## 3. Dust shell models## 3.1. Spectral energy distributionThe spectral energy distrubion (SED) of IRC +10 420 with 9.7 and
18
IRC +10 420 is highly reddened due to an extinction of by the interstellar medium and the circumstellar shell. From polarization studies Craine et al. (1976) estimated an interstellar extinction of . Jones et al. (1993) derived from their polarization data to . Based on the strength of the diffuse interstellar bands Oudmaijer (1998) inferred for the interstellar contribution compared to a total of . We will use an interstellar of as in Oudmaijer et al. (1996). This interstellar reddening was taken into account by adopting the method of Savage & Mathis (1979) with . ## 3.2. The radiative transfer codeIn order to model both the observed SED and m visibility, we performed radiative transfer calculations for dust shells assuming spherical symmetry. We used the code DUSTY developed by Ivezi et al. (1997), which solves the spherical radiative transfer problem utilizing the self-similarity and scaling behaviour of IR emission from radiatively heated dust (Ivezi & Elitzur 1997). To solve the radiative transfer problem including absorption, emission and scattering several properties of the central source and its surrounding envelope are required, viz. (i) the spectral shape of the central source's radiation; (ii) the dust properties, i.e. the envelope's chemical composition and grain size distribution as well as the dust temperature at the inner boundary; (iii) the relative thickness of the envelope, i.e. the ratio of outer to inner shell radius, and the density distribution; and (iv) the total optical depth at a given reference wavelength. The code has been expanded for the calculation of synthetic visibilities as described by Gauger et al. (1999). ## 3.3. Single-shell modelsWe calculated various models considering the following parameters
within the radiative transfer calculations: SED and visibility were
modelled for to 9000 K, black bodies
and Kurucz (1992) model atmospheres as central sources of radiation,
different silicates (Draine & Lee 1984, Ossenkopf et al. 1992,
David & Pegourie 1995), single-sized grains with
to 0.6 Fig. 3 shows the SED calculated for
=7000 K,
=,
Draine & Lee (1984) silicates, MRN grain size distribution
(m) and different dust temperatures.
It illustrates that the long-wavelength range is sufficiently well
fitted for cool dust with K, optical
wavelengths and silicate features require
. The inner radius of the dust shell
is at
(: stellar radius), the equilibrium
temperature at the outer boundary amounts to
K. However, the fit fails in the
near-infrared underestimating the flux between 2 and
5 We found this behaviour of single-shell SEDs to be almost
independent of various input parameters. Increasing
from
to
leads to somewhat higher fluxes, but only for
m. The equilibrium temperature at the
outer boundary decreases by a factor of two if the shell's thickness
is increased by one order of magnitude. Larger
gives slightly less flux in the
near-infrared, larger wavelengths (m)
are almost unaffected. The Draine & Lee (1984) and David &
Pegourie (1995) silicates give almost identical results, the optical
constants of Ossenkopf et al. (1992) lead to a larger
9.7 The 2.11
## 3.4. Multiple dust-shell compenents## 3.4.1. Two component shells
Since we failed to model the SED with the assumptions made so far, we
introduced a two-component shell as Oudmaijer et al. (1996). For that
purpose, we assume that IRC +10 420 had passed through a superwind
phase in its history as can be expected from its evolutionary status
(see Schaller et al. 1992, Garcia-Segura et al. 1996). This is in line
with the conclusions drawn from the Oudmaijer et al. (1996) model and
recent interpretations of HST data (Humphreys et al. 1997). A previous
superwind phase leads to changes in the density distribution, i.e.
there is a region in the dust shell which shows a density enhancement
over the normal distribution. The
radial density distribution may also change within this superwind
shell. For more details, see Suh & Jones (1997). Since dust
formation operates on very short timescales in OH/IR stars, we assume
a constant outflow velocity for most of the superwind phase and thus a
density distribution. For
simplicity, we consider only single jumps with enhancement factors, or
amplitudes,
Concerning the grains we stay with Draine & Lee (1984) silicates and an MRN grain size distribution with m and m as in the case of the single shell models. The influence of different grain-size distributions will be discussed later. We calculated a grid of models for
K with superwinds at
to 8.5 with amplitudes
Fig. 7 shows the fractional contributions of the direct stellar
radiation, the scattered radiation and the dust emission to the total
emerging flux. The stellar contribution has its maximum at
2.2
## 3.4.2. Influence of the grain-size distributionAs in the case of the single-shell models we also studied other
grain size distributions. The MRN distribution derived for the
interstellar medium gives a continuous decrease of the number density
with increasing grain sizes. On the other hand, the distribution of
grains in dust-shells of evolved stars rather appears to be peaked at
a dominant size (e.g. Krüger & Sedlmayr 1997, Winters et al.
1997). It is noteworthy that even in the case of a sharply peaked size
distribution the few larger particles can contribute significantly to
the absorption and scattering coefficients (see Winters et al. 1997).
Accordingly, the 2.11 Concerning the visibility, a larger (smaller) negative exponent in
the distribution function can, in principle, be compensated by
increasing (decreasing) the maximum grain size. For instance,
requires
m to fit the 2.11 For a given exponent in the grain-size distribution function of
we arrive at the same maximum grain
size as in the case of the one-component model, viz.
0.45
If sufficiently small, the lower cut-off grain size can be changed moderately (within a factor of two) without any significant change for SED and visibility. If exceeds, say, m, the fits of the observations begin to become worse. For instance, the curvature of the visibility at low spatial frequencies and the flux-peak ratio of the silicate features are then overestimated. Finally, we repeated the calculations under the assumption of
single-sized grains. In order to model the visibility a grain size
## 3.4.3. Influence of the density distributionInspection of the best fits derived so far reveals that there are still some shortcomings of the models. First, although being within the observational error bars, the model visibilities always show a larger curvature at low spatial frequencies. This seems to be almost independent of the chosen grain-size distribution. Second, the flux beyond m is somewhat too low. This may be due to our choice of a density distribution for both shells. We recalculated the model grid for different density distributions for both shells with x ranging between 1 and 4. A flatter distribution in the outer shell increases the flux in the long-wavelength range as required but leads also to a drop of the flux in the near-infrared. The plateau in the visibility curve remains unaffected but the curvature at low spatial frequencies is increased. To take advantage of the better far-infrared properties of cool shells with flatter density distributions, but to counteract their disadvantage in the near-infrared and at low spatial frequencies, the density distribution of the inner shell also has to be changed. It should be somewhat steeper than the normal distribution. Then the near-infrared flux is raised and the visibility shows a smaller curvature in the low-frequency range. It should be noted that the curvature is most affected for superwinds of low amplitudes. However, the steeper density decrease in the inner shell leads to increasingly low visibility values in the high frequency range. Since this has to be compensated by an increase of the superwind amplitude the advantages of the steeper distribution are almost cancelled. Thus, we can stay with a density distrubution in the inner shell and moderate superwind amplitudes (). The then best suited models we found are those with superwinds at and a distribution in the outer shell. The corresponding SED and visibility are shown in Fig. 10 for different superwind amplitudes. We note again that the quality of the fits is in particular determined by the outer shell, whereas the inner shell's exponent is less constrained. A distribution in the inner shell and large superwind amplitudes () give similar results.
The radii of the inner and outer shell are and (with K), resp., corresponding to angular diameters of mas and mas. Adopting the same assumptions for outflow velocity, dust-to-gas ratio and specific dust density as in the previous section, the expansion ages are and , for the mass-loss rate of the inner component one gets . In the outer component either the outflow velocity has increased or the mass-loss rate has decreased with time due to the more shallow density distribution. Provided the outflow velocity has kept constant, the mass-loss rate at the end of the superwind phase, 92 yr ago, was , and, for instance, amounted to 200 yr ago. Since the flatter density distribution provides a better fit for
the long-wavelength range of the SED, while the visibility is equally
well fitted compared to the standard density distribution, it is
superior to the model of Sect. 3.4.1. Fig. 11 gives the fractional
flux contributions (stellar, dust, scattering) for the same model as
shown in Fig. 7 but with an
distribution in the outer shell. The various flux contributions at
2.11
## 3.4.4. Influence of the dust temperatureFinally, we studied the influence of the dust temperature at the
inner boundary of the hot shell. For that purpose we recalculated the
previous model grids for dust temperatures of 800 and 1200 K. As
already shown for the single-shell models, an increase of the
temperature at the inner boundary increases the flux in the
near-infrared and substantially lowers the flux in the long-wavelength
range. On the other hand, the higher the temperature the less is the
curvature of the visibility at low spatial frequencies, the plateau is
only significantly affected for low-amplitude superwinds. The shape of
SED and 2.11
However, the improvement of the 2.11
The radii of the inner and outer shell are now considerably smaller than those of the previous models due to the higher temperature of the hot shell. The radiative transfer calculations give here and (with K), resp., resulting in angular diameters of mas and mas. Accordingly, the expansion ages are and , for the mass-loss rate of the inner component one gets . Provided the outflow velocity has kept constant, the mass-loss rate at end of the superwind phase, 63 yr ago, was , and, for instance, amounted to 200 yr ago. ## 3.4.5. Intensity distributionsFig. 14 displays the spatial distribution of the obtained normalized model intensity for the model shown in Fig. 10 ( K, , , and density distribution in the inner and outer shell, resp.) The (unresolved) central peak belongs to the central star, and the two local intensity maxima to the loci of the inner rims of the two shells at 35 mas and 157 mas, resp. The m intensity shows a ring-like distribution with a steep decline with increasing distance from the inner boundary of the circumstellar shell. Similarly shaped intensity distributions have also been found by Ivezi & Elitzur (1996) for optically thin shells.
We recall that this intensity distribution is based on radiative transfer models taking into account both the SED and the m visibility. Fig. 15 shows the model visibilities for much higher spatial frequencies than covered by the present observations. The required baselines would correpond to and 440 m instead to 6 m (13.6 cycles/arcsec). Since the dust-shell's diameter is mas a plateau is only reached for spatial frequencies larger than, say, 15 cycles/arcsec depending on the strength of the superwind. The central star is resolved at spatial frequencies of cycles/arcsec. At frequencies cycles/arcsec the shape of the observed and the modelled visibility function is triangle-shaped, which is a consequence of the ring-like intensity distribution of the dust shell.
Visibility observations are often characterized by fits with Gaussian intensity distributions. The resulting Gaussian FWHM diameter is then assumed to give roughly the typical size of the dust shell. A Gauß fit to the observed visibility would yield a FWHM dust-shell diameter of (219 30) mas in agreement with the one given by Christou et al. (1990). However, radiative transfer models show that a ring-like intensity distributions appears to be more appropriate than a Gaussian one for the dust shell of IRC +10 420. The distribution shows a limb-brightenend dust condensation zone and a ring diameter of 70 mas. © European Southern Observatory (ESO) 1999 Online publication: August 13, 199 |