Models of the Mass-ejection Histories of Pre-planetary Nebulae. II. The Formation of Minkowski's Butterfly and its Proboscis in M2–9

M2–9 is a poster child for all types of bipolar nebulae created by symmetric pairs of flows from a central source. The observations that serve as the foundation of our simulations are compiled in Figure 1. (Please refer to this figure in subsequent discussions.) Special spatial features of M2–9 that constrain the models are described in Section 2.1. The tools of our hydro modeling are discussed in Section 2.2. The outcomes of models that match the current structure of M2–9 are presented in Sections 2.3 and 2.4.

2.1.1. Knots

2.1.2. Large-scale Features

Hydro simulations were made using AstroBEAR, an adaptive-mesh-refinement code that uses a Riemann solver to follow the varying Eulerian state variables for various types of flows over time (Cunningham et al. 2009; Carroll-Nellenback et al. 2013). The spatial resolution of the mesh adjusts as needed wherever pressures change rapidly. Here we mention a few highlights of the code. See the more detailed description of AstroBEAR in B+13 and B+17.

Our simulations are restricted to a two-dimensional slice through the nebula that contains the symmetry axis. In previous papers (B+13 and B+17) we noted that a comparison of 2D and 3D models showed only slight changes on model outcomes unless substantial thin-shell or shear instabilities are present. Such instabilities are so disruptive to the low-density outer structures of M2–9 that the observed knots N4/S4 cannot form. Moreover, each 3D model of adequate spatial resolution requires prohibitive amounts of computer time. We did not attempt such models.

The 2D computational grid is 32 × 128 kau on a side with the outflow nozzle at the origin and the nebular symmetry axis along the y-axis. Note that we adopt spatial units of kilo au ("kau") since these are the natural units of the computations. 1 kau ≈ 0.06 pc ≈ 1.5 × 10¹¹ km. The units of flow speed, density, and other quantities are standard. The grid is filled with static, stratified ambient gas at time t = 0, where "ambient gas" refers to the surrounding pre-flow AGB wind whose density distribution is presumed to decline as 1/r^m (where 1.8 ≤ m ≤ 2.3) and whose speed is negligible compared to that of the gas injected at the nozzle. The cell size is 500 au/2ⁿ, where n is an integer ≤6 determined by an adaptive grid algorithm. Radiative cooling is computed from the coronal cooling rates of Dalgarno & McCray (1972).

Each 2D run required 2–24 hr to complete. So, in order to enhance computational speed, n is set to 6 for the first 300 years, 5 for the following 900 years, and 4 thereafter. Thus the maximum resolution of the simulations is matched to the scale of the growing nebular structure. The time steps in the model are of the order of days or weeks. Synthetic images of the hydro state variables are created every 50 years and compiled in large output files for display using standard display tools. It is important to note that AstroBEAR does not generate emission-line images or synthetic position–velocity diagrams for comparison to observations of speed patterns.

The grid is initially filled with a stationary "ambient" AGB wind of the form n_amb(r) = n_amb(r₀)(r₀/r)^m (cm⁻³), where r₀ is the radius of the spherical nozzle, n_amb(r₀) is the AGB wind at the nozzle, and m is a free parameter. A steady tapered conical jet, or "spray," flows radially from the boundary of a spherical nozzle starting at time t = 0. The radius of the nozzle is 500 au. The spray at the nozzle boundary is characterized at all times by its hydro parameters at the nozzle, n_spray(r₀), speed v_spray(r₀), temperature T_spray(r₀), and its geometry ϕ. All of these are fixed throughout the run. ϕ is the 1/e width of a Gaussian that is used to modulate the polar distribution of n_spray(r, ϕ) and v_spray(r, ϕ) at all r. This tapered momentum flow means that the momentum of the spray, proportional to n_spray(r, ϕ)v_spray(r, ϕ), is concentrated along its symmetry axis.

For reasons made clear in Section 2.3, we adopt a very "light" spray in which n_spray(r₀) ≪ n_amb(r₀). Thus the low-density spray (like a steady puff) interacts strongly with the ambient AGB wind right from the onset of spray injection. Lobes slowly form and grow over the course of 2500 yr. Their shapes and dimensions mimic those of the present inner lobes of M2–9 when the initial conditions of the ambient gas and the flow are properly chosen (Section 2.4).

The starting point for the simulations is this: a spray exits the nozzle with mass density n_spray(r₀) ≪ n_amb(r₀), highly supersonic speed (Mach number ≳100), and opening taper angle ≈40° and encounters a much denser and static AGB wind near the nozzle. The best values of n_amb(r₀), n_spray(r₀), v_spray(r₀), m, and ϕ are found from the model that best fits the observations at t = 2500 yr.

We ran over 70 simulations before finding the most satisfactory (albeit imperfect) fit shown in Figure 2. The most successful of the models at t = 2500 yr is shown in Figure 2. The locations of a few key features of the model are identified in the panels. In this model r₀ is 0.5 kau, n_amb(r₀) = 1 × 10⁴ cm⁻³, and the descriptors of the injected gas are n_spray(r₀) = 400 cm⁻³, v_spray(r₀) = 200 km s⁻¹, T_spray(r₀) = 100 K, and ϕ = 40°. The corresponding mass flow rate through the nozzle is $\dot{M}\approx {10}^{-8}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}$, the total injected mass is 2 × 10⁻⁵ M_⊙ after 2500 yr, and the total injected momentum is 0.004 M_⊙ km s⁻¹ (10³⁶ g cm s⁻¹).

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Figure 2(a) shows the density distribution at t = 2500 yr. The inner and outer lobes are found in the lower and upper halves of this panel, respectively. Overall, the model nicely reflects the shape of M2–9 seen at present. The low-density interior of the bulb is filled by low-density streamlines of injected gas from the nozzle. The expected rim of ambient gas that they have displaced is found at its outer perimeter. Dense clumps at y = 20 and 45 kau represent knot pairs N3/S3 and N4/S4, respectively; however, knots N4/S4 are only twice as far from the nucleus as N3/S3, not a factor of four as observed. The sheath has the appropriate width; however, its interior density is uniform, not edge-brightened, as images suggest. (Other shortcomings of the model are discussed later.) Finally, we find that increasing the value of r₀ by more than 50% smears the spray (and increases the momentum injection rate) such that N4/S4 cannot form.

The streamlines contact the rim of the inner lobes at a reverse shock along the edges of the bulb (see lime-green line in Figure 2(b)). The normal component of their momentum is transmitted into the gas. The local gas temperatures and speeds change abruptly across this shock (Figures 2(b)–(d)).

The speed of the inner shock is of the order of the flow speed at the point of contact (≤200 km s⁻¹) times the cosine of the polar angle of the streamline. Thus the speed of the inner shock varies steadily along its length. Nonetheless, the shock speeds are sufficiently large to excite the bright tracers in the post-shock zone such as [Fe ii], [S ii], and [N ii].

The outer edges of the sheath expand horizontally (Figure 2(c)) and supersonically into the cold ambient gas, preceded by a leading shock in which H₂ is excited. The locus of this shock is indicated in Figure 2(e). A contact discontinuity ("CD") lies between the pair of shocks. There is no pressure gradient across the CD to force gas through it. On the other hand, over time the embedded CD drifts outward (x-direction) into ambient gas near the outer edge of the sheath. Gas formerly upstream of the CD intrudes through the CD. As we discuss in Section 4, this slow but steady intrusion of ambient gas leads to complex speed gradients between the CD and the inner shock. This is illustrated to the immediate right of the lime-green contour in Figure 2(b).

The bulb and the surrounding sheath are approximate analogues of the wind-heated bubble and rim in the inner zones of classical round and elliptical PNe, respectively; see Kwok et al. (1978) and Toalá & Arthur (2014). The latter paper shows that the radial streamlines can induce highly disruptive thin-shell instabilities along the inner shock rim at the bulb's surface. No such instabilities are observed in M2–9. In practice this places an upper limit on the flow injection speed at 225 km s⁻¹. The minimum injection speed, 180 km s⁻¹, is set by the requirement that the knots reach their observed positions.

The formation of the outer lobe, or the proboscis, of M2–9 is a very different story from that of the inner lobes. The proboscis is powered by fast, injected gas that has bypassed N3/S3 and re-converges further upstream to form N4/S4. This bypass flow traverses the inner shock, where it is heated to >10⁴ K as it starts to converge to the y-axis. An invisible, low-density, high-speed "hot plume" forms. The plume is easily seen above N3/S3 in Figures 2(b) and (d).

The faint shock along the outside edge of the hot plume forms shocked vertical "feathers" where conditions resemble those in the shock on the inside edge of the bulb. Morphologically, the feathers appear to be outward extensions of the bulb. Line ratios of [N ii], [S ii], and [Fe ii] in the bulb and feathers mimic one another. The vertical striations along the surface of the feathers appear to be the result of surface shears that we cannot simulate on a 2D grid. Beyond the feathers the proboscis is seen only in starlight that was scattered from the displaced ambient gas and dust.

The knots N4/S4 contain a mixture of injected gas delivered by the plume and displaced ambient gas. In our simulations the fraction of fast gas in N4/S4 is about 60%. So the speeds of N4/S4 are of the order of 120 km s⁻¹ whereas observations of N4/S4 by C+11 show a greater speed. This fraction (and the knot speed) steadily increases as the downstream density of slow ambient gas declines over time (see Section 4).

As noted above, the most glaring shortcoming of the hydro model on M2–9 is that N4/S4 are not four times as far from the core as N3/S3. We explored remedial initial conditions without any success. For example, increasing the flow speed from 200 to 225 or 250 km s⁻¹ resulted in highly disruptive instabilities throughout the proboscis beyond N3/S3. As a consequence, the hot plumes do not develop and knots N4/S4 cannot form. We reran the "baseline" simulation of Figure 2 with the same ambient density structure except that the decrease with radius scales as (r₀/r)^2.2. This had no impact on the location of N4/S4 at 2500 yr; however, N3/S3 were further from the nucleus relative to the baseline result and the outer lobes were too narrow. Decreasing n to 1.8 or less led to an open proboscis in which N4/S4 cannot form.

The most applicable kinematic data consist of the long-slit observations shown in Figure 1(b). The optical spectra of [N ii] and [O iii] were obtained by A. R. Hajian & B. Balick (2018, private communication) using the Echelle Spectrograph on the 4 m Blanco Telescope at Cerro Tololo Interamerican Observatory ("CTIO").⁵ The observations were made in 15 seeing during 1998 July though a 1'' wide slit oriented north–south. The spectra have spectral resolution of 7 km s⁻¹. The near-IR spectra of [Fe ii] and H₂ were obtained 15 months earlier by S+05 at CTIO with about the same spectral resolution. See that paper for technical information.

Ignoring N1/S1 and N2/S2, the emission-line spectra north and south of the nucleus are mirror images of one another except that the northern lines are slightly redshifted and the south lines are slightly blueshifted with respect to the average Doppler shift of the broad nuclear emission lines. Thus the south lobe is approaching. See Solf (2000) and S+05 for a much more detailed description of the large-scale Doppler patterns in the optical and infrared spectra, respectively.

Recall that the [Fe ii], [N ii], and [O iii] lines trace the locus of the inner shock at the edge of the bulb. Their respective kinematic patterns should be (and are) very similar. All of the long-slit spectra show a small but clearly positive overall gradient in their Doppler shifts in the y-direction, dv_y/dr, from the base of the bulbs through N3/S3 and the feathers. This is the "kinematic gradient" whose observed magnitude is 10 km s⁻¹ from the core to N3/S3 before any correction for inclination (a factor of 3.2). In addition to this "vertical" speed gradient in the y-direction, the lobes expand "horizontally" in the x-direction at v_x ≈ ±15 km s⁻¹ with very little gradient along the slit. These results agree very well with the heuristic inclination-corrected fit to Echelle data by C+15.

In contrast, the H₂ lines are seen through the slit on each side of the nucleus from the front and rear edges of the sheath (Figure 1) at Doppler shifts of ±15 km s along the line of sight with no kinematic gradient (i.e., no overall spectral tilt). No mass moves vertically along the edge of the sheath as it slowly opens horizontally. The models show that the gas near the H₂ emission region is entirely ambient gas that was initially pushed aside by the tips of the outer lobes as they moved vertically. The present motions of the outer edge of the sheath are a perpetuation of the same horizontal flow pattern because the nearby ambient gas in their path is unable to slow the flow.

The most obvious feature of the long-slit spectra of [Fe ii], [N ii], and [O ii] is the y-gradient in speed from the nucleus through the tips of the feathers (Figure 1(b)). The positive vertical speed gradient arises as follows. As noted earlier, the streamlines strike the rim and the inner shock obliquely. The locally transverse component of momentum is unaffected by the impact. Given the lobe's flame-shaped geometry, the post-shock flow speeds will be larger at smaller polar angles (near the top of the lobe)—where the flow vectors are nearly parallel to the shock—than at the base. Accordingly, there is an easily discernible positive gradient in the vertical speed of the gas where the emission lines of [Fe ii], [N ii], and [O iii] arise (adjacent to the lime-green line in Figure 2(b)). (This explanation for the vertical speed gradient along the walls is robust for most lobe geometries, e.g., Lee & Sahai 2003, hereafter "LS03.")

As noted in Section 2.2, the slow horizontal drift of the CD allows slow ambient gas to intrude through the CD, whence it mixes with faster gas that passed through the inner shock earlier. This causes a sharp speed gradient between the inner shock and the CD. The duration of this penetration of ambient gas is longer at the (older) base of the sheath than at the top. The outcome is that the region between the inner shock and the CD is a region of complex yet systematic kinematics with substantial speed gradients in the x and y directions.

For several technical reasons we are unable to make a detailed quantitative comparison of the kinematic data and predictions of the model. First, as noted earlier, AstroBEAR does not compute local line emissivities. Second, the visualization program that we used, VisIt, does not accommodate the rendering of a position–velocity diagram for 3D volumes constructed from 2D models. (This limitation will be fixed in the near future.)

However, VisIt can display a trace of the hydro state variables along a single straight "data readout line" that can be placed interactively on the panels of model outcomes, including those presented in Figure 2. The value of this tool is limited since the locus of the line tool is straight and the inner shock is curved. Even so, it is clear that the pattern of flow speeds just beyond the lime-green line in Figure 2(b)—the locus of the inner shock—is dominated by a gradient in v_y of the appropriate slope. On the other hand, the absolute values of the speeds vary dramatically when the data readout line is translated only slightly. The net result is that the tilts of the lines in Figure 1(b) are easy to understand qualitatively but tricky to replicate in detail. We are forced to accept a qualified victory.

The model successfully predicts that the sheath is steadily widening (Figures 2(a) and (c)). However, the predicted speed of the sheath, v_x ≈ 8 km s⁻¹, is half of the observed value. This discrepancy is stubbornly difficult to resolve in our simulations without degrading the fit of the model elsewhere. For example (as noted earlier), an increase in the injection speed at the nozzle results in unacceptably disruptive instabilities throughout the entire proboscis.

Recall that N3 and S3 are not launched from the core but rather originate in the plugs that develop with the rims of the lobes at very early times. These plugs quickly become much denser than the gas through which they move, so the motions of N3 and S3 are soon ballistic. Their speeds can be read directly from Figure 2(b) or, better still, derived from the knots' positions measured directly from the AstroBEAR density panels in 100 yr intervals starting at t = 300 yr. We have applied both procedures. The results of the two methods are consistent: N3 and S3 have steady vertical speeds of 30 km s⁻¹, in excellent agreement with their current values (C+15).

Spectra of [Fe ii], [N ii], and [O iii] show that N3/S3 have slightly more extreme Doppler shifts than the edges of the bulbs in the same line of sight. The small "anomalous" component of knot velocity is of the order of 15 km s⁻¹ after deprojection. The anomalous speeds are not predicted by the model. We can only speculate on a cause. A series of almost annual images of M2–9 published by C+11 show clearly that N3/S3 are immersed within the collimated and precessing "lighthouse" jet whose flow speed exceeds 10⁴ km s⁻¹. Perhaps the denser knots N3/S3 deflect the beams so that the post-deflection speeds of the locally shocked gas are anomalous.⁶

The present model tells an unusual story about the historical movements of N4/S4. The knots are merely ephemeral patterns of high density at the tips of a complex flow for the first 2000 yr. During this time fast gas arriving through the converging hot plumes continuously roils the gas at the leading tip of the outer lobe. Blobs of compressed gas form and then dissolve, only to be occasionally replaced by new leading blobs. The tips of the outer lobes advance in occasional lurches. The average speed of the tips is about 100 km s⁻¹. After 2000 yr the last of the blobs becomes a long-lived entity embedded in hot swirling gas. Its terminal speed is 120 km s⁻¹, i.e., ≈20% larger than its average speed.

A highly scientific value of the model is that it allows us to trace the evolution of the various structures of M2–9 back to within 100 yr of the onset of spray injection (Section 3). The process of forming the unusually complex structure of M2–9 is actually quite straightforward. Geometrically, the low-density spray burrows into a far slower AGB wind of far higher density and immediately inflates the nascent hollow bulbs. A thin rim of slow displaced and compressed ambient gas immediately forms along the leading edge of the bulb along with a dense leading "plug" of compressed ambient gas at its tip. The pair of plugs (analogous to the water wave at the prow of a speeding boat) become N3/S3.

The slow plug is soon overtaken and bypassed by the ongoing spray. Thus begins the process of forming the proboscis and knots N4/S4. The plugs immediately become and remain an obstacle within the much faster spray in which they are immersed. Under suitable conditions the bypass flow converges back to the symmetry axis downstream from N3/S3, where ephemeral blobs such as N4/S4 can form. At the same time, the gas initially displaced in the x-direction by the expanding (but decelerating) edges of the bulb forms a mildly compressed sheath of slow ambient gas. Both the bulb and sheath expand ballistically and supersonically.

The interaction of the streamlines with the rim of displaced gas creates shocks of speed ≲200 km s⁻¹ along the perimeter of the bulb. These shocks excite various emission-line tracers. The obliquity of the impact and the effective shock speed vary somewhat with height (that is, polar angle) along the curved walls of the bulb. Thus the emission lines exhibit a systematic gradient in Doppler shift when observed through a slit placed along the symmetry axis.

Once formed, the plugs move ballistically at 30 km s⁻¹ as they morph into N3/S3. The off-axis streamlines flow around the plug, creating a pair high-speed "plumes" on a two-dimensional grid (Figures 2(b) and (d)). The plumes contain fast, hot shocked gas that is constrained by the inertial pressure of the static gas that they displace. They sweep around the plug, and owing to the confining pressure of the ambient gas near the core they slowly converge toward the symmetry axis. (The model also predicts a thin axial column of cold gas released from N3/S3 that ascends toward and supplies mass to N4/S4. This column is not observed.)

After 2500 yr a relatively dense and stable pair of blobs (N4/S4) has formed at the outermost tip of the converging hot plumes (Figure 2(a)). Each knot consists of a 40:60 mixture of swept-up ambient and fast injected gas, so its vertical speed is ≈120 km s⁻¹. The speeds of the blobs can steadily increase as the density of ambient gas declines and as high-speed injected gas is delivered from behind. At 2500 yr the forward speed of N4/S4 is 20% larger than the average growth speed of the tip of the outer lobe. A larger fraction of injected gas would increase the distance and speed of N4/S4 from the nucleus.

The values of the initial conditions adopted for the model in Figure 2 were obviously adjusted to provide a good fit to the extant spatial observations of M2–9. New observations (such as proper motions) are needed to test the model's predictions of its evolution. In principle, images in soft X-rays would also be useful, but their feasibility is a formidable challenge. ALMA images of the proboscis in CO or the cold dust distribution would also be helpful for evaluating the model.

The present model, like all models, is a physics-based yet still hypothetical story of the evolution of M2–9. Any model of a nebula as complex as M2–9 has its limitations, and ours is no exception. To begin with there are eight free, non-orthogonal parameters that describe the nozzle, the ambient gas, and the injected spray that need to be adjusted for a good fit, somewhat rendering the optimization of the model a matter of informed guesswork. More free parameters might improve the quality of the model—at the expense of the complexity of running simulations.

The shapes of the lobes, the width of the sheath, the convergence of the plumes to form blobs, and the locations of the knots demand very specific combinations of r_0, n_amb(r₀), n_spray(r₀), and v_spray(r₀). We cannot be certain that our selection of initial conditions is optimum. Tests show that our parameter choices cannot be perturbed by more than a factor ranging from 10% to 50% and still yield satisfactory results. That makes the selected parameters robust choices. However, our model may not be the best possible.

There are many other limitations of the present model. The geometric structure of the "spray" and the ambient gas are obviously idealizations. Also, the ambient gas was presumed to be static for reasons of convenience. (Tests show that employing AGB wind speeds of 15 km s⁻¹ has little effect on the outcomes at the cost of hugely increased computational time.) We did not consider magnetic fields nor did we attempt any three-dimensional simulations. AstroBEAR has its own limitations; for example, it uses a very simplified, numerically efficient cooling curve that is vital for computing the structures of shocks and the formation of dense knots. Finally, we were unable to synthesize P–V diagrams from models for direct comparison to the kinematic observations.

In addition, we find some troublesome outcomes of the simulations. Foremost among these are the precise locations of the knots N3/S3 and especially N4/S4. The latter are observed to be considerably further from N3/S3 than predicted (twice as far). Other shortcomings are these:

• 1.  
  The speed of the edges of the sheath, v_x, is observed to be 50% larger than predicted.
• 2.  
  The model predicts that the zone between the bulbs and the sheath is filled with ambient gas at densities ranging from 10 to 100 cm⁻³ (Figure 2(a)). Yet the images of M2–9 in Figure 1 show that this zone is really a gap. A filled sheath should have been visible in reflected starlight in F547M and F814W images in the HST archives (NB: this concern does not apply to lobe edges seen in shock-excited H₂).
• 3.  
  Any major change in n_amb(r₀) from 10⁴ cm⁻³ yields a poor fit to the width of the sheath and the locations of the knots at t = 2500 yr. Similarly, an increase in v_spray(r₀) from 200 km s⁻¹ by 15% triggers disruptive thin-shell instabilities throughout the proboscis. Lower flow injection speeds result in unacceptable knot locations and average knot speeds.
• 4.  
  The model does not account for asymmetries in the distribution of nebular brightness on opposites side of the symmetry axis. (NB: only the surface brightness and not the nebular structure exhibits these asymmetries. So it appears that the uneven illumination of the nebula rather than an asymmetric distribution of internal pressures is the cause.)

Spatial structure. Some of the striking features of M2–9 have counterparts in other pPNe or very young PNe. For example, like M2–9, Hen 3–1475 has a pairs of inner bulbs and proboscises that converge to high-speed knots and fuzzy leading blobs with measurable proper motions. Hen 2–320 has bulbs, a faint sheath, and feathers, like M2–9, but no knots or blobs. Mz3 has a bulb with a cylindrical extension that resembles the feathers of M2–9. Hen 3–401, IRAS 13208–6020, and IRAS 17340–3757 are bipolar nebulae showing feathers on the outer edges of hollow cylinders, like M2–9, but with no knots, bulbs, or sheaths in their interiors. CRL 618, IRAS 22036+5306, and OH 231.8+04.2 each have pairs of lobes with knots at their tips but no bulb, sheath, or outer lobe. Two low-inclination bipolar pPNe, IRAS 17150–3224 and perhaps Hen 2–166, exhibit dusty plugs at the leading edges of their relatively wide lobes. The physical connections among these objects are not clear. (Future hydro models would be very helpful to this end.)

The model presented here is obviously very specific to M2–9. Yet the processes that govern its early evolution may apply in some form to many other pPNe with feathers and coaxial pairs of lobes. In the earlier papers in this series, B+13 and B+17, we found that steady flows had to be light at the nozzle surface (n_flow(r = r₀) < n_amb(r = r₀)) in order for the lobes to be fashioned by the inertial pressure of their environment into the shapes that are observed. Light outflows are mandatory for creating lobes that appear hollow as they grow.

There are other interesting and general lessons to be learned about the density structures of pPNe from the hydro simulations. For example, the expansion speed at the tips of lobes formed by light stellar flows will always be less (and often substantially less) than the speed of injected gas because its speed is retarded by the displacement of ambient gas. Thus a general caution is in order: the speeds of the tips of hollow lobes will underestimate the injection speed at the nozzle. This applies to many well-observed pPNe such as M1–92, M2–56, Hen 2–166, Hen 3–401, IRAS 17150–3224, and Frosty Leo (Roberts 22).

Sprays (tapered conical flows) have the flexibility to fit a wide range of lobe shapes whereas uniform-density cylindrical (parallel streamlines) and conical (diverging streamlines) flows do not. Specifically, steady, uniformly filled cylindrical jets can only create pole-shaped lobes similar to those of Hen 3–401, IRAS 13208–6020, and IRAS 17340–3757. Additionally, forming a cylinder wider than a close stellar binary orbit (≈20 au) is tricky unless some sort of carefully shaped exterior collimator deflects diverging streamlines into parallel ones. Similarly, light conical flows will create leading arc-shaped plugs that act as pistons that displace ambient gas. The growing arcs must rapidly slow down as their area increases and they displace a wedge of downstream gas. Such a plug is prone to highly destructive instabilities, as shown by simulations of spherical winds by Toalá & Arthur (2014).

It is very awkward for steady flows of any geometry, whether uniform or tapered, to create pairs of on-axis knots that are not the compressed remnants of plugs. N3/S3 in M2–9 are rare exceptions. The basic reason is that knots are prone to crush unless they are much denser than the ambient gas (see B+17 for more about the crushing of clumps). The process of forming knot-like structures in situ was discussed by Akashi & Soker (2008) and Akashi et al. (2015).

Kinematics. Earlier in this section we summarized our explanation for the seemingly linear rise of Doppler shift with distance seen in P–V diagrams in long-slit observations of M2–9 (Sections 2 and 4.1; Figure 1(b)). This explanation easily generalizes to the edges of all closed, expanding lobe sheaths of any width that are bounded by shocks.