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Appendix A: Theoretical model

First we devised a method for calculating the inner disk radius. The traditional approach was to use the dust sublimation temperature T_sub as a unique well-defined boundary condition, based on the assumption that the temperature drops with the distance from the star. However, it has been shown numerically (Kama et al. 2009; Vinković 2012) and theoretically (Vinković 2006) that the temperature of the big grains that populate the surface of the inner disk actually increases with distance within the optically thin dusty zone (see sketch in Fig. 1) before it starts to monotonically decrease within the optically thick dust. This temperature inversion effect results in a local temperature maximum within the dust cloud (see Fig. 3 in Kama et al. 2009), where dust can overheat. A precise calculation of the dust distribution that includes this effect requires information on the spatial dust distribution within the optically thin zone. Since the density structure of the optically thin dust zone cannot be geometrically constrained, we are left with a highly ambiguous concept of the inner disk rim.

Vinković (2006) proposed using two inner disk radii that distinguish between two coexisting – optically thin and thick – disk zones around a star of luminosity L_∗: (A.1)where R_in is the disk radius in the midplane, L_acc is the accretion luminosity described below and Ψ = 2 and Ψ ~ 1.2 are used for the optically thick and thin inner radii, respectively. Here we considered only optically thick disks with T_sub as the dust sublimation temperature of the inner disk rim surface. The sublimation temperature depends on the gas density ρ_gas. Based on the fit to the data from Pollack et al. (1994), we used: (A.2)This is sufficient for the range of gas densities used in our model. A more detailed description of T_sub is given by Kama et al. (2009).

The surface of the inner disk is directly exposed to stellar heating, which creates a vertical temperature structure very close to isothermal in the narrow spatial region of our interest. The high-resolution radiative transfer calculation by Vinković (2012) showed that this is a good approximation for the inner rim of the disk. This approximation can only overestimate the dusty disk height because in reality the temperature slightly drops with the height above midplane. The radial temperature change is approximated with (A.3)The gas is in vertical hydrostatic equilibrium when the gas density structure is described as

where ρ₀ = ρ_gas(R_in,0) is the midplane gas density at the inner disk rim, H_p is the disk scale height, c_s is the isothermal sound speed, Ω_K is the Keplerian angular velocity at the midplane, M_∗ is the stellar mass, μ_gas = 2.33 is the mean molecular weight in units of the proton mass m_p, and k is Boltzmann’s constant.

Takeuchi & Lin (2002) showed that the vertical component of the gas velocity is much smaller than the horizontal component v_r,gas, given by (A.8)Accretion of the gas onto the star contributes with its luminosity to the total energy that heats the inner rim. To accommodate for this effect we used the accretion luminosity (Calvet & Gullbring 1998; Muzerolle et al. 2004) in Eq. (A.1) (A.9)where M_∗ is the stellar mass, ξ ~ 0.8 is the correction factor for the magnetospheric truncation of the disk, G is the gravitational constant, and R_∗ is the stellar radius.

The accretion rate Ṁ_acc can be calculated from the vertical profile of the gas density (Eq. (A.4)) and the horizontal velocity (Eq. (A.8)) (A.10)which exists as inflow only if (A.11)The accretion rate depends on the distance from the star, which drives the disk evolution, except for q = −0.5 when the radial velocity is independent of r and the disk is in steady-state. We focused our attention on a very small part of the disk around R_in, where Ṁ_acc varies slowly. Hence, we define the accretion rate as Ṁ_acc(R_in). Equation (A.10) also shows that Ṁ_acc ∝ ρ₀. We used this to derive ρ₀ from Ṁ_acc, which is in turn derived from L_acc ∝ Ṁ_acc in Eq. (A.9).

The local viscous dissipation of energy in the accretion disks contributes to the local disk temperature, and we checked for this effect. The net temperature at R_in is the combination (Ciesla & Cuzzi 2006) of the stellar irradiation (Eq. (A.1), with T_sub replaced by T_irr) and viscous heating (A.12)where σ is the Stefan-Boltzmann constant. If , we used Eq. (A.1) to derive R_in and if , we used Eq. (A.12). These conditions have to be checked iteratively when calculating R_in.

The long-term stability of the inner disk rim depends on the radial migration of the dust particles. We followed the prescription by Takeuchi & Lin (2002) where dust particles are driven by gas drag and their radial velocity depends on the distance from the midplane. The source of this vertical velocity variation is the gas pressure gradient, which causes two effects: the gas rotation is different from Keplerian rotation above the midplane, and the gas drag force depends on gas density. Without the gas drag the dust particles would follow Keplerian orbits. We considered particles smaller than the mean free path of gas molecules. This condition is described with Epstein’s gas drag law that gives the gas drag stopping time:

where ρ_grain is the dust grain solid density, a is the dust grain radius, and is the mean thermal velocity of gas.

We considered turbulent disks where the gas turbulent motion stirs up dust particles to higher altitudes to prevent dust sedimentation. The equilibrium distribution of dust particles is reached when the sedimentation due to gas drag and the diffusion due to turbulence are balanced. Using the α-prescription for the viscous effect of turbulence (Shakura & Sunyaev 1973; Pringle 1981), the dust density distribution is (Takeuchi & Lin 2002) (A.15)where ρ_d(r) is the midplane dust density profile. The Schmidt number S_c is a measure of the coupling between the particles and the gas. S_c ~ 1 for the small particles used in our paper and increases to infinity for very big particles. This means that very big particles are not influenced by the gas turbulence and they accumulate in the midplane. The expression of S_c is a matter of debate, and we adopted the expression suggested by Youdin & Lithwick (2007) for diffusion driven by anisotropic turbulence (A.16)Takeuchi & Lin (2003) showed that the stellar radiation pressure force slowly erodes the dusty disk surface, but they did not investigate the impact of this force on the surface of inner disk rim. We used their methodology to include the radiation pressure force in the dust velocity profiles. We ignored the complicated radiation pressure force from the near-infrared photons from the disk interior (Vinković 2009), but this force can only enhance the disk erosion and reduce the height of the optically thick disk. The stellar radiation pressure force compares to the stellar gravity by a factor of (Vinković 2009) (A.17)where ⟨ Q^ext ⟩ is the dust extinction coefficient averaged over the stellar spectrum. We used a stellar black-body spectrum to calculate ⟨ Q^ext ⟩ from olivine dust grains. However, the result is close

to ⟨ Q^ext ⟩ ~ 2 because the stellar radiation peaks at wavelengths shorter than the size of big grains that populate the inner disk surface. The radial component of the dust velocity with included radiation pressure was derived by Takeuchi & Lin (2003) as

Our goal is to find the relative height z_max/R_in that separates the inflow below this height from the outflow v_r,dust(R_in,z_max) > 0above z_max. We varied the parameters listed in Table 1 and iteratively searched for ρ₀ consistent with Ṁ_acc, which in turn has to be consistent with L_acc. The pseudocode describing the computational procedure is the following:

© ESO, 2014