5 Discussion and conclusions

We have presented in this paper state-of-the-art SEDs of four HAe stars, and discussed them in the context of circumstellar disk models. We have used the simplified description of the disk structure and emission outlined by Chiang & Goldreich (1997). The disk is in hydrostatic equilibrium in the vertical direction at the temperature of the disk midplane (flared disk). Its emission is computed as the sum of the emission of the optically thick midplane and of the optically thin, overheated surface layers. The properties of grains in the midplane and atmosphere can be independently specified.

These models provide a good fit to the observations, if the disk is truncated at an inner radius of a few tenths of AU. This roughly corresponds to typical dust sublimation temperatures (1500-2000 K), if we consider that the inner "wall" of the flared disk is exposed directly to the stellar radiation, and is therefore much hotter than the disk midplane, which receives the stellar radiation at a small grazing angle. The inner wall is then puffed-up with respect to the disk, and intercepts a significant fraction of the stellar radiation, which increases with the inner radius $R_{\rm i}$. The emission of the inner wall is mostly in the near-infrared and can in principle account for the observed shape and luminosity at these wavelengths, which has been a long-standing puzzle. We suggest that the recent interferometric observations of AB Aur obtained by Millan-Gabet et al. (2001) in H and Kdetect exactly the emission of this wall, which at these wavelengths looks like a ring on the plane of the sky.

One of the implications of the above disk model is that there is no significant absorption between the star and the inner wall. In other words, any gas within $R_{\rm i}$ must be optically thin to the stellar radiation. This is not the case for the disk models considered in Sect. 4, where we have adopted a surface density profile $\Sigma\propto R^{-1.5}$ and derived the disk mass required to fit the long-wavelength fluxes. With the usual assumption of a gas-to-dust mass ratio of 100, the optical depth of the inner gaseous disk is very large (about 500-1000 at $R_{\rm i}$, assuming that the opacity is mostly due to H^-; see Hartmann et al. 1993). The gas absorbs efficiently the stellar radiation, so that dust sublimation occurs much closer to the star (at about 2-3 $R_\star$), where the disk midplane temperature, which is significantly lower that that of the unscreened wall, reaches dust sublimation values. However, the surface density of the inner disk is not constrained by the data, as long as the disk remains optically thick at $R>R_{\rm i}$. The CG97 model with $\Sigma\propto R^{-15/14}$(Fig. 3) has surface density at $R_{\rm i}$ 40 times lower than our standard model, and an optical depth to the stellar radiation of only about 10-20. This model corresponds to an accretion rate $\dot M \sim 10^{-8}$$M_\odot$ (for a viscosity coefficient $\alpha=0.01$), in agreement with the low accretion rates expected in HAe stars (Hartmann et al. 1993; Ghandour et al. 1994). It is possible that the amount of gas in the inner disk is even lower (if, for example, the viscosity coefficient $\alpha$ is not constant). Before speculating further, however, the effect of the sudden increase of the opacity due to dust sublimation (by more than a factor 1000) in disks where the gas opacity is not extremely large, as well as the stability of the resulting geometry, should be addressed in detail.

As far as the emission at longer wavelengths is concerned, we confirm previous results that CG97 disk models can account for the observed SEDs not only of TTS but also of HAe stars. The mid-IR, as well as the emission in any dust feature, is dominated by the emission of the optically thin surface layers. Although we have not addressed the dust mineralogy in any detail, we find that energetically CG models can account for the observations. The dust in the disk atmosphere is probably a mixture of largish silicates and carbonaceous materials. At even longer wavelengths, dust in the disk midplane dominates the emission; however, the degeneracy of the SED with respect to dust and disk parameters does not allow more than a check of consistency between models and observations.

In summary, most of the known properties of Herbig Ae stars can be well explained by models where the circumstellar matter is confined to a disk. We suggest here that a proper treatement of the structure of the inner disk may solve the long-standing puzzle of the near-infrared properties of these stars. Namely, we notice that for low accretion rates the inner gaseous disk is optically thin, the optically thick dust wall produced by dust condensation is exposed directly to the stellar radiation and is therefore puffed-up with respect to the cooler disk midplane. The SED is the result of the emission of three different parts of the disk, namely the inner wall in the near-infrared, the optically thin atmosphere in the mid-infrared, the midplane at sub-millimeter and millimeter wavelengths.

Disks in TTS must be very similar. The lack of the "3 $\mu$m" bump in stars of later spectral type can be easily understood if we consider that the effects of the inner wall become less and less relevant as the stellar luminosity decreases and $R_{\rm i}$ approaches $R_\star$. Moreover, if, as we suspect, the accretion rates of the classical TTS are on average higher than in HAe stars, the inner gas disk is more optically thick, with the effect of decreasing $R_{\rm i}$ even further. The dependence of the wall emission on the spectral type of the central star is currently under investigation (Dullemond et al. 2001).

Of the goals stated in Sect. 1, two concerned more specifically the nature of UXORs. There is no significant difference in the inferred disk structure between the three UXORs and AB Aur, with the single exception that grains in AB Aur are probably smaller than in the other three stars. The significance of this last point, however, is not clear, since the degree of variability of AB Aur is dubious. It needs to be confirmed in a larger sample of photometrically stable HAe stars before we can interpret it as a hint of an evolutionary difference between the two groups.

We do not find any clue to the nature of the dust enhancements that occasionally occult the three UXORs. However, we note that the inner wall is potentially a promising location for the "clumps''. At $R_{\rm i}$, the sound speed is a few km s^-1 so that density enhancements of stellar size can grow and dissipate on timescales of a few days, which are typical of UXOR minima. The fact that we expect that such walls will be less relevant in cooler stars may explain the relative lower fraction of UXORs among TTS. However, in order for the line of sight to intercept the wall, one needs to see the disk at large inclination, larger than about 75 deg in our examples. This is much more edge-on than found by Natta & Whitney (2000) from the polarization of the visual light during photospheric minima. For such values of the inclination we expect that the outer disk will obscure the star at all times (Bertout 2000). In spite of these difficulties, we feel that the role of density fluctuations near the dust sublimation radius should be investigated further.

Acknowledgements

The discussion of the disk structure in HAe stars has involved many of our collegues, without whom this paper would have not been possible. Among them, I (AN) want to thank in particular Jeroen Bouwman, Carsten Dominik, Kees Dullemond, Alex de Koter and Rens Waters, who have made my visit in Amsterdam most pleasant and fruitful. Very interesting conversations with Nuria Calvet, Claude Bertout and Caroline Terquem have helped in clarifying a number of issues addressed in this paper. We gratefully aknowledge the support of ASI (grants ARS-99-15 and 1/R/27/00) to the Osservatorio Astrofisico di Arcetri. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France.