4 Disk models

We have computed for the four stars the SEDs predicted by disk models, following Chiang & Goldreich (1997; CG97 in the following). In these models, the disk is isothermal in the vertical direction and in hydrostatic equilibrium so that its pressure scale height is a growing function of the distance from the star (flared disk; see Kenyon & Hartmann 1987). The flux at each wavelength is the sum of the emission of the disk midplane and of an optically thin, hotter surface layer, often referred to as the disk atmosphere (Calvet et al. 1991), which does not affect the physical structure of the disk, but dominates the SED in the mid-infrared.

The CG97 models need to specify a large number of parameters which are not well constrained by the SED alone (Chiang et al. 2001). However, some of these parameters can be derived from independent observations, namely the stellar properties, the disk inclination $\theta$ (Table  1) and the disk outer radius, which we have taken equal to the observed millimeter continuum size. In fact, this is only a lower limit to the physical size (Dutrey et al. 1996), which can be much larger if the disk surface density is a weak function of radius. Of the remaining disk parameters, we have fixed the surface density profile to be $\Sigma \propto r^{-p}$ with p=1.5. The disk inner radius $R_{\rm i}$ is defined by the condition $T(R_{\rm i})=1500$ K, where T is the temperature of a black body directly exposed to the stellar radiation (see Sect. 4.3 for further discussion of this point). The CG97 models allow us to use different grain properties in the disk midplane and atmosphere. For the midplane, we make the usual assumption $k_{\rm 1.3\,mm}=0.01$ cm² g^-1, $\kappa\propto \lambda^{-\beta}$ and varied $\beta$. Since the disks we consider are optically thin only at long wavelengths, the description of the opacity as a power-law function of $\lambda$ is adequate. The grains in the disk atmosphere are likely to be significantly smaller than those in the disk midplane. We have assumed a mixture of graphite and astronomical silicates (Draine & Lee 1984) with a size distribution as in MRN between 0.01 and 1 $\mu$m for both species. About 1/3 of the cosmic abundance of C and all Si is locked into grains. We have followed Chiang et al. (2001) in computing the heating of the disk midplane caused by a mixture of grains in the atmosphere.

The adopted parameters are summarized in Table 4. Note that the disk masses derived from the fits are within a factor $\sim$2 of those computed using Eq. (1) and given in Table 2. The model-predicted SEDs are shown in Fig. 1; the solid lines show the total emission and the thin dotted lines the separate contribution of the disk midplane and atmosphere.

 

 

Table 4: Disk parameters

(1)	(2)	(3)	(4)	(5)	(6)
Star	$M_{\rm D}$	$R_{\rm i}$	$R_{\rm D}$	p	$\beta$
	($M_\odot$)	(AU)	(AU)
AB Aur	0.01	0.4	100	1.5	2.0
CQ Tau	0.02	0.1	50	1.5	1.0
UX Ori	0.08	0.3	50	1.5	1.0
WW Vul	0.02	0.3	50	1.5	1.0

4.1 The long-wavelength part of the SEDs

The disk models provide a rather good fit to the long-wavelength part of the SEDs. The choice p=1.5, $\beta=1$ works well for three of our four stars, the only exception being AB Aur, which has an extremely steep spectrum at long wavelengths and requires $\beta\sim 2$. This suggests that the grains in the AB Aur disk midplane are smaller than those in the three UXORs. The possibility of grain evolution (i.e., coagulation) among HAe stars is suggested by Meeus et al. (2001). These authors find that the sub-millimeter spectral index in a sample of 12 stars decreases from values 3 in AB Aur and HD 179218 (which according to its location on the HR diagram is only 10⁵ yr old), to values of $\sim$2 in a number of isolated HAe stars. It is tempting to associate "large grains" to UXORs and interpret their variability as a result of aging disks. However, this may be premature, since there are a number of counterexamples. Just to quote one, HD 163296 has spectral index $\sim$2 and no evidence of large variability (Herbst & Shevchenko 1998). This issue needs to be addressed again with the help of observations at longer wavelengths (several HAe stars with flat spectral index may be detected at 7 mm with the VLA) and spatially resolved continuum maps.

Figure 3: Disk models for AB Aur with different surface density profiles. The solid line is for p=1.5, $M_{\rm D}=0.01$ $M_\odot$ (same as Fig. 1), the dashed line for p=15/14, $M_{\rm D}=0.007$ $M_\odot$. All the other parameters are as in Fig. 1

In the far-infrared, there is room for improvements, especially in the case of CQ Tau, where the model predicted flux is lower than observed by a factor $\sim$2-3 in the interval $\sim$30-150 $\mu$m. Chiang et al. (2001) have recently presented an improved version of the CG97 disk models, where a more realistic dust model is adopted. They show that ices are important contributors to the outer disk opacity, and that taking into account intermolecular translational modes of water ice at 45, 62 and 154 $\mu$m improves the fit to the CQ Tau SED. However, the dependence of the disk midplane emission on $\theta$ may also play a role, since at the very high inclination of CQ Tau a small change in $\theta$may change significantly the predicted emission of the optically thick disk midplane.

In spite of the rather good fit, it is important to remember that the law $\Sigma\propto R^{-1.5}$ is an ad hoc law derived by Hayashi (1981) for the Solar system. In a steady accretion disk, it is $\Sigma= \dot M/(\alpha c_{\rm s} h)$, where $\dot M$ is the accretion rate through the disk and one makes the usual assumption that the viscosity is proportional to the sound speed $c_{\rm s}$ and pressure scale height h via the coefficient $\alpha$. Using for $c_{\rm s}$ and h the radial dependence derived by CG97 (for optically thick disks), one obtains p=15/14, assuming that $\alpha$ does not depend on R. More realistic disk models predict more complex surface density profiles, generally flatter than a p=1.5 power-law (D'Alessio et al. 1998; Papaloizou & Terquem 1999). Models with p=15/14fit the data well, as shown for the case of AB Aur in Fig. 3. The p=15/14 disk is slightly less massive ( $M_{\rm d}=0.007$ $M_\odot$) than the p=1.5 disk and has lower surface density in its inner parts.

4.2 Mid-infrared and silicate emission

In the mid-infrared, the SED is dominated by the emission of the disk atmosphere. The models of Fig. 1 reproduce the observations reasonably well. In general, disk models which include the optically thin disk atmosphere can reproduce the intensity of the observed dust emission features and mid-infrared emission (see also Chiang et al. 2001). They can also account for the observed variations from star to star and among stars of similar properties in terms of relatively small changes of the characteristics of grains in the disk atmospheres. In the CG97 models, the emission at each wavelength $\lambda$ is proportional to the ratio $\kappa_\lambda/\kappa_\star$, where $\kappa_\star$ is the opacity at the frequency where the stellar radiation peaks, i.e., about 0.3-0.4 $\mu$m for the HAe stars. In many dust models, the mid-infrared opacity is dominated by silicates, while $\kappa_\star$ is due to carbonaceous grains (graphite in our case). One can vary the intensity of the mid-infrared emission by changing $\kappa_\star$, either by varying the fraction of carbon in grains or the size distribution of graphite. Figure 4 shows, for example, the intensity of the 10 $\mu$m feature when the amount of cosmic carbon locked into graphite varies from 100% (long-dashed line) to 3% (solid line). The relative intensity of the feature $F_{\rm peak}/F_{7.7\,\mu {\rm m}}$ increases from about 2 to 5.4. Our assumption that 30% of cosmic carbon is in graphite gives $F_{\rm peak}/F_{7.7\,\mu {\rm m}}$ $\;\sim 3.2$. One could reproduce the range of observed values of $F_{\rm peak}/F_{7.7\,\mu {\rm m}}$ by varying the fraction of carbon in grains between 10% and 50%. Alternatively, if the maximum size of graphite grains is 0.12 $\mu$m, rather than 1 $\mu$m, $\kappa_{10\rm\mu m}/\kappa_\star$ decreases from 0.52 to 0.32, and $F_{\rm peak}/F_{7.7\,\mu {\rm m}}$decreases from 3.2 to 2.0. Changes in $F_{\rm peak}/F_{7.7\,\mu {\rm m}}$ can also be due to variations in the composition and size distribution of the silicates themselves. The low value of $F_{\rm peak}/F_{7.7\,\mu {\rm m}}$ and of the overall mid-infrared emission in AB Aur can be understood if the grains in the disk atmosphere (as well as those in the midplane) are somewhat smaller than in the other stars. Note that graphite can be replaced wholly or in part by other grain species, such as metallic Fe, whithout changing our results. Variations in the grain size and in the proportion of the various materials from star to star are seen in the ISO spectra of HAe stars (Bouwman et al. 2001).

Figure 4: Model-predicted silicate feature for UX Ori for increasing fraction of carbon locked in graphite grains, from 3% (solid line) to 10% (short-dashed line), 30% (dotted line) and 100% (long-dashed line). All other parameters as in Fig. 1

The cross section of "astronomical'' silicates we have used is a more accurate description of the silicates in the diffuse ISM than in pre-main-sequence circumstellar disks, where a mixture of largish pyroxene and olivine grains provides a better fit to the shape of the 10 and 20 $\mu$m features (Reimann et al. 1997; Natta et al. 1999, 2000b; Bouwman et al. 2001). However, a study of the mineralogy of grains in the disk is not the scope of this paper. We just want to point out that CG97 models can be used to investigate the mineralogy of dust, and, in particular, the nature of the silicates present at the disk atmosphere.

4.3 The near-infrared excess

The only serious failure of the CG disk models is their inability to account for the large excess luminosity in the near-infrared seen in all the stars. This old problem (Hartmann et al. 1993) has been recently rediscussed by Natta et al. (1999) in their study of UX Ori and noted by Chiang et al. (2001) in all the HAe stars in their sample. No satisfactory solution has so far been proposed. We will discuss it here again in more detail, and propose a modification of the structure of the inner disk which may explain the observations.

We focus in this section on the emission in the range of wavelengths 1.25-7 $\mu$m, (which we will call "near infrared" in the following) which is shown in detail in Fig. 5. For each star we have subtracted from the observed emission the stellar contribution, as shown in Fig. 1. This is a large fraction of the total in the near-infrared, about 70% in J, 30-50% in H, 10-20% in K for the three A stars. For CQ Tau, the star contributes 100% of the H flux, and is still 45% of the total in K. As a consequence, the disk fluxes at shorter wavelengths are very uncertain, since they are strongly affected by uncertainties on the stellar parameters and extinction and by the variability of the star in the visual and near-infrared. This is shown, for example, by the two sets of points for UX Ori, which correspond to two observations at different epochs.

Figure 5: Observed near and mid-IR spectra of the four stars, as labelled, after subtraction of the photospheric emission

In spite of these caveats, there are a number of characteristics worth noticing. First of all, the four stars have very similar SEDs, with a peak in $\nu F_\nu$ around 2 $\mu$m and a sharp decrease at shorter wavelengths (the $\sim$3 $\mu$m bump; Hillenbrand et al. 1995; Hartmann et al. 1993). Between $\sim$3 and 7 $\mu$m the SEDs have an almost identical shape. The dependence of $\nu F_\nu$ on wavelength is flat, roughly as $\lambda^{-(1.1-1.3)}$. This is typical of the SED of geometrically flat, optically thick disks, where $F_\nu\propto \lambda^{-4/3}$.

The excess luminosity in the near-infrared is rather large, ranging between 12 and 25% of the stellar luminosity (see Table 3). These values are comparable to the fraction of stellar light intercepted and re-radiated in the near-infrared by the inner regions of a flat disk. However, the conclusion that flat disks can account for the observed near-infrared properties of HAe stars is very likely incorrect.

A flat disk intercepts a maximum 25% of the stellar radiation only if it extends inward all the way to the stellar radius. This cannot be the case in HAe stars, as illustrated in Fig. 6. Flat disks that extend to the stellar radius reproduce the observed value of $L_{\rm NIR}$, but their emission peaks at wavelengths which are too short, typically 0.8 rather than 2 $\mu$m. In the range 3-7 $\mu$m, even a model with $R_{\rm i}$ = $R_\star$ predicts values of $\nu F_\nu$ a factor of 2 lower than observed. Disks with large inner holes have a SED that peaks at longer wavelengths, but the fraction of stellar light intercepted and re-emitted is far too small.

Figure 6: Disk model predictions for AB Aur. Dashed lines are for flat disks, solid lines for flared ones. The inner radius $R_{\rm i}$ increases from $R_\star$ to 25 $R_\star$, as labelled. All other parameters are the same. The dots and the thick solid line show the observed spectrum, after subtraction of the photospheric component (same as Fig. 5)

Figure 7: Fraction of the stellar luminosity intercepted by disks with outer radius $R_{\rm D}=200$ AU and increasing inner radius $R_{\rm i}$. Both flat and flared disks are plotted (dashed curves). The solid lines show the fraction of stellar light intercepted by the inner disk wall at $R_{\rm i}$ computed as $H_{\rm i}/R_{\rm i}$, i.e., assuming that the wall has the shape of a cylinder of height $H_{\rm i}$. In curve (1) $H_{\rm i}$ is equal to the pressure scale height h, in curve (2) to the height of the disk photosphere. The vertical thin dashed lines show the location of T=1500 and 2000 K. The stellar parameters are those of AB Aur

Figure 7 shows the fraction of stellar radiation intercepted by flat and flared disks of increasing inner radius. Flat disks with $R_{\rm i}$ = $R_\star$ ($\sim$0.01 AU) intercept 25% of the stellar radiation (independently on stellar and disk parameters as long as $R_{\rm D}\gg R_\star$), while flared disks intercept significantly more. The exact value depends on the disk flaring and outer radius; it is about 50% in the case shown in the figure. If $R_{\rm i}$ increases, the fraction of intercepted stellar light decreases rapidly. For $R_{\rm i} \sim 0.3$ AU ($\sim$25 $R_\star$), it is almost zero in the case of flat disks and of the order of 25% for flared disks. This is also significantly less than observed; it increases to 40% only for $R_{\rm D}=1000$ AU. Moreover, as shown in Fig. 6, the emission of flared disks with large $R_{\rm i}$ is always peaked at longer wavelengths, not in the near-IR. We find no way in which either flat or flared disk models can account for the observed SEDs over the whole range of wavelengths.

The models discussed so far consider only the fraction of stellar radiation intercepted by the disk surface. In fact, if the disk has a large inner radius, there is an additional region, roughly shaped as a cylinder of radius $R_{\rm i}$ and height $H_{\rm i}$, which also intercepts and re-radiates a fraction of the stellar light. This inner wall is much hotter and has therefore a much bigger scale height than a CG disk at the same distance from the star, because it sees the stellar surface almost perpendicularly. The exact amount of intercepted radiation may be significant. As a zero order indication, we show in Fig. 7 the values for a wall at $R_{\rm i}$ with $H_{\rm i}=h$, where h is the scale height of the disk at the temperature of the wall (curve 1) and $H_{\rm i}=H$, where H is the photospheric height ($\sim$4-5 h; curve 2).

Figure 8: Model-predicted SED of AB Aur, same as Fig. 1 (dotted line). We have added the emission of the inner disk wall, as described in the text (dashed line). The total is shown by the solid line

The vertical dashed lines in Fig. 7 show the approximate region where we expect dust evaporation to occur. Thus, if the inner radius of the disk is determined by dust evaporation, then $R_{\rm i}\sim$ 0.3 AU, and the fraction of stellar radiation intercepted by the inner wall can be as high as 20%, while the disk surface will intercept about 25% of the stellar radiation. These numbers are roughly consistent with the values observed in AB Aur and in the other HAe stars in our sample. We show for example in Fig. 8 the AB Aur SED where we have added to the disk emission shown in Fig. 1, a BB component at T=1700 K corresponding to an inner wall at $R\sim$ 0.27 AU which intercepts about 30% of the stellar radiation. While the fit to the data is not perfect, it is clear that the model we propose decreases the discrepancy between models and observations. However, more detailed models are in order, which include, among others, the effect of the shade that the puffed-up inner wall projects over the disk at larger radii (Dullemond et al. 2001).

This scenario, of a flared disk with a large inner hole determined by dust evaporation, appears consistent with the recent results of near-IR interferometry of Millan-Gabet et al. (2001). They find that the visibility data in J, H, K for AB Aur are best fit by a model where the emission is coming from a ring of radius $\sim$0.3 AU, seen almost face-on. In our scenario, the Millan-Gabet et al. ring is just the inner wall of the disk, which dominates the observed near-infrared emission.