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2140 2160 2180 2200 wavelength (nanometers)

2.14e-0a 2.15e-0a 2.16e-0a wavelength (m)

Fig. 5. AMBER/VLTI high angular resolution observations of the inner disk in the HD104237 HAe system. The top-left panel shows that the spectral energy distribution of the system, the near-infrared excess exceeding the photospheric emission (dotted,) is due to the disk inner rim; the top-left panel shows the AMBER/VLTI total power spectrum (dashed) and interferometric differential visibilities (red points with errorbars), which show no variation across the prominent Br7 line; in the bottom panels visibilities predicted by different models of the Br7-emitting regions are compared to the observed differential visibility, the models consistent with the observations predict an emitting region essentially coincident with the disk inner dusty rim (see sketch on the bottom-left panel). The figure has been adapted from [29]

2.14e-0a 2.15e-0a 2.16e-0a wavelength (m)

Fig. 5. AMBER/VLTI high angular resolution observations of the inner disk in the HD104237 HAe system. The top-left panel shows that the spectral energy distribution of the system, the near-infrared excess exceeding the photospheric emission (dotted,) is due to the disk inner rim; the top-left panel shows the AMBER/VLTI total power spectrum (dashed) and interferometric differential visibilities (red points with errorbars), which show no variation across the prominent Br7 line; in the bottom panels visibilities predicted by different models of the Br7-emitting regions are compared to the observed differential visibility, the models consistent with the observations predict an emitting region essentially coincident with the disk inner dusty rim (see sketch on the bottom-left panel). The figure has been adapted from [29]

where Td and Md are the dust temperature and total mass, D is the distance to the observer, Bv (Td) the Planck function at the appropriate frequency and temperature, and kv is the dust opacity per unit mass. At millimeter wavelengths the dust opacity as a function of frequency can be approximated with a power law kv ~ v3, and the Planck function can be well approximated by the Rayleigh-Jeans function, hence

with a = 2 + [. This implies that the shape of the spectral energy distribution at millimeter wavelengths can be used to derive the dust opacity power law index ¡3, while the total flux measured at a given wavelength is proportional to the product of temperature and mass.

The value of 3 depends on the type of dust grains in the ensemble: composition, shape, size, and combination of these. The mixture of grains that fits the properties of the interstellar medium correspond to a value of ¡ close to two (a ~ 4). If, however, the dust grains become much larger that the wavelength at which the fluxes are measured, then the dust opacity becomes gray (as only the geometrical cross section of the grains is relevant) and a value approaches 2 (3 = 0). Even if the exact value of beta depends on a variety of dust properties that are hard to constrain (see Fig. 6 for some examples of 3 computations for different properties of the grain population), the general result that a low value of 3 is only consistent with the presence of large grains is a solid one (se also [9]).

Obviously this is a powerful probe for the presence of very large grains in circumstellar disks as discussed in [2]. The results of the first millimeter and submillimeter survey for grain growth in circumstellar disks is reported in this

Fig. 6. Dust opacities per unit mass (bottom panel) and power law exponent (top panel) as a function of the maximum size for the dust grains and for various dust size distributions. The histogram on the right side of the top panel illustrates the values of the index beta measured in a sample of protoplanetary disks around T Tauri and HAe stars. Adapted from [25]

Fig. 6. Dust opacities per unit mass (bottom panel) and power law exponent (top panel) as a function of the maximum size for the dust grains and for various dust size distributions. The histogram on the right side of the top panel illustrates the values of the index beta measured in a sample of protoplanetary disks around T Tauri and HAe stars. Adapted from [25]

paper. In practice, as already discussed in [2], the situation is more complex, as disks are not isothermal ensembles of optically thin dust.

Even if the assumption of an average temperature, as discussed in [23], is not a poor approximation of (1), to give a rough estimate of the disk mass and to obtain an accurate determination of the value of 3, it is necessary to use more sophisticated disk models that take into account the temperature profile and the presence of an optically thick inner region of the disk. As discussed in [30], low values of a approaching 2 may be an indication of low values of 3, hence grain growth, or may be the consequence of unexpectedly high optical depth disks (see Fig. 7).

To resolve these ambiguities, fit proper disk models, and derive accurate values of 3, it is necessary to resolve the disk emission at millimeter wavelengths and to use this additional constraint to solve the ambiguities. Tho achieve this, it is necessary to obtain angular resolutions of the order of arcsec or beter, corresponding to linear resolutions of the order of 100 AU in the nearest star forming regions. These angular resolution at millimeter wavelengths can only be achieved by large radio interferometers (see also the lecture by Beuther). An example of such a study is the one on the CQ Tau system by [31] (see also Fig. 8). The combination of the millimeter spectral index from 1 to 7 mm and high angular resolution VLA observations at this wavelength allow to obtain an accurate measurement of 3 and to derive the presence of very large (centimeter size) grains in the disk midplane.

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