7. The H2 images
7.1. Image production technique
The 2D- and 3D-calculations enable H2 -emission maps to be generated. We present images of our models in the 1-0 S(1) and 2-1 S(1) lines of molecular hydrogen and associated integral spectra. By coupling the abundance of carbon monoxide to molecular hydrogen we also estimate the emission of CO-rotation lines.
Using the model data we calculate the emissivities of every cell in the two transitions assuming only statistical equilibrium for the vibrational states (non-LTE). This is necessary since LTE is reached only at densities above 108 cm-3. So we solve the full rate equations for the three level system with which we approximate the vibrational states of the hydrogen molecule to get the populations .
The collisional rates are adapted from Draine et al. (1983), the are the Einstein coefficients of the transitions averaged over the rotational states (Turner et al. 1977). Finally the volume emissivity is given by
where the exponential is due to the level population of the rotational states for which we assume thermal equilibrium. An approximate partition function for the rotational levels of molecular hydrogen is taken from Brand et al. (1988). Other H2 lines can be calculated in the same way. The CO abundance is assumed to be 10-4 of the number of hydrogen molecules. The emissivities are given by an analytic expression derived from detailed numerical calculations of the non-thermal level populations of CO (McKee et al. 1982). We have chosen to map the 0-0 R(1) line of CO which indicates relatively cool gas in contrast to the 1-0 S(1) line of H2. We map only zones where the velocity is non-zero to exclude the ambient medium.
The next procedure is a simple geometrical mapping of these emissivities
to a detector array (with spatial resolution 1015 cm) in the
plane of the sky. Different orientations of the flow with respect to the
observer are studied. This is achieved by assuming the medium to be optical
thin to the line emission (for the infrared H2 lines this is
a consequence of the forbidden quadrupole transitions). To calculate spectra
we use the velocity information together with the orientation of the flow
to get the red shift. Additionally applying thermal broadening (due to
the local temperature) the spectral intensity is defined for every fluid
parcel. Mapping that to a numerical spectrograph (with a velocity resolution
of 1 km s-1) we get spatially resolved spectra which can be
directly compared with the observations.
7.2. The emission maps
The subsequent diagram displays the emission structure of models 2D_1 and 2D_2 in the 1-0 S(1) line of molecular hydrogen (Fig. 8). We have chosen an inclination angle of with respect to the plane of the sky. One clearly recognizes the emission from the dense cone in model 2D_1 and the multiple ring structure in model 2D_2 which resembles limb brightened features seen in many observations.
Fig. 9 shows a molecular hydrogen map of Model 3D_1. The important 1-0 S(1) and 2-1 S(1) vibration-rotation lines of molecular hydrogen and the 0-0 R(1) rotation line of carbon monoxide are displayed. The H2 -lines peak at the bow shock, 1-0 S(1) being somewhat more extended than 2-1 S(1). At the tip the emission originates mainly from shocked jet gas (the parameters of the model are such that the jet shock does not dissociate H2), in the wings ambient H2 survives and radiates. CO-emission is seen from the whole cavity due to the low-lying rotational level of the transition. So this line indicates the location of the cool swept-up gas generated in the weak bow shock at the wall of the cavity. An additional contribution comes from the jet beam where the gas has a temperature of 100 K.
We draw attention to the bow-like arc upstream of the main bow shock. Similar features have been identified in many protostellar jets (HH 288 West (McCaughrean & Dent 1996), L1448, DR 21 West (Davis & Smith 1996a,b)). Seen from a position rotated about around the jet axis (Fig. 11), this arc looks like an elongated knot (similar to HH 212 (Zinnecker et al. 1996), HH 211 West (McCaughrean et al. 1994)). So features which seem to be multiple bow shocks are generated naturally by continuous jets.
Fig. 10 is a study of the 1-0 S(1) line for several inclination angles of the jet with respect to the plane of the sky. The source is located at the left margin of each frame. The length scales are the same as for Fig. 9. The next series (Fig. 11) shows the same but now we look at the jet from a position rotated by around the jet axis. As stated above, the comparison with Fig. 10 makes clear the origin of the second bow shock: it is dense material from the main bow shock which broke off and was advected upstream at one side. The maps in Fig. 11 are reminiscent of the chaotic shock structures in the red lobe of L1448, L1157 (Davis & Eislöffel 1995) and HH 288 East (McCaughrean & Dent 1996). Highly complex disordered lobes have also been discovered through sub-arcsecond observations of the shocked H2 gas in high-luminosity stars (DR 21 East, Smith & Davis 1996b; Cepheus A, Hoare & McCaughrean 1996; OMC-1, McCaughrean, priv. comm.). While global 3D-modelling remains impractical, high-resolution simulations have demonstrated that dense layers can be shattered by disturbed shocks, resulting in a wealth of fine-scale structure (Strickland & Blondin 1995).
Dense molecular outflows into uniform environments can clearly produce highly collimated H2 and CO line-emission distributions provided the jet is directed near to the plane of the sky. However, even at away from this plane, we find a well-collimated source. Note that in this case, besides the less streamlined appearance, the bows appear less arclike - more comma-shaped than parabolic. In general, however, all bows are predicted to be asymmetric due to the spike oscillations (with the exception of catching the jet head midway in the transition between spikes).
Figs. 14, 15, 16 and 17 display the emission maps of the pulsed jet of model 3D_2. The H2 emission in the bow looks cone-like and more extended than in the continuous jet. Again limb-brightening and asymmetric features are commonplace. The extended multiple-bow structures are a feature of protostellar jets (e.g L1634, Hodapp & Ladd 1996; HH 211 East, McCaughrean et al. 1994). The pulses in the beam are bright in H2 near to the source and fade when they travel down the flow.
7.3. Momentum and energetics
The outflow momentum can be extracted from the CO maps, which will closely follow the compressed swept-up material. The general location we predict here is consistent with the location observed in deeply-embedded sources (see Davis & Eislöffel 1995). The fact that the CO is also quite well collimated is consistent with recent findings for HH 211 (McCaughrean, priv.comm) as well as other very young sources observed with sufficient resolution (e.g. L1448, Bachiller et al. 1995a).
The relationship of the total luminosity to the source power is now derived (for one side of a bipolar flow). The 1-0 S(1) luminosity is for the non-pulsed jet model, from which a total H2 luminosity of is estimated (taking into account the 2-1 emission of and using the data of Smith (1995)). The jet power is , most of which is transferred into the atomic gas.
The 0-0 R(1) CO luminosity is . This corresponds to a radiating mass of order of on employing a conversion factor of erg s-1 molecule-1. The CO-derived mechanical luminosity is then where is an averaged velocity component along the line of sight. We thus find a wide range in the estimate of the ratio of mechanical to radiated luminosity. This is consistent with the analysis of several outflows by Davis & Eislöffel (1995), but more detailed and stringent conclusions will require extensive exploration of the velocity and density parameter space.
The total molecular hydrogen luminosity of the pulsed model is about three times higher than the non-pulsed model. The 1-0 S(1) is and the 2-1 S(1) is . Hence in this case about 20% of the total available energy is radiated in the H2 lines. By contrast, the 0-0 R(1) CO luminosity is almost unchanged at .
7.4. Vibrational excitation
The excitation within the hot molecular gas is probed by the 2-1 S(1) transition which has the upper energy 12,550K, in contrast with its warm cousin the 1-0 S(1) transition at 6,956K. The ratio provides an excitation temperature which is the equivalent temperature of a constant temperature slab in LTE. We here find 2300 K in the uniform jet and 2200 K in the pulsed jet model, derived from the integrated 1-0/2-1 ratios of 8.7 and 9.6 respectively. Both these values are in the range of the majority of observations (Gredel 1994, Davis & Smith 1995). Note, however, that the present simulations ignore H2 O formation and cooling which would be expected to increase the excitation to 3,500 K (Smith 1994).
The predicted excitation is not constant (Figs. 12, 17). Knots exist which are of relatively low excitation. This is of course most clearly visible within the jet itself where shocks lose their excitation as they propagate downstream (Fig. 17). It remains to be seen if the jet knots in HH 212, for example, will exhibit this behaviour. Also note the following predictions:
7.5. High-resolution spectroscopy
The 1-0 S(1) line profile from the non-pulsed jet model (Fig. 13) clearly consists of two components. The strongest component arises from the weakly accelerated external gas and peaks at 20 km s-1 (for the orientation of ). The secondary peak at 60 km s-1 originates from the jet gas. We can compare this to observed low spatial-resolution profiles centered on well-known Herbig-Haro bow shocks, which generally show a shift of the peak of 10 km s-1 from the local cloud radial velocity (Zinnecker et al. 1989). A secondary peak is occasionally observed. High-resolution spectroscopy of the deeply embedded sources is required.
The pulsed jet model yields a more complex profile (Fig. 19). Several components are present and a line width approaching 100 km s-1 is predicted. Hence there is a considerable transverse acceleration of molecules in this case.
A deeper understanding is achieved by assigning spectra to distinct parts of the flow. Therefore in Fig. 18 we present a spatially resolved spectrogram of model 3D_1 (orientation angle . Here shocked jet and cloud gas can be distinguished easily by their velocity. Double-peaked emission at the main bow shock (knot A) as well as low velocity features in the wings are predicted (see L1448, Davis et al. 1996). The second bow (knot D) has only a low velocity. A close look at an ensemble of spectra over the whole jet shows a weak tendency: higher radial speeds are expected near the flow axis, although low velocity emission is also possible due to the superposition of local condensations at the cavity walls.
© European Southern Observatory (ESO) 1997
Online publication: July 8, 1998