Mon. Not. R. Astron. Soc. 336, 283–290 (2002)

The importance of new rate coefficients for deuterium fractionationreactions in interstellar chemistry

Helen Roberts,1� Eric Herbst2 and T. J. Millar3

1Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA2Departments of Physics and Astronomy, The Ohio State University, Columbus, Ohio 43210, USA3Department of Physics, UMIST, PO Box 88, Manchester M60 1QD

Accepted 2002 May 31. Received 2002 May 28; in original form 2002 April 3

ABSTRACTRate coefficients for deuterium fractionation reactions that are important at low temperatureshave recently been studied at 10 K in the laboratory for the first time. In this paper, weincorporate these newly measured rates, or values based on them, into existing models ofgas-phase interstellar deuterium chemistry. We then compare our results with those obtainedpreviously and with observations of deuterated molecules in dark clouds, specifically TMC-1, L134N and L1544. In general, the new rates tend to reduce the amount of fractionationthat can occur in the gas, which may present problems in regions where high moleculardeuterium/hydrogen ratios have been observed.

Key words: astrochemistry – molecular processes – ISM: abundances – ISM: clouds – ISM:molecules.

1 I N T RO D U C T I O N

Observations of deuterated molecules have long been recognizedas important tracers of the physical and chemical processes that areoccurring in the interstellar medium (ISM). Deuterium in the ISMis mostly locked up in the form of HD, with an abundance relativeto H2 of only ∼10−5. Under certain conditions, however, deuteriumcan be preferentially brought out of this interstellar reservoir andinto an active chemistry, resulting in greatly enhanced abundancesof deuterium-bearing molecules.

It is now generally accepted that the most important precursorsto fractionation in dark clouds are the reactions

H+3 + HD ⇀↽ H2D+ + H2 + �E1 (1)

CH+3 + HD ⇀↽ CH2D+ + H2 + �E2 (2)

C2H+2 + HD ⇀↽ C2HD+ + H2 + �E3 (3)

(Millar, Bennett & Herbst 1989; Roberts & Millar 2000a). In eachcase, the forward reactions are exothermic, with specific exother-micities �E1 = 232 K, �E2 ∼ 370 K and �E3 ∼ 550 K. Conse-quently, the deuterated molecular ions are destroyed very slowlyby endothermic reactions with H2 at temperatures <100 K. Despiteits relatively low exothermicity, reaction (1) is the most importantreaction for fractionation at typical dark cloud temperatures of 10 Kor so because H+

3 is more abundant than the two carbonaceous ions.As temperatures rise, however, the greater exothermicities of reac-tions (2) and (3) play an increasingly important role since CH2D+

and C2HD+ are destroyed far more slowly than H2D+.

�E-mail: hroberts@mps.ohio-state.edu

The rates of the forward reactions were measured by Adams& Smith (1981), Smith, Adams & Alge (1982) and Herbst et al.(1987) at temperatures down to 80 K. At this temperature, the for-ward rate coefficients for the first two reactions are two-thirds tothree-quarters of the Langevin rate of 1.7 × 10−9 cm3 s−1, while theforward rate coefficient for the third reaction is 3.2 × 10−10 cm3 s−1.According to theoretical estimates (e.g. Herbst et al. 1987), theserate coefficients should gradually approach the Langevin limit as thetemperature is reduced to 10 K. The rate coefficients for the back-ward reactions can then be determined under equilibrium conditionsfrom knowledge of the equilibrium coefficient (Herbst 1982). Al-though this coefficient depends rigorously on the Gibbs free energychange between reactants and products, it is normally acceptableto ignore the entropy term and the temperature dependence of thethermodynamic parameters, so that the Gibbs free energy can bereplaced by the energy change at 0 K (i.e. the reaction exothermic-ity). With these approximations, the equilibrium fractionation ofCH2D+ and C2HD+ can be estimated by assuming that the back-ward rate coefficients are smaller than their forward counterpartsby a Boltzmann factor, exp (−�E/T ), where �E is the (positive)exothermicity of the forward reaction in units of kelvin. Reaction(1), on the other hand, requires a more detailed analysis (Herbst1982). Equilibrium constants have been calculated for this system,most recently by Sidhu, Miller & Tennyson (1992); these suggestthat the rate coefficient of the backward reaction is ∼10−18 cm3

s−1 at 10 K. At such low temperatures, when the reactions be-come essentially irreversible under equilibrium conditions, otherdestruction reactions for the deuterated ions become important. It isthese destruction reactions which propagate deuterium throughoutthe system and transfer the fractionation of the primary ions to otherspecies.

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284 H. Roberts, E. Herbst and T. J. Millar

The ion H+3 has long been known to be crucial in ion–molecule

chemical networks (Herbst & Klemperer 1973; Herbst 2000), sinceits low proton affinity means that it reacts rapidly with most otherneutral molecules by donating a proton. The ion H2D+ also has alow proton/deuteron affinity, and reacts with the same species asH+

3 . Thus, for reactions such as

H+3 + X → XH+ + H2, (4)

there are the analogous reactions

H2D+ + X → XH+ + HD (5)

→ XD+ + H2. (6)

The enhanced H2D+/H+3 ratio, caused by reaction (1) – a primary

fractionation – will be reflected in the resulting XD+/XH+ ratio (asecondary fractionation), albeit at a reduced value since there aretwo channels for the reaction of H2D+ with X.

The ions H+3 and H2D+ have only recently been directly observed

in the ISM [H+3 by Geballe & Oka (1996) and H2D+ by Stark, van

der Tak & van Dishoek (1999)]. However, since the first detection ofDCN in the Orion nebula (Jefferts, Penzias & Wilson 1973), manyother deuterated molecules have been observed, and molecular D/Hratios measured. These observations have largely supported the pro-cess of deuterium fractionation set out above, and chemical modelshave been fairly successful in explaining the levels of deuteriumfractionation in various molecular clouds (e.g. Millar et al. 1989;Roberts & Millar 2000a,b).

Recently, Gerlich, Herbst & Roueff (2002) studied the forwardand backward reactions for system (1) in an ion trap at 10 K, whileGerlich & Schlemmer (2002) studied the forward and backwardreactions for systems (2) and (3) in the same trap. Regarding theforward reactions, the experimental results show that the rate co-efficients lie considerably below the Langevin value. These valuesare compared with the so-called ‘standard’ extrapolated values usedpreviously in models in Table 1. For example, the rate coefficientfor the H+

3 + HD reaction has been measured to be 3.5 × 10−10 cm3

s−1, a value approximately one-fifth of the previously used value.While the laboratory measurements of the forward reactions canbe put into chemical models directly, further analysis is needed forthe reverse, endothermic reactions. For the reaction between H2D+

and H2, the hydrogen used comes from a para-H2 generator, buthas a probable impurity of 1 per cent ortho-hydrogen. The ortho-hydrogen has sufficient rotational energy to react much more rapidlythan the para-hydrogen (Gerlich et al. 2002). In addition, the extrarotational energy from the ortho-H2 heats up the H2D+ ion, whichfurther enhances the backward rate coefficient. The net result is thatthe backward rate measured in the laboratory is probably larger thanthe value relevant to dense interstellar clouds at 10 K.

Table 1. A comparison of rate coefficientsa for three fractionation reactionsat 10 K.

Reaction Standard rate New rate(cm3 s−1) (cm3 s−1)

H+3 + HD → H2D+ + H2 1.7(−09) 3.5(−10)

H2D+ + H2 → H+3 + HD 2.5(−18) 7(−15)–7(−14)

CH+3 + HD → CH2D+ + H2 1.3(−09) 2.6(−10)

CH2D+ + H2 → CH+3 + HD 7.4(−26) 2.2(−15)

C2H+2 + HD → C2HD+ + H2 1.0(−09) 7.5(−10)

C2HD+ + H2 → C2H+2 + HD 3.2(−33) <8(−16)

aThe notation x(−z) implies x × 10−z ; see text for discussion of standardand new rates.

To estimate the backward rate in cold clouds, Gerlich et al. (2002)started with the result of Le Bourlot (1991) that the abundance ofortho-H2 is approximately 0.001 that of para-H2. They then calcu-lated the effect of this amount of ortho-H2 on H2D+, and reacheda conclusion similar to Pagani, Salez & Wannier (1992) that ortho-and para-H2D+ are in roughly equal abundance. With these results,they then estimated a rate coefficient at most 10 times smaller thanthe laboratory value and possibly smaller than this. A reasonablerange of values is shown in Table 1. These values are orders ofmagnitude greater than the previously used standard values.

The backward reactions for systems (2) and (3) have not yet beenstudied in as great detail as that for system (1). For system (3),only an upper limit, shown in Table 1, has been obtained, whilefor system (2), the laboratory value is derived in a complex seriesof reactions. We have estimated the relevant interstellar rate coef-ficient for CH2D+ + H2 at 10 K based on the laboratory work andon analogy with the better-studied reaction between H2D+ and H2.Our value is in Table 1.

As discussed by Gerlich et al. (2002) for the H+3 + HD system,

the new laboratory results, suitably interpreted, can have importantconsequences for our understanding of deuterium chemistry in thecold ISM. It is the purpose of this paper to explore these conse-quences in some detail. In Section 2, we consider the effects of thenew rates for 10 K clouds and present new predictions for deuteriumfractionation, both for purely gas-phase models and for models inwhich depletion of heavy materials on to dust particles is allowedto occur. In Section 3, we compare our new results with observa-tions, while in Section 4 we discuss these results. Conclusions arepresented in Section 5.

2 L I M I T S O N D E U T E R I U M F R AC T I O NAT I O N

It can be shown that the steady-state [H2D+]/[H+3 ] ratio is given by

[H2D+]

[H+3 ]

= kf

kr + ke[e−] + ∑ki [Mi ]

[HD]

[H2], (7)

and there are similar equations for the [CH2D+]/[CH+3 ] and

[C2HD+]/[C2H+2 ] ratios (e.g. Millar 2002). In equation (7), kf and

kr are the rate coefficients for formation and destruction of H2D+

via reaction (1), ke is the rate coefficient for dissociative recombi-nation of H2D+ with electrons, and [e−] is the fractional abundanceof electrons, while

∑ki [Mi ] represents the destruction of H2D+

by other species (such as CO, C, O and N2), [Mi ] is the fractionalabundance of species i, and ki is the rate coefficient for its reactionwith H2D+.

This equation can also be written as

[H2D+]

[H+3 ]

= S(H2D+)[HD]

[H2], (8)

where S(H2D+) is the enhancement factor, defined as the enhance-ment in H2D+ fractionation over the [HD]/[H2] ratio. One gen-eral rule arising from the fractionation processes we assume isthat secondary fractionation occurs at a lower level than primaryfractionation (Millar 2002). Therefore, the enhancement factor inequation (8) also represents the maximum enhancement that canoccur in any molecule fractionated via reaction of H2D+. Similarly,S(CH2D+) and S(C2HD+) would be the maximum enhancement infractionation possible for deuterated species formed from CH2D+

and C2HD+, respectively.

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Rate coefficients for deuterium fractionation 285

2.1 Pure gas-phase chemistry

We can now investigate the effect that the new rates for reaction (1)will have on the expected level of deuterium fractionation. The mostabundant neutral species that destroy H+

3 and deuterated analoguesare CO and O (∼ a few times 10−4 relative to H2), which react ata rate of ∼10−9 cm3 s−1, so that we can estimate

∑ki [Mi ] to be

∼5 × 10−13 cm3 s−1. The mean fractional abundance of electronsobserved in a range of molecular clouds is ∼10−7, relative to H2

(Williams et al. 1998), and the most recently determined rate forH2D+ recombination with electrons is 3 × 10−7 cm3 s−1 at 10 K(Larsson et al. 1996).

Substituting these numbers into equation (7), and using the stan-dard rates for reaction (1), yields an enhancement factor of

S(H2D+) = 1.7 × 10−9

2.5 × 10−18 + 3.0 × 10−14 + 5.0 × 10−13

= 3.2 × 103. (9)

In equation (9), the term for destruction via neutral species is at leastan order of magnitude larger than the other destruction terms, andso the enhancement in H2D+ is limited by the rate of its formationand by the abundances of the neutral species.

Using the new rate coefficient for kf and the slower of the esti-mated rate coefficients for kr (∼7 × 10−15 cm3 s−1), we find that Sis reduced, although destruction by neutral species still dominates:

S(H2D+) = 3.5 × 10−10

7.0 × 10−15 + 3.0 × 10−14 + 5.0 × 10−13

= 6.5 × 102. (10)

If kr is an order of magnitude faster (∼7 × 10−14 cm3 s−1), it doeshave a small effect:

S(H2D+) = 3.5 × 10−10

7.0 × 10−14 + 3.0 × 10−14 + 5.0 × 10−13

= 5.8 × 102, (11)

but the major effect on S(H2D+) in both equations (10) and (11)arises from the fact that the rate of formation of H2D+ has decreasedby a factor of 5.

Calculating S(CH2D+) and S(C2HD+) using the new rates listedin Table 1, we see a similar result, since the rates of formation ofthese species have fallen by factors of 5 and ∼1.3, respectively.Thus, the calculated fractionation of species is expected to fall atsteady state (reached at ∼106 yr) and, most likely, at earlier timesas well.

The specific effects of incorporating the new rates into a gas-phase model are illustrated in Fig. 1 for the fractionation of H2D+

and a few other deuterated species as a function of time. The modelused is the gas-phase model described in Roberts & Millar (2000b)and represents a dark cloud at 10 K with constant physical condi-tions. The chemical network includes reactions that produce singlyand doubly deuterated species. Results with the rate coefficient fordestruction of H2D+ by H2 being ∼7 × 10−14 cm3 s−1 are almostidentical to the case where it is ∼7 × 10−15 cm3 s−1, and so wehave only plotted the latter case. For an underlying [HD]/[H2] ratioof 3 × 10−5 (Linsky et al. 1995; Wood, Redfield & Linsky 2002),the steady-state [H2D+]/[H+

3 ] ratio has fallen from ∼0.1 to ∼0.02,when the new rates are adopted, as predicted by equations (9)–(11),leading to a reduction in the fractionation of the other deuteratedspecies.

Butner, Lada & Loren (1995) measured DCO+/HCO+ ratios to-wards several starless cores, finding ratios between 0.027 and 0.07.

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Figure 1. Predicted evolution of selected molecular D/H ratios from gas-phase models using the standard (top) and new (bottom) rate coefficientslisted in Table 1. Tkin = 10 K, n(H2) = 104 cm−3.

These are consistent with model predictions using the standard ratesfor reactions (1)–(3), but are all higher than the DCO+ fractionationthat the new rates predict.

Table 2 shows the difference that changing these rate coefficientsmakes for a wider variety of species at steady state. The model resultsare also compared with observations of the cyanopolyyne peak inTMC-1, the archetypal ‘dark cloud’ source, in which the widestvariety of molecular D/H ratios have been measured. As expected,adopting the new rate coefficients for reactions (1)–(3) has loweredthe molecular D/H ratios. Although the agreement with observationshas improved for three species (DCO+, C4D and HDCS), the newrates are somewhat worse, overall, at explaining the fractionation inTMC-1, in the sense that the predicted fractionation is typically toolow. Moreover, the deuterium fractionation at times before steadystate is reached, which is more appropriate, is somewhat smallerstill. So what does this mean for our models of interstellar deuteriumchemistry?

Turner (2001) has recently recalculated the deuterium fraction-ation of several molecules in TMC-1, using detailed radiativetransfer models, which account for many hyperfine componentsof the observed transitions. For NH2D, DCO+, N2D+, c-C3HDand C2D he quotes ratios of 0.00085, 0.012, 0.0064, 0.0475 and0.048, respectively. With the exception of C2D, this fractionation issignificantly lower than listed in Table 2. The new rates do givebetter agreement than the old for these DCO+, N2D+ and NH2Dratios (although, for NH2D, even the new rate gives a ratio al-most an order of magnitude too high), but the C2D/C2H ratio is

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286 H. Roberts, E. Herbst and T. J. Millar

Table 2. Comparison of molecular D/H ratios at the cyanopolyynepeak in TMC-1 with steady-state theoretical values [Tkin = 10 K,n(H2) = 104 cm−3].

Species Fractionation Modelledb

in TMC-1a Standard rates New rates

NH2D 0.0111 0.020 0.007HDCO 0.0592 0.055 0.013DCN 0.0111,2 0.025 0.005DNC 0.0152,3 0.015 0.003C2D 0.014 0.027 0.006C4D 0.0045 0.017 0.004DCO+ 0.021 0.062 0.016N2D+ 0.081 0.052 0.012c-C3HD 0.16 0.020 0.004DC3N 0.03–0.17 0.026 0.005DC5N 0.0138 0.026 0.006HDCS 0.029 0.046 0.012CH3OD 0.02652 0.030 0.009

aReferences: 1, Tine et al. (2000); 2, Turner (2001); 3, Guelin,Langer & Wilson (1982); 4, Millar et al. (1989); 5, Turner (1989);6, Bell et al. (1988); 7, Howe et al. (1994); 8, MacLeod, Avery &Broten (1981); 9, Minowa et al. (1997).bItalic face indicates agreement to within a factor of 3 betweenmodelled and observed ratios.

almost five times higher than the previous observation, and eighttimes higher than the new model prediction. Turner’s observedC3HD/C3H2 ratio is also 2–10 times higher than either of the modelspredict.

The observations made by Turner present a serious problem toall of the models of deuteration. N2H+ and N2D+ are thought tohave very simple chemistries, forming via proton/deuteron trans-fer from H+

3 and H2D+ to N2, and being destroyed by dissociativerecombination with electrons and reaction with CO. The very lowN2D+/N2H+ ratio observed by Turner would, therefore, imply thatsome process is acting to suppress the H2D+/H+

3 ratio in TMC-1. Ifthis were, indeed, the case, we could postulate that the fractionationin the other deuterated species in TMC-1 comes from the CH2D+

and C2HD+ ions. There is a route to fractionation of DCO+, for ex-ample, via CH2D+ + O, albeit at a reduced level, since S(CH2D+)is limited to ∼200 at 10 K using the standard fractionation reac-tion rates, and is only 40 with the new rates. However, the fact thatNH2D/NH3 is also so low is puzzling. Ammonia has one of thehighest proton affinities of any interstellar molecule, meaning thatit will react rapidly with species such as DCO+ and HCO+ to formNH+

4 and NH3D+. We then expect that the recombination of NH3D+

with electrons will produce significant amounts of NH2D (Vikoret al. 1999), and so it is difficult to see how the NH2D/NH3

ratio could be so much lower than DCO+/HCO+. It wouldbe interesting to see if there is any evidence that suchlow N2D+ and NH2D fractionation coexist in any othersources.

Whatever the actual molecular D/H ratios in TMC-1, it is be-coming increasingly apparent that this source is not the quiescent,dark cloud that has previously been assumed, and that a homo-geneous gas-phase chemistry may be too simplistic to model theprocesses occurring there. There are desorption mechanisms thatcan be important even at low temperatures (e.g. Willacy & Millar1998; Markwick, Millar & Charnley 2000), so it may be that gas–grain interactions cannot be neglected in dark clouds. Depletion ofheavy species on to grains would increase the predicted molecu-

lar D/H ratios (Roberts & Millar 2000b), an effect that we nowconsider.

2.2 The effects of depletion

As discussed above, the steady-state fractionation of H2D+, CH2D+

and C2HD+ is limited by the abundance of neutral species in the gasphase (

∑[Mi ]). In cold, dense clouds, however, gas-phase species

that collide with dust grains are likely to stick, and it is expected that,over time, dust grains will build up icy mantles, while the abundanceof heavy species in the gas will fall.

Equation (7) suggests that the removal of species such as COand O will lead to a further enhancement in the [H2D+]/[H+

3 ] ratio.Although this equation is formally correct only under steady-stateconditions, it is a reasonable approximation as long as changes inthe abundances of the species occur slowly. As

∑[Mi ] decreases,

S(H2D+) will rise. Secondary fractionation processes will then resultin an increase in all molecular D/H ratios for the period beforeeverything freezes out.

This theoretical expectation was confirmed by the detection ofvery large DCO+/HCO+ ratios in L1544 in clumps in which COis significantly depleted (Caselli et al. 1999). It has also been usedto explain the high abundances of deuterated species, in particu-lar the doubly deuterated species, NHD2, observed towards L134N(Roberts & Millar 2000b; Rodgers & Charnley 2001).

With the standard rate coefficients for formation and destructionof H2D+, and assuming that the electron abundance remains roughlyconstant, depletion of neutral species by factors of up to an order ofmagnitude or a little more will cause a direct increase in H2D+ frac-tionation (see equation 9). The enhancement factor can become verylarge, although it is limited by the electron abundance. In this limit,S(H2D+) reaches a value of 6 × 104 and leads to an [H2D+]/[H+

3 ]ratio of almost 2.

Now, assuming kf = 3.5 × 10−10 cm3 s−1 and kr = 7 × 10−15 cm3

s−1, the increase in fractionation as neutral species are depleted isstill limited by the electron abundance rather than by kr (see equa-tion 10). However, because H2D+ is forming more slowly, the maxi-mum enhancement factor is reduced to ∼104, and so the [H2D+]/[H+

3 ] ratio peaks at ∼0.3. We note that if the electron abundancewere significantly lower, or if H2D+ recombined more slowly withelectrons, then this ratio could become higher. The value of the ratecoefficient for the H+

3 dissociative recombination with electrons hasbeen debated for many years. In recent years, the so-called ‘large’rate obtained from storage ring experiments has gained popularity(Larsson 1995; Sundstrom et al. 1994), and is used in our networks,as is the analogous measurement of the recombination of H2D+

(Larsson et al. 1996). Recent experiments on varying source con-ditions for the storage ring experiments on H+

3 recombination indi-cate that the reaction may be a factor of ∼4 slower than previouslythought because of rotational excitation in previous experiments(Larsson, private communication). Similar experiments for H2D+

have not yet been carried out.If destruction of H2D+ by H2 occurs an order of magnitude more

quickly (kr = 7 × 10−14 cm3 s−1), then this, rather than destructionby electrons, is the term which limits the maximum enhancementof the [H2D+]/[H+

3 ] ratio (see equation 11). In this case, regardlessof the level of depletion, or ke[e−], the enhancement factor cannotexceed 5000, so that the maximum value of [H2D+]/[H+

3 ] is 0.15.These three scenarios are illustrated in Fig. 2, which shows the

evolution of [H2D+]/[H+3 ] and other, related, molecular D/H ratios,

for the three sets of reaction rates using the accretion model ofRoberts & Millar (2000b). By comparing the results in Fig. 2 with

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Figure 2. Evolution of selected molecular D/H ratios from accretion modelsusing standard rate coefficients for formation and destruction of H2D+ (top),compared with using rates of 3.5 × 10−10 cm3 s−1 for its formation andeither 7 × 10−15 cm3 s−1 (middle) or 7 × 10−14 cm3 s−1 (bottom) for itsdestruction.

those in Fig. 1, one can see that depletion becomes important at atime of∼1 × 105 yr, after which enhancements are seen in deuteriumfractionation for all three cases with respect to their purely gas-phasecounterparts.

These models do not include any desorption mechanisms, and soonce the time exceeds 2 × 106 yr, gas-phase abundances are typ-ically <10−20 with respect to H2. The drop in the D/H ratios asthe gas phase becomes bereft of heavy species is caused in the fol-

lowing manner. First, the H2D+/H+3 ratio drops off before reaching

steady state because the electron abundance increases. The electronabundance in turn increases because the H+ abundance increases(we assume overall charge neutrality and H+ is the dominant iononce the heavy species have frozen out). In undepleted gas, H+ isdestroyed by neutral species such as H2O, CH4, CH and OH, whoseabundances decrease sharply in the final stages of accretion.

3 A C O M PA R I S O N W I T H O B S E RVAT I O N S

Table 3 lists some of the molecular D/H ratios that have been ob-served towards dark clouds in the local ISM. TMC-1 was discussedin Section 2.1, but we include a few selected ratios here for compari-son with the other sources. NGC 1333 is included as the only sourceto date in which H2D+ has been directly measured. As mentionedabove, L134N and L1544 both show enhanced D/H ratios, whichindicate some level of depletion.

Table 3 also compares selected molecular D/H ratios predicted bysteady-state gas-phase and accretion models at selected times usingthe different rate coefficients shown in Table 1. Information at othertimes for both sets of models is shown in Figs 1 and 2. Model Arefers to the use of standard rates, model B to the use of new ratesbut with the slower of the two new possibilities for the backwardrate in reaction (1), while model C contains the faster of these twopossibilities. For the gas-phase case, models B and C yield the sameresults. The specific times chosen for accretion models A and C aredefined so as to obtain the best agreement for the doubly deuteratedisotopomer NHD2, which is detected in L134N (see below). As canbe seen in Fig. 2, the time for model C, 6 × 105 yr, corresponds to thepeak fractionation time, whilst the time chosen for model A, 3 × 105

yr, is well before this time. For accretion model B, results for twotimes are listed; the later time, 8 × 105 yr, is both the time of bestagreement for NHD2 and the time of maximum fractionation, whilethe earlier time, 4 × 104 yr, is included to show how sensitive thefractionation is to time in an accretion model. For the times shown,which are all in excess of 1 × 105 yr, the accretion models generallypredict molecular D/H ratios significantly higher than the steady-state gas-phase models. In addition to the calculated molecular D/Hratios, it is important in accretion models to consider the absoluteabundances of heavy gas-phase species. For the times in Table 3, COis depleted by factors of 10–100 from its steady-state abundance.

It is also interesting to note that predicted values for the ratioof atomic deuterium to atomic hydrogen for all models shown inTable 3 do not come close to reaching the high values (0.1–1.0)needed for models in which grain surface reactions produce largeabundances of deuterated molecules from atomic D (Caselli et al.2002). Atomic deuterium is destroyed more rapidly than atomichydrogen in the models, since it is also involved in fractionation re-actions (see, for example, table 1 in Roberts & Millar 2000a). One ofthese is the reaction OH + D ⇀↽ OD + H, which has an exoergicityof ∼810 K (Crosswell & Dalgarno 1985). Although the atomic D/Hratio is small, the abundance of atomic D at low temperatures anddensities is relatively large. Thus, the OD fractionation predictedby both the gas-phase and accretion models is high. OD/OH ratiosfrom the models that we have discussed are also listed in Table 3. Itmay be interesting to consider a grain surface chemistry using ODand OH as precursors, rather than D and H.

3.1 TMC-1 and NGC 1333

The accretion results are generally too high for TMC-1, which isreasonable since this source is not known for strong depletion of

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Table 3. A selection of molecular D/H ratios observed towards several interstellar sources, along with model predictions assumingTkin = 10 K, n(H2) = 104 cm−3.

Observationsa Model predictionsTMC-1b NGC 1333c L134N L1544 Gas-phase models Accretion models

Rates usedd A B, C A B CTime (yr)e Steady state 3 × 105 (4–8) × 105 6 × 105

H2D+/H+3 – 0.022 – – 0.104 0.023 0.543 0.145–0.359 0.108

DCO+/HCO+ 0.021 0.012 0.181 0.125 0.062 0.015 0.135 0.045–0.084 0.032N2D+/N2H+ 0.081 – 0.351 – 0.046 0.011 0.138 0.046–0.090 0.034DCN/HCN 0.0111 – 0.053 – 0.017 0.004 0.126 0.038–0.079 0.029NH2D/NH3 0.0111 – 0.11 0.136 0.020 0.006 0.106 0.047–0.088 0.038NHD2/NH3 – – 0.0054 – 1.5(−4) 1.4(−5) 0.005 0.001–0.003 5.0(−4)D/H – – – – 0.003 8.6(−4) 0.017 0.006–0.012 0.004OD/OH – – – – 0.358 0.093 0.304 0.104–0.147 0.063

aReferences: 1, Tine et al. (2000); 2, Stark et al. (1999); 3, Turner (2001); 4, Roueff et al. (2000); 5, Caselli et al. (1999); 6, Shah &Wootten (2001).bCyanopolyyne peak.cNGC 1333 IRAS 4A.d See text.eSee text for discussion of times for accretion models.

heavy neutral species. Using gas-phase model B, but including amodest depletion, where CO is 2–5 times lower than its steady-state abundance, would increase the ratios to the level observed inTMC-1. In addition, slightly higher depletions could account forthe somewhat higher deuterium fractionation seen for HCO+ in thecores studied by Butner et al. (1995).

For NGC 1333, the fractionation is far too small to require accre-tion models. Moreover, the new rate coefficients at 10 K used in apurely gas-phase model do give significantly better agreement withthe observed H2D+/H+

3 and DCO+/HCO+ ratios. However, NGC1333 is a young stellar object, and Stark et al. (1999) find that theH2D+ emission they observed arises from gas at a temperature of25–35 K. This paper has only considered chemistry at 10 K, but atthese higher temperatures we expect kr to increase, anyway, and be-gin to dominate the destruction of H2D+, suppressing fractionation(Millar et al. 1989; Roberts & Millar 2000a). The standard reactionrates for reaction (1) give good agreement with the ratios observedin NGC 1333 for Tkin = 20–30 K.

3.2 L1544 and L134N

The molecular D/H ratios predicted by the gas-phase models aregenerally lower than those observed towards L1544 and L134N, sothat some level of depletion must be assumed. With the standardrates (model A), the accretion model is in good agreement with theobservations of L1544, after ∼3 × 105 yr have passed. The samecan be said for model B at the later time of 8 × 105 yr. For L1544, amore detailed accretion model, including collapse, was calculatedby Aikawa et al. (2001), who obtained much higher densities thanutilized here and also found very high deuterium fractionation. Theeffect of increasing the density in our models would primarily be toshorten the time-scale for freeze-out, since species collide with andstick to the grains more frequently in a denser medium. It would notaffect the peak level of fractionation.

For L134N, accretion model A at a time of 3 × 105 yr showsexcellent agreement with observed D/H ratios including that fordoubly deuterated ammonia. Moreover, the degree of depletion atthis time is not very far advanced. Table 4 compares abundancesof a variety of non-deuterated species seen towards L134N withpredictions from accretion models A and B at the same timesas shown in Table 3. It can be seen that the results of model A

Table 4. Comparisonaof molecular abundances observed in L134N (Dickenset al. 2000) with values from accretion models A and B.

Observed Model A Model B3 × 105 yr 4 × 105 yr 8 × 105 yr

CO 7.0(−5) 1.79(−5) 1.23(−5) 1.61(−6)NH3 5.9(−8) 2.08(−8) 1.89(−8) 3.50(−9)C2H 4.0(−9) 1.19(−8) 9.68(−9) 4.90(−9)HC3N 8.7(−10) 2.90(−9) 3.77(−10) 2.38(−12)CH3OH 3.7(−9) 3.18(−11) 1.10(−11) 7.34(−13)N2H+ 6.1(−10) 3.51(−10) 3.17(−10) 3.93(−11)SO 3.2(−9) 3.41(−10) 2.74(−10) 3.69(−13)SO2 <1.6(−9) 1.90(−10) 1.77(−10) 2.78(−14)HCO+ 7.9(−9) 2.50(−9) 3.08(−9) 5.15(−10)CS 9.9(−10) 3.16(−9) 1.11(−9) 5.17(−12)

aItalic face indicates agreement to within a factor of 5.

match most of the observations to within a reasonable degree ofuncertainty.

For model B, the peak NHD2/NH3 abundance ratio (0.003, whichoccurs at 8 × 105 yr) is slightly lower than that observed in L134N,but, with the exception of N2D+, the predicted ratios for the otherspecies at that time are reasonably close to those observed. Theproblem is that, because the rate of formation of H2D+ has beenreduced by a factor of 5, a more severe depletion than in model Ais required to increase the molecular D/H ratios to the levels thatwere observed. Table 4 shows that the molecular abundances fornon-deuterated species predicted by model B, at 8 × 105 yr, are sig-nificantly lower than those observed in L134N such that only twoare in reasonable agreement with observation. For CO, for exam-ple, the predicted value is more than 40 times lower than what isobserved. At the earlier time shown, accretion model B is in rea-sonable agreement for non-deuterated species but shows too smalla degree of fractionation.

For model C, even the peak fractionation is too low to explainthe molecular D/H ratios that have been observed in L134N as wellas L1544. In particular, the peak calculated abundance ratio forNHD2/NH3 is an order of magnitude lower than observed in L134N.

Reducing the electron abundance, or the rate for recombinationof H2D+ with electrons, would allow S(H2D+) to increase further in

C© 2002 RAS, MNRAS 336, 283–290

Rate coefficients for deuterium fractionation 289

accretion model B (see Section 2.2). Our previous rates for the reac-tions H+

3 + e− and H2D+ + e− come from Sundstrom et al. (1994)and Larsson et al. (1996), respectively, but we have also carried outa calculation where we assume that the reactions occur four timesmore slowly than this. This does lengthen the time period wheremolecular D/H ratios are large, and increases their enhancement atlate times. For example, instead of peaking at 0.003 after 8 × 105 yr,the NHD2/NH3 ratio peaks at 0.013 after 106 yr, and best matchesthe observation in L134N (0.005) after ∼6 × 105 yr. However, theke[e−] term in equation (10) only becomes the limiting factor in de-termining H2D+ fractionation after substantial depletion of neutralspecies has already occurred. This means that, once the deuteriumfractionation has risen to the levels observed in L134N, this modelwill still predict gas-phase molecular abundances that are much toolow.

4 D I S C U S S I O N

We have examined the effects of new rate coefficients (Gerlichet al. 2002; Gerlich & Schlemmer 2002) for fractionation reactionsthat are important in modelling deuterium chemistry in interstel-lar clouds. In particular, we adopt newly measured ‘forward’ ratesat 10 K that are five times lower than the Langevin value for theformation of H2D+ and CH2D+, and ∼1.3 times lower than theprevious value for C2HD+. We also adopt new values for the back-ward rates, which do involve simulations since they are dependenton the fraction of molecular hydrogen in its ortho-form (Gerlichet al. 2002), and cannot be taken directly from laboratory values.The new forward and backward rates lead to lower molecular D/Hratios for all species in dark interstellar clouds, whether we utilizepurely gas-phase or accretion models for the chemistry.

These results could be affected by variations in the underlyingD/H ratio, but it is unlikely that such variations could reconcile allthe model results with the observations. Jenkins et al. (1999) andSonneborn et al. (2000) measured the range of D/H in the local ISMto be (0.74–2.2)×10−5; their highest value is less than twice as highas the ratio we use.

For some sources, particularly TMC-1, the consequences of thecalculated reduction in deuterium fractionation are not drastic; rea-sonable if somewhat worse agreement with observations was shown,and this could be improved by assuming slight depletion of heavyspecies. A similar case can be made to fit the DCO+ observationsof Butner et al. (1995) in a variety of cold cloud cores. This slightdepletion would occur shortly after a time of 105 yr in our simple ac-cretion model, or somewhat longer in gas–grain models that includenon-negligible desorption mechanisms.

For the perhaps unusual case of L134N, the situation is far worse.Although the degree of deuteration and the absolute abundancesin this source were previously well fitted by an accretion model(Roberts & Millar 2000b), this can no longer be said to be the case.In Table 3 the peak molecular D/H ratios from accretion modelsusing the new reaction rates are listed under labels B (at the latertime shown) and C, and these are generally lower than those ob-served in L134N. The discrepancy between observed ratios and theresults for accretion model B is not large given the uncertainties inobservations, but it must be remembered that this level of agreementonly occurs for a very short period of time near the peak calculatedD/H ratios. A lengthening of this time period can be achieved witha lowered electron abundance, or a lower rate for recombinationof H2D+ with electrons. However, whether or not the rate coeffi-cient for recombination is lowered, the times at which the deuteriumfractionation predicted by accretion model B can be said to be in

reasonable agreement with observation are so late that the absoluteabundances for all accreting species are generally far too low. Ifmodel C pertains, then the accretion model of deuterium fraction-ation can no longer explain even the ratios observed in L134N towithin a reasonable degree of uncertainty regardless of the overalldegree of depletion.

The situation regarding L1544 is nowhere near as dire since strongCO depletion is observed. More detailed model calculations for thiscollapsing source (such as that carried out by Aikawa et al. 2001)with the new rates utilized here are required.

5 C O N C L U S I O N S

Observations of molecular D/H ratios in dark clouds are currentlyused to probe physical conditions, such as temperature, cosmicray ionization rate and ionization fraction (e.g. Caselli et al. 1998;Williams et al. 1998), and to determine whether gas–grain interac-tions are significant (e.g. Caselli et al. 1999; Markwick, Charnley &Millar 2001). In star-forming regions, where ice mantles have evapo-rated and released the products of grain surface chemistry, deuteriumfractionation can provide valuable information on the temperaturesunder which the mantles formed and how species have been pro-cessed in the ice (e.g. Hatchell, Millar & Rodgers 1998; Hatchell,Roberts & Millar 1999; Ceccarelli et al. 2001). Deuterium fraction-ation has also been used to probe the connection between interstellarand cometary ices (e.g. Meier et al. 1998a,b). The underlying abun-dance of deuterium is an important cosmological parameter, but itis not easily determined outside of the local ISM; observations ofdeuterated molecules can be made, however, and chemical mod-els used to infer the underlying D/H ratio (e.g. Chin et al. 1996;Lubowich et al. 2000).

All of the above studies rely on our understanding of the mostbasic deuterium fractionation processes. As this paper has shown,the new and lower rate for reaction of H+

3 with HD at 10 K has castdoubt on this understanding.

To date, TMC-1 is the only source in which a large number ofdeuterated species have been observed, and it is becoming increas-ingly clear that this source is far more complex than was previouslyassumed. Observations of a range of molecular D/H ratios in otherdark clouds may be able to shed further light on gas-phase deuteriumchemistry.

As well as the implications for studies of gas-phase deuteration,this work could also have important consequences for studies ofgrain surface deuterium chemistry. The high fractionation in D2COtowards the protostellar sources IRAS 16293−2422 and 16293 E(Ceccarelli et al. 2001; Loinard et al. 2001) has been attributed tosurface hydrogenation/deuteriation of CO, which requires a ratherlarge fractionation in atomic deuterium; D/H ∼ 0.1–1.0 (Charnley,Tielens & Rodgers 1997; Caselli et al. 2002). Yet none of our modelsis able to achieve such a high atomic D/H ratio.

Finally, as the rate for the reaction of H+3 with HD is so important,

it is now crucial that this reaction be studied over the entire temper-ature range from 10 to 300 K. It would also be useful if a theoreticalunderstanding of the rate coefficients for reaction system (1) wereavailable.

AC K N OW L E D G M E N T S

EH and HR acknowledge the support of the National Science Foun-dation (USA) for their research in astrochemistry. Astrophysics atUMIST is supported by PPARC.

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