3 Steady state chemical models and bistability

3.1 General considerations

The same gas phase chemical network has been used for determining the steady state solutions of the atomic and molecular species in the gas phase. This is performed by solving d$\vec{X}$/dt =0 where $\vec{X}$ is a vector representing the abundances of all the gas phase species considered in the chemical network. No surface reaction, except for the formation of H₂, has been introduced. Le Bourlot et al. (1993) have shown that the intrinsic non linearities present in the chemical equations may lead to a non linear behaviour of the phase space of the chemical solutions. For a given value of the control parameters (density, temperature, cosmic ionization rate, elemental abundances, etc.), the steady state solutions depend on the initial conditions, (on the atomic, ionic and molecular fractional abundances of oxygen, carbon, sulphur etc.). Then, two stable and one unstable steady state solutions can appear for a given set of control parameters, corresponding to an identical physical state of the gas. This behaviour is known as bistability and is a well known property of open thermodynamical systems. The two stable states have very different chemical properties. The Low Ionization Phase (LIP) solution corresponds to the standard solution found in most chemical models where the fractional ionization is low ( 10^-8 - 10^-7) and is provided mainly by molecular and metallic ions. On the other hand, the High Ionization Phase (HIP) solution involves a larger fractional ionization ( 10^-5 - 10^-6) dominated by atomic ions (H⁺, C⁺, etc.). Comparison between the LIP solutions and the time-dependent results obtained in the previous section (in the FR = 0 model), integrated to late times leads to identical results, within the numerical accuracy. In fact, the steady state solutions depend only on the $n_{\rm H}$ to $\zeta$ ratio (Lee et al. 1998). So, we may display the stable steady state solutions for two different values of the cosmic ionization rate on the lower and upper abscissae axis by simply scaling the proton density abscissae axis by the same factor as the cosmic ionization rate if the temperature is assumed to be identical. Figure 5 shows the stable steady state solutions found for molecular oxygen and water within a range of densities between 10²and 10⁵ cm^-3 with a fixed temperature of 10 K for a cosmic ionization rate of 1.3 10^-17 s^-1 on the bottom x axis whereas the same values correspond to a range of densities comprised between 400 and 4 10⁵ cm^-3 for a cosmic ionization rate that is four times larger, as shown on the upper x axis. Figure 5a displays the results obtained with the standard depletion of oxygen whereas Fig. 5b refers to the non standard depletion of oxygen (see previous section).

The two cases show a specific range of densities where bistability does occur. The LIP phase corresponds to the high density regions and leads to large fractional abundances of molecular oxygen and water. Such a result is in agreement with the various model results found in the literature (cf. Bergin et al. 2000). However, we find a range of densities where a high ionization phase is simultaneously present: the range is between 200 and 1000 cm^-3 with the standard depletion of oxygen (respectively 800-4000 cm^-3 for a cosmic ionization rate of 5.2 10^-17 s^-1) whereas it is shifted in the range 1300-5200 cm^-3 (respectively 5200-16000 cm^-3 for a cosmic ionization rate of 5.2 10^-17 s^-1) with the non standard depletion of molecular oxygen. This latter situation offers a possible solution to the SWAS observations if the initial molecular state of the cloud (and the control parameters) are such that a HIP solution is obtained.

Since the beam of the SWAS satellite is as large as 4 arcminutes, it should also be kept in mind that the two chemical phases can be spanned in the same beam. Then, the resulting averaged fractional abundance could be a mixture of the two phases. It is, however, not possible to describe precisely the occurrence of such cases since various physical effects are neglected in these isothermal models, such as the possible coupling of the two phases via radiative transfer. Higher spatial resolution is then clearly required to derive more definitive conclusions. A preliminary suggestion of the presence of two chemical phases has indeed been found by Gerin et al. (1997) from observations of CS, SO and deuterated species in the high latitude cloud MCLD 123.5+24.9 at much higher spatial resolution.

3.2 Chemical discussion

At this point, a chemical discussion on the formation and destruction processes of both molecular oxygen and water is interesting. Table 5 displays the main formation and destruction processes of H₃⁺, O₂ and H₂O which occur in both phases. In Table 5, we consider the solutions obtained for $n_{\rm H}$ = 5000 cm^-3 where both LIP and HIP solutions coexist, using the non standard value of oxygen depletion (O/$n_{\rm H}$ = 2.14).

 

 

Table 5: Formation and destruction reactions relevant to H₃⁺, O₂ and H₂O

Phase		Formation reaction	percentage	Destruction reaction	percentage
LIP	H3+	H2+ + H2	100%	H3+ + CO	44%
				H3+ + O	15%
				H3+ + N2	13%
				H3+ + electron	12%
	O2	O + OH	100%	O2 + secondary photons	44% (dissociation)
				O2 + C+	21%
				O2 + S+	7%
				O2 + secondary photons	7% (ionization)
				O2 + S	7%
				O2 + H+	5%
	H2O	H3O+ + electron	92%	H2O + H3+	47%
		H3O+ + CS	8%	H2O + HCO+	17%
				H2O + secondary photons	13%
				H2O + C+	9%
HIP	H3+	H2+ + H2	100%	H3+ + electron	77%
				H3+ + CO	10%
				H3+ + O	6%
	O2	O + OH	100%	C + O2	82%
				S+ + O2	8%
	H2O	H3O+ + electron	100%	H2O + C+	71%
				H2O + secondary photons	11%

A first conclusion of our analysis is that very few reactions, amongst the 4000 reactions involved in the chemical network, are really important for a given molecule.

3.2.1 H₃⁺

H₃⁺ plays a key role in the chemistry of oxygen compounds. Indeed, H₃⁺ + O gives OH⁺ which subsequently reacts with H₂ giving H₂O⁺. In dense cloud conditions H₂O⁺reacts immediately with H₂ giving H₃O⁺. Then H₃O⁺dissociatively recombines with electrons giving both OH and H₂O, as discussed in the previous section. The channel leading to H₂O represents the entire gas-phase formation process for water whereas the subsequent reaction of OH with atomic oxygen gives molecular oxygen with a 100% contribution. The formation of H₃⁺ is only due to the reaction between H₂⁺ and H₂. However H₂⁺ is formed from the impact of cosmic rays (CR) on molecular hydrogen through a direct photoionization process:
H₂ + CR $\Rightarrow$H₂⁺ + electron with a ionization rate $\alpha$$\times$$\zeta$. There is another possible channel for this reaction involving dissociative ionization of H₂:
H₂ + CR $\Rightarrow$ H⁺ + H + electron with a ionization rate ( $1 - \alpha$) $\times$ $\zeta$.

The branching ratio between the two channels has been measured by Shah & Gilbody (1982) for keV protons and the relevant astrophysical value has been calculated by Cravens & Dalgarno (1975). The latter value of $\alpha$, 0.97, is well within the range of measured values. We find it worthwhile to recall the difficulty of interpreting with standard models the low fractional abundance of H₃⁺detected in dense interstellar clouds by McCall et al. (1999). We have to keep in mind that cosmic rays are also critical for destruction of both O₂ and H₂O. Indeed, the analysis reported in Table 5 involves secondary UV photons (generated by the excitation of H₂ by secondary electrons produced after the interaction of cosmic rays and molecular hydrogen) which contribute significantly to the destruction of O₂ and H₂O. The difficulties encountered in interpreting the SWAS data may then be linked to the H₃⁺ puzzle.

3.2.2 The O + OH reaction

The O + OH reaction is the dominant O₂ formation process. Davidsson & Stenholm (1990) predict, on the basis of an extended Langevin model for the calculation of the rate coefficient, that the reaction is rapid at very low temperatures. However, the reaction rate coefficient is not known experimentally at very low temperatures. Smith & Stewart (1994) have found that the reaction proceeds rapidly for temperatures down to 158 K. To extrapolate the value at low temperatures as is done in the present model is however highly uncertain. To this purpose, we have studied the possible influence of an activation barrier in the reaction rate coefficient. Figure 6 displays the fractional abundances of O₂ and H₂O for a proton density $n_{\rm H}$ of 10⁴ cm^-3 with different values of the activation barrier.

Figure 6: Steady state fractional abundances of O2, H2O and OH for a proton density of 104 cm-3 as a function of the activation barrier in the O + OH reaction. The vertical line corresponds to the actual value used in the chemical network. The dotted horizontal lines correspond to the observed upper limit for O2 and H2O towards starless cores. The full horizontal line corresponds to the observed abundance of water towards Sgr B2

When the activation barrier is of the order of 80 K, the fractional abundance of O₂ drops to values lower than 10^-6 whereas no sensible influence is found for the H₂O fractional abundance. It is clearly highly desirable that new experiments or calculations address this reaction at very low temperatures. This would also affect the fractional abundance of molecular oxygen predicted in the models presented in Sect. 2.

3.2.3 The role of the C/O ratio

The C/O ratio available in the gas phase is subject to large uncertainties. However, infrared observations with the ISO satellite have allowed the sampling of the abundances of oxygen and carbon found on the ice mantles towards some specific sources like W33A and the Galactic Centre (Gibbs et al. 2000). Combining these results with gas phase observations we can deduce the C/O ratio which is found to be between 0.4 and 0.67. The largest uncertainty lies in the actual gas phase oxygen abundance. We have performed several steady state calculations for various values of the C/O ratio by keeping the gas phase abundance of total carbon constant and varying only the oxygen elemental abundance. Figure 7 presents the corresponding fractional abundances of O₂ and H₂O for a proton density $n_{\rm H}$ of 5000 cm^-3. We find a region of bistability in the range 0.49-0.66 of the C/O ratio for the considered density. It is clearly seen that the HIP solutions lead to low values of both O₂ and H₂O fractional abundances.

Figure 7: Steady state stable solutions of O2 and H2O as a function of the C/O ratio for a proton density of 5000 cm-3. The dotted horizontal lines correspond to the observed upper limit for O2 and H2O. The full horizontal line corresponds to the observed abundance of water towards Sgr B2

3.2.4 Comparison with the observations

The HIP solutions offer a natural means to interpret, within steady state models, the low observed upper limit of molecular oxygen towards the different lines of sight. Concerning water, we find that the observed high abundance towards Sgr B2, a region of relatively low density, is equally compatible with some steady state solutions. Finally, it appears that there is no possibility to explain the low abundance of water observed in high density starless cores. However, this sole interpretation is far too naive as different chemical phases are likely to be present in the large beam of the SWAS telescope. Then a combination of the different chemical phases can be considered, and would allow an explanation from the steady-state point of view of the low abundance of O₂ and the large abundance of water towards Sgr B2. Another point is that HIP and LIP solutions exist, not only at steady state, but also for "early times": the two solutions usually diverge from t>10⁶ yr. However, the actual temporal dependence is very dependent on the chosen initial conditions and no general conclusions may be derived. Clumpy models, such as those described by Spaans & van Dishoeck (2001) may offer an alternative explanation although they would fail for starless cores and for sources with weak UV fields (e.g. Ophiuchus).

Concerning the lack of observations of gas phase water towards starless cores, we should keep in mind that only the ortho component of water is probed by SWAS and that no definitive conclusion about water abundance should be derived before being able to detect para-H₂O. Unfortunately, the lowest transition of para-H₂O is at 1113 GHz, a range which is outside present capabilities of space observatories.