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

3. Results and discussion

3.1. C-type shocks

3.1.1. Comparison with Timmermann (1998)

Timmermann (1998) presented profiles for 3 models of C-type shocks with speed = 20 km s^-1 and initial densities = and cm^-3. In Fig. 2, we compare his and our corresponding results in the case of = cm^-3, for which the initial magnetic induction B = 100 µG; the initial ortho:para ratio is 1. Collisional dissociation of H₂ by a non-reactive ion I⁺ of mass 32 was included in both calculations, adopting the rate coefficient of Draine et al. (1983) (see Sect. 2.1).

Fig. 2. A comparison of the shock profiles of (left) Timmermann (1998, model 2) and (right) ourselves, using the same rate coefficient for dissociation by a non-reactive ion I+ of mass 32. The temperature of the neutral fluid and the ortho:para-H2 ratio are plotted in the upper panels, the densities of selected species in the lower panels.

Whilst the shock widths and temperature profiles are similar in the two calculations shown in Fig. 2, important differences also emerge. The increase in the atomic hydrogen density, owing to dissociation of H₂ in the shock wave, is much smaller in our model than in that of Timmermann and, consequently, the maximum ortho:para ratio that is attained is also less. The behaviour of the ion (H⁺, ) densities is very different. Timmermann's calculation shows an initial rise in n(H⁺) and n(), presumably due to the compression of the ionized fluid, which precedes that of the neutrals. In our model, this rise is quenched, even reversed, by chemical reactions which remove ions, as discussed in Sect. 2 above. The total density of ions in our shock model is only cm^-3, an order of magnitude smaller than the density alone in Timmermann's model. Perhaps the most puzzling aspect of Timmermann's results is the rapid rise in the density of H⁺ as the gas cools towards its postshock temperature; this increase coincides with the compression of the neutrals. We know of no reason for such an increase in the H⁺ density: cosmic ray ionization occurs on a distance scale which is orders of magnitude larger. We tentatively conclude that the treatment, in Timmermann's model, of the chemistry of these ions is incomplete or incorrect. The higher ion abundance, calculated by Timmermann, has direct and indirect (through the rate of formation of H by ion impact dissociation of H₂) consequences for the ortho:para ratio; the net effect is an overestimation of the ortho:para ratio in Timmermann's calculations.

3.1.2. A grid of models

In what follows, we continue to include the dissociation of H₂ by ions, but only real and chemically active ions are considered and the rate coefficient is derived assuming a constant cross section, as described in Sect. 2. Models of C-type shocks were calculated for

• initial densities = n(H) + 2n(H₂) = , , , cm^-3,

• initial ortho:para ratios = 0.01,1.0, 2.0, 3.0,

• shock speeds = 10, 20, 30, 40 km s^-1 (except for = cm^-3, when = 10, 15, 20, 25, 30 km s^-1).

We obtain the initial magnetic induction from B(µG) = .

The left-hand column of Fig. 3 shows the variations of the temperature of the neutrals, , of the fractional abundance of H, n(H)/, and of the local ortho:para ratio, n(ortho)/n(para), through C-type shock waves with 10 40 km s^-1, and, initially, = cm^-3 and ortho:para = 0.01. We note that allowance for the chemical reactivity of the ions has important consequences, notably for the width of the shock wave (see Pineau des Forêts et al. 1997). The reduction in the density of ions, owing to their removal in chemical reactions, results in a wider shock wave; the increase in width is a factor of 2 for = cm^-3 and = 20 km s^-1 (compare Fig. 3a with right panel of Fig. 2).

Fig. 3a-h. The variation of (a ) the temperature of the neutrals, (K), (b ) the fractional abundance of H, /, and (c ) the ortho:para ratio through C-type shock waves with 10 40 km s-1; initially, = cm-3, B = 100 µG, and the ortho:para ratio is 0.01. (d ) The rates (s-1) of the processes which contribute to the dissociation of H2, and production of H, in a C-type shock with = 40 km s-1, and, initially, = cm-3 and B = 100 µG. (e ) Temperature profiles, (f ) fractional abundance of H, n(H)/, and (g ) n(ortho)/n(para) for J-type shocks with speeds 10 25 km s-1; the initial gas density is = cm-3, the initial ortho:para ratio is 0.01, and B = 10 µG. (h ) The rates (s-1) of the processes which contribute to the dissociation of H2, and production of H, in a J-type shock in which = 25 km s-1, and, initially, = cm-3 and B = 10 µG.

Fig. 3d compares, for = cm^-3 and = 40 km s^-1, the rates (in s^-1) of the various processes which contribute to the dissociation of H₂: collisional dissociation by electrons, by hydrogen atoms and molecules, and by ions; chemical dissociation, including neutral-neutral reactions of the type O(H₂, H)OH, OH(H₂, H)H₂ O. At this high shock (and hence ion-neutral drift) speed, the rate of dissociation by ion impact exceeds the rate of production of H by chemical conversion of oxygen into water. The dissociation rate is of the order of s^-1 over much of the shock width. Estimating the flow time of the neutrals through the shock wave as cm/ cm s^-1 3 s, the fraction of H₂ that is dissociated is given approximately by s/ s = 3; this estimate is confirmed by the numerical results (Fig. 3b). At a lower shock speed, = 20 km s^-1, collisional dissociation by ions becomes insignificant, and neutral-neutral reactions initiated by O turn out to be the most important; the resulting fraction of atomic H is of order . For = 10 km s^-1, the temperature remains too low to activate these reactions and the atomic H fraction does not increase from its initial (preshock) value of (Table 1).

3.1.3. Conversion of para- to ortho-H₂

The activation energy of approximately 5000 K for the reaction of H with H₂ makes the efficiency of para to ortho conversion strongly dependent on the maximum neutral temperature reached in the shock wave, . As shown in Fig. 3c, for = cm^-3, conversion is negligible for = 10 km s^-1 ( = 200 K), only partial at 20 km s^-1 ( = 700 K), and complete for km s^-1 ( 1300 K). We have found that these threshold temperatures (700 K to start conversion, and 1300 K to reach the statistical value of 3) are valid across the whole grid of models, irrespective of the initial values of the ortho:para ratio and .

The weak dependence on can be understood as follows. The number of para to ortho conversion per H₂ molecule across the shock is given by the integral of over the shock profile, where is the reaction rate coefficient at temperature T (averaged over all H₂ levels). We have seen that, even allowing for the contribution of collisional dissociation by ions, the fraction of atomic H in C-type shocks remains always close to (Fig. 3b). Since is approximately constant on the hot plateau, where conversion takes place, n(H) can be considered to be roughly proportional to for all the models. The flow time, on the other hand, is found to be inversely proportional to (see Table 2). The dependence on cancels, leaving as the main parameter controlling para to ortho conversion in our models. The weak dependence on is due to the sudden onset of conversion, once the temperature is sufficiently close to the reaction barrier.

3.1.4. Electronic tables

We provide in Table 2a-f (electronic form only, available at CDS and from http://ccp7.dur.ac.uk/ ) an extensive set of parameters, derived from our grid of C-type shock models. Table 3 contains a subset of the results in Table 2a-b, by way of illustration, for = cm^-3 and = 0.01 and 1.

Table 3. Shock structure, H₂ ortho:para ratios, and line fluxes (in erg cm^-2 s^-1 sr^-1) for = cm^-3.
Column contents: is the shock speed (in km s^-1); L is the width of the region in which 50 K; t is the time required for the neutral fluid to traverse the width of the shock wave; n(H)/ is the maximum H abundance in the shock wave; is the maximum temperature of the neutrals; is the mean of the excitation temperature for levels to 8; is the mean of the excitation temperature for levels to 11; is n(ortho)/n(para), and is N(ortho)/N(para), evaluated at the rear of the shock wave (where has fallen to 50 K); is the mean of the empirical ortho:para ratio, derived for levels to 8; is the mean of the empirical ortho:para ratio, derived for levels to 11; is the percentage of the initial mechanical energy flux (/2) radiated in all the lines of H₂ included in the model; is the percentage of the initial mechanical energy flux radiated in the transitions 0-0 S(2) to S(7); S(J) is the flux radiated in the H₂ 0-0 S(J) line in a direction perpendicular to the shock surface, in erg cm^-2 s^-1 sr^-1 (values below are replaced by ...). In all columns, parentheses indicate powers of ten.

Table 2a lists for each model: the shock width L (of the region in which 50 K); flow time t (time required for the neutral fluid to traverse the width of the shock wave); (50 K) (postshock density at = 50 K); (maximum ion-neutral drift velocity in the shock wave); [n(H)/]_max; (maximum temperature of the neutrals); (mean of the excitation temperature , defined by Eq. 6, for levels to 8); (mean of the excitation temperature , defined by Eq. 6, for levels to 11); n(ortho)/n(para) and N(ortho)/N(para) at the rear of the shock wave (where has fallen to 50 K); (mean of the empirical ortho:para ratio, derived from Eq. 5, for levels to 8); (mean of the empirical ortho:para ratio, derived from Eq. 5, for levels to 11).

Table 2b lists for each model: the initial mechanical energy flux, /2 (in erg cm^-2 s^-1); (percentage of the initial mechanical energy flux radiated in all the lines of H₂ included in the model); (percentage of the initial mechanical energy flux radiated in the transitions 0-0 S(2) to S(7)); flux radiated in each of the H₂ lines: 0-0 S(0) to S(11); 1-0 S(0) to S(11); and 2-1 S(1) (in erg cm^-2 s^-1 sr^-1).

Table 2c contains, for all of our models with = cm^-3, the column densities of each of the 49 levels of H₂, divided by their statistical weights and sorted by increasing energy above the ground level.

Tables 2d,e,f are the same as Table 2c but for = , , and cm^-3.

3.2. J-type shocks

It is well known that C-type shocks occur only if the transverse magnetic field strength in the preshock gas exceeds a critical value (cf. Draine 1980). In practice, the magnetic field strength in specific objects is often unknown and must be derived from energy-equipartition arguments, which lead to density-scaling relations of the type used above. However, even if the initial magnetic field strength is sufficiently high for a C-type shock wave to obtain in steady-state, time-dependent calculations (Chièze et al. 1998) show that an initially J-type shock evolves through a mixed C- and J-type structure to ultimately become C-type. Furthermore, the time of evolution to steady state can be comparable with or exceed the estimated lifetimes of molecular outflows. Given the uncertainty in the initial value of the transverse component of the magnetic induction, and the fact that the shock will have a J-type component at early times, we have additionally studied the variation of the ortho:para-H₂ ratio in J-type shocks.

3.2.1. Shock structure and H₂ emission

A grid of J-type shock models was computed for

• = , , , cm^-3,

• = 0.01, 1.0, 2.0, 3.0,

• = 5, 10, 15, 20, 25 km s^-1.

A small but finite initial magnetic induction was assumed, B(µG) = 0.1[(cm^-3)], which is an order of magnitude smaller than adopted in Sect. 3.1 above for the same initial gas density. The other initial conditions were the same as in the corresponding C-type shock model. When computing the J-type models, it was assumed that the neutral and ionized fluids were fully coupled.

In the right-hand column of Fig. 3 are plotted the temperature profile, fractional abundance of H, n(H)/, and n(ortho)/n(para) for J-type shocks with a range of shock speeds, 10 25 km s^-1. As for the C-type shocks in Fig. 3, the initial gas density is = cm^-3, and the initial ortho:para ratio is 0.01.

Both C- and J-type shocks involve an initial rise in the temperature, followed by radiative cooling and compression of the gas towards its postshock state. However, the quasi-discontinuous temperature rise associated with a J-type shock ensures that the maximum temperature is attained adiabatically, and it is consequently much higher than in a C-type shock of the same speed. Thus, whilst a C-type shock with = 20 km s^-1 gives rise to a maximum temperature = 800 K of the neutral gas (see Fig. 3a), = K in the corresponding J-type shock (Fig. 3e).

The high maximum temperatures in J-type shocks give rise to much more rapid radiative cooling, by rovibrational transitions of molecular hydrogen, than in the corresponding C-type shocks. Thus, the width of a J-type shock with = 20 km s^-1 and initial = cm^-3 is of the order of cm, as compared with cm for a C-type shock of the same speed. The corresponding times of flow through the shocks are 13 yr and 4000 yr, respectively. Evidently, H₂ emission lines of high excitation energy are much more intense, relative to those of low excitation energy, under J-type shock conditions.

The treatment of radiative cooling is crucial when determining the structure of and H₂ line emission from the postshock flow. Chang & Martin (1991) made predictions of H₂ line intensities in J-type shocks, considering H₂ cooling only. With the same parameters as in their model 10k5 ( cm^-3, K, = 15 km s^-1), and with B = 45 µG, we find that H₂ cooling dominates down to about only 1000 K. The approximation of neglecting coolants other than H₂ is, therefore, inadequate when the preshock gas density is so high. We have also compared our results for = cm^-3 and = 10 km s^-1 with those of Burton et al. (1992) obtained with the J-type shock code of Hollenbach & McKee (1989). We find good agreement (within 10%) for the 0-0 S(1) to S(3) lines but predict 3 times more flux in the 1-0 S(1) line. The intensities of the pure rotational lines agree because the populations of the low rotational levels are close to thermal equilibrium.

3.2.2. Dissociation of H₂ and para to ortho conversion

The process of para to ortho conversion is more rapid in J- than in C-type shocks, owing to the higher postshock temperatures. However, the time available for conversion is much shorter, typically by a factor of 100, owing to the rapid cooling in the dense postshock flow. As a result, we find that para to ortho conversion does not occur for = 5 km s^-1 ( = 1000 K) and that it remains incomplete in J-shocks where the maximum temperature is below  K, i.e. km s^-1 (Fig. 3g).

When the postshock temperature exceeds K, the rate coefficients for dissociation of H₂ in H and H₂ collisions become sufficiently large for these processes to dominate the dissociation of molecular hydrogen (Fig. 3h). Indeed, for = cm^-3 and = 25 km s^-1, n(H)/ 0.4 in the postshock gas (Fig. 3f). The high abundance of atomic H then ensures that conversion of para- to ortho-H₂ proceeds rapidly and that the statistical value of 3 is attained. Similar results were obtained for the other values of that we investigated, with complete conversion requiring  K, i.e. 15 km s^-1.

3.2.3. Electronic tables

We provide in Table 4a-f (electronic form only, available at CDS and from http://ccp7.dur.ac.uk/) an extensive set of parameters, derived from our grid of J-type shock models, in the same format as Tables 2a-f for C-type shocks. By way of illustration, Table 3 contains a subset of Table 4a-b, for = cm^-3 and = 0.01 and 1.

© European Southern Observatory (ESO) 2000

Online publication: April 17, 2000
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