## 3. The diagnostic technique## 3.1. Physical foundation and validityHere, 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, the [SII] doublet at 6716,6731 Å, the [OI] 6300,6363 and the [NII] 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 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 . As a consequence, one could determine 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 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 only (see below), the ionization state of O and N can also be expressed as a function of and 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 , 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 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. The method allows, however, to identify regions of both high and low ionization. ## 3.2. The diagnostic diagramsThe 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 = , O/H =
and S/H =
. Now the intensity ratio of any two
of the observed lines can be expressed as a function of the electron
density and the temperature
, which determine the population of
the higher levels, and of the hydrogen ionization fraction
, which, together with the
temperature, regulates the relative abundances of H Under the assumption of local ionization equilibrium with respect to the fractional ionization of hydrogen, the following relationship holds for both oxygen and nitrogen: where are the collisional ionization rates for these species, are the direct plus dielectronic recombination rates, and 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 . 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 () diagram shows a strip along the loci of the values for which the predicted line ratio equals the observed one including a error. With several different line ratios observed, the intersections of the strips define the values of the local () and their uncertainty. An example of such a diagnostic diagram, for one position in the HH 34 jet, is shown in Fig. 1.
Originally, the diagnostic procedure used the
[SII]/H, [SII]/[OI] and
[NII]/H 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 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 emission can be produced both by
collisional excitation, arising at high temperatures (several
10
The line emission arises on a scale of 10 ## 3.3. Validation of the techniqueWe compared the results of this procedure to the predictions of the radiative shock models in HMR94. First, we tested if similar average values for and will be obtained for the regions in which the different species radiate. For doing this, we determined the average values of and weighted by the flux of the various lines from Figs. 2 and 3 as: where is the flux in the
different lines and
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
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
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 and 0.052 respectively. With the same observed parameters, our diagnostic provides 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 , 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 "" 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 and higher
ionization. As with H, 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 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 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 and 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. © European Southern Observatory (ESO) 1999 Online publication: February 23, 1999 |