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

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4. Comparison with observations and diagnostic diagrams

4.1. Observed value of the ortho:para ratio

The ortho:para ratio, n(ortho)/n(para), plotted in Fig. 3c and Fig. 3g, is the local value at each position in a planar C- or J-type shock wave. An observer, on the other hand, measures H2 populations integrated over the shock width and, as we shall now see, deduces an ortho:para ratio which is always lower than the final value, reached behind the shock wave.

The method by which the ortho:para ratio is determined observationally is illustrated in Fig. 4a. This method makes use of the excitation diagram, which is a plot of [FORMULA] against [FORMULA], where [FORMULA] is the column density measured in rotational level J, and [FORMULA] is the excitation energy of this level, relative to [FORMULA]; [FORMULA] and [FORMULA] (J even) or 3 (J odd). If the rotational sublevels are populated according to a Boltzmann distribution at temperature [FORMULA], and if the ortho:para ratio is equal to its LTE value at [FORMULA], then


and the excitation diagram is a straight line of slope [FORMULA]. The downwards displacement of the line joining points corresponding to odd J (ortho-H2), relative to even J (para-H2), provides a measure of the departure of the ortho:para ratio from its value in thermal equilibrium at [FORMULA]. For each ortho level J, the ortho:para ratio is obtained from


where the excitation temperature between levels [FORMULA] and [FORMULA], [FORMULA], is given by


and [FORMULA] is the column density obtained by linear interpolation between the points [FORMULA] and [FORMULA] in the excitation diagram (see Fig. 4a). In practice, the excitation diagram may exhibit curvature, indicative of the emitting gas having a range of kinetic temperatures. The curvature is small for C-type and greater for J-type shocks, owing to the very rapid variation of the kinetic temperature in the post J-shock flow. Curvature can introduce systematic errors in the ortho:para ratio determined from Eq. (5), but we find that averaging over an equal number of ortho and para levels yields a reliable value (with error [FORMULA]%).

[FIGURE] Fig. 4. a An excitation diagram, obtained using the column densities of the rotational levels [FORMULA] from a model of a C-type shock in which [FORMULA] = 30 km s-1 and, initially, [FORMULA] = [FORMULA] cm-3 and ortho:para = 0.01; triangles show the LTE populations [FORMULA] used to estimate the ortho:para ratio (see text, Sect. 4.1, Eq. (5)). b The local ortho:para density ratio, n(ortho-H2)/n(para-H2), and the ratio of the corresponding column densities, N(ortho-H2)/N(para-H2), plotted as functions of distance through a model of a C-type shock in which [FORMULA] = 30 km s-1 and, initially, [FORMULA] = [FORMULA] cm-3 and ortho:para = 0.01. Also plotted in this figure are the values of the ortho:para ratio derived, as described in Sect. 4.1, from the column densities of the rotational levels v = 0, J = 5 to J = 8 [[FORMULA]] and v = 1, J = 1 to J = 11 [[FORMULA]].

It is important to realize that the ortho:para ratio derived empirically depends on the lines that are observed: levels with differing excitation energies are populated in different parts of the shock wave, where the (local) values of the ortho:para ratio may also differ. To illustrate this important point, we plot in Fig. 4b the variations of measures of the ortho:para ratio through a C-type shock wave in which [FORMULA] = 30 km s-1 and, initially, [FORMULA] = [FORMULA] cm-3 and ortho:para = 0.01. In this model, the local ortho:para ratio, n(ortho)/n(para), reaches its final (and maximum) value of 2.85 at a distance [FORMULA] cm (flow time [FORMULA] 2000 yr); the shock pulse is delineated by the neutral temperature profile. The ratio of total column densities in the ortho- and para-H2 levels, N(ortho)/N(para), integrated up to the current point in the profile, is seen to lag behind the local ortho:para ratio, attaining its final (maximum) value only for sufficiently long integration times after the shock pulse. This ratio is difficult to determine observationally, as most of the cool postshock H2 lies in the J = 0 and J = 1 levels, the populations of which are measurable only in absorption; foreground and background cold molecular material add further, unknown contributions to the column densities of these levels.

In emission line studies, the column densities of levels with [FORMULA] 2 are measured, and the inferred ortho:para ratio differs from N(ortho)/N(para). Additionally plotted in Fig. 4b is [FORMULA], the mean value of the empirical ortho:para ratios derived from Eq. (5-6) for levels [FORMULA] (using e.g. the H2 0-0 S(2) to S(7) lines falling in the ISOCAM spectral range), integrated up to the current point in the profile. This ratio first follows closely the total column density ratio, N(ortho)/N(para) but attains a limiting value when the gas has cooled to about 500 K. The levels considered, which have [FORMULA] 1600 K, are not significantly populated at low temperatures, and their integrated column densities become effectively "frozen". As a consequence, this empirical ratio does not depend on the (unknown) amount of cool preshock and postshock gas along the line of sight, enabling a meaningful comparison to be made between ISOCAM observations and the shock models.

Finally, Fig. 4b also plots [FORMULA], the mean value of the ortho:para ratios empirically derived from column densities of the levels [FORMULA] (corresponding to the upper levels of the ro-vibrational [FORMULA] lines observable from the ground). These levels have higher energies (6000 to 14000 K) than the [FORMULA] levels; thus, the column density ratios become "frozen" even sooner after the temperature maximum, and [FORMULA] remains smaller than [FORMULA]. We conclude that an object where [FORMULA] [FORMULA] [FORMULA] cannot be explained by the presence of a single stationary-state planar shock wave within the observing beam.

4.2. Diagnostic diagrams and application to HH 54

4.2.1. Pure rotational lines in C-type shocks

Neufeld et al. (1998) have used ISO SWS02 (grating mode) observations of the E,K emission knots in HH 54, together with the C-shock models of Timmermann (1998), to constrain the shock speed and preshock ortho:para ratio in this object. They have shown that a useful diagnostic diagram is obtained by plotting the line intensity ratio S(3)/S(1) (a measure of the excitation temperature [FORMULA]) as a function of S(2)/S(1) (a measure of both [FORMULA] and the ortho:para ratio) for a range of shock models with fixed preshock density. With an arbitrarily chosen [FORMULA] = [FORMULA] cm-3, the corresponding observational point indicates a shock speed of 22 km s-1 and an initial ortho:para ratio of [FORMULA], accepting some extrapolation of Timmermann's results below [FORMULA] = 1.

In view of the improvements that we have made, in the treatment of H2 excitation, for example, and of the discrepancies with the results of Timmermann (1998), we present in this section diagnostic diagrams which derive from our own grid of models and which cover a wider range in [FORMULA] and [FORMULA]. Our diagnostic diagram is analogous to that in Neufeld et al. (1998), but we focus on the levels emitting lines in the ISOCAM range (0-0 S(2) to S(7)). Along the x-axis, we plot [FORMULA], the mean value of the excitation temperature, [FORMULA], defined by Eq. (6) above and deduced for [FORMULA], J = 5, 6, 7 and 8 from the ISOCAM lines. Along the y-axis, we plot [FORMULA], the corresponding mean value of the ortho:para ratios, deduced for these same levels using Eq. (5).

Fig. 5a is such a diagnostic diagram, for [FORMULA] = [FORMULA], [FORMULA], [FORMULA], and [FORMULA] cm-3. Solid curves correspond to an initial ortho:para ratio of 1.0, dashed curves to 0.01; an initial ortho:para ratio [FORMULA] 0.01 yields the same results as 0.01. Each point on a given curve represents a different shock speed. As the shock speed (and temperature) increases, reactions of H with H2 begin to occur, and the ortho:para ratio deduced from the ISOCAM lines rises, reaching values in excess of 2.0 for [FORMULA] km s-1. Being an average over the shock thickness, the ortho:para ratio never attains 3.0 (see Sect. 4.1).

[FIGURE] Fig. 5a and b. The mean observed ortho:para ratio, derived from levels [FORMULA] 5 to 8, plotted against the mean excitation temperature of these same levels, for a range of shock speeds, preshock densities, and initial ortho:para ratios (see text, Sect. 4.2): a C-type shocks; b J-type shocks. Solid curves correspond to [FORMULA] = 1.0, dashed curves to [FORMULA] = 0.01; each point on a given curve represents a different shock speed. The point plotted as a star derives from SWS02 observations of HH 54 E,K (Neufeld et al. 1998).

Fig. 5a shows that the rise in the ortho:para ratio occurs at lower excitation temperatures as [FORMULA] decreases. This shift of the diagnostic curves with [FORMULA] has two important consequences. First, any given observed point, when plotted in this diagram, does not correspond to unique values of both the initial ortho:para ratio and [FORMULA]. Adopting a lower [FORMULA] yields a lower initial ortho:para ratio and a higher [FORMULA]; this introduces an intrinsic uncertainty in the determination of the preshock parameters. Second, the curve corresponding to a very low initial ortho:para ratio (0.01 in Fig. 5a) defines for each [FORMULA] the minimum ortho:para ratio at a given excitation temperature. An observed point located below this curve is incompatible with the models at the corresponding value of [FORMULA] and implies a higher preshock density.

As an illustration, we plot as a star symbol in Fig. 5a the point which derives from the SWS observations of HH 54 (Neufeld et al. 1998). We see that values of [FORMULA] [FORMULA] cm-3 are excluded by this observation. For [FORMULA] = [FORMULA], [FORMULA], and [FORMULA] cm-3, the inferred pairs of values of [FORMULA] and [FORMULA] are, respectively, (30 km s-1, [FORMULA]), (20 km s-1, 0.5), and (15 km s-1, 0.8). In order to determine the initial ortho:para ratio, we require an independent constraint on [FORMULA] but the observed H2 line fluxes are not sufficient for this purpose. As Fig. 6 shows, for a given excitation temperature and value of [FORMULA] (= 1.0 in Fig. 6), the intensity of the 0-0 S(5) line differs by at most a factor of 3 among C-shock models with [FORMULA] ranging from [FORMULA] to [FORMULA] cm-3. This weak dependence on [FORMULA] stems from the inverse proportionality of the shock thickness to [FORMULA], which results in an almost invariant total H2 column density through the shock wave ([FORMULA] cm-2, to within 50%). Additional uncertainties in the filling factor of the shock layer in the instrument beam make it difficult to constrain [FORMULA] from line fluxes alone. The shock thickness might provide a more stringent constraint (cf. Table 2a). In HH 54, high [FORMULA] values would be favored if the rotational emission arises from small knots of [FORMULA] 5" (1.5 [FORMULA] cm), as observed in rovibrational lines. However, ISOCAM images suggest that the H2 emission is more widespread (Cabrit et al. 1999).

[FIGURE] Fig. 6. The computed intensity of the 0-0 S(5) line, plotted against the mean excitation temperature, for a range of C- and J-type shock models with an initial ortho:para ratio of 1 (see text, Sect. 4.2). The point plotted as a star derives from SWS02 observations of HH 54 E,K (Neufeld et al. 1998).

We conclude that the initial ortho:para ratio [FORMULA] [FORMULA] in the region of HH 54 observed by SWS but that the uncertainties in [FORMULA] are such that it is not possible to decide whether the ratio has reached thermal equilibrium in the preshock gas.

4.2.2. Pure rotational lines in J-type shocks

Fig. 5b is the diagnostic diagram for our grid of J-type shock models with [FORMULA] = [FORMULA], [FORMULA], and [FORMULA] cm-3. The results are broadly similar to those for C-type shocks, but the rise in the ortho:para ratio occurs at much higher excitation temperatures, [FORMULA] 1000 K. The shifts between curves corresponding to different values of [FORMULA] are smaller than in the case of C-type shocks. It follows that the inferred values of the initial ortho:para ratio and shock speed are less sensitive to the assumed [FORMULA]. From the observations of HH 54 by Neufeld et al. (1998), we deduce [FORMULA] [FORMULA] 1.0 and [FORMULA] [FORMULA] 5 km s-1.

In Fig. 6 are plotted the fluxes in the 0-0 S(5) line, as a function of the excitation temperature, for our J-type shock models. The line flux is slightly more sensitive to [FORMULA] than is the case for C-type shocks. However, the predicted line fluxes for J-type shocks are at least an order of magnitude below the flux observed in HH 54 by ISO SWS. A J-type shock could account for the observed flux only if the preshock gas is dense ([FORMULA] = [FORMULA] cm-3) and the shock front is curved over a scale comparable to the beam size.

4.2.3. Rovibrational transitions as diagnostics

Prior to the launch of ISO, H2 emission lines from shocks driven by young stellar objects were observable only from the ground by means of the [FORMULA] transitions in the H and K atmospheric bands. Relative populations in the [FORMULA] levels were used to deduce a mean rotational excitation temperature, typically in the range 2000-3000 K, and a mean ortho:para ratio in the range 2.5 to 3.0 (e.g. Gredel 1994; Smith et al. 1997).

In principle, such observations may also be used to constrain the initial ortho:para ratio in the pre-shock gas. We present in Fig. 7 diagnostic diagrams equivalent to those in Fig. 5, but for the [FORMULA] levels of H2. We plot along the x-axis [FORMULA], the mean value of the excitation temperature [FORMULA] of the [FORMULA] levels, and along the y-axis [FORMULA], the corresponding mean value of the ortho:para ratio, deduced for the same levels, as described in Sect. 4.1. The symbols and curves are the same as in Fig. 5.

[FIGURE] Fig. 7a and b. The mean observed ortho:para ratio in levels [FORMULA] 1 to 11, plotted against the mean excitation temperature of these same levels, for a range of shock speeds, preshock densities, and initial ortho:para ratios (see text, Sect. 4.2.3): a C-type shocks; b J-type shocks. The dashed rectangle derives from H- and K-band spectroscopic observations of HH 54 E,K (Gredel 1994).

Once again, we use the knots E,K in HH 54 as an illustration. The dashed rectangle in Fig. 7 indicates the range of parameters compatible with near infrared observations of these emission knots by Gredel (1994). We first note that [FORMULA] is larger than [FORMULA], which suggests that a single planar shock wave within the SWS beam cannot account for all of the observations (cf. Sect. 4.1). Indeed, the rotational excitation temperature in [FORMULA], [FORMULA] [FORMULA] 2900 K, implies a faster C-type shock ([FORMULA] [FORMULA] 35 km s-1) than the pure rotational transitions observed from HH 54. The corresponding initial ortho:para ratio is ill-determined. At such high shock speeds, para to ortho conversion is fast enough to yield an observed ratio close to 3.0, regardless of the initial value. In the case of a J-type shock, we infer [FORMULA] = 10 km s-1 and a better constrained initial ortho:para ratio [FORMULA] 2.0, higher than deduced from the pure rotational lines ([FORMULA] for both C-type and J-type shocks).

The situation that vibrational transitions imply a higher shock speed than pure rotational transitions is also found in other protostellar outflows observed with ISO (e.g. Cep A; Wright et al. 1996). Similar difficulties were already encountered when trying to reconcile all H2 lines observed across the near-IR range with a single planar shock (e.g. Smith & Brand 1990). A solution proposed by previous authors (e.g. Smith et al. 1991) is that the shock front is curved within the beam (bow shock): rovibrational emission probes regions closer to the bow apex, where the shock speed is higher, while pure rotational lines probe further out in the bow shock wings. The curved front could be entirely of one (C- or J-) type or contain both C-type and J-type surfaces. Studies of the ortho:para ratio using both sets of lines could impose additional constraints on this model. In the case of HH 54 E,K, the initial ortho:para ratio [FORMULA] 2.0 deduced from the vibrational transitions, assuming a J-type shock, is incompatible with the value [FORMULA] from the pure rotational lines. A C-type bow shock with an initial ortho:para ratio [FORMULA] 0.8 and [FORMULA] [FORMULA] 35 km s-1 remains a possibility.

An alternative explanation of the rotational and vibrational line emission is that the shock wave has not reached steady-state (Chièze et al. 1998). In this model, the rotational lines arise mainly in the C-type component, whilst the vibrational lines come from the following J-type component (Flower & Pineau des Forêts 1999). The initial ortho:para ratio, ahead of the shock wave, could be low ([FORMULA] 0.8), as required by the pure rotational lines, but the ratio could increase in the magnetic precursor to a local value of 2.0 just ahead of the J-discontinuity, as required by the vibrational lines. A more detailed investigation of these possibilities in relation to HH 54 and other objects is deferred to a subsequent paper.

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Online publication: April 17, 2000