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).

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.

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.

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.

Precession Resonance for Mayan and Barbarossa Periods

Above, I found that the best convergence of the seven (five solar system planetary orbital, and Lunar apse & node, first or second) harmonics, was 5124.58yr, close to the Mayan Long Count. In another post above, I found that the precession period of Venus' orbit about Earth's, if Earth's were held fixed, is 21.99558 * 6340yr, thus showing resonance with Barbarossa's orbital period.

I find also that the precession period of Earth's orbit about Jupiter's, if Jupiter's were held fixed, is 27.983 * 5124.58yr, showing resonance with the Mayan Long Count. The precession period of Jupiter's orbit about Saturn's, if Saturn's were held fixed, is 21.487 * 6340yr, similar to Venus about fixed Earth, but near the half cycle of Barbarossa's period. The precession period of Mercury's orbit about Venus', if Venus' were held fixed, is 27.965 * 2 * 5124.58yr, again showing resonance with the Mayan Long Count.

I saw your article in the current issue of "Skeptic" magazine, arguing that 2012 will not be a catastrophe. I think it will be a catastrophe.

I'll debate you, or any Ph.D. employed by either NASA or the IAU, on either or both of the subjects, "2012 will be a catastrophe" or "Planet X exists" (I'll take the "pro" sides). I'll debate any or all of you, anywhere, anytime, in any format, though for lack of money, I'll have to phone in.

A year ago, Dr. Neil deGrasse Tyson politely declined to debate me about "Planet X". Dr. Phil Plait ignored my challenge, and censored me from his messageboard. Dr. Clay Sherrod led the successful effort to censor me from the ALPO messageboards.

Sincerely, Joseph C. Keller, M. D. (B. A., cumlaude, Mathematics, Harvard)

My most complete and accurate estimate of the period of the proto-Jupiter, now is 11.871464 Julian yr = 6339.501 tropical yr divided by 534 (this precision is limited by, among other things, the seven-digit precision of Jupiter's semimajor axis). As before, I assume that all semimajor axes were preserved, but that proto-Jupiter's mass was distributed symmetrically.

In this estimate, I account accurately for the influence of all other planets and moons, including Barbarossa (a small influence, due to its great distance). Orbital eccentricities are assumed to be zero, except that, for Barbarossa, I use the eccentricity, 0.6106, that I find from the sky surveys.

The most important change in my calculation, is that instead of assuming that the proto-Jupiter mass was at three equally spaced Lagrange points, I assume that it was spread into a wide ring (an infinitesimally narrow ring gives an infinite effect). I estimate the effect of this ring, by assuming that it is wide enough that the net effect is zero nearby. The effect is proportional to 1/2/sqrt(2) at 90deg, and 1/4 at 180deg. The effect, as a function of theta, is periodic with local minima at 0 and 180. So, the best numerical integration, using values at 0, 90, 180, & 270, is with equal weights. This, and some smaller accurizations, decrease the implied interval from 6339.637 tropical yr (with the Lagrange point distribution) to *6339.501 trop yr* (with my new preferred wide ring estimate).

If I were a year off, and the Egyptian calendar began at the summer solstice 4328BC instead of 4329BC, then the time to the winter solstice 2012AD, is (assuming today's precession rate) *6339.5032 trop yr*, accounting for Earth's longitude of perihelion in 4328BC and in 2012AD (est. advance of Earth's perihelion: 360deg in 112,000 yr; est. effective eccentricity, 0.017). The discrepancy, between this time interval, and the interval resonant with the proto-Jupiter, is only 0.002yr = 0.7day.

Another approach to Barbarossa's period, is through the Bessel functions of the second kind (Neumann functions). The ratio of the Mayan Long Count, or rather, more precisely, the best grand visible solar system cycle, 5124.58 Julian yr, which I discuss above, to Barbarossa's period (assumed to be 6339.5032 modern tropical yr) is 1::1.23705. This is close to the ratio, 1.24113, of the first peaks, of the first and third Neumann functions. It is even closer to the ratio, 1.23422, of the first peaks of (Neumann function / (1/sqrt(x) ), that is, the approximate ordinates where the first and third Neumann functions first touch their common envelope, if one approximates that envelope by a function A/sqrt(x). The precise calculation of these envelope points, gives the ratio, 1.23113.

(Neumann functions can be found from rapidly convergent series in Apostol's Calculus, vol. 2, or in Franklin's Calculus. The series in Apostol omits the factor 2/pi which Franklin includes. The formula in Franklin contains a grossly erroneous definition of the "psi" function, which contradicts both Apostol, and Jahnke & Emde.

Franklin gives the method for finding envelope curves. This requires finding the derivative of the Neumann function with respect to its order. Difficulties arise from the definition, of the integer-order Neumann function, as a limit. I use Hankel's complex-valued integral formula in Jahnke & Emde, for H1, with k=1, to overcome these difficulties. The formula ultimately requires values of the "factorial function" near x = -1/2; Jahnke & Emde give the exact values of the factorial function, and its logarithmic derivative and second derivative, at x = -1/2.)

The Mayan Long Count, Barbarossa Interval and Legendre Polynomials

The Mayan Long Count really is based (see my previous posts) on what arguably is the most practical approximate least common multiple, or grand cycle, of the main periods of the (unaided eye observable) inner and outer solar system. This is 5124.58 Julian yr according to my reasonable mathematical definition using modern data.

Imitating the Maya or their predecessors, Joseph Scaliger apparently used Ptolemy's (less accurate) Lunar node and apse periods, to find a completely analogous grand cycle, but of periods of the inner solar system only, plus Barbarossa. The best such grand cycle would have been 6295yr, which is the interval between Julian Day Zero, and the first New Year of the Gregorian calendar.

Apparently Scaliger disguised his Hermetic calculation, in the guise of an arbitrary multiplication of trivial calendric cycles, whose end product, 7980yr, has little importance. The Maya adopted 360days * 20 * 20 * 13 = 5125.257 Julian yr, not for disguise, but for popularity and convenience. Scaliger, on the other hand, needed a believable excuse for his cycle; he had survived the Huguenot massacre, and eventually outlived Giordano Bruno.

Also there is the Barbarossa interval of cataclysmic Earth change: defined either as Barbarossa's orbital period according to my estimate from the four sky survey images (6339.93 Julian yr), or as exactly 534x the period of the proto-Jupiter (6339.362 Julian yr), or as the interval from the beginning of the Egyptian calendar (according to my explanation of "Sothic dates") to the end of the Mayan Long Count (6340.361 Julian yr +/- one whole tropical yr), or as half the interval from Brauer's Younger Dryas Onset to the end of the Mayan Long Count (~6341.5yr). Choosing the 6339.361 Julian yr = 6339.5000 tropical yr as most accurate and reliable, the ratio of this interval, to the Mayan Long Count, is 1.237050.

Above, I equated this ratio, approximately, to the ratio of ordinates of first peaks, or, approximately, of ordinates of first envelope points, of the first and third Neumann functions (Bessel functions of the second kind). Today I find a more accurate equation: it is a general resonance of Legendre polynomials.

Let's find the point on the unit interval, where the ratios of the Legendre polynomials oftenest approach (+/-) a power of two. From a table of the first few Legendre polynomials, it appears that one, is such a point (by definition they all equal one, at one) but also 0.81 is such a point.

Using an IBM 486 computer, and finding the values of the Legendre polynomials by the recurrence relation, I find that for the first 60 Legendre polynomials, the peak of this resonance, i.e., the maximum within the *interior* of the unit interval (excluding the portion near one) of the sum of cos(2*pi*(log(abs(Pi))-log(abs(Pj)))/log(2)) over all i,j, is 0.81173 = 1/1.23194. For the first 70, 80, 90, or 100 polynomials, this local maximum of the resonance, 0.81173, decreases steadily, by monotonically smaller decrements, to 0.81050 = 1/1.23381. Assuming that the tail is proportional to 1/n, the value 1.23194 at 60 and the value 1.23381 at 100, imply a limiting value, for all the Legendre polynomials, of 1.236615. This corresponds to a Barbarossa interval of 5124.58 * 1.236615 = 6337.13 Julian yr.

The Mayan Long Count, Barbarossa Interval and Legendre Polynomials (cont.)

(In the previous post, "first 60 polynomials", etc., means first 60 pairs of even & odd polynomials.)

Applying the trigonometry formula for the cosine of a difference, then Fubini's theorem, the sum in the previous post, becomes a convolution as for a periodogram. With the first 5000 pairs (i.e. first 10,000) of Legendre polynomials, the maximum periodicity of log(abs(Pi(x))), for log(2), is at x = 0.809047, 1/x = 1.236022. Multiplying by the presumed underlying Mayan Long Count interval, 5124.58 Julian yr (this is the solar system resonance, which is approximated in Mayan-style round numbers, by the actual Mayan calendar Long Count) gives 6334.09 Julian yr.

Maybe this "Legendre2resonance" = "Leg2" = approx. 0.809047, must be divided, in this physical application, by "g/2" where g is the electron gyromagnetic ratio, 2.0023193044. That would give 6341.44 Julian yr = 6341.58 tropical yr, as the Barbarossa period, exactly half the time, 12683 yr, from Brauer's Younger Dryas onset to Dec. 2012AD.

Another approach to adjustment, is to note that 1 + Leg2 + Leg2^2 +... = 1/(1-Leg2), nearly equals the ratio of Jupiter's and Earth's major axes. Taking Jupiter's semimajor axis as 5.204267 AU, and assuming that the foregoing equation is exact, gives a slightly corrected value for Leg2, which gives 6343.480 Julian yr as the Barbarossa period.

Likewise, I note that 1/(1-Leg2)^2 nearly equals the number of mean solar days, 27.32166, in one sidereal Lunar month. Assuming the equation is exact, refines Leg2, and gives 6336.921 Julian yr as Barbarossa's period.

Finally, with the draconitic Lunar month, 27.21222 d, instead of the sidereal, the refinement of Leg2, to satisfy the equation 1/(1-Leg2)^2 = draconitic month::mean solar day, gives 6339.934 Julian yr as Barbarossa's period. Early this year, my fitting of an orbit to the sky survey detections, gave 6339.93 Julian yr.

My estimates of the catastrophic "Barbarossa period" seem consistent:

1. Barbarossa's sidereal orbital period from the sky surveys (6340.07 tropical yr) has a "sigma" error of > 1 yr and < 9 yr. Fitting an orbit to the sky survey positions, I found the radial speed only to the nearest 1 part in 1000, because it seemed that smaller differences surely were insignificant. This implies a sigma, for the radial speed, of > 1/4000, and a sigma for the total kinetic energy and hence (by the virial theorem) total energy and major axis, of > 1/10,000. By Kepler's 1.5 power law, the sigma for the period thus is > 1/7000. The sigma error in estimating the centroids of the light patterns on the sky survey scans, seems to be < 1 arcsec, and the observed path about five deg, so the error in initial and final angular speeds is < sqrt(2)*1arcsec/2.5deg = 1/6400. With about a 1/40 increase in radius during the 1/20 radian travel (between the centers of the first and second segments), the angular speed decreases 1/20, so the sigma of this decrease is < sqrt(2)/6400*20 = 1/225, hence the sigma of the radial speed is < 1/450, and by the previous paragraph, the sigma of the orbital period is < 9yr.

2. Brauer's lake varves, together with the Mayan calendar, imply 12683/2 = 6341.5 yr, but Brauer's earlier articles contend with uncertainties of several years, so it seems likely that, somehow or other, such uncertainties still exist.

3. My reconstructed Sothic/Arcturian Egyptian calendar gives 6340.503 tropical yr before the winter solstice 2012AD (it must begin on a summer solstice, and Earth's orbital eccentricity has a small effect). A year corresponds to a quarter-day error in heliacal rising, which is reported only to the nearest day, so 6339.503 and 6341.503 tropical yr also are likely.

4. The "Legendre 2 Resonance" estimated from the first 25,000 pairs of Legendre polynomials, is 0.809023. From the first 5000 pairs it's 0.809047. From the first 1000 pairs it's 0.809168 (all these are by direct computations without interpolation, using an Intel Pentium CPU). This shows an accurate 1/n tail. Adding the tail to the 25,000 pair computation, gives x = 0.809017 = cos(36) = 0.809016994. Multiplication of sec(36) by my best Mayan Long Count general solar system resonance (5124.58yr) gives 6334.47 tropical yr. However, I don't know whether the polynomials should be weighted equally, as I did, or not.

5. Resonance with the proto-Jupiter implies 6339.501 tropical yr. The main uncertainty here, is the configuration of the proto-Jupiter's mass ring. The chief alternative, equal masses at 120 deg intervals, gives only 4/5 as much centripetal force as the approximation I used. So the uncertainty in orbital period is roughly 1/5 * 1/2 * 1/1000 * 1/4 = 1/6 yr.

6. Resonance with half (never whole) orbital periods of the seven largest planets, defined as minimum sum of squared deviations from odd half-periods, gives 6339.522 tropical yr. The resonance with Mercury is so sensitive to tiny changes in the Barbarossa period, that it would dominate everything else, so, I omitted it. Pluto's long-term average orbital period seems uncertain due to its eccentricity and interaction with Neptune and other planets. I took the usual (e.g., 2007 World Almanac) sidereal orbital periods of Jupiter (11.862 Julian yr), Saturn (29.458 yr), Uranus (84.01 yr) and Neptune (164.79 yr). I calculated yesterday that the oldest observation of Jupiter recorded by Ptolemy (occultation, 3rd cent. BC) and Ptolemy's own opposition observation of Jupiter (2nd century AD) both are consistent, vis-a-vis modern ephemerides, with orbital period 11.862 yr (I found 11.8624 and 11.8619 yr, resp.)(data from Olaf Pedersen's book on Ptolemy). I used Wikipedia's sidereal orbital periods for Venus (224.70069 day) and Mars (686.971 d). I used 365.25636 d for Earth's sidereal period. For this calculation, the inner planets need greater accuracy than does Jupiter, so it seems unlikely that Ptolemy's observations would help much. Mars, Jupiter, Uranus and Neptune had nearly n+0.5 cycle (range: n+0.43 to n+0.54). The sum of squared deviations from the nearest odd half-cycle, was 0.2247 at 6339.264 Julian yr, 0.1711 at 6339.364 Jul yr, and 0.1957 at 6339.464 Jul yr. Quadratic interpolation gave the minimum at 6339.383 Julian yr = 6339.522 trop yr.

Summarizing: the only time interval consistent with all my estimates and their uncertainties, is 6339.503 tropical year, corresponding to an Egyptian Year One at 4328BC.

In the previous post, I mention correlations of the "Legendre 2 Resonance", in our solar system. More such correlations are:

1. The mass ratio Venus / (Earth + Luna) = 0.8050950.

2. Using Luna's average eccentricity of 0.0549, the angular momentum ratio (Earth rotation)::(Earth-Luna orbit) is 1::4.877 (Earth's dimensionless polar moment of inertia = 0.3307, per YV Barkin, 2009; vs. 0.4 for a homogeneous sphere). If the ratio were 1::3 instead, then the number of Earth rotations per sidereal month would (to first order, that is, until the ratio began to deviate appreciably from 1::3) remain constant as rotational and orbital angular momentum exchange. That the ratio is near 1::5 instead of 1::3 suggests that somehow other quantities are involved.

From a 1977 review of Robert Gentry's work on radio-halos (today I find this review, of uncertain authorship, posted on Dr. Gentry's own website):

"Gentry believes the evidence points to one or more great 'singularities' that have affected Earth in the past, representing physical processes which we do not now observe. ...Further (as we will explore in a subsequent review), Gentry concludes that the most recent singularity may have occurred only several thousand years ago."

Apparently Gentry doesn't have a Ph.D., but is primary author of many items about radio-halos, that appeared in Science, Nature and other leading journals in the 1970s. Today I scanned most of these that are on Gentry's website. I found no calculations of the age or epoch of anything, only lengthy discussions of laboratory measurements and possible chemical processes of radio-halos. Gentry's later writings often mention 6000 yrs, but I find no evidence that this figure comes from any radio-decay calculation, rather than Ussher's chronology of the books of Moses. Afterwards I searched RV Gentry in Web of Science (Science Citation Index), found almost all of the 35 articles that seemed relevant (i.e., about actual radio-halo studies, not cosmology or creationism)(especially his major articles, which mostly were in Science or Nature), on the shelves in the Iowa State Univ. libraries, skimmed them, and found no actual date computation more recent than a lead/uranium date of 250,000yr for coalified wood (Science 194(4262):315+, 1976).

Nonetheless, it appears from the above that at least one reviewer, sanctioned by Gentry himself, concluded that Gentry's work suggests, not any creation event, recent or otherwise, but rather, an unknown cyclical atomic phenomenon of period several thousand years. This supports my theory, of Barbarossa events every 6340yr.

These two especially active "Ring of Fire" volcanos, in Washington state USA and in Kamchatka Russia, resp., erupted frequently c. 6300-5900 yr "BP" (i.e., yr ago), with a few hundred years uncertainty of C14 calibration.

These volcanos might be "canaries in the coal mine". Let's watch them (more than Yellowstone, the rare, though huge, eruptor featured in the recent movie "2012").

(Avachinsky is smaller than, and not the same as, Ichinskaya.)

More About the "Legendre 2 Resonance" (generally, "Jacobi 2 Resonance")

This is a fundamental mathematical relationship that I have discovered. Choose a real number x between 0 and 1, and a large whole number n. Evaluate the first n Legendre polynomials, at x (this is fast, using the "recurrence relation"). Take the logarithm of the absolute values of these evaluations, and multiply that logarithm, by 2*pi/log(2). The result is, a set of n negative real numbers.

The periodicity of this set, can be found by convolution with cosine and sine. That is, add the cosines of the numbers (all the cosines will be 1, if all the Legendre polynomials have values of the form +/- 1/2^k) to get a sum "S1", add the sines to get a sum "S2", and consider f = S1^2 + S2^2. The absolute maximum of f is at x=1, because there all the Legendre polynomials are 1, and all the logs are 0. However, there is, in the limit as n approaches infinity, an important relative maximum at x = cos(36 degrees) (or at least very close to cos(36) ). As detailed in my recent post, I used an Intel Pentium CPU (in a Hewlett-Packard desktop computer) to find "f" for n=1000 (i. e., the first 1000 pairs of Legendre polynomials), n=5000 and n=25,000; then used the accurate 1/n tail to estimate that the local maximum occurs for x = 0.809017 = cos(36) = (1 + sqrt(5))/4 = "golden ratio"/2.

If one convolutes with the zeroth and first Bessel functions, J0 and J1, instead of with cosine and sine, that relative maximum occurs at the same abscissa, x = cos(36). That is, find the values of all the Legendre polynomials, take the logs of their absolute values, multiply by 2*pi/log(2), and find the values of J0 and J1, instead of cosine and sine, for this set.

For n=400 (that is, the first 400 pairs of Legendre polynomials) I found (using an IBM 486 computer, programming with BASIC in double precision) this relative maximum at 0.80939, for n=800 at 0.80921, for n=1600 at 0.809113, and for n=3200 at 0.809063. (For all these, the last digit is by rough interpolation "by eye".) The successive differences are 0.000180, 0.000097, and 0.000050. This shows a fairly accurate 1/n tail, so the abscissa for n = infinity can be estimated by the geometric series, as 0.809013. This hardly differs from cos(36) = 0.809017. (I evaluated the Bessel functions using their symmetry or antisymmetry to get a positive abscissa, then used Hankel's semiconvergent series to fifth degree, per Jahnke & Emde, for abscissas > 16; and 27 terms of the power series, for abscissas < 16. Experiment showed that the cutoffs 16 and 27 were plenty big to make the error negligible.)

In the limit of large n, the Legendre polynomials' recurrence relation gives the sequence defined by P(n+1) = r*P(n) - P(n-1), where r = 2*x. Thus I find that when r equals the golden ratio, this sequence tends especially to lie near powers of 0.5, if it starts as the Legendre recurrence relation.

The Legendre and the Chebyshev polynomials are special families within a two-parameter collection of families of orthogonal polynomials called Jacobi polynomials. For all families of Jacobi polynomials, the recurrence relation, in the limit of large n, approaches the simple relation, P(n+1) = 2*x*P(n) - P(n-1). The zeroth and first Chebyshev polynomials are the same as the zeroth and first Legendre polynomials (1 and x, resp.). Though the Chebyshev polynomials have this simple recurrence relation from the beginning, and the Legendre polynomials only approach it in the limit of large n, one might expect the Chebyshev polynomials likewise to have a resonance peak (i.e., relative maximum of "f") at cos(36). Surprisingly, the Chebyshev polynomials have a resonance trough (relative minimum of "f"), not peak, at cos(36).

(With the IBM 486 CPU and double precision, I find that for n=800 pairs of Chebyshev polynomials, the trough is at 0.8090076; for n=1600, 0.80901217; and for n=3200, 0.80901458, where all the last digits are rough interpolations "by eye". The successive differences are 0.00000457 and 0.00000241, showing a fairly accurate 1/n tail, though here instead of a peak at decreasing abscissas, there is a trough at increasing abscissas. The geometric series estimate for n = infinity, is 0.80901458 + 0.00000241 = 0.80901699 = cos(36).)

Approximately, the Mayan Long Count (an approximation to a 5124.58yr resonance of solar system periods), times 4/(golden ratio), equals twice the Barbarossa period. The Long Count contains approximate whole numbers of periods of Jupiter, Saturn, Uranus and Neptune, and of Luna's apsidal advance. For some reason, so likewise does the Long Count * 4 / (golden ratio).

Thus the Barbarossa period contains approximate whole or half-whole numbers of these periods. The five year difference between the actual Barbarossa period, est. 6339.364 Julian yr, and the 5124.58 yr resonance basis of the Mayan Long Count * 2/(golden ratio) = 6334.33, causes the actual Barbarossa period to lack the Lunar apsidal and Saturn resonances, but to gain a Mars (half) resonance.

"I do not take kindly to the argument that because certain working hypotheses may not possess eternal validity or may possibly be erroneous, they must be withheld from the public."

- C. G. Jung, "Symbols of Transformation", quoted in "Psychological Reflections" (Princeton, 1970), p. 183.

Hi Joe, I think that's okay if you happen to be a Carl Jung, or a Einstein, Gell Mann, etc. I think it's counterproductive here. People are just going to wait until 2012, they're not going to allocate telescope time to it, for fear of looking like idiots if nothing happens in that year. If something horrendous does happen, then nobody is going to be that bothered about a possible brown dwarf, we'll all be far too busy burying the dead to worry about observational astronomy. Something of a Cassandra situation for you, and that has to be frustrating but people are just going to wait and see.

What did you make of the recent huge tsunami on the sun? So unusual that they thought it was a computer glitch.

Hi Joe, ...People are just going to wait until 2012, they're not going to allocate telescope time to it, for fear of looking like idiots if nothing happens in that year. If something horrendous does happen, then nobody is going to be that bothered about a possible brown dwarf, we'll all be far too busy burying the dead to worry about observational astronomy. ...

Thanks for the post! You've given not only a springboard for discussion, but also a wake-up call.

I don't know exactly what will happen. Maybe all of a sudden one day, 90% of Earth's surface is baked by meteor swarms, or solar flares, or something that we haven't had at all since 6000 or 12000 years ago, then the survivors go back to the Bronze Age again. On the other hand, maybe there will be a warning, like cataclysmic brightening of Barbarossa, or solar behavior unprecedented in our history, or comets big enough to detect a few hours or days before impact, or preliminary jolts before "the big one".

My prediction? Known and/or unknown physical forces slay almost all humans on almost all of the world's surface. This will happen on some or all of the days within three months either side of Dec. 21, 2012. Patches will be accidentally spared with only moderate destruction (i.e. < 50% death rate). I wish I knew more about where these patches will be, but scraps of historical and archaeological evidence suggest that large rivers and midsize lakes offer some kind of protection. Maybe it's as simple as getting in the water when a fireball comes, yet not such a big body of water that it makes a tsunami.

Shortly before the U. S. Civil War, U. S. newspapers ran articles denouncing "Crazy Sherman" because Sherman publicly predicted high casualties. It turned out that Sherman's prediction was way low, but his was the only prediction that even got it in the ballpark (source: Ken Burns' TV series, "The Civil War").

There's some evidence that undergroud shelters won't work: the Pyramid Texts say "the people of the mountains were exterminated", yet these are the people with readiest access to cliffs and caves. Our ancestors might have known that megalithic structures somehow sufficed where natural caves or digging in, didn't. These megalithic shelters might have been on hilltops (maybe even, mountaintops, e.g. Mt. Ararat or the Andes) to avoid some kind of toxic miasma mistranslated as a "flood".

It would have been nice if I had been able to persuade some of the government astronomers to lift a finger (two astronomers employed by the government, one at Harvard and one at the U. of Iowa, actually did lift a finger many times to help me, in practical ways)(Dr. Van Flandern's help, was at his own expense, long after he had ceased to be a government astronomer), but I think most were too corrupted by the tax money. They have bought into the poisonous attitude that it is more about turf than truth.

The "climate change" ethics scandal, though only marginally related to my topic (I don't know whether 2012 will bring warming, cooling, or neither; I'm betting on cooling, though CO2 reduction might be important anyway), might weaken the ability of the academic establishment to stonewall. If their stonewall cracks a day sooner - on Dec. 19, 2012 instead of Dec. 20, 2012 - it could make a big difference. The Wall Street Journal hasn't printed the short letter I mailed them the day after Thanksgiving, in which I point out that the science guild is unreliable, that anyone with a World Almanac and a checkbook calculator can prove for himself in five minutes that the Mayan calendar embodies profound astronomical and mathematical knowledge (e.g., the Long Count is an exact multiple of Uranus' orbital period), and that mainstream scientists had better get on the case.

I and several government (state and "private", i.e. semi-state, university) astronomers spoke at the North Central Region (NCRAL) amateur astronomers' conference in May 2009. I attended most of their talks but I'm pretty sure none of them attended mine. I also spent the previous two days hiking many miles around several nearby college campuses trying to recruit diverse faculty members, to attend my talk, but I'm pretty sure none of them attended either.

Maybe it's like this every time: 6340 yrs ago, 12680 yrs ago, etc. Maybe our ancestors (anyway, more or less our ancestors) left the Giza pyramids, Mayan calendar and other clues so that "next time it can be different".

It's said that only one, small, plain statue of Khufu is known. Khufu wasn't an egomaniac: he didn't even bother to have anyone make his statue. If he was buried in the pyramid at all, it was only because it was dear to him as his life's work. What is Khufu's pose in that statue? It's like the movie, "The Ten Commandments": "So let it be written, so let it be done!" and he's smiling.

Hi Joe, about eighty thousand years ago, the human race was almost wiped out. Favourite culprit being the super volcano Toba. If anything happened about six thousand years ago, it was a massive cultural leap forward.

I remember once a lecturer reading a fragment of a love poem y Sappho, you could have heard a pin drop in the room. Everyone felt that they had touched the mind of a woman from a bygone age. yet we are all painfully aware of of modern idiocies of interpreting the mind set, world view of an Alexander for instance. He did it for the chicks and fast cars school of thought. We can see him in red braces and a striped made to measure shirt.

Bearing that in mind, let's try and think like an ancient celt for instance. "There was a time of great troubles and tribulations. Those that went and made offerings to the sacred lakes; doorways to the other world; survived and those that didn't perished." Any standing water meant a great deal to people who had never seen a mirror, and who obviously knew that water had powerful magic. The point is, that we now simply cannot comprehend their world view. The same goes for baleful Sirius. It seems to change colour all the time.

On this board most people would have to say, that some change in the core of a planet, would e instantaneously transmitted to the core of the sun. How that information might effect things would be open to debate. But things are going to take time in the region of tens of kilometres per second. So, stuff must be already happening. Low mass stuff at first, changes in the magnetosphere of the sun springs to mind. That's why I thought of the tsunami on the sun, pretty major event that. Now if such changes have been noted, then the powers that be have decided not to tell anyone. To tell the truth, I wouldn't say anything either. If our time is up, why worry people?

...If anything happened about six thousand years ago, it was a massive cultural leap forward. ...

...If our time is up, why worry people?

It hasn't yet become canonical academic dogma, but recent articles in mainstream archaeology journals say that there was a drastic reduction in human cranial diversity in N. America and probably in Europe too, roughly 6000 yr ago (I cited the most important article - a lengthy, careful analysis of the N. American crania - in one of my posts earlier this year). The date is uncertain because of the small number of N. American skulls and the uncertainty in dating them. It was really a drastic reduction: prior to roughly 6000 yr ago, the cranial diversity in N. America alone, was much greater than in the entire world today.

The old theory, pretty much German National Socialist anthropology c. 1937, was, that some genetic mutation(s) occurred among Europeans c. 6000 yr ago, which allowed the mutant group, which happened to speak proto-Indo-European, to outproduce or outfight their neighbors, kill them off and/or seize or buy the neighbors' women (so that after many generations, only the neighbors' mitochondrial DNA was left, as modern English have mainly Danish nuclear DNA and British mitochondrial DNA), spread proto-Indo-European language all over Europe and west Asia, and greatly upgrade civilization. Alfred Rosenberg (even today, his text sits in the open stacks of most big university libraries) cited the obvious similarity of some common ancient Egyptian words to Indo-European, and the obvious European appearance of many Old Kingdom aristocrats, as evidence that even ancient Egypt was part of this evolutionary explosion. Sumer and very early China (six-foot red-haired folks in Mongolia) can be worked into the theory somewhat. (Even such mainstream sources as the American Heritage Dictionary, say that the Indo-European languages mysteriously exploded from a common root c. 6500 yr ago.) The new cranial data from N. America, contradict the foregoing National Socialist theory of recent human evolution. It's no longer necessary to rebut the quasi-National Socialist anthropologists with shaky nurture-over-nature arguments (Franz Boas et al) or by throwing them in prison (outlawing "racism").

It's unlikely that in N. America, too, and at the same time, hugely advantageous sporadic mutation(s) drastically reduced the human diversity on the continent. It's likelier that there was a worldwide cataclysm, with a survivor effect on linguistic and cranial diversity on both continents. If something randomly wiped out, say, England and Germany and Russia, then almost all northern Europeans would have French culture, language and physiognomy. About 6500 yrs ago, I think, this did happen, and the survivors happened mainly to be Indo-European in Europe, and Athabascan (e.g. Navaho, Apache) in western N. America. High civilization suddenly appeared almost everywhere simultaneously c. 5000 yr ago, not because of a brain mutation in Nordics or Semites or somebody, but because people everywhere had to rebuild for a thousand years after being almost wiped out. About the only things left of the "antediluvian" civilization, are some megaliths. The antediluvians, judging by the megaliths, by the hints left about astronomy, Barbarossa's orbit, etc., and by the lack of artifacts on the scale of, say, Hoover Dam or the battleship Missouri, seem to have been about as advanced as 16th century Europe.

There was no drastic climate change 6000 yr ago. Central N. America became only moderately drier, transitioning to the same forest types it has today.

The reason to worry people, is, as I said, so "this time it can be different". The more I "worry" people, the likelier that powerful instruments will be brought to bear, research will be done, and survival techniques improvised. None of these cyclic cataclysms exterminated mankind. As on the advertisement for last month's hit movie, "2012", the question is: who will survive?

I speak of a 6339.5 year, cataclysmic Barbarossa period. I and many others speak of a 5125 year Mayan Long Count period. Exceptionally many solar system resonances (i.e., common multiples) occur for both these periods.

There is a "third period" for which exceptionally many solar system resonances occur. This is: the Mayan Long Count, times two, divided by the golden ratio; i.e. 5124.58 Julian yr / cos(36), where I use as the "essential" Mayan Long Count, the best harmonic resonance with the main solar system periods, according to their modern values. This third period is 6334.329 Julian yr.

There is a significant tendency for asteroid orbital periods to resonate with this "third period", that is, for their orbital periods to divide it. Those asteroids which have attracted the most attention (for reasons other than bizarre orbits) should have the most accurately known long-term average periods. Also, massive asteroids should have less chaotic orbits (perturbing most of their interactors through many cycles, before they themselves are perturbed); and very slowly rotating asteroids should be most immune to unknown forces involving rotation.

So, I chose the eight asteroids on Wikipedia's list of the most massive:

To gain statistical sensitivity, I augmented these, with ten presumably very massive asteroids from Wikipedia's list of those largest in dimension (but cautiously, only those on that list, for which all three body axes are given):

7 Iris 16 Psyche 19 Fortuna 29 Amphitrite 45 Eugenia 52 Europa (note allowed conflict with name of Jupiter's moon) 65 Cybele 87 Sylvia 88 Thisbe 107 Camilla

Lastly I added five from Wikipedia's list of slow rotators:

Wikipedia's list of slow rotators, had three more "asteroids", named only by a more modern system, but two of these had orbital periods that I couldn't find quickly searching the Web, and the other of the three had an orbit resembling that of Halley's comet. So, I didn't use these three.

My statistic, "f", was the sum of the natural logarithms of the fractions of an orbit, by which the "third period", 6334.329 yr, failed to contain a whole number of periods of the asteroid. Thus the most significant result, is that for which the product obtained by multiplying all the leftover fractional periods (e.g., 0.201 period too much, 0.102 period too little, etc.), is the smallest. The distribution of "f" is nearly normal, because "f" is the sum of many (23) terms, each of which has, the same distribution which differs not too extremely from a normal one. Elementary calculus gives the mean and variance of the distribution for one term, and from that, the mean and variance of f are found by very elementary statistics.

According to this, the tendency for these 23 asteroids' periods to divide wholly into 6334.329 Julian yr, is significant at sigma = 2.2, i.e. p = 1.4%. Also, nine of the 23 quotients lie within 0.0845 period of a whole number; the binomial significance of that is p = 0.94%.

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(update Dec. 17, 2009)

In my above set of 18 very massive asteroids, another significant feature, is that the asteroids whose rotation periods are less than, but most nearly equal to, half that of the giant planets, also show the best orbital resonance with the 6334.329 yr period.

The harmonic average sidereal rotation period of Jupiter+Saturn+Neptune, weighted by mass, = 2*5.1185h (data from 2007 World Almanac); same, including Uranus, = 2*5.1874h. So, the theoretical breakpoint is either 5.12h, 5.19h, or a compromise.

Seven fastest rotators in my set of 18 very massive or very large asteroids:

29 Amphitrite: 5.39h, fractional excess orbital period 0.706 4 Vesta: 5.34h, fractional excess period 0.926 87 Sylvia: 5.18h, fractional excess period 0.414 511 Davida: 5.13h, fractional excess period 0.003 107 Camilla: 4.84h, fractional excess 0.012 16 Psyche: 4.20h, fractional excess 0.044 65 Cybele: 4.04h, fractional excess 0.521

I gleaned three more, massive asteroids from Wikipedia's list of asteroids of large dimension, by accepting members of that list which have only one or two dimensions given (624 Hektor was excluded because it is a Trojan asteroid with period equal to Jupiter's):

48 Doris 13 Egeria 94 Aurora

An article on the USNO website, by Dr. James L. Hilton, lists some of the most massive asteroids. Here I learned of four more, massive asteroids (all at least 1/400 the mass of Ceres):

None of these seven, have orbital periods close to resonance with 6334.329 yr, but the shortest rotation period among them, is 5.385h. So, none would be expected to show such resonance, according to my hypothesis.

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(update Dec. 20, 2009)

From the online IRAS asteroid catalog at VizieR, I got a list of all asteroids of estimated average diameter > 150km. Of these, 22 already are among the 25 most massive and/or largest asteroids considered above. Three more, I exclude because they are Trojans (5.0 < a < 5.4). Of the 72 new large asteroids remaining on the list, Wikipedia gives the rotation period and a sufficiently precise orbital period (i.e., to 0.1 day or better) for only 15. Changes in the layout (authorship or vintage) of the Wikipedia articles as I went down the list, correlated with blocks of asteroid articles consistently giving or not giving the rotation period, or giving this or that precision for the orbital period. This suggests that many more of these asteroids might have adequate data elsewhere.

Anyway, I used the 15 additional asteroids with adequate data on Wikipedia:

6 Hebe 22 Kalliope 31 Euphrosyne 39 Laetitia 49 Pales 85 Io (note allowed conflict with name of Jupiter's moon) 96 Aegle 130 Elektra 238 Hypatia 241 Germania 259 Aletheia 283 Emma 324 Bamberga 423 Diotima 444 Gyptis

The shortest rotation periods are 4.148h (22 Kalliope), 4.775h (423 Diotima), and 5.138h (39 Laetitia; this is the same rotation period as 511 Davida, q.v. above). The other 12 rotation periods all are > 5.52h. Laetitia's orbital period is second-closest (remainder, 0.931) to a whole divisor of 6334.329 Julian yr.

The closest (remainder, 0.936) is 49 Pales, whose rotation period is said to be 10.42 +/- 0.02h, but probably really is 5.21 +/- 0.01h. Pales is the only asteroid (of the 26, among the 72, for which any rotation period is given) whose rotation period is given with error bars. Only three of the 26 rotation periods, fail to be stated to the nearest 0.01 or 0.001h. Apparently the reason for Pales' low precision, is reliance on the rather difficult determination in the rather old study of Schober et al, Astron Astrophys Suppl 36:1-8, 1979. Schober gives the period of 92 Undina, another of his three studied asteroids, to 0.02h accuracy also; though of 88 Thisbe, to 0.0006h accuracy.

Schober's lightcurve shows that 49 Pales' true rotation period likely is 10.42/2 = 5.21h. Pales has a deep secondary minimum, noted by Schober. In contrast to 88 Thisbe, 49 Pales' secondary is much less deep than its primary. However, like 88 Thisbe, 49 Pales' secondary, according to Schober's plot (p. 4, upper left especially) occurs exactly 180deg from its primary. A much weaker secondary at 180deg, can be due only to reflectance difference or to axis wobble. Yet "...reflectance for most [asteroids]...is close to constant over the whole surface." (Zwitter et al, A&A 462:795-799, 2007, p. 797) and if reflectance varies considerably, the peak reflectance will be at some random phase which will destroy the 180deg phase of the secondary minimum. So, the "secondary" minimum really is a weaker recurrence of the primary minimum, affected by wobble.