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Astron. Astrophys. 327, 1123-1136 (1997)

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5. Two component models

The temperature of the grains decreases continuously with increasing distance from the star. Within the disk of [FORMULA] Pictori s, water may exist in the form of vapor below the sublimation limit. When the temperature of the grains is low enough (below 110-120 K), any existing water vapor would condense preferably on the grains, thus creating icy particles. Alternatively, the grains could contain already some ice (e.g. fragmentation of comets) that sublimates if a large grain is broken into small pieces, which are hotter, or when the grains approaches the star. The optical properties of the grains containing ice are very different from the properties of the non-icy grains. As a consequence, the temperatures of the grains are lowered (the iced grains absorb less of the visible emission of the star than pure olivine, and a part of the absorbed energy is converted into sublimation heat), and some characteristic ice features appear in the spectrum of the disk.

Having calculated the temperature of the iced grains as a function of from distance to the star, we found that the "ice boundary" for the small grains (i.e. smaller than 10 [FORMULA] m) where the temperature is between 100 and 120 K (for which the sublimation timescales are about 1 million years and 50 years respectively for a grain of radius 1 [FORMULA] m, according to Backman and Paresce, 93), is located between 85 and 145 UA for non-icy grains, and between 55 and 80 UA for grains containing ice (see Fig. 14).

[FIGURE] Fig. 14. A plot of the distances at which a dust grain reaches the temperatures of 100 K, 110 K, and 120 K, for two compositions: core-mantle silicate grains (NIG) and core-mantle silicate grains containing ice (IG) (14% of porosity replaced by ice, see below)

However, as suggested by Artymowicz, 1994, the efficient UV photosputtering of the ice in the [FORMULA] Pictori s system may destroy the ice up to large distances from the star. Therefore, we have kept in our models the sublimation limit as a free parameter. In the models there is transition zone between 90 and 100 AU in which iced grains coexist with non-iced ones (the particles may be on eccentric orbits, or have a distance to the star that changes more rapidly than the sublimation/freezing timescales). In these models, the "naked" grains produce the 10 [FORMULA] m flux that we detect up 100 AU, and the iced ones are responsible for the 60 and 100 [FORMULA] m IRAS fluxes.

There are other elements in favor of the hypothesis of iced grains. Artymowicz et al., 1989, evaluated a model dependant upper bound to the albedo of the outer disk grains between 0.66 and 0.76 with amodel independent lower bound of 0.5. Core-mantle silicate grains have an albedo around 0.4 which a bit low compared to the previous values, but if they are covered with ice, their albedo rises to a value of 0.65.

In the solar system the ice boundary is located at Jupiter's orbit. The grains of Saturn's rings are composed essentially of ice (Fink and Larson, 1975).

We took a power law of the iced component density (with an index between -1.7 (Artymowicz et al., 1989) and -2.5) connected (through a multiplicative scale factor, a free parameter in the range [1,10]) to the "hot" component at a distance from the star between 90 and 100 AU. The particle size distribution was assumed to be the same ashe inner component. We used the same scheme as before to calculate the optical constants of the iced grains by using the Maxwell-Garnett mixing rule (we assumed that some part of the vaccum of the porous grains is filled by ice), and by calculating the absorption coefficients of the iced grains using the Mie theory. We tried then to fit the IRAS fluxes using the two component model, varying the percentage of ice in the grains, the ice boundary location and the step value. The size distribution could be slightly modified but we tried to keep it the same as in the inner regions.

The best fit was obtained with particles that are 14 % (in volume) ice, replacing some part of the vacuum (porosity). We have found that this value is constrained by the 100 to 60 [FORMULA] m flux ratio. In this model, the "ice boundary" is located at a distance of 90 AU from the star. We used a "step" value of 1.8, and an outer component density decreasing like. In Table 3 we report the results obtained when trying to fit IRAS data. As mentioned before, the size distribution is strongly constrained by the IRAS fluxes (see Table 3), but the influence of the size distribution index on their fit is not as critical as in the one component model.


Table 3. Comparison between IRAS Fluxes and thr two component models (st= step used, ind. = index(es) of the size distribution).

Looking at the Table 3, one notices that the two components model gives a very satisfactory fit of IRAS data when taking a size distribution ranging from 0.1 to 10 [FORMULA] m with a power law index of -3.0 and -3.3 from 10 to 1000 [FORMULA] m ( [FORMULA] ).

The "step" value we have used in our models could in fact reflect a change in the sizes of the particles, or more simply an abrupt change in the number of particles, some of them (those composed of pure ice), for example disappearing because of sublimation.

5.1. Comparison to previous measurements in the visible

We checked that the two component model is compatible with visible coronographic observations. We again used the Henyey-Greenstein or Leinert phase functions for both the inner and outer components; however the grains in the outer component have a different porosity). The scattering profiles obtained are are plotted in Figs. 15 and 16, with the coronographic data of Kalas and Jewitt (1995) superimposed. All the profiles appear to be compatible with their data.


Table 4. Comparison with Backman's fluxes

[FIGURE] Fig. 15. The observed and reconstructed surface brightness of the N-E extension in [FORMULA] Pic. Dashed line: non-icy grains contribution to the scattered flux (Henyey-Greenstein phase function), Dot-dashed line: icy grains contribution to the scattered flux (Henyey-Greenstein phase function). Plain line represents the total scattered flux calculated using Henyey-Greenstein phase function. For comparison, we have also plotted (-...) the total scattered flux using the empirical phase function of Leinert. The rounds are from the measurements of Kalas and Jewitt (1995).

[FIGURE] Fig. 16. Same as Fig. 15 but for the S-W extension.

As seen in Fig. 9, the scattered light comes essentially from a half-ring located at a distance of 100 AU from the star. Looking at Figs. 15 and 16, we also notice that the scattered flux coming from the outer component (grains with ice) is higher than the flux from the non-icy grains, down to a distance of 10 to 20 AU.

This may also explain the fact that the scattered flux shows only small asymmetries (meanwhile the thermal flux shows asymmetries up to a factor of 2.3), even very close to the star (Lecavelier des Etangs et al., 1993).

Calculating the albedo of the outer material (iced grains) from the Mie theory, we have shown that it is roughly constant over the visible range, with a value slightly above 0.6 (see Fig. 17). This is consistent with the results of Paresce and Burrows, 1987 who observed no significant color effects in the disk (within uncertainties of 20%), and the albedo estimated by Artymowicz et al., 1989 (a value between 0.66 and 0.76).

[FIGURE] Fig. 17. Mean albedo (averaged over the size distribution)of the particles as a function of the wavelength. Plain line: outer component i.e. particles containing ice, dashed line: inner component
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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998