Astron. Astrophys. 343, 571-584 (1999)

4. Modelling the cavity shape

The remarkably well defined shape of the eastern lobe of HH 211 allows detailed comparisons with morphologies predicted by theoretical modelisations of molecular outflows. We shall consider models in which the outflow is identified with the wake of a bow-shock propagating in a protostellar jet (Raga & Cabrit 1993, Masson & Chernin 1993, Chernin et al. 1994; see also the discussion in Sect. 5). This formation mechanism is strongly suggested by the overall structure of the eastern lobe of HH 211, where the CO cavity is in the wake of an H₂ bow-shock, located at the extremity of a collimated jet emanating from the protostar position.

4.1. Bow-shocks

A bow-shock can be created by the impact of the high-velocity jet onto the ambient interstellar medium ("terminal shock"), but also by the formation of an "internal working surface" within the jet, at the position of a strong velocity discontinuity (Raga & Cabrit 1993). In both cases, a double-shock structure is developing within the jet, with a high density region trapped between the two shocks. This high pressure gas is ejected sideways and interacts with the ambient medium to form a bow-shaped double-shock structure (see Fig. 10). The physical and kinematical structures of such bow-shocks have been studied in detail by many authors (see e.g. Hartigan et al. 1987, Raga & Cabrit 1993, Wilkin 1996 and references therein). We refer to Raga & Cabrit (1993) for a discussion of the differences between terminal bow-shocks and those created by internal working surfaces.

Fig. 10. Sketch of the structure of a bow-shock (velocities are indicated in the reference frame of the shock). A high density region in the jet ejects material in the transverse direction at a velocity , and with a mass loss rate . The impact of this material onto the ambient medium (density ) creates a bow-shock, which is propagating at a velocity with respect to the protostar. The ejection zone within the jet is supposed to be small enough to be identified with a point, marked by a cross, which is at the origin of the cylindrical coordinates system used in the text.

Dyson (1975) derived the geometry of a momentum-driven bow-shock, assuming the shape is determined by normal ram-pressure equilibrium between the ejected material and the ambient medium (i.e. neglecting the centrifugal pressure term). The resulting shape is:

where and denotes cylindrical coordinates with origin at the ejection point (see Fig. 10) and is the stand-off distance between the ejection point and the bow-shock apex. This characteristic size of the shock is given by (Baranov et al. 1971):

where and are the mass-loss rate and matter-ejection velocity within the bow shock, is the density of the ambient medium, and the bow propagation velocity. Recently, Wilkin (1996) presented a complete, analytical solution of the geometry, velocity distribution, and surface density distribution of a momentum-driven bow-shock. The shape of the bow is given by:

This relation is in good agreement with the Dyson's solution for small , i.e. close to the shock axis, but differs significantly for large angles, because of the inclusion of the centrifugal pressure term.

The Eqs. 1, 2 and 3 assume that the ejection of matter from the jet is isotropic. Zhang & Zheng (1997) and Wilkin et al. (1997) have investigated the case where the ejection is confined in a cone of solid angle surrounding the jet axis. Not only the definition of has to be modified ( is replaced by ), but the whole shape of the bow is affected and appears more collimated. In addition, the velocities are more forwardly directed. The structure of an internal working surface is actually even more complex, since in this case the ejection of matter within the shock probably does not occur around the axis but rather in a cone oriented perpendicular to the jet axis. Numerical simulations indicate that the ejection extends from to (Falle & Raga 1993). In the following, we shall only consider the simplest case (isotropic ejection) and therefore the bow shapes given by Eqs. 1 or 3.

4.2. Shock propagation

As the bow-shock is travelling down within the jet, its typical size is most probably changing, since the parameters in Eq. 2 can vary. For instance, the ambient density, , is expected to decrease with distance from the central source. Following Raga & Cabrit (1993), we will assume a simple power-law relation between and the distance x between the protostar and the shock:

If is the only varying parameter, this last relation simply results from a density stratification of the form . In this case, would correspond to the classical paradigm of the protostellar envelope density decreasing as .

Pending an analytical solution for a propagating bow-shock, we will use the more empirical, approximate method proposed by Raga & Cabrit (1993) to determine the shape of the resulting cavity by integrating the relations describing the shape of the (steady-state) bow-shock (Eqs. 1 or 3) and its propagation (Eq. 4). The origin of the angles is fixed at the present position of the shock, at a distance d from the protostar. Any point on the curve we are trying to determine has thus a projection on the jet axis. To take into account the continuous variation of the shock position, and therefore of the value of , we will assume that follows the same function of as in the steady-state case (Eqs. 1 or 3), but with evaluated at the projected distance x of the point under consideration. This hypothesis is probably justified for the case of a slowly-varying . The overall shape of the propagating bow-shock can thereby be computed from the following set of equations:

There are two free parameters in this simple modelisation: , which describes the evolution of the characteristic size of the bow, and d, the present position of the shock. In addition, the present size of the bow shock, , is a scaling factor for the whole model. In order not to add another free parameter, we did not consider any projection effect, and thus assumed the flow to be in the plane of the sky.

Raga & Cabrit (1993) presented the shapes resulting from the Dyson's description of the bow, and noted that they reproduced typical flow morphologies in a satisfactory way. These curves are presented in Fig. 11, together with the shapes we obtained by using the more realistic bow geometry given by Wilkin (Eq. 3). The latter curves have similar behaviour to those calculated by Raga & Cabrit, but are "rounder" in the part of the flow close to the exciting source.

Fig. 11. Shapes of the cavities created by the propagation (from right to left) of a bow-shock, assuming the bow geometry is described by Eq. 1 (dashed lines - see Raga & Cabrit 1993) or Eq. 3 (continuous lines). In both cases, the characteristic size of the bow varies as , where x is the distance from the protostar. All distances are in units of , where d is the present position of the shock. In this figure, .

4.3. Comparison with the observations

Fig. 12 shows the extremely good agreement between the HH 211 eastern lobe and one of the shapes we obtained with the model described in the previous section, assuming that the bow-shock geometry is given by Eq. 3. It is actually surprising to find such good agreement between the observations and the result of our very simple calculations. It may be worth to remind here that very strong hypothesis (isotropic wind into the bow-shock, approximate method used to take into account the shock propagation; see previous sections) have been made in the calculation of the curves presented in Figs. 11 and 12. This toy-model is more intended to provide a first guess of the cavity shape created by the propagation of a shock rather than to be a realistic modelisation. Several physical or chemical effects (e.g. opacity effects along the line of sight, dissociation of the molecules within the shocks) have not been taken into account and could also affect the observed morphology, even if they seem not to play a crucial role for HH 211.

Fig. 12. Comparison between the shape of the cavity predicted by our model (thick line) and the observations: CO  low-velocity emission (see Fig. 4; note that the shape of the cavity is not significantly modified if other integration limits are used) and 230 GHz continuum emission (see Fig. 9). The image has been rotated for clarity. The parameters of the model are: , , and  AU.

The curve presented in Fig. 12 corresponds to (i.e. to a conservative value) and to a ratio (which reflects the high collimation of the flow). Our simulations show that, for a given , the value of is constrained within a few unities to obtain a satisfactory qualitative agreement between the model and the observations. Note that the small, highly collimated zone in the closest vicinity from the protostar is not reproduced: this points to a higher (absolute) value of at this position (see Fig. 11, case ). This is actually consistent with the protostar being located at the edge of a molecular filament (Fig. 7), since in this case one expects the density to decrease rapidly near the source, and then more slowly in the interstellar medium.

The spatial scale of the model presented here is  AU (for a distance of 315 pc). is given by Eq. 2, in which the poorest known parameter is the mass loss rate . Assuming typical values for the other parameters, we can very roughly estimate:

where is the ambient number density. This estimate of the mass loss rate within the bow-shock is a lower limit of the actual mass loss rate in the jet, since part of the material ejected by the protostar is not lost in the bow-shock but can be accumulated at the extremity of the jet. The mass of the jet was estimated to be (Sect. 3.3). With the above mass loss rate, we can thus derive a timescale of the order of 1000 years. The kinematical age of the HH 211 jet is of the order of 750 years (note that these two ages are both proportional to , assuming that ). Obviously, these timescales are only extremely crude estimates of the actual age of HH 211: they nevertheless indicate that this object is most certainly extremely young.

© European Southern Observatory (ESO) 1999

Online publication: March 1, 1999
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