3 Cold clouds

Figure 1: Predicted gas-phase molecular abundances vs. time, from models P1 (left) and P2 (right). T = 10 K; n(H2) = 104 cm-3.

Dark, quiescent clouds, such as TMC-1 and L134N, have temperatures $\sim$10 K and H₂ densities of $\sim$10⁴ cm^-3 (Pratap et al. 1997; Dickens et al. 2000). Figure 1 shows the evolution of selected gas-phase abundances from gas-grain models under these conditions. Only grain-surface chemistry on olivine is considered here, since experiments show that diffusion on amorphous carbon occurs so slowly at 10 K that the chemistry will not proceed. As well as H₂O and O₂, we have plotted the abundances of CO, NH₃ and HC₃N over time, since one of the observational constraints is that the low H₂O and O₂ abundances inferred by SWAS are towards regions where significant abundances of other molecules are present. Carbon monoxide is included as the most abundant interstellar molecule (after H₂) and provides some measure of the overall depletion, while cyanoacetylene (HC₃N) is included as a representative of organic molecules.

The major differences between models P1 and P2 occur at times >10⁷ yr. The slow diffusion rates for all species in model P2 mean that grain surface chemistry occurs slowly and there are more small species on the grain surfaces which desorb more easily. Overall, therefore, freeze-out is not so efficient at late times. Even though both accretion and desorption mechanisms act in these models, a steady-state chemistry is not reached. The abundances of many gas-phase species, including CO, H₂O and O₂, rise until somewhere between 10⁵ and 10⁶ yr, at which time freeze-out begins to dominate and their abundances decline. With such behaviour, molecules such as O₂, that take a significantly long time to reach a high abundance in gas-phase models, are never able to reach this abundance in gas-grain models. A closer look at the temporal dependence of the O₂ abundance shows a peak that lasts a relatively short time.

Let us look in somewhat more detail at the processes governing the abundances of gas-phase water and molecular oxygen. Water is formed in the gas phase mainly through a sequence of reactions starting with that between atomic oxygen and the ion H₃⁺ and ending with the dissociative recombination of the ion H₃O⁺. Meanwhile, a far larger amount of water is formed on grain surfaces via hydrogenation of O atoms by H atoms. Molecular oxygen is formed in the gas phase via a variety of neutral-neutral reactions (e.g. O + OH) and on the surfaces of grains via recombination between surface O atoms. Once accreted onto dust grains, water is inefficiently desorbed in our models via both cosmic ray desorption and thermal evaporation (Fraser et al. 2001). Surface molecular oxygen is desorbed at about the same slow rate as CO via cosmic rays (Hasegawa & Herbst 1993) and is actually depleted more rapidly by reactions in model P1, so that even less returns to the gas than in model P2.

The increase in gaseous abundance followed by a decrease when accretion becomes important is not universal. The NH₃ abundance, for example, remains relatively high (10^-7 with respect to H₂) until $3\times10^7$ yr in model P1 and at least 10⁸ yr in model P2, while for HC₃N the accretion actually causes a secondary "late-time'' abundance peak of (2-5) $~\times~10^{-9}$ after $\sim$10⁷ yr. This result, which holds for all cyanopolyynes, was first noted by Ruffle et al. (1997).

Looking specifically at water and molecular oxygen, we see that H₂O falls below the limits set by SWAS towards cold clouds for times >10⁶ yr, at which times the O₂ abundances are also well below the observed upper limits. A simple explanation of the SWAS results, therefore, lies in adopting time scales somewhat larger than usually considered in gas-phase models.

The timescale for accretion in gas-grain models is largely dependent on the density, which governs the number of collisions between gas-phase molecules and grains, and on the so-called "sticking coefficient'', the probability that a molecule colliding with a grain will freeze onto it. In the models discussed here, we have adopted a sticking coefficient of 0.5, but we have investigated the effects of varying its value between 0.1 and 1. For times up to a few times 10⁵ yrs, the results are the same. At later times, the higher the sticking coefficient, the more rapidly the gas-phase species freeze out and the less pronounced the secondary peak in the cyanopolyyne abundances. However, for times <10⁷ yr, varying the sticking coefficient does not change the abundances of the species discussed here by more than an order of magnitude.

In the following sections we compare our model results with observations of the sources TMC-1 and L134N, which have previously been studied in some detail by ground-based telescopes, and investigate whether the timescales necessary to reproduce the SWAS results are reasonable.

3.1 TMC-1

Figure 2: The left-hand axis and solid line with triangles represent the percentage agreement for 43 species, to within a factor of 10, between a pure gas-phase chemistry (T=10 K, n(H2)  =104 cm-3) and the TMC-1 observations of Ohishi et al. (1992) and Ohishi & Kaifu (1998). The right-hand axis and the solid, dashed and dash-dotted lines represent the ratios of the predicted abundances of H2O, O2 and CO, respectively, to the limits set/observed abundances towards TMC-1.

In order to illustrate that the limits on H₂O and O₂ abundance set by SWAS were surprisingly low, Fig. 2 shows how the predicted abundances from a pure gas-phase chemistry (the so-called new standard model; see Terzieva & Herbst 1998) match observations of the "cyanopolyyne peak'' in TMC-1 as a function of time. The comparison for H₂O, O₂, and CO is plotted explicitly, while other species are considered together. As noted by previous authors (Terzieva & Herbst 1998; Bettens et al. 1995), best agreement for most species is at "early time'' ($\sim$10⁵ yr). Here, almost 80% of the abundances observed by Ohishi et al. (1992) and Ohishi & Kaifu (1998) are matched by the gas-phase model to within an order of magnitude. However, although the O₂ abundance is well below the upper limit set by SWAS, the water abundance is at least ten times too high. Once steady state is reached, the overall agreement has dropped to $\sim$20%, and both O₂ and H₂O abundances are 7-8 times higher than the observational limits.

Figure 3: As in Fig. 2, but from a gas-phase model using a C/O elemental ratio of 1.

Bergin et al. (2000) found that an increase in the elemental C/O ratio from the solar value of 0.4 to >0.9 allows H₂O and O₂ abundances to agree with the SWAS observations at later, more relevant times. Terzieva & Herbst (1998) showed that varying the C/O ratio in their gas-phase model affects the agreement with observations of TMC-1. In particular, they found that the time of best agreement tends to increase with increasing C/O ratio, but presented no specific predictions for H₂O and O₂.

Figure 3, therefore, shows the agreement over time for the gas-phase model in which elemental oxygen has been depleted so that $\rm C/O=1$. Again, the agreement for H₂O, O₂ and CO is plotted explicitly, while the percentage agreement is given for the less abundant species. The time of best agreement (nearing the 80% level) is now between $5\times10^6$ and 10⁷ yr, when O₂ and CO are also in good agreement with the observations, and the H₂O abundance has dropped slightly below the SWAS limit.

Figure 4: Top: The percentage agreement of a gas-grain chemistry (T=10 K, n(H2)  =104 cm-3) with the observations of Ohishi et al. (1992) and Ohishi & Kaifu (1998). BOTTOM: The ratios of the predicted abundances of H2O, O2 and CO to the limits set/observations made towards TMC-1.

Experimental evidence does suggest that H₂O is more tightly bound to grains than CO, and this might provide some explanation as to why there should be a differential depletion between carbon and oxygen. Nevertheless, a more realistic treatment should provide some reason for depletions as a function of time. Earlier models of TMC-1 (e.g. Ruffle et al. 1997; Markwick et al. 2000; Hartquist et al. 2001) have included accretion onto and desorption from grain surfaces. Our gas-grain model does provide a detailed description for differential depletion rates through cosmic ray desorption (Hasegawa & Herbst 1993), which yields varying desorption rates for individual species based on the strength of their attachment to grain particles. The physics of this mechanism for desorption is, however, treated only in an approximate fashion, and the desorption rates may not be sufficiently accurate for our purposes.

RH1 note that the overall level of agreement between the gas-grain models with initial low-metal gas-phase abundances and the abundances of the gas-phase molecules detected in TMC-1 does not decrease monotonically from the early-time peak, as occurs in the oxygen-rich gas-phase model, but shows a second peak of considerable agreement between a few times 10⁶ and 10⁷ yr. This is illustrated in the top panel of Fig. 4, for gas-grain models P1 and P2. Model P2 is in better agreement than P1 at late times because gas-phase abundances are generally higher (see Fig. 1). Although cosmic ray desorption undoubtedly results in a C/O gas-phase elemental abundance ratio that increases with time, the effect does not reproduce the late-time results in Fig. 3.

The bottom panel of Fig. 4 compares the H₂O and O₂abundances from the models with the upper limits set by SWAS, and also compares the modelled and observed CO abundances. Interestingly, the predicted amount of O₂ is almost always below the limit observed by SWAS. However, the low H₂O limit is only reproduced for times $\geq$2- $3\times10^6$ yr, at which time the CO abundance is at least 2-3 times lower than is observed, and decreases monotonically with increasing time. At a time of $1 \times 10^{7}$ yr, where model P2 shows a secondary peak for overall agreement with observation, the predicted CO abundance is $\sim$6 times too low. This level of agreement is probably acceptable given observational uncertainties, although the deviation becomes greater for still later times.

   

Table 1: A comparison of molecular abundances observed towards TMC-1 with predictions from gas-grain models at 10 K.

	Observation	Model P1	Model P2
		( $3\times10^6$ yr)	(107 yr)
H2O	<7.0(-8)1	2.8(-8)	4.0(-8)
O2	<3.2(-6)2	1.4(-7)	4.2(-8)
CO	8.0(-5)3	1.5(-5)	1.3(-5)
NH3	2.0(-8)4	4.3(-8)	1.2(-7)
HC3N	6.0(-8)4	1.5(-9)	4.8(-9)

Note: a(-b) implies $a\times10^{-b}$
Refs.: ¹ Snell et al. (2000b), ² Goldsmith et al. (2000), ³ Ohishi et al. (1992), ⁴ Ohishi & Kaifu (1998).

In Table 1, we compare observations with gas-grain model predictions for the five important species shown in Fig. 1 at the secondary times of best general agreement for models P1 and P2 (see Fig. 4). There is reasonable agreement for four of the species, including CO, but the predicted HC₃N abundance is more than 13 times lower than the observation. This poor agreement is not any better at early times; it derives from the recent order-of-magnitude increase in its estimated abundance based on hyperfine analysis (Ohishi & Kaifu 1998). We also note that the predicted H₂O abundances are only just below the SWAS upper limit; if this limit were to be revised downwards by any significant degree, we would require later times at which CO and HC₃N abundances would need to be even further depleted. Finally, the general agreement at these times is not as good as it is at the early-time peak in two senses: (1) fewer molecules are fit to within an order of magnitude, and (2) some molecules, such as most sulphur-bearing species, are off by several orders of magnitude.

If we vary the sticking coefficient of 0.5 used in the models, the level of best agreement for most species is still $\sim$80%, after 10⁵ yrs, but the secondary peak in the percentage agreement becomes less pronounced the higher the value of the sticking coefficient. With a sticking coefficient of 0.1 the overall agreement reaches 60-70% after 10⁷ yrs, but if the sticking coefficient is 1, the secondary peak, between $3\times10^6$ and 10⁷ yrs, is only 45-55%, a range close to that of our standard models. Thus, with a sticking coefficient of 0.1, both models P1 and P2 are in reasonable agreement with the observations of TMC-1 at a time of 10⁷ yrs. The H₂O and O₂ abundances lie below the SWAS upper limits, while the predicted CO and NH₃ abundances are within a factor of 2 of what is observed. The predicted HC₃N abundance, however, is still more than an order of magnitude too low.

In summary, our current gas-grain models do not optimally match all the observations of TMC-1 at one time, although a rather unphysical, low sticking coefficient results in better agreement than does our standard model. Considering that gas-phase models with $\rm C/O=1$ are able to match the observations fairly accurately at a single time, the discrepancy between observations and the standard gas-grain models may be due to deficiencies in these models, especially in the differential desorption rates. But, then again, the discrepancy may also be astronomical in origin. For example, different parts of TMC-1 may have different evolutionary times and the observations may not be sufficiently resolved to deconvolute contributions from cloud parts with differing ages.

There is some observational evidence that different parts of the TMC-1 cloud are indeed at different evolutionary stages. Several previous studies have noted the chemical differentiation between the "cyanopolyyne'' (CP) and "ammonia'' peaks (e.g. Olano et al. 1988; Pratap et al. 1997), which are separated by $\sim$0.32 pc, or 7.9 arcmin, only somewhat larger than the SWAS beam. Cyanopolyynes have their peak abundances towards the former, whereas emission from ammonia is brightest at the latter. A variety of explanations for this variation have been given (e.g. van Dishoeck et al. 1993; Bergin et al. 1996; Markwick et al. 2000), but perhaps the simplest, based on pure gas-phase models, is that the NH₃-peak is chemically more evolved than the CP-peak (Saito et al. 2002). To fully explain the SWAS results might require a convolution of predicted abundances throughout TMC-1 (Lee et al. 1996b).

3.2 L134N

Figure 5: Top: The percentage agreement of gas-grain chemical models (T=10 K, n(H2)  =104 cm-3) with the 27 species observed by Ohishi et al. (1992) and Dickens et al. (2000). Bottom: The ratios of the predicted abundances of H2O, O2 and CO to the limits set/observations made towards L134N.

If we now compare the results of our gas-grain models with ground-based observations of L134N (Fig. 5), we find that the early-time solution is no longer clearly preferred. This result is in agreement with pure gas-phase model calculations, in which a C/O elemental abundance of unity does not have the same effect as in TMC-1 (Terzieva & Herbst 1998). Instead, the time of best agreement for both models P1 and P2 is at 10⁶ yr, although the peak percentage agreement is not as high as the early-time peak in TMC-1. A few predicted abundances at the time of peak agreement in L134N are listed in Table 2 for comparison with the observations.

   

Table 2: A comparison of molecular abundances observed towards L134N with predictions from gas-grain models at 10 K.

	Observation	Model P1	Model P2
		(106 yr)	(106 yr)
H2O	<3.0(-7)1	8.6(-8)	3.2(-7)
O2	<3.4(-6)2	4.2(-6)	5.9(-6)
CO	8.0(-5)3	5.0(-5)	5.1(-5)
NH3	2.0(-7)4	6.1(-8)	7.3(-7)
HC3N	2.0(-10)4	3.9(-10)	5.1(-10)

Note: a(-b) implies $a\times10^{-b}$
Refs.: ¹ Snell et al. (2000b), ² Goldsmith et al. (2000), ³ Ohishi et al. (1992), ⁴ Dickens et al. (2000).

As can be seen in Table 2, NH₃ and HC₃N abundances from both models are within a factor of 4 of the observations at 10⁶ yr. The predicted H₂O abundance from model P2 is similar to the upper limit seen in L134N, but the O₂ abundance is almost twice as high, while in model P1 the H₂O abundance is $\sim$3 times lower than the upper limit and the O₂ abundance is similar to the SWAS limit. These abundances are falling rapidly, so that after $2\times10^6$ yr the predicted amount of O₂ is $\sim$10 times lower than the SWAS limit, while H₂O is between 2.5 and 7 times lower, depending on the model.

Although the CO abundance is a few times lower than observed towards L134N by a time of $2~ \times~ 10^{6}$ yr, it is in reasonable agreement, given the uncertainties inherent in obtaining accurate CO abundances observationally. We note, again, though, that if the upper limits on H₂O and O₂ abundance were more stringent, then matching the observations would require later times and, therefore, a lower CO abundance. In addition, the general agreement would be lower.

3.3 Ice observations

   

Table 3: A comparison of solid-state observations towards Elias 16 (Nummelin et al. 2001) with predictions from gas-grain models at 10 K.

	Observed	Model P1		Model P2
		105 yr	106 yr	105 yr	106 yr
H2O	$\sim$10-4	4.0(-5)	1.8(-4)	4.2(-5)	1.8(-4)
CO	26%	12.1	26.7	12.2	39.2
CO2	23%	0.2	0.3	3.3(-3)	0.1
CH3OH	<3.2%	1.5	2.8	2.0(-4)	9.5(-4)

Notes: a(-b) implies $a\times10^{-b}$; H₂O abundance is relative to H₂, the other ice abundances are listed as a percentage of H₂O.

Figure 6: Predicted abundances of grain surface species vs time, from models P1 (left) and P2 (right). T = 10 K; n(H2)   = 104 cm-3.

Can the hypothesis that large portions of TMC-1 and L134N are at late times chemically be constrained by ice observations? One basic problem is that there is no suitable infrared continuum source behind either cloud core to allow absorption spectra of ice mantle components to be made. So, for the majority of ice components, one must consider the analogous source in front of the field star Elias 16, which lies behind the Taurus molecular cloud. The observed abundances or upper limits are listed in Table 3 for four species along with theoretical values at both early and late times, while theoretical values are plotted as a function of time in Fig. 6. What must be explained by theory are a large abundance of water ice, equal to much of the elemental oxygen abundance, amounts of CO and CO₂ ice equal to roughly 1/4 the water ice abundance, and a low upper limit for methanol ice.

In both models P1 and P2, the water ice abundance simply increases over time as oxygen atoms accrete onto the grains and are hydrogenated. To reach the observed level requires $\sim$10⁶ yr, although a significant fraction of this level is reached somewhat earlier in time. The observed abundance of CO ice is fit well by both models P1 and P2 at the late times listed in the table, but the predicted abundance is not significantly smaller at somewhat earlier times. We conclude that the comparison between theory and observation for these two ices gives some support to the late time hypothesis.

The O₂ ice abundance is not well determined, with only a high upper limit of 20% with respect to water ice, a limit set by Vandenbussche et al. (1999) towards NGC 7538 IRS9. Both models P1 and P2 show very low abundances of molecular oxygen ice, although the abundance of O₂-ice becomes much higher in model P2, where the diffusion of all species is slowed. This may seem counterintuitive, since slowing the diffusion of heavy species slows the rates of reaction between them. However, the majority of O₂ in model P2 comes from accretion, and so is not greatly affected by any surface reactions. In model P1, it is the destruction of O₂ by O to make O₃, or by H in the series of reactions:

$${\rm O_2 \chemlk{H} HO_2 \chemlk{H} H_2O_2 \chemlk{H} H_2O + OH, }$$ (1)

making water ice, which keeps its surface abundance at 10^-11 or less, with respect to H₂.

The abundance of methanol ice is low towards Elias 16, and this low abundance is easily reproduced by both models at all times. Note that CH₃OH ice has a much lower abundance in model P2 than P1, because it is produced by hydrogenation of CO on grain surfaces, a process which occurs much more slowly in model P2.

The only major problem with these models is the under-production of CO₂. This was investigated and discussed in detail by Ruffle & Herbst (2001b), who found that, as long as diffusion rates are relatively rapid (model P1), a large CO₂-ice abundance can be obtained by a slight increase in either temperature (from 10 to 12 K) or H₂density (from 10⁴ to 10⁶ cm^-3). The large CO₂ ice abundance can be achieved by $3 \times 10^{5}$ yr, however, so that CO₂ is not a good indicator of the age of the source.

Thus, in conclusion, ice abundances offer some evidence in favor of source ages equal to or in excess of 10⁶ yr, although a source age of $3 \times 10^{5}$ yr or even somewhat younger is certainly not ruled out.