial velocities of standards HD 131156, HD146233, HD161096, HD161797, and HD171391 from this run agreed with published values to ±2 km s⁻¹.

Figure 1 shows a portion of the Cygnus-X region surrounding its most massive and energetic constituent, Cygnus OB2. This region encompasses a complex array of nebulae and star-forming regions located at various distances along the line of sight, with most of the activity located in the local or "Orion" spiral arm at a distance of 1–2 kpc. Odenwald & Schwartz (1993) present a schematic three-dimensional representation of the Cygnus-X region, while Schneider et al. (2006) provide velocity-resolved molecular maps of Cygnus-X and individual star-forming objects nearby. One apparent feature of this region is the relative absence of dust (red) and PAH emission (green) at the location of Cyg OB2 itself, except at the southern edge. This apparent "hole" is consistent with a cavity cleared by the winds of >100 evolving massive stars. Some of the prominent star-forming regions labeled in Figure 1 clearly constitute the heads of gaseous pillars, most of which point toward Cyg OB2, the dominant source of luminous and mechanical energy in this vicinity. These pillars are reminiscent of those in M 16 (Indebetouw et al. 2007; Hester et al. 1996), RCS 49 (Whitney et al. 2004), and other massive star-forming regions.

The seven BS candidates labeled in Figure 1 lie around the periphery of Cyg OB2, none falling within the main concentration of Cyg OB2 stars but all within ∼5–8 pc of the canonical Cyg OB2 center. The orientations of the features, seen more clearly in Figures 2–11, appear random, having no preferential alignment with respect to Cyg OB2. Generically, BS orientation reveals the relative motion between the star and surrounding matter. We might have expected, if the stars were all runaways from Cyg OB2, that the BSs would point away from the Association in the direction of the stars' motions. On the other hand, if the hydrodynamics of material surrounding Cyg OB2 were dominated by a hot outflow from an over-pressured region, we might have expected many of the BSs to point back toward the source of mechanical energy, as do three BSs near M 17 and at least one near RCW 49 (Povich et al. 2008). The absence of either signature suggests that (1) the central stars driving the BSs, if they are runaways, have different origins, and (2) there is no large-scale supersonic wind emanating from Cyg OB2. Only candidates 2 and 4 have orientations consistent with their stars being runaways having an origin within Cyg OB2.

The colors of the candidate BS nebulae, best seen in the individual zoomed images in Figures 2–11, at first glance reveal two classes of objects. Candidates 1, 3, 4, 7 are seen exclusively at 24 μm and longer wavelengths while the rest exhibit strong emission in the IRAC bandpasses with some additional contribution at longer wavelengths.

We performed aperture photometry of each BS candidate in each of the six bandpasses using the Spitzer Science Center (SSC) post-basic calibrated data mosaic images by assigning irregular crescent-shaped polygonal apertures defined visually on the [8.0] and/or [24] images. We also manually defined background regions surrounding each BS. Because these objects lie in complex regions of diffuse emission, care was taken to select multiple background regions judged to be representative of the local background away from the nebula but within several arcminutes. We applied aperture corrections, as recommended in the IRAC and MIPS calibration Web sites,⁶ using values appropriate to a circular aperture of equivalent area to the polygonal apertures. These corrections are less than 1.3 in all cases and are sometimes less than unity for the IRAC bands. Given the imprecision of the aperture corrections and the high and variable background levels, we adopt a minimum uncertainty of 20% in all bands. Table 2 gives background-subtracted aperture fluxes in mJy for all of the nebulae. In some cases, only 1σ upper limits are given, mostly in the IRAC bands where some objects are not detected.

For each BS candidate we were able to identify a symmetrically placed point source, presumed to be the energizing star, behind the apsis. Figure 12 is a 2MASS JHK color–color diagram showing each of the ten stars (diamonds) associated with the BSs. The solid and dashed curves (asterisks and crosses, respectively) show a fiducial main sequence and supergiant sequence. A solid line illustrates the reddening vector for A_V = 5 mag. The majority of stars lie in a close group near H − K = 0.3, J − H = 0.6, consistent with early-type stars seen behind 4–7 mag of visual extinction. This range is similar to other OB stars in Cyg OB2 (Massey & Thompson 1991), suggesting that they lie at a similar distance. Stars 4 and 6 are much redder, suggesting 12–15 mag of visual extinction, under the assumption that these are also early-type stars. The K-band magnitudes for these two stars are 2–3 mag fainter than the other targets, indicating either a larger distance or the effect of localized regions of high extinction. Star 6, in particular, is surrounded by a more substantial region of warm dust and photo-excited material, consistent with localized extinction. Star 3 (HD195229), a known B0.2III, is much less red, consistent with ∼1 mag of visual extinction, and consistent with the known variability of extinction across Cyg OB2 (Massey & Thompson 1991).

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We performed aperture photometry on the central star of each BS at the mid-IR IRAC bandpasses using the SSC basic calibrated data frames, a 10 pixel circular aperture for which the aperture correction is minimal, and a much larger annulus to measure the diffuse background levels. Most of the stars were saturated at IRAC [3.6] and [4.5] bands in the 10.4 s exposures, so the high-dynamic-range (HDR) 0.4 s exposures were used. Table 1 lists these fluxes and their uncertainties.

Figure 13 shows the spectral energy distributions (SEDs) of the BS candidates and central stars at the 2MASS, IRAC, and two MIPS bandpasses. Asterisks denote photometry of the BS alone, while diamonds denote the star plus BS. The BSs are not detected at 2MASS JHK bands and are sometimes undetected in the IRAC bandpasses. In only three cases (stars 6, 8, and 9) are pointlike sources detected at [24], indicative of dust in the immediate vicinity of the stars. Uncertainties are always smaller than the plotted points. The solid curves are blackbody fits to the JHK photometry, while the dotted curves are fits to the IRAC and MIPS data using scaled (Draine & Li 2007) dust emission models. In Sections 3.4 and 4, we discuss the details of the fitting, the models, and model results for each object individually.

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The shape of a momentum-driven BS is a universal function that scales as the "standoff distance," R₀, between the star and the BS apsis, as shown analytically by Baranov et al. (1971), Dyson (1975), van Buren et al. (1990), and Wilkin (1996). Using the formulation of Wilkin (1996, Equation (9)), we generated synthetic BS "images," applying arbitrary size, rotation, and intensity scale factors to facilitate comparison with the data. Figure 14 shows a selected subset of BS candidates (1, 2, 3, 5, 6, and 7) at [8.0] (green), [24] (red) and the scaled synthetic BS (blue). The simulated BSs are all shown at zero degree inclination, while the inclinations of the BS candidates are unknown.

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The morphologies of candidates 5 and 7 (central stars O9V and O5V) show striking similarity to the theoretical shape. The agreement is best at [24] where the BS can be traced to larger radial distances than at [8.0] where the BS is less extended and exhibits a more irregular morphology. For object 2, there is also good agreement, although the BS itself is faint and difficult to see because of the large dynamic range in the images. For objects 1, 3, and 6 the morphological agreement is considerably less good. We take this as evidence for a physical origin other than a BS. The morphologies of the other objects 4, 8, 9, and 10 (not pictured) also differ from the canonical BS shape, appearing more irregular and clumpy. We interpret the good agreement between the model shape and objects 2, 5, and 7 as support for the momentum-driven BS nature of these objects.

3.4.1. BS Candidate 1 and Star 1 (GSC03161 − 01188)

BS candidate 1, pictured in Figure 2, is associated with the V = 13.3 star GSC03161 − 01188 from the Guide Star Catalog, also known as 1288 from the work of Reddish et al. (1966). Figure 15 (bottom) shows the WIRO spectrum of GSC03161 − 01188. We classify GSC03161 − 01188 as O9V on the basis of the He i/He ii line ratios and by comparison to a Jacoby et al. (1984) spectral library O9V star (top). We compute a spectrophotometric distance for the K = 8.21 star GSC03161 − 01188 by adopting a visual extinction of 4.5 mag from Figure 12 (A_K = 0.51 mag), an intrinsic color V − K = −0.85, and assuming an absolute magnitude for an O9V star of M_V = 4.05 ± 0.2 (Martins et al. 2005). This yields a spectrophotometric distance of 1500 ± 150 pc for GSC03161 − 01188, consistent with the adopted distance to Cyg OB2, ∼1.6 kpc (Hanson 2003; Massey & Thompson 1991; Schulte 1958).

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Cross-correlating the spectrum of GSC03161 − 01188 with a synthetic model atmosphere O9V spectrum from Lanz & Hubeny (2003), as described more fully in Kiminki et al. (2007), yields a heliocentric radial velocity of −17 ± 6 km s⁻¹. This velocity is consistent with, but more negative than, the average radial velocities of Cyg OB2 members, −10 ±  6 km s⁻¹ surveyed by Kiminki et al. (2007).

The SED in Figure 13 shows that this nebula is detected only at [24] and [70]. Stellar flux dominates the 2MASS and IRAC bandpasses. The solid line shows the SED of an O9.5V star, approximated as a blackbody with T_eff = 30,000 K and R = 8 R_☉ (Martins et al. 2005) which, at a distance of 1.6 kpc with A_V = 4.5 mag, provides a good fit to the near-IR data. This is consistent with our spectral classification based in Figure 15.

Given the lack of similarity to the theoretical shape and the modest radial velocity of the central star, this nebula is unlikely to be a BS. We assign this object an ambiguous classification. Possibilities include a dusty asymmetric bubble or a partial shell.

3.4.2. BS Candidate 2 and Star 2 (G80.9020+0.9828)

Figure 2 shows BS candidate 2 and Star 2 (G80.9020+0.9828). Figure 16 (bottom) shows the WIRO spectrum of this star and a comparison B2 dwarf equivalent from the model atmospheres of Lanz & Hubeny (2003, top). The relative strengths of the H i 6563 Å and the He i 6678 Å and He i 5876 Å lines lead us to classify this star as B2V, plus or minus several spectral subtypes, owing to the low signal-to-noise spectrum. Its heliocentric radial velocity is −12 ± 15 km s⁻¹, consistent with the range of radial velocities observed for Cyg OB2 members (Kiminki et al. 2007). We compute a spectrophotometric distance from the 2MASS photometry (K = 10.5) by adopting a visual extinction (Cardelli et al. 1989) of 4.5 mag (A_K = 0.51), an absolute magnitude for an B2V star of M_V = −2.4 ± 0.4, an intrinsic V − K color of −0.4. This yields a spectrophotometric distance of 2500 ± 500 pc, where the error is driven primarily by the uncertainty on the spectral-type/absolute magnitude. A high-surface-brightness apsis lies 9'' (0.07 pc projected separation) from the star toward the upper left in Figure 2, and there is also a more extended region of 8 μm emission preceding the apsis to the upper left. The [24] lima-bean shaped nebulosity and the arc-like structure seen at [8.0] do not share a common symmetry axis, so it is difficult to ascertain which wavelength best traces the putative BS.

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Figure 13 shows that BS 2 is a strong detection at the IRAC and MIPS bandpasses. The relative faintness of the BS in [4.5] relative to the other three IRAC bandpasses is consistent with strong PAH emission in the shock. The central star dominates the 2MASS bandpasses. Star 2 is fainter than Star 1 and is well fit by a B2V–B3V star (T_eff = 20,000 K, R = 4 R_☉) at a distance of 1.6 kpc with an extinction of A_V = 4.5 mag (solid line).

Although this candidate BS nebula is faint and difficult to represent owing to a large dynamic range in the images, the morphological agreement with the theoretical shape is reasonably good. The apsis of this nebula is bright at [8.0] and [24] and may indicate an interaction or illumination of a nearby cloud. We retain this object as a probable BS.

3.4.3. BS Candidate 3 and Star 3 (HD195229)

BS candidate 3 appears to be generated by its attendant star, HD195229, listed as B0.2III in SIMBAD. Figures 17 and 18 show our WIRO and WIYN spectra of this star, from which we confirm a B0.2III spectral type. From these spectra we find a heliocentric radial velocity of −3 ± 2 km s⁻¹, slightly less negative than other massive stars near Cyg OB2. The luminosity for a B0.2 giant is somewhat uncertain because the evolutionary tracks for massive giant and supergiant stars are nearly vertical in an H–R diagram (Marigo et al. 2008). Assuming M_V = −5 ±  0.5, V − K = −0.8 and A_V = 1.0 from Figure 12 for this K = 7.3 star lead to a spectrophotometric distance of 1800 ± 500 pc, consistent with the distances of massive stars in Cyg OB2.

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HD195229 is also known as a high-proper-motion star with μ_R.A. = 0.14 ± 0.55 mas yr⁻¹, and μ_decl. = −2.89 ± 0.47 mas yr⁻¹ (Perryman et al. 1997). Its Hipparcos proper motion corresponds to a velocity of 28 ± 8 km s⁻¹ in the plane of the sky at 2000 pc. Combining this with the star's radial velocity gives a space velocity of 32 ± 8 km s⁻¹, placing it above the nominal 30 km s⁻¹ threshold for runaway status. The yellow bar in Figure 3 shows the implied sky motion (from left to right) over 10,000 yr and has position angle uncertainty of about 15°. The orientation of the candidate BS implies a velocity vector toward the lower left of Figure 3, while the proper-motion data imply a velocity that is toward the lower right.

Figure 13 shows that BS 3 is detected with certainty only at [24] and longer wavelengths, with a marginal detection at [3.6] and [4.5]. The stellar SED is well fit by an early-B giant (T_eff = 25,000 K, R = 10 R_☉), consistent with its spectral type, at a distance of 1.6 kpc with an extinction A_V = 1.0 mag. Curiously, HD195229 has a much lower extinction than the rest of the stars in Cyg OB2 (4–6 mag), requiring either that this star is seen through a local minimum in the obscuring dust, or that HD 195229 is actually on the near side of a heavy veil of extinction that enshrouds most of Cygnus X.

The lack of similarity between the [24] appearance and the theoretical shape in Figure 14, coupled with the misalignment between the proper motion and putative BS morphology, casts doubt on a BS interpretation. We categorize the nature of this nebula as ambiguous.

3.4.4. BS Candidate 4 and Star 4 (G79.4171+1.2703)

3.4.5. BS Candidate 5 and Star 5 (A10)

Figure 5 shows BS Candidate 5 to be a stunning combination of emission at the IRAC and MIPS bandpasses. The central star is A10 in the notation of Comerón et al. (2002). Our new spectrum of A10, shown in Figure 19, is consistent with a late-O star, although the uncertainty is several spectral subclasses. Its heliocentric radial velocity is 10 ± 10 km s⁻¹, consistent with other Cyg OB2 members. Its JHK colors and magnitudes are similar to the central star of BS Candidate 1, suggesting a similar reddening and late-O spectral type if at the distance of Cyg OB2. The angular distance between the star and BS apsis is ∼25'' or 0.19 pc projected separation.

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Figure 13 shows that the 2MASS data are well fit by an O9V (T_eff = 31,000 K, R = 8 R_☉) at a distance of 1.6 kpc with an extinction of A_V = 7.5 mag. The nebula dominates the SED at the IRAC and MIPS bandpasses.

Figure 14 shows that the [8.0] and [24] appearance closely resembles the theoretical BS shape. The orientation is consistent with either motion toward Cyg OB2 or interaction with a flow of material emanating from Cyg OB2. We classify this object as a probable BS.

3.4.6. BS Candidate 6 and Star 6 (G80.7657+0.4966)

3.4.7. BS Candidate 7 and Star 7 (A37)

3.4.8. BS Candidate 8 and Star 8 (G77.5168+1.9047)

3.4.9. BS Candidate 9 and Star 9 (G76.8437+0.1231)

3.4.10. BS 10 and Star 10 (GG77.0511 − 0.6092)

3.4.11. BD+43°3654

The BS candidates presented here comprise a mixed bag of phenomena. Objects 2, 5, and 7 (and BD +43°3654) are most likely to be genuine BSs powered by runaway stars on the basis of having early spectral types and nebular morphologies most similar to the classical BS appearance. However, none of stars 2, 5, or 7 have remarkable radial velocities. Objects 1, 3 (the high proper-motion star HD195299), 4, and 10 are ambiguous, having nebular morphologies less similar to the classical BS shape, but, nevertheless, appear to be early-type stars. Objects 6, 8, and 9 have central stars with excess IR emission, suggesting circumstellar material. Stars 6 and 9 also have high extinction and/or SEDs that rise toward longer wavelengths, consistent their being YSOs hosting disks or circumstellar envelopes. The nebula associated with Star 6 shows evidence for being the head of a gaseous pillar being ablated by radiation from the direction of Cyg OB2. Hence, objects 6, 8, and 9 are the least likely to be high velocity runaway stars.

In a magnitude-selected sample of 195 bright O stars, Gies & Bolton (1986) found that 16 (8%) could be classified as runaways either on the basis of radial velocities (6; 3%), proper motions (5; 2.5%), or extreme distance from the Galactic Plane (5; 2.5%). By contrast the radial velocity survey of 146 Cyg OB2 stars by Kiminki et al. (2007) did not find any runaways. The Spitzer Cygnus X Legacy Survey covers a much larger region (∼23 deg²) than the sample of core Cyg OB2 members studied by (Kiminki et al. 2007; ∼0.35 deg²), which was selected from the optical imaging survey of Massey & Thompson (1991). The ratio of areas in these two surveys is roughly 65, so one might conclude that the number of early-type stars in Cygnus X is as many as 146 ×65 = 9500. However, the earliest and most massive stars appear to be contained within the core Cyg OB2 region covered by Massey & Thompson (1991), so that this is a gross overestimate. We very conservatively adopt that the Spitzer  Cygnus X Legacy Survey includes ∼600 OB stars earlier than B2, or about 4 times the number of OB stars surveyed by Kiminki et al. (2007). In this region we find only three candidates for runaways—four if BD +43°3654 is included. At face value, this is 3/600 = 0.5%. Of course, high space velocities are a necessary but not a sufficient condition for the formation of BSs. van Buren et al. (1995) found that 58 out of 188 high velocity stars evince BSs at IRAS sensitivity levels, and of these, 25 have well resolved structure. We expect that Spitzer would detect even fainter and smaller BSs. If we assume that 1/4 of runaway stars produce BSs visible at Spitzer sensitivity levels and that 8% of all OB stars are runaways, we would expect to find on the order of 12 BSs in the Cygnus X region. Why do we detect so few?

We identify only three probable BS runaways, and perhaps only one of these (BS 2) has an orientation suggesting an origin near Cyg OB2. It is certainly possible that most of the runaways, like BD +43°3654, could fall outside the survey area if they have high space velocities and were probable ejected from the Association very early in its formation. Stars with a tangential velocity of 30 km s⁻¹ would travel beyond the boundaries of the Spitzer Cygnus X Legacy Survey (∼26 pc diameter) in about 0.5 Myr—a small fraction of the 2–4 Myr age of Cyg OB2. Dynamical ejection scenarios wherein N-body interactions in close multiple systems expel stars would predict this—that most of the runaways are produced early in the formation of a young massive star cluster and have already traveled beyond the immediate cluster vicinity. Supernova ejection scenarios, on the other hand, predict that the start of the ejection process is delayed several Myr (until the explosion of the most massive stars), and then it continues at a constant or increasing rate as stars from lower-mass OB binaries complete their evolution. Although the arguments are mostly qualitative, our findings would seem to be more consistent with the dynamical ejection scenario.

Because stellar mass loss rates are probably the most uncertain parameter in Equation (5) above, we invert the approach taken by Povich et al. (2008) and exploit the phenomenon to yield an estimate of the mass loss rates for the central stars. Inverting Equation (5) yields the mass loss rate of the star as a function of the relative motion between the str and ISM, the ambient density, the stellar wind speed, and the standoff distance,

$$\begin{equation} \dot{M}_{w,-6} = {{0.67 [R_0({\rm pc})]^2 [V_a ({\rm km\;s^{-1}})]^2 n_{a,3}} \over { V_{w,8}}}. \end{equation} \tag{ 6 }$$

R₀ is readily observed from the BS images, assuming a 1.6 kpc distance to Cygnus X and assuming cos i ≃ 1. We assume a minimum star–ISM speed of V_a = 30 km s⁻¹ for runaway stars (Gies & Bolton 1986), and we adopt the stellar wind speeds of Mokiem et al. (2007) which, for our sample of O5–B0 stars, range from 3000 km s⁻¹ to ∼800 km s⁻¹ (V_w,8 = 3 to <0.8). The ambient number density, n_a,3, is difficult to measure directly, but may be estimated from n_s,3, the post-shock density within the luminous BS nebulae. For strong, highly supersonic shocks, n_s,3 ≃ 4n_a,3 (Landau & Lifshitz 1987). Without knowing the Mach number of each individual case, we adopt n_s,3 = 2n_a,3, and consider the uncertainties to be factors of 2–4. We estimate n_s,3 by fitting the BS SEDs in Figure 13 with the dust/PAH models of Draine & Li (2007) which give the dust emissivity per H nucleon as a function of the radiation field intensity, parameterized as a multiple, U, times the mean interstellar radiation field of U_ISRF = 2.3 × 10⁻² erg s⁻¹ cm⁻² (Mathis et al. 1983). We make the motivated assumption that the BS luminosity stems from reprocessed photons from the central star rather than thermalized mechanical energy from the stellar wind (R. Benjamin 2009, private communication). The appropriate choice of model radiation field intensity, U, is then the ratio of the flux from the central star at the standoff distance over the mean interstellar field, estimated as,

$$\begin{equation} U = {{R_*^2 \sigma T_*^4}\over {R_0^2}} U_{{\rm ISRF}}^{-1}, \end{equation} \tag{ 7 }$$

where R_* is the stellar radius, T_* is the effective stellar temperature, and σ is the usual Stefan–Boltzmann constant. The postshock number density can then be estimated as

$$\begin{equation} n_{s,3} = 10^{-3} {{N} \over {V}}, \end{equation} \tag{ 8 }$$

where V is the BS volume which we estimate by approximating the BSs as hollow cones at a known distance, D, and N = L_obs/L₀, the ratio of the observed luminosity at a given bandpass to the theoretical luminosity per nucleon in the Draine & Li (2007) models. Effectively, n_s,3 is found empirically because it is the scale factor by which we multiply the Draine & Li (2007) models to match the observed SED. Because V ∝ D³ and the inferred luminosity L_obs ∝ D², our estimate for n_s,3 varies as D⁻¹, inversely with the adopted distance.

Draine & Li (2007) parameterize the model SEDs in terms of a maximum and minimum radiation field incident on the dust, U_max and U_min. While dust in the typical ISM is exposed to a range of intensities, we expect that material in the BSs is irradiated by a single intensity dictated by the stellar luminosity and standoff distance. Therefore, we adopt the models with U_max = U_min ≡ U, where U is chosen according to Equation (7). Figure 13 of Draine & Li (2007) shows that the flux ratio [24]/[70] is sensitive to U over the range 1 < U < 10⁷, and we make use of this sensitivity in fitting the BS SEDs below. We adopt the models appropriate to Milky Way dust and leave the PAH fraction, q_PAH, as a free parameter. Larger values of q_PAH raise the flux in the IRAC bandpasses where PAH emission is strong, but have no effect on the SED at [24] and [70].

Dotted lines in Figure 13 show illustrative dust models that provide reasonable fits to the mid- and far-IR data. The slope of the SED between [24] and [70] allows us to guess the appropriate value for U without any prior knowledge of the stellar luminosity or R₀. Interestingly, the values for U in each case calculated from Equation (7) are generally within factors of 2–3 of the initial guess. This correspondence serves as a kind of corroborating evidence that the BSs are, in fact, powered by the radiant energy from their central stars. The fits of the models to the data in Figure 13 are done by eye to maximize the agreement at [24] and [70] by adjusting n_s,3, and then the parameter q_PAH is varied to achieve a best fit at the IRAC bands. The models provide reasonably good fits to the data for objects 3, 5, and 8. In other cases, the overall fit is poor at the IRAC bandpasses because the model PAH features are too weak, even for the maximum model q_PAH, or too weak, even for the minimum model q_PAH. We take these discrepancies as indications that PAHs may be destroyed in the BSs (former case; objects 1, 4, and 7), and that PAH excitation may be elevated and dominate the SED in some objects (latter case; objects 2 and 6). The adopted values for U and n_s,3 appear in each panel.

Armed with estimates of the ambient number densities, n_a,3 = 1/2n_s,3, we proceed to calculate mass loss rates, $\dot{M}_{w,-6}$, using the method outlined above. Again, we consider only the four objects with the most reliable spectroscopic types where the BS nature is more likely to be correct. Table 4 summarizes the relevant parameters adopted for each of objects (2, 5, 7, 11). Derived mass loss rates vary from <0.18 × 10⁻⁶ M_☉ yr⁻¹ for the early-B star (Star 2) to ∼1 ×  10⁻⁶ M_☉ yr⁻¹ for the O9V star (Star 5), to ∼3 × 10⁻⁶ M_☉ yr⁻¹ for the O5V star (A37). The O4If star BD+43°3654 stands out, having an implied mass loss rate of 160× 10⁻⁶ M_☉ yr⁻¹. This large value results from the rather large standoff distance, R₀ = 1.4 pc, implied by the MSX IR image from Comerón & Pasquali (2007). We note that we have no direct measure of the ambient ISM density for this object, so we have assumed the average value from the other four early-type BS stars. Results for BD+43°3654 should be regarded as particularly uncertain. These results are broadly consistent with the mean $\dot{M}$ values for mid- to late-O stars estimated by Lamer & Leitherer (1993) and Fullerton et al. (2006) but a factor of several or more larger than the more recent compilation of Mokiem et al. (2007).

Table 4. Derived Mass Loss Rates for Most Reliable Objects

No.	S.T.	ns,3	$\dot{M}_{w,-6}$
		(103 cm−3)	(10−6 M☉ yr−1)
2	B1V–B3V	0.1	<0.18
5	O9V	0.1	0.75–1.3
7	O5V	0.1	2.5–3
11	O4If	0.1a	160

Note. ^aNot measured; value adopted from the average of other stars.

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We consider the uncertainties on mass loss rates to be factors of 2–4, dominated by the uncertainties on n_a,3, which are of the same magnitude. The other parameters that appear in Equation (6) are likely known to ∼30%, so that uncertainties in the standoff distance, which enter as R²₀ and V²_a, do not have a major impact on the result. We present the calculation of mass loss rates in this section as an outline of a novel method without claiming that this approach is in any sense superior to other well established techniques.