Issue 
A&A
Volume 529, May 2011



Article Number  A105  
Number of page(s)  41  
Section  Interstellar and circumstellar matter  
DOI  https://doi.org/10.1051/00046361/201015209  
Published online  12 April 2011 
Online material
Appendix A: Optical depth vs. variable β
Because a direct inversion of the brightness temperature profile is impossible, the determination of the parameters is fully implicit. Figure A.1 illustrates two possible types of brightness temperature profiles that can occur in our analysis. The continuous and dashed lines represent brightness at 1.3 mm and 2.7 mm for a typical power law distribution, with constant dust properties β(r) = β_{m}. The outer region is optically thin, and the slope constrains p + q. β_{m} is derived from the brightness ratio O. The inner region is optically thick, and constrains the exponent q as well as the temperature at 20 AU, T_{20}. The small brightness difference between the two frequencies is caused by the RayleighJeans correction. The dotted and dotdashed lines represent an optically thinner disk at 1.3 and 2.7 mm respectively, with a viscous type profile with R_{c} = 150 AU. In addition, β(r) is assumed to vary with radius following Eq. (17) with R_{b} = 60 AU and R_{w} = 20 AU, κ(1.3 mm) being constant. Here, the inner region is optically thin, and its slope is q + γ. Note that if the temperature of that disk would be 4 times higher, it would mimic reasonably well the previous power law, optically thick case, provided γ is not too large. Accordingly, sources displaying a wavelengthindependent flattened (apparent exponent ≃ 0.4 − 0.7) inner brightness distribution can be interpreted either as optically thick sources, or as variable β(r) with β(r) ≃ 0 in the inner region. Steeper apparent exponents are not realistic for the temperature dependence. Note that the typical noise level is around 0.05 − 0.1 K in our observations at both wavelengths.
Fig. A.1
Sample result illustrating the shape of the brightness distribution for our disk models. Thick line: constant β power law model at 1.3 mm, dashed line: same model at 2.7 mm. Dotted line: constant κ(1.3 mm) but variable β(r), taperededge model at 1.3 mm; dashdotted line: same model at 2.7 mm. 

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Appendix B: Sampling effects and best model
Because of the fully implicit derivation of the model parameters, an objective determination of the “best” model is difficult. The same source may be (nearly) equally well represented by either Model 1 or Model 2. We use a χ^{2} criterium to determine the best matching model. However, it is important to realize that our data consists in a large (several 10^{4}) number of statistically independent visibilities, each with very little (essentially zero) signaltonoise. The χ^{2} is given by (B.1)where O_{i} are the (complex) observed visibilities (O^{2} actually being used to note O × O^{ ∗ }, O^{ ∗ } being the complex conjugate of O), M_{i} the modeled visibilities. The weights are derived from the theoretical noise using the system temperature, antenna gain, observing bandwidth and integration time. In general, , where M^{b} is the bestfit model, so even the null model M_{i} = 0 yields a χ^{2} on the order of N, the number of visibilities, as W_{i} is the inverse of the variance of . Thus, the reduced χ^{2}, is a poor evaluation of the fit quality, which is close to 1 even for a very poor (null) model. Only the relative differences Δχ^{2} between models of equivalent number of parameters can reveal whether one is better than the other.
Another subtle effect in comparing absolute values of χ^{2} is the impact of discretization. A numerical model M is an approximation of the theoretical model T, M = T + E, where E is a numerical error term. So Because the model fit the observations and the numerical errors are not correlated with the observations, the last term is negligible, consequently the final χ^{2} is a sum of the true (no numerical errors) term plus an offset cause by numerical effects. To make numerical errors negligible requires to be much less than 1. This is especially important when comparing different theoretical models. However, within a given model, the bestfit parameters may be determined with sufficient precision even if the numerical error term is not small.
Appendix C: Impact of the assumed temperature law
In this appendix, we investigate the impact of the dust temperature profile on the derived disk parameters. We consider two different profiles. Profile (i) is a power law T(r) = T_{100}(r/100AU)^{ − q}. Profile (ii) is a broken power law: it has a constant temperature between R_{i} = 40 AU and R_{f}, R_{f} being a variable parameter, while for r < R_{i} or r > R_{f}, the temperature is a power law with exponent q = 0.5, with to T(r) = T_{1} at r = 1 AU. The temperature law is continuous as a function of r, and we used T_{1} = 200 K by default. We analyzed the observations of a few sources (DL Tau, DM Tau and MWC 480) to explore the dependency of the derived surface density parameters on T_{0},q and R_{f}. Figure C.1 illustrates the main impact of the temperature law on the surface density parameters, which is applicable to all optically thin sources. Figure C.1 is for Model 2 (so p is to be interpreted as γ), but similar results are obtained for Model 1.
Fig. C.1
Sample results illustrating the main dependency of the surface density profile on the temperature law. Top left: R_{c} versus q. Top right: R_{c} versus R_{f}; bottom left: Σ.T versus T; bottom right: p + q versus q. The observed source is DL Tau. 

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Fig. C.2
Sample results illustrating the weak dependency of the dust parameters R_{b} and R_{w} on the assumed temperature law. 

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For Profile (i):

Σ_{0} is nearly proportional to 1/T, with small corrections at low T owing to deviations from the RayleighJeans behavior.

Similarly, p + q is nearly constant. This is equally valid for Model 1 (power law) and Model 2 (tapered edge).

In Model 1, R_{out} is only weakly affected by the changes in p

In Model 2, R_{c} increases by 20 to 30% when q increases from 0 to 0.5.
For Profile (ii):

In Model 1, R_{out} slightly decreases with R_{f} (by about 10%), and p changes by about 0.1. Variations are not fully monotonic, however.

In Model 2, R_{c} decreases by about 20 to 30%, when R_{f} goes from 50 to 200 AU. This is similar to the effect of q in Profile (i), as increasing R_{t} flattens the temperature distribution.
For more optically thick sources, like MWC 480, the effect on p is larger, because of the opacity corrections. However, in this case, q can be determined from the observations, because the χ^{2} significantly depends on its value. Restricting the range of q to within its typical uncertainty limits the impact on p to about 0.2.
Except for the absolute scaling of the density as 1/T_{100} (or 1/T_{1} in Profile (ii)), the derived density distribution are thus not significantly affected by the assumed temperature law.
More importantly, R_{c} and p are affected in the same proportions at both wavelengths. Thus, the uncertainties on the temperature law have no significant effect on the derivation of the radial dependence of β(r) (see Fig. C.2). Incidentally, we note that in DL Tau, a better fit to the observations is obtained using Profile (ii) with R_{f} = 100 AU.
Appendix D: Disk parameters from observable quantities in the viscous model
The shape of the surface density profile used in Model 2 corresponds to the selfsimilar solution of the viscous evolution of a disk under the assumption that the viscosity is constant in time and a power law of radius (see LyndenBell & Pringle 1974; Pringle 1981). Under these assumptions, the surface density as a function of time and radius is given by (Eq. (17) of Hartmann et al. 1998) (D.1)where r = R/R_{1}, T = (1 + t_{∗}/t_{s}) is a dimensionless time, t_{∗} the disk/star age and t_{s} is the viscous timescale at R_{1}, defined by (D.2)Our observations (at unknown time T) are characterized by the surface density law described by our Eq. (5) (D.3)So by identification, we obtain (D.4)and (D.5)which, eliminating R_{1} usinq Eq. (D.4) (D.6)A time derivative of Eq. (D.1) (taken for r = 0) further indicates that the mass accretion rate is (D.7)We have in principle five unknowns (C,R_{1},T,ν_{1},γ), and five measurements: three from our study (M_{d} or Σ_{0},R_{c},γ), the stellar age t_{∗} from evolutionary tracks and the mass accretion Ṁ, usually derived from the accretion luminosity (see Gullbring et al. 1998). Although this formally yields a solution, it is nearly degenerate when one considers the uncertainties on the measured quantities. This can be realized by noting that the mass accretion rate can be rewritten as (Eq. (14) from Isella et al. 2009) (D.8)while from Eqs. (6) and (D.5), the time dependency of the disk mass is simply (Eq. (A7) from Andrews et al. 2009) (D.9)so, by simple elimination (D.10)which simply gives (D.11)This is the only equation involving t_{s}. R_{c}, and thus R_{1} does not appear in this expression because R_{1} only reflects the initial condition of disk size, not its future evolution.
The (time independent) viscosity at any arbitrary radius is given by
(D.12)which, using the expression of R_{1} in Eq. (D.4), can be expressed in terms of the observable quantities as
(D.13)It is customary to express it in terms of the α parameter, ν(r) = α(r)c_{s}(r)H(r), where c_{s} is the sound speed, and H(r) the scale height (D.14)In hydrostatic equilibrium, (D.15)T_{g} being the gas temperature in the disk midplane. Approximating T_{g}(r) by a power law of exponent − q (q ≃ 0 − 0.6), we derive R_{c} and γ are directly constrained by our observations, while t_{∗} is derived from evolutionary tracks. The last term c_{s}(R_{0})H(R_{0}) depends on the stellar properties. We note from Eq. (D.15) that , and thus αC_{s} scales to first order as .
Appendix E: Alternate disk models
With the alpha prescription of the viscosity (radially uniform and constant in time α) and a (time independent) power law temperature T_{k} = T_{0}(r/r_{0})^{ − q}, , so γ = 3/2 − q, Eq. (D.1) can also be written as (E.1)At long times, T ≫ 1, the density profile evolves as p = 3/2 − q, or p + q = 3/2.
A similar formula can also be recovered for the β prescription of the viscosity, (Richard & Zahn 1999). It is equivalent to setting q = 1 in Eq. (E.1), and thus corresponds to γ = 0.5.
The selfsimilar solutions of the evolution equation for the disk surface density were obtained under several simplifying assumptions. Desch (2007) pointed out that accounting for the early planet migration as predicted by the Nice model (Tsiganis et al. 2005; Gomes et al. 2005), the initial exponent of the surface density for the Solar Nebula would be very close to p = 2.2. To explain this slope, Desch (2007) recovered a different shape for the surface density in steady state configuration. The general form of the surface density in the Desch (2007) solution is (E.2)where r_{u} is the radius at which the disk has an apparent slope p and x_{u} = (2 − p − q)/(p + q − 3/2). For p + q > 2, x_{u} < 0 and the surface density vanishes at radius . Note that the classical steadystate result Σ(r) ∝ r^{ − (3/2 − q)} corresponds to the asymptotic limit x_{u} → ∞, and is obtained by imposing different boundary conditions on the evolution equation of angular momentum.
Appendix F: Unresolved, possibly thick, sources
For unresolved sources, the outer radius can only indirectly be constrained from the observed flux. Assuming uniform opacity τ, and a standard power law for the temperature T(r) = T_{0}(r/R_{0})^{ − q}, the outer radius is given by (i being the inclination) (F.1)
A lower limit is recovered or i = 0 and τ → ∞ (F.2)The disk mass is given by (F.3)With q ~ 0 − 0.5, a lower limit on M_{d} is obtained for τ ≃ 0.5. Solutions with density/opacity decreasing with radius will lead to higher masses.
Appendix G: Figures for individual sources
We display here the figures for individual sources.
Fig. G.1
As Fig. 5 but for BP Tau. Contour level is 3 mJy/beam (6σ) at 1.3 mm, and 0.4 mJy/beam (3σ) at 3.4 mm. 

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Fig. G.2
As Fig. 5 but for CI Tau. Contour level is 2.2 mJy/beam (3.5σ) at 1.3 mm, and 0.86 mJy/beam (2σ) at 2.7 mm. 

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Fig. G.3
As Fig. 5 but for CQ Tau. 

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Fig. G.4
As Fig. 5 but for CY Tau. Contour level is 3.3 mJy/beam (4σ) at 1.3 mm, and 1.6 mJy/beam (4σ) at 2.7 mm. 

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Fig. G.5
As Fig. 5 but for DG Tau. Contour level is 16 mJy/beam (5σ) at 1.3 mm, and 4.3 mJy/beam (4.3σ) at 2.7 mm. 

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Fig. G.6
As Fig. 5 but for DG Tau b. Contour level is 7.4 mJy/beam (3.7σ) at 1.3 mm, and 3.2 mJy/beam (3.2σ) at 2.7 mm. 

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Fig. G.7
As Fig. 5 but for DL Tau. Contour level is 4.3 mJy/beam (5.5σ) at 1.3 mm, and 1.4 mJy/beam (3.5σ) at 2.7 mm. 

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Fig. G.8
As Fig. 5 but for DQ Tau. Contour level is 3.6 mJy/beam (4.5σ) at 1.3 mm, and 0.8 mJy/beam (1.6σ) at 3.4 mm. 

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Fig. G.9
As Fig. 5 but for FT Tau. Contour level is 2.6 mJy/beam (8σ) at 1.3 mm, and 1.3 mJy/beam (6σ) at 2.7 mm. 

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Fig. G.10
As Fig. 5 but for GM Aur. Contour level is 10 mJy/beam (65σ) at 1.3 mm, and 1.9 mJy/beam (3.2σ) at 2.8 mm. 

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Fig. G.11
As Fig. 5 but for Haro 610 N. Contour level is 3.5 mJy/beam (4σ) at 1.3 mm, and 1.2 mJy/beam (3σ) at 2.8 mm. 

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Fig. G.12
As Fig. 5 but for Haro 610 S. Contour level is 3.5 mJy/beam (4σ) at 1.3 mm, and 1.2 mJy/beam (3σ) at 2.8 mm. 

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Fig. G.13
As Fig. 5 but for Haro 613. Contour level is 8.3 mJy/beam (5σ) at 1.3 mm, and 2.8 mJy/beam (5.6σ) at 2.6 mm. 

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Fig. G.14
As Fig. 5 but for Haro 633. Contour level is 2 mJy/beam (3.3σ) at 1.3 mm, and 1.5 mJy/beam (1.7σ) at 2.6 mm. 

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Fig. G.15
As Fig. 5 but for HH 30. Contour level is 0.4 mJy/beam (2σ) at 1.3 mm, 0.5 mJy/beam (1.2σ) at 3.4 mm, and 0.36 mJy/beam (2.2σ) at 2.8 mm. 

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Fig. G.16
As Fig. 5 but for HL Tau. Contour level is 32 mJy/beam (4.5σ) at 1.3 mm, and 7.3 mJy/beam (9σ) at 2.8 mm. 

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Fig. G.17
As Fig. 5 but for MWC 758. Contour level is 4 mJy/beam (2.7σ) at 1.3 mm, and 0.8 mJy/beam (1.3σ) at 2.6 mm. 

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Fig. G.18
As Fig. 5 but for Lk Ca 15. Contour level is 2.6 mJy/beam (4σ) at 1.4 mm, 1.0 mJy/beam (3σ) at 2.8 mm, and 7.9 mJy/beam (5σ) at 1.3 mm. 

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Fig. G.19
As Fig. 5 but for MWC 480. Contour level is 15 mJy/beam (5.8σ) at 1.3 mm, 2.9 mJy/beam (7σ) at 2.8 mm, and 12 mJy/beam (5.5σ) at 1.4 mm. 

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Fig. G.20
As Fig. 5 but for T Tau. Contour level is 16 mJy/beam (3σ) at 1.4 mm, and 4.8 mJy/beam (4.8σ) at 2.8 mm. 

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Fig. G.21
As Fig. 5 but for UZ Tau E. Contour level is 8.6 mJy/beam (6σ) at 1.3 mm, 1.9 mJy/beam (4.7σ) at 2.8 mm, and 1.3 mJy/beam (4σ) at 3.4 mm. 

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Fig. G.22
As Fig. 5 but for UZ Tau W. Contour level is 2.6 mJy/beam (1.9σ) at 1.3 mm, 0.6 mJy/beam (1.5σ) at 2.8 mm, and 0.7 mJy/beam (2σ) at 3.4 mm. 

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© ESO, 2011
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