Cholula/Teotihuacan: an accurate, robust "precessional alarm clock" (phrase due to J. M. Jenkins)
Summary. If the line between the Pyramid of the Moon and the Pyramid of the Sun at Teotihuacan represents the N-S line at the time of construction, then not only do the present latitudes of the pyramids of Teotihuacan and Cholula roughly equal the present Declinations of Algieba and Arcturus: the original difference in latitude between the Pyramid of the Moon and the Pyramid of Cholula, precisely equaled the present difference in apparent Declination of date, between Algieba and Arcturus. Furthermore, let us consider the point P, where the present ecliptic intersects Arcturus' present parallel of Declination. The present difference in Right Ascension between Arcturus and P, equals 1.6126 times the angle between the Avenue of Teotihuacan (which Millon and Dow say is oriented 15deg25' E of N) and the line between the Pyramids of the Moon and Cholula. The present difference in Right Ascension between Arcturus and Algieba, equals 1.6124 times the angle between the Pyramid of Moon - Pyramid of Sun line, and the Pyramid of Moon - Pyramid of Cholula line. This ratio is determined by the difference in Right Ascension between Algieba and Antares: 1.6123 radian. If Teotihuacan's "other" orientation, 16deg30' +/- 30' E of N (inferred from streets, etc., aligned S of E) is used instead of the Avenue, it corresponds to the RA difference between Arcturus and Avior, with ratio 1.60307.
I use the Wikipedia coordinates of the Pyramids of Cholula & of the Moon. These are copied correctly from the cited reference, a Google service called "Geohack" at toolserver.org, on which they seem to be given to the nearest 0.0001deg, roughly the accuracy of GPS (which is affected by atmospheric phenomena). Millon gives as central coordinates for his map, the coordinates of the Pyramid of the Sun, but both his longitude and his latitude end in 30", suggesting that the difference between Millon's and Wikipedia could be mostly the rounding error of the former.
Let us suppose that the Pyramid of Cholula originally were at a geographic latitude equal to the time-local minimum of Arcturus' apparent Declination of date, which for this year occurs at March 25.162, 2013AD (based on Hipparcos catalog data and including correction for Proper Motion, parallax, precession, nutation and aberration). Then if the line between the Pyramids of Moon & Sun represents geographic north at the time of construction, it would need to be oriented (according to spherical trigonometry) 2.0987deg W of N, for Algieba's apparent Declination of date, on March 25.162, to equal the original geographic latitude of the Pyramid of the Moon. On Millon's map I measure this orientation as 2.1064deg and estimate the one-sigma error resulting from my contributing ruler measurements as roughly 0.034, so even if ruler measurement is the only error, the discrepancy between measured and theoretical orientation is only 0.2 sigma.
If the Pyramid of the Moon - Pyramid of the Sun line really is at 2.0987deg W of N as required above, then the angle PyrSun-PyrMoon-PyrCholula is 1/1.61241 times the difference in Right Ascensions of date, of Arcturus & Algieba. Though this happens to be near the Golden Ratio = 1.618034..., it is much nearer the Right Ascension difference of Algieba & Antares measured in radians, 1.612326. Dividing the Right Ascension differences, by a factor such that the Antares-Algieba RA difference becomes 1.0 radian, keeps all the angles less than 90 degrees, even with the inclusion of Antares in the set, and also allows the inclusion of Antares, by means of this hint, without building a monument for it. Thus the layout expresses not only the Declination difference but also the Right Ascension difference between Arcturus & Algieba. The consistent time interval is still several years, because in both coordinates, the two stars change at nearly the same rate: in their change in apparent Declination they differ from each other only by a fraction of an arcsecond per year, and their change in Right Ascension differs only Arcturus-Algieba = 42-49-1+0 = -8"/yr, including precession and Proper Motion; -8"/59deg = -1/27000. The change in Right Ascension Antares-Algieba = 55-49-0+0 = +6"/yr; +6"/92deg = +1/55000. The pyramid coordinate rounding error is roughly 0.00005/0.5deg = 1/10000, but if the pyramid coordinates were exact, then the difference in the ratios 1.612326 for Antares-Algieba, and 1.61241 for Arcturus-Algieba, a part in 19200, would disappear in 1/19200 / (1/27000 + 1/55000) = 0.9 yr.
Let us also consider the point P where Arcturus' present parallel of Declination intersects the present ecliptic. The angle between the Avenue of Teotihuacan, and the line PyrMoon-PyrCholula, is 1/1.61263 times the difference in Right Ascensions of date, of Arcturus and P. The Right Ascension of date, of P, does not change with precession, but Arcturus' large Proper Motion in Declination affects it indirectly by +9"/yr due to the small angle, between the ecliptic and the Declination parallel. The difference in Right Ascension, of Arcturus and P, changes 42-1-9 = +32"/yr; +32"/87deg = +1/9800. The rounding error in the orientation of the Avenue of Teotihuacan, might be 2.5' --> 1/1300 or 0.5' --> 1/6500. Millon adopted Dow's 1964 astronomical measurement of 15deg25', but gave honorable mention to another researcher's measurement, 15deg28'. A value of 15deg25.6' would produce the abovementioned monument-free standard ratio, RA Antares minus Algieba expressed in radians.
The Right Ascension of Avior, a.k.a. epsilon Carinae (reaches more than 10deg altitude at Teotihuacan now, and was farther north in the past)(not to be confused with the famous eta Carinae) changes little with precession, only 18.5"/yr, because it is near the south ecliptic pole. (This is even less than Canopus, which changes 20"/yr.) The angle between 16deg30' E of N (the "other" Teotihuacan angle, based on the 16deg30' +/- 30' S of E figure quoted by Dow in American Antiquity, 1967, for an EW street that had been measured precisely) and the line PyrMoon-PyrCholula, is 1/1.60307 times the difference in Right Ascensions of date, of Arcturus and Avior, at the same date, Mar 25.162, used above. If the correct "other" Teotihuacan angle were 16deg11' instead of 16deg30', the ratio would be the same as the Antares-Algieba standard. As with P, Avior's difference in RA with Arcturus changes rapidly with precession, 42-1-18.5 = 22.5"/yr. Millon's 1972 book accompanying his map, says that S of E orientations at Teotihuacan generally range from 16.5 to 17deg. However, the most prominent long EW structure at Teotihuacan, is the EW wall shown on Millon's large photogrammetric paper map, forming the northern boundary of the Citadel (the large courtyard of the Pyramid of Quetzalcoatl). My own measurement Feb. 27 on Iowa State University's copy of Millon's map, shows that this wall's orientation (along its northern, outside edge) is 90deg + 55' +/- 1' clockwise from the Avenue, i.e. 16deg20' +/- 1' S of E. The needed "other" Teotihuacan angle is seen even more accurately in the angle which, according to spherical trigonometry, the great circle from the El Castillo (at Kukulkan) to Mundo Perdido (at Tikal) pyramids, again using 0.0001deg precision Wikipedia coordinates, crosses the parallel of latitude of the Pyramid of the Moon: the complement of 16deg12'. Accounting for pole shift, this would have been equal to an ancient angle at Teotihuacan that would be 16deg13' now.
From my measurements on Millon's map, the distance between the Pyramids of the Moon and Sun is equivalent to 25.02" geographic latitude, where the last digit is doubtful due to ruler error. Perhaps this distance was chosen for additional redundancy in the code. The difference in RA between Arcturus and the point P is 87.146deg. The geometric mean of these two angles is 46.7'; 16deg11' - 15deg25' = 46' is just the increment needed, to make the "other" Teotihuacan angle, 16+ deg E of N, perfectly correspond to Avior in this theory. Another interpretation of the PyrMoon-PyrSun distance, is that according to a formula of Newcomb for the historical rate of precession, the 25.019" corresponds to exactly half the mean precession rate since 130 BC +/- 80 (or maybe more) yr, where the uncertainty is from my estimated ruler error: this matches the dates of the earliest known major construction at these pyramid sites.
The Tycho-2 catalog reveals several times as many stars of Vmag < 8.00, as does the Skiff spectral type catalog. I've not yet had time to assess the Tycho stars, but can see from their number that the correlation of early K giant stars (Arcturus, Algieba, Botein and others) whose reduced Declinations are geographic latitudes of tetrahedral corners, as discussed in my previous investigations, might or might not be statistically significant.
Let us consider the Right Ascension of Luna. At 01:15 GMT Mar 25 = Mar 25.052, according to the JPL ephemeris, the J2000 RA of Luna becomes equal to the J2000 RA of Algieba according to the Bright Star catalog (corrected for proper motion only). Luna's RA changes 31"/minute, so aberration, or the use of J2013.23 instead of J2000 meridians, hardly affect the result. (The time derivative of the nutation displacement of Earth's pole, which I got online from celnav.de, is suspiciously large though perhaps consistent with the amplitude of the 0.5 yr nutation term. If I omit the nutation altogether, i.e. use the mean pole corrected for precession and secular obliquity change only, I find that the minimum Declination of Arcturus is Mar 21.706. I calculated nutation by interpolating 60 day intervals; if I use much smaller intervals, the date of Arcturus' minimum Declination can change by several weeks.)
If the North Pole position is changed to its former position suggested by Petrie's survey of Giza (i.e. 340" W of N as seen from the Great Pyramid, per Petrie's value 5'40" +/- 10"; but no closer nor farther from Giza) then Arcturus' minimum Declination on Mar 25, 2013, would differ only 0.30" (not much more than coordinate rounding error) from the former geographic latitude of the Cholula pyramid. However, if the Pole position were the same when Teotihuacan/Cholula was surveyed, as when Giza was, then the 2.1deg W of N orientation of the PyrMoon-PyrSun line, did not indicate the Pole. Instead, this angle gives more evidence for the c. 130 BC construction date. The time midpoint between 130 BC and 2013 AD is 942 AD; denote the mean celestial pole then, according to the online NASA Lambda utility, by B. Denote the mean celestial pole at 2013 AD by C. Denote the position of Algieba at 2013 AD, according to the online Hipparcos catalog, by A. Spherical trigonometry gives the angle BAC = 1.612326 (see above for this number's importance) * 2.137244deg. Including the effect of Petrie's 5'40", the angle between the PyrMoon-PyrSun line and the original Earth pole would have been 2.0987 + 200.99"/3600 = 2.1545deg. The difference from 2.1372, 1.0', is only 0.5 sigma times my paper map measurement error (see above) and implies only 0.7 sigma deviation from the measured PyrMoon-PyrSun angle (vs. 1.0 sigma & opposite sign, without the 201" correction). These coincidences suggest that the builders of Giza in 2500 BC, knew that the small upcoming pole shift would be perpendicular to their chosen meridian. The result for Arcturus suggests that the pole shift the Giza builders foresaw, mostly occurred between 130 BC and the present.
These are the reasons Algieba was chosen:
1. It is one of the 50 brightest stars in the sky, and the brightest star in the mane, or sickle, of Leo.
2. Algieba's Right Ascension (J2000 10h20m) interpolates with almost equal steps that of Antares (16h29m), Arcturus (14h16m), the Sun, or rather its antipode, at Mar 25.162 (0h16m = 12h16m - 180deg) & Avior (8h23m), all of which together span only about 8h RA, so are simultaneously visible in the Teotihuacan night sky in late March.
3. Suggestively, Arcturus, Algieba and Avior have the same color; all are, or have primaries that are, early K giants, i.e. K0 to K3, type III (possibly K7 for Avior) while Antares is M1.5 Ib. Algieba has a bright, late G secondary. Avior & Antares have moderately dim, early B secondaries, so the subjective colors of these stars might be less red than the primary's spectral type would suggest. Similar color is another clue that the stars all belong to the same monument scheme.
4. Algieba's Declination is very near that of Arcturus.
5. When Arcturus' Declination is minimum March 25.16, 2013, Luna is roughly south of Algieba.
|the real Teotihuacan alignments||921||Anonymous User||10-Mar-13 00:52|
|Re: the real Teotihuacan alignments||112||Paoputzu||22-Mar-13 19:36|