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Astron. Astrophys. 361, 1095-1111 (2000)

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3. Global properties of RAFGL7009S

3.1. Mass distribution

RAFGL7009S is classified as an ultra compact HII region, which implies small extensions and high densities. The observations gathered by McCutcheon et al. (1991) on this source indicate the presence of a molecular outflow associated with the source. The H2 column density is extremely high ([FORMULA]5.3[FORMULA]1022cm-2).

We propose, in this section, to obtain a global overview of the mass distribution at a given temperature in this object using the combination of IRAS data (at 12, 25, 60 and 100 µm) and the one obtained by McCutcheon et al. (1995) at larger wavelengths (450, 800 and 1100 µm). The flux emitted by the grains is related by an integral equation to the mass distribution. Under some assumptions, this equation can be inverted.

According to Kirchoff 's law, the monochromatic emission [FORMULA] of dust grains is related to their mean absorption coefficient per unit mass (g-1 cm2) [FORMULA] (Xie et al. 1993):


where [FORMULA] (g cm-3)is the grain mass density and [FORMULA] is the blackbody Planck function. We can define [FORMULA] (g cm-3 K-1) the dust mass per unit volume and at temperature T located at [FORMULA] in the cloud under consideration. The emerging flux is then given by:




where L is the spatial extent of the cloud along the line of sight, [FORMULA] is the solid angle subtended by the cloud, [FORMULA] is the density at the position [FORMULA].

The first part of the above equation represents the cosmological background contribution to the observed flux.

We can define:


where [FORMULA] is the dust mass distribution at a given temperature and D the distance to the source ([FORMULA] 3.3 kpc).

In the optically thin limit, [FORMULA], hence [FORMULA] and the thermal radiation escapes freely from the whole cloud. In this limit, the flux equation becomes independent of the cloud geometry and mass spatial distribution. With a position switched observation we can get rid of the background term (this one is however negligible if the source is a strong emitter). We have then:


We look for [FORMULA], the mass distribution at the temperature T, in the surrounding envelope. To do this, we ignore the source geometry which no longer enters explicitly in the calculation. We must therefore invert the previous equation. Following Xie et al. (1993), we use Chen's Möbius modified theorem (1990). This theorem expresses the fact that [FORMULA] is given by:


where [FORMULA] is the Möbius function that equals 1 if n=1, (-1)r if n possesses r distinct prime factors and otherwise [FORMULA]=0. We assume that the variation of the emissivity with frequency follows a power law, [FORMULA]. Writing [FORMULA], we define the function f(x) as the inverse Laplace transform of the flux:




To fully cover a range of possible values for the grain emissivity (e.g. Koike et al. 1995), we use two different absorption coefficients for the grains.

The first is given by Xie et al. (1993) and equals [FORMULA](cm2g-1) = [FORMULA] = 7.9[FORMULA]10[FORMULA]. The second is provided by more recent measurements done by Boulanger et al. (1996) leading to [FORMULA]/[FORMULA][FORMULA]10-25[FORMULA]([FORMULA]/250µm)-2. Assuming that the ratio [FORMULA]/[FORMULA]100, we obtain [FORMULA](cm2g-1) = [FORMULA] = 4.15[FORMULA]10[FORMULA] in the second case.

Observations in the far infrared/millimeter range are fitted using the functional form [FORMULA] (proposed by Xie et al. 1993) which seems adapted to the observations in this wavelength region and is analytically invertible by the Laplace Transform. The best fit parameters to the data leads to A = 3.258[FORMULA]10-78 and B = 8.673[FORMULA]10-6, [FORMULA] = 7 where [FORMULA] and [FORMULA] are given in Jansky and Hertz, respectively (see Fig. 1).

[FIGURE] Fig. 1. Spectral energy distribution in the line of sight of RAFGL7009S. The diamonds are measurements by McCutcheon et al., the stars are the IRAS filters measurements (at 12, 25, 60 and 100µm) and the continuous spectrum was obtained with the Short Wavelength Spectrometer on board ISO. The dotted line corresponds to the functional form [FORMULA] fitted to the long wavelength range data, and used in the temperature inversion (see text). The dashed line represent an estimate of the flux at higher frequency not used in the inversion as it is not in the optically thin region, and the dot-dashed line is the sum of the two contributions plus a higher temperature blackbody ([FORMULA]500K) as discussed in the text.

With these parameters we must then use the inverse Laplace Transform of the function [FORMULA] (we replaced [FORMULA] by the functional form [FORMULA] in the previous equation). Using the Laplace transform properties 1, the special functions [FORMULA] and [FORMULA], the inverse transforms we need to build [FORMULA] with [FORMULA]=3/2 and 2 are respectively:


The results of this inversion are shown in Fig. 2 for the two different values used for [FORMULA].

[FIGURE] Fig. 2. Mass distribution at a given temperature resulting from the inversion on the long wavelength range data: solid line [FORMULA]=2, dashed line [FORMULA]=3/2.

Integrating [FORMULA] over the temperature, the derived dust mass in the envelope is equal to 21.8[FORMULA] ([FORMULA]=3/2) and 43.4[FORMULA] ([FORMULA]=2). For both grain models, the bulk of the dust ([FORMULA] 95%) is very cold ([FORMULA] 33 and 29 K) for [FORMULA]and [FORMULA] respectively.

The law of variation of the emissivity with frequency drives the efficiency by which a grain absorbs and emits radiation. The higher the emissivity index is at long wavelength, the lower is the equilibrium temperature of the dust grains. We must therefore invoke more mass of matter to account for the same flux with a steeper emissivity law if we only fit the optically thin part of the emission. The observed flux is related to the emissivity by an integral equation containing a given fixed point ([FORMULA]) and a slope. Although the fixed point has to be determined carefully, [FORMULA], is a fundamental parameter to constrain if we want to derive a reliable mass estimate.

The continuous spectrum presented in Fig. 1 in the high frequency domain is the one measured by the Infrared Space Observatory (ISO). It shows an infrared excess. The IRAS 12µm measurement is considerably lower than the continuum. This is due to the presence of the strong silicate absorption as well as the water ice librational mode (hindered rotation in the solid phase). The presence of other molecules in the solid phase does not allow the spectrum to be fitted with a simple curve. However, the residual emission after subtraction of the fitted profile in the long wavelength part of the spectrum corresponds roughly to a blackbody of 80-120K assuming an emissivity [FORMULA] (dashed line in Fig. 1)).

The numerous absorptions, from both gas and solid phases, in the near and mid-infrared are discussed elsewhere (d'Hendecourt et al. 1996; Dartois et al. 1998). They appear on a continuum which can be attributed to a blackbody of 500K-1000K, emitted by dust grains located near the protostar(s).

When we performed the inversion, we did not take into account the short wavelength part of the spectrum (represented by the dashed fit in the Fig. 1) as the optically thin hypothesis is no longer valid. The inversion of the complete spectrum would require solving a more complex integral equation which needs to have source geometry well specified. Solving the equation in its general form will enhance the contribution of higher temperatures in the dust mass distribution presented in Fig. 2, lowering the proportion of very cold dust. However, the major part will still reside in the low temperature regime.

The continuum emission at 110.2 GHz is shown in Fig. 3. The integrated flux is 100 mJy, with an accuracy of 10%, assuming no loss due to the interferometer filtering (Half Power Beamsize [FORMULA] 44"  ). The dust envelope appears to indicate a simple power law density distribution. This finding contrasts with the CO gas phase observation of this source.

[FIGURE] Fig. 3. Continuum map of RAFGL7009S at 110.2 GHz, as derived by interferometer observations. The contours levels are 3mJy/Beam, representing the 3[FORMULA] level. The lower left corner represents the half power beam size of the clean beam

3.2. Mean H2 density

Using the 30m telescope, we observed C34S(2[FORMULA]1), [FORMULA](3[FORMULA]2) and CS(5[FORMULA]4) towards RAFGL7009s. The intensity ratio is C34S(2[FORMULA]1)/C34S(3[FORMULA]2) = 0.6. Using an LVG model (P. Schilke, private communication), we performed a statistical equilibrium calculation, using a C34S column density of 3.3[FORMULA]1013cm-2, and a line width of 5.5 km/s. The observed intensities are consistent with a high density (n(H2) [FORMULA] 106 cm-3). Assuming a terrestrial isotopic ratio for sulfur, we obtained a constraint on the kinetic temperature by comparing the CS(5-4) and C34S(3-2) data: the gas is quite cold, with TK [FORMULA] 20 K. We also observed two lines of para-formaldehyde, which lead to the same conclusion.

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© European Southern Observatory (ESO) 2000

Online publication: October 10, 2000