An example TTauri ProDiMo model

Peter Woitke, Feb. 2013

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ProDiMo stands "PROtoplanetary DIsc MOdel", it's a F90 software package to simulate protoplanetary discs including astro-chemistry, detailed gas heating and cooling balance, and continuum and line radiative transfer in 2D symmetry. The source code is available on a collabortaive basis. You can apply here for a ProDiMo user account.

The following ProDiMo model of a “typical TTauri star” is designed to predict continuum and line fluxes that roughly resemble the observations of real class II TTauri stars. The effective stellar temperature is chosen as Teff=4000K, and the stellar luminosity L⋆ = 1 L⊙; these values correspond to spectral type K7, a stellar mass of M⋆ = 0.7 M⊙ and an age of about 1.6 Myrs.

1. Spectral Energy Distribution

The calculated spectral energy distribution (SED, see Fig. 1) is featured by

2. Disc Shape and Dust Settling

The density setup is parametric in this model, see Fig. 2, i. e. the disc shape is fixed by powerlaws for the column density and scale height as function of radius. However, we use a modified powerlaw here, with exponential tapering off, for the column density as

which can naturally explain the often somewhat larger spectral appearance of the disc in (sub-)mm molecular lines, because the lines remain optically thick even at large radii where the continuum is already optically thin and vanishes in the background.

The quite tall inner disc, and a very modest increase of the height z where the radial Av reaches 1 and 0.1 (as function of radius, see red dashed lines in Fig. 2) are key to produce the desired SED features (Fig. 1). The tall inner disc is needed to intercept enough star light to re-radiate it as prominent near-IR excess, and the very modest “disc flaring” produces the desired SED-slope around 100 μm. In contrast, hydrostatic disc models have very thin inner discs, and strongly flaring outer discs.

Dust settling is included according to Dubrulle et al. (1995), assuming an equilibrium between upward turbulent mixing and downward gravitational settling. This results in a size-dependent reduction of the dust scale-heights H(r, a) with respect to the gas scale-height H(r), dependent on turbulent mixing parameter α, as

where Omega is the Keplerian orbital frequency, γ ≈ 2 and τf =(ρd a)/(ρ cs) is the frictional timescale, ρd is the dust material density, ρ is the midplane gas density, and cs is the midplane sound speed.

The resulting local dust/gas mass ratios and mean dust particle sizes <a> are shown in Fig. 3. Note that the dust settling according to (Dubrulle et al. 1995) is density-dependent, and so the effects on local dust/gas and size distribution are much more pronounced in the tenuous outer layers. The regions important for line emission (roughly Av ≈ 0.01...1) can easily have gas/dust ratios that are larger by several orders of magnitude as compared to the overall (volume integrated) gas/dust ratio, here assumed to be 100. The following table summarizes the parameters of the model

3. Gas and Dust Temperatures

The resulting dust and gas temperatures in this model are shown in Fig. 4. Ignore the red regions on the r.h.s. of Fig. 4, the gas densities are so low here, that these regions are completely irrelevant for both continuum and line emission (the X-rays cause an HII region here). Important are the warm disc surface layers below (down to about Av = 1) where typically Tgas > Tdust. This is key to produce the emission lines as observed (negative temperature gradients would result in absorption lines!), and it is important that the temperature contrast between dust and gas is modest to fit the observed line flux magnitudes. In contrast, Tgas = Tdust can safely be assumed in the midplane regions (Av > 10), where the densities are large (inelastic dust-gas collisions are frequent) and where the UV and X-ray radiation fields, which cause the temperature differences, cannot penetrate into.

Figure 5 shows the leading heating and cooling processes. Particularly important for the line formation regions are PAH-heating, exothermic chemical reactions, several follow-up heating processes after photo-excitation or photo-dissociation of H2, and neutral carbon ionization. The most important cooling processes are thermal accommodation, H2O ro-vibrational lines, CO ro-vibrational lines, [OI] 63 μm and [CII] 157 μm line emission. It is important to have ro-vibrational lines, not only rotational, actually for both cooling and heating.

4. Chemical Structure

Figure 6 shows some results concerning the chemical composition in the disc. The disc is mainly made of H2, however the uppermost layers are atomic, or even fully ionized (by X-rays). The free electrons are donated by H+ in the uppermost layers, by C+ in a transition layer, then by a small fraction of metal atoms not bound in refractory dust materials like Fe+, Mg+ and S+. In the deeper layers, the free electron concentration is very small, like 1.E-12 to 1.E-20, creating a “dead zone”, where the few electrons created by cosmic rays are absorbed by PAH particles, creating PAH−.

The chemistry shows a typical PDR/XDR like layered structures, with transitions H → H2, C+ → C → CO → COice, and O → OH → H2O → H2Oice. However, in the close and very dense midplane, conditions are more like in planetary atmospheres, with very abundant CH4, CO2, H2O and NH3 gas, but not so much CO. The CO-poor midplane may be an artifact of the assumption of chemical equilibrium, because the chemical relaxation timescales here (and only here) can exceed the age of the star.

The chemistry in the outer layers can show quite unusual pathways as compared to standard astro-chemical models for interstellar clouds, because the the dust settling can lead to quite dense yet almost dust-free local conditions, where one of the most important reactions dust + H + H → dust + H2 becomes very slow because of the lacking dust surface.

The model predicts the snowline (transition between gaseous and frozen water) to be situated at about 0.5 AU, but viscous heating in the midplane (ignored in this model) might shift it further outward. Anyway, at higher altitudes, the snowline bends and becomes almost a horizontal line, because the water ice is UV photo-desorbed in the directly irradiated layers.

5. Predicted Continuum Observations

The model predicts continuum fluxes (SED), images, and visibilities at various wavelengths. Figure 7 indicates the disc regions mainly responsible for the continuum emission at different wavelengths. Figure 8 shows two examples of calculated images, one in the IR and one at mm-wavelengths. The spectral appearance at 3.8 μm is dominated by the inner rim, whereas the disc has an apparent size of about 50AU at λ=1.3mm. Note that the scattering is treated as isotropic in this ProDiMo model, so the preferred forward scattering by the surface of the close disc half in the l.h.s. image is not properly accounted for by the model.

6. Predicted Line Observations

The model makes detailed predictions about various emission line fluxes (actually thousands of them, ranging from the optical to mm-wavelengths), as well as line velocity-profiles, molecular maps and channel maps for selected lines on demand. In Fig. 9 we see the results for the J =2 → 1 lines of the three isotopologues CO, 13CO, C18O. Since these molecules have different abundances (assumed to be 1.0, 0.014, and 0.0020 with respect to CO, respectively), the lines in the series become less optically thick, are formed deeper and closer in, their FWHM and peak separation increases. The less abundant isotopologues form their lines in deeper layers, which makes the line ratio dependent on the vertical temperature gradients. These gradients, in return, depend on the assumption about the dust settling ...

Figure 10 shows similar results for three selected water lines, which according to the model are emitted by completely different spatial regions of the disc. Trying a nebular analysis on these fluxes (e.g. deriving “the” rotational excitation temperature by a rotational diagram, assuming that all lines are emitted by the same gas with the same temperature) would obviously be quite misleading.

Figure 11 shows calculated channel maps for 13CO J =2 → 1. One can see the much larger apparent size of the CO-disc (∼ 180AU) as compared to the continuum (∼ 50AU, upper left), although the local (column-integrated) gas/dust ratio is constant and equal to 100 by assumption in this model. The simple truth is that the CO molecular lines are still optically thick, even 13CO at 180AU, where the continuum is already optically thin and vanishes in the background.