3. Physical parameters and results
The calculations have been carried out in the scheme described in Paper III for the nonlinear evolution of axisymmetric pinching (body) modes for a cylindrical jet. We have set, at , the jet Mach number and the density ratio . In order to set the remaining parameters, and , we recall from Paper II that a choice of K yields post-shock temperatures K, consistent with the low excitation spectra observed. We recall also that since the initial cooling time is large with respect to the dynamical time scale, choices of that differ up to 40% have shown in tests calculations to have very little effect on the nonlinear evolution of the instability. Concerning the choice of , observations of HH34 and HH111 are consistent with jet radii cm and particle density (Bürkhe et al. 1988, Morse et al 1993a). We have therefore adopted for the column density ; this choice implies for the jet a mass flux and a momentum flux , with .
In Fig. 1 we show a contour plot of the morphology of the emission flux, for , at the time when the train of shocks has reached a distance from the origin corresponding to about 400 radii. The results for yield similar morphologies. We note however that the general morphology is weakly dependent on the particular time chosen, as can be seen, for this set of parameters, in Fig. 1a,b ( (Fig. 1a) and (Fig. 1b). In this figure we show also the details of the morphology of the leading shock, with a clear bow-like form, and of preceding ones (i.e. at smaller z) that appear to have a more compact structure. The line fluxes are obtained by integrating the emissivity along the line-of-sight under the hypothesis that this is perpendicular to the jet longitudinal axis. From Fig. 1 we see that one of the leading shocks, being a result of shock merging processes, is somewhat more distant from the preceding one, is wider and has a distinct bow-shock like morphology. The knots result quasi-equally spaced, with mean intra-knot distance of jet diameters, and with an initial gap whose length partly depends on the amplitude of the initial perturbation imposed perpendicularly to the equilibrium velocity ( in the present case). We note also that the shocks most distant from the source weaken with time (compare Fig. 1a and b).
The spectral and kinematical characteristics of the shocks resulting from K-H instability are shown in Fig. 2 where we plot, at the time and for , the on-axis behavior against z of the electron density in units of the initial density (panel a) and the fluxes of (panel b) and (panel c); in panel d) we plot the flux ratio , obtained averaging the fluxes over the emission volume of each shock, and in panel e) we show the shock pattern speeds . As a comparison, we show in Fig. 3 the behavior of the same quantities of Fig. 2 but for and at time . In both cases of the ionization fraction attains maximum values that do not exceed , and the different shock strengths yield values of reaching (Fig. 2a) and (Fig. 3a), respectively for and . Figs. 2b,c and 3b,c show how the flux in the two lines increases, reaches a maximum and then decreases, in qualitative agreement with observations; from Figs. 2d and 3d we notice also that the line ratio attains a high value for the leading, widest shock. A further comparison of Fig. 2 with Fig. 3 show in the latter case wider initial gap and intra-shock spacings, and lower values of the ratio (Figs. 2d and 3d), consistently with the increasing of strength and excitation level of the shocks at a higher Mach number. Finally Figs. 2e and 3e show that the proper motions of knots increase with distance from the origin from up to 0.8, in the case of , and from 0.5 up to 0.7, for . These results are in good qualitative agreement with the findings of Eislöffel & Mundt (1992) for HH34. We have taken Figs. 2 and 3 as representative snapshots of typical morphologies; in fact, calculations show that the shock train, after a time scale depending on the physical parameters, reaches a asymptotic configuration that remains quasi-steady and simply shifts forward.
About the possibility of shock merging effects, that lead to bow-shock like features, we note that the condition for these processes to set in is that, locally, a shock at larger z must have proper motion smaller than the following one,i.e. at smaller z (see the discussion in the companion Paper III). Looking at Figs. 2e and 3e we can see that this condition can be verified in several positions along the jet, therefore one may expect that bow-shock like knots, originated by K-H modes, can be a more common feature than actually shown in our calculations, limited in time. We remark finally the internal consistency between the increasing of shock pattern motions with distance z along the jet (Figs. 2e and 3e), which should give a lower shock strength, and the increasing of the ratio (Figs. 2d and 3d).
Table 2. Model results, to be compared with Table 1
We leave distances and densities in units of a and , with , bearing in mind that the values must obey . In Table 2, the results for are in general agreement with observations; setting for a the values of HH34 and HH111, the major discrepancies are: i) the length of the initial invisible part of the jet (the 'gap') that is larger by a factor of with respect to observations, ii) the widths of the knots in tend to be smaller by a similar factor, and iii) the post-shock electron densities are smaller than the observed values, especially in the case , by a factor . However, being the choice of the set of parameters by no means unique, one can expect, at best, only a broad agreement from the comparison with observations. What is important to stress is the trend brought about by the variation of the Mach number, i.e. higher values of M cause an increase of the intra-knot spacing and a decrease of the ratio. Also the temporal evolution plays a role, increasing the relative length of the visible part of the jet (see Fig. 1).
Observations indicate jet velocities , implying . Unfortunately, our capability of carrying out calculations with , on the same grid (), was greatly hampered by the growing size of the integration domain. In order to gain some insight on the behavior of the instability at different Mach numbers we have carried out additional calculations on a coarser grid () for , and with a larger longitudinal size of the domain (800 jet radii). The simulation of the case has been carried out up to , when the leading perturbation reached the right boundary; therefore we cannot represent a quasi-steady situation and in Table 2, there are missing data for this case: is larger than the computational domain, thus we cannot estimate , , and the knot number; the age of the jet is also missing and the electron density is, quite likely, severely underestimated. The remaining quantities: , knot width and separation, jet velocity and knot speed are instead reasonably well defined by the simulation.
We recall that we defined M as the ratio of the jet velocity with respect to the external medium to the internal sound speed, and that is reported in Table 2. Observations of shock velocities in bow-shocks (Morse et al. 1992, 1993b, 1994) show values lower than those consistent with the measured proper motions of the bow-shocks themselves. This may suggest that the pre-shock ambient medium is not steady with respect to the central source, but may be drifting along the jet due to, perhaps, the effect of previous outburst of the source, thus lowering the actual velocity jump jet-to-ambient and the effective Mach number. In particular, Morse et al. (1992) found for HH34 a velocity of the pre-shock medium, at the bow-shock, , and for HH111 (Morse et al. 1993b).
Following Hardee & Norman (1988), it is possible to derive that, in the linear and adiabatic regime, this mode has a resonance frequency and a corresponding resonance wavelength
where is a coefficient that depends on the geometry (Cartesian or cylindrical) and on the particular mode (symmetric or asymmetric).
The numerical results for the nonlinear evolution are reported in Fig. 4a,b. In Fig. 4a symbols represent the mean intra-knot spacing, in units of a, as a function of M and the error bars indicate the difference between maximum and minimum spacing. The dashed line shows the resonance wavelength of the fastest growing mode in the linear and adiabatic regime, according to (1), and the dot-dashed line interpolates our nonlinear results but with coefficient instead of of (1). In Fig. 4b we plot intensity ratio against M. To avoid ambiguity in the choice of a particular shock, we have selected one of the first shocks of the chain that have the advantage of being nearly of constant strength for a given M, as time elapses. The behavior of the intensity ratio is well represented by a power-law fit (solid line). Therefore, in the K-H scenario, the larger is the mean intra-knot spacing the smaller must be the line ratio.
These last results show clearly how two of the main observable features of stellar jets, such as the intra-knot spacings and the line ratios, result connected if knots originate from the K-H instability, and this represents a test on the mechanisms proposed and a prediction for further observations (cf., point 7 in Sect. 2).
© European Southern Observatory (ESO) 1998
Online publication: April 28, 1998