3 The transport-diffusion-reaction equations

In a protoplanetary accretion disc the matter within the disc slowly migrates inwards by viscous accretion with a drift velocity $v_{\rm r}$. Additionally, during the intermediate phase where (i) mass infall has ceased and the formation of the central star is nearly finished while the remaining disc gradually is accreted onto the star and (ii) massive formation of planetesimals has not yet begun, the protoplanetary disc is convectively unstable over most parts of the disc (cf. Ruden & Lin 1986; D'Alessio et al. 1998). The turbulent convective flows result in an efficient mixing of the disc material both in the radial and the vertical direction. In Paper I we have discussed this mixing processes and its implications for the dust components in the accretion disc. Here we concentrate on the mixing of combustion products of carbon oxidation into the cold outer disc zones where these combustion products later may be incorporated into the ice component of the planetesimals. We assume as in Paper I that the disc is vertically well mixed such that there are no significant concentration gradients of gas phase species in the vertical direction. Radial mixing, operating on a much longer timescale, in conjunction with the radial inwards drift results in radial concentration gradients of gas phase species within the disc. The transport-diffusion-reaction equation describing the temporal and radial variation of the concentration c_i of species i with respect to hydrogen nuclei is (cf. Paper I)

$$% ~{\partial~c_i\over\partial~t}~+v_{{\rm r},i}~{\partial~c_i... ...~r}~~rnD_{i}~{\partial~c_i\over\partial~r}~ +{R_i\over n}\cdot$$ (64)

$v_{{\rm r},i}$ is the radial drift velocity of species i and D_i its diffusion coefficient, induced by turbulent mixing. Since we are considering gas phase species, the drift velocity and the diffusion coefficient are the same for all species. n is the particle density of H nuclei and R_i is the rate of creation or destruction of species i, as described in the preceding section.

The drift velocity $v_{\rm r}$ for all species is given by the solution of the equations for the structure of the accretion disc. The diffusion coefficient Dis related by

$$% D={\nu\over S\!c}$$ (65)

to the coefficient of turbulent viscosity $\nu$ in the disc. This quantity also is obtained from a solution of the equations for the disc structure in the $\alpha$-approximation of Shakura & Sunyaev (1973). The Schmidt-number $S\!c$ is an empirical constant which usually is determined from laboratory measurements of mixing in turbulent flows. We choose in the model calculation a value of $S\!c=1$ which yields a lower limit for D which in turn results in a slight underestimation of the efficiency of the diffusion process. This choice is preferred over the probably more realistic choice of $S\!c=0.7$ in order not to overestimate the role of diffusional mixing. For more details with respect to the choice of $S\!c$ see Paper I.

For calculating the mixing of the combustion products of the carbon grains within the protostellar disc, we have to solve six such equations for the groups 0, ... 3 defined above and two for the concentrations of CO and CO₂molecules. These equations are coupled to the equation for the destruction and mixing of carbon dust grains as described in Paper I. They cannot be solved independently from the equations for the concentration and mixing of carbon grains, but there is no back coupling to that equations because in an oxygen rich environment a growth of carbon grains is not possible.

In this paper we consider stationary accretion discs, which are a reasonable approximation of the inner region up to approximately 30 AU for protoplanetary accretion discs around solar like protostars at an age of about 10⁶ years (e.g. Ruden & Lin 1986; Wehrstedt & Gail 2002). In the stationary case the solution of the diffusion Eq. (64) requires the prescription of appropriate boundary conditions. For molecules of the groups 0, 1, 2, and 3 it seems plausible to assume that (i) such molecules are completely converted into CO in the hot innermost regions of the disc and (ii) that such molecules have negligible abundances in the outermost disc regions where material is present which essentially is material from the parent molecular cloud. This material has passed through the accretion shock at the disc surface but in the outer region of a disc temperatures behind the shock never become high enough in order to strongly modify the composition of the infalling material (cf. Mitchell 1984; Neufeld & Hollenbach 1994). Though one observes in molecular clouds a lot of C-H-compounds, their density is usually low (e.g. van Dishoek et al. 1993). Thus we prescribe the following boundary conditions

$$% c_i=0 \quad (i=0,1,2,3) \quad\mbox{at} \quad r=r_i~,\ r=r_{\rm a}$$ (66)

where r_i and $r_{\rm a}$ are the inner and outer radii of our disc model, respectively. This does not consider that some of the gases are supplied to the inner disc in non-negligible quantities by means of interstellar ices. With this choice of initial conditions we intend to show, however, that even in the absence of an interstellar contribution there result high abundances of hydrocarbons in the regions where comets are thought to have formed.

For the CO molecule we have the following boundary conditions: at the outer radius $r_{\rm a}$ the matter essentially has the composition of the molecular cloud. For the dust we assume in this calculation the dust model of Pollack et al. (1994) according to which about 60% of the total carbon is in carbon grains and the remaining 40% are in the gas (+ice) phase, essentially as CO. Thus, we prescribe (counting CO as group 4 in our calculation)

$$% c_4=0.4\epsilon_{\rm C} \quad \mbox{at} \quad r=r_{\rm a}.$$ (67)

$\epsilon_{\rm C}$ is the abundance of C relative to H by number. The inner boundary of our disc model is chosen close to the star. Temperatures are high in this region and all carbon compounds are converted into CO. The inner boundary condition then is

$$% c_4=1.0\epsilon_{\rm C} \quad \mbox{at} \quad r=r_i.$$ (68)

For the CO₂ molecule we prescribe the boundary conditions (66).

The chemistry considered in this paper only considers gas phase reactions between neutral molecules, which is appropriate for the warm inner regions of the protoplanetary disc where dust oxidation and the chemical follow up reactions of this process occur. It completely neglects the ion-molecule chemistry triggered by UV radiation, X-rays, cosmic rays, and long-lived radioactive nuclei and the role played by reactions on dust surfaces. The molecular species formed during carbon combustion in the inner part of the disc and mixed into the outer parts may be subject to further chemical reactions in this disc region. Since many of the processes due to the finite penetration depth of the ionising radiation or particles in the region $r~{\hskip 1pt}{\raise 1pt \hbox{$<$ }}{\hskip- 7.5pt}{\lower 3pt \hbox{$\sim$ }}{\hskip 2pt}\ 30$ AU are limited to a surface layer extending over only some fraction of the total disc height, an appropriate treatment of such processes requires a consideration of vertical mixing in the disc, which is not included in the present disc model. For this reason such processes are neglected in the present calculation.

 

 

Table 3: Parameters used for the calculation of the disc model.

accretion rate	$\dot M$	$10^{-6},\ 10^{-7},\ 10^{-8}$	$M_\odot$ yr-1
stellar mass	M*	1	$M_\odot$
effective temperature	$T_{\rm eff}$	4500	K
stellar luminosity	L*	5	$L_\odot$
viscosity parameter	$\alpha$	$3 \times 10^{-3}$
molecular cloud temp.	$T_{\rm mol}$	20	K
inner disc radius	$r_{\rm in}$	5	R*
outer disc radius	$r_{\rm out}$	100	AU