2 Model

2.1 Disk model

As in Paper I, we have adopted simple static disk models - the Kyoto model (Hayashi 1981) extrapolated to 700 AU, and a model in which the mass and density are lowered by an order of magnitude. The lower mass model is similar to that adopted by Dutrey et al. (1997), with which they estimate the averaged molecular abundances in DM Tau. The total disk mass is about $6\; 10^{-2} ~M_{\odot}$ for the extended Kyoto model, and $6\; 10^{-3}~ M_{\odot}$ for the lower mass model. Although much work has been undertaken to formulate the structure and physical evolution of protoplanetary disks (Cameron 1973; Hayashi 1981; Adams & Lin 1993, and references therein), there is still no obvious standard model. The main goal of this paper is to comprehend the essential characteristics of deuterium chemistry in a protoplanetary disk using simple disk models.

The details of the Kyoto model are described in Paper I. The column density of hydrogen nuclei $\Sigma_{\rm H}$ (cm^-2) and the temperature T(K) as functions of R are given by the equations

  

$$\Sigma_{\rm H}(R) 7.2\; 10^{23} \left(\frac{R}{100~ \rm AU}\right)^{-3/2}$$ = (1)

$$28\left(\frac{R}{100~ {\rm AU}} \right)^{-1/2}\left(\frac{L_{\ast}} {L_{\odot}}\right)^{1/4} .$$ T(R) = (2)

The luminosity of the central star $L_{\ast}$ is assumed to be 1 $L_{\odot}$ in this paper. It is assumed for simplicity that the disk is isothermal at each radius.

The gas is in hydrostatic equilibrium in the vertical direction. From the mass distribution given by Eq. (1) and the temperature distribution given by Eq. (2), the density distribution by number of hydrogen nuclei (cm^-3) can be shown to be

 

$$n_{\rm H}(R,Z)=1.9\; 10^9 \left(\frac{R}{100 ~{\rm AU}}\right)^{-... ...}\right) {\rm exp}\left[\frac{GM_{\ast}\mu m_{\rm H}}{kT(R^2+Z^2)^{1/2}}\right]$$ (3)

where G is the gravitational constant, $\mu$ is the mean molecular weight of gas, $m_{\rm H}$ is the mass of a hydrogen atom, and k is the Boltzmann constant. Since the disk mostly consists of H₂ and He, the mean molecular weight $\mu$ is 2.37, with the elemental abundances of Anders & Grevesse (1989). The mass of the central star $M_{\ast}$ is assumed to be  $1~M_{\odot}$.

2.2 Reaction network

The basic equations for molecular evolution are given by

$$\frac{{\rm d}n(i)}{{\rm d}t}=\sum_{j} \alpha_{ij}n(j) +\sum_{j,k} \beta_{ijk}n(j)n(k),$$ (4)

where n(i) is the number density of species i, and the $\alpha_{ij}$ and $\beta_{ijk}$ are rate coefficients. We use the "new standard model'' network of chemical reactions for the gas-phase chemistry (Terzieva & Herbst 1998; Osamura et al. 1999). The ionization rate by cosmic rays is assumed to be the "standard'' value for molecular clouds, $\zeta =1.3\; 10^{-17}$ s^-1 (e.g. Millar et al. 1997), because the attenuation length of cosmic ray ionization is much larger than the column density for $R\ge 50$ AU. The ionization rate is uncertain by a factor of a few at least (e.g. van der Tak & van Dishoeck 2000). Calculations with $\zeta =2.6\; 10^{-17}$ s^-1 yield molecular column densities that differ by less than a factor of 3 from those with our standard $\zeta$.

We have extended the network to include mono-deuterated analogues of hydrogen-bearing species (Millar et al. 1989; Aikawa & Herbst 1999b). For normal exothermic reactions and dissociative recombination reactions, we have assumed that the total rate coefficient is unchanged for deuterated analogues, and have also assumed statistical branching ratios. There are some exceptions to the statistical rules; for example, the dissociative recombination of HCND⁺ does not produce DCN but HCN. Similar rules are set for the hydrogenation of HCN and DCN, i.e., HCN + H₂DO⁺ produces HCND⁺ or HCNH⁺, but not DCNH⁺. Another important exception is

$${\rm H_2CN} + {\rm D} \to {\rm HCN} + {\rm HD} ~ ~ (a)$$

$$\to {\rm DCN} + {\rm H_{2}} ~ ~ ~ (b).$$

The lower branch (b) is one of the main formation paths for DCN. If we apply the statistical rule, the branching ratio a/b is 2. However, the experiment by Nesbitt et al. (1990) shows the ratio $a/b=5\pm 3$. In this paper we assume the ratio a/b to be 5.

We have included those deuterium exchange reactions for molecular ions and HD that are known to proceed in the laboratory or have been studied in detail theoretically (Millar et al. 1989); such reactions drive the fractionation yet are known to occur for only a few ions since activation energies are common (Henchman et al. 1988).

Although our model does not contain surface chemistry, as does the latest model of Willacy & Langer (2000), we do include the surface formation of H₂ molecules, the surface recombination of ions and electrons, and formation and desorption of ice mantles. As in Paper I, we adopt an artificially low sticking probability S=0.03 for adsorbing species on the grain surface, in order to mimic the effect of non-thermal desorption. The sticking coefficient was originally chosen to fit the observed spectrum of CO emission lines in GG Tau. For thermal desorption from ice mantles, we adopt the same rate coefficients as in Aikawa et al. (1997). The total numbers of species and reactions included in our network are 773 and 10539, respectively.

The elemental abundances used here are the so-called "low-metal'' values (e.g. Lee et al. 1998; Aikawa et al. 1999). The initial molecular abundances are determined by following molecular evolution in a precursor molecular cloud core with physical conditions $n_{\rm H}=2\; 10^4$ cm^-3 and T=10 K up to $3\; 10^5$ yr, at which time observed abundances in molecular clouds are reasonably reproduced (Terzieva & Herbst 1998).

2.3 X-rays

Results from the X-ray observation satellites, ROSAT and ASCA, show that T Tauri stars are strong X-ray emitters, with X-ray luminosities in the range 10²⁹-10³¹ erg s^-1 (Montmerle et al. 1993; Glassgold et al. 1997). We include chemical processes caused by X-rays following Maloney et al. (1996) and Glassgold et al. (1997). In Paper I, we showed that X-rays affect the disk chemistry via ionization and induced UV radiation. Direct and secondary ionization of heavy elements by X-rays, which were not taken into account in Paper I, are included in this paper. We find that inclusion of these new processes does not much alter the results of Paper I. For completeness, however, we discuss their inclusion. We adopt an X-ray luminosity of 10³¹ erg s^-1, which is almost the upper limit of the observed luminosity, in order to examine the upper limit of the X-ray effects.

2.3.1 Secondary ionization

X-rays produce electrons with energies of several hundred eV by ionizing atoms and molecules. These electrons cause secondary ionization of about 30 molecules and atoms per keV of primary electron energy, most of which are hydrogen molecules. Secondary ionization is more effective than direct ionization by X-rays in a hydrogen-dominated gas, so that the overall ionization rate per unit time is given approximately by

$$\zeta_{\rm x}=N_{\rm sec}\int \sigma(E)F(E){\rm d}E,$$ (5)

where $N_{\rm sec}$ is the number of secondary ionizations per unit primary photoelectron energy, $\sigma$ is the cross section for direct X-ray ionization for all elements weighted by their solar abundances, and the X-ray photon flux F(E), in units of per unit area per unit time, is given by

$$F(E)=F_{\rm o}(E){\rm e}^{-\tau(E)}$$ (6)

where $\tau$ is the optical depth along a path from the central star. The actual value of $\zeta_{\rm x}$ at a radius of R=700 AU in the Kyoto model is shown as a function of height from the midplane in Fig. 2 of Paper I. The ionization rate via X-rays is higher than that via cosmic rays in regions roughly above the scale height of the disk.

Although the photoelectrons ionize mostly H₂ and H, the secondary ionization of heavy elements may be important for their chemistry. Secondary ionization rate coefficients for heavy elements are estimated by

$$\zeta_{\rm x}^{\rm m}=\zeta_{\rm x} \frac{\sigma_{\rm ei, m}(E)}{\sigma_{\rm ei, H}(E)}$$ (7)

where $\sigma_{\rm ei, m}(E)$ is the electron-impact ionization cross section of element m at energy E. We used the cross sections given by Lennon et al. (1988), and averaged the ratio $\sigma_{\rm ei, m}(E)/ \sigma_{\rm ei, H}(E)$ over the energy range 0.1-1 keV, because the energy of the primary electron is a few hundred eV. Typically, the ratio is $\sim$3-7 for elements such as C (4.29) and Mg (7.09).

2.3.2 Direct ionization

The direct X-ray ionization rates for C, N, O, Si, S, Fe, Na, Mg, Cl and P are calculated at each point of the disk using the X-ray flux obtained from Eq. (6) and the ionization cross sections given by Verner et al. (1993). We assume primary ionization of heavy elements in a molecule leads to a doubly ionized species because of the Auger effect; the doubly ionized species then dissociate into two singly charged ions. We consider this destructive reaction only for simple diatomic molecules, because we do not know the products when polyatomic molecules are dissociated by X-rays. This simplification does not affect our results, because X-ray induced photolysis is much more efficient than direct ionization in terms of molecular destruction.

2.3.3 UV photolysis induced by X-rays

Energetic photoelectrons produced by X-rays collide with hydrogen atoms and molecules, and generate UV photons, just as cosmic rays do. Although this process was considered in Paper I, it was not discussed. In H₂-dominated regions, the photoreaction rate coefficient for a species m is given by

$$R_{\rm m}=2.6 \psi p_{\rm m} x_{\rm H2} \zeta_{\rm x} (1-\omega)^{-1} ~ ~ {\rm s}^{-1}$$ (8)

where $\psi = 1.4$ is the ratio of number of Lyman-Werner photons produced per H₂ ionization (Maloney et al. 1996). Since hydrogen is mostly in the molecular form in the region we are interested in, we adopt the constant value $x_{\rm H2}=0.5$ in this formula. We can ignore the contribution from hydrogen atoms because atomic hydrogen only dominates in the surface regions, which do not contain many molecules. The coefficient $p_{\rm m}$ is given by Gredel et al. (1989), and $\omega$is the grain albedo, which is assumed to be 0.5. For molecules which are not listed in Gredel et al. (1989), we estimate the value of $p_{\rm m}$from similar molecules. Following Gredel et al. (1987), we adopt $p_{\rm m}=10$ for CO. The deuterated hydrogen molecule HD is also subject to photodissociation by induced UV. Because HD is dissociated by lines, just like CO, we assume $p_{\rm m}=10$ for HD.

In fact, Eq. (8) assumes that induced photons are absorbed by molecules locally; in other words, a spatial gradient of $\zeta_{\rm X}$ is not considered. Since $\zeta_{\rm X}$ is higher at larger heights in our model, and since induced photons can be emitted in the vertical direction of the disk, the induced photons could penetrate deeper into the disk than we estimate, especially in the outer radius ( $R\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$>$ }}}500$ AU) in our lower mass disk, in which the total column density is low ( $A_{\rm v}\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$<$ }}} 1$ mag). But a quantitative calculation of this effect requires a 2-D radiation transfer calculation, which is beyond the scope of this paper.

2.4 UV radiation

A protoplanetary disk is irradiated by UV radiation from the external interstellar field and from radiation due to the central star. The UV flux from the central star varies temporally, and at R=100 AU the unattenuated UV flux can reach a value 10⁴ times higher than the interstellar flux (Herbig & Goodrich 1986; Imhoff & Appenzeller 1987; Montmerle et al. 1993). As in Paper I, we utilize this maximum value. The radiation fields from the two sources strike the disk from different directions, and thus suffer different degrees of attenuation. We obtain the attenuation of interstellar UV in terms of the visual extinction $A_{\rm v}$, by calculating the vertical column density from the disk surface to the height we are interested in, using the relation $A_{\rm v}= N_{\rm H}/[1.8 \; 10^{21}$ cm^-2mag^-1]. The attenuation of the stellar UV, $A_{\rm v}^{\rm star}$, is obtained by calculating the column density from the central star. Values of $A_{\rm v}$ and $A_{\rm v}^{\rm star}$ are given in Table 1 of Paper I, as a function of height at R=700 AU for the Kyoto model. These values of extinction are then put into the photo-rate equations of our network.

In Sect. 4, where we report calculated values for molecular distributions in embedded disks, we consider an additional attenuation of 1-2 mag via ambient gas, while the attenuation of the stellar radiation is not modified.

Although the UV radiation is mainly attenuated by dust, self- and mutual-shielding must be considered for H₂ and CO (van Dishoeck & Black 1988; Lee et al. 1996), so that we must solve for the molecular abundances and the UV attenuation self-consistently. As in Paper I, we solve a one-dimensional slab model at each radius of the disk and utilize modified shielding factors from Lee et al. (1996).

Figure 1: Vertical distribution of normal (dashed lines) and deuterated (solid lines) molecules at R=500 AU. The disk age is assumed to be $t=9.5\; 10^5$ yr and the disk mass is less massive than the Kyoto model by an order of magnitude