OCULAR SHOCK FRONT IN THE COLLIDING GALAXY IC 2163

The top panel in Figure 1 displays a composite color image of IC 2163/NGC 2207 with the HST B image from Elmegreen et al. (2000) in cyan and the surface brightness image from our ALMA CO data in red. Features of interest in IC 2163 are marked in this figure. In the bottom panel of Figure 1, the color-coded I(CO) image is overlaid with contours of Spitzer 8 μm emission at 4 and 8 MJy sr⁻¹. This figure shows that along the eyelids in IC 2163 and along the spiral arms of NGC 2207, the CO emission occupies the same narrow ridge as the 8 μm emission. For clarity, higher level and lower level contours of 8 μm surface brightness are omitted. The molecular gas in IC 2163 is strongly concentrated in the northern and southern eyelids, with fainter CO emission from the inner spiral arms, the portion of the tidal tail included in this field, and part of the tidal bridge extension of the northen eyelid, until it blends, in projection, with a spiral arm of NGC 2207. The eyelids are generally much brighter in CO than the spiral arms of NGC 2207; the highest molecular column density in the galaxy pair (including the nucleus of NGC 2207) is on the northern eyelid.

In the outer northeastern part of the field in Figure 1 (R.A. east of 06 16 28 with decl. north of −21 21 50), no H i emission is detected (see Section 3.2 below and Elmegreen et al. 2016), so the CO emission there is probably not real. At this outlying location and at an analogous outlying location in the southwestern part of the field in Figure 1, the correction for the CO primary beam exceeds a factor of 1.3.

Figure 2 displays the CO velocity field of IC 2163/NGC 2207. We note that the CO isovelocity contours on the eyelids make a shallow angle with respect to the outer edge of each eyelid. The velocity field image reveals that over a wide range of azimuth, there is a large velocity gradient across the ∼1 kpc width of the eyelid at any given azimuth. We define the line-of-sight velocity difference between the outer and the inner edge of the eyelid as

$$\begin{eqnarray}&&{\rm{\Delta }}v\equiv {v}_{\mathrm{outer}}-{v}_{\mathrm{inner}}.\end{eqnarray} \tag{ 1 }$$

Zoom In Zoom Out Reset image size

The values of ${\rm{\Delta }}v$ for the eyelids are much greater than the velocity differences across the width of the spiral arms in NGC 2207. At some locations along the eyelids, cuts across the eyelid width perpendicular to the eyelid ridge-line yield values of ${\rm{\Delta }}v$ in excess of 100 km s⁻¹.

For measurement and analysis of the velocities at the eyelids, we transformed the images of IC 2163 to face-on polar coordinates. We adopted values of the projection parameters of IC 2163 from Elmegreen et al. (1995a, 1995b), i.e., we take the intersection of the plane of the disk with the plane of sky on the receding side as the H i kinematic major axis at a position angle of PA = 65° and the disk inclination i as 40° (where i = 0 for face-on). The ocular is intrinsically more oval than it appears in the sky-plane (Elmegreen et al. 1995a). For the transformation to polar coordinates, we take the center as the nucleus, which is at the center of the oval. The receding, approaching, near, and far sides of IC 2163 are identified in the sky-plane image of the CO surface brightness in Figure 3. The long lines in this figure are labeled with the value of the azimuthal angle θ in face-on polar coordinates, i.e., the kinematic major axis at θ = 0° and 180° and the kinematic minor axis at θ = 90° and 270°. The azimuth θ is measured counterclockwise from the receding kinematic major axis at PA = 65 °.

Zoom In Zoom Out Reset image size

Figure 4 displays the CO surface brightness image and the CO velocity field in face-on polar coordinates. The H i kinematic minor axis on the side nearest us is at $\theta =270^\circ$ (as indicated in Figure 3). Then the observed line-of-sight (LOS) velocity

$$\begin{eqnarray}{v}_{\mathrm{obs}}(R,\theta ) & = & {v}_{\mathrm{sys}}+[{v}_{R}(R,\theta )\sin \theta \\ & & +{v}_{\theta }(R,\theta )\cos \theta ]\sin i+{v}_{z}\cos i,\end{eqnarray} \tag{ 2 }$$

where R is the face-on galactocentric radius, v_R is positive for expansion, and v_θ is the sum of the circular velocity v_c and tangential streaming v_t. Because the tidal peturbation to IC 2163 is nearly in-plane, we assume that across the eyelid width, ${\rm{\Delta }}{v}_{z}$ for the molecular gas is negligible compared to ${\rm{\Delta }}{v}_{R}$ or ${\rm{\Delta }}{v}_{t}$. In Figure 4, the southern eyelid starts at $\theta \approx 75^\circ$, and the northern eyelid starts at $\theta \approx 250^\circ$.

Zoom In Zoom Out Reset image size

For the eyelid measurements of each parameter, i.e., the LOS velocity, the face-on radius R, the difference in face-on radius ${\rm{\Delta }}R$ = ${R}_{\mathrm{outer}}-{R}_{\mathrm{inner}}$ between the outer and inner edges of the eyelid, and the LOS column density $N({{\rm{H}}}_{2})$, we defined the edge of the eyelid as the closest pixel position to where I(CO) = 200 Jy beam⁻¹ m s⁻¹ except for the following two transition regions at the ends of the eyelids. Specifically, to distinguish between CO emission from the southern eyelid and that from the inner spiral at $\theta =167^\circ \mbox{--}174^\circ$ and to distinguish between CO emission from the northern eyelid and that from the transition to the tidal tail at $\theta =28^\circ \mbox{--}37^\circ$, we used the location of IRAC 4 clumps as a guide. The transformation from plane-of-sky to face-on polar coordinates resulted in a pixel size of $2\buildrel{\circ}\over{.} 3$ for the azimuthal angle θ. We measured the values of ${\rm{\Delta }}v$ at one pixel ($2\buildrel{\circ}\over{.} 3$) intervals of θ in the polar coordinate image, with the line-of-sight velocity measured at the same value of θ at both edges. We then averaged our measurements over four pixels ($9\buildrel{\circ}\over{.} 3$) in θ as that gives an arclength $R{\rm{\Delta }}\theta$ approximately equal to the FWHM of the PSF at the smallest value of R on the inner edge of an eyelid.

Table 2 lists the values of the LOS velocity difference ${\rm{\Delta }}v$ and the difference in face-on radius ${\rm{\Delta }}R$ between the outer and inner edges of the eyelid. The values of $N({{\rm{H}}}_{2})$ in this table are averages for the area between the inner and outer edges of the eyelid. For ${\rm{\Delta }}R$ and $N({{\rm{H}}}_{2})$, we used the same values of θ and averaged over the same 93 in azimuth as for ${\rm{\Delta }}v$. Given the channel width, the minimum uncertainty in ${\rm{\Delta }}v$ is 7 km s⁻¹. The uncertainties listed for ${\rm{\Delta }}v$ in Table 2 are either 7 km s⁻¹ or the standard error of the mean in averaging over 93 in θ, whichever is greater. The rms noise in the CO channel maps times the channel width is 37 Jy beam⁻¹ m s⁻¹, equivalent to $N({{\rm{H}}}_{2})$ = 3.3 M_☉ pc⁻². The uncertainties listed for $N({{\rm{H}}}_{2})$ in Table 2 are either 3.3 M_☉ pc⁻² or the standard error of the mean in averaging over 93 in θ, whichever is larger.

Table 2.  Eyelid Velocity Differences and $N({{\rm{H}}}_{2})$ ^a,b

θ	${\rm{\Delta }}v$	$N({{\rm{H}}}_{2})$	${\rm{\Delta }}R$	dv/dR
(°)	(km s−1)	(M☉ pc−2)	(arcsec)	(km s−1 kpc−1)
(1)	(2)	(3)	(4)	(5)
Northern Eyelid
263.6	31 ± 7	63.9 ± 4.9	10.4	27 ± 6
272.9	110 ± 26	91.7 ± 3.4	8.1	124 ± 29
282.2	127 ± 22	114.6 ± 3.3	7.2	161 ± 28
291.5	179 ± 20	153.3 ± 16.1	6.5	252 ± 28
300.8	148 ± 28	170.8 ± 5.6	7.3	186 ± 35
310.1	102 ± 24	151.3 ± 3.8	4.5	207 ± 49
319.4	117 ± 12	142.9 ± 4.3	4.2	255 ± 26
328.7	71 ± 7	113.9 ± 14.2	4.3	151 ± 15
338.0	40 ± 9	71.9 ± 3.3	4.5	81 ± 18
347.2	58 ± 8	84.5 ± 4.7	5.6	95 ± 13
356.5	63 ± 7	76.4 ± 4.0	7.4	78 ± 9
5.8	61 ± 7	63.8 ± 3.3	5.8	96 ± 11
15.1	57 ± 7	62.0 ± 3.3	4.0	130 ± 16
24.4	48 ± 7	65.7 ± 3.3	4.6	95 ± 14
33.7	37 ± 7	48.6 ± 4.0	3.8	89 ± 17
Southern Eyelid
75.5	9 ± 20	40.3 ± 7.0	10.0	8 ± 18
84.8	−59 ± 14	84.5 ± 6.0	8.8	−61 ± 14
94.1	−59 ± 7	86.3 ± 6.7	6.5	−83 ± 10
103.4	−60 ± 11	86.4 ± 5.0	6.7	−82 ± 15
112.7	−63 ± 7	90.1 ± 4.4	6.5	−89 ± 10
122.0	−69 ± 7	54.0 ± 3.6	6.9	−92 ± 9
131.2	−57 ± 7	67.7 ± 3.3	6.5	−80 ± 10
140.5	−55 ± 7	80.3 ± 6.0	5.1	−99 ± 13
151.1	⋯	⋯	⋯	⋯
161.4	−28 ± 7	105.1 ± 5.6	3.0	−85 ± 21
170.7	2 ± 10	65.8 ± 9.0	3.5	5 ± 25

Notes.

^aFrom measurements made at fixed values of θ in face-on polar coordinate images and averaged over $9\buildrel{\circ}\over{.} 3$ in θ. The azimuth θ is measured ccw from the receding kinematic major axis at PA = 65°. ^b ${\rm{\Delta }}v$ is the LOS velocity difference between the outer and inner edges of the eyelid, $N({{\rm{H}}}_{2})$ is the LOS molecular column density averaged over the area between the same outer and inner edges for the same fixed azimuth as ${\rm{\Delta }}v$, ${\rm{\Delta }}R$ is the difference in face-on radius, and dv/dR is the difference approximation to the directional derivative in the disk-plane of v with respect to R for θ fixed and inclination $i=40^\circ$.

Download table as:  ASCIITypeset image

The observed large values of ${\rm{\Delta }}v$ over a wide range of azimuth (see Table 2) argue for a mixture of radial and azimuthal streaming of gas at the outer edge of the eyelid with respect to gas at the inner edge of the eyelid. With the adopted orientation of IC 2163, the outer disk gas that streams radially inward, slows down radially in the ocular ridge, and speeds up azimuthally there, would produce ${\rm{\Delta }}v$ positive on much of the northern eyelid, i.e., from $\theta =270^\circ \mbox{--}360^\circ$, and ${\rm{\Delta }}v$ negative on much of the southern eyelid, i.e., from $\theta =90^\circ \mbox{--}180^\circ$. This is in the same sense as the observed values. Thus the observed CO velocity field is consistent with the idea that gas from the outer part of IC 2163 flows inward until its radial streaming slows down abruptly to produce the thin pileup zone in the eyelids.

In Table 2, the positive values of ${\rm{\Delta }}v$ for $\theta =6^\circ \mbox{--}34^\circ$, as the gas in the northern eyelid approaches the H i tidal tail, probably mean that the ${\rm{\Delta }}{v}_{\theta }\cos \theta$ term in Equation (2) dominates in this range. The values of $| {\rm{\Delta }}v|$ are smaller at the extreme eastern and extreme western ends of each eyelid than elsewhere on the eyelid; according to the encounter simulations, the material there originated in regions that experienced weaker tidal torque.

On the northern eyelid at $\theta =270^\circ$, ${\rm{\Delta }}v$ results from radial streaming only and has an LOS value of 110 ± 26 km s⁻¹, equivalent to ${\rm{\Delta }}v/\sin i$ = 171 ± 40 km s⁻¹ in the plane of the disk. At $\theta =0^\circ$ on the northern eyelid, ${\rm{\Delta }}v$ results from azimuthal motions only, and the value of ${\rm{\Delta }}v/\sin i$ is 96 ± 11 km s⁻¹. Thus there is considerable shear associated with tangential streaming since it is unlikely that the circular velocity changes by an amount this large across the 1 kpc width of the eyelid. The tangential streaming at $\theta =0^\circ$ has an appreciably smaller magnitude than the radial streaming at $\theta =270^\circ$, but the values of v_R and v_t probably vary with θ. On the southern eyelid at $\theta =90^\circ$, $| {\rm{\Delta }}v|$ results from radial streaming only, and the value of $| {\rm{\Delta }}v| /\sin i$ = 92 ± 11 km s⁻¹.

Column (5) of Table 2 lists the difference approximation to the directional derivative in the disk-plane of the velocity with respect to R across the width of the eyelids for θ fixed:

$$\begin{eqnarray}&&\displaystyle \frac{{dv}}{{dR}}=\displaystyle \frac{{\rm{\Delta }}v{(\sin i)}^{-1}}{{\rm{\Delta }}R},\end{eqnarray} \tag{ 3 }$$

where ${\rm{\Delta }}v$ is the LOS velocity difference from Column (2) and ${\rm{\Delta }}R$ is the difference in face-on radius in kiloparsecs. Note that this includes radial and tangential streaming. Except at the extreme eastern and extreme western ends of the southern eyelid and at the extreme western end of the northern eyelid, the values of $| {dv}/{dR}|$ are all quite large, with values in excess of 150 km s⁻¹ kpc⁻¹ on a significant part of the northern eyelid and 80–90 km s⁻¹ kpc⁻¹ on the southern eyelid. The velocity gradient is greater than or equal to the directional derivative, and thus the velocity gradients across the eyelids are large. Such large values are qualitatively consistent with the encounter simulation predictions that the eyelid compression region is a shock zone.

For the rest of this paper, we shall assume that the tangential streaming component ${\rm{\Delta }}{v}_{t}$ dominates ${\rm{\Delta }}{v}_{c}$ at the eyelids.

If we take the duration of a specific ocular disturbance to be of the order of the ocular radius divided by the streaming motions at the ocular, then the present ocular structure in IC 2163 should last for a timescale of the order of several tens of millions of years.

Elmegreen et al. (1995b) separate the H i associated with IC 2163 from that associated with NGC 2207. For IC 2163, the top panel of Figure 5 displays H i isovelocity contours overlaid on the CO velocity field, the bottom panel of Figure 5 has H i column densiy contours overlaid on I(CO), and Figure 6 has the same N(H i) contours overlaid on the HST B-band image. The lowest N(H i) contour level in these figures is 5 M_☉ pc⁻², which is equivalent to $3\times$ the rms noise over two channel widths. The beam symbol represents the H i resolution; the H i PSF has an FWHM equal to $13\buildrel{\prime\prime}\over{.} 5\times 12^{\prime\prime}$, PA = 90°. Thus the H i images provide a considerably less detailed view than our CO data with a resolution of $2^{\prime\prime} \times 1\buildrel{\prime\prime}\over{.} 5$.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In the CO velocity field, the kinematic major axis is at a position angle of PA = 65° ± 5°: on the outer edge of the eyelids, the highest velocity molecular gas is at PA = 65° on the northern eyelid and diametrically opposite the lowest velocity molecular gas on the southern eyelid. This is consistent with the value of PA = 65° ± 10° that Elmegreen et al. (1995b) obtain for the kinematic major axis in H i.

Because of tidal effects, the velocity fields in H i and CO are not regular, and in H i the dynamic center appears to be $\sim 5^{\prime\prime}$ northeast of the nucleus. The largest amplitude wiggles in the H i isovelocity contours tend to be displaced toward the outer side of the CO eyelids. Not all of the wiggles in H i velocity contours just north of the northern eyelid have the same sense of curvature; in the vicinity of the massive H i cloud I5 (Elmegreen et al. 1995b), labeled in the bottom panel of Figure 5, the curvature of the wiggles in the H i velocity contours reverses sense. This indicates complex flows in the H i gas related to this massive H i cloud. There are also pronounced wiggles in the H i velocity contours near the massive H i cloud I3, which is centered at the start of the tidal tail.

At the CO eyelids, the CO and H i isovelocity contours have a similar tilt at some locations but not at other locations, particularly on the southern eyelid. Some differences between the tilt of the CO and H i isovelocity contours at the eyelids may result from differences in spatial resolution and differences in where the column densities of molecular and atomic gas have local maxima.

The H i velocity dispersion at the massive H i clouds I5 and I3 is ∼50 km s⁻¹ after correction for the velocity gradient across the ∼2 kpc synthesized beam (Elmegreen et al. 1995b). Elmegreen et al. (2000) suggest that these high values for the H i velocity dispersion could result from (a) high turbulence in the H i gas and lead to unusually large H i scale heights or from (b) several streams with different velocities within the H i synthesized beam. If the H i disk is thicker than normal, the H i velocities at higher altitudes may differ from those in the disk. For comparison, Section 5 below presents our measured values of the CO velocity dispersion in various parts of the galaxy pair.

The massive H i cloud I5, whose center lies outside of the CO eyelid and outside of the B-band ridge, is located northeast of the region of greatest $N({{\rm{H}}}_{2})$ on the northern eyelid. Based on Hα and 24 μm emission, Elmegreen et al. (2016) find a very low star formation rate in their aperture A10, which covers most of I5. Thus the H i here is not a product of photodissociation. I5 may be along (or part of) the stream producing the greatest concentration of molecular gas on the eyelids, and some of its H i may get converted into molecular gas in the eyelid compression zone.

We compare the CO velocity field values of $| {\rm{\Delta }}v| /(\sin i)$ and $| {dv}/{dR}|$ on the northern eyelid with those on the southern eyelid.

For $\theta =273^\circ \mbox{--}310^\circ$ on the northern eyelid, the streaming motions in the disk ${\rm{\Delta }}v/(\sin i)$ range in magnitude from 159 km s⁻¹ to 278 km s⁻¹, and the values of dv/dR are 124–252 km s⁻¹ kpc⁻¹. Since this range of θ lies within $40^\circ$ of the H i kinematic minor axis, these velocities represent mainly radially inward motion of molecular gas at the outer edge of the northern eyelid relative to its inner edge. On the southern eyelid for $\theta =94^\circ \mbox{--}131^\circ$, there is little variation of ${\rm{\Delta }}v$ or of dv/dR with azimuth; the values of ${\rm{\Delta }}v/(\sin i)$ here are negative and have a mean magnitude of 93 ± 11 km s⁻¹, and dv/dR has a mean magnitude of 85 ± 11 km s⁻¹ kpc⁻¹. Since this range of θ is within $41^\circ$ of the kinematic minor axis, the observed negative values of dv/dR on the southern eyelid here represent mainly radially inward motion of molecular gas at the outer edge of the southern eyelid relative to its inner edge. Thus in the CO velocity field the magnitude of the radial streaming motions and the values of $| {dv}/{dR}|$ near the H i kinematic minor axis are considerably higher on the northern eyelid and vary much more with azimuth θ than on the southern eyelid.

Our measurements find a relative velocity difference between the outer and inner edges of each eyelid in the direction that produces a compression zone at the eyelids. Although the sense of this velocity difference is consistent with having gas from the outer part of the galaxy flow inward until it reaches the eyelids, an alternative interpretation would have the inner edge of the eyelids moving outward with respect to the outer edge. To see if the inner edge of the eyelids is moving radially relative to the center of the disk, we compare the CO value of ${v}_{\mathrm{inner}}$ (the velocity at the inner edge of the eyelid) on the kinematic minor axis with v_sys. The values of v_sys (heliocentric, optical definition) listed by Elmegreen et al. (1995b) are 2775 ± 10 km s⁻¹ from the kinematic minor axis of their H i velocity field and 2756 ± 15 km s⁻¹ from optical observations at the nucleus. The CO emission at the IC 2163 nucleus is faint and thus not useful for measuring v_sys.

After our CO velocities are converted to a heliocentric, optical definition, the northern eyelid has ${v}_{\mathrm{inner}}$ = 2777 ± 25 km s⁻¹ at $\theta =270^\circ \pm 5^\circ$, consistent with the H i and optical values of v_sys. However, the southern eyelid has ${v}_{\mathrm{inner}}$ = 2842 ± 9 km s⁻¹ at $\theta =90^\circ \pm 3^\circ$, which is appreciably higher than v_sys. The difference between the values of ${v}_{\mathrm{inner}}$ on the northern and southern eyelids at the kinematic minor axis can be seen in the color displays of the CO velocity field in Figures 2 and 5, where the northern eyelid is dark green and the southern eyelid is light green at these positions. Relative to the center of the disk, the inner edge of the northern eyelid does not seem to be moving radially but the inner edge of the southern eyelid is moving radially outward. After the gas streamed radially inward to form the eyelids, the inner edge of the southern eyelid may have started to move outward in a second ocular wave. Under certain conditions in the models by Struck et al. (2005), a second ocular wave can form and propagate outward.

As noted in Section 3.2 above, the H i velocity field reveals a small apparent shift of the dynamic center of IC 2163 to the northeast of the center of the galaxy.

These various asymmetries may be evidence of a dynamic situation in which the evolution of the ocular is not symmetric about the center of the galaxy because of differences in the near-field tidal forces on the two eyelids as discussed in Section 3.4 below.

Section 3.3 points out certain velocity asymmetries in the ocular. Relative to us, the near side of IC 2163 is at PA = 335°. According to the model by Elmegreen et al. (1995a), this is also the side of IC 2163 that was closest to NGC 2207 as IC 2163 traveled behind NGC 2207 from the north and west to its present position partially behind the outer eastern side of NGC 2207. This means that most of the northern eyelid, in particular, the part with values of ${\rm{\Delta }}v\gt 100$ km s⁻¹, is now on the side closest to NGC 2207. Thus the velocity asymmetries may result from differences in the near-field tidal forces on the two eyelids. The western end of the northern eyelid connects smoothly in shape and velocity to the tidal bridge arm between the galaxies. The eastern end of the southern eyelid connects smoothly in shape and velocity to the shocked gas returning to the main disk from the southern side (leading edge) of the broad H i tidal tail (see Figure 1 and the CO and H i velocity fields in Figure 5), which is on the side of IC 2163 farthest from NGC 2207.

We consider what effect the large differences in velocity across the ∼1 kpc width of each eyelid have had on the molecular column densities there. In Figure 7, we plot the LOS molecular column density $N({{\rm{H}}}_{2})$ from Column (3) of Table 2 versus the magnitude of the directional derivative $| {dv}/{dR}|$ of the velocity with respect to R from Column (5) of that table. The measurements for the 15 values of azimuth θ on the northern eyelid are plotted as red points; those for the 10 values of θ on the southern eyelid are plotted as blue points. A linear regression of $N({{\rm{H}}}_{2})$ on $| {dv}/{dR}|$ for all 25 points in this figure gives

$$\begin{eqnarray}&&N({{\rm{H}}}_{2})=41+(0.45\pm 0.06)\left|\displaystyle \frac{{dv}}{{dR}}\right|,\end{eqnarray} \tag{ 4 }$$

for $N({{\rm{H}}}_{2})$ in M_☉ pc⁻² and dv/dR in km s⁻¹ kpc⁻¹. The correlation coefficient is 0.84 with 23 degrees of freedom, and thus there is a strong correlation. Because the northern eyelid has a substantially larger range of $| {dv}/{dR}|$ than the southern eyelid, the regression tends to be dominated by the data on the northern eyelid. A linear regression for the 15 values of θ on the northern eyelid (the red points in Figure 7) gives

$$\begin{eqnarray}&&N({{\rm{H}}}_{2})=28+(0.52\pm 0.08)\left|\displaystyle \frac{{dv}}{{dR}}\right|\end{eqnarray} \tag{ 5 }$$

and a correlation coefficient of 0.87 with 13 degrees of freedom.

Zoom In Zoom Out Reset image size

The $| {dv}/{dR}|$ values in this figure involve a combination of radial and tangential streaming. We assume that at the eyelids the tangential streaming is ${\rm{\Delta }}{v}_{t}\simeq {\rm{\Delta }}{v}_{\theta }$. For the northern eyelid, the point for $\theta \simeq 270^\circ$ (where the measured $| {dv}/{dR}|$ represents pure radial streaming) and the point for $\theta \simeq 0^\circ$ (where the measured $| {dv}/{dR}|$ represents pure tangential streaming) lie on or very close to the fitted line in Figure 7. We do not measure the density contrast (eyelid/pre-eyelid) directly since our interferometric observations do not detect the pre-eyelid molecular column density.

Our choice of value for ${X}_{\mathrm{CO}}$ is listed in Section 2. If we were to choose a different value of ${X}_{\mathrm{CO}}$ for the eyelids, without varying its value with azimuth along the eyelid, this would change the slope of the fitted line in Figure 7, but there would still be a strong correlation between $N({{\rm{H}}}_{2})$ and $| {dv}/{dR}|$.

The following factors may lead to some of the scatter seen in Figure 7 and/or decrease the slope of the line. (a) After losing some azimuthal velocity in the shock/shear zone in the eyelids, the gas will start to fall inward slowly. This would tend to decrease its density at the inner edge of the eyelid and should have a stronger effect for stronger shocks. In Figure 4, the CO emission from the southern eyelid appears more ratty at the inner edge than at the outer edge, which may be a consequence of shocks generating turbulence. (b) According to the encounter simulations, the gas reaching the eyelid comes in from a range of radii R.

The radial compression at the eyelids causes an increase in the gas column density by direct radial impact and also leads to a high rate of shear there. One expects shocks and compression from the radial streaming and also from the shear associated with tangential streaming; both types of streaming can lead to intersecting gas streams.

The following argument based on angular momentum conservation computes the rate of shear arising from radial streaming into the eyelids. Consider particles in an annulus bounded by radii R₂ and R₁ with ${R}_{1}\lt {R}_{2}$ and initial azimuthal velocities ${v}_{\theta ,2}$ = ${v}_{\theta ,1}$ = ${v}_{\theta }^{* }$ (for a flat rotation curve). If a perturbation kicks particles inward from R₂ to radius ${R}_{\mathrm{outer}}$ (partway between R₁ and R₂) and kicks particles outward from R₁ to radius ${R}_{\mathrm{inner}}$ to form a thinner annulus bounded by ${R}_{\mathrm{inner}}$ and ${R}_{\mathrm{outer}}$, with ${R}_{\mathrm{inner}}\lt {R}_{\mathrm{outer}}$, then by angular momentum conservation, the azimuthal velocity of particles initially at R₂ speeds up to

$$\begin{eqnarray}&&{v}_{\theta ,\mathrm{outer}}={v}_{\theta }^{* }\left(\displaystyle \frac{{R}_{2}}{{R}_{\mathrm{outer}}}\right),\end{eqnarray} \tag{ 6 }$$

and the azimuthal velocity of particles initially at R₁ slows down to

$$\begin{eqnarray}&&{v}_{\theta ,\mathrm{inner}}={v}_{\theta }^{* }\left(\displaystyle \frac{{R}_{1}}{{R}_{\mathrm{inner}}}\right).\end{eqnarray} \tag{ 7 }$$

The rate of shear between ${R}_{\mathrm{outer}}$ and ${R}_{\mathrm{inner}}$

$$\begin{eqnarray}&&\gamma =\displaystyle \frac{{v}_{\theta ,\mathrm{outer}}-{v}_{\theta ,\mathrm{inner}}}{{R}_{\mathrm{outer}}-{R}_{\mathrm{inner}}}.\end{eqnarray} \tag{ 8 }$$

Substituting for ${v}_{\theta ,\mathrm{outer}}$ and ${v}_{\theta ,\mathrm{inner}}$ gives

$$\begin{eqnarray}\gamma & = & {v}_{\theta }^{* }\displaystyle \frac{({R}_{2}/{R}_{\mathrm{outer}}-{R}_{1}/{R}_{\mathrm{inner}})}{{R}_{\mathrm{outer}}-{R}_{\mathrm{inner}}}\\ & = & {v}_{\theta }^{* }\displaystyle \frac{({R}_{2}{R}_{\mathrm{inner}}-{R}_{1}{R}_{\mathrm{outer}})}{{R}_{\mathrm{outer}}{R}_{\mathrm{inner}}({R}_{\mathrm{outer}}-{R}_{\mathrm{inner}})}.\end{eqnarray} \tag{ 9 }$$

Then setting ${R}_{\mathrm{outer}}\approx {R}_{\mathrm{inner}}\approx R$, this reduces to

$$\begin{eqnarray}&&\gamma =({v}_{\theta }^{* }/R)\displaystyle \frac{{R}_{2}-{R}_{1}}{{R}_{\mathrm{outer}}-{R}_{\mathrm{inner}}}.\end{eqnarray} \tag{ 10 }$$

The density contrast is

$$\begin{eqnarray}&&\displaystyle \frac{{\rho }_{\mathrm{compressed}}}{{\rho }_{\mathrm{initial}}}=\displaystyle \frac{{R}_{2}-{R}_{1}}{{R}_{\mathrm{outer}}-{R}_{\mathrm{inner}}}.\end{eqnarray} \tag{ 11 }$$

Comparison of Equations (10) and (11) shows the proportionality between the rate of shear and the density contrast:

$$\begin{eqnarray}&&\displaystyle \frac{{\rho }_{\mathrm{compressed}}}{{\rho }_{\mathrm{initial}}}=(R/{v}_{\theta }^{* })\gamma ,\end{eqnarray} \tag{ 12 }$$

where γ = ${\rm{\Delta }}{v}_{\theta }/{\rm{\Delta }}R$ is the rate of shear.

The rate of shear and the column density contrast correlate with each other because both result from the radial compression of the eyelid.

From our data, we are unable to disentangle how much of the observed correlation in Figure 7 is due to direct radial impact and how much is a subsequent consequence of the shocks in the intersecting gas streams that arise from shear.