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Astron. Astrophys. 342, 717-735 (1999)

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5. General discussion

We would like to emphasize once more that our results on jet ionization fraction and average excitation temperature do not depend on a model of the heating mechanism and/or the jet evolution. On the other hand, restrictions of our analysis should also be pointed out. First, our sample is rather small, so that individual characteristics of the examined objects may somewhat limit our ability to derive generalized properties of HH jets. Secondly, it should be kept in mind that our spatial resolution usually is insufficient to resolve single knots. Also, the use of line ratios from intensities integrated over the full line width does not allow to disentangle contributions from different velocity components of the jet. Therefore, the result for each position along the beam is an average over a relatively large parcel of gas, which could contain regions of very different conditions, e.g. near an unresolved shock front. This affects especially our derived temperatures, which only roughly indicate the average excitation temperature of the forbidden lines. Thirdly, the technique is capable of identifying regions of high ionization, but there the application of the procedure may become critical due to the neglect of photoionization in the regulation of the ionization state of O and N (see Sect. 3.1). Therefore, if a value of [FORMULA] is obtained, the result should be taken with caution.

Despite these difficulties the application of our technique to spatially resolved spectra of `classical' Herbig-Haro jets gives us a number of interesting insights into the physics of these objects, which we discuss in the following.

5.1. Ionization in HH jets

For all examined objects, the gas in the beams turns out to be partially ionized . In Table 2 we summarize the values of [FORMULA] derived in each jet, together with the average temperatures, and the electron and total hydrogen densities. In most cases the hydrogen ionization fraction ranges between 0.02 and 0.35. It is generally lower for less excited and heavier jets, and higher for more excited and lighter ones. Qualitatively, the ionization fraction is also higher on average in regions where the jet beam violently interacts with its surroundings, as in the faint section of the HH 46/47 beam. Of the jets studied, the least ionized appears to be HH 34 ([FORMULA] in the bright beam), which also presents the highest [SII]/H[FORMULA] ratio and the highest densities.


Table 2. Physical conditions along the beam of the examined jets
[FORMULA] Approximate limiting values from the beginning to the end of the indicated region. An intermediate value refers to a localized peak or a marked decay in the selected region. See the plots for details.
[FORMULA] Projected distance from the source.
[FORMULA] The inclination angles for HH 34, HH 46/47 and Th 28 are from Eislöffel & Mundt 1992, Eislöffel & Mundt 1994, and Krautter 1986, respectively. The values for the other jets are estimated from our recombination model.
[FORMULA] Red lobe.

The ionization degrees we find in our objects are generally higher than those diagnosed by HMR94. These authors derive the hydrogen ionization fraction comparing observed line ratios with a grid of planar shock models, varying the pre-shock density, shock speed, and parallel magnetic field. The extremely low ionization fraction in these models comes as a natural consequence of the fact that the observed ratios should be produced by low excitation (i.e. low velocity) shocks, ranging from 20 to 40 km s-1. As HMR94 illustrate in their Fig. 1, a 35 km s-1 shock propagating in a neutral medium with pre-shock density [FORMULA] cm-3 is not able to produce in situ an ionization fraction greater then [FORMULA]. It is worth noting, however, that the average ionization given by HMR94 is highly influenced by the results from line ratios involving Balmer lines, like [OI]/H[FORMULA], [SII]/H[FORMULA], [NII]/H[FORMULA], that lead systematically to ionization fractions a factor 2 lower than those derived from [NII]/[OI] and [NI]/[NII] (see their Table 3). The authors interpret this discrepancy as arising from the simplyfying assumption of planar shock geometry. On the other hand, the [FORMULA] values derived in HMR94 from the [NII]/[OI] ratio are very close to the ones we find from analogous observational data. Since the [NII]/[OI] ratio is extremely sensitive to the ambient ionization (see the orientation of our [OI]/[NII] curve in Figs. 4 and 5), one might ask if HMR94 possibly underestimate the average ionization using line ratios involving Balmer lines. This can occur if processes other than shock excitation at a fixed temperature suppress H[FORMULA] in comparison with forbidden lines. In this case, the use of the curves in Figs. 3, 4 and 5 of HMR94, that provide [FORMULA] as a function of [SII]/H[FORMULA], [OI]/H[FORMULA], [NII]/H[FORMULA] would lead to exceedingly low ionization fractions (and shock speeds). Such a situation may arise if the shock propagates in an already ionized medium. A pre-ionization would reduce the number of neutral H atoms available to produce collisionally excited H[FORMULA] at a given temperature. An additional ionization would increase also the amount of the recombination contribution, but for weak shocks this would be minor in comparison with collisional H[FORMULA] (see Fig. 14 of HMR94), unless the pre-ionization is substantial. As a consequence, in jets like the ones examined by HMR94, where the ionization is moderate, one would underestimate [FORMULA] using ratios involving Balmer lines.

The necessity of taking the pre-ionization history of the gas into account is suggested by the fact that in most of our observed jets the ionization fraction slowly decreases along the jet or at least sections of it. This is in accord with the idea of slow time dependent recombination along the beam suggested by BCO95: since the typical recombination time of the jet gas is of the order of the travel time through the bright jet section, partially ionized material can still be observed at large distances from the source if the jet gas gets almost completely ionized in the acceleration region. In their model the initial ionization is produced by a violent shock which is heavily shielded from view by circumstellar material. The ionization state soon decouples from the local thermodynamic conditions due to the sudden expansion of the jet beam (such a rapid expansion has been suggested, for example, in MRR91); the jet recollimates and the ionization fraction gently decreases along the jet axis on spatial scales determined by the product of recombination time and the flow velocity. This scenario is not necessarily in contradiction with the presence of shocks in the flow if these are too weak to produce further ionization. Jet shock calculations, however, should take the fact into account that the fronts may form in a medium that already has a considerable ionization degree.

In order to test the BCO95 recombination model, we calculated recombination curves, and superimposed them on the ionization data. Our model, described in more detail in the Appendix, assumes that the jet gas flows along surfaces shaped as nested cones that diverge or converge monotonically over a finite length. We point out that by taking the average flow radius, flow speed, and electron density from optical images, we implicitly assume that the average properties of the flow are those of the zones of maximum emission, which in turn are those of maximum compression (see Sect. 5.3 for a discussion). The curves are calculated by varying the opening angle of the flow until the best fit is obtained. In HH 34 and in the bright section of HH 46/47 the overall behavior can be described by a single family of self-similar cones. In HH 24C, HH 24E and HH 24G, however, marked jumps in the ionization degree, each followed by a well defined independent decay are seen. The ionization fraction in these jets is best reproduced by a series of independent flow cones of different opening angles (see below). With the exception of the HH 24G flow, which shows a diffuse appearance, the resulting best-fit opening (positive or negative) angles are small, scattering around values of a few degrees. This is in good agreement with the results of MRR91, who measured the apparent flow diameters on optical images (which might not be the actual dynamical flow surfaces, if cold, unobservable gas was moving at the true boundary of the flow). Our estimates, on the other hand, come from a fluid model, albeit assuming an initial jet diameter based on observations. The agreement between the two opening angles is not fortuitous if one considers that inertia alone provides a free-flowing opening angle of at most a few degrees for a Mach 20-30 jet.

A plausible interpretation for the jumps in the ionization fraction in the HH 24C/E/G jets seems to be the presence of shocks strong enough to reionize the gas in the beam. The [OI]/[NII] ratio invariably presents a minimum in these positions. Moreover, the subsequent smooth decay of the ionization fraction always occurs downstream of the jump in the HH 24C/E/G jets. This is in apparent conflict with a bow shock interpretation of these knots: a bow shock would produce a jump in the ionization degree with a decay upstream of the jump. If a Mach disk strong enough to ionize the gas was associated with the bow shock, the ionization fraction should increase at the location of that Mach disk, then decrease inside the working surface, and peak again at the front shock (such a feature is indeed observed inside HH 47A (see Fig. 7)). At our spatial resolution, we may not be able to resolve this structure in an internal bow shock. Instead, we would only observe a jump in [FORMULA] moving along the beam at about the jet speed. The fact that the ionization fraction decreases downstream of the jump is consistent with the flow passing a throat that compresses it. An oblique shock may form there, capable of reionizing the gas. Downstream of the shock the jet material then progressively looses its ionization.

While the shape of the ionization decay clearly indicates that most of the knots in our jet beams do not show the behavior expected for mini-bow shocks, this investigation alone may not be sufficient to discard the formation of such internal working surfaces. It seems possible that the interaction of the central beam with the surrounding medium comes into play. It is clear from HST images that the high ionization of the faint section in the HH 46/47 flow is due to the formation of shocks along the boundary of the wiggling flow, which also propagate in the surrounding medium. On the other hand, the jumps in HH 24E and HH 24G probably are better interpreted as the interaction between the jet beam and dense clumps in the medium. The shocks disturbing the beam may be generated when the jet collides with and/or perforates one of these dense clumps. The comprehension of this phenomenon would be greatly improved by proper motion measurements of the individual condensations. This could help disentangling if the knots are actually associated with steady nebular clumps or if they are nearly comoving with the jet: in the latter case the excitation of the beam gas could be related to the nonlinear development of Kelvin-Helmholtz instabilities arising in the interaction with a comoving jet cocoon.

5.2. Temperature

Apart from the ionization fraction, our diagnostic diagrams indicate an average excitation temperature. We generally find values of about 9000 to 12000 K, with minima and maxima of about 5000 and 24000 K in isolated positions. If the knots do represent shocks, even if weak, the temperature actually rises after the shock front well above these values and then decreases rapidly up to two orders of magnitude in the post shock cooling region. We are unable to observe such features here, since the typical length of a post-shock cooling layer (10-100 AU) is much smaller than our spatial resolution, which for example at the distance of HH 34 corresponds to [FORMULA] 1.2 [FORMULA] cm. Therefore, even if weak shocks are present in the flow we measure only an average temperature of the emission regions of the considered forbidden lines. Taking this into account, the results of our diagnostic are not in conflict with the results of shock calculations. The application of our technique to data of much higher spatial resolution could in principle allow us to retrieve the temperature variations predicted by shock models in the forbidden line emission region. Although at the present resolution our temperatures have only a limited physical meaning, we note that they can nevertheless give a qualitative information concerning the characteristics of the shocks exciting the gas. In fact, where a moderate to high ionization fraction is observed simultaneously with a low temperature, either a relatively high velocity of the shock, or a shock front forming in a region of high pre-ionization are indicated.

5.3. Total jet density

One of the main goals of this work was the determination of the total jet density. The total hydrogen density [FORMULA] calculated from [FORMULA] ranges between about 103 cm-3 and a few 104 cm-3 for a heavier jet like HH 34. Due to our limited spatial resolution we can only derive average densities over large parcels of gas, and we may ask ourselves if we under- or overestimate the total density, and if a correction factor should be applied as in HMR94. These authors suggest that any determination of [FORMULA] involving line ratios from post-shock cooling layers should take into account the compression of the emitting gas. Thus, in order to determine the average density, they take a geometric mean of the pre-shock and post-shock values. This in turn involves the multiplication of the [FORMULA] determined by line diagnostics by a factor [FORMULA], where C is the shock compression defined as the ratio between the post- and pre-shock densities averaged over the [SII] emission region. That way, they find the densities in the HH 34 and HH 47 jets lower than those directly determined as the ratio [FORMULA] by a factor 1/4 and 1/5, respectively. HMR94 also observe that [FORMULA] determinations from the observed luminosity in a suitable forbidden line, like [OI][FORMULA] 6300, should not be corrected for compression, since even if the emission comes from the densest part of the flow, it is diluted over the observing aperture, which tends to cancel the estimate error.

Since our results come from line ratios, filling factor effects are unimportant. On the other hand, our derived values are also weighted strongly towards regions of maximum emission, which in turn are those of maximum compression. In this respect our values may overestimate the average density, too. If shock excitation is responsible for the heating of the gas, we should apply a correction factor as well. This may also be suggested by the fact that including shock compression HMR94 find mass loss rates almost identical to those provided by the luminosity in the [OI] line, while ours turn out to be larger. However, since our results do not derive from a shock model, there are large uncertainties associated with the adoption of correction factors. Thus, we prefer here to give total densities as direct ratios between the electron density and the ionization fraction. It should be kept in mind that while [FORMULA] determinations are unaffected by shock compression, both the electron and the total density might be overestimated, possibly by a factor 3 to 5. A comparison between the jet surface brightness predicted by our results and refined spectrophotometric measurements will help clarifying this aspect in the future.

5.4. Jet mass loss and momentum transfer rates

In Table 3, we summarize the derived values for the mass loss and momentum supply rates in the examined jets, under the assumption that the density is constant over the jet section, and equal to the average over the indicated region. Since we do not apply a correction factor for shock compression, [FORMULA] and [FORMULA] may be overestimated by the same factor as the derived densities. Also, the estimate of [FORMULA] and [FORMULA] is highly dependent on the flow radius and the assumed spatial jet velocity, which is not always known since the inclination angle of the jet axis is not known. We find the average mass loss rate to vary from 3.8 [FORMULA] (HL Tau jet) to 1.2 [FORMULA] M[FORMULA] yr-1 (HH24 G), while momentum transfer rates vary between 1.6 [FORMULA] (HL Tau jet) and 3.1 [FORMULA] M[FORMULA] yr- 1 km s-1 (HH 24G). In three of our objects a molecular outflow is known to be associated with the optical jets, and it is interesting to compare the corresponding momentum transfer rates. For the HH 34 jet, we find [FORMULA] M[FORMULA] yr- 1 km s-1, while Chernin & Masson (1995) find [FORMULA] M[FORMULA] yr- 1 km s-1 for the weak blue lobe of the molecular outflow, assuming a dynamical time of [FORMULA] yr. For the HL Tau jet [FORMULA] M[FORMULA] yr- 1 km s-1 (assuming an inclination angle of [FORMULA]), while for the blueshifted part of the associated molecular outflow Monin et al. (1996) derive [FORMULA] M[FORMULA] yr- 1 km s-1. These results suggest that the jets supply enough momentum per unit time to drive the surrounding molecular flow. In HH 46/47, however, the blue lobe of the molecular outflow possesses a momentum transfer rate of [FORMULA] M[FORMULA] yr- 1 km s-1 (Chernin & Masson 1991, 1995), while we find a lower value, [FORMULA] M[FORMULA] yr-1 km s- 1 for the jet, even without correcting for shock compression.


Table 3. Average mass loss and momentum transfer rates
[FORMULA] blueshifted lobe
[FORMULA] 1 [FORMULA] this paper; 2 [FORMULA] Eislöffel & Mundt 1992; 3 [FORMULA] Heathcote & Reipurth 1992; 4 [FORMULA] Raga et al. 1991; 5 [FORMULA] Ray et al. 1996; 6 [FORMULA] Chernin & Masson 1995; 7 [FORMULA] Eislöffel & Mundt 1994; 8 [FORMULA] Heathcote et al. 1996; 9 [FORMULA] Chernin & Masson, 1991; 10 [FORMULA] Olberg et al. 1992; 11 [FORMULA] MRR91; 12 [FORMULA] Eislöffel & Mundt 1997; 13 [FORMULA] Mundt et al. 1990; 14 [FORMULA] Lopéz et al. 1995; 15 [FORMULA] Cabrit et al. 1996; 16 [FORMULA] Monin et al. 1996
[FORMULA] selected region: [FORMULA] z [FORMULA] from the source.
[FORMULA] for an inclination angle to the plane of the sky of [FORMULA].
[FORMULA] assuming an inclination angle to the plane of the sky of [FORMULA].
[FORMULA] selected region: [FORMULA] z [FORMULA] from HL Tau.

5.5. Further consequences for jet models

We already mentioned the importance of taking partial ionization into account in reliable shock models of the internal beam excitation. Moreover, partial ionization may introduce important differences in the modelling of magnetic acceleration and collimation of jets. When a substantial fraction of neutrals is present, ambipolar diffusion and other drift effects must be taken into account, introducing additional terms to the usual set of magnetohydrodynamic equations. Bacciotti et al. (1997) suggest that in a plasma with 10% ionization a term corresponding to the Hall effect should be introduced in the magnetic induction equation. The dispersive nature of this term leads to a weakening of the shocks that may eventually form in the beam.

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Online publication: February 23, 1999