Requiem for Relativity

The Mayan Long Count: an Astronomical Cycle

(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.

Barbarossa's Period Resonates with Inner Solar System


Like the Mayan Long Count, Barbarossa's period (the actual period between cataclysmic events) resonates with the known solar system. Barbarossa's period, 6340yr, approximately divides:

the present period of precession of Venus' orbit about Earth's, if Earth's orbit were held fixed: 139451.970yr = 6340 * 21.99558;

and also, more roughly, the others of the nine periods defined as:

the period of precession of the orbit of one inner planet (Mercury, Venus, Earth, Mars) about the orbit of another adjacent inner planet with the other's orbit held fixed, or with neither held fixed.

(Technical note: I averaged all possible orbital radii independently, as though apses were thoroughly scrambled, time-weighting the integral over theta, according to r^2. The integral over the half-orbit always was in 40 equally weighted Riemannian steps. The 1-dimensional integral over the possible arguments of one planet from its node on the other's orbit, and the 2-dimensional integral over the possible anomalies of each planet, also were in equally weighted Riemannian steps, using Romberg's method and increasing the number of steps until the value estimated by the Romberg geometric series, converged accurately enough.

Planet positions were from averaged osculating elements in the 2010 Astronomical Almanac, except for Mars which was from Wikipedia. Planet masses were from the most accurate authoritative sources I found, and were consistent with the values in the 2007 Astronomical Almanac. Inclinations were large enough, that their approximation as infinitesimal, was inadequate.)

Of intervals equal to a whole number of Julian years < 30000, a local minimum of the sum of squared deviations from the nearest whole number of intervals in a period, occurred at 6346yr, but there were 16 lower local minima, one at 27812yr and 15 at intervals < 6300. A local minimum of the sum of (deviation/period)^2, i.e. (relative deviation)^2, occurred at 6353yr, but there were 18 lower local minima, one at 27740yr and 17 at intervals < 6300.

Something could occur in our lifetimes, that occurs once in a hundred human lifetimes. I suggest moving to a medium-sized body of water such as a large river (there is evidence that the Nile, Ohio and Mississippi were large enough) or lake (presumably European lake dwellings were on lakes that were large enough). The Egyptian priest told Solon, according to Plato's Timaeus, that bodies of water protected from the "fire", but that "floods" also were a risk (tsunamis make ocean beaches too dangerous).

"Crouching Tiger, Hidden Dragon" Revisited

In an earlier post, I considered the period that Jupiter (the "crouching tiger") would have, if "reduced mass" weren't used, in the calculation of Jupiter's period from Jupiter's semimajor axis and the Sun's mass; that is, if Jupiter's matter were dispersed into a ring (like Saturn's), or into three equal clumps (like Lagrange points with leading and trailing Trojan asteroids) at 120 deg intervals. This essentially is the period of the proto-Jupiter.

Likewise in that earlier post, I considered the merged proto- (Jupiter + Saturn), proto- (J+S+Uranus) and proto-(J+S+U+Neptune) (the "hidden dragons"). For these, I used the semimajor axis that would give the same total orbital energy as today.

Neglecting the gravitational effect of the ring (or triangle corners) of matter on itself, but considering the effect of all sunward planets and moons, I found that the periods are:

J: 6339.974 Julian yr divided by exactly 534
J+S: 6340.900 / 452
J+S+U: 6340.632 / 436
J+S+U+N: 6341.703 / 416

I found the mean of these four resonant intervals, 6340.802 Julian yr +/- SEM 0.358. The proto-periods range from < 12 to > 15 yr, and the phases are randomized by > 400 cycles, so it is significant that any number so small as 6340 or 6341 should be so close to a whole multiple of all of them. For 6341, the significance calculation is:

1.026*2/11.87 * 0.100*2/14.03 * 0.368*2/14.54 * 0.703*2/15.24
= p = 0.00115 %, i.e. a number in 90,000. For 6340, the significance calculation is:

0.026*2/11.87 * 0.900*2/14.03 * 0.632*2/14.54 * 1.703*2/15.24
= p = 0.00109 %, also about one in 90,000.

Today, I accurized the above by including the effect, on the period, of the proto-planet's own gravity. The centripetal force of an equilateral triangle distribution is sqrt(3)/9 times what there would be if the entire mass were at the Sun. Choosing this configuration because of the evidence given by the Trojan asteriods (a remnant of the configuration?) I find that 6340.143 Julian yr / 452 is the period of the proto - (J+S). This effect is about the mean effect for all four protoplanets, so the accurized mean is about 6340.045 Julian yr = 6340.184 tropical yr.

This is very close to the orbital period I fitted this spring, to the four sky survey images of Barbarossa: 6339.93 Julian yr = 6340.07 tropical yr. Thus an event is to be expected not at the winter solstice 2012AD, but likelier at the summer solstice 2012AD, because of the evidence from Egyptian "Sothic" dates, that the main event last time, was at the summer solstice 4329BC.

"Crouching Tiger, Hidden Dragon" Details

Previously, I also neglected the effect of kinematical "reduced mass" on Earth's year. Correcting for this, would increase the interval by 0.010yr.

Triton's retrograde orbit, suggests that Triton should not be included in Neptune's mass contribution to the protoplanet. Excluding Triton, decreases J+S+U+N by 0.061yr, thus decreases the mean interval for the four protoplanets, by 0.015yr.

Intra-body potentials were disregarded. Only Sun-body potentials, proportional to 1/a, were considered.

The proto-Jupiter would be affected by significant centrifugal force from the proto-Saturn. As a fraction of the Sun's force, it's about

1/3000 * 1/4 * 1/2 * 1/2 = 1/48000

so increases Barbarossa's period, as inferred from Jupiter's, by 1/96000, i.e. 0.066yr, increasing the mean of the four inferred intervals, by 0.0165yr.

These small corrections scarcely affect the main conclusion.

Quadrants of Destiny

Ptolemy's catalog, if its stated epoch is accurate, implies an average precession for Earth's axis, corresponding to a precession cycle of 24601yr (see Pedersen). The Sirius-Arcturus alignment over Menkaure-Khufu, is (according to modern precession rates) 6170yr after the Sirius-Crater(Barbarossa 2012 point) alignment over Menkaure-Khafre, which is near 6150yr = 24601/4. This seems to memorialize a quarter of the precession cycle; Barbarossa's period is 6340yr = 25360/4.

Muller & Rohde at Berkeley, with the substantial concurrence of Kiessling at Humboldt Univ., said in 2005 that mass extinctions occur every 60-65 Myr, with the last one (the dinosaurs) 65Myr ago. The Sun's orbital period around the galaxy is given by various estimates, as 225 to 250 Myr; let's adopt 250 Myr.

The Sun's "vertex motion", i.e. motion relative to the average of nearby stars, is roughly 8% of its speed around the galaxy, and in roughly the same direction. On the other hand, the Sun's "ether drift motion" according to Dayton Miller (roughly corroborated by Michelson-Morley) is roughly the same direction but about half the magnitude, i.e. 4%.

Suppose the Sun's galactic orbit is slightly elliptical, with eccentricity 0.04, and the Sun now is near its periapse, thus 8% faster than its mean speed, and 4% faster than adjacent stars in circular orbits (the local circular orbit frame, i.e. "ether"?). The previous quadrant would have been traversed 8%/2 = 4% faster than average, i.e. 250 Myr/4*96% = 60 Myr. Thus fast & slow quadrants, counting from the present, are 62.5 +/- 2.5 Myr, i.e. 60 & 65 Myr, resp.

We are then 5 Myr overdue for a mass extinction. Maybe the (relatively mild) mass extinction already happened 5 Myr ago, during the time when apes are thought to have been evolving rapidly. "Sagittarius A* " (in a sense the center of the galaxy, and near the center by any reasonable definition) would have crossed the "invariable plane" (i.e. "principal plane" orthogonal to the total angular momentum vector, excluding Barbarossa) of our solar system, 4.5 Myr ago. It would have crossed the ecliptic 4.1 Myr ago.