Part 64- The Evidence for Long-Axis Stretch Along the Head Lobe Shear Line at Hapi/Babi


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

INTRODUCTION 

The header is the same photo as photo 2 in Part 62 and this post is a follow-on from Parts 62 and 63 which dealt with the delamination of the three Ma’at pits when the head lobe was still clamped to the body lobe. This could therefore be considered the third part in an ongoing mini-series whose underlying theme is always the delamination of the Ma’at pits. More posts will follow, totalling at least five for the mini-series. They build up evidence for the collateral mechanism for the delamination of the holes: long-axis stretch along the head lobe shear line and the consequent sympathetic delamination of layers along the shear line. 

A NOTE ON THE SETH PROTRUSION

It should be noted that although the term Hapi/Babi is used for the Ma’at 01, 02 and 03 pit delamination line, some of that line was along the short finger-protrusion of Seth. Since it’s confusing to say Hapi/Babi/Seth, this short finger is counted as being part of Babi in this part and in Parts 62 and 63. This expedient is all the more compelling when we consider that the Seth finger is very much a part of the Babi morphological evolution by virtue of finding itself on the Babi side of the mirror-symmetry line described in Part 63. 

The Seth finger was included as part of Seth in the ESA regional map only because of visual similarities on its surface with no known reason for that similarity. The reason is that the head lobe originally sat on the finger, and this caused its characteristic rectangular shape by virtue of the furious outgassing escaping from underneath it (Parts 1, 5, 7, 8, 20). And apart from a tiny extra area next to the finger, described further below, the head lobe sat nowhere else on the official Babi area.

Photo 2-The ESA regional map. The finger is the pink protrusion coming towards us along the Hapi rim (next to the Hapi label). This is counted in this blog as being very much a part of the Babi morphology. 

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

PHOTOS

The originals aren’t part of the numbering scheme. Dots should act as a guide only before verifying the dotted features on the originals. This is because the dots can sometimes partially obscure the detail they’re trying to show. 

Some keys are narrative keys and so some key colours get divided into paragraphs. These longer keys end with ‘/////’.

Photo 3- the header reproduced. 


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

Key 

Green- these two lines are the same shape, a translational match. They show a massif on the right that has slid from its seating on the left. The seating and massif correspond to the second and third bumps respectively. Those are the bumps or mini-volcanos in Part 62, that both delaminated along the shear line. They delaminated along with their respective layers, both delaminating ultimately from the first bump. The left hand green line here is sitting on top of the second bump or mini-volcano. The right hand green line is kissing the bottom rim of the third bump. In this view, it’s the bottom end of the line that’s kissing the upper perimeter of the bump rim. In this sense, the slid massif is only notionally related to the third bump. It will be shown in future parts that it probably was sitting on the third bump originally, but the main point here is to show that it’s a slid massif and that it slid in a direction that’s along the long axis of the comet. Furthermore, it will be shown that it ground along the head lobe rim as it slid i.e. it sheared from the head rim due to the shear gradient set up across the long-axis tensile forces. This shear gradient was described in Part 62.

Red- these aren’t exactly slide tracks but are a proxy for the slide track of the massif. The bottom red line is in line with the section of head lobe that’s 1000 metres directly above it in reality but slightly below it and to the right in this view due to Rosetta’s slight parallax. In other words, Rosetta wasn’t quite vertical over the head rim and shear line when this photo was taken. So the lower red line in question is in line with that straight section of head rim just before it curves round and downwards somewhat towards the right. This match will be depicted in lower photos, as it was in Part 62. 

The reason the bottom red line is a proxy for a slide track is that a) it’s in line with the bottom ends of both the matching green lines and b) it’s actually the ragged line that the matching section of head rim tore away from. So the massif that slid to the right was kissing the future head lobe rim along this line when the head was still seated on the body. It then tore away from the future head rim under the influence of long-axis stretch along the shear line. And of course, the shear line is, by definition, where the head rim originally sat. So the massif slid along, or ground its way along the head rim before the head fully detached ‘upwards’ as viewed in upright duck mode. Upwards is towards us in this overhead view. 

This means the massif didn’t exactly tear ‘away’ as mentioned above but tore ‘along’, i.e. it sheared under the shear force, described in Part 62, which was focussed along the shear line. The massif would have slid only during head herniation and before head detachment because all the tensile forces of stretch would have been transferred almost instantaneously to the incipient neck once the head sheared. That left little or no residual tensile force in the Babi layers to continue fuelling the long-axis stretch and further widen the delaminations. And there was certainly no residual tensile force at all once angular momentum had been conserved at today’s head lobe displacement (T= circa 5.8 hrs, see ‘Spin-up Calcs’ page).

The upper red line is certainly in line with the slid massif and its seating and may be a simple slide track. However, it may have delaminated from the other red line on head shear i.e. at 90° to the long-axis stretch (see Part 62 for this orthogonal delamination scenario). So it would have delaminated from the lower red line towards the top of the frame, which is along the same vector as the Babi slide (Part 40). 

/////

Photo 4- same as photo 3 with additions.


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

Yellow- on the right, it depicts the back of the slid massif. On the left it depicts the bottom of the second bump. Together with their respective green lines, the yellow lines enclose similar shapes. 

Mauve- a mini match within the slid massif and within its seating. 

Photo 5- the delaminated layers. All annotations except green have been temporarily removed, just for this photo. 


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

Green- these are just three of the five delaminated layers that are depicted in the header of Part 38. The green line along the front of of the slid massif is the same as in photo 3. So it turns out that the front edge of the massif constitutes the front edge of the third delaminated layer. Its green line is extended round in a dog-leg although the real layer may be under that other terraced or bunched-up detritus beyond it (above it in the frame). That’s the material from the Babi slide (Part 40). 

The seating of the slid massif isn’t shown here because we know where it is. It’s right next to the very obvious line running along the front of the second bump which is at the front edge of the second delaminated layer. The seating is so close to the second layer front edge that we can say that the massif, i.e. the third layer, delaminated from the second layer along the direction of the long axis. This is in keeping with what was stated in Part 62. The second layer’s front edge does a fairly sharp turn and continues down Babi (upwards in the frame) to join another green line on the left. 

That left hand line leads back up to the Hapi rim at which point it describes two waves. Those are actually up/down waves and from the front (viewing from the left and very low down), they look like gull wings. So these are the gull wings which Part 62 describes as the first bump or mini-volcano from which the second and third bumps delaminated. But the bumps are just the ends of their respective layers, perched on the Hapi rim. So if the bumps delaminated, it means the entire layers delaminated too: second from first and third from second. 

The fourth layer is off-frame to the right and technically beyond the scope of this post. However, in Part 63, it’s implied as the southern perimeter of the rift at this end of the long-axis delaminations along Hapi. So it tore rather than delaminating, just as happened at the Aswan/Seth end of the long axis delaminations.

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We can see from photo 5 that the bumps delaminated along the long axis in a straight line and yet their respective layers seem to fan out from one point. The first line even has a three-pronged trident shape pointing at the gull wing waves that is replicated at the second bump. The trident staffs are the delaminated layers marked green but the two sets of three prongs are a definitive match (see them annotated much more clearly in Part 38, photo 6). The replication on the second layer is at the place you’d expect, which is the with the prongs pointing at the base of the second bump. And they’re even slightly flipped round by the differential forces within the fan because the prongs are that much closer to the true shear line with the much steeper shear gradient. 

The trident match is less obvious here but is annotated and described in detail in Part 38, photo 6. Here’s a link to Part 38:

https://scute1133site.wordpress.com/2016/01/20/67pchuryumov-gerasimenko-a-single-body-thats-been-stretched-part-38/

The trident movement, arcing over on its staff from the axis at bottom of the fan is strong evidence for the fanned delaminations. It seems highly unlikely that a trident shape would be reproduced at both bumps. 

Photo 6 below is the Part 38 header (not Part 38’s photo 6 with the trident annotated but you can see it here in its header, unannotated). It shows the full fan effect. The delaminated fronts making up the fan in this photo are in light blue because ridges were being denoted in light blue back then):
Photo 6- Part 38 header, reproduced. 


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

Light blue- these are the ridges that betray the the fanned-out delaminations. 

Dark green dot pairs- the three delaminated bumps (lower-left) and two more that were presented in Part 38 (upper-right). 

Photo 7, below, is reproduced from Part 62. It shows the Ma’at pits and their sources on the body as well as many matches from head to body. However, the main point in reproducing it here is to show that the shear line on the body (terracotta) runs contiguously with the lower of the two red slide lines. We’ve already established that the slid massif was originally attached to this line. Therefore, it must have also been attached to the portion of head rim that was seated there. That portion of head rim is that straight section of rim just before it curves round and downwards somewhat towards the right. It’s offset just to the right and downwards from the bottom red line in this view due to parallax (as mentioned above in the ‘red’ description in photo 3). If in any doubt, you can use the additional version with two lines of arrows showing the match or the beige dots that align in one line when the head is reseated. 

Photo 7- the head rim was seated along the lower red seating line for the slid massif. Therefore the slid massif was originally attached to the head rim before shearing along it and sliding to its current position. 



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

Dark green- as for photo 3 but there’s an extra fan rib or delaminated layer on the left. It’s one of the two extra ones shown in the Part 38 header above. The other one is off-frame to the left. This extra one that’s shown also extends to the common ‘stalk’ of the four fanned lines while, if you recall, the fifth one is detached from the fan in the form of our slid massif. The first line, the off-frame one, branches to the left from a point that’s a fraction off-frame in the top corner. 

Notice that the faint dark green dots on the head rim match to the wavy line on the first green line on the body. This is the gull-wing match, matched in 3D (from the side) in minute detail in Part 5. Except, this is a slight cheat because what we’re seeing on the head rim is a second, curved-up gull wing delamination that sits directly above the true set it delaminated from. It bent upwards during head lobe herniation and now exhibits a gentle curve. It constitutes the upper component of the first zig zag in Part 62. So there are actually two sets of gull wings on the head that sandwiched down onto one set on the body. This was described in detail in Parts 5 and 7. 

Although the sideways, 3D match of the gull wings isn’t strictly related to this post, they were sitting right on top of the Ma’at 01/01A source (Part 62) and that’s why they were raised into two gull wings. You can even see the two slurry piles that oozed neatly out in front of each wing on the body’s set of wings. They’re semicircular piles, each one the same width as its wing. They exhibit more concentric semicircles within their semicircular perimeter. Any engineer would identify that as classic signature of slump. 

So, since the 3D gull wing match is somewhat related to this post and intimately related to Ma’at 01, various links to the gull wing 3D match have been put at the end of this post along with a photo of the match with exquisitely detailed fiduciary-point matches. Please see the appendix. 

Red- still with photo 7, this is the massif’s seating against the head lobe which doubles as its implied slide track. The slide track betrays the long-axis shear force vector causing the layers to delaminate along the long axis. This led to the pits delaminating along with their gas sources being emitted from each of the layer fronts. The gas sources uplifted the layer fronts into bumps or mini-volcanos before exiting through the actual pits. One of those uplifting zones was the two gull wings constituting the first mini-volcano. On head shear, the pits recoiled up the head and down the body (Part 62; also Parts 40 and 41). Their gas sources were left marooned at the shear line and are clearly visible today directly below Ma’at 01, 02 and 03 (Part 62). 

Orange- this is the now-bunched-up layer that slid away from the shear line after head lobe shear. This layer used to sit over the bumps and the fan shape. It fully delaminated as opposed to undergoing a fan delamination. That’s how it was loosened enough to slide. The very fact it sat over the fanned layer meant that it was overburden that increased the tensile strain resistance of the fanned layer. That’s why the fanned layer resisted full delamination and didn’t end up joining its overburden layer behind the orange line. 

Other colours- these are as for this same photo in Part 62 (photo 17).

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Photos 8 and 9- these are a comparison of the header and a close-up of the Part 38 header. They exhibit exactly the same annotations and so are presented together with the same key. The focus is on the intricacies of the attachment of the massif to the red line. 




Photo 8: Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/A.COOPER

Photo 9: Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0

Red- the attachment and implied slide line.

Dark blue- the outside curve of the massif (not annotated in other photos further above) and its seating on the ‘outside’ of the red line. In reality, it’s simply at the bottom of a long, thin sloping area that constitutes a widened version of the red line. The green line runs along the top of that widened line. 

Green- the front of the slid massif and its seating on the second bump (as for photo 1).

Light blue- fiduciary positions of certain larger boulders.

APPENDIX- THE 3D GULL WING MATCH

Photo 10- the 3D gull wing match. Fullest zoom required for all detail.
Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0

The photo above is the finer-detailed match from Marco Parigi’s summary post of my Part 5. 

My Part 5 is the first link below, which shows how I traced the sideways view of the gull wings on head and body and checked meticulously that they were indeed the same features as those we see for the green gull wing match in photo 7 above in this post.

Marco’s summary post is the second link for which I did the more detailed version of the match shown above. A quote from his post sums up why this match alone, made in December 2014, proves almost beyond doubt that the head lobe sheared from the body. This, despite it being around 1% of the evidence to date for head lobe shear. 

https://scute1133site.wordpress.com/2014/12/20/67pchuryumov-gerasimenko-a-single-body-that-has-been-stretched-part-5/

http://miny3dmatches67p.blogspot.co.uk/2015/10/mini-matches-and-3d-matches-on-67p.html?m=1

“These mini matches make the original match conclusive, because if the large scale match was a coincidence based on large features, there is extremely low expectation of the constrained smaller section also matching in the small scale and/or in the third dimension.”

Quote from above link by Marco Parigi, October 24th 2015.

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 63-The Mirror-Image Pit Delaminations Along Hapi/Seth


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

This is the second part in a mini-series regarding the long-axis delamination of pits along the shear line. Stretch theory is not a mainstream theory and is not entertained by the Rosetta mission scientists. Therefore, the pit delaminations described below are at odds with the current thinking on pit morphological evolution. If you are interested only in peer-reviewed literature this will save you further reading. 

The two header photos are reproduced further below with their keys and originals. ESA’s regional map is at the bottom.

Everything applying to the long-axis stretch and delamination of pits along the Hapi/Babi rim (Part 62) also applies to pit delamination along the Hapi/Seth rim at Aswan. These two sections of the head lobe shear line underwent the same process in mirror fashion. It will be some time before this Aswan section is written up in full hence this overview to show how the Ma’at 01, 02 and 03 pit delaminations at Hapi/Babi were consistent with the overall picture across this north pole side of the comet. 

Three pits delaminated along the Hapi/Seth rim in a mirror image to the delamination of Ma’at 01, 02 and 03 along the Hapi/Babi rim. That’s why there were two half-pits identified along the Aswan rim at Hapi/Seth in the original sinkhole paper by Vincent et al. (July 2015).

The large pit in the Seth region is the mirror image of the Ma’at 03 source described in Part 62. That source is the third mini-volcano and it was itself once an outgassing pit on the body. It has the largest source diameter of the three Ma’at pit sources. It’s that majestic, undercut curve overhanging Hapi. That diameter is about the same as the main Seth pit. And the Seth pit is the largest of the three pits that delaminated in the mirror image of Ma’at 01, 02, 03. The other two are the half-pits on the Aswan rim. 

The Seth pit and the Ma’at 03 source pit lie at either end of the very straight long-axis stretch line along Hapi that will be described in the next part. Both pits are larger than any of the others because they share a unique morphological evolution as directed by the long-axis tensile forces of stretch. 

For both pits, this evolution proceeded in an identical and mirrored fashion. Both pits are sited where the long-axis stretch vector turned to make its way in a straight line along the side of the herniating head lobe. The stretch vector did this in an attempt to make its way around the herniating head lobe. This ‘line of least resistance’ behaviour is patently clear, stamped on the comet’s surface in the form of the red triangle (Part 26, signature 6) and is the raison d’être for Serqet and Nut. So both pits are sited at the exact point where the long-axis tensile force vector joined the herniating head lobe rim. 

Both pits opened up as a result of 200-metre-wide rifts careening up the long axis tensile force vector for 1.6 kilometres and into the future head lobe rim, passing underneath it before the rim sheared from the body. For the Seth rift see Parts 48 and 49. The two points at which the two rifts went under the head lobe opened up into a 200-metre-wide notional square with the Hapi rim acting as a third side and seated Hathor matrix as the fourth side. The head lobe acted as a loose lid. 

So the Seth pit and Ma’at 03 pit are sited where there was both head lobe shear and a 200-metre rift opening up. Only the Hapi rim stayed (relatively) still while the three-way movement of two rift perimeters and head rim induced furious outgassing. This is why the Seth pit is by far the largest identified pit on the comet and the same 200 metres width as the rift it sat inside. The Ma’at 03 pit is the same diameter and is effectively an unidentified, dormant half-pit like the two along the Aswan rim.

Both pits, Seth and Ma’at 03 source pit, then delaminated away from the shear line after head lobe shear. The Ma’at 03 pit suffered a single delamination across Babi (Part 40) while its original Ma’at 03 source pit stayed put at the shear line as you might expect a hole to do. 

However, the Aswan pit actually slid on the onion layer below it. To repeat, this hole did actually slide: a sliding hole within its sliding matrix which was the entire Aswan basin onion layer. This is why the Seth pit is flat at its base. The base is the next onion layer down, looking up at us and it’s the layer the Seth pit slid on. It’s also why the base of the pit is at exactly the same level as the lower lip in Hapi at the base of the Aswan cliff. That’s the layer over which Aswan along with its pit slid. This sliding scenario is absolutely dictated by the conclusions drawn from photo 7 in Part 49. There’s a line of boulders on this lower lip that is a translational match to the base of the Aswan cliff line. This is strong corroborative evidence for the Aswan-plus-pit slide but photo 7 in Part 49 is even more compelling:


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

Larger Red- the 200-metre-wide rift

Bright green- the extension of the rift that went under the head rim before the head sheared. 

Light Blue- the Seth pit and the quasi square location where it first opened up before sliding (between the bright green lines). 

Small red- red line at right: the Aswan cliff line (including its line in the shadow from other photos); at middle: a red line showing a boulder deposition line that matches to the cliff base; at left: the original seating of the cliff base. The Aswan cliff slide is almost the same distance as the Seth pit slide but certainly apparently less. In reality it has to be the same because the cliff is joined to the pit. The difference will be explained in a future part because the full extent of the sliding behaviour in this small area is beyond the scope of this post. What we see annotated and explained in this photo is 90% of the full picture anyway. 

Slate blue- at top: a pair showing the upper half pit (or second half-pit if equating it to its mirror image Ma’at 02 source pit) along with a very obvious match to its seating next to the blue quasi-square; at bottom: the first half pit which equates to Ma’at 01’s source pit. 

 After the Seth pit slid, it delaminated into three across Aswan (Part 32).

There’s much more to say on the Aswan delaminations, along with copious evidence in the form of translational matches on the body and head and mirrored matches to the head rim underside. Some of this evidence has been noted in passing in Parts 37, the 48/49 twins and Part 50. The unavoidable conclusion is that all these pits that are today on Aswan, Seth, Babi and Ma’at were once delaminating their way in opposite directions along the Hapi shear line. 

Most tellingly of all, this mirrored, long-axis delamination behaviour along the Hapi rim is symmetrical about the north pole point in Hapi. Since the north pole is where the rotation axis pierces the comet’s surface, this is a significant signature that the mirrored delamination process was driven by spin-up of the comet:

Photo 2- the mirror-image delamination of the holes across Seth/Hapi. 


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

Brown- this is the paleo rotation axis adjusted up the comet’s x axis by 700 metres so as to make a comparison with today’s rotation axis (dark blue). The paleo rotation axis held sway before head shear. Only then did the axis precess 12°-15° to today’s axis location. The paleo axis is derived by taking the line that’s orthogonal to the inferred paleo rotation plane (see the Paleo Rotation Plane Adjustment page). 

Notice how the paleo axis runs down the middle of those two horseshoe craters and also directly between Aswan and Babi. It’s the line of symmetry between them. It’s also the line of symmetry between the two red rifts running from the original Seth pit seating and from the Ma’at 03 source. 

And most importantly of all, it represents an almost perfect line of symmetry between the two sets of three delaminating pits (light blue). You can see the three Ma’at pit sources delaminating one way (left) along the long axis from the brown line. And you can see the three Seth pits delaminating (and in their case, tearing and stretching) to the right along the same long-axis line at Seth. 

The paleo pole is the larger brown end dot. It’s equidistant from the original Seth pit seating and the Ma’at 03 source pit. The original Seth pit seating is where the symmetry of tensile forces originally held sway before the pit slid. The paleo pole is therefore the centre point origin along the symmetry line for all the symmetrical features described above. This is strong evidence that spin-up of the comet was the mechanism by which this mirror-symmetrical pattern arose and the consequent long-axis delamination at Hapi occurred.

Photo 3- the ESA regions


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

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 62- The Morphological Evolution of Ma’at 01, 02 and 03


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

This post has been written with the hope that one or some of the end-of-mission planners might be able to read it. It concerns the morphological evolution of the pits that will be examined by Rosetta as she descends to her final resting place on 67P. That will happen at Ma’at on September 30th 2016 and the moment she lands will be the end of the two-year mission. 

The evolution of the pits as laid out below would perhaps point to the consideration of tweaks in the way the data is taken on the way in to land. 

The pit evolution scenario described below is according to stretch theory which isn’t accepted by the mainstream. It proposes that 67P was spun up via asymmetrical outgassing which itself isn’t controversial. The controversial part is the suggestion that this spin-up caused the comet to stretch into an ellipsoid, then allow the head lobe to herniate from the body and finally, shear away from the body, rising on a growing neck (the neck we see today).

The following is related as if it’s fact so as to spare the reader endless qualifications and conditional statements. It is, of course, a hypothesis. However, there’s an abundance of evidence supporting it.

PHOTOS 

The originals aren’t part of the photo numbering system. The main post follows the photos. It’s a summary of the photo explanations including further background explanation and links to other stretch blog posts. If you have any doubts or objections to what is presented in this ‘photos’ section, they should hopefully be resolved in the further explanation lower down. This is a very complex area so it would be a surprise if you got through the photos without any further questions. Hopefully the photos will be compelling enough to make you read on. 

The ESA regional maps are at the end of this photo section if you need to get orientated (photos 19 and 20).

Photo 1- the Ma’at pits 01, 02 and 03. 
Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA

Photo 2- An overhead view with the three pits (larger blue dots) and the exact outgassing sources for them, dotted blue and sited on the body below.


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

In photo 2, the majestic curve of the source for 03 is half hidden behind the head rim so it’s extrapolated with smaller dots. The evidence for outgassing from these three sources is documented in many blog parts, mainly Parts 5 and 7 and 8. There are also numerous head/body mirrored matches for this small area in Parts 1, 2, 5, 7, 8 and many more parts up to 52 and this Part. These matches completely constrain the body outgassing sources to match the 01, 02 and 03 hole positions (when the head was seated on the body). This is in addition to each source exhibiting signs of furious outgassing. 

Photo 3- same as photo 2 but with the pits numbered and a new one added, dubbed 01A for convenience, and just a little bit hidden here hence the arrow. The left hand outgassing source (as viewed here from above) fed 01A as well as 01. 01 then delaminated from 01A upwards towards us and to the right. Notice how 01 is facing its body source.

Photo 4- the bumps on the body (four dark green dots) that are a tell-tale sign of uplift from outgassing along the Hapi/Babi cliff rim. 
Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0/A.COOPER

These bumps are just next to the body outgassing sources depicted in photos 2 and 3. The bumps supplied the outgassing sources (see their relationship to the sources in photo 11). The bumps for the 01A/01, 02 and 03 sources are the three on the right. The fourth sitting on the left is beyond the scope of this post. Known as the ‘gull wings’ in this blog, each bump is neatly sited at the join of separated layers: four bumps betraying five layers. The separated layers are long-axis delaminations and the gases emerged from under each join hence their translational symmetry and equidistant separation. 

All the bumps are more pronounced in this photo. They are slightly less so in the following photos due to a less favourable angle. However, if they’re pronounced here then they really are very prominent even if apparently less so in other photos. 

Photo 5- a long shot of the head and body lobes showing the three relevant bumps and the Ma’at pits they supplied. The pits are the larger blue dots with their perimeters dotted in smaller blue (except 02 shows an extension which matches the body, shown later).


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

Photo 6- as above but with the pits numbered and 01A arrowed. Again 01A isn’t very obvious but will become so in subsequent photos.

Photo 7- a close-up of the so-called cove on the head lobe showing 01A and 01.


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

In photo 7, 01A and 01 are dotted light blue. It can be seen that 01A is just the floor of the original pit before it delaminated. Pale orange denotes a massif attached to 01 that slid from its seating with 01. The seating is also pale orange. The 01A floor can be extrapolated from the pale orange massif slide, attached to 01, which means 01 slid too by the same distance and on the same direction vector. So the massif dragged 01 from the extrapolated 01A position to its current position. The extrapolation is also fully constrained by the dark green and mauve annotations. Green are delaminated layers which, when sandwiched back together, follow the pale orange massif slide vector. Sandwiching them back together is reversing the stretch movie, so to speak. Once clamped back together, they form the lower perimeter of 01A when combined with the ridge above the upper green line that was dragged with it (see also photo 8). Mauve is a suspected dyke which supplied 01A/01 when 01 sat on 01A. The gasses forced their way between the two green layers and out into the 01A hole when 01 sat there. This may be why 01A and the area between the green lines (the mauve gas track) looks scoured and not typical of the smooth dust that dominates Ma’at. 01A’s upper perimeter is constrained by the cove delamination (Part 34) and is beyond the scope of this post. 

Photo 8- As photo 7 but with the slide/delamination vector arrows added. 

In photo 8, the right hand arrow depicts the orange massif/blue 01 slide vector in reverse, that is, if you reversed the stretch movie. So it’s pointing from where the pale orange massif is today to where it used to be. When the massif slid up and back, it dragged its 01 hole with it from the original 01A floor. The middle arrow shows the end of the upper green delamination being seated to the corresponding end of its lower twin, again in reverse. The green dotted line has now been extended further up into a zig zag. That zig zag can be seen to have concertinaed upwards in an extended version of the two green lines described above. This zig zag is due to the cove delamination in Part 34. The left hand arrow is pointing downwards, showing that the tip of the upper zig zag point sat on the corresponding tip of the lower zig zag point. And the upper two lines of the zig zag sandwich onto themselves like the lower two described above. Therefore, the whole zig zag collapses down when you reverse the stretch movie. The two sandwiched zig zags used to be clamped to their body match. This body clamping therefore constrains the body gas sources either side of the sandwiched tip to supply the holes either side of the sandwiched tip on the head (i.e. source 01A/01 feeding both 01A and 01 and source 02 feeding 02). The zig zag delamination of course implies that the entire layer hosting Ma’at 01, 02 and 03 delaminated upwards from the lower (head rim) layer too. The zig zag is, after all, the end-on cross-section of those two layers. These two layers indeed exhibit mini-matches along from their zig zag points as well as to their expected body seating (see photos further down).

Photo 9- a simple version of photos 7 and 8. 

Photo 10- this is the same as photos 5/6. It has red lines depicting the notional paths of gasses from the uplifted body bumps to their respective pits on the head. 


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

You can see that the right hand bump supplied both 01A and 01 which delaminated upwards from 01A as described above. The 02 source actually delaminated from the 01A/01 source. There’s lots of evidence for this in Parts 37-39: the opening up of the so-called shallow crater, as well as the ‘gull wing slide’ and the ‘India shape slide’ in those parts absolutely constrain the 02 source to delaminate that distance along that same slide vector from the 01A/01 source. The 03 source then delaminated in turn from the 02 source. Only then did the pits that the 02 and 03 sources were supplying, delaminate upwards with the zig zag described above. Those were of course, the Ma’at 02 and 03 pits. That’s how three sources supplied four holes (or supplied three holes and the original hole base, 01A, that they delaminated from). Of course, the gas supply happened only when the head and its pits sat right down on the bumps. It’s now 1000 metres above the bumps so most of the red lines, across the shadowed expanse of the neck, are just notional links. The red lines kissing the bump tips is also notional because the exact gas sources are just to the right of the bumps (in this view). This is because the gases were exiting to the right of the bumps i.e. out from between the delaminated layers in the reverse direction to that in which they delaminated. The exact sources are shown in the next photo. 

This large amount of outgassing was along the shear line where the head rim was herniating upwards under the tensile forces of stretch and about to shear away. That’s what caused the zig zag: upward delaminating layers. The main source of gas was along the shear line (the Babi/Hapi rim) and it found its way out between the long axis delaminations hence the bumps being like mini-volcanos that are sited both on the shear line and the long-axis delamination interfaces. The difference between long-axis delaminations and upward head herniation delaminations is laid out after this photo section. 

Photo 11- this is the same as photo 10 but with the exact source positions added next to the bumps or mini-volcanos. The sources are actually just out of sight because the gasses ran up the side of the Babi cliff that drops into Hapi. So we’re seeing the very tops of the sources where the bottoms of the holes or pits were originally clamped.

Photo 12- this is the same as the above two but it has a small addition. The beige dots curve up the volcano bump of the 03 source and continue up that curved half-bell shape on the head. The head rim bell shape nests perfectly over the 03 source. The 03 source looks like a scoured-out cave under the volcano bump. The bell shape on the head rim is a chimney leading to 03 and it was supplied from the cave albeit in a much more concertinaed-down configuration. The current chimney length is due to head lobe herniation which stretched the chimney. This is possibly responsible for the pockmark holes dotted up the chimney side that may be collateral gas escape routes. 

Photo 13- the Part 38 header, adjusted. The pits are numbered as above but so too are their sources, dotted with blue lines on the body. 


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

In photo 13, all four bumps are annotated in dark green. They’re pairs of green dots because the first set at the top is the first set from which the other four delaminated. That’s the classic set of so-called gull wings so there’s one dot for each gull wing. The other three sets retain vestigial double bumps in sympathy. The fourth set, almost in shadow is of course irrelevant here. Yellow denotes key fiduciary points that match from head to body. You can now see the bell-shaped chimney from above (albeit in shadow but it is there) and see how it clamps over the 03 source. The full head rim to body shear line match is shown in photo 17 and the chimney is fully visible.

Photo 14- the same as 13 but without the numbering. 

Photo 15- this is the same as photo 2 but with additional matches between the first two layers of the head lobe that delaminated with the zig zag in photo 8. It also shows matches between the head rim and body including triple matches for both layers to the body (in yellow). 


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

Yellow nearest to us (lower right) is the Ma’at 02 pit’s perimeter extension that was marked in light blue in photo 5. This has a mauve feature adjoining it. The same yellow and mauve lines are seen on the head rim layer below the upper layer and from which the upper layer with the pits delaminated. This is a bled match (features repeated through layers). It’s not very apparent on the head rim layer in this photo (see original). But it is there in many photos. It’s most robust for the mauve feature and for the yellow line up to half way which is why its upper portion has smaller yellow dots. They depict a gentle trough rather than a defined line. The upper-left long spidery triangle is on the body and matches both head layer yellow lines including that extra short line on the head rim. Notice how the body yellow match kisses its light blue source semicircle in the same manner as the yellow extension meets 02 on the head. All the yellow lines are angled the same way. 

The larger bright green dot on the body is the 02 source bump or gull wing set and the small green dots show the extent of the bump. The bump nested into the niche shown in bright green on the head rim. There’s a rope-like ridge running down the body from the 02 source bump. It turns and runs towards top-right in this view and casts a sharp shadow along its length. This is the rest of the delaminated layer of which the 02 source bump is a part. In part 38 this is confirmed by noticing that this line matches to the sharper ridge leading to the upper left yellow lines (the ‘trident shape’, not quite so obvious here). These are mini-delaminations of a thicker onion layer. They’re perhaps 15 metres thick. The others are visible here but beyond the scope of this post. 

Photo 16- this is the same as photo 15 but includes the head rim to body shear line match in various colours. This match is traditionally done in terracotta but with yellow for the cove match. In addition here, we have the bright green 02 source bump which would normally be terracotta and also dark green going up and down over both gull wings for the first set (bump for 01A/01 source). The dark green head match is a cheat- it’s the upper line of the first zig zag, not the lower line on the true head rim. But they’re perfectly in line from this above view. The cove and its seating is yellow. The 01A/01 source is blue as before but since it was squeezing up at shear line inside the cove on its seating, it’s usually yellow. The head match for the 01A/01 source is 01A itself and that’s hidden behind the pale orange massif attached to 01 (visible here but not dotted pale orange). 

Photo 17- this is the same as 15 and 16 with two more additions. It shows the chimney for Ma’at 03 in beige dots (tracing its centreline) and the continuation of the beige line down the 03 source bump on the body. You can now see how the 03 chimney’s curved base nested over the curved base of the bump on the body. The three beige dots on the body run upwards towards us from the base of the bump (a mini volcano, like the other bumps, through being uplifted). The three dots run up to the true rim of Babi overlooking Hapi. This is a spectacular, curved overhang over Hapi. The area under the three beige dots is deeply undercut into a cave which we can’t see here. That’s more evidence for this being a gas source for 03. And of course the smaller blue dots extend, as in photo 2, to beyond where the head rim obscures the majestic, overhanging curve. That curve neatly matches (is concentric with) the curved base of its volcano as well as the curve of its chimney base, now 1000 metres directly above. 

Still with photo 17, the apparent offset of the head lobe rim to its body match here is almost entirely due to Rosetta’s viewpoint parallax but technically speaking there’s a fraction of offset due to head tip. Head tip also contributes to the head terracotta/bright green line being 4% shorter due to foreshortening. However some of that 4% may be due to the fact that the dark green/terracotta corner on the head rim was frilled upwards and backwards which is why its two yellow lines don’t progress to a joined-together point like their body match. The frill needs ironing out so that it extends out to match the green/terracotta corner on the body. Careful analysis of the mauve match shows that the second layer pinned the back of the frill down on the body while the outgassing from under the first bump (the main gull wings) frilled up the edge on that corner. Even Ma’at 01’s perimeter did some of the pinning because it’s part of the zig zag. There much about the frill in Parts 5 and 7. It extends along the head rim further than is visible here. 

There’s also a suggestion of a collateral hole, supplied by the beige 03 chimney. It’s nested up under the second head layer and in line with the chimney and Ma’at 03. This suggests Ma’at 03 delaminated upwards from this larger hole so that both were once one hole, clamped over the 03 source. That would be like 01 sitting on 01A and both being clamped over their respective source. 

Photo 18- this is the same as 5-6 and 10-12 with a few alterations.


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

The red lines are now running more faithfully from their true sources on the body to their respective pits on the head. Before, they were running notionally from the green-dotted bumps. The long lines across the shadow are largely dispensed with in this version. There are still short lines running up from the body sources but they’re just the rough distance of gas travel from the sources to the holes when they were seated on the body and herniating upwards. The 03 source line has three dots running down its volcano bump. They depict the gas flow, hidden on the other side and flowing out of the undercut cave. The 02 and 01A/01 sources aren’t undercut. 

Photo 18 also shows the 03 gas path through the chimney more faithfully and a short offshoot to the intermediate hole tucked under the second layer rim. It also shows a tentative position for an intermediate hole along the 02 gas path. This is only because both the 01 and 03 paths have intermediate holes. Seeing as this comet is almost obsessively symmetrical when delaminating the suggested hole is annotated just as a suggestion. It’s not readily apparent as a hole but may make itself apparent as a narrow collateral dyke. There’s already a tiny collateral dyke within the yellow match line just below the suggested position and that tiny dyke has the 02 path running right under it. 

Finally, the dimple just above the head rim, midway between the 02 and 03 paths is the position of the bright green dot that fits over the 02 bump in photo 15. The 02 bump is dark green here, in keeping with the others as that’s their usual colour. 

Photos 19 and 20- the ESA regional maps.
Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA

INTRODUCTION

This Part should be regarded as a twin to Part 52. Reading that part will give more background as would Parts 38-41. However, this part can still be read and understood in isolation. As is often stated here on the stretch blog, the process by which stretch theory explains the delamination of the pits known as Ma’at 01, 02 and 03 is described below as if it’s fact so as to spare the reader endless qualifications and conditional statements. Of course, it’s a hypothesis but after two years of scrutinising this small area (hundreds of photos for very many hours) I’m sure the evidence is highly compelling. It’s so extensive that only a small amount of that evidence can be presented in this post with the rest left to links to other posts. 

BACKGROUND

The Rosetta mission is going to land the Rosetta orbiter on 67P/C-G on 30th September 2016. At that instant, all transmissions will stop and the mission will end. 

On the way in to land, Rosetta will analyse the pits at Ma’at on the head lobe near the head rim. They’re named Ma’at 01, Ma’at 02 and Ma’at 03 although Ma’at 02 has recently been renamed ‘Deir El-Medina’. Rosetta is slated to land between 02 and 03. I’ll generally refer to the three pits as 01, 02, and 03 for brevity here with occasional reference to their longer names. 

I’m posting this because I think the largely accepted view that the pits are probably sinkholes might affect the decisions on how to take the data on the way in to land. For instance, it might affect the decisions regarding attitude orientation (where Rosetta is pointing) and which instruments are switched on for taking data. At the time of posting this, there are intense discussions as to which instruments can be allowed to operate and to what extent because of acute power and telemetry constraints. This Rosetta blog post published recently outlines the constraints:

http://blogs.esa.int/rosetta/2016/09/16/beginning-of-the-end/

Of course, the mission scientists will be attempting to point the instruments into the pits so you might think the pointing issue is academic. But the pits have delaminated from each other. The knowledge of their delamination might prompt the pinpointing of certain specific areas, perhaps at the stratum of the delamination in successive pits. Or extra scrutiny of the boulders in Ma’at 02, knowing some might’ve fallen from the sides of Ma’at 03 as it delaminated. 

THE FOURTH PIT AT MA’AT

There’s also a fourth pit on the head, next to the other three, that hasn’t been identified as such because doesn’t look like pit. It’s the most most important of the four because it’s the original one the other three delaminated from and today it’s just the old, scoured-looking floor with no walls. It’s lying right under the landing flight path (I believe) and data will probably be taken from above it. But surely it would be a good thing to know in advance that it’s not the comparatively boring flat expanse it seems to be, but key to everything Rosetta is analysing in the last moments of her 2-year mission. 

THE LONG-AXIS DELAMINATION PROCESS

The long-axis delaminations at Babi/Hapi are presented in more detail in Part 38:

https://scute1133site.wordpress.com/2016/01/20/67pchuryumov-gerasimenko-a-single-body-thats-been-stretched-part-38/

There’s additional information on the long-axis delaminations in Part 52:

https://scute1133site.wordpress.com/2016/08/08/part-52-the-body-lobe-match-to-rosettas-landing-site-includes-suggested-flyby/

Part 52 was linked in a comment I made on the “Celebrating Two Years at the Comet” post in the hope that a low flyby of the Babi slide would provide data on the jet there that’s related to the 02 source and therefore to Ma’at 02 itself. The Babi slide itself (Part 40) is beyond the scope of this post but puts it in perspective. 

According to stretch theory, pits 01, 02 and 03 came about as a result of onion layers delaminating along the Babi/Hapi rim under the influence of long-axis stretching of the comet. 67P stretched into a quasi ellipsoid and continued stretching even more along the long axis of that ellipsoid. This eventually caused the diamond shape we see today that defines the body and, partially so, the head lobe. The long-axis delaminations along Babi (and Seth) were therefore trying to accommodate the stretch. Long-axis stretch is why Hapi and Sobek are longer than Bastet and Anuket, resulting in a neck with an elongated cross-section that’s aligned with the long axis of the comet. The cross-section and its long-axis alignment can be seen in Hirabayashi, M., et al 2015 extended data figure 2:

http://www.nature.com/nature/journal/v534/n7607/fig_tab/nature17670_SF2.html

The tensile force vectors operating during the stretching process would have been brought about by spin-up via asymmetrical outgassing. 

The long-axis stretch and delamination process happened when the head lobe was still attached to the body, prior to shearing. The shear line runs along the Babi/Hapi rim where the delaminations are most apparent. 

Since the layers were delaminating along Babi, any particular feature on the original layer that delaminated would be reproduced in the successive delaminations. This would include the pits/holes and the uplifted mini-volcanos supplying them. There are also other features such as a an obvious trident shape. This is why the mini-volcanos in photo 4 are so similar and near to being equidistant:

Photo 21 (4 reproduced).
Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0/A.COOPER

Since the tensile force vector for the long-axis delaminations was in line with today’s Babi/Hapi rim and the head was still attached, it meant that the delamination layer fronts would be expected to be at 90° to that vector, or at least notionally so, all other factors being equal. However, this is notwithstanding the initial layering configuration prior to delamination and indeed this appears to make the Babi delaminations angled forward towards Bastet somewhat. This may be a primordial arrangement but it might instead be due to the pinning of layers at the proto-Anuket region, causing the V-shaped configuration we see on the head today and, rather disguised, on the body too (see Part 27, ‘Signature 3’). 

However, notionally speaking, the delaminated layer fronts were ‘draped’ across the long axis tensile stress vector. Since the head was still seated on the body during long-axis stretch, these draped layers ran from proto-head to the body, across the future head lobe shear line at the Hapi rim. 

THE REASON FOR HATHOR BEING FAN-SHAPED

The head lobe shear line itself runs in the long axis plane or to be more exact, in a plane that is parallel to the long-axis plane. The shear line is also the line along which the proto-head started herniating. So it was a weak spot. The head chose to herniate along, and eventually shear away from this particular line because it was literally a shear line: a sharp tensile strain gradient occurred across it causing differential strain forces. Whilst the long-axis tensile forces were the same across this line (no gradient) the differing tensile force resistance (strain resistance) of the matrix either side of the line resulted in the strain gradient. This caused shear along the Hapi border with both Babi and Seth and it’s indeed the very reason for all three regions exhibiting different morphologies. Furthermore, it’s the reason for Seth and Babi being rough mirror images of each other fanning out from the straight portion of the Hapi rim. That straight portion was where the shear force was most potent, causing the most defined head shear. At Bastet and Anuket the shear turns into a yanking-up which is why the neck looks similar at both ends (Bastet: Part 21 and Anuket: Parts 24,25).

On one side of the shear line there were the more susceptible Babi delaminations; on the other, the proto-Hathor cliff, still clamped face-down on Hapi, was delaminating more subtly: it delaminated into narrower layers (the strong striations we see today) and did so in a fan shape. This is why both Hathor and Wosret are fan-shaped.

The greater resistance to strain of the Hathor cliff matrix when seated on Hapi was presumably due to the matrix being colder and more brittle. This property would become more pronounced with the depth of Hathor under the seated head and lead to increasing strain resistance towards the central area. This translates to less stretch at depth and would be an elegant explanation for the fan shapes at Hathor and Wosret: less stretch at the bottom of the fans and more stretch at the top couldn’t help but form a fan shape. 

This gradation of strain resistance with depth would also account for the apparent directing of the long axis tensile forces around the proto head lobe and the consequential sudden strain gradient at the Hapi rim where the fan-shape strain mechanism gave way to full-blown layer delamination across Babi. That would explain the location of the shear line and by extrapolation, the shape the head rim traces. Thus, the symmetrical shape of the head when you look down on it is explained (yet again, aligned with the long axis).

Moreover, the nodule sitting centrally at the base of the Hathor cliff could represent the threshold at which no stretch at all was possible (100% strain resistance). It would be like trying to stretch and deform an avocado. The flesh would stretch and deform around the unyielding stone and the brittle skin would crack. If made of several layers, the skin would delaminate as well or instead. The Hathor cliff matrix would then represent an intermediate phase of reluctant, partial stretch as would happen if the avocado stone had a thick, semi-ductile shell. For 67P, this translates to less stretch at depth with the tensile forces being directed around whatever doesn’t want to stretch or wants to stretch less. 

If 67P was a perfect ellipsoid the tensile forces wouldn’t have a proto-head to get forced around and work free via shearing along Hapi and Sobek. However as soon as any proto head lump started to herniate under stretch, it could be visualised as an avocado with two stones or a peanut-shaped stone. The tensile forces would start to be directed around the girdle of the peanut, this causing shear at the surface in line with the girdle i.e. at the shear line/head rim line we see today. 

This is the first mention of the Hathor fan-shape explanation on this blog as well as the fact of the shear line being an actual shear line where shear forces acted either side of it. These will of course get their own posts. They’ve been brought forward here because of their relevance to the long-axis delaminations. 

THE PITS

The Ma’at pits were caused by outgassing, yes, but in a shorter, sharper catastrophic episode than through gentle sublimation. They never had roofs that fell in after a cavity had gently eroded away underneath. They acted just like volcano calderas and even had their uplifted mini-volcanos that supplied them with the fast-sublimating gasses emerging at the shear line during head lobe herniation. These are clearly visible on the body today, aligned 1000 metres below 01, 02 and 03. When the head was still attached, the already-delaminated pits sat on (or rather, just next to) the mini-volcanos.

Whilst the Ma’at 02 pit probably did sit on or very near its bump at the time of being supplied, the bump delaminated a little further from the outgassing source signature we see today. So it continued delaminating longways after 02 delaminated upwards. This is evidence that a small amount of long-axis delamination continued even as the two ‘zig zag’ layers were delaminating upwards under the influence of head lobe herniation. 

This slight continuation of long-axis delamination after head herniation commenced is perhaps corroborated by the 03 path. 03’s outgassing source (the undercut cave) hugs its volcano very faithfully but, together, they are slightly further displaced along the body from today’s 03 pit on the head. The 03 source has the strongest 3D match to 03 itself: a bell-shaped chimney, starting at the third volcano which is seen to curve up across the first layer of the head lobe to meet 03.

So the gas source for 02 is slightly displaced from its volcano. The other two sources are hugging theirs, just to one side. This is consistent with the gases at the shear line finding their way out firstly from the shear line itself but also from under the draped ‘lasagne layer’ in the reverse direction of delamination as you’d expect. 

THE EVIDENCE FOR THE VOLCANOS OUTGASSING

The mini-volcanos are what are known as the ‘gull-wings’ in this blog because the first set that was matched from body to head looked like a pair of gull wings. That was in Part 5, December 2014. The other three sets of gull wings delaminated from it, along the long axis, so they partially preserve the gull wing shape. It wasn’t known in 2014 that they had delaminated but there was much evidence of former outgassing from in front of all three sets that was noted at the time. I noted the outgassing evidence from in front of sets one and two, suggesting it was the reason for the uplift. Rosetta blog commenter, Robin Sherman, often referred to the possible ejection of material from the cave under the third set because it looked scoured-out somehow. Part 5 talks of the first set being uplifted, hence their gull wing shape. Parts 7 and 8 discuss the path the gasses took from the soon-to-be Hapi that was still trapped under the head lobe. That discussion incorporated the second set of gull wings. Hence, all three sets were identified as experiencing catastrophic outgassing and therefore being uplifted. And of course, it has to be remembered that this entire section of body had already been matched to the head rim underside anyway. That was done in Parts 1 and 2. But still, it wasn’t realised that the gull wings or volcanos were delaminated from each other. That came in Part 38:

https://scute1133site.wordpress.com/2016/01/20/67pchuryumov-gerasimenko-a-single-body-thats-been-stretched-part-38/

You can click through to Parts 39-41 from Part 38 for the recommended extra background reading. 

The long-axis delaminations and how they match to the head layers is explained in much more detail in Parts 38-41. Regular readers may notice that the numbering scheme for gull wings above is different from in Part 38. The numbering above is for clarity regarding the matching of mini-volcanos to Ma’at holes in this post.

THE TWO-WAY DELAMINATION (LONG AXIS VS HEAD HERNIATION)

Photo 22- the cross-cut strata. 


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

Key:

Yellow- true strata. The body components look wiggly due to being flaccid after recoiling backwards along their length. 

Red on the head- quasi strata. These are the head lobe herniation ‘strata’ which are really just passive chunks of matrix that yielded to head lobe herniation. They just happened to yield to that upward tensile stress at a ~50-metre depth, tore from the shear line and recoiled up the head lobe. In so doing they hid the real strata within them (see below). The two upper red lines are the layers that recoiled. The third bottom one is the head rim. The top red line is the one at which the true head lobe strata (yellow) appear to stop dead. In fact, the head strata carry on within and across the red quasi strata but are almost hidden except in the head cove. Red and yellow can be seen cross-cutting in the cove. The uppermost red line hasn’t been discussed so far in this part because it was the very top head herniation layer and it slid up past the Ma’at holes (Ma’at 02 is dotted blue, just below this line). The layers above the first (rim) line and second line are the two that we’ve been discussing and which have the zig zag at their ends. The uppermost line appears to have been set back from the shear line when on the body which is why there’s a blue dot on the body. That’s the Ma’at 02 body delamination. So the head took the bottom two layers of the Ma’at pits that sat on their Ma’at sources. And the body kept the very top layer. That blue dot on the body (or rather its surrounding walls) sat on top of where Ma’at 02 is today.

Red on the body- these are quasi-strata as well. The same principle applies as for the lines for the red quasi strata on the head but the signature of the layers is less obvious. The red lines on the body were originally joined to the lines on the head: bottom body to top red; top body to bottom head; and middle to middle. 

Blue- top blue is Ma’at 02; middle is the 02 source; bottom, on the body, is the jet that originally sat on top of Ma’at 02. Its adjacent red line wrapped round it and clamped to the red line above Ma’at 02 when they were sitting on their common 02 source at the shear line. 

The two-way delamination explains the paradoxical, different-angled ‘strata’ in this location at eastern Ma’at. They delaminated in two directions: along the long-axis stretch vector and along the upwards head herniation vector. They’re at a notional 90° angle to each other as mentioned above. That’s the best way to visualise it close-up. From a distance it’s a bit more nuanced.

The two-way delamination is the reason the strata appear to cross-cut inside the head lobe ‘cove’ above the border between Seth and Babi. And it’s why head strata appear to stop dead in their tracks at a straight line running all the way from western Ma’at, through the cove, and on round to Bastet in the east. That impressive line is the upper layer of the head herniation delaminations. These delaminations could be viewed as sliding or even recoiling upwards after having torn at the shear line due to the herniating head pulling on them. 

Together, they form that impressive green zig zag at their ends (photo 8). The zig zag betrays the delamination. But these two upward herniation layers are not real strata. They’re just layers of passive comet matrix which were torn at the shear line by head lobe herniation and recoiled (Part 41). They just happened to fail at that particular ~50-metre thickness, one after the other, as the herniation progressed. But the most important thing to remember is that both upward-recoiled layers contain all four true strata that originally delaminated along the long axis before tearing right across their width at the shear line and recoiling upwards. Those are the true strata and they supplied all the material for the two upward-recoiling layers which appear in photo 22 as quasi-strata. The quasi strata almost completely hide the true, long-axis-delaminated strata. The true strata became almost totally hidden due to tearing across their widths, succumbing to the quasi strata’s 50-metre-thickness requirement and becoming flaccid as they recoiled up the head and along their length.

But these hidden, true strata are traced in Part 41. And most crucial of all for this post, the true strata are the long-axis-delaminated layers that contain the delaminated holes, Ma’at 01, Ma’at 02 and Ma’at 03. 

CONCLUSION

I hope the mission scientists can take this into consideration in their deliberations on how to take the pit data as Rosetta descends to the surface on 30th September 2016. 

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 61- Close-up gifs of the Bastet Pancakes Matching to Aker

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

 ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DyASP/IDA/

Part 51 header for context. The two left-hand mauve features on Bastet and Aker (head and body respectively) are the same as the much closer-up mauve features in the “head” and “body” gif components. 
Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0

INTRODUCTION

This post is a follow-on part to Part 51 ‘The Bastet Pancakes and Their Relationship to Neighbouring Slides’. That part showed how the pancakes, which are thought to be splatted cometesimals are in fact slid pieces of crust which used to to sit in the two dips at Aker. 

Part 51 showed the matches in several photos and included two gifs. One was a bit crudely dotted and they weren’t close-ups showing the multitude of subtle matches. This post shows a gif that’s a closer-up version of those two gifs. 

For the sake of absolute transparency, the nature of the Part 51 header, a NAVCAM photo, didn’t lend itself well to cropping for the gifs. Aker was the problem. It was to do with it enlarging when rotating it to horizontal and moving out of frame. So an OSIRIS picture was found which was from the same angle. However, small discrepancies in distance meant it had to be resized by about 10% to fit the NAVCAM Bastet component. Seeing as the matches had already been made on the single header photo where no resizing was needed, this seemed a reasonable course of action because we know that the matches are the same size in that photo. Any objection to this approach would have to contend with the fact that the head lobe is showing a multitude of matches that are all 90% the size of their body counterparts. 

The viewing angle issue was negligible and that’s betrayed by the fact that the matches fit in two dimensions when resized. If there was a perspective discrepancy, they would be the same length on resizing but not the same width. 

OSIRIS still used for gif:

Credits: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DyASP/IDA/

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/

FOR OSIRIS

ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DyASP/IDA/

All dotted annotations by Andrew Cooper

Part 60- 150m Massif Slides 250m on Ma’at 


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

(It’s just this one photo in this post with crops, annotations and gifs. Full credits at the bottom). 

INTRODUCTION

Just one of many small-scale slid layers that are sitting in the queue of posts. This one is on Ma’at, just above Nut. It does betray matches from the perimeter of the slid layer to its seating if you look carefully (especially a saw-tooth line down the left perimeter). However it betrays bled matches very well. Bled matches are features that are repeated down through the layers in the third dimension. Somehow they’ve bled through to the surface. There’s a lot on bled matches in Part 45. It’s akin to spilling wine on a newspaper and the stain bleeding through the pages- except it presumably happened in reverse through sublimating gases finding the line of least resistance through the layers and possibly depositing refractory material on the way through. 

THE PHOTOS

The original NAVCAM photo is first, then a closer version zooming into the area in question, then a succession of stills and different speed gifs constructed from the stills. It’s fairly self-explanatory.

The coloured versions are fiduciary points to direct you to where the bled matches are. They’re by no means exhaustive e.g. the yellow line continues all the way round that lighter feature and there are two more v shapes below the mauve-dotted one. Also, once you’ve familiarised yourself, you can return to the main non-gif photo and see the subtler matches to the right, along that wavy finger extension. The two bright green lines are not strictly on the main slid section but on another delaminated layer under it because everything was delaminating at Ma’at (see Part 29). 

It’s pointless just looking at the dotted versions only. They are just an initial guide. 

It’s advisable to hold the tip of a pen to any point of interest. It’s surprising how little it moves from gif image to gif image when you do this. The shadowing and surrounding features make it look as if it’s not aligned so well but it’s an illusion. The residual non-alignment is due to the slid section being at a slightly different profile angle (over the curving head) and human error in constructing the gif. You can even trace your pen tip down a certain line as the gif is running. 

Using the fast gifs is recommended for getting your bearings. Then slow for main analysis then fast again for better confirmation of what you’ve found. Then back to slow, and so on. 

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CLOSER UP:​​​​

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. 

Part 59- The Dare-Devil Apis Flyby, Escape, and 2020 Reacqusiton


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

Green- Apis

Red- Orbit line

Blue- Equator ground track of orbit line

CONTENTS

1) Introduction 

2) Assumptions for inputs for calculations 1

3) Notes on the flyby calculation

4) Terms in the flyby calculation 

5)The flyby scenario (simple version)

6) Assumptions for inputs 2

7) The flyby scenario (longer version)

8) The numbered stages in detail

9) Flyby relative speeds and duration

10) Delta v budget for flyby and escape

11) After escape: hibernation and return via the quasi orbit

12) Flyby vs crash means more data transmission

13) Conclusion

14) Appendix- glossary 

1) INTRODUCTION

On September 30th 2016, the Rosetta mission team are going to crash-land the Rosetta orbiter on 67P. Marco Parigi suggested on the Rosetta blog that the Rosetta orbiter should do a close flyby and then boost to escape, hibernate, and return when the comet approaches perihelion again in 2021. I’ve suggested several times on the Rosetta blog about studying Apis at very close quarters as I believe it to be the the most primordial region on the comet, based on the findings in stretch theory. 

I’ve now put together a dare-devil flyby scenario for September 30th 2016 that would satisfy both objectives. Instead of crashing, Rosetta would overfly Apis at 200 metres’ altitude, taking 26 minutes and 40 seconds to complete the Apis ground track. She would then boost to escape and return in 2020. She would return largely of her own accord because she’d be injected into a quasi orbit of 67P on escape. 

Of course, nothing in this post would be new to anyone looking at the comet and calculating potential landing or flyby scenarios. It’s basically a Hohmann transfer injection from a circular orbit. The injection orbit is an ellipse. A full Hohmann transfer would circularise the orbit again at the lower radius but we would fly on and perform the escape burn from the comet. However, the way in which it’s done requires only 1 m/sec of delta v, including return and reacquisition in 2021. This proves that despite fuel being low, the delta v budget for all these manoeuvres is a fraction of a per cent of the delta v fuel supply that Rosetta arrived with in 2014. 

There may be legitimate objections to the height of the initial circular orbit suggested here or the fact that Rosetta’s approach takes more than 16 hours from such a low height. This is done to minimise the orbital speed at flyby without having to do correction burns to slow Rosetta at the last minute (correction at 1.7 km altitude is planned for the controlled crash slated for September 30th 2016). Slow approach from a low altitude increases perturbations from the uneven gravity field but the characterisation of those perturbations is addressed in detail in the twin post to this part. It’s Part 58, linked here:

https://scute1133site.wordpress.com/2016/08/31/part-58-the-case-for-a-dare-devil-flyby-of-apis/

The data from the characterisation would be fed into this flyby set up, adjusting the size of the flyby ellipse by a small amount. Any residual, uncharacterised anomalies would result in a flyby that was a few metres higher or lower than the 200-metre nominal altitude and a slightly earlier or later arrival time over Apis. 

2) ASSUMPTIONS FOR INPUTS FOR CALCULATIONS

67P’s mass: 1e13 kg.

67P’s rotational period: 12.7000 hours.

67P’s obliquity: 52° (angle of its rotation axis to its orbital plane). 

Apis radius of rotation: 2600 metres.

Apis width: 400 metres (as defined by the equator ground track across Apis, which is essentially across its width). 

These inputs vary in their accuracy but this doesn’t detract from the basic setup for the flyby scenario. Any adjustment to inputs simply trickles down through the calculations rather like for a spreadsheet. This changes the orbital elements involved subtly and the timeline slightly but the scenario remains essentially the same. 

The twin post also addresses a wide range of objections that may be raised to the flyby scenario. These range from not taking into consideration 67P’s gravitational anomalies (which would be overlaid on the flyby scenario) to the fact that escape to a hibernation orbit at a higher aphelion would be too cold for the panels/batteries to cope. Also, transmission of data and time needed on the Deep Space Network. If you have any objections, it would be as well to read the twin post too, linked above. 

3) NOTES ON THE FLYBY CALCULATION. 

All calculations were done on paper using a calculator. No computer software of any sort was used. This is stated only to emphasise that I established in advance what the best flyby scenario was, with recourse to stretch theory, and worked backwards from there. There’s just one scenario that stands out above all others, dictating strict parameters for Rosetta’s position and velocity at all times during flyby and escape, so having established that, I worked backwards systematically from there. That’s as opposed to playing around with orbital software until something looks roughly doable after ten thousand Monte Carlo runs. The strict parameters were: 

1) approaching the comet along the extended equatorial plane (the rotation plane).

2) overflying Apis at 200 metres’ altitude and at the lowest possible relative speed (along an equator ground track). Apis is chosen for a large number of reasons but “with recourse to stretch theory”, it is established as being the most primordial region on the comet’s surface. This is due to it being the only section of the 29km^2 surface that hasn’t been, torn, scraped, delaminated, stretched, sheared or flung from the comet. It sits in splendid isolation, comprising the flattened tip of the long axis, as if that was its role during the stretching process. And that’s because it really was its role during stretch. It escaped harm by keeping out of trouble, riding on the stretching tip. Everything else snapped away from Apis via flexion forces rounding the stretching tip and left a passive rectangular tablet experiencing no stresses at all, just stuck on the tip (see the Paleo Rotation Plane Adjustment page for more details on this). 

3) a 52° plane change burn (a noncoplanar transfer burn). 

4) followed soon after by a parabolic escape orbit burn to ensure Rosetta is sent exactly south (or north) of 67P’s orbital plane. Her direction vector velocity component would therefore be orthogonal to 67P’s orbital plane.

5) followed by a very small correction burn to push the parabolic escape speed residual (which is actually zero) a fraction higher to ensure true escape at extremely slow speed and still orthogonal to 67P’s orbital plane. Rosetta would thus enter a heliocentric orbit that doubles up as a quasi orbit of 67P. This ensures automatic return to the comet for reacquisition in January 2021. The automatic return means Rosetta does this with a theoretical zero delta v budget after the small correction burn stated above. 67P’s heliocentric position (its true anomaly) at the time of acquisition will be the same as for August 2014, in other words, the perfect position for Rosetta to resume scientific analysis. 

I may post up the actual calculations in due course but it’s rather tedious to type up let alone read equations without proper maths notation symbols. That was the case for the “Spin-up Calcs” page with about one tenth of the calculations as here. In the meantime, the flyby scenario outlined below just gives the resultant outputs: orbital elements, orbital speeds, escape speeds, delta v burns and delta v budgets. In other words, it contains all the actions that Rosetta would need to perform to execute the flyby, escape and 2021 return. 

4) TERMS IN THE FLYBY CALCULATION (a full glossary of terms is at the bottom).

Zero point- this is the periapsis (closest approach point) of the flyby ellipse. It’s positioned such that it will be directly above the mid point of Apis rotating under it when the timeline reads T=0. The zero point is 200 metres above the Apis mid point when the mid point rotates instantaneously under it. Rosetta will fly through the zero point at T=0 as well so both the Apis midpoint and Rosetta meet up at/below the zero point at T=0. The zero point is planned as a fixed point in space with respect to the cometcentric reference frame as well as the heliocentric, ecliptic reference frame. It’s determined in advance and is positioned on the line along which 67P’s equatorial plane intersects 67P’s orbital plane. The heliocentric ecliptic reference frame constraint is only relevant to the escape burns and return to 67P in 2021. You can forget about it being of any importance at all to the actual flyby setup. It’s only well after flyby that it comes into its own. 

T minus (number of hours) is time before T=0 

T plus (number of hours) is time after T=0. The timeline runs from T= minus 79.9810 hours to T plus 300 hours. 
T = 0. This is the time at which Rosetta flies through the periapsis of the flyby ellipse. It’s also the time of Rosetta’s ground track crossing of the mid-point of Apis, in a prograde direction (west to east) along the equator. The equator is therefore Rosetta’s ground track. This is because she’ll be orbiting in the equatorial plane. Thus, Rosetta crosses through the zero point (periapsis) at T = 0 and at that same instant the Apis mid point is vertically below her as it also rotates west to east. 

Periapsis- the closest point of approach in the flyby orbital ellipse. It’s at one end of the long axis (semimajor axis) of the ellipse. It constitutes the zero point that’s described above. So the periapsis and zero point are one and the same, at the same fixed point in the cometcentric reference frame (more on this further below).

Apoapsis- the highest point of the flyby ellipse. This is by definition at the opposite end of the ellipse from the periapsis and so a line drawn between periapsis and apoapsis will run through the centre of gravity of 67P. From the point of view of the centre of gravity, the two ends of the ellipse are 180° apart, in opposite directions. Since the centre of gravity will be used for the origin of a polar coordinate system, below, it’s useful to remember this, even when Rosetta is in its preparatory circular orbit that doesn’t posses an apoapsis (constant altitude/radius). When on this circular orbit, Rosetta does kiss the future flyby ellipse’s apoapsis twice, once on circular orbit insertion and a second time (after one orbit) at flyby injection. 

5) THE FLYBY AND ESCAPE SCENARIO TIMELINE (LESS DETAILED VERSION WITH LESS EXPLANATION- FULL VERSION LOWER DOWN).

All times are measured from the T=0 time when Rosetta flies through the flyby ellipse periapsis, directly above the Apis midpoint. 

At T minus 79.9810 hours (timed at engine burn shut-off):

Circular orbit insertion. Radius 9594 metres. Orbital period 63.5 hours (exactly five times 67P’s 12.7-hour rotation). Orbital speed 0.2637 m/sec. Apis mid point is at 252.828° from the zero point measured in the prograde direction of rotation. All angles around 67P’s 360° rotation circle and Rosetta’s orbit are measured in the prograde direction of rotation from the zero point acting as 0°. The origin of this polar coordinate system is the centre of the comet’s rotation axis i.e. its centre of gravity. The polar coordinate system is sufficient for characterising Rosetta’s position in relation to 67P’s rotation and specifically her ground track position along the equator in relation to the angle of Apis from the zero point. The polar coordinate system is also sufficient for pinpointing the periapsis and apoapsis of Rosetta’s orbital ellipse because their polar angle is the same as the true anomaly angle used for orbital position: 0° for periapsis and 180° for apoapsis. Thus we can dispense with referring to the true anomaly and just refer to the angle from the zero point. It’s basically just a 360° protractor with the centre of gravity at the centre and “0°” stamped at the zero point.

At circular orbit insertion, Rosetta is at 180° from the zero point at burn shut-off. So a straight line between Rosetta and the zero point will run through 67P’s centre of gravity. Crucially, the orbital plane is in the extended equatorial plane of 67P so its ground track will always run along the equator from this point onwards until well after the flyby. Rosetta’s orbit is prograde i.e. with the comet’s rotation. 

From T minus 79.9810 hours to T minus 16.4810 hours: 

67P rotates five times (or an apparent four times) under Rosetta during her 63.5-hour orbit. ‘Dry run’ soundings are taken when Rosetta is at 270°, 0° and 90° from the zero point. These are dummy runs for flyby orbit injection. The Apis angle and orbit altitude are measured at those three locations in the orbit. On each occasion, the Apis mid point should be 72.828° ahead of Rosetta and Rosetta’s altitude should be 9594 m. If there’s any drift, actual flyby injection at 180° from zero point is performed a few seconds early or late according to the anomaly. This small advance/retard operation is called a reset here. The reset means the zero point is thus shunted forwards/backwards a few metres and and timeline is reset accordingly to 16.4810 hours with seconds being added or taken away. The Apis angle is determined with fiduciary points along the equator vertically below Rosetta (as was done for sub-second rotation period determination). 

At T minus 16.4810 hours (timed at engine burn shut-off):

Flyby ellipse orbit injection. Delta v burn is 0.0864 m/sec retrograde (i.e. directed tangential to prograde orbit direction to achieve a retrograde delta v). This is done to reduce Rosetta’s orbital speed from 0.2637 m/sec to 0.1773 m/sec. Rosetta is at 180° from the zero point at burn shut-off i.e. she’s at the apoapsis of the flyby orbit which commences at that instant. Apoapsis radius is 9594 metres. This is because she was injected from the circular orbit. Periapsis is 2800 metres, 200 metres more than the 2600-m radius of rotation for Apis. Periapsis is at the zero point which was fixed long before the flyby scenario was initiated. Ellipse semimajor axis is 6197 metres, semiminor axis is 5183 metres, ellipse eccentricity is 0.548. Apis is again at 252.828° from the zero point because Rosetta orbited once during which period 67P rotated under her 5 times.

T = 0: Apis midpoint flyover. The Apis flyby commences at T = minus 0.2222 hours (13 minutes and 19.5 seconds) as the Rosetta ground track acquires the western rim of Apis. Rosetta passes over the mid point at T = 0 and her ground track crosses the eastern rim of Apis at T plus 0.2222 hours. Total Apis flyby time is 26 minutes and 39 seconds. Rosetta orbital speed at zero point (T = 0) is 0.6075 m/sec. Apis tangential speed is 0.3574 m/sec so the transit speed of Rosetta across Apis, west to east, is 0.2501 m/sec. This speed ignores a very slight slowing of Rosetta in the 13 minutes either side of periapsis which has the effect of slowing the stated transit speed and lengthening the transit time by a few seconds. However, this balances either side of the zero point so all values right on the zero point at T = 0 remain nominal and Rosetta overflies the Apis mid point on time. 

T plus 16.4810 hours (timed at burn shut-off). Note ‘plus’ meaning after flyby zero point crossing:

Noncoplanar transfer burn to change Rosetta’s orbital inclination by 52°. Delta v burn is 0.1554 m/sec orthogonal to Rosetta’s orbital plane. This results in a 52° inclination change only, while prograde orbital speed of 0.1773 m/sec remains the same (apoapsis speed as for flyby ellipse injection). The 52° inclination change takes Rosetta’s orbital inclination from 0° (around the equator line) to, unsurprisingly, 52°. This is the same as saying her new orbital plane is now at 52° to the old one around the equator. 

Since 67P’s equatorial plane is already at 38° to the orbital plane on which 67P orbits the sun, by adding 52°, Rosetta is orbiting 67P in such a way that her orbital plane is now at exactly 90° to 67P’s orbital plane. The obliquity of 67P is 52° and is by definition the angle its rotation axis makes with the 67P orbital plane around the sun. This means the 67P equatorial plane is at 38° (90°-52°) to its orbital plane around the sun. Hence the need for an inclination change that’s the same as the obliquity value in order get Rosetta orbiting at 90° to 67P’s orbital plane. The noncoplanar transfer burn is performed when Rosetta is back at apoapsis which is why the orbital speed is 0.1773 m/sec, as it was exactly one orbit ago when flyby injection shut-off occurred. The orbital speed can’t increase or decrease if the delta v burn is a sideways push at 90° (orthogonal) to the orbital plane. A sideways push just gives rise to a change of direction, which is an angle change, which is an inclination change in this case of orbits. In the case of supermarket trolleys travelling at 0.1773 m/sec and a child pushing on it exactly sideways, it will eventually give rise to a 52° aisle change with no change in forward speed. It’s an exact read-across, there’s nothing very exotic about this.

T plus 27.5960 hours (timed at burn shut-off at the semiminor axis vertex at orbit radius, 5183 metres):

Inject to parabolic ‘escape’. Delta v is 0.1204 m/sec prograde i.e. directed backwards along a tangent to the retrograde orbit line. This gives rise to an increase in prograde orbital speed from 0.3871 m/sec to 0.5075 m/sec, escape speed for this altitude. Escape is put in inverted commas above because a parabola puts Rosetta on the cusp of escape only. A parabola is used for escape for a reason, instead of a hyperbola which implies excess speed for true escape. This reason is explained in the more detailed flyby explanation under the relevant subheading below.  

T plus 300 hours (hopefully after a long window for flyby data transmission):

Escape injection. A delta v burn of 0.03 m/sec is performed. It needs to be the slowest possible speed that ensures escape with no unintended return, decaying orbit and eventual crash during hibernation. That’s why 3 cm/sec is chosen. It’s much easier to get it right when several hundred km from the comet because it’s a larger vector value in relation to any perturbing gravitational accelerations. It’s a prograde burn in the same direction of travel away from 67P as for the parabola but with any necessary adjustments. Those adjustments are made after assessing that speed vector component away from 67P (which should be near zero) and the direction vector component which should be 90° to 67P’s orbital plane. In effect it would be a stand-alone burn from an almost stationary point in cometcentric space. But it would in effect be directed down a continuation of the parabola arm. There will be a celestial sphere reference point for this direction vector that the star trackers can lock onto for the burn. 

6) ASSUMPTIONS FOR INPUTS

1) The rotation period of 67P is assumed to be 12.7000hours. The four decimal places implies sub-second accuracy so as to be in keeping with other inputs like the Apis angle measured to three decimal places. This rotation period value may change slightly. It’s changing all the time but ESOC, with the help of JPL, are characterising these rotation period changes right up to the last day. Any change in rotation period can simply be fed into the calculations above and we watch it trickle through like a spreadsheet input and output. The outputs would change by very little and this flyby scenario would remain essentially unchanged. The output changes would point to where the intitial conditions needed tweaking, mainly the initial orbit height and delta v burn for flyby orbit injection (slowdown rocket burn to start the flyby ellipse).

2) The comet’s mass is assumed to be 1E1013 kg. That’s 1 x 10^13 kg or 10,000,000,000,000 kg. It’s actually been updated recently to 9.982E12 kg, which is 99.8% of the value above. But the calculations were already done with the old value before noticing this change. Again, trickle-down would barely change the output.

3) The surface of Apis is assumed to be at a radius of 2600 metres from the rotation axis. 2600 metres is therefore its radius of rotation. This is an informed guess based on measuring the shape model. It’s the least accurate of the estimated inputs. It could be 2500 metres or 2700 metres. Adjustment of the radius of rotation for Apis affects the size and eccentricity (fatness) of the flyby ellipse. This in turn affects the initial orbit height and delta v burn for flyby orbit injection. Again, the effect is minimal though noticeable when compared to the possible changes in (1) and (2). 

4) The west to east width of Apis is estimated at 400 metres. The flyby would have its ground track exactly along 67P’s equator, travelling west to east. Apis straddles the equator with the actual equator line just to one side (south) of a line that would bisect Apis. The stated midpoint of Apis is the midpoint of the section of Apis along which the equator line runs. It’s roughly the same width as the rest of Apis, which is a quasi rectangle. Rosetta crosses the width of that rectangle, not its length. Any adjustment to the Apis width would not affect the flyby ellipse calculations or timeline. It would simply lengthen or shorten the flyby. 

7) THE FLYBY SCENARIO DESCRIBED IN GREATER DETAIL

There’s a glossary in the appendix at the bottom (numbered “14”). If you find the terms used here difficult to follow, you’ll need to read that section first and then come back. Since this sub-heading is explaining the flyby in greater detail, it necessarily repeats what is presented in the shorter version. 

The point in the elliptical flyover orbit where Rosetta will pass over the midpoint of Apis is the periapsis point of that orbit. Since periapsis is a fixed point in space in a cometcentric reference frame (ignoring orbital speed around the sun) this point can be described as our target point. It’s the target for Rosetta to pass through and for the Apis midpoint to pass under together at exactly the same time. It will be called the zero point because it’s also designated as the point where the timeline is zero as well. All times before the zero point passing are called “T minus [time in hours]”. Times after passing through the zero point are called “T plus [time]”. So to summarise:

So the zero point is the periapsis of the flyby ellipse and it’s the point which Rosetta and Apis pass through (Rosetta) and 200 metres under (Apis midpoint) simultaneously.

T minus zero is the time at which Rosetta and the Apis midpoint fly/rotate together through and under the zero point respectively. 

The zero point is also used for a related measurement. It’s defined as the zero angle line for the midpoint of Apis as it rotates in its 360° circular path. As the midpoint passes under the zero point we can start counting 1°, 2°, 3°, 90°, 180°, 270° etc. It will be at 359.99°, 1.27 seconds before passing through the zero point again and starting a new cycle. Since two decimal places don’t get us under one-second accuracy, three decimal places are used so as to be in keeping with other calculations. Hence, as you’ll see below, Apis needs to be at 252.828° when flyby orbit injection is performed. The Apis midpoint angle is sometimes just called the Apis angle for brevity. 

The zero point is fixed in the celestial reference frame so a line from 67P’s centre of gravity to the zero point is permanently directed at a particular star or point on the celestial sphere, regardless of comet rotation or orbit speed/direction. The celestial reference frame is the heliocentric ecliptic reference frame as opposed to the Equatorial reference frame which is used by astronomers who routinely characterise 67P’s rotation axis angle in relation to the Earth’s Equatorial plane and not the ecliptic plane. All notions of Equatorial RA and declination should be dropped for this flyby and escape. Even the ecliptic plane can confuse somewhat because the key plane throughout this scenario is 67P’s own sun-centred orbital plane. And coming a very close second is its own equatorial plane (rotation plane) that is at 38° to its orbital plane. We’ll see soon that the flyby ellipse’s long axis gets locked along the intersection line of these two planes prior to the whole flyby scenario. This is of course entirely dependent on 67P’s orbital plane and rotation plane. 67P doesn’t care where the Earth is, what angle it’s tilted at or what plane it orbits in. It only cares about where the sun is and consequently, its own orbital plane, along with the angle to that plane in which it chooses to rotate. To 67P, the Earth is just another rock like asteroids are for us. 67P’s orbital plane just happens to be at 7° to the ecliptic which is the orbital plane of our own rock so we use our orbital plane, the ecliptic plane, to define 67P’s plane. 

So even though we must start getting into the mindset of 67P’s orbital plane (and equatorial plane) as reigning supreme for this scenario, we do have to define it in some reference frame. And from that definition we will use the same reference frame for fixing the the zero point (periapsis) and the apoapsis. Once that’s done, we leave the ecliptic reference frame and return to the cometcentric reference frame for all our flyby/escape burns, safe in the knowledge that, behind the scenes, the flyby ellipse is configured correctly with respect to 67P’s orbital plane and equatorial plane and that the escape burns will fling Rosetta out at 90° to the orbital plane. That 90° vector puts Rosetta into her own separate heliocentric hibernation orbit. But 67P’s orbital plane is wholly directing the escape vector. That’s because in order for Rosetta’s orbit to be heliocentric and also act as a quasi orbit of 67P she has to escape at exactly 90° to 67P’s orbital plane. 

Both 67P’s heliocentric orbital plane and Rosetta’s post-escape, heliocentric orbital plane have an inclination to the ecliptic plane (they’re virtually identical values around 7°) and the ecliptic plane is just a common plane to which they can both be referred to hence its being called a reference frame. The ecliptic is the 2D reference plane in the 3D ecliptic reference frame. The ecliptic plane is the x,y plane and adding the third axis, z, above and below the x,y plane, we have a 3D reference frame. It’s the preferred choice because all solar system orbiting bodies and their orbital elements are based on the ecliptic plane (specifically their orbital inclination and longitude of ascending node). 

The ecliptic reference frame is used simply to define the coordinates of the zero point (periapsis), the apoapsis, and consequently, the flyby ellipse’s major axis that runs between them. These are all locked onto 67P’s orbital plane. But this locking is done along the line where its equatorial (rotation) plane intersects that orbital plane. This intersection line is a fixed line in space. It is entirely dependent on what 67P is doing and is established for the flyby ellipse well in advance of the whole flyby scenario. Since it’s fixed in space, it remains fixed throughout the flyby scenario, which is why we can call the zero point a fixed point as well. It’s because the zero point can be placed on this line, 2800 metres from 67P’s centre of gravity. The coordinates of the line get measured in the ecliptic reference frame and that measurement amounts to a fixed celestial coordinate value on the celestial sphere in that reference frame (or two coordinate values, one for each end of the line, produced to infinity). The zero point can then be fixed in space, 2800 metres along the plane intersection line from 67P’s centre of gravity. Then we place the apoapsis 12,742 metres along the intersection line, on the other side of the centre of gravity. 12,742 metres is the length of the flyby ellipse (its major axis) so this second point is its apoapsis. The centre of gravity is on the major axis by definition anyway, so the major axis of the flyby ellipse therefore gets locked exactly along our intersection line of 67P’s orbit plane and equatorial plane. 

At that point we can return to the cometcentric reference frame using the zero point as our periapsis target for Apis flyby and the flyby ellipse plane as our main reference plane. Because we fixed the flyby ellipse major axis on the plane intersection line from the outset, we know that the delta v burns we perform in this reference frame will eventually lead to Rosetta escaping at 90° to 67P’s orbital plane. 

Since the flyby ellipse is in the rotation plane of the comet and the rotation plane is the same as the equatorial plane, the reference plane for flyby is the equatorial plane. As mentioned above, angular measurements are made around the flyby ellipse from the zero point as 0° and the centre of gravity as the origin. 

Of course, 67P is moving along its orbit all this time with the equatorial plane intersection line sliding along the orbital plane but when projected onto the celestial sphere which is considered as infinitely far away, there’s no change in the intersection line’s celestial coordinates. This is how we obtain a zero point that’s fixed in relation to the comet even though the comet and zero point are in reality, moving round 67P’s orbit. It’s a constructed cometcentric reference frame for the purposes of the flyby which simply strips out the heliocentric orbital speed of 67P’s centre of gravity, leaving just the excess speeds of Rosetta going round and round the centre of gravity. This is just like here on Earth where we have a geocentric reference frame for watching NEO’s fly past or changing the orbits of satellites. We’ve stripped out the Earth’s orbital speed of 30km/sec leaving just the relative excess speed and direction of the approaching NEO or orbiting satellites. 

There are actually two choices for the placement of the zero point along the line from the centre of gravity: 2800 metres in one direction or 2800 metres in the other. It doesn’t matter which. One choice will result in sending Rosetta exactly north of 67P’s orbital plane after escape. The other choice will send her exactly south. 

Whilst Rosetta could be sent north or south anyway via the use of inefficient brute force burns, we assume here that fuel is very low so this very careful pinpointing of the zero point/periapsis and apoapsis in advance minimises the delta v and therefore the fuel needed for the 52° plane change after flyby and prior to escape. With the major axis of the flyby ellipse (running between the two fixed apsides) being aligned along the plane intersection line, we ensure that both the 52° plane change burn and the parabolic escape burn are indeed performed with a minimum of delta v. This is because plane changes are best carried out at the lowest speed possible and apoapsis is where Rosetta is slowest. But plane changes are also best carried out at the node crossing of the two planes. So by placing the apoapsis on the node, which is another way of saying the plane intersection line, we kill two birds with one stone. 

By the same token, if the ellipse is aligned with its semimajor axis aligned along 67P’s orbital plane and equatorial plane intersection line, it means the minor vertices (midpoints on the ellipse between the two apsides) are uniquely placed. A tangent to the post flyby ellipse curve (after the 52° plane change) at those vertices runs exactly parallel to 67P’s orbital plane. It means the parabolic escape orbit can be performed here with the knowledge that the burn is at 90° to the required escape direction north/south of 67P’s orbital plane. Rosetta is already orbiting in a plane at 90° to 67P’s orbital plane after the plane change so there’s only one direction vector within that plane that needs fixing rather than any direction in 3D space. So by injecting to escape at the semiminor vertex, we ensure the escape isn’t simply in a plane at 90° to 67P’s orbital plane but in a direction that is completely orthogonal to it. The key is that the long arms of the parabola run at 90° to the apex of the parabolic curve. If Rosetta instigates the injection to parabolic escape when travelling parallel to 67P’s orbital plane, she is at the apex of the parabola she’s about to embark on. So she has to end up travelling at 90° to that parallel direction i.e. orthogonal to 67P’s orbital plane. The parabolic trajectory will take her through 67P’s orbital plane a few hours after parabolic injection and she’d end up travelling away on the opposite side of the plane from the side where injection took place. 

The zero point would have to be fixed somewhere precise before instigating flyby anyway so placing it and the apoapsis along the plane intersection line doesn’t cost us anything. 

Distances for the flyby are in metres and were calculated to two decimal places so as to be consistent with the precision to one second for the time stamps. However, they are rounded to the nearest metre for easy reading. The same goes for orbital speeds that would be to 6 decimal places for sub-second accuracy. This sort of precision won’t be possible in the real life scenario anyway but its easy enough to do so that any errors drift from values that are known to be exact to the second. Despite such precision, an error of 60 seconds wouldn’t compromise the flyby much and the rounded distances and speeds here imply between a 3- and 20-second error. Reinstating the extra decimal places returns us to one-second or sub-second accuracy. 

8) THE NUMBERED STAGES IN DETAIL

1) T minus 79.9810 hours: 

Rosetta is injected into a circular orbit of radius 9594 metres. This injection would be from whatever orbit Rosetta was on prior to this. At the time of writing she’s performing orbits that go through the 9km altitude point but these are probably ellipses. It would require somewhere around a maximum of 0.2 m/sec for the delta v burn to circularise at 9594 metres as she passes through that point, assuming the new orbit is coplanar with the former orbit. In this flyby scenario, there would be much preparatory work for several days and weeks, all performed in the extended equatorial plane which is the rotation plane. This work, involving characterising the gravity field, would mean insertion to circular orbit at 9594 metres would indeed be coplanar. That’s because, as stated above, the circular orbit and flyby ellipse are both in the extended equatorial plane.

The burn for circular orbit insertion is performed (burn shut-off time to be precise) at T minus 79.9810 hours. Since all calculations are done in seconds, the precision is to the second. Therefore, four decimal places are used for hours to bring the precision to less than a second. So the ‘0’ at the end isn’t superfluous- it could be a ‘1’ or a ‘3’ etc.. Precision doesn’t necessarily mean accuracy. Accuracy depends on other inputs (e.g. Apis radius), calibration and accuracy of measurements (e.g. Rosetta speed, Apis angle) and assumptions such as characterising the gravity field sufficiently. 

Crucially, the new circular orbit for Rosetta needs to be in the same plane as the 67P rotation plane (i.e. the extended equatorial plane) and be a prograde orbit, that is, with the comet’s rotation. 

The orbital speed is 0.2637 m/sec and is notionally constant because it’s a circular orbit. In reality it would vary a bit due to the gravitational anomalies. However it’s assumed these are largely characterised for the equatorial plane (see below). 

The Apis midpoint must be at 252.828° when the circular orbit injection is made and Rosetta has to be at 180°, i.e. watching Apis rotating ahead of her from that position at that moment. 

Even though the flyby orbit injection is 63.5 hours away, its zero point has been fixed already as being 180° away from this, the circular orbit injection point. The zero point is along a line from Rosetta at circular orbit injection, through 67P’s centre of gravity and at 2800 metres altitude on the other side. It’s a fixed point (in the cometcentric reference frame and as stated above, the celestial reference frame. Both reference frames strip out the movement/displacement effect of 67P’s orbital speed of circa 15km/sec. So the zero point is fixed, with Rosetta and the comet doing an elaborate dance through and under it for 78.9810 hours before they pass through and under it at the same time. 

The zero point is sited where 67P’s extended equatorial plane intersects its heliocentric orbital plane. Hence the zero point which is also the periapsis of the flyby orbit sits in 67P’s orbital plane. The apoapsis point at the other end of the ellipse is also specifically locked on the plane intersection too so it too is sitting in 67P’s orbital plane and that means the line that joins periapsis to apoapsis runs along 67P’s orbital plane too. This is the semimajor axis of the ellipse. This neat geometrical setup proves very useful for the escape injection burn later on. 

An orbit of 9594 metres (9.594km) has a period of 63.5 hours which is 5 times the 12.7-hour comet rotation period. This means that the comet rotates four times under Rosetta in one Rosetta orbit at that height. All altitudinal perturbations brought about by the changing comet configuration below Rosetta are thus symmetrical around the orbit. Rosetta should therefore return to the same 9594-metre height even if perturbed by a few metres up and down through the orbit. More importantly, she will arrive at the same point on the circle as where the circular orbit injection was made, 180° from the zero point. This is therefore also a fixed point in the cometcentric and celestial reference frames. When Rosetta passes through it, Apis will again be at 252.828°. Everything will be just as it was 63.5 hours before. That’s because the comet rotated exactly four times under Rosetta while she completed exactly one orbit. 

Metres instead of kilometres are used from now on because they are the SI unit in which the calculations are made. 

2) T minus 79.9810 hours to T minus 16.4810 hours:

Final checks. These include orbit height stability; final comet rotation rate analysis; Apis angle. 

The Apis angle can be ascertained from fiduciary points on the surface rotating beneath Rosetta. Small boulders were used for calculating the 67P rotation rate slowdown and speed-up before and after perihelion in 2015. This was done with sub-second accuracy, which means that the Apis angle can be measured to at least two decimal places if not the three used here for sub-second time precision consistency. 

As mentioned above, the Apis angle has to be at 252.828° at flyby orbit injection just as it needed to be when circular orbit injection was performed 63.5 hours earlier. The reason for this will become clear. 

Rosetta is injecting to flyby orbit at 180° from the zero point exactly. This is because the injection burn location is the apoapsis of the flyby orbit. Apoapsis is always 180° from periapsis- together they define the two ends of any orbital ellipse. We already know that the periapsis of our flyby ellipse is defined as the zero angle point for the Apis midpoint angle (as well as the zero point ‘target’ for flyby). So we can use the same 360° protractor for our flyby ellipse. Hence its apoapsis is at 180°, exactly opposite the periapsis zero point as you’d expect for any orbital ellipse. It’s the point from which Rosetta is dropped into the flyby ellipse (orbit) and it’s the highest point to which she will return after flyby. 

Since Rosetta is dropped from the circular orbit, the apoapsis is the same height as the circular orbit radius, 9594 metres. Since we’ve chosen a periapsis zero point before we even injected to the circular orbit way back at T minus 79.9810 hours, we must inject Rosetta into its flyby ellipse at the correct time and place. That place is the point at which we started with the original circular orbit injection burn. If we’re a few seconds late with flyby injection we’ll still overfly Apis but we won’t overfly the Apis midpoint as we fly through the periapsis zero point. The midpoint will arrive under the zero point ‘on time’ for Apis as only Apis can do. Rosetta will fly through the zero point late. This “few seconds late” scenario is referring to an actual unwanted error of however many seconds as opposed to the carefully reset zero time and zero point described further up if the Apis angle is found to be wrong. The reset is predicted and accounted for but an unwanted error might be due to say, still more uncharacterised remaining residuals in the Apis angle and altitude or the burn duration/shut-off time. 

Even though the later injection shunts the zero point round slightly beyond its planned position as well, Apis has a rotational angular velocity that’s five times that of Rosetta, orbiting at 9594 metres. So late injection at 9594 m translates to a painfully slow creep-round of the zero point 180° away while Apis races on under and beyond it. So a few seconds late with injection means Rosetta arrives at this shunted-round zero point late and then overflies the Apis midpoint a few seconds later. It will still reach its 200-metre altitude (just before the Apis centre point overfly) but it will be a few metres higher than need be on departure at the Apis eastern edge. None of that matters at all for a few seconds’ delay but it starts to matter when it’s over a minute and is rather unsatisfactory if it’s 5 minutes. If it’s 20 minutes, Rosetta will be surveying eastern Imhotep from 1000-metres-plus instead.

Each of the first three of four comet rotations under Rosetta during the 63.5-hour circular orbit is used as a dry run. The comet rotates 1.25 sidereal (celestial reference frame) rotations for every time it rotates under Rosetta. The apparent rotation period will be 16.6250 hours. Rosetta will be at 270°, 0°, and 90° each time the Apis midpoint is (supposed to be) in the correct angle position ahead. On each dry run, Apis should be 72.828° ahead of Rosetta’s ground track point along the equator (252.828°-180°). This can be checked by using a fiduciary boulder that should pass directly under Rosetta on each dry run. This is made slightly easier since Rosetta is orbiting in the same plane as the rotation plane and its ground track is the equator of the comet. The Apis midpoint is on the equator anyway (this is one of many reasons why Apis is a star candidate for flyby). The fiduciary boulder would also be on the equator. It would be 72.828° of comet longitude behind (west of) the Apis midpoint. 

When Rosetta reaches 180° again, where the circular orbit insertion was performed, it’s no longer a dry run but the real thing. If the Apis midpoint was seen to be creeping ahead or slipping behind 72.828° from Rosetta on the dry runs we can ‘go for burn’ a few seconds early or late. That would mean that our zero point that was supposed to be so rigidly fixed is itself shunted backwards or forwards a few metres or tens of metres. That’s the reset if it’s needed because Apis is found to be ahead/behind and dragging the aimed-for zero point with it. 

In contrast to the reset, the potential unwanted error at injection, described further above is an unwanted error even though Apis is in the predicted position. So an off-nominal burn when Apis is where we predicted it to be is the is error as opposed to the carefully planned reset. An off-nominal burn means a slightly off-nominal orbital speed which brings Rosetta in early/low or high/late. Late/early burn shut-off with the correct orbital speed means much the same thing but with lesser altitude anomalies. 

The reset requires the timeline to have a few seconds added to it or taken away. If Rosetta’s orbit altitude isn’t quite right on the dry run observations, a similar operation is done as for the Apis angle with minor adjustments of a few metres to the size of the flyby ellipse, which affects its orbital period. That would again mean going for burn a few seconds earlier or later. 

3) T minus 16.4810 hours (assuming no dry run adjustments or a reset, adjusting the timeline to 16.4810 hours anyway):

Flyby ellipse injection. This is 63.5 hours after circular orbit injection. The comet/Rosetta configuration is identical to the configuration at circular orbit insertion time at T minus 79.9810 hours. This is because 67P has rotated five times under that exact cometcentric point in space (5 x 12.7 =63.5). This is why the 9594-metre orbit radius was chosen. That’s the only radius that would bring Rosetta back to this same point after exactly five comet rotations (and four apparent rotations under Rosetta as she completes her single orbit). 

If all is well, Apis is at 252.828° again, and Rosetta is at 9594 metres again. 

If the height is correct and with the orbit being circular, the orbital speed is as predicted as well: 0.2637 m/sec. 

We know we want to pass over Apis at 200 metres’ altitude. That’s 2800 metres from the centre of gravity (c of g). 2600 metres for Apis radius of rotation plus 200 metres altitude of flyby makes 2800 metres. The c of g is the focus of the new ellipse, the flyby ellipse, just as it was the centre of the circular orbit. Circles are a special case ellipse with both foci merged at the centre so the circular orbit was technically orbiting the focus of an ellipse as well- and the c of g is where the focus is located in both cases. 

So at T minus zero, flyby midpoint, we want to be 2800 metres on the opposite side of 67P from where we are now. We’re now at 180°, 9594 metres, T minus 16.4810 hours and ready to inject to flyby. That 2800 metre point is of course the zero point, known and planned-for 63.5 hours ago. But now we’re thinking aloud, trying to work out what size and shape our flyby ellipse will have to be in order for Rosetta to fly through that zero point at all, let alone do it on time. 

We already know how far we are from the c of g, or focus, on this side of the comet: 9594 metres. We know the 2800m-radius zero point is diametrically opposite, through the c of g on a straight line (periapsis to apoapsis, 0° to 180°, through the c of g focus). So we just have to add 9594 to 2800 to get that length. That’s the long axis or major axis of the flyby ellipse. So that length is 12394 metres. Dividing in two gives us the semimajor axis, 6197 metres and that acts like a radius as an input for the orbital speed equation.

There’s only one ellipse that can have a 12394-metre major axis and a periapsis of 2800 metres. It has to be fat enough to satisfy those parameters and still be an ellipse (after all, you could fit a circle, which is a very fat ellipse, to those two parameters on graph paper but nothing would orbit along it in space). The fatness of the ellipse is the eccentricy and the eccentricity for this ellipse with this major axis and periapsis is 0.548. The semiminor axis is 5183 metres. The eccentricity and semiminor axis aren’t necessary as inputs but it shows we can define a viable orbital ellipse with just the two main parameters: the major axis 12,394m, and periapsis distance, 2800m.

We know the flyby ellipse apoapsis and the original circular orbit kiss at 9594 metres. They have to because that’s where the flyby orbit injection is performed and Rosetta starts ‘dropping’ away from its circular path into the required ellipse. The fact they kiss means that if you left Rosetta to orbit in that ellipse indefinitely, it would keep kissing the apoapsis on every orbit at 9594 metres’ altitude. It has to do this at some set orbital speed. The apoapsis speed is the slowest speed in the orbit. It’s calculated using just the 6197-metre semimajor axis value and the 9594-metre apoapsis value, along with the comet’s mass and the gravitational constant. That speed is 0.1773 m/sec. 

If 0.1773 m/sec is the speed at which Rosetta would always pass through apoapsis then that’s the speed we need her to have when the injection burn shuts off. That speed will send her into the correct ellipse for a 2800-metre periapsis and a 200-metre Apis flyby. So the delta v needed (the amount of slowdown) is the circular orbit speed minus the flyby ellipse speed at apoapsis (when the orbit is momentarily kissing the same 9594 radius as the circle). So delta v for flyby injection is 0.2637 m/sec – 0.1773 m/sec = 0.0864 m/sec. This is a retrograde burn (directed tangentially to the orbit line and prograde (forwards)- see glossary). 

So now Rosetta has been injected onto her flyby ellipse and the retrograde burn was shut off when Apis was at 252.828°. The shutdown was at T minus 16.4810 hours. Rosetta is at 180°, Apis is 72.828° ahead of her in the rotational reference frame as measured from the zero point (180° + 72.828° = 253.828°). Apis is also rotating with an angular velocity 7.43 times that of Rosetta when orbiting at 0.1773 m/sec at 9594m and there’s no way Rosetta is going to catch up with it. So that looks like big a problem but it’s not. 

The flyby ellipse has an orbital period of 32.9620 hours. We can derive this from the semimajor axis of 6197 metres. Therefore half an orbit is 16.4810 hours. That’s why the flyby injection was performed at T minus 16.4810 hours. It was already known that it would take that long from orbit injection to the flyby zero point, half the ellipse, apoapsis to periapsis. 67P rotates at 12.7 hours per full rotation (that should technically read 12.7000 hours to the four decimal places to show its sub-second precision but it’s a bit pedantic to keep writing it. So if Apis was at the zero point when orbit injection was performed, it would be 3.7810 hours ahead of the zero point when Rosetta flew through after 16.4810 hours of flyby approach. Apis needs to be wound back past the zero point by that same amount, 3.7810 hours so that it is rotating for 12.7 plus 3.7810 hours before passing through the zero point. That means 16.4810 hours of rotation for Apis and 16.4810 hours of orbiting for Rosetta prior to passing through the zero point. So they both pass through together. The 3.7810-hour winding-back for Apis can be expressed as an angle. One full comet rotation takes 12.7 hours. 3.7810 hours as an angle is 3.7810/12.7 x 360° = 107.172°. But this is wind-back in the opposite direction from the angle measurement that was always with the direction of orbit or rotation. When measured in the proper, conventional way it’s an angle of 360°- 107.172° = 252.828°. That angle is measured from the zero point in the direction of rotation and of course was already known 79.9180 hours ago at circular orbit injection. 

So it simply means that at flyby orbit injection, Apis is ahead of Rosetta because it still has to rotate 107.172° to the zero point and then another full 360°. Rosetta only has to do half an orbit on its ellipse. That way, they pass through the zero point together, 16.4810 hours after flyby injection. 

4) T plus 27.5960 hours (timed at burn shut-off at the semiminor axis vertex at orbit radius, 5183 metres):

Inject to parabolic ‘escape’. This was described above and though it’s repeated here, it’s described in a subtly different way which may aid the understanding of the parabolic injection burn. Delta v for the burn is is 0.1204 m/sec prograde i.e. directed backwards along a tangent to the retrograde orbit line. This gives rise to an increase in orbital speed from 0.3871 m/sec to 0.5075 m/sec, escape speed for this altitude. Escape was put in inverted commas further above because a parabola puts Rosetta on the cusp of escape only. A parabola is used for escape for a reason, instead of a hyperbola which implies excess speed for true escape. That reason is the unique properties of its two open ends which are parallel at infinity (and are so, in effect, at 200-300 km from 67P). Hyperbolas’ arms aren’t parallel and fan out by definition. Those arms, when at some distance from the perturbing body, are essentially straight lines called the asymptotes. If Rosetta were sent along one of them, she’d end up travelling away from 67P at less than 90° to 67P’s orbital plane around the sun. This is not permitted for easy return to 67P in 2021. 

Since Rosetta is now orbiting in a plane that’s at 90° to 67P’s heliocentric orbital plane we can exploit this property of the parabola having parallel arms. At the very beginning, we chose both the zero point and the apoapsis to be sited on the intersection line of 67P’s orbital plane and its equatorial plane. That meant both the apsides (periapsis and apoapsis) sat on that plane intersection line. So it also means they both sit on 67P’s orbital plane. They are the semimajor vertices, and their positioning has implications for this semiminor vertex we’re currently at for the parabolic injection burn. Since they both sit on 67P’s orbital plane, it means the long axis of Rosetta’s flyby ellipse and of this new (52° plane changed) ellipse also sits on that plane. The long axis is the semimajor axis. So when Rosetta went through apoapsis on her inclination change burn, the 52° adjustment meant she had to end up travelling at 90° to 67P’s orbital plane. That in turn means this semiminor vertex where the parabolic escape burn is performed is running exactly parallel to 67P’s orbital plane around the sun. Or to be exact a tangent to the orbital ellipse at the vertex point is parallel to 67P’s orbital plane. Now, the new parabola that Rosetta is sent on by the burn, wraps round 67P with the burn point at the apex of the turn. The burn point or semiminor vertex is the point of symmetry for the parabolic curve. A tangent to the parabola curve at this point is at 90° to the two infinitely long, parallel arms. Such a tangent is also parallel to the orbital plane of 67P. Thus the parabola arms are both at 90° to 67P’s orbital plane. Rosetta is immediately set on a path to travel down one of the arms after the burn. Thus, we can be absolutely sure that Rosetta ends up travelling away from 67P on a trajectory that’s at exactly 90° (orthogonal) to 67P’s orbital plane. Only when she’s 200-300 km away down that arm would we take stock, see that she is indeed travelling at 90° and give her a tiny push of 3cm/sec to be sure of escape. If travelling at 88° or 89°, the escape burn of 3cm/sec would get a minuscule sideways vector to correct this. This puts Rosetta on a truly heliocentric orbit which is nevertheless a quasi orbit of 67P. She will, in theory, return to this point above 67P four years and three months later in January 2021- and arrive at 3cm/sec.

5) T plus 300 hours. This time stamp is somewhat arbitrary but it’s at a point where flyby data transmission is hopefully complete and the parabola arm location of Rosetta is now far enough from 67P to be pointing away at effectively 90° to 67P’s orbital plane around the sun. The following narrative for the T plus 300 hours time stamp is just a copy and paste from the simple version of the timeline above. It has no elaborations like 1 to 4 have:

Escape injection. A delta v burn of 0.03 m/sec is performed. It needs to be the slowest possible speed that ensures escape with no unintended return, decaying orbit and eventual crash during hibernation. That’s why 3 cm/sec is chosen. It’s much easier to get it right when several hundred km from the comet because it’s a larger vector value in relation to any perturbing gravitational accelerations. It’s a prograde burn in the same direction of travel away from 67P as for the parabola but with any necessary adjustments. Those adjustments are made after assessing that speed vector component away from 67P (which should be near zero) and the direction vector component which should be 90° to 67P’s orbital plane. In effect it would be a stand-alone burn from an almost stationary point in cometcentric space. But it would in effect be directed down a continuation of the parabola arm. There will be a celestial sphere reference point for this direction vector that the star trackers can lock onto for the burn. 

9) RELATIVE SPEEDS DURING FLYBY AND FLYBY DURATION. 

Apis has a tangential speed of 0.3574 m/sec. That’s because it rotates in a circle of 2piR in 12.7 hours. 2 pi times 2600m is 16338.4 metres. 12.7 hours is 45,720 seconds. Speed is distance over time which is 16338.4/45,720 = 0.3574 metres per second. 

The orbital speed of Rosetta along its flyby ellipse at periapsis is 0.6075 m/sec (quite a lot faster than the 0.1773 m/sec at apoapsis due to falling deeper into 67P’s gravity well). The difference between the Apis tangential speed and the orbital speed of Rosetta is 0.6075 – 0.3574 = 0.2501 m/sec. Apis is about 400 metres wide from east to west. Rosetta is orbiting exactly along and over the equator because if you recall, the original circular orbit was in the same plane as 67P’s plane of rotation, which is the extended equatorial plane. So is the flyby orbit because the flyby injection was directed forwards (prograde) within the circular plane- so there was no sideways push to change planes. That means Rosetta has to come in over and along the equator from west to east, flying over Imhotep. The equator runs through and across Apis from west to east as well, of course. So Rosetta traverses the 400-metre width of Apis at 0.2501 m/sec. This makes the flyby time 1599 seconds, or 26 minutes and 39 seconds. And it should take place at just above 200 metres’ altitude all the way, kissing 200 metres’ altitude at the mid point. It’s just above 200 metres either side because Rosetta’s flyby ellipse is fatter than the rotation circle of Apis and therefore fatter than the rotation circle of the 200 metre altitude point as well. The circle kisses the ellipse periapsis point, i.e. nested inside one end of the flyby ellipse. It kisses that one point only and the ellipse diverges either side of it, but not by much when considering such a small distance as the 400-metre Apis ground track on a 12km long and 0.548 eccentricity ellipse. In reality it’s over 900 metres of circle/ellipse nesting due to the rotation movement of Apis over 1599 seconds. But it’s still essentially just over a 200-metre-altitude flyby all the way. 

200 metres was chosen because if there were a small error in the delta v burn to flyby injection or there were unresolved gravitational anomalies, it would hopefully not cause an error of more than 200 metres in altitude so there would just be an even closer flyby and all data would still be sent back after escape. Besides, if there were such a negative altitude anomaly, it would also speed up the approach time (because of being a smaller ellipse). Rosetta would arrive early and overfly Atum. Atum’s radius of rotation is somewhat less than that of Apis, stuck out on the long axis tip as Apis is, so we’d be safe anyway. The opposite scenario of a positive altitude anomaly means Rosetta would be safe anyway. She’d arrive late and overfly eastern Imhotep at an even higher altitude because Imhotep drops away too. This is yet another advantage of targeting Apis: negative altitude anomalies from the nominal 200 metres translate into early arrival and comfortable flyby altitudes as the terrain height drops away from Apis anyway. And positive altitude anomalies can’t translate into a crash risk because Apis has the highest radius of rotation of the whole comet. 

In reality, flyby would be a few tens of seconds longer because Rosetta would be just a tad slower than 0.6075 m/sec at the beginning and end of the flyby. The 0.6075 m/sec applies only for an instant at the zero point. 13 minutes before zero point and 13 minutes afterwards Rosetta’s speed would still probably be just above 0.6000 m/sec though. 

Apis would rotate through 12.590° during the 26 minute 39 second flyby. As mentioned above, the circle it described would nest quite well with the tighter curve of the flyby orbit ellipse at periapsis, keeping the two curves somewhat close to parallel and minimising Rosetta’s altitude above the nominal 200m lowest point when either side of it. 

10) THE DELTA V BUDGET FOR THE APIS FLYBY AND ESCAPE

Flyby orbit injection: 0.0864 m/sec.

Plane change burn at apoapsis and T plus 16.4180 hours: 0.1554 m/sec.

Parabolic injection burn from semiminor vertex: 0.1204 m/sec.

Full escape burn along parabola arm: 0.0300m/sec.

Total: 0.3922 m/sec. 

To this, we could add about 0.1500 m/sec for any preliminary burn for the initial circular orbit insertion, depending on where Rosetta had come from. This would increase the total to 0.5422 m/sec. With about 0.45 m/sec for the best reacquisition scenario in 2021, the entire budget remains at below 1 m/sec. 

11) AFTER ESCAPE: HIBERNATION AND RETURN VIA THE QUASI ORBIT.

0.03 m/sec is 3 centimetres per second. This would still translate to a few thousand kilometres of drift over the 4-year hibernation period. However, with the escape burn performed orthogonal, either north or south with reference to 67P’s orbital plane, Rosetta could be teased into being a quasi-orbiting satellite of 67P. This would greatly minimise the delta v necessary for return to 67P in four years’ time. 

The key thing to remember here is that when any perturbation is made on a small body by another larger one, the smaller one always, in theory, crosses back through the point where it was perturbed. In the case of NEO’s it’s the node where their orbit crosses the Earth’s orbit and if the Earth happens to be near that point in its orbit at the same time, the NEO will be perturbed into a new ellipse. But the new ellipse still goes through the node and the NEO will keep passing through the node, once per orbit. It has a memory for that point because that’s where its newly perturbed orbit started. There are glaring exceptions to this, the main one being later perturbations by Venus, Mars and Jupiter. These remote perturbations (remote from the node) cause the otherwise faithful node crossing to drift. 

In the case of Rosetta, 67P mimics the perturbing body and Rosetta is mimicking a perturbed body. In reality, the escape burn is doing most of the apparent perturbing as if Rosetta was on nearly the same heliocentric orbit as 67P and 67P perturbed her by 3 cm per second. Wherever it is along the shared heliocentric orbit of 67P and Rosetta that the escape burn is performed, that is the node where their two heliocentric orbits start diverging and will cross next time round. In the case of Rosetta, the burn to escape means 67P no longer affects her, only the sun. She’s in a very slightly inclined heliocentric orbit from 67P and those two orbits cross at the node. If Rosetta sails away at 3 cm per second and does so directly north or south of 67P’s orbital plane on September 30th 2016, then she would be drifting towards this same perturbation point in 6.4 years’ time after one full orbit. She’d be approaching directly from the south of 67P’s orbital plane if she left directly northwards, and vice versa if she had left directly southwards. Of course, 67P would be approaching the perturbation point as well, to cross at the same time, because it’s performed an identical ellipse around the sun to Rosetta’s slightly inclined version of the same orbital ellipse. 

Now, hibernation would be for around four years, not 6.4 years. That takes us to September 2020 and 67P/Rosetta will be on the other side of their virtually identical orbital ellipses. Three months after that, their two heliocentric orbits cross a second time. They cross at the diametrically opposite crossing to their September 2016 node. In those three months, Rosetta would be drifting towards 67P from directly above or below its orbital plane and doing so at 3cm per second. She’d be only 500km away at the start point, in theory, but probably a few thousand km away in practice due to escape velocity residuals. That crossover point, reached in January 2021 just happens to be the exact position in the orbit where Rosetta acquired 67P in 2014. This is the sweet spot for comet acquisition as we know from last time around and 67P/Rosetta would be travelling back towards the sun. 

So it’s simply a happy coincidence that the 2021 comet acquisition sweet spot is where the two orbits cross back over and Rosetta is passing straight back past 67P to embark on the half of her orbit that’s on the other side of 67P’s orbital plane. As stated above, they would both be on the opposite side of their heliocentric orbital ellipses to the man-made node due to the escape burn. However, this is the opposite node and is defined by the point where the two orbital ellipses cross back over on the opposite side. There are two nodes: an ascending node and a descending node. They’re diametrically opposite in the orbit. 

If Rosetta is sent exactly north or south of 67P’s orbital plane she executes an identical ellipse (same semimajor axis, same eccentricity) around the sun and in an identical time. The only difference is that Rosetta’s orbit has a tiny inclination to 67P’s. It crosses 67P’s two nodes where the orbital planes of both bodies cross. From 67P’s point of view, Rosetta appears to be orbiting in a highly elliptical orbit with 67P in the middle, not at one focus as would be the case if Rosetta were truly orbiting. The ellipse would extend about 2000km above and below 67P. In theory, it would be an almost a straight line, straight up above the 67P orbital plane at 90° for about two years and then tracing the same path all the way back again for another two years or so. If that happened, Rosetta would fly straight past the comet at the opposite node, in January 2021, right on the acquisition sweet spot. But the tiniest errors in escape delta v (speed and direction residuals) would turn this into hundreds of kilometres at least. Jupiter’s differential gravitational acceleration adds up to 2000km more but probably somewhat less. These are all small distance anomalies though. It’s why Rosetta would be woken up in at least September 2020 after four years so as to take her bearings, asses the distance anomaly and do an acquisition delta v burn. If woken early enough, the burn would need to be very small, 0.2 m/sec for a 3000km distance anomaly. This saves on delta v budget which means fuel. 

The escape vector has to be exactly north or south of 67P’s orbital plane. If it’s angled at all, it throws Rosetta outwards from or inwards to the sun. This changes the orbital eccentricity. This wouldn’t matter if a) they were meeting back up at the perturbation node 6.4 years later and b) the semimajor axis was the same (allows same orbital period) But for meeting up at the opposite node, even if the orbital period is the same, the eccentricity difference fattens the ellipse from the original node point and Rosetta finds herself thousands of kilometres inside or outside 67P’s orbit. 

The only way to get the quasi orbit we want is with Rosetta approaching 67P in September 2020 from exactly above or below 67P’s orbital plane. That approach vector is itself a signature that Rosetta is in an identical ellipse around the sun, with an identical period but with an orbital plane that’s at a slight inclination to 67P’s orbital plane. She would approach from above if she’d been sent above 67P’s orbital plane in September 2016 and she’d approach from below if sent below in September 2016. In both cases, she would have spent half an orbit above or below the 67P orbital plane. 

One easy part in this scenario is that it doesn’t matter all that much how fast the escape is so long as it’s directly north or south of 67P’s orbital plane. The faster it rises from the plane, the faster it will return on the other side of the orbit in mirror fashion. It won’t change the ellipse size or shape or its orbital period, just its inclination to 67P’s orbit. However, a faster escape and consequent return, say 0.1 or 0.2 m/sec does mean more delta v needed to slow it down on return. It also means that Jupiter can tease them even further apart if their initial separation is bigger. That’s why 3 cm per second was chosen. It reduces the possibility that Rosetta ends up 30,000km away on wake up and having to give her a 2 m/sec delta v just to get her back to the comet in six months and then another 2 m/sec to slow her down. This worst case eventuality is actually an argument for wake up a little earlier in May or June 2020. This would place Rosetta at the same heliocentric orbital position as for the 2014 wake up. 

With the quasi orbit, Rosetta might be close enough to 67P to do some useful science without even being given delta v. That’s if it was low on fuel and needed reserves for attitude control to point the antenna at the Earth and then turn back to 67P repeatedly. 

12) FLYBY VERSUS CRASH MEANS MORE TRANSMISSION OF DATA.

If the dare-devil flyby gets overly perturbed by the uneven gravity field and crashes, we lose nothing. It won’t be any different from the controlled crash scenario. It will gather and transmit data right up to the point of crashing and transmit no data thereafter. However, I suspect the Rosetta transmitter can’t transmit, in real time, all the data that could be collected before crashing. It has to do it in real time because the mission scientists have ruled out transmission after the crash. I’m not sure, but the low transmission bit rate in September must be a lot less than the data that could be recorded and stored over that last hour or so and then transmitted later over a period of days or weeks. The Deep Space Network (receiving dishes) will be busy with other missions in October, not least of all because Rosetta hasn’t asked for any time on it after the crash. But if it’s just the last last hour or so of very useful close-up data with as many instruments as possible taking data, surely it would be possible to send it back in October and November. It would just be sent in dribs and drabs after the flyby scenario outlined below. That’s what New Horizons is still doing: it’s taking it 16 months to send back a few hours’ worth of data. 

13) CONCLUSION

This post concerns itself with the nuts and bolts of the flyby, orbital elements, orbital speeds and delta v burns etc. The twin post expounds on the numerous advantages of opting for a flyby rather than a crash and explains why Apis is a star candidate for that flyby. Its credentials run to a long list that doesn’t just focus on its status as being probably the least processed region on the comet and therefore the closest to being primordial. However, the primordial surface argument, that can only be arrived at by invoking stretch theory, is the most pressing reason to go for the Apis flyby. 

14) APPENDIX- GLOSSARY

This glossary, like the keys to the photos in other parts, is a narrative glossary. It complements the overall explanation outlined above.

Vector- for a velocity vector it’s the speed and direction as opposed to just the speed. For a force vector, e.g. a rocket thrust, it’s just the direction. Velocity is used when a change of orbital speed is brought about by a rocket burn because the change can be faster or slower implying an adding in the forward direction or a subtracting in the negative direction. However, often just ‘speed’ is stated to denote the orbital speed. This is because its direction isn’t needed in many instances. It’s just the prograde speed of Rosetta on its orbit at that point of the orbit and instant in time. It still has a direction of course and we would soon be talking of velocity if we were on a collision course with the comet. The idea it to just miss it though. 

Prograde- forwards direction along the orbit line (strictly speaking, the tangent to the curved orbit line at a particular point on the orbit and instant in time). 

Retrograde- backwards direction along the orbit line (strictly, a tangent too).

Prograde burn- a rocket engine boost to attain an increase in velocity along the direction of the orbit line. The thrust vector (direction) for a prograde burn is backwards along the retrograde line.

Retrograde burn- a rocket engine boost to attain a decrease in velocity along the direction of the orbit line. The thrust vector for a retrograde burn is forwards along the prograde line so as to slow down the orbiter. This fancy talk of vectors and thrust is no different from what you have to do when pushing your loaded trolley round the supermarket. And the speeds along the orbit (relative to the comet) are identical to trolley speeds: 20 to 70 centimetres per second. The velocity changes to speed up and slow down are no different either. In both cases, it involves a force acting for a finite period over a short distance to bring about a change in velocity. 

Flyby Injection burn- this is the retrograde burn needed to slow Rosetta enough for 67P’s gravity to pull it into an elliptical orbit from a circular starting orbit. The burn ‘injects’ it from the circle into the ellipse. It in effect slows Rosetta enough to make her start dropping towards 67P but since there’s a residual element of speed that’s tangential to the radius at that point (the residual prograde speed along the circular orbit), Rosetta can’t drop vertically. She’s drawn into an ellipse. Since the vertical component of gravitational acceleration comes to dominate the shape of the orbit, the residual tangential speed has less influence than it did for a circular orbit. That’s how the circle gets distorted into an ellipse. Notice the word “trajectory” is hardly used at all here or anywhere else in the blog. It’s woolly and tells us nothing about the shape of the path of Rosetta unless preceded with “elliptical”, “circular”, “parabolic” “hyperbolic” etc. These are all curved paths of one sort or another and all are types of orbit. As soon as we use the word ‘trajectory’, we think in terms of a start point, an end point and some path, whatever path, usually straight, between the two. By visualising the elliptical path of Rosetta you can start to see that as it approaches 67P the increasingly curved shape of the ellipse almost meshes with the circular path of the rotating Apis section of crust. If we talk vaguely of the ‘flyby trajectory’, this subtlety will go right over our heads.  

Escape injection burn- this is the prograde burn needed to increase Rosetta’s prograde orbital speed to escape speed. In the scenario outlined above, it’s actually the parabolic escape burn, sending Rosetta down one of the parabola’s arms and only to the cusp of escape. It would be executed 27.5960 hours after the Apis flyby and occur at the semiminor vertex of the newly inclined version of the flyby orbit ellipse. Rosetta needs to escape 67P in order for Marco Parigi’s hibernation scenario and subsequent return to take place. Rosetta is given a 0.03 m/sec surplus speed in the escape injection described here. That’s just three centimetres per second. That means it doesn’t drift too far from 67P during hibernation and can still return via the quasi orbit. If she orbits 67P in a true, conventional orbit during hibernation, the orbit may decay and cause Rosetta to crash on the surface. 

Delta v- the increase or decrease in velocity attained due to prograde or retrograde burns. Velocity is used because it has direction, either forwards or backwards. The backwards scenario would be because a decrease in velocity is a negative (backwards) change. Often, sideways delta v’s occur to take the orbit out of plane with the current orbital plane (a non-coplanar transfer burn) as was the case for the 52° inclination change delta v at apoapsis, described above. 

Delta v budget- the sum of all delta v’s both positive and negative (prograde and retrograde burns) for a particular orbit change scenario or set of changes over time. Delta v budget is important because it’s constrained by available fuel supplies. For this scenario there are two delta v budgets, one for the Apis close flyby, including the inclination change burn and two escape burns after that. Then there’s another budget for reacquisition in 2021. This is nominally a single burn to capture Rosetta in a true orbit around 67P assuming she flies straight back into its arms. However this is so unlikely as to be dismissed. It’s assumed Rosetta will have undergone at least 3,000 km of drift and require 0.2 m/sec delta v to get back in 3 months along with a 0.24 m/sec retro delta v to go into orbit. The two budgets, September 2016 budget and September 2020 to January 2021 budget are added to make one budget. That’s because the flyby is a stand-alone proposition which needs a certain amount of fuel. The reacquisition is another bonus scenario, a rather long shot attempt at waking up Rosetta. Once woken up, it can’t do much without the enough fuel for the reacquisition delta v budget. Both budgets are around 0.5 metres per second, total 1 metre per second. 

“c of g” – this is used for brevity for the the centre of gravity of 67P. It’s also the centre of rotation i.e. central along the rotation axis. So all radii of surface features such as Apis are measured from it. 

Radius of rotation- this is the same as the radius described above. It’s just a reminder that mentioning radius in this blog usually immediately leads on to rotation and the tangential speed of that surface feature. For Apis, the radius of rotation is estimated at 2600 metres from the shape model dimensions. Its tangential speed is therefore 0.357 metres per second, just about the fastest value for anywhere on the comet, which makes the relative flyby speed more sedate. That’s obviously a big advantage, one of several for targeting Apis. If its radius of rotation is actually 2500 or 2700 metres, we just change the 2600 metre input here and watch the output change by a tiny amount. The scenario remains unchanged in its basic principle. The outputs that would change are periapsis/apoapsis distances (see next two glossary entries below), the size and eccentricity of the ellipse, the orbital speed at different points round the orbit, the orbital period, and the delta v’s required. But they’d all change by very small amounts for a 100-metre change in the rotation radius of Apis, or for that matter, for a 10 to 20-minute adjustment in 67P’s rotation rate. The rotation rate has been seen to vary by that much already over a few months. 

Periapsis- the closest point of an elliptical orbit to the comet’s c of g. The c of g of the comet is also the focus of the ellipse that defines the orbit. Satellites always orbit around one of the foci of the ellipse because that’s the point where the gravitational force is directed towards throughout the orbit (being the c of g, of course). We’ll be constructing an elliptical orbit for Rosetta with a 2800-metre periapsis. What we want to do here is make Apis pass right under that 2800-metre periapsis point exactly when Rosetta is going through periapsis as well. That gives us a 200-metre-high flyby. That assumes my informed guess that Apis is 2600 metres from the c of g is correct. It’s important here to imagine the periapsis as one end of the orbital ellipse we construct and also as a target point for Rosetta to arrive at and the centre of Apis to arrive under at the same time. If we strip out the motion of 67P around the sun we have a comet-centric view of the orbit. That way the periapsis can be seen as a point in space that stays locked, just as the ellipse it’s on is locked. The periapsis point can then also be used as a zero point for measuring the angle of Apis from the target (so the angle zero point, the flyby target and the periapsis point are all one and the same thing). Apis rotates in a perfect circle of course, 360°, and passes under Rosetta’s proposed periapsis point every 12.7 hours. We’ll see that Rosetta acquires the western edge of Apis when Apis is about 6° or 13 minutes from the periapsis zero point. Rosetta passes over the centre of Apis when they both go through the periapsis zero point (so they both hit the target together as planned). Rosetta then crosses the eastern edge of Apis when Apis is 6° past the zero point. Rosetta burns to escape a few minutes later. 

Apoapsis- the farthest point of an elliptical orbit from the c of g. This is an important point too because it’s the point from which the Apis flyby ellipse is started. It’s the the same radius as the circle Rosetta had been orbiting before the retrograde burn slowed her orbital velocity. This negative delta v means Rosetta slows down along its circular orbital path. That in turn means the downward pull from 67P’s gravity overcomes the ‘centrifugal’ force of the former circular path speed. Rosetta immediately drops away from the circular path and assumes an elliptical path that stays inside the circle. But the ellipse kisses the circle at its highest point from 67P because that’s where the burn happened and so that’s where it started dropping away. That point is the apoapsis. It’s fixed like the periapsis and is exactly 180° opposite the periapsis, at the other end of the ellipse. 

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. 

Part 58- The Case For a Dare-Devil Flyby of Apis


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

CONTENTS
1) Introduction 
2) The crash avoidance trick
3) Data acquisition, orbital speed and ground track speed
4) Objections to the escape, hibernation and reacquisition scenario
5 to 10) Gravitational anomaly characterisation [various headings, not required reading]. 
11) Rotation plane approach makes it a 2D problem
12) Proving the wobble
13) Advantages of rotation plane approach
14) Transmission of data
15) Conclusion
16) Appendix 1- more on the paleo rotation plane affecting gravitational anomalies
17) Appendix 2- the sounding orbits

SHAPE MODEL VIDEOS FOR THIS POST
Credits: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA

Here’s a link to ESA’s rotating shape model of 67P from which the still above is taken. 

http://sci.esa.int/rosetta/54728-shape-model-of-comet-67p/


Credits: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA

This second still is from another video which shows the rotation more head-on (looking down the rotation plane) which is key to this post and the flyby’s rotation plane approach. However, it’s just one rotation, not continuous. 

1) INTRODUCTION

The Rosetta mission scientists have decided to land the Rosetta orbiter on 67P on 30th September 2016, on the head lobe at Ma’at. It will take data at unprecedented low altitudes. However, after landing there’s no realistic chance of any data return so Rosetta can transmit its data only during the descent. The Apis flyby proposed here gets very low as well, lasts much longer and includes a low-delta-v escape burn to transmit more data back to Earth. This then leaves the slim chance that the orbiter can return in 2020-21 to do more science. 

The next part, Part 59, will lay out a scenario for Rosetta to perform a dare-devil flyby of Apis on September 30th 2016 instead of crash-landing on 67P and sending back no data thereafter. This part is a preparatory post which lays out the advantages of the Apis flyby while addressing certain criticisms that might be raised. Otherwise, the flyby post might look rather naive, as if these things hadn’t been thought of. It’s too much to put in one post hence this being a preparatory post. If you want to skip straight to the flyby post (click next above), you don’t have to read this post. But if you have objections regarding anything such as solar panel power generation during the higher aphelion hibernation phase till 2020 or characterisation of the gravitational anomalies arising from the duck shape, they’re addresed here. 

This post should be regarded as a one-stop shop for the nuts and bolts behind the scenes for the flyby. It isn’t essential reading but you may want to delve into certain areas of interest, hence the list of contents at the top. 

The flyby lasts 27 minutes for the 200-metre altitude part across the 400-metre width of Apis. Data collection an hour either side of closest approach would still be taken at or below the previous closest approach distance ever attained from the comet. That previous record will be 1 kilometre from the surface, achieved in the preceding weeks. 

The flyby description will include the relevant orbital elements and delta v burns required for both the flyby ellipse and the orbit from which the flyby is injected. Delta v burns are velocity changes via rocket burns. The timeline includes burn to escape after flyby in preparation for flyby data transmission. 

After escape from 67P, Rosetta would go into hibernation and return in 2020-2021. The characterisation of the burn for the hibernation orbit is also addressed. It’s a heliocentric orbit but the burn to get into that orbit is done on one of two particular vectors that would result in Rosetta being in a quasi orbit of 67P at the same time as being in a heliocentric orbit. This ensures automatic return in 2020, which minimises delta v, saves fuel and keeps the 67P reacquisition time to within a 3 to 6 month window. That’s the window between hibernation wake-up and the ideal comet acquisition point in the 67P orbit. This corresponds to the same portion of its orbit that was traversed during the January to August 2014 wake up and acquisition.

2) THE CRASH AVOIDANCE TRICK

Glossary for this section:

Apoapsis- furthest orbital distance from 67P along any chosen orbital ellipse. 

Periapsis- closest orbital distance from 67P along any chosen orbital ellipse. 
Apoapsis and periapsis are at opposite ends of the long axis (major axis) of the orbital ellipse. So they’re at either end of the oval with 67P tucked in on the inside of the oval at the the periapsis end. That’s useful for visualising what follows and for the next part. 

One reason for this preliminary post is that the shape of 67P means that both the landing being planned and the suggested flyby approach are fraught with difficulties. This is owing to the rotating duck shape causing all manner of gravitational acceleration anomalies on approach. These problems are discussed below. It has to be stressed that the flyby scenario outlined in the next post is the nominal version based on 67P’s gravity field being even, as it would be if it were a sphere and its centre of gravity were acting as the point source average for spheres at all times. The anomalies have to be characterised and the resulting gravitational field values used to adjust the nominal flyby calculations. So, at first glance, the flyby calculations would seem rather naive and optimistic but they would be nominal and awaiting these adjustments. 

The different types of gravitational anomaly are identified here as well as the reasons for their existence and how to characterise them, Although I’m sure ESOC can manage this for themselves it’s an interesting exercise that sheds light on just why it is that rotation plane approach (through the extended equatorial plane) is so advantageous. 

It may still seem optimistic to go for a 200-metre flyby once the nominal scenario has been updated to account for the gravitational anomalies. However, the flyby target, Apis, is unique in that it’s forgiving if Rosetta comes in too low. 

Since Apis is stuck out on the long-axis tip, Rosetta will have a good chance of ducking in front of it, thereby avoiding a crash if she comes in too low. There’s a strict orbital parameter correlation between coming in too low and arriving too early. A lower altitude than nominal means the flyby orbital ellipse is smaller than nominal. A smaller ellipse has a shorter orbital period and therefore has earlier time steps all around that ellipse, from apoapsis to periapsis, compared with the larger, nominal ellipse. Since we’ll be injecting to the flyby ellipse from the apoapsis of that ellipse, any uncharacterised anomaly that causes Rosetta to come in too low (smaller than expected ellipse) will also cause all the time steps from that point on, around the ellipse, to be ahead of nominal. That’s because Rosetta would be pulled in a bit lower by the anomaly, essentially jumping tracks from the nominal ellipse to the smaller ellipse. In practice, it will probably be lots of jumps, up and down and the average ellipse would be smaller (or larger but that’s not a crash hazard). 

This is why if Rosetta comes in low, she’ll arrive early. And being early, she will overfly the lower-radius area of Atum instead. 

Why Atum? Since the arrival time of Rosetta is early, Apis won’t have arrived at the appointed meeting spot. Atum rotates ahead of Apis because it’s eastward along the equator. Since Rosetta will be using the equator as a ground track, Atum will be there to greet Rosetta at the meeting spot instead. Atum is at a lower radius and so Rosetta’s off-nominal lower radius will be offset by a similar amount. This cheat applies to negative altitude anomalies of up to 200 metres, hence the suggested nominal flyby altitude of 200 metres. 

If Rosetta comes in late, she will be higher than nominal anyway and overfly eastern Imhotep at a comfortable but still very low altitude. 

It’s a win-win for errors either side of the nominal arrival time and it’s only achievable because we’re exploiting the high radius of Apis at the relatively pointed end of the comet. No other surface location can be so forgiving by offering this crash-avoidance trick and its because Apis is right on the tip of the long axis. If the flyby altitude really did end up lower than nominal, Apis would still be surveyed, possibly at an even lower altitude than nominal and so would Atum. If the altitude were so low as to cause a crash, we’d be no worse off than the current scenario of intentionally crashing and receiving no data thereafter. 

The other long-axis tip at the at Hatmehit/Bastet border is more prone to allowing crashes owing to its more rounded profile over a 2-kilometre ground track. This is good for landings (which are a bit like crashes) but it’s not good for flybys trying to avoid crashing because there isn’t anywhere of a lower radius if Rosetta came in too low and therefore early. If it was like Apis/Atum then we’d have to compress Bastet down by circa 300 metres to get it to a lower radius and definitely get rid of that cliff at the back of Hatmehit that acts like a giant hurdle. Bastet, east of Hatmehit along the equator would then be profiled like Atum beyond Apis and be ready to receive Rosetta ducking in low and early. In the absence of this landscaping option, Apis is the star candidate for a flyby. 

Moreover, since Apis is the only chunk of crust on the entire comet whose surface wasn’t reworked by the stretch vectors (as far as one can tell when running the stretch movie) it must be the oldest crust surface that’s available for close scrutiny. 

This post also lays out some of the other advantages of the flyby scenario as opposed to crashing.

3) DATA ACQUISITION, ORBITAL SPEED AND GROUND TRACK SPEED 

Just to recap, if the flyby ellipse were misjudged, because it’s such a weird gravity field, then Rosetta might crash at roughly the same speed as it would have done anyway in the controlled crash. And no data would be sent back after impact but again, that’s just the same as with the controlled crash. So we’re risking nothing with the close flyby. 

I suspect that collection of data and transmission in real time before crashing means a tiny data return. With a flyby at a slower relative speed and the data return sent back later over a period of time, the data volume would surely be far greater and so more data collection during the flyby could be done. The flyby would be slower because it’s at a 200-metre altitude whereas crashing brings Rosetta deeper and faster into the gravity well. 

The controlled crash scenario, as it stands, allows for a retrograde burn (a slowdown burn) at 1.7 km altitude but the proposed crash speed relative to the surface is nevertheless cited as being about 0.5 metres per second. Here’s a JPL pdf describing the mission, including the crash landing scenario. JPL are working closely with ESOC in the last few weeks to characterise the anomalies hence the inclusion of the landing (from page 29 onwards):

http://www.lpi.usra.edu/sbag/meetings/jun2016/presentations/Rosetta.pdf

The 200-metre altitude flyby proposed in this post has a relative speed to Apis of 0.251 m/sec meaning half the speed, twice the time and potentially four times the data as for the crash scenario. That’s four times the data because crashing requires the approach orbit’s periapsis to be under the comet’s surface (otherwise it won’t crash) whereas flyby has its periapsis midway through the flyby. Periapsis is the closest approach point in the orbital ellipse. The flyby is timed so that Rosetta flies through periapsis at the same time that the Apis midpoint rotates under the periapsis point, 200 metres below Rosetta. 

So with half the speed and double the distance, the flyby ground track takes around four times longer than the crash ground track time when measured from a particular altitude on approach. The proposed flyby takes 27 minutes to cross Apis and, as mentioned above, data could be taken at similarly low altitudes either side of Apis. 

Taking four times longer could mean four times the data return in raw bit terms but perhaps even more in terms of the overall value of that data i.e. a more holistic overview of a large area at very close quarters rather than a snapshot of a smaller one. And, as suggested above, the sending of real time data only during the crash trajectory probably means a trickle of data anyway during that period. That reduces the overall data volume even more. 

In reality, the flyby would take a bit less than four times longer because the controlled crash maximum ground track speed of 0.5 m/sec varies more than the flyby maximum ground track speed of 0.251 m/sec due to dipping deeper into the gravity well. 

The reason the Rosetta mission’s crash scenario has a retro-burn at 1.7km altitude is that the approach is instigated from a higher altitude than for the suggested flyby scenario (see the height of the orbit from which she’s injected to crash trajectory in the JPL pdf linked above). A higher altitude of injection means a higher orbital speed in the vicinity of 67P. And the crash trajectory is still an orbital ellipse, just a very eccentric one that intersects the surface. Rosetta would probably be travelling at around 1 m/sec by the time of the retro-burn and so the burn would slow her to perhaps 0.6m/sec. Then she’d speed up again to around 0.85 m/sec to 0.95 m/sec in the last few minutes of dropping towards the surface. Once you strip out the rotational motion of the comet’s surface (0.25 m/sec to 0.35 m/sec) relative to Rosetta’s speed, it works out at 0.5 m/sec as stated by the mission scientists.

The actual landing site was chosen while drafting this post and the Rosetta blog post on it is here:

http://blogs.esa.int/rosetta/2016/07/21/final-destination-maat-region/

The chosen landing site isn’t near the equator, meaning at least the last stages of the approach can’t be through the equatorial plane. However, the JPL diagram may still pertain to their chosen site. The 1.7 km retro burn would just include an out-of-plane vector component of just a few centimetres per second. That would take it out of the equatorial plane and into another orbital plane that intersects the landing site.

The rotational speed (tangential speed of the surface) at the landing site is probably around 0.2 metres per second for that distance from the 67P rotation axis so Rosetta would be speeding up to 0.7 metres per second again after the retro burn in this case. 0.7-0.2 = 0.5 metres per second, the stated crash speed. And yes, the approach to the landing site is with the landing site’s rotational vector, or thereabouts. You’d expect that so that the comet’s rotational speed offsets some of the approach speed. The Apis flyby makes 100% use of this phenomenon. 

The flyby scenario outlined in the next post is initiated from a lower altitude than the crash scenario so even the fastest orbital speed when skimming past Apis is just 0.6 m/sec. And the relative speed, after stripping out the rotational speed of Apis, is just 0.251 m/sec or half the value of the controlled crash speed. 

Injecting to flyby ellipse from a lower point than the crash ellipse injection does admittedly mean that Rosetta gets tossed about more by the gravitational anomalies on approach. This is because the slower she is on approach, the longer she’s hanging around in each successive vicinity where the anomalies are tugging on her. To take the extreme opposite end of the scale, a 50 metre per second flyby would cut through the anomalies (and the entire underlying gravitational field) so fast that she’d hardly deviate from a straight line. At circa 0.2 to 0.6 metres per second over the descent she gets tugged about a lot more. 

She even gets tugged somewhat more than the crash approach which is doing around 0.3 to 1 metre per second through the same vicinity (that speed envelope is an informed guess). Indeed, that’s probably the very reason the crash ellipse is injected from a high altitude with a retro burn just above the comet. It’s also therefore why this post is at pains to characterise the anomalies and exploit the crash-avoidance trick if Rosetta is dragged in too low despite that characterisation. 

The whole point of injecting low for flyby is to pass Apis at low speed. That injection height is just under 10 km, the height of the close mapping orbits in 2014. But if the orbital speed is 0.9 m/sec, say, instead of 0.6 m/sec, the speed relative to Apis at flyby is 0.551 m/sec as opposed to 0.251 m/sec, over double. Moreover, the orbit geometry would be pretty well a hyperbola fanning out either side and off into space. 0.6 m/sec when above Apis means the flyby is doing its very best to hug the rotational path swept by the Apis surface. It does this fairly well by orbiting in an ellipse of eccentricity 0.548. That’s an ellipse that hosts 67P rotating in one half and wraps round the circle that Apis describes quite well. 

4) OBJECTIONS TO THE ESCAPE, HIBERNATION AND REACQUISITION SCENARIO

Marco Parigi has suggested both on the Rosetta blog comments and on his blog that we shouldn’t just do the flyby. We should escape from 67P, put Rosetta into hibernation and return in 2020 to 2021. So I was prompted by his suggestion to look at the delta v and orbit options for hibernation and return. Here’s his blog post advocating his stance:

http://marcoparigi.blogspot.co.uk/2016/06/rosetta-lament.html?m=1

Incidentally, while we’re with Marco, he found this recent (2015) rockfall on Anuket, as well as other interesting discoveries of recent erosion in follow-on posts:

http://marcoparigi.blogspot.co.uk/2016/06/more-on-rockfall-area.html?m=1

Several concerns and objections to the hibernation and return option have been laid out in various Rosetta blog posts on the subject. But no single one of those concerns seems to be an absolute deal-breaker. It just seems that it’s a long shot owing to such things as ailing instruments, a deeper, colder aphelion distance of 850 million km, dwindling fuel supplies for any future manoeuvres, and interplanetary probe disposal concerns. These four concerns are looked at in sequence below. 

1) Instruments often prove to be surprisingly resilient. Witness the Opportunity rover on Mars, the Mars Odyssey orbiter, the Kepler exoplanet hunter and NEOWISE, all going strong well after their mission time frame and not without their own heart-stopping moments requiring ingenious solutions. 

2) The aphelion distance, which is now the same as 67P’s aphelion isn’t much further out than the hibernation aphelion that Rosetta reached before wake-up and comet acquisition in 2014. 

Photo 2- Rosetta’s approach and acquisition orbit in comparison with 67P’s orbit.
Copyright ESA/ESOC

Measuring Rosetta’s acquisition aphelion against 67P’s aphelion, it appears that Rosetta’s was 800 million km, possibly a shade under, to 67P’s 850 million km. Perhaps it was 795 million km.

The Rosetta blog post on the decision to land Rosetta said:

“67P/Churyumov-Gerasimenko’s maximum distance from the Sun (over 850 million km) is more than Rosetta has journeyed before. The result is that there is not enough power at its most distant point to guarantee that Rosetta’s heaters would be able to keep it warm enough to survive.”

The post is here:

http://blogs.esa.int/rosetta/2016/06/30/rosetta-finale-set-for-30-september/

If the previous aphelion for Rosetta was at 795 million km, that’s 93.5% of 67P’s 850 million km. The sun’s radiative flux at 850 million km compared with 795 million km is (applying the 1/r^2 rule) 0.935 x 0.935 = 87.5%. So flux is 87.5% of what it was for Rosetta’s acquisition orbit aphelion. This is a noticeable reduction in flux but not huge either. 

The reduction will affect both the temperature of the instruments and the ability of the solar panels to charge the batteries so it is tricky. But their words were “…not enough power…to guarantee…”, and that applies to a reduction in generative capacity to 0.875 of the former level. I’d suspect that 0.9 would be around the built-in safety margin anyway. It’s the equivalent of the panels going out of sun alignment by 26° which might have been a possible predicted issue. 

Moreover, Rosetta’s solar panel efficiency increases with temperature drop so electricity generation isn’t quite decreasing along the 1/r^2 slope with distance, r. Perhaps that keeps the effective generation capacity at or above 0.9 anyway. 

Dust was mentioned in one of the Rosetta posts describing the landing option but not directly concerning dust on the panels affecting electricity generation, just that she would have been flying in a dusty environment for 2 years. It was presumably referring to the instruments but possibly to the panels getting a dusting as well. Panel dusting has never been stated as an issue. 

So it looks as though there’s a chance of getting through hibernation with 90% of the solar power Rosetta had available last time round and arriving back at 67P in 2020 with €1.4 billion worth of kit, ready for work, and essentially doing that for free. 

3) Regarding fuel, Rosetta carried enough fuel for 2300 m/sec of delta v. A large amount of that was used during swing-bys on the way to 67P and for acquistion in 2014. This left several hundred m/sec of delta v in the tank. The entire flyby, escape, hibernation, return and 2020 orbit insertion scenario would use around 1 m/sec of delta v. This would be partly because of sending Rosetta into the quasi orbit of 67P for four years after the close flyby of Apis. She would return in 2020 largely of her own accord, requiring minimal delta v to reacquire the comet. If she were off target by 30,000 km, instead of the predicted few thousand km it would require a maximum of 4.5 m/sec delta v to get her back in 6 months and inserted into a 10 km orbit. At 3,000 km it would be about 0.5 m/sec delta v. Both scenarios are in addition to around 0.35 m/sec delta v for the September 2016 flyby and escape. In both cases, Rosetta would be close enough to get to work on data collection during the reacquisition approach phase. 

4) As for disposal concerns, Rosetta could be crashed in 2021, once she had observed further changes from a second perihelion approach. It would be an invaluable extra corpus of data and would be exploiting a €1.4 billion probe for almost no cost compared with the original mission. It would just require mission control, probably a pared-down team. Perhaps it’s a candidate for a crowd-sourcing volunteer effort or a Kickstarter. I would contribute for sure. 

5) GRAVITATIONAL ACCELERATION ANOMALIES ON APPROACH (IN-PLANE AND OUT-OF-PLANE TUGS ARE DICTATED BY THE CURRENT AND PALEO ROTATION PLANES)

The flyby orbit presented in the next part doesn’t take into account the gravitational anomalies presented by the duck shape or peanut shape of 67P. Of course, these have to be characterised and superimposed on the flyby orbit calculations. The characterisation process is outlined below. It’s detailed but not exhaustive. This certainly isn’t required reading. It’s here for completeness. If you’re not interested in such detail you can skip the next five sub-headings and resume at the one entitled “Rotation Plane Approach Makes it a 2D problem…”

We temporarily dispense with the duck analogy here, in favour of a peanut or dumbbell shape so as to visualise the gravity field better. There are a few references below to 67P rotating under Rosetta. The tendency might be to think of Rosetta orbiting the comet while the comet stays relatively still but the low gravity means that the orbital speed is really slow. So the comet rotating under Rosetta is indeed the case for all but the closest orbits. The theoretical rotationally synchronous circular orbit for 67P is 3.3 km from the centre of gravity, less than 1 km from most of the surface. Such an orbit would decay very quickly though, owing to the gravitational anomalies caused by the weird shape. Any higher than that and 67P rotates under Rosetta as she orbits even if, as in all these scenarios, she is orbiting with the prograde rotation vector. With elliptical orbits, she can whizz past a little higher than 3.3 km and still keep up with the rotation, but not much higher. On a 10 km circular orbit, 67P rotates more than five times per single Rosetta orbit so Rosetta sees the comet rotate a little over four times in each orbit. 

If Rosetta approaches along and through the rotation plane, her orbital plane will be one and the same as the rotation plane of 67P. The rotation plane is the extended equatorial plane and so the flyby orbital ellipse’s ground track is along the equator. As Rosetta gets nearer and nearer, any chosen close approach flyby track will be along the equator. Since Apis is on the equator it was the easiest choice for that track to run along for all the reasons outlined in this post. And of course one of those reasons is that it’s on the equator at all and therefore in the rotation plane of the comet. 

6) THE OUT-OF-PLANE ANOMALIES

The bilobed peanut shape of 67P will always present gravitational anomalies on approach, whichever plane Rosetta approaches along. However, the rotation plane approach constrains those anomalies to being predominantly within the plane of Rosetta’s flyby orbit ellipse. This is due to the gravitational field, along with its anomalies, rotating in lock-step with the comet under Rosetta, and indeed through her, as she approaches. The shape of 67P is most symmetrical about the rotation plane. The the volumetric asymmetries that are biased to Rosetta’s right or left as she approaches are thus kept to a minimum and therefore the gravitational anomalies that might pull her sideways are also kept to a minimum. And the left or right directions are, of course, not straight ahead and so they’re out of plane tugs. The volumetric asymmetries, such as they do exist even on rotation plane approach, cause 67P to wobble visibly. If there were no wobble there would be no sideways tugs and we’d only have to deal with the in-plane tugs that affect Rosetta’s altitude. 

Using the rotation plane for the flyby ellipse therefore minimises out-of-plane, gravitational anomalies which cause sideways tugs. They’re much smaller than for any other approach plane, especially an approach plane at 90° to the rotation plane. In that instance, the peanut would be rotating like a Catherine wheel in front of Rosetta for much or most of the time. This would cause all manner of varying sideways tugs that were out of plane with Rosetta’s orbital ellipse plane as she approached. 

It appears that rotation plane approach has been exploited already for the Valentine’s Day flyby. And perhaps for the Philae landing too, although the approach photos appear to have him somewhat left the plane, looking in the prograde direction. Closer in, he seems to be running much closer to the equator ground track. The September 2016 landing scenario is off-plane by definition because the landing site is just above the head rim at Ma’at so as to overfly the sink holes there. That’s a long way from the equator but it’s already been explained above that the JPL pdf diagram showing Rosetta going down the rotation plane till the last minute may still apply. So rotation plane approach isn’t a new idea but an obvious solution being exploited already. 

7) RECOGNITION OF THE PALEO ROTATION PLANE’S ROLE

What is new, however, is the recognition of the existence of the paleo rotation plane. The stretching along the paleo rotation plane caused today’s symmetrical, diamond-shaped body and the symmetrical head, which mirrors the diamond at the back and is neatly rounded at the front due to herniating from the body. So the whole comet is symmetrical about the paleo plane. But the rotation plane precessed 12°-15° to its current position after the head sheared from the body and rose on the stretching neck. This is the reason for the wobble, mentioned above. It’s therefore the reason for the out of plane gravitational anomalies as they sweep through Rosetta while she approaches down the current rotation plane. If the paleo plane had never precessed, the approach would be wobble-free because the symmetrical shape would also be symmetrical about the rotation plane. The current rotation plane ‘slices’ through the comet at a 12°-15° angle and so, when set to rotate, the symmetrical shape wobbles. 

8) THE IN-PLANE ANOMALIES

There are two causes of in-plane anomalies when approaching through 67P’s rotation plane. The first is the the constantly varying radius values of Rosetta from the two lobes. This is exacerbated by the large lobe exerting more gravitational acceleration than the small one. If they were the same mass, we could at least have a single value for the acceleration on Rosetta for a given radius when the lobes are aligned under her (one behind the other). As it stands, the distance between the lobes when aligned is enough to make the two values rather different when Rosetta is close to the comet. This is because the distance between the aligned lobes is a large proportion of Rosetta’s orbital radius. At 30km orbits the proportion is much less. Even with two equal-mass lobes, this aspect of the in-plane anomalies would still give a constant variation in gravitational acceleration at a given radius as 67P rotated under Rosetta. But at least it would follow an even, simple harmonic motion curve. As it is, the SHM curve has a higher amplitude when the large lobe is nearest. 

The above in-plane anomaly problem has to be superimposed on the angle anomaly that the two lobes present to Rosetta. In truth, the SHM curve due to lobe alignment is a simplified argument that is treating the two lobes as if they’re sliding up and down a pole and somehow sliding through each other twice per ‘rotation’. We know that in the real world they align twice per rotation, yes, but they get past each other by rotating round each other and presenting the peanut shape sideways-on to Rosetta, twice per orbit as well. 

When the peanut/dumbbell presents itself side-on (90° to when the lobes are aligned) the centre of gravity is still in the middle between the two lobes, roughly speaking. However, in this case, each lobe pulls Rosetta in a slightly different direction. The two lobes work against each other just a tad and cancel out each other’s force slightly in the process. The degree to which the lobes do this is dependent on the angle their separate centres of gravity subtend with Rosetta and the radius line between her and the centre of gravity of the comet. For any given orbital radius of Rosetta, these two angles reach a maximum when the peanut is side-on to Rosetta and reach zero when the two lobes are aligned. Their rate of change is in simple harmonic motion as well, dictated by the circular path of the two lobes’ centres of gravity around the actual centre of gravity of the comet. The angle each lobe’s centre of gravity makes sits notionally within Rosetta’s flyby orbital plane (“notionally” because we know there are slight out-of-plane anomalies too). And her flyby plane is of course at one with the current rotation plane. The degree of tug depends on the sin of the subtended angle. Since the angle and the sin diminish precipitously with distance, the anomalies due to the peanut configuration changing under Rosetta diminish quite rapidly with an increase in Rosetta’s orbital radius. This is similar to the lobe alignment anomaly’s relationship with radius but the rate of diminution is different. 

The resultant force after vector summing the two lobes’ separate sin-derived tuggings is a force that is tangential to Rosetta’s orbital path at that instant (the vector summing is with respect to a tangent to Rosetta’s orbital path). But the radial force, downwards towards the centre of gravity is reduced by the same vector summing. The radial force is dependent on the cosine of the same angles the two lobes subtend with the radius line. The reduction results in an effective drop in overall gravitational acceleration when the peanut presents itself sideways on. This is set against the resultant tangential force from vector summing the sin component. 

This gives a slight change in direction to the overall force but the overall force remains diminished for a side-on peanut configuration. That’s because the two cosine components really are summed (two positive elements working in the same direction down the radius) whereas the two sin components are in effect a subtraction (one working on a prograde tangent along the orbit and the other working along a retrograde tangent). A vector summing can be a subtraction just like 4 minus 2 is a sum. The result is either a small prograde force on Rosetta or a small retrograde force. But when we further sum this resultant force to the new, reduced radial force towards the true centre of gravity between the two lobes, the final result is that the overall force (sin resultant vector summed with the cosine resultant vector) is reduced. It’s reduced but its direction is shunted slightly forwards or backwards in relation to the actual centre of gravity. That effective c of g shunting is a proxy for the vector-summed prograde or retrograde force anomaly. The tangential force anomaly (one should really say gravitational acceleration anomaly timesed by Rosetta’s mass) either accelerates or retards Rosetta along her orbit. And the overall reduction due to the vector summing is superimposed on this. The SHM lobe alignment anomaly is then also superimposed on it to complete the in-plane characterisation of the gravitational field. 

The centre of gravity from Rosetta’s point of view effectively gets shunted forwards or backwards as the lobes rotate under her and this sends her on a very slightly different ellipse via prograde or retrograde accelerations. It’s a smooth change as the comet rotates under her so the result is a constantly varying ellipse. This results in a slightly wavy path for Rosetta along her otherwise perfect flyby approach ellipse that’s calculated with the true, central centre of gravity in mind. The wave is up and down i.e. altitude and therefore within her orbital plane. 

The anomalies from peanut rotation diminish greatly with distance and are almost negligible at 30km orbits. But since a flyby at 200 metres has to come in low, the peanut rotation problem makes itself felt far more. Both types of in-plane anomaly (lobe alignment and subtended lobe angle) are superimposed on the average gravitational acceleration for any given orbital radius. That’s the acceleration based on the true centre of gravity without invoking the subtended angles and vector summing. 

The main thing to remember here is that however much the gravitational anomalies may be problematical, the tugs are predominantly within Rosetta’s flyby orbital plane and so they affect her altitude only. Even the prograde and retrograde accelerations from sin vector summing result in a raising or lowering of altitude. 

9) SUPERIMPOSING THE TWO IN-PLANE ANOMALIES, LOBE ALIGNMENT AND SIN VECTOR SUMMING

This would all be modelled using the shape model, the relevant Newtonian equation, GM/r^2 and the trig outlined above. Both types diminish with radius from 67P but at different rates. Lobe alignment anomalies diminish with the ratio between Rosetta’s radius from c of g and the distance between the lobes. Sin vector summing diminishes with sin which means sin is at a value of 1 if Rosetta sat between the lobes and zero at an infinite radius. It diminishes very fast at first as Rosetta moves to higher radii than sitting between the lobes because it depends on the sin wave which curves as a quadrant of a circle. The ‘sounding orbits’ described below would sound out any anomalies that diverged from the simple shape model derived analysis. 

10) RECAP ON THE DIFFERENCE BETWEEN IN-PLANE AND OUT-OF-PLANE TUGS

The out-of-plane tugs are fairly small due to 67P being a remarkably symmetrical shape rotating in a plane that’s precessed only 12°-15° from its plane of symmetry. The extent to which they are there is due to volumetric asymmetries that are either side of Rosetta’s orbital plane. The tugs due to lobe alignment and sin vector summing are not directed out-of-plane even though they’re brought about by the peanut presenting volumetric asymmetries. The altitude tugs are indeed due to the rotating peanut configurations but crucially, that peanut shape looks fairly symmetrical either side of Rosetta’s orbital plane at all times hence the relatively small out-of-plane tugs while characterising quite large in-plane tugs. Those in-plane tugs are caused by Rosetta being pulled up and down (by lobe alignment) or forwards and backwards (sin vector summing). Neither is out of plane despite being caused by what Rosetta sees as volumetric asymmetry. The reason is that the resultant summation is always an in plane force due to the different volumes (lobes) straddling the orbital plane whatever direction they happen to be tugging from. The out-of-plane tugs are still there, betraying the fact that the lobes aren’t actually straddling quite perfectly but they are much smaller anomalies than the in plane anomalies. 

11) ROTATION PLANE APPROACH MAKES IT A 2D PROBLEM, NOT A 3D PROBLEM

Though the in-plane anomalies are larger, the minimisation of out-of-plane anomalies makes the problem of characterising the gravitational acceleration much easier. It becomes a two-dimensional problem and not a three-dimensional one. The gravity field can be characterised by executing sounding orbits in the rotation plane and observing perturbations in Rosetta’s altitude which are always within her orbital plane. Her orbital plane is two dimensional of course, so that’s why it becomes a 2D problem. These sounding orbits would characterise the perturbations in the rotatation plane only and would be of limited use for orbiting in significantly different planes. But it’s the only plane in which the soundings can be done in 2D (or minimising 3D effects if trying to characterise the out-of-plane tugs too). Once characterised, it then behoves us to make full use the data on the flyby by approaching in that plane. This is the main reason for approaching down the rotation plane. 

There’s more on sounding orbits in appendix 2. They would be conducted to characterise anomalies over and above those derived via the shape model calculations. 

Photo 3- the paleo rotation plane (brown) is the plane of symmetry for the diamond-shaped comet. 
Credits: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA

Brown- the paleo equator. This lies in one plane, which is the paleo rotation plane. The paleo plane is the plane of symmetry for the symmetrical diamonds that are visible in the head and body lobes, more noticeably so for the body. In other words, both diamond shapes are volumetrically symmetrical either side of the paleo rotation plane. This is because spin-up i.e. fast paleo plane rotation, brought about a single diamond shape before the head herniated from the body, sheared and rose on the neck. 

Blue- (very small dots) the current rotation which is the current equator. The current plane precessed 12°-15° from the paleo plane. They cross at Apis and Hatmehit, the comet’s long axis tips, because they acted as gimbals for the precession about the long axis. So the two equator lines cross at the tips of the long axis and the two rotation planes intersect along the entirety of the long axis that runs through the comet. Lines always cross at points and planes always intersect along lines.

Red- lines that show the diamond shapes. The body diamond is remarkably similar at the other, partially obscured end as you’d expect in a body that stretched due to spin-up. The other end of the head lobe isn’t so similar, probably because of the head lobe herniation. However, the v shapes at this visible end of the head lobe match the configuration of the v shape of this end of the body lobe . And of course, there’s copious evidence for the head being attached at both ends anyway (Parts 17, 24, 29, 54 and 57 for this end, Serqet-to-Seth, and Parts 21 and 51 for the Bastet-to-Aker end).

12) PROVING THE WOBBLE

As things stand today, the current rotation plane and the paleo rotation plane conspire to ensure that some out-of-plane anomalies can’t be avoided. Even though the comet is volumetrically symmetrical about the paleo plane, the current plane causes that symmetrical shape to wobble in such a way as to present volumetric asymmetry to Rosetta as it rotates under her. This wobble is apparent in the fact that the duck shape (or peanut) of 67P doesn’t rotate head-over-heels but in an awkward right shoulder over left flipper manner. 

The ESA video of 67P rotating is linked below. Rosetta is actually moving slightly in relation to the rotation axis as she approaches, especially towards the end so you need to bear that in mind. But after a couple of viewings, you can tease out the rotation itself and see that it’s frustratingly skewed from a neat head-over-heels movement. That’s the wobble. 

http://m.esa.int/spaceinimages/Images/2014/08/NavCam_animation_6_August

Thus a symmetrical shape, rotating on a plane that is not its symmetry plane, causes a non-symmetrical gravitational field to sweep through Rosetta’s flyby orbital plane and through Rosetta herself. This causes the unwanted sideways tugs on Rosetta, taking her out of the plane of her orbit. This is in addition to the altitude anomalies described above that are due to the peanut configuration being end-on or side-on. And being altitude anomalies, those are within her orbital plane, not out-of-plane. 

The above phenomenon is elaborated on in the appendix to this post, which comes after the conclusion. It contains more information about the paleo plane as well.  

13) ADVANTAGES OF ROTATION PLANE APPROACH 

Although the above sub-heading outlines what a nightmare the gravitational field is for the dynamicists, it does conclude on a brighter note with the assertion that rotation plane approach for the flyby orbit is the best option. As such, it’s at point 3 in this list.

1) Rotation plane approach minimises the speed of the approach relative to the comet’s surface. This is because Rosetta is travelling with the rotation vector so wherever she skims past, the tangential speed of the comet’s surface at that point is moving in the same direction as Rosetta. The rotation plane is the same as the extended equatorial plane. This being the case, the flyby ground track has to be along the equator. Any and every section along the equator, if chosen as flyby close approach ground track, will be behaving in this manner of rotating in the same direction as Rosetta. 

However, not all the points around the equator behave equally in their exploitation of this phenomenon. It depends on their radius from the rotation axis: the further any point is from the rotation axis, the faster its tangential speed. The tangential speed is the instantaneous straight line speed at any instant for that point on the surface. Although it describes a circle as it rotates, it has a straight line speed at any infinitely small instant. If Rosetta is flying just overhead at that instant, the tangential speed is all-important. The tangential speed can be subtracted from Rosetta’s speed to minimise the effective flyby speed. 

 2) Approaching within the extended equatorial plane or rotation plane is the only reasonable strategy for flying over the long axis tips which rotate in that plane. Since the long axis tips are by definition at the highest radius from the centre of gravity they have the highest tangential speed. So, combined with point (1) this minimises the relative speed to Rosetta still further when comparing the long axis tips with any other candidate site along the equator. 

These factors, rotation plane and high radius, are two of the reasons I’ve suggested Apis as a candidate on several occasions. It’s at the very tip of the long axis and the equator runs straight across it. Since the equator defines the equatorial plane and the equatorial plane is also the rotation plane, it means that rotation plane approach will overfly Apis if we time it right. This means timing our flyby so that Apis rotates under the periapsis point of the flyby orbit at the same time that Rosetta flies through the periapsis point. The other reason for advocating Apis for flyby is that, according to the tenets of stretch theory, it’s probably the most primordial section of crust on the comet and so it would be a prime candidate for close scrutiny. 

Another major advantage related to choosing Apis with respect to its high radius of rotation is that orbiting at a higher radius from the comet reduces Rosetta’s orbital speed, thereby reducing the relative speed even further. This isn’t strictly related to rotation plane approach. However, the only way you can do a close flyby at a high radius and thus exploit this slower orbital speed is to approach one of the long axis tips. And the only way we can reasonably do this is by rotation plane approach. Approaching from the side in a plane at 90° to the rotation plane would mean approaching the Catherine wheel and timing it so that Apis passed under Rosetta from one side to the other while she flew on ahead. This would be fraught with difficulties regarding the gravitational anomalies and wouldn’t exploit the relative speed advantage of orbital speed being in line with tangential speed. That’s why rotation plane approach is the most sensible option for exploiting the lower orbital speed phenomenon even though it’s indirectly related. 

3) Approaching along the rotation plane also minimises the perturbations from the bilobed gravity field, by keeping them largely in one plane, the orbital plane of the flyby ellipse. This results in the anomalies causing largely altitude changes rather than lateral, out-of-plane perturbations. This is discussed more fully in the ‘gravitational anomalies’ sub-headings above. The effect of the anomalies being predominantly in one plane is that complex 3D modelling is largely avoided. However, the 3D component becomes more apparent as Rosetta gets to lower orbital radii (but see number 7, below).

4) This is the crash avoidance trick with a few other points added. Another advantage of choosing rotation plane approach and specifically Apis as a landing site is as follows. Any off-nominal approach that brought Rosetta too low would also bring her in early, because the flyby ellipse would be commensurately smaller (orbital period depends on the semimajor axis, or the ‘long radius’, of the ellipse. Apis wouldn’t yet have rotated to its planned position under the planned periapsis point. The now-lower periapsis would mean the Apis circle of rotation would intersect the flyby ellipse meaning a potential crash scenario. But because Apis wouldn’t have arrived yet, a lower-radius area would present itself under the new periapsis point. This exploits the fact that Apis is at the long-axis tip and therefore at the highest radius of rotation. The lower-radius area presenting itself would be the Atum region so Rosetta would overfly Atum safely even though she was at a radius that would have crashed her into Apis. She would have ‘ducked’ round and under Apis, so to speak, and done so earlier than the planned overfly time. She would see Apis following behind her at a greater radius than her periapsis point over Atum. She’d be travelling 0.251 metres per second (or a little more) faster than Apis so she’d escape unharmed. Prior to that, she would have overflown Apis anyway, taking data. She would just be at a non-nominal altitude, possibly even lower than 200 metres. The most likely place for an unintended crash would be clipping the eastern rim of Apis where it drops away to Atum. 

5) Gravitational anomalies make themselves felt more, the closer you get to the comet. So the most critical part of the orbit as far as being shunted off course is the Apis flyby at periapsis. However, when Rosetta is above Apis, the peanut-shape configuration is presenting itself end-on to her and so the anomalies are far more predictable than when her ground track is above the neck. When above the neck (peanut sideways on) the anomalies are potentially greater because the body lobe is accelerating Rosetta forwards and that would be quite problematical if she was at 200 metres. However, in this flyby scenario, Rosetta is a few kilometres from the comet when her ground track runs across the neck for the last time (between Bastet and Aker). So this pulling-forward anomaly, though unavoidable and hopefully largely characterised by the time flyby day arrives, is minimised. Apis is stuck out on the long-axis tip with no gigantic lobes nearby to pull Rosetta forwards or sideways. The head lobe is hidden behind the body lobe at flyby periapsis above Apis, so it’s pulling along the same vector as the body. That’s what makes it more predictable. Even the body lobe itself with Apis at its tip is very symmetrically positioned below Rosetta as she simultaneously flies through her periapsis point and over the midpoint of Apis . The body lobe is pulling vertically, yes, but it’s a much more predictable gravity field just at the low altitude where Rosetta needs predictability the most. See also number 7 which is of a similar nature. The main point here in point 5 is the head lobe being hidden behind the body and exerting its gravitational acceleration along the same vector as the body. Point 7 addresses the reason for the body being at its most symmetrical to the orbiting Rosetta at almost the exact point of periapsis. 

6) We’ve established that 67P’s gravitational anomalies are minimised and are largely within one plane if Rosetta approaches through the rotation plane for her flyby. This in turn means that if she’s sent on a succession of higher orbits within the rotation plane, they can be used as ‘sounding’ orbits to characterise those very same anomalies that will act at lower radii within the orbital plane of the flyby. Those sounding orbits could then be gradually reduced in radius and the changes in flux at each radius compared. It has to be remembered that it’s not the change in flux around the orbit per se but, strictly speaking, the change in flux for that radius as 67P rotates under Rosetta. 

This iterative process would work its way lower and lower, always characterising the anomalies below the lowest orbit i.e. modelling the uncharted territory below by using data from higher orbits. Thus, there would always be information on the in-plane gravitational field flux below the lowest actual sounding orbit. 

That’s a somewhat simplistic description of the sounding orbits. There’s more information as well as algorithm inputs in appendix 2, below, after the conclusion. 

7) Since the current rotation plane precessed from the paleo rotation plane, the two planes have to cross in two places on the comet. Those are the gimbals of the precession and they’re located where the current equator crosses the paleo equator. In theory, they should cross exactly on the long-axis tips, which is the V shape in the Hatmehit cliff and the midpoint of Apis. In practice they both cross about 200-300 metres west of these points, so that’s pretty close. This means that when Rosetta arrives over Apis, she is both over the current equator, as dictated by her orbit, and over the paleo equator that is crossing the current equator at that very point. The paleo equator is on the paleo plane and the paleo plane ensured almost perfect symmetry for the diamond-shaped body lobe (because the body lobe stretched into the diamond along that plane). Thus, just when Rosetta arrives over her target at 200 metres’ altitude and it’s most crucial not to get an anomalous sideways tug, she finds herself sweeping in over the most favourable place on the comet for offering a volumetrically symmetrical shape either side of her position. So there will be no out-of-plane tugs during the 27-minute Apis flyby despite being so close. 

14) TRANSMISSION OF DATA FROM THE FLYBY

The data collected in the flyby could be sent back in dribs and drabs over the Deep Space Network (DSN) after telemetry occultation, which is due to the sun’s passing between Earth and 67P in early October. I realise there are other pressures on the DSN (Juno etc.) hence my saying “dribs and drabs”. The only problem is that the controlled crash-landing has been decided on already and so no time on the DSN has been requested for late October or November 2016.  

15) CONCLUSION 

The next part will describe the proposed 200-metre flyby orbit timeline, including orbital elements, orbital speeds, delta v burns and flyby duration. 

It will also describe the delta v burn for escape into to a quasi-orbit of 67P, allowing Rosetta to return four years later, largely of her own accord. Jupiter is unfavourably placed for this set-up but the differential gravitational perturbations it brings about are nevertheless very small. 

16) APPENDIX 1- THE CURRENT AND PALEO ROTATION PLANES CONSPIRE TO BRING ABOUT THE GRAVITATIONAL ANOMALIES EXPERIENCED ON APPROACH.  

This appendix recapitulates some of the points made in the main post but with a number of additions, especially regarding why the paleo plane is so important. 

The current and paleo plane interaction is discussed here because the anomalies this interaction gives rise to are minimised if Rosetta approaches 67P down the extended equatorial plane (today’s rotation plane). As regular readers will know, this blog mentions the equatorial plane ad nauseam but calls it the rotation plane so as to emphasise the rotation of the comet being within that plane (and the infinite number of parallel planes either side of it). 

Although this continual emphasis is applied to the current rotation plane it’s applied even more to the paleo rotation plane. It’s mentioned so much because the comet stretched along the comet’s long-axis vector and the long axis was aligned within the paleo rotation plane. That’s how it became the long axis, elongating more and more, pulling 67P into a diamond shape. This was before the head lobe herniated and sheared from the body. The diamond is still very visible in the body lobe shape and partially evident in the head lobe shape. The stretching came about via spin-up, which was in turn due to asymmetrical outgassing. 

It has to be borne in mind that the comet stretched along the paleo rotation plane and not the current plane. The paleo plane then precessed by 12°-15° after the head lobe sheared from the body. So the paleo rotation plane is now at 12° to 15° to the current plane. The precession was about the long axis, meaning that Apis and the Hatmehit cliff acted as the gimbals. 

The paleo equator is perfectly visible as a line that runs through 17 stretch signatures all round the comet. It bisects each one which is the same as saying that each signature straddles the paleo equator symmetrically. Their common line of symmetry is the paleo equator and that line lies in one plane of course. The plane is the paleo rotation plane. The paleo rotation plane caused the stretch to occur along the long axis, which lies within the plane. The long-axis tips therefore straddle the paleo equator and constitute two of the 17 signatures (Apis and the Hatmehit cliff ‘V’ apex). They’re the only two signatures that remain in today’s rotation plane as well as the paleo plane by virtue of acting as the gimbals for the precession from one plane to the other. 

The stretch caused the 17 stretch signatures to appear where the tensile stress was greatest: along the paleo equator. The 17 signatures are described in the ‘Paleo Rotation Plane Adjustment’ page in the menu bar. 

The paleo equator is also recognised in this blog as being the line of symmetry for the diamond shapes of both head and body. If the paleo plane still held sway, it would be somewhat easier to approach the comet for a close approach. That’s because the volumetric symmetry of the comet either side of the paleo plane is so even. It’s even, owing to the symmetrical tensile forces of stretch that formed the symmetrical diamond shape. This would lead to a more evenly rotating gravity field as that field rotated in lock-step with the comet. In such a paleo scenario, the gravitational field would be rotating with the paleo rotation plane. This would be because the symmetrical shape would be rotating with the paleo plane whilst also observing the paleo plane as its plane of symmetry. The gravitational anomalies for Rosetta on approach would be largely in one plane, that is, her approach orbital plane, if she approached along the paleo rotation plane. The two planes would be in one plane. 

But of course, that could happen only if the paleo plane still held sway. As things stand today, the current rotation plane makes the old plane wobble like a wobbly wheel so we can never approach along the paleo plane. It refuses to rotate in one plane anymore and instead, the wobble sweeps out a volume. That volume is an 800-metre wide disc running through the comet. This is apparent in the slightly awkward rotation of the duck shape. Instead of going neatly head over heels, it rotates right shoulder over left flipper. Here’s the same ESA video of the rotation that was linked in the main body of this post:

 http://m.esa.int/spaceinimages/Images/2014/08/NavCam_animation_6_August

The wobble means Rosetta can’t approach down any plane without experiencing some lateral (out of plane) perturbations. However, since the current plane is only 12°-15° off the paleo plane, it’s not a bad approximation to the old plane and so it keeps the lack of rotating volumetric symmetry to a minimum. It’s certainly our best option by far for approaching the comet while ensuring a minimum of out-of-plane perturbations. 

The extent to which those perturbations are minimised is related to the extent to which the volumetric asymmetry is minimised. Thus, if Rosetta approaches the comet for flyby down today’s rotation plane it’s largely altitude anomalies that need characterising as the comet rotates under Rosetta on approach. The reason altitude anomalies prevail is that ‘up’ and ‘down’ are within the orbital plane of the approach orbit and ‘left/right’ are out-of-plane, sideways tugs. The volumetric asymmetries causing the ‘right shoulder over left flipper’ wobble do cause slight out-of-plane tugs but because the current plane is only 12°-15° off the paleo plane we’re able to get as close as possible to the ideal that would prevail if we could precess the plane back to where it used to be. 

If 67P still rotated in the paleo plane, the out-of-plane tugs would be tiny because the stretching in that plane produced such a symmetrical diamond shape, balanced either side of the plane. So although the comet’s shape is just as symmetrical as the day it stretched, it’s the precession of the rotation plane that brought about the wobble of that symmetrical shape. And crucially, it’s the wobble of the rotating symmetrical shape that brings about the volumetric anomalies, not the symmetrical shape itself. So it’s the wobble and not the shape that is responsible for the out-of-plane gravitational anomalies as well. 

Approximating to the paleo plane and minimising the out-of-plane gravitational anomalies means we can characterise the anomalies in a two-dimensional plane rather than in three dimensions. That plane is the current rotation plane which is also the plane chosen for the orbit of the flyby ellipse. 

I’m sure the people at ESOC and DLR will have worked out everything stated above in terms of how the comet rotates and how to approach it. However, the difficulties as they stand arise from a paleo plane that has precessed by 12°-15° and is now causing an otherwise symmetrical shape to wobble. This would at least help them to see why it is that they’re seeing this frustrating wobble that’s so close to being a neat, symmetrical, head-over-heels rotation. 

17) APPENDIX 2- THE ‘SOUNDING’ ORBITS

It’s as well to remember here that “flux” applies to the changing gravitational acceleration at a given radius from the comet’s centre of gravity as 67P rotates under Rosetta. It’s not the more familiar flux that applies to a changing gravitational acceleration with increasing radius away from a sphere. The reason the flux changes for a given radius is that 67P is a peanut shape and that shape generates a weird field that rotates with it, sweeping through Rosetta. The sweep brings varying gravitational acceleration values with it as it sweeps through that particular orbit radius she happens to be at. 

This appendix elaborates on point 6 in the ‘advantages of rotation plane approach’ heading, above. Point 6 outlined the way in which sounding orbits at a higher radius than the flyby could be used to characterise the gravitational anomalies within the orbital plane of the flyby. This is because the sounding orbits would be performed within the same plane as the flyby orbit (which is the current rotation plane) and from the anomalies experienced within those higher orbits, we could extrapolate the anomalies to a lower radius. 

There would be changes in gravitational acceleration flux as 67P rotated under Rosetta for any given orbital radius. The actual flux signature (as opposed to absolute value of the gravitational flux) would be largely similar for different orbital radii. The flux signature is the field change signature i.e. the degree to which the field changes for Rosetta at a particular radius as the field passes through her. There would be field changes that were the same in character at different radii from the comet’s centre of gravity (c of g) for any chosen latitude/longitude point on the surface below Rosetta. That surface point would be on a line between the centre of gravity and Rosetta. This would also be the case for any given ground track section on the comet, which is after all a string of lat/long values. And for the sounding orbits and flyby, the ground track would always be the present-day equator because the equator defines today’s rotation plane and flyby ellipse plane. 

Any ground track section corresponds to a particular orbital ellipse section above it describing that ground track section. Thus, for circular sounding orbits at any particular radius, the comet’s field rotating through Rosetta will change for each circumference section of the orbit. The orbital angular velocity has to be subtracted from the comet’s rotational angular velocity first though but that’s an easy procedure for circular orbits. A circumference section is the same as an ellipse section because a circle is a special case ellipse with its two foci merged at the centre. 

The trick is to characterise the gravitational anomalies for each circumference section and try to do so with the smallest possible sections so as to have the highest possible resolution for the gravitational field. This would have been largely done without sounding orbits and just using a good shape model of assumed even density. You could then just crunch the g’ value (a particular unique gravitational force value) for any orbital radius above any particular lat/long point. Then you’d move along a virtual orbit line within the rotation plane to the next lat/long point along the equator and repeat. This would be done by using Newton’s law and integrating the gravitational attraction of successive, thinly sliced comet layers at right angles to the Rosetta-to-c of g line. But the density isn’t exactly even hence the need for sounding orbits. 

The sounding orbits need to use some means of detecting the anomalous movement of Rosetta which would be the perturbations from the predicted idealised line using the shape model. And the gravitational anomalies would be the sole cause of that anomalous movement. The method used for detecting the perturbations is probably Doppler effects on the radio transmissions received at the Earth. This was done for characterising the potential cavities inside 67P and that procedure presumably had to start with a good shape model and assume it had an even density. But on top of that, you can superimpose a hypothetical density that varies smoothly say, from body to head and which has no cavities. And it seems the head does have a lesser density than the body. No papers seem to have emerged on this but there’s been talk of the head being 10% less dense for nearly a year. That’s in keeping with stretch theory (material drawn from the head and body equally to supply the neck results in the head supplying proportionately more hence becoming less dense).

The flux (the change in g’ with rotation of the field at a given radius) would be smaller the higher Rosetta orbits. The overall field strength would be smaller both in absolute magnitude anyway. But the gradient of the flux would be smaller too, that is, the rate of change of gravitational acceleration experienced by Rosetta as the field swept through her at that radius would be smaller. The ground track section corresponding to that section of sweep through her would always be a section of the current equator. 

So for a given equator ground track, the flux along that sweep for any given sounding orbit radius is going to be pulling Rosetta mostly up or down in terms of altitude and altitude is of course, orbital radius. The degree to which this up and down wave happens at, say, 7 km altitude will be noticeably similar to the wave at 10 km altitude but not exactly the same. let’s take the particularly awkward stretch across Bastet towards Aker. If you mapped the two orbits, 7 and 10 km, over the equator ground track for Bastet to Aker and looked at them from the side, you’d see the two waves stacked one above the other. The waves would be physical paths through space traced by Rosetta as she was tugged higher or lower off her otherwise smooth circular (or elliptical) path. But the upper wave would be more subtle, less exaggerated than the lower one. 

Characterising the relationship between the two waves involves the 1/r^2 inverse square law for gravitational acceleration with respect to radius, r. It also involves the 1/r^1/2 (‘root one on r’) relationship between radius and Rosetta’s orbital velocity. Superimposed on this is the actual rotation of the comet which is very steady and linear over short periods. These three relationships mesh to give a diminishing flux signature with altitude above the equator ground track section. But the crucial thing is that the flux at a lower orbit can be divined by taking the flux of the upper orbit and churning it through the algorithm. 

Other parameters would go into the algorithm, for instance, the orbital ellipse size and eccentricity. Orbiting in an ellipse affects the r value as Rosetta cuts through the concentric radii while trying to perform sounding ostensibly at one radius. The above is based on circular orbits only (notwithstanding the wave perturbations being characterised). Ellipses will be favoured for the lower flybys during September. With knowledge of the notional elliptical pathway for each sounding segment, software can correct for the varying r value while still assessing the perturbation. With successively stacked ellipses, the varying r values stack into a mesh and build up a picture in a slightly rougher version of the ideal concentric circular orbits. 

So the algorithm would use higher altitudes to predict the gravitational acceleration flux at lower altitudes. Rosetta could then fly those lower altitudes and check the prediction. When the out-of-plane anomalies really start to be felt (see below) we would be relying more on the predictions from higher orbits than the actual soundings from the lower orbits. The lower orbits would still be useful but mostly for the out-of-plane anomalies (these are the sideways tugs due to volumetric anomalies of a wobbling symmetrical shape).

As Rosetta gets to lower radii in her sounding orbits, the dreaded out-of-plane anomalies make themselves felt more. These aren’t so apparent when she’s at a distance. That’s because they’re dependent on the angle that the anomalous volume doing the tugging makes with Rosetta’s orbital plane. 

Specifically, it depends on the sin of that angle and the sin diminishes dramatically with distance, r. This is similar to the sin relationship described in the main post for the altitude anomalies brought about by the rotating peanut configuration. But in this case, the angle subtended with Rosetta is off to the side, outside the orbital plane. In the former case, the angle was within the orbital plane or at least notionally so. So the soundings at, say, 6 km might be difficult to extrapolate to 2.8 km i.e. the Apis flyby radius. This would be because the sideways tugs would enroach on the up/down orbital plane signature. 

It is however, fortunate that the main region presenting out of plane anomalies in the last stages of Apis approach is Imhotep and Imhotep is a near-perfect diamond shape. Rosetta would admittedly be off the centre line of the diamond by virtue of tracking along the current equator and not the old equator down the centreline 12°-15° away. But at least it starts off as a predictably symmetrical shape that’s just offset slightly from Rosetta’s like of approach to Apis. This is yet another advantage of choosing Apis as a flyby candidate. 

It does get trickier with the sideways tugs as you get closer but the principle of sounding in this manner is very much easier when done in the rotation plane as opposed to in any other plane. The above example of Imhotep is proof of this as is the near perfect volumetric symmetry either side of Apis due to Rosetta crossing the true paleo equator just at the crucial, lowest point (see point 7, above). 

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