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Astron. Astrophys. 346, 505­519 (1999)

ASTRONOMY AND ASTROPHYSICS

High­resolution speckle masking interferometry and radiative transfer modeling of the oxygen­rich AGB star AFGL 2290
A. Gauger1 , Y.Y. Balega2 , P. Irrgang1 , R. Osterbart1 , and G. Weigelt1
1 2

¨ ¨ Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, D-53121 Bonn, Germany Special Astrophysical Observatory, Nizhnij Arkhyz, Karachai-Cherkesia, 357147, Russia

Received 28 September 1998 / Accepted 8 March 1999

Abstract. We present the first diffraction­limited speckle masking observations of the oxygen­rich AGB star AFGL 2290. The speckle interferograms were recorded with the Russian 6 m SAO telescope. At the wavelength 2.11 µm a resolution of 75 milli­arcsec (mas) was obtained. The reconstructed diffraction­ limited image reveals that the circumstellar dust shell (CDS) of AFGL 2290 is at least slightly non­spherical. The visibility function shows that the stellar contribution to the total 2.11 µm flux is less than 40%, indicating a rather large optical depth of the circumstellar dust shell. The 2­dimensional Gaussian visibility fit yields a diameter of AFGL 2290 at 2.11 µm of 43 masâ51 mas, which corresponds to a diameter of 42 AUâ50 AU for an adopted distance of 0.98 kpc. Our new observational results provide additional constraints on the CDS of AFGL 2290, which supplement the information from the spectral energy distribution (SED). To determine the structure and the properties of the CDS we have performed radiative transfer calculations for spherically symmetric dust shell models. The observed SED approximately at phase 0.2 can be well reproduced at all wavelengths by a model with Teff = 2000 K, a dust temperature of 800 K at the inner boundary r1 , an optical depth V = 100 and a radius for the single­sized grains of agr = 0.1 µm. However, the 2.11 µm visibility of the model does not match the observation. Exploring the parameter space, we found that grain size is the key parameter in achieving a fit of the observed visibility while retaining the match of the SED, at least partially. Both the slope and the curvature of the visibility strongly constrain the possible grain radii. On the other hand, the SED at longer wavelengths, the silicate feature in particular, determines the dust mass loss rate and, thereby, restricts the possible optical depths of the model. With a larger grain size of 0.16 µm and a higher V = 150, the observed visibility can be reproduced preserving the match of the SED at longer wavelengths. Nevertheless, the model shows a deficiency of flux at short wavelengths, which is attributed to the model assumption of a spherically symmetric dust distribution, whereas the actual structure of the CDS around AFGL 2290 is in fact non­spherical. Our study demonstrates the
Send offprint requests to: R. Osterbart (osterbart@mpifr-bonn.mpg.de) Based on data collected at the 6 m telescope of the Special Astrophysical Observatory in Russia

possible limitations of dust shell models which are constrained solely by the spectral energy distribution, and emphasizes the importance of high spatial resolution observations for the determination of the structure and the properties of circumstellar dust shells around evolved stars. Key words: stars: imaging ­ stars: individual: AFGL 2290 ­ stars: AGB and post-AGB ­ stars: mass-loss ­ stars: circumstellar matter ­ infrared: stars

1. Introduction AFGL 2290 (OH 39.7+1.5, IRAS 18560+0638, V1366 Aql) belongs to the group of type II OH/IR stars, which can be defined as infrared point sources with a maximum of the spectral energy distribution (SED) around 6­10 µm, with the 9.7 µm silicate band in absorption, and with OH maser emission in the 1612 MHz line (Habing 1996). Most of these objects show a long-period variability in the infrared and the OH maser emission (Engels 1982; Herman & Habing 1985), although also a small fraction either varies irregularly with small amplitude or does not vary at all. OH/IR stars are surrounded by massive circumstellar envelopes composed of gas and small solid particles (dust, grains). These circumstellar dust shells (CDS) are produced by the ejection of matter at large rates (M > 10-7 -1 -1 M yr ) and low velocities ( 15 kms ), and in some cases they totally obscur the underlying star. Based on the luminosities ( 104 L ), the periods (500d to 3000d) and bolometric amplitudes ( 1 mag), the kinematical properties and galactic distribution, the majority of OH/IR stars are highly evolved low­ and intermediate­mass stars populating the asymptotic giant branch (AGB) (Habing 1996). They extend the sequence of optical Mira variables to longer periods, larger optical depths and higher mass loss rates (Engels et al. 1983; Habing 1990; Lepine et al. 1995). The improvements of the observational techniques, especially at infrared wavelengths, and the elaboration of increasingly sophisticated theoretical models have provided a wealth of new information on the structure, the dynamics, and the evolution of the atmospheres and circumstellar shells of AGB stars, although many details still remain to be clarified (see the review


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by Habing 1996). A general picture has become widely accepted in which both the large amplitude pulsations and the acceleration by radiation pressure on dust contribute to the mass loss phenomenon for AGB stars. From observations, correlations are found between the period and the infrared excess (indicating the mass loss rate) (DeGioia­Eastwood et al. 1981; Jura 1986), and between the period and the terminal outflow velocity (Heske 1990). On the theoretical side, hydrodynamical models showed that due to the passage of shocks generated by the stellar pulsation the atmosphere is highly extended, thus enabling dust formation and the subsequent acceleration of the matter (Wood 1979; Bowen 1988). The inclusion of a detailed treatment of dust formation revealed a complex interaction between pulsation and dust formation, which results e.g. in a layered dust distribution and affects the derived optical appearance (Fleischer et al. 1992, 1995; Winters et al. 1994, 1995). Until now most interpretations of observations as well as most theoretical models are based on the assumption of a spherically symmetric dust shell, often motivated by the circularity of the OH maser maps. However, observations show that some objects have substantial deviations from spherical symmetry (e.g. Dyck et al. 1984; Kastner & Weintraub 1994; Weigelt et al. 1998). This suggests that the asymmetries observed in many post­AGB objects and planetary nebulae (cf. Iben 1995) may already start to develop during the preceding AGB phase, which provides new challenges for the modeling of the mechanisms and processes determining the structure of the dust shells around AGB stars. High spatial resolution observations can yield direct information on important properties of the dust shells around AGB stars, such as the dimensions and geometry of the shell. Therefore, such observations contribute additional strong constraints for the modeling of these circumstellar environments, which supplement the information from the spectral energy distribution. Measurements of the visibility at near-IR wavelengths, for example, can be used to determine the radius of the onset of dust formation as well as to constrain the dominant grain size (Groenewegen 1997). To gain information on details of the spatial structure, in particular on asymmetries and inhomogeneties of the CDS, the interferometric imaging with large single­dish telescopes is especially well suited because one observation provides all spatial frequencies up to the diffraction limit of the telescope and for all position angles simultaneously, allowing the reconstruction of true images of the object. We have chosen AFGL 2290 for our study because it represents a typical obscured OH/IR star with a high mass loss rate, whose location is not too far away from us. The distance to AFGL 2290 can be determined directly with the phase lag method (cf. Jewell et al. 1979), which gives D = 0.98 kpc (van Langevelde et al. 1990). For the bolometric flux at earth a value of fb 2.410-10 Wm-2 is derived by van der Veen & Rugers (1989) from infrared photometry between 1 µm and 12 µm and the IRAS fluxes. At 0.98 kpc the luminosity is L = 7200 L , which is within the typical range for an oxygen­rich AGB star. The long period of P = 1424 d determined from the variation of

the OH maser (Herman & Habing 1985) and the high mass loss rate suggest, that the star is in a late phase of its AGB evolution. So far, Chapman & Wolstencroft (1987) reported the only high angular­resolution infrared observations of AFGL 2290. From 1­dimensional slit­scan speckle interferometry with the UKIRT 3.8 m telescope at 3.8 µm and 4.8 µm they derive 1­ dimensional visibilities and determine Gaussian FWHM diameters. Radiative transfer models for the AFGL 2290 dust shell have been presented by Rowan­Robinson (1982), Bedijn (1987), Suh (1991) and recently by Bressan et al. (1998). These models yield dust shell properties within the typical range of OH/IR stars, e.g. a dust mass loss rate of about 410-7 M yr-1 (Bedijn 1987; Bressan et al. 1998), or an optical depth at 9.7 µm of about 10 (Bedijn 1987; Suh 1991). However, none of these studies includes constraints from high spatial resolution infrared measurements. In Sect. 2 we present the results of our speckle masking observations of AFGL 2290. The approach for the radiative transfer modeling is described in Sect. 3 comprising a short description of the code, the selection of the photometric data and a discussion of input parameters for the models. In Sect. 4 we present the results of the radiative transfer modeling starting with the discussion of a model, which yields a good fit of the observed SED at all wavelengths but does not reproduce the observed 2.11 µm visibility. In search of an improved model the changes of the resulting SED and visibility under variations of the input parameters are investigated in the following sections. We finish the paper with a summary of the results and our conclusions in Sect. 5. 2. Speckle masking observations The AFGL 2290 speckle data presented here were obtained with the Russian 6 m telescope at the Special Astrophysical Observatory (SAO) on June 14 and 16, 1998. We recorded a total number of 1200 speckle interferograms of AFGL 2290 (600 on June 14 and 600 on June 16) and 2400 speckle interferograms of the unresolved reference star HIP 93260 (1200 on each of the two nights) with our 256â256 pixel NICMOS 3 camera through an interference filter with center wavelength 2.11 µm and a bandwidth of 0.192 µm. The exposure time per frame was 100 ms, the pixel size was 30.61 mas and the field of view 7. 8 â 7. 8. The 2.11 µm seeing was about 1. 2. A diffraction-limited image of AFGL 2290 was reconstructed from the speckle data by the speckle masking bispectrum method (Weigelt 1977; Lohmann et al. 1983; Weigelt 1991). The process includes the calculation of the average power spectrum and of the average bi­spectrum and the subtraction of the detector noise terms from those. The modulus of the object Fourier transform was determined with the speckle interferometry method (Labeyrie 1970). The Fourier phase was derived from the bias­compensated average bispectrum. Figs. 1 and 2 show the visibility function of AFGL 2290 at 2.11 µm. The azimuthally averaged visibility decreases steadily to values below 0.40 of the peak visibility at the diffraction cut­off frequency (13.5 arcsec-1 ). Thus, the circumstellar


A. Gauger et al.: Speckle masking interferometry and radiative transfer modeling of AFGL 2290

507

Fig. 1. Two­dimensional 2.11 µm visibility function of AFGL 2290 derived from the speckle interferograms. The contour levels are plotted from 20% to 80% of the peak value in steps of 10%.

Fig. 3. Diffraction­limited 2.11 µm speckle masking image of the AFGL 2290. North is at the top and east to the left. The contours level intervals are 0.25 mag. The lowest contour level is 3.25 mag fainter than the peak intensity.
1

1

Visibility V(2.11 µm)

0.8

0.8

0.6

Normalized Intensity
0 2 4 6 8 10
-1

0.6

0.4

0.4

0.2

0.2
0 12

0 0 20 40 60 80 100 Distance from center [mas]

Spatial frequency q [arcsec ]

Fig. 2. Azimuthally averaged 2.11 µm visibility function of AFGL 2290 and errorbars.

Fig. 4. Azimuthally averaged image of AFGL 2290 (solid line) and of the unresolved reference star HIP 93260 (dashed line).

dust shell is almost totally resolved, and the contribution of the unresolved stellar component to the monochromatic flux at 2.11 µm must be less than 40 %, suggesting a rather high optical depth at this wavelength. In order to derive diameters for the dust shell, the object visibility function was fitted with an elliptical Gaussian model visibility function within a range of 1.5 arcsec-1 up to 7.5 arcsec-1 . We obtain a Gaussian fit diameter of 43 masâ51 mas for AFGL 2290 corresponding to 42 AUâ50 AU for an adopted distance of 0.98 kpc or 5.7 r â6.8 r for an adopted distance of 0.98 kpc and an adopted stellar radius of r = 7.5 mas (cf. Sect. 4.1), respectively. Fig. 3 shows the reconstructed 2.11 µm speckle masking image of AFGL 2290. The resolution is 75 mas. Fig. 4 shows the azimuthally averaged images derived from the reconstructed 2­dimensional images of AFGL 2290 and the reference star

HIP 93260. In the 2-dimensional AFGL 2290 image a deviation from spherical symmetry can be recognized. The intensity contours are elongated in the south­eastern direction along an axis with a position angle of 130 . 3. The radiative transfer modeling approach 3.1. The radiative transfer code The radiative transfer calculations are performed with the code ´ DUSTY developed by Ivezic et al. (1997), which is publicly available. The program solves the radiative transfer problem for a spherically symmetric dust distribution around a central source of radiation and takes full advantage of the scaling properties inherent in the formulation of the problem. The formulation of


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the radiative transfer problem, the model assumptions and the ´ scaling properties are described in detail by Ivezic & Elitzur (1997). Therefore, we give only a brief discussion here. The problem under consideration is a spherically symmetric dust envelope with a dust free inner cavity surrounding a central source of radiation. This geometry is not restricted to the dust shell of a single star. It can as well describe a dust envelope around a group of stars (e.g a binary) or even around a galactic nucleus. The radial dependence of the dust density between the inner and outer boundary can be chosen arbitrarily. To arrive at a scale invariant formulation two assumptions are introduced: i) the grains are in radiative equilibrium with the radiation field, and ii) the location of the inner boundary r1 of the dust envelope is controlled by a fixed temperature T1 of the grains at r1 . Due to radiative equilibrium this temperature is determined by the energy flux at r1 , which in turn is controlled by the energy flux from the central source via the radiative transfer through the dusty envelope. Then prescribing the dust temperature at r1 is equivalent to specifying the bolometric flux at the inner boundary, and the only relevant property of the input radiation ´ is its spectral shape (Ivezic & Elitzur 1997). Similarly, if the overall optical depth of the dust envelope at some reference wavelength is prescribed, only dimensionless, normalized distributions describing the spatial variation of the dust density and the wavelength dependence of the grain optical properties enter into the problem. This formulation of the radiative transfer problem for a dusty envelope is well suited for model fits of IR observations, because it minimizes the number of independent model parameters. The input consists of: ­ the spectral shape of the central source of radiation, i.e. the variation of the normalized monochromatic flux with wavelength, ­ the absorption and scattering efficiencies of the grains, ­ the normalized density distribution of the dust, ­ the radius of the outer boundary in units of the inner boundary, ­ the dust temperature at the inner boundary, ­ the overall optical depth at a reference wavelength. For a given set of parameters, DUSTY iteratively determines the radiation field and the dust temperature distribution by solving an integral equation for the energy density, which is derived from a formal integration of the radiative transfer equation. For a prescribed radial grid the numerical integrations of radial functions are transformed into multiplications with a matrix of weight factors determined purely by the geometry. Then, the energy density at every point is determined by matrix inversion, which avoids iterations over the energy density itself and allows a direct solution of the pure scattering problem. Typically fewer than 30 grid points are needed to achieve a relative error of flux conservation of less than 1%. The number of points used in angular integrations is 2­3 times the number of radial grid points, and the build­in wavelength grid has 98 points in the range from ´ 0.01 µm to 3.6 cm (see Appendix C in Ivezic & Elitzur 1997).

The distributed version of the code provides a variety of quantities of interest including the monochromatic fluxes and the spatial intensity distribution at wavelengths selected by the user, but not the corresponding visibilities. Since we want to employ the visibilities obtained from our high spatial resolution measurements as constraints for the radiative transfer models, we have supplemented the code with routines for the calculation of synthetic visibility functions. 3.2. Selection of photometric data An important ingredient for the radiative transfer modeling of circumstellar dust shells around evolved stars is the spectral energy distribution (SED). Due to the variability of Miras and OH/IR stars, the SED of such objects ideally has to be determined from coeval observations covering all wavelengths of interest. Unfortunately, no such coeval photometric data set for the wavelength region from 1 µm to 20 µm is available in the literature for AFGL 2290. Thus, we have to define a `composite' SED, which is derived from observations made by different authors at different epochs, but at about the same photometric phase (Griffin 1993). From the infrared photometry of AFGL 2290 available in the literature, we selected those publications which specify the date of observation and present the fluxes in tabulated form, either in physical units (e.g. Jy) or in magnitudes with given conversion factors (at least as a reference). Table 1 lists the references, the date and phase of observation and the wavelengths. The phases were determined from the period P = 1424d and the epoch of maximum, JD = 244 4860.8, which has been derived from the monitoring of the OH maser emission by Herman & Habing (1985). Engels et al. (1983) determined periods of OH/IR stars from infrared observations and found that the periods and phases are in agreement for objects in common with the sample Herman & Habing (1985). It can be seen from the entries in Table 1 that the wavelength range from = 1.2 µm to = 20 µm is only fully covered by observations around phase 0.2 (see entries 2, 4, 9, 12).The respective fluxes are shown in Fig. 5. The measurements of Herman et al. (1984) and Nyman et al. (1993) match each other quite well at = 3.8 and = 4.8 µm, although the observations are separated by two periods. The fluxes of Price & Murdock (1976) and Gehrz et al. (1985) agree with the Herman et al. and Nyman et al. data within the errors given by the authors. To represent the SED of AFGL 2290 we adopt the data of Herman et al. (1984), Gehrz et al. (1985), and Nyman et al. (1993). The scatter between the different data sets gives a rough estimate of the uncertainty of the `composite' SED at phase 0.2 of 0.25. We do not correct for interstellar extinction because the corrections are less than, or of the same order as, the uncertainty estimated above. For AFGL 2290 Herman et al. (1984) give a value of AV = 1.6 for an adopted distance of 1.19 kpc which reduces to AV = 1.3 at a distance of 0.98 kpc. With the wavelength dependence of the interstellar extinction in the infrared from Becklin et al. (1978) one obtains a correction


A. Gauger et al.: Speckle masking interferometry and radiative transfer modeling of AFGL 2290 Table 1. Infrared photometry of AFGL 2290 ordered by the date of observation No. 1 2 3 4 5 6 7 8 9 10 11 12 Julian Date 244 0000+ 1045 2295 2725 3726 3972 4082 4348 4533 5146 7816 7832 8041 Phase P = 1424 d 0.320 0.198 0.500 0.203 0.376 0.453 0.640 0.770 0.200 0.075 0.087 0.233 (-3) (-2) (-2) (-1) (-1) (-1) (-1) (-1) (+0) (+2) (+2) (+2) Ref. 1 1 2 3 4 3 3 4 5 6 7 8 4.2 2.2 2.3 2.2 2.3 2.3 2.2 2.23 2.28 2.19 3.6 3.6 3.7 3.6 3.6 3.7 3.8 3.79 3.80 3.79 5.0 4.9 4.8 4.9 4.9 4.8 4.8 8.4, 8.8 8.7 8.7 8.7 8.2 8.7 10.4 10.6 10.0 10.0 10.0 10.2 10.5 Wavelengths [ µm ] 11.0 11.0 11.6 11.4 11.4 11.4 11.5 19.8 19.8 12.6 12.6 12.6 12.6 12.2 12.5 19.5 19.5 19.5 19.6 20

509

1.25, 1.65

1.25, 1.65 1.63 1.26, 1.68 1.24, 1.63

9.6 9.7

4.64

References: 1) Price & Murdock 1976, 2) Lebofsky et al. 1976, 3) Gehrz et al. 1985, 4) Engels 1982, 5) Herman et al. 1984, 6) Noguchi et al. 1993, 7) Xiong et al. 1994, 8) Nyman et al. 1993. Numbers in parantheses give the cycle with respect to epoch JD 244 4860.8.

-10

-11

-12 -12 -14

given by van der Veen et al. (1995) and the IRAS low resolution spectra from the IRAS Catalog of Low Resolution Spectra (1987). The latter are corrected according to Cohen et al. (1992). Since the broadband fluxes and spectra are averages of several measurements taken at different phases (IRAS Explanatory Supplement 1985), the flux levels, e.g. at 12 µm, are lower than the fluxes from ground based observations around phase 0.2. Therefore, we multiply the IRAS data with a factor of 1.95 to join them with the ground based data. Finally, observations at mm wavelengths were reported by Walmsley et al. (1991) who measured a flux of 0.025 Jy at = 1.25 mm at phase 0.18 (JD 2447960), and by van der Veen et al. (1995) who derived 3 upper limits of 0.13 Jy and 0.14 Jy at 0.76 mm and 1.1 mm respectively for phase 0.25 (JD 2448069), which are consistent with the 1.25 mm flux. 3.3. Selection of input parameters
0.1mm 1mm 100

Log F [Wm ]

-2

-16 -13 1 10 Wavelength [µm]

Fig. 5. IR­fluxes of AFGL 2290 observed about phase 0.2 and at phase 0.77. The data are taken from Price & Murdock 1976 (+), Herman et al. 1984 ( ), Gehrz et al. 1985 ( ), and Nyman et al. 1993 ( ). Also shown are the colour corrected IRAS fluxes adopted from van der Veen et al. 1995 (filled sqares), and the IRAS low resolution spectra (dashed line). The IRAS data are multiplied by a factor of 1.95 in order to match the photometric data at 12 µm. The insert shows mm measurements by Walmsley et al. 1991 (·) and van der Veen et al. 1995 ( , 3 upper limits).

factor of 1.35 at = 1.25 µm, 1.03 at = 4.8 µm, and 1.15 at = 9.5 µm. AFGL 2290 was observed by IRAS (IRAS Point Source Catalog 1985). We adopt the colour corrected broadband fluxes

We represent the central star by a blackbody with an effective temperature Teff . In contrast to the visible M type Mira variables with Teff < 3500 K, the effective temperature of OH/IR stars with optically thick dust shells cannot be directly determined. However, if OH/IR stars can be considered as an extension of the Mira sequence to longer periods and larger optical depths, one might extrapolate the period­Teff relation for Mira variables derived by Alvarez & Mennessier (1997) to P > 650 d, which yields Teff < 2500 K in agreement with the values expected for the tip of the AGB. The dust density distribution is obtained from the velocity law, which results from an approximate analytic solution for a stationary dust driven wind with constant mass loss rate (e.g. Schutte & Tielens 1989). If the gas pressure force is neglected and the flux averaged absorption coefficient is assumed to be constant with the radius r in the wind, the velocity distribution is given by


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v (r) = v

1-

r1 r

1-

v1 v

2

Table 2. Parameters and resulting properties of model A.

(1)
Teff [K] 2000 Md [M yr-1 ] 2.7 10-7 T1 [K] 800 r1 [R ] 7.80 a
gr

[µm] 0.10

rout /r1 104 [mas] 7.50

0.55 100 10 7.49

where v1 denotes the velocity at the inner boundary r1 , and v is the velocity at infinity. The relevant free parameter is the ratio of these velocities = v1 /v , because only the normalized density distribution enters into the calculation. We adopt this velocity law, because it accounts for the changing density gradient due to the acceleration of the matter by radiation pressure on dust in the innermost parts of the dust shell. Compared to a dust shell with a 1/r2 density distribution and equal optical depth the dust density at r1 is higher by a factor of 0.5(1 + )/ and the mass loss rate is lower by a factor of 0.5(1 + ) (cf. Le Sidaner & Le Bertre 1993). According to the theory of dust driven winds the velocity at the inner boundary, where efficient grain condensation takes place and the acceleration of the matter by radiation pressure on dust starts, is close to local sound velocity cs (see Gail 1990). This is supported by observations of the velocity separation of the SiO maser emission in OH/IR stars (Jewell et al. 1984), which presumably originates from the dust forming region. With cs < 2 kms-1 for temper atures of about 1000 K and with the measured outflow velocity of AFGL 2290 of v = 16 kms-1 (Herman & Habing 1985) one obtains 0.12, which we adopt as the standard value for . As described in the previous section, the location of the inner boundary r1 of the dust shell is determined by the choice of the dust temperature T1 at r1 . For the outer boundary rout we adopt a default value of rout = 103 r1 . As shown in the following section a larger outer boundary affects only the far infrared fluxes for = 100 µm without altering the other properties of the model. We consider spherical grains of equal size described by the grain radius agr . This is certainly a simplification because based on theoretical and observational arguments, one expects the presence of a grain size distribution n(agr ). Therefore, a size distribution similiar to the one observed in the ISM - (n(agr ) agr3.5 ) is often assumed for radiative transfer models of circumstellar dust shells (e.g. Justtanont & Tielens 1992, Griffin 1993). Consistent models for stationary dust driven winds, which include a detailed treatment of (carbon) grain formation and growth, in fact yield a broad size distribution which can well be approximated by a power law (Dominik et al. 1989). However, in circumstellar shells around pulsating AGB stars the conditions determining the condensation of grains change periodically. The time available for the growth of the particles is restricted by the periodic variations of the temperature and density. This results in a narrower size distribution (Gauger et al. 1990, Winters et al. 1997) which might roughly be approximated by a single dominant grain size. For the dust optical properties we adopt the complex refractive index given by Ossenkopf et al. (1992) for `warm, oxygen­ deficient' silicates. The authors consider observational determinations of opacities of circumstellar silicates as well as laboratory data and discuss quantitatively the effects of inclusions on

fb [Wm-2 ] 3.0 10-10

the complex refractive index, especially at shorter wavelengths ( < 8 µm). These constants yield a good match of the overall spectral shape of the observed SED of AFGL 2290, especially of the 9.7 µm silicate feature. However, we will also discuss the effects on the radiative transfer models resulting from different optical constants in the Appendix. With the tabulated values of the complex refractive index the extinction and scattering efficiencies are calculated from Mie theory for spherical particles assuming isotropic scattering. Once a satisfactory fit of the spectral shape is achieved with suitably chosen values for the remaining input parameters Teff , T1 , agr , and 0.55 (the optical depth at the reference wavelength 0.55 µm) the match of the normalized synthetic SED with the observed SED determines the bolometric flux at earth fb . Combined with the effective temperature one obtains the angular stellar diameter and thereby the spatial scale of the system. With an assumed distance D to the object, the luminosity L , the radius of the inner boundary in cm, and the dust mass los