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Astron. Astrophys. 322, 73-85 (1997)

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4. Model output - Result example

The model output consists of a file containing the physical (temperature, density, and line emissivities) and geometrical (position, velocity and volume) characteristics of each (temporal) cell of each (flow) particle (i.e. typically 700-800 particles of 2000-3000 cells each). This allows us to derive not only the global properties of the bowshock (total luminosities, line profiles) but also construct 3D data cubes similar to those obtained with bidimensionnal spectrographs. These results (obtained for various shock velocities, ambient gas densities, ionization and magnetic parameters) will be presented in a second forthcoming paper. In what follows, we will focus on the analysis of the luminosity profiles in the case of three examples (models #1, 2 and 3; see Tab. 1). Note that the model #2 has the same bowshock parameters as in TDA's model E.


Table 1. Parameters defining the three models.

4.1. Tuning of the tolerance parameters

The setting of the tolerance parameters of the code (see Sect. 2.6.3) has been made to comply with the following three main constraints: smooth changes of the ionic populations from one fluid cell to the other; falling well within the targetted spectral and spatial resolutions; limiting the computation time. For each model run, the choice of tolerance parameters have been checked against a grid of test particles in which we could trace the accuracy in the evolution of the mean ionization levels of the elements and of the cooling positions [FORMULA]. An example of the change in [FORMULA] as a function of the maximum allowed rates of change is given in Table 2.


Table 2. Example of the evolution of [FORMULA] as a function of the maximum allowed change rates for our test particle. Note the increase of accuracy along with the number of fluid cells. Changes of less than 0.1 pc fall within the numerical noise of the model.

Except for the thermal ionization stage (defined as the first 5.103 yr of the particle's evolution) where a high accuracy is required (maximum allowed change rates of 0.01 %), we used the following set of tolerance parameters: maximum allowed change rates of 0.5 % (for T, [FORMULA] and [FORMULA]) and 0.25 % (for pressure [FORMULA]); velocity sampling of 5 km s-1 ; Z sampling of 1 pc. The maximum pressure change rate must be lower than the others because it governs the sampling of the pressure driven stage Sect. 3.2) for which a lowering of the accuracy quickly affects the derived cooling position.

4.2. Cooling length

Fig. 10 shows the evolution of the cooling position [FORMULA] as a function of the entrance position [FORMULA] of the test particle. The overall shape of this curve for model #1 is very similar to that of TDA, although displaying much larger cooling lengths for particles with [FORMULA] smaller than a few parsecs. Curves of models #1 and #3 are only slightly different as can be gathered from Fig. 10 while the corresponding curve for model #2 (not shown for clarity) is almost identical to #1. This indicates that neither the nuclear ionizing radiation, nor the magnetic field, significantly affect the evolution of the gas until it has cooled and reached the photionization stage.

[FIGURE] Fig. 10. Cooling distance [FORMULA] of different test particles as a function of their entrance position [FORMULA] for our models # 1 (solid line), # 3 (dashed line) and for TDA's (dotted line). Cooling distances for the model # 2 are almost identical to those of model # 1 and are not shown.

Luminosity profiles (for model #1) of lines with very different excitation levels are displayed in Fig. 11 (panels A, C and D). The curves differ markedly in position and shape according to the degree of excitation. Indeed, extremely high excitation lines like [Fe XIV ] [FORMULA] 5303 are emitted during the pressure driven stage and show a profile concentrated towards low Z (i.e. higher temperature gas) while somewhat lesser excited lines like [N II ] [FORMULA] 6583 or [O III ] [FORMULA] 5007 are emitted either during the catastrophic cooling or the photoionization stages. In general, lower excitation lines can be expected to display broader profiles than the high excitation lines (e.g. [O II ] [FORMULA] 3727, [N II ] [FORMULA] 6583).

[FIGURE] Fig. 11. a [O III ] [FORMULA] 5007 (solid line), [O II ] [FORMULA] 3727 (dotted line) and [O I ] [FORMULA] 6300 (dashed line) luminosity profile s (luminosity per unit of Z after integration across an entire bowshock transversal slice) for model #1. b Comparison of the [O III ] [FORMULA] 5007 luminosity profile of the three models, #1 (solid line), #2 (dotted line) and #3 (dashed line). c [Fe XIV ] [FORMULA] 5303 (solid line), [Fe VII ] [FORMULA] 6086 (dotted line) and [O III ] [FORMULA] 4363 (dashed line) luminosity profiles for model #1. d [C IV ] [FORMULA] 1549 (solid line), [O VI ] [FORMULA] 1035 (dotted line) and [N II ] [FORMULA] 6583 (dashed line) luminosity profiles for model # 1.

It is interesting to note the presence of two 'peaks' in the luminosity profiles of [O III ]. These are even more striking in the [O III ] [FORMULA] 5007 luminosity profile of model #3 (see Fig. 11, panel B) since the contribution of the photoionized gas to the [O III ] [FORMULA] 5007 luminosity is much lower than in models #1 or 2. Such peaks take place at the corresponding distances where the [FORMULA] vs [FORMULA] curve shown in Fig. 10 presents two extrema (e.g. [FORMULA] 1 and 20 pc corresponding to [FORMULA] 40 and 140 pc, respectively). This was first pointed out by TDA.

Towards larger Z's, the gas is unable to recombine but tends instead towards photoionization equilibrium as a result of the presence of the external ionizing field. Model #1 ([FORMULA]  = 0.012) shows that at larger Z the line emission due to photoionization comes to dominate the profiles of the low and intermediate excitation lines (e.g. [O III ] [FORMULA] 5007, [O II ] [FORMULA] 3727 and [O I ] [FORMULA] 6300). A comparison between the [O III ] [FORMULA] 5007 luminosity profile of model #1 with #3 (of much smaller [FORMULA]  = 0.001; see Fig. 11, pannel B) illustrates well the role played by the external ionizing field in our model. This suggests that for a spatially unresolved bowshock, it is not technically possible in our model to distinguish the spectral signature (in terms of line ratios) of the shock from that of photoionization. On the other hand, for spatially resolved cases, as in large scale ionized gas in Radio-Galaxies, comparison of the positions of the high and low excitation lines will allow us to put strong constraints on our model as well as on its input parameters.

4.3. Integrated luminosities and line ratios

The integrated luminosities beeing dominated by the emission of the denser and cooler downstream photoionized gas, the bowshock models display rather low excitation spectra as compared to that of the diffuse (preshock) ambient gas. As our model clearly presents two distinct emission zones: shock excited and photoionized, it conforms with the current trend of recent models to invoke more than one component to account for the observed ENLR emission (see review by Morse et al. 1996). For instance, Dopita & Sutherland (1996) have proposed a model in which the photoionized precursor can account for up to half of the Balmer line luminosity while the recent photoionization models of Binette et al. (1996) propose a mixture of matter-bounded and ionization-bounded clouds.

4.4. Bowshock luminosity profiles

A detailed analysis of the behavior of the line ratios as a function of the input parameters of the model will be presented in a forthcoming paper. However, comparison between the three models shown above already reveals interesting trends. First, the overall excitation of the spectra increases with magnetic field (see model #1). Indeed, by lowering the compression factor of the gas, a larger field has the effect of increasing the ionization parameter of the gas during the photionization stage. Also interesting is that as the ionization parameter of the ambient medium is lowered (e.g. model #3), the temperature indicated by the [O III ] [FORMULA] 4363/[O III ] [FORMULA] 5007 line ratio becomes more representative of collisional excitation (e.g., [FORMULA] K and [FORMULA] K for models #1 and #3, respectively; c.f. Tab. 3).


Table 3. Integrated total luminosities (over the range 0-350 pc) relative to H [FORMULA] 4861 for a set of astrophysically interesting lines. Absolute H [FORMULA] 4861 luminosities are: 1.96 1039 erg s-1 (# 1); 2.83 1039 erg s-1 (# 2); 1.56 1039 erg s-1 (# 3).

4.5. Model limitations

A good review of the limitations of this type model has been already given by TDA in their paper. They are related to the assumptions made either in the hydrodynamical description of the bowshock, or in the computation of the atomic and transfer processes. In the following, we only review the major limitations in regards to our model.

4.5.1. Hydrodynamical description

First, fixing the geometry of the bowshock and assuming its stationarity completely hides the complexity of the flow which could well rapidly become turbulent (as a result for instance of the growth of Kelvin-Helmoltz instabilities at the tangential discontinuity surface between the shocked ambient medium and the radio material). This is apparent if one compares with hydrodynamical simulations (e.g. Steffen et al. 1996), especially for the tails of the bowshock (high Z) where an expanding cocoon model is much more relevant. Finally, the assumption that the particles do not interact thermally must break down close to the apex where the flow is very slow (stagnation zone).

4.5.2. Internal ionizing sources

By incorporating MAPPINGS IC to the TDA model, we have eliminated most of the drawbacks related to their approximate description of the atomic processes (CIE assumption, absence of charge transfer reactions, approximative cooling rate...). In the present model, however, we still assume that the only ionizing radiation is that produced by the active nucleus, therefore neglecting the contribution of the diffuse ionizing field generated in situ within the shock (hard UV line and continuum radiation from the very hot gas). This diffuse radiation could well affect the ionization balance of a fraction of the gas. Dopita & Sutherland (1996) found for instance that a shock velocity of 500 km s-1 internally generates enough radiation to reproduce an ionization parameter of [FORMULA] 8 10-3, which is comparable to the values inferred from photoionization models of AGN. We emphasize, however, that the infinite slab geometry which they adopted is very different from the thin laminar flow which takes place along our bowshock structure. In practice, the cooler, denser gas layers (i.e. which have already cooled) are effectively exposed only to the nearby outer layers which flow around it. Indeed, we are justified in considering that the photons generated by the hot gas when it is more distant than a few shocked gas layer widths (typically less than ten parsecs) away, either downstream or upstream, can be safely neglected as a result of the large geometrical dilution affecting such UV sources.

On the other hand, even within a few shocked gas layer widths, the amount of diffuse radiation generated appears not to be negligible. As inferred from Fig. 12, the fraction of very hot ([FORMULA] 106 K) gas within nearby layers and coexisting with the warm emitting line gas, is substantial. To estimate its impact on the ionization balance, we have computed with MAPPINGS IC the hot gas column density, at 106 K, which is required to change by ten percent the population of the dominant OIII and OII ions of the warm (104 K) gas. The results indicate column densities of order 3 1018 and 1019  cm-2 in the case of OIII with [FORMULA] =10-2 and of OII with [FORMULA] =10-3, respectively (where [FORMULA] is an estimate of the ionization parameter imposed by the external AGN radiation). These should be compared with the 1019 cm-2 hot gas column density shown in Fig. 12. To conclude, although the amount of diffuse field is certainly not as overbearing as in the infinite slab case, neglecting it altogether is probably an excessive stance and can be considered a caveat of the current models. Having said this, since our warm gas which produces most of the strong optical lines is already photoionized from an infinite supply of external photons, adding additional sources is probably inconsequential to the profile and luminosities of those lines.

[FIGURE] Fig. 12. Temperature stratification of the bowshock layer (model # 1) at three different positions ([FORMULA] pc, solid line; [FORMULA] pc, dashed line; [FORMULA] pc, dotted line). The amount of gas at each temperature is expressed in terms of column density. Note the rapid vanishing of the hot gas component as Z increases.
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© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998