1 Introduction

The material of a protoplanetary accretion discs contains a rich mixture of gases, ices, and dust. In the outer parts of the disc one encounters material which originates from the parent molecular cloud which fell onto the protostellar disc during the star formation process and was incorporated into the disc material after having passed through the low Mach-number accretion shock standing atop the disc surface. The dust component is not likely to have significantly changed its composition by passing through the accretion shock and it is assumed that the dust composition of the outer protostellar disc essentially equals that of the parent molecular cloud. How this mixture of dust components might look like in the outer disk is described, for instance, in Pollack et al. (1994).

One of the major components of this mixture is some kind of carbon dust. Since the mixture of elements in the interstellar medium and in protoplanetary discs is oxygen rich, the carbon dust component is not stable in this environment. It would be converted into CO if chemical reactions between the carbon dust grains and oxygen bearing species from the gas phase were possible. The sole reason why carbon dust grains survived in the interstellar medium between their formation in the carbon rich environment of a circumstellar dust shell around a carbon star and their ultimate incorporation into the protoplanetary accretion disc of our Solar System is, that during their residence time in the interstellar medium the grains have never been heated sufficiently that the activation energy barriers involved in the carbon oxidation process can be surmounted.

As the central star accretes material from its protostellar disc, the hot inner disc zone close to the star is continuously replenished by mass accretion from the outer disc region. Material from the outer disc slowly spirals inwards into the inner disc regions and gradually heats up as it comes closer to the star. In the inner region of the disc ( $r~{\hskip 1pt}{\raise 1pt \hbox{$<$ }}{\hskip- 7.5pt}{\lower 3pt \hbox{$\sim$ }}{\hskip 2pt}\ 1$ AU) the discs midplane temperature exceeds 1000 K for at least the first $5 \times 10^5$ years of disc evolution (cf. Ruden & Lin 1986). In this region the carbon dust is rapidly destroyed by oxidation, as is shown in Gail (2001, henceforth called Paper I).

Since the disc is convectively unstable (cf. Ruden & Lin 1986; D'Alessio et al. 1998), the revolution of the disc matter around the central star is superposed by convectively driven turbulent motions. This turbulent motion induces a radial and vertical mixing of disc material from different zones of the protoplanetary disc. The observational finding of crystalline silicate dust material in comet Hale-Bopp indicates that such large-scale mixing actually took place in the Solar Nebula (cf. Hanner et al. 1994; Nuth 1999; Nuth et al. 2000). In Paper I we have shown that by mixing processes crystalline dust material from the zone where the initially amorphous dust of interstellar origin is annealed and develops a crystalline structure (at $T\approx 800$ K) is transported into the cold outer disc regions where the cometesimals are formed.

In this paper we concentrate on the radial mixing of the products of combustion of carbon grains. As was shown in Paper I the carbon dust is oxidised by reactions with OH radicals in a zone of the disc where the midplane temperature roughly equals 1100 K. This temperature is considerably lower than the temperature where carbonaceous material burns under terrestrial conditions. This lower temperature results from the slow increase of temperature as the disc matter drifts inwards by accretion, which occurs on timescales of the order of $10^4\dots10^5$ yrs, depending on the distance r from the star. The carbon grains do not burn as in a flame but they are very slowly oxidised once the concentration of OH radicals resulting from water vapour dissociation has reached a level where the oxidation timescale becomes shorter than diffusional mixing timescales. An immediate consequence of this low oxidation temperature is that only part of the solid carbon is immediately converted into CO while a considerable fraction forms CH₄ and some C₂H₂, as has been found in Finocchi et al. (1997a). The methane then is converted in a sequence of reactions into CO which involve reactions with free O atoms. The abundance of free oxygen atoms in the gas phase (by dissociation of H₂O) reaches a level sufficient for converting the methane into CO only at a much higher temperature ($\approx$1350 K) as that required for oxidation of solid carbon by OH. As a consequence of the delayed final oxidation of the carbon compounds in the oxygen rich environment one obtains a considerable concentration of the intermediate products methane and acetylene of carbon oxidation (Finocchi et al. 1997a).

These intermediate products, CH₄, C₂H₂ and some other hydrocarbons, if mixed into the cold outer part of the disc, cannot be oxidised in this region because there exist no gas phase species which can react with hydrocarbons to form CO. The hydrocarbons are metastable in this region. Only in the far outer regions of the disc where the disc becomes transparent for cosmic rays or in a surface layer of the disc subject to irradiation by UV and X-rays from the protosun these molecules may become involved in the ion molecule chemistry triggered by cosmic rays and ionising radiation. Thus, radial mixing in the protoplanetary accretion disc builds up a certain level of concentration of hydrocarbon compounds in the outer disc regions up to about 30 AU, resulting from carbon combustion in the region r<1 AU. Beyond this region the products of carbon oxidation cannot by mixed during the first 10⁶ years of disc evolution (Wehrstedt & Gail 2002).

Observations of comets C/1996 B2 (Hyakutake), C/1995 O1 (Hale-Bopp) and C/1999 H1 (Lee) have shown that cometary ices contain a significant amount of hydrocarbons (cf. the review by Crovisier & Bockelée-Morvan 1999). Especially CH₄, C₂H₆ and C₂H₂ are quite abundant. This suggests that at least some fraction of the hydrocarbons observed to exist in cometary nuclei ices are products of the oxidation of the carbon dust which have been mixed into the cold regions of the Solar Nebula.

In this paper we study the interplay between carbon dust oxidation, gas phase reactions of the product molecules, and radial mixing in the disc in order to determine the abundance of carbon compounds from carbon combustion in the cold regions of the Solar Nebula where the nuclei of the comets were formed 4.6 Gyrs ago. The model calculations are done for a simple stationary Keplerian $\alpha$-disc model in the one-zone approximation. The restriction of the disc model to the one-zone approximation only allows to treat radial mixing processes and excludes a treatment of vertical mixing processes in the disc, though these might be important for the gas-phase chemistry because of the strong UV and X-ray irradiation of the disc surface due to the young proto-sun (Aikawa & Herbst 1999,1999; Willacy & Langer 2000). With this caveat, we shall show that radial mixing of the combustion products of carbon grains may be responsible for some of the hydrocarbons observed to exist in cometary ices.

The plan of this paper is as follows: in Sect. 2 we describe the chemical reactions considered and the method used for their solution, Sect. 3 discusses the system of transport-diffusion-reaction equations, Sect. 4 presents the results of a model calculation and Sect. 5. gives some final remarks.