Astron. Astrophys. 356, 1010-1022 (2000)

2. Theoretical models

We consider both C- and J-type shocks. The equations for the H₂ rovibrational level populations are integrated in parallel with the hydrodynamic equations (Flower & Pineau des Forêts 1999), as in the studies of Timmermann (1998) and Chang & Martin (1991). We take advantage of the more complete and reliable data that now exist for the collisional excitation of H₂ by H, He, and H₂ (Le Bourlot et al. 1999), including the 49 rovibrational levels of H₂ with energies up to 20 000 K. The treatment of ortho:para interconversion was adopted from Le Bourlot et al. (1999). We pay careful attention to the chemistry of the gas, particularly the ion chemistry, incorporating a total of 120 species and a network of 864 reactions (see Schilke et al. 1997 and references therein). The MHD and chemical rate equations are solved in parallel in order to allow correctly for the effects of changes in the degree of ionization on the structure of the shock wave: see Sect. 2.2. The drag on the neutral fluid owing to collisions with charged grains is also included, adopting a standard gas to dust ratio and grain size distribution (Mathis et al. 1977).

Our treatment of the thermal balance of the gas includes the contributions to the cooling of species other than H₂ (the only coolant considered by Chang & Martin), namely C, O, C⁺, H₂O, CO, OH, and NH₃. More detailed information on the models may be found in the papers of Flower & Pineau des Forêts (1999), Chièze et al. (1998), Schilke et al. (1997), and Flower et al. (1996).

2.1. Collisional dissociation of H₂

Dissociation of H₂ by electron impact, which proceeds mainly via the b state of H₂ (Stibbe & Tennyson 1998), and by H atom impact (Dove & Mandy 1986) have been taken into account. The rate of collisional dissociation of H₂ by H₂ impact, under the low-density conditions prevailing in the interstellar medium, is poorly known. Measurements in shock tubes (Jacobs et al. 1967; Breshears & Bird 1973) yielded a rate coefficient for dissociation of H₂ by H₂ which is approximately an order of magnitude smaller than in H₂ - H collisions. However, the densities in laboratory shock tubes are vastly greater than in the interstellar gas, and the relevance of such measurements to interstellar conditions, where only the few lowest rotational states are significantly populated, has been questioned by Dove & Mandy (1986). Ultimately, we decided to include dissociation in H₂ - H₂ collisions, with a rate coefficient 8 times smaller than for H - H₂, as indicated by the shock tube measurements; but we recognize that this may yield an overestimate of the contribution of this process.

Not included in our previous studies, but incorporated here, is the collisional dissociation of H₂ by ion impact. Timmermann (1998) allowed for dissociation of H₂ by a representative non-reactive positive ion I⁺ of mass 32, adopting the rate coefficient for collisional dissociation from Draine et al. (1983). These authors allowed for the thermal contribution and, more important in practice, that of ion-neutral drift, to this process. For the range of drift speeds of interest here, the rate coefficient of Draine et al. is well reproduced by the functional form (shown as a solid line in Fig. 1)

where

and m is a mass, T a kinetic temperature, V a flow velocity, is Boltzmann's constant, and `i' refers to an ion, `n' to a neutral (in practice, H₂); the reduced mass, . Note that, for a heavy molecular ion with , , and the first term on the rhs of Eq. (2) is approximately equal to .

Fig. 1. Rate coefficient (in cm3 s-1) for the collisional dissociation of H2 by heavy ion impact. Full curve: Langevin model (Draine et al. 1983); broken curve: assuming a constant cross section; see text, Sect. 2.1.

We shall see below that, even when we include collisional dissociation of H₂ by I⁺, with the above rate coefficient, our results differ from those of Timmermann (1998). Specifically, the degree of dissociation of the gas (fraction of atomic hydrogen) which we obtain is less than Timmermann predicted, with important consequences for the ortho:para-H₂ ratio.

Draine et al. (1983) derived the rate coefficient for dissociation of H₂ by ion impact using the Langevin (`orbiting') model of ion-neutral collisions, assuming that dissociation occurs following each orbiting collision; this procedure yields an upper bound to the actual rate of dissociation. The orbiting model is appropriate at low collision energies, comparable with the well depth of the ion-neutral interaction, which is unlikely to exceed 0.1 eV. However, at collision velocities 10 km s^-1, required for the rate of dissociation by ion impact to become significant, the collision energy 1 eV, and the orbiting model is inappropriate. We believe that a better approximation to the rate coefficient at these high energies may be obtained by assuming a constant cross section, = , where a has been taken equal to the H₂ internuclear distance ( cm), whence

where 52 000 K is the dissociation energy of H₂ (4.48 eV), expressed in K through division by . The rate coefficient which derives from Eq. (3) is plotted in Fig. 1 as a dashed line. As expected, it lies below the prediction of the orbiting model. Of course, the correct rate of dissociation by ion impact could prove to be even smaller than this revised estimate.

2.2. Initial conditions and chemistry

As we wish to study ortho:para H₂ conversion in shocks, starting from values 3, we shall consider a cold molecular cloud shielded from the external UV field ( mag), as appropriate for regions of moderate star formation efficiency located away from photodissociation fronts. In Table 1, we list the initial species densities that we calculate, in equilibrium, for = , , , and cm^-3, assuming T = 10 K and a cosmic ray ionization rate s^-1. For comparison, we also list the initial species densities adopted by Timmermann (1998; Table 1, model 2) for = cm^-3. There are surprisingly large differences, between our initial values and those of Timmermann, for some of the species listed in Table 1, particularly C, C⁺, and H⁺. These differences are not attributable to the different cosmic ray ionization rates and initial temperatures. I⁺ is the non-reactive ion of mass 32, introduced by Timmermann. S⁺ has the same mass, and its equilibrium abundance is also given in Table 1. Note that S⁺ is reactive in the shock wave, where a sequence of hydrogen abstractions is initiated by the endothermic reaction S⁺(H₂, H)SH⁺, which is driven by ion-neutral streaming. The abundances of other ions, such as and H⁺, also tend to decrease, owing to the compression of the gas by the shock wave. This compression enhances the rates of reactions such as (O, H₂)OH⁺ and leads ultimately to a decrease in the fractional ionization of the gas, through dissociative recombination of the molecular ions with electrons. Also important are the reactions O(H₂, H)OH and OH(H₂, H)H₂ O, which produce two hydrogen atoms for each oxygen atom removed. These reactions have activation energies, which can be thermally surmounted in the shock-heated gas. As we shall see below, they are important contributors of atomic hydrogen and thence influence the variation of the ortho:para-H₂ ratio through the shock wave.

Table 1. Initial values of the densities (cm^-3) of the listed species as calculated here, in equilibrium, for = , , , and cm^-3, and as given by Timmermann (1998) for = cm^-3. Timmermann adopted an initial gas kinetic temperature T = 35 K and cosmic ray ionization rate = 2 s^-1; we chose T = 10 K and s^-1. Numbers in parentheses are powers of 10.

As can be judged from Table 1, the equilibrium values of both the ion and atomic hydrogen densities are initially low under the dense dark cloud conditions considered here. As the (proton-transferring) ions and H atoms modify the ortho:para-H₂ ratio in the shock wave, the timescale for this process to occur may be comparable with the duration of the shock pulse. Under these circumstances, the initial value of the ortho:para ratio is significant in determining the integral value of this parameter. In view of this fact, the initial value of the ortho:para ratio has been taken to be a free parameter of the model. The time required for this ratio to reach equilibrium in gas of density = cm^-3, in which n(H⁺) + n() cm^-3, approaches yr. If the gas is cold, the equilibrium ortho:para ratio is small ( at ). Passage through a shock wave with a maximum temperature exceeding about 500 K will, as we shall see below, cause this ratio to increase rapidly to a value of the order of 1, and a further - yr are then required for the ratio to relax to its equilibrium value in the postshock gas. We may therefore conclude that, if a low ( 1) initial value of the ortho:para ratio is implied by observation, e.g. of an outflow source, then the gas is unlikely to have been significantly heated by the shock wave associated with an earlier outflow in the same jet, given that the time between such episodes is believed to be much less than that required for the ortho:para ratio to return to equilibrium. In the present calculations, we consider initial ortho:para ratios between 0.01 and 3.0.

© European Southern Observatory (ESO) 2000

Online publication: April 17, 2000
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