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Astron. Astrophys. 342, 542-550 (1999)

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2. Model

Our approach is similar to that adopted, for different purposes, by Elitzur and de Jong (1978) and, more recently, by Störzer and Hollenbach (1998). We consider a unit volume of gas which is "approaching" the ionization front of an HII region at a velocity [FORMULA] which is of order 1 [FORMULA] for a D-type ionization front (see the discussion of Störzer and Hollenbach). Thus, material is advected from deep in the molecular cloud to the PDR region. Advection can be of importance for species with large abundance gradients along the normal to the ionization front. The timescale for advection of a species with abundance [FORMULA] should be of order [FORMULA] and becomes comparable to chemical timescales for sufficiently large abundance gradients [FORMULA]. We expect large abundance gradients to be caused by processes, such as grain mantle evaporation, which have an exponential grain temperature dependence.

In this context, we solve the time dependent equations for the chemical abundances, in both the gas and solid phases, using a radiation field whose strength is determined by a time dependent extinction [FORMULA]. The latter is given by:


Here, we have supposed that the initial extinction is [FORMULA] (which we have usually taken to be 10 magnitudes) and [FORMULA] is the corresponding initial separation from the ionization front. The ratio [FORMULA] is taken to be [FORMULA]. The rate coefficients for photoprocesses involving Si-bearing species were taken from Sternberg and Dalgarno (1995). Fits to the photodissociation rates of [FORMULA] and CO as functions of [FORMULA] were obtained separately, using the steady state PDR model of Le Bourlot et al. (1993), with the same density and external radiation field. The resulting time dependent (ordinary) differential equations were solved by the Gear method.

The chemical network is similar to that used by Schilke et al. (1997) in their study of C-type shocks. Thus, we have in general (for exceptions, see below) adopted the rate constants for reactions involving Si-bearing species given in Table 2 of Schilke et al. (1997; this is available via anonymous ftp from cdsarc.u-strasbg.fr/cats/J/A+A/321/293). The network considers formation of SiO from either Si or Si+ although the former dominates. It also includes build-up of silicon hydrides via reactions for example of Si+ with [FORMULA]. We note, in particular, that we have used a rate coefficient for the Si(O2, O)SiO reaction ([FORMULA] cm3 s-1) which derives from the room temperature measurement of Husain and Norris (1978) and which exceeds that measured by Swearengen, Davies and Niemczyk (1978) by more than an order of magnitude. Recent measurements of Le Picard et al. (1998) confirm the higher value of Husain and Norris. Le Picard et al. find that the rate coefficient for this reaction may be fitted by the form [FORMULA] cm3 s-1. We have, as a precaution, verified that the results presented below are essentially unmodified if the lower value of Swearengen et al. is adopted. The most critical reaction for the results presented in this paper is, in fact, the Si(OH, H)SiO reaction, for which we have adopted a rate coefficient of [FORMULA] cm3 s-1. Previous work (Langer and Glassgold 1990, Schilke et al. 1997) has assumed that the rate coefficient for this and analogous reactions is proportional to the population of Si atoms in the J = 1 excited fine structure state and consequently has an [FORMULA] dependence. However, studies of analogous reactions involving atomic carbon (Clary 1993, Husain 1993) show that the dominant long-range potential involves the polarizability and not the quadrupole moment of the atom (which is non-zero only for non-zero values of the total angular momentum J). We have therefore adopted a rate coefficient varying only with the thermal speed of the reactants.

The importance of the reaction of Si with OH has the consequence that the silicon chemistry in a PDR is sensitive to the OH abundance and consequently to the kinetic temperature variation with depth. We have adopted temperatures consistent with observations of molecular hydrogen in the Orion Bar (Parmar et al. 1991, Luhman et al. 1998). In our standard model, the temperature is assumed to vary with extinction according to the the formula [FORMULA]. Luhman et al. show that this profile is consistent with their NIR molecular hydrogen spectra towards the Orion Bar. We also consider constant temperature (500 K) models for comparison.

In contrast to the case of shocks, sputtering is expected to play no role in a PDR. Instead, we consider direct photodesorption (see below). Thermal evaporation of silicon-containing grain mantles occurs on a time scale [FORMULA]


In the above, [FORMULA] is the grain temperature and [FORMULA] is the binding energy in kelvin of the Si-bearing ice to the grain surface (Watson and Salpeter 1972). [FORMULA] is a characteristic frequency given by:


where [FORMULA] is the surface per molecule, of mass M, in the external layer of the grain mantle, and [FORMULA] is the vapour pressure of the considered Si-bearing species at zero kelvin (Leger, Jura and Omont 1985). For SiO, we have adopted the vapour pressure of CO, which gives


[FORMULA] is an important but uncertain parameter. We have assumed that Si is present in the form of "dirt" and is a minor constituent of a solid but relatively volatile material. Thus, its evaporation rate is determined by that of the (presumed more abundant) volatile species. We attempt in the following to determine a reasonable value for [FORMULA] by considering the available observations of Si+ and SiO towards the Orion Bar PDR. For this purpose, we allow [FORMULA] to vary between the binding energy for water ice ([FORMULA] 6000 K) and that for ices consisting of non-polar species ([FORMULA] 1000 K) (see Leger, Jura and Omont 1985).

Also critical in this procedure is the determination of the grain temperature [FORMULA] as a function of depth in the PDR. We have chosen to adopt for this purpose the "Ansatz" of Hollenbach et al. (1991), who estimated the "secondary" heating due to hot grains within the PDR; a more realistic treatment would require consideration of different geometries. However, given the uncertainty in [FORMULA], a more accurate treatment of the dust temperature does not seem warranted.

The silicon which is assumed to be present as a minor consituent of of the grain mantles may also be phodesorbed. Estimates of the photodesorption yield Y (no of Si atoms ejected per incident UV photon) for different substances vary widely (see e.g. Draine and Salpeter 1979, d'Hendecourt et al. 1985). A more recent study of water ice by Westley et al. (1995) indicates, moreover, that the process is efficient only after the ice has been exposed to a certain amount of radiation. Thus, it appears that Y depends upon the processing of the ices, including the formation of radicals in the ice matrix, which makes it difficult to estimate the photodesorption yield in the situation of interest to us here. The rate of photodesorption of a Si atom (or SiO molecule) from a grain may be estimated from


in cm3 s-1, where [FORMULA] is the UV (912-2000 Å) photon flux, [FORMULA] is the grain number density, [FORMULA] is the photodesorption cross section and [FORMULA] is the fraction of sites occupied by species X. Setting the number of sites [FORMULA], where [FORMULA] is the mean distance between adjacent sites in the ice mantle, which we take to be 2.6 Å, we obtain


in s-1 for the inverse of the lifetime [FORMULA] of an atom or molecule against photodesorption. In this formula, [FORMULA] is assumed to vary as [FORMULA], and hence [FORMULA] increases rapidly with depth. We note that [FORMULA] is independent of the grain size distribution.

In the models of the following section, we set Y=0 when considering mantle evaporation. When photodesorption is included, we approximate the Westley et al. saturated rate by taking [FORMULA] in terms of the grain temperature [FORMULA]. We assume the [FORMULA] abundance in the grain mantles is [FORMULA] that of hydrogen (Whittet 1993) and that the water molecules are ejected intact. The SiO photodesorption yield is taken equal to that of water. However, we also consider the case that silicon comes back into the gas phase due to photodesorption from a "more refractory" grain surface component (this could for example be the form of refractory Si which Tielens (1998) has postulated to explain the silicon depletion pattern). In this case, we put Y(Si) equal to a constant value of [FORMULA]. This yield has been deduced from the observations of Turner (1998) of SiO in translucent clouds. We find that this value of Y gives agreement with the fractional abundances of gas phase SiO, of the order of [FORMULA], deduced by Turner from his observations, for the conditions (of radiation field, in particular) that Turner believes to be appropriate to these regions. The value of [FORMULA] that we have derived is much larger than those listed by Turner.

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Online publication: February 22, 1999