Spin-up Calcs

22nd March 2016 (Reproducing calculations published on Rosetta blog comments on 8th October 2015).

THE PROCESS FOR STRETCH

The scenario is as follows: asymmetrical outgassing causing spin-up to the point where the comet undergoes some fragmentation, shedding slabs from its extremities due to their tangential speed being > 0.8 m/sec. This release is followed by the sub-escape-velocity core stretching into two lobes rather than releasing a lobe into orbit if it was a brittle-break release. All such spin-up scenarios would harbour a radius of rotation domain with tangential speeds that are conducive to throw material within the domain into orbit and not escape. There would be actual core matrix in a thick, hollow cylinder between the inner and outer radii of the domain. This matrix would be susceptible to going into orbit if shearing away in a brittle break or stretching the comet if ductile and subject to tensile resistance (the neck) attenuating the ‘lift-off’. The rotation period necessary for this to happen on 67P is 2-3 hours. The models say the comet is too brittle to stretch but the multiple head-body matches and crust-sliding signatures show that it must have done. The symmetrical diamond-shaped body is a give-away for the stretching process with the head lobe undergoing an excess of stretch via herniation.

THE CALCULATIONS

The following calculations are culled from a comment I made on the Rosetta blog on 8th October 2015. The blog post is here:

http://blogs.esa.int/rosetta/2015/09/28/how-rosettas-comet-got-its-shape/

The comment thread for that post was long with considerable debate on stretch. You would need to read that thread to see the full comment and its context. This is simply the calculations for 67P’s spin-up; slab ejection above escape velocity; nucleus stretch into two lobes; and its subsequent slowdown in rotation period due to angular momentum (AM) conservation. The calculations themselves characterise the process required for the comet stretching into two lobes. The headings beyond the calculations involve musings on whether the stretch event was indeed a short, sharp shear and head lobe rise or a slow ratcheting-up over time via the same process. Since both scenarios rely on the same process of spin-up leading to negative g, stretch and AM conservation slowdown, the calculations are sufficient and the further headings are an optional extra.

The reason for not publishing these calculations earlier was that I was waiting on more refined measurements of the head, body and especially the neck. That was in order to refine the radius (r) values for the head lobe and body lobe about the centre of gravity (c of g). Also, a more accurate value for the mass of the head lobe which affects the tensile stress along the shear line.

These revised figures haven’t been published yet so the calculations in my October 2015 comment still stand.

Comment excerpt:

“…I may as well paste my latest figures with the most stringent possible r value and comet extremities under significant negative g.

I used the published comet dimensions and this rotating model for measurements:

http://rosetta.jpl.nasa.gov/news/measuring-comet-67p/c-g

I took many screenshots and then used the 4.1 km base as a calibration measurement for all other measurements. I came up with a figure of 2.460km between the two lobe c of g’s today and 1.460km between them when pressed together as a single body. So the stretch would therefore be 1000m.

I then shifted a quarter of the way along the 1460m axis (of the pressed-together comet) from the body c of g to get the barycentre/comet c of g when pressed together. This assumes the head being 1/4 the mass of the comet. This gave me 1095 metres as a distance between the whole-comet c of g and the head c of g. In other words the orbital radius of head lobe c of g about the barycentre. This is needed as the axis of rotation for measurement of the circumference distance moved by the head lobe c of g and calculating the various tangential velocities for weightlessness, orbit and escape.

Some calculations are rather precise considering the inputs but that’s just to keep them tight as they sometimes get used as inputs and are then rounded in other places.

The circumference swung by the c of g at 1095m is 6881m (1095 x 2 x 3.142). I then took a tangential speed at head c of g radius that would lift the head lobe off into a low orbit. That speed is 0.94m/sec which is exactly halfway between the cusp of weightlessness, 0.780m/sec, and  escape velocity, 0.780 x 1.414= 1.104 m/sec. Those two figures are specific to r=1095m, head c of g. Here is the calc for the weightless scenario:

Comet mass 1E13kg
G constant 6.674E-11Nm^2/kg^2
’x’ means ‘multiplied by’

Using the vis viva orbital velocity equation:

v^2= GM(2/r-1/a) but r=a due to circular orbit. 
v^2=6.674E-11 x 1E13 (2/1095-1/1095)
v^2=667.4 x 9.1324E-4
v^2=0.609
v= root 0.609
v= 0.780 metres per second tangential speed at a 1095-metre radius to induce weightlessness of the head lobe.

This is rough due to the bilobed gravity field but the c of g is a good enough approximation for the mass in negative g above the c of g to cancel the mass in positive g below it. In fact, it’s pessimistic because the ‘centrifugal’ forces are dependent on r^2 so mass at a higher r value carries more clout.

Escape velocity for that same radius of 1095m is 1.104 m/sec due to the root 2 rule.

So the head lobe would be ejected to orbit, but not escape, between 0.778 m/sec and 1.1 m/sec. That’s why the mid point of 0.94 m/sec was chosen as a value for ejecting the head lobe to low orbit. Meanwhile, the slabs at the extremities are well past escape v (see below).

Circumference of ‘orbit’ for seated head lobe is 1095 x 2 x 3.142 = 6881 metres. This corresponds to the still-attached head lobe rotating in a circle about the barycentre at 0.94m/sec (at its c of g), in negative g, and about to shear off.

The comet rotation period for the proposed tangential speed at r= 1095 m just before head lobe ejection is:

6881/0.94= 7320 secs or 2.033 hours.

For angular momentum conservation calculation between the pre-stretch and stretched comet, the c of g distance between head and body lobe can be used for the reduced mass version of coefficient of inertia. The reduced mass doesn’t need to be calculated because it’s the same for both versions but the r value does change and that goes from 1460m to 2460m.

The decrease in rotation period is proportional to r^2 as all AM calcs have an r^2 component multiplied by the coefficient of inertia (reduced mass in this case, which cancels).

1460m to 2460m is a 1.685 increase. 1.685 squared is 2.839. So omega (rotation period) slows by a factor of 2.839. It goes from 2.033 hours at shear to 5.772 hours at fullest stretch [today’s comet]. Today’s comet rotates at 12.4 hrs so it’s slowed by an extra 6.6 hours from 5.772 hrs (if the above 2.033 hrs is correct).

The 2.033-hr rotation period is probably getting close to the absolute maximum rotation speed needed due to the rotation radius being perhaps pessimistically low, the escape speeds reasonably high and the bilobed gravity field not being corrected for.

Varying inputs gives slightly varying but consistent outputs, all slower rotations than this example. They are up to 3 hours, or some rather more optimistic ones at 3H 30. The result is that it’s not out by an order of magnitude, requiring spin-up to 0.2 hours. It all looks plausible and most likely between 2 and 3 hours.

What makes this scenario very neat indeed is that the 0.94m /sec chosen for the head lobe to go into orbit means that Hatmehit and Imhotep had already gone through escape velocity threshold of 0.9 m/sec (using vis viva @ r=1695m for Hatmehit) and were now at 1.45 m/sec tangential speed- in highly negative g. 1.45 m/sec is derived from the 0.94 m/sec at 1095m and subjecting it to the ratio of radii, 1695/1095. So the slabs had to escape if they detached and yet the head lobe was destined never to escape. That answers the question as to why slabs should fly away while the head lobe conveniently remained. Its low r value and consequent higher g influence (baked into the velocity vis viva equation above) meant it needed a very high rotation rate to escape. The slabs didn’t so much.

To cap things off, the 5.772-hr rotation rate after stretch and spin-down corresponds to .558m/sec tangential speed for the head lobe c of g at 2460m from the body c of g:

Circumference=1845 x 2 x 3.142= 11,594m
Time= 5.772 x 3600 = 20,779 secs
Tangential v= 11,594/20,779= 0.558m/sec

This is very close to the 0.601 m/sec orbital speed for that r (r= 1845m from barycentre, 2460m x 0.75). The 0.601 is calculated using vis viva too, as per first calculation. That would mean that the head lobe really did go into orbit at that height and physically couldn’t get any higher, even if the neck hadn’t been helping to hold it back. That fact, by definition, ended the stretch event.

Note that the AM conservation value of r is 2460m due to being stipulated as the ‘r’ value in the AM calculation using the reduced mass coefficient of inertia. The barycentre is ignored due to being implied in the reduced mass. The barycentre has to be used for vis viva, giving r=1845m. However, both equations are AM conservation equations (hence the name vis viva, based on Keplers 3rd law) and so I’m not quite sure why that small discrepancy is there of 0.043m/sec (0.601-0.558).

This means we’d have a scenario that seems right. The head lobe was released some time after it went into negative g and had built up some excess tangential speed to break cohesive bonds. It then sheared and simply went into a low orbit. It also just happened to drag up some neck material that caused it to orbit slightly lower than it might have. But it was all very fluid and weightless after stretch, just orbiting with this hour-glass neck that was hourglass due to being stuck between the lobes and not being at orbital speed. So it wasn’t supporting the head, just dangling between the two lobes.

It might seem obvious that the head needed to be weightless to stretch or indeed be in negative g during the stretch, and there would also be a point where it would go from negative g to positive g and be weightless by definition through the sign change.

However, in this scenario we have a spin up to a particular required rate for slabs to have escaped and the head to coincidentally be just past the cusp to break away and go into orbit, not escape. Then we have a spun-down rate that’s completely constrained by that initial required spin rate for everything to work properly and that spun down rate is dictated via AM-conservation. That final spin rate just so happens to be orbital speed for the radius we see the head lobe at today. That’s three finely tuned coincidences, all tied to one intial spin rate: escape speed threshold surpassed for slabs; orbital speed threshold surpassed for head lobe to detach but not escape and final dictated spin rate is the orbital speed required for today’s head lobe height.

So I think the head lobe orbited for some time afterwards with the neck dangling in tension between the two lobes- hence the hourglass. I think the neck probably helped with the final spin-down to 12.4 hours in some way. It would have been constantly trying to move backwards like the drifting rocks.

After spin-down from 5.772 hours to 12.4 hours, the head wanted to drop lower but simply sat down on the neck. Of course, it probably dropped a bit by compressing the neck a bit but that’s the finer detail we know must’ve happened because it really is sitting on the neck in a positive g field.

//// End of comment excerpt.

WAS IT A CATASTROPHIC SHEAR AND HEAD LOBE RISE OR WAS IT A SLOW, RATCHETING PROCESS?

The calculations for the tensile force along the head lobe shear line and across its shear plane (seating area) are in a comment that follows a little way down from the 10th October 2015 comment in the thread linked above. This force (at the 2-3 hour rotation period) greatly exceeds the comet’s estimated tensile resistance as given by Thomas et al, 2015.

So we already know that, regarding orbital calculations alone, the head would have gone into orbit but not escaped. But we still need to consider the possibility that a high tensile resistance might allow the head to cling to the body despite being subject to the negative g required for lift-off and orbit.

However, the tensile force due to the negative g was so much greater that the head would have sheared. Moreover, it would have lost little of the energy required to go into orbit. It would have simply ended up orbiting a little lower. The energy lost is equal to force x distance which is equal to tensile resistance x head seating area x neck stretch distance.
However, if the negative g force is so much greater than the tensile stress of the comet, the head wouldn’t have waited obligingly for the spin-up to be sufficient to throw it into orbit 1000 metres above the body where it is today. It would simply have sheared when the negative g surpassed the tensile strength value by a small amount. It would then have moved up a small distance until the excess spin energy was converted to potential energy and heat energy. This was overlooked in that second comment. The potential energy increase would be in the form of a much smaller head lobe rise than in the excerpted comment. The heat energy expended would be via attenuation by the incipient neck, that is, work done by the neck stretching against the tensile force. Force x distance= energy.

This is the same process as described in the excerpted comment above but it’s not a one-off catastrophic shear and head lobe rise. It would be a series of smaller versions of the same phenomenon. That would result in a ratcheting up of the head on the neck. The AM conservation element would work the same way too. The head wouldn’t collapse back down owing to the fact that it was largely weightless for a long period after each ratcheting. Then, if there happened to be a slowdown in rotation, the compressive strength of the neck would support the head’s net gravitational component anyway (as it clearly does today with the head nowhere near weightless on the full-grown neck). A slowdown in rotation would be because of random changes in the net torque delivered by random asymmetrical outgassing. This would be in addition to the totally predictable slowdown due to AM conservation during each of these mini stretch events.

The head lobe would therefore wait for the next cranking up of the spin period to a point where the process could repeat and ratchet the head up a little more. By this process, the higher spin rates delivering the ratcheting-up would simply be unremarkable high points in a random walk of fast and slow rates over the centuries and millennia. The head lobe would nevertheless rise relentlessly over that period.

THE ASTEROID COROLLARY

The only question mark remaining might have been whether the asymmetrical outgassing would repeatedly spin the comet up to the required spin rates. But there are now several documented cases of fragmenting comets and at least one of an asteroid that was thought to have fragmented due to spin-up alone. At the very hour of posting this page, the Earth was flying between two cometary fragments that are thought to be related fragments:

http://www.jpl.nasa.gov/news/news.php?release=2016-078

There’s a sharp cut-off for spin rates of asteroids at around 2 hours which suggests those spinning faster than a two-hour rate are liable to fragment and so no longer exist to be observed. For comets it seems to be around 3 or 4 hours which is consistent with their lower density. However, the argument below suggests that they don’t fly apart, thereby disappearing forever. Rather, they shed some material, with the remaining rubble pile core simply stretching at a lower radius of rotation than escape velocity. The stretched core therefore slows via AM conservation and so these ‘disappeared’ fast rotators are still with us, just reworked, smaller and rotating sedately. The slower rotation is due to the very spin-up rate that reworked them- it’s baked into the AM conservation process and is dependent on the degree of stretch. These would be the dog bone shaped asteroids that litter the skies.

If comets can spin up to rates that allow them to shed slabs at their extremities, they will by definition shed slabs and rocks into orbit below a certain radius of rotation. And the lowest of those might never truly escape but simply start to rise whilst pulling up some nucleus material up behind them. That would especially be the case if they were large like 67P’s head lobe. That would, by definition, be a stretch event.

Meanwhile, the orbiting debris would be scrubbed away by any moderately close approach to Jupiter (the orbiting cloud behaving like an extremely low-density single comet and thus having a very much higher Roche limit).

What hasn’t been posited for asteroids (or comets) as far as I know, is the scenario in the excerpted comment above whereby the nucleus slows down via stretching after shedding slabs or large boulders. This would mean that a large number of the peanut/dog bone shaped asteroids we see today used to be spinning at above two hours (spun up by the YORP effect) and are simply the stretched core of what was a larger asteroid that threw off multiple slabs from its extremities. It stands to reason that the core wouldn’t shed slabs below a certain radius of rotation because at that radius the tangential speed wouldn’t be at either escape velocity or even orbital velocity (weightless slabs).

However, those slabs (or boulders) in a rubble pile that were spinning at say, low orbital velocity would tend to rise from the surface. There would be a cusp at which they would rise a little way and stop, drawing rubble from the pile up behind them in the process. This would happen in both directions along the long axis, thereby extending that long axis noticeably. The gravity vector would run down the centre from both ends, keeping the process symmetrical and in check. The result is the dog bone shaped asteroids that are common in the asteroid belt and even more so in the NEO population. They constitute about 10% of NEO’s.

I’m suggesting that many of these dog bone asteroids used to be bigger and were rotating at sub 2-hour rates. They fragmented due to excess spin-up and the core stretched at the same time via the process described in the excerpted comment. The stretch would slow the core via AM conservation. And so all these sedate-looking slow-rotating dog bones used to be rounder and bigger and rotating fast. Their current shape and rotation rate are the smoking gun for this novel core stretching process.

APPLYING THE ASTEROID COROLLARY TO 67P

And of course, exactly the same process would apply to 67P if it had been spun up to fragmentation. The only problem is the models that say comets are too brittle to stretch. They can only fragment. Visible evidence on 67P clearly indicates that it did stretch and was therefore not too brittle to stretch. The models therefore need to be revisited.

MORE EVIDENCE FOR THE SLOW, RATCHETING PROCESS

Another indication that the stretching process was a slow, ratcheting-up is that the gull wing delamination and other delaminations and tears in Hapi show that the head had largely sheared already before actually lifting off. We know this because the gull wing delaminations match to the head rim so both body and head stretched and delaminated along the Hapi shear line together when clamped together (Parts 38-39). Only after that did the head lift off. The matches run a long way down the Hathor cliff and so the whole Hathor cliff stretched along the long axis when clamped. However, that very process must have unclamped it from the body in a very short time. That’s why Hathor is fan-shaped and looks cleaved. By the time it had stretched with the body, it offered no resistance to the head lifting off and all the tensile resistance was transferred to Anuket. That’s why Anuket looks extruded (and indeed is extruded as evidenced by the four tell-tale ‘ticker-tape’ lines at Anuket, Part 25). The Bastet neck surface looks like Anuket for the same reason- it’s at the other end of the cleaved Hathor cliff.

The upshot of Hathor not offering resistance to shear and stretch means that the catastrophic shear and 1000m head lobe rise looks even less likely. That catastrophic scenario requires a high intitial tensile resistance that’s comparable to the negative g force required to throw the head lobe up 1000m. The resistance would likely drop markedly on shear as the neck thinned and perhaps organised its hypothesised marbles into laminar flow. Anuket’s appearance is consistent with this happening (a worked-over look to its matrix) but if Hathor wasn’t helping out, nor its fan-shaped twin at the south pole, Anuket’s resistance was paltry. That’s why it looks as if it may well have been a ratcheting process. The negative g would have overcome this small tensile resistance very readily, resulting in the multi-staged ratcheting.

I should think Marco Parigi will be delighted with that analysis since he’s always said that the stretch was a slow, punctuated, ratcheting-up of the head and that it may still be happening. The system would be ‘pumped’ via random spin-up events over time.

However, despite providing all this circumstantial evidence, it still looks to me like a one-off catastrophic stretch event with slurry signatures and crust slides all over the comet. Then again, there’s no reason why these couldn’t have been punctuated events too. Indeed, a few of the crust slides show tide marks as if they stopped and started four or five times from seating to their current position.

STATEMENTS ABOVE THAT PREFIGURE FUTURE PARTS

This page contains several observations that haven’t been mentioned yet but will get their own post in the distant future. This is mentioned here in case regular readers thought they’d missed posts on these observations. They are:

1- The Hathor cliff stretched, cleaved and is fan-shaped as a result of this.

2- The south pole has a similarly shaped and sized fan bounded by Anuket and Bastet in the same manner as Hathor. It too shows signs of stretch along the long axis (vertical lines running up it as well as at its seating on the south pole body).

3- Anuket was yanked out of the body and reworked as a result of this upheaval (although this is touched on in Part 25).

4- The Bastet neck behaved like Anuket.

5- crust slides show tide marks.

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