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


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

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

The blue arrow is 67P’s spin axis

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

CONTENTS

1) The photos

2) Introduction

3) The error

4) The Latitude/Longitude analogy

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

6) The relationship between figure 2 and the celestial sphere

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

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

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

10) Calculations

11) Implications of the RA/Dec anomaly

12) Conclusion

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

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

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

1) THE PHOTOS

Photos 1 to 9 follow, along with narrative captions. 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Photos 8 and 9

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

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

Photo 9 credit: 

Creative Commons Attribution 3

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

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

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

2) INTRODUCTION

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

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

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

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

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

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

3) THE ERROR

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

4) THE LATITUDE/LONGITUDE ANALOGY

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

10) CALCULATIONS

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

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

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

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

Minor axis divided by major axis = 0.469

Cos-1 0.469 = 62.03°.

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

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

Minor axis divided by major axis = 0.478. 

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

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

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

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

11) IMPLICATIONS OF THE RA/ Dec ANOMALY

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

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

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

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

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

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

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

12) CONCLUSION

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

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

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Part 67- Ma’at 02 Shows no Changes (Contrary to OSIRIS Findings)


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

UPDATE 12th April 2017

The lead author of the paper in question, Jean-Baptiste Vincent, responded to my email notifying him of this blog post. I’ve pasted it below. I’d refrained from doing so before now because it mentioned new discoveries that were as-yet unpublished but as of the date of this update two recent papers appear to have covered them. The original blog post begins after “///END”.

Dear Andrew,

I finally found some time to read your blog post and think about your criticism of the claims we made in the paper I published last year. As mentioned before, I strongly recommend that you publish your work in a scientific journal. MNRAS (Monthly Notices of the Royal Astronomical Society) is planning a new Rosetta special issue, with a submission deadline of 31 March 2017. I think you should submit a paper there.
Now regarding blog post #67. I would like to first clear some misunderstanding. 

The images you mention (Ma’at pit #2), are published in far lower resolution than the raw data we used for the analysis, and the conversion to jpg often introduces artifacts. Therefore, you should consider all these images as qualitative data only, and refer to the raw data for quantitative measurements. The raw images are publicly available on the ESA server (http://www.cosmos.esa.int/web/psa/psa-introduction), with a resolution of 50 cm/px. This is the typical resolution at which we studied the comet morphology.

The ellipses overlayed on the images are not intended to mark precisely the edge of the “flow”, but rather show the region of interest where we have identified changes. It is done in this way as to not force our interpretation on the reader. It’s up to you to look at the region of interest and see for yourself whether you are convinced that the flow pattern has changed, or not.

The “after” image has been rotated in the paper, as to appear with roughly the same viewing angle as the “before” image, even though it was acquired from a very different direction due to orbital constraints (we needed to keep Rosetta in the terminator plane). After rotation,the azimuth of both images is comparable, but the elevation is still different, and this introduces a perspective distortion, which must be considered when comparing the images.

Because of the change of seasons, and different local time, the solar illumination is completely different in both images, by almost 180 degrees ! This is unfortunate, but there is nothing we could do to prevent it.

Therefore, as you noted in your post, both changes in perspective and illumination may fool us into seeing changes when there are none. I must admit that there was a heated debate in our team regarding this specific pair of images. Still today not everyone is convinced that the flow has changed, although the majority of our team members supports this interpretation, which is why we published it in this way. 

And there is an other complication: we have observed in nearby areas that a significant amount of dust has moved around (a couple of meters of dust thickness removed over 100m surface, and deposited elsewhere). This large scale resurfacing is changing the local albedo of the surface and complicates even more the interpretation.

I stand by my arguments that granular flows occur on the comet, and can repeat on a short time scale. We have multiple evidence for this and have recently submitted new papers with showing avalanches on >100m scale. The areas we mention in these papers were poorly observed by the NavCam, and the OSIRIS data is not yet public, which is why you may not have seen it already.

But how to really be certain about the changes I mention in the paper? I think the best way to get rid of most uncertainties is to work in full 3D by projecting both images on the shape model and then measure precisely the edges of the features of interest. Simply comparing features from one image to the next is not sufficient because of all the problems discussed above.

Such comparison was was not possible at the time of writing this paper. Today, we have performed a 3D “before and after” in a few specific areas of the comet, and hope to apply it to Ma’at in the coming months. 

It is also interesting to note that the crash site of Rosetta was selected especially because it would give us the chance to acquire higher resolution images of the pit and flows, with illumination conditions and viewing angle closer to what we had in September 2014. This new data set should also help us understand better if we are right with our original interpretation.
I’ll be happy to send you an update when we have progressed on this topic.

With best regards,

Jean-Baptiste

[End of email]

Here is the hi res version of the post-perihelion photo (Zoom plus original).

///END

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

The pair of photos in the header is showing the pit known as Ma’at 02, located on the head lobe at Ma’at and near the head rim. It’s also called Deir El-Medina. It will be referred to here as Ma’at 02 or just 02 where it’s clear. That’s in keeping with Parts 62-65 where it’s called 02 so as to keep focussed on its middling position between the pits Ma’at 01 and Ma’at 03. This will become increasingly important as we see the significance of the fact that 01, 02 and 03 each sit on their own delaminated layers. That will be dealt with in another part soon. 

The header pair is from the paper, J.B. Vincent et al. 2015 entitled, “Are fractured cliffs the source of cometary dust jets? Insights from OSIRIS/Rosetta at 67P”. They constitute one set of before/after observations in figure 8 of that paper, which is in section 4.3.2 “Granular flows” on page 6. There are four photos in figure 8 showing two pairs of before/after observations. The full figure is reproduced below. 

Photo 2- J.B. Vincent et al 2015 figure 8 (hereinafter referred to as figure 8). 

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

The lower pair of photos will be dealt with in a future post. These two before/after pairs are the only two examples cited in Vincent et al. 2015 that claim visible evidence for changes in the supposed granular flow structures. We shall be focussing in detail on the upper pair, Ma’at 02, in this post. As you can see, the before photo is dated September 2014 and the after photo is dated March 2015. The caption claims that there are visible changes between the two photos. It says:

“Top panel: flows from Ma’at regions between two of the active pits have changed; their outline is different and they seem to have expanded laterally.”

This post will show that their outline isn’t different and that they haven’t expanded laterally. Therefore, the so-called flows in the the before/after pair haven’t changed, or at least, haven’t changed in any discernible way.

The claim of changes between the two photos was already called into question by Marco Parigi on his blog:

http://marcoparigi.blogspot.co.uk/2016/09/67p-nucleus-changes-citizens-vs.html?m=1

In his post, Marco reproduces the figure 8 photo and says, “Little effort is made to connect the dots for the reader to try to work out for themselves exactly what is happening and why.”

This post, Part 67, uses 34 photos, meticulously annotated in close-up, in order to define the same detailed features in the two figure 8 photos and does so at the 5- to 15-metre scale. This analysis is at a scale that’s an order of magnitude more detailed than the 130-metre ellipse placed over the entire flow in figure 8, without any further guidance. 

Since there was no discernible change at Ma’at 02 between the two photos, it follows that evidence for ongoing changes, less still erosion, at Ma’at 02 is not forthcoming. The reason this is important is that if the pits are virtually dormant it says something about their morphological evolution: that they perhaps had a more active period in the past but that their current very low activity is insufficient to produce any discernible changes over one perihelion passage. 

Alternatively, it’s possible that their current low level of activity is indicative of past activity and the pits were therefore formed over many thousands of years. This second scenario might be at odds with the assumed 13,000-year inner solar system dynamical history for 67P. Lack of erosion evidence also has implications for the heterogeneity of 67P: if a few pits are capable of being self-excavated slowly over a long period rather than by some other means, there must be pockets of volatiles stashed under the surface. 

Of course, this blog has its own explanation for all the pits, which is laid out comprehensively in Parts 62 to 64 as well as informing Parts 32, 41, 52 and an upcoming part. It’s beyond the scope of this post to bring that hypothesis to bear on the lack of ongoing erosion at Ma’at 02, or at least, to do so in any substantive way. However, the upcoming part will deal with this and there’s also a very brief overview in the ‘stretch explanation for the flows’ sub-heading further down. 

Links to Parts 52 and 62 to 64 are at the bottom of this post. 

THE AIM OF THIS POST

Firstly, the aim of this post is certainly not an attempt to disprove the J.B. Vincent et al. theory that granular material may once have flowed down gravitational slopes, here or anywhere else on 67P, at some time in the past. Nor is it tasked with disproving that cliffs may have collapsed due to sublimation-induced erosion in the past (they almost certainly have to some small extent). This cliff-collapsing is cited in Vincent et al. 2015 as a precursor to granular material no longer being supported near the cliff edge. It’s claimed that the granular material therefore flows towards and over the edge as a result of the support being removed. 

Furthermore it’s acknowledged here that cliff collapse and granular flow may be ongoing processes but happening at a very slow rate. Indeed, I may have found one such collapse at Aswan but I’m waiting for better photos. If it’s borne out, it will be the sixth separate discovery of changes over the 2-year mission discovered by Marco and me (three each). All these changes show shifted material; none proves erosion in the sense of mass-wasting. This is entirely consistent with stretch theory. 

What this post does aim to do is lay out an analysis that shows there are no discernible changes between the before and after photos of Ma’at 02 in figure 8. There may have been changes which are not discernible but the apparent changes cited in the caption to figure 8 are due to the confounding effects of 1) a different viewing angle and 2) different lighting throwing different structures into more or less relief. 

THE SO-CALLED FLOWS

The flows are referred to as “so-called flows” above, despite the theoretical possibility that these features might have experienced granular flow. The modifiers are added because this post (and the subsequent one on the second photo pair) will show that figure 8 does not offer evidence that they have flowed either during perihelion 2015 or at any time in the past. Therefore, nothing of any substance can be said about these features actually being flows. Only speculative deduction based on their flow-like form can be made. Since they may be flows and look like flows, it would be reasonable to include the modifiers but it’s not reasonable to state as fact that they definitely are granular flows. However, now that this distinction has been made very clear here, they will be sometimes be referred to as flows for brevity from now. That’s because that’s what they are commonly known as and it’s unwieldy to keep saying “so-called flow”. 

It seems ‘flows’ and ‘flow’ are used interchangeably for Ma’at 02 and that’s continued here. It conveys the idea that there are several separate flow-like features within the entire area that’s itself deemed to be a flow. 

THE STRETCH THEORY EXPLANATION FOR THE FLOWS

It would be appropriate to give this explanation at this point because it would seem strange to be exhibiting manifest scepticism for the theory that these might be ongoing granular flows without explaining why they’re thought not to be. It’s true that they’re tentatively entertained here as possibly being flows that might even have flowed recently. But stretch theory points to them being a one-off drawing out of surface material during stretch. That does admit the possibility of some small amounts of more recent, vestigial flow due to light, ongoing erosion, which would be solar-radiation induced. 

The photographic evidence for stretch being the cause of the flows and any ongoing flow being negligible to non-existent will be presented in the post mentioned as being published soon.

THE CONFOUNDING ISSUES REGARDING FIGURE 8

Header reproduced

The three confounding issues regarding the comparison of the Ma’at 02 pair in figure 8 are explained briefly below before being laid out in detail under separate headings further down. 

1- the viewing angle of the before photo is about 100° displaced from the viewing angle of the after photo. This has the effect of enhancing the apparent area of certain gentle slopes in the crumpled terrain of the flow when viewed from one direction and diminishing them when viewing from the other direction. 

2- the shadowing is noticeably different between the two photos with a lighting azimuth change of around 130° relative to the local comet surface. This has the effect of enlarging or diminishing the apparent size of (presumably) otherwise unchanged features. It also results in a substantial area of the actual pit being illuminated so brilliantly that it appears not to be in the pit but on the rim. This affects 3, below. 

3- the ellipses placed over the flow that’s claimed to exhibit changes aren’t placed over the same area. The illusion in 2 has led to a misidentification of the pit rim in the after photo. This means the ellipse in the after photo has been shunted further over into the pit so as to incorporate this fake rim in the same manner as the true rim was incorporated into the ellipse in the ‘before’ photo. This has in turn led to a misidentification of the fiduciary features around the outer perimeter of the flow because the opposite side of the ellipse has been shunted over by a commensurate amount. The result is that the entire ellipse in the after photo is shunted a few tens of metres over in a direction that is towards the centre of the pit. 

1) THE VIEWING ANGLE
Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0/A.COOPER

Photo 3- (taken from Part 66). The red view on the right is the ‘before’ photo viewing angle in Vincent et al., Figure 8. The yellow view is the ‘after’ photo’s viewing angle. The circa 100° change in viewing angle means that, in the ‘after’ photo, we’re viewing Ma’at 02 from the other side and from an ‘upside-down’ position. It should be noted that the right hand, red view is, confusingly, related to the left hand ‘before’ photo of Ma’at 02 in figure 8. And the left hand yellow view corresponds to the right hand ‘after’ photo in figure 8. 

The left hand ‘before’ version in figure 8 is what one might call the classic view, taken from above the head lobe. The right hand ‘after’ version is clearly from a different angle. One might at first think that the camera has been shunted towards the top of the frame with respect to the ‘before’ version. This is indeed the case but it’s been shunted so far that it’s gone right past the vertical point over the pit and on to a similar angle on the other side. 

So we’re looking at the pit from a completely different angle in the ‘after’ photo. Both views are around 40° from the local horizontal across the pit top. This was calculated from the fact that the pit is near-circular and the apparent length/width ratio is 0.6 or thereabouts in both photos. Sin-1 of 0.6 is 38° so let’s round it to 40° because it looks around 45° or just under. So the viewpoint has been rotated by around 180°- (40° + 40°) = 100° through a plane that arcs almost over the centre of pit to the other side. 

Since it ends up on the opposite side, the two views are in almost opposite directions, about 150° to each other when looking vertically down from above and into the pit. The 150° would therefore be the azimuth change which is rather like seeing the two viewing directions as two angled clock hands against the surface without being able to see that they also slope up towards us at around 40°, that is, sloping up in the vertical plane above the pit. 

Photo 4- the azimuth change. Red and yellow are at about 150° to each other when looked down on from above Ma’at 02. This isn’t quite directly above but high enough to show the azimuth change. 
Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0

This upside-down illusion was dealt with in more detail in Part 66. That part used two very similar photos of Ma’at 02, taken from almost the same two angles as the ‘before’ and ‘after’ photos in figure 8. One was the Ma’at 02 mosaic, taken by Rosetta on its final trajectory to its controlled crash-landing. This was stitched together by Rosetta blog commenter, Gerald. The other photo used was a classic NAVCAM view. Part 66 was primarily for context for people trying to make out the viewing angle for the stitched mosaic and alerting them to the fact that there was something of an illusion at play. It was pure coincidence that Part 66 uses two Ma’at 02 photos that are almost identical views to the Vincent et al. 2015 figure 8 pair, so Part 66 acts as a perfect primer to this part. It’s worth reading that part for an extra grounding although you can still follow this part fairly well without it. Neither photo 3 nor photo 4 above is from the pair used in Part 66 to illustrate the illusion. They’re old photos showing sideways and above views simply to demonstrate the 100° angle difference for the Part 66 pair and, by coincidence, for the figure 8 pair in this part. 

Returning to the figure 8 pair, the orientation of the ‘after’ photo has been kept the same way up in the frame i.e. no rotation has been allowed. This would be eminently practical for small movements in viewpoint but with a hike of 100° right over the pit, it means that the view is essentially upside-down. It could be argued that there’s no upside-down in space but this pair of photos is being presented to humans for comparison and humans have a built-in difficulty in recognising things that are upside-down or when they themselves have to turn upside-down to keep the subject the ‘right’ way up. This is what we are being asked to do before we start comparing the two views.

And of course, strictly speaking, on 67P there is an upside-down because there is some gravity. It’s probably around 2.5e-4 m/sec^2 at Ma’at 02. So, in photo 3, the lady’s hanging-down hair really would hang down if she were standing on a tower at the required viewing point. It would just take about a minute or a bit less for her hair to succumb to the weak gravity. 

Now, let’s say for argument’s sake that we were already used to looking at Ma’at 02 from this angle because we had all been schooled in viewing the whole duck-shaped comet in upside-down mode. That would be fine as far as it goes. But then the left hand photo of the pair would be unfamiliar instead. That’s our usual, familiar view and it would be deemed to be upside-down. Because we’re human and don’t like comparing things upside-down we object and try to flip them round (rotate them 180°). Part 66 goes into more detail on this. The only problem with rotating the ‘after’ version is that the two versions then face in opposite directions when put side by side for comparison. But at least that way we can start to discern the fact that we’re now viewing the pit from the other side, which has significant implications for the apparent changes described in the figure 8 caption. 

Photos 5 and 6- the figure 8 pair with the ‘after’ version (March 2015) rotated 180°.  
Photo 7- another photo with roughly the same viewing angle as the rotated ‘after’ version. This is photo 4 without the annotations. It gives more context of the surroundings showing that we are indeed now on the other side of the pit. Ma’at 02 is dotted blue
Copyright ESA/Rosetta/NAVCAM – CC BY-SA IGO 3.0

Photos 8 and 9- these are the same as photos 5 and 6 but they’re festooned with fiduciary points that match across between the two. 
Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/A.COOPER

By toggling, you can start to see you’re looking at the pit from a completely different direction in the two photos. The biggest giveaway is the two red dips at the back of the pit in the ‘after’ (rotated) version. They can be seen in full 3D sloping down into the pit. In the ‘before’ photo, all you can see is the profile of the top rims and can’t see anything of the majestic troughs sloping down into the pit below the rim. The same goes for the pink dip. In the ‘after’ version, we’re looking directly at the back wall of the pit. In the ‘before’ version, we can’t even see the back wall of the pit because the viewpoint is way back behind that back wall. And of course, the reverse situation applies with the other side of the pit and this contributes to one of the errors explained below (the inclusion of the rugged terrace and cutting off the end of the flow).

Consequences of the two different viewing angles:

The flow in question undulates as it progresses away from the viewer in both versions of the figure 8 pair. It’s now easier to see this using photo 5 with the ‘after’ version rotated 180°. Any slope that was angled towards the viewer in the ‘before’ photo (September 2014) is liable to be somewhat foreshortened in the ‘after’ photo (March 2015) and vice versa. In fact, there’s one quite severe slope that runs across the field of view from left to right in both photos. Its left hand end in the ‘before’ photo is its right hand end in the ‘after’ photo and vice versa. The fact it runs across the field of view means that it’s very noticeable as a long, wide slope in the ‘before’ photo and much narrower (but the same length) in the ‘after’ photo. It’s angled very much towards us in the before photo so it’s somewhere near to square-on to the viewer. In the after photo it’s angled very much away from us. That’s what makes it look narrower. 

Photos 10 and 11- the sloping ridge that gets foreshortened. Outlined in white. 

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

Photo 12- shadowing of the sloping ridge proving the slope is quite marked. Just the two ends of the ridge are dotted white. This is Gerald’s stitch of the 30th September 2016 landing trajectory photo. It will be presented again below for another reason.
Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/GERALD

If the viewer doesn’t know about the angling of this slope they don’t realise that the entire flow is foreshortened along its length in the ‘after’ photo with respect to the ‘before’ photo despite the actual flow being unchanged. Without that knowledge, it’s reasonable to think that the flow has changed shape between the two photos. More importantly, if the flow as a whole has been foreshortened along its length in the ‘after’ photo due to the foreshortened ridge, it means by definition that the length-to-width ratio has been reduced which is another way of saying that the flow now appears wider. This is what the figure 8 caption says: “they seem to have expanded laterally”. We shall see below that there are two other factors that contribute far more to the lateral expansion illusion, but this foreshortening of the length due to the 100° difference in viewing angle also makes its own contribution, exacerbating the illusion still further. 

The foreshortened slope shown above is the most obvious one in the flow but there are several other less dramatic slopes. Combining them makes for a flow that appears to change shape and texture before our eyes over 6 months when in fact it hasn’t changed in any discernible way at all. If you laid tin foil on a table and crumpled it to a similar degree as the flow, it too would appear to change shape and texture if you viewed it from either side of the table. 

2) THE LIGHTING ANGLE

Header photos reproduced for easy reference. The ‘after’ photo is rotated 180° (not numbered).

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

The azimuth angle of the sun to the local surface across Ma’at 02 swings by around 130° between the ‘before’ and ‘after’ photos. This makes the shadowing noticeably different between the two photos and it has the effect of enlarging or diminishing the apparent size of otherwise unchanged features. This exacerbates the foreshortened slope issue described above, which leads to the flow being slightly foreshortened along its length dimension in the ‘after’ photo. 

Moreover, the lighting angle change allows the flow’s outer perimeter to creep, so to speak. Its edge is sharply delineated by shadowing in the ‘before’ photo because there’s a fairly steep-sided trough running along behind the perimeter’s leading edge. This is thrown into shadow in that photo. However, this trough is rather whited-out in the ‘after’ photo. This makes the flow and the smooth, dusty area beyond it look to be roughly on the same level as if the trough that had been shadowed in September 2014 has been smoothed out into the dust of Ma’at by a few metres. It would have to be at least a few metres in order to smooth the sharp, steep perimeter slope of the trough into a shallow, blended slope. This blending is again, an illusion. The reason we can be sure of this is that photos from much later, including Gerald’s stitch* of the mosaic (September 30th 2016, 18 months later) show that the flow perimeter has apparently returned to its old September 2014 shape and position, i.e. contracting from the apparently spread-out, washed-out version of March 2016 and remembering its old shape. This is of course impossible so if September 2014 and September 2016 look the same it means the peregrinations of March 2015 never really happened and are a trick of the lighting. 

*To allay any fears that Gerald’s stitching process may have caused anomalous artefacts in the outer perimeter shown, we can see it along with the relevant unstitched mosaic component below it. This shows that no stitching was required in the actual flow area because the entire flow was caught within one mosaic frame. 

Photos 13 and 14- photo 13 is photo 12 reproduced: Gerald’s stitch of the 30th September 2016 mosaic component of Ma’at 02. Photo 14 is the relevant mosaic frame. Gerald’s stitch shows the flow looking essentially the same as it did in the Vincent et al ‘before’ photo that was dated September 2014, two years before.
Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/GERALD

In photo 13 (September 2016) the flow may be completely unchanged from two years before, in September 2014 or there may be some as-yet unidentified changes. But just like the Vincent et al ‘after’ photo, the viewing angle is on the wrong side of the pit for making subtle comparisons with the ‘before’ photo. However, the point being made here is that seeing as the view is on the same side as the ‘after’ photo (March 2015) it can be compared with that photo and it looks substantially different from it. This might suggest at first glance that change really was afoot but, on closer inspection, it’s different in such a way that makes it strikingly similar to the September 2014 ‘before’ photo. This suggests the flow never did change up to March 2015 (the ‘after’ photo) and that the changes in that photo are an illusion. 

The extent to which the 2016 mosaic frame is similar to the ‘before’ photo and different from the ‘after’ photo is entirely consistent with the illusions caused by the three points being presented here: viewing angle, lighting angle and ellipse placement. These three factors aren’t simply presented here as general confounding issues. Further below, we shall carve out the exact anomalous chunk from the ‘after’ photo that arises from the lighting illusion and also add in a section that arises from the foreshortening illusion. As a result, we shall end up with the true shape of the flow itself in the ‘after’ photo, which will turn out to be the same shape and size as depicted in the ‘before’ photo. 

The creep issue, related to the lighting angle change, also applies to other lumps and dips within the flow and its this creep that exacerbates the foreshortened slope issue, extending and retracting the periphery of each lump and dip. The outer flow perimeter just happens to be the most obvious example of this because it’s a continuous dip or trough, meandering along the perimeter and is set against the smooth, untouched dust of Ma’at. It’s easy for the lighting angle (from above the head lobe) to illuminate the trough in the ‘after’ photo and thus blend it into the smooth dust as if it has spread out. When lit from the other side, as in the ‘before’ photo and also the mosaic photo, the trough reappears markedly. It reappears either side of March 2015 (i.e. in September 2014 and September 2016) proving the blending illusion in the ‘after’ photo. Nothing has changed in any discernible way over the two-year mission. 

Photos 15 to 17- the outer perimeter of the flow is well-defined by the same lighting angle in both September 2014 and September 2016 but whited and smoothed out in March 2015. 

Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA. For photo 16 (Sept 2016) stitching credit: GERALD

In photos 15 and 16, the trough running along just inside the perimeter is in shadow. However, the perimeter is whited out in photo 17 (March 2015) due to lighting from circa 130° towards the opposite direction. This illuminates the trough, giving the impression that the flow perimeter has spread out further into the dust of Ma’at.  

The illusion of the blending of the perimeter into the Ma’at dust is greatly exacerbated by the misplacing of the white, dotted ellipse in the figure 8 ‘after’ photo. The ellipse perimeter ends up around 20 metres closer to this apparently blended perimeter line, thus giving the illusion that the already non-existent blending extent is 20 metres and not a few metres. This ellipse placement error is discussed in more detail under its own heading below. 

The circa 130° lighting angle change also results in a substantial area of the actual pit being illuminated so brilliantly that it appears not to be in the pit but on the rim. This is by far the greatest illusion between the ‘before’ and ‘after’ photos. From the figure 8 caption statement, “and they [the flows] seem to have expanded laterally”, it appears that this illusion has gone unnoticed. The authors have added the section that’s inside the pit in the ‘before’ photo to the rest of the flow in the ‘after’ photo. The section in the pit clearly isn’t part of the flow in the ‘before’ photo but again, because of white-out from much higher illumination, the lumps and dips in this area of the pit get smoothed together in the ‘after’ photo. In addition they’re apparently smoothed into the flow, increasing its width. But the smooth area of the flow actually starts on the true rim, about twenty metres above all these rugged bumps and dips. Their rugged nature betrays their true nature: they’re consolidated material sitting on a terrace, below the pit rim and therefore very much inside the pit. 

Photos 18 to 20- the rugged terrace perimeter is pale blue and the true pit rim is pale yellow. Other colours are fiduciary points: three light blue boulders, a fuchsia crack (not visible in the ‘before’ photo due to the 100° viewing angle change) and a dark blue ridge. These fiduciary points aren’t in photo 18 but are in its reproduced version in photo 25. 

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

Much work went into identifying the pale yellow rim across the apparently featureless ‘after’ photo. This was done using the ‘before’ photo as a reference for finding fiduciary points along the rim line (those fiduciary points are not shown here). Also, these fiduciary points on the ‘before’ photo were cross-referenced with other Ma’at 02 photos for confirmation. Photos were annotated with the yellow rim fiduciary points including the ‘before’ and ‘after’ photos. The after photo does exhibit enough faint features to catch the rim line. These photos will be added in an appendix in due course for completeness. However, if you look hard enough at the originals (using the annotations only as an initial guide), you can see the rim on the after photo. It takes some time though. 

UPDATE (INLINE APPENDIX- 5th November 2016)

As promised above, here are the photos with the fiduciary points for the pale yellow rim line. They’re in the following order with an ‘A’ for Appendix. 

A1-‘before’ photo with different, coloured dots acting as fiduciary points sitting on specific features (shape vertices and lines) near the pale yellow rim and across the flow area. 

A2- before photo ‘original’ but this is actually a zoomed in crop of an earlier version which still has the other annotations but is missing the ones dotted near the rim and on the flow as in A1 above. 

A3- the ‘after’ photo with the same coloured fiduciary dots as A1, i.e. placed on the exactly the same features near the rim and on the flow. 

A4- the ‘after’ original in the same manner as A2 is presented for the ‘before’ photo. 

A5- Gerald’s stitch with the same fiduciary points.

A6- Gerald’s stitch (original).

Key for the fiduciary dots on all three photos follow.

Light grey, brown, beige and slate blue single dots denote tips of pointed sections on the outer flow perimeter. Brown is more elusive than the rest.

Yellow- this nestles at the tip of a triangular set-back at the top of the slope and at the back end of one of the ‘fat tip’ areas. See further down for the fat tips. 

Green- this sits at one end of a small, smooth, straight section. The section is a narrow rectangle and other end of this rectangle is the slate blue tip. The width of the rectangle separates the fat tip area (whose apex is the slate dot) from the slope. 

Two dark blue dots- these sit on the two tips of a a swallow-tail feature at the back end of the area that leads across to the beige-dotted tip. The after photo shows the whole length of this area, from blue pair to beige as being one distinguishable area due to the apparent sweeping direction of the flow between them. This is partially apparent in the other two photos as well. 

Terracotta- a single dot along the sweeping flow between the blue pair and beige. This denotes a dip. 

Curving slate blue line- this is an obvious majestic curve. 

Light mauve- adjoining the slate blue line, this is a zig zag line. 

Fuchsia outcrop- this is actually part of the original annotations. The pale yellow line shows it consistently sitting inside the pit on the correct side of the rim. However, without any annotations, the outcrop apparently jumps out of the pit in the ‘after’ photo and sits perched on the rim with two sides bordering the flow. In the ‘before’ photo, just the tip of the fuchsia outcrop can be seen and it’s clearly in the pit. In Gerald’s photo, more of it is visible because it’s the same viewpoint as the after photo. It certainly looks more in the pit than out. The tip is in the pit, as for the ‘before’ photo and the rest is forming the pit sidewall, i.e. it’s inside the rim. 

The crucial point here is that Gerald’s photo and the after photo are from the same viewpoint and yet the everything left of the pale yellow line in Gerald’s is in shadow beyond the rim. This is in complete contrast to the ‘after’ photo that apparently shows the same area to the left of the pale yellow line as being just an ordinary continuation of the flow. It appears to be at about the same level as the rest of the flow, continuing on to a rim that would be about a third of the way across Gerald’s pit. The obvious drop-away into the pit in this portion of the ‘before’ photo shows that this can’t possibly be part of the flow. It therefore corroborates Gerald’s shadowed area that suggests a sharp drop-away too. So it’s clear that the area to the left of the pale yellow line in the ‘after’ photo is in the pit, beyond the rim, and yet appears to be part of the flow above the rim. 

All the fiduciary points marked in A1, A3 and A5 are apparently much further from the pit rim in the ‘after’ photo even though they have been very carefully placed on the same features. The most obvious example is the pair of dark blue dots on the swallow tail. The swallow tail is definitely right on the pit rim. And yet it’s apparently marooned in the middle of the flow in the ‘after’ photo. This is further corroborated in photo 18. The blue dots aren’t marked but the swallow tail tips can be made out fairly easily. They’re at the 8th and 13th pale yellow dots from the bottom. According to the ‘after’ photo, these two dots should be right out in the middle of the flow. And yet they’re on the rim in photo 18 as well.

UPDATE 2 (same date)- I decided to annotate photo 18 with the swallow tail dots after all. It’s important because identifying the shape of the swallow tail precisely, allows us to clinch the argument for the tail being apparently in mid-flow in the ‘after’ photo but on the rim in the before photo.
Photos A7 to A10 are variations on photo 18. 
A7-shows two larger dark blue dots on the central areas of the two fins of the tail and two small blue dots at the fin tips. It also shows the beige pointed tip and the terracotta dip (whited out). 
A8- shows the same as A7 but with the perimeter of the shape that’s obvious in the ‘before’ and ‘after’ photos as an identifiable shape in the flow- almost like an embedded mini-flow. With a swallow tail and a beak, it doesn’t quite look bird-like but it is a characteristic shape that can also be discerned in the ‘before’ and ‘after’ photos. Notice the characteristic, curved dip between the two fins that is especially visible in the ‘before’ photo, sitting right on the rim and just visible in the ‘after’ photo. 
A9- photo 18 reproduced to allow a dot-free analysis of the swallow tail while still having the pale yellow rim for guidance. 
A10- the true original with no dots for a completely dot-free analysis of the swallow tail and rim. 
The upper fin in A7 to A10 stays resolutely on the rim in all photos. The lower fin in A7-A10 appears to drift from the rim and across the flow by a few metres when viewed in ‘before’, ‘after’ and Gerald’s stitch. This is probably due to the A7-A10 photo being very well-lit and the other photos being subject to the same shadow-drift or ‘creep’ anomaly as described further above. However, the drift is small and it’s just one part of the whole swallow tail, which as a whole, is on the rim in the ‘before’ photo and halfway across the flow in the ‘after’ flow. 

Photos A7 to A10
And finally for this update, the ‘after’ photo with the same shape in beige with original (A11 and A12). Fins are in the centre of the flow.

END OF UPDATES/////

Photo 18 is another one of the last photos of Ma’at 02, taken during the landing phase and it’s a very good overhead shot. Photos 19 and 20 are our usual ‘before’ and ‘after’ photos and the originals for all three follow them. The area enclosed by the pale blue and pale yellow lines is the same in all three photos and is the rugged terrace area. It’s the same area even though it changes shape due to the very different perspectives. There’s also some shadowing of the area in the ‘before’ photo. In that photo, the light blue dots turn to a darker blue in the shadow to show it’s an estimated line. And there’s some obscuration of the rugged terrace by the rim in the ‘after’ photo. This area enclosed by the pale blue and pale yellow lines, the rugged terrace, has been added to the flow area, in error, in the ‘after’ photo, March 2015. 

The spurious addition of the rugged terrace to the flow greatly increases the apparent width of the flow in the ‘after’ photo when compared with the ‘before’ photo. This illusion certainly would lead us to believe that the flow had “expanded laterally” when in fact it remained unchanged. 

The additional area taken up by the rugged terrace is on the opposite side to the outer perimeter that appears to be blended by 20 metres due to the ellipse border being placed 20 metres closer to the perimeter than in the ‘before’ photo. This means apparent width has been added to both sides of the flow. So with the inclusion of the rugged terrace, the apparent extension of the width of the flow is even greater. 

3) THE MISPLACING OF THE ‘AFTER’ PHOTO ELLIPSE

Photos 21 to 23- the misplacing of the ellipse. Photo 21 shows the ‘after’ ellipse superimposed over the ‘before’ photo, showing the discrepancy. Photos 22 and 23 are the originals with their respective ellipses for comparison. 

Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/A.COOPER (photo 21)

Further below we shall see how the ellipse crosses the outer flow perimeter in a different place in the ‘after’ photo from its crossing point in the ‘before’ photo. However, the most important things to note in the meantime are:

a) the inclusion of the rugged terrace by shunting the ellipse over into the pit in the ‘after’ photo.

b) the noticeably narrower area of smooth dust enclosed between the outer flow perimeter and the ellipse perimeter in the ‘after’ photo. as compared with the wider area of smooth dust enclosed in the ‘before’ photo. You can use the terracotta dots as a guide so you know what it is you’re looking for but then it’s best to toggle between the two original white-dashed ellipses to see this different-size area of smooth dust outside the flow perimeter. The markedly narrower area in the ‘after’ photo is enclosed between the 9 o’clock point and (almost) the 12 o’clock point on the white ellipse.

Photos 24 and 25- photo 24 is photo 18 reproduced. It shows the rugged terrace from overhead. Photo 25 shows the crucial part of the ‘after’ photo ellipse projected onto photo 24. The original follows. No apology is made for the weird ellipse shape. It follows the fiduciary points impeccably and is thus shaped due to the fact that the ‘after’ photo ellipse was projected onto different topographical layers (pit rim and pit base) causing parallax, and projected from a substantially different direction. 
Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/A.COOPER

Key:

Pale yellow- the pit rim as depicted very obviously by shadowing and topography in the ‘before’ photo (see the fuller description further below). 

Pale blue (medium size)- the perimeter of the rugged terrace, which is low down in the pit. The official term in the OSIRIS papers for ‘rugged’ is ‘consolidated material’ which is considered as being rocky in appearance, presumably comparatively solid (consolidated) and without any dust covering. In other words, consolidated material is wholly different in nature from granular flow that has to comprise dust and grit by definition. 

Pale blue (small)- the very small pale blue dots denote the best assessment of the bottom of the slope that runs between terrace and rim. This slope is also rugged and is shown here to be part of the area that’s been added to the flow area in the ‘after’ photo. In that sense it should be considered as being part of the rugged terrace for our purposes: a rugged area that’s been mistaken for a smooth, flowing area. 

In photo 24, the true pit perimeter is dotted pale yellow. This is the same line for the rim as is strongly suggested in the Vincent et al. 2015 ‘before’ photo. In that photo, this line is the dividing line between the obvious flow-like features bordering the rim and the steeply sloping, rugged terrain that drops away into full or partial shadow. The difference in terrain type either side of the rim (flow versus rugged) is very obvious in both photo 24 photo and the ‘before’ photo. However, it’s not at all obvious in the ‘after’ photo where the rugged terrace, including the rugged slope, is included and is assumed to be part of the flow. 

The spurious addition of the rugged terrace in the ‘after’ photo means that the rim of the Ma’at 02 pit has been misidentified in that photo whilst it’s been correctly identified in the ‘before’ photo. This leads to the ellipse being shunted over by about 20 metres, possibly 30 metres, in the ‘after’ photo. It’s shunted right into the pit area in the belief that the perimeter of the ellipse is just about enclosing the pit rim and therefore the flow perimeter that starts at the rim. But the ellipse perimeter is in fact enclosing the rugged terrace that’s well inside the pit.

Since the ellipse gets shunted by 20 or so metres on the pit side, it also gets shunted by the same amount on the other side at the outer perimeter of the flow. Otherwise, the ellipse would get fattened into a circle and the shunting error would be noticed immediately. 

This shunting means that the point at which the ellipse crosses the outer perimeter of the flow is further along the perimeter in the ‘after’ photo than in the ‘before’ photo. This discrepancy is about 20 metres which is roughly commensurate with the rugged terrace width of around 30 metres. 

The ellipse doesn’t only cross the outer perimeter of the flow 20 metres too far along. It also gets placed about 20 metres closer to the unchanged flow perimeter. This means there’s a 20-metre-narrower area of unblemished dust between the flow perimeter and the ellipse than in the ‘before’ photo. This is done in error and for no good reason. The viewer can’t help but look at the flow perimeter being substantially closer to the ellipse perimeter in the ‘after’ photo and assume that it has crept towards the ellipse perimeter by that amount between September 2014 and March 2015. But the simple truth is that it’s the ellipse perimeter that has crept towards the unchanging flow perimeter by virtue of being misplaced. 

This apparent movement towards the ellipse perimeter is exacerbated by the blending illusion described above where white-out in the ‘after’ photo seems to smooth the trough along the perimeter further out into the dust. 

When trying to identify changes with a +\- error of 2 metres, a 20-metre ellipse placement error is an order of magnitude greater than the error bars. This is in addition to the mistaken inclusion of the rugged terrace. Including the circa 30-metre-wide rugged terrace in the flow adds up to 50% to the flow’s width (50% across its central width, 10-40% either side- see below). This completely skews its apparent shape, rendering the comparison of the ‘before’ and ‘after’ photos in figure 8 completely useless. This is of course unacceptable.

Recapping the figure 8 caption in Vincent et al. 2015, it says:

“Top panel: flows from Ma’at regions between two of the active pits have changed; their outline is different and they seem to have expanded laterally.”

The changes in the flow are stated as fact in the caption whereas the analysis in this post shows there are no discernible changes in the flow at all. 

The statement above should be withdrawn along with the photos and an erratum issued for this paper regarding the certainty of ongoing granular flow or indeed any granular flow, ongoing or historical, on the comet. 

We’re still left with the hypotheses that granular flow may well have happened in the past but that is all. 

QUANTIFYING THE EXACT ANOMALOUS CHANGE IN THE SHAPE OF THE FLOW DUE TO THE RUGGED TERRACE INCLUSION AND ELLIPSE MISPLACEMENT

Photos 26 to 28. This is a sequence of three annotations showing essentially the same two things: (1) the extent of the rugged terrace area that was added and (2) the circa 20-metre section of the outer flow perimeter that was cut off the end. This sequence of pairs is the original header without the ‘after’ photo flipped. This is so as to compare the flow shape more easily in the context in which it was originally intended to be appreciated in figure 8, along with its caption. 

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

Photo 26 shows the before/after pair with the flow shape perimeters dotted in red as the viewer can only be expected to perceive them, given the visual cues and the explanation in the figure 8 caption. 

Photo 27 has the ‘after’ photo showing the true perimeter of the flow without the rugged terrace included. The ‘before’ photo is unchanged because it never included the rugged terrace nor was it truncated along its outer perimeter. You can now see the extra chunk of the flow that was cut off due to the 20-metre error. It’s encroaching into the ‘before’ photo because it was cut right out of the ‘after’ photo. 

Photo 28 shows the rugged terrace area in pale blue just to show how much area had to be removed in order to show the correct area of the flow. The measurement across the pinched central dimension of the true flow area is the same as the widest part of the rugged terrace area. This means the flow area’s width was doubled, no less, across its central dimension. It’s no wonder the flow looks like a fat rectangle in the ‘after’ photo. Its true, pinched shape is hiding in plain sight within the whited-out area. 

The 20-metre cut-out is shown in pale green in photo 28 as well. 

It should be stressed that, as with the rim perimeter of the flow, the true outer perimeter isn’t guessed. It’s been meticulously researched using several photos and by matching fiduciary points between them. This was needed to understand exactly where the ellipse crosses the outer perimeter in the ‘before’ and ‘after’ photos. 

IDENTIFYING THE EXACT DRIFT ANOMALY OF THE ELLIPSE PLACEMENT ALONG THE OUTER FLOW PERIMETER

You may wonder how different points on the the outer flow perimeter could be misidentified, since certain wavy sections look fairly obviously the same in both photos. However, there are also two different sections that look uncannily similar in both photos. The manifold illusions described above have resulted in a particular fat, pointed area on the flow perimeter in the ‘after’ photo being mistaken for the next fat, pointed area along in the before photo. That next area is highly visible in the ‘before’ photo and is at the true end of the flow. It’s the fat point that should have been chosen in the ‘after’ photo but it’s hardly visible there for reasons laid out below. The incorrect fat point that was ultimately chosen is the first one along from the true end point. It was probably chosen because it was:

a) the first discernible point along the perimeter in the ‘after’ photo. This was due to white-out and foreshortening of the true, first point.

b) this second point is roughly the same shape in the ‘after’ photo as the first point in the ‘before’ photo. 

Photos 29 to 31- the five fat points along the outer perimeter of the flow. The dots in photo 29 are just outside the tips of the points. Photo 30 shows the notional fat bodies of the points. Photo 31 shows them named ‘a’ to ‘e’. It should be noted that the light grey dot is sitting just inside its fat body and on the inside of its point tip. That’s because the rounded point tip is off-frame. The rounded tip is very obvious in all the other photos so it’s easy to judge that it’s only just off-frame. 
Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/A.COOPER

Incidentally, this being another September 2016 landing trajectory photo, it was taken within about an hour of Gerald’s stitched mosaic. Again, you can see that the outer perimeter, as defined by the coloured dots, is remarkably similar to the September 2014 ‘before’ photo perimeter. There are white-out issues in places but you can just about see the same line. 

It was d in the ‘after’ photo that was mistaken for e in the ‘before’ photo. e is the true last point at the end of the flow. e’s point is pink but e has a white perimeter because it comprises the sloping ridge mentioned above that’s already dotted white. Yellow and green are fiduciary points for other photos that may get added later. 

Incidentally, there’s a triangle between c and d that exhibits slight flow-like appearance but it’s less marked than the fat points along the rest of the perimeter. It was left out from being annotated in all three photos because it’s confusing in that it’s more apparent in some photos than others. The main thing is that it is there in all three photos and doesn’t change in any discernible way. That is itself one small additional piece of evidence for the flow not expanding laterally as claimed. 

This mistaking of d for e along the perimeter was possibly an error that was forced by the exigencies of moving the ellipse further over into the pit which was itself an error. In other words, it’s an error that might not otherwise have been made. However, if you were ever going to mix up two of the points along the outer perimeter, it would be these two. This is because the misidentified point, d, sits right at the top of the well-defined slope described in (1). That’s the white-dotted slope that’s much-diminished in the ‘after’ photo due to foreshortening. The correct fat point, e, is at the end of the flow perimeter. It’s perfectly visible in the ‘before’ photo and, crucially, its fat area (next to its actual pointed tip) comprises the very slope in question, the foreshortened slope. So, when trying to identify point e in the ‘after’ photo, it’s not there or it’s so hidden by foreshortening that it’s not very easy to see at all. This is compounded by the different lighting angle that whites out the sloping ridge anyway (it’s directly facing the sun) and smudges any vestige of the ‘e’ point because the ‘e’ point itself comprises the sloping ridge. In fact, the e point was so elusive that it was cut right out of the ‘after’ photo.

Photos 32 to 34- photos 32 and 33 show the foreshortened white ridge in relation to the fat points numbered a to e. You can see how the ridge comprises e and also that e’s shape in the fig 8 ‘before’ photo looks similar in shape to d’s shape in the ‘after’ photo. Photo 34 is Gerald’s stitch again, showing a similar angle to the ‘after’ photo but from a bit lower down.
Copyright: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/A.COOPER/GERALD

Gerald’s photo shows the e fat-point perimeter in white (i.e. the end of the foreshortened ridge). This is somewhat guessed but that upper, white portion is nevertheless fairly accurate because there’s a faint white streak beyond it, leading to a double-bumped, pointed outcrop beyond. Those fiduciary features are just about visible in the ‘before’ photo and are foreshortened in that photo due to the 100° viewing angle change. In the ‘before’ photo, the white streak is almost invisible to the right of the dotted ellipse (and may admittedly be an artefact of the ellipse-dotting). The double bump, however, definitely does have highly foreshortened terrain between it and the bottom of the sloping ridge. Since we’re looking into a U-shaped dip in both photos, the white ridge is fairly square-on in the before photo while the white streak and outcrop are foreshortened. And conversely, in the after photo, the white ridge is foreshortened while the streak and double bump are more square-on. This apparently mind-numbing detail will help us greatly in understanding the stretch theory explanation for the flows, coming out soon. 

One might say that the features either side of this wrongly chosen point (which is point d at the top of the slope) should have shown up the error. However, the white-out and creep issues described above for the ‘after’ photo cause the fat area of the d point in that photo to mimic the shape of the e point area in the ‘before’ photo. The e area (sloping area) is perfectly visible in the before photo but now all but disappeared in the after photo, with its tip actually disappeared, off frame. If you hunt for it without knowing it’s disappeared you are liable to pick this same-shaped d area that’s actually the next area along, starting at the top of the slope. 

Just to complicate matters further, the somewhat larger, flat area that’s visible on the other side of this wrongly chosen d area in the ‘after’ photo, does apparently match between the two photos and is apparently correct. How can this be if the correct area (area e) is between the two? Again, it’s because area e has all but disappeared due to the foreshortening and white-out effects. This allows the area at the top of the slope to appear to join seamlessly to the larger area at the bottom of the slope. But the large area at the bottom of the slope has area e between it and area d at the top of the slope. 

The ‘after’ photo ellipse therefore cuts the outer flow perimeter off at the second-to-last fat point tip (the tip of area d), which is about 20% of the way along the perimeter length from e, thus shortening the flow’s outer perimeter length dimension by 20%. But since the ellipse is supposed to be showing the same area (because it crosses what is thought to be the same place as in the ‘before’ photo, point e), we’re led to believe that the outer perimeter length presented within both ellipses is the same. But the the flow perimeter in the ‘after’ photo is missing 20% of its length while masquerading as the full-length flow. This might seem automatically to result in an apparent 20% apparent extension in width by virtue of the law of proportionality. This is without even the addition of the rugged terrace. It’s probably more like 10% owing to the vicissitudes of the 100° difference in viewing angle.

Once this notional 10% is factored in, we still have the fact that the ellipse perimeter is 20 metres closer to the flow perimeter in the ‘after’ photo leading us to believe that it’s the flow perimeter that has crept towards a stationary ellipse by 20 metres. But the flow perimeter has stayed rock solid within the bounds of discernibility. 

The 10% is a length-to-width proportionality illusion. The 20-metre creep towards the ellipse is a translational shunt illusion. The foreshortened length due to the foreshortened slope is a also a length-to-width illusion. The rugged terrace is an actual inclusion of a real area, widening the flow even more. All four illusions work together to widen the flow substantially between September 2014 and March 2015. But in reality it remained unchanged. 

And indeed, it appears to have remained the same right up to the very last photo taken of Ma’at 02 in September 2016. 

This is notwithstanding further scrutiny of all the available photos on a much finer scale (at the same level of resolution as was conducted in this post) but using photos with similar viewing angles and lighting. I haven’t had the time to do this because of having a backlog of posts on other areas of the comet.  

CONCLUSION

This post has studied the ‘before’ and ‘after’ photos in the upper pair in figure 8 of Vincent et al. 2015. By analysing them at the 5- to 15-metre scale, it has been found that there were no discernible changes to the so-called flows between September 2014 and March 2015. This is in direct contradiction to the figure 8 caption which states:  

“Top panel: flows from Ma’at regions between two of the active pits have changed; their outline is different and they seem to have expanded laterally.”

It would be appropriate here to make a comment about the scale at which OSIRIS appear to be analysing the comet and the scale at which Marco Parigi and I are analysing it. Although OSIRIS have a shape model that’s accurate to the 5-metre scale, the photos in the OSIRIS papers are rarely analysed or annotated at a scale of less than 50 to 100 metres. In contrast, Marco and I consistently analyse the comet at the 5- to 15-metre scale, zooming right out to the 100m scale or more for context, then zooming back in to the 5- to 15-metre scale again for continuing the analysis on the adjacent area. This is reflected in our blog posts and our photo annotations. 

It would have been impossible to find the fiduciary points necessary to ascertain no change in the Vincent et al 2015 photo pair without employing this method and doing so using multiple photos from different angles (including many not actually reproduced in this post). This necessitated 34 annotated photos and 6000 words. This is because the flow area was scrutinised at the ~10 x 10 metre scale, meaning the flow was potentially divided into about 100 small areas instead of one big area (i.e. what was enclosed by the ellipse in figure 8). Around 20 of these smaller areas were shown via annotation and at least as many more were ascertained for orientation purposes but not used in the post. 

It’s therefore impossible to describe the shear intricacy of the Ma’at 02 pit in 23 words and two photos as figure 8 attempted to do. 

This assiduous approach to our analysis is why the four corrections I’ve made to OSIRIS papers to date (including this post) aren’t serendipitous finds. It’s borne of a rigorous analysis of almost the entire northern hemisphere down to the 5- to 15-metre scale. The same goes for Marco’s recent correction of the El Maarry et al. 2016 OSIRIS paper regarding the placing of the Anuket/Sobek border and notifying them of a cliff collapse. I made two further corrections in that paper (counting together as one of the four corrections mentioned above). All these corrections in El Maarry et al. 2016 were kindly acknowledged and corrected. One of the other two corrections I made for other OSIRIS papers was also acknowledged and corrected. 

I’m therefore loath to criticise sharply because Marco and I want to continue to be of help. However, it has to be said that, until the comet is scrutinised assiduously at the 5- to 15-metre scale, its morphological evolution won’t be understood. Analysing at this scale doesn’t furnish us with the last 5% of understanding; it furnishes the first 95%. It’s why this, the stretch blog, and Marco’s blog are currently well-advanced in understanding the evolutionary morphology of 67P. There are many additional discoveries of sub-mechanisms that drive morphological change and that have followed on naturally from what is now a nuanced understanding of the main stretch mechanism. A number of these sub-mechanisms are already documented in the previous 66 parts. Several more are still to come, along with dozens of parts showing examples of how these sub-mechanisms dramatically reshaped specific features. 

LINKS REGARDING MA’AT 02 EVOLUTION AND DELAMINATION

Part 52 

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

Part 62 (click through to Parts 63 and 64) 

https://scute1133site.wordpress.com/2016/09/18/part-62-the-morphological-evolution-of-maat-01-02-and-03/

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/GERALD