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

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3. The diagnostic technique

3.1. Physical foundation and validity

Here, we briefly describe the essential features of the spectroscopic diagnostic procedure and introduce some recent improvements that reduce considerably the errors in the determination of the searched physical quantities. The most commonly observed lines in Herbig-Haro jets are H[FORMULA], the [SII] doublet at 6716,6731 Å, the [OI][FORMULA] 6300,6363 and the [NII][FORMULA] 6548,6584 lines. The only physical parameter that can be directly determined from the intensity of these lines in a model-independent way is the electron density, from the [SII] doublet (e.g., Osterbrock 1989). Following BCO95 it is possible, however, to determine as well the average ionization fraction [FORMULA] in the particularly low excitation conditions which apply in the beam section of stellar jets. The procedure is based only on some very general assumptions about the processes that regulate the ionization state of the atomic species involved. The results for the ionization fraction do not depend on any specific mechanism of jet formation and/or evolution.

In jet beams, as in all low-excitation nebulae, sulphur can be considered to be all singly ionized. In its original form, the procedure assumed that the ionization fractions of oxygen and nitrogen were primarily determined by charge exchange reactions with hydrogen. This allowed one to express the population of neutral O and singly ionized N as a function of the hydrogen ionization fraction [FORMULA]. As a consequence, one could determine [FORMULA] from the comparison of computed and observed line ratios. Charge-exchange as the dominant process controlling the ionization of O and N is suggested by the fact that in the linear section of HH jets there is little observational evidence for high electron temperature or for local or nearby sources of energetic radiation; in addition, previous investigations indicated that a substantial fraction of neutral hydrogen should be present in the beams (see, e.g., Raga 1991, HMR94).

In a second inspection, however, we recognized that although radiative ionization processes are probably not very important in these regions, collisional ionization terms should be implemented in the equations for the ionization state of O and N, given the presence of shocks along the beams. Moreover radiative plus dielectronic recombination can in principle compete with charge exchange. In fact comparing the corresponding reaction rates (see, e.g., Osterbrock 1989) one finds that while for oxygen charge exchange rates are four orders of magnitude larger than the recombination rates (and hence absolutely dominant), for nitrogen the charge exchange rate coefficients are three orders of magnitude smaller than for oxygen, and only slightly larger than the recombination rates. The recombination time scale for nitrogen is, however, about three times smaller than for hydrogen. So, as a first approach it appeared reasonable to assume that after the initial ionization in the jet acceleration region, recombination of N decreases its ionized fraction rapidly to the point where charge exchange becomes dominant, as assumed in the original version of our technique. Quantitatively, comparing the recombination and charge exchange reaction rates at the temperatures of interest, we found that for nitrogen the exclusion of any other mechanism besides charge exchange is a valid approximation only as long as the hydrogen ionization fraction [FORMULA] is lower than about 0.5.

To overcome these limitations, we recently included in the O and N ionization equations terms describing both collisional ionization, and radiative and dielectronic recombination. Since all the implemented rates depend on the electronic temperature [FORMULA] only (see below), the ionization state of O and N can also be expressed as a function of [FORMULA] and [FORMULA] in this case, and the procedure can be applied as well.

We do not treat photoionization processes. We checked the validity of this approximation in a case in which a stellar source of mildly energetic photons locally maintains a partial degree of ionization in the gas, as a result of the competing effects of photoionization and hydrogen radiative recombination. This physical situation seems appropriate for the outer edges of H ii regions, but is unlikely in the beams of stellar jets. Expressing the ionizing flux in terms of the standard ionization parameter [FORMULA], i.e. the ratio between the ionizing photons and the free electrons at a given distance from the source, it is again possible to evaluate the relative importance of photoionization and the other effects as a function of temperature and hydrogen ionization fraction for oxygen and nitrogen. We find here that oxygen is again completely regulated by charge exchange, while for nitrogen the neglect of photoionization is justified as long as [FORMULA] is lower than 0.5-0.6.

In conclusion, our procedure can be applied to the beams of all those jets in which low excitation and ionization conditions are expected to hold, so that the plasma is mostly neutral. The objects will be selected on the basis of the the absence of lines of high excitation or from highly ionized species as e.g. O[FORMULA]. The method allows, however, to identify regions of both high and low ionization.

3.2. The diagnostic diagrams

The technique uses line ratios from different species, therefore we have to assume a set of relative abundances to compute them. Here, the abundances of nitrogen, oxygen and sulphur relative to hydrogen are taken to be N/H = [FORMULA], O/H = [FORMULA] and S/H = [FORMULA]. Now the intensity ratio of any two of the observed lines can be expressed as a function of the electron density [FORMULA] and the temperature [FORMULA], which determine the population of the higher levels, and of the hydrogen ionization fraction [FORMULA], which, together with the temperature, regulates the relative abundances of H+, N+ and O0.

Under the assumption of local ionization equilibrium with respect to the fractional ionization of hydrogen, the following relationship holds for both oxygen and nitrogen:

[EQUATION]

where [FORMULA] are the collisional ionization rates for these species, [FORMULA] are the direct plus dielectronic recombination rates, [FORMULA] and [FORMULA] are the direct and inverse charge exchange ionization rates, respectively. Following Raga et al. (1997), we take the collisional ionization and the radiative recombination rates from Landini & Monsignori-Fossi (1990), the dielectronic recombination rates from Landini & Monsignori-Fossi (1990) and Nussbaumer & Storey (1983), and the charge exchange rates from Kingdon & Ferland (1996) and Arnaud & Rothenflug (1985). For each considered ion the emissivity in the lines of interest is found using a code by A. Raga (priv. comm.) that calculates the statistical equilibrium populations in the excited levels of the various species as a function of the electron density and the temperature (for details see BCO95, BHN96). Once the electron density is determined from standard methods, any line ratio can be regarded as a known function of [FORMULA]. The ionization fraction is then determined numerically together with the average excitation temperature, comparing calculated and observed line ratios.

In practice, for each position along the jet for which we have measured the relative intensities of the forbidden lines, a diagnostic ([FORMULA]) diagram shows a strip along the loci of the [FORMULA] values for which the predicted line ratio equals the observed one including a [FORMULA] error. With several different line ratios observed, the intersections of the strips define the values of the local ([FORMULA]) and their uncertainty. An example of such a diagnostic diagram, for one position in the HH 34 jet, is shown in Fig. 1.

[FIGURE] Fig. 1. Example of a diagnostic diagram calculated for the position at 12:005 in the HH 34 jet. Each strip defines the loci of the ([FORMULA]) values for which the predicted line ratio equals the observed one [FORMULA] a 1[FORMULA] rms error. The intersection of the crossing stripes provides [FORMULA] K and [FORMULA] with uncertainties of [FORMULA]250 K and [FORMULA]0.007, respectively.

Originally, the diagnostic procedure used the [SII]/H[FORMULA], [SII]/[OI] and [NII]/H[FORMULA] ratios, where the simplified notations [NII], [OI] and [SII] stand for the sum of the two [NII] lines at 6548 and 6584 Å, the sum of the two [OI] lines at 6300 and 6363 Å, and the sum of the sulphur lines at 6716 and 6731 Å, respectively. As recently demonstrated by Pat Hartigan (priv. comm.), the H[FORMULA] line, however, cannot be used for a determination of the physical parameters of the emitting gas in this procedure. This is due to the fact that H[FORMULA] emission can be produced both by collisional excitation, arising at high temperatures (several 104 K) and moderate ionization, and by recombination, mainly occurring at low temperatures (less than 6-7000 K). In the cooling region behind a shock front steep temperature gradients are present, and both emission mechanisms contribute to the observed H[FORMULA] line. This is evident in Figs. 2 and 3, that show the post-shock peak-normalized intensity for each line of interest (collisional H[FORMULA], recombination H[FORMULA], [SII] 6716+6731, [OI] 6300+6363, [NII] 6583+6548) for a 35 km s-1 and a 70 km s-1 shock. We calculated the intensity profiles with our radiative code, extracting [FORMULA], [FORMULA] and the compression factor from the plots in Fig. 1 of HMR94.

[FIGURE] Fig. 2. Post-shock line intensities relative to peak values for the 35 km s-1 shock in Fig. 1 of HMR94, with superposed profiles of the temperature and ionization fraction (the former divided by 105 and the latter multiplied by a factor 10 for clarity)

[FIGURE] Fig. 3. Post-shock line intensities relative to peak values, temperature (divided by a factor 2 105) and ionization fraction for the 70 km s-1 shock in Fig. 1 of HMR94

The line emission arises on a scale of 1013 - 1015 cm, which at the typical distance of HH jets (at least 100 pc) is not spatially resolved in ground-based observations. Thus, it is not possible to separate the two H[FORMULA] contributions, and the use of the H[FORMULA] line in the diagnostic may lead to misinterpretations of the physical conditions of the emitting gas. The forbidden lines, on the contrary, come from regions characterized by intermediate temperatures and ionization fraction, which are rather similar one to each other. Therefore, the H[FORMULA] line is not used in our diagnostic procedure, and the diagrams are constructed from the ratios [SII]/[OI] (sensitive to both [FORMULA] and [FORMULA]), [OI]/[NII] (sensitive mainly to [FORMULA]) and [NII]/[SII] (to check consistency).

3.3. Validation of the technique

We compared the results of this procedure to the predictions of the radiative shock models in HMR94. First, we tested if similar average values for [FORMULA] and [FORMULA] will be obtained for the regions in which the different species radiate. For doing this, we determined the average values of [FORMULA] and [FORMULA] weighted by the flux of the various lines from Figs. 2 and 3 as:

[EQUATION]

where [FORMULA] is the flux in the different lines and z is the (linear) distance from the shock front. The resulting average properties for the emission regions of the various ions are listed in Table 1. They show that for both shock speeds, the average [FORMULA] values are remarkably similar for the forbidden-line emitting species, and close to the value of the [FORMULA] plateau of Figs. 2 and 3. Therefore, they can be considered representative of the conditions of the emitting gas. The temperature, however, varies more over distance, though the average values are similar for the various forbidden lines. The "average temperature" can therefore only be considered as a rough indication of the excitation temperature of the various ions.


[TABLE]

Table 1. Flux-weighted average [FORMULA] and [FORMULA] in the shock cooling region.


In a second step, we checked if at a given "finite" spatial resolution similar results will be obtained through our diagnostic procedure. This time, we integrated the line emission given in Figs. 2 and 3, because these region would not be resolved in our spectra. For the integrated line intensities we constructed then our diagnostic diagrams (Figs. 4 and 5), using the line ratios [SII]/[OI], [OI]/[NII] (and [NII]/[SII] to check consistency). We take as [FORMULA] the post-shock electron density weighted by the [SII] lines (as it would be if one would derive it from the observations): this turns out to be 212 cm-3 for the 35 km s-1 shock and 4700 cm-3 for the 70 km s-1 shock. The diagnostic then provides [FORMULA] and [FORMULA] K for the 35 km s-1 shock, while for the 70 km s-1 shock we derive [FORMULA] and [FORMULA] K. Indeed, the derived values agree very well with the values quoted in Table 1, i.e. our method reproduces the ionization fractions and the excitation temperature of a shock code averaged over the forbidden-line emission region well. In particular, the resulting values are close to the [SII]-weighted averages. This is desirable, since also the electron density is measured from the [SII] lines.

[FIGURE] Fig. 4. Diagnostic diagram obtained for the integrated line ratios of the 35 km s-1 shock in HMR94. The curves cross at [FORMULA] and [FORMULA]. The values are error-free here, since the input line ratios are calculated numerically.

[FIGURE] Fig. 5. Diagnostic diagram for the integrated line ratios of the 70 km s-1 shock in HMR94. One obtains here [FORMULA] and [FORMULA].

We also tested if our technique gives results similar to those of HMR94 when applied to observed data. In that paper the authors examine three prototypical jets, and find the average ionization fractions comparing observed spectra integrated over the brightest part of the beams with their grid of shock models. From the observed [NII]/[OI] ratios HMR94 find for HH 34, HH 46/47 and HH 111 [FORMULA] and 0.052 respectively. With the same observed parameters, our diagnostic provides [FORMULA] 0.027, 0.070 and 0.064. These results are consistent within about 20% with the ones given in HMR94. The difference could come from different elemental abundances, the peculiar preshock density or ambient magnetic field adopted in the shock models. Therefore, we conclude that the new version of our method gives results in agreement with the HMR94 shock calculations (see also Sect. 5.1). While the improved technique in most cases provides ionization fractions similar to those found with the original version, major differences are found for the average [FORMULA], which can be substantially higher than the 5000-6000 K that were usually derived. Higher values of the excitation temperature are, however, much more reasonable in the context of shock excitation. It must be remembered that both the ionization fraction and the temperature we derive are averages weighted by the flux of the considered forbidden lines. On the other hand, the shock models show that the gas ionization fraction maintains an almost constant value along a large portion of the cooling region, while temperature varies rapidly by two orders of magnitude with distance from the shock front. As a consequence, the ionization fraction we find can be considered highly representative of the emitting gas as a whole, whereas the provided "[FORMULA]" is only a rough indication of the local excitation temperature.

One could ask if our diagnostic procedure is also applicable if the gas entering the shock has a substantial pre-ionization. Then, a certain amount of O and N could be ionized through charge exchange and collisional ionization just behind the front. Since the excitation of the forbidden lines grows rapidly with temperature, this might produce two separate regions of emission: one close to the shock front at high temperature and moderate ionization and the other in the intermediate layers with lower [FORMULA] and higher ionization. As with H[FORMULA], we would have the problem of not being able to distinguish the two regions at our resolution. A proper answer to this question would require the determination of the ionization structure of O and N running a shock model into a partially ionized medium. As a zero-order approach, however, one can add a predetermined value to the shock ionization fraction profile in HMR94, and evaluate the line emission profiles as a function of the distance from the front (as in Figs. 2 and 3). We examined the cases of the 35 km s-1 shock with additional ionizations of [FORMULA] 0.2 and 0.4, and of the 70 km s-1 shock with an additional ionization [FORMULA]. As expected, close to shock front the emission in the forbidden lines is enhanced with respect to the case of propagation into a neutral medium; the peak emission, however, is again localized in the intermediate region, thanks to the increase of the ionization in that zone and, more important, to the compression of the postshock gas. Therefore, a determination of the [FORMULA] plateau value is still possible: the [FORMULA] and [FORMULA] averages weighted by the emission of the forbidden lines are similar to each other, and close to the values determined by the diagnostics within better than 10-15%.

The amount of dust extinction towards the various positions along the jets is generally unknown, and may even vary along the beam of a single object. Therefore, we did not apply any dereddening correction to the relative intensity of the lines in our analysis. On the other hand, reddening is not expected to have a big affect on our diagnostic results, due to the proximity in wavelength of the lines used. We estimated the influence of reddening assuming a fiducial value for the visual extinction of [FORMULA] mag (such an extinction has been estimated towards a few T Tauri stars, including T Tau itself). Using this value and the standard interstellar extinction curve of Savage & Mathis (1979), one finds that the [SII]/[OI] ratio calculated from the observed values would be overestimated by about 18% with respect to the emitted ratio, while both the [OI]/[NII] and the [NII]/[SII] ratio would be underestimated, by about 9%. As a consequence, in a `dereddened' diagnostic diagram the [OI]/[NII] contours would be slightly shifted towards lower ionization, while the [SII]/[OI] and the [NII]/[SII] contours would be shifted toward higher temperatures. Recalculation of the jet parameters in several selected positions in various jets assuming such a reddening has confirmed that due to the limited wavelength range of the used lines and to our choice of the line ratios, the uncertainty in the determination of [FORMULA] and [FORMULA] because of uncorrected reddening is in any case not larger than the measurement error, being at most about 8-10% for the ionization fraction, and about 15% for the temperature.

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© European Southern Observatory (ESO) 1999

Online publication: February 23, 1999
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