1 Introduction

Herbig-Haro (HH) jets sometimes show sudden changes of direction, which have been generally attributed to collisions with dense obstacles. The best studied example of this effect is the HH 110 outflow.

HH 110 was discovered by ^{Reipurth & Olberg (1991)}. It was soon realized that this object corresponds to a jet which becomes brighter and less collimated after a sudden change in direction (Reipurth et al. 1996; ^{Rodríguez et al. 1998}). Proper motions (e.g., ^{Kajdic et al. 2012}) and spatially resolved spectroscopy (e.g., ^{Riera et al. 2003}; ^{López et al. 2010}) can be interpreted in this jet deflection context. This object has even been compared to laboratory experiments of jets deflected by collisions with an obstacle (^{Hartigan et al. 2009}).

Though HH 110 is the most notable example, qualitatively similar situations are found in regions with high spatial concentrations of YSO outflows. For example, ^{Walawander et al. (2005)} found some notable deflected jets in the Perseus molecular cloud, and ^{Hayashi & Pyo (2009)} in the L1551 dark cloud. These objects mostly await more detailed studies.

^{Raga & Cantó (1995)} modeled analytically and numerically (with 2D gas dynamics) the initial deflec tion of a jet by an oblique collision with a dense obstacle, and suggested that HH 110 might correspond to such a flow. However, at later times the jet penetrates the cloud, and has a curved trajectory before reemerging (^{Raga & Cantó 1996}), not resembling HH 110. 3D simulations of this type of jet/cloud interaction were presented by ^{de Gouveia Dal Pino (1999)} and ^{Raga et al. (2002)}.

In the present paper, we study the passage of a jet through a slip boundary, in which the environment is stationary (with respect to the jet source) on one side, and in supersonic sideways motion on the other side. This situation could represent a jet impacting on a supersonically travelling cloud (which would be a reasonable scenario in a region with supersonic turbulence), or on material disturbed by the previous passage of another YSO outflow.

We first present analytic considerations for evaluating the deflection of the jet as a function of the environment/jet ram pressure balance ratio p and the incidence angle ϕ of the interaction (§2). Finding that there are two regimes (a jet deflection and a stagnation case), we then compute two 3D simulations illustrating the very different resulting flows (§3). The results are summarized in § 4, and we discuss the possible application of these models to observed deflected HH jets in § 5.

2 Analytic considerations

Let us consider the problem of a steady jet flow emerging from a stationary environment into a region with a moving environment. For the sake of simplicity, we assume that the jet emerges at an angle ϕ with respect to the (infinitely narrow) environmental velocity shear surface, and that the plane containing the jet and the environmental velocities is perpendicular to this shear surface. This situation is shown schematically in Figure 1.

Figure 1: Schematic diagram showing an interface (dashed horizontal line) between a stationary environment (below) and a streaming environment (above) of velocity va. A jet (of velocity vj impacts on the interface at angle ϕ. Two shocks are formed: one that deflects the jet (at an angle β to the interface) and a second one that deflects the environment (at an angle α). The contact discontinuity separating the deflected jet and environmental gas lies in between these two shocks. The deflected environment has a velocity var and the deflected jet a velocity vjr, both parallel to the shocked environment/jet contact discontinuity. The jet beam goes through a deflection δ=ϕ-α. The xz-reference system used in the numerical simulations (in which the jet travels along the x-axis) is shown with the arrows in the center left region of the schematic diagram. 

In a cut through the midplane of the flow, two oblique shocks are formed: one of them refracting the jet beam, and the other one deflecting the flowing environment. These shocks lie at angles β and α, respectively, with respect to the environmental shear layer (see Figure 1). The velocities behind these two shocks (vjr and var) are parallel, forming a deflected environment/jet flow. For the case of a 2D problem, the two shocks are straight, and we will assume that this is also a reasonable approximation for the mid-plane of a cylindrical jet/plane shear layer interaction.

The postshock flow velocities form angles α1 and β1> with the shock surfaces. The tangents of these angles are inversely proportional to the compression in each shock, so that for the highly radiative HH flow case (in which the compressions have typical values of ≈100) they will have values ≪1. It is therefore a good approximation to set α≈β (see Figure 1).

With this condition, the balance between the two post (strong) shock pressures is:

with

where ρj and ρa are the densities and vj and va the velocities (with components normal to the shocks van and vjn) of the jet and the environment, respectively. Clearly, p is the square root of the environment-to-jet ram pressure ratio. From equation (2) we then obtain the shock angle α as a function of the incidence angle ϕ:

In the α1, β1≪1 high compression approximation, the deflection angle between the impinging (vj) and refracted (vjr) jet velocities is δ=ϕ-α (see Figure 1). Using equation (2) we then obtain:

The shock angle α and the deflection angle δ (equations 2 and 5, respectively) are shown as a function of p and ϕ (see equation 3) in Figure 2. Using equation (5) we can now calculate the velocity of the deflected jet:

see Figure 1.

Figure 2: The α (solid lines) and δ (dashed line) angles (i.e., the angle of the jet shock with respect to the environmental shear layer and the jet deflection angle, respectively, see Figure 1) as a function of the incidence angle ϕ. The results shown in the three frames correspond to p=0.5, 1 and 2 (see equation 3). For p = 1, α and ϕ coincide. 

It is clear that α and δ switch their values under a p→1/p transformation (see equations 4 and 5), as can be seen in Figure 2. This property is an expected result due to the symmetry of the jet/streaming environment interaction (see Figure 1).

Also, the p = 1 case (equal jet and environmental ram pressures, see equation 3) produces the straightforward result:

deduced from equations (4-5).

In Figure 3, we show α (or δ) for a number of different p (or 1/p) values. Two families of solutions (having values of either 1 or 0 for ϕ=180∘) are separated by the linear, p = 1 solution (see equation 7).

Figure 3: The displayed curves correspond to α as a function of ϕ for p = 0.1, 0.2, .... 1, 1/0.9, 1/0.8, 1/0.7, ... 1/0.1 for (top to bottom curves) or to δ as a function of ϕ for 1/p=0.1, 0.2, ... 1, 1/0.9, 1/0.8, 1/0.7, ... 1/0.1 (top to bottom curves). The thick, central straight line is the p = 1 solution. 

It is clear that for p<1 there is a maximum possible value δm for the jet deflection (see Figures 2 and 3). From equation (5) it is straightforward to show that this maximum occurs for an incidence angle

and has a maximum deflection

Figure 4 shows these two angles as a function of p.

Figure 4: Maximum jet deflection angle δm and incidence angle ϕm at which the maximum deflection occurs as a function of p. 

Another interesting feature of the jet deflection is that for p>1 one can have δ>90∘. For

the jet flow has a stagnation (δ=90∘) at a shock perpendicular to the jet flow (this result being straightforwardly obtained from equation 5). Figure 5 shows ϕs as a function of p.

Figure 5: Incidence angle for jet stagnation ϕs as a function of p. 

For ϕ>ϕs, we have deflections δ>90∘. This corresponds to a regime in which the jet is deflected into a direction that goes against the streaming environment. Because this occurs only in the p>1 regime of high environmental ram pressure, this deflected material will quickly slow down and remain close to the jet/streaming environment interaction region. This “stalled jet” regime is explored below with a gas dynamic simulation.

Finally, we note that the mixed jet/streaming environment flow emerging from the two-flow interaction region has an environment to jet mass ratio

the second equality being a direct result of the ram-pressure balance condition (see equation 1). As the flow travels away from the interaction region (shown in Figure 1), more environmental mass is incorporated into the beam of the curved flow.

3 Numerical simulations

3.3 The Deflected Jet and the Stalled Jets

Figure 6 shows the flow resulting from model M1 after a 2000 yr time-integration. The first panel shows an xz-cut through the middle of the jet beam, displaying the density stratification and the flow field. The other three panels show the column density calculated for three orientations (θ=0, 30 and 60°) between the z-axis and the plane of the sky.

Figure 6: Results obtained for model M1 (see the text) after a t=2000 yr time-integration. The left panel shows the zx-midplane density map (given in g cm^-3 by the bar at the right of the frame), and (white) arrows with lengths and directions corresponding to the local flow velocities. The three right panels show column density maps (given in 10¹⁹ cm^-2 by the bar at the right) computed assuming θ=0, 30 and 60° angles between the x-axis and the plane of the sky (as shown on the top of the three panels). The z- and x-axes are given in units of 10¹⁷ cm. The color figure can be viewed online. 

We find that the jet shows the deflection shock of our analytic model, and that after the end of the deflection shock, the locus of the jet beam curves due to the ram pressure of the streaming environment, The column density maps (three right panels of Figure 6) show that the deflected flow has a curved, collimated jet morphology for all chosen orientation angles (including the more head-on, θ=60∘ observational perspective).

The curved jet beyond the deflection shock is the jet/sidewind interaction regime that has been studied by ^{Cantó & Raga (1995)}, in which the jet develops a “plasmon shaped” cross section. This plasmon cross section is seen in the yz-density cuts shown in Figure 7, corresponding to planes perpendicular to the initial direction of the jet flow.

Figure 7: Results obtained for model M1 (see the text) after a t = 2000 yr time-integration. The three panels show the density stratifications on yz-cuts (see Figure 6) at three different values of x (1, 2 and 3 × 10¹⁷ cm), as labeled on the top of each frame). The color figure can be viewed online. 

Figure 8 shows the flow resulting from model M2 after a 2000 yr time-integration. The mid-plane density stratification shows that the jet terminates at a shock with a complex, time-dependent shape. The post-shock material forms a series of clumps or eddies that are advected by the impinging environment, forming a turbulent trail.

Figure 8: The same as Figure 6 but for model M2. The color figure can be viewed online. 

The column density maps show that this turbulent trail has a complex 3D structure, with considerably higher column densities when observed in more head-on directions (see the θ=60∘ frame of Figure 8). The complexity of the structure is illustrated by the yz-density cut (taken at x=1.38×1017 cm) shown in Figure 9.

Figure 9: Density yz-cut at x=1.38×1017 cm obtained from model M2 (see the text) after a t = 2000 yr time-integration. The color figure can be viewed online. 

4 Summary

We developed an analytic model (§2) of a radiative jet going through an environmental shear surface. We obtain a full solution in the limiting case of high compression in the two radiative shocks produced in the interaction.

For given values of the dimensionless parameter p (equal to the square root of the streaming environment to jet ram-pressure ratio, see equation 3), we find expressions for the jet deflection angle δ (see equation 5) and the angle α between the interaction shocks and the environmental shear surface (see equation 4) as a function of the incidence angle ϕ (see Figures 1, 2 and 3).

Two interesting results are that:

The interactions with ϕ>ϕs correspond to stalled jets, in which the post-interaction shock jet is stopped and then entrained by the streaming environment.

In order to illustrate the “deflected” and “stalled” jet interaction regimes, we computed two 3D simulations with values of p (see equation 3) and incidence angle ϕ (see Figure 1) clearly placing the flows in each of these two regimes. These simulations are described in § 3, and the results are shown in Figures 6-9.

We find that:

We are not aware of an observed HH jet deflection that could be in this latter regime.

5 Conclusions

We have presented analytic and numerical models for a radiative jet that travels through an environmental shear surface, which divides an internal region around the jet source (which shares the motion of the jet source) and an external region with a non-zero velocity (with respect to the jet source).

This flow configuration could correspond to different situations. For example:

These two cases are relevant for regions with many bipolar jet systems, such as the L1551 cloud (e.g., ^{Hayashi & Pyo 2009}), the Perseus molecular cloud (^{Walawander et al. 2005}) and NGC 1333 (^{Raga et al, 2013}), some of which show several jets with sudden changes in projected direction. Both jet-cloud and jet-molecular outflow collisions are possibly occurring.

Our present model is also relevant for the case of the remarkable HH 110 deflected jet system (e.g., ^{Kajdic et al. 2012}). This object has been modeled as an oblique collision of a jet with a dense cloud (e.g., ^{de Gouveia Dal Pino 1999}; ^{Raga & Cantó 2005}). ^{Raga et al. (2002)} carried out a more extended numerical exploration of the jet/cloud collision problem and obtained deflected jets that qualitatively resemble HH 110. They noted, however, that for reasonable jet/cloud density contrasts it was not possible to maintain the jet deflection at the cloud surface for a long enough timescale, as the jet starts boring through the cloud.

The addition of a relative motion between the jet source and the cloud directly solves this problem. The models of the present paper show that this relative motion results in an approximately stationary jet deflection, which will last until the collision region reaches the edge of the cloud. The relative jet source/cloud motion is probably the missing element in all of the models that have been previously calculated for the HH 110 flow.