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Astron. Astrophys. 357, 637-650 (2000)

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4. Modelling the dust emission

Previously, good fits to the SEDs of similar young high mass sources have been produced using spherically symmetric dust shell models (Chini et al. 1987; Churchwell et al. 1990a (hereafter CWW90); Wolfire & Churchwell 1994; Hoare et al. 1991; Faison et al. 1998). The most successful models have used constant density shells (Chini et al. 1987; CWW90; Faison et al. 1998). These models have large outer radii ([FORMULA]) and large inner cavities ([FORMULA]) with a dust temperature at the inner edge of [FORMULA]. The grain mixture used is a mixture of crystalline carbon and silicate, with the graphite/silicate ratio reduced to about half the standard Mathis et al. (1977) abundances (MRN) in order to fit both the 9.7µm silicate feature and infrared optical depths. Unfortunately there is no overlap between our sample and the list of sources modelled previously; however, WC89 showed that the SEDs for all these UCHII region sources are remarkably similar, and Faison et al. (1998) modelled all 10 sources for which they had sufficient data with shells matching the above description, so we would expect that the SEDs of our sources could also be fitted with this model.

The submm observations presented here provide not only submm points on the SEDs but also images whch provide structural information, and these additional constraints lead us to try some further modelling of the dust distribution. Following previous work, we model the sources using spherically symmetric shells heated by one or more embedded massive stars. As well as SEDs and submm images, further constraints on the models come from physical conditions derived from molecular line studies and infrared optical depths.

To solve the radiative transfer for spherically symmetric dust shells we used the DUSTY program (Ivezic & Elitzur 1997; Ivezic et al. 1997), which has kindly been made available by the authors. DUSTY requires as input the properties of the central source and the surrounding dust shell - including grain properties, density distribution, temperature at inner radius, and optical depth at specified wavelength - and solves the equation of radiative transfer to find the temperature and the emission as function of radius. It then calculates the resulting spectral energy distribution and radial surface brightness profile.

In order to compare the radial profiles from DUSTY with our observations, we turned the 1D model profiles into 2D surface brightness maps and convolved with maps of Uranus which we take as representative of the telescope beam. Images of Uranus are given in Fig. 2 and the average radial profile is shown in Fig. 3. In creating the 2D maps we assumed a distance to match the source with which we wanted to compare the model. We extracted the 1D average radial flux profiles of the beam-convolved models, and did the same for the observed sources.

[FIGURE] Fig. 3. Uranus radial profiles (azimuthal averages) at 450µm (dashed) and 850µm (dotted). The two profiles at each wavelength correspond to images taken on August 13 and 14.

The data to which the SEDs are fitted comes from IRAS (12-100µm), the SCUBA observations presented here (450-2000µm), and Chini et al. (1986a,b) (1300µm). We have not attempted to fit the short wavelength (1-5µm) fluxes for these sources (Moorwood & Salinari 1981; Chini et al. 1987). High resolution near-IR observations show clusters of stars surrounding UCHII regions, and the flux in a 10-[FORMULA] beam is dominated by stars which are not embedded or powering the UCHII region itself (Feldt et al. 1998, 1999). Spherically symmetric models tend to be optically thick at these wavelengths, whereas models of flattened envelopes or with outflow cavities allow some scattered starlight to reach the observer. Although the near-IR fluxes may be confused by nearby stars, measurements of extinction of the hydrogen recombination lines in the 1-5µm range are a valid constraint, as these are emitted from the UCHII region itself.

In DUSTY the shape of the SED is independent of the luminosity of the central source but to compare with the measured fluxes we scaled the model SEDs to the estimated luminosity of each source. For luminosities we initially used the values in Churchwell et al. (1990b), that is, the bolometric fluxes from IRAS data as calculated by WC89 corrected for the revised distances of Churchwell et al. (1990b). We later adjusted some of the luminosities in order to fit the SEDs better (see Sect. 5).

4.1. Constant density models

In Fig. 4 we show a DUSTY reproduction of the CWW90 best-fit large radius constant density model together with SED data and radial profiles for two sources in our sample, G13.87 and G10.47. The CWW90 model is scaled to appropriate fluxes for our sources (it was chosen to fit another UCHII region source, G5.89). Parameters for this and other models are given in Table 4. These sources were selected for comparison because they have comparable distances (4.4 and 5.8 kpc) but different radial profiles: whereas G10.47 is typical of the strongly peaked sources, with a radial distribution which falls off quickly on small scales, G13.87 has a much flatter distribution, as can be seen in Fig. 4. The fit to the SEDs is reasonable, remembering that the CWW90 model is actually for G5.89 and a better fit to our sources could be obtained by adjusting the radius or column density. However, a large radius constant density shell fails badly to fit the radial profiles in the submillimetre: within the shell ([FORMULA]) the profile is too flat; further out, it falls off too quickly with radius at 850µm (at 450µm the flux at large radius is a result of the large error beam).

[FIGURE] Fig. 4. SED (left) and radial profiles (right) for a flat density distribution source following CWW90, compared with observed fluxes and profiles for G10.47 and G13.87 (errorbars). Profiles at 850µm and 450µm are shown solid and dashed lines respectively. In the plot of profiles, the lower and upper of each pair of curves with the same origin are for G10.47 and G13.87 respectively. The CWW90-type model parameters are given in Table 4


Table 4. Model parameters as applied to each source: density index [FORMULA], optical depth [FORMULA], inner and outer radii [FORMULA] and [FORMULA], and derived column density and mass. Bolometric luminosities for each source are given in Table 5.
a) Masses for the [FORMULA] models are for the combined low plus high optical depth combinations described in the text.
b) CWW90-style flat density model (see Sect. 4.1)

We cannot see how to reconcile the large radius constant density shells which fit the SEDs so well with our submillimetre radial profiles. The form of the radial profiles is largely determined by the chosen density distribution, as this in turn determines the radial temperature distribution. The flat density distributions that have been used to explain the SEDs result in submillimetre profiles that are too shallow. Of the other parameters that can be altered within the range of existing flat density models, increasing the temperature at the inner boundary results in SEDs which peak shortward of 100µm and does not solve the flat profile problem; increasing the outer radius extends the flat profile section further out, making the fit worse; and reducing it produces a sharp falloff which underpredicts the fluxes at large radius (though it produces a good fit to the inner parts of the strongly peaked sources, as we show later).

4.2. [FORMULA] models

In order to model the submm radial profiles we need to use a steeper density distribution, but this adversely affects the fits to the SEDs. The steeper density distributions have relatively more material close to the star and produce more emission at shorter wavelengths. For example, Hoare et al. (1991) favoured an [FORMULA] density distribution which was consistent with their 800µm scans and with densities at large radius but produced an SED which peaked at 70-80µm. As noted by CWW90, in order to have the SED peak at [FORMULA], a high optical depth is required which produces too strong a silicate feature and near-IR optical depths at the Br[FORMULA], Br[FORMULA] and Pf[FORMULA] wavelengths which are also higher than observed. Conversely, if the near-infrared part of the spectrum is fitted, then the submm emission is too low and the SED peaks at too short a wavelength.

One way to produce both long and short-wavelength emission is to combine SEDs of different optical depth, considering the sources to have different column densities of dust in different directions, breaking the spherical symmetry. In this way, the observed silicate feature and NIR optical depths can be produced by regions of low opacity and the submm emission by higher optical depth paths. As molecular clouds are observed to be clumpy, and a popular explanation for cometary UCHII regions is that they have formed on the edge of molecular clouds and are expanding into regions of different densities on each side (Tenorio-Tagle et al. 1977; Dyson et al. 1995; Redman et al. 1996; Williams et al. 1996; Lizano et al. 1996), it seems reasonable that the dust optical depth should vary with direction.

The radiative transfer code calculates for spherically symmetric geometry only. In order to model a non-spherical case, we have had to make an approximation: for the purpose of modelling, we assume that the different optical depth components do not obscure each other. This allows us to add their SEDs without modification. What we have modelled is a geometry where these components do not shadow each other - strictly, where the components are like segments of an orange viewed from the stem. A more realistic simple geometry would be a high optical depth sphere/torus with conical regions of low optical depth (perhaps cleared by a bipolar flow). In an orientation where one cone is forward-facing, there is little shadowing by the surrounding material, and our model is a good approximation to this scenario.

We first consider the sources with the flattest distributions in our sample - the non-peaked sources G13.87 and G43.89 - and look at the radial profiles to determine the density distribution. In Sect. 4.4 we shall see that neither [FORMULA] or [FORMULA] models are consistent with the radial profiles. Here we note that a [FORMULA] density distribution fits very well. Fig. 5 shows the submm profiles for [FORMULA] shells compared to the measured profiles for G13.87 and G43.89. Fig. 5 also shows that by combining two [FORMULA] partial shells with 100µm optical depths of 0.1 and 0.9, we can also produce acceptable fits to the SEDs. The composition of the SED is shown for G13.87.

[FIGURE] Fig. 5. SEDs and radial profiles for the non-peaked sources G13.87 (top) and G43.89 (bottom), fitted by [FORMULA] density distributions. The SED for both G13.87 and G43.89 is the combination of [FORMULA] and [FORMULA] models given in Table 4. These components are shown individually for G13.87. SCUBA fluxes are shown as squares: fluxes at 450 and 850µm are integrated over the map, whereas the 1350 and 2000µm fluxes are single pixel measurements and effectively lower limits. Other datapoints are near-IR (+) (Chini et al. 1987), IRAS 12-100µm (triangles); and 1300µm ([FORMULA]) (Chini et al. 1986a,b).

We are not concerned about the small discrepancies that remain between the models and the data, as several further sources of uncertainty are not represented in the errorbars. The errorbars on the SED do not take into account the fact that different measurements were taken with different pointing centres and with different beam sizes, though the [FORMULA] field of view over which the SCUBA fluxes were integrated is comparable to the IRAS beam at longer wavelengths. The 1350 and 2000µm points were single beam measurements and are effectively lower limits. Uncertainties on the radial profiles are from the noise on the map only and do not take into account strong asymmetries (such as multiple cores) in the SCUBA maps, or variation in the beam shape between the Uranus observations used to convolve the models and the actual beam shape for the source observations.

The model parameters are given in Table 4. The central radiation field is black body with temperature 42,000 K, equivalent to an 06 star (the exact temperature of the star has little effect on the SED, which is all reprocessed flux). We assume MRN grains throughout. The shell inner radius is set by its temperature, which we take as 300 K throughout (similar to Faison et al. (1998) and CWW90). The significance of this inner temperature and the effect of changing it is discussed further below. The temperatures at the outer edge are also given in Table 4, and at 10-20 K are typical of ambient molecular cloud material. For each source, the shell thickness (or outer radius) is chosen to fit the observed radial profile. The fits were carried out by eye from a limited range of models: current computing resources preclude running DUSTY for more than a few carefully chosen sets of input parameters. The SEDs for G13.87 and G43.89 were fitted by a combination of [FORMULA] and 0.9 [FORMULA] SEDs contributing to the bolometric luminosity in proportion 1:1 (G43.89) and 2:3 (G13.87).

4.3. [FORMULA] models with cores

The peaked sources - G10.47, G12.21, and G31.41 - cannot be fitted simply by [FORMULA] distributions. At small radii, the observed emission falls off too steeply. Comparison with Uranus profiles suggests it is more consistent with a compact central source of dimensions much smaller than the beam. Also motivated by the knowledge that these sources have molecular hot cores, we consider the addition of a compact high column density core component to an [FORMULA] outer envelope similar to the [FORMULA] distributions used to fit the non-peaked sources.

These cores must be compact (much smaller than the beam) from the radial profiles, but as the cores are unresolved, the exact radial distribution is poorly constrained. From the relative fluxes in the peak/extended components of the images, the core contributes less than [FORMULA] as much submm flux as the total [FORMULA] contribution to the SED. This enables us to rule out cores with SEDs which peak shorter than 100µm, as these produce FIR fluxes in excess of the IRAS measurements when scaled to contribute enough flux in the submm. To produce a long-wavelength dominated SED from a compact core requires very high optical depths as the material must lie close to the heating source in order to emit enough flux; the more centrally condensed the distribution, the higher the optical depth.

If we identify the submm cores with hot molecular cores, then the molecular line observations provide some constraints on the parameters of the cores. From molecular line observations, column densities in hot cores exceed [FORMULA] and may reach [FORMULA] (Cesaroni et al. 1992, 1994a; Hatchell et al. 1998a); assuming a 100µm absorption coefficient [FORMULA] (Wolfire & Churchwell 1994) this corresponds to [FORMULA]. For our compact core component, as the radial distribution is poorly constrained from the profiles, we use a constant density model with [FORMULA] and a core radius of [FORMULA], consistent with the measured molecular line column densities and source sizes (Cesaroni et al. 1992, 1994a; Hatchell et al. 1998a).

Fig. 6 shows the radial profile of this core (convolved with the beam) together with the observed profiles of G10.47 which is a peaked source. The core model fits the central peak well but falls away too fast; an additional extended component is needed. An obvious candidate is an [FORMULA] envelope like that used to fit the non-peaked sources. However, a spherically symmetric model with a single dense central core, modelled as a break in the radial density distribution, cannot reproduce the SEDs. With all the radiation from the star processed through the core, such a model produces a shortfall of flux at short wavelengths. In order to avoid this, we assume that the cores are not centrally positioned in the envelopes and are independently heated from the envelopes for the purposes of the modelling. The geometry we envisage places the core within a few arcseconds of the star which heats the envelope, at or near the inner edge of the envelope. The cores must be close to the envelope heating sources, and for the radial distribution modelling, we have assumed the centres are indistinguishable with our [FORMULA]/[FORMULA] beam. This geometry leaves low optical depth paths from the envelope heating source which can account for the mid-IR fluxes.

[FIGURE] Fig. 6. Core model (without envelope) at 450µm (dashed line) and 850µm (solid line) compared with observed profiles (errorbars) for the peaked source G10.47.

Again, we are limited to combining spherical solutions for our models. By forming linear combinations of the SEDs, effectively we neglect the effect on the core SED of obscuration by envelope material along the line of sight. The amount of obscuration would depend on the exact (unknown) position of the core relative to the envelope heating star, and this would modify the exact shape of the core SED. However, our core SEDs are very approximate in shape anyway as we know very little about the size and density distribution of the cores. The important characteristics of the core SED - that it peaks longward of [FORMULA]m and contributes strongly at submm wavelengths - will not be altered even by obscuration by an envelope that is optically thick out to [FORMULA]m.

Fig. 7 shows the SED and radial profile fits to the peaked sources G10.47, G12.21 and G31.41. In each case the fit is a combination of two [FORMULA] envelope components with optical depths of [FORMULA], plus a compact core with [FORMULA] as described above. The composition of the fit is shown for G10.47. As for the non-peaked sources, the outer radii of the [FORMULA] distributions were chosen to match the radial profiles. Again, the model parameters are given in Table 4. For each source, the [FORMULA] and 0.9 components were combined in ratio 2:3. The proportions of luminosity attributed to core and envelope, which are strongly constrained by the radial profiles, were (core:envelope) 1:3 (G10.47), 3:17 (G12.21) and 7:13 (G31.41). The mismatch between the model and observed radial profiles 30-[FORMULA] from the central peak in G10.47 can be explained by the additional dust cores (Fig. 1).

[FIGURE] Fig. 7. SEDs and radial profiles for the peaked sources G10.47 (top), G12.21 (middle) and G31.41 (bottom), fitted by [FORMULA] envelopes plus compact cores. The SED for G10.47 shows the combination of [FORMULA] and [FORMULA] models to form the envelope SED, plus a compact core, used to form the final SED. SCUBA fluxes are shown as squares: fluxes at 450 and 850µm are integrated over the map, whereas the 1350 and 2000µm fluxes are single pixel measurements and effectively lower limits. Other datapoints are near-IR (+) (Chini et al. 1987), IRAS 12-100µm (triangles); and 1300µm ([FORMULA]) (Chini et al. 1986a,b).

The opacities from the 2-4µm Br[FORMULA], Br[FORMULA] and Pf[FORMULA] lines as measured for G5.89 (CWW90), which lie in the range 2.5-5.6, are consistent with those of the low optical depth [FORMULA] models with which we fitted the SEDs, which have 2-4µm opacities of 1.0-3.1 ([FORMULA]) and 2.9-9.4 ([FORMULA]).

4.4. Other models

The [FORMULA] and [FORMULA] plus core combinations that we have shown in Figs. 5 and 7 are chosen to produce a reasonable fit to the data with as few components as possible. The combinations are possibly not unique. Also the models have been chosen by eye by combining a limited number of models, and not by iterating through the parameter space to select models according to some best fit criterion. Therefore we hesitate to place too much weight on the exact values of parameters that we used, as there is a risk of overinterpretation. However, in our experience it is quite difficult to produce fits to both the SED and the radial distributions simultaneously, and while a few parameters are poorly constrained, certain areas of parameter space can be ruled out.

In Sect. 4.1 we noted that flat density distribution envelopes fail to fit the submm radial profiles. Fig. 8 shows that [FORMULA] and [FORMULA] density distributions have too steep and too shallow radial profiles, respectively, for the envelope.

[FIGURE] Fig. 8. [FORMULA] (solid) and [FORMULA] (dashed) density models compared with 450 and 850µm radial profiles of the two non-peaked sources G13.87 and G43.89.

For both envelope and core, we have taken the dust temperature at the inner radius of the shell to be 300 K. The exact temperature is not well constrained but we can rule out inner temperatures consistent with dust sublimation temperatures of [FORMULA], or too much short wavelength emission is produced. Similarly low temperatures were required by CWW90 and Faison et al. (1998) for their constant density fits to the SED, and the problem is worse for more centrally condensed [FORMULA] density distributions. Fig. 9 shows SEDs for a [FORMULA] envelope with an inner temperature of 1000 K and two optical depths, and data from G13.87 and G10.47. 1-20µm data for these sources would provide better constraints, but even for the lower optical depth model the silicate feature is much stronger than other similar sources (Faison et al. 1998), and the 2-4µm optical depths of 10-40 are much larger than the 2-6 observed in G5.89 (CWW90).

[FIGURE] Fig. 9. SED of [FORMULA] model with 1000 K inner temperature for [FORMULA] (solid) and 3.0 (dashed) compared with observed fluxes for G10.47 and G13.87 (errorbars).

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© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000