text of Joseph C. Keller's poster for Oct. 10, 2009 CPAK conf. (U. of California - Irvine), p. 16

(schematics of Egyptian official & agricultural calendars, and of heliacal rising of Arcturus, here)

THE EGYPTIAN/MAYAN CALENDAR GIVES BARBAROSSA'S PERIOD

With modern precession formulas, I reinterpreted Eduard Meyer's Sothic and other dates. I found that the four most famous dates - those of Amenhotep, Thutmose, Seti, and Sesostris - are all, with reasonable interpretation, consistent with an Egyptian calendar beginning at the summer solstice, 4329BC. On this day, both Arcturus and Canopus rose heliacally in Egypt, Arcturus at 30N latitude, and Canopus at the Tropic of Cancer.

The beginning of the current Mayan "Long Count", 3114BC, matches the beginning of Egyptian dynastic chronology, 3110BC, even though there is little evidence of advanced Mesoamerican civilization that old. If Egyptian astronomers somehow knew Barbarossa's period, then the Egyptian/Mayan calendar could have designed to span the 6340.5 yr between the summer solstice, 4329BC, and the winter solstice, 2012AD.

At these times, Barbarossa is at its outgoing latus rectum, a geometrically and perhaps also physically significant point of its elliptical orbit.

text of Joseph C. Keller's poster for Oct. 10, 2009 CPAK conf. (U. of California - Irvine), p. 17

(here, Barbarossa's orbit with times of cardinal points: outgoing latus rectum 2012AD, 4329BC outgoing semiminor axis 3794BC aphelion 1593BC incoming seminor axis 609AD incoming latus rectum 1144AD perihelion 1578AD )

POPE GREGORY AND JOSEPH SCALIGER GIVE BARBAROSSA'S PERIOD

From my analysis of the four sky survey positions, I find that Barbarossa's perihelion occurred in 1578. In that age when heresy was fatal but "Hermetic knowledge" was the rage, who knows what motivated Pope Gregory or Joseph Scaliger, or what fragments of the lost 90% of classical literature they possessed? After a decade of high-priority study by Pope Gregory's elite astronomers, including Clavius, the Gregorian calendar was instituted in 1582. Simultaneously, Scaliger instituted Julian Day Zero, 6295yr earlier; this differs only 0.7% from Barbarossa's 6340yr period.

BISHOP USSHER GIVES THE MAYAN LONG COUNT?

Barbarossa's incoming latus rectum was 1144AD. The interval after Ussher's creation date, 4004BC, is 5147yr, differing only 0.4% from the Mayan Long Count. Ussher's 4004BC is to Barbarossa's incoming latus rectum, about what the Mayan 3114BC is to Barbarossa's outgoing latus rectum.

text of Joseph C. Keller's poster for Oct. 10, 2009 CPAK conf. (U. of California - Irvine), p. 18

HEINRICH EVENTS: BARBAROSSA LAPS PRECESSION

Small climate changes called "Heinrich events" reportedly happen at variable intervals, every seven to ten thousand years. (See: Bond & Lotti, Science 267:1605, 1995)

Let's do this math ourselves. From the sky surveys, I found Barbarossa's orbital period, 6340yr. Newcomb found Earth's (rotational) precession period, 25785yr. While Barbarossa makes one orbit, Earth's precession makes about one quarter cycle. Barbarossa must gain another 90 degrees to catch up to, and lap, Earth. Barbarossa might be fast, moving from perihelion to latus rectum in 434yr, while Earth precesses little. So, almost the shortest Heinrich event/interval, is 6340+434=6774yr.

Barbarossa might be slow, moving from latus rectum to aphelion in 2736yr. Meanwhile Earth precesses about 1/8 cycle, so Barbarossa must move an extra 1/8 cycle, but then Earth has time to precess another 1/16 cycle, etc.; 1/4 + 1/8 + 1/16 + ... = 1/2, so this Heinrich interval, about the longest possible, is 6340 + 2736*2 = 11812.

text of Joseph C. Keller's poster for Oct. 10, 2009 CPAK conf. (U. of California - Irvine), p. 19

SOMETHING HAPPENS EVERY 6340 YEARS

Most say the "Younger Dryas" began 12900yr ago, but from lake varves, Brauer et al find that the sudden climate change in Europe was exactly 12683yr before 2012. At the onset of the "Younger Dryas" a "black layer" was deposited on North America, where every animal bigger than the bison became extinct, as did, more or less, the Clovis culture, all within a span of about a century or less.

Climate change c. 6000yr ago was not drastic, but human cranial diversity in North America and Europe, which had been greater than in the entire world today, drastically decreased, indicating near-extinction of the human species. The only known simultaneous bicoastal megatsunami (Australia, at least) occurred 6000-6500 years ago, and the oldest known megalithic observatory, Nabta in Egypt, was built then too. Pyramid texts say that in early times, Hathor wielding the Eye of Re, and the Cobra Snake of Re, nearly exterminated mankind. According to Plato's "Timaeus", Egyptian priests told Solon, that the Hellenes (i.e. Indo-Europeans) had lost almost everything but the myth that the son of Helios burned up the world with his father's chariot. Thus the Indo-European languages diverged from their common (surviving) root in the 5th millenium BC.

In the North American Midwest, the climate became only moderately drier, changing to the forest types seen today. Yet this archaeological period, c. 6000yr ago, is characterized by a sparse population living on large rivers. Europeans lived in lake dwellings. The priest told Solon, that living near water saved people from the fire.

text of Joseph C. Keller's poster for Oct. 10, 2009 CPAK conf. (U. of California - Irvine), p. 20

(here, diagram showing Sirius sighted over a pyramid to the south; and Arcturus, 2520BC, and Barbarossa, 8690BC, sighted over the other pyramids)

BRIGHT STARS OVER THE PYRAMIDS

At 2520BC (about the time the Giza pyramids were built) one could stand on the pavement north of the base of Menkaure's pyramid, and see Sirius on the meridian above it, just at its peak. While doing this, one could turn one's head, and see Arcturus simultaneously at the vertex of Khufu's pyramid.

According to modern precession formulas, at 8690BC one could stand north of Menkaure's, sighting Sirius at Menkaure's peak just as I described, and turn one's head to Khafre's. There, at the peak of Khafre's, one would see a point in the constellation Crater: a point only 0.8 deg from the Dec. 21, 2012 location of Barbarossa.

Though the time interval, 6170yr, differs 2.7% from Barbarossa's orbital period, and the location differs 0.8deg from Barbarossa's Dec. 2012 position, it would seem that the Egyptians used their freedom, in the design of the Giza pyramids, to memorialize information about the orbital period and critical sidereal location of Barbarossa. As everyone can see from the example of Old English, even in the best of times languages become incomprehensible in a mere thousand years. The numerical agreement is all the more remarkable because of the hints that Earth's rotation has been disturbed in the meanwhile.

(My poster also has five full- or nearly full-page color illustrations. Stonehenge is in the upper left. To the lower right, are a predynastic Hathor head with horns & stars; statues of Khufu & Khafre; Menkaure with Goddesses; and the Giza pyramids.)

In my CPAK poster, I assume that Barbarossa's latus rectum occurs within a few days of Dec. 21, 2012. This assumption amounts to rotating the shape of the orbital ellipse a degree counterclockwise, from what I actually calculated. It allows the radial acceleration, rdotdot, to equal small multiples of H*c (where H is the Hubble parameter and c the speed of light), near Dec. 2012. (The Dec. 21, 2012 coordinates I gave on the poster, were extrapolated directly from the data, not calculated from orbital elements, so aren't affected by my assumption.)

My actual calculation in early 2009, fitting an orbit to the data, was that the latus rectum occurs 2003.92 AD, and the perihelion in 1569 AD. At Dec. 21, 2012, the radial acceleration rdotdot, is approx. negative (the minus sign conventionally means that rdot is decreasing, the radial acceleration is toward us) sqrt(3)*pi*H*c/(2*alpha), where alpha is the fine structure constant, 1/137.

Our Giant Planets Lie on a Circle Dec. 26, 2012

I accurized my extrapolation of the positions of Jupiter, Saturn, Uranus and Neptune. For each planet, I used the last four time points, of those that were separated by 40-day intervals, in the 2010 Astronomical Almanac. I used the full accuracy of the almanac (0.1 arcsec). This gave at least a third order Newton backward formula for each planet.

For Jupiter and Saturn, I used the fourth order difference too, estimated assuming that the quantity of interest was a linear function, plus a sinusoidal one of period equal to the planet's orbital period. Thus the fourth derivative is (-omega^2) times the second derivative. From the differentiation formulas in the CRC Math tables, comes an expression for the fourth backward difference in terms of the second and third.

This gave the least-squares best fit (errors ~ 0.001deg) to Euclid's III.20-21 (i.e., angles ABC = ADC when all points lie on a circle) for the outer planet positions of approx. 18h GMT Dec. 26, 2012. My analysis of six different pairwise planetary configurations on the recent 2012 crop circle (see earlier post) had given the date Dec. 25.0 +/- SEM 0.4.

Has any of you read, or have you preliminary thoughts on the new book by Michael Strauss, "Requiem for Relativity: the Collapse of Special Relativity?"

Has any of you read, or have you preliminary thoughts on the new book by Michael Strauss, "Requiem for Relativity: the Collapse of Special Relativity?"

Re: Our Giant Planets Lie on a Circle Dec. 26, 2012

Two posts ago, I reported that my careful extrapolation, by actual and estimated higher-order differences, of the printed 2010 Astronomical Almanac, showed that our four giant planets most nearly (by sum of squared errors) satisfy the angle equations of Euclid III.20-21 (i.e., lie on a circle) at approx. 18h GMT, Dec. 26, 2012. (This theorem of Euclid's gives a definition of "nearly on a circle" which applies to noncoplanar points.)

Tonight I confirmed this by getting the heliocentric J2000.0 (not equinox & ecliptic of date) celestial (not ecliptic) coordinates for 0h 12/20/2012, 12/23 & 12/26, online from the JPL DE405 ephemeris (the JPL ephemerides long have been the basis of the Astronomical Almanac) at ssd.jpl.nasa.gov, through the JPL Horizons Ephemeris Generator linked to ephemeris.com. I chose the ephemeris values for the barycenter of the planet's system, i.e. including its moons.

The spherical coordinates then were interpolated quadratically between these three time points. Despite doing almost everything differently, I got about the same result as two posts ago: 16h18m GMT 12/26/2012 (vs. my last attempt, 17h37m GMT 12/26/2012).

I've noticed two more unusual things about the giant planets' positions in 2012. The night before last, I fitted ellipses of various inclinations (dy/dx along an axis), to the projections of J, S, U, and N, onto the ecliptic at the time when, according to my rather accurate extrapolation, the actual J, S, U, and N best fit a circle (by the Euclid III. 20-21 criterion). Because the projections onto the ecliptic also closely approximate a circle, most of these ellipses were nearly circles. However, for inclination 79.767deg, the solution is a parabola with its axis at that inclination. Within 0.4deg of that inclination, the solutions are ellipses with eccentricity > 0.1. At 79.79deg, the ellipse has e = 0.6, Barbarossa's eccentricity. Barbarossa's perihelion, according to my calculation from the sky surveys, is at ecliptic longitude 85. So, at Dec. 2012, the four giant planets nearly lie on a circle, but they also nearly lie on an ellipse of the same shape and orientation as Barbarossa's orbit. The inclination of the parabola (or any ellipse of e > 0.1) comes to equal 85, 1.3yr later.

This morning I noticed another unusual thing. At the time of best fit to a circle, the sum of the giant planets' linear (not angular) momenta (assuming their moons have the same velocity as the planets)(with a rough correction for orbital eccentricity) relative to the sun, projected onto the ecliptic, is toward longitude 165deg. Barbarossa's ecliptic longitude then is 176.

(for convenience, also posted to thread, "Quantized Redshift Anomaly")

[quote]Originally posted by Joe Keller [on Mar. 1, 2006, in thread, "Quantized Redshift Anomaly"]

...Blitz, Fich & Stark (Astrophysical J. Supplement 49:183-206, 1982) used radio emissions of carbon monoxide to measure RV's of HII regions in our galaxy. Using my same sky windows, I made a periodogram for the 31 HII regions whose RV's were stated to < 1.0 km/s accuracy. (One trio and two duos of regions were so close in position and RV, that I merged them, netting 27 regions.) Then I augmented these by including the 22 regions whose RV's were less accurate, and made another periodogram. The...most valid [peak was] 2.36 ... km/s.[quote]

My determination of Barbarossa's orbit, early this year, showed that Barbarossa's (scalar) speed relative to the Sun, at the last sky survey detection point, 1997.2 AD, was 2.4044 km/s; and at the latus rectum, 2003.9 AD, was 2.3900 km/s. By extrapolation, Barbarossa's speed is 2.3650 km/s at 2015.5 AD and 2.3550 km/s at 2020.2 AD.

Tifft's small extragalactic redshift quanta varied somewhat with type of galaxy and with location, so redshift quanta of gas clouds in our own sector of our own galaxy might tell us more about our local ether. Tifft's most definite small extragalactic redshift quanta were 2.31 and 2.88 km/s (Tifft, ApJ 485:465-483, 1997; pp. 467b, 473a).

Instead of having, near the latus rectum, a tiny critical radial acceleration that is some small multiple of the Hubble parameter times the speed of light, Barbarossa might have a critical speed that equals the local small Tifft quantum. The catastrophe might be like a sonic boom in the ether. Halley's comet already has slowed to 2.19 km/s, so it's too late to look at it to see if anything happened to it when its speed slowed to 2.36 km/s.

Above, I remark that the angle equations JUS=JNS, etc., which should be satisfied if the four points lie on a circle, are most nearly satisfied (according to the online JPL ephemeris) at 16h GMT, Dec. 26, 2012. The three smallest angles that arise, are 23.13, 29.73, and 31.19deg.

There are twelve angles in all, but on a circle these have to be equal in pairs, leaving only six independent angles. The three biggest of the independent angles, are functions of the three smallest (or vice versa). So essentially, these three angles describe the configuration of our giant planets at Dec. 26, 2012.

The three angles are twice as big (Euclid III.20), if subtended from the center of the circle (not the Sun) instead of the middle of the three listed planets: i.e., J-Center-S instead of J-N-S. Thus they become 46.26, 59.46, and 62.38 deg, resp.

At least roughly, these are the angles found at the bases of the triangles that compose the D & M pyramid at the Cydonia region of Mars. According to the estimate of professional cartographer Torun, the bases of the top triangles are 60deg, of the middle triangles about 49.6 +/- 0.2 or 45.1 +/- 0.2, and the bases of the bottom triangle 55.3 +/- 0.2 deg.

For the entire data set of Blitz et al, 1982, the highest periodogram peaks, for the 194 data points I used, found in the interval between 1.5 & 3.5 km/s, are, from highest to lowest, 3.22, 2.87 and 2.35 km/s. One of these is near Tifft's generally strongest period in this range, 2.88 km/s. My 2.35 period is near another found by Tifft in his extragalactic studies, namely 2.31 (see yesterday's post).

I broke Blitz's Table A into two parts:

1. The middle three, of Table A's seven pages. These cover roughly the nine clock hours from RA 20h, counterclockwise to RA 5h, and include 85 points. Using these alone, the 2.350 km/s periodogram peak was found instead at 2.406 and was rather low.

2. The first two and last two, of the seven pages. These basically amount to 16h to 20h, and 5h to 9h. (Because the galactic north pole is at almost 13h RA, Blitz et al, based in the northern hemisphere, hardly could observe clouds between 9h & 16h.) These include 109 points. Using these alone, the 2.350 peak was found instead at 2.324.

The regional difference, in the period, of at least 0.08 km/s, causes the result to be too inaccurate to predict the date of Barbarossa's "sonic boom" with assurance. If the regional difference is due to the relationship between the galactic and ecliptic planes, then the seven missing clock hours from 9h to 16h, would have increased the result 7/9 as much as the nine clock hours from 20h to 5h, giving the estimate:

2.324 + (2.350 - 2.324) * (1 + 7/9) = 2.370 km/s

Barbarossa achieves this speed, relative to the sun (composition of radial and tangential speeds) at:

Details of handling the data. Blitz's Table A, ApJ Supp 1982, gives carbon monoxide radial velocities in HII interstellar clouds within a few kiloparsec of the Sun, in our galaxy. For eight points, the association, of the CO line with the HII cloud, was uncertain; Blitz gave these in parentheses. I didn't use these: I wanted a homogeneous data set without special objects.

I used all the remaining 194 points. One point, #93, was missing its one-sigma error; I attributed to it the error of a similar nearby point, #97.

The points included two pairs of twins (#81&82, #172&173) and one set of quadruplets (#153-156), i.e. clouds at almost the same coordinates (adjacent in the Table) with identical radial velocities and one-sigma values. I effectively merged a pair, or a quadruple, into one, by halving or quartering the weight of each point, and dividing the one-sigma value by sqrt(2) or sqrt(4). This procedure affected the result by only 0.001 km/s.

Lastly, I replaced each point by ten points, one at each of the 5th, 15th, 25th,...,95th percentile values assuming a normal distribution about its central value, with the given sigma. This was my way of accounting for the much greater sigma of some points.

More proof that the Tifft periods are local phenomena

Today, I corrected Blitz's catalog (of intragalactic HII cloud, CO line, Radial Velocities) for solar apex motion. I defined the solar apex motion as (negative of) the vector whose direction gives the highest correlation between cos(theta) and RV, and whose magnitude v0 then gives the slope of the best fitting linear relationship between cos(theta) and RV. (This is standard elementary statistics; see Dixon or Snedecor.)

The apparent solar apex velocity vector, from the HII cloud RVs, was toward rectangular celestial (epoch 1950) coords. (x,y,z) = (+0.102,-0.002,+0.995), i.e. RA 358.9deg, Decl +84.1deg. The magnitude of the apparent solar apex velocity was 44.2 km/s.

(In this analysis, I didn't use all the abovementioned 194 points. I used only those of the 194, that were on the first six of the seven sheets of Blitz's Table A, because I'd run out of change at the library and hadn't been able to photocopy the last sheet, instead copying only the two columns I needed for the previous post. Roughly this amounts to excluding points with 6h < RA < 10h, so the RA of the 165 utilized points, ranges from 17h, counterclockwise to 6h: roughly a hemisphere. This sample symmetry gained by omitting the last sheet, might help, by canceling even order harmonic effects.)

Before subtracting the effect of the apparent apex motion (i.e., subtracting the first order Laplace term of the gas cloud radial velocity function on the celestial sphere) the highest periodogram peaks between 1.5 & 3.5 km/s, in order from strongest to weakest, were 2.87, 2.35, and 3.21 km/s (vs. 2.87, 2.35, and 3.22 for the full 194 point data set according to the previous post). However, the 3.21 peak had switched from strongest to weakest, of these three.

After correction for apex motion, the 2.874 peak became 2.877 and was only slightly weaker. The 2.350 peak became moderately weaker and moved to 2.313. The 3.21 peak was all but obliterated; maybe it originated as mainly a spacing regularity.

The two observed extragalactic redshift frequency quanta in this range, most prominently displayed in Tifft's papers, are 2.88 and 2.31. Remarkably, intragalactic HII clouds, adjusted for apparent apex motion, show as the most prominent redshift quantization frequencies, in this range 1.5-3.5 km/s, also 2.877 and 2.313!

The two Tifft periods I discovered by the methods above, in the radial velocities of carbon monoxide radicals in Milky Way HII clouds (Blitz's catalog, published 1982), were 2.3127 and 2.8770 km/s (the last digits aren't significant). Tifft mentions 2.31 and 2.88 most prominently, in my range, 1.5-3.5. (Some of Tifft's most prominent larger periods are multiples of these.)

Some will say, "I don't know enough, about the vast and often somewhat debatable mathematical dogma of Fourier analysis, to say whether Keller analyzed it correctly." Others will say, "I know more than Keller, about the vast and often somewhat debatable mathematical dogma of Fourier analysis, and I think Keller made this or that methodological error, therefore I don't have to pay any more attention." Both would be wrong.

As lawyers say, "The thing speaks for itself." How could such elementary and straightforward methods as mine, get intragalactic numbers, so close to Tifft's extragalactic numbers, however "significant" these might otherwise theoretically be?

If you don't know how to make a periodogram, but got an "A" in high school trigonometry, then get a book and learn how (try any book or article by a statistician named Tukey). If you think you know more about making a periodogram, than I do, then verify my work yourself (email me and I'll send you my computer program, so you can copy the data and save the work of typing it in).

Some say, "Creationism is incorrect and Tifft periods would prove Creation, therefore Tifft periods do not exist and I don't have to pay any more attention." Tifft periods have no more to do with Creation, than do periods in the population of rabbits in Saskatchewan.

Some say, "Joe Keller is claiming this, and Joe Keller believes in Planet X, therefore I don't have to pay any more attention." It's not about me, though, except that I do know how to write a BASIC computer program that adds a few data.

Before correcting the data for the presumed solar apex motion (i.e., removing the first order Laplace term) I found periods 2.35, 2.87, and 3.21. The phase of the best fitting sinusoid, was only 11deg from "+ sine" for the 3.21 period, but even farther from any such multiple of 90deg, for the other periods. That is, the phases were not significantly near multiples of 90deg.

On the other hand, after correcting the data for the presumed solar apex motion (i.e., removing the first order Laplace term) I found that the two surviving strong periods, 2.31 and 2.88, gave best fitting sinusoids only 1.3deg, and 8deg, resp., from the phase, "+ sine" (p = 0.26%, to have one or the other sinusoid within 1.3deg of a multiple of 90 phase, and the other within 8deg of the same multiple of 90).

That is, observed intragalactic gas redshift values cluster at ...-0.75*2.31, +0.25*2.31, +1.25*2.31,...,etc., and at ...-0.75*2.88, +0.25*2.88, +1.25*2.88, etc. This regular phase, is further evidence that the correlation is not accidental.

The ratio of the small Tifft periods, is 2.8770/2.3127 = 1.244. This 5::4 ratio likely arises because in a body made of electrically charged deBroglie waves, the outward pressure due to electric repulsion, varies as 1/radius^4, but the outward pressure due to the momentum of the (lowest harmonic) standing deBroglie waves, varies as 1/radius^5.

Suppose conditions require that the pressure increase or decrease 1% per unit time (i.e., that the logarithmic time derivative of the pressure equal +/- 0.01). To achieve this for the electric pressure, will require 5/4 the rate of change, v, in the radius, r, as to achieve this for the deBroglie pressure. Saying this with algebra: the condition, for electric (leads to 2.88 km/s) and deBroglie (leads to 2.31 km/s) pressures, leads respectively to omega = 4*v/r and to omega = 5*v/r. If omega is fixed, then whatever r is, the "v" solving the former equation will be 1.25 times the "v" solving the latter.

The necessary speed of electron expansion or contraction, v, depends on omega and r. If, following Planck, omega = E/hbar, and, following Einstein, E = m*c^2, then r = 1.49/10^15 cm for the electron.

Suppose further that E = 0.5 * alpha^2 * m * c^2, where alpha is the fine structure constant. This is the (absolute value of) the total (potential plus kinetic) orbital energy of a Bohr electron in the ground state of a Hydrogen atom. Then r = 5.59/10^11 cm. This might be practically the same as the classical quantum mechanical radius of the electron, rQMC = sqrt(3) * rCompton = sqrt(3) * hbar /(m*c) = 6.69/10^11 cm (MH MacGregor, The Enigmatic Electron, Kluwer, 1992; p. 5. eqn. 1.3). So, the smallest Tifft redshift periods, are the quantum mechanical and electrostatic radial pulsation speeds, of electrons in Hydrogen atoms.

More correlations of Dec. 2012 planetary positions

As discussed above, our four giant planets most perfectly lie on a circle at approx. 16h (JPL ephemeris) or 18h (extrapolation from 2010 Astronomical Almanac heliocentric data), Dec. 26, 2012. Carefully extrapolating the 2010 Astronomical Almanac heliocentric data, the exact time is 17h37m, Dec. 26, 2012. If the giant planets' centers' positions at that time, according to this extrapolation, are projected onto the ecliptic of date (with the equinox of date) then the parabola through the four points has axis at 79.767deg.

Interpolating the JPL ephemeris positions for 0h Dec. 20, 23 & 26, the linear momenta (using the masses of the planets plus their moons, and finding the velocities by second order numerical differentiation) of all eight major planets are added vectorially and projected onto the ecliptic of date (with equinox of date): the direction is longitude 167.60deg, at 17h37m Dec. 26. If only the four giant planets are used, it's 167.21. Barbarossa's ecliptic longitude at that time, with that ecliptic and equinox, is 176.55.

In complex analysis, three points on a line are considered four points on a circle, with infinity as the fourth point. Just as JSUN lie on a circle Dec. 26, 2012, also SunMercurySaturn and SunVenusS lie on lines near this date (making the circles SunMeSInfinity & SunVSInf). Interpolating the JPL ephemeris and rotating to ecliptic coordinates for the equinox and ecliptic of 2000.0, I find that Mercury and Saturn have the same longitude at GMT 0h0m Dec. 20, and Venus and Saturn have the same longitude at GMT 16h24m Dec. 21.

For both Mercury and Venus, Saturn gets unusually close to the inner planet in latitude as well, so that transits of the Sun occur as seen from Saturn. Transits of Mercury and Venus (especially Venus) are less rare from Saturn than from Earth, but still unusual. Not only is a transit of Venus visible from Earth in June 2012, also there is a transit of Venus visible from Saturn on Dec. 21, 2012, only five hours after Earth's solstice!

Including the Sun, the Infinity point, and the eight major planets, there are 10*9*8*7/(1*2*3*4) = 210 possible "circles of four" that can occur. Some of these are rarer than others, but remarkably, three of them (see above) occur within a week, 0h Dec. 20 to 18h Dec. 26, 2012.

Today also I made a better estimate of the time represented by the "2012 crop circle" which appeared in England in 2008. Again using Nick Nicholson's photo printed from the internet, with my same longitudes of the planets (measured from the photo and corrected for its oblique projection) this time I find that the minimum variance of (expected longitude minus observed longitude) occurs if the date is GMT 1h57m, Dec. 22, 2012. The standard deviation is almost 3deg, so since Mercury moves 4deg/day, the confidence interval is a day or two.

Yet more correlations of Dec. 2012 planetary positions

According to the online JPL ephemeris: as seen from the center of mass of Saturn's system (practically the same thing as the center of Saturn) the center of Venus crosses the boundary of the Sun's disk (i.e., the transit of Venus begins, as seen from Saturn) at 11h35m UT, 12/21/12. This is only 15 minutes after Earth's solstice.

Monte Carlo simulation, with random positions on circular orbits using the giant planets' semimajor axes as radii, shows that the circle SaturnUranusNeptune, will intersect Jupiter's orbit 57% of the time. So, half the time, Jupiter has two points at which it can lie on that circle. The circle JSUN is a configuration that occurs on the average once a decade.

Using the planetary axes of rotation according to the 2009 Astronomical Almanac, p. E3 (based on the 2006 IAU report), and the JPL ephemeris positions, I find that at 13h51m UT, 12/24/12, the axes of Mars and of Uranus make the same angle, 19.4333deg = arcsin(0.33271), with the planes perpendicular to their vectors from the Sun. Mars' precession (thought to be about 1/7 of Earth's) would make this 19.4355, using Mars' 2013.0 axis. (The tetrahedral angle is arcsin(1/3) = 19.471deg.)

At 2013.0, Neptune's axis is 27deg from the perpendicular plane, but that is a few years from Neptune's solstice.

Overall, as of today, Oct. 22, 2009, the best book I've seen on 2012, in any bookstore, is "Beyond 2012 - Catastrophe or Awakening" by Geoff Stray. It has a transcription error on p. 39: the oldest Egyptian megalithic observatory (that at Nabta) was dated 6000-6500 BP (Before Present, i.e. 4000-4500 BC) not 6000 BC.

In this paragraph, I paraphrase and augment Stray's explanation of the Venus cycles which seem to have been important to the Maya. On the average (i.e. using "mean motion") Venus laps Earth in 1.598683 yr (I use Earth's sidereal year of 365.25636d; and the mean of the 15 osculating mean motions of Venus listed among its orbital elements on p. E5 of the 2010 Astronomical Almanac: these oscillate roughly with period about 110d, which is the differential period of Mercury and Earth). This is nearly 8/5 yr, so Venus and Earth return approximately to their same alignment relative to the stars, after 5 cycles, i.e. 8 yrs, with an orbital period resonance of about 13::8. The orbital period resonance is more precisely (13*30 + 5)::(8*30 + 3) = 395::243. So, there are three important intervals: the 1.6 yr "morning star/evening star cycle" which constitutes one lap; the 8 yr "short Venus cycle" which returns Earth & Venus to nearly the same relative position; and the 243 yr "long Venus cycle" which also returns Earth & Venus to the same relative position but ten times more precisely. (End of paragraph.)

I gather that some think the Mayans knew the equation:

243 = 8*30 + 1.6*2 = 8*26 + 1.6*22.

That is, the long Venus cycle is 26 times the short Venus cycle, plus 22 morning/evening cycles. The numbers 26 and 13 were prominent in Mayan calendars. Thirteen also appears in the 13::8 Venus::Earth period resonance.

Might there be an "ultralong Venus cycle"? Consider:

243*26 + 1*22 = 6340yr = Barbarossa's period, analogous to an "ultralong Venus cycle".

That is, Barbarossa's period is found from the long Venus cycle and Earth's year, by the same Diophantine linear formula by which the long Venus cycle is found from the short Venus cycle and the morning/evening cycle.

Some Mayan/2012 lore, says that the last 25 yrs of the long count, i.e. the interval 1987-2012, are special. These 25 yrs might be identical with the extra 22 yr in the above formula, either through imperfect recollection or through small errors in Barbarossa's or Venus' period.

(In my previous post, I mention that the long Venus::Earth cycle, 395::243, is ten times more precise than the short Venus cycle, 13::8. That's based on a conservative estimate of Venus' orbital period precision based on the significant figures usually quoted. According to my optimistic estimate of the precision based on the Astronomical Almanac orbital elements, the 395::243 long cycle is at least 50x as precise, in discrepancy per unit time, as the 13::8.)

The Mayan Long Count is a mnemonic for an actual astronomical cycle, approx. 5124.58 Julian yr, at which seven harmonics converge. Using orbital periods 11.862, 29.458 and 84.01 yr for Jupiter, Saturn and Uranus (World Almanac 2007), gives 432.0165 (5.9deg discrepancy, from whole number), 173.962 (13.7deg discrepancy) and 60.9996 (0.14 deg discrepancy) periods, resp. (p = 0.00005%). Not only these three; four more harmonics converge here.

Enter the dragon. According to Wikipedia's scientific biography of Prof. Rawlins, "Babylonian System A" gave Luna's apsidal period as 6695 anomalistic months :: 6247 synodic months, i.e. 8.85108 yr. The modern value (Wikipedia) is 8.8504 yr. Ptolemy gave 3512 anomalistic months :: 3277 synodic months, i.e. 8.84867 yr.

Ptolemy gave Luna's draconic (sun-eating eclipse dragon; i.e. nodal regression) period especially accurately: 5923 draconic months :: 5458 synodic months, i.e. 18.59867 yr. The modern value (Wikipedia) is 18.5996 yr.

The period of the difference between the apse (prograde) and node (retrograde), is 1/(1/8.8504 + 1/18.5996) = 5.9969 Julian yr = 5.9968 sidereal yr. This is a 6::1 resonance with Earth's orbit.

Let's search for time intervals equal to whole multiples or half multiples of both Luna's apsidal and draconic periods, using the modern values of those periods. Let's also require that the intervals nearly equal whole or half multiples of both Earth's and Venus' sidereal orbital periods. As modern values, Earth's sidereal year is 365.25636 day, and Venus' mean motion (2010 Astronomical Almanac, mean of 15 listed osculating elements) is 1.60213287deg/day.

The apsis, line of nodes, and orbital diameters are lines which recur in the same place each half cycle. In my search, I required that the line of nodes be within 9deg of an exact whole or half cycle. I then required that the root-mean-square discrepancy, of the four cycles, from the nearest whole or half cycle, be less than about 12.7deg.

With a search interval of 0.0001 draconic cycles (less than 0.002 yr, i.e. ~1deg of Venus' orbit) most qualifying intervals were represented by two or three consecutive values; so, the search was acceptably fine. There were eleven intervals shorter than 5124.58 yr, whose rms discrepancy was less. The best of these, 1916.03 yr, had about half the rms discrepancy as for 5124.58.

However, none of the eleven shorter intervals came close to whole period resonance with more than one of Jupiter, Saturn, or Uranus; nine of the eleven came within 23 deg or closer, of whole period resonance with one of them. Eight of the eleven came within 20deg, of half period resonance with at least one of J, S, or U; and three of those eight, with two of them (one, missed half period resonances with Jupiter and Saturn by only 0.8 & 3deg, resp.). The binomial significance of having 20 of 11*3=33, fall within 23deg of a whole or half period, is p << 0.005%, sigma = 4.6.

So, time intervals corresponding to approximate whole or half periods for all of Venus orbit, Earth orbit, Lunar apse, and Lunar node, tend also to correspond to approximate whole or half periods of Jupiter, Saturn or Uranus orbit. The interval, 5124.58 Julian yr, excels all these, because though its four inner solar system resonances are slightly less exact than those of eleven shorter intervals, it has whole period resonance with all three of J, S, and U. It has whole period resonance with the Lunar apse, and half period resonance with the Lunar node and with Venus' and Earth's orbits.

Ancient astronomers designed the Mayan calendar to memorialize this interval, and to memorialize their own skill. This interval, to less than one year error, could be approximated as either 5124 or 5125 Julian (or sidereal or tropical) yr, or as either 5128 or 5129 365-day years (e.g. Egyptian official calendar as used by Ptolemy) or as either 5199 or 5200 360-day years (twelve-month lunar calendar, also widely used). Of these six whole numbers, five factor involving prime numbers of 41 or greater. By far the easiest to remember, is 5200, which factors involving no prime greater than 13. Hence the Mayan Long Count is 360 days * 20 * 20 * 13.

Another resonant (for the inner solar system) interval from the search, is 6323.59 Julian yr. If the Lunar cycle modern values are compromised with Ptolemy's, with coefficients 0.2, 0.4, 0.6, or -0.2, this peak deteriorates in rms fit, but does not change position. So, it is not the same as Scaliger's 6295 yr interval, but it might be akin to Barbarossa's period, 6340 yr.

If Ptolemy's Lunar apsidal and draconic periods, are substituted for the modern, but with modern Venus and Earth orbital periods, and a new search is made for intervals in resonance with the same four inner solar system periods (whole or half, Lunar apsidal & draconic, and Venus & Earth sidereal) the new resonance peaks are of similar height and spacing to the old discussed above. One of the new peaks is 6295.61 Julian yr. This interval equals the difference 1583.0AD (first whole year of Gregorian calendar) minus 4713.0BC (Julian Day Zero) = 6295 yr. Joseph Scaliger instituted the Julian Date calendar in 1582AD. The real cyclic basis for it, thus seems to have been the draconic and apsidal cycles according to Ptolemy, Earth's sidereal year, and Venus' sidereal year.

The Mayan Long Count: an Astronomical Cycle (cont.)

The long Venus cycle is 243 Earth years:

5*243 + 5125 = 6340. Five long Venus cycles followed by a Mayan Long Count, equal Barbarossa's period.

The Mayan Long Count contains an exact whole number of Uranus periods. This is especially plausible because with good conditions, Uranus, though discovered in Europe with a telescope, can be seen with the unaided eye. The Mayan Long Count (if assumed to be the actual harmonic convergence interval, 5124.58 yr, that I found) contains 31.098 Neptune periods, not exact enough to qualify in my search above.

The eleven shorter intervals I identified, whose harmonic convergence for the Venus, Earth and Luna (node and apse) periods was as good or better than the Mayan Long Count, had only borderline significant resonance with whole or half periods of Saturn or of Uranus. They had no Neptune resonance. Almost all their giant planet resonance was with Jupiter, which would be expected from Jupiter's mass and nearness.