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.