TWO-DIMENSIONAL MODELS OF PROTOPLANETARY DISK CHEMISTRY T.J. MILLAR, H. NOMURA and A.J. MARKWICK Physics Department, UMIST, PO Box 88, Manchester M60 1QD, UK
Abstract. We have developed a two-dimensional model of a flared protoplanetary disk (PPD) incorporating a self-consistent treatment of gas and dust temperature, and a detailed treatment of the gas-phase chemistry as well as the freeze-out and desorption of material from dust grains. The results show that, in the inner 10 AU of the disk, the gas-phase abundances are dominated by material evaporated from dust grains. The surface layer of the disk shows many of the characteristics of photon-dominated regions. Keywords: Protoplanetary disks, astrochemistry
1. Introduction The increasing sensitivity of interferometers and the advent of new interferometric facilities such as the SMA and ALMA have provided a stimulus for the development of chemical models of protoplanetary disks. These models have been based, to a large extent, on the α disk model, where α is the constant of proportionality which relates the turbulent stress tensor to the local gas pressure (Shakura and Sunyaev, 1973). In this case the origin of the turbulence is not specified. For the simplest model of such a viscous disk one ignores the vertical structure and solves the conservation equations for mass and momentum to find the radial profiles of density and temperature in the mid-plane. Such profiles can then be coupled to the chemical kinetic equations which describe the chemical evolution of the accreting gas, but only in the mid-plane. Early studies were performed by Duschl et al. (1996), Bauer et al. (1997), and Finocchi et al. (1997) and, used the minimum-mass solar nebula model – in which the mass of the present Solar System is distributed over the disk up to a cut-off radius taken to be 800 AU – by Aikawa et al. (1996, 1997, 1999). These different models lead to different temperatures and densities. For example, Aikawa et al. find the temperature to be 280 K at 1AU compared to 1000 K which Duschl and colleagues find. These differing temperature profiles have a significant effect on the molecular distributions since these turn out to be sensitive to two particular temperature regimes: (i) the ‘snow line’, the distance at which ice mantles evaporate and load the gas with hydrogenated molecules, and (ii) the dust destruction zone, in which refractory grain cores are destroyed and alter gas-phase elemental abundances. Astrophysics and Space Science 285: 761–768, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands.
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In this paper, we report on the results of a combined chemical-hydrodynamic model of a flared 2D disk. We have made a self-consistent model of the disk, described in Section 2, taking into account heating both by viscous dissipation and by the central object, and with a full treatment of the radiative transfer in 2D. We have used the outputs from the physical model as inputs for the chemical model which describes gas-phase chemistry, the gas-grain interaction, and the return of material to the gas-phase by the thermal evaporation of grain mantles (Markwick et al., 2002). 2. Physical Structure of the Disk Irradiation by a central star has a strong influence on the structure of a protoplanetary disk especially in the outer region, where the flared disk intercepts stellar radiation efficiently (Kenyon and Hartmann, 1987). This reprocessing disk model has been improved, for example, by Chiang and Goldreich (1997), taking account of the optically thin layer of the disk, and by D’Alessio et al. (1998), treating the radiative transfer process in optically thick and thin regions continuously and the turbulent viscous heating as well in the vertical direction of the disk (dealing with radiation from the central star separately). Here we obtain the self-consistent profile of a protoplanetary disk, making use of a full two-dimensional radiative transfer calculation. We consider an axisymmetric disk surrounding a central star with the physical parameters of typical T Tauri stars: a mass M∗ = 1M , a radius R∗ = 2R , and a temperature T∗ = 4000K. For the disk model, we assume vertical hydrostatic equilibrium: dρ dP = cs 2 = −ρgz , (1) dz dz and radiative equilibrium between emitted and absorbed radiation at each point: ∞ ∞ dνκν Bν [T (r, )] = dνκν dµdφIν (r, ; µ, φ), (2) 4π 0
0
where P , ρ, cs , gz = z, T , κν , and Iν represent the pressure, the density, the sound speed, the vertical gravitational force, the temperature, the monochromatic opacity, and the specific intensity at a frequency ν, respectively. In equation (2), local thermodynamic equilibrium is assumed between thermal emission and absorption – that is ην = κν Bν (T ), where Bν (T ) is the Planck function for blackbody radiation. The surface density distribution of the disk is determined by equating the gravitational energy of accreting mass to thermal heating via viscous dissipation at each radius: 1/2 3GM∗ M˙ R∗ 9 2
αcs 0 = 1− , (3) 3 4 r 8π r 2
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where , cs0 , and G represent the surface density, the sound speed at the midplane, and the gravitational constant, respectively. We adopt α = 0.01 here. Also, the disk is considered to rotate with the Keplerian frequency: = (GM∗ /r 3 )1/2 and have a constant mass accretion rate of M˙ = 10−7 M yr−1 . Thus, we obtain selfconsistent density and temperature distributions in the disk, through solving the above equations iteratively (see Nomura, 2002 for details). In order to calculate the specific intensity Iν in equation (2), we solve the axisymmetric two-dimensional radiative transfer equation, s κν (r , )ρ(r , )Bν [T (r , )]e−τν (r , ) ds , (4) Iν (r, ; µ, φ) = 0
where τν (r , ) is the specific optical depth from a point (r , ) to (r, ). We do this by means of the short characteristic method in spherical coordinates (Dullemond and Turolla, 2000). As heating sources, we consider viscous dissipation at the 2 midplane of the disk (Q+ vis = 9 αcs 0 /4) and the gravitational energy released 4 at the central stellar surface via its contraction (Q+ star = σ T∗ , where σ is the Stefan-Boltzmann constant). For the opacity model, we adopt the frequency dependent absorption coefficient κν of Adams and Shu (1985, 1986) for the dusty region (T < 2300K) and the analytical model of the Rosseland mean opacity κR (ρ, T ) by Bell and Lin (1994) for the gaseous region (T > 2300K), which represents the Alexander/Cox/Stewart opacity. Figure 1 shows the resulting temperature and density profiles over the whole disk obtained by means of our two-dimensional radiative transfer calculation. The results show major differences from the one-dimensional calculations as follows: the temperature at the surface layer rises due to direct irradiation from the central star. Meanwhile, the temperature at the equatorial plane beyond r ∼ 20AU also rises although this region is optically thick to the direct radiation from the central star. This result is due to our full two-dimensional treatment of radiative transfer: the reprocessed radiative heating from the inner disk dominates the viscous heating at these distances, owing to the large flared disk height and a hot surface layer. For comparison with our detailed chemical calculations, Figure 2 shows the temperature and density profiles in the inner 10 AU of the disk. The isotherms show clearly the presence of the hot surface layer.
3. Chemical Structure of the Disk The chemical model was described in detail by Markwick et al. (2002). Briefly, it consists of around 3000 reactions connecting 241 species. The chemical network includes the gas-grain interaction (freeze-out, thermal desorption) and photoprocesses. The output from the physical disk model described above provided the physical parameters (temperature, density, visual extinction) for the chemical
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Figure 1. The temperature (solid lines) and density (dotted lines) contour plots for the whole disk as a result of our two-dimensional radiative transfer calculation. Higher temperature profiles are obtained at the surface layer and near the midplane beyond x ∼ 20 AU, compared with a one-dimensional calculation.
Figure 2. The temperature (solid lines) and density (dotted lines) contour plots in the inner disk.
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model runs, of which there were 2236 to cover the region shown in Figures 3 and 4. These contour plots show the molecular distributions of H2 CO and C2 H in the disk, two molecules that are among those currently observed in protoplanetary disks, although current observations are sensitive only to the outer regions of the disk. A comparison of Figure 3 with Figure 2 shows that the abundance of formaldehyde is very sensitive to the temperature distribution. This is because H2 CO is produced in the gas phase mainly through its thermal desorption from grain surfaces in the inner disk. At large radii, H2 CO freezes out on to dust grains and is transported by the accretion flow to the inner disk. The region where the formaldehyde abundance is highest is, therefore, the region where the temperature becomes high enough to release it into the gas phase. Further inwards of these regions, the gaseous H2 CO is processed chemically into other molecules. On approaching the surface of the disk, the fractional abundance of formaldehyde increases to ∼ 10−12 10−11 . The temperature is still increasing in this region, but now at the same time the density is decreasing (see Figure 2), so that grain effects become less important and chemistry is dominated by gas-phase processes. C2 H is also stored on dust grains at large distances and transported inwards, as are many molecules. While thermal desorption does cause the gas phase abundance of C2 H to increase, the highest fractional abundances occur close to the surface of the disk. In this region, C2 H is produced by the photodissociation of C2 H2 . Hydrocarbon molecules are relatively abundant in the surface layer because the high radiation flux produces reactive atoms, radicals and atomic ions and collision times are short since the gas density is relatively large, ∼ 1012 cm−3 . An additional factor is that near the surface T ∼ 300K, a temperature which allows some reactions which possess small activation energy barriers, and are inefficient at low temperatures, to proceed. An example important in the formation of C2 H2 is the reaction C2 H2+ + H2 −→ C2 H3+ + H which competes with the radiative association reaction C2 H2+ + H2 −→ C2 H4+ + hν At low temperatures, the radiative association dominates but at room temperature, the hydrogen abstraction reaction dominates because of the steep inverse temperature of the radiative rate coefficient (T−1.5 ). Thus in the photon-dominated layer in the disk, C2 H+ 3 formation is favoured and hence C2 H2 from dissociative recombination of this ion.
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Figure 3. The distribution of formaldehyde, H2 CO, in the inner disk. The colour scale shows the log of the fractional abundance relative to total hydrogen nuclei.
Figure 4. As Figure 3 but for C2 H.
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4. Summary Our calculations using a self-consistent 2-D model of radiative transfer show that the temperature and density structures of protoplanetary disks are significantly different from those found in 1-D calculations. These structures, particularly temperature, have a significant effect on gas-phase molecular distributions which tend to peak where grain mantles evaporate. Once molecules are in the gas phase, the abundances can be altered by gas-phase chemistry. We find that the surface layers of the disk act like high density photon-dominated regions and that the abundances of certain species, particularly atoms, atomic ions and simple radicals, peak here. Although the results presented in this paper have concentrated on the inner 10 AU, a future article will present results out to 100 AU and compare our detailed results with previous 1-D models. Future work will include the effects of self-shielding on the H2 and CO distributions, as well as the effects of grain sedimentation and coagulation on the structure and chemistry.
Acknowledgements Astrophysics at UMIST is supported by a grant from PPARC.
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