A&A 418, 1035-1043 (2004)
DOI: 10.1051/0004-6361:20035718
C. M. Walmsley^{1} - D. R. Flower^{2} - G. Pineau des Forêts^{3,4}
1 - INAF, Osservatorio Astrofisico di Arcetri,
Largo Enrico Fermi 5, 50125 Firenze, Italy
2 -
Physics Department, The University,
Durham DH1 3LE, UK
3 -
IAS, Université de Paris-Sud, 92405 Orsay Cedex, France
4 - LUTH, Observatoire de Paris, 92195 Meudon Cedex, France
Received 20 November 2003 / Accepted 9 February 2004
Abstract
We have carried out calculations of ionization
equilibrium and deuterium fractionation for conditions
appropriate to a completely depleted, low mass pre-protostellar
core, where
heavy elements such as C, N, and O have vanished from the gas
phase and are incorporated in ice mantles frozen on dust grain
surfaces. We put particular emphasis on the interpretation of
recent observations of H_{2}D^{+} towards the centre of the
prestellar core L 1544 (Caselli et al. 2003) and also
compute the ambipolar diffusion timescale. We
consider explicitly the ortho and para forms of H_{2},
H_{3}^{+}, and H_{2}D^{+}. Our results
show that the ionization degree under such conditions depends
sensitively on the grain size distribution or, more precisely,
on the mean grain surface area per hydrogen nucleus. Depending
upon this parameter and upon density, the major ion may be
H^{+}, H_{3}^{+}, or D_{3}^{+}. We show
that the abundance of ortho-H_{2}D^{+} observed towards L 1544
can be explained satisfactorily in terms of a complete
depletion model and that this species is, as a consequence,
an important tracer of the kinematics of prestellar cores.
Key words: submillimeter - molecular processes - radio lines: ISM
An additional result from recent observational studies of prestellar cores is that depletion of molecular species on to dust grain surfaces is also a marker of relatively evolved cores (i.e. close to the pivotal state) with high central densities and column densities (e.g. Caselli et al. 2002; Tafalla et al. 2002). Tafalla et al. show that CO becomes depleted by at least a factor of 10, relative to its canonical abundance, above densities of cm^{-3}. CS shows similar behaviour, whereas nitrogen containing species such as NH_{3} and N_{2}H^{+} have abundances which are either constant or (in the case of ammonia) increase somewhat in the high density gas surrounding the dust emission peak. Their interpretation of these observations is that molecular nitrogen (N_{2}), which is the source of the observed NH_{3} and N_{2}H^{+}, is sufficiently volatile to remain in the gas phase at densities of order 10^{5}cm^{-3} (essentially because of spot heating of the grain surfaces by cosmic rays), whereas CO (the source of carbon for most C-containing species) is not.
What happens at still higher densities? It seems likely that, at sufficiently high densities, all species containing heavy elements will condense out. The difference in sublimation energies for CO and N_{2} is not great (e.g. Bergin & Langer 1997) and thus one might expect N_{2} and other N-containing species to disappear from the gas phase at densities only somewhat higher (or dust temperature slightly lower) than found for CO. Whilst it is true that the laboratory experiments may not accurately simulate interstellar ice surfaces, it seems probable that N_{2} disappears from the gas phase at densities above 10^{6} cm^{-3}.
In fact, the densities inferred from mm dust continuum emission in several objects are of order 10^{6} cm^{-3} and there is some evidence for "holes'' in the N_{2}H^{+} distribution at densities above cm^{-3}. For example, Bergin et al. (2002) find a flattening of the N_{2}H^{+} column density distribution in B68 which could be interpreted as being due to an absence of N_{2} towards the dust peak. Belloche (2002) finds a "N_{2}H^{+} hole'' around the peak of the Class 0 source IRAM04191, which is most easily interpreted in terms of N_{2} condensing out at the highest densities in the prestellar core. Last, but not least, Caselli et al. (2003, in what follows CvTCB) have detected compact emission (2000 AU in size) in the 1_{10}-1_{11} H_{2}D^{+} line towards the dust emission peak of L 1544. This latter discovery is significant in view of the fact that CvTCB infer an abundance of order 10^{-9} relative to H_{2}. Estimates of the ionization degree in L 1544 (Caselli et al. 2002) are of order . Taken together, these results imply that H_{2}D^{+}(and hence probably H_{3}^{+}) is a major ion. It seems likely that this can be the case only if ions containing heavy elements, such as N_{2}H^{+} and HCO^{+} are absent, which in turn can be true only if the heavy element content of the gas has condensed on to grain surfaces. If this is the case, one must rely on species lacking heavy elements (such as H_{2}D^{+}) to trace the kinematics of the high density nucleus of the pre-protostellar core. Also, it becomes relevant to establish the ionization degree and other physical parameters of a region where species containing heavy elements have condensed out.
In this paper, we study some of the consequences of complete depletion in an isolated, low mass prestellar core. In particular, we compute the ionization degree and its dependence upon physical parameters such as the gas density and the grain size. We consider in detail deuterium fractionation in these conditions and try to explain the observed H_{2}D^{+} (or, more precisely, the ortho-H_{2}D^{+}) abundance in objects such as L 1544. In Sect. 2, we discuss the model assumptions and, in Sect. 3, we summarize the results. Then, in Sect. 4, we consider some of the implications of our calculations and in Sect. 5, we briefly summarize our conclusions. Some preliminary results of our study have been presented elsewhere (Walmsley et al. 2004).
Our model derives from the work of Flower et al. (2003) and Flower & Pineau des Forêts (2003). The time-dependent chemical rate equations are solved numerically for a situation in which heavy elements have condensed out on to dust grain surfaces. As in the case of the primordial gas, the only significant gas-phase species are compounds containing the elements H, D and He. An important role is played by the grains which influence the charge distribution (ions, charged grains, and electrons) and enable the formation of H_{2} and HD. Apart from these latter two processes, we neglect surface chemistry in the present study. We discuss below our assumptions concerning reaction rate coefficients.
First, we consider briefly what is implied by the statement that "no heavy elements are present in the gas phase''. For the purpose of the present study, this means in essence that ions such as H_{3}^{+}, produced as a consequence of cosmic ray ionization of molecular hydrogen, are destroyed by dissociative recombination or recombination on grain surfaces, rather than by transferring a proton to species such as CO and N_{2} which have larger proton affinities than H_{2}. Assuming a rate coefficient k_{e} for dissociative recombination of H_{3}^{+} with electrons of cm^{3} s^{-1}(McCall et al. 2002, extrapolated to 10 K with a T^{-0.52}dependence: see Appendix A) and a rate coefficient for proton transfer reactions between H_{3}^{+} and species X such as CO or N_{2} of the order of the Langevin value, cm^{3} s^{-1}, the condition for "complete depletion'' is [X] , where the square brackets denote a fractional abundance; this implies that, for example, the CO abundance [CO] in the "complete depletion limit'' should be less than 400[e], or roughly in cores such as L 1544, where . Thus, the calculations reported here will be relevant when species such as CO have abundances much less than 10^{-6}.
In what follows, we report the steady-state conditions predicted by our calculations. The timescales to reach chemical equilibrium are somewhat shorter than the dynamical timescales in cores such as L 1544. The free-fall time is approximately y for a density of 10^{6} cm^{-3}; but, when magnetic fields are present, the collapse time is about an order of magnitude larger (see Ciolek & Basu 2000; Aikawa et al. 2003). As shown below, the equilibrium timescale for the entire chemistry is roughly 10^{5} y, although ionization equilibrium (with a characteristic time , which is of the order of a few years) is reached much more rapidly. We conclude that the steady-state assumption is a valid first approximation but that some of our results may need to be reconsidered in the light of their time dependence.
Recombination of ions with electrons on negatively charged grain surfaces is important because, if the available grain surface area is large enough, this process can dominate the destruction of ions such as H_{3}^{+}. Recombination on grains is certainly the main destruction mechanism for protons, which recombine only slowly (radiatively) with electrons in the gas phase, and hence our results depend on the surface area of negatively charged grains. Accordingly, we have considered carefully the distribution of grain charge, adopting the rate coefficients of FPdesF. We assumed that H_{2} and HD form on grain surfaces, at rates which scale with the total grain surface area.
A key role is played by the ortho/para ratio of molecular hydrogen, which is expected to be much higher than in LTE (see the discussion of Le Bourlot 1991); this has consequences for the ortho/para ratios of species such as H_{2}D^{+} (see Gerlich et al. 2002, hereafter GHR; Pagani et al. 1992), which become enhanced roughly in proportion to that of H_{2}. Also important in this context are the spin selection rules for ortho-para conversion, discussed, for example, by Uy et al. (1997).
Caution has to be exercised when using the data of Uy et al. (1997). These authors considered conditions in which the ortho and para forms are statistically populated, i.e. in proportion to their nuclear spin statistical weights, (2I + 1). This situation obtains at temperatures and densities much higher than those considered here. In prestellar cores, only the low-lying states of the ortho and para species are significantly populated. Under these circumstances, the statistical weights of the rotational levels, (2J + 1), must be taken explicitly into account. When compiling the rate coefficients for reactions involving the ortho and para forms of H_{2}, H_{2}^{+}, H_{3}^{+}, and H_{2}D^{+}, we made the simplifying assumption that only the lowest rotational level is populated in each case and treated the corresponding ortho and para forms as distinct species, which interconvert through chemical reactions. The complete set of reactions included in the model and their rate coefficients are given in Appendix A.
In the limit of complete depletion, deuterium fractionation occurs mainly in the reactions
Even in the absence of heavy elements, there are several processes which limit deuterium substitution in species such as H_{3}^{+}. For example, it has been known for a long time that dissociative recombination with electrons, at a rate , competes with deuterium enrichment in H_{2}D^{+}, for example (Watson 1978). It is also clear that recombination of H_{2}D^{+}, D_{2}H^{+} and D_{3}^{+} on grain surfaces can have a similar effect, and so models with a large grain surface area tend to have moderate D-fractionation. The relative importance of these two processes is roughly the same for D-fractionation and for ion neutralization: in models where the D-fractionation is limited by recombination on grain surfaces, the degree of ionization is limited by the same process.
D-fractionation in an ion such as H_{2}D^{+} is determined also by the transfer of the deuterium to the higher members of the chain, D_{2}H^{+} and D_{3}^{+}. Thus reactions 2 and 3 restrict the fractionation in H_{2}D^{+} and D_{2}H^{+}, respectively. As we shall see below, the effect of these reactions is to limit ratios such as [H_{2}D^{+}]/[H_{3}^{+}] and [D_{2}H^{+}]/[H_{2}D^{+}] to values of the order of unity.
GHR pointed out that ortho-H_{2} can restrict deuterium fractionation because the reaction between ortho-H_{2} and ortho-H_{2}D^{+} is exothermic and believed to be fast. Depending on internal excitation, this process may also be important for D_{2}H^{+}. We find that the reaction with ortho-H_{2} is not usually dominant but is significant because the abundance of ortho-H_{2} is high.
The ratio of the H_{2}D^{+} and H_{3}^{+} abundances
is given approximately by:
The code described by Flower et al. (2003)
has been used to calculate the steady-state abundances of the
chemical species, X_{i}.
The program integrates the coupled differential equations
Figure 1: Sample result for our reference model ( cm^{-3}, T = 10 K, = 0.1 m, s^{-1}) showing the approach to steady-state conditions for ortho/para H_{2}, D and H. | |
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In Fig. 1, we show the results of a sample calculation in which, initially, the hydrogen and deuterium are in molecular form, with an elemental abundance ratio . The initial ortho/para H_{2} ratio was , the equilibrium value at the gas kinetic temperature T = 10 K (assumed constant). We find that the time required to reach steady state for all species is of the order of 10^{5} y (almost independent of number density). However, as noted above, the time to attain ionization equilibrium (a few years) is much shorter than that required to attain equilibrium between ortho- and para-H_{2} (the longest chemical timescale). Atomic hydrogen and deuterium are intermediate, reflecting the timescale for the formation of H_{2} and HD on grains.
The time characterizing chemical steady state is comparable with the free-fall time, , at the lowest density ( cm^{-3}) and is an order of magnitude larger than at the highest density ( = cm^{-3}) considered. However, magnetic support to the cloud increases the time for core formation and, in practice, the timescale for chemical equilibrium is likely to be comparable with that for contraction and collapse of the protostellar core.
Figure 2: Abundances of major ions and electrons in our standard model ( = 0.1 m, T = 10 K, ) as functions of the density of molecular hydrogen. | |
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Figure 3: Ortho/para ratios of H_{2} (multiplied by 10^{4}), H_{3}^{+}, and H_{2}D^{+} as functions of density for our reference model (cf. Fig. 2). The approximate expression of Gerlich et al. (2002) for H_{2}D^{+} is shown for comparison. | |
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In Fig. 2 are plotted the fractional abundances of the principal ions and of the electrons as functions of ; the results of computations of ortho/para ratios for the same model are presented in Fig. 3 and will be discussed in Sect. 3.4 below. At the lowest density considered, n(H_{2}) = 10^{5} cm^{-3}, H^{+} is the major ion and the removal of molecular ions by dissociative recombination is significant. H^{+}, on the other hand, is removed by recombination on grain surfaces. As n(H_{2}) increases, recombination on grains dominates dissociative recombination for the molecular ions. H^{+} becomes a relatively minor species at high density.
The balance between the various deuterated forms of H_{3}^{+} is determined approximately by Eq. (4) and its analogues for [D_{2}H^{+}]/[H_{2}D^{+}] and [D_{3}^{+}]/[D_{2}H^{+}]. At densities below 10^{6} cm^{-3}, the dissociative recombination term k_{e} [e] is dominant in the denominator of the right hand side of Eq. (4), whereas, at higher densities, the term due to the reaction with HD becomes more important and results in [H_{2}D^{+}]/[H_{3}^{+}] (and [D_{2}H^{+}]/[H_{2}D^{+}]) being close to unity. On the other hand, D_{3}^{+} cannot be destroyed by HD and becomes the dominant ion.
Figure 4: Abundances of major ions and electrons in a model with a smaller grain size of 0.025 m. | |
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Figure 5: The free electron abundance computed as a function of density for grain sizes between 0.025 and 0.2 m. The result of McKee (1989) is shown for comparison (broken curve). Note the decrease in the level of ionization for small grain sizes (large grain surface areas). | |
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In Fig. 5, the free electron abundance, n(e)/ , computed as a function of density, is compared with the relationship of McKee (1989), , which is often used in discussions of core evolution. It may be seen that our model generally predicts lower degrees of ionization. We find that n(e)/ varies approximately as between and cm^{-3}. As already noted, the degree of ionization of the gas is sensitive to the assumed grain size, owing to the importance of the recombination of ions with electrons on grain surfaces. The fraction of electrons attached to grains remains small for all the grain sizes considered here: the ratio of negatively charged grains to free electrons attains 0.20 for for m and cm^{-3}. However, we note that this ratio becomes larger for smaller grain sizes, becoming 1 for m. The fraction of grains which is neutral decreases from 0.68 for m to 0.20 for m and for cm^{-3} (but this ratio is insensitive to the gas density). The fraction of positively charged grains is always negligible.
The ambipolar diffusion timescale,
,
depends
on the ion abundance, n_{i}/n_{n},
the rate coefficient for momentum transfer between the ions and
the neutrals (essentially H_{2}),
cm^{3} s^{-1}, and the masses m_{i} and m_{n} of the charged and neutral
collision partners. We have taken
Figure 6: Ratio of ambipolar diffusion time (Eq. (5)) to free fall time as function of molecular hydrogen density for values of the grain radius between 0.025 and 0.2 m. | |
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Figure 3 shows that the ortho/para ratios considered here are relatively insensitive to density. For molecular hydrogen, we derive a steady-state ortho/para ratio of approximately for the conditions in the nucleus of L 1544, and this value is reflected in the H_{2}D^{+} ortho/para ratio, as shown by GHR. In fact, we find that the H_{2}D^{+} ortho/para ratio predicted by Eq. (7) of GHR is a good approximation to our numerical results.
The expression of Le Bourlot (1991) for ortho/para H_{2} was derived for different conditions (lower density and high abundances of molecular ions such as HCO^{+}) and is not a good approximation in the conditions that we have considered. In our situation, the ortho/para H_{2} ratio is given approximately by , where represents the sum of the number densities of ions such as H^{+} and H_{3}^{+} which convert ortho- to para-H_{2} through proton transfer reactions, with a rate coefficient for ortho to para conversion which is of order 10^{-10} cm^{-3} s^{-1}.
The ortho/para H_{2}D^{+} ratio has a flat dependence on density, reflecting the behaviour of ortho/para H_{2}. The ortho forms of both H_{2}D^{+} and H_{3}^{+} are produced mainly in reactions of the para forms with ortho-H_{2} and so their high ortho/para ratios are attributable to the relatively high ortho-H_{2} abundance; Fig. 7 illustrates this point. Because the ortho/para H_{2} ratio is not thermalized under the conditions which we consider, the ortho/para H_{2}D^{+} and H_{3}^{+} ratios are not thermalized either. It follows that measurements of the ortho/para H_{3}^{+} ratio should not be used to infer a temperature.
Figure 7: Ortho/para ratios of H_{3}^{+} and H_{2}D^{+} as functions of the molecular hydrogen ortho/para ratio. The conditions are those of our reference model: n(H_{2}) = 10^{6} cm^{-3}, T = 10 K, = 0.1 m, s^{-1}. | |
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The basic question that we wish to answer is whether the ortho-H_{2}D^{+} emission observed in L 1544 can be reproduced by a model which assumes complete heavy element depletion. The abundance of ortho-H_{2}D^{+} relative to H_{2} observed towards the peak of L 1544 is according to CvTCB. In Fig. 8 are shown the computed values of [ortho-H_{2}D^{+}]/[H_{2}] as a function of density for various assumed grain sizes. There is a tendency for [ortho-H_{2}D^{+}]/[H_{2}] to decrease with increasing density, owing to the fact that deuteration proceeds further down the chain (towards D_{3}^{+}) at high density. Our results are within a factor of 2 of the observed peak ortho-H_{2}D^{+} abundance, for grain radii below 0.1 m. Given the observational uncertainties, this level of agreement might be considered satisfactory. It is also significant that computations for a = 0.01 m show that the computed ortho-H_{2}D^{+} abundance decreases for all values of n(H_{2}) relative to the results in Fig. 8 for a = 0.025 m, and thus grain radii in the range 0.025-0.05 m maximize the ortho-H_{2}D^{+} abundance. We note that the "observed'' abundance assumes a rate of collisional excitation of H_{2}D^{+}, a temperature, and a density distribution, of which the latter two are controversial (see, for example, Galli et al. 2002). Higher angular resolution studies of the mm-submm continuum are needed for further progress.
One result from our study is that a relatively high rate of cosmic ray ionization is required in the nucleus of L 1544 to account for the observed ortho-H_{2}D^{+} emission; this implies the penetration of low energy (100 MeV) cosmic rays into the high density core. Cosmic ray fluxes higher than assumed here would enhance the computed ortho-H_{2}D^{+} abundance; but to a higher flux would correspond an increased rate of heat input to the gas and temperatures in excess of those observed in NH_{3} and other tracers.
Figure 8: The computed ortho-H_{2}D^{+} abundance from our reference model (T = 10 K and s^{-1}) as a function of density for several values of the assumed grain radius. The values observed by Caselli et al. (2003) are shown for comparison. | |
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Figure 9: Atomic H (full lines) and atomic D (broken lines) densities for four values of the grain radius (increasing from bottom to top) for densities between 10^{5} and 10^{7} cm^{-3}. | |
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The grain size distribution in prestellar cores is not known. It depends on the timescales for grain coagulation and mantle accretion, relative to the dynamical and chemical timescales. In view of this uncertainty, we compared the results of computations assuming a unique grain radius with the corresponding calculations assuming a MRN power-law size distribution; we retained the same value of the grain surface area per H-nucleus, , in both cases. In calculations with a unique value of , (m)) cm^{2}. When a power law dependence is adopted, providing the exponent r > 3 (MRN derived r = 3.5), the lower limit to the radius essentially determines the grain surface area. When r = 3.5, the lower limit to is 600 Å for cm^{2}. We found that the results of calculations with the same value of grain surface area per H-nucleus were quantitatively similar for the cases of a unique grain radius and a MRN power-law size distribution.
As shown above, the detection of H_{2}D^{+} in L 1544 by CvTCB can be explained if m or, equivalently, if the grain surface area per H-nucleus is at least cm^{2}, which is close to the value for grains in diffuse clouds. This conclusion is consistent with deductions which have been made from the mm-infrared extinction curve in prestellar regions: Bianchi et al. (2003) found that the ratio of mm to NIR dust absorption is marginally larger in prestellar environments than in diffuse clouds; it is also consistent with the simulations of Ossenkopf & Henning (1994).
Under conditions of complete heavy element depletion, the thickness of the grain mantle is expected to be roughly 40 percent of the grain radius, or 400 Å for 0.1 m grains. Then, 45 percent of the grain mass is in the refractory core and 55 percent in the ice mantle. A MRN size distribution yields an ice mantle 260 Å thick, independent of the grain radius: see Appendix B.
Unfortunately, the 1_{01}-0_{00} transition to the ground state of para-H_{2}D^{+} (at 1.37 THz) will be difficult to excite at the temperatures of prestellar cores and may be observable only in absorption towards a background continuum source. On the other hand, H_{3}^{+} has no allowed submm transitions and is best observed in the NIR at 3.5 m in absorption towards background sources (McCall et al. 1999). It has been assumed that the H_{3}^{+} which has been observed exists in regions whose composition is similar to that of "normal'' interstellar molecular clouds and, furthermore, that the ortho/para forms are in LTE. In our view, both of these assumptions may prove to be invalid. Observations of H_{2}D^{+} and the other members of the family may help to shed light on this issue.
Our results can be compared with those of Roberts et al. (2003) who carried out a time dependent calculation of the chemistry in a prestellar core with density cm^{-3} including freeze-out. They found, as do we, that D_{3}^{+} can be the most abundant ion, though their results suggest that H_{3}^{+} becomes the major ion when all heavy species freeze out; this may be attributable to a different way of dealing with grain neutralization of ions.
Further progress in this area will depend on our capacity to sample
transitions other than the the
1_{10}-1_{11} line of H_{2}D^{+}. H_{3}^{+} and
D_{3}^{+} do not have permanent dipole moments, and hence their detection is
feasible only in absorption in the near infrared.
Our calculations suggest that searches for D_{3}^{+} should be directed
towards regions
where high heavy element depletion is believed
to have occurred. Searches for
emission in low-lying transitions of D_{2}H^{+} should also be made: the
abundance of D_{2}H^{+} in conditions of complete depletion is predicted to
be comparable to that of H_{2}D^{+}. Under conditions of LTE at temperature
T, the relative populations of the lowest ortho (0_{00}) and para
(1_{01}) levels of D_{2}H^{+} would be
Acknowledgements
It is a pleasure to thank Evelyne Roueff and Eric Herbst for informative and helpful discussions, Paola Caselli for her comments on the text, and the referee, Ted Bergin, for a constructive and thought-provoking report. CMW thanks the Ecole Normale Supérieure in Paris for hospitality during a stay when part of the work described here was carried out as well as the Max Planck Institute in Bonn for hospitality on diverse occasions.
In this Appendix are specified the reactions included and the rate coefficients adopted in our chemical model.
Protons have nuclear spin I = 1/2 and are fermions. The ortho and para forms of H_{2}, H_{2}^{+}, H_{3}^{+} and H_{2}D^{+} have been treated as distinct species. Ortho- and para-H_{2} were assumed to form hot, on grains, in the ratio 3:1 of their nuclear spin statistical weights. Subsequent proton-exchange reactions interconvert the ortho and para forms. Such reactions were assumed to involve only the rotational ground states of the ortho and para modifications, and the corresponding rate coefficients take account of the relative statistical weights (2I+1)(2J+1) of the products and reactants.
Deuteration of H_{3}^{+} in reactions with HD may be viewed as proton transfer (from H_{3}^{+} to HD) or deuteron-proton exchange; we allowed for both viewpoints in our reaction set. Rate coefficients for dissociative recombination, with electrons in the gas phase or on negatively charged grains, were assumed to be the same for the ortho and para forms of the recombining ion.
We note that deuterons have nuclear spin I = 1 and are bosons. Multiply deuterated species have ortho and para forms. In the present study, we have not attempted to allow for the consequences of Bose-Einstein statistics when evaluating the rates of reactions involving multiply-deuterated species.
Table A.1: Reactions and rate coefficients adopted in our chemical model. The parameters , , and define the rate coefficients k cm^{3} s^{-1} at temperature T through the relation k = J ; J allows for Coulomb focusing in reactions of positive ions and negatively charged grains (Draine & Sutin 1987, Eq. (3.4)); = 0.1 m is adopted in the table. The rates s^{-1} of reactions induced directly by cosmic rays (crp) or indirectly by the secondary electrons, which generate H_{2}fluorescence photons (secpho), are given by , where is the rate of cosmic ray ionization of H_{2}. In the first three reactions (formation of H_{2} or HD on grain surfaces), denotes the fraction of such reactions which form the specified product; the rate is calculated internally in the program from the grain parameters. Key reactions are in bold face. Numbers in parentheses are powers of 10.
In this Appendix, we derive estimates of the ice-mantle thickness in completely depleted cores. We follow the approach of FPdesF.
Following FPdesF, we assume the refractory cores to be composed of silicates (in practice, olivine, MgFeSiO_{4}) or graphite. The mass of the cores, per unit volume of gas, is , where the summation extends over the elements X, and M denotes the elemental mass. We define through the relation , and , which corresponds to a fraction of 0.0056 by mass relative to the gas-phase H and He.
Assuming that all elements heavier than He are in the grain cores or frozen in the mantles, the total dust-to-gas mass density ratio is 0.0125, and the mass per unit volume of the gas of the ice in the mantle, is , where .
The volume of a grain ice mantle is determined by the relation
(B.1) |
(B.2) |
(B.3) |
This analysis can be generalized to the case of a MRN power-law
size distribution, using the formalism of Le Bourlot et al. (1995). If the
lower limit of the size distribution is taken to be a_{m} = 0.01 m (and the upper limit is a_{M} = 0.3 m), then the thickness of the mantle is obtained by solving the equation
(B.4) |
(B.5) |