Astron. Astrophys. 345, 977-985 (1999)

3. Results

Fig. 1 shows plots of the distribution of number density for simulations C and G. The cocoon of the varying jet has many bow-shaped shocks travelling through it as a result of the internal working surfaces in the jet forcing gas and momentum out of the jet beam. It is interesting to note that the bow shock of the steady jet is more irregular than that of the pulsed jet. This irregularity is probably due to the Vishniac instability (e.g. Dgani et al. 1996) growing at the head of the jet. Presumably the variation of the conditions at the head of the varying jet dampens the growth of this instability. It should be noted that the enforced axial symmetry in these calculations makes the bow shock appear more smooth than it would in 3D simulations. However, the phenomenon of the bow shock breaking up occurs in both 2D and 3D simulations.

Fig. 1a and b. Log-scale plots of the distribution of number density for a , simulation C, and b , simulation G at yrs. The scales are in units of cm-3

We will discuss each of the properties mentioned in Sect. 1 below.

3.1. Momentum transfer

We make use of the jet tracer to track how much momentum has been transferred from the jet to the ambient medium. The fraction of momentum transferred from jet gas to ambient molecules is

where and are the number density of molecular hydrogen and the total number density in cell ij respectively, i and j are cell indices in the z and R directions, and is the volume of cell ij. This equation is valid since the only momentum on the grid originated in the jet, and since the simulations are stopped before any gas flows off the grid. Note that we only consider momentum transferred to ambient molecules because we are only interested in how efficient YSO jets are at accelerating molecules, not atoms. Thus we ignore ambient molecules which have been dissociated in the acceleration process. Table 2 shows the fraction of momentum in ambient molecules for all the simulations after 300 yrs. For completeness we also show , the total fraction of momentum transferred to ambient gas, whether molecular or atomic.

Table 2. The proportion of momentum on the grid residing in ambient molecules () and in all ambient gas () at yrs. Also given is the value of , the power-law index for the mass-velocity relationship

The first interesting point to note from Table 2 is that the amount of momentum residing in ambient molecules in these simulations is typically an order of magnitude less than the total momentum contained on the grid. Note, however, how significant amounts of momentum are transferred to the ambient medium as a whole , especially in those cases where the jet density matches that of its environment. This is as one would expect. What is perhaps surprising at first is the low efficiency of momentum transfer to ambient gas that remains in molecular form in the post-bow shock zone.

Comparison between and for models A and C and models B and G clearly shows that the momentum transfer efficiency from the jet to the ambient medium decreases with increasing density. This result is particularly marked in the case of post-shock ambient molecules. Although further simulations should be performed to confirm this finding, it is physically plausible. Cooling causes the bow shock to be narrower (i.e. more aerodynamic) than in the adiabatic case, thus reducing its cross-sectional area. Obviously this leads to a reduction in rate at which momentum is transferred from the jet to its surroundings. The fact that the effect is more marked for post-shock ambient molecules must reflect changes in the shape of the bow (as opposed to pure changes in its cross sectional area) with increased cooling.

Since we are simulating systems here which are probably of low density in comparison to typical YSO jets, our results suggest that radiative bow shocks, from at least heavy and equal density jets (with respect to the environment), are not very good at accelerating ambient molecules without causing dissociation. This result also points to the fact that in the case of such jets, the jet may carry much more momentum than one might naively estimate based on a rough balance with the momentum in any associated observed molecular flow.

We now turn to differences in the efficiency of momentum transfer in pulsed versus steady jets. The topic of differences in entrainment rates will be discussed more fully in Sect. 3.5. Fig. 2 shows grey-scale plots of the distribution of and of jet gas for models C () and G (). Comparison between the steady jet and the varying velocity jet suggests that momentum is indeed being forced out of the beam of the varying jet by the internal working surfaces as predicted for example by Raga et al. (1993). It is also interesting to note the similarity between the distribution of velocity and the distribution of jet gas in both simulations. Moreover it is clear that the momentum leaving the jet beam is dumped into jet gas which has been processed through the jet-shock and internal working surfaces and now forms a cocoon around the jet itself. Since this gas is largely atomic (most of it having passed through strong shocks), this effect does not directly lead to extra acceleration of molecular gas. However, the ejected momentum could conceiveably pass through the cocoon of jet gas eventually and go on to accelerate ambient molecules. The wings of the shocks caused by the internal working surfaces in the jet have encountered the edge of the cocoon by the end of these simulations but, even so, the fraction of momentum in ambient molecules varies by not more than 3% as a result of the velocity variations. This result was noted by Downes (1996) for slab symmetric jets, but here we extend this result to cylindrical jets with a variety of strengths of velocity variations. It is also interesting to note that, from comparisons between simulations G and I, the efficiency of momentum transfer (in particular to molecular gas) is not very sensitive to .

Fig. 2. Plots a and b show the distribution of the jet tracer variable and for simulation C respectively. Plots c and d show the same quantities for simulation G. Plots a and c use a linear scale and a value of 1 indicates pure jet gas. Plots b and d also use a linear scale and are in units of km s-1

3.2. The mass-velocity relationship

We do find a power-law relationship between mass of molecular gas and velocity. If we write

then we find that lies between 1.58 and 3.75 and that tends to increase with time, in agreement with Smith et al. (1997). Table 2 shows the values of at yrs for all the models. These values are consistent with observations (e.g. Davis et al. 1998), and also with the analytical model presented in the appendix of Smith et al. (1997) for the variations of mass with velocity. However, it is important to emphasise that what is actually observed is a variation in CO line intensity with velocity. CO line intensity is directly proportional to mass, in the relevant velocity channel, providing we are in the optically thin regime and the temperature of the gas is higher than the excitation temperature of the line (see, e.g. McKee et al. 1982). Note that Smith et al. (1997) incorrectly state that the channel line brightness scales with . Fig. 3 shows a sample plot of the molecular mass versus velocity for the jet moving at an angle of to the plane of the sky. We do not see the jet contribution in the velocity range chosen here.

Fig. 3. Plot of the relationship between the molecular mass and velocity for simulation G at yrs assuming the jet moves at an angle of to the plane of the sky. Note how a power-law (dashed line) fits the data quite well

The molecular fraction in the jet has a marked influence on the value of predicted by these models as we can see by comparing the results for simulations G and G2. In fact, increases with decreasing molecular abundance in the jet. This is due to the reduction in strength of the high velocity jet component.

It appears that does not depend in a systematic way on the amplitude of the velocity variations. Note also that the introduction of a wide shear layer dramatically reduces . This is due to the fact that more gas is ejected out of the jet beam (because of the more strongly paraboloid shape of the internal working surfaces) and this accelerates the cocoon gas, leading to a stronger high velocity component. In addition, a wide shear layer causes the bow shock to be more blunt. It can be seen from the analytic model of Smith et al. (1997) that this also leads to a lower value of . It is interesting to speculate that lower values of gamma, which may be more common in molecular outflows from lower luminosity sources (see Davis et al. 1998) could result from such flows having a higher molecular fraction in their jets and perhaps a wide shear layer.

The behaviour of with viewing angle is the same as that noted by Smith et al. (1997). The actual values of obtained by these authors are somewhat lower than those obtained here. However, since our initial conditions are so different, and since is dependent on the shape of the bow shock, this discrepancy is not disturbing.

3.3. H₂ proper motions and emissions

We measured the apparent motion of the emission from the internal working surfaces. Near the axis of the jet this emission moves with the average jet speed (i.e. ), as would be expected from momentum balance arguments. However, there are knots of emission arising from the bow shock itself and these move much more slowly (5-15% of the average jet speed) with the faster moving knots being closer to the apex of the bow. This is in agreement with the observations of Micono et al. (1998).

Fig. 4 shows the emission from the S(1)1-0 line of H₂ for model G. There is little emission from the cocoon since the cocoon gas has been strongly shocked in the jet shock and so is mostly atomic. We can also see that the emission becomes more intense as we move away from the apex of the bow shock, as reported by many authors (e.g. Eislöffel et al. 1994). It is also clear that the internal working surfaces in the jet are giving rise to emission in this line. We can see that the emission begins to die away as we move away from the jet source. This is in agreement with observations of, for example, HH 46/47 (Eislöffel et al. 1994) where the emission from the knots appears close to the jet source and then fades away.

Fig. 4. Log-scale plot of the distribution of emission from the S(1)1-0 2.12µ line of H2 for simulation G after 300 yrs. The scale is in units of erg cm-3 s-1

This decrease in emission happens for two reasons. The first is that the shocks in the jet become weaker as they move away from the source simply because the velocity variations, which give rise to the shocks in the first place, are smoothed out by the shocks (see, for example Whitham (1974)). In addition, the mass flux through an individual shock decreases with time due to the divergent nature of the flow ahead of each internal working surface. This means that the emission will decrease because there is less gas being heated by the shock.

3.4. The `Hubble law'

We have found that the so-called `Hubble law' (e.g. Lada & Fich 1996) is reproduced in these simulations. Fig. 5 shows a position velocity diagram calculated from simulation G assuming that the jet makes an angle of to the plane of the sky. This diagram is based on the mass of H₂ rather than intensity of CO emission. There is a gradual, virtually monotonic, rise in the maximum velocity. It is also worth noting that near the apex of the bow shock the rise in the maximum velocity present becomes steeper. These properties are related to the shape of the bow shock as gas near the apex of the shock is moving away from the jet axis at higher speed than that far from the apex.

Fig. 5. Contour plot of the position-velocity diagram (of mass rather than emission) for simulation G after 300 yrs assuming the jet moves at an angle of to the plane of the sky. Note the gradual rise of the maximum velocity as we move away from the source. The contours are logarithmic running from to g. The contribution from the jet is removed from this diagram to make the effect clearer

As a very basic model of this, suppose we represent the contact discontinuity between the post-shock jet and ambient gas to be an impermeable body moving with velocity v through a fluid whose streamlines will follow the surface of the body. See Fig. 6 for a schematic diagram of the system. Let this surface be described by the equation

where and a is the position of the apex of the bow shock on the z axis (see, e.g., Smith et al. 1997). Since the contact discontinuity is a streamline of the flow we get that the ratio of the z-component to the r-component of the velocity is simply

Note that this is the negative of the slope of the bow shock. This is because of our choice of the bow shock pointing to the right, and hence the z component of the velocity will be negative. If we assume the post-shock velocity to be (related to v by the shock jump conditions), it is simple to show that

Finally, after some simple algebra, we can write down the velocity along the line of sight as a function of z by

where is the angle the bow shock makes to the plane of the sky. In fact we can write down if we assume that the bow shock is a strong shock everywhere, thus yielding a compression ratio of 4 (from the shock conditions). It is easy to derive that

This formula yields a shape for the position-velocity diagram which suggests the Hubble-law and is similar to diagrams generated from these simulations assuming that the flow is in the plane of the sky. This indicates that the `Hubble-law' effect is, at least partly, an artifact of the geometry of the bow shock.

Fig. 6. Schematic diagram of the setup used to derive the Hubble law for position-velocity diagrams. The fluid motion is shown by vectors. After contact with the bow shock, the fluid is assumed to flow along the contact discontinuity with a smaller, but constant, velocity. See text

3.5. Entrainment

As is clear from Fig. 2 there is not much extra entrainment of ambient gas resulting from the velocity variations. If the velocity variations were to involve the jet `switching off ' for a time comparable to the sound crossing time of the cocoon then we would expect ambient gas to move toward the jet axis and probably be driven into the cocoon when the jet switches on again. This does not happen in these simulations where the maximum period of the variations is 50 yrs and the amplitude is at most 60% of the jet velocity. However, there is a small amount of acceleration of molecular gas at the left-hand boundary of the grid. This effect is quite small, but may grow over time. It is also possible, however, that this effect is due simply to the reflecting boundary conditions.

These simulations show very little mixing between jet and ambient gas except very close to the apex of the bow shock. This means that the high velocity component of CO outflows often observed along the main lobe axis (Bachiller 1996) is difficult to explain without invoking the presence of CO gas in the jet beam itself just after collimation.

© European Southern Observatory (ESO) 1999

Online publication: April 28, 1999
helpdesk.link@springer.de