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Spectral energy distribution

Details of the computer code used for solution of the radiative transfer in dusty envelopes can be found in Szczerba et al. (1997). In brief: the frequency-dependent radiative transfer equation is solved for a dust under assumption of spherically-symmetric geometry for its distribution taking into account particle size distribution and quantum heating effects for the very small dust particles.

The modelled source is certainly C-rich (see Omont et al. 1995 and Kwok et al. 1995). Therefore, for modelling of its spectral energy distribution (SED) we assumed that dust is composed of: policyclic aromatic hydrocarbons (PAH) for dust sizes a between 5 and 10Å (see Szczerba et al. (1997) for details concerning PAH properties), amorphous carbon grains (of AC type from Rouleau & Martin 1991) for a>50Å and dust with an opacity obtained from averaging of the absorption efficiences for PAH and AC grains according to the formula:

$$Q_{{\rm abs},\,\nu}\,=\, {\rm f}\,\cdot\,Q_{{\rm abs},\,\nu... ...}(a)\, +\,(1-{\rm f})\,\cdot\,Q_{{\rm abs},\,\nu}^{\rm AC}(a),$$

for grain sizes between 10 and 50Å. Here: f=1 for a= 10Å and f=0 for a=50Å. Dust with opacity values constructed in this way allow us to use a continous distribution of dust grain sizes and fill the gap between properties of carbon-bearing molecules and small carbon grains.

The ${\rm 21\,\mu}$m feature was approximated by a gaussian with parameters determined from modelling of IRAS07134+1005 (centre wavelength equal to ${\rm 20.6\,\mu}$m, and width of ${\rm 1.5\,\mu}$m) which has the strongest feature among the known ${\rm 21\,\mu}$m sources. In the case of ${\rm 30\,\mu}$m band we used the addition of two half-gausians with the same strength and different width. Initial fit was done to IRAS 22272+5435 and its parameters were: width for short wavelength side $\sigma_{\rm L}\,=\,{\rm 4\,\mu}$m, width for long wavelength side ${\sigma_{\rm R}\,=\,9\,\mu}$m and central wavelength ${\rm 27.2\,\mu}$m (see Szczerba et al. 1997). For modelling of IRAS04296 we have reduced the strength of this feature by 50%. Superposition of the 21 and ${\rm 30\,\mu}$m features was added to the absorption properties of amorphous carbon in order to construct an empirical opacity function (EOF).

In Fig.7 the best fit obtained from the solution of the radiative transfer problem including quantum heating effects for the PAH grains is shown together with observational data which will be described in detail elsewhere. Note, however, that we present also two sets of photometry (from B to M band) corrected for interstellar extinction (open symbols) according to the average extinction law of Cardelli et al. (1989), assuming that total extinction at V is 1.0 or 2.0 magnitudes and plotting only the smallest and largest value of corrected fluxes at given band.

 

This estimate of the total extinction range can be inferred from the analysis of data presented by Burstein & Heiles (1982).

 

$$L_{\rm star}$$ 3.92

d 5.4kpc

$$R_{\rm out}$$ 0.5pc

$$V_{\rm exp}$$ 12kms^-1

$$R_{\rm in}$$ 6.410^-4pc

$$\overline{T}_{\rm d} _{\rm in}$$ 870K

$$\rho_{\rm gas} \sim$$

$$_{\rm post-AGB}$$

$$R_{\rm in}$$ 7.0610^-3pc

$$\overline{T}_{\rm d} _{\rm in}$$ 270K

$$\rho_{\rm gas} \sim$$

$$_{\rm AGB}^{\rm min}$$

$$_{\rm AGB}^{\rm max}$$

a_- 5Å

$${\mu}$$ a₊

p 3.5

$$t_{\rm dyn}$$ 575yr

$$M_{\rm dust}$$

The best fit to the spectral energy distribution of IRAS04296 is shown by heavy solid line (see Table2 for details concerning parameters of the model). Our modelling procedure was such that we tried to get fits to SED which fall in between the extinction corrected fluxes. In this way, we have taken into account not only the effect of the circumstellar extinction but also of interstellar extinction. The thin long-dashed line represents the input energy distribution of the central star for logg=0.5 and =6500K according to model atmosphere calculations of Kurucz (private communication). The heavy short-dashed line shows the fit which was obtained with the same assumptions but changing the effective temperature of the star to 6000K. As one can immediately see in the IR range of the spectrum the quality of the fits are very similar. However, in the optical and ultraviolet (UV) part of the spectrum the fit assuming =6000K is not able to explain extinction corrected data. In consequence, we are quite convinced that our estimation of for IRAS04296 close to 6500K is reasonable and, what is even more important, agrees pretty well with the spectroscopic estimation (6300K). Note that spectral type of this source was found to be G0 Ia from the low resolution spectrum (Hrivnak 1995) which implies an effective temperature of around 5500K for the star if we asume that the same relationship applies for post-AGB supergiants as for ``normal'' ones (see Schmidt-Kaler 1982). For such a low temperature we were not able to fit even the reddenned data in the UV.

The thin solid line in the wavelength range from about 18 to ${\rm 48\,\mu}$m represents the model continuum level found after solution of radiative transfer equation for dust without using the EOF parameters as in Tab.2 while keeping the dust temperature (or probability distribution of dust temperature) the same as for the case of dust with EOF. Taking into account the estimated continuum level and assuming that 21${\mu}$m feature extends from 18 to 22 ${\mu}$m we estimate the energy emitted in 21 ${\mu}$m band as about 5.7 % of the total IR flux (251 for $\lambda$'s from 5 to 300 ${\mu}$m assuming a distance to the source of 1 kpc). With the dotted line for wavelengths longer than 18 ${\mu}$m we present the fit which was obtained using an opacity function with the EOF for only 21 ${\mu}$m component. It is clear that such fit is not able to explain IRAS photometry at 25 ${\mu}$m. Our recent ISO observations show that this source is also a 30${\mu}$m emitter. In the forthcoming paper (Szczerba et al. 1998, in preparation) we will discuss this finding in detail.