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

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3. Inner radial dust density

The flux observed in the mid-IR originates from the thermal emission of grains. Thus, in order to derive the dust density we need to know the grain temperature and emissivity. In the case of a disk seen edge-on, as in the case with the [FORMULA] Pictori s system, an addditional difficulty arises from the fact that the observed flux is integrated along the line of sight.

3.1. Equations

Given the relatively small opening angle of the wedge disk (30 degrees), we can assume in first approximation that the dust temperature is constant in the direction perpendicular to the plane of the disk. The three-dimensional problem is then reduced to a two-dimensional problem. Under these conditions, the observed flux to be considered is the one-dimensional flux obtained by the integration of the signal in the direction perpendicular to the disk axis in the deconvolved image. The axes of the problem are defined as follows; the y-axis is parallel to the line of sight, and the x-axis is perpendicular to the line of sight. The radial surface density [FORMULA] (r), where i stands for N-E or S-W, is related to the observed flux profile, [FORMULA], via the equation:



where [FORMULA] is the number of grains having a radius between a and [FORMULA] and this distribution is normalized such that [FORMULA]; [FORMULA] is the absorption coefficient of the grains and [FORMULA] their temperature; [FORMULA] is the Planck function at wavelength [FORMULA] (i.e. 12µm for our mid-IR observations); d is the Earth- [FORMULA] Pictori s distance, [FORMULA] is the spatial resolution of our image. Given the asymmetries found, we have considered the N-E and S-W sides separately. For each side, we have assumed invariance by rotation (see discussion in Sect. 3.5.2). For reasons of simplicity, we have also assumed, in a first step, that the grain properties are independent of the distance to the star.

In estimating IRAS fluxes, we have integrated the spectral energy density over each IRAS filter.

The temperature of the grains is calculated on the basis of energy conservation principle:



where [FORMULA] is the star monochromatic emissive power of the A5V star [FORMULA] Pictori s and [FORMULA] is the stellar radius of [FORMULA] Pictori s i.e. 1.29 [FORMULA].

In the case of grains containing ice, we have to take in account the heat used to vaporize ice and we use instead the equation (Lamy, 1974):



where [FORMULA] is the mass sublimation rate of the ice at temperature T and [FORMULA] is the latent heat of sublimation at the same temperature. As noted by Lamy, 74, we have found that this term is important only for temperatures higher than 150 K.

The dust properties (optical constants, size distribution) are key inputs to the model. We discuss hereafter how these inputs were determined.

3.2. Dust properties

3.2.1. Composition and porosity

Key information about the nature of the dust in the inner part of the [FORMULA] Pictori s system arises from the observation of the characteristic silicate feature around 10 microns (Telesco et al., 1991; Knacke et al., 1993; Aitken et al., 1993). In LP94, following Knacke et al., 1993, we used Draine and Lee astronomical silicates (DL hereafter) as the constitutive material of the dust grains. However, when making a refined model of the disk, we can no longer use this kind of silicates. Indeed, they are not real silicates, but a "mixture" introduced to reproduce observations of the insterstellar medium, whereas the spectrum observed in [FORMULA] Pictori s is more similar to that of comet Halley. If we examine carefully the spectrum, we can notice very sharp structures at 9.7 and 11.3µm, showing that the silicates must be partly crystalline (contrarily to DL silicates, which are only amorphous). However, the relatively broad feature around 10µm suggests that amorphous silicates represent a relatively important part of the grains composition. We may also use porous grains, as the grains of the solar system are sometimes porous (e.g. Greenberg and Hage, 1990 and references therein). The absorption coefficient of a mixture of different kinds of silicates (eventually porous) has been calculated according to the Maxwell-Garnett effective medium theory (Bohren and Huffmann, 1983) combined with Mie theory, an approach that has been shown to be useful in deriving the absorption coefficients of porous particles (Hage and Greenberg 1990; Kozasa, Blum and Mukai, 1992). Additionally, we used the optical constants deduced from laboratory measurements (Dorschner et al., 1995, for amorphous olivines and pyroxenes; Mukai and Koike, 1990, for crystalline olivine).

Greenberg and Li, 1995, have put forward the hypothesis that the [FORMULA] Pictori s dust was created by evaporating comets. Thus, grain properties follow a model of cometary dust proposed by Greenberg and Hage (1990). In this model, they suggest that the comets, the most primitive bodies of the solar system, are made of unmodified protosolar nebula interstellar dust. This interstellar dust, according to the models developed by Greenberg, 1985, is mostly composed of particles with a silicate core and with an "organic refractory" mantle. While we do not describe in detail the optical properties of this material, we notice several interesting points:
- This mantle adds a feature in the spectrum around 8µm, leading to a better fit of the 10µm spectrum than in the case of silicate grains without a mantle.
- They have absorption coefficients that are relatively high in the optical range (albedo around 0.4). As a consequence, these particles have a higher temperature than the "naked" silicate grains which generally have an albedo around 0.8. Their high temperature is also partly due to the fact that these particles are very porous (P=0.984). This makes these type of grains an interesting candidate to explain why we are able to detect in our images at 11.9µm the disk up to distances as far as 100 AU from the star. If the particles were relatively "cold", this would imply a large number of particles hardly compatible with the measured scattered flux at 100 AU.

Greenberg and Li claim they can obtain a better fit of the 10µm spectrum when using these type of grains instead of simple silicate grains. Thus we have also considered this type of grain in our models.

3.2.2. Size distribution

There is limited information about the particle size distribution in [FORMULA] Pictori s. In the inner part of the disk, the only information comes from the presence of the silicate feature in the spectrum, which requires particles as small as a few µm. In the outer part of the disk (r [FORMULA] 100 AU), the absence of significant colour excess in visible observations (Paresce and Burrows, 1987), implies that the dominant particles for the scattering are larger than a few µm.

From the theoretical point of view, we know that compact grains smaller than about 2µm are very efficiently expelled, by radiation pressure (Artymovicz, 1988). Larger grains are also removed from the system for instance by Poynting-Robertson effect or by collision. Besides, we can expect that processes like collisions and dust release by [FORMULA] Pictori s grazing comets may replenish the medium with small particles. The resulting size distribution is thus difficult to predict. In the following we have taken a pragmatic approach, i.e. a power law for the size distribution (with two cut-off sizes, the maximal one being fixed in all the models to 1000µm) where we allow a change in the power law index at a given size. It gives three free parameters that will be found when trying to fit the different observables of the disk (10µm spectrum, IRAS fluxes, scattered fluxes).

3.3. Inversion algorithm

Eq. 1 is an ill-posed problem because of the additive noise in the data, [FORMULA], and makes the Abel transform method inneficient (Craig, 1986). However, several techniques can be used to solve these kind of problems, whose most popular application is the image restoration. We chose the Maximum Entropy method (Gull and Skilling, 1984), but using the cross-entropy instead of absolute entropy to avoid about the problem of determining the value of the parameter "m" ("model" in Gull and Skilling algorithm). This parameter represents the value of the background in the absence of any object in the data (e.g. the background of an image). In our problem, there is no real background value, thus we prefer a method in which the amount of information is not calculated by comparison to a reference value (m), but in which the information provided by a data value is quantified by comparison to its nearby values. This method is very robust in the cases of noisy problems with lack of data, and has proved to be efficient in giving the simpliest solution compatible with the data for a wide field of inverse problems. This procedure was applied for each side (NE and SW), assuming invariance of all quantities by rotation around the z-axis perpendicular to the disk plane.

The deduced surface density; using core-mantle silicate grains according to Greenberg et al. models; is shown in Fig. 5a and the normal optical thickness [FORMULA] is plotted in Fig. 6. Assuming a volumic mass of the grains of 2.5 g cm-3, the total mass of dust is 2.4 [FORMULA] g inside a radius of 10 AU, that is compatible with the total dust mass obtained previously in that region (Aitken et al., 1991; Backman et al. 1993; Knacke et al., 1993). The surface density again shows an inner void of matter.

[FIGURE] Fig. 5. The surface density of the inner [FORMULA] Pictori s dust disk deduced from our 10µm data. We have used core-mantle silicate particles following the Greenberg et al. models.
[FIGURE] Fig. 6. The normal optical thickness of the disk (i.e. [FORMULA]) that gives an indication on the number of particles independent from the size distribution.

3.4. Fit of the 10µm spectrum

We constructed the disk spectrum inside a 4 arcsec beam deduced from the surface density found above and compared it to the spectrum obtained by Knacke et al. (1993). We have tested various compositions of the grains based on basic silicate materials (olivines, pyroxenes, crystalline olivine). Using this assumption, we were unable to fit correctly the 10µm spectrum of the disk measured by Knacke et al. when using any mixture of silicates or when varying the size distribution index. We only noticed (according to the remark of Knacke et al., 1993) that the fit was slightly better around 8µm when we used a mixture of olivine and pyroxene instead of pure olivine.

As said previously, Greenberg and Li proposed their model of core-mantle (mantle of refractory organics) silicate grains to reproduce the [FORMULA] Pictori s spectrum. We have also attempted to use this material. We found that we could obtain a good fit of the 10µm spectrum with this material ([FORMULA], number of "free" parameters = 35), even better than the fit obtained by Greenberg and Li, because, contrary to them, we used the density of particles in the inner regions deduced from our observations (instead of taking a power law), and also because we took for the core silicate a mixture of pyroxene and olivine instead of olivine only (we took a mixture of 55% of amorphous olivine MgFeSiO4, 35% of amorphous pyroxene Mg0.6 Fe0.4 SiO3, and 10% of crystalline olivine: this composition is similar to the one found by Sanford, 1988, for interplanetary particles of our solar system). One builds a macro, porous grain (P=0.98) from elementary 0.1µm sized particles (see Greenberg and Hage, 1990). For such an elementary grain, the mantle of organic refractories has a volume fraction twice that of the silicate component (or roughly equal in mass since the volumic mass of the refractory organics is the half that of the silicate one). The indexes of the size distribution were: -3.0 from 0.1 to 10µm and -3.3 from 10 to 1000µm. These values are not crucial for the fit of the 10µm spectrum but as we will show in the section concerning the fit of the IRAS data, the IRAS fluxes constrain strongly these values. It is also interesting to note here that this type of size distribution is very close to the estimated size distribution of the interplanetary particles in our solar system (cf Mukai, 1990). The spectrum obtained and the data of Knacke et al., 1993, are superimposed in Fig. 7.

[FIGURE] Fig. 7. The observed and reconstructed spectra of the inner [FORMULA] Pic dust disk (inside a 3.7 arcsec beam). Plain line: the reconstructed spectrum using core-mantle silicate grains, the stars represent the data of Knacke et al., 1993, combining IRTF and Big Mac data. The corresponding [FORMULA] is 18 (this must be compared to the number of measurement points; 35). We have normalized the spectra at 11.9µm since our density is deduced from an averaged flux over the 10.5-13.3µm filter.

It must also be noted that DL silicates give a relatively acceptable fit to the 10µm spectrum ([FORMULA] using an index of -4.0 for all sizes from 0.1 to 1000µm, see Fig. 8) so we cannot rule them out as possible candidate even if, as explained above, we refrain to use them because their optical constants are not deduced from laboratory measurements.

[FIGURE] Fig. 8. Same as in Fig. 7, but using DL silicates grains, with a size distibution following a power-law with an index of -4.0, ranging from 0.1 to 1000µm. The corresponding [FORMULA] is 30.

3.5. Discussion

3.5.1. Influence of the dust properties on the inner profile

To show that the result of an inner depleted region is robust if the assumptions on the sizes of the grains (composition, index of distribution and cut-off values) are changed, we have tested different size distributions by varying the index of the power-law within the range {-2,-4}; changed the lower cut-off radius from 0.1µm to 1µm, and varied the composition of the grains i.e. taking Draine and Lee astronomical silicate grains, olivine porous grains or even blackbody grains instead of core-mantle silicate grains. In every case, the inner depleted region is still here, confirming strongly the results obtained in LP94.

3.5.2. Invariance by rotation

In Eq. 1, we have assumed that the flux was radially axi-symmetrical on each side of the disk. However, we show here that this assumption is not crucial. Indeed when observing at 12µm, we are probing only a small part of the disk, located in an opening angle of about 30 degrees perpendicular to the line of sight if the density were radially constant, and a distorded cone when we modulate the cone using the "real" radial density (see Fig. 9 and its caption for the construction of the normalized map). This is due to the flux rapidly a fast decreasing with the distance (the flux is essentially fixed by the temperature of the grain). Then, the invariance by rotation assumed in the following section is likely to be verified in that region inside a 30 degrees angle.

[FIGURE] Fig. 9a and b. The 2D normalized maps showing the area where the thermal emission and the scattered light are emitted along a line of sight (the observer is assumed to be in [FORMULA], vertical axis). The normalization is such that at each point in x, the maximum emission is 1. Therefore, at a given x position, a "wide" emission corresponds to a local maximum in the density. We used the surface density plotted in Fig. 5. A bright ring has been overplotted in both maps to illustrate a 250 AU disk perimeter. Upper (a): thermal flux at 12µm; lower (b): scattered flux in the R band showing the anisotropical effects of the scattering phase function. At each abscissae value, we have normalized the flux to a value of 1. The discontinuities in the maps are due to shape of the radial density; each time the density has a local maximum, the maps show a discontinuity.

3.5.3. Scattering contribution at 12 microns

We have evaluated also the contribution of the scattered starlight at 12µm. The result show that it can be neglected within 100 AU, beyond which the scattered flux, varying as r-2, supercedes over the thermal emission which varies much more rapidly with distance, because of the decreasing temperature of the grains. However, this contribution should be taken into account for observations with a more sensitive instrument like the ISO spacecraft, (we have even shown that the images made with the ISOCAM 10µm camera should reveal scattered starlight beyond 150 AU).

3.5.4. Comparison to Backman's models

Backmann et al. (1993) have developed a series of models in order to reproduce their 10 and 20µm observations of the disk within two beams of 4 and 8 arcsec in diameter, and to fit the IRAS measurements. As shown in Fig. 10, their best model does not agree with our data.

[FIGURE] Fig. 10. The 12µm profile deduced from Backman's density in model 11 (dashed line), compared to profile deduced from our 10µm image of the [FORMULA] Pictori s disk (plain line).

However, it must be noted that they used crude absorption coefficients for the dust grains. Indeed, the coefficients were approximated by two power-laws as a function of the wavelength: an index of 0 (constant) for the lowest wavelengths and an index of -1 or -2 for the longer ones (where [FORMULA] is a parameter depending on the size of the grains). If we carefully examine the real coefficients (see Fig. 11) we find out that for a fixed grain size (1µm in the case of Fig. 11), the shape of the real absorption curve with the wavelength is very different from the approximated one. Thus one must be critical of the results that use approximated absorption coefficients.

[FIGURE] Fig. 11. The absorption coefficients of a grain of 1µm radius. We have plotted the absorption deduced from the Mie theory for porous olivine grains (plain line), and for core-mantle silicate porous grains (dotted line). For comparison, we have overplotted the approximated coefficients used by Backmann et al. in their models (1993): dashed line for a 1µm sized grain, dot-dashed line for a grain having a radius of 0.04µm. The high porosity of the grains we used in our models "shifts" the absorption coefficient towards smaller wavelength, corresponding to a smaller "effective" grain size.

One key result of the Backmann et al. paper is the need of a two component model to be able to fit the IRAS fluxes. We have tried to check if such a conclusion remains valid when using less crude optical properties for the grains.

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998