Part 73- The 4.5-Kilometre-Long Rift From The Northern Long-Axis Tensile Force Line

Original:

Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0/A.COOPER

The headers are reproduced below with their keys and explanations. The ESA regions map is at the end of this post for those who are unfamiliar with the region names and locations. 

INTRODUCTION

In Part 72, we saw how dominant the northern and southern long-axis tensile force vectors were. This part continues that theme but dwells on the fact that the northern tensile force line gave rise to a rift running the whole length of the body lobe. It should be regarded as the entire area of Seth, Babi and Ash, below the bottom red line, shunting away en masse from the northern, long-axis tensile force line. It involves at least two onion layer thickness. 

THE PHOTOS

Photo 1- the long-axis tensile force lines. This is reproduced from Part 72.

Red- the northern and southern long-axis tensile force lines. The northern one is the nearer one. They run from long-axis tip to long-axis tip on the body. They passed either side of the proto-head lobe before it sheared from the body. They now pass either side of the neck which is itself elongated along the long axis for the very reason that these two force vectors were stretching it along that vector. 

Bright green- the Apis region at the long-axis tip. The two tensile force lines join just above Apis and in line with its centre, the exact long-axis tip. 

Photo 2- the lower onion layers that rifted from the northern long-axis tensile force line. In Hapi, it’s the third layer down from the paleo surface. The paleo surface is the original surface of the single body-  see Part 41, scroll to ‘the three levels’. This third layer down appears as the second layer down today because the old top layer slid to the back of Aswan (the terraced cliff) and Babi (the Cliffs of Aten). The two lines angled away, either side of Hapi, are today’s top layer rifted from the same northern tensile force line. This corresponds to the second paleo layer down except near Apis where it’s also the top paleo layer (see the Ash recoil, below). This completes the rift running the whole length of the body lobe.  

This is essentially a rift that runs from one end of the body lobe to the other. It runs between the two long-axis tips because the tensile force line it rifted from also runs between the two long-axis tips. It should be visualised as being the entire section of surface crust incorporating Seth, Ash and Babi doing a one-off shunt away from the tensile force line. The shunt was between 150 metres (at Aker/Babi) and 400 metres (central Hapi). It incorporates the 1.6km x 200m rift (Parts 48 and 49) running through Seth and Ash. It involves at least two layers. The upper layer slid even further on as described in Parts 32, 33, 40.

The wider, Hapi section of the rift corresponds to the shunt of the Hapi cliff line from the line of boulders along Hapi. This was presented in Part 47. It could be possible that, to some extent, it was the boulder line that rifted away from the Hapi cliff line when the head lobe sheared and drew neck material up with it and out of Hapi. The boulder line would in that case have been drawn across Hapi in a translational movement from the cliff whilst maintaining the shape of the cliff line along its length. Part 47 shows how that translational match across 350 to 400 metres is still discernible today. This sweeping up of neck material and dragging the boulder line back from the cliff in the process would explain the rift being wider at this point. This was touched on in the previous part. The principle of the head lobe drawing up neck material with it as it rose on the incipient neck after shearing is dealt with in Part 25. 

What is new in this part, regarding this much longer rift, is that the two rifts from Parts 48/9 and Part 47 have been linked as being one long rift. This was achievable because the line from which they rifted, the northern tensile force line, was identified as continuing along the Hapi boulder line from the southern perimeter of the 1.6km x 200m rift (Part 72). And in order to join those two lines, the southern perimeter of the 1.6km x 200m rift had to be identified as extending further, past the mauve anchor and into Hapi. That identification was done via the four mauve features that delaminated along the northern tensile force line, thereby betraying its existence in Hapi. That proved that the northern tensile force line continues from the mauve anchor, right up to the beginning of the boulder line. This discovery was presented in Part 71 and so it links the southern perimeter of the 1.6km x 200m rift to the Hapi boulder line. This means the rift runs from Apis to at least the other end of Hapi. 

The final piece in the puzzle is that the Babi slide (Part 40) incorporates a 150-metre-wide rift along the border of Aker and Khepry. Since this rift runs from the end of the boulder line in Hapi to the other long-axis tip, it completes the rift running the entire length of the body lobe, as depicted above. This rift hasn’t been blogged yet but it was responsible for getting the slide track of the fourth Babi cuboid wrong in the original Part 40 post. It was the discovery of the cuboid’s true track (and matched seating at the end of Hapi) that betrayed the rift. See the update at the end of Part 40 showing this seating match and slide track in detail on an OSIRIS image. The 150m rift is implied by the new track but wasn’t explicitly pointed out. So the rift now runs the entire length of the body lobe from long-axis tip to long-axis tip as shown in the photos above and photo 3 below. 

Photo 3- the v’s show the direction of the rifting crust away from the northern long-axis tensile force line. The v’s are therefore slide vectors. 

Photo 4- this shows how there were two overall stretching and sliding vectors at play: (1) long-axis, core-directed stretch (running between and parallel to the two tensile force lines) and (2) radial sliding of crust (outside the tensile force lines). 

Although the long axis stretch and radial slide vectors appear somewhat schematic in photo 4, they are real slide vectors that have been identified via translational matches and were blogged long ago. See photos 7 to 11 which build on the above slides with several more arrow vectors. Photo 11 shows the various part numbers for each slide. 

The reason photo 4 has fewer arrows is because the intention is to make it appear schematic so as to emphasise the obviously different direction of the long arrow between the tensile force lines. That arrow is running along the long axis i.e. parallel to the tensile force lines while all the others are directed away from it in a radial pattern. Clearly, there were two different mechanisms at play either side of the northern tensile force line. We saw this very much in close-up in Part 71 with the mauve delaminations sliding along the length of the tensile force line, kissing one side of the line as they slid along it. Meanwhile, the Aswan slide rifted away from the other side of the line at 90°. The northern tensile force line is a very strong demarcation line between these two different slide vectors. This was also shown as far back as Part 50 with a really crisp depiction of the orthogonal nature of the slide vectors in the vicinity of the ‘blocky rectangle’, which is the #4 mauve delamination. That was a detailed OSIRIS photo.

Photo 5- this shows the continuation of the long-axis stretch vector going under the neck and along its centreline.

In photo 5, the head lobe is in the way of the neck so we’re looking through the head, and the neck as well, to the arrow running along the base of the neck. It has points at both ends, depicting the fact that the neck was stretching both ways, causing it to elongate along the long axis. In reality, all points along the long axis were stretching ‘both ways’ with an equal and opposite tension along the lines at any given point. However, the double-pointed arrow helps to show that equal and opposite tension averaged across the middle of the comet, while the two ends are intuitively seen as stretching away from each other in opposite directions. That’s why the short arrow at Aker (top-left) is pointing in the opposite direction to the ones on the red triangle that points at Apis. 

Now that all the long-axis stretch lines are in place, you can see that they are following the core-directed stretch vector which is completely different from the radial vector for the sliding surface crust. The only reason the surface crust slid radially is that it had been sheared by the shear gradient across the northern tensile force line. That meant the crust actually sheared along the length of the tensile force line. This was the initial stage for allowing the rift being described in this part to happen. The crust was now free to slide and it slid radially, en masse, to a higher radius because the comet was spinning so fast: a 2- to 3-hour rotation period on head shear. 

The shearing of the lower layer of crust by the northern tensile force line isn’t quite the same as the classic shear line itself as matched in the very early parts of the blog. The classic shear line was the exact body matches (to the head rim) on the next layer above. This rift of the lower onion layer seems to be a sympathetic shearing slightly further in and under the head lobe. However, the second layer up that sits on this deeper layer does host the true shear line. It’s slightly further back and must’ve dragged the head rim out with it or the two wouldn’t exhibit the matches we see today. 

This sympathetic inner/lower layer was dealt with in more detail in Parts 39 and 41 as part of the “three levels”. It’s level 3 which is this inner/lower level; level 2 is the main Aswan terrace and also the smooth, riven-looking area of Babi; and level 1, is the slid Babi cuboids (the Cliffs of Aten) and the stacked up cliff creating the rim around Aswan. The photo of these layers, annotated, is in Part 41. 

The three levels described above have nothing to do with the four layers on the other side of the tensile force line that are within the red triangle. Such is the strong demarcation line either side of the tensile force line. Three of the four layers in the red triangle probably do correspond to the three levels the other side because they were once attached prior to the northern tensile force line holding sway and shearing them apart. However, the sliding and delaminating processes either side of the line are so markedly different that it’s difficult to trace the layers across the gap of the rift. This will be a future project but is not considered worthwhile at the moment.  

In the final analysis, it makes little difference distinguishing the line of the lower layer from the one above it with the shear line matches. This is because we’ll come to see that they were nested together at the northern tensile force line just prior to shear and the shear went through both layers along that line. The distinction between the line of the layers is however a useful concept because the second layer with the shear line matches slid back from the tensile force line somewhat further than its lower companion. The classic example of this is the Aswan slide in Part 69. The bright green matches in that part are etched onto the lower layer that didn’t slide as far. And when the Aswan layer is slid back to its seating on that lower layer, the lips or cliffs of the two layers would nest to form one big cliff. That cliff is the original tear line- and both were sheared together along the northern tensile force line when they were seated. Their location at the time of the shear was exactly between the end of the boulders and mauve feature #1. After shearing, they then slid together. Then Aswan slid on still further. This hasn’t been blogged yet but has been implied repeatedly in the last few parts. It will have its own post soon. 

Photo 6- this shows the slides and delaminations from Parts 69, 70 and 71

Photo 6 is shown for context so that you can start to see how all three parts, 69 to 71, rely very heavily on this major rift running from long-axis tip to long-axis tip of the body. This is despite the fact that the slide in Part 69 looks to be completely independent from the four layers when viewed in close-up. When all the layers are eventually slid back to the tensile force line in future Parts, we’ll see that the four layers and mauve features were nested and kissing the right hand end of the upper green wavy line. 

Key:

Small bright green wavy lines- the Part 69 translational matches that show the entire Aswan layer slid over this lower layer to where it is today. The front rim or cliff of Aswan therefore used to be nested to the front rim of this quasi rectangular lump of lower layer. This is the layer described above as nesting below Aswan and the two together being sheared along the tensile force line between the boulders and the #1 mauve feature. 

Large red- the northern and southern tensile force lines. 

Four short lines in smaller red dots- these run between the northern and southern tensile force lines and stop abruptly at the tensile force lines. They represent the four delaminated layers, #1 to #4, in Part 70. #1 is the farthest and lowest one of the four. They were sheared at either end by the tensile force lines leaving four short lengths of layers to delaminate towards Apis within the red triangle. They delaminated towards Apis because that was the direction of the long-axis stretch vector. 

Mauve dots- these are the four mauve features, #1 to #4, as described in Part 71. The farthest and lowest one is #1. Each mauve feature sits on its layer of the same number. 

Photo 7- the slide vectors. Basic version without additional annotations. 

Photo 8- same as photo 7 with additional annotations.

Added bright green curve- this is the Ash recoil, first described in Part 32. It’s curved because it’s betraying the radial nature of the layer slides. This is the main layer front in this vicinity and so it will be at 90° to the slide vectors. Since the slide vectors are radial, this line can’t help but be curved. 

The dusty Ash surface between the Ash recoil and Apis (the other green line) is both today’s top layer and the paleo top layer i.e. nothing slid away from above it. It did all the sliding itself beyond the Ash recoil line hence the recoil curve itself and the flaccid, blanket-like look of Ash beyond it.  The recoil curve is the loose edge of the blanket, curving according to the exigencies of the radial force vectors. 

Notice the recoil curve has a gap across the 200-metre width of the 1.6km x 200m rift before resuming within the red triangle. In reality, you can make out its line across the rift because it dragged material across the rift in its wake (see original). But it’s not very noticeable here. It is noticeable in the photos of Part 49 though. 

Photo 9- with extra slide vector arrows
Two more slide arrow vectors are added to photo 8. One is further up the red triangle and in line with the long axis stretch vector between the two tensile force lines. This vector represents the delamination vector for the four layers of Part 70. 

The other added arrow is at the end of the 1.6km x 200m rift. Although the rift is predominantly associated with rifting across its 200m width, it also stretched along its length. This is implied by the two stretch vectors either side of the rift which means the two perimeters and floor of the rift couldn’t escape the stretching along its length. The matches on opposite sides all along the rift show this assumption to be correct. 

Photo 10- with the mauve delaminations from part 71. 

Mauve- the mauve feature delaminations, #1 to #4. The two middle ones are slightly obscured by their red arrow. 

Dark blue- the north pole. 

Brown- the paleo north pole preliminary adjustment position (see Part 37). 

Photo 11- the fully annotated photo with yellow numbers showing the blog part that describes that particular slide vector arrow in that area. The parts show the relevant translational matches. 

Photo 12- the ESA regions. 
Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/A.COOPER

PHOTO CREDITS

FOR NAVCAM:

Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0

To view a copy of this licence please visit:

http://creativecommons.org/licenses/by-sa/3.0/igo/

All dotted annotations by A. Cooper. 

FOR OSIRIS:

Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/A.COOPER

Part 72- The North and South Long-Axis Tensile Force Lines

Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0/A.COOPER

Red- the two long-axis tensile forces. They’re almost perfectly mirrored, passing either side of the head lobe and from long-axis tip to long-axis tip. This symmetrical mirroring is helped by the fact that the shape of 67P itself is symmetrical about its long axis. 

Bright green- Apis at the long axis tip. It looks offset from the red triangle tip (the sharp vertex where the two lines join) from this view. But if we were to drift round so we were looking straight down on the tip, it would point straight at the central bright green dot of the five. This is because Apis is at the long-axis tip and the stretching of 67P, before the head lobe sheared, was directed from the centre of Apis to the centre of Khepry along the red lines. The symmetry of the tensile forces dictates that they have to join at the centre of Apis. In reality, they join just above Apis but the join is exactly in line with the Apis centre.

The stretching of 67P was due to spin-up to a 2- to 3-hour rotation rate and the spin-up torque would have been from random, asymmetrical outgassing. 

The two lines follow the same mirrored crustal features on either side of their centreline which itself traces the long axis of the body. This symmetry both of the features and the lines is because the lines are tensile force lines which had a shear gradient across them when 67P was stretching as a single body. The shear gradient sheared the crust along the tensile force lines, thus forever leaving their stamp on the comet’s surface. Since the forces were arranged symmetrically either side of the long axis centreline it follows that the crustal patterns they created (rifts and delaminations) are also symmetrical about the long-axis centreline. It’s a mirrored symmetry with the centreline being the reflection line. 

The centreline/reflection line is contiguous with the paleo equator for the two sections of the two lines running from Hapi to where they join just above Apis (see the Paleo Rotation Plane Adjustment page in the menu bar). 

At Hapi, the centreline runs through the centre of the neck, longways. So it’s actually ~400m below Hapi and at a level between where the northern and southern tensile force lines run i.e. in the same plane that’s spread between them. 

At the other end of the neck, the centreline emerges at Bastet/Aker. It then drops over the centre of the V-shaped Aker and traces the central ‘prow’ of Aker. The prow is the aforementioned V-shape translated down Aker into 3D. It runs down the centre of Aker and is also contiguous with the paleo equator like the centreline of the red triangle at the opposite end of the neck. We’ll see in later photos that the two red lines run parallel to each other down either side of Aker while remaining parallel to and equidistant from the prow. In other words the two tensile force lines maintain their symmetry past the end of the neck and down Aker. This also applies to Khepry as the two red lines run down its two outside edges which continue on from the Aker edges. By the time we reach the other long-axis tip where Khepry bends round sharply to the base of 67P the two red lines have maintained their symmetry for 4.5 kilometres, from long-axis tip to long-axis tip (Apis to Khepry). 

The apparent flaccidity of the red line through Hapi is due to the slide of the Babi/Hapi cliff line which occurred to a slightly greater extent than it did at Aswan/Hapi. The slide of the whole Hapi cliff line across Hapi was presented in Part 47. The extra movement of the Babi portion (radially away from the north pole as always) hasn’t been blogged yet. However, anyone who’s familiar with the dark green ‘gull wings’ (the classic third set of wings) and the fuchsia ‘India shapes’ nearby would realise this entire cliff line has to have shunted in sympathy with their shunting because they sit on its rim. This extra shunt southwards, which is essentially a shunting of the entire Babi layer, is the reason for that stunning dog-leg in the ridge running down from Hapi to the Cliffs of Aten. That ridge was formerly dead straight, under tension, and pointing directly at the north pole. It went flaccid when the tension was released by the Babi layer shunt. 

PHOTOS

Photo 2- header reproduced.

Photo 3- a view from the other side. The southerly line is nearest to us here.
Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0/A.COOPER

Photo 4- the shape model. You can see the sharp red vertex pointing at the centre of Apis here. 
Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/A.COOPER

Photo 5- view from the other end of the body. 
Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/A.COOPER

Red- the two tensile force lines. The right hand one is the northern one, running through Hapi, along the line of the boulders. You can see the largest boulders on the horizon. These are at the beginning of the boulder line so you can match these to the above photos. 

Bright green- the long-axis tip of the body at Khepry. 

The ‘prow’ at Aker is the faint central ridge running down from the top which borders the neck. The prow is faintly shadowed and has three boulders chipped away from its bottom end. Its top end starts in the middle between where the two red lines turn sharply over the edge of the top rim of Aker to follow its two rugged side perimeters. The prow defines the paleo equator. Today’s equator runs about 400 metres to its north, down the right hand side of Aker and Khepry as viewed here. 

The two tensile force lines running down either side of Aker and Khepry had a shear gradient too, just like at the red triangle. This sheared the crust via slip-shear, thus actually creating the perimeters of Aker and Khepry. This shearing left Anhur on the south side and Babi on the north side, free and loosened from Aker. They were therefore now free to slide radially across the surface from their respective poles. They slid to a higher radius due to the high spin rate on shearing of the head lobe (2- to 3-hour rotation rate. See Spin Up Calcs in the menu bar). The slip-shear on the Babi side left a discernible rift of around 150 metres wide running from Hapi to the Cliffs of Aten. This is the corollary to the 1.6km x 200m rift at the opposite end of the body, caused by the same tensile force line. 

If you look from head-on in front of Aker/Khepry (or from above) you can see the symmetry of the Babi and Anhur slides either side. 

For more context on the morphological evolution at this end of the body, see the Paleo Rotation Plane Adjustment page in the menu bar (description after photo 7 on that page). Also Part 51 which matches the two Bastet pancakes to the depressions either side of the prow. And Part 61, which has close-up gifs for the same pancake/Aker match. 

Photo 6- south pole shot showing the same Aker and Khepry end on the right as in photo 5. Also the path of the southerly tensile force line along the southern side of the neck at Sobek. Few close-ups exist of this line at the moment.
Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0/A.COOPER

The Sobek path of the southerly tensile force line is the mirror image of the force line in Hapi following the boulder line. It keeps up this mirror image faithfully along the top of Geb (Geb/Sobek) and part of Anhur but becomes blurred for a few hundred metres across the Anhur slide. After that it comes into view as in photo 5, above. 

RELATIONSHIP BETWEEN THE TENSILE FORCE LINES AND THE DISTINCT MORPHOLOGIES THEY CREATED. 

The two tensile force lines are directly or indirectly responsible for the greater part of the morphological diversity on 67P. Luckily, these different areas exhibiting the different morphologies were noted and given names in the sub-series (Parts 22-29). This was before the tensile force lines were fully understood. It was simply noted that these lines seemed to divide off the areas. 

However, the shear gradient across the lines at the two red triangle long sides was noted in Part 26. This was the ‘wind-tail’ analogy i.e. the red triangle being protected by the proto-head and later, the neck, from the full force of stretch. This caused the shear gradient across both tensile force lines.

The shear gradient was an increase in tensile force from a small value inside the triangle to a much larger value outside the triangle. The shear gradient was steep, over just a ~20m width. So it was like a lot of parallel ropes under tension along the length of the triangle sides and across a band 20 metres wide. The ropes would be under greater tension on the outside of the band than on the inside, thus causing shear. That explains the inevitable slip-shearing of the crust along the tensile force lines. This caused the 1.6km x 200m rift (Parts 48 and 49) on one side of the red triangle and the Anubis tear and slide on the other side (a less neat and obvious rift). These rifts occurred outside the relative calm that prevailed inside the triangle whose shape actually represents the lee from the tensile forces sitting behind the neck. So the triangle is a visible representation of the of the ‘wind tail’, stamped onto the comet’s surface. 

The following photos show the areas noted in the sub-series, Parts 22-29, and they’re culled from those parts. They apply only to the Seth/Anubis end of the comet and how the two tensile force lines divided up the different morphologies at this end. The two tensile force lines also sheared and rifted the other end at Aker and Khepry. They therefore actually formed those two regions by tearing them away from Babi and Anhur either side of them. However, this dividing up of Aker, Khepry, Babi and Anhur by the tensile force lines at that end of the comet was explained amply above, along with the photos and parts suggested for further reading. 

Photo 7- the areas at the Seth/Anubis end of the comet that are named in this blog and are related to the two tensile force lines.
Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0/A.COOPER

Yellow- Aswan, formerly known as site A. This doesn’t kiss the northern line today but used to (to be blogged soon).

Bright green- the slab A extension. So named because it appeared to be related to site A (and its missing slab). It looked related by virtue of sharing the same curved back as Site A and also by adjoining site A. Its southern perimeter is contiguous with the northern tensile force line in this post. That would be the upper-right line, the very straight one leading up to the mauve dot. It follows the tensile force line because the tensile force line sheared the crust along this line. This caused the 1.6km x 200m rift which explains most of the Slab A extension’s ‘flayed’ look. So the southern perimeter of the 1.6km x 200m rift is the southern perimeter of the slab A extension and the rift sits wholly within the extension. 

Incidentally, the subsequent discoveries that do indeed relate the slab A extension to Site A are beyond the scope of this post but are to be found in Part 32 (the Ash recoil), also 37 and 69.

Red- the red triangle. The red triangle includes the four bright green dots running up to the mauve dot. They’re only green so as to show the perimeter of the slab A extension above, which is contiguous with the red triangle. It’s contiguous because the northern tensile force line sheared the crust, thereby separating the red triangle from the slab A extension. It did so by causing the 1.6km x 200m rift. So what used to be attached to the red triangle is now 200m away and parallel to it (the opposite rift perimeter). That’s why the slab A extension looks flayed, being the floor of the rift. The red triangle extends beyond the slab A extension by three red dots, the last one being at the tip. This is an old photo- the last two red dots at the bottom, placed in shadow, are too low. They should be raised to kiss the edge of the shadow and thus form a sharper triangle tip. They should follow the feature with two dark eyes and a bright droopy nose. This one place on 67P where recognising a face in the rocks is useful. It holds up well at almost all angles and is a boon for locating the red triangle tip.

Orange- the ‘missing’ Babi slab which is now known not to be missing. It slid radially from the north pole and concertinaed up forming the Cliffs of Aten (Part 40). That’s why the ‘dog-leg’ ridge was described above as being under tension. The first Babi cuboid that had slid 800 metres was pulling it tight before the Babi layer dislodged to ease the tension. The ridge is attached to the cuboid to this day and the cuboid is unequivocally matched to the Hapi shear line via its jet source and slide tracks (Part 52).

Photo 8- shows the Anubis slide which was sheared from the red triangle by the very straight southern tensile force line. 
Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0/A.COOPER

T- red triangle (swamped by other colours).

Fuchsia- this is the perimeter of what was supposed as being the missing Anubis slab in Part 23. It has since been established that even if some slab material was flung from the comet, much of it slid instead. This was established in Part 54 and other translational slide matches have since been found but not blogged as of the date of this part. The main point for our purposes in this part is that there was material that used to be attached to the red triangle but was sheared away from it along that very straight, southern tensile force line. You can see it here except it’s dotted fuchsia because it’s a Part 23 photo. The traditional red triangle southern, long side runs from the dark green dot to the last small red dot at the sharp end of the triangle. There are five fuchsia dots running between them along that line, including the one kissing the dark green dot. The second and third fuchsia dots to the left of dark green are on the really straight part of the tensile force line, betraying the fact that it is indeed a tensile force line with a very steep shear gradient. 

We now know from Parts 70 and 71 that the red triangle wasn’t quite as undisturbed as originally thought and contains delaminated layers. Those strange floppy bits overhanging Anubis are the Part 71 layers that were sheared across their widths by the southern tensile force line and are now drooping over under the influence of gravity. They can be translationally matched to Atum, 600 metres away. The delaminated layer lines are depicted in Part 71 with red dots. The lines meander across the width of the red triangle and three of them arrive at one or other end of each floppy piece. This proves that the floppy pieces are just the delaminations sliced across their widths like lasagne strips. 

This concludes the relationship between the two tensile force lines and the morphologically distinct areas they created that were identified in the sub-series: the red triangle; the slab A extension; the Anubis slide/rift; Aswan. All these areas were caused by the slip-shearing along the two tensile force lines and the subsequent rifting from or sliding along the lines.

The two lines caused more rifting and delamination as they dropped down into Hapi and did so on both the northern and southern sides of the neck. Both force lines were responsible for the first and second delaminated layers in Hapi as described in Part 70 and the first and second mauve features as described in Part 71. They were responsible in the sense that their shear component slip-sheared the layers, allowing them to delaminate. Then the tensile component of the force lines delaminated them. This created the red triangle extension as described in Part 70. 

Inside the red triangle extension, the mauve features delaminated along the northern tensile force line, kissing it faithfully all the way and directly outside the line everything slid away from it at 90°. This was already the case for the classic red triangle with the 1.6km x 200m rift opening up at 90° and parallel to the triangle long side. But it also occurred in Hapi with the Aswan slide of Part 69. This will be elaborated on soon because Part 69 didn’t ever show Aswan attached to the northern tensile force line. But ultimately it was, and Part 69 just shows the last stage of the slide. 

This takes us to the beginning of the boulders in Hapi which run along Hapi’s length. They define the northern tensile force line between the #1 mauve delamination (on layer #1) and the point where the line dives down the front, northern side of Aker. We know from Part 47 that the Hapi cliff rim recoiled or slid from the boulder line and is a translational match to the line. It may conversely be the case that the boulder line slid from the Hapi rim as the neck was extruded from the body (see Part 25). But the translational match is sound in both cases, whichever way round they slid, and so one or other had to be the true tensile force line. 

Apart from the Sobek morphology along the south pole side of the neck, this completes the description of how the two tensile force lines divided up the comet into morphologically distinct areas. Sobek will have to wait until better photos come along. 

PHOTO CREDITS

FOR NAVCAM:

Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0

To view a copy of this licence please visit:

http://creativecommons.org/licenses/by-sa/3.0/igo/

All dotted annotations by A. Cooper. 

FOR OSIRIS:

Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/A.COOPER

Part 71- The Four Mauve Anchor Delaminations Betray The Four Layer Delaminations below Anuket

Your attention is drawn to the new “PARTS MENU (quick links)” in the menu bar above. It allows much quicker sequential-number access to the numbered parts than the year/month chronology that’s so favoured by blogging platforms. 

Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/A.COOPER

Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0


Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0/A.COOPER

The first header is photo 14  reproduced. The second header is unnumbered as it’s a reminder from Part 70 and will be inserted in a few places to save scrolling up and down to the top. Photo 14 is at the end of the post. It shows a summary of the various layer slides and delaminations laid out in this part and Parts 69 and 70. Photo 14 has a description which, along with other preceding photo keys, explains all the slide vectors in the first header. 

The other three header photos are designated as being photos 1 to 3. Photo 3’s description is divided into several paragraphs ending with ‘/////’.

It’s advisable to read Part 70 as a primer to this part because it shows two views of the four layers. It’s much shorter than usual, mostly photos. 

Your attention is also drawn to Appendix 2, which has a selection of photos showing how the four mauve delaminations each sit on their own respective delaminated layer. 

Photo 1

The four mauve dots in the third header are sitting on four similar shapes that once nested together. They have delaminated along with the onion layers they’re sitting on and they’ve delaminated along the long-axis direction of the comet which is consistent with the tensile forces of stretch. This is elaborated on further down. The delamination was rather like the opening up of one side of a cantilever tool box where the trays slide over each other (except the trays are filled to make them solid slabs or layers sliding over each other). We’re looking at four mauve features and each one sat on its own respective tray or layer, making four features on four layers. Each mauve feature is at the northern end of its respective layer and perched on the front lip of that layer. Each front lip is stepped up above the next layer and not very obviously so in the case of the farthest two layers towards Hathor. Each layer is deeper as we progress from the one nearest us, and towards the Hathor cliff (from layer #4 to layer #1 in photo 2). The features are dotted mauve because they were all once nested to the classic mauve anchor (Part 24) which is the third feature on the third layer per photo 2. 

Photo 2

This is a close up of the photo above. It shows the exact outlines of the four mauve features and numbers them, #1 to #4, from the Hathor cliff end back towards us. The classic mauve anchor from Part 24 is #3. The blocky rectangle from Part 50 is #4. If you’ve been reading the blog, these features will be familiar and things will make sense more readily. The #1 and #2 mauve features will get their own names in due course because they’ll be referred to a lot for several more parts. Their northern perimeters (left hand perimeters in photo 2) define where the Aswan layers were originally attached before they slid across Hapi (Part 47). 

In summarising photos 1 and 2, we have four layers enclosing three delaminations: #2 from #1; #3 from #2; #4 from #3.

The four layers, #1 to #4 shouldn’t be confused with ‘levels 1 to 3’ in Part 39. Those three levels ran along Babi and Aswan. The four delaminated layers in this part are a different set of delaminations subject to wholly different shear force and tensile force vectors. This different force vector set-up is by virtue of their being inside the red triangle, or rather, inside the newly extended red triangle presented further below. 

Photo 3

This shows the four delaminations in close-up. It also shows the 1.6km x 200m rift in red (Parts 48 and 49). This rift is key to understanding the shear forces and tensile forces of stretch that caused the mauve delaminations.

Light blue denotes the so-called fracture plane which is in Part 26, signature 2. The fracture plane is now known to comprise exactly one of the delaminations, layer #2, and hosts the #2 mauve feature perched on its front lip, towards the #1 delamination. Its length and width are indicative of the other similarly sized delaminations either side of it which are less obvious until you use this shape as a guide. The fracture plane is therefore the area of one cantilever tool box tray. In reality of course, the layer containing the exposed delamination (fracture plane) extends beneath the other layer towards us i.e. under layer #3 that hosts mauve feature #3 and possibly under layer #4 as well. That’s where the toolbox analogy breaks down, unless it’s conceived as having very wide trays with the exposed delaminations being the first open apertures of each tray appearing. 

The four dots, mauve, yellow, orange and dark green are the four coloured anchors first presented in Part 24. Mauve yellow and orange sit along the front lip of the #3 layer. Dark green sits on the #2 layer. 

If you’re schooled in the Part 24 narrative, it will come as a surprise that the dark green anchor isn’t on the same layer as the other three because the head lobe shear line is supposed to run through all four. It does run through all four but the simple fact is that orange delaminated from dark green while the head lobe was still attached to the body. This happened as the #3 layer delaminated from the #2 layer. This delamination of orange from dark green was tweeted with photos long ago but is still awaiting its own blog post. The mirror image of the body delamination is even discernible on the head rim underside. They’re both in the same tweet:

WordPress has reproduced this as a tweet stream (with missing tweets) instead of just the link I typed so apologies to Dr. Nick Attree for being dragged in to this blog post. You can click on it to see the relevant tweet he’s quoting. The photos and their originals (which are more compelling) are reproduced at the bottom as a mini-appendix. 

As if this isn’t enough proof, this same delamination is betrayed even on the upper side of the head rim, that is, the flared, upper side of the rim at Serqet. It’s the two curved, blue ridges in Part 24. Those two ridges are situated directly on the other side of the rim from the ‘underside’ tweet photo. They used to be nested. 67P simply couldn’t be trying any harder to let us know it stretched. 

The tweet photos prove that the dark green-to-orange delamination took place before the head lobe actually lifted off the body. The head had to be clamped to the body for them to exhibit the same mirrored features let alone the same sliding signature. And since the head must therefore have been attached to the body during the delamination, and it also involves a whole layer delaminating by 300 metres along the comet’s long axis, it represents ongoing stretch of the single body due to spin-up. That would therefore be proof of 67P stretching even before the head sheared. 

The reason this delamination can be extrapolated to a whole layer is because the mauve and yellow anchors moved back on the same front lip of the layer with the orange anchor. That brings us back to where we were: the #3 layer delaminated from the #2 layer, taking the mauve, yellow and orange anchors with it. Meanwhile, the green anchor remained on the #2 layer as the vestige of the progenitor to the orange anchor. Much more evidence of stretch before head lobe shear is available in Parts 26 to 29. 

/////

THE MAUVE DELAMINATIONS 

Photo 4- this is photo 2 reproduced

Below is a description of the mauve delaminations. More photos follow, further down. 

This part is important in its own right but it’s also placed here in preparation for explaining the Aswan/Hapi layer slides. Those slides happened right next door to the line of mauve delaminations in this part. They went off in a different direction. This part and the next few parts will explain why the mauve delaminations behaved so differently from the Aswan/Babi slides even though all the layers concerned were joined together before they were torn apart, slid and delaminated. They were joined as continuous layers running across that very straight line we see running down the northern side of the mauve delaminations. That would be the straight line kissing this side of all four mauve features in the header photos. It is the southern perimeter of the 1.6km x 200m rift. 

So far, only one of the Aswan/Babi slides has been explained. It was the Aswan layer slide in Part 69. The southern perimeter of the 1.6 km x 200m rift is crucial to understanding why the Aswan layer and the one below it broke away from where they did and why they slid in the direction they did. They broke away from the southern rift perimeter line and went in one particular direction, 90°, from it. Meanwhile the four mauve delaminations shown in the headers slid along the other side of the southern perimeter. They slid along it, kissing it all the way, and didn’t move away from it at 90° at all. This is completely different behaviour for the two sections of crust that were once joined and then momentarily kissing each other on being sheared from each other. This seemingly bizarre behaviour is easily explained when viewed from the perspective of a stretching comet as will be gradually laid out in this part and the ones following it.

The 1.6 km x 200m rift was annotated in part 69 and invoked for explaining the main Aswan terrace slide. However, the exact mechanism for how the rift sheared along today’s southern perimeter and opened up has not been fully explained before, not even in Parts 48/49. This also implies that Part 69 isn’t the whole story and that the supposedly immobile, lower layer in that part also slid from the southern rift perimeter. It will be shown to have done so in a future part but you don’t have to wait for that part. All the matches are there, it just requires looking at a few photos from different angles to see them. The photos are all in Part 69 and this part. 

It has been hinted at before that the southern rift perimeter line was very important and that the comet’s morphology differs greatly on either side of it but this part starts to unravel why the line is such a strong demarcation line between two distinct areas. It won’t be the full explanation but the photos below make a start in showing how the mauve delaminations were concertinaed out along this line, hugging it all the way. 

The fact that the mauve delaminations extend right into Hapi from the classic mauve anchor (Part 24) means that we can now extend the northern long side of the red triangle as far as the furthest delamination (#1). That’s almost as far as the Hathor cliff. It also implies that the other long side of the red triangle, the southern side can be extended in a similar manner. The northern extension is shown in photo 5, below, and it shows the southern extension in the background. 

Photo 5- the red triangle extensions with original
Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0/A.COOPER

The northern long side of the original red triangle is shown running from the sharp, red triangle tip at bottom-right, up to the mauve anchor. The southern long side also runs from the sharp tip and runs to the dark green anchor. The base of this original triangle, also drawn in red, is between the mauve and dark green dots. The new extensions run on past both the mauve and dark green dots. On the mauve side, the extension to the northern long side kisses all four mauve delamination features as it descends into Hapi and ends at a location that’s in line with the strangely squared-off northern end of the Anuket neck. Much the same thing happens for the southern extension but that’s beyond the scope of this post. If we were to join the two ends of the two extensions (not shown) this would be the new, larger red triangle base. It runs under Anuket and along the front lip of layer #1. Thus, the new triangle’s footprint takes in most of the footprint of the Anuket neck’s base. This should come as no surprise for anyone who’s read Part 57. Layer #1 is where the large, flat expanse of Serqet used to sit before tipping up to its current, more vertical position. It tipped up because it was herniating prematurely and through the layer above. That’s why it’s dubbed ‘the vertical wall’ in part 26. Serqet extruded the Anuket neck from the body after the head sheared and rose on the stretching neck. Layer #1 is therefore the base of the Anuket neck by definition. This was even stated as far back as Part 25 (the three-sided box shaping the squared-off form of the Anuket neck through having material extruded from it and through it).

The southern extension doesn’t need to rely on any supposed symmetry with the northern extension to be invoked. It was already implied in Part 57 but wasn’t explicitly pointed out because the northern extension is more obvious and should be aired first. But the southern extension is symmetrical with its northern twin, just like the rest of the isosceles triangle (the red triangle) that the two lines extend from. This is owing to the fact that both the red triangle and its two extensions are straddling the paleo rotation plane (see the ‘Paleo Rotation Plane Adjustment’ page in the menu bar). The paleo rotation plane caused this symmetry when 67P was stretching as a single body. This was before the head lobe sheared and rose on the stretching neck. It’s explained in the page cited above and in Part 26. 

The southern red triangle extension line exhibits the same layer delaminations as the ones on which the northern, mauve delaminations sit. In other words, those delaminated layers are strips that run across Hapi or Seth. They run in front of Anuket but in the case of layer #1, its middle section is largely hidden under Anuket. The delaminated layers, #1 to #4, therefore kiss the red-dotted lines at each end. That’s where they were sheared at both ends by the shear gradient running down either long side of the red triangle (Part 26 and upcoming parts). All crust layers outside these two lines were also sheared, by definition, and recoiled from the lines. In the case of the northern line, it was the 1.6km x 200m rift in Parts 48/49. In the the case of the southern line, it was layers at Anubis that recoiled from that exact line (translationally matched to the line in Part 54). Put another way, the flat expanse of Anubis is another large rift. Specifically, it’s the floor of a much wider rift than the 1.6km x 200m rift. This hasn’t been mentioned before and will get its own full post in the future. The northern and southern red lines are of course the long sides of the red triangle including its new extensions. 

One of the delaminations runs directly in front of the neck in Hapi (the so-called fracture plane, signature 2 in Part 26). This is layer #2 containing mauve feature #2 on its front lip. And a narrow part of the width of the next layer down, layer #1 is also visible across the front of the base of the Anuket neck but, as mentioned above, a large part of its central portion goes under the neck leaving its full 300-metre width visible only at either end. 

It should be mentioned that there is also a supposed layer #0 beyond layer #1. That’s beyond the scope of this post and will be presented later. It’s inferred rather than self-evident and that inference is via the four clear delaminations we see in the headers. Its importance arises from the fact that it runs entirely under the neck behind Anuket and is probably not actually a layer but unlayered (or less obviously layered) core matrix material. It probably therefore also contributed material to the Anuket neck along with layer #1.

All of the four delaminated layers are about 800 metres to a kilometre long and about 300 metres wide. They are successively longer as we move from layer #4, the shortest, to layer #1, the longest. This is owing to them fitting into and across an ever-widening triangle as they progress towards its base along the front lip of layer #1. The layer widths are determined by the distance between the mauve-dotted delaminations shown in this part because each mauve feature sits on the front lip of its respective layer. But the so-called “fracture plane” (layer #2) presents itself in its entirety as being about 800m by 300m and it also contains the #2 delaminated mauve feature. It was called a fracture plane because it wasn’t recognised as a delamination in Part 26. It was only recognised as the top of a deeper layer than the coloured anchors’ layer (#3) and it was assumed the layer above it had cleaved away cleanly from it and in doing so, sheared along the line of the coloured anchors. Instead, the layer above slid away across it. That was the #3 layer. #3 is the one with three of the four coloured anchors: mauve, yellow and orange. This means the layer with the anchors delaminated from the fracture plane around the time of head shear. They delaminated while being integral to the front lip of layer #3 and did so from where they had sat along and on top of the the front lip of layer #2, the fracture plane. So the two lips were set one above the other as you’d expect them to be before the delamination occurred. 

So the mauve, yellow and orange anchors used to sit along the front lip of the fracture plane. They delaminated from it on their own layer, layer #3 and are now 300 metres set back from that front lip of layer #2, the fracture plane. There are additional translational matches across this 300-metre width for the yellow and orange anchors as well as mauve, which prove this. 

Moreover, the very flared-out rim of the lowest Serqet layer (i.e. the head lobe rim above the anchors) fits to the fracture plane directly below it. It’s part of the same layer that the mauve, yellow and orange anchors are a part of (layer #3). It was married up to the four anchors when the head was clamped to the body and still stretching. That’s why it’s so flared out- it’s one of the most stretched parts of the comet. It’s also why the head-body matches in Part 24 are so faithful. They match the flared rim to the anchors as well as the apparently softer material between them. It’s a 1km-long continuous match. 

The green anchor remained on layer #2 while layer #3 delaminated from layer #2 along with the other three anchors. However, the same scenario applies to the green anchor and head rim above it albeit with the Serqet rim being stretched down more than flaring out (Part 27). This was due to the lack of horizontal movement of the dark green anchor in contrast to the mauve, yellow and orange anchors sliding on layer #3 and giving rise to the flared head rim just before the head sheared. 

The four mauve features are denoted mauve because they were all originally nested under or over the mauve anchor, which is the mauve feature #3 on layer #3. Features #1 and #2 were nested under it and #4, the blocky rectangle, was notionally nested over it but in practice it was attached to the back of it (see subsequent photos and the Part 50 header). The slide of the mauve, yellow and orange anchors within their layer and across the 300-metre-wide fracture plane is in keeping with the well-documented red triangle recoil (signature 5 in Part 26) which was the next delamination back towards Apis. The red triangle recoil was the last delamination i.e. layer #4 from layer #3. 

It has to be remembered that these ~800m-long, 300m-wide strips are simply the visible parts of the now-exposed, delaminated layers. Each layer carries on under the higher-numbered layer that slid back across it. Presumably, they extend under for quite a long way. 

The layer delaminations along the southern perimeter of the red triangle are slightly less obvious but clear once you’ve checked the dots and then gone to the original to trace the line for yourself. Perhaps the most obvious southern extension delamination is the second one, which is the one in the tweet linked above. It shows the green-to-orange delamination which is layer #3 delaminating from layer #2. The third delamination is pretty evident as well. That’s in photo 9, below. It’s the ‘red triangle recoil’ from Part 26. It shows layer #4 delaminating from layer #3. So we have four layers enclosing three delaminations: #2 from #1; #3 from #2; #4 from #3. This means there are three strips or cantilever tool box trays, each one representing a delamination. 

Since the three delaminations of the four layers are bounded by very straight extensions of the straight red triangle sides, it means the triangle has grown beyond its former base along which the four anchors are spread (Part 24). The sides are now longer and the base is wider. Since it’s an extension of the original triangle and bigger, it’s ‘similar’ per the strict geometrical definition of similar which means the same shape of triangle with the same proportions and angles but of a different size. The classic red triangle is isosceles in nature and nested inside its larger, new-found companion which is of course, also isosceles. So the sharp end towards Apis is common to both triangles. And since the triangle owes its existence to spin-up and stretch, it’s aligned exactly along the long axis. Its line of symmetry is running down the middle and that line is contiguous with the paleo equator which is the paleo rotation plane. The sharp vertex is therefore on the paleo equator and the base of the triangle is bisected by it.

The extended red triangle’s base extends under the Anuket neck. In fact, the base encloses about half of the Anuket neck’s footprint on the notional Hapi plane that extends under Anuket. This is hugely significant for anyone who’s read Part 57 (also Part 25). Anuket was extruded out of the body through this area by Serqet as Serqet lifted from the body, specifically, by the sliced vertical wall layer that’s now a part of Serqet along with the flared rim layer. The vertical wall was probably only able to do this because the mauve delaminations described in this part slid back to reveal much deeper material. This would be non-layered core material or less obviously layered core material that was either three or four layer thicknesses deep. Layer #1’s surface is three layer thicknesses down. Layer “0’s” surface, beyond layer #1 and beyond our triangle (for now), is four layers down and appears to be possible core material. That’s why the Anuket neck looks so different from Hathor and Sobek next door on either side. They were cleaved, Anuket was extruded because it’s at the back of the neck with respect to the long axis and the rotation plane. Thus, it couldn’t avail itself of the cleaving process and was unceremoniously wrenched out of the body. This is why large chunks of icy material are falling from the join between Anuket and Hathor. They’re coming from the Anuket side of the join because they were yanked out from the core. 

Photo 6- Simple close up of the four mauve features.
Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0/A.COOPER
Photo 7- A more detailed close up. ‘Original’ from part 25.
Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0/A.COOPER

Mauve- the delaminated features.

Light blue- the so-called ‘fracture plane’ which is delaminated layer #2 containing mauve feature #2.

Larger mauve dot plus yellow, orange and dark green- the four coloured anchors from Part 24. 

Photo 8- Same as photo 7 but with the 1.6 km x 200m rift shown in red. 
Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0/A.COOPER

Red- this is not the whole rift. Roughly half of it is off-screen to the right, towards Apis. The upper line, kissing the four mauve features is both the southern rift perimeter and the northern perimeter of the red triangle. The red triangle’s northern long side used to extend as far as the larger mauve dot (the mauve anchor and feature #3) and its base was then formed by drawing a line from the mauve dot to the dark green dot. 

Strictly speaking, the old triangle base goes all the way along the front lip of layer #3 without jumping across layer 2 to the green anchor. However, when the red triangle was first presented, it wasn’t known that orange and green sat on different delaminated layers so the shear line was assumed to be one delaminated layer lip, not jumping across layers to green at the last moment, near the southern long side of the triangle. It was known to look slightly awkward but put down to the vicissitudes of uneven surfaces. However, 67P is continually showing up even these tiny apparent excursions as having an explanation via the tensile forces of stretch, in this case, the #3 from #2 delamination. Another one is the messy-looking ‘tell-tale line’ (Part 25) at the green anchor that spreads out like a river delta. It’s now been matched to the shape of the green anchor (which has existing matches either side of the delta) and it was matched only a few days ago, bringing eighteen months of head-scratching to an end (see part 70, photos 5/6). 

The new delaminations presented here are #2 from #1 and #3 from #2. #4 from #3 was already known as the red triangle recoil in Part 26 (see photo 9, below). Because of the two extra delaminations, the northern long side of the triangle now extends a long way past the mauve dot and into Hapi. It extends to the edge of the frame as shown and then about five red dots off-frame before turning right to go along the new, larger triangle’s base. In doing so the base traces the far end of the #1 mauve feature before diving under the Anuket neck. And the far end of the #1 mauve feature is front lip of the #1 layer. In photo 8, the neck is that isolated feature at the top of the frame that looks like a gnarled tree trunk. It’s the northern end of the neck that looks very squared-off. It protrudes a bit into Hapi thus creating an alcove with the Hathor cliff whose base is at the very top of the frame, in shadow. 

Photo 9- The red triangle recoil from Part 26. This shows the matches betraying the delamination of the #4 layer from the #3 layer. 
Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0/A.COOPER

This is a long, narrative key with some colour sections broken into paragraphs.

In photo 9, the front lip of the #3 layer is the head lobe shear line and therefore the #4 layer recoiled (delaminated) when the head sheared from the front lip of #3. The head lobe shear line curves across from layer #3 to layer #2, reaching the green anchor which is just off-screen to the right. It then continues on round the body from there (see Parts 17, 19, 30, 21, 2, 1, 3 which match the shear line in sequence all round the head/body and almost back to the mauve anchor). The crossover point from #2 to #3 is exactly at the orange dot where there’s a step-up that’s mirrored on the head rim. You can see this step-up in the tweet photos linked above. 

Red arrows- direction of the recoil (layer #4’s slide) directly away from the head lobe shear line (layer #3’s front lip) and towards the long-axis tip at Apis. The recoil distance is about 200 metres and in line with the 67P long axis as are the other delaminated layers’ slide vectors. This is a strong signature of stretch via spin-up being the cause of the delaminations. 

Mauve- this shows the face of the entire mauve anchor i.e. the section that clamped directly onto its head lobe match 1000 metres directly above. It’s a mirrored match. The mauve anchor is the #3 mauve feature in the header photos for this part. This shape and its head lobe match are shown together in close up in Part 24 (photos 17, 18 and 19). They of course look remarkably similar. Notice how in photo 9 the mauve dots go round three sides of a square, which is in profile to us. These three sides define the two sides of the match and the top between them. That would be in ‘upright duck mode’ with the dust of Hapi sitting below the match i.e. the open end of the square facing down to the dust. 

The three-sided square comprises the entire blocky massif that comprises the anchor itself. The classic mauve anchor location denoted by a single mauve dot is always at the centre of the top perimeter. There’s a reason the bottom side of the notional square isn’t marked. It’s because this photo is from Part 26 and at the time of writing that part, it was suspected but not known for sure that the mirrored match to the head rim ends across this line. The direct match does indeed end across this line. It can’t match any lower because the curved shadow below it is the seating for the blocky massif that’s off-frame to the right here. That massif is the #2 mauve feature and it’s shown in photos 10 and 11. In those two photos it will be shown to nest into this curved, shadowed area. 

There also appears to be a match on the head rim to the #2 mauve feature and not to its curved seating under the mauve match. That match is a continuation of the head rim’s direct match to the main body mauve anchor in this photo, photo 9. This proves the head rim sheared from the mauve anchor on the body when the #2 feature was still nested at the mauve anchor (#3). Otherwise the #2 feature couldn’t match to the head rim where it does.

In part 24 photos 17, 18 and 19, the shadowed curve was the one actually matched to the head rim instead of the #2 mauve feature. This is because it looked remarkably similar from that angle because the seating matches the mauve feature anyway. It was also due to not knowing that the #2 mauve feature nested here or even that layer #3 had delaminated from layer #2. However this extra match below the direct mauve match was dotted light yellow because it looked as if it was essentially a match but there was more to the story. The delaminated #2 mauve feature nesting in the shadowed curve is the rest of that story- another slight anomaly resolved by invoking the tensile force vectors of stretch. 

The shadowed curve is the line that’s dotted mauve in the close up photos above because that’s the exact nesting line on #3, the anchor, for #2 to nest to. 

Bright green- this annotation is bright green instead of mauve and red. It’s owing to the fact that this photo is from Part 26 when features spread along this line behind and in front of the mauve anchor were in bright green. This was in deference to the slab A extension perimeter line. That very straight line is exactly the same one as the southern rift perimeter which itself is the demarcation line for the slab A extension. They are one and the same. Thus these bright green annotations can be translated to the mauve/red language of this part as follows. The left hand bright green line is the front end of the blocky rectangle and so would be the #4 mauve feature in the header. As mentioned above, it notionally nested to to the mauve anchor but in practice it was clamped to the back of it. This seating for the blocky rectangle on the back of the mauve anchor is the middle bright green line. The seating hasn’t been pointed out or annotated in any colour in above photos. The right hand bright green line is just a small portion of the southern rift perimeter extending into Hapi and so is part of the red-dotted rift line in photos above. It should be said that the blocky rectangle line and its seating line are both a bit too long. The match has been refined since Part 26 (see Part 50). 

Larger yellow dot- the classic position of the pointed tip of the yellow anchor. This matches to the tip of the pointed head rim section above. That point on the head rim is the very obvious southern pillar of the C. Alexander Gate. The anchor on the body itself spreads either side of this dot as described in Part 24. 

Larger orange dots- the right hand one is the classic position of the orange anchor. It corresponds to the apex of a sharp, V-shaped dent in the head lobe rim above. The left hand one is the position of its slid match on the red triangle recoil layer which is the #4 layer for this part. So the right hand orange dot is on the front lip of layer #3 and the left hand one is on the front lip of layer #4.

Small yellow- the three concatenated curves running down from the larger yellow dot are sitting exactly on the head lobe shear line. The three-sided feature below the three curves and incorporating the orange dot is largely following the V-shaped match on the shear line but extends past the exact match at either open end. This feature marks the solid massif comprising the main body of the orange anchor. The isolated yellow curve to the right of the larger yellow dot marks the solid massif comprising the main body of the yellow anchor. Only its top edge follows the shear line. 

Very small yellow- mini matches that map over to very small red dots (see below). 

Red (large and small)- all these features match to their respective yellow matches along the shear line. They are a translational match, not a mirrored match as described for the case for the mauve anchor and its match. They are translational because they slid (recoiled), in the direction of the arrows, from their yellow matching features. The mauve match broke in two hence its mirrored character. 

/////

Regarding the red triangle recoil, It’s interesting to recall that layer #3 delaminated from layer #2 while the head was still clamped to the body. We know this from the tweeted orange-from-green (#3 from #2) delamination which the head rim obeyed in lock step with the body. We also know this from the fact that the head rim matches to the coloured anchors on layer #2 (dark green) and layer #3 (mauve, yellow and orange). So the #3 from #2 delamination on the body dragged the yet-to-shear head with it because head and body were essentially a single stretching body at that time. But the #4 from #3 delamination (the red triangle recoil) was a genuine recoil when the head sheared. It didn’t drag head lobe layers across with it like #3 from #2 did. 

Of course, since #3 from #2 dragged future head layers with it then #2 from #1 absolutely had to do so as well. This is key to understanding the behaviour of the Serqet ‘vertical wall’ as it was herniating prior to head shear. It’s why it looks the way they do as described above (sheared by the red triangle extensions at either end; tip up along the front lip of layer #2 acting as a long hinge). Parts 57 and 29 provide much extra information and future parts will elaborate on this. 

And finally, for photo 9, the layer #2 from layer #1 delamination is why Serqet (and Nut) are constrained to be directly above and exactly within the extended red triangle sides i.e. the width of its new, extended base. That delamination caused slip shear by definition. The slip shear happened along the ends of the triangle long sides at either end of the new base, cutting the future vertical wall into a flat, rectangular tablet, one layer thick, which tipped up on herniation. The thickness of that layer is the width of Nut. The slip-shear event is the reason for saying “sheared by the red triangle extensions” above. 

Photo 10- A close up, viewed from above the head lobe. 
Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/A.COOPER

Yellow- these lines show various outlines of the head rim and the Anuket neck. The right hand one at a 45° angle is the actual head rim. The dark feature outlined in the middle is the obvious lump that can be seen in many photos, sticking out at the top of the neck. There’s a small, brighter part of neck beyond it and to the right. To the left of the lump, we see the Anuket neck running steeply down to Hapi. This section is in extreme profile from this viewpoint, looking down on the head lobe. 

Red- the 1.6km x 200m rift. The important southern rift perimeter is the one nearest to us. It’s the one kissing all the mauve features that delaminated from each other along it. Much of the rift is out of view at top left. This shows roughly half its length at most. The southern perimeter is shown dropping down into Hapi along the red triangle extension. In doing so, it runs down the centre of the mauve anchor and along a very straight line that continues out of view behind the head lobe. It kisses the shadowed curve mentioned in photo 9 and so this is the definitive #3 mauve feature to which #2 nests. The shadowed curve has been called the mauve anchor above because it is for all intents and purposes, the mauve anchor. It’s just that it doesn’t match directly to the head because the #2 feature matched to the head when it was seated in the shadowed curve. This was mentioned above. The area of the mauve anchor above the shadowed curve does match directly to the head so the very top rim of the cavern that forms the shadowed curve is the bottom perimeter of the mauve anchor. But the cavern itself isn’t a direct match to the head. 

You may recognise the track of this drop-down into Hapi of the red, southern perimeter. It looks similar to the bright green curve in Part 69. Strictly speaking, it’s not: it’s a straight line (when viewed from directly above) that runs down the centre of the mauve anchor. That bright green curve in Part 69 was representing the northern edge of the mauve anchor which is indeed curved. We are now in territory where the matches and mini-matches are highly nuanced. The shadowed curve, representing the #3 feature extends almost from the anchor centre line (the red line) to the southern perimeter of the mauve anchor and a little beyond it. Although the red line notionally represents the top of the mauve anchor as it passes it, it’s more to do with the fact that it defines the anchor’s central rib. The rib appears offset to the north of centre somewhat in this photo but it’s an optical illusion in this photo and photo 9. Photo 12 shows it as being central. 

Mauve- the four mauve features. They run from #4 at the top of the frame to #1 at the bottom. Only three are actually visible in this view: #2, #3 and #4. #1 at the bottom is denoted as a single dot where it would be if we could see through the head lobe to where it sits in Hapi. We can now see how feature #2 nests into the shadowed curve of #3 below the classic mauve anchor. The best ‘mini-match’ for them in this photo is the curved top of #2 that fits to the curved roof of the #3 cavern that causes the shadow of the eponymous shadowed curve. However, #2 has a triangular shape (very small mauve dots) and that is what fits to the actual recess of the cavern (dotted the same way in the recess shadow- for guidance, not very accurately). 

Photo 11- this shows #2 and its seating at #3 in the shadowed curve. 
Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/A.COOPER

From this angle both features look less curved, more angular. You can see how the seating at #3 uses the southern edge of the central rib (just to the left of the red line) as the northern perimeter of the seating line (mauve). The red line goes down the centre of the rib so the mauve seating line is the southern rib edge and the bright green curve of part 69 is the northern rib edge. The rib will become highly significant in the next few parts because it’s a very strong demarcation line between these mauve delaminations along it and the Hapi/Aswan slides away from it at 90°. It’s much straighter and more symmetrical when viewed from directly above than when viewed slightly from the side as in the last two photos.

Photos 12/13/14- context for photos 10 and 11. These are culled from Part 69 but have added annotations. Photo 14 is the main header for this part. Photo 13 is the same as 12 but with the mauve feature delamination vectors shown along with the rift/Aswan layer slide vectors. The vectors are the red arrows. 14 numbers the four mauve features that delaminated on the four layers.
Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/A.COOPER

Photos 12/13/14 show another view of the mauve delaminations. They’re a zoomed-out version of photo 11 in which you can see all four mauve features. Their mauve outlines are dotted with small mauve dots while the main mauve anchor (direct match to the head rim) has three larger dots denoting its top and southern side. These three dots betray the width of the rib that is itself dotted bright green. The smoothly curving bright green lines show how symmetrical the rib is. Again, you can see how it’s almost acting like a wedge between the mauve delaminations along it and the Aswan/Hapi slide away from it. Of course it never acted as an actual wedge- it’s the physical manifestation of the tensile and shear forces that gave rise to slip-shear and ultimately to those two different layer movement vectors. It’s the force vectors stamped on the comet for us to see and the two bright green ends are flared out because they were yanked in opposite directions along the shear line while the pointed part remained intact under the head lobe. More on that in ensuing parts. 

THE DUAL IDENTITY OF THE RIB AND ITS CONSEQUENT DUAL COLOURING. 

The rib is dotted bright green in photo 12 and not red because there’s a history of denoting this feature as a whole in bright green, dating right back to part 22, as well as noting its close relationship to a particular ridge on the head that’s also always marked bright green (since Part 24). 

Although the rib centreline defines the red triangle extension and so was dotted red in photos 10 and 11, that centreline also denotes the slab A extension perimeter from part 22. The two areas share this border precisely because it was such a strong tensile force line with such a steep shear gradient. The steep shear gradient was what made the line narrow. And the shear caused the 1.6km x 200m rift which means the southern perimeter of the rift is also contiguous with the red triangle and the slab A extension perimeters. That’s because the rift caused the divide between the two areas and their two distinct morphologies. The 200-metre rift floor is wholly within the slab A extension area and the mauve delaminations are wholly within the red triangle. The two morphologies are divided by this startlingly straight slip-shear line that’s no more than 20 metres in width. That’s the southern rift perimeter, including the bright green rib in photo 12. This is how we know the shear gradient was so steep across the line. The tensile force vector diminished suddenly across the 20-metre width and this diminishing set up the shear gradient which led to slip-shearing of the crust. The reason for the sudden diminishing of tensile force across the line was presented in Part 26 (red triangle likened to a wind-tail in the lee of a rock) but will be elaborated on in the next few parts. 

When we’re dwelling on the red triangle, as in this part, the rib centreline is marked red because it’s related to the vast area of the triangle behind it. If we’re dwelling on the slab A extension it has to be marked bright green in order to see its relationship to that area and to the ridge on the head directly above it. Since the Aswan/Hapi slides were entirely on the slab A extension side of the rib, it gets marked bright green for anything to do with the Aswan/Hapi slides. This is why the northern curved edge of the rib was marked bright green in Part 69. And it’s thus coloured in photos 12/13 because they are preview photos from a future part concerning the Aswan/Hapi slides. 

Since photos 12/13 are from a future part, you can see the bright green line extending further into Hapi beyond the #1 mauve feature and right up to the boulders. This is the line along which the Aswan layers were once attached as we’ll see in due course. 

APPENDIX- tweet photos and originals.

APPENDIX 2- various photos showing the four mauve delaminations on their respective delaminated layers. 

Photo A1

Small red- the layers, #1 to #4.

Large red- (in the second photo) the red triangle. Notice how it drops down into Hapi at the mauve anchor which is a dot that’s almost obscured by the red triangle dots. The red triangle line does a similar thing at the green anchor but with a small dog-leg round the solid, front corner of the anchor before the drop-down. 

Photo A2

Small red and large red- as for A1

Brown- a small portion of the paleo rotation plane (paleo equator). Notice how it runs straight through the sharp vertex of the red triangle and then across the centre of Apis at the long axis tip (bright green). It also bisects the red triangle longways as well as Anuket longways and Serqet widthways. This is because all these features were caused by the tensile forces and slip-shear forces of stretch via spin-up. That’s why they’re all symmetrical across the line that runs around the comet from long-axis tip to long-axis tip. That line is the paleo equator and the rotation plane that caused the stretching along the long-axis line.

Dark blue- a nearby portion of today’s equator/rotation plane for comparison. 

Bright green- Apis on the horizon. This is one flattened end of the highly symmetrical, diamond-shaped body. It’s centre is the exact long-axis tip of the comet.

In some photos in Part 70, the red-dotted layer lines traced their way round the shape of mauve features #1 and #2. However, this was for convenience to show where the mauve features were as they were somewhat whited out. The actual layer lips run across the front edges of these two mauve features i.e. the #1 layer lip and #1 mauve feature front edge are contiguous. And the same goes for #2. So the mauve features sit on their respective layers, not in front of them. This correct, straighter line for the layers is shown in the above photos and because of this, the mauve dots showing the delaminations sit directly behind the red lines. 

Photos A3 and A4 below are from Part 70 with the red layer lines going round the back of the mauve features #1 and #2. They’re reproduced here because the mauve features weren’t presented yet in Part 70 and so weren’t marked. So they’ve been added here to show again that there’s one delaminated mauve feature for each delaminated layer. The four mauve feature delaminations therefore serve to corroborate the four layer delaminations. This is because they’re more obvious in themselves than the layer delaminations as a whole and each one sits on a layer lip. This is the lead-in for looking for other translational matches between the layers. These matches are indeed there for all four layers and have been presented for #4 from #3 (red triangle recoil) and part of #3 from #2 (the tweet photos). Several more such translational matches exist, completing the matches between all four layers but these will be presented in a future post. 

But you don’t have to wait for those posts. All you have to do is look at enough close-up photos that include the layer front lips shown in this part and see the matches for yourself. Some of those matches are evident in the close ups in this part but weren’t noted because there’s enough evidence to prove the delaminations and its just too much discussion for one Part. 

Photo A3- This is photo 2 from Part 70 and one of the headers in this part. It has its key annotated on the frame itself. 

Photo A4- this is photo 6 from part 70.

Continuous mauve line- the classic mauve anchor (Part 24). This is the outline of the direct match to the head rim above. You can see that head match in this photo. Look for the mauve dot on the head rim and then trace the much smaller mauve dots either side of it to trace the same shape as on the body. The close up matches are in Part 24. 

Four separated-out mauve dots- these are the mauve feature delaminations, one for each layer delamination. each one is sitting on the front lip of its respective layer but #1 and #2 have the red layer line going round the back of them. It should really go across their fronts so that #1 and #2 are included on and within their respective #1 and #2 layers instead of sitting just in front of them. 

Yellow and green- annotations from Part 25. See that part for a full description but these two colours essentially trace the shear line where the head rim once sat. The continuous mauve line is also part of the shear line because of being a direct match to the head.  

PHOTO CREDITS

FOR NAVCAM:

Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0

To view a copy of this licence please visit:

http://creativecommons.org/licenses/by-sa/3.0/igo/

All dotted annotations by A. Cooper. 

FOR OSIRIS:

Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/A.COOPER

Part 70- The Four Delaminated Layers Across the Width of Anuket.

Original- to track the lines without dots obscuring them:


Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0/A.COOPER

Key

Red- four delaminated layers sitting along the front of the Anuket neck. They’re the same width as Anuket. 

Mauve, yellow, orange- these depict the head lobe shear line. In other words, the continuous line drawn by these colours matches the shape of the head rim, 1000 metres above. This photo was culled from part 24 which explains these matches in great detail. Hence the ‘original’ still has the colours from that part. 

Photo 2- the layers, numbered, ready for Part 71 which will have a much fuller explanation. 

Photo 3- as photo 2 but with the red triangle sides (Part 26) added.

Large red- the red triangle (Part 26, signature 5). These are just the two long sides of what is a long, thin isosceles triangle. The sharp vertex is off-frame to the left. 

These two larger-red dotted lines look bumpy from this side view. From above they are both dead straight. This is because their straightness was brought about by the tensile forces of stretch. This was when 67P was stretching as a single body before the head sheared from the body. They are tensile force lines etched on the surface. 

Specifically, they are tensile force lines which had a steep shear gradient across their width (diminishing parallel tensile force vectors across a very narrow width of ~20 metres. This caused slip-shearing of the crust on either side. Hence these delaminated layers, #1 to #4, being perfectly enclosed within the two long sides and not encroaching past them to the areas beyond the triangle. 

The base of the triangle isn’t marked in large red here because it runs along the front lip of layer #3, which is essentially the head lobe shear line. The upper (northern) perimeter of the red triangle goes through the mauve dot and continues down into Hapi. In Part 26 it stopped at the mauve dot and turned to run along the base which was then the layer #3 front lip. 

Now that the layer #1 and #2 delaminations have been found it means the base of the red triangle runs along the front lip of layer #1 which means it runs under the Anuket neck. 

Photo 4- with mini-matches added in yellow. 

Yellow- These have been matched in other photos including in Part 26. However, the pair either side of the ‘2’ and the pair above it haven’t been mentioned on the blog before. They will eventually get the close-up treatment to prove these matches do exist in finer detail. 

Photo 5- a different view of the delaminated layers. 

Original:


Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0/A.COOPER

In photo 5, we can see the front lip of layer #1 which was in shadow in the above photos. We can also see the Anuket neck and how these four layers are arranged across its width which is highly significant (see Part 57, also Part 29). You can now see how the coloured dots and lines on the body match to the head rim (Part 24).

Photo 6- as for photo 5 but with the layers numbered in the same sequence as in the other view above. 

Notice how the head lobe shear line cuts across layer #2 as the former head rim seating curves round to match the path of today’s curving head rim. 

PHOTO CREDITS

Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0

To view a copy of this licence please visit:

http://creativecommons.org/licenses/by-sa/3.0/igo/

All dotted annotations by A. Cooper. 

Part 69- The Entire Aswan Terrace Slid 200 Metres

Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0/A.COOPER

Green- the Aswan terrace cliff base (furthest away) and its original seating (nearest). The middle line is a line of boulders that also mirrors the shape of the cliff base. 

Mauve- a feature set into the cliff face and its seating nearer to us. 

Arrows- direction of the 200-metre slide. Note the small arrow off to the right on a nearby section that experienced the same degree of sliding. The arrow is smaller only because the head lobe is in the way. 

THE SLIDE

The header photos show the Aswan terrace, which is the main flat area. The front edge, nearest to the viewer, drops away in a cliff that’s about 100 metres or so high. It drops down to another remarkably flat terrace. That terrace is smaller and although it’s officially part of the Seth region, it looks rather as if it’s sitting in the Hapi region with Hapi dust skirting round its rectangular base. 

The bright green lines trace the bottom of the ~100 metre Aswan terrace cliff as well as its original seating along the front of the smaller terrace below. The cliff base slid 200 metres across the lower terrace. This means the entire Aswan terrace slid 200 metres because the cliff is the front edge of the terrace. The slide was brought about by the centrifugal forces induced by spin-up of the comet to a 2 to 3 hour rotation period. 

This should come as no surprise to regular readers. There are longer slides elsewhere on the comet and they involve larger slabs of the same thickness (e.g. Part 43, the ‘red slide’ at Imhotep). Moreover, Part 32 described the delamination of the main sink hole into the three holes we see today. The three holes are next door to this slide and they went in the same direction with a small radial difference in keeping with the radial nature of all the slides around the north pole (the radial pattern is shown in Part 37). 

This slide of the entire Aswan terrace layer is one layer lower than than the sink hole delamination. The two extra sink holes essentially slid (i.e. delaminated from the main hole) across the top of the Aswan terrace layer. That’s why the base of the second sink hole is at exactly the same level as the Aswan terrace. So it was the next layer up, sitting on the Aswan layer, that was clamped around the main sink hole, lost shear resistance at its base and slid back. The main Aswan layer in this part succumbed to the same loss of shear resistance. The loss of shear resistance was due to centrifugal forces. 

There’s an intermediate green line between the cliff base and its seating. That line traces a line of boulders. The boulder line mirrors the line of both the cliff base and its original seating. This phenomenon of leaving parallel lines of boulders in the wake of a slide is most obvious at Imhotep in one of the green slides. It’s shown in the Part 42 overview of Imhotep slides. The Imhotep green slides are not fully blogged yet. That Imhotep slide has left multiple lines of boulders that are parallel to the particular cliff base that slid. This suggests a stop-start component to the slide and supports Marco Parigi’s hypothesis that the spin-up of the comet is ‘pumped’ with several bouts of spin-up leading to several bouts of sliding along the same vector. Each time the slide starts anew, it involves a sudden loss of shear resistance at the base of the layer. The jerking motion as the layer sets off again is apt to dislodge boulders from the cliff face. The dislodged boulders of course trace the shape of the cliff base where they fall because that’s all they can do. This explains the green line of boulders in this part. They mirror the shape of the cliff base perfectly despite being 150 metres from it. Vincent et al. (2015) couldn’t explain this line of boulders as resulting from erosion of the Aswan cliff. This was because it seemed unlikely that these large boulders could travel over 100 metres from the cliff without breaking up. Link to the paper:

https://arxiv.org/abs/1512.03193

The full title of the paper is at the bottom of this post with the link repeated. 

There’s also a curved feature in the header, marked in mauve, along with its seating. It’s set into the cliff wall at its base. It’s fairly rectangular as well as being curved. It resembles a cave entrance or fireplace though in reality it probably has little depth into the cliff face. It will be referred to as ‘the fireplace’ at times in this post and in future. This blog is replete with “features” of all shapes and sizes but unless they have a pithy, unofficial name, it’s difficult to remind the reader 40 parts into the future. 

It’s difficult to tell how recessed the fireplace is but it certainly has some depth into the cliff. It’s rather dingy in that concave part of the cliff base, being in shadow, but this is one of the best photos for seeing into this recess. We can see that the curved base of the fireplace matches to the curve of its seating. 

THE 80-METRE CRACK ON THE CLIFFTOP MAY NOT BE PART OF A SLOW, ONGOING EROSION PROCESS. 
Photo 2- Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0/A.COOPER

Mauve- fireplace.

Yellow- the crack, directly above the fireplace. 

The fireplace is sitting directly below a 80-metre crack that runs along the top of the Aswan terrace cliff. The crack is within the terrace dust and notionally parallel to the cliff edge, 10 metres from it and arced to the cliff edge at either end. It was noted in Vincent et al. (2015) and it was hypothesised that the 10-metre strip might crumble away as part of ongoing cliff erosion. It may well do so but its location directly above the fireplace, and being the same length, suggests a structural weakness running up the cliff that’s an extension of the fireplace structure. This, coupled with the stresses of a 200-metre slide across the lower terrace would very likely give rise to the 80-metre crack. I therefore hypothesise that this crack is yet another artefact of the stretching process, is a one-off structural failing and is not part of an ongoing erosion process brought about by sublimation.

It’s acknowledged here, as always, that sublimation is happening and leading to erosion to some very small extent. It isn’t responsible for the gaping chasms across the 67P landscape though. They were caused by delamination, rifting and sliding. 

THE SIDEWAYS SHUNT OF THE FIREPLACE
Photo 3- fig 5B from Vincent et al. (2015) Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/J.B. VINCENT ET AL. (2015)/A.COOPER

Red- the slide track of the fireplace from seating to its present day position. The track shunts left (as viewed when looking directly towards the cliff face). 

Yellow- the 80-metre crack sitting directly above the fireplace.

Bright green- As in the header (note the very end of the cliff base line peeping out of the shadow). The head lobe rim obscures part of the original seating line but it’s visible up to the central ‘nose’. 

You may notice that the mauve seating is closer to the central ‘nose’ of the cliff seating as compared with the actual fireplace and cliff nose that are further apart. This is explicable via tracing the path of the fireplace, which really means tracing the path of the section of cliff that contains the fireplace as it executed its 200-metre slide. There’s a track line joining the right hand end of the fireplace to the right hand end of its seating (from the viewpoint in the header, facing the cliff). These track lines have been documented elsewhere on this blog especially for the ‘orange slide’ at Imhotep (Parts 44 and 45). This particular track line takes a dive to the left at a certain point. 

The sideways shunt is in keeping with the long-axis stretch vector which is evidenced elsewhere along the Seth/Hapi rim (see next Part). The long-axis stretch has been documented in Parts 38-41 and various parts thereafter.

The stretching of the cliff itself might seem far-fetched but it will be seen in the next post that components nearby did indeed stretch along the long axis instead of undergoing the usual delaminating and rifting in response to the tensile forces of spin-up. 

THE PART 49 PRECEDENT FOR THE ASWAN SLIDE
Photo 4- Part 49’s 200m rift photo: (a) as originally shown without the Aswan slide (b) with the Aswan slide and slide vector arrows for the slide and rift (c) Unannotated original. 

ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/JB VINCENT ET AL. (2015)/A.COOPER

Part 49 presented the evidence for the 1.6 km x 200m rift across Seth and Ash. The 200m slide of the entire Aswan terrace is really just the large section that had to slide in order to open up that rift. That’s why both the rift and the slide are about the same distance of 200 metres. The Aswan portion is a bit less in photo 4- perhaps it overhung the seating. More likely, it dragged the supposedly stationary layer below it just a tad. We’ll eventually come to see that this lower layer did indeed slide as well and has its own seating. 

So the Aswan slide is very closely related to the 1.6km x 200m rift. The additional information in this post is:

1) the areal extent of the slid layer on one side of the rift. That area is the main Aswan terrace. The sliding of the main Aswan terrace automatically implicates the wide, terraced cliff above the main terrace as sliding along with it.

2) the layer on which the Aswan terrace slid which is the small, lower terrace with the bright green lines on it. 

3) the thickness of the layer that slid as implied by (1) and (2) i.e. the thickness of the main Aswan terrace. 

4) the fireplace identification and the fact that it may inform the evolution of the crack directly above it. 

5) the jerk to the left of the fireplace slide, implying a stretching of the cliff face and, by extension, the actual terrace area (long-axis stretch in keeping with the Babi/Hapi delaminations in Parts 38 and 39). This is further evidenced by the fact that the bright green boulder line is offset slightly from the straight-line translational symmetry between the cliff base line and seating line in photo 4 and the previous photos. This implies a shunt to the right of the central ‘nose’ from its seating while the fireplace shunted to the left.

6) the deposition of the boulder line mirrors the cliff base line, which would be due, perhaps, to a stop-start sliding history. This implies a possible ‘pumped’ sliding and therefore intermittent spin-up to the rotation period necessary for shear resistance failure. 

THE J.B. VINCENT PAPER 

Are fractured cliffs the source of cometary dust jets ? insights from OSIRIS/Rosetta at 67P 
JB Vincent et al. (2015)
https://arxiv.org/abs/1512.03193

 PHOTO CREDITS

FOR NAVCAM:

Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0

To view a copy of this licence please visit:

http://creativecommons.org/licenses/by-sa/3.0/igo/

All dotted annotations by A. Cooper. 

FOR OSIRIS:
Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/JB VINCENT ET AL. (2015)/A.COOPER

Part 68- RA/Dec Anomaly in 67P Spin-axis Precession Papers


Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/A.COOPER

Key to the header photo which is not part of the numbered sequence:

The blue arrow is 67P’s spin axis

The short red line represents a circle which is at right angles to the pointed tip and so it’s at quite an oblique angle, hence being depicted here as a line just showing the diameter. The circle represents the approximate area that’s covered by the precession of the spin axis. So the point of the blue arrow precesses or rotates around and within this circle. The 225 data points in photo 1 are all squeezed into this red circle which is itself probably a bit bigger than the actual precession circle of around 0.5° diameter. The graph in photo 1 has stretched this circle into an ellipse. 

CONTENTS

1) The photos

2) Introduction

3) The error

4) The Latitude/Longitude analogy

5) Applying the RA/Dec anomaly to 67P’s spin axis RA/Dec data

6) The relationship between figure 2 and the celestial sphere

7) The effects of the RA/Dec anomaly and why separated RA and Dec components show useful patterns

8) Information from Gutiérrez et al. (2016) that sheds more light on the RA/Dec anomaly

9) Three pieces of evidence for the RA/Dec anomaly

10) Calculations

11) Implications of the RA/Dec anomaly

12) Conclusion

This part isn’t designed to be read from start to finish if you have reasonable knowledge of the RA/Dec reference frame, precession/nutation and 67P in general. In fact, the first section, ‘The Photos’, should be enough. The rest of the sections are included for thoroughness and do include some useful extra analysis as to the reason the separated RA and Dec components of 67P’s spin axis show patterns, as well as implications for modelling the true circular precession pattern and ultimately, the inhomogeneity or otherwise of 67P. Sections 2 to 12 also include a more detailed explanation of what’s being described in the photos section. 

The most important sections are the odd-numbered ones as well as section 10. All the other even-numbered sections are generally weighted towards extra explanation for those not quite so well versed in the subject. That’s only a rough categorisation and there are useful snippets of information in the even sections too. 

Mathematical and physics terms/notations are repeated quite a lot as well as geometrical relationships and analogies. This is because a newcomer can’t be expected to remember all these things once, at the beginning, and juggle them all the way to the end. They’re repeated as gentle reminders at every stage so as not to sow confusion where one might be derailed by the subtler aspects of the arguments. 

1) THE PHOTOS

Photos 1 to 9 follow, along with narrative captions. 

Photo 1- figure 2 from Gutiérrez et al. (2016). 

Credit: Gutiérrez et al. (2016)Astronomy and Astrophysics590, A46 (2016)DOI: 10.1051/0004-6361/201528029Copyright ESO 2016

This is an isotropic plot of the RA/Dec values for 67P’s spin axis that were measured over 125 days in late 2014. An isotropic plot is one which has units that are of the same size along the x axis as they are along the y axis. In this case the RA axis is the x axis and Dec is the y axis. They both exhibit degree units of the same size. This is the source of the error described in this part. The error is that the RA axis degree units should be 0.438 of the length of those on the Dec axis but they are of equal length. This causes the elliptical shape of the data that is really a circle in the real-world precession behaviour of the spin axis. The elliptical shape wasn’t amenable to modelling of the precession behaviour, presumably because it doesn’t represent the real-world, circular precession of the spin axis. 

Photo 2- figure 5 from Gutiérrez et al. (2016). 
Credit: Gutiérrez et al. (2016)Astronomy and Astrophysics590, A46 (2016)DOI: 10.1051/0004-6361/201528029Copyright ESO 2016

This is the attempted modelled fit to the elliptical data. It fits to the outside perimeter of the data points which is why Figure 5 is used for the calculations sections below. However, it has a hole in the middle which leads to difficulties filling it whilst constraining the inertia moments and excitation to reasonable values. The hole doesn’t affect the calculations below but the calculations will probably have an indirect effect on the hole by making the circularised data set easier to model. 

Photos 3 to 7- these show the celestial sphere in the Equatorial reference frame which is the RA/Dec reference frame. The right ascension (RA) lines are the longitude-type lines that join at the celestial north pole. They’re 15° apart. The declination (Dec) lines are 10° apart and are akin to latitude lines. The celestial reference frame can have its origin at the centre of the Earth (geocentre) or at the centre of gravity of 67P. The several-hundred-million km shift is immaterial since they’re both deemed to be in the same place as set against the quasi-infinite distance to the reference stars on the celestial sphere. 

Photo 3- this is from Wikipedia. It’s a portion of the celestial sphere where 67P spin axis points to, i.e. its spin pole coordinates. This is for general orientation and for referring back to. The spin pole isn’t marked in this photo.
Credit: IAU and Sky & Telescope magazine (Roger Sinnott & Rick Fienberg) Creative Commons Attribution 3

https://commons.m.wikimedia.org/wiki/File:Draco_IAU.svg#mw-jump-to-license

https://creativecommons.org/licenses/by/3.0/deed.en

Photo 4- a close-up with the RA/Dec position of 67P’s spin pole, marked in orange. It’s in the middle, next to the alpha symbol. This is at RA, Dec = 69.57° , +64.01°, placed with an accuracy of about 0.5°. Each hour line of RA is 15° so 5hr is 75° hence the dot being 2/3 between 4hr and 5hr for 69.57°. In this case, the RA/Dec reference frame origin is at the centre of gravity of 67P.

Credit: IAU and Sky & Telescope magazine (Roger Sinnott & Rick Fienberg)/A.COOPER
(same credit for photos 5 to 7 which are similar crops). 

Photo 5- this is closer still. The celestial pole is kept in view at top-left for orientation. You can always refer back to photos 3 and 4 as well for orientation. 
In photo 5, the mauve-dotted line shows a track of 10° as measured along the RA axis. This means the line is drawn along the Dec = +64.01° line. It straddles the 67P spin pole dot, so it runs 5° of RA either side. 

The fuchsia line shows a track of 10° as measured along the Dec axis. This means the line is drawn along the RA = 69.57° line. Notice how the fuchsia line along the Dec axis is more than twice the length of the mauve line along the RA axis, even though they both define 10° along their respective axes. Only the Dec value of 10° represents the true angle subtended by the fuchsia line, as measured from the reference frame origin at the centre of 67P. The mauve RA value of 10° does not represent the true angle as measured from the reference frame origin. It represents the number of degrees around the circle represented by Dec = +64.01°. That circle is cos 64.01 = 0.438 of the size of the circle along which Dec is being measured. Dec, however, is measuring round an RA line, 69.57°, which is a great circle or circumference line. RA lines are always great circles. Dec lines are not great circles (barring just one, the celestial equator). Dec lines are diminishing hoops like latitude lines. Thus the RA degree increments along Dec = +64.01° are 0.438 times smaller than the Dec increments along RA = 69.57°. 

To be clear, RA is always defined as the RA component of the coordinates as viewed from the origin. It’s not viewed from the centre of the small Dec line circle that the RA coordinate is sitting on even though RA ticks off degree units around that circle. The observer sits at the RA/Dec origin and observes these ticked-off degree units from below as they arc around in a small circle above him. The ticked-off degree units allow the observer to locate the RA position of an object along that circle and the circle is itself the Dec component of the coordinates. But that’s as far as it goes: the ticked-off RA degrees (in raw form) have nothing to do with the true angular distance travelled by the object as viewed from the origin. Those degree units are simply too small because they’re ticking their way around a smaller circle than the great circle that the Dec degree units are ticking round. Being too small, they tick off an angle that’s greater than the true angle of travel as measured at the origin. This means the raw RA measurement has to be multiplied by the cosine of the Dec line they’re measuring along in order to give the true angular distance of travel along the RA axis. And it should be stated that this is only applicable to short distances such as the figure 2 data plot with an RA spread of just 1.2°. Any greater RA spread for a straight-line angular distance and the track climbs the Dec lines, introducing non-linear effects.

The small Dec circle and the sphere it’s sitting on, as described above, aren’t usually thought about very much. Their relative proportions remain the same at whatever r value from the origin they are measured whether a tiny sphere at the reference frame origin or the infinitely large celestial sphere. The infinite number of congruent cases between those two extremes proves that RA degree increments are always smaller than the Dec increments to the tune of the cosine of the Dec value along which they’re measured. For the 67P spin axis, the radius of the Dec circle is cos 64.01 of the sphere radius at which Dec is measured. RA units at Dec = +64.01° are therefore smaller than the Dec units by a factor of cos 64.01 = 0.438. 

As another example, at Dec = +45°, RA degree increments would be 0.707 (cos 45) of the Dec value. Dec increments always stay the same and at parity with the angle as measured at the origin but RA varies and never represents the angle as measured at the origin except along the celestial equator where Dec = 0° and cos 0° = 1 i.e. unity.

Photo 6- an uncluttered version of photo 5 with just the end points of the RA 10°-long line and the Dec 10°-long line. The orange spin axis location for 67P is in the middle, as before. The spin axis points at this spot on the celestial sphere.

Photo 7- this is the same as photo 6 but with a large, yellow circle drawn around the orange spin axis dot. 
In photo 7, the yellow circle has a diameter of 10° of RA because it kisses both mauve dots at either end of the 10°-long RA line. However despite being a circle which, by definition, has a constant diameter, that diameter only measures 5° along the 10°-long Dec line. We know this because it reaches only halfway to the two fuchsia end points. If the dotting were more professional the Dec-axis diameter, measured in Dec degrees would be around 0.438 of the RA-axis diameter measured in RA degrees. It wouldn’t be quite exactly that proportion because the 0.438 coefficient corresponds only to the +64.01° Dec value at the orange dot. But this illustrates a principle: circles drawn on the celestial sphere at high Dec values measure ‘longer’ as measured in RA degrees along the RA axis than they do in Dec degrees as measured along the Dec axis. This is despite having constant diameters by definition. 

It follows from the above, that if the RA and Dec coordinates of the circle are transposed to an isotropic plot, that is, a Cartesian graph with RA and Dec axes with equally spaced degree units, the circle will be stretched along the RA axis into an ellipse. This is what has happened in figure 2, above, and it’s the essence of the error to be described below in this post.  

The bright green dotted line in photo 7 shows a hypothetical asteroid track across the sky close to the celestial pole. This could apply to an asteroid passing 67P but perhaps it’s easier to unclick the RA/Dec reference frame origin from 67P’s centre of gravity and clip it into place at the Earth’s geocentre. Now the bright green track is an NEO flying almost over the Earth’s North Pole and projecting a track onto the celestial sphere, tracking past the celestial pole. The celestial pole is where the Earth’s spin axis or North Pole points, like 67P’s orange dot. The bright green line subtends an angle of 10° at the geocentre. We can ascertain this because it’s the same length as the 10° distance between the Dec lines. And yet it travels through 105° of RA. This is a more exaggerated version of the yellow circle with the 0.438 coefficient. The ‘RA degree measurement’ of the NEO track is 10.5 times the real value of the angular travel as measured from the geocentre. So it’s a coefficient of 1/10.5 = 0.095.

Photos 8 and 9

Photo 8 credit: IAU and Sky & Telescope magazine (Roger Sinnott & Rick Fienberg)Creative Commons Attribution 3

https://commons.m.wikimedia.org/wiki/File:Draco_IAU.svg#mw-jump-to-license

Photo 9 credit: 

Creative Commons Attribution 3

https://commons.wikimedia.org/wiki/File:MessierStarChart.svg#mw-jump-to-license

Photo 8 is a celestial star chart similar to those above. It shows Draco in the middle, a long, sinuous constellation at around the same average Declination as the yellow circle around 67P’s spin axis in photo 7. Notice how Draco bends up and down quite markedly. 

Photo 9 is an isotropic RA/Dec plot of the whole sky (RA and Dec plotted with equal degree units). In this plot, all constellations at high Dec values have to be stretched along the RA axis. You can see Draco stretched into a long line that’s much less wavy than its real shape (it at the top and says “Dra” near its left-hand end). This proves that if you transpose raw RA and Dec values to an isotropic plot, the real-world shape you’re trying to depict will be stretched drastically along the RA axis if that shape resides at around Dec = +64.01°. This is what happened in the case of the ‘elliptical’ spin axis data pattern in figure 2 of Gutiérrez et al. (2016). That pattern should in fact be circular. The attempts at modelling the spin axis to fit the ellipse led to difficulties in constraining the moment of inertia and excitation values. Those difficulties arose from the misapprehension that the spin axis was describing an elliptical pattern as it precessed, instead of the real-world circular pattern. 

2) INTRODUCTION

This part concerns an erroneous interpretation of 67P’s spin-axis precession data in Jorda et al. 2016. The paper’s title is, ‘The global shape, density and rotation of Comet 67P/Churyumov-Gerasimenko from preperihelion Rosetta/OSIRIS observations’, by L. Jorda et al. published in October 2016. 

This paper has been cited eight times as of the date of this blog post (November 2016), including pre-publication citations made when it was in the submission phase. 

One of the citing papers, Gutiérezz et al. (2016), relies heavily on the erroneously interpreted spin-axis data and it attempts to model the spin axis precession accordingly but this means it’s labouring under the assumption that the RA and Dec data points locating the spin axis movement are compatible with each other. They’re not compatible in their current form as plotted in that paper’s isotropic plot (figure 2) and without the RA component of each data point being adjusted for the isotropic nature of that plot. The paper’s conclusion says that when RA and Dec data are considered together they “do not allow constraining the inertia moments and excitation level” that characterise the spin axis precession. However, when RA and Dec are considered separately, there is some success in detecting “significant combinations of parameters”. It’s argued here that this is because the RA component of each data point isn’t compatible with its corresponding Dec component due to not correcting the RA component for the isotropic plot. The full title of the citing paper is ‘Possible interpretation of the precession of comet 67P/Churyumov-Gerasimenko’ by P.J. Gutiérezz et al. (2016). 

It was the Gutiérezz et al. (2016) paper that prompted me to realise there was a problem with the precession data, specifically their isotropic graphs showing the spin axis data points plotted with right ascension (RA) and declination (Dec) for the two axes. Since it’s this paper’s graphs that allowed me to prove the data misinterpretation, this part will focus on the Gutiérezz et al. paper and not the Jorda et al. paper. 

In pursuit of full transparency, I have not read the Jorda et al. paper. It’s paywalled, whereas I was able to get access to the Gutiérezz et al. paper. Since Gutiérezz et al. cites the Jorda et al. findings very clearly and then plots them, it follows that the critique below of Gutiérezz et al. must also apply to Jorda et al., the original source of the misinterpreted precession data. If this reasoning is somehow misinformed, I shall be happy to make a correction regarding Jorda et al. but the data and graphing as presented in Gutiérezz et al. would still be at fault. Since both lead authors are co-authors on the other’s paper, and the error is common to both papers, it seems appropriate to critique the error itself and apply it to both papers.

Another reason for focussing on the citing paper, Gutiérezz et al. (2016) is that their modelled ellipse in figure 5, that best fits the observed data, is used for the calculations below (see the calculations heading). Figure 5 is the second header image.

3) THE ERROR

In the following analysis “the observed data” is the term used for the 232 RA/Dec coordinate data points for the 67P spin axis position in Jorda et al. (2016). This data set is called the “observationally derived data” in Gutiérrez et al. (2016), although they stripped out 7 outliers leaving 225 data points. The 232 data points were observed over 125 days in late 2014. They were taken in successive 10-hour blocks. Each point is therefore the average RA/Dec position of the 67P spin axis during each 10-hour block.

In essence, the error could be described as an artefact of the RA/Dec coordinate system finding its way into the isotropic precession graphs of Gutiérezz et al. (2016) without being corrected for. It is this artefact that has produced the ellipses in those figures. They should not be ellipses, they should be near-perfect circles.

Keeping this in mind, the last paragraph in the ‘Summary and conclusions’ of the paper is revealing:

“To evaluate whether it is possible to constrain the inertia moments and excitation level, a systematic search of the probability of compatibility between simulated and actual RA/Dec patterns by means of two-sided K-S tests was performed. Even if it is possible to find very significant combinations of parameters [Iy, Iz, EI] when RA and Dec coordinates are considered separately, K-S probabilities when RA and Dec data are considered together do not allow constraining the inertia moments and excitation level.”

It’s proposed here that the reason significant combinations of parameters can be found for RA and Dec coordinates, when considered separately, is that, at Dec = +64.01°, the RA and Dec values in the observed data are each measuring different real-world angular distance increments of the nutation angle, theta. Specifically, the RA values in the observed data cover more RA degree units for the same angle as the Dec degree units do. We are referring here to when measuring an angle at the origin of the RA/Dec reference frame (such as theta) first along the RA axis and then measuring the same angle along the Dec axis. The same angle spans 2.282 times as many RA degree units as Dec units. The same applies to a theta angle that is not aligned along either axis and thus is composed of an RA and a Dec component. These two-component data points for theta constitute almost all, if not all, of the 225 data points in figure 2. The RA component stretches the circular precession pattern into an ellipse via the 2.282 coefficient. This is why the ellipse’s major axis is aligned with the RA axis. 

The angle, theta, is the angle of the spin axis nutation as measured from the origin of the RA/Dec reference frame. The origin is at the centre of gravity of the comet. The angle of nutation is the angle between the angular momentum vector and the spin axis. The angular momentum vector is the putative average of the 225 observed data points in figure 2. So it’s sitting in the middle and can be seen more clearly in figure 5 as the centre of the modelled/fitted ellipse. It’s also the centre of the circle when the ellipse is corrected. 

The angular momentum vector is at one and the same with the inertial axis, Z, about which the intrinsic spin axis, z, precesses. We can keep the terms Z and z in the back of our minds after the short explanation below. After the explanation, they’ll be referred to by their familiar names: the angular momentum vector (which is Z) and the comet’s spin axis (which is z). 

Z and z are used for transforming (or relating) the precessing cometcentric reference frame to an unmoving inertial frame. Z is one axis of the XYZ fixed reference frame that is outside the xyz intrinsic comet frame of which z is one axis. Z and z are axes which means they are both one-dimensional lines. 

Both reference frames have a common origin at the centre of gravity of 67P and thus z joins Z at the origin. All we have to remember here is that Z is the angular momentum vector and it’s a line fixed in space. And the spin axis, z, is a line that moves around Z, while joined to it at the origin. z moves around Z only if it happens to be precessing, which it is for 67P. 

Since z is locked to Z at the origin it moves round Z, sweeping a cone, with its base locked in one place. Z is then the average central axis within the long, thin cone that’s swept out. If the spin axis isn’t precessing, it merges with the angular momentum vector (z merges with Z). They then become one line pointing from the centre of gravity of the comet, along the spin axis/angular momentum vector to a point on the celestial sphere at RA, Dec = 69.57°, +64.01°. 

This RA/Dec value is the Jorda et al (2016) value, as defined by their 232 spin axis data points. Gutiérrez et al. (2016) removed 7 outliers to arrive at the 225 points, as stated above. This shifts the RA/Dec value for Gutiérezz et al. (2016) by a very small amount. However, since the original Jorda et al. (2016) data interpretation was the input for Gutiérrez et al. (2016) it would be best to stay with the Jorda et al. (2016) angular momentum vector coordinates even though the graphs in the header show it to be a fraction off due to stripping the 7 outliers. You have to look hard to see the difference anyway because it’s a judged average centre-point of all the dots, and it makes no difference at all to this analysis. 

The offset angle, theta, is the nutation of the spin axis from the angular momentum vector. Theta is the traditional term for nutation in any discussions about precession. Theta is shown in figure 4 in Gutiérrez et al. (2016). It’s not shown here but nutation is described below. 

Much focus is placed here on the actual origin of the RA/Dec reference frame, which is placed at the centre of gravity of 67P, and the fact that theta, the nutation, is measured at the origin. This is fundamental to understanding the nature of the figure 2 and figure 5 RA/Dec anomaly.

As the spin axis precesses around the angular momentum vector with any given theta value for the nutation, it describes a circle. That circle may become a smaller circle as the theta angle is reduced or a spiral as theta is in the process of growing or diminishing. The result of taking 225 10-hour averages of the spin axis coordinates along these circles and spirals of varying radius, results in a pattern or shape that is a notional circle filled with 225 dots. This has been stretched into a notional ellipse by the RA anomaly. Notional, because they aren’t those exact shapes but appear strongly to suggest them.

The angular momentum vector can be thought of as a laser beam pointing from the RA/Dec origin at 67P’s centre of gravity and out to the celestial sphere at RA, Dec = 69.57°, +64.01°. It stays rigidly pointing at that spot, a single laser point. Meanwhile, the spin axis of the comet can be thought of as another laser beam, pointing from the origin as well, and describing the circles and spirals of different radii on the celestial sphere. These circle around the fixed laser point of the angular momentum vector and as the radius of the circles change, they betray the change in the nutation angle, theta. 

The spin axis therefore sweeps a cone as mentioned above. These cones are of varying sizes according to the radius of sweep (the described circles) or become deformed cones when the radius is spiralling. The radius of the described circles is very small, just a few tenths of a degree. Theta is therefore the angle between the two lasers and it’s measured right down at the origin of the RA/Dec reference frame, which is the common end point of both lasers.

Theta is determined by the combination of the RA and Dec readings, specifically, their vector product. These are also measured from the origin of the RA/Dec reference frame, which is placed at the centre of gravity. So far, so good, but we should take pause to note that it is this common origin for measuring theta and also RA/Dec is the source of the confusion causing the RA anomaly.  

We should be able to ‘sit at the origin’ look up, along the line of the angular momentum vector, and see the angular displacement (nutation/theta) of the spin axis away from the fixed angular momentum vector. That angle may or may not vary over time as the spin axis precesses round the angular momentum vector over time. In the case of 67P, theta does vary, causing the spread of the data points and that’s why they fill the ellipse in figure 2 (which should be a filled circle).

At any instant in time, the angle theta can be taken by reading forwards/backwards along the RA axis a certain number of RA degree units from the fixed angular momentum vector; and then reading up/down along the Dec axis a certain number of Dec degree units. However, the RA and Dec values in the observed data are each using different real-world angular increments distances along the RA and Dec lines as they ascertain the angle theta. More specifically, the RA degree units do not measure the same angular distances as those angular distances used for ascertaining theta. This is despite the fact both RA and theta are using the same RA/Dec origin. This is crucial and it is the source of the anomalistic interpretation of the data, which in turn, leads to the ellipses in the figure 2 and figure 5 graphs, which should be circles. 

This phenomenon of different angular measurement scales for RA and Dec is simply an idiosyncrasy of the RA/Dec system, one which is especially apparent at high Dec values where the incremental RA degree units along the Dec lines do not represent the actual angle in degrees as measured for theta at the origin. This is because the RA degree units are doing there own thing: they’re measuring their way around a smaller circle, akin to a latitude line. The radius of this circle is smaller than the radius of the sphere it’s a part of. It’s smaller by a factor of the cosine of the Dec angle of the ‘latitude’ circle that’s being measuring around. 

Meanwhile, each Dec degree angle increment really is measuring the true Dec axis angular component of the theta angle, as measured from the origin. This is because the Dec lines are like latitude lines and, by definition, they’re measuring the true angle between the equatorial plane and that Dec value. So the Dec data in figure 2 presents no problems at all. But each and every Dec value in the figure has been slid backwards or forwards along the RA axis by its rogue RA counterpart while remaining at the correct Dec value. This results in the area that the 225 data points define being stretched to the left and right, either side of the central RA value of 69.57°. As the data points are sliding right and left too far along the RA axis, they’re maintaining their correct Dec position and so the area represented by the 225 data points doesn’t get stretched from top to bottom, only from left to right. 

The +64.01 Dec value for the angular momentum vector at the centre of the observed data is a fairly high Dec value (two-thirds of the way towards +90°) so the anomaly is very significant. The value of the anomaly is a coefficient of cos 64.01, which is 0.438. This leads to a correction factor of 1/0.438 that’s needed when transposing the RA data to the isotropic plot in figure 2. The reciprocal of the cosine, 1/0.438 is 2.282 so the data we see in figure 2 is stretched along the RA axis by a factor of 2.282. It needs to be de-stretched by the correction factor of 0.438. When that’s carried out, the ellipse de-stretches to become a perfect circle. This circle will be the true circular pattern that defines the precession of the spin axis around the angular momentum vector. 

4) THE LATITUDE/LONGITUDE ANALOGY

In contrast to the RA degree units, the incremental Dec degree units always represent the actual angle as measured at the origin. As stated, it is just an idiosyncrasy of the RA/Dec system and is an exact analogy to the longitude degree lines on the Earth at 64° latitude being bunched together much more than the latitude lines. The correction factor in this analogy is again 2.282. 

One degree of latitude always covers 111.2 km whether at 64°N, 15°N or 85°S. However, one degree of longitude always varies according to the latitude line along which it’s being measured. At 64.01°N, it happens to cover 48.729 km. The latitude value of 111.2 km for one degree really does subtend an angle of one degree at the centre of the (Lat/Long) reference frame at the centre of the Earth. However, the longitude value of 48.729 km for one degree does not subtend an angle of one degree at the origin. It subtends an angle of 0.438° at the origin and 0.438 is cos 64.01. 

Thus, any real-world distance in km, measured in degrees of latitude or longitude across the earth presents us with a problem: at 64.01° latitude the same distance in km will be 2.282 times ‘longer’ in longitude degrees than it is in latitude degrees. A distance of 1.112 km measured South to North (up the latitude ‘axis’) will measure as being 0.01° but the same distance measured West to East (along the longitude ‘axis’) will measure as being 0.0282°. And yet both measurements are measuring the same distance across the surface and along each axis. And that distance subtends an angle of 0.01° at the centre of the Earth. So a 0.0282° longitude measurement of a 1.112 km distance along the surface corresponds to a 0.01° angle when that distance subtends an angle at the geocentre. Measuring the 1.112 km distance from the geocentre is the same manner which theta would be measured in the RA/Dec reference frame i.e. from the origin of the reference frame. The reason for this 0.438 factor at 64.01° latitude is that the centre of the 64.01° latitude circle isn’t at the centre of the Earth. It’s at the centre of the plane defined by the 64.01° latitude line. The radius of that circle is the cosine of 64.01 timesed by the radius of the Earth i.e. 0.438 of the radius of the Earth. Consequently the circumference is 0.438 of the circumference of the Earth. Therefore, each longitude degree increment around that circumference measures a distance that is 0.438 of the distance measured by each latitude degree increment around the circumference of the Earth. And the only way to measure the angle at the centre of the Earth as subtended by a distance at the surface is to measure it in degrees along a great circle i.e. along the circumference of the Earth as latitude angles always do. Longitude angles do this only along the equator. Above and below the equator, they start bunching together. They’re not then measuring the angle subtended at the centre of the Earth but at the centre of the latitudinal, cross-sectional plane at which the measurement is being made. 

There’s one crucial difference between this lat/long analogy and the RA/Dec system. Despite the RA (‘longitude-type’) increments being concertinaed together and not measuring the true angle as measured from the RA/Dec origin, they are still used as an angular measurement from that origin. This leads to apparently large angle swings when measuring at high Dec values, near to the poles, even when the real angle swing is actually very small. NEO’s that pass over the poles show near-180° swings in RA in the space of a few minutes when in fact they’ve only moved comparatively slowly and by an angle of 10° or 15°. They move through all the bunched-up RA lines near the poles at apparent break-neck speed. But it’s the very small distances between the RA degree lines that are causing this phenomenon (see the bright green line in photo 7).

This phenomenon was nicely illustrated with the well-documented close approach of 2012 DA14 in February 2013 when it swung to Dec = -87°, right under the South Pole. I’m very familiar with this phenomenon from reading the ephemerides of hundreds of close-approaching NEO’s: a small angular distance travelled (akin to theta) as measured from the RA/Dec origin can cover five or ten times that angular distance in RA if it’s at a high Dec value. The real angle is the small one travelled, akin to theta, and the RA angle component is just a rather clumsy and confusing way of representing it. The Dec component isn’t a problem: it records the true angle travelled up/down the Dec axis as measured at the origin. 

The RA value bears no relation to the actual theta angle as measured from the origin unless it gets crunched through the cosine of the Dec angle at which the RA is being measured. And to be precise, it’s the inverse of the cosine that operates on the RA value to give the true theta angle at the origin. The single case where the RA value is the same as theta is when the RA is being measured around the celestial equator. In the case of the NEO approaches, the 10°-15° of angular travel, if in line with the equator, would show up as 10°-15° in RA, i.e. at parity with the real angle as measured from the geocentre. That would be as opposed to 180° of RA when measured as it goes 10°-15° over the pole from one side to the other (or, say, 150° of RA if a little offset from the pole).

5) APPLYING THE RA CORRECTION TO 67P’s RA/Dec SPIN AXIS DATA

For the 67P case, we can keep the lat/long analogy in mind while substituting RA for longitude and Dec for latitude. We can set the angular momentum vector so it’s fixed to the origin at one end (as it always is) and pointing rigidly at RA, Dec = 69.57°, +64.01°. Then we can measure the deviation of the spin axis from the rigid angular momentum line (which is the nutation of the spin axis, theta) and measure it in RA degrees and Dec degrees as Jorda et al. (2016) did and Gutiérrez et al. (2016) reproduced. 

In this case, any Dec degree value measured up and down the Dec axis from the angular momentum vector really does correspond to one degree as measured from the origin of the RA/Dec reference frame. 

Conversely, any RA degree value measured backwards and forwards along the RA axis from the angular momentum vector does not correspond to one degree as measured from the origin of the RA/Dec reference frame. It corresponds to just 0.438° when measured along the 64.01° Dec line. And 0.438 is cos 64.01.

Thus, all RA degree unit measurement values for the nutation angle, theta, as measured from the origin are 2.282 times greater than the Dec degree unit measurements for the same angular displacement (at Dec = +64.01 or very close to it, as the 225 data points are). The 2.282 coefficient is 1/cos 64.01. 

Dec is always in parity with the real theta angle component as measured along the Dec axis but RA is always 2.282 times greater than the real theta angle component as measured along the RA axis. This means all the RA components in each and every RA/Dec data point (the observed data in figures 2 and 5) have to be divided by 2.282 when transferred to an isotropic plot (i.e. multiplied by 0.438 = cos 64.01). The resultant shape of the corrected data spread will be a filled circle.

With the adjustment made, there will be no additional bias of the nutation angle, theta, along the RA axis in both directions either side of the angular momentum vector. The angular momentum vector is positioned at the centre of the data points in figure 2. 

6) FIGURE 2’s RELATIONSHIP TO THE CELESTIAL SPHERE AND THE RA/Dec ORIGIN AT THE CENTRE OF THE SPHERE

Since we’ve looked at the RA/Dec increments in such detail, you may have noticed the RA scale increases to the right in figure 2 but to the left in the conventional manner for the star charts in photos 3 to 9. This is because in figure 2, we’re looking down the angular momentum vector towards the 67P RA/Dec origin at its centre of gravity. So we’re looking down from the celestial sphere and figure 2 has its RA scale back-to-front as if we’re behind a screen that’s got the celestial sphere projected onto it from 67P on the other side. The angular momentum vector, at the centre of all the dots, is pointing directly out of the screen at us- we’re looking straight down it towards the RA/Dec origin and that’s sitting behind the screen and behind the data points. The plane of the data points (the screen) is at right angles to the line of the angular momentum vector and all the data points are piercing holes in the screen where the spin axis was when measured. If we use the laser analogy and shine 225 lasers from the origin, through the holes to represent the 225 spin axis positions, those lasers would all shoot past us to our right, left, etc., just missing us and would describe the circular precession pattern on the celestial sphere ‘just behind’ us. We’re sitting right in the middle of that circular pattern of circles and spirals drawn on the celestial sphere. 

7) THE EFFECTS OF THE RA/Dec ANOMALY AND WHY THE SEPARATED RA AND DEC COMPONENTS SHOW PATTERNS

The RA-axis stretch is a strong clue that it’s an artefact of the transposition of raw, uncorrected data from RA/Dec to an isotropic plot. It also explains why, according to the quote above from Gutiérrez et al. (2016), the inertia and excitation parameters couldn’t be constrained when RA and Dec were analysed together. They could only be constrained when the RA and Dec components for each of the 225 data points were separated out and analysed as two different data sets. 

When they remain unseparated, the RA/Dec taken as a single data set, are smudging the circle into an ellipse and this shape can’t be modelled because it’s an ersatz precession pattern. When the RA and Dec components for each and every one of the 225 data points are treated separately as two sets of 225 data points, the data is internally consistent and patterns reflecting the true, hidden, circular-shaped pattern are betrayed. When the RA and Dec components are merged for each data point, as we would normally assume we can do in order to model useful precession patterns, they represent a vector product. The vector product is the diagonal product of the distance along the RA scale (the RA component) and the distance along the Dec scale (Dec component). But because of the 2.282 stretch in RA, the vector product is smudged and the data points become ever-more difficult to reconcile as one progresses from the central RA value to the left extremity and right extremity of the ellipse. Put another way, the vector-summed data is not internally consistent, and is therefore representing an ersatz precession pattern. This means it can’t be modelled with reasonable inertia moment and excitation levels. This explains the failure in Gutiérrez et al. (2016) to constrain the inertia moments and excitation values for RA and Dec when analysed together as a single data set.

8) INFORMATION FROM GUTIÉREZZ ET AL. (2016) THAT SHEDS MORE LIGHT ON THE RA/ Dec ANOMALY 

Regarding Gutiérezz et al. (2016), on page 2 of the paper, it says:

“By applying the SPC method [stereo-photoclinometry*], Jorda et al. (2016) retrieved a spin axis that moves around (RA, Dec) = (69.57°, +64.01°). Jorda et al. (2016) obtained that the spin axis does not describe a circumference, but approximately fills an ellipse in an isotropic plot (Fig 2).” 

This quote is a citation of the precession data interpretation in Jorda et al. (2016). The figure 2 graph they’re referring to is stated as being that data from Jorda et al. (2016) and indeed the caption of figure 2 cites the data, if not the graph itself as coming from Jorda et al. (2016). This is the reason for including Jorda et al. (2016) in this analysis.

*stereo-photoclinometry is a method by which a shape model of 67P was constructed using stereo landmarks of the comet’s position and attitude in OSIRIS photos.
Gutiérezz et al. (2016) go on to say that this “ellipse in an isotropic plot” is at odds with the circular precession plot determined in Preusker et al. (2015). So Gutiérrez et al. (2015) certainly regard the Jorda et al. (2016) ellipse in figure 2 as being the physical shape of the precession and not just a shape on the graph that somehow represents a different shape for the actual precession. The difference between the Preusker et al. (2015) circle and the Jorda et al. (2016) ellipse seems important enough to be noted. And yet, as we’ve seen above, the ellipse is indeed just a shape on the graph that represents a different shape for the actual precession which is a circle. 

The key term here is “isotropic plot”. This is the source of the ellipse anomaly. The plotted data describe a filled ellipse but they should describe a filled circle. 

The conclusion of the paper has already been quoted above as saying that the RA/Dec data couldn’t allow the inertia moments and excitation to be constrained when RA and Dec were analysed together. It only showed “significant combinations of parameters” (inertia moments and excitation levels) when RA and Dec were separated. This was also apparently the case for the Lomb periodograms (various figures in the paper). These are presented with separated RA and Dec in all cases. The peaks show remarkable correlation between the RA and Dec values- their respective peaks nest into each other very well. However periodograms of the mixed RA and Dec data are not considered. Gutiérezz et al. (2016) also say of Jorda et al. (2016):

“Interestingly, Jorda et al. (2016) analyzed the spin axis orientation by means of the phase-dispersion minimization technique [to obtain a periodicity of 276 hours] from separately considering the RA and Dec coordinates.”

Thus, RA and Dec were separated wherever possible, for Lomb periodograms, K-S probabilities and phase-dispersion minimization. The only instance where RA and Dec were mixed was in the unavoidable situation where the physical shape of the precession data had to be modelled. For the shape to exist at all, it required the vector product of both RA and Dec for each data point in order to spread out into the ellipse shape in figure 2. When this was modelled using the Euler equations cited in the paper, difficulties arose resulting in the inability to constrain the parameters due to closing the central hole while using reasonable parameter values. As the conclusion states:

” K-S probabilities when RA and Dec data are considered together do not allow constraining the inertia moments and excitation level.”

Since the K-S probabilities are “a function of Iy and El” (figure 9 caption) and Iy and El are the y inertia moment and excitation level, it follows that the above quote is referring indirectly to the modelling of the inertia moments and excitation level via the Euler equations and with RA and Dec considered together. 

Furthermore, figure 9 shows this indirectly stated anomaly in the form of two white lines, one continuous and one dotted, in both of its two frames. These lines represent the “excitation level for each Iy associated with the highest K-S probability” when RA distributions are compared (continuous white) and Dec distributions are compared (dotted white). Please see figure 9 and its caption below). I would suggest that when the 0.438 coefficient is applied to the RA data in figure 2 and the Euler equations applied once again to model what is now a circle, the two white lines in figure 9 will automatically merge. This would still involve separated RA and Dec components but it would indicate that RA and Dec can indeed be mixed in a similar K-S probability graph. And when this is done the K-S probabilities, when considering RA and Dec together, should after all allow constraining of the inertia moments and excitation level.

Photo 10- Figure 9 from Gutiérrez et al. (2016) 

Credit: Gutiérrez et al. (2016)Astronomy and Astrophysics590, A46 (2016)DOI: 10.1051/0004-6361/201528029Copyright ESO 2016

9) THREE KEY PIECES OF EVIDENCE FOR AN RA, Dec ANOMALY

It’s interesting to note that the ellipse described by the observed data is orientated exactly along the RA axis when characterised by the modelled ellipse in figure 5. This is a very strong indicator that if there were any anomaly, it’s entirely to do with the RA axis. This is a smoking gun for the phenomenon described above: the artefact resides entirely in the squashing-together of the RA degree units; the Dec degree units remain the same size in the Equatorial reference frame, from -90° to +90° and can therefore be transposed to the Cartesian graph without distortion. This means that the y-axis (Dec axis) spread of the data points in figure 2 really do represent the angular diameter of the precession circle as measured from the RA/Dec origin at 67P’s centre of gravity. All the stretch is along the x-axis (RA axis). 

The second thing of note is that it is an ellipse and not, say, a notional square with rounded sides or an amorphous shape. An ellipse is by definition a circle that’s been stretched along one axis only. Again, this reinforces the idea that there’s an artefact operating along just one axis. This, coupled with the fact that this one axis is aligned with the RA axis, is very strong evidence for the RA anomaly.

The third piece of evidence that shows it’s a circle stretched into an ellipse is that the major axis of the ellipse is very close indeed to 2.282 times longer than the minor axis. Since this is 1/cos 64.01 and the observed data are centred on Dec = 64.01°, it means that when the RA degree units are reduced to 0.438 of there current figure 2 size, a circle will be obtained.*

*Please note, only figure 2 is fully isotropic and amenable to the 0.438 coefficient operation. Figure 5 and thereafter have an RA axis that is actually slightly squashed- by a factor of just 0.916 though, not 0.438. To adjust these graphs correctly the 0.916 factor has to be taken into account. The 0.916 factor is akin to applying only some of the 0.438 coefficient: squashing together the RA degree units a bit but nowhere near enough. 

10) CALCULATIONS

On-screen measurements of the ellipse major and minor axes in figure 5 were subject to perhaps a 2% error. For figure 2, estimating an ellipse that’s not drawn in, it was probably a 5% error. Actual measurement values aren’t shown, just the ratios they imply. This is because they were taken at arbitrary levels of zoom. The proportions remain valid for any given zoom value. 

Calculations are taken to three decimal places for precision but that precision is greater than the error bars. Despite this, the measurements of the ellipse in figure 5 (along with the necessary 0.916 adjustment) produced an inferred Dec figure of 64.01°. This was pure chance and is purely down to a) luck in the measurement within the 2% error bars and b) various roundings up and down to 3 decimal places. It’s definitely not indicative of a systematic error in the methodology as one might be tempted to think.

The target value to look for in the ratio of minor axis divided by major axis of the ellipses is cos 64.01° which is 0.438. 

A measurement of the raw data in figure 2, on the assumption it fits to an estimated ellipse produced the following figures:

Minor axis divided by major axis = 0.469

Cos-1 0.469 = 62.03°.

So from the ellipse in figure 2, we can infer a Dec value of ~+62° which is close to the +64.01° value, which is the central RA, Dec figure around which the observed data is spread. 

A measurement for the modelled ellipse in figure 5 produced the following figures (including the 0.916 adjustment for the slightly squashed RA axis):

Minor axis divided by major axis = 0.478. 

But the RA axis needs to be stretched by 1/0.916 so as to obtain a truly isotropic relationship before applying the cos-1 rule. This is done here a slightly different way. It’s done by reducing the minor axis to 0.916 of its measured value, which is the same as multiplying the 0.478 minor/major ratio by 0.916. 

0.478 x 0.916 = 0.438 (this is the exact target value). 

Cos-1 0.438 = 64.01° (this is exactly the same as the observed central Dec value)

Thus, the inferred Dec value for the figure 5 modelled ellipse is 64.01°, which is exactly the same as the the actual central figure in the modelled data. With the RA anomaly corrected for, and the fact that the resultant inferred Dec value is the same as the observed Dec value, it means the spin axis was indeed precessing in a circular pattern around RA, Dec = 69.57°, +64.01° and not in an ellipse. 

11) IMPLICATIONS OF THE RA/ Dec ANOMALY

Once the observed data was plotted onto an isotropic axis, it meant that the circular precession was depicted as an ellipse. This ellipse was then taken as the basis for modelling the precession. 

The modelling software would have modelled an elliptical precession in order to fit it to the apparent elliptical precession in the observed data. This would have affected the values chosen for the excitation level and inertia moments. If modelling a circular precession, however, these values would presumably need to be changed. This in turn would have implications for ascertaining the homogeneity of 67P. 

Figure 5 had a hole in the middle of the modelled ellipse that fitted the elliptical version of the observed data. On page 5, it’s stated that this hole could be reduced only by reducing the nutation and therefore the excitation level. This would in turn mean introducing some inhomogeneity for 67P. However, the hole in the modelled version is based on trying to fit an ellipse to what is actually a circular pattern to the precession. 

By rerunning the modelling and basing it on the true circular pattern, it may show that the hole can indeed be filled in and with no invoking of inhomogeneity. 

The true circular pattern of the observed data may imply a subtly varying nutation describing a spiral. That seems an intuitively possible dance for a comet’s spin axis to perform, but I suspect it would be difficult for 67P to perform that trick in ever decreasing/increasing ellipses. This is because it implies a simple harmonic motion (SHM) component to the nutation value on each and every rotation about Z, the angular momentum vector. 

The fact that the one axis along which nutation variations are happening corresponds to the RA axis is a sign that something is amiss. This is further illustrated in the related issue of finding a better fit by separating out all the RA and Dec values and plotting them separately. There is nothing inherent in the geometry of space that should show up patterns (better/worse fits) that align with a man-made coordinate system based on the random value of the Earth’s tilt and the randomly chosen First Point of Aries (RA = 0°). This better fit of the separated RA and Dec values betrays something ersatz about the data. It’s showing the RA stretch anomaly when the RA data is mixed in with the Dec data. It’s distorting the overall data via the stretched vector product.

If the true circular pattern of the precession is used to model against, only the gradual linear reduction and increase in nutation over time is needed to fill the central hole. That way, no SHM in the nutation value needs to be invoked to describe an ellipse while trying to close the hole at the same time. Thus the hole would be closed and opened over a longer circular/spiralling cycle.

12) CONCLUSION

In summary, the precession that was measured in Jorda et al around the central RA, Dec value (69.57°, +64.01°) had to describe some sort of shape whether an ellipse or a circle. However, if it was indeed a circle, it would be stretched into an ellipse if it were transposed in raw form from the Equatorial RA, Dec system into a Cartesian system using equal axis increments (an isotropic plot). The only way the shape of the precession can be faithfully reproduced on the Cartesian graph is if the Cartesian axis units are proportioned in such a way as to reflect the RA, Dec proportions at Dec = 64.01°. This means bunching up the RA units so that they are only 0.438 (1/cos 64.01) of the Dec units. When this correction is done in the case of figure 2, a perfect circle is obtained.

The ellipses in figure 2 and figure 5 in Gutiérrez et al. (2016) are therefore depicting an anomalous artefact of the RA, Dec system. The same applies to the other ellipses in the subsequent figures of the observed data.