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Astron. Astrophys. 325, 1264-1279 (1997)

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4. Model assumptions

4.1. The dust model

We assume that three dust components are present in the disk matter: olivine, carbon dust and troilite. For a more complete description of our dust model, we refer to Paper II. The main features of our dust and ice model including the new assumptions concerning troilite are the following:

Ices We suppose, that different sorts of ices are frozen out on the grains surface at large distances from the protostar ([FORMULA] 25 AU). H2 O and CO are the most abundant species but small amounts of CH3 OH, NH3, and some other species are present, too (Pollack et al. 1994). We will neglect the species with low abundances and assume that only the two dominating components water and carbon monoxide form coatings on the grains. We have separated the H2 O and the CO ice in two distinct layers. CO should be the outer one because it has the lowest vapourisation temperature ([FORMULA] 30 K, e.g. Sandford and Allamandola 1993). H2 O vapourises at higher temperatures ([FORMULA] 150 K) and forms the inner layer. This model is of course too simple: CO and water ice can be mixed and CO molecules are able even at rather low temperatures to diffuse through the H2 O matrix and/ or they are trapped in a clathrate structure within the water ice (Sandford and Allamandola 1988, 1990a, 1990b, 1993, Lunine and Stephenson 1985), depending on the lattice position. Laboratory experiments show that CO and other volatiles are released at several distinct temperatures (e.g. Hudson and Donn 1991). Part of the CO is released not until the water ice starts to evaporate at about 150 K. We neglect such complications in this paper and simply treat H2 O and CO as two separate ice components. The corresponding rate terms for H2 O and CO evaporation and recombination with the surface, respectively, are added to the continuity equation (16) for H2 O and CO.

Carbon dust The carbon dust is simply assumed to be graphite, as in Paper I and II, though this certainly is an oversimplification. The exact nature of interstellar carbon grains is unclear. The carbon found in meteorites (the kerogenes) is a chemically processed material which probably formed in the solar nebula (Cronin et al. 1988) though an at least partial extra solar nebular origin of the carbon material found in meteorites cannot be excluded (Wright and Gilmour 1990) 4.

For carbon dust we use the thermodynamical properties of graphite as in Paper II. According to the assumed MRN distribution for the grains (see below), 70% of the available carbon is bound in graphite. The remaining fraction is supposed to be present as CO.

Silicate dust According to the results of Mathis et al. (1977), silicon bearing dust of the form [FORMULA] Fe [FORMULA] SiO4 is present in the interstellar medium. The value of x is unclear. We will then suppose that [FORMULA], i.e., no iron is present in this dust. We take Mg2 SiO4 as model for the silicon dust. The omission of iron atoms in the dust is not important since iron doesn't participate in the chemistry in our reaction network. Furthermore the thermodynamical properties of olivine (Mg2 SiO4) are known and thus we can calculate the vapour pressure (Duschl et al 1996).

Troilite The iron sulfide FeS (Troilite) has been added to the calculation as an additional dust component. It is known, that troilite particles are present in meteorites and, thus, in the early solar system. It seems likely that they are already present in the parent molecular cloud and that they constitute a major sulfur bearing material. For details see the discussion in Pollack et al. (1994). They propose the following model for the abundance of S bearing species: the abundance of FeS in the disk corresponds to 75% of the solar system abundance of S, about 20% of S should be present as S or H2 S in the gas phase and 5% as part of the ice coatings (H2 S, SO, SO2). They argue that a certain fraction of the iron contained in the molecular cloud dust is converted into FeS during the passage of the accretion shock and that the abundance of FeS in the molecular cloud material corresponds to roughly 50% of the sulfur abundance. Since at present there is no definite hint that part of the Fe is really converted to FeS in passing the accretion shock, we prefer to assume that one half of the available sulfur in the disk is bound in these troilite particles.

4.2. Destruction of silicate dust

Silicate dust is destroyed by thermal decomposition above [FORMULA] 1 650 K according to the same process as described in detail in Paper II

[EQUATION]

In the calculation presented in Paper I, the SiO does not undergo further chemical reactions, until collisional dissociation becomes possible. The newly added gas phase silicon chemistry leads to supplementary silicon bearing species. The different pathways of the silicon chemistry are explained in Sect. 5.2.

The dust can partially or completely be destroyed in passing the accretion shock for infall velocities [FORMULA] km [FORMULA] s-1 (Shull 1977, Draine and Salpeter 1979) and in this case is later reformed somewhere behind the shock. We consider in this paper a late phase of the star formation process where most of the mass is already concentrated in the star and the remaining infall of matter occurs at large distances (Cassen 1994) where the accretion shock is likely to be too weak to destroy the dust particles. Only the ice coatings first vapourise and later re-condense in passing the shock (Lunine et al. 1991). Thus, we simply assume that the dust equals the unmodified interstellar dust. We, then, assume the standard MRN (Mathis et al. 1977) size distribution for the ensemble of dust particles including the ice layers

[EQUATION]

a is the particle radius and C is a constant (log [FORMULA] for silicate dust and log [FORMULA] for carbon dust). The method for calculating the decomposition of an ensemble of grains with such a size distribution is described in Paper II.

The rate terms corresponding to the different destruction, evaporation, and re-condensation processes are added as source terms to the continuity equation (16) for the corresponding species.

4.3. Oxidation of the carbon dust

Carbon dust can be destroyed by different mechanisms. In Paper I we treated carbon destruction by pure evaporation. The evaporation starts to be effective at a temperature of [FORMULA] 1 500 K and [FORMULA] molecules (mainly i =1, 2, 3) are injected into the gas phase. These molecules undergo a lot of gas phase reactions until they are finally converted into CO (see Paper I for a detailed description of the pathways). There exist two additional possibilities for carbon destruction: chemical sputtering with H atoms and oxidation by O atoms or oxygen bearing molecules. Sputtering by H is discussed in Lenzuni et al. 1995. We believe this process to be not important for the protoplanetary disk (see Paper II) but this process may be important in hot molecular cores or for the material directly entering the protostar in the early collapse phase. In this paper we consider the oxidation process.

Different oxygen bearing species are able to react with C atoms on the surface of the carbon grains. From flame chemistry under laboratory conditions it is known that the most significant molecules for the oxidation of graphite are OH and O2 (El-Gamal 1995). Under the conditions encountered in the protoplanetary accretion disk the molecular oxygen has a much too low abundance as to be important (any O2 would be rapidly converted to H2 O in reactions with H and H2 in the hydrogen rich environment). However, OH becomes moderately abundant at elevated temperatures when H2 starts to dissociate and the H atom reacts with H2 O

[EQUATION]

to form OH.

The essential first reaction step in the oxidation of soot (carbon dust) by OH radicals is (El-Gamal 1995)

[EQUATION]

The reaction has a measured probability of 0.1 ... 0.2 to occur during a collision (Neoh et al. 1981, 1984, Garo et al. 1990, Roth et al. 1990, 1992). This key reaction cracks a six-ring at the periphery of the huge PAH's forming the carbon dust particles and transfers two carbon atoms into the gas phase:

[FORMULA]

The ketyl radical HCCO reacts then in the gas phase with H as follows

[EQUATION]

One of the carbon atoms is immediately converted by this reaction into CO and the other one forms a methylene radical CH2. The CH2 radical resulting from this reaction then reacts with the molecular hydrogen to form a methyl radical, that in turn reacts with H2 to form ultimately methane

[EQUATION]

As we shall see in Sect. 5, this process is of considerable significance for the chemical composition of the disk matter.

This process of carbon grain destruction starts to operate in an accretion disk already at a temperature of about 1 000 K.

Also, the reaction with free oxygen atoms has a significant reaction probability (Roth et al. 1990, von Gersum S. and Roth P. 1992). The oxidation process of graphite particles by free O atoms is (Warnatz, personal communication)

[EQUATION]

In this case the liberated C atom is promptly locked in CO and is lost for the hydrocarbon chemistry. We have included this process in the calculation but it becomes never important in the protoplanetary disk because the only source of noticeable amounts of free O atoms is the decomposition of the silicates particles which starts at about 1 650 K. At this temperature the carbon dust has already disappeared because of erosion by OH.

FeS particles in a hydrogen rich environment start at a temperature of [FORMULA] K to be reduced to pure iron particles according to the reaction

[EQUATION]

(e.g. Fegley and Prinn 1989). On the other hand, at a temperature slightly above [FORMULA] 700 K the FeS particles start to vapourise and they inject in this case mainly free sulfur and iron atoms into the gas phase. It can be shown from thermodynamical calculations that small amounts of FeS molecules are present in the vapour, too, but their abundance in the gas phase is about a factor of 10 lower than the Fe and S abundances. So we will neglect them. The vapourisation process of troilite is therefore assumed to be

[EQUATION]

The sulfur atoms in the gas phase are rapidly converted into H2 S. Since both processes, vapourisation and reduction to Fe, occur at approximately the same temperature we neglected in this calculation the process of conversion of FeS into Fe particles in order to avoid to solve an additional equation for iron particles. The result for the gas phase sulfur chemistry is identical for all practical purposes.

The equilibrium vapour pressures [FORMULA] and [FORMULA] of Fe and S, respectively, over solid FeS satisfy

[EQUATION]

according to the law of mass action. If only Fe and S atoms are injected into the gas phase, we have [FORMULA] and then

[EQUATION]

[FORMULA] is the free enthalpy of formation of the troilite particles from their constituents in the gas phase. We calculate [FORMULA] for troilite from the polynomial fit given in Sharp and Huebner (1990).

The radius change of a given grain is

[EQUATION]

and the variation in the particle densities of S and Fe is

[EQUATION]

[FORMULA] is the thermal velocity of the FeS molecules, [FORMULA] is the volume occupied by the nominal molecule in the condensate and [FORMULA] is the so called sticking coefficient which is assumed to be [FORMULA] =0.1.

For troilite grains we assume that they all have the same initial radius since they don't provide an essential contribution to the total opacity and, thus, do not significantly modify the disk structure due to their opacity. Usually, for a major dust component the finite width of the size distribution has to be considered in order to avoid that all the grains disappear instantaneously. A test calculation has shown that this causes a discontinuous slope of the opacity which leads to serious numerical difficulties for high order solution schemes for ODE's, like that used in our calculation.

4.4. Initial conditions

The chemical composition of the material at the outer disk's edge should not be very different from the composition of the parent molecular cloud. The abundances [FORMULA] used in the calculation are solar system abundances as given by Anders and Grevesse (1989) and Grevesse and Noels (1993) (c.f. Tab 2). If we define [FORMULA] (total amount of hydrogen nuclei), we set initially

[EQUATION]

the remaining fraction being mostly H2. This assumption is somewhat arbitrary. It has been chosen to allow a rapid settling of chemical equilibrium between H2 and H.

Helium and nitrogen are assumed to enter the accretion disk as He atoms and N2 molecules. These two gases have very low condensation temperatures and are assumed not to be condensed onto grains (e.g. Yamamoto 1985). Their abundances in the gas phase are

[EQUATION]

and

[EQUATION]

The small amounts of NH3, NO, and HCN observed to be present in star forming regions (van Dishoeck et al. 1993) are neglected in this calculation.


[TABLE]

Table 2. Solar system element abundances (Anders and Grevesse 1989, Grevesse and Noels 1993)


Since 70% of the carbon atoms are bound in carbon dust and since the remaining fraction is bound in CO, we have

[EQUATION]

This CO is assumed to be frozen onto the surface of the dust grains. A small fraction of the carbon is bound in less abundant molecules, for instance CH4, CH3 OH, H2 CO, HCN, CS (van Dishoeck et al. 1993). Most of them will initially be condensed in the ice mantles. We neglect them in the present model calculation.

The fraction of oxygen that is not bound in CO or in silicate grains is supposed to be present as water ice. We have in this case

[EQUATION]

The thicknesses of the resulting ice layers are calculated from [FORMULA] and [FORMULA] as described in Paper I.

The assumption that all the remaining oxygen is present as water ice is somewhat doubtful because in molecular clouds the available oxygen never is completely bound in water molecules but is partially bound in O or O2. However, the accretion shock at the disk edge modifies the chemical composition of the infalling material. The material from the surrounding cloud "falls" onto the disk with velocities ranging from 10 km s-1 at [FORMULA] AU to 100 km s-1 at [FORMULA] AU (Cassen 1994). Immediately behind the shock the temperature can raise up to several 1 000 K (Neufeld and Hollenbach 1994) and our calculations show that most of the neutral-neutral chemistry becomes active at about such temperatures. The model of Neufeld and Hollenbach (1994) has shown that any atomic or molecular oxygen efficiently is converted into water molecules which later will freeze out on the surface of the grains (Lunine et al. 1991).

The available sulfur is assumed to be partially bound into FeS grains (50%). The remaining fraction stays in the gas phase. In the parent molecular cloud much of the sulfur will be bound into SO, SO2 or is present as the free S atom and some of the gases may be frozen onto grains. As in the case of oxygen, it is to be expected that ice mantles evaporate in passing the accretion shock and that most of the sulfur contained in the gas phase is converted into H2 S. The H2 S has a rather low condensation temperature (cf. Yamamoto 1985 or Sandford and Allamandola 1993). For simplicity we assume that the H2 S remains completely in the gas phase and choose

[EQUATION]

We prescribe the above defined chemical composition of the gas parcel at the outer edge of the accretion disk. We have chosen an outer disk radius of 40 AU since this is approximately the size of most of the observed disks (Ruden and Pollack 1991) and of our solar system. This parcel moves inwards with the velocity (5) and our model calculation follows the gas parcel and calculates its chemical composition at each distance s.

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© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998

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