The Bent Pyramid: a bigger winner than I thought
The declination of Arcturus at 4327.5 BC, was +55.2307deg. To make this calculation today, I used:
the "rigorous" precession formula from the 1990 Astronomical Almanac, p. B18
the mean of the J2000.5 and J1999.5 coords. of Arcturus from the respective Almanacs
Wikipedia's proper motion values, -1.09345 & -1.99940 "/yr in RA & Decl
Wikipedia's RV, +5 km/s
I used the usual first-order proper motion extrapolation, correcting the change in RA according to the secant of the midrange J2000.0 declination. My first order correction for RV, which I've never used before, added only 0.0050deg, which I've included in the result. Porter, Astronomical Journal 12:28, 1892, gives PM -0.80 & -1.98 "/yr in RA & Decl; Porter's PM values would have lessened the result by only 0.0034deg. The consensus of published astronomical opinion seems to be that Ptolemy's catalog confirms Arcturus' proper motion as very roughly constant for 2000 yr, and that Brahe's catalog gives much more accurate confirmation going back 400 yr.
This declination, would have been the geographic latitude of a place where Arcturus reached the zenith, according to a plumb line, at the start of the Egyptian calendar one "Barbarossa period" (6339.5 tropical yr) before the end of the Mayan Long Count. (The start of the Egyptian calendar, was at a summer solstice when Arcturus rose heliacally at Giza; but this was changed slightly, when the Upper Egypt calendar, which began a few days later when Arcturus rose heliacally at Elephantine Island the same year, replaced that of Lower Egypt.)
In a 2009 post, accurizing an observation by Bert Janssen, I explain that, considering Earth's oblateness, and assuming a few miles shift in Earth's pole suggested by Petrie (the Great Pyramid aligns 5 arcminutes off present true north, and is about a mile from the 30th parallel today, despite evidence that a relatively awkward building site was chosen because for some unknown reason, fractions of a mile were important in its placement), the 3-space angle, pole-Giza(Menkaure)-Stonehenge equals the angle pole-Earthcenter-Stonehenge with only 100 meters error. Menkaure's slope, well preserved by its granite siding, also equals (the complement of) this angle, with only 4 arcmin error.
So, Menkaure's pyramid memorializes the geocentric latitude of Stonehenge, not the geographic latitude. Likewise the Bent Pyramid apparently memorializes the geocentric latitude of any place where at 4327.5 BC, Arcturus reached the zenith defined by a plumb line. The conversion to geocentric latitude gives 55.0494deg = 55deg 3'. According to an online discussion by John Legon, Petrie's and Dorner's measured values for the mean angle of the "lower part" of the "lower slope" of the Bent Pyramid, are 55deg 1' and 55deg 5', resp., in excellent agreement with this theory.
The builders knew all along, they couldn't maintain that slope all the way to the vertex, so they fulfilled their mission, by making it the lower slope, or at least the lower part of the lower slope, as it is today (in the mean). What about the upper slope?
The "Sothic cycles" of Prof. Eduard Meyer, are the time intervals between heliacal risings of Sirius on a given date in the official (leapyear-less) 365-day Egyptian calendar. In my earlier posts, I explained that the Egyptian calendar, and "Sothic" date of Amenhotep I, are based rather on "Arcturian" cycles. The date of Amenhotep I is two Arcturian cycles after the beginning of the calendar: the average length of these cycles, I found as (4329-1533)/2 = 1398 yr. The Barbarossa period, 6340 yr, nearly equals 4.5 Arcturian cycles = 4.5*1398 = 6291 yr. Perhaps knowing some of this, Joseph Scaliger chose "Julian Day 1" to be about 6295 yr prior to the beginning of the Gregorian calendar. Also, Hindu texts say there are 6333 Gandharvas, asociated with cyclical divisions of the sky.
The Egyptians would have been likelier to memorialize 1/3 Barbarossa cycle, than to memorialize, say, 1/2 or 1/4, because 1/3 Barbarossa cycle is 1.5 Arcturian cycles, not 2.25 or 1.125. A third of a Barbarossa period, is 6339/3 = 2113 yr, that is, 4329 BC + 2113 = 2216 BC. Maybe the Bent and Red pyramids' slopes (upper slope for the Bent) were chosen in anticipation of a timeline milestone almost 400 yr in their future.
At this milestone (i.e., 1/3 Barbarossa cycle = approx. 1.5 Arcturian cycle) the geocentric latitude, where Arcturus is at the plumbline zenith (i.e., geographic latitude = Arcturus' declination in coordinates of the ecliptic of date), would be 43deg 45', with precise enough accounting for proper motion, radial velocity, and precession, similar to my previous calculation. The most often seen estimates of the slopes of the Red (a.k.a. Northern Dahshur), and (upper) Bent (a.k.a. Southern Dahshur), Pyramids, are both 43deg 22'. However, Dr. Ahmed Fakhry, "The Pyramids" (U. of Chicago Press, 1961), p. 95, gives 43deg 40' for the Red Pyramid. Dr. I. E. S. Edwards (Keeper of Egyptian Antiquities at the British Museum 1955-1974), "The Pyramids of Egypt", p. 89, says the Red Pyramid is "the earliest tomb known to have been completed as a true pyramid", and that its slope is 43deg 36' 11".
