## 3. Physical parameters and resultsThe 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
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
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 ## 3.1. DiscussionIn Table 2 we list the main results of our model for (the reader should make comparisons with observations of HH34 and HH111 in Table 1).
We leave distances and densities in units of 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 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
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 |