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

3. Analysis of a single particle evolution

This section is dedicated to a careful examination of the evolution of a test particle.

3.1. Parameters defining the shock

For comparison purposes, we adopt the same particle as in TDA, i.e. a particle entering their model labeled E at 1 pc. The bowshock parameters are then: ,  pc,  km s^-1,  cm^-3 and . The ionization parameter we inferred from TDA is derived from their value of the nuclear ionizing flux (i.e., a luminosity of  erg s^-1) in the range 0.5-4.5 keV, a continuum with and an apex distance to the nucleus of 500 pc). The magnetic parameter of the ambient medium has been set to  = 3 µG cm .

3.2. The evolution scheme

The evolution of a particle along the bowshock can be divided into several stages according to the locally prevailing physical process. Note that the relative lengths and importances of each stage vary from particle to particle and depend as well on the bowshock parameters. We distinguish the following stages:

3.2.1. Thermal ionization stage.

Just after being shocked, the particle still has the same ionization state as the ambient gas while having reached a temperature beyond 10⁶  K. A fraction of the thermal energy of the gas is being used to raise the ionization level. This leads to an extremely high initial cooling rate (see Fig. 4). This stage is very short (less than 0.5 pc in curvilinear abscissa, i.e. less than 5.10³  yr) as the particle rapidly reaches ionization levels consistent with its temperature (see Fig. 5).

Fig. 4. Evolution tracks of the temperature, pressures (total, gas and magnetic), hydrogen number density and radiative cooling rate per volume unit for our test particle. Note the steep increase in hydrogen number density and magnetic pressure during the catastrophic cooling stage (  pc).

Fig. 5. Top diagrams: mean ionization levels of various elements for our test particle (left: carbon, nitrogen and oxygen; right: iron, calcium and magnesium) as a function of the curvilinear abscissa. Lower, left panel: ionic fraction of Fe XIV, Fe VII and O III as a function of the curvilinear abscissa. Lower, right panel: luminosity per unit of Z (integrated over the whole perimeter of a bowshock transversal slice) in the [O III ] 5007, [O II ] 3727 and [O I ] 6300 lines (zoom over the catastrophic cooling zone).

3.2.2. Pressure driven stage.

Once the ionization stage ends, the gas retrieves the low radiative cooling rates typical of a hot, low density plasma. Therefore, the cooling of the particle is ruled by the decreasing pressure field. In this pressure driven stage, the gas cools slowly as its pressure and density decrease. The length of this region depends strongly on the initial temperature and density of the particle. Note, that for the extreme case of very high velocity bowshocks in low density surrounding gas, the density decrease can actually inhibate the radiative cooling term (which varies roughly as the square of the density), leading to very extended pressure driven stages.

3.2.3. Catastrophic cooling stage.

As the temperature falls and reaches a few 10⁵  K, the radiative cooling engine races. The temperature decrease is not balanced any more by the pressure decrease, which leads to a fast increase in density which itself accelerates the cooling and so depresses ever more rapidly the temperature. The final compression factor of the gas is determined by the magnetic field whose pressure (which varies as the square of the density) can in some cases dominate the gas pressure. During this catastrophic cooling stage, the gas cools down to temperatures around 5 10³ -10⁴  K within a few parsecs only, releasing radiatively within a very thin zone what remains of its thermal energy. The salient features of this stage are seen in the T, , P and Q curves of Fig. 4 (around  pc).

3.2.4. Photoionization stage.

The cool, high density gas leaving the catastrophic cooling zone rapidely reaches thermal equilibrium (balance between radiative cooling and photoionization heating by the nuclear radiation). The density decreases slowly ( tightly follows the pressure evolution as the temperature is set by the thermal balance). Therefore, the ionization parameter of the gas increases and the thermal equilibrium moves slowly toward higher temperatures and higher excitation populations (quasi-static evolution).

3.3. Ionic populations and line emissivities

The 'mean ionization' ¹ degree of several elements (O, N, C, Fe, Ca and Mg) as a funtion of the curvilinear abscissa is given in Fig. 5, together with the individual ionic fraction of three ions of astrophysical interest covering a wide range of excitation (Fe XIV, Fe VII and O III). Prominent features in these curves are: the spike (at the beginning) associated with the thermal ionisation stage and the fast change of population related to the catastrophic cooling stage (  pc). It is worth noticing the importance of the photoionization stage in the luminosity profile (see Fig. 5) of intermediate and low excitation lines.

3.4. Cooling length

As the catastrophic cooling zone (see Sect. 3.2.3) is associated with a large increase in density, we have defined the cooling position of a particle, , as the location of its maximum density. The cooling length is then derived using where is the position of the shock discontinuity (projected on Z) for the particle under consideration. As shown in the previous section, corresponds to a zone of enhanced emission which influences the final position of the bulk of the optical emission (and therefore the shift between radio and optical emission).

To make a more direct comparison with TDA, we have configured another test model which conforms strictly to TDA's model by using the same cooling rates, post-shock assumptions and the pure, fully ionized hydrogen assumption (i.e., ). The derived temperature and hydrogen number density tracks are displayed in Fig. 6. Note that we exactly recover their curves (see their Fig. 4; model E, particle with  pc), successfully testing the hydrodynamical routines of our code.

Fig. 6. Temperature and hydrogen number density as a function of Z for a particle with  pc in the case of a simplified model which strictly conforms to TDA's model E (see their Fig. 4). Bowshock parameters: ,  pc,  km s-1,  cm-3, and  µG cm .

The striking difference between the evolutionary tracks of our complete model (Figs. 4 and 5) and those of TDA (Fig. 6) in which simplified physical assumptions have been used is noteworthy. For instance, for the same test particle, our complete model and TDA's one lead to values as different as 133.2 and 57.4 pc, respectively. This important difference in a key quantity is due to the cumulative effects of modified post-shock conditions (see Sect. 3.5) and different cooling rates.

3.5. Influence of post-shock conditions equations

Our set of post-shock condition equations differs from the one of TDA (see Sect. 2.4). This leads, for a given Mach number, to a higher temperature and a lower density. In order to estimate the impact of such a change on the fate of the particles, we now compare the above model (Fig. 6) which closely reproduces TDA's model E, with one which uses our post-shock conditions but keep TDA's simplified cooling function (allowing us to get rid of the influence of using different cooling functions). Temperature and density evolutions for the same test particle as before are displayed in Fig. 7. We derive cooling lengths more than twice as long than those of TDA. We emphasize this difference in since it is one of the main observationnal constraints of any bowshock model. Our comparison illustrates well the essential role played by the pressure driven zone (see Sect. 3.2) in the evolution of the particle as well as the high sensitivity of the model to the choice of post-shock conditions.

Fig. 7. Comparison between TDA's model E (dashed line; same model as in Fig. 6) and our model when configured to use TDA's cooling function (solid line). The magnetic parameter of the ambient medium has been set to zero as in the model of Fig. 6.

3.6. Out of equilibrium versus equilibrium ionization

TDA assumed 'Collisional Ionization Equilibrium' (hereafter CIE) for the ionization balance of oxygen. This can be contrasted with the more physical non CIE assumption as implemented in our model (for each element listed in Sect. 2.1) and in which any particle's evolution depends on its past history and on the rates of change of each ionic specie (see Sect. 2.6.2). Within the zones of high radiative cooling (i.e., in the range 2.10⁵  K to 10⁵), when non CIE effects are taken into account, we find a mean ionization degree of the ions which is systematically higher (memory effect) than in the case of CIE (see Fig. 9). This can at times strongly reduce the efficiency of radiative cooling, lowering the cooling rate of the gas by more than a factor two as, shown in Fig. 8 (see also Sutherland & Dopita 1993). The cooling lengths of the particle for CIE and non CIE are 124.9 and 133.2 pc, respectively, and are consistent with the reduced cooling rate. In the non CIE case, this lag behind of the ionization degree has also the important implication that, for the same temperature, the ionization fractions and, therefore, the emissivities differ in the non CIE case from that of CIE. Although TDA used a cooling function which, to a first order, corrects for the main effect on the total cooling rate of using CIE (see dashed line in Fig. 8), at the time of calculating the emissivities of each line, only a proper treatment of non CIE effects as done here can ensure deriving an emission line spectrum fully consistent with the more accurate non CIE assumption.

Fig. 8. Plot of the cooling rates ( in erg s-1 cm-3) as a function of the temperature, for the Collisional Ionization Equilibrium (CIE) case (dotted line) and for the non CIE case (solid line). The dashed line curve is a fit to a non CIE cooling curve (Kafatos 1973) used by TDA in their model.

Fig. 9. Mean ionization level of oxygen as a function of temperature, for the Collisional Ionization Equilibrium (CIE) case and for the non CIE case. Note the rapid change in the non CIE mean ionization level just after the shock discontinuity (high temperature end) which results from the rapid ionization of moderately preionized gas (see Sect. 3.2.1.

© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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