THE KINEMATICS OF HH 34 FROM HST IMAGES WITH A NINE-YEAR TIME BASELINE

Figure 2 shows the rotated and aligned 1998.71 and 2007.83 [S ii] images of the region around HH 34S. We define a 19'' × 21'' box (shown on the two images) including HH 34S, and carry out a cross-correlation of the emission (within this box) of the two epochs.

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From a paraboloidal fit to the peak of the cross-correlation function, we find a Δx = 0923 offset along the outflow axis and Δy = −0043. The error resulting from the alignment of the two frames and the fitting to the peak of the cross-correlation function is of ≈002. The observed displacement corresponds to a proper motion of PM_34S = (0101 ± 0002) yr⁻¹, corresponding to a plane-of-the-sky velocity v_{T, 34S} = (215 ± 5) km s⁻¹ at a distance of 414 pc (see Menten et al. 2007). An arrow illustrating this motion is shown in the bottom frame of Figure 2.

In an analogous way we have calculated the proper motion of HH 34S from the 1998.71 and 2007.83 Hα images. A plane-of-the-sky velocity of (222 ± 5) km s⁻¹ is obtained, which is not significantly different from the velocity deduced from the [S ii] images (see above). The Hα proper-motion vector is shown in Figure 2.

From the observed proper motion and a distance x_34S ≈ 105'' between the source and HH 34S, we obtain a dynamical time t_{dyn, 34S} = x_34S/v_{T, 34S} ≈ 1000 ± 50 yr. Therefore, if HH 34S travels ballistically, the ejection velocity had a value v_j ≈ v_{T, 34S}/cos  35° ≈ 270 km s⁻¹ (where ϕ = 35° is the approximate angle between the HH 34 outflow axis and the plane of the sky, see Section 4.3) at a time t = −1000 yr (t = 0 corresponding to 2007.83).

In order to study the relative motions of the substructure of HH 34S, we first displace the 1998.71 [S ii] image using the proper motion of the whole HH 34S bow shock (see Section 3.1). In this way, we have two images (the 2007.83 map, and the displaced 1998.71 map) in which the cross-correlation of the whole HH 34S structure has a peak at zero displacement.

We now take these two maps and cover them with 24 side square boxes, with displacements of 12 between the centers of the boxes (so that the successive boxes have 12 overlaps). For all of the boxes in which the peak flux in the two images is larger than 0.01 photons pixel⁻¹ cm⁻² s⁻¹, we then carry out cross-correlations in order to calculate a proper motion for the emission of the material within the boxes. In this way, we calculate the "kinematical map" shown in Figure 3.

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Kinematical maps such as the one shown in Figure 3 can be computed for arbitrary box sizes. We find that for boxes smaller than ≈2'' an increasing number of spurious proper motions (identified by their disproportionally large values and arbitrary directions) are obtained, particularly in relatively low intensity regions. For larger box sizes, maps similar to the one shown in Figure 3 are obtained, but with a lower angular resolution. In this way, the size that we have chosen for the cross-correlation boxes is close to the smallest size possible for calculating a coherent kinematical map (therefore resulting in about the highest possible angular resolution for the map).

The [S ii] kinematical map (see Figure 3) shows regions with velocities close to zero, indicating that they have a proper motion similar to the average proper motion of the whole HH 34S structure. Not surprisingly, the brighter condensations have relative velocities close to zero (as they are the ones that determine the average motion of HH 34S).

Interestingly, we see that in the region toward the outflow source, we have material with trailing velocities (the largest of these in the extended bow shock wing on the top of Figure 3). Also, there are regions of trailing velocity within the central region of HH 34S. Finally, in the region farthest away from the outflow source, we see a cross-axis expansion motion. These qualitative features of the [S ii] kinematical map are summarized in the schematic diagram shown in Figure 4.

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In Figure 5, we show the relative proper motions which result from the Hα images. As is clear from the top frame, the Hα maps of HH 34S show a structure of superposed "wisps" (described in detail by Reipurth et al. 2002), which differs in a qualitative way from the "clumpy" morphology of the [S ii] maps (see Figure 3). Despite this clear morphological difference, the Hα kinematical map (Figure 4) shows characteristics which qualitatively agree with the [S ii] motions (Figure 3). Clear differences are seen in the extended bow shock wing on the top of the frames, which shows lower Hα relative proper motions (higher velocities, almost parallel to the wing, are seen in the [S ii] kinematical map). This difference might be due to the fact that the cross-correlation method is not sensitive to displacements along filamentary structures, and therefore does not detect motions along the Hα bow shock wing. However, such motions are seen along the [S ii] bow shock wing because it shows a more clumpy structure.

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From the [S ii] kinematical map shown in Figure 3, we determine that the maximum velocity away from the outflow axis is v_⊥ = 55 km s⁻¹. The maximum relative velocity along the outflow axis is v_|| = 87 km s⁻¹ (along the extended outflow wing on the top of Figure 3).

With these values, we can then apply the relations deduced from a kinematical bow shock model by Raga et al. (1997), which can be written as

$$\begin{equation} 2v_\perp =v_{{\rm bs}};\,\,\,\,\,v_{||}=v_{{\rm bs}}\cos \phi , \end{equation} \tag{ 1 }$$

where v_bs is the velocity with which the bow shock travels with respect to the surrounding medium, and ϕ is the angle between the outflow axis and the plane of the sky. With the values of v_⊥ and v_|| given above, from Equation (1) we then obtain v_bs = 110 km s⁻¹ and ϕ = 38°. Additionally, the measured proper motions imply that the environment into which HH 34S is travelling is moving away from the outflow source at a velocity v_e = v_{T, 34S} − v_|| = 121 km s⁻¹.

For calculating the proper motions for the chain of aligned knots within ∼30'' from the outflow source we restrict ourselves to the [S ii] frames, since the knots are considerably fainter in Hα, and the kinematics deduced from the [S ii] and Hα images do not differ significantly (see Reipurth et al. 2002).

The region with the knots within ∼30'' from the outflow source is shown in Figure 6. We see that within x < 11'' (7 × 10¹⁶ cm at 414 pc) from the source there are clear, marginally resolved emission peaks, and that for x > 11'' the knots are much more extended. Because of this qualitative difference, we have computed the proper motions in different ways.

• 1.  
  Knots with x  <  11''. We have carried out paraboloidal fits to the emission peaks of the successive knots, and then computed proper motions from the difference in knot positions. The identification chosen for the corresponding knots in the two time frames is given in the bottom frame of Figure 6.
• 2.  
  Knots with x  >  11''. We have defined boxes including the emission from the successive knots in the 1998.71 image (top frame of Figure 6), and boxes with the same size but with a displacement of 6 pixels (06) along the outflow axis in the 2007.83 image. The proper motion of these knots was determined from a paraboloidal fit to the peak of the cross-correlation function of the emission within the pairs of (displaced) boxes.

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The resulting proper motions are given in Table 1.

For the x > 11'' knots, we have also carried out paraboloidal fits to the emission peaks in the 2007.83 image. The offsets x (along the outflow axis) and y (perpendicular to the outflow axis) from the source position in the 2007.83 frame are given in Table 1 (for all of the knots). Two of the boxes defined for the x > 11'' knots actually contain two emission peaks (see Figure 6). For these boxes, in Table 1 we have made pairs of entries (entries 10/11 and 13/14) which share the same values of v_x and v_y (obtained from the cross-correlation within the box that includes the two peaks) but different x and y values (obtained from the fits to the peaks in the 2007.83 frame).

The top frame of Figure 7 shows the total plane-of-the-sky velocity $v_T=\sqrt{\vphantom{A^A}\smash{\hbox{${v_x^2+v_y^2}$}}}$ as a function of the position x of the knots, showing a general trend of decreasing v_T for increasing distances from the source. It is clear that two knots show peculiar deviations from the general trend.

• 1.  
  The second knot (knot 2 of Table 1) has v_T ≈ 250 km s⁻¹, approximately 40 km s⁻¹ higher than the velocities of all of the other knots.
• 2.  
  The seventh knot (knot 7 of Table 1) has a velocity v_T ≈ 170 km s⁻¹, about 30 km s⁻¹ lower than the motions of its neighbors.

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Reipurth et al. (2002) obtained anomalously large motions for the knots closer to the HH 34 source. The fact that we are obtaining an apparently similar effect probably indicates that in this region there are strong changes in the morphologies of the knots, which result in large proper motions that do not represent the physical motion of the gas within the jet.

It is also not completely surprising that knot 7 (see Table 1) has a velocity lower than its neighboring knots. As can be seen in the two bottom frames of Figure 6, knot 7 shows an axially extended structure. This structure has an emission peak close to its leading edge in the 1998.71 frame, which partially disappears in the 2007.83 frame. This morphological change of course results in a lower value for the measured proper motion. For this knot, we have also computed proper motions using the cross-correlation function method (see above), and basically the same results (as with the peak emission fitting) are obtained.

From Table 1 we see that the values of v_x are 1–2 orders of magnitude larger than the v_y values for the corresponding knots. This result indicates that the proper motions are well aligned with the outflow axis. In order to quantify this, we define two angles:

$$\begin{equation} \alpha =\tan ^{-1}\left(\frac{y}{x}\right);\,\,\,\,\, \alpha _v=\tan ^{-1}\left(\frac{v_y}{v_x}\right), \end{equation} \tag{ 2 }$$

where the angles α (between the source–knot direction and the outflow axis) and α_v (between the proper-motion vector and the outflow axis) are defined in the schematic diagram shown in Figure 8. It is clear that for knots with time-independent morphology moving in ballistic trajectories, one will have α = α_v.

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In Figure 7, we show the values of α and α_v (see Equation (2)) as a function of x computed from the knot positions and proper motions shown in Table 1. In the x ∼ 3 × 10¹⁶ cm region, there is a lateral excursion of the jet beam (reaching values of α ≈ −5°), which can be clearly seen in the bottom image of Figure 6. This excursion to larger deviations from the outflow axis can also be seen in the orientation angle α_v of the proper motions. A comparison between the second and third panels of Figure 7 shows that in the x < 1.2 × 10¹⁷ cm region we have α ≈ α_v to within the measurement errors, indicating that the motions are consistent with a ballistic motion of knots ejected with an ejection direction (on the plane of the sky) that changes by up to ∼5°.

For x > 1.2 × 10¹⁷ cm we have a different situation, with low values of α as a function of x, and with α_v having an excursion to larger negative values (of up to ∼ − 7°). This result can be interpreted in two ways.

• 1.  
  The knot motions in this region have measurable deviations from ballistic trajectories.
• 2.  
  The spatially resolved knots in this region develop systematic side-to-side asymmetries which result in larger measured values for v_y.

With the values of the proper-motion velocities and the distance from the source, we can compute a dynamical time

$$\begin{equation} t_{{\rm dyn}}=\sqrt{\frac{x^2+y^2}{v_x^2+v_y^2}}\,, \end{equation} \tag{ 3 }$$

for each of the knots along the jet. The values of t_dyn as a function of x are shown in the bottom frame of Figure 7 (also see the last column of Table 1).

We obtain an almost monotonically increasing t_dyn versus x dependence, which indicates that the motions of the knots do not deviate substantially from ballistic motions. The only exception is knot 7 (indicated by an arrow in the bottom frame of Figure 7), which has a value of t_dyn marginally higher than the dynamical times of knots 8 and 9 (see Table 1). This is again an indication that the measured proper motion of knot 7 might not correspond to the motion of the gas within this knot (see the discussion in Section 4.1).

In order to determine the spatial velocity of the knots, it is of course necessary to determine the orientation angle of the outflow axis with respect to the plane of the sky. To this effect, we combine the tangential velocity v_T (determined in Section 4.1) with radial velocities (with respect to the velocity of the local CO emission) for the HH 34 jet knots.

In Figure 9, we show the absolute values of the radial velocities determined by García López et al. (2008) from the [Fe ii] 1.64 μm emission of the HH 34 jet and counterjet. Given the remarkable jet/counterjet knot-to-knot symmetry seen in Spitzer images of the HH 34 outflow (Raga et al. 2011b), we would expect the jet and the counterjet to have very similar radial velocity versus position signatures.

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Also shown in Figure 9 is the Hα radial velocity versus the position obtained for the high-velocity component of this line as a function of position (obtained by integrating the emission of the successive knots within a 1'' diameter diaphragm) from the two-dimensional spectra of Beck et al. (2007). These Hα velocities show a reasonably good agreement with the [Fe ii] 1.64 μm HH 34 jet radial velocities, but unfortunately the data of Beck et al. (2007) do not cover the x ∼ 2–8 × 10¹⁷ cm region (see Figure 9).

As can be seen in Figure 9, the [Fe ii] 1.64 μm jet/counterjet radial velocities agree well for the knots with x > 7 × 10¹⁶ cm, but show differences of up to ∼50 km s⁻¹ for the knots closer to the outflow source. Possible interpretations for this discrepancy in the jet/counterjet radial velocities are as follows.

• 1.  
  The jet and the counterjet have different orientations with respect to the plane of the sky. This could easily be obtained if this outflow is ejected from a source in orbital motion around a binary companion, resulting in a mirror symmetric jet/counterjet spiral structure (such as appears to be observed in the HH 111 outflow; see Noriega-Crespo et al. 2011).
• 2.  
  The observed line profiles could have a contribution from jet emission scattered on dust present in the surrounding environment. A stronger scattered component in the counterjet could be responsible for the difference in the jet/counterjet radial velocities (such effects have been studied, e.g., by Feldman & Raga 1991, Calvet et al. 1992, and Henney 1994). Such a jet/counterjet asymmetry in the dust scattering might be expected because of the orientation (with respect to the plane of the sky) of a flattened, scattering structure perpendicular to the outflow axis.

Let us explore further the first of these possibilities. If we consider the jet/counterjet [Fe ii] 1.64 μm radial velocities at x ≈ 5 × 10¹⁶ cm (see Figure 9), using the corresponding proper-motion velocities of the knots along the jet, we would deduce ϕ_j ≈ 29° and ϕ_cj ≈ 40° orientation angles between the plane of the sky and the jet/counterjet (respectively). If we consider the radial velocities at x ≈ 1.4 × 10¹⁷ cm (in which the absolute values of the jet and counterjet radial velocities coincide), we obtain an orientation angle (with respect to the plane of the sky) of ϕ = 34° for both the jet and the counterjet.

From this, we would conclude that at x = 1.4 × 10¹⁷ cm, the jet and the counterjet lie along an axis with a ϕ = 34° orientation with respect to the plane of the sky, and that closer to the source (at x ≈ 5 × 10¹⁷ cm) the jet and counterjet have mirror symmetric deviations (along the line of sight) of ∼5°–6° from this axis. As can be seen in Figure 7 (in the α versus x plot), the jet shows plane-of-the-sky deviations from the average outflow axis of ∼5°, although the maximum deviations occur at x ≈ 3 × 10¹⁶ cm.

Assuming ballistic motions for the knots, we can then calculate the past ejection velocity v_j as a function of ejection time t = −t_dyn (t = 0 corresponding to 2007.83; see Equation (3)). We calculate the velocity at which each of the knots was ejected in two ways.

• 1.  
  v_j = v_T/cos 35°; in other words, we take the proper-motion velocity $v_T=\sqrt{\vphantom{A^A}\smash{\hbox{${v_x^2+v_y^2}$}}}$ and assume that all of the knots move at a ϕ = 35° angle with respect to the plane of the sky.
• 2.  
  $v_j=\sqrt{\vphantom{A^A}\smash{\hbox{${v_T^2+v_r^2}$}}}$, where v_r is the [Fe ii] 1.64 μm radial velocity measured by García López et al. (2008) for the HH 34 jet knots. To do this, we take the positions at which the radial velocities were measured, and choose the proper motion of the optical knot closest to each of these positions.

The results of this exercise are shown in Figure 10 (where, following the discussion of Sections 4.1 and 4.2, we have eliminated knots 1, 2, and 7). It is clear that the ejection velocity time histories computed in the ways described above both give a general trend of increasing velocities from ∼400 yr ago to the present time.

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If we take the proper-motion velocities v_{T, k}, and assume that the knots move ballistically, the plane-of-the-sky position x_k(t) of an arbitrary knot k has a time dependence of the form

$$\begin{equation} x_k(t)=x_{k,0}+v_{T,k}\,t, \end{equation} \tag{ 4 }$$

where x_{k, 0} is the 2007.83 position of knot k (the measured values of x_{k, 0} and v_{T, k} are given in Table 1).

Figure 11 shows the trajectories in the (x, t)-plane given by Equation (4) for the HH 34 jet knots. In the two frames of this figure, the 2007.83 positions of the knots correspond to the intersection of the knot trajectories with the x-axis. We show the trajectories for all of the measured knots (see Table 1) except knots 1, 2, and 7 (see Sections 4.1 and 4.2).

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In the top frame of Figure 11, we see that the fact that the successive knots do not have identical velocities results in trajectory crossings (i.e., in collisions between successive knots, given the fact that the motions are quite well aligned). The first trajectory crossings will occur at t ∼ 200 yr and at a distance of ∼2 × 10¹⁷ cm away from the source, down the HH 34 outflow axis.

The bottom frame of Figure 11 shows a more extended time evolution of the knot positions. This plot shows that the knot trajectories have a remarkable convergence at the point (x_c, t_c) ≈ (5 × 10¹⁷cm, 900 yr). This is a direct indication that at this position and time the knots will merge in order to form a major mass concentration.

As shown in the bottom frame of Figure 11, the region of convergence of the knot trajectories lies ∼10¹⁶ cm upstream of the present-day position of HH 34S. Therefore, we conclude that in the proper motions of the knots along the HH 34 jet we see direct evidence for the formation of a massive structure (possibly resembling the present-day HH 34S) some ∼900 yr in the future.