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Requiem for Relativity
- Joe Keller
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15 years 5 months ago #22813
by Joe Keller
Replied by Joe Keller on topic Reply from
The Ektachrome slides of my May 1, 2009, NCRAL speech: text and description of images
Slide 1. My advertising flier with excerpt from convention schedule, my email address josephkeller100@hotmail.com, the statement "The speaker is not paid & does not profit from registration fees", words "SKY SURVEY ORBIT - PLANETARY PROPERTIES", Shaltout 2007 Fig. 6 graph with my note "temples orient to Lowell's Planet X", statue of Pharaoh Menkaure & Queen Khamerernebty.
Slide 2. Pg. 1 of convention schedule.
Slide 3. Title page "Discovery of planet X 'Barbarossa' by Joseph C. Keller, M. D., B. A., Harvard, cumlaude, Mathematics".
[Comment at talk: as for my qualifications to do this, that's about it; but majoring in Math at Harvard was Percival Lowell's qualification too.]
Slide 4. Picture of Urbain LeVerrier, David Todd, Percival Lowell, Wallace J. Eckert, Robert Sutton Harrington. [Comment at talk: LeVerrier (co-discoverer, with Adams, of Neptune - Galle was first to observe it) said another large planet must exist beyond Neptune; Todd found the direction; Lowell found about the same direction and also about the right mass/distance^3; Eckert was an IBM executive and one of the first to use mainframe computers for orbits, finding a big discrepancy in Neptune's; Harrington, who passed away in his early 50s from esophageal cancer, was head of this department at the USNO, and found about the same direction and mass/distance^3 as Lowell.]
Slide 5. Picture captioned THOMAS VAN FLANDERN, Ph.D., ASTRONOMY, YALE, www.metaresearch.org . [Comment at talk: Dr. Van Flandern, who recently passed away from cancer also, was a supporter of Harrington. You don't need to write down what I say, because all of it and more, everything I know about this, is on Dr. Van Flandern's messageboard.]
Slide 6. (with yellow disks as Type G binary) p. 1
Paul Wesson (1981):
diagram illustrating angular momentum vector as J=r*vperp*m
p = angular momentum / mass^2
is about the same for spinning planets, binary stars, and galaxies.
Hence, the angular momentum of the KNOWN Solar System is subnormal.
Average Type G binary (Aitken, The Binary Stars):
T=84yr, epsilon=0.518
If Planet X has epsilon=0.518, what mass and semimajor axis, give the same "p", as two Suns in such an orbit? m*sqrt(a) = const.#1
Slide 7. p. 2
This author (2008):
diagram of two nonintersecting confocal coplanar ellipses randomly oriented
Following Gauss, consider the mutual torque of concentric, almost-coplanar circles.
If confocal ellipses, weight a*(1-epsilon), a, and a*(1+epsilon) by dwell time = r^2, and average 3x3, by 1 4 1, 4 16 4, 1 4 1 Simpson's Rule.
Omit all of Pluto's perihelion interaction with Neptune (3:2 resonance with avoidance).
Use median epsilon = 0.1 for classical Edgeworth-Kuiper Belt, a = 44 AU.
If Planet X has epsilon = 0.518, what mass and semimajor axis, give the ratios 1::1, 1::2, and 1:
for outer::inner torque per degree inclination, for the classical Kuiper Belt, Pluto, and Neptune, resp.? (This replaces Bode's Law beyond Uranus.)
m*a^(-3) = const.#2
Slide 8. p. 3
Wesson (1981) + this author (2008) -->
m = 0.054 solar mass
a = 335 AU
but this needn't be one body.
This author's discovery has epsilon = 0.611, a = 344, r(1997) = 214, ecliptic (lat,long)(1997) = (-12deg, 175deg)
It's a binary with mass ratio Barbarossa::Frey = 50::1.
[Comment at talk: the name is from the preface of a Berry Fleming - also a Harvard graduate - novel, a political satire.]
Slide 9. p. 4
The "disappearing dots" on all four Red and Optical Infrared sky surveys (1954-1997), if corrected for Earth parallax, move along a constant speed Great Circle. Choose r(1997), rdot(1997) and mass ratio so the four c.o.m.'s lie on curve rvector(t) with rvectordotdotdot = 0 identically, then rvector and rvectordot determine orbit. [I said in my May 1 talk, that I "went one better" and parametrized x,y,z, not quadratically in time, but by individually fitted constant term + sinusoid, with all the sinusoids of period, corresponding to the average angular speed during the interval of observation.]
Slide 10. p. 5
Anomalous Pioneer 10/11/Galileo Acceleration
Guess: accel. = H*c*exp(-r^2/r0^2) =(approx.) 7.0*10^(-
*exp(-r^2/r0^2)
(H = 72 km/sec/Mpc)
Use Galileo value 8*10^(- cm/s^2 (Anderson)
and Pioneer 11/10 8.41, 8.02, 8.78, 7.84 (Olsen)
What mass and ecliptic long., at r=212, ecl. lat.=-12deg, minimize the unexplained relative standard deviation of
(ln(accel(i)/accel(j))/(r(j)^2-r(i)^2))^(-1/2) ?
Answer: m =approx. 0.028 solar mass, long. =approx. 173deg, giving r0 =approx. 46 AU.
diagram of Sun in center with Barbarossa @ 1:00 position, P10 @ 4:00 position, anti-Barb.? @ 7:00, P11 @ 10:00
Solution duality allows Barbarossa plus anti-Barbarossa (like the "anti-Pluto").
Slide 11. p. 6
Why I looked at the (+) CMB dipole:
(1) The sun's gravitational field is the only thing we KNOW of, big enough, strong enough, symmetrical enough, to cause the "Cosmic" Microwave Background
(2) The Maxwellian moments (i.e. spherical harmonic poles) of anisotropy correlate with the ecliptic.
(3) (3/2)*k*T(2.726) = 3*G*M(sun,including planets) *m(electron)/R --> R = 43 AU (Kuiper Belt)
(3/2)*k*T(2.726) = (3 + 2/3, i.e. includes electron spin d.o.f.)*G*M(sun,including planets)*m(electron)/R --> R = 52.56 AU (the "other" Pioneer anomaly of Anderson)(Kuiper Belt ends)
drawing of bell curve superposed on circle, to connote spherically symmetric Gaussian
localized proton sqrt(timeave(kx^2)) = m(proton)*c/2/hbar = (1/2)/sigma
sigma^(-2)*0.2682(i.e. max, at r/sigma=1.369)/sqrt(pi/2) * G*m(proton) = G*M(sun,including planets)/R^2 --> R = 52.43 AU
Slide 12. p. 7
(4) "Interstellar" spectral line absorption of 69 Leonis and Theta Crateris;
discordance of catalog magnitudes: USNO-B R1 vs. R2, B1 vs. B2, Harvard vs. Johnson.
(5) This model gives the CMB dipole, though the model's multipoles decrease as ~5^(-n), so the 2nd & 3rd are too big.
diagram of circle with Sun @ ctr, r0=52.6/52.6=1, r=212/52.6=4.07, note "Temperature proportional to Grav. potential", angle theta between r & r0, Barbarossa outside circle @ r=4.07
caption "52.6 AU = locus of Grav(macro) = Grav(proton)"
"CMB" dipole = (4*pi/3)^(-1) * (integral from 0 to pi)(dtheta*2*pi*sin(theta)*cos(theta)*M(Barbarossa)/M(Sun) *((r^2+1-2*r*cos(theta))^(-1/2) - (1/2)*((cos(theta),sin(theta))(vector dot product)(cos(theta) - r, sin(theta)))*(r^2+1-2*r*cos(theta))^(-3/2)) = 3358microK / 2.726K --> M(Barbarossa) = 0.0137 solar mass
Slide 13. p. 8
The CMB dipole is non-Doppler.
Galaxies > ~ 100Mpc distant, seem redshifted (after correction for the Hubble relation) about as if the CMB dipole were Doppler ("convergence depth"). Yet the Michelson-Morley and Miller interferometry (consistent through many variations in equipment, technique and laboratory site) roughly indicates a
"pseudodrift" = k * Vvectordot (vector cross product) Vvector
where
Vvector = Earth orbital motion, Uvector
+ Earth rotation motion at lab, Wvector (Miller @ Mt. Wilson, 34.2N)
+ Solar apex motion, Pvector (use small value 7km/s toward RA 270, Decl +20)
The main terms are k * Uvectordot cross Uvector = -10.4 zhat km/s (ecliptic coords.) --> k = 0.588 /(cm/s^2)
k * Uvectordot cross 7*sin(20+23.5)*zhat km/s = 0.0564 Uvector (vs. obs. 0.0514);
Miller's unexplained azimuth now can be explained:
norm(k * Wvectordot) * 7*sin(20) km/s /(10.4*cos(34.2) km/s) = tan(24.5E) (vs. obs., Miller [1933] p. 235, averaging tangents, 0.5*(Aug/Sep weighted by # of obs. + Feb/Apr weighted by # of obs.) = tan(23.0E).
Slide 14. p. 9
Gravitational Collapse - Sooner and Stranger than I thought
For MJup, RJup, the surface minus center potential for a hydrogen atom =approx. G*MJup*MHydrogenatom / 2 / RJup = 9.5 ev < ionization potential, but for a carbon atom it's 113 ev > IVth ionization potential. For a Jovian planet not mainly hydrogen, transitional gravitational collapse starts early and entails many ionic species.
If my guess about "pseudodrift" is right, the "pseudodrift" at the equator of an Earth-size, 3000 Earth-mass body with rotation period 6 min (which would give it the same equatorial bulge omega^2*r/(m/r^2) as Jupiter) is
5 km/s * (24hr/6min)^2 =approx. 300,000 km/s
Slide 15. My Keuffel & Esser graph "Red and Optical Infrared Sky Survey Positions" showing for the 1954, 1986, 1987 (with rough correction for Earth parallax) and 1997 Barbarossa/Frey c.o.m. positions, uncorrected for Earth parallax (because near opposition)(except for 1987), alignment on constant angular speed (0.122deg/yr) great circle (29.6deg slope S of E) to ~1 arcminute accuracy.
Note: with all Earth parallax corrections, 1987-1997 is 6% slower than 1954-1986.
Slide 16. My Keuffel & Esser graph "Plausible Barbarossa/Frey Orbit", showing equal areas in equal times for one possible orbit of Barbarossa's chief moon Frey, with epsilon=0.89 but circular as seen from the Sun, with a=6.5 AU. The dimensions of the orbit were determined by moving the 1954 Frey, westward parallel to the solar orbital path, until it fell on the circle determined by the other three Frey positions. Then the 1954 rearward displacement of Frey which would necessitate such a correction to 1986.5 position, was calculated. It was assumed that the 1997 Frey needed little correction, i.e. that it was not much nearer us, nor farther, than Barbarossa. This particular simple graphical solution, gives a Barbarossa/Frey period of 159yr, and a Barbarossa mass of 0.0109 solar mass.
Slide 17. (Slides 17 through 29 are Ektachrome photos of a flatscreen computer monitor with online scans of photographic DSS sky survey plates, showing Barbarossa or Frey on the Left, & the control plate on the Right. Red, "R" band, is approx. Wratten 25. Optical IR is "I" band. 1997 is Optical IR, the other years all Red. The 1954 sky survey scan is 1.7" resolution, the others 1". The exposure time of these plates is 45 to 60 min, i.e. ~ 0.5" motion EW). Barbarossa 1954 is elongated EW. English amateur astronomer Robert Turner suggested rings in such a situation (when it arose also for Frey in 1987).
Slide 18. Frey 1954 + 2 submoons.
Slide 19. Barbarossa 1986.
Slide 20. Barbarossa 1986, by cross, close-up of Eberhard effect (micro - edge contrast enhancement, characteristic of photographic development; intermittent luminosity, if on timescale < 30ms, might simulate adaptive optics and cause smaller, sharper image, accentuating the Eberhard effect).
Slide 21. Frey 1986 + 2 submoons.
Slide 22. Same as #21, close-up.
Slide 23. Barbarossa 1987. Control 1986 plate on Right, has unrelated "disappearing dot" by cross.
Slide 24. Barbarossa 1987, close-up, by cross, Eberhard effect again.
Slide 25. Frey 1987, elongated EW. This was 1st object I discovered, and is ~1 mag brighter than other Freys or Barbarossas, which average Red mag ~ +18.7 by comparison with nearby USNO-B catalog stars. This Frey might be a juxtaposition of Frey with submoon(s).
Slide 26. Frey 1987, close-up, Eberhard effect again. Nearby catalog star shows less Eberhard effect. Intermittent luminance might be the reason Barbarossa & Frey don't appear on the four relevant Blue sky survey plates (exposure >~ 10min, vs. 45-60 min). Red color (as for KBOs per Wickramasinghe & Hoyle) might also be the reason.
Slide 27. Barbarossa 1997, by cross. Control on Right is 1987 Red plate, but the 1997 "I" plate also was compared to the DENIS CCD "I" band survey, nominal I mag limit 18.5, for all 1997 objects. (The small DENIS patches happened to miss the predicted Barbarossa & Frey positions for their epochs.)
Slide 28. Barbarossa 1997, close-up by cross, with Eberhard effect.
Slide 29. Frey 1997, by cross, with three possible submoons.
Slide 30. My Keuffel & Esser graph of Ptolemy's version of Crater (7 stars: alpha, gamma, delta, epsilon, zeta, eta, theta; excludes beta). Mercator projection. 2000AD and 2525BC positions, latter marked by colored disks. Only alpha moved considerably; constellation was more symmetrical in earlier time. Barbarossa orbit shown soon to pass between upper "horns".
Slide 31. Tirion's star map facing S in N hemisphere springtime.
Slide 32. Tirion's map of Hydra & Crater.
Slide 33 through Slide 36. My 2009 Ektachromes, 1 or 2 min exposures, 35mm Leica on tripod, of, altogether, all of Hydra W to E.
Slide 37. Pharaohs of the Great Pyramids. (Severely damaged Cheops giving "So let it be done!" salute as in the movie, "The Ten Commandments"; Chephren with falcon Horus on shoulders; Menkaure with Khamerernebty [comment at talk: note how equal they are; she's not kneeling at his feet or anything like that].)
Slide 38. Socrates quote from Plato's Republic: "because we do not know the truth about ancient times..."
[Comment at talk: we have to be detectives.]
Slide 39. p. 1H
History and Archaeology
Mayan calendar cycle ends on winter solstice, 2012 AD. [Comment at talk: the chance of randomly ending on a solstice or equinox is only ~1/100; those people knew something! Europeans couldn't predict solstices so accurately, until Pope Gregory's calendar reform in the late 16th cent.]
Surviving texts suggest Hellenistic and/or Hindu astronomers knew precession accurately enough to predict solstice 2000yr ahead.
German lake varves (Brauer, 2008) date sudden onset of "Younger Dryas" cooling to 12683yr before 2012.
Sky survey positions imply Barbarossa ("Planet X") orbital period = 6340yr.
This author's compendium of reported climatic, oceanic and volcanic events dates a sudden Earth change to 6299 +/- 32 (1 sigma) yrs. before 2012.
Perturbation conserves orbital period, hence comets in 1::1 resonance with Barbarossa are likely. Lescarbault / LeVerrier's "Vulcan" had same orbital inclination as Barbarossa.
A new physical force might manifest when rdotdot = -2*H*c in Dec. 2012. Atmospheric CO2 --> nanodiamonds? Barbarossa brightens?
Slide 40. Menkaure between Hathor and local Goddess. Sun disk between Hathor's horns. (In another statue, Menkaure humbly was shown much smaller than Goddesses.)
Slide 41. Typed version of #39.
Slide 42. p. 2H
They recorded when it happened:
Egyptian records (preserved by Solon, Critias, Plato) say "Atlantis" fell "9000 yr" before Plato, i.e. ~11,000 to 12,500 BP.
Pope Gregory XIII acceded 1572, began decade-long calendar reform effort (why so urgent?). Barbarossa's perihelion was 1569. Joseph Scaliger [thought by many to be the greatest scholar of his generation] established "JD zero", 6295yr before 1582 start of Gregorian calendar: Hermetic knowledge? fear of heresy?
Did archaic Egyptian (or "Atlantean") astronomers SEE IT and base a calendar on its perihelion? [Comment at talk: the perihelion year could have been determined by taking third differences of the angle of anomaly. Brightening might have been due to matter infall, as for cataclysmic variables, or due to some unknown force. Recently it was reported that a brown dwarf briefly brightened 40x, really for unknown reasons; though I'm considering 1,000,000x brightening.]
Slide 43. Sun wrapped with uraeus serpent (constellation Hydra?), tomb of Nefertari (Queen of Ramses II), 19th Dynasty, New Kingdom.
Slide 44. Modern construction by leading Iowa contractor. Note that pillars are prefab concrete, halves bolted together, with protective caulk over joint. Even the best modern construction, compared to ancient construction, reveals that much art has been lost due to its replacement by "more efficient" new techniques.
Slide 45. Archaic Egyptian flint knife, with carefully scalloped blade. Jean Capart said archaic Egyptian stonecraft exceeded any by mankind before or since.
Slide 46. Close-up of handle of #45, with sun(?) entwined by caduceus serpents (Hydra?).
Slide 47. Typed version of #42.
Slide 48. p. 3H
They recorded where it happened:
THE DECLINATION:
Temples' azimuths, converted to the declination of the stars rising or setting there, give in Egypt a histogram peak at Declination +12.7deg (Shaltout, Belmonte & Fekri, 2007, Fig. 6, "Peak 3"). Excluding "Pyramid Temples" and NSEW temples, this is their highest histogram peak: higher than the peaks for summer solstice sun or (corrected to date of temple foundation) for Sirius and Canopus.
Barbarossa's Declination one sidereal orbit before 2012, was +12.1 degrees.
This author found examples:
1. The "bastion" of the Hermopolis Magna Komasterion (Procession House); an earlier door at the other end, would have faced Barbarossa-rise.
2. The Great Basilica at Abu Mina; again, an earlier door at the foundation's other end, would have faced rising.
3. The Hibis temple at el-Kharga; texts indicate an earlier, New Kingdom, temple here, but the door still faces the rising.
Not only temples, but also streets and megaliths:
Slide 49. Shaltout et al, Fig. 6; discussed in #48.
Slide 50. Typed version of #48.
Slide 51. p. 4H
At azimuths corrresponding, for their latitude, to risings or settings at ~ +/- 15 deg Declination:
the causeway from the Sphinx to the Pyramid of Chephren (Khafre)
"Alignment V" of Nabta (Egypt c. 6000-6500 BP) megaliths
some Hellenic temples, and some megaliths in Britain and Brittany (Lockyer, "Stonehenge", 1906)
the city of Teotihuacan
The declinations range from 12 to 17 deg ("the so-called 17 deg family ... in Mesoamerica" - Sprajc, 2005). One quarter precession cycle ago, this could have been caused by a 5 deg decrease in Earth's obliquity.
Legion (Archaeological Soc. of Staten I., 1979) noted that for reasons unknown, the Great Pyramids avoid better, flatter, cheaper bedrock sites only yards away. If their layout is geometrically similar to (the EW reflection of) Orion's Belt projected by the top of a stick (with Epsilon Orionis on the meridian) then azimuths 1--> 2 and 2--> 3 (Prof. Petrie's 1880 survey, online per Ronald Birdsall) are correct for ~7723 BC, with the modern precession rate, but obliquity 27.95deg (my calculation). The two archaic azimuths, preserved by IVth Dynasty builders, uniquely determine Earth's pole at the founding epoch.
Slide 52. Giza plan.
Slide 53. Typed version of #51.
Slide 54. p. 5H
THE CONSTELLATION:
Egyptians called the Pleiades "the seven Hathors", but Ptolemy lists seven stars for Crater, the constellation which (then more than now, due to Alpha Crateris' proper motion) resembles Hathor's cow horns in Egyptian art. Crater was linked poetically to Nile floods (it nears the Sun in late summer) and lies near another Ptolemy constellation, the water snake, Hydra.
Crater earlier might have been called Hathor, the wild cow of the Nile delta, who also was linked to the Milky Way and to Nile floods. Upside-down in the east, Crater resembles a house that archaic Egyptians might have built when they had rain and timber. "Hathor" equals "Hwt Hor", "house of Horus".
Starting soon after 2012, Barbarossa's orbit will straddle Crater's (Hathor's) horns. Hathor was pictured as a cow decorated with stars; what constellation was it? Geraldine Pinch (Folklore, 1982) describes New Kingdom dolls belonging to Egyptian girls: one pictured wears a tiara with seven large holes. "The prehistoric Hathor head mentioned ... has stars at the points of the horns, at the ears and on the forehead" (Bleeker, 1973).
Slide 55. Hathor, Menkaure, and local Goddess.
Slide 56. Late Period Hathor, from tomb of contemporary of Nectanebo I. Note straightened horns, detailed cobra-like uraeus.
Slide 57. Sun wrapped in serpent, New Kingdom (19th Dyn.) (similar to that in tomb of Nefertari).
Slide 58. 11th Dynasty (early Middle Kingdom) Hathor with uraeus resembling protrusion from sun disk.
Slide 59. Close-up of #58.
Slide 60. Close-up of #55. Old Kingdom uraeus is only a dot near bottom of sun disk.
Slide 61. Close-up of #60.
Slide 62. Typed version of #54.
Slide 63. p. 6H
WHAT HAPPENED:
The various climate change time frequencies arise from Earth's axial precession frequency y, the ecliptic plane's precession frequency u, and the beats x +/- 4*y of Barbarossa's orbital frequency x, with 4*y:
( y or u ) +/- ( 1 or 2 )*( x +/- 4*y )
(details posted on Dr. Van Flandern's message board, by this author).
Tortuguero Monument 6 announces that in 2012 AD, "Bolon descends". Bolon ( = ball, bolus?) is a name linked to war, "the nine colleagues" (moons?), and to seeds (in Egypt the -15 deg Declination is associated with the Sun during autumnal Nile floods). Indeed Barbarossa in 2012 descends in Declination and ecliptic latitude; it will have descended in Declination also from Earth's precession during the last orbital period; and possibly, a period ago, an unknown Earth-Barbaorssa interaction decreased Earth's obliquity 4.5 deg, thus decreasing Barbarossa's Declination about that much.
Slide 64. Independent researcher's best fit of my Crater chart to Hathor's horns (faint because it's colored pencil, not colored paper disks).
Slide 65. My fit of #64, using colored paper disks for stars.
Slide 66. Typed version of #63.
Slide 67. p. 7H
IN THE WORDS OF THE EGYPTIANS:
Egyptians called Mars the "Red Horus" (Lockyer, "The Dawn of Astronomy") but Hathor was pictured as a reddish-brown cow with starlike white spots, Hathor's priestesses wore red patterned dresses and red scarves (Lesko, 1999), and Hathor was said to be " 'THE ONE WHO REMEMBERS HORUS' ... Hathor's motherhood [of Horus] is ... symbolical ... this Horus is the sky-God ... at Dendera [Horus-Somtus, solar Horus] ... sitting ... above the water [or] as a serpent ..." (Bleeker, 1973). Hathor also was associated with gold and with various semiprecious blue minerals (Bleeker), having association with mining: " ... the cult statue of Hathor consistently appears as if emerging from a hillside ..." (Lesko). In sum: Hathor was the house of Horus.
Other archaic Egyptians seem to have recorded the event alternatively as, in sum: Hathor had the eye of Re. " THAT EYE OF [RE] WHICH IS ON THE HORNS OF HATHOR,... " - Pyramid Text utterance 705 (these were inscribed on the inner walls of 5th & 6th Dynasty pyramids)(Lesko). " ... the sun-eye becomes separated from Re ... becomes an independent numen ... the uraeus." (Bleeker II.D.f). [Apollo defeats a snake but Re reconciles it.] " ... Hathor as sun-eye was sent by Re to chastise mankind ... " (Bleeker III.D.6.b).
[Added May 3: Early Egyptian texts (see Budge) say "the eye of Re" "exterminated" "the enemies ... the people of the mountains". This might refer to the peoples of mountainous Iran, Turkey, and SE & Central Europe. The Indo-European languages are said to diverge from c. 6000-7000yr Before Present. One surviving tribe with a common language, ancestors of the Persians, Hittites, Hellenes, Latins, etc., might have filled the vacant region.]
Slide 68. Hathor cow with sun disk, New Kingdom (19th Dynasty).
Slides 69-78. Repetitive material including several projections of Hathor's horns onto my Crater chart, under various lighting conditions.
Slide 74. Typed copy of #67.
Slide 1. My advertising flier with excerpt from convention schedule, my email address josephkeller100@hotmail.com, the statement "The speaker is not paid & does not profit from registration fees", words "SKY SURVEY ORBIT - PLANETARY PROPERTIES", Shaltout 2007 Fig. 6 graph with my note "temples orient to Lowell's Planet X", statue of Pharaoh Menkaure & Queen Khamerernebty.
Slide 2. Pg. 1 of convention schedule.
Slide 3. Title page "Discovery of planet X 'Barbarossa' by Joseph C. Keller, M. D., B. A., Harvard, cumlaude, Mathematics".
[Comment at talk: as for my qualifications to do this, that's about it; but majoring in Math at Harvard was Percival Lowell's qualification too.]
Slide 4. Picture of Urbain LeVerrier, David Todd, Percival Lowell, Wallace J. Eckert, Robert Sutton Harrington. [Comment at talk: LeVerrier (co-discoverer, with Adams, of Neptune - Galle was first to observe it) said another large planet must exist beyond Neptune; Todd found the direction; Lowell found about the same direction and also about the right mass/distance^3; Eckert was an IBM executive and one of the first to use mainframe computers for orbits, finding a big discrepancy in Neptune's; Harrington, who passed away in his early 50s from esophageal cancer, was head of this department at the USNO, and found about the same direction and mass/distance^3 as Lowell.]
Slide 5. Picture captioned THOMAS VAN FLANDERN, Ph.D., ASTRONOMY, YALE, www.metaresearch.org . [Comment at talk: Dr. Van Flandern, who recently passed away from cancer also, was a supporter of Harrington. You don't need to write down what I say, because all of it and more, everything I know about this, is on Dr. Van Flandern's messageboard.]
Slide 6. (with yellow disks as Type G binary) p. 1
Paul Wesson (1981):
diagram illustrating angular momentum vector as J=r*vperp*m
p = angular momentum / mass^2
is about the same for spinning planets, binary stars, and galaxies.
Hence, the angular momentum of the KNOWN Solar System is subnormal.
Average Type G binary (Aitken, The Binary Stars):
T=84yr, epsilon=0.518
If Planet X has epsilon=0.518, what mass and semimajor axis, give the same "p", as two Suns in such an orbit? m*sqrt(a) = const.#1
Slide 7. p. 2
This author (2008):
diagram of two nonintersecting confocal coplanar ellipses randomly oriented
Following Gauss, consider the mutual torque of concentric, almost-coplanar circles.
If confocal ellipses, weight a*(1-epsilon), a, and a*(1+epsilon) by dwell time = r^2, and average 3x3, by 1 4 1, 4 16 4, 1 4 1 Simpson's Rule.
Omit all of Pluto's perihelion interaction with Neptune (3:2 resonance with avoidance).
Use median epsilon = 0.1 for classical Edgeworth-Kuiper Belt, a = 44 AU.
If Planet X has epsilon = 0.518, what mass and semimajor axis, give the ratios 1::1, 1::2, and 1:
for outer::inner torque per degree inclination, for the classical Kuiper Belt, Pluto, and Neptune, resp.? (This replaces Bode's Law beyond Uranus.)m*a^(-3) = const.#2
Slide 8. p. 3
Wesson (1981) + this author (2008) -->
m = 0.054 solar mass
a = 335 AU
but this needn't be one body.
This author's discovery has epsilon = 0.611, a = 344, r(1997) = 214, ecliptic (lat,long)(1997) = (-12deg, 175deg)
It's a binary with mass ratio Barbarossa::Frey = 50::1.
[Comment at talk: the name is from the preface of a Berry Fleming - also a Harvard graduate - novel, a political satire.]
Slide 9. p. 4
The "disappearing dots" on all four Red and Optical Infrared sky surveys (1954-1997), if corrected for Earth parallax, move along a constant speed Great Circle. Choose r(1997), rdot(1997) and mass ratio so the four c.o.m.'s lie on curve rvector(t) with rvectordotdotdot = 0 identically, then rvector and rvectordot determine orbit. [I said in my May 1 talk, that I "went one better" and parametrized x,y,z, not quadratically in time, but by individually fitted constant term + sinusoid, with all the sinusoids of period, corresponding to the average angular speed during the interval of observation.]
Slide 10. p. 5
Anomalous Pioneer 10/11/Galileo Acceleration
Guess: accel. = H*c*exp(-r^2/r0^2) =(approx.) 7.0*10^(-
*exp(-r^2/r0^2)(H = 72 km/sec/Mpc)
Use Galileo value 8*10^(-
cm/s^2 (Anderson)and Pioneer 11/10 8.41, 8.02, 8.78, 7.84 (Olsen)
What mass and ecliptic long., at r=212, ecl. lat.=-12deg, minimize the unexplained relative standard deviation of
(ln(accel(i)/accel(j))/(r(j)^2-r(i)^2))^(-1/2) ?
Answer: m =approx. 0.028 solar mass, long. =approx. 173deg, giving r0 =approx. 46 AU.
diagram of Sun in center with Barbarossa @ 1:00 position, P10 @ 4:00 position, anti-Barb.? @ 7:00, P11 @ 10:00
Solution duality allows Barbarossa plus anti-Barbarossa (like the "anti-Pluto").
Slide 11. p. 6
Why I looked at the (+) CMB dipole:
(1) The sun's gravitational field is the only thing we KNOW of, big enough, strong enough, symmetrical enough, to cause the "Cosmic" Microwave Background
(2) The Maxwellian moments (i.e. spherical harmonic poles) of anisotropy correlate with the ecliptic.
(3) (3/2)*k*T(2.726) = 3*G*M(sun,including planets) *m(electron)/R --> R = 43 AU (Kuiper Belt)
(3/2)*k*T(2.726) = (3 + 2/3, i.e. includes electron spin d.o.f.)*G*M(sun,including planets)*m(electron)/R --> R = 52.56 AU (the "other" Pioneer anomaly of Anderson)(Kuiper Belt ends)
drawing of bell curve superposed on circle, to connote spherically symmetric Gaussian
localized proton sqrt(timeave(kx^2)) = m(proton)*c/2/hbar = (1/2)/sigma
sigma^(-2)*0.2682(i.e. max, at r/sigma=1.369)/sqrt(pi/2) * G*m(proton) = G*M(sun,including planets)/R^2 --> R = 52.43 AU
Slide 12. p. 7
(4) "Interstellar" spectral line absorption of 69 Leonis and Theta Crateris;
discordance of catalog magnitudes: USNO-B R1 vs. R2, B1 vs. B2, Harvard vs. Johnson.
(5) This model gives the CMB dipole, though the model's multipoles decrease as ~5^(-n), so the 2nd & 3rd are too big.
diagram of circle with Sun @ ctr, r0=52.6/52.6=1, r=212/52.6=4.07, note "Temperature proportional to Grav. potential", angle theta between r & r0, Barbarossa outside circle @ r=4.07
caption "52.6 AU = locus of Grav(macro) = Grav(proton)"
"CMB" dipole = (4*pi/3)^(-1) * (integral from 0 to pi)(dtheta*2*pi*sin(theta)*cos(theta)*M(Barbarossa)/M(Sun) *((r^2+1-2*r*cos(theta))^(-1/2) - (1/2)*((cos(theta),sin(theta))(vector dot product)(cos(theta) - r, sin(theta)))*(r^2+1-2*r*cos(theta))^(-3/2)) = 3358microK / 2.726K --> M(Barbarossa) = 0.0137 solar mass
Slide 13. p. 8
The CMB dipole is non-Doppler.
Galaxies > ~ 100Mpc distant, seem redshifted (after correction for the Hubble relation) about as if the CMB dipole were Doppler ("convergence depth"). Yet the Michelson-Morley and Miller interferometry (consistent through many variations in equipment, technique and laboratory site) roughly indicates a
"pseudodrift" = k * Vvectordot (vector cross product) Vvector
where
Vvector = Earth orbital motion, Uvector
+ Earth rotation motion at lab, Wvector (Miller @ Mt. Wilson, 34.2N)
+ Solar apex motion, Pvector (use small value 7km/s toward RA 270, Decl +20)
The main terms are k * Uvectordot cross Uvector = -10.4 zhat km/s (ecliptic coords.) --> k = 0.588 /(cm/s^2)
k * Uvectordot cross 7*sin(20+23.5)*zhat km/s = 0.0564 Uvector (vs. obs. 0.0514);
Miller's unexplained azimuth now can be explained:
norm(k * Wvectordot) * 7*sin(20) km/s /(10.4*cos(34.2) km/s) = tan(24.5E) (vs. obs., Miller [1933] p. 235, averaging tangents, 0.5*(Aug/Sep weighted by # of obs. + Feb/Apr weighted by # of obs.) = tan(23.0E).
Slide 14. p. 9
Gravitational Collapse - Sooner and Stranger than I thought
For MJup, RJup, the surface minus center potential for a hydrogen atom =approx. G*MJup*MHydrogenatom / 2 / RJup = 9.5 ev < ionization potential, but for a carbon atom it's 113 ev > IVth ionization potential. For a Jovian planet not mainly hydrogen, transitional gravitational collapse starts early and entails many ionic species.
If my guess about "pseudodrift" is right, the "pseudodrift" at the equator of an Earth-size, 3000 Earth-mass body with rotation period 6 min (which would give it the same equatorial bulge omega^2*r/(m/r^2) as Jupiter) is
5 km/s * (24hr/6min)^2 =approx. 300,000 km/s
Slide 15. My Keuffel & Esser graph "Red and Optical Infrared Sky Survey Positions" showing for the 1954, 1986, 1987 (with rough correction for Earth parallax) and 1997 Barbarossa/Frey c.o.m. positions, uncorrected for Earth parallax (because near opposition)(except for 1987), alignment on constant angular speed (0.122deg/yr) great circle (29.6deg slope S of E) to ~1 arcminute accuracy.
Note: with all Earth parallax corrections, 1987-1997 is 6% slower than 1954-1986.
Slide 16. My Keuffel & Esser graph "Plausible Barbarossa/Frey Orbit", showing equal areas in equal times for one possible orbit of Barbarossa's chief moon Frey, with epsilon=0.89 but circular as seen from the Sun, with a=6.5 AU. The dimensions of the orbit were determined by moving the 1954 Frey, westward parallel to the solar orbital path, until it fell on the circle determined by the other three Frey positions. Then the 1954 rearward displacement of Frey which would necessitate such a correction to 1986.5 position, was calculated. It was assumed that the 1997 Frey needed little correction, i.e. that it was not much nearer us, nor farther, than Barbarossa. This particular simple graphical solution, gives a Barbarossa/Frey period of 159yr, and a Barbarossa mass of 0.0109 solar mass.
Slide 17. (Slides 17 through 29 are Ektachrome photos of a flatscreen computer monitor with online scans of photographic DSS sky survey plates, showing Barbarossa or Frey on the Left, & the control plate on the Right. Red, "R" band, is approx. Wratten 25. Optical IR is "I" band. 1997 is Optical IR, the other years all Red. The 1954 sky survey scan is 1.7" resolution, the others 1". The exposure time of these plates is 45 to 60 min, i.e. ~ 0.5" motion EW). Barbarossa 1954 is elongated EW. English amateur astronomer Robert Turner suggested rings in such a situation (when it arose also for Frey in 1987).
Slide 18. Frey 1954 + 2 submoons.
Slide 19. Barbarossa 1986.
Slide 20. Barbarossa 1986, by cross, close-up of Eberhard effect (micro - edge contrast enhancement, characteristic of photographic development; intermittent luminosity, if on timescale < 30ms, might simulate adaptive optics and cause smaller, sharper image, accentuating the Eberhard effect).
Slide 21. Frey 1986 + 2 submoons.
Slide 22. Same as #21, close-up.
Slide 23. Barbarossa 1987. Control 1986 plate on Right, has unrelated "disappearing dot" by cross.
Slide 24. Barbarossa 1987, close-up, by cross, Eberhard effect again.
Slide 25. Frey 1987, elongated EW. This was 1st object I discovered, and is ~1 mag brighter than other Freys or Barbarossas, which average Red mag ~ +18.7 by comparison with nearby USNO-B catalog stars. This Frey might be a juxtaposition of Frey with submoon(s).
Slide 26. Frey 1987, close-up, Eberhard effect again. Nearby catalog star shows less Eberhard effect. Intermittent luminance might be the reason Barbarossa & Frey don't appear on the four relevant Blue sky survey plates (exposure >~ 10min, vs. 45-60 min). Red color (as for KBOs per Wickramasinghe & Hoyle) might also be the reason.
Slide 27. Barbarossa 1997, by cross. Control on Right is 1987 Red plate, but the 1997 "I" plate also was compared to the DENIS CCD "I" band survey, nominal I mag limit 18.5, for all 1997 objects. (The small DENIS patches happened to miss the predicted Barbarossa & Frey positions for their epochs.)
Slide 28. Barbarossa 1997, close-up by cross, with Eberhard effect.
Slide 29. Frey 1997, by cross, with three possible submoons.
Slide 30. My Keuffel & Esser graph of Ptolemy's version of Crater (7 stars: alpha, gamma, delta, epsilon, zeta, eta, theta; excludes beta). Mercator projection. 2000AD and 2525BC positions, latter marked by colored disks. Only alpha moved considerably; constellation was more symmetrical in earlier time. Barbarossa orbit shown soon to pass between upper "horns".
Slide 31. Tirion's star map facing S in N hemisphere springtime.
Slide 32. Tirion's map of Hydra & Crater.
Slide 33 through Slide 36. My 2009 Ektachromes, 1 or 2 min exposures, 35mm Leica on tripod, of, altogether, all of Hydra W to E.
Slide 37. Pharaohs of the Great Pyramids. (Severely damaged Cheops giving "So let it be done!" salute as in the movie, "The Ten Commandments"; Chephren with falcon Horus on shoulders; Menkaure with Khamerernebty [comment at talk: note how equal they are; she's not kneeling at his feet or anything like that].)
Slide 38. Socrates quote from Plato's Republic: "because we do not know the truth about ancient times..."
[Comment at talk: we have to be detectives.]
Slide 39. p. 1H
History and Archaeology
Mayan calendar cycle ends on winter solstice, 2012 AD. [Comment at talk: the chance of randomly ending on a solstice or equinox is only ~1/100; those people knew something! Europeans couldn't predict solstices so accurately, until Pope Gregory's calendar reform in the late 16th cent.]
Surviving texts suggest Hellenistic and/or Hindu astronomers knew precession accurately enough to predict solstice 2000yr ahead.
German lake varves (Brauer, 2008) date sudden onset of "Younger Dryas" cooling to 12683yr before 2012.
Sky survey positions imply Barbarossa ("Planet X") orbital period = 6340yr.
This author's compendium of reported climatic, oceanic and volcanic events dates a sudden Earth change to 6299 +/- 32 (1 sigma) yrs. before 2012.
Perturbation conserves orbital period, hence comets in 1::1 resonance with Barbarossa are likely. Lescarbault / LeVerrier's "Vulcan" had same orbital inclination as Barbarossa.
A new physical force might manifest when rdotdot = -2*H*c in Dec. 2012. Atmospheric CO2 --> nanodiamonds? Barbarossa brightens?
Slide 40. Menkaure between Hathor and local Goddess. Sun disk between Hathor's horns. (In another statue, Menkaure humbly was shown much smaller than Goddesses.)
Slide 41. Typed version of #39.
Slide 42. p. 2H
They recorded when it happened:
Egyptian records (preserved by Solon, Critias, Plato) say "Atlantis" fell "9000 yr" before Plato, i.e. ~11,000 to 12,500 BP.
Pope Gregory XIII acceded 1572, began decade-long calendar reform effort (why so urgent?). Barbarossa's perihelion was 1569. Joseph Scaliger [thought by many to be the greatest scholar of his generation] established "JD zero", 6295yr before 1582 start of Gregorian calendar: Hermetic knowledge? fear of heresy?
Did archaic Egyptian (or "Atlantean") astronomers SEE IT and base a calendar on its perihelion? [Comment at talk: the perihelion year could have been determined by taking third differences of the angle of anomaly. Brightening might have been due to matter infall, as for cataclysmic variables, or due to some unknown force. Recently it was reported that a brown dwarf briefly brightened 40x, really for unknown reasons; though I'm considering 1,000,000x brightening.]
Slide 43. Sun wrapped with uraeus serpent (constellation Hydra?), tomb of Nefertari (Queen of Ramses II), 19th Dynasty, New Kingdom.
Slide 44. Modern construction by leading Iowa contractor. Note that pillars are prefab concrete, halves bolted together, with protective caulk over joint. Even the best modern construction, compared to ancient construction, reveals that much art has been lost due to its replacement by "more efficient" new techniques.
Slide 45. Archaic Egyptian flint knife, with carefully scalloped blade. Jean Capart said archaic Egyptian stonecraft exceeded any by mankind before or since.
Slide 46. Close-up of handle of #45, with sun(?) entwined by caduceus serpents (Hydra?).
Slide 47. Typed version of #42.
Slide 48. p. 3H
They recorded where it happened:
THE DECLINATION:
Temples' azimuths, converted to the declination of the stars rising or setting there, give in Egypt a histogram peak at Declination +12.7deg (Shaltout, Belmonte & Fekri, 2007, Fig. 6, "Peak 3"). Excluding "Pyramid Temples" and NSEW temples, this is their highest histogram peak: higher than the peaks for summer solstice sun or (corrected to date of temple foundation) for Sirius and Canopus.
Barbarossa's Declination one sidereal orbit before 2012, was +12.1 degrees.
This author found examples:
1. The "bastion" of the Hermopolis Magna Komasterion (Procession House); an earlier door at the other end, would have faced Barbarossa-rise.
2. The Great Basilica at Abu Mina; again, an earlier door at the foundation's other end, would have faced rising.
3. The Hibis temple at el-Kharga; texts indicate an earlier, New Kingdom, temple here, but the door still faces the rising.
Not only temples, but also streets and megaliths:
Slide 49. Shaltout et al, Fig. 6; discussed in #48.
Slide 50. Typed version of #48.
Slide 51. p. 4H
At azimuths corrresponding, for their latitude, to risings or settings at ~ +/- 15 deg Declination:
the causeway from the Sphinx to the Pyramid of Chephren (Khafre)
"Alignment V" of Nabta (Egypt c. 6000-6500 BP) megaliths
some Hellenic temples, and some megaliths in Britain and Brittany (Lockyer, "Stonehenge", 1906)
the city of Teotihuacan
The declinations range from 12 to 17 deg ("the so-called 17 deg family ... in Mesoamerica" - Sprajc, 2005). One quarter precession cycle ago, this could have been caused by a 5 deg decrease in Earth's obliquity.
Legion (Archaeological Soc. of Staten I., 1979) noted that for reasons unknown, the Great Pyramids avoid better, flatter, cheaper bedrock sites only yards away. If their layout is geometrically similar to (the EW reflection of) Orion's Belt projected by the top of a stick (with Epsilon Orionis on the meridian) then azimuths 1--> 2 and 2--> 3 (Prof. Petrie's 1880 survey, online per Ronald Birdsall) are correct for ~7723 BC, with the modern precession rate, but obliquity 27.95deg (my calculation). The two archaic azimuths, preserved by IVth Dynasty builders, uniquely determine Earth's pole at the founding epoch.
Slide 52. Giza plan.
Slide 53. Typed version of #51.
Slide 54. p. 5H
THE CONSTELLATION:
Egyptians called the Pleiades "the seven Hathors", but Ptolemy lists seven stars for Crater, the constellation which (then more than now, due to Alpha Crateris' proper motion) resembles Hathor's cow horns in Egyptian art. Crater was linked poetically to Nile floods (it nears the Sun in late summer) and lies near another Ptolemy constellation, the water snake, Hydra.
Crater earlier might have been called Hathor, the wild cow of the Nile delta, who also was linked to the Milky Way and to Nile floods. Upside-down in the east, Crater resembles a house that archaic Egyptians might have built when they had rain and timber. "Hathor" equals "Hwt Hor", "house of Horus".
Starting soon after 2012, Barbarossa's orbit will straddle Crater's (Hathor's) horns. Hathor was pictured as a cow decorated with stars; what constellation was it? Geraldine Pinch (Folklore, 1982) describes New Kingdom dolls belonging to Egyptian girls: one pictured wears a tiara with seven large holes. "The prehistoric Hathor head mentioned ... has stars at the points of the horns, at the ears and on the forehead" (Bleeker, 1973).
Slide 55. Hathor, Menkaure, and local Goddess.
Slide 56. Late Period Hathor, from tomb of contemporary of Nectanebo I. Note straightened horns, detailed cobra-like uraeus.
Slide 57. Sun wrapped in serpent, New Kingdom (19th Dyn.) (similar to that in tomb of Nefertari).
Slide 58. 11th Dynasty (early Middle Kingdom) Hathor with uraeus resembling protrusion from sun disk.
Slide 59. Close-up of #58.
Slide 60. Close-up of #55. Old Kingdom uraeus is only a dot near bottom of sun disk.
Slide 61. Close-up of #60.
Slide 62. Typed version of #54.
Slide 63. p. 6H
WHAT HAPPENED:
The various climate change time frequencies arise from Earth's axial precession frequency y, the ecliptic plane's precession frequency u, and the beats x +/- 4*y of Barbarossa's orbital frequency x, with 4*y:
( y or u ) +/- ( 1 or 2 )*( x +/- 4*y )
(details posted on Dr. Van Flandern's message board, by this author).
Tortuguero Monument 6 announces that in 2012 AD, "Bolon descends". Bolon ( = ball, bolus?) is a name linked to war, "the nine colleagues" (moons?), and to seeds (in Egypt the -15 deg Declination is associated with the Sun during autumnal Nile floods). Indeed Barbarossa in 2012 descends in Declination and ecliptic latitude; it will have descended in Declination also from Earth's precession during the last orbital period; and possibly, a period ago, an unknown Earth-Barbaorssa interaction decreased Earth's obliquity 4.5 deg, thus decreasing Barbarossa's Declination about that much.
Slide 64. Independent researcher's best fit of my Crater chart to Hathor's horns (faint because it's colored pencil, not colored paper disks).
Slide 65. My fit of #64, using colored paper disks for stars.
Slide 66. Typed version of #63.
Slide 67. p. 7H
IN THE WORDS OF THE EGYPTIANS:
Egyptians called Mars the "Red Horus" (Lockyer, "The Dawn of Astronomy") but Hathor was pictured as a reddish-brown cow with starlike white spots, Hathor's priestesses wore red patterned dresses and red scarves (Lesko, 1999), and Hathor was said to be " 'THE ONE WHO REMEMBERS HORUS' ... Hathor's motherhood [of Horus] is ... symbolical ... this Horus is the sky-God ... at Dendera [Horus-Somtus, solar Horus] ... sitting ... above the water [or] as a serpent ..." (Bleeker, 1973). Hathor also was associated with gold and with various semiprecious blue minerals (Bleeker), having association with mining: " ... the cult statue of Hathor consistently appears as if emerging from a hillside ..." (Lesko). In sum: Hathor was the house of Horus.
Other archaic Egyptians seem to have recorded the event alternatively as, in sum: Hathor had the eye of Re. " THAT EYE OF [RE] WHICH IS ON THE HORNS OF HATHOR,... " - Pyramid Text utterance 705 (these were inscribed on the inner walls of 5th & 6th Dynasty pyramids)(Lesko). " ... the sun-eye becomes separated from Re ... becomes an independent numen ... the uraeus." (Bleeker II.D.f). [Apollo defeats a snake but Re reconciles it.] " ... Hathor as sun-eye was sent by Re to chastise mankind ... " (Bleeker III.D.6.b).
[Added May 3: Early Egyptian texts (see Budge) say "the eye of Re" "exterminated" "the enemies ... the people of the mountains". This might refer to the peoples of mountainous Iran, Turkey, and SE & Central Europe. The Indo-European languages are said to diverge from c. 6000-7000yr Before Present. One surviving tribe with a common language, ancestors of the Persians, Hittites, Hellenes, Latins, etc., might have filled the vacant region.]
Slide 68. Hathor cow with sun disk, New Kingdom (19th Dynasty).
Slides 69-78. Repetitive material including several projections of Hathor's horns onto my Crater chart, under various lighting conditions.
Slide 74. Typed copy of #67.
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15 years 5 months ago #23476
by Joe Keller
Replied by Joe Keller on topic Reply from
When did Egyptian chronology begin?
Summary. The position of New Year's Day, "1 Thoth", was maintained accurately in the 365-day old Egyptian calendar, at least from 1537BC to 26BC (and beyond, due to resistance to the Alexandrian calendar). At any given latitude, Sirius and the Sun, rise simultaneously once a year, but this "heliacal rising of Sirius" occurs on 1 Thoth, only approx. on, and about, every 1442yr. Allegedly, according to records of Amenhotep I, at 6376yr before 2012, the heliacal rising of Sirius, at the equator, occurred on 1 Thoth; that is, Sirius and the Sun lay on the same meridian, on 1 Thoth. This compares to the orbital period of Barbarossa, 6340yr, and to half Brauer's time from Younger Dryas onset to 2012, 12683/2=6341.5yr. The average of 10 methods of calculating this interval from Egyptian calendars and archaeoastronomical references, is 6242 yr before 2012, range 6436 to 6052.
The Work of Eduard Meyer. I found Eduard Meyer's "Geschichte des Altertums" free online in German, at www.zeno.org . Band (Vol.) 1, Abteilung (Part) 2, Buch (Book) 1, has in its introduction, a section called "Chronologie" which says:
"Als Anfangstag des wahren Sonnenjahrs (Anfang des Jahres, geschieden von dem Neujahrstag des buergerlichen Jahres) gilt ihnen der Fruehaufgang der Sothis, des Siriussterns, der unter dem Parallel von Memphis am 19. Juli julianisch (d.i. im Jahre 4241 v. Chr. am 15. Juni gregorianisch, zur Zeit der Sommersonnenwende) eintrat."
I gather that Meyer thought Egyptian chronology began 4241 BC, i.e. 4241+2012-1 = 6252 yrs before 2012. Can Meyer be corroborated?
Julian calendar date of Amenhotep I's heliacal rising. It is convenient to use the Julian calendar, and to correct its dates of equinoxes/solstices, and to convert from it to the old 365-day Egyptian calendar. According to the Columbia Encyclopedia, the Council of Nicaea, 325AD, considered March 21 to be the spring equinox.
According to www.absoluteastronomy.com , the 9th year of Amenhotep I (18th Dynasty, New Kingdom) had a heliacal rising of Sirius which occurred on the "9th day of the 3rd month of summer". The difference between the sothic (the time between heliacal risings of Sirius) and tropical years, is only a day per century, so though these figures hardly could be accurate to better than +/- 50yr, it's been calculated by others, that this relation, of the heliacal rising of Sirius, to the summer solstice, at Thebes, dates the event to 1517BC; but if at Memphis or Heliopolis, to 1537BC. Assuming Earth's precession period has been 25771.5yr (the present value), the tropical year would have been 365.24219d, and the Julian calendar extended backward would have had the spring equinox, in 1517BC, on March 21 + (1517+325-1)*(365.25 - 365.24219) = April 4, and summer solstice June 22 + 14d = July 6. Hence I suppose Amenhotep I's heliacal rising was July 6 + 29.5*2 + (9-1) = Sept 12 (Julian calendar).
Heliacal risings on 1 Thoth. Censorinus (see Appendix suggested that a heliacal rising of Sirius in Egypt, had occurred on the old, pre-Alexandrian Egyptian 365-day, leapyear-less New Year, "1 Thoth", at Julian calendar date July 20, 139AD (Wikipedia, "Egyptian calendar"). The Alexandrian calendar of Augustus (basically a Julian calendar for Egyptians) in 26BC had approximately fixed the Egyptian New Year, "1 Thoth", at the Julian calendar's August 29; but without the leap year's "epagomenal day", 1 Thoth in 139AD would have been 0.25*(139+26-1) = 41d earlier, i.e. July 19 (close enough, to July 20). Again, the old leapyear-less Egyptian "1 Thoth" in 1517BC would have been 0.25*(1517-26) = 373 = (365+8)d later than Aug 29, i.e., on a Sept 6; so, Amenhotep I's heliacal rising was on 7 Thoth, not 1 Thoth, if the observation refers to 1517BC, the supposed value for Thebes observation. If the observation refers to Memphis or Heliopolis, then it supposedly was a 1537BC heliacal rising, which indeed was on 7 - 20/4 = 2 Thoth! Demanding that the heliacal rising occur on 1 Thoth, gives 1541BC for its date, and 1549BC for the start of Amenhotep I's reign.
The sothic year is ~365.25306d (see Appendix A). It is so close to the Julian year, 365.25d, that the (average) Julian calendar date of heliacal rising, advances only a day in 327yr. The Julian calendar date of heliacal rising should have been (1537-26)/327 = 4.6d later, than Sept 12, in 26BC, and (1537+139-1)/327 = 5.1d later, in 139AD. Though Aug 29, 26BC, and July 20, 139AD, both are 1 Thoth, neither could be a heliacal rising, because other astronomers' calculations found that ~Sept 12 (Julian calendar) was Sirius' heliacal rising in Egypt, c. 1500BC. The actual heliacal rising on the old Egyptian 1 Thoth, would have been 1541-1442 = 99BC (see Appendix A).
Rising vs. meridian. The only authentic date, for heliacal rising of Sirius on the old Egyptian 1 Thoth, is Amenhotep I's 1541BC, and the authentic period is 1442yr. So, the Egyptian calendar might have started with 1 Thoth on the heliacal rising 1541+2*1442 = 4425BC = 6436yr before 2012.
The date is later, if, instead of heliacal rising seen from an observatory at 30N lat., one considers heliacal rising seen from the hypothetical observatory on the equator, i.e., equality of the meridians (Right Ascensions) of Sirius and the Sun. In the time of Amenhotep I, the heliacal rising at 30N, occurred 68d after the summer solstice. Because the rising Sun, and rising Sirius, make there about a 60deg angle with the horizon, and are about (17 + 23.5*cos(68*90/91)) = 26deg apart in declination, equality of the meridians would have occurred 26/sqrt(3)*365/360 = 15d earlier. This date would have fallen on 1 Thoth, in 1541 - 15*4 = 1481BC. The Egyptian calendar, if its very first day were 1 Thoth when the Sun was on the meridian of Sirius, would have started 6436 - 60 = 6376 before 2012.
Independent calculation of heliacal rising. Now we need only make two historical assumptions: suppose August 29, 25BC was 1 Thoth, and that the Alexandrian calendar of Augustus (essentially the Julian calendar) made essentially no miscount between then, and Pope Gregory's reform. For 25BC through 2012AD our Alexandrian/Gregorian calendar includes approx. (first leap year was 22BC) integer[(2012+25)/4] = 509 leap days, minus the 10 skipped days of Gregorian reform, minus 3 Gregorian skipped century leap days; so 1 Thoth is Aug 29, 2012 - (496-365 = 131)d = Apr 20, 2012, on which the Sun's RA is approx. 30. Sirius' RA is 101.
Sothic cycles, from the hypothetical equatorial observatory, start prior to Apr 20, 2012, by intervals ((101-30)/360 + n)*1442. For n=4 this is 6052yr before 2012.
Amenhotep I vs. Censorinus vs. Theon vs. Meyer vs. this author. My 6052 yr based on the Julian calendar & an observatory on the equator, differs from the 6376 based on Amenhotep I & an observatory on the equator. From Censorinus derives yet another figure, (2012AD - 139AD) + 3*1442 = 6199yr (6139 if at equator) before 2012. From Theon derives yet another, (2012AD - 26BC) + 3*1442 = 6363yr (6303 if at equator) before 2012. Meyer's is 6252 (6192 if on the equator). So, 5x2 = 10 methods give 10 estimates (mean = 6242yr before 2012) of the origin of the Egyptian calendar, ranging from 6436 to 6052yr before 2012.
Appendix A.
The ecliptic latitude of Sirius is unaffected by Earth's axis precession, and only slightly affected by the slower, smaller-amplitude ecliptic precession. Jean-Marc Baillard (2004, online example in "HP [Hewlett Packard] museum") gives Sirius' ecliptic latitude as -39deg39'18" in 1600AD & -39deg35'19" in 2134AD. So, in 772BC (chosen as the midpoint between two dates below, 26BC & 1526-8=1518BC) it would be -39.95deg.
If Sirius were on the ecliptic, the time between heliacal risings would be simply a sidereal year, 365.25636d. If Sirius were at the ecliptic pole, the time would be a tropical year (length of tropical year is from sidereal yr & 25771.5yr precession period), 365.24219d. Interpolating with a sine curve, between these presumed maxima and minima, we find the time between heliacal risings is 365.25306d. This gives a Sothic cycle of
365.25306/(365.25306-365)*365/365.2425 = 1442 Gregorian years (vs. the 1460 usually stated).
Appendix B.
Damien F. Mackey, "Fall of the Sothic theory: Egyptian chronology revisited", TJ 17(3):7073, December 2003 (online @ www.answersingenesis.org ) says:
"Theon, an Alexandrian astronomer of the late 4th century AD...according to whom the conclusion of a 1,460-year period had occurred in the 5th year of the emperor Augustus - 26 BC...
"...the 3rd century AD Roman author, Censorinus. According to Meyers interpretation of the Sothic data as provided by Censorinus, a coincidence had occurred between the heliacal rising of Sirius and New Years Day in the 100th year before Censorinus wrote...
"Of the various major Egyptian Sothic documents, such as the Illahn Papyrus, the Elephantine Stele, and the Ebers Papyrus, the latter famous for its information about medical practices in Egypt - also contains reference to a Sothic rising in the 9th year of another un-named king, who has been identified as Amenhotep I of the 18th dynasty."
Summary. The position of New Year's Day, "1 Thoth", was maintained accurately in the 365-day old Egyptian calendar, at least from 1537BC to 26BC (and beyond, due to resistance to the Alexandrian calendar). At any given latitude, Sirius and the Sun, rise simultaneously once a year, but this "heliacal rising of Sirius" occurs on 1 Thoth, only approx. on, and about, every 1442yr. Allegedly, according to records of Amenhotep I, at 6376yr before 2012, the heliacal rising of Sirius, at the equator, occurred on 1 Thoth; that is, Sirius and the Sun lay on the same meridian, on 1 Thoth. This compares to the orbital period of Barbarossa, 6340yr, and to half Brauer's time from Younger Dryas onset to 2012, 12683/2=6341.5yr. The average of 10 methods of calculating this interval from Egyptian calendars and archaeoastronomical references, is 6242 yr before 2012, range 6436 to 6052.
The Work of Eduard Meyer. I found Eduard Meyer's "Geschichte des Altertums" free online in German, at www.zeno.org . Band (Vol.) 1, Abteilung (Part) 2, Buch (Book) 1, has in its introduction, a section called "Chronologie" which says:
"Als Anfangstag des wahren Sonnenjahrs (Anfang des Jahres, geschieden von dem Neujahrstag des buergerlichen Jahres) gilt ihnen der Fruehaufgang der Sothis, des Siriussterns, der unter dem Parallel von Memphis am 19. Juli julianisch (d.i. im Jahre 4241 v. Chr. am 15. Juni gregorianisch, zur Zeit der Sommersonnenwende) eintrat."
I gather that Meyer thought Egyptian chronology began 4241 BC, i.e. 4241+2012-1 = 6252 yrs before 2012. Can Meyer be corroborated?
Julian calendar date of Amenhotep I's heliacal rising. It is convenient to use the Julian calendar, and to correct its dates of equinoxes/solstices, and to convert from it to the old 365-day Egyptian calendar. According to the Columbia Encyclopedia, the Council of Nicaea, 325AD, considered March 21 to be the spring equinox.
According to www.absoluteastronomy.com , the 9th year of Amenhotep I (18th Dynasty, New Kingdom) had a heliacal rising of Sirius which occurred on the "9th day of the 3rd month of summer". The difference between the sothic (the time between heliacal risings of Sirius) and tropical years, is only a day per century, so though these figures hardly could be accurate to better than +/- 50yr, it's been calculated by others, that this relation, of the heliacal rising of Sirius, to the summer solstice, at Thebes, dates the event to 1517BC; but if at Memphis or Heliopolis, to 1537BC. Assuming Earth's precession period has been 25771.5yr (the present value), the tropical year would have been 365.24219d, and the Julian calendar extended backward would have had the spring equinox, in 1517BC, on March 21 + (1517+325-1)*(365.25 - 365.24219) = April 4, and summer solstice June 22 + 14d = July 6. Hence I suppose Amenhotep I's heliacal rising was July 6 + 29.5*2 + (9-1) = Sept 12 (Julian calendar).
Heliacal risings on 1 Thoth. Censorinus (see Appendix
suggested that a heliacal rising of Sirius in Egypt, had occurred on the old, pre-Alexandrian Egyptian 365-day, leapyear-less New Year, "1 Thoth", at Julian calendar date July 20, 139AD (Wikipedia, "Egyptian calendar"). The Alexandrian calendar of Augustus (basically a Julian calendar for Egyptians) in 26BC had approximately fixed the Egyptian New Year, "1 Thoth", at the Julian calendar's August 29; but without the leap year's "epagomenal day", 1 Thoth in 139AD would have been 0.25*(139+26-1) = 41d earlier, i.e. July 19 (close enough, to July 20). Again, the old leapyear-less Egyptian "1 Thoth" in 1517BC would have been 0.25*(1517-26) = 373 = (365+8)d later than Aug 29, i.e., on a Sept 6; so, Amenhotep I's heliacal rising was on 7 Thoth, not 1 Thoth, if the observation refers to 1517BC, the supposed value for Thebes observation. If the observation refers to Memphis or Heliopolis, then it supposedly was a 1537BC heliacal rising, which indeed was on 7 - 20/4 = 2 Thoth! Demanding that the heliacal rising occur on 1 Thoth, gives 1541BC for its date, and 1549BC for the start of Amenhotep I's reign.The sothic year is ~365.25306d (see Appendix A). It is so close to the Julian year, 365.25d, that the (average) Julian calendar date of heliacal rising, advances only a day in 327yr. The Julian calendar date of heliacal rising should have been (1537-26)/327 = 4.6d later, than Sept 12, in 26BC, and (1537+139-1)/327 = 5.1d later, in 139AD. Though Aug 29, 26BC, and July 20, 139AD, both are 1 Thoth, neither could be a heliacal rising, because other astronomers' calculations found that ~Sept 12 (Julian calendar) was Sirius' heliacal rising in Egypt, c. 1500BC. The actual heliacal rising on the old Egyptian 1 Thoth, would have been 1541-1442 = 99BC (see Appendix A).
Rising vs. meridian. The only authentic date, for heliacal rising of Sirius on the old Egyptian 1 Thoth, is Amenhotep I's 1541BC, and the authentic period is 1442yr. So, the Egyptian calendar might have started with 1 Thoth on the heliacal rising 1541+2*1442 = 4425BC = 6436yr before 2012.
The date is later, if, instead of heliacal rising seen from an observatory at 30N lat., one considers heliacal rising seen from the hypothetical observatory on the equator, i.e., equality of the meridians (Right Ascensions) of Sirius and the Sun. In the time of Amenhotep I, the heliacal rising at 30N, occurred 68d after the summer solstice. Because the rising Sun, and rising Sirius, make there about a 60deg angle with the horizon, and are about (17 + 23.5*cos(68*90/91)) = 26deg apart in declination, equality of the meridians would have occurred 26/sqrt(3)*365/360 = 15d earlier. This date would have fallen on 1 Thoth, in 1541 - 15*4 = 1481BC. The Egyptian calendar, if its very first day were 1 Thoth when the Sun was on the meridian of Sirius, would have started 6436 - 60 = 6376 before 2012.
Independent calculation of heliacal rising. Now we need only make two historical assumptions: suppose August 29, 25BC was 1 Thoth, and that the Alexandrian calendar of Augustus (essentially the Julian calendar) made essentially no miscount between then, and Pope Gregory's reform. For 25BC through 2012AD our Alexandrian/Gregorian calendar includes approx. (first leap year was 22BC) integer[(2012+25)/4] = 509 leap days, minus the 10 skipped days of Gregorian reform, minus 3 Gregorian skipped century leap days; so 1 Thoth is Aug 29, 2012 - (496-365 = 131)d = Apr 20, 2012, on which the Sun's RA is approx. 30. Sirius' RA is 101.
Sothic cycles, from the hypothetical equatorial observatory, start prior to Apr 20, 2012, by intervals ((101-30)/360 + n)*1442. For n=4 this is 6052yr before 2012.
Amenhotep I vs. Censorinus vs. Theon vs. Meyer vs. this author. My 6052 yr based on the Julian calendar & an observatory on the equator, differs from the 6376 based on Amenhotep I & an observatory on the equator. From Censorinus derives yet another figure, (2012AD - 139AD) + 3*1442 = 6199yr (6139 if at equator) before 2012. From Theon derives yet another, (2012AD - 26BC) + 3*1442 = 6363yr (6303 if at equator) before 2012. Meyer's is 6252 (6192 if on the equator). So, 5x2 = 10 methods give 10 estimates (mean = 6242yr before 2012) of the origin of the Egyptian calendar, ranging from 6436 to 6052yr before 2012.
Appendix A.
The ecliptic latitude of Sirius is unaffected by Earth's axis precession, and only slightly affected by the slower, smaller-amplitude ecliptic precession. Jean-Marc Baillard (2004, online example in "HP [Hewlett Packard] museum") gives Sirius' ecliptic latitude as -39deg39'18" in 1600AD & -39deg35'19" in 2134AD. So, in 772BC (chosen as the midpoint between two dates below, 26BC & 1526-8=1518BC) it would be -39.95deg.
If Sirius were on the ecliptic, the time between heliacal risings would be simply a sidereal year, 365.25636d. If Sirius were at the ecliptic pole, the time would be a tropical year (length of tropical year is from sidereal yr & 25771.5yr precession period), 365.24219d. Interpolating with a sine curve, between these presumed maxima and minima, we find the time between heliacal risings is 365.25306d. This gives a Sothic cycle of
365.25306/(365.25306-365)*365/365.2425 = 1442 Gregorian years (vs. the 1460 usually stated).
Appendix B.
Damien F. Mackey, "Fall of the Sothic theory: Egyptian chronology revisited", TJ 17(3):7073, December 2003 (online @ www.answersingenesis.org ) says:
"Theon, an Alexandrian astronomer of the late 4th century AD...according to whom the conclusion of a 1,460-year period had occurred in the 5th year of the emperor Augustus - 26 BC...
"...the 3rd century AD Roman author, Censorinus. According to Meyers interpretation of the Sothic data as provided by Censorinus, a coincidence had occurred between the heliacal rising of Sirius and New Years Day in the 100th year before Censorinus wrote...
"Of the various major Egyptian Sothic documents, such as the Illahn Papyrus, the Elephantine Stele, and the Ebers Papyrus, the latter famous for its information about medical practices in Egypt - also contains reference to a Sothic rising in the 9th year of another un-named king, who has been identified as Amenhotep I of the 18th dynasty."
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Arcturus in 4328 BC Egypt: Hermetic knowledge
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 todays 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 Vegas 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 five, or sometimes six extra days at the end. I call this the tropical calendar because its 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 todays tropical year of 365.24219 days. The length of the tropical year is nearly constant, because under perturbation, an orbits major axis distance is stable to third order (conjecture of Lagrange & Poisson, proved by Tisserand) and also because Earths precession rate has only higher-order dependency, on Earths or Lunas small eccentricity or on Earths 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 Ptolemys 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. Wikipedias article, Julian Calendar, says:
the [Roman] pontificesadded 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 Ptolemys, i.e., March 20, so the solstice was June 21 (considering the +/- 2d effect of Earths orbital eccentricity). Since the first leap year in the Alexandrian calendar was 22BC, it would be most accurate to consider (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 Amenhoteps 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 Amenhoteps date, 1 Thoth would have been 69d past the solstice, if Amenhoteps 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. Amenhoteps 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 Years festivalthe heliacal rising of the Sirius [Mistranslated? Should say Arcturus? - JK] star. (Toffteen, p. 179). This year is comparable to the other estimates of Amenhoteps antiquity made by Egyptologists.
Sec. 4. Amenhotep I, Nabta, Giza and the heliacal risings of Arcturus.
Neglecting Earths orbital eccentricity, the suns ecliptic longitude (in the coordinates of the ecliptic of date) on Amenhoteps date, would be 90+67*360/365 = 156.1. To correct for eccentricity, I find Earths longitude of perihelion at 103 (todays value) - (2000+1524)*(1/25785 + 1/112000) * 360 = 42. Sin(156-180-42) = -0.91, so the eccentricity correction to the suns longitude, is -1.8deg, and the corrected longitude becomes 154.3.
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. Ill start by assuming its Elephantine Island: that was an important commercial, military and religious site during the New Kingdom, and its 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 153.15, a degree 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 suns longitude at the heliacal rising, from 153.15 to 154.51; that is, on 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 suns 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..
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 suns 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 Earths orbital eccentricity, the suns 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 Earths longitude of perihelion at 103 (todays 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 suns longitude at the apparent (top, of sun, at horizon) heliacal rising of Procyon, for 1780BC, is 74.53deg. Interpolating the year, in the table at the end of Sec. 9, the difference Giza - Elephantine is 2.40, so for Cynopolis at 28.30N (see Sec. 6) the heliacal rising of Procyon in 1780BC occurs when the sun's longitude is 76.25.
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 IIIs 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 & OConnor, 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 Thutmoses 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 its 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, Thutmoses date is decades away from exact coincidence, of heliacal rising with solstice, anywhere in Egypt.
Like Amenhoteps rising of Arcturus, Thutmoses 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' = 28.3deg 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, was negligible because that sun longitude ranged from 96 (in 4328BC) to 96-10(perihelion gain vs. sidereal)+7(Arcturus' rising's gain vs. sidereal) more than the perihelion. For Sirius, the corresponding figure is 46 (in 1422BC) to 46+10+3, which adds about 24deg*0.034*cos(53) = 0.50d of shift (because Earth has slower angular speed near aphelion). This makes the length of time needed, 2yr less. So, the more accurate date is 1424BC, and the sun's longitude on the Sothic date is 89.1 + 0.5 = 89.6.
Sec. 7. The Festival Year of Seti I.
Cerny, Journal of Egyptian Archaeology 47:150+, 1961, quotes Setis inscription:
Year 1, 1st month of winter, day 1, beginning of perpetuity.
Amenhoteps date proves that originally in 4328BC, 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.
Setis 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 Setis year as 4328BC + 2*1508 = 1312BC.
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 1312BC to 143AD (the eccentricity correction of Sec. 6, makes this 1453yr and 142AD). 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 Setis cycle, two tropical cycles led to one Sirius cycle, all accurate enough that the commemorative Roman coin (dated according to Wilbur Jones) was exact to the year. 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 4328BC, is 6339yr 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". 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 (Newcombs determination of the period corresponding to Earths 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 4328BC) differ only 90.19 - 90.32 = -0.13, though correction to exactly 30N and to the predicted Tropic, degrades this to -0.32.
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 89.93 & 90.06 (though correction to "exactly 30N" and to *our predicted* Tropic, changes this to 89.89 & 90.21).
The summer solstice heliacal risings of Arcturus & Canopus at two different Egypt sites in 4328BC, 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 latitudes of Hawaii, Olympus on Mars, and Jupiter's Red Spot, already is well known.)
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 ~6364yr ago: 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 Setis 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. 4328BC, 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 +0.12deg to account for eccentricity. The result for Nabta is 1614.34 "mean degrees", which gives 6343.5yr 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. 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 Earths 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 suns 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 stars codeclination, the observers colatitude, and the distance between observers zenith and the stars 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 observers zenith.
2) The triangle, the lengths of whose sides, are the segment from Earths N pole to the ecliptic N pole, the observers colatitude, and the distance between observers zenith and the point beneath the N ecliptic pole (observers ecliptic colatitude); here I found the last of these, thence the ecliptic longitude of the observers zenith, mindful of the signs of angles.
3) The triangle, the lengths of whose sides, are the suns ecliptic colatitude (always exactly 90degrees), the observers ecliptic colatitude, and the distance between observers zenith and the sub-rising point of the top of the sun (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 Earths polar flattening, is negligible. The final correction to the suns longitude, for the parallax caused by Earths 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., Stirlings 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
Arcturus in 4328 BC Egypt: Hermetic knowledge
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 todays 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 Vegas 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 five, or sometimes six extra days at the end. I call this the tropical calendar because its 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 todays tropical year of 365.24219 days. The length of the tropical year is nearly constant, because under perturbation, an orbits major axis distance is stable to third order (conjecture of Lagrange & Poisson, proved by Tisserand) and also because Earths precession rate has only higher-order dependency, on Earths or Lunas small eccentricity or on Earths 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 Ptolemys 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. Wikipedias article, Julian Calendar, says:
the [Roman] pontificesadded 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 Ptolemys, i.e., March 20, so the solstice was June 21 (considering the +/- 2d effect of Earths orbital eccentricity). Since the first leap year in the Alexandrian calendar was 22BC, it would be most accurate to consider (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 Amenhoteps 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 Amenhoteps date, 1 Thoth would have been 69d past the solstice, if Amenhoteps 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. Amenhoteps 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 Years festivalthe heliacal rising of the Sirius [Mistranslated? Should say Arcturus? - JK] star. (Toffteen, p. 179). This year is comparable to the other estimates of Amenhoteps antiquity made by Egyptologists.
Sec. 4. Amenhotep I, Nabta, Giza and the heliacal risings of Arcturus.
Neglecting Earths orbital eccentricity, the suns ecliptic longitude (in the coordinates of the ecliptic of date) on Amenhoteps date, would be 90+67*360/365 = 156.1. To correct for eccentricity, I find Earths longitude of perihelion at 103 (todays value) - (2000+1524)*(1/25785 + 1/112000) * 360 = 42. Sin(156-180-42) = -0.91, so the eccentricity correction to the suns longitude, is -1.8deg, and the corrected longitude becomes 154.3.
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. Ill start by assuming its Elephantine Island: that was an important commercial, military and religious site during the New Kingdom, and its 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 153.15, a degree 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 suns longitude at the heliacal rising, from 153.15 to 154.51; that is, on 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 suns 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.
.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 suns 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 Earths orbital eccentricity, the suns 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 Earths longitude of perihelion at 103 (todays 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 suns longitude at the apparent (top, of sun, at horizon) heliacal rising of Procyon, for 1780BC, is 74.53deg. Interpolating the year, in the table at the end of Sec. 9, the difference Giza - Elephantine is 2.40, so for Cynopolis at 28.30N (see Sec. 6) the heliacal rising of Procyon in 1780BC occurs when the sun's longitude is 76.25.
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 IIIs 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 & OConnor, 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 Thutmoses 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 its 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, Thutmoses date is decades away from exact coincidence, of heliacal rising with solstice, anywhere in Egypt.
Like Amenhoteps rising of Arcturus, Thutmoses 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' = 28.3deg 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, was negligible because that sun longitude ranged from 96 (in 4328BC) to 96-10(perihelion gain vs. sidereal)+7(Arcturus' rising's gain vs. sidereal) more than the perihelion. For Sirius, the corresponding figure is 46 (in 1422BC) to 46+10+3, which adds about 24deg*0.034*cos(53) = 0.50d of shift (because Earth has slower angular speed near aphelion). This makes the length of time needed, 2yr less. So, the more accurate date is 1424BC, and the sun's longitude on the Sothic date is 89.1 + 0.5 = 89.6.
Sec. 7. The Festival Year of Seti I.
Cerny, Journal of Egyptian Archaeology 47:150+, 1961, quotes Setis inscription:
Year 1, 1st month of winter, day 1, beginning of perpetuity.
Amenhoteps date proves that originally in 4328BC, 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.
Setis 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 Setis year as 4328BC + 2*1508 = 1312BC.
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 1312BC to 143AD (the eccentricity correction of Sec. 6, makes this 1453yr and 142AD). 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 Setis cycle, two tropical cycles led to one Sirius cycle, all accurate enough that the commemorative Roman coin (dated according to Wilbur Jones) was exact to the year. 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 4328BC, is 6339yr 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". 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 (Newcombs determination of the period corresponding to Earths 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 4328BC) differ only 90.19 - 90.32 = -0.13, though correction to exactly 30N and to the predicted Tropic, degrades this to -0.32.
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 89.93 & 90.06 (though correction to "exactly 30N" and to *our predicted* Tropic, changes this to 89.89 & 90.21).
The summer solstice heliacal risings of Arcturus & Canopus at two different Egypt sites in 4328BC, 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 latitudes of Hawaii, Olympus on Mars, and Jupiter's Red Spot, already is well known.)
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 ~6364yr ago: 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 Setis 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. 4328BC, 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 +0.12deg to account for eccentricity. The result for Nabta is 1614.34 "mean degrees", which gives 6343.5yr 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. 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 Earths 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 suns 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 stars codeclination, the observers colatitude, and the distance between observers zenith and the stars 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 observers zenith.
2) The triangle, the lengths of whose sides, are the segment from Earths N pole to the ecliptic N pole, the observers colatitude, and the distance between observers zenith and the point beneath the N ecliptic pole (observers ecliptic colatitude); here I found the last of these, thence the ecliptic longitude of the observers zenith, mindful of the signs of angles.
3) The triangle, the lengths of whose sides, are the suns ecliptic colatitude (always exactly 90degrees), the observers ecliptic colatitude, and the distance between observers zenith and the sub-rising point of the top of the sun (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 Earths polar flattening, is negligible. The final correction to the suns longitude, for the parallax caused by Earths 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., Stirlings 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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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 todays 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 Vegas 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 its 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 todays tropical year of 365.24219 days. The length of the tropical year is nearly constant, because under perturbation, an orbits major axis distance is stable to third order (conjecture of Lagrange & Poisson, proved by Tisserand) and also because Earths precession rate has only higher-order dependency, on Earths or Lunas small eccentricity or on Earths 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 Ptolemys 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. Wikipedias article, Julian Calendar, says:
the [Roman] pontificesadded 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 Ptolemys, i.e., March 20, so the solstice was, most likely, June 21 (considering the effect of Earths 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 Amenhoteps 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 Amenhoteps date, 1 Thoth would have been 69d past the solstice, if Amenhoteps 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. Amenhoteps 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 Years festivalthe heliacal rising of the Sirius [Mistranslated? Should say Arcturus? - JK] star. (Toffteen, p. 179). This year is comparable to the other estimates of Amenhoteps antiquity made by Egyptologists.
Sec. 4. Amenhotep I, Nabta, Giza and the heliacal risings of Arcturus.
Neglecting Earths orbital eccentricity, the suns ecliptic longitude (in the coordinates of the ecliptic of date) on Amenhoteps date, would be 90+67*360/365.25 = 156.0. To correct for eccentricity, I find Earths longitude of perihelion at 103 (todays value) - (2000+1524)*(1/25785 + 1/112000) * 360 = 42. The eccentricity correction to the suns 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. Ill start by assuming its Elephantine Island: that was an important commercial, military and religious site during the New Kingdom, and its 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 suns 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 suns 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..
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 suns 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 Earths orbital eccentricity, the suns 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 Earths longitude of perihelion at 103 (todays 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 suns 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 IIIs 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 & OConnor, 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 Thutmoses 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 its 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, Thutmoses date is decades away from exact coincidence, of heliacal rising with solstice, anywhere in Egypt.
Like Amenhoteps rising of Arcturus, Thutmoses 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 Setis inscription:
Year 1, 1st month of winter, day 1, beginning of perpetuity.
Amenhoteps 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.
Setis 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 Setis 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 Setis 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 (Newcombs determination of the period corresponding to Earths 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 Setis 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 Earths 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 suns 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 stars codeclination, the observers colatitude, and the distance between observers zenith and the stars 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 observers zenith.
2) The triangle, the lengths of whose sides, are the segment from Earths N pole to the ecliptic N pole, the observers colatitude, and the distance between observers zenith and the point beneath the N ecliptic pole (observers ecliptic colatitude); here I found the last of these, thence the ecliptic longitude of the observers zenith, mindful of the signs of angles.
3) The triangle, the lengths of whose sides, are the suns ecliptic colatitude (always exactly 90degrees), the observers ecliptic colatitude, and the distance between observers 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 Earths polar flattening, is negligible. The final correction to the suns longitude, for the parallax caused by Earths 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., Stirlings 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
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 todays 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 Vegas 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 its 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 todays tropical year of 365.24219 days. The length of the tropical year is nearly constant, because under perturbation, an orbits major axis distance is stable to third order (conjecture of Lagrange & Poisson, proved by Tisserand) and also because Earths precession rate has only higher-order dependency, on Earths or Lunas small eccentricity or on Earths 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 Ptolemys 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. Wikipedias article, Julian Calendar, says:
the [Roman] pontificesadded 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 Ptolemys, i.e., March 20, so the solstice was, most likely, June 21 (considering the effect of Earths 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 Amenhoteps 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 Amenhoteps date, 1 Thoth would have been 69d past the solstice, if Amenhoteps 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. Amenhoteps 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 Years festivalthe heliacal rising of the Sirius [Mistranslated? Should say Arcturus? - JK] star. (Toffteen, p. 179). This year is comparable to the other estimates of Amenhoteps antiquity made by Egyptologists.
Sec. 4. Amenhotep I, Nabta, Giza and the heliacal risings of Arcturus.
Neglecting Earths orbital eccentricity, the suns ecliptic longitude (in the coordinates of the ecliptic of date) on Amenhoteps date, would be 90+67*360/365.25 = 156.0. To correct for eccentricity, I find Earths longitude of perihelion at 103 (todays value) - (2000+1524)*(1/25785 + 1/112000) * 360 = 42. The eccentricity correction to the suns 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. Ill start by assuming its Elephantine Island: that was an important commercial, military and religious site during the New Kingdom, and its 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 suns 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 suns 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.
.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 suns 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 Earths orbital eccentricity, the suns 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 Earths longitude of perihelion at 103 (todays 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 suns 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 IIIs 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 & OConnor, 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 Thutmoses 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 its 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, Thutmoses date is decades away from exact coincidence, of heliacal rising with solstice, anywhere in Egypt.
Like Amenhoteps rising of Arcturus, Thutmoses 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 Setis inscription:
Year 1, 1st month of winter, day 1, beginning of perpetuity.
Amenhoteps 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.
Setis 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 Setis 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 Setis 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 (Newcombs determination of the period corresponding to Earths 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 Setis 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 Earths 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 suns 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 stars codeclination, the observers colatitude, and the distance between observers zenith and the stars 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 observers zenith.
2) The triangle, the lengths of whose sides, are the segment from Earths N pole to the ecliptic N pole, the observers colatitude, and the distance between observers zenith and the point beneath the N ecliptic pole (observers ecliptic colatitude); here I found the last of these, thence the ecliptic longitude of the observers zenith, mindful of the signs of angles.
3) The triangle, the lengths of whose sides, are the suns ecliptic colatitude (always exactly 90degrees), the observers ecliptic colatitude, and the distance between observers 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 Earths polar flattening, is negligible. The final correction to the suns longitude, for the parallax caused by Earths 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., Stirlings 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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