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

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4. Other observables

4.1. Outer density profile

Since we have no spectral signature of the dust in the outer regions, we have adopted, as a first attempt, the simpliest solution, that is, no change change in the grain composition and no change in the size distribution between the inner (r [FORMULA] 100 AU) and the outer regions. In order to be able to deduce the consequences of our derived density on other observables and check the compatibility with them (IRAS measurements, visible scattering profiles) we have had to extrapolate our density beyond 100 AU. The visible observations of the scattered starlight showed that the density should decrease for distances larger than 100 AU following a power-law with an index in the range [-1.5,-2.3] (Artymowicz et al., 1989; Kalas and Jewitt, 1995; Smith and Terrile, 1984). We have therefore extrapolated our density profile using a power-law density with an index of -1.7 index beyond 100 AU (Artymowicz et al. 1989).

4.2. Thermal emission

We tried to reproduce the photometric fluxes of the dust at 10.1 and 20 [FORMULA] m measured by Backmann et al., 1993, with the IRTF, in 4 and 8 arcsec beams, including the effects of the convolution with the IRTF PSF. The results are given in Table 1.


Table 1. Comparison with Backman's measured photometrical fluxes at 10.1 and 20 [FORMULA] m. CM stands for core-mantle silicate grains (size distribution: power law index of -3.0 from 0.1 to 10 [FORMULA] m and -3.3 from 10 to 1000 [FORMULA] m). DL stands for Draine and Lee silicates with size distribution ranging from 0.1 to 1000 [FORMULA] m and a power law index of -4.0.

The fluxes deduced from our surface density are in disagreement with Backmann et al. observations even taking into account the effects of the IRTF PSF. The Backman et al. data were taken at 3 air masses and used of the star [FORMULA] Car as a calibrator. This leads to an underestimation of the measured flux in the Q band because the atmospheric transmission falls with increasing wavelength along the Q filter. In reality, the [FORMULA] Pic flux increases with wavelength (Backman, private communication). Secondly, the density grows rapidly with increasing distance to the star. It turns out that the 20 [FORMULA] m flux is produced in a very large domain of distance to the star with two dominating regions: inside 10 AU and between 50 and 100 AU. As our data have large uncertainties in the last region, it may also explain the discrepancy with Backman et al. data.

We also tried to reproduce the fluxes in the IRAS bands of 12, 25, 60 and 100 microns. The results are reported in Table 2 where we indicate the predicted flux using the surface density deduced from our observations (r [FORMULA] 100AU) extended with a radial power-law with an index of -1.7 (Artymowicz et al. 1989; Backmann et al., 1993), and the total flux in a 100 arcsec beam measured by IRAS. The best results were obtained using DL silicates with an index of -3.5 for the size distribution but we have also list the fluxes given by our model that uses a smaller index more appropriate for fitting the 10 [FORMULA] m spectrum (-4.0).


Table 2. Comparison between IRAS fluxes and one component models. Since the size distributions used to fit the 10 [FORMULA] m spectrum gave very poor fits, we have tried to modified the size distribution indexes, but the resulting model fluxes agree poorly with the IRAS measurements.

The results in Table 2 are in relatively poor agreement with IRAS fluxes except at 12 [FORMULA] m, which is the wavelength at which density profile was deduced (all the 12 [FORMULA] m comes the inner disk, i.e. inside 50 AU). This can be understood because, as explained in Sect.5, the 10 [FORMULA] m data probes the dust in a narrow cone with an opening angle of about 30 degrees, while at longer wavelength the data are sensitive to much wider cones (the opening angle grows with wavelength). If there are azimuthal anisotropies in the density, the 25 [FORMULA] m flux may not correspond to the density deduced from 12 [FORMULA] m measurements. As seen in Table 2, the reproductibility of the IRAS fluxes depends dramatically on the chosen size distribution indexes. This means that the IRAS fluxes put the strongest constraints on the size distribution.

Finally, we have checked that our density and our dust mineralogy choice are consistent with sub-mm and mm observations. We found a flux of 97 mJy at 800 [FORMULA] m compatible with Zuckerman and Becklin's observations (1993) (115 [FORMULA] 30 mJy), and 22 mJy at 1300 [FORMULA] m (Chini et al, 1991 found 24 [FORMULA] 2.6 mJy) without assuming a size of the disk limited to 500 AU (since the disk has been traced as far as 800 AU from the star by Kalas and Jewitt, 1995).

4.3. Comparison with visible observations

As coronographic techniques improve with the use of adaptative optics, the limitations to observe the inner part of the disc will become less restrictive, and one should be soon able to observe the scattered starlight by dust particles in regions only accessible to infrared cameras up until now (Burrows et al., 1995, using the Hubble Space Telescope have succeeded very recently in observing the scattered light down to 20 AU). In order to assist these observations, we have constructed, using our surface density, a profile showing what one could see in the visible.

4.3.1. Scattering models

As the Mie theory (to derive the scattering phase function) applies only to compact spherical grains (but it can be used to calculate the scattering coefficients in the case of porous grains, see Mukai et al., 1992), we used the different kinds of phase functions one can find in the litterature. To modelize the scattering profile due to porous grains we used the Henyey-Greenstein phase function which is an approximation of the real phase function. It been used very often to fit the scattering properties of grains in various environments such as the interstellar medium (Mattila, 1970), planetary atmospheres (Irvine, 1975), and planetary rings (Cuzzi et al., 1984). We have also used the Leinert empirical phase function deduced from measurements of the zodiacal light in our solar system.

We have tried to reproduce the flux profiles in the R band measured by Kalas and Jewitt (1995). As shown in Figs. 12 and 13 the calculated scattered flux is slighly lower than the observed flux.

[FIGURE] Fig. 12. The observed and reconstructed surface brightness profile of the N-E extension in [FORMULA] Pic using a one component model. Plain line: using Henyey-Greenstein phase function, dashed line: using Leinert empirical phase function (found in the Solar System). The stars are from the measurements of Kalas and Jewitt (1995).

[FIGURE] Fig. 13. Same as in Fig. 12 but for the S-W extension.

The observed scattered light at a given projected distance to the star are most sensitive to grains located further because (1) for the largest grains, the phase function is narrowly peaked in the forward scattering direction and (2) because the radial density peaks at 75 AU. If the disk is non-axisymmetric, observing the thermal radiation at 12 [FORMULA] m and the scattered light at 0.6 [FORMULA] m will not lead to the same density profile since they do not probe the same regions in the disk (see Fig. 9). This fact may explain the discrepancy about the S-W/N-E asymmetries which are very weak in visible, but can reach a factor of two in our data. For the same reasons, a discontinuity in the radial density will create a discontinuity in the derivative of the scattering profile at a distance somewhat closer to the star than the radial discontinuity. So if the break in scattered profile observed at 80 AU by many observers (Golimowsky et al., 1993; Kalas and Jewitt, 1995) is real, the corresponding radial discontinuity is located around 85-90 AU.

4.4. Conclusion about the one component models

As shown above, one component the model allows only marginally to fit the IRAS data whose observed flux is, in majority, produced in the range [80-100 AU] where the density is maximum (it must be noted that it is also the case for all the longer wavelength, even for the 1.3 mm observations).

The results so far suggest that we should remove only one assumption, i.e. that the grains are composed of the same material everywhere in the disk.

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998