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
16) Appendix 1- more on the paleo rotation plane affecting gravitational anomalies
17) Appendix 2- the sounding orbits
Here’s a link to ESA’s rotating shape model of 67P from which the still above is taken.
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.
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):
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:
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:
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:
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.
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:
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.
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.
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.
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:
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).
Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0
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All dotted annotations by A. Cooper.
Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/A.COOPER