4.1. Steady state
Fig. 3 indicates why the gas-phase elemental abundance of sulfur plays such an important role in bistability, especially at high densities. Moderate to high abundances of sulfur lead to moderate electron abundances; it is these moderate electron abundances that are most conducive to bistability. If the electron abundance is too large, as tends to happen with "high metal" elemental abundances (Table 1), bistability cannot exist. A perusal of some "high metal" solutions (Lee et al. 1996) shows that although the abundances of molecular ions such as are driven to lower levels by dissociative recombination, they are still somewhat greater than the abundances of the atomic ions, such as . In this and in other ways (such as the C abundance), the "high metal" solutions are more akin to the LIP solution in the bistability region despite their high ionization fraction. If the electron abundance is very low, which occurs (except at very low gas densities) when "low metal" abundances (including low sulfur) are used, then the molecular ions strongly dominate. Starting with "low metal" elemental abundances at moderate to high densities, and adding sulfur increases the fractional ionization gently, driving down the molecular ionic abundances until a regime is reached in which both the HIP (fractional ionization greater; atomic ions more prominent) and the LIP (fractional ionization smaller; atomic ions less prominent) can seemingly coexist. (Remember that bistability does exist for "low metal" solutions with the sulfur abundance in Table 1 or even with no sulfur at all, but the region of bistability lies at very low densities.) If the sulfur elemental fractional abundance is increased past a value of 3.5 10-6 while holding other abundances at the "low metal" values, then bistability can no longer occur, presumably due to too high an ionization fraction. Bistability at low densities can exist with significantly higher sulfur elemental abundances, since it exists for the primary "dense core" abundances; here, presumably, the excess sulfur is balanced in some manner by the low C/O abundance ratio.
The role of sulfur in providing regimes of moderate ionization seems central to bistability. Can other elements take the place of sulfur? Certainly, at very low densities (right side of Fig. 3), there is sufficient ionization from elements such as C that no sulfur at all is needed for bistability. At higher densities, we have varied the abundance of Si and of the metals Na, Mg, and Fe to determine the effect on bistability. We find that an increase of one order of magnitude of the true metals from their "low metal" abundances permits bistability if and only if the sulfur fractional abundance is kept below 8 10-7. In addition, the range of bistability is decreased relative to the low metal models and is shifted to somewhat higher densities. No truly high density regions of bistability exist. An increase of two orders of magnitude, as noted above, (which brings the metallic abundance to their "high metal" values) kills bistability completely. If one starts with the "low metal" abundances, and increases the abundance of Si, a similar effect occurs to an increase in the true metals; Si is also more efficient at ionization than is S.
The specific results for the high density LIP and HIP solutions in Table 2, obtained with "low metal" abundances except for a sulfur fractional abundance of 2 10-6, show that neither phase exhibits large abundances of polyatomic molecules. The abundances of some molecules are higher in the HIP and of others are higher in the LIP. Those higher in the HIP tend to be small carbon-bearing radicals. In addition, for many molecules, the differences are quite small (a factor of or less). The phases are, as usual, distinguished by differences in ionization; the HIP level of ionization is due mainly to ionized sulfur. A major result is that the steady-state fractional abundance of atomic C in the HIP is high (2 10-5 with respect to H2), despite the high density of the gas, for normal values of . Such a solution may indeed be applicable to observations of large abundances of C in some physical situations. In particular, Schilke et al. (1995) have shown that at least some fraction of the atomic C detected towards TMC-1 is not just in an outer boundary layer and is reasonably explained by an HIP solution, albeit at densities only up to = 2000 cm-3. The spatial regions modelled by the HIP phase would then not be the higher density region where the large molecules tend to have peak abundances (e.g. TMC-1(CP)). The solutions listed in Table 2 refer to a higher density region (assuming a standard cosmic-ray ionization rate); nevertheless, it is instructive to compare the solutions with some observed abundances.
In Table 3 we have listed observed fractional abundances in the dark clouds TMC-1 and L134N for some small molecules as well as the complex species HC3 N and HC7 N and, in the case of TMC-1, atomic C. We have also listed the analogous calculated abundances for the steady-state (SS) HIP and LIP solutions in Table 2. In past work, we have found that discriminators between the HIP and LIP solutions include the simple carbon-bearing radicals (CH, C2 H, C3 H), sulphur-containing molecules (CS, SO, SO2) and molecular ions such as and , in the sense that abundance differences between the two phases are usually noticeable. In addition to these species, we have also included NO, OH, and NH3 in Table 3. As can be seen in the table, neither set of steady-state solutions to the new neutral-neutral model network represents the small molecule abundances observed in TMC-1 or L134N particularly well. The strongest argument in favor of the HIP solution remains the high atomic carbon abundance in TMC-1 observed by Schilke et al. (1995); this and other solutions favoring the HIP are listed in boldface in Table 3. Although the HIP shows higher abundances of the carbon-bearing radicals (CH, C2 H, C3 H) than does the LIP, presumably due to the influence of atomic C, the calculated abundances are still below the observed values. For the other molecules on the list, only perhaps SO is better fit by the HIP phase since the LIP phase overproduces it.
Table 3. Comparison of steady-state (SS) and early-time (ET) HIP and LIP abundances with respect to H2 with observed fractional abundances in TMC-1 and L134N.
Unfortunately, calculated molecular abundances are model and time dependent; the abundances of polyatomic molecules are generally higher in the new standard model, especially at early time. Also, we have only listed the results of one bistable model; many others are available. Still, the qualitative conclusion that the abundances of many polyatomic molecules are low in bistable regions seems secure.
The results of the new standard model generally show smaller regions of bistability, proving that the phenomenon is, at this stage of understanding, dependent on model network. The region of bistability is particularly reduced for elemental abundances based on the "low metal" abundances (Table 1). In fact, with these primary abundances, no bistability is seen at all. Even when the sulfur abundance is increased, bistability is seen only if the C/O elemental abundance ratio is set lower than the 0.4 value of the "low metal" case (in analogy with the "dense core" abundances). In the cases where bistability is seen, its range in terms of is considerably smaller than with the new neutral-neutral model. There is no simple explanation for the difference between networks in terms of differing fractional ionizations.
Ruffle et al. (1997) have recently pointed out that depletions of elements such as C, N, and O from the range of values considered here can actually increase abundances of complex molecules such as HC3 N at late times. With depletions for C, N, and O of a factor of 5 from the low metal abundances, we confirm this effect, and, using the new neutral-neutral model, note that the bistability at 10 K vanishes.
4.2. Time dependence
No mention so far has been made of early-time bistable solutions in our time-dependent calculations. Early-time solutions show large enhancements for large molecules with "low metal" abundances and with atomic starting conditions for all species other than H2 (Herbst & Leung 1989; Lee et al. 1996). We have looked at early-time solutions in some detail. For solutions showing bistability at steady state, there are no large early-time enhancements in either the LIP or the HIP with our given atomic and molecular initial abundances. The situation is depicted in Fig. 4 for the molecule HC3 N with the new standard model and the "dense core" abundances. With a value of low enough (4 10-21 cm3 s-1) such that only a low ionization solution is found, it can be seen that a large early time solution exists for atomic initial conditions but not for molecular initial conditions (high in CO and O2) since in the latter case, no free carbon is available to make complex species. Both solutions are the same at steady state. When is increased into the bistable region (8 10-21 cm3 s-1), atomic initial conditions lead to the HIP and molecular initial conditions lead to the LIP; both sets of solutions show no large peak at early time although the atomic initial conditions lead to a small early-time enhancement. When is increased still further (2 10-20 cm3 s-1) so that only a high ionization solution can be reached, both atomic and molecular initial conditions show no early-time enhancement at all.
The slight enhancement for HC3 N at early time in the bistable region when atomic initial conditions are used suggests that the excess C in this solution is indeed feeding through partially to carbon-bearing species, and that carbon-bearing species might in general be more abundant at early time. For the species listed in Table 3, we have also tabulated their calculated early-time abundances for solutions leading to both the HIP (atomic initial conditions) and LIP (molecular initial conditions). The model used is the same one used to produce Table 2; viz, the new neutral-neutral model at high density with a standard , plus "low metal" abundances except for enhanced sulfur. It can be seen that there is an enhancement in the abundances of the carbon-bearing species relative to the steady-state HIP values. With the enhancement, the agreement between these species and observation in both TMC-1 and L134N improves somewhat, but is still not as good as obtained with that obtained using low ionization models outside of the zone of bistability (see Fig. 3).
We have also considered the time-dependent chemistry for bistable solutions starting from a variety of initial abundances in between our standard atomic and molecular initial conditions used to derive the results in Tables 2 and 3. We reemphasize the point that we are not varying the elemental abundances but are varying the initial atomic and molecular forms in which the "low metal" + high sulfur elemental abundances are grouped. There are two issues here to be investigated: (a) which sets of initial abundances lead to which phase at steady state, and (b) whether or not early-time abundances can be significantly changed by changing the initial abundances.
Regarding issue (a) we have found the initial presence of at least some molecular oxygen to be a necessary condition for an LIP solution at steady state. Specifically, such a solution is obtained even for initial abundances rich in heavy atoms except oxygen, if 82.3% or more of the oxygen is molecular. For initial abundances with molecules other than O2 such as CO and/or N2, the necessary amount of O2 to achieve an LIP solution can be significantly smaller. If all C is initially in the form of CO and N in the form of N2, the LIP is obtained with at least 2% of the remaining O in the form of O2. If 10% of the carbon is in the form of C and 90% in the form of CO, it takes 12% of the remaining oxygen in the form of O2 to obtain the LIP solution.
Regarding issue (b) we have found the early-time abundances to be very sensitive to changes in the initial abundances, as previously discussed by Flower & Pineau des Forêts (1996). Despite the complexity, a large initial abundance of atomic C generally leads to some enhancement in the early-time abundances of carbon-bearing species, as is to be expected. To give a flavor for the variability obtainable in early-time abundances, we have considered two sets of initial conditions close to the so-called "separatrix", which divides those sets of initial conditions leading to different (bistable) results. Specifically, we started with initial abundances rich in atoms with the exception of oxygen, for which we used 82% (leading to the HIP) and 83% (leading to the LIP) respectively in the form of molecular oxygen. Initial conditions close to the separatrix are known to increase the time needed to achieve bifurcated results. Some early-time abundances from these two sets of initial conditions are shown in Table 3; the columns for these particular solutions are labelled by primes and the designations HIP and LIP, which refer to the final results. Although the two solutions do bifurcate later into the standard LIP and HIP solutions, they show very little difference at all at early time (5.6 104 yr). In both of the early-time solutions, there are moderate enhancements in the abundances of carbon-bearing species, due presumably to the high abundance of C in both phases. As regards the agreement with observed abundances in TMC-1 and L134N, these high density, early-time solutions do not lead to sufficient cyanopolyynes, carbon-bearing species, and ammonia, but do yield high atomic C values.
A final point of discussion concerns the effect of including dust chemistry on the phenomenon of bistability. This has already been discussed in the literature by Shalabiea & Greenberg (1995) and by Le Bourlot et al. (1995b). In the interim, we have confirmed the result of Tielens (lecture, 1995; see also Charnley et al. 1997; Tielens & Charnley 1997) that the rate equations used in standard models of interstellar dust chemistry do not adequately take the finite size of dust particles into account. We have modified the rate equations (Caselli et al. 1998), and tested their effects on standard gas-grain models (Shalabiea, Caselli, & Herbst 1998). The question of bistability can now be tackled.
© European Southern Observatory (ESO) 1998
Online publication: June 2, 1998