5 The chemical models

Our observational results have been interpreted using detailed, chemical kinetic models of deuterium chemistry in dark clouds. The models, which are fully described in Roberts & Millar (2000a,b), now involve 298 species (135 of them containing deuterium) and $\sim$5550 reactions.

There are two basic models; the "gas-phase'' model considers only reactions between gaseous species (with the exception that H₂ and HD can form on grain surfaces),

   

Table 9: Column densities of c-C₃HD and c-C₃H₂ and [C₃HD]/[C₃H₂] ratios.

Source	N(c-C3H2)a	N(c-C3HD)	${\rm\frac{[C_3HD]}{[C_3H_2]}}$
	($\times$1012cm-2)	($\times$1012cm-2)
B5IRS1	--	0.77 ($\pm$0.33)	--
L1448mms	>11.2	1.58 ($\pm$0.63)	<0.14 ($\pm$0.06)
L1448NW	>9.9	1.59 ($\pm$0.69)	<0.16 ($\pm$0.07)
L1527	>8.4	2.30 ($\pm$0.31)	<0.27 ($\pm$0.04)

^a From Buckle & Fuller (2001).

while the "accretion'' model includes the possibility that gas-phase species can collide with and stick to grains, essentially being removed from the gas.

In cold, dense gas, species which collide with dust grains are likely to stick; indeed, it is now well known that interstellar grains become encased in mantles of ice on a similar timescale to the dynamical and chemical evolution of molecular clouds (Willacy & Millar 1998; Rawlings 1999).

The deuterium fractionation, in particular, will be strongly affected. In cold clouds deuterium is extracted from its reservoir in HD by exothermic reactions such as

$${\rm H_3^+ + HD} \rightleftharpoons {\rm H_2D^+ + H_2 + \Delta} E$$ (9)

which leads to an enhancement in the [H₂D⁺]/[H₃⁺] ratio. Since both H₂D⁺ and H₃⁺ have extremely low proton affinities they rapidly react with species such as CO, C and H₂O, enhancing molecular D/H ratios throughout the system. In undepleted gas, reaction (9) represents a minor loss for H₃⁺, however, as other species begin to freeze onto grain surfaces, reaction with HD becomes the dominant loss mechanism for H₃⁺. As this produces H₂D⁺, the [H₂D⁺]/[H₃⁺] ratio increases further and results in an increase in all molecular D/H ratios for a time before most of the heavy species are removed from the gas.

This theoretical expectation, first predicted by Brown & Millar (1989), has recently been confirmed observationally by the detection of large [DCO⁺]/[HCO⁺] ratios in L1544 in clumps in which CO is significantly depleted (Caselli et al. 1999). It may also prove important in explaining the large molecular D/H ratios observed towards IRAS16293 and L134N.

All models presented in this paper use standard depleted solar elemental abundances for dark clouds, as listed in Table 10.

 

 

Table 10: The initial fractional abundances, relative to the total hydrogen density, used in our models.

Species	Abundance	Species	Abundance
H2	5.00 $\times$ 10-1	H3+	1.00 $\times$ 10-11
He	1.40 $\times$ 10-1	Si	2.00 $\times$ 10-8
C+	7.30 $\times$ 10-5	Fe+	1.00 $\times$ 10-8
N	2.14 $\times$ 10-5	S	1.00 $\times$ 10-7
O	1.76 $\times$ 10-4	HD	3.20 $\times$ 10-5

We have adopted the cosmic D/H ratio measured by Linsky et al. (1995), 1.6$\times$10^-5, a visual extinction of 10 mags and a cosmic ray ionisation rate of $1.3 \times 10 ^{-17}$ s^-1.

5.1 A comparison with model results

Figure 5 shows the variation in [HDCO]/[H₂CO] and [DCN]/[HCN] with temperature and density,

Figure 5: Steady-state [HDCO]/[H2CO] (top) and [DCN]/[HCN] (bottom) ratios, from our gas-phase models, varying over a temperature range 10-50 K (in intervals of 10 K) and a $\log_{10} n$(H2) range of $\sim$3.5-6.5 (in intervals of 0.5), the dashed planes indicate the observed [HDCO]/[H2CO] ratios.

as predicted by our gas-phase models. Regardless of density, the model matches the observed [HDCO]/[H₂CO] ratios, 0.05-0.07, for a kinetic temperature between 10 and 20 K. However, the observed [DCN]/[HCN] ratios, of 0.04, are at least one and a half times larger than the model predicts, regardless of the physical conditions we assume, and $\sim$twice as large for reasonable density estimates (10⁴-10⁵ cm^-3).

It appears, therefore, that the gas-phase model alone is insufficient to explain the observations. There are, however, uncertainties inherent in any chemical modelling of this type. In particular, HCN, HNC and CN are all formed via dissociative recombination of HCNH⁺ with electrons, where our model assumes that 50% of the reactions form CN, while 25% form HCN and 25% form HNC (Herbst 1978). Recent experimental and observational work now suggests that HCN and HNC will be formed in equal amounts, but CN will not be formed by this reaction (Semaniak et al. 2001; Dickens et al. 2000). As there are analogue reactions for forming DCN and DNC, with same rate coefficients and branching ratios, this means that the abundances of HCN, HNC, DCN and DNC would all increase if we adopted these new branching ratios. However, we are primarily interested in abundance ratios, rather than absolute abundances, and as the deuterated and non-deuterated species are being affected in the same way, the molecular D/H ratios will be largely unchanged. We estimate that altering the branching ratios for recombination of HCNH⁺ (and deuterated analogues) will not alter the predicted [DCN]/[HCN] ratio by more than 50%. It would still be too small to explain all the observations.

The accretion model, on the other hand, produces increased D/H ratios over the gas-phase model, as heavy species freeze out onto grain surfaces. Figure 6 compares results from the gas-phase and accretion models for [HDCO]/[H₂CO], [DCN]/[HCN] and other key molecular D/H ratios.

The [DCN]/[HCN] ratio is particularly sensitive to accretion, rising from its steady-state gas-phase value of 0.016 to match the observed ratios of 0.04 in $\sim$ $5\times10^4$ yrs. The DCN abundance is enhanced by constant cycling between DCN and DCNH⁺, as DCNH⁺ forms via proton transfer from H₃⁺ and H₂D⁺ to DCN, and via deuteron transfer from H₂D⁺ to HNC. There are similar reactions which form HCN, and so, as the abundance of H₂D⁺ increases, relative to H₃⁺, there is an increased chance of forming DCN rather than HCN.

HDCO and H₂CO also react with H₂D⁺ and H₃⁺, forming the ions H₃CO⁺ and H₂DCO⁺ which may then recombine to H₂CO and HDCO, the increasing [H₂D⁺]/[H₃⁺] ratio being reflected by the [HDCO]/[H₂CO] ratio. However, this is not the only route to forming H₂CO and HDCO, therefore, over the same time period the [HDCO]/[H₂CO] ratio, is not so sensitive to accretion processes. The ratio rises from 0.05 to 0.063, and so remains consistent with the ratios we measured towards the low-mass cores.

These results assume a kinetic temperature of 10 K and an H₂ density of $3 \times 10 ^4$ cm^-3. [DCN]/[HCN] ratios are sensitive to temperature (see Fig. 5), so if we assumed $T_{\rm kin} = 20$ K, the initial [DCN]/[HCN] ratio would be lower and it would take twice as long (10⁵ yrs) for the predicted ratios to reach the levels we observed. As [HDCO]/[H₂CO] ratios increase with temperature for $T_{\rm kin}<50$ K (see Fig. 5), predicted HDCO fractionation at this time would be much higher than was observed.

Changing the density in the accretion model primarily affects the timescale on which freeze-out occurs; increasing n(H₂) from $3 \times 10 ^4$ to 10⁵ cm^-3 decreases the time it takes for the predicted [DCN]/[HCN] ratios to rise to agree with the observed values from $5\times10^4$ to $\sim$10⁴ yrs,

Figure 6: Molecular D/H ratios evolving over time from our gas-phase (left) and accretion (right) models; $T_{\rm kin}=10$ K, n(H $_2)=3\times 10 ^{4}$ cm-3. The best fits to the [HDCO]/[H2CO] and [DCN]/[HCN] ratios we observed are also indicated on the plots.

and the time taken for everything to freeze out from $\sim$10⁶ to $\sim$ $3 \times 10 ^5$ yrs. On the other hand, if the density were lower, $\sim$10⁴ cm^-3, the predicted [DCN]/[HCN] ratio would not agree with the observed value until $\sim$ $2 \times 10 ^5$ yrs of accretion had occurred.