Free Access
Volume 529, May 2011
Article Number A105
Number of page(s) 41
Section Interstellar and circumstellar matter
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) = βm. The outer region is optically thin, and the slope constrains p + q. βm 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, T20. The small brightness difference between the two frequencies is caused by the Rayleigh-Jeans correction. The dotted and dot-dashed lines represent an optically thinner disk at 1.3 and 2.7 mm respectively, with a viscous type profile with Rc = 150 AU. In addition, β(r) is assumed to vary with radius following Eq. (17) with Rb = 60 AU and Rw = 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 wavelength-independent 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.

thumbnail 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), tapered-edge model at 1.3 mm; dash-dotted 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 χ2 criterium to determine the best matching model. However, it is important to realize that our data consists in a large (several 104) number of statistically independent visibilities, each with very little (essentially zero) signal-to-noise. The χ2 is given by (B.1)where Oi are the (complex) observed visibilities (O2 actually being used to note O × O ∗ , O ∗  being the complex conjugate of O), Mi 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 Mb is the best-fit model, so even the null model Mi = 0 yields a χ2 on the order of N, the number of visibilities, as Wi is the inverse of the variance of . Thus, the reduced χ2, is a poor evaluation of the fit quality, which is close to 1 even for a very poor (null) model. Only the relative differences Δχ2 between models of equivalent number of parameters can reveal whether one is better than the other.

Another subtle effect in comparing absolute values of χ2 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 χ2 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 best-fit 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) = T100(r/100AU) − q. Profile (ii) is a broken power law: it has a constant temperature between Ri = 40 AU and Rf, Rf being a variable parameter, while for r < Ri or r > Rf, the temperature is a power law with exponent q = 0.5, with to T(r) = T1 at r = 1 AU. The temperature law is continuous as a function of r, and we used T1 = 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 T0,q and Rf. 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.

thumbnail Fig. C.1

Sample results illustrating the main dependency of the surface density profile on the temperature law. Top left: Rc versus q. Top right: Rc versus Rf; bottom left: Σ.T versus T; bottom right: p + q versus q. The observed source is DL Tau.

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thumbnail Fig. C.2

Sample results illustrating the weak dependency of the dust parameters Rb and Rw on the assumed temperature law.

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For Profile (i):

  • Σ0 is nearly proportional to 1/T, with small corrections at low T owing to deviations from the Rayleigh-Jeans behavior.

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

  • In Model 1, Rout is only weakly affected by the changes in p

  • In Model 2, Rc increases by 20 to 30% when q increases from 0 to 0.5.

For Profile (ii):

  • In Model 1, Rout slightly decreases with Rf (by about 10%), and p changes by about 0.1. Variations are not fully monotonic, however.

  • In Model 2, Rc decreases by about 20 to 30%, when Rf goes from 50 to 200 AU. This is similar to the effect of q in Profile (i), as increasing Rt 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 χ2 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/T100 (or 1/T1 in Profile (ii)), the derived density distribution are thus not significantly affected by the assumed temperature law.

More importantly, Rc 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 Rf = 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 self-similar 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 Lynden-Bell & 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/R1, T = (1 + t/ts) is a dimensionless time, t the disk/star age and ts is the viscous timescale at R1, 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 R1 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,R1,T,ν1), and five measurements: three from our study (Md   or   Σ0,Rc), the stellar age t from evolutionary tracks and the mass accretion , 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 ts. Rc, and thus R1 does not appear in this expression because R1 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 R1 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)cs(r)H(r), where cs is the sound speed, and H(r) the scale height (D.14)In hydrostatic equilibrium, (D.15)Tg being the gas temperature in the disk mid-plane. Approximating Tg(r) by a power law of exponent  − q (q ≃ 0 − 0.6), we derive Rc and γ are directly constrained by our observations, while t is derived from evolutionary tracks. The last term cs(R0)H(R0) depends on the stellar properties. We note from Eq. (D.15) that , and thus αCs 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 Tk = T0(r/r0) − 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 self-similar 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 ru is the radius at which the disk has an apparent slope p and xu = (2 − p − q)/(p + q − 3/2). For p + q > 2, xu < 0 and the surface density vanishes at radius . Note that the classical steady-state result Σ(r) ∝ r − (3/2 − q) corresponds to the asymptotic limit xu → ∞, 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) = T0(r/R0) − 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 Md 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.

thumbnail 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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thumbnail 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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thumbnail Fig. G.3

As Fig. 5 but for CQ Tau.

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thumbnail 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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thumbnail 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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thumbnail 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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thumbnail 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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thumbnail 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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thumbnail 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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thumbnail 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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thumbnail Fig. G.11

As Fig. 5 but for Haro 6-10 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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thumbnail Fig. G.12

As Fig. 5 but for Haro 6-10 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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thumbnail Fig. G.13

As Fig. 5 but for Haro 6-13. 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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thumbnail Fig. G.14

As Fig. 5 but for Haro 6-33. 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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thumbnail 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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thumbnail 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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thumbnail 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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thumbnail 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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thumbnail 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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thumbnail 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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thumbnail 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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thumbnail 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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