I've been improving my calculations, and the latest edits to my posts contain those improvements, but here's another summary that I think is my best yet:

1. Torque-free precession is when there is practically no external torque (i.e. the tiny torques due to Luna & the Sun are negligible) . Because there is no external torque, Earth's spin angular momentum vector is constant. But that doesn't mean that the same piece of turf has to be under the pole all the time. Earth can move so that different pieces of turf are under the pole. The "pole" (i.e. latitude & longitude of the piece of turf) can move (i.e. "precess") a lot, relative to Earth coordinates, even though the angular momentum vector is unchanged in "absolute" coordinates. (When this happens there is also a slight daily wiggling of Earth's spin vector in absolute coordinates, due to Earth's slight asphericity and Earth's being somewhat askew, which causes the spin vector to have a direction slightly different from the angular momentum vector, but that's pretty negligible.)

2. Earth has almost perfect symmetry about the z-axis of its spheroid. So, two of Earth's moments of inertia are equal, both slightly smaller than the z-axis moment of inertia. The equality of those two x & y-axis moments, simplifies "Euler's equations" (Euler's equations amount to a statement of conservation of angular momentum, expressed in rotating Earth coordinates by way of an expression for "Coriolis force" applied to the angular momentum vector itself). Then it's pretty simple to rearrange Euler's equations to show that Earth can "precess" so that the pole moves slowly along a line of Earth latitude. Even without Euler's equations, just considering conservation of energy and angular momentum, it is evident that two variables, (1) the latitude of this precession line, and (2) the period of Earth's rotation, are determined by two equations involving angular momentum and kinetic energy of rotation and Earth's moments of inertia. Angular momentum is a known constant, the same as it is today, but I need to know the kinetic energy (see #4) and the moments of inertia (i.e. the flattening) (see #5).

3. As long as the changes in kinetic energy and angular momentum are small, the instantaneous period of Earth's sidereal rotation changes only by a few seconds: e.g. 56 sec shorter, in the 18.90deg precession solution discussed below. But the latitude of the precession line, i.e. angle of precession, can become large with only small changes in kinetic energy and no change in angular momentum. Basically this is because Earth's flattening is small. When Earth's kinetic energy is exactly what it is now, all Earth can do is rotate with nearly perfect symmetry about its z-axis, as it does now; but if Earth somehow acquired more kinetic energy of rotation, it could precess at a large angle.

4. Earth's gravitational self-energy is large; about 1000x Earth's kinetic energy of rotation. The energy released by, say, a major decrease in Earth's flattening (like energy released by a falling weight or a skater pulling in his arms) is of the order of magnitude to enable large-angle torque-free precession. My best effort to calculate this energy of flattening, is that it is

8/5 * (k/3)^2 * Earth's total gravitational self-energy

"k" is the flattening ratio, 1/297. (In practice I make small adjustments to account for my belief that the core does not participate in this torque-free precession.) Half of the above energy, a "4/5" is simply due to generally greater distances between mass particles, when Earth is flattened. Half of the energy, i.e. a "4/5" is a "monatomic adiabatic perfect gas response" to the other half, assuming Earth isn't truly "incompressible": when Earth is less flattened, forces are greater, pressures are greater, and Earth shrinks slightly to a new equilibrium, releasing yet more potential gravitational energy, namely an "8/5" but due to "the virial theorem", half of that energy goes into heat, and half, another "4/5" is available for mechanical work, so 4/5 + 1/2 * 8/5 = 8/5 is the answer.

5. I also need to know how much Earth flattens. I simply averaged all the spheroids one would get with the poles at different points along the line of latitude. The answer is that the flattening is lessened by the factor

1 - 3/2 * sin^2 (alpha)

where alpha is the colatitude of the precession, i.e. the central half-angle of the precession cone.

6. Using my estimates of flattening and gravitational energy release, I have all that I need to solve those two equations in two unknowns, for Earth's spin rate and precession angle. I find that there are two solutions:

I. What is happening today.

II. Torque-free precession along the 71st (lat 71.10) parallel, with less than a minute difference in day length.

Between these two solutions are "energy trough" states, i.e., states that are easily traversible. I & II are like the two extreme swings of the pendulum.

7. Now here are the two reasons to believe that I am right in #4 and #5:

a) There is another equation, obtained from Euler's equations, that gives the period of the torque-free precession. The period turns out to be a year; if my estimate of the gravitational potential energy release, is increased by only 2% (believable, considering the crudity of my approximations) then the precession period is exactly one tropical year.

b) The torque-free precession is at such a latitude that it would explain Hapgood's estimates of the poles. With a precession period of one year, having the pole at Hudson's Bay in winter has about the same effect on ice cap position, as having the pole at Hudson's Bay year-round. Hapgood estimated the last three Ice Age pole latitudes (in today's Earth coordinates) as, most recent to least recent, 60, 72, & 63deg latitude.

The Cholula pyramid in Mexico is the world's largest manmade structure, more voluminous than the Great Pyramid of Giza, though only half as tall. Today it is covered in soil and vegetation and resembles a hill with a Spanish church on top. Arcturus is the brightest star in the northern hemisphere (though some authorities say Vega, depending on photometric details).

I used VizieR's online Bright Star Catalog for the J2000 coordinates of Arcturus, got the Proper Motion (-1.1"/yr RA & -2.0"/yr Decl) from Wikipedia, and used the online NASA Lambda utility to convert to 2013.0 celestial coordinates (mean equinox and ecliptic of date). I neglect Earth nutation and the aberration of starlight; neither effect amounts to more than a few arcsec. I find the Declination of Arcturus thus to be 19deg06'55".

According to Wikipedia, the geographic latitude of the Cholula pyramid is 19deg03'27"; another online source gives 19deg03'29.8". According to the geodesy teaching website plone.itc.nl/geometrics ("Geometric Aspects of Mapping, 3. Reference Surfaces for Mapping") the plumb line (i.e. perpendicular to "geodetic" surface) commonly deviates from the perpendicular to Earth's reference ellipsoid by up to 50" in mountainous regions, the "deflection of the vertical"; this phenomenon might explain 1' of the 3' error above. Also even with perfect knowledge of Earth's precession, the presumed 3.5' error in Arcturus' Declination could be caused by only a 5% overestimate of Arcturus' Proper Motion in Declination over 2200 yrs (the time since the pyramid's construction began, according to Wikipedia, though maybe there were even older predecessor structures)(2" * 5% * 2200 = 220").

The modern distance and radial velocity estimates for Arcturus, 11pc and 5km/s, imply that Arcturus' motion is so nearly tangential that its Proper Motion never has been more than 0.2% greater than it is now. On the other hand a 1' Declination observation error in 600 years would be 1*60/(600*2) = 5%.

According to Wikipedia, the pyramid is part of a flood myth:

"According to myth, the pyramid was built by a giant named Xelhua of adobe bricks, after he escaped a flood..."

The Great Pyramid, the Cholula Pyramid, and the Pyramids of the Sun and Moon: working together to prove that "2012" isn't nonsense

Abstract. Following Petrie, the position of the pole at the time of the pyramid builders is determined two ways. At their oppositions in early 2013, Arcturus and Algieba reach within an arcminute of the zenith, at the pyramids of Cholula, and of the Sun and Moon, resp., using those poles. The distance between the pyramids of Cholula and the Moon, tells the correct precession rate. The distance and angle between the pyramids of Moon and Sun, and the sizes and shapes of the pyramids themselves, tell us that the pyramid builders assumed an inexact second derivative of the ecliptic pole position, apparently for the sake of using round numbers, and correcting this improves the accuracy to a few arcseconds for Arcturus. Similar plans explain the positions of the three main Giza pyramids, Monk's mound at Cahokia, and the Bosnian pyramids.

I. Introduction to the stars.

Arcturus (Greek, "the bearkeeper") is the brightest star in the northern hemisphere, though some sources say Vega, depending on photometric details. Algieba (Arabic, perhaps originally "the mane" of Leo, according to Burnham) is a famous second magnitude double star, one of the fifty brightest stars in the sky. These stars are in the spring sky, about a radian apart. Both stars are nearby orange giants, with large proper motions.

Based on their negative radial velocities and on trigonometry, increases in their proper motions during the last 5000 yr, would cause 45" & 5" overestimates of declination for Arcturus & Algieba resp., if only the instantaneous proper motion, arcsec/yr, and not its time derivative, arcsec/yr^2 (and in the case of Arcturus, the significant time second derivative arcsec/yr^3) were known 5000 yrs ago. The accuracy of placement of the Cholula pyramid with respect to Arcturus, explained in Sec. VI, implies that not only accurate proper motions, but also first and second derivatives of proper motions, or the mathematical equivalent (e.g. a presumption of oblique rectilinear motion) were known to the builders.

The orbit of the double star, Algieba A & B, is highly eccentric and highly inclined to our line of sight, so it would have been difficult for the builders to average the center-of-light motion. This would have been easier with Castor AB; see the Addendum re Giza. Algieba AB's orbit is vague and ambiguous, but by measuring and quadratically extrapolating positions on Burnham's graph, "Celestial Handbook", vol. 2, p. 1063, and using mass and magnitude estimates from Kaler's stars.astro.illinois.edu webpage, I find that Algieba AB's epoch 2013.0 difference between center-of-light and center-of-mass motion is, in declination, +2.02 mas/yr.

II. Positions of the stars in early 2013.

From their 2000.0 AD positions (Bright Star Catalog) in 2000.0 AD celestial coordinates, and the "rigorous formulae" on p. B18 of the 1990 Astronomical Almanac, I found their declinations in 2013.0 AD celestial coordinates, including proper motions. These declinations are:

Arcturus +19deg06'55"

Algieba +19deg46'32" (primary component)

There is no atmospheric refraction correction at the zenith, but the nutation and starlight aberration corrections will affect the accuracy considerably. At opposition, the change in declination due to aberration and to nutation in longitude (chart at www.pietro.org) can be estimated as

(-20.5"-15") * sin(23.44) * cos(34, resp. 25)

= -12" & -13" for Arcturus & Algieba, resp.

For nutation in obliquity, the effects are

-6" * sin( -34, resp. +25)

= +3" & -3" for Arcturus & Algieba, resp.

The oppositions occur near 2013.32 & 2013.15, resp., giving by precession from 2013.0 another

-5" & -3" in declination, resp. (omitting -0.6" Proper Motion for Arcturus)

This gives declinations at their oppositions in early 2013, as observed when at the zenith:

Arcturus +19deg06'41"

Algieba +19deg46'13"

III. Introduction to the pyramids.

The Great Pyramid of Giza, the pyramid of Cholula, and the pyramid of the Moon are among the world's largest known pyramids both in height and volume. The Great Pyramid of Giza is a close #2 (to Giza's pyramid of Khafre) in the world for height. The pyramid of Cholula, thought to have been begun in the 3rd century BC, is #1 in the world for volume and #2 in Mexico for height. The pyramid of the Moon is somewhat smaller than its more famous, nearby, larger teammate in Teotihuacan, the pyramid of the Sun; the pyramid of the Sun is #3 in the world for volume and #1 in Mexico for height. Although most findings of this paper apply almost as well to the pyramid of the Sun as to the nearby pyramid of the Moon, I choose the pyramid of the Moon because according to Wikipedia, the pyramid of the Moon "covers a structure older than the pyramid of the Sun".

According to the geodesy teaching website plone.itc.nl/geometrics ("Geometric Aspects of Mapping, 3. Reference Surfaces for Mapping") the plumb line (i.e. perpendicular to the, "geodetic" surface i.e. "geoid") commonly deviates from the perpendicular to Earth's reference ellipsoid by up to 50" near mountains, but in flat regions uncommonly more than 10". A low-resolution world map of the geoid (Uotila, 1962; cited in Heiskanen & Moritz, "Physical Geodesy", 1967, Fig. 21, p. 157) shows that both Giza and Teotihuacan/Cholula are in regions where the geoid is extraordinarily close to the reference ellipsoid, about -6m and +2m, resp. At Giza, the gradient of the geoid height is relatively large but lies EW so would affect observed declinations little. At Teotihuacan/Cholula, there is a moderate NS geoid gradient equivalent to about 2" northward tilt of the plumb line.

IV. The latitudes of the pyramids.

The geographic latitudes are:

19deg03'27" (pyramid of Cholula)

19deg41'59" (pyramid of the Moon)

A large error, is the motion of Earth's pole over thousands of years. Flinders Petrie (p. 125; Ch. 13, Sec. 93 in Birdsall's online edition) says that the most reliable structures in the Giza pyramids indicate that the pole at the time of their foundation, lay 5'40" +/- 10" west of the present true north. Petrie remarks that this indicates a rate of geographic pole migration only a few times greater than measured in recent centuries, and that physically, quantitatively, it is consistent with major changes in ocean currents. Likely, a pole shift would change Giza's latitude a comparable amount, so I assume that the Great Pyramid was originally at exactly 30N. If the Great Pyramid serves as a monitor of the pole change since the foundation of the greatest Mexican pyramids, then Cholula, and the pyramids of Sun and Moon, would have been 180" and 177" nearer the pole, resp., when founded. Their geographic latitudes then would have been:

19deg06'27" (pyramid of Cholula)

19deg44'56" (pyramid of the Moon)

With this ancient pole, Arcturus would have been observed only 14" too far north to match this original Cholula perfectly, and Algieba 77" too far north to match this original pyramid of the Moon perfectly, at their respective oppositions in early 2013.

If no latitude change happened at Giza (maybe that site was chosen for this reason, determining its longitude; then the site away from 30N was chosen because of the architectural convenience of the Giza plateau) then these pyramids would have been 227" and 226" nearer the pole. Their geographic latitudes would have been:

19deg07'14" (pyramid of Cholula)

19deg45'45" (pyramid of the Moon)

Arcturus & Algieba would have been 33" too far south & 28" too far north, resp. This is a better fit: 14^2+77^2 = 6125 > 1873 = 33^2+28^2. The declinations of these stars change about -50" * sin(23.44)*cos(34 or 25) = -16" or -18"/yr, resp. Using the oppositions in early 2012 o

The "17degree" (or "15degree") angle(s) at Teotihuacan: proof of torque-free Earth precession

Many authors discuss the offset angles from true north, of the buildings and streets of Teotihuacan and some other Central American pyramids. These angles are variously given as 15, 16, or 17deg, sometimes plus fractions. Four different angles are being discussed:

According to Giulio Magli of Milan (ArXiv.org) the Teotihuacan angles all are either about 15deg25' (15.417deg)(angle "Nord") or 16deg30' (16.500deg)(angle "Est") east of true geographic north. (The former angle also is attested by Rene Millon's 1970 photographic map of Teotihuacan's 1962 appearance.) This makes two angles under discussion. I would add, that the line between the pyramids of Moon and Sun, is offset west of true north, according to the coordinates found at Wikipedia, by roughly arctan(0.002/0.0071) or, precisely accounting for geocentric latitude, by arctan(0.0002*cos(19.6)/(0.0071*(296/297)^2) = 1.530deg; the 1-sigma rounding error is 0.00005/sqrt(3)/0.0002*1.53 = 0.22deg. This interpyramid line, may well be more important than the present true north. Teotihuacan's prominent lines are rotated clockwise from the interpyramid line, by about 15.417+1.530 = 16.947deg, and 16.500+1.530=18.030deg.

Suppose the interpyramid line is, really, exactly 1.68deg west of north (this is consistent with the given 1.53 +/- 0.22; though measuring with a ruler on the large paper map of Rene Millon & Armando Cerda, from the centers of the topmost rectangles drawn on the pyramids, I find 2.106deg). Maybe this is the line to the ancient pole of torque-free precession (i.e. spheroid pole); suppose Teotihuacan's latitude relative to this ancient pole, was about the same as now (the expression below is an order of magnitude less sensitive to this factor). Let a great circle extend from Teotihuacan, in the direction 16.500+1.68deg east of this ancient north pole, tangent to the latitude circle of torque-free precession. Then by Napier's rules for spherical triangles, the opening angle of the torque-free precession cone is

But where else do we see this angle? We see it in 15.417+1.68=17.097. So, the interpyramid line signifies the direction to the (Ice Age) torque-free precession pole. The 16.5deg E of N angle, signifies the line to the farthest eastward deviation of the ancient pole which was precessing yearly. The 15.417deg E of N angle, signifies that the precession occurred along a cone of opening angle 15.417+1.68 = 17.1deg, not much less than the 18.9deg I calculated.

Go:bekli Tepe Vulture Star Map dates catastrophe 12900 years ago

Turkish archaeologist Haldun Aydingu:n, is credited for a photo of the Vulture Map on a column at Go:bekli Tepe. The version from which I measure, I printed from an online photo of that column, presumably Haldun's, found by Google Images on the website timothystephany.com (Timothy presents an alternative star map theory).

The bundles and baskets at the top of the column, represent a calendar [note July 6: a number; see addendum below]. The vulture on our left, holds an egg on his left wing, signifying a catastrophe which produced many vultures.

The large hole or eye in the center of the vulture's circular head, is Denebola (beta Leonis, a magnitude +2.14 Vega-like star in a multiple star system). The line between the horizontal and vertical parts of this T-shaped column, is the celestial equator of epoch. This celestial equator is shown most precisely by the horizontal straight line between the vulture's legs. The large long-necked bird at the column's bottom, has a profile resembling a parabola opening downward; the hole or eye in its head is near the focus of that parabola. This hole is nu Hydrae, a magnitude +3.11 orange giant.

Denebola and nu Hydrae define the epoch of this map, by having equal right ascensions. I measure 89.8deg counterclockwise from the celestial equator defined by the line between the vulture's legs, to the straight line between Denebola and nu Hydrae on the map. Using the proper motions given in the online Bright Star catalog (without correction for coordinate curvature; the proper motion arcs are < 2deg) and the rigorous precession formulas from the 1990 Astronomical Almanac, this angle occurs in the sky, 12,900 yr (to the year!) prior to 2000AD. An exact 90deg angle occurs 12,928 yr prior to 2000AD. Because Denebola & nu Hydrae move tangentially (their radial velocities are 0 & -1 km/sec, resp.) the geometric change in the angular speed of their proper motion amounts to < 1 mas/yr in 13kyr.

I measure the ratio of upper to lower distances from the equator, conveniently marked by the vulture's legs, as 1.0846. This ratio of declinations occurs 12,331 yr prior to 2000AD. There seems to be some minification at the bottom of the photo, because a bulky "view camera" apparently was not used. Because the photographer found it impractical to aim exactly perpendicularly, the film was not parallel to the stone. If the lower segment is minified 10% relative to the upper and this were corrected, then the date becomes 12,443 yr before 2000AD. The distance ratio is much more time-sensitive than the angle with the equator: at 12,400 yr before 2000AD, the angle with the equator is still 85.76deg; but at 12,900 yr before 2000AD, the upper::lower ratio is only 0.60865.

The hole or eye in the center of the circular bird's head in the lower right corner of the upper part, is Regulus (alpha Leonis). Above and to the right of Regulus is another pair of bird's legs with a long straight line between them; this is the ecliptic. Because this star map is for 1/2 precession cycle ago, the tilt of the ecliptic here, near longitude 180, was the reverse of what it is now. Though the ecliptic presently is slightly south of Regulus, 12,900 yr ago it was about one degree north of Regulus, according to the dubiously convergent formula in the 1990 Astronomical Almanac which I confirmed by extrapolating NASA Lambda values. The bird to the right of the ecliptic-leg bird, whose neck, head and breast appear near the right edge of the column, has a hole or eye above its cheek which represents gamma Leonis (Algieba). On the map, the ecliptic is about three degrees from Regulus, if my identification of Algieba is correct.

The bird whose legs form the ecliptic, lacks obvious eyes. Instead there is a manifoldly right-angled shape resembling a bold "I" in the upper right. Such an abstract shape, rather than a realistic bird as elsewhere, would imply that a hole signifies not a real star, but only the direction to a star. The lower left corner of the upper capital of the "I" is collinear with Regulus and Algieba, but was deliberately defaced to signify that this perfect collinearity was merely an earlier situation. Near the center of the upper capital itself is a hole which signified the then-current position angle of Adhafera (zeta Leonis). Using proper motions from the online Bright Star Catalog, I find that the line Algieba-Adhafera, was 2.87deg (my measurement to about 10% accuracy) clockwise from perfect collinearity with the line Regulus-Algieba, 12,615 yr before 2000AD; perfect collinearity occurred 13,761 yr before 2000AD. Here, the geometric correction for time change in proper motion, mainly for Algieba, is a few mas/yr; this changes the former date to 12,713 yr before 2000AD. The actual distance 12 kyr ago, between Algieba & Adhafera, in proper proportion to the distance between Regulus & Algieba, would have been about halfway between the defaced corner, and the hole in the capital "I".

The smaller eccentric hole or second eye, in the vulture's head, at about 2:20 position from Denebola, is 95 Leo. This magnitude +5.53 line-of-sight neighbor has the same spectral type as Denebola, main sequence A3V, but is much more distant. With proper motion to 12,900 yr before 2000AD, 95 Leo would lie 40.68' from Denebola, at azimuth (clockwise from north) 13.13deg in J2000 celestial coordinates; after rotation to the celestial coordinates of date, 12,900 yr earlier, the azimuth is 62.55deg; with a protractor I measure the azimuth according to the vulture's legs' equator, as 72deg; but I can do much better:

Earth's obliquity at that epoch was 24.03deg, which multiplied by cos(right ascension) = cos(-12.08) gives approx. 23.50deg as the local inclination of ecliptic to celestial coordinates. So the expected azimuth from ecliptic north, is 62.55-23.50=39.05. The straight line separating the upper and lower parts of the vulture's beak, lies parallel to the birdslegs' ecliptic depicted near Regulus, so I assume it is a local line of ecliptic latitude. Thus with a protractor, I estimate the ecliptic azimuth of the line Denebola - 95 Leo, as 40.1deg and by orthogonal ruler measurements, as 39.8deg, averaging 39.95. Conveniently, the proper motion of Denebola relative to 95 Leo, 0.502"/yr, is almost perfectly perpendicular to their line 12,900 yr ago; the additional 39.95-39.05=0.90deg azimuth requires only 76 yr, giving a date 12,976 yr before 2000AD.

Let's compare actual inter-star great circle distances 11,000 yr prior to 2000AD, to distances on my photo of the Vulture Map. The distances are normalized to the Denebola-nuHydrae distance. Vulture Map distances are in parentheses:

In sum, Regulus was scooted left, to make room for a magnified inset of western Leo. The scorpion in the southern hemisphere, assuming that it is part of the larger map including Denebola and nu Hydrae, coincides roughly with the positive Cosmic Microwave Background dipole.

Aside from the Vulture Stone Star Map, the Younger Dryas catastrophe date also is corroborated by Earth's rate of precession. Newcomb's cubic formula (as cited by Clemence, Astronomical Journal 52:89, p. 90, 1946) implies that 1/2 precession cycle before 2013.0AD, is 13,257 yr before 2000AD. Newcomb's successors have proposed small corrections to the linear term: one typical correction (Van Flandern, IAU Colloquium #9, Heidelberg, August 1970) changes this to 13,253.

Listing all the date estimates:

1/2 precession cycle: 13,253 yr before 2000AD

from Vulture Map angle of Denebola - nu Hydrae line to equator: 12,900 +/- ? ratio of upper and lower segments o

John Major Jenkins popularized the phrase "precessional alarm clock". Thus I was led to the right trees, by other hounds who, though barking up the wrong trees (Venus, the Pleiades, the galactic center) at least were in the right forest.

In my previous article, I noted that Arcturus' actual declination, including nutation, aberration and proper motion, at its opposition in the spring of 2013AD, equals the geographic latitude of the pyramid of Cholula with an error of only 194". This error decreases to 14", if I follow Petrie in believing that the small, consistent misalignment of the Giza pyramids indicates a small pole shift, and if I also assume that the pole shift was such that the Great Pyramid originally was at exactly 30N.

Most authorities list Arcturus as the brightest star in the northern hemisphere, though some list Vega, depending on photometric details. Likewise Cholula is not just any pyramid; most authorities call it the largest pyramid, by volume, ever built. The chance that the brightest star in the northern hemisphere would happen to lie within 194" of the 19.06deg latitude of (according to most authorities) the most voluminous pyramid ever built, is p=0.18%. If correction for the pole shift suggested by the Giza layout is admitted, this becomes 14" and p=0.013%.

As I explain in my previous article, the analogous alignment of Algieba (the brightest star in the "sickle" of Leo and one of the 50 brightest stars in the sky) over the pyramids of Teotihuacan, at its own opposition in late winter 2013, is almost as accurate as that of Arcturus over Cholula. The pyramids of Cholula and Teotihuacan are archaeologically similar. Arcturus and Algieba also are similar: both are orange, high proper motion stars.

If the Great Pyramid of Giza were intended to mark the pole by lying at exactly 30N, it hardly could lie exactly under any very bright star at a specific time like 2012AD. So, the pyramid builders resorted to the next best thing. The most prominent pair of bright stars in the northern hemisphere, Castor and Pollux (no pair of stars in the northern hemisphere are both brighter and nearer together) happen to straddle the 30th parallel now. In an addendum to my previous article I note that the ratio of distances Cheops-Chephren : Chephren-Mycerinus (sometimes with and sometimes without projection of the latter onto the line of the former) nearly equals various practical, precise definitions of the current break ratio of Castor and Pollux over the 30th parallel.

This leaves another degree of freedom: if the relative distances and break angle at Giza are determined by Castor, the 30th parallel, and Pollux, still the Cheops-Chephren slope still can be chosen freely. The Cheops-Chephren slope has been chosen to equal the azimuth of the rising of Vega: not then, but rather, now. It is another "precessional alarm clock".

The rising of Vega.

Vega is the second brightest star in the northern hemisphere, and according to some authorities, the brightest. At its opposition in the summer of 2013, Vega's actual observed declination exceeds the geographic latitude of Monk's Mound (near Cahokia, Illinois) by 523". The pole shift correction used above, reduces this to 309". Ptolemy's star positions within the constellation Bootes, are consistent with modern estimates of the proper motion of Arcturus; but his positions within Lyra, are not consistent with the modern theory of the constant rectilinear proper motion of Vega. In my previous article I further found similar evidence of a large higher order term in the proper motion of Vega, from positions of Flamsteed and Bradley. So, maybe the inaccurate placement of the Cahokia Mounds is due partly to the pole shift indicated at Giza, and partly to poor prediction of Vega's proper motion.

There are three competing definitions of the azimuth of rise of a star:

Definition 1) the azimuth as would be observed from the surface of a perfect airless spheroid. This is competitive, because it can be calculated from the great-circle path of the star, observed far above the horizon, where atmospheric refraction is negligible. Vega's actual observed declination, including nutation, aberration and proper motion (but after correction for atmospheric refraction, if any) at its opposition in the summer of 2013, is 38deg47'57". This implies an azimuth of rise, from geographic latitude 30.0deg on the surface of an airless 296/297 spheroid, of

43.653651deg.

If the builders wished to notify future people of an important date (what other motive could they have had?) it seems they would not have chosen a method which could not be directly observed. So, definitions (2) or (3) are likelier.

Definition 2) the actual observed rising, when the star first appears above the ground below, as seen from the top of the Great Pyramid. This is impaired by the faintness of the star at rising, the large atmospheric refraction, and the likelihood of considerable erosion of the hillside across the Nile, over millenia. Even modern astronomers hardly can measure accurately, apparent altitudes less than one degree (Greenbaum, Astronomical Journal 59:17-19, 1954, Fig. 1, p. 19). I use the original Great Pyramid height of 146m above its base (Fakhry, "The Pyramids", p. 115), its base elevation above sea level of 59.85m (Butler, "Egyptian Pyramid Geometry", p. p. 125, as cited on a messageboard; citing Maragioglio & Rinaldi, citing Vyse), and the elevation of 100m (Manley, "Penguin Historical Atlas of Ancient Egypt", map, p. 29) of the north slope of the bluff across the Nile, on which Vega would rise about 30 miles to the northeast. I assume that the line from Cheops' peak, to the bluff, is perpendicular to Earth's radius, where it intersects the bluff, and use the radius of curvature of Earth's spheroid, along this approximate azimuth, to find the angle of depression at Giza neglecting refraction. By the formula 1.24/10^6 deg/meter (converted from civil engineer Haseeb Jamal, teaching website www.aboutcivil.com) I find half this, as the angle of depression of the light ray where it grazes the bluff. A slightly different alternative would have been to use 1.17/10^6 deg/m, the practice of British surveyors in Egypt in the 1920s, who prescribed 0.13 times Earth's curvature and made observations "in the afternoon hours when refraction is at its minimum and steadiest value..." (Ball, Nature 109:8, 1922).

Then I used the time-honored Astronomical Almanac formula for astronomical atmospheric refraction (appears in most if not all of the 1985 through current Almanacs) with atmospheric pressure appropriate to 100m elevation according to the standard atmosphere table in the 1987 Handbook of Chemistry & Physics, and temperature 37C roughly appropriate to a sunset in contemporary Cairo in July (and a natural reference temperature for summer, because it equals human body temperature). Then I added all three terms to find that this rising of Vega seen from the top of the original Great Pyramid, would occur when Vega was geometrically 90.872937 deg from the zenith. This implies an azimuth

42.911458deg.

Definition 3) this definition is most practical because it is observable, yet the star is brighter at "rising" and can be seen for awhile prior, the refraction is less, and the "horizon" is indestructible: the star appears to cross, not the dirt horizon, but rather the horizon circle 90.0deg from the zenith. The Astronomical Almanac refraction correction (using now, standard atmospheric pressure at the original top of the Great Pyramid, 206m above sea level; and still 37 C) implies a geometric zenith angle of 90.508584. This implies an azimuth

43.224214deg.

The actual Cheops-Chephren azimuth, relative to Petrie's estimate of ancient Giza's true N (5'40" W of modern true N) is

July 25, 2012 crop circle (Windmill Hill, near Avebury, Wiltshire) says doomsday date Oct. 8, 2012

As others already have suggested, the 16 overlapping disks signify days. (Nyako Nakar and also Chuen Xul show 16 on their diagrams, and that's my count from the online photos; but Bertold Zugelder shows 17 on his diagram.) The non-overlapping disks each signify consecutive "afternoon half moons" (in more exact astronomical language, the "first quarter").

The biggest disk, the one near the center, signifies the afternoon half moon of July 26 (9h GMT), the day after discovery of this crop circle. The long, monotonically diminishing sequence of seven disks, on one side of the largest disk, signifies the past seven "first quarters", those that have occurred since the winter solstice of 2011.

On the other side of the biggest disk, the next disk signifies the afternoon half moon of August 24 (14h GMT). The thin circle next to (just outside) it, signifies the ascending node, which the moon reaches that same day, Aug. 24 (12h GMT).

The next and final disk signifies the afternoon half moon of September 22 (19h GMT). That very same day, at 15h GMT, is the equinox, which is symbolized by the thick cresent-like circle just outside the moon disk.

Summarizing, we have:

1. The afternoon half moon of the day after the crop circle, shown by the biggest disk.

2. The seven prior first quarter moons since the winter solstice, shown in diminishing size behind it.

3. The next afternoon half moon, which the thin line shows to be near (only two hours after) the ascending node.

4. The final afternoon half moon, which the thick line (crescent) shows to be near (only four hours after) the autumnal equinox.

The 16 overlapping circles then show us how many days until the big event. Sixteen days after Sep. 22, is Oct. 8. That seems to be the day. Prepare as you would for an earthquake, wildfire, flood or other natural disaster.

Others (e.g. Zugelder's diagram) have noted the break in the axis of symmetry: the 16 overlapping disks lie on a greater azimuth than the non-overlapping disks. These azimuths correspond to the azimuths of July 25 & 26 moonrise, with the later rise (July 26, azimuth 117.1820deg at the coordinates of Stonehenge and zero altitude, according to the airless model of the online JPL ephemeris) corresponding to the larger azimuth of the disks representing later time (the overlapping "day" disks). The July 25 moonrise (azimuth 109.9716deg) corresponds to the azimuth of the disks representing earlier time (the non-overlapping disks).

The most nearly perpendicular aerial photo I've found, is Bert Janssen's, posted on www.cropcircleconnector.com . Tracing it from the screen and comparing its two axes, to the nearest (presumed NS) tractor lines, I find that the overlapping disks are at azimuth 125.70 and the non-overlapping at azimuth 120.365. The camera angle should increase these angles, so this constitutes at least rough agreement with the azimuths of moonrise.

July 28 crop circle (Jubilee Copse, near Hannington, Wiltshire) confirms Oct. 8 +/- 7 date

This crop circle contains a 12x12 grid; the crop in 65 of the squares in unflattened, and 79 are flattened. This indicates 144/2 = 72 +/- 7 days. July 28 + 72 days, is October 8.

July 22 crop circle (Aldbourne, Oxfordshire) confirms Oct. 8 date

This is a square inscribed in a circle, but altered so one side of the square is short. If each side of the square should have been 365.25/4 days, then as I measure the slanted aerial photo on the screen, the short side is 77.26d. July 22 + 77d = October 7. The three dots draw attention to the "moving" end of the short side: they signify past, present, and, the biggest, fanciest dot, the future.

Correcting for Earth's eccentric orbit using the trapezoidal rule for integration, a more exact time for the short quadrant, which if complete would have ended about Oct. 7 + (91.31 - 77.26), is

June 12 (reported) crop circle (Silbury Hill, near Avebury, Wiltshire) confirms Oct. 8 date

This small formation was discovered June 12, but may well have been formed a day or two earlier. It shows a "sun" with five Earth / full moon alignments in a regular pentagon. There is a "3rd quarter" moon (geocentric) at 11h GMT on June 11, and another at 08h GMT on Oct. 8. This suggests that these circlemakers had some familiarity with our calendar and clocks. Be that as it may, the interval for formation of the crop circle, June 11 (?), to Oct. 8, thus covers five "3rd quarter" moons exactly.

Werewolves of London again?

Even before I made this second post about crop circles, a Boston banker who reads this messageboard, emailed me that he thought it might be a signal to co-conspirators in a rigged stock market crash. The circles I discuss here, not only indicate Oct. 8, they refer to the number of days in a week, dates of the month and hours of the day; if extraterrestrials knew about those details of our language, they wouldn't need to be cryptic.

This conspiracy would have to be London-based with inside men in the RAF. October is the month most associated with big stock market drops, and Monday (e.g. Oct. 8) is the day of the week most associated with them. The crop circles might be made at night by aircraft emitting microwave beams.

Jubilee Copse July 28 crop circle confirms October 8 date

This 12x12 checkerboard crop circle has 12 words of 12 "bits" of information each. We know which are the rows and which the columns, and which end begins the word, because the first "bit" of each word is a broader rectangle. These broader rectangles are found along one side of the checkerboard, the righthand side as the circle usually is shown on the computer monitor.

Each row is thus read from right to left, and is a twelve-digit binary representation of a real number between 0 and 1. The rounding error cannot exceed 1 part in 2^13 = 8192. This corresponds to 1.07 hour if the unit is a year, or 0.044deg if the unit is a circle.

The seventh row (number) is the easiest to interpret. This number, multiplied by the 365.24219 days of the current tropical year, gives 291.498 days. The latest winter solstice was Dec. 22, 2011, at 05:30 GMT. Adding that many days, gives Oct. 8, 2012, 17:27 GMT.

Seven and eleven are the largest prime numbers < 12. This might be a hint that the 11th row (number) is akin to the 7th, and indeed they are nearly equal. However, the 11th number does not seem to correspond to the great circle arc swept by the sun between 05:30 GMT Dec. 22, 2011 and Oct. 8, 2012. Instead, it seems to correspond to the distance traveled by the sun in space (relative to the Earth) as a fraction of the distance covered in one sidereal year.

My first-order approximate calculation gives an end date Oct. 8, 2012, 15:48 GMT. Because the rounding error from these 12-digit binary numbers can correspond to more than an hour, this and the previous date are consistent with each other, up to rounding. In my approximation, I found the arc from the JPL ephemeris apparent RA & Declination, which refer to the equinox and ecliptic of date, then subtracted the precession. I then approximated the Earth-sun radius variation, with perihelion at 11deg longitude, as proportional to the negative cosine of (longitude minus 11deg), and integrated the tangential motion, neglecting the radial motion.

My second-order calculation uses eccentricity 0.016723 (from data in the 1987 CRC Handbook of Chemistry & Physics) and a perihelion at sun longitude 13.56+270 = 283.56, which is the mean of the apparent geocentric sun longitudes, at the perihelion times given to the nearest hour by the USNO for 2011 & 2012. I retained all terms of second order in the eccentricity, and found that the 11th number corresponds to the fraction of distance traveled to 16:41 GMT Oct. 8, less than half as far from the other value, as the first-order calculation was.

The 12th number, seems to correspond to the change in Luna's right ascension, between Luna's crossing the ecliptic Oct. 4, and "the event", Oct. 8. This number is a fraction of 180deg, not 360deg; this halves the rounding error. The difference between ascending and descending nodes is arbitrary; one or the other always will lie within 180deg. Because of the thirteen times greater angular speed of Luna vs. the sun, and the halving of the unit interval, this number could give the time of "the event" to within two minutes, instead of an hour. The time given by this twelfth number is Oct. 8, 2012, 17:23, differing only four minutes from the time implied by the seventh number. Maybe this is luck; or maybe even the circlemakers don't know the time to within minutes, but decided to make the twelfth time close to the seventh, rounding error and all, to impress us with their skill and with the importance of the message.

The numerology of the June 12 Silbury Hill crop circle (see previous post) suggests that the circlemakers know about the hours of our day and the days of our month. On the other hand, the numerological relation could have been mere luck, or Dr. Jung's "synchronicity" (an organizing principle perhaps above the circlemakers). Logically, the June 12 crop circle message stands, even without the numerological extras.

Today, August 2, I find that the 8th of the base-2 numbers in the crop formation, again assuming that the unit is 360degrees, equals the difference in right ascension (apparent; equinox of date, per JPL Horizons) between Luna's center and Mars, at 16:51 GMT on Oct. 8.

Assuming they signify fractions of a 180deg arc, the 1st & 2nd numbers differ by 1741". The geocentric angular diameter of Luna at its apogee Oct. 5, is 1769". Also, the mean of the 1st & 2nd numbers, is the reciprocal of 2.71888, close to the base of natural logarithms, 2.71828... .

At 2012.8AD, according to the 1990 Astronomical Almanac, p. B18, Earth's mean obliquity is 23.43763deg. According to Simon et al, Astronomy & Astrophysics 1994, Luna's orbital inclination then, to the ecliptic of date, is 5.15668deg. So Luna's maximum declination would be 28.5943deg. The 3rd number, assuming it is a fraction of a 360deg arc, is 28.652deg.

For the angle between Mars and Venus on Oct. 8, the light-time correction happens to be small, only 4". The 5th binary number in the Jubilee Copse crop formation, if it represents a fraction of 180deg, would represent 84.7265deg. The JPL ephemeris gives this as the difference in ecliptic longitude (apparent, using ecliptic of date) between Mars and Venus, at 19:42 GMT, Oct. 8.

More natural, would be to project Mars' position onto the great circle tangent to Venus' sky path (tangent at, say, 17:23 GMT) rather than Earth's ecliptic. The 5th number (times 180deg) equals this difference in "longitude", at 18:22 GMT.

*********

The August 2 crop formation at Chalk Pit, near Wootton Rivers, Wiltshire, is an Archimedes spiral of approx. six turns. This crop formation would have been made at almost exactly the time of the full moon, 03:27 GMT. Two full moons in the future, would have been Sept. 30.

The arclength of an Archimedes spiral is given exactly by an elementary integral (CRC tables #156). The ratio of the length of six turns, to the length of one turn, is 33.5441. What if the mean synodic month, 29.5306 d, were replaced by this number of days? We would have Aug. 2 + 2*33.5441 = Oct. 8.

August 1 crop circle (Avebury Stone Circle, Wiltshire) gives Oct. 12 date; July 31 (Windmill Hill, Wiltshire) gives Oct. 8; July 29 (Andechs Abbey, Bavaria) gives Oct. 9 (8?)

This formation shows 13 disks trailing Jupiter, 7 trailing Venus and 6 trailing some other planet. By interpolating in the JPL ephemeris (which gives positions apparent at the Sun) I found that this crop circle indicates, by one method of analysis, the time T = 19:04 GMT, Oct. 12, 2012.

A planet's "synodic period" is defined as the time for recurrence of the same Earth-Sun-Planet angle. I approximate this angle by its projection onto Earth's ecliptic of date. Thirteen synodic Jupiter periods after the above specific time T, Jupiter's longitude, corrected to a fixed equinox, has changed the same amount, as has Venus' longitude after seven synodic Venus periods. That is, Earth's longitude is the same after 13 synodic Jupiter periods as after 7 synodic Venus periods.

The angle by which both the Jupiter's and Venus' and Earth's longitudes change, is 70.939deg. This is close to the 69deg angle given by Jocelyn Cazes on her diagram, which is posted on the website

cropcircleconnector.com

Addendum #1, Aug. 7: If I choose T = Nov. 17, approximately, then the change in Earth's longitude after six synodic Mars cycles, equals that after seven synodic Venus cycles, except for sign. So, the smallest planet depicted in the crop formation, might represent Mars. Venus was placed between Mars and Jupiter, to hint that it is the planet involved in both comparisons.

Addendum #2, Aug. 7: The July 31, Windmill Hill, Wiltshire crop circle, shows five binary numbers of eight digits each, which I list from the inside to the outside of the spiral:

The first "0" of flattened crop, is to enhance the legibility of the second (first meaningful) digit, always a "1". The only number which appears twice, is 69 and

July 31 + 69 = Oct. 8

The full moon is Aug. 2, so 67 might be the count from then to Oct. 8, a backup in case the formation was not discovered the day it was made.

The following times of that month also are noteworthy:

max speed of Luna toward Earth: July 19, 17:27 GMT max " away from " : Aug. 4, 16:00 GMT

If Luna's orbit were not perturbed by the sun, that is, if Luna's orbit were a perfect ellipse, these times would be for the latera recta, 90 deg from the perigee and apogee, as measured from Earth's center. The former time is 81 d (not 80 d) before Oct. 8; the latter time is 65 d before Oct. 8. The fastest speed of the Earth-Luna barycenter toward the sun, occurs Oct. 5, 23:39.

Addendum #3, Aug. 8: The Andechs Abbey, Bavaria crop formation is listed as discovered July 29, but always there is the possibility that it was formed slightly earlier, maybe before midnight July 28, or even before dawn July 28 and not noticed. July 28 + 72 = Oct. 8. This formation consists of 12 nested tetrahedrons. Some faces of the tetrahedrons are missing, to allow the nesting. This suggests that the meaning is in the number of edges (6) not the number of faces (4). The bold lines on the intact faces, also suggest attention to the edges. Twelve tetrahedrons times six edges per tetrahedron = 72.

Have you ever noticed that when two objects that have the property of mass approach each other the space between them shrinks? Could it really be that simple? Does space condense in the presence of mass?

I'd procrastinate, but I can't seem to find the time

August 13, 2005 crop circle, Woolstone Hill, near Uffington, Oxfordshire, indicates Oct. 7 or 8, 2012

I'll explain this one according to the train of thought, that the circlemakers seem to want us to have. The most emphasized number in this crop formation, is six (which can be seen either as flat, or as faces of stacked cubes).

The next most emphasized number, is sixteen (the number of small squares forming the large perimeter square). There is a subtle ambiguity, though: 16, or 15? Each of the small squares contains a spiral-like sequence of monotonically lengthening tiny rectangles. The longest tiny rectangle is on a side of the small square, which rotates 90 degrees from one square to the adjacent square, except for one step, where it does not rotate (the next rotation is 180 deg to compensate). So, there are 16 small squares, but these rotate only 15 times.

This ambiguity, "15 or 16?", parallels the ambiguity, "sidereal or synodic month?". For the length of the mean sidereal month, I use polynomial 3.4.a.1 of Simon et al, Astronomy & Astrophysics 282:663 (1994) p. 669; it gives 27.3216616 d during the entire interval Aug 2005 - Oct 2012. For the mean synodic month, I modify the mean sidereal month according to a sidereal year of 365.25636 d, getting 29.5305889 d. The ratio of synodic to sidereal month is

16/15 * 1.0133 = 16/15 * (1 + 1/75)

so the "15 or 16?" ambiguity hints that the circle deals with both sidereal and synodic months.

There is another kind of month often used by astronomers, especially ancient astronomers predicting eclipses: the draconic month, averaging 27.21222085 d during the interval. Also, there is the anomalistic month, averaging 27.5545498 d.

Since the 16 and the 6 in the crop formation are qualitatively different in design, we are supposed to multiply them as factors, rather than add them as we would objects of the same kind.

16*6 = 96 draconic months = 2612.373 d = Aug 13.0, 2005 + 2612.373 d = Oct 7.373, 2012

Repeating the above calculation with sidereal months, gives

Oct 17.880

With anomalistic months, it gives

Nov 9.237

and repeating it with synodic months, but using 15 instead of 16, gives

Nov 21.753

What about the fine radial segments, called "feathers" by many who study this crop formation? Of the innermost group of feathers, I count blocks of 11, 11, 11, 10, 10, 10. Of the outermost group of feathers, I count 18, 18, 18, 19. Of the middle group of feathers, I count 11, 11, 11, 11.

The average value of blocks of the innermost group of feathers, is 10.5. For the time estimate based on the sidereal months, I have

Oct 17.880 - 10.5 = Oct 7.380

perfectly agreeing with the time based on the draconic month. The total number of feathers in the middle group is 44. Thus based on the synodic month, I have

Nov 21.753 - 44 = Oct 8.753

and finally for the anomalistic month, using half the total number of feathers in the inner group, 63/2 = 31.5, I have

July 15, 2008 crop formation near Avebury (the Mayan End Date solar system): a fake?

I've discussed this crop formation before on this thread, but today I used my old measurements from the photo by Nick Nicholson which I printed out from www.ConspiracyPlanet.com, and my old corrections for the obliquity of the photo, together with accurate quadratic interpolations of ecliptic longitudes given by the JPL online ephemeris. The longitude differences between the planets and Earth are consistent with

Mercury Dec 20, 2012, 17h GMT Venus Dec 14 15h Mars Jan 08, 2013, 10h Jupiter Dec 24, 2012, 15h Saturn Dec 27 1h Uranus Dec 20 19h Neptune Dec 27 15h

The other planet isn't Pluto: it's a schematic indication of our moon, Luna, with geocentric longitude equal to the heliocentric longitude shown for the object in the crop formation. Simon (sec. 3.4.a.2) gives the mean lunar perigee in coordinates of date, and the lunar perigee shown in the crop formation is consistent with Jan 13, 2013, 07h. Luna ("Pluto") itself is shown at longitude consistent with Dec 16, 2012, 07:30 GMT.

Such inaccuracy is not characteristic of the circlemakers. Luna's rapid change in longitude would have made it easy to depict Dec. 21 rather than Dec. 16 (60 deg difference). Be that as it may, the mean of the nine indicators is

June 2, 2012 crop formation at Manton Grove, near Marlborough, Wiltshire: Oct. 8 again

This crop formation, sometimes called the "polar clock", has been noted by crop circle experts to show signs of authenticity in its details of construction. I also show here that in the ingenuity and precision of its astronomical plan, it is typical of other excellent crop formations. Indeed it is an astounding astrometric tour de force.

It indicates the date Oct. 8, 2012. June 2 is 128 days prior to Oct. 8. This is a round number, 128 = 2^7, in the most universal arithmetic, binary arithmetic, so should be dignified by a crop circle. The number 128 participates in these astronomical resonances:

128 tropical Earth years (of 365.242188 d, determined for 2012.6AD from a sidereal Earth year of 365.25636 d, together with Clemence's version of Newcomb's quadratic precession rate polynomial, with Van Flandern's suggested dp1 = +1.26"/century correction)

= 68.053 sidereal Mars years = 208.07 sidereal Venus years = 46751.00006 d

On the other hand, 128 sidereal Earth years (of 365.25636 d)

= 46752.8141 d = 46753 - 66.9/360 d

There is another kind of Earth year, analogous to the draconic month: the time for Earth to make one revolution relative to the regressing node of Earth's orbit on some fixed copy of Earth's orbit such as the J2000 ecliptic. I approximate the frequency of revolution of this node, by numerically differentiating the formula for the node of the ecliptic of date on the 2000.0AD ecliptic (cubic forward, or backward, difference formula from 2000.0AD at 2 yr or smaller intervals) at the bottom of p. B18 of the 1990 Astronomical Almanac. I find that this "nodical year" is

365.25636 * 360deg /(360deg + 8.700") = 365.253908 d

and 128 such years is

46752.5002 d = 46753 - 179.9/360 d

Earth's year relative to its perihelion, i.e. the anomalistic year, is

365.259635 d (resp. 365.259549 d)

for 1994-2000AD according to Weisstein's online World of Astronomy (resp. adding 25 min to the tropical year, according to the USNO website); 128 such anomalistic years is

46753.2333 d (resp. 46753.2223 d) = 46753 + 84.0/360 d (resp. 80.0 d) = 46754 - 276.0/360 d (resp. 280.0 d)

and I'll adopt the mean of Weisstein and USNO, 278.0.

The actual perihelia and aphelia of the Earth-Luna barycenter vary considerably due to planetary perturbations; let's find them from the JPL online Horizons ephemeris for Jan 2012AD, Jan 2012-128, July 2011AD and July 2011-128:

perihelia 1884AD Jan 01 23:38.9GMT 2012AD Jan 04 02:59.4

At the time of this crop formation, the immediate past block of 128 actual (i.e. according to actual Earth-Luna barycenter) perihelion-to-perihelion anomalistic years 1884-2012AD equals

46754 - 309.9/360 d

and the immediate past 128 actual aphelion-to-aphelion anomalistic years 1883-2011AD equals

46754 - 259.9/360 d

Now let's compare these angles to the observed. I measure on a printout of the most perpendicular aerial photo I found online, a photo stamped "WCCSG.COM" and with the crescent "APS" logo. One way (method A) to measure the arcs, is to center the protractor on the center of the central disk, and measure to, say, the centers of the end-segments of the arcs. Another way (method B) is to extend lines from the end-segments of the arcs, and with a protractor measure the angle between those lines. Evidently these two methods are not equivalent: for example, the best fitting line through the common end segments near 5:00 in my photo, considerably misses the center of the central disk.

I used both methods, and will denote the measurements A and B. In order from inside to outside, the lengths of the arcs, in degrees, are

inside the whole, third, arc (symbolize less than 46753 d):

A 210 B 220.5 expected 179.9 ("nodical")

A 61.67 B 69.67 expected 66.9 (sidereal)

outside the whole, third, arc (symbolize less than 46754 d):

A 247 B 250 expected 259.9 (actual previous 128 aphelion-to-aphelion)

A 273.5 B 274 expected 278.0 (approx. mean anomalistic)

A 320.5 B 319.67 expected 309.9 (actual previous 128 perihelion-to-perihelion)

The "expected" values are according to the various kinds of years giving something less than a whole number of days, when multiplied by 128. The third arc, or ring, of the crop formation symbolizes the tropical year, which gives a whole number of days, when multiplied by 128. Four of the five arcs agree closely with theory, within my evident error (or perhaps rather, ambiguity) of measurement. I hope to make more accurate measurements soon, and to compare my arc measurements with those of other investigators.

Not only are crop circle messages generally vulnerable to random degradation, they are vulnerable to intentional jamming and destruction by adversaries of the circlemakers. The complexity of the codes gives redundancy, and also makes it difficult for adversaries to guess quickly enough, while the circle is being made, how to damage the message efficiently.

Thanks for asking! Unfortunately, I don't have the time or money, and no one else is looking to my knowledge. Of course there already are the sky survey photos of it.

So you are confident it exists and that you have its orbit pretty well figured out? I'm just an interested follower of the story - no ability to search for it myself.