## 3. Analysis of a single particle evolutionThis section is dedicated to a careful examination of the evolution of a test particle. ## 3.1. Parameters defining the shockFor 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 ## 3.2. The evolution schemeThe 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 10
## 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 10 ## 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 emissivitiesThe 'mean ionization'
## 3.4. Cooling lengthAs 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 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.
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 equationsOur 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.
## 3.6. Out of equilibrium versus equilibrium ionizationTDA 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 2.10
© European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |