Astron. Astrophys. 325, 450-456 (1997)

4. The radiative transfer model

4.1. Model description

In the KS94 model, dust can be heated by i) a central point source, ii) stars in an extended stellar cluster iii) a mean interstellar radiation field and iv) locally in hot spots. These different models of the principle heating sources for the dust are best distinguished by the observed distribution of the IR emission. Hot spot models, which are particularly suited for star-burst galaxies such as M82, show order of magnitude larger mid-IR sizes compared to models having a central point source. In addition to the continuum emission, this model also includes the Si-O stretching vibration at 9.7 and the PAH emission features which can be well explained by vibrational excitation of C-H, C=C fundamental modes of polycyclic aromatic hydrocarbons (PAH; Allamandola et al., 1989; Puget & Léger, 1989). Beside the specific resonances of the PAHs, very small dust particles also have to be taken into account since they enhance the near and mid IR emission.

For Circinus, the observations presented here show that at least the mid-infrared luminosity originates in a compact central source which is much smaller than the starburst ring. The photoionization modelling of the high excitation emission lines by M96 and the relative weakness of Br in the starburst ring also suggest that re-processed EUV photons from the AGN probably dominate the total infrared luminosity. Our main aim in applying this radiative transfer code, therefore, was to test if a central heating source could adequately reproduce the observed size versus wavelength dependence and overall spectral energy distribution, including the presence of PAH features generally attributed to star forming regions. Having assumed a central power law heating source the remaining `free' parameters in the model are the:

These parameters have either been fixed or varied within the observational constraints to achieve the best fit as follows:

i) The intrinsic luminosity of the central source has been assumed equal to the observed total infrared luminosity i.e. .

ii) Following M96 we adopt a spectrum extending from the near infrared to 300 Å below which the dust properties are unknown (Zubko et al., 1996, Zubko et al., 1997). The resulting infrared spectra are actually relatively insensitive to the exact value in the range 1 and the predicted spectral energy distribution alone cannot be used to definitively distinguish a galactic nucleus powered by a super-massive object from a compact star formation region (Krügel et al., 1983). Both the compact size of the infrared emitting region and the EUV spectrum inferred from the high excitation emission lines, however, argue against a star cluster contributing a significant fraction of the total luminosity.

iii) The inner boundary has been assumed equal to the evaporation temperature () of the large grains (Churchwell et al., 1990). Since the chemistry of the individual dust populations and other processes involved such as grain sputtering, destruction in shock waves, interaction with charged particles, etc. are not well understood, this parameter is not very precise but sets a lower limit on the inner boundary. We use as evaporation temperature for the silicates and for the large carbon particles C K and find that the grains melt at a distance pc.

iv) We have adopted the upper limit of 23" for the FWHM in our 1.3mm continuum map as the best estimate for the outer radius pc of the dust emitting region.

v) The adopted gas mass is that deduced from the 1.3mm continuum observations. It depends on the grain properties (see Eq. 1) and is for the standard dust model used here.

vi) A resolved far IR or millimeter continuum map is required to derive a good measure of the dust density distribution . Here we have assumed that . Both parameters and are formally constrained by the two conditions required to fit the total gas mass and the depth of the Si-O absorption . Within physically meaningful limits (e.g. ), we derive for a spherically symmetric dust density distribution:

where

and

where denotes the proton mass. The Eq. (2-4), limit the parameter space of the models.

In our standard dust model, we assume spherical dust particles, made of pure carbon or astronomical silicate material and having a power-law size distribution (Mathis et al., 1977). For such a grain mixture and with size parameters as given in Table 2, g/cm² is derived. For comparison if one adopts only astronomical silicates for the large grains we find g/cm². Both values of are uncertain, as it is known that inhomogeneities, impurities, non-spherically symmetric grain shapes and "fluffiness" have a large influence on the emissivities (Ossenkopf, 1991, 1993, Krügel & Siebenmorgen, 1994b, Stognienko et al., 1995, Siebenmorgen & Gredel, 1997). The depth of the Si-O band was originally estimated by Moorwood & Glass (1984). These authors found assuming an underlying black-body spectrum with T = 292 K or assuming an underlying optically thin silicate emission spectrum. We find a slightly better fit to the SED for . Applying parameters as specified in Table 2 into Eq. (2-4) we find .

Table 2. Best fit parameters

4.2. Mathematical formulation

Following the notation by KS94, the radiative transfer equation is written:

which is solved for a spherical geometry using a ray tracing method described by Siebenmorgen et al. (1992). The source function is

where the dust emission is calculated from

Equilibrium temperatures of the large grains are calculated self-consistently for every grain size at every location. The temperature distribution function accounts for the quantum heating process of the small grains and is computed as discussed by Siebenmorgen et al. (1992). The photo-destruction of the PAHs is calculated as in Siebenmorgen (1993).

The central heating source is introduced by the inner boundary condition:

where

The cut off frequency is not well defined. By using we ensure that the emission in the IR is due to dust.

© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998

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