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Egyptian calendar & 2012

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  • appiusclaudiuspriscus
    Arcturus in 4328 BC Egypt: Hermetic knowledge by Joseph C. Keller, M. D. May 19, 2009; last revision May 31, 2009 Sec. 1. Calendars. The calendar of every
    Message 1 of 1 , Jun 3, 2009
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      Arcturus in 4328 BC Egypt: Hermetic knowledge

      by Joseph C. Keller, M. D.
      May 19, 2009; last revision May 31, 2009

      Sec. 1. Calendars.

      The calendar of every civilization changes. Our calendar changed (Julian to Gregorian) in 1582AD, skipping ten days and introducing a more complicated leap year formula. Julius Caesar changed our calendar much more. Because calendars change, king lists with reign lengths always have supplemented the calendar.

      Our year basically starts at the winter solstice, though off by eleven days. The ancient Greek year explicitly started anew at every summer solstice, with Year 1, of the first four-year "Olympiad", occurring in 776BC.

      The Egyptian calendar had features enabling it to preserve dates for millenia. One of these features was that the day count, instead of being corrected by a complicated leap-day formula to fit the tropical year, either started anew in each tropical year (what I call the "tropical calendar"; basically it was like our ordinary calendar), or remained simply 365, ignoring discrepancy with the tropical year (what I call the "365-day calendar"; basically it did the job of today's astronomical Julian Date). This gave two different calendars which could correct each other. The phase between the two different "years" is a kind of year count.

      Another Egyptian feature was that the dates of "heliacal" rising of bright stars (dates of risings nearest sunrise) including at least Arcturus, Canopus, Sirius and Procyon, were recorded, alongside the 365-day calendar date or tropical date, and reign years. For a star on the ecliptic (with zero proper motion), this would be the same as recording the phase between the 365-day year and sidereal year; the phase of the 365-day year would shift with period 1425yr. For a star at the ecliptic pole, the tropical date of heliacal rising would be constant; heliacal rising, like tropical date, would shift relative to the 365-day year, with period 1508yr.

      The heliacal rising of Canopus is used in calendars even today:

      "…claims Allen, [Canopus] was known as Karbana in the writings of an Egyptian priestly poet in the time of [Thutmose III]…

      "…Allen claimed that the heliacal rising of Canopus `even now [1899] used in computing their [the Arabs'] year,…"

      - Fred Schaaf, "The Brightest Stars", pp. 107, 109 (Google book online)

      "In the Gulf region the Canopus calendar,…10-day units from the late summer heliacal rising of Canopus, has long been a traditional calendrical system for Bedouins and sailors."

      - Gary D. Thompson, 2007-2009 (online, members.optusnet.com.au)

      Stars far from the ecliptic (e.g., Arcturus and Canopus) rise far from the point of sunrise, thus are easier to see at sunrise. Arcturus about equals Vega in brightness, i.e. "Visual magnitude" (some sources, e.g. the 1997 Pulkovo Spectrophotometric catalog, say Vega is brightest, some, e.g. the 5th ed. Bright Star Catalog, Arcturus; the slight difference depends on photometric details) but Arcturus' orange color penetrates better than Vega`s blue-white, when observed at low altitude angle. Stars near the equator also might rise far from the sun if near the tropic opposite the sun (e.g., Sirius).

      The ancient Egyptians had two calendars. One calendar had three seasons of four lunar months each, alternating 30 and 29 days, so the average would equal the synodic lunar month. Apparently the first day of the third season ("summer") was set at the summer solstice. There would have been eleven, or sometimes twelve extra days at the end. I call this the "tropical calendar" because it's based on the solstice (Tropics of Cancer & Capricorn).

      The other calendar, apparently, was structured the same as the first, but always with a total of 365 days, whatever the current summer solstice date. I call this the "365-day calendar". The first day of this calendar was "1 Thoth" (see OA Toffteen, "Ancient Chronology", p. 180; online Google book), which shifted around the tropical year with period 1508 yrs, assuming today`s tropical year of 365.24219 days. The length of the tropical year is nearly constant, because under perturbation, an orbit's major axis distance is stable to third order (conjecture of Lagrange & Poisson, proved by Tisserand) and also because Earth's precession rate has only higher-order dependency, on Earth's or Luna's small eccentricity or on Earth's nearly constant axial tilt.

      Sec. 2. Finding the solstice.

      The Alexandrian calendar (26/25BC; its first leap year was 22BC) of Augustus, was a Julian calendar for Egyptians. With its leap year, it fixed the old "1 Thoth" New Year of the ancient Egyptian 365-day calendar, at August 29 of the Julian calendar.

      The Council of Nicaea, 325 AD, which deliberated the date of Easter, considered March 21 to be the spring equinox, hence December 21 the winter solstice. Perhaps surprisingly, this was not shifted the expected two days (one day per 128 yr) earlier than Ptolemy's winter solstice date, current in the 2nd century AD. Ptolemy (the astronomer) said, the Egyptian calendar day, 26 Choiak, is the winter solstice (James Evans, "The History and Practice of Ancient Astronomy", p. 180; online Google book). Because 1 Thoth had been standardized to August 29, 26 Choiak was December 21 (recalling that the 1st & 3rd months of any four-month Egyptian season had 30d, and the 2nd & 4th, 29).

      The Julian calendar (45BC) suffered some irregular extra days at first. Wikipedia's article, "Julian Calendar", says:

      "…the [Roman] pontifices…added a leap day every three years, instead of every four years. According to Macrobius, the error was the result of counting inclusively, so that the four-year cycle was considered as including both the first and fourth years. This resulted in too many leap days. Augustus remedied this discrepancy after 36 years by restoring the correct frequency. He also skipped several leap days in order to realign the year. The historic sequence of leap years in this period is not given explicitly by any ancient source, although the existence of the triennial leap year cycle is confirmed by an inscription that dates from 9 or 8 BC. The chronologist Joseph Scaliger established in 1583 that the Augustan reform was instituted in 8 BC, and inferred that the sequence of leap years was 42, 39, 36, 33, 30, 27, 24, 21, 18, 15, 12, 9 BC, AD 8, 12 etc. This proposal is still the most widely accepted solution."

      Between 22.0 BC and 11.0 AD (after which, leap years were correct) there were 6 Julian calendar leap days instead of the necessary (22+11-1)/4 = 8. The Alexandrian calendar would have had to keep pace with the Julian calendar.

      Ptolemy wrote a little later than 128yr (the time for the Julian calendar's equinox to lag a day) after the adoption of the Alexandrian calendar, so the equinox in 22BC would have been dated a net 8-6-1?=1 days earlier than Ptolemy's, i.e., March 20, so the solstice was, most likely, June 21 (considering the effect of Earth's orbital eccentricity). Since the first leap year in the Alexandrian calendar was 22BC, it would be most accurate to consider August 29 of (22+4/2)=24BC, as 69d past the solstice.

      Sec. 3. Finding "1 Thoth".

      Most authors list only the three main "Sothic dates" (those of Sesostris III, Amenhotep I, and Thutmose III), though Brein (2000) lists six. One of the three main "Sothic dates" (so-called because Prof. Eduard Meyer, of the Univ. of Berlin, hypothesized in 1904 that they pertained to Sirius; which indeed one does) is:

      (Ebers papyrus) Year 9 of the reign of 18th Dynasty (New Kingdom) Pharaoh Amenhotep I. It is the 9th day of the 3rd month of summer (i.e. 3rd season), i.e. 11th month of the Egyptian year.

      If this refers to what I call the "tropical calendar", and if the first day of the summer season had been adjusted to be on the solstice, then Amenhotep's date is 30+29+(9-1) = 67d past the solstice. In Sec. 2, I found that 1 Thoth in 24BC, was 69d past the summer solstice, when 1 Thoth was fixed by the Alexandrian calendar. Eight years earlier than Amenhotep`s date, 1 Thoth would have been 69d past the solstice, if Amenhotep`s date is 1 Thoth. (Amenhotep I was the pharaoh, who decided to stop building pyramids, and instead cut tombs into rock cliffs.)

      According to Damien Mackey, the 4th century AD Alexandrian astronomer Theon (father of the famous pagan martyr Hypatia) said that the Alexandrian calendar began (approximately) at the end of a cycle of some kind. Now we know, what that cycle was. It was the 1508yr cycle of the Egyptian 365-day year vs. the tropical year. Amenhotep's date is 1 Thoth, in approximately the year 24+1508-8 = 1524BC (so Amenhotep I's reign started 1532BC). Indeed it was recorded as "New Year's festival…the heliacal rising of the Sirius [Mistranslated? Should say Arcturus? - JK] star." (Toffteen, p. 179). This year is comparable to the other estimates of Amenhotep's antiquity made by Egyptologists.

      Sec. 4. Amenhotep I, Nabta, Giza and the heliacal risings of Arcturus.

      Neglecting Earth's orbital eccentricity, the sun's ecliptic longitude (in the "coordinates of the ecliptic of date") on Amenhotep's date, would be 90+67*360/365.25 = 156.0. To correct for eccentricity, I find Earth's longitude of perihelion at 103 (today's value) - (2000+1524)*(1/25785 + 1/112000) * 360 = 42. The eccentricity correction to the sun's longitude is, by Simpson's rule,

      (156-90)*2*0.017*(cos(90+42)/6+cos(90+42-66)/6+cos(90+42-33)/1.5) = -0.3deg

      and the corrected longitude is 155.7.

      The Egyptians are said to have observed the stars, from observatories in Memphis and Heliopolis (both near Giza, 29.979N), from the desert near Hierakonpolis, from Thebes, from various shrines in upper Egypt, and from Elephantine Island. I'll start by assuming it's Elephantine Island: that was an important commercial, military and religious site during the New Kingdom, and it's on the Tropic of Cancer (exactly so, at ~ 4000BC, according to the formula in the 1990 Astronomical Almanac). Considering refraction, and measuring to the top of the sun disk, in 1524BC at Elephantine I., 24.1N, the apparent heliacal rising of Arcturus would occur with the sun at longitude 153.16, 2.5deg too little. (The details of my calculation are in Sec. 9.)

      My calculated difference in Arcturus' heliacal rising at Giza vs. Elephantine, in 1524BC, is -5.01 degrees of sun longitude, for the +5.88 degrees of latitude separation on the ground. So, moving 1.60 degrees farther south, would change the sun's longitude at the heliacal rising, from 153.16 to 154.52; only a day before 1 Thoth. So, Egyptian astronomers consistently made this observation from a point about 1.6 degrees of latitude south of Elephantine, near the northern edge of the Nubian desert: namely, Nabta, 22.5N (warning: the Nature article describing Nabta, though excellent in substance and highly recommended overall, contains some numerical errors involving an erroneous distance scale on their main map).

      The farther south the observer, the more vertical the rising of the sun and Arcturus, so the smaller the gross or net effects of atmospheric refraction. The Nabta megaliths are a Stonehenge-like structure, dated 4000-4500BC, over a mile in size, whose alignment even now displays sometimes as good as one-arcminute accuracy.

      Thus for Amenhotep the cycle is marked, not by the seasons, but by a kind of sidereal year based on one star: not Sirius as for the Sothic year of Eduard Meyer, but the "Arcturian year". Expressing the sun's longitude at heliacal rising, as a quadratic function of time (and assuming the Egyptians made no refraction correction, nor did Arcturus' proper motion change), and integrating, I find that Arcturus' heliacal rising, on 1 Thoth, two cycles before Amenhotep, was 4328BC, if the observation was made at Nabta.

      Because of Arcturus' large southward proper motion, the two cycles before Amenhotep average a mere 1402yr apiece. The match, to what follows, is precise enough to prove that Arcturus' average proper motion was about the same from 4328BC to 1524BC, as from 1524BC to 2000AD, and as today at 2000AD. This "Year 1" is 6339yr before 2012 (see Sec. 8).

      I find that the heliacal rising of Arcturus from Giza (not Elephantine I.) at 4328BC occurred with the sun at longitude 90.19, assuming the Egyptians used the top of the sun and made no refraction correction. The heliacal rising of Canopus from Elephantine in the same year, was with the sun at longitude 90.32. If the Egyptians had used the center of the sun, both these longitudes would be ~ 1/4 deg less: that is, in the founding year of the Egyptian calendar, on the summer solstice, Arcturus rose heliacally on-center with the sun, on the 30N parallel, and Canopus rose heliacally on-center with the sun, on the Tropic of Cancer.

      At 4328BC, the sun longitude at heliacal rising for Arcturus differed 10 deg between Upper and Lower Egypt; Canopus differed 20deg. Upper Egypt apparently avoided this latitude-associatied ambiguity altogether, by making their calendar's Day 1 (1 Thoth), simply the heliacal rising of Arcturus at Nabta, 14 days after the summer solstice.

      Sec. 5. Sesostris III and the heliacal rising of Procyon.

      Generally, any bright star will rise heliacally nearest 1 Thoth, only for four consecutive years, about every 1400 or 1500 yr. There are only a few convenient first-magnitude stars: some are too near the ecliptic, in the sun`s glare; and at any epoch of precession, some are too far south ever to rise in Egypt. Hence some observations would be made on days of the 365-day calendar, other than 1 Thoth.

      The earliest of the three main known "Sothic dates" is:

      (Kahun papyrus, a.k.a. Lahun papyrus) Year 7 of the reign of 12th Dynasty (Middle Kingdom) Pharaoh Sesostris III. It is the 16th day of the 4th month of "winter" (i.e. 2nd season), i.e. 8th month of the Egyptian year.

      Suppose this observation were made, not on any special day (like 1 Thoth) of the 365-day calendar, but rather at a special epoch, namely, when 1 Pachons, the eventual nominal first day of summer of the 365-day calendar, fell on the summer solstice. (Likewise our Christmas Day falls on the eventual nominal first day of winter as fixed in the Alexandrian calendar: 1 Tybi = 1 Thoth + 4*29.5d = August 29 + 118d = December 25.) Sesostris' 1 Thoth would be 4*29.5 + (365 - 12*29.5) = 129d past the summer solstice. So, by comparison with Amenhotep, the year of Sesostris' "Sothic" (or rather, Procyonic) date is 1524 + (129-67)/365.25*1508 = 1780BC (so, the reign of Sesostris III began 1786BC). This is a century later than usually supposed, but Egyptologists think Middle Kingdom dates are uncertain anyway, and they largely have been based on the supposed Sirius risings, which the research in this article supplants.

      Suppose that, like Amenhotep's date given in the Ebers papyrus, Sesostris' date given in the Kahun papyrus, is a tropical date with 1 Pachons set to the summer solstice. The date would be (29+1-16) = 14d before the summer solstice. Neglecting Earth's orbital eccentricity, the sun's ecliptic longitude (in the "coordinates of the ecliptic of date") on this date, would be 90-14*360/365 = 76.2. To correct for eccentricity, find Earth's longitude of perihelion at 103 (today's value) - (2000+1780)*(1/25785 for equinox precession + 1/112000 for perihelion advance) * 360 = 38. Cos(76 + 14/2 -180 - 38)*2*0.017*13.8 = -0.3, so the corrected longitude is 76.5.

      From Elephantine I., the sun's longitude at the apparent (top, of sun, at horizon) heliacal rising of Procyon, for 1780BC, is 74.52deg. Interpolating the year, in the table at the end of Sec. 9, the difference Giza - Elephantine is 2.41, so for Cynopolis at 28.30N (see Sec. 6) the heliacal rising of Procyon in 1780BC occurs when the sun's longitude is 76.24.

      Sec. 6. Thutmose III and the heliacal rising of Sirius.

      The latest of the three main "Sothic dates" is:

      Year ? (the reign year has been broken off the tablet, according to Toffteen, p. 181) of the reign of 18th Dynasty (New Kingdom) Pharaoh Thutmose III. It is the 28th day of the 3rd month of "summer" (i.e. 3rd season), i.e. 11th month of the Egyptian year.

      Toffteen translates the text as "Sirius festival". This time, the star really was Sirius, if the date refers not to the tropical calendar (as for the other two main Sothic dates) but instead, to the shifting 365-day calendar. (The same emendation might repair the two contradictory lunar dates from this reign, which have produced three schools of thought among Egyptologists, that Thutmose III's 54-year reign began 1504, 1490 or 1479BC. See RE Parker, JNES 16:39+, 1957, cited by LW Casperson, "The Lunar Dates of Thutmose III", in the JSTOR online archives; and, Cline & O'Connor, "Thutmose III", Google book online.)

      This Sothic date is about 1422BC (more precisely, 1424BC; see below): it is 29.5*10+27 = 322d past, not the the first day of the tropical calendar, but rather, past 1 Thoth. In turn, 1 Thoth is 67 - (1524-1422)/1508*365.25d past the summer solstice, so Thutmose's date is net 364.3d past the summer solstice, and the sun is at longitude 89.1.

      Luckily, near this century, the longitude of the sun, at the heliacal rising of Sirius, is practically independent of latitude in Egypt. For Giza in 1422BC it's 89.66 and at Elephantine, 89.69. This date in 1422BC is a heliacal rising of Sirius on the summer solstice throughout Egypt, an event even rarer than the heliacal rising of Arcturus on the summer solstice on the 30N parallel in 4328BC.

      This heliacal rising of Sirius was recorded so carefully, in part because it occurred on the solstice, and because it was simultaneous throughout Egypt; but, those both are conditions that last about a century (due to the small difference between the tropical and sidereal years). Unlike the 4328BC rising of Arcturus, Thutmose`s date is decades away from exact coincidence, of heliacal rising with solstice, anywhere in Egypt.

      Like Amenhotep's rising of Arcturus, Thutmose's rising of Sirius was recorded because it occurred on the shifting 365-day calendar date, of the heliacal rising of Sirius somewhere in Egypt in the foundation year 4328BC. Using the same quadratic approximation and integral calculus method as for Arcturus, I find that for Giza observations, two Sothic cycles before this 1422BC rising, would be 4338BC ( 2 * average 1458 yr Sothic cycles ago). For Elephantine, I find 4302BC ( 2 * average 1440 yr Sothic cycles ago). By interpolation, consistency with the 4328BC founding date, occurs for an original "Dog Star" observatory at 28.35N.

      Indeed the latitude of Cynopolis (literally "Dog City")(Egyptian name: Hardai) is 28deg18' = 28.30deg N (Rawlinson, "History of Ancient Egypt", p. 16; online Google book). This was the city of the divine jackal Anubis and his wife Anput. It had a cemetery for dogs. It is said to be the original breeding site of dogs like the Great Dane.

      When integrating the shift, in days, of the 365-day year, I simply multiply the change in sun longitude of the star's heliacal rising, by 365.25/360. For Arcturus, the eccentricity correction needed for this, because that sun longitude ranged from 96 (in 4328BC) to 96-9(perihelion gain vs. sidereal)+11(Arcturus' rising's gain vs. sidereal) more than the perihelion, is 0.3d, so the calendar's Year 1 would be, more precisely, 4329BC.

      For Sirius, the corresponding figure is 46 (in 1422BC) to 46+9+17, which adds 24deg*0.034*cos(59) = 0.4d of shift (because Earth has slower angular speed near aphelion). This makes the length of time needed, ~ 2yr less. So, the more precise year is 1422+2+(4329-4328) = 1425BC, and the sun's longitude on this 365-day calendar date is 89.1 + 0.7 = 89.8.

      Sec. 7. The Festival Year of Seti I.

      Cerny, Journal of Egyptian Archaeology 47:150+, 1961, quotes Seti's inscription:

      "Year 1, 1st month of winter, day 1, beginning of perpetuity."

      Amenhotep's date proves that originally in ~ 4329BC, 1 Thoth was 14 days past the summer solstice at Nabta, thus the 15th day of what eventually was called the third season. As it shifted backward through the tropical year, it seems subsequently to have been renamed the *first* day of the third season, then the first day of the *second* season (winter, to which Seti refers). Still later it was renamed the first day of the so-called *first* season. In the shifting 365-day calendar, 1 Thoth migrates through all the seasons anyway.

      Seti's festival year might have been two tropical cycles after the founding year. That is, 1 Thoth returned yet again to its original ecliptic longitude, giving Seti's year as 4329BC + 2*1508 = 1313BC.

      This is near the date oftenest given by Egyptologists, because it is about one of their standardized 1460yr (or 1463yr) Sothic cycles before the ~ 139AD cycle end mentioned by Censorinus. More precisely, using quadratic approximation and integral calculus again, I find that this Sothic (i.e., of Sirius) cycle really is 1454yr, from 1313BC to 142AD (the eccentricity correction of Sec. 6, makes this 1453yr and 141AD). Censorinus had defined the cycle end, only as "100yr" (generally for Romans, time intervals were inclusive, e.g., "4 yr before 143AD" meant "140AD") before 238AD, the time of his writing; the round number suggests it was known only vaguely or ambiguously. The Romans commemorated this cycle with a coin in 143AD (Wilbur Jones, "Venus and Sothis", p. 79).

      For Seti's cycle, two tropical cycles led to one Sirius cycle; then came oblivion, until now! For Thutmose, it was two Sirius cycles and then oblivion. For Amenhotep, it was two Arcturus cycles, one tropical cycle and then oblivion.

      Sec. 8. Barbarossa.

      The planet I discovered, Barbarossa, according to my best estimate based on all four sky survey points 1954-1997, has orbital period 6340yr. The Egyptian calendar foundation year 4329BC, is 6340yr before 2012AD, the end of the Mayan calendar cycle. The end of this longest Mayan (5125 year) cycle, falls on the winter solstice (such a long-range solstice forecast would require astronomical knowledge about as good as that of 16th century Europe). A Mayan monument announces that then "Bolon descends" who has "nine colleagues" (Barbarossa's Declination and ecliptic latitude will be decreasing, and its Declination, will be less than last time due to Earth's axis precession; another meaning of "descent" might be the sudden decrease in Declination due to a decrease in Earth's obliquity last time; see below). From lake varves, Brauer et al have determined the sudden onset of the Younger Dryas cooling, at least in Europe, to the year, at 12683 = 6341.5 * 2 yr before 2012.

      The heliacal rising of Arcturus, the most convenient star for such observations, at the summer solstice, would occur at intervals of approx. 25,785 yr (Newcomb's determination of the period corresponding to Earth's tropical minus sidereal orbital frequency). This occurrence on the 30th parallel, centrally located at the apex of the Nile delta, would seem enough reason to found a calendar, especially with Canopus rising heliacally on the solstice at Elephantine the same year.

      Usually the sun's longitude at a star's heliacal rising, differs 5 or 10 deg between Upper & Lower Egypt. The recorded dates discussed above, show that at least four favorite stars were observed on or near the summer solstice during the 42deg of precession from 4300 to 1300 BC. With a favorite star, on the average every ten degrees of longitude, often as not at least one would be rising heliacally on the summer solstice somewhere in Egypt. For two to do so, is only mildly unusual, but for their locations to be the 30N parallel (Arcturus) and the Tropic of Cancer (Canopus) is suspicious.

      If two stars happen to rise heliacally on the same day (i.e., both rise heliacally with sun longitude in the range, say, 89.50 to 90.50), the root-mean-square difference in the sun longitudes is 0.4deg. Arcturus at Giza (0.021deg S of 30N) and Canopus at Elephantine (est. from Astronomical Almanac obliquity polynomial, 0.04deg S of Tropic of Cancer in 4329BC) differ, in the calendar's Year 1, only 90.17 - 90.31 = -0.14, though correction to exactly 30N and to the predicted Tropic, degrades this to -0.33.

      Either star in the preceding paragraph, if the sun longitude is corrected to sun-center rising rather than sun-top rising, should have its sun longitude randomly in the range ~89.24 to ~90.24 (assuming the sun and star atmospheric corrections roughly cancel). Yet these correct to 89.91 & 90.05 (though correction to "exactly 30N" and to *our predicted* Tropic, changes this to 89.87 & 90.20).

      The summer solstice heliacal risings of Arcturus & Canopus at two different Egypt sites in 4329BC, might be luck were it not for the nearly exact equality of the sun's longitude at sun-center heliacal rising, to 90.0deg, for Arcturus & Canopus at the 30N parallel and the Tropic of Cancer, resp. This seems to reveal a new physical phenomenon. Maybe an unknown (not inverse square law) force exerted by these nearby giant stars, stabilized Earth's axis, after some disturbance, in a new position constrained by geometric relations described as heliacal rising at 30th parallel and Tropic. Another such unexplained geometric relation, the equal (in absolute value) latitudes of Hawaii, Mt. Olympus on Mars, and Jupiter's Great Red Spot, already is known (mentioned by Richard Hoagland on the "Coast to Coast AM" radio show).

      I find that the azimuths of the two lines connecting the three Great Pyramids (according to Petrie's survey) match the projection of Orion's Belt (reflected EW) from the top of a stick, for one direction of Earth's axis. The obliquity of this direction is 27.95deg, not 23.45 as today.

      Maybe stabilization by Arcturus and Canopus, happened to Mars too. As an approximation, let's assume Mars' orbital plane is the same as Earth's, take the Declination of Mars' N pole as 52.92deg (2009 Astronomical Almanac), take the ecliptic latitude of Mars' N pole 90deg minus 23deg59', and find the difference in ecliptic longitude of Earth's and Mars' axes by spherical trigonometry. Then the net precession 1/25771.5 - 1/171000, implies opposite axis tilts ~6375yr before 2012: Arcturus and Canopus would have risen heliacally near Mars' 30N & Tropic of Cancer, but at Mars' winter, not summer, solstice.

      I propose that these solstice heliacal risings were not luck, but rather were the result of an unknown physical force associated with the orbital period of my distant hyperjovian planet, Barbarossa (which never comes near Earth; see my posts to the messageboard of Dr. Tom Van Flandern, for details). Comet swarms in 1::1 orbital resonance with Barbarossa might also have a role.

      Arcturus' proper motion is greater than that of any other first-magnitude star visible from Egypt. The most important component of its proper motion, is the decreasing Declination, which increases the sun longitude of heliacal rising, so the difference between the 365-day and Arcturian years is abnormally large, and the length of the Arcturian vs. 365-day cycle abnormally short. Another suspicious fact about Arcturus, is that it is almost the closest "giant" (i.e., Spectral Type III) star to the sun, only slightly farther than Pollux, and believed to be considerably more massive than Pollux. Similarly, Canopus is the second-closest "supergiant" (i.e., Type I or II) star to the sun.

      Above, I suggested that Seti's reference to "1st month of winter, day 1" revealed only that 1 Thoth, originally the first day of the "third" or summer season, later was called the first day of the "second" or winter season, and then the first day of the "first" season which was neither winter nor summer, where it stayed, having slipped backward to the beginning of the calendar. In the 365-day calendar, there would be a tendency over the centuries to change the name of the season, of which 1 Thoth was first day, as its season in the tropical year moved backward. This would seem to be enough reason for Seti, in some context or other, to call 1 Thoth the first day of winter, but there is another, more speculative, explanation.

      This other explanation is, that c. 4329BC, when Barbarossa presumably attained a special position in its orbit, an unknown physical interaction occurred which caused Earth changes and also illuminated Barbarossa. If Barbarossa were visible for many years, the Egyptians (or Atlanteans?) might have been able to determine its orbital period. They would notice and record any calendrical markers of the year of recurrence. Maybe the marker, is when the heliacal rising of Arcturus at Nabta occurs on the original 365-day calendar's first (in some sense) day of winter.

      When 1 Thoth originally, in Upper Egypt, equalled our June 22 summer solstice + 14d, 1 Phamenoth equalled our December 30, almost New Year's Eve. At the Egyptian calendar's foundation, 1 Phamenoth was a rough approximation of the winter solstice. Suppose Arcturus rises heliacally on 1 Phamenoth in 2012AD. Relative to Arcturus' heliacal rising, 1 Thoth must regress 4*365.24 + 177 d since the foundation. The Arcturus term in my integral must be adjusted only -0.09deg to account for eccentricity. The result for Nabta is 1614.55 "mean degrees", which gives 6344yr at Nabta.

      So, at Nabta in 2012AD, Arcturus rises heliacally only a day before 1 Phamenoth, the original rough winter solstice of the ancient Egyptian 365-day calendar. (Likewise Arcturus rose a day before 1 Thoth in 1524BC; these discrepancies might be due to the same error.) This is the origin of the association of "Bolon" (Barbarossa? or merely Arcturus?) with the winter solstice, in 2012, in the Mayan calendar. The Egyptian calendar contains Hermetic knowledge, about the period of Barbarossa, and the period of climate change cycles, including the Younger Dryas.

      Sec. 9. Calculation of heliacal risings and settings.

      I took star positions and proper motions from the current (May, 2009) online VizieR version of the Bright Star Catalog, 5th ed. The proper motions in RA refer to linear arcseconds, not fifteenths of seconds of Right Ascension; so in extrapolating RA, they need to be multiplied by sec(phi), where phi is the midrange of the base and extrapolated Declinations. After extrapolating the proper motion, I found the celestial coordinates in the equinoxes of date for 2000.0AD, for 1001.0BC, and for 4001.0BC, using the rigorous formulas of the 1990 Astronomical Almanac, p. B18. I also found Earth's obliquity for those years, using the polynomial on the same page.

      No computer program was used. All calculations were on one Texas Instruments 30X IIS calculator, made in China.

      The remaining calculations, to find the sun's longitude at heliacal rising of the star, consisted of using one spherical trigonometry formula (see, e.g., the CRC Math tables):

      cos( a ) = cos( b )*cos( c ) + sin( b )*sin( c )*cos( A )

      (this is the vector dot product in spherical coordinates) four times in three spherical triangles:

      1) The triangle, the lengths of whose sides, are the star's codeclination, the observer's colatitude, and the distance between observer's zenith and the star's sub-rising point (90.5667deg, using the 34 arcminute average atmospheric refraction given by Stephen Daniels, www.astronomy.net, 1999, online). Here I found the vertex angle, and hence the Right Ascension of the observer's zenith.

      2) The triangle, the lengths of whose sides, are the segment from Earth`s N pole to the ecliptic N pole, the observer's colatitude, and the distance between observer's zenith and the point beneath the N ecliptic pole (observer`s ecliptic colatitude); here I found the last of these, thence the ecliptic longitude of the observer`s zenith, mindful of the signs of angles.

      3) The triangle, the lengths of whose sides, are the sun's ecliptic colatitude (always exactly 90degrees), the observer's ecliptic colatitude, and the distance between observer's zenith and the sub-suncenter point for suntop rise (90.8250deg, using the 34 arcminute average atmospheric refraction cited above, and also an average sun semidiameter of 0.5*31 arcmin).

      All observer terrestrial latitudes are geographic. I estimate that the effect of Earth's polar flattening, is negligible. The final correction to the sun's longitude, for the parallax caused by Earth's radius, was -0.0019deg at Giza and -0.0020deg at Elephantine Island.

      The sun longitudes for 2000.0AD, 1001.0BC & 4001.0BC always were interpolated with the unique second degree polynomial, with time as abscissa (see, e.g., Stirling's interpolation formula, CRC Math tables). Other, less important estimates throughout this paper, were made with sufficient accuracy using convenient linear or quadratic interpolation schemes. I found "Sothic"-type rising cycle periods, by integrating, with variable limits, the increase in sun longitude at rising, in coordinates of equinox of date (given by a quadratic interpolant), plus the gain of the equinox vs. the 365-day point. Sun longitudes are for exact heliacal rising somewhere on the relevant parallel.

      The format below lists the longitudes for 2000.0AD, 1001.0BC, 4001.0BC:

      Sun ecliptic longitudes at apparent heliacal rising of Arcturus:

      Latitude of Giza (29.979N, bronze plaque on Cheops' pyramid, per GPS by Lehner)
      199.6386, 157.2941, 97.7240

      Latitude of Elephantine Island (24.10N, according to Tompkins, "Secrets of the Great Pyramid", p. 179)
      202.9072, 161.6498, 107.6906

      Sun ecliptic longitudes at apparent heliacal rising of Procyon:

      Latitude of Giza (29.979N)
      121.9500, 86.1111, 51.0502

      Latitude of Elephantine Island (24.10N)
      120.1393, 83.8767, 48.0433

      Sun ecliptic longitudes at apparent heliacal rising of Sirius:

      Latitude of Giza (29.979N)
      121.3902, 93.3560, 68.3317

      Latitude of Elephantine Island (24.10N)
      117.0431, 93.7134, 61.0702

      Sun ecliptic longitudes at apparent heliacal rising of Canopus:

      Latitude of Giza (29.979N)
      149.3027, 136.9634, 116.0721

      Latitude of Elephantine Island (24.10N)
      136.4925, 123.3116, 94.4229
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